# Pseudogap phases, chiral anomaly and topological order with quantum loop
entanglement

Predrag Nikolić  Department of Physics and Astronomy,
George Mason University, Fairfax, VA 22030, USA  Institute for Quantum Matter
at Johns Hopkins University, Baltimore, MD 21218, USA

###### Abstract

A many-body quantum system whose topological defects are conserved, abundant
and mobile is a correlated quantum liquid. Since topological defects can be
classified by homotopy groups, each homotopy identifies a class of quantum
liquids. Here we explore the quantum liquids based on the $\pi_{3}(S^{2})$
homotopy group, i.e. Hopf fibration. Their topologically non-trivial dynamics
emerges from the interlinking between magnetic flux or skyrmion loops in the
charge and spin sectors respectively. We lay down a field theory foundation
for analyzing such states by naturally incorporating the well-known framing
regularization into the theory, and constructing the appropriate topological
Lagrangian terms. We show that at least two strongly correlated phases of
interlinked loops can exist in $d=3$ spatial dimensions at zero and low finite
temperatures. These phases are closely related to the chiral quantum anomaly
and do not have an obvious topological order, but they are distinguished from
the trivial disordered phase with a generalization of the Wilson loop
operator. In $d=4$ spatial dimensions, interlinked loops are able to produce
topological order at zero temperature, featuring charge, angular momentum and
braiding fractionalization. We discuss some possible experimental signatures
of loop entanglement in the quantum noise of charge currents.

##  I Introduction

Topological defects profoundly influence the dynamics because they are
resilient to perturbations. The most intricate example is topological order, a
non-local many-body quantum entanglement which produces a ground state
degeneracy on topological manifolds without spontaneous symmetry breaking Wen
and Niu (1990); Wen  _et al._ (1990); Wen (1995, 2004). Topological order
based upon charge current vortices is realized in two-dimensional fractional
quantum Hall liquids, and exhibits charge and exchange statistics
fractionalization. Other kinds of topological defects from charge or spin
degrees of freedom can support topological orders in higher dimensions Nikolić
(2020a). A more subtle aspect of topological dynamics concerns the quantum
fluctuations of a “classical” topological invariant on topologically trivial
manifolds Nikolic (2023). No smooth transformations of fields can alter the
topological invariant formulated in the continuum limit, but quantum tunneling
events, called instantons, can. Instantons may be strongly suppressed. The
topological protection that defects enjoy normally amounts to a linear
confining action potential between an instanton and an anti-instanton. This
neutralizes the instantons and leaves only the topological index preserving
fluctuations to shape the dynamics at large scales. However, even in that
case, sufficiently strong thermal fluctuations can deconfine the instantons
and cause a phase transition to a phase without any topological protection
mechanism Nikolic (2023).

In this paper, we study the topological dynamics associated with “hopfions”,
the topological defects classified by the $\pi_{3}(S^{2})$ homotopy group
Faddeev ; Faddeev (1976); Faddeev and Niemi (1997). The simplest realization
of such defects are interlinked loops and knots in three spatial dimensions
Stone and Goldbart (2009), but their mathematical structure and quantum
fluctuations introduce various complications. The loops can be built with
either charge or spin currents. The former are vortices and the latter are
skyrmions. Both are theoretically described as interlinked quantized flux
loops of a U(1) gauge field. One of our main results is that hopfions can
exist in two entangled loop phases with a conserved Hopf topological
invariant. The phase transition to a disentangled state occurs at a finite
temperature Nikolic (2023). There is no requirement that latent heat,
spontaneous symmetry breaking or topological order be associated with such a
transition; instead, the loop disentanglement is similar to the Berezinsky-
Kosterlitz-Thouless transition Berezinsky (1972); Kosterlitz and Thouless
(1973); Kosterlitz (1974), but occurring in three spatial dimensions.

One strategy for the detection of loop disentanglement, which we explore here,
is based on the quantum noise spectrum of electron currents. We will show that
the chiral quantum anomaly Adler (1969); Bell and Jackiw (1969); Fujikawa
(1980); Nielsen and Ninomiya (1981, 1983); Peskin and Schroeder (1995); Zinn-
Justin (2005) is directly related to the Hopf index, so correlated Dirac and
Weyl semimetals might provide a measurement platform. But, the correlation
between the Hopf index and current fluctuations is more general. We expect
that this correlation, and the transition itself, would be much more prominent
with emergent U(1) gauge fields in solids instead of the natural gauge field.
Vortex fluctuations in superconductors are one possible source of such a gauge
field. There are experimental indications via Nernst effect Wang  _et al._
(2001, 2006); Li  _et al._ (2010) that vortices are present even in the
pseudogap normal phase of cuprates, so they could live in an entangled phase.
Related indications of incoherent Cooper pairing are found in other studies
Corson  _et al._ (1999); Crane  _et al._ (2007). Another type of an emergent
U(1) gauge field can arise from the magnetism in heavy-fermion materials or
chiral magnets, especially quantum-disordered magnets Machida  _et al._ (2007,
2010); Cheng  _et al._ (2017). If local magnetic moments are thermally or
quantum-mechanically disordered but retain fluctuating skyrmion loops, the
emergent gauge field, which one would associate with the “topological” Hall
effect, can carry a Hopf index and undergo the disentanglement topological
phase transition at a certain temperature.

Our main goal is to develop a physical insight and theoretical formalism for
exploring these possibilities. A particular challenge is that multiple types
of topological defects can drive phase transitions, in addition to many
conventional mechanisms. Monopole fluctuations in $d\geq 2$ dimensional
compact U(1) gauge theories are understood fairly well Polyakov (1975, 1977);
Fisher  _et al._ (1989); Fisher (1990); Nagaosa (1993); Nagaosa and Lee
(2000); Sachdev (1999); Herbut  _et al._ (2003); Case  _et al._ (2004);
Hermele  _et al._ (2004); Nogueira and Kleinert (2008), but hopfions have been
observed and considered so far only as topological defects without significant
dynamics Babaev  _et al._ (2002); Dennis  _et al._ (2010); Hall  _et al._
(2016); Sutcliffe (2017); Rybakov  _et al._ (2019a, b); Voinescu  _et al._
(2020); Kent  _et al._ (2021); Sugic  _et al._ (2021), or as a background
Berry curvature for Hopf band insulators Moore  _et al._ (2008); Deng  _et
al._ (2013, 2014, 2018); Chang  _et al._ (2017); Volovik and Zhang (2017);
Schuster  _et al._ (2019); Ünal  _et al._ (2019); Alexandradinata  _et al._
(2021); Lapierre  _et al._ (2021); Graf and Piéchon (2023); Wang  _et al._
(2023). We emphasize here that hopfions in $d=3$ and $d=4$ spatial dimensions
can drive independent topological phase transitions across which the flux
loops survive (i.e. monopoles are suppressed) but lose a well-defined
structure of interlinks. This requires at least some dedicated length scale in
the problem other than the lattice constant. The lack of such a scale keeps
the phase diagram simple and without any unconventional entangled states, as
in the case of plain bosonic lattice models Fisher (1990); Sachdev (1999). If
a quantum liquid has a conserved topological invariant, it can exhibit
topological order Nikolić (2020a). However, this is not implied. We will argue
that the conserved Hopf index induces topological order in $d=4$ spatial
dimensions, generalizable to $d=4n$ ($n\in\mathbb{N}$). In contrast, Hopf
index conservation stabilizes a pseudogap phase in $d=3$, which is perhaps
related to correlated symmetry-protected topological (SPT) phases Chen  _et
al._ (2013, 2012) by the likely ability to host gapless states on the system
boundary.

The only topological defects of SU(2) spin configurations in $d=3$ spatial
dimensions are hedgehogs and hopfions. They are classified by $\pi_{2}(S^{2})$
and $\pi_{3}(S^{2})$ homotopy groups respectively. Hedgehogs are equipped with
a point singularity and enjoy topological protection against quantum
fluctuations Nikolić (2020a, b); Nikolic (2023), while hopfions are non-
singular configurations similar to $d=2$ skyrmions. Since electrons carry both
charge and spin, their charge and spin currents are closely intertwined. If
electron’s spin current exhibits a topological structure, its charge current
will reflect the same structure unless spin and charge are deconfined. The
charge analogue to a hedgehog is a monopole. The binding of monopoles to
hedgehogs is a “topological” magnetoelectric effect, and the equivalent
phenomenon in two spatial dimensions is a “topological” Hall effect. There are
also charge current configurations analogous to hopfions. They are usually
classified by the second Chern number of the U(1) bundle Peskin and Schroeder
(1995); Stone and Goldbart (2009); Zinn-Justin (2005), and their association
with the Hopf index of the $\pi_{3}(S^{2})$ homotopy group is rarely mentioned
Bandyopadhyay (2012). But, here we wish to explore the topological connections
between the charge and spin dynamics, so we will adopt the classification of
spin defects for the charge sector, even though the fundamental charge-related
fields may not have the appropriate structure to mathematically merit such a
classification. The analogy between charge and spin defects is established
only though a hierarchy of emergent gauge fields Nikolić (2020a). For example,
a vortex in $d=3$ dimensions is a line singularity which can be captured by a
phase $\theta$ winding of a superconductor’s order parameter, but when
vortices become delocalized we can track their dynamics only with an emergent
gauge field $a_{\mu}$ derived from $\partial_{\mu}\theta$. This gauge field is
the first hierarchical emergent field. Monopoles are singular configurations
of $a_{\mu}$, and if they were to become delocalized we would need the second
hierarchical emergent field $a_{\mu\nu}$ derived from
$\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}$. The flux of $a_{\mu\nu}$
through a sphere reads out the topological index or charge of all monopoles
enclosed within the sphere. The same type of an antisymmetric gauge field can
be directly constructed from the spin configuration; this is the familiar spin
chirality. The equivalence between the charge and spin topological dynamics is
visible via $a_{\mu\nu}$.

The rest of the paper is organized as follows. Section II introduces the Hopf
invariant and explains its quantization and topological protection in physical
terms. We first review the construction of the Hopf index from gauge fields in
Section II.1, and then connect it to the Gauss’ linking number in Section II.2
where it becomes apparent that matter must be coupled to the gauge field in
order to resolve the framing regularization problem. In Section III we study
the quantum liquids of Hopf topological defects in $d=3$ spatial dimensions,
which can be potentially realized in some correlated materials. We demonstrate
in Section III.1 the conservation of the Hopf index despite the natural
quantum processes which create, annihilate and link or unlink the flux loops.
A further analysis of instanton events with renormalization group, presented
elsewhere Nikolic (2023), establishes the topological stability of a
correlated entangled-loop phase at low temperatures. We in fact identify two
Hopf indexes for every gauge field coupled to matter, and thereby two
corresponding loop disentanglement transitions. In Section III.2, we formulate
and explore the field equations for a prototype system that can host hopfions.
This reveals the natural coupling between the charge and spin dynamics from
which the “topological” Hall and magnetoelectric effects, as well as the
chiral magnetic interaction, emerge. We also seek manifestations of
topological order but find none, at least in the simple considered theory
framework. Nevertheless, we argue in Section III.3 that certain signatures of
the conserved Hopf entanglement could be observable in the quantum noise
correlations and specific heat. We pay a special attention to the chiral
quantum anomaly and explain in a simple manner how it is related to the Hopf
index; from that point of view, the chiral anomaly might provide a window into
the loop-entanglement transition in materials with Dirac or Weyl
quasiparticles. By analogy with topological order, we briefly discuss a
mechanism for anomaly fractionalization.

Four-dimensional topological order based on hopfions is finally explored in
Section IV. It features charge and angular momentum fractionalization (Section
IV.1), fractional braiding statistics between particles and 2-spheres (Section
IV.2), topological ground state degeneracy (Section IV.3), and a correlation
between the charge and spin sectors (Section IV.4). All conclussions are again
summarized in Section V, and discussed in the view of remaining open problems.
A generalization of the Hopf invariant to higher dimensions is presented in
Appendices B for the spin sector and C for the charge sector.

In this paper, we use units $\hbar=c=1$ and Einstein’s convention for the
summation over repeated indices. Spatial directions are denoted by Latin
letters $i,j,k,\dots\in\\{1,\dots,d\\}$, and space-time directions by Greek
letters $\mu,\nu,\lambda,\cdots\in\\{0,1,\dots,d\\}$.

##  II The topology of loop entanglement

Thermodynamic phases of mobile topological defects are sharply distinct from
fully disordered phases as long as a relevant topological charge (invariant)
is conserved. Fractional quantum Hall liquids are examples of such phases.
They can be generalized to higher spatial dimensions $d>2$ by utilizing
$\pi_{d-1}(S^{d-1})$ homotopy groups where the relevant topological defects
are monopoles and hedgehogs in the charge and spin sectors respectively.

Here we study the analogous unconventional dynamics of the topological defects
characterized by the $\pi_{3}(S^{2})$ homotopy group. The relevant topological
invariant is the Hopf index. Physically, the quantum liquids in this class are
made of entangled and interlinked flux loops. While a classical snapshot of
such a state may look chaotic like a bowl of spaghetti (with each spaghetti
strand wrapped into a loop), we will show that quantum dynamics combined with
loop entanglement stabilizes unconventional strongly correlated topological
phases which need not be classifiable as topological order.

###  II.1 Hopf index

Hopf index $N\in\mathbb{Z}$ is a topological invariant which characterizes
configurations of a three-component unit-vector field $\hat{\bf n}({\bf x})$
in a three-dimensional space without boundaries, ${\bf x}\in S^{3}$. Since the
set of $\hat{\bf n}$ vectors is equivalent to a unit-sphere $S^{2}$, and a
simple $d=3$ space without boundaries is a 3-sphere $S^{3}$, the Hopf
invariant distinguishes equivalence classes of the $\pi_{3}(S^{2})$ homotopy
group. Two configurations $\hat{\bf n}({\bf x})$ with different Hopf
invariants cannot be smoothly transformed into each other. We will construct
the Hopf index and demonstrate its topological invariance by physical
arguments in several stages.

Using Einstein’s notation $\hat{n}^{a}$ for the components of the vector
$\hat{\bf n}$, we define the chirality

$J_{i}^{\phantom{x}}=\frac{1}{2}\epsilon_{ijk}^{\phantom{x}}\epsilon^{abc}\hat{n}^{a}(\partial_{j}\hat{n}^{b})(\partial_{k}\hat{n}^{c})$
(1)

and observe that its flux

$\int\limits_{S^{2}}d^{2}x\,\hat{\eta}_{i}J_{i}=n\phi_{0}\quad,\quad
n\in\mathbb{Z}$ (2)

is quantized in the units of $\phi_{0}=4\pi$ on any closed surface $S^{2}$.
The integer $n$ is the topological invariant of the $\pi_{2}(S^{2})$ homotopy
group, also known as Pontryagin index, and $\hat{\eta}_{i}$ is a unit-vector
locally perpendicular to the surface $S^{2}$. A field $\hat{n}^{a}(x,y)$ which
becomes uniform at large distances $r=(x^{2}+y^{2})^{1/2}$ can carry a
quantized Pontryagin index on every $z=\textrm{const.}$ plane because all
points at infinite $r$ can be effectively identified, turning each plane into
a sphere $S^{2}=\mathbb{R}^{2}\cup\\{\infty\\}$. Such a field configuration is
a skyrmion line. We may smoothly deform the field $\hat{n}^{a}({\bf x})$ to
focus its non-zero flux (2) into a finite region $r<R$ and leave a uniform
chirality-free configuration $\hat{n}^{a}\to\textrm{const.}$ at all points
$r>R$; the skyrmion thickness $R$ can become arbitrarily small, yet the
topological Pontryagin index cannot change under these smooth transformations.
We may also arbitrarily reshape the skyrmion line. Generally, skyrmion lines
can form closed loops; line terminations are not allowed because the ensuing
Pontryagin index change across the termination point cannot be obtained from a
smooth spatial variation of $\hat{n}^{a}$.

In the first step of Hopf index construction, we specialize to the vector
fields $\hat{n}^{a}$ which carry an arbitrary set of _infinitely thin_
skyrmions loops $C_{l}$. The corresponding chirality (1) is a singular
function of coordinates

$J_{i}({\bf x})=\phi_{0}\sum_{l}n_{l}\oint\limits_{C_{l}}dx_{l,i}\,\delta({\bf
x}-{\bf x}_{l})\ ,$ (3)

where $n_{l}$ are integers, ${\bf x}_{l}\in C_{l}$ is the set of points on the
flux filament $C_{l}$, and $\delta$ is the Dirac $\delta$-function of a three-
dimensional vector. The flux (2) is quantized with $n=n_{l}$ on any _open_
manifold $S^{2}\to A$ which intersects the filament $l$. Let us interpret the
chirality as an “electric current density”; the “current” carried by each loop
$C_{l}$ is quantized, $I_{l}=n_{l}\phi_{0}$. The “magnetic field” ${\bf B}$
which obtains from all “currents” is given by Ampere’s law (up to an
unimportant factor):

${\bf J}=\boldsymbol{\nabla}\times{\bf B}\ .$ (4)

If the loops are interlinked, then the total current passing through the loop
$C_{l}$ is

$\sum_{m}N_{lm}I_{m}=\int\limits_{A_{l}}d^{2}x\,\hat{\boldsymbol{\eta}}{\bf
J}=\oint\limits_{C_{l}}d{\bf x}\,{\bf B}\ .$ (5)

Here, $A_{l}$ is a surface bounded by $C_{l}$, $\hat{\boldsymbol{\eta}}$ is
the unit vector locally orthogonal to it, and $N_{lm}$ is the number of times
the loop $m$ wraps around the loop $l$. We allow $l=m$ on the left-hand side
and denote by $N_{ll}$ the self-linking number; if $N_{ll}\neq 0$, then the
loop $l$ is a knot. Note that inside the filament $l$ and along its contour
$C_{l}$ we have $I_{l}d{\bf x}={\bf J}dxda_{l}$, where $da_{l}$ is the
infinitesimal area element of the $l$-filament’s cross-section. So, after
multiplying (5) by $I_{l}$ and summing over all loops $l$, we get

$\sum_{l,m}N_{lm}I_{l}I_{m}=\int d^{3}x\,{\bf B}{\bf J}\ .$ (6)

Finally, we recall the “current” quantization $I_{l}=n_{l}\phi_{0}$; the last
equation becomes

$N=\sum_{l,m}N_{lm}n_{l}n_{m}=\frac{1}{\phi_{0}^{2}}\int d^{3}x\,{\bf B}{\bf
J}\ .$ (7)

The number $N$ is evidently an integer.

From this point on, we will interpret the “current density” ${\bf J}$ as a
curl of a gauge field ${\bf A}$:

${\bf J}=\boldsymbol{\nabla}\times{\bf A}\ .$ (8)

The integer (7) becomes the topological Hopf index

$N_{A}=\frac{1}{\phi_{0}^{2}}\int\limits_{S^{3}}d^{3}x\,{\bf
A}(\boldsymbol{\nabla}\times{\bf
A})=\frac{1}{\phi_{0}^{2}}\int\limits_{S^{3}}d^{3}x\,{\bf A}{\bf J}$ (9)

after showing that the flux of ${\bf J}$ need not be focused into singular
filaments. The above simple argument about integer quantization of the Hopf
index holds against all smooth transformations of $\hat{\bf n}({\bf x})$ which
reshape the flux loops and turn them into thick flux tubes without introducing
overlaps. Going beyond this restriction is less transparent. We will present a
rigorous direct argument about the topological invariance of the Hopf index
only in Section IV.4 using SU(2) spinor representations. Until then, we will
rely on a physical point of view. Let us regard an arbitrary gauge field ${\bf
A}({\bf x})$ as the coarse-grained representation of the quantum superposition
between many filament configurations (3). In this picture, quantum
fluctuations deform and move the quantized flux filaments until the flux
diffuses into a continuous and non-singular structure. We allow, in principle,
all quantum processes. Some processes, which we call instantons, can convert a
filament loop configuration into another one with a different Hopf index. We
will see in Section III.1 that instantons cannot occur with smooth $\hat{\bf
n}({\bf x},t)$ fields, and argue that they are “confined” at low temperatures,
i.e. every instanton is accompanied by an anti-instanton no further that a
certain finite space-time interval away. This dynamically protects the Hopf
index at macroscopic scales against quantum fluctuations, and gives (9) the
meaning of a topological invariant.

We will work with two physical contexts in which the Hopf index is relevant.
One context is the spin context we already introduced. The emergent gauge
field ${\bf A}$ is derived from the chirality

$\epsilon_{ijk}\partial_{j}A_{k}=\frac{1}{2}\epsilon_{ijk}\epsilon^{abc}\hat{n}^{a}(\partial_{j}\hat{n}^{b})(\partial_{k}\hat{n}^{c})$
(10)

of the spin field $\hat{n}^{a}$. This is the effective gauge field of the
“topological Hall effect”. It captures the effective Lorentz force which
spinful particles experience when moving through a topologically non-trivial
magnetization background, assuming adiabatic limit (particle spin locked to
the background magnetization). The topological quantization of the skyrmion
number determines $\phi_{0}=4\pi$, so the Hopf index characterizes the
interlinking of skyrmion loops. The second context is tied to charge
fluctuations of a spinor field $\psi$. We extract the gauge field using

$A_{i}\psi^{\dagger}\psi=-i\psi^{\dagger}\partial_{i}\psi\ .$ (11)

This gauge field captures a set of Dirac string loops in any coherent $\psi$
configuration, carrying $\phi_{0}=2\pi$ flux quantum. Fluctuations may
continuously distribute this flux, but the Hopf index remains well defined and
quantized as long as monopoles of ${\bf A}$ are not present.

The purpose of this paper is to explore unconventional correlated states of
matter whose dynamics quantizes and topologically protects the Hopf index. The
above demonstration of the Hopf quantization reveals the physical requirements
on such states. The degrees of freedom must be able to support appropriate
flux currents which carry a globally quantized flux. Monopoles must be absent,
i.e. the flux must be closed into loops. We will show that stable non-trivial
phases with a quantized Hopf index exhibit subtle correlation phenomena in
$d=3$ spatial dimensions or topological order in $d=4$.

###  II.2 Gauss’ linking number, framing and twist

If the gauge flux ${\bf J}$ forms singular quantized loops $C_{l}$ with a
fixed quantum $n_{l}=1$ according to (3), then the Hopf index (9) is simply
the sum of loops’ linking numbers,

$N=\sum_{i,j}N_{ij}=\sum_{i}N_{ii}+2\sum_{i<j}N_{ij}\ .$ (12)

The linking number $N_{ij}$ of two loops (introduced by Gauss) obtains by
substituting Biot-Savart law

${\bf A}({\bf x})=\frac{1}{4\pi}\int d^{3}x^{\prime}\,\frac{{\bf J}({\bf
x}^{\prime})\times({\bf x}-{\bf x}^{\prime})}{|{\bf x}-{\bf x}^{\prime}|^{3}}$
(13)

into (5) with interpretation ${\bf B}\equiv{\bf A}$:

$N_{ij}=N_{ji}=\frac{1}{4\pi}\oint\limits_{C_{i}}\oint\limits_{C_{j}}\frac{({\bf
r}_{i}-{\bf{\bf r}}_{j})(d{\bf r}_{i}\times d{\bf r}_{j})}{|{\bf
r}_{i}-{\bf{\bf r}}_{j}|^{3}}\ .$ (14)

Here, ${\bf r}_{i}$ represents the set of points on the loop $C_{i}$ and
$d{\bf r}_{i}$ is the tangential infinitesimal displacement along $C_{i}$.
Introducing a unit vector

$\hat{{\bf m}}=\frac{{\bf r}_{i}-{\bf r}_{j}}{|{\bf r}_{i}-{\bf r}_{j}|}$ (15)

quickly demonstrates the quantization of the linking number

$N_{ij}=-\frac{1}{4\pi}\oint\oint dx_{1}dx_{2}\,\hat{{\bf
m}}\left(\frac{\partial\hat{{\bf m}}}{\partial
x_{1}}\times\frac{\partial\hat{{\bf m}}}{\partial x_{2}}\right)\in\mathbb{Z}\
.$ (16)

There is a problem, however, when one uses (14) to compute the self-linking
number $N_{ii}$ of a knot. If $C_{i}=C_{j}$, then a singularity at ${\bf
r}_{i}={\bf r}_{j}$ is integrated along the knotted loop. The regularization
of this singularity is known as framing. One constructs an auxiliary image
$C^{\prime}_{i}$ of the knotted loop $C^{\phantom{x}}_{i}$ and displaces it
everywhere from the original by a framing vector field ${\bf f}({\bf
r}_{i})\neq 0$. The self-linking number $N_{ii}$ is then computed as the well-
defined interlinking number of the image and the original. The framing field
can be infinitesimal and it is chosen, for concreteness, to be locally
perpendicular to the original loop. As good as it gets, this procedure
unavoidably introduces an ambiguity in the definition of self-linking. Since
the framing field is perpendicular to a looped string in a three-dimensional
space, it can complete $n\in\mathbb{Z}$ full rotations about the string as we
traverse the loop. This clearly wraps the image $n$ extra times around the
original loop, and hence affects the computed inter-linking number. Introduced
purely by the regularization, $n$ has nothing to do with the shape of the
original loop. Physically, the framing field must be rooted in some
microscopic degrees of freedom which we might have not encountered before.

Framing finds a very simple and natural representation in field theory. The
gauge field ${\bf A}$ featured in the Hopf index is a dynamical degree of
freedom with a certain Maxwell Lagrangian density
$(\boldsymbol{\nabla}\times{\bf A})^{2}$. If we couple it to a matter field
$\chi$ of charge $q$, we also introduce a gradient term
$(\boldsymbol{\nabla}\chi+q{\bf A})^{2}$ into the Lagrangian density. For
simplicity, we regard $\chi$ as an angle-valued field in an XY model context.
Now, gauge invariance suggests an adaptation of the expression (9) for the
Hopf index

$N_{\chi}=\frac{1}{q\phi_{0}^{2}}\int\limits_{S^{3}}d^{3}x\,(\boldsymbol{\nabla}\chi+q{\bf
A})(\boldsymbol{\nabla}\times{\bf A})\ .$ (17)

Suppose we want to calculate the Hopf index of a self-linked flux loop $C$.
The gauge field of a quantized flux filament has the curl given by (3), so
that

$N_{\chi}=\frac{1}{q\phi_{0}}\oint\limits_{C}d{\bf
r}^{\prime}\,(\boldsymbol{\nabla}\chi+q{\bf A})\ .$ (18)

The $\chi$ angle is free to wind $n$ times along the loop’s path $C$,
contributing $n$ to the Hopf index when the charge is quantized by
$q\phi_{0}=2\pi$. This is different than the previously considered
contribution of the pure gauge field, and requires $\chi$ to carry a singular
vortex line somewhere in space, passing through $C$. In the charge context,
the field $\chi$ is simply the quantum mechanical phase of the physical
charged matter field. The microscopic origin of $\chi$ is not obvious in the
spin context, but skyrmions are certainly complex objects which have enough
structure to support another degree of freedom. The construction (18) gives
$\chi$ the role of a “twist” field which captures the internal twisting of the
flux filament about its own axis as it forms a loop. The framing field ${\bf
f}$ describes the same, so we can relate it to $\chi$ with

$\hat{{\bf f}}\left(\frac{d\hat{{\bf f}}}{dr}\times d{\bf r}\right)=d\chi\ .$
(19)

In a coarse-grained field theory, $\chi$ captures the microscopic flux
filament twist which cannot be resolved at macroscopic length scales.

It should be pointed out that (17) is invariant under smooth gauge
transformations ${\bf A}\to{\bf A}+\boldsymbol{\nabla}\theta$. However, it is
not invariant under singular gauge transformations in which $\theta$ contains
vortex singularities. This is essential for capturing the twist. It will
become apparent shortly that quantum dynamics protects (17) better that the
pure gauge-field Hopf index (9). Furthermore, the Hopf index is invariant
under time-reversal but changes sign under mirror inversion transformations.

From now on, we will call a “hopfion” any field configuration that carries a
non-zero Hopf index (17). Hopfions are topological defects classified by the
$\pi_{3}(S^{2})$ homotopy group. The simplest hopfion configurations are shown
in Fig.1.

Figure 1: Simplest Hopf topological defects: (a) simple loop $N=0$, (b)
twisted loop $N=1$, (c) linked simple loops $N=2$, and (d) simplest (trefoil)
knot $N=3$. Loops are oriented by the direction of the flux they carry. Even
though time reversal changes the direction of flux, it does not affect the
Hopf index. Mirror inversions change the sign of the Hopf index.

##  III Hopfions in three dimensions: Instanton confinement and quantum
anomaly

###  III.1 Hopf index conservation

Hopf index is topologically protected from all smooth transformations of the
classical field configuration. However, quantum tunneling introduces local
abrupt changes of the classical states (in a regularized theory). A high
action cost of such tunneling doesn’t necessarily make it statistically
impossible, so quantum events, which we will refer to as instantons, are able
to change the classical topological invariant. The question of topological
protection, then, becomes the question of instanton confinement. If every
instanton event is immediately followed by a nearby anti-instanton, we say
that the instantons are confined into neutral dipoles and the topological
index is globally conserved. A thermodynamic instanton confinement-
deconfinement transition can occur at finite temperatures Nikolic (2023).

Quantum dynamics routinely generates virtual low-energy fluctuations of
neutral topological defect clusters. These are vortex-antivortex pairs in two-
dimensional superconductors, and small vortex loops in three-dimensional
superconductors. An isolated flux loop is also the simplest neutral
topological cluster of hopfions. Cheap fluctuations of small Hopf-neutral
rings can gradually reshape the segments of a larger loop and move it through
space. The interaction between loops is a simple superposition of their
quantized fluxes; two flux filaments cancel out if they overlap exactly but
have opposite directions. Importantly, fluctuating rings can also unlink and
re-link two loops, as shown in Fig.2. The events which do not significantly
modify the gradient energy

$E_{\chi}\propto\int d^{3}x\,(\boldsymbol{\nabla}\chi+q{\bf A})^{2}$ (20)

preserve the Hopf index (17). The examples illustrated in Fig.3 include twist-
compensated unlinking and twist exchange. Other events which alter the Hopf
index of the system are instantons. Interlinked and self-linked loops can be
mathematically characterized by various other topological invariants Alexander
(1928); Jones (1985, 1987); Kauffman (1987); Witten (1989); Livingston ;
Morishita (2012); Purcell (2020), but these cannot be conserved in the
presence of low-energy unlinking fluctuations.

Figure 2: Quantum and thermal fluctuations of small flux rings (green) can
unlink and re-link flux loops (A and B). Such a ring has lower energy
$(\boldsymbol{\nabla}\chi+q{\bf A})^{2}$ when it is twisted, because it needs
a winding twist field $\chi$ to compensate the circular gauge field ${\bf A}$
of the B string. Its net Hopf index is zero, so the unlinked loops A and B
have the same Hopf index as when they were linked; unlinking imparts a twist
on the A string, as expected from the electromotive force induced by Faraday
effect. Alternatively, if the small ring excitation is untwisted, then it
costs more energy and carries a non-trivial Hopf index by being interlinked
with B; the resulting untwisted unlinked A and B have a different Hopf index
than before. The second unlinking event, in which the Hopf index changes, is
an instanton.

Figure 3: Local processes which preserve the Hopf index (17). (a)-(b) A
symmetric unlinking and re-linking of two loops compensated by twist. (c)-(d)
An exchange of twist between two nearby loops. If the $\chi$ field is not
condensed (as we assume), then its gradient is not strictly tied to the gauge
field ${\bf A}$. This makes it possible for the loop-singularity of
$\boldsymbol{\nabla}\chi$, represented by the dashed ring, to tunnel from one
flux loop to another without a classical energy penalty. Note that the
untwisted loop can continuously shrink to a point. Therefore, flux loops can
be created and annihilated, then linked and unlinked, without changing the
Hopf index.

Hopf index can be protected only if instantons and anti-instantons attract
each other with sufficiently strong “forces” that grow with the distance. Let
us first analyze the protection of the pure gauge-field Hopf index $N_{A}$
given by (9). Instantons must be local events in order to minimize action, so
consider a unit instanton at the origin of space-time; this corresponds to a
change of the Hopf index by $1$ between the times $t<0$ and $t>0$. Then, set
up a 3-sphere manifold $S^{3}$ centered at the origin in the $D=4$ dimensional
space-time. The Hopf index $N_{A}$ calculated on this manifold is $1$, the
number of enclosed instantons (easily seen by stretching the manifold into an
infinite slab with one flat boundary at $t<0$ and another at $t>0$). If
$S^{3}$ has a finite space-time radius $r$, then the naively optimal gauge
field configuration which realizes $N_{A}=1$ qualitatively behaves as
$A_{\mu}\propto 1/|{\bf r}|$, with a dimensionless proportionality constant
and a non-zero curl spread evenly across $S^{3}$. The fixed value of $N_{A}$
in the $r\to 0$ limit cannot be obtained without a singularity of ${\bf A}$,
so, by (10), an instanton corresponds to a singular configuration of $\hat{\bf
n}({\bf x},t)$. This constitutes the topological protection of the Hopf index
under smooth transformations of $\hat{\bf n}$. The imaginary-time action cost
of the instanton’s ${\bf A}$ field up to the distance $R$ from the origin is
at best

$\displaystyle V(R)$ $\displaystyle\propto$ $\displaystyle\int\limits_{|{\bf
x}|<R}\\!\\!\\!d^{4}x\,\left[\left(\frac{\partial{\bf A}}{\partial
t}-\boldsymbol{\nabla}A_{0}\right)^{2}+(\boldsymbol{\nabla}\times{\bf
A})^{2}\right]$ (21) $\displaystyle=$ $\displaystyle\int\limits_{|{\bf
x}|<R}\\!\\!\\!d^{4}x\,\frac{1}{2}(\epsilon_{\mu\nu\alpha\beta}\partial_{\alpha}A_{\beta})^{2}$
$\displaystyle\sim$ $\displaystyle
K\int\limits_{0}^{R}dr\,r^{3}\left(\frac{1}{r^{2}}\right)^{2}=K\log\left(\frac{R}{R_{0}}\right)\
.$

We neglected the cost due to the gradient Lagrangian density
$(\boldsymbol{\nabla}\chi+q{\bf A})^{2}$ because $\chi$ can compensate the
gauge field of the instanton (at expense of becoming singular). When we
introduce an anti-instanton at a distance $R$ from the origin, the behavior
$A_{\mu}\sim 1/r$ becomes neutralized beyond $r>R$ as qualitatively
anticipated in the integral. Therefore, $V(R)$ estimates the weakest possible
interaction potential between two instantons a space-time distance $R$ apart.

Renormalization group analysis Nikolic (2023) shows that the logarithmic
interaction is marginally confining in $D=4$. If such an interaction applies
to every pair of instantons, then the partition function of the instanton gas
suffers from an infra-red divergence. As a consequence, instanton fluctuations
can proceed only if the interaction reconstructs into an exclusive form at
large distances, where every instanton interacts with exactly one anti-
instanton, and vice versa. The gauge flux $\boldsymbol{\nabla}\times{\bf A}$
spreads evenly only near the instantons, and then focuses into narrow tubes
which connect the opposite-charge instantons across larger distances. The
reconstructed potential is eventually linear, $V(R)\propto R$, and instanton
confinement is guarantied at zero temperature. The gauge-field Hopf index (9)
is globally conserved below a finite critical temperature $T_{\textrm{c}A}$
for instanton deconfinement Nikolic (2023).

Now we turn our attention to the enhanced Hopf index $N_{\chi}$ in (17) which
takes the framing regularization into account. Instantons of $N_{\chi}$
require an uncompensated gauge field and cost an extra gradient energy
$(\boldsymbol{\nabla}\chi+q{\bf A})^{2}$. The previous “best-case” scenario
$\partial_{\mu}\chi+qA_{\mu}\sim 1/r$ yields an extremely large gradient
energy at large distances,

$V^{\prime}(R)=\int\limits_{|{\bf
x}|<R}\\!\\!\\!d^{4}x\,(\boldsymbol{\nabla}\chi+q{\bf
A})^{2}\sim\int\limits_{0}^{R}dr\,r^{3}\frac{1}{r^{2}}\propto R^{2}\ .$ (22)

Again, there is a better way. The disturbance of the gradient focuses into
tubes and reduces the instanton interaction to a more manageable linear form,
$V^{\prime}(R)\propto R$. The Hopf index $N_{\chi}$ is topologically protected
below a finite critical temperature $T_{\textrm{c}\chi}$ for instanton
deconfinement.

The protection of $N_{A}$ implies the protection of $N_{\chi}$, but not the
other way round. The more dramatic large-distance behavior of (22) relative to
(21) suggests that instantons of the twist field have a shorter confinement
length than the instantons of the gauge field. Hence, we may expect two
deconfinement phase transitions with $T_{\textrm{c}\chi}>T_{\textrm{c}A}$. The
lower-temperature confined phase preserves the flux loop entanglement and
possibly conserves some knot invariants Alexander (1928); Jones (1985, 1987);
Kauffman (1987); Witten (1989); Livingston ; Morishita (2012); Purcell (2020)
other than the linking number $N_{A}$. This is the only confined phase of Hopf
instantons in a pure gauge theory, realized in the continuum non-compact
limit. If matter is coupled to the gauge field, then the conservation of
$N_{A}$ requires at least a short-range Higgs mechanism across a coherence
length scale $\zeta$ which exceeds the instanton confinement length
$\lambda_{A}$. This insight comes from the renormalization group Nikolic
(2023). The flow into the fixed point $\zeta>\lambda_{A},\lambda_{\chi}\to 0$
indicates instanton confinement even in the absence of long-range order
($\zeta<\infty$), according to a generalized Wilson loop correlation. We
suspect that the conservation of $N_{A}$ is a necessary (probably not
sufficient) condition for the stability of topological orders defined by loop
braiding Walker and Wang (2012); Wang and Levin (2014); Jiang  _et al._
(2014); Jian and Qi (2014); Fuji and Furusaki (2019); Lan  _et al._ (2018);
Lan and Wen (2019); Wang  _et al._ (2020); Thakurathi and Burkov (2020).

The higher-temperature confined phase in the temperature range
$T_{\textrm{c}\chi}>T>T_{\textrm{c}A}$ is possible only when the gauge field
is coupled to an independent matter field. It preserves a single knot
invariant, the Hopf index $N_{\chi}$. There is no surviving loop coherence.
Note that (17) is not the same as (9) with a redefined gauge field ${\bf A}\to
q^{-1}\boldsymbol{\nabla}\chi+{\bf A}$; the curl of $\boldsymbol{\nabla}\chi$
is not included in the Hopf index definition because we assume that an
underlying lattice regularization renders the quantized $\chi$-vortex
filaments physically unobservable. This enables $N_{\chi}$ to acquire value
$1$ from the twist of a single loop, while $N_{A}$ without framing is always
even.

Incompressible quantum liquids with topological order are a broad class of
phases whose confined instantons cannot ruin the conservation of delocalized
topological charge. The realizations of topological order in all
$\pi_{n}(S^{n})$ homotopy groups are based on singular topological defects
(monopoles and hedgehogs). The number conservation of mobile singularities is
captured by a topological Lagrangian term in the effective Lagrangian density,
i.e. a Chern-Simons or background-field coupling that arises from some
microscopic Berry phase mechanism. This implements particle-singularity
attachment, a fractional quantization of charge and exchange statistics, as
well as ground state degeneracy on topological manifolds at the level of
classical field equations. The dynamics of hopfions is fundamentally different
because the field configurations of hopfions do not have singularities in
$d=3$ spatial dimensions. A full topological Lagrangian term for hopfions can
be constructed only in $d=4$ spatial dimensions, leading to topological order
which we discuss in Section IV. Its remnant in $d=3$ dimensions is only a
coupling that regulates the dynamics of instantons. The instanton term can
indirectly drive instanton confinement transitions by controlling the coupling
constants in (21) and (22), but the absence of symmetry breaking hides this
transition in the classical field equations. Furthermore, there is no
mechanism for fractionalization because this depends upon the ability of
particles with a conserved number to bind singular topological defects with a
conserved number.

We can compare hopfions to skyrmions in $d=2$, which are also classical
topological defects without singularities. Skyrmions are the only topological
defects of the three-component unit-vector field $\hat{\bf n}$ in $d=2$. While
the skyrmion number is a topological invariant given by (1) and (2), it is not
protected against instantons which take form of $\hat{\bf n}$-hedgehogs in the
$D=3$ space-time. The interaction between a hedgehog and anti-hedgehog is
given by the Coulomb potential, $V(R)\sim 1/R$ in $D=3$. This is not a
confining potential, and the renormalization group Kosterlitz (1977) reveals
that instantons are unavoidably deconfined. Therefore, the skyrmion number
necessarily fluctuates in quantum magnets. A physical consequence is that the
topological Hall effect shaped by skyrmions cannot be topologically ordered
and fractional, and there is no “pseudogap” state with confined instantons
below a finite critical temperature.

###  III.2 Field equations for hopfion dynamics

Here we study the classical dynamics of coupled charge and spin degrees of
freedom. We will find that the dynamics of hopfions is left out and entirely
expressed in quantum processes. Without having to worry about the particle
exchange statistics, we will derive the classical equations of motion from a
simple bosonic XY-like model. A basic Hamiltonian density of charged particles
with field operator $\psi=\psi_{0}e^{i\theta}$ and local magnetic moments
(spins) $\hat{\bf n}$ is:

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\frac{\kappa_{c}}{2}\left(\partial_{i}\theta\\!+\\!a_{i}\\!+\\!qA_{i}\right)^{2}+\frac{\kappa_{s}}{2}(\partial_{i}\hat{n}^{a})^{2}-\mu
b_{a}\hat{n}^{a}$
$\displaystyle+\frac{\kappa_{t}}{2}\left(\partial_{i}\chi\\!+\\!qA_{i}\right)^{2}+\frac{1}{2e^{2}}(\epsilon_{ijk}\partial_{j}a_{k})^{2}$
$\displaystyle+C\left(\epsilon_{ijk}\partial_{j}A_{k}-\frac{1}{2}\epsilon_{ijk}\epsilon^{abc}\hat{n}^{a}(\partial_{j}\hat{n}^{b})(\partial_{k}\hat{n}^{c})\right)^{2}\
.$

The particles are coupled as usual to the electromagnetic gauge field ${\bf
a}$, and the spins experience Zeeman coupling to the magnetic field
$b_{i}=\epsilon_{ijk}\partial_{j}a_{k}$. We introduced an auxiliary gauge
field ${\bf A}$ and pinned it to the spin chirality with the $C$ term in order
to track the Hopf index of the spin configuration. ${\bf A}$ also couples to
the charged particles in order to capture topological Hall effect (THE) and
the chiral spin interaction induced by the magnetic field. THE arises because
electrons carry both charge and spin; we assume adiabatic limit in which the
electron spin is strictly aligned with the background magnetization $\hat{\bf
n}$ by a strong Kondo-type coupling, so that a non-zero spin chirality (1)
generates an effective Lorentz force which acts exactly as an external
magnetic field. The factor $q=1/2$ converts the $4\pi$ skyrmion quantum of the
chirality flux to the corresponding $2\pi$ magnetic flux quantum. Integrating
out ${\bf A}$ reveals the chiral interaction $(\boldsymbol{\nabla}\times{\bf
a}){\bf J}$ between the external magnetic field and spin chirality ${\bf J}$.
Together, these physical effects are a topological interaction between
magnetic moments and the electromagnetic field. By the virtue of THE, the
charged matter field $\theta$ could play the role of a framing or twist field
for skyrmion flux loops. Nevertheless, we must couple ${\bf A}$ to a separate
dedicated twist field $\chi$ in order to provide independent framing in the
charge and spin sectors; $\chi$ physically implements a restoring torsion
force against the skyrmion twist. Hopfions in the charge sector are indirectly
governed by the Maxwell term for ${\bf a}$.

The Lagrangian density in real time is obtained by promoting the spatial
indices $i,j,k$ into upper and lower space-tine indices $\mu,\nu,\lambda$ and
relating them via a metric tensor $g=\textrm{diag}(1,-1,-1,-1)$ which
implements a sign reversal of all Hamiltonian terms:

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{\kappa_{c}}{2}\left(\partial_{\mu}\theta\\!+\\!a_{\mu}+qA_{\mu}\right)^{2}+\frac{\kappa_{s}}{2}(\partial_{\mu}\hat{n}^{a})^{2}+\mu
b_{a}\hat{n}^{a}+\mathcal{L}_{\textrm{B}}$
$\displaystyle+\frac{\kappa_{t}}{2}\left(\partial_{\mu}\chi\\!+\\!qA_{\mu}\right)^{2}+\frac{1}{4e^{2}}(\epsilon^{\mu\nu\alpha\beta}\partial_{\alpha}a_{\beta})^{2}$
$\displaystyle+\frac{C}{2}\left(\epsilon^{\mu\nu\alpha\beta}\partial_{\alpha}A_{\beta}-\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}\epsilon_{abc}\hat{n}^{a}(\partial_{\alpha}\hat{n}^{b})(\partial_{\beta}\hat{n}^{c})\right)^{2}$

Some coupling constants have been adjusted in order to retain the original
spatial-index content, and it is understood that the unpaired indices are
contracted by squaring, e.g. $(f_{\mu\nu\lambda}g^{\lambda})^{2}\equiv
f_{\mu\nu\alpha}g^{\alpha}f^{\mu\nu\beta}g_{\beta}$. Ferromagnetic spin
dynamics requires special care as it involves a Berry phase term
$\mathcal{L}_{\textrm{B}}$; analogous theory can be written for
antiferromagnetic spins represented by a smooth “staggered magnetization”
$\hat{\bf n}$, but then the Berry phase, Zeeman interaction and the coupling
of ${\bf A}$ to charged currents would all be gone. We will derive the field
equations from the stationary action condition and express them using the
current densities of charge $j_{\mu}^{\phantom{x}}$, twist
$J_{\mu}^{\phantom{x}}$ and spin $J_{\mu}^{a}$

$\displaystyle j_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\theta+a_{\mu}+qA_{\mu}$ (25) $\displaystyle
J_{\mu}$ $\displaystyle=$ $\displaystyle\partial_{\mu}\chi+qA_{\mu}$
$\displaystyle J_{\mu}^{a}$ $\displaystyle=$
$\displaystyle\epsilon_{abc}^{\phantom{x}}\hat{n}^{b}\partial_{\mu}^{\phantom{x}}\hat{n}^{c}\
,$

as well as the topological currents

$\displaystyle J^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}\epsilon_{abc}^{\phantom{x}}\hat{n}^{a}(\partial_{\alpha}^{\phantom{x}}\hat{n}^{b})(\partial_{\beta}^{\phantom{x}}\hat{n}^{c})$
(26) $\displaystyle\mathcal{I}^{\mu}_{\textrm{s}}$ $\displaystyle=$
$\displaystyle\frac{1}{q\phi_{0}^{2}}\epsilon^{\mu\nu\alpha\beta}(\partial_{\nu}^{\phantom{x}}\chi+qA_{\nu}^{\phantom{x}})\partial_{\alpha}A_{\beta}^{\phantom{x}}$
$\displaystyle\mathcal{I}^{\mu}_{\textrm{c}}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}}\epsilon^{\mu\nu\alpha\beta}(\partial_{\nu}^{\phantom{x}}\theta+a_{\nu}^{\phantom{x}}+qA_{\nu}^{\phantom{x}})\partial_{\alpha}a_{\beta}^{\phantom{x}}\
.$

The spin chirality tensor $J^{\mu\nu}$ characterizes topological textures of
the spins in space-time, such as skyrmions, while
$\mathcal{I}^{\mu}_{\textrm{s}}$ and $\mathcal{I}^{\mu}_{\textrm{c}}$ are the
current densities of the Hopf index in the spin and charge sectors
respectively. Generally, there is one independent Hopf index for every U(1)
gauge transformation in the theory. We will also use the field tensors

$f_{\mu\nu}=\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}\quad,\quad
F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\ .$ (27)

Without the Hopf currents, the Lagrangian density constructed so far makes no
connection to the Hopf index. Nevertheless, its classical dynamics is very
rich and it is useful to explore it before introducing additional couplings to
$\mathcal{I}^{\mu}_{\textrm{s}}$ and $\mathcal{I}^{\mu}_{\textrm{c}}$. The
field equations obtained from the stationary action include current
conservation laws ($\theta$, $\chi$ variations)

$\partial_{\mu}j^{\mu}=0\quad,\quad\partial_{\mu}J^{\mu}=0\quad,$ (28)

Maxwell equation (${\bf a}$, ${\bf A}$ variations)

$\kappa_{c}j^{\mu}-\frac{1}{e^{2}}\partial_{\nu}f^{\mu\nu}=0\ ,$ (29)
$\kappa_{c}\,q\,j^{\mu}-2C\left(\partial_{\nu}F^{\mu\nu}+\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}\partial_{\nu}J_{\alpha\beta}\right)=0\
,$ (30)

and the spin wave equation ($\hat{\bf n}$ variations)

$\displaystyle
S\epsilon^{abc}\hat{n}^{b}\partial_{0}\hat{n}^{c}+(\delta^{ab}-\hat{n}^{a}\hat{n}^{b})\biggl{\\{}-\kappa_{s}^{\phantom{x}}\partial_{\mu}\partial^{\mu}\hat{n}^{b}+\mu
b^{b}$ (31)
$\displaystyle\qquad+\frac{3C}{2}\left(F^{\alpha\beta}+\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}J_{\mu\nu}^{\phantom{x}}\right)(\partial_{\alpha}^{\phantom{x}}J_{\beta}^{b}-\partial_{\beta}^{\phantom{x}}J_{\alpha}^{b})\biggr{\\}}=0$

The first term comes from the Berry phase $\mathcal{L}_{\textrm{B}}$, where
$S$ is the quantum spin magnitude in the units of $\hbar$. This captures the
ferromagnetic spin precession in the internal and external magnetic field.

Perhaps the most interesting aspect of the dynamics in this system is the
transfer of gauge flux between the charge and spin sectors. If we express the
charge current curl $\epsilon^{\mu\nu\alpha\beta}\partial_{\alpha}j_{\beta}$
with (29) and substitute in it the $j_{\mu}$ definition (25), we reveal a
correlation between the electromagnetic and spin-chirality gauge fields
(assuming the absence of monopoles)

$f_{\mu\nu}+qF_{\mu\nu}=-\frac{1}{\kappa_{c}e^{2}}\partial_{\lambda}\partial^{\lambda}f_{\mu\nu}\approx
0\ .$ (32)

We will mainly consider slowly and smoothly varying electromagnetic fields,
and neglect the right-hand side in this formula. At the same time,
$F_{\mu\nu}$ shapes the spin chirality $J_{\mu\nu}$. Defining

$W_{\mu\nu}=F_{\mu\nu}+\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}J^{\alpha\beta}\
,$ (33)

the field equation (30)

$\displaystyle\partial^{\nu}W_{\mu\nu}=\frac{q\kappa_{c}}{2C}j_{\mu}$ (34)

can be easily solved in the absence of charge currents with an arbitrary
vector field $w_{\mu}$:

$j_{\mu}=0\quad\Rightarrow\quad
W_{\mu\nu}=\epsilon_{\mu\nu\alpha\beta}\partial^{\alpha}w^{\beta}\ .$ (35)

In these conditions ($q=1/2$),

$J_{\mu\nu}=-\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}F^{\alpha\beta}+\partial_{\mu}w_{\nu}-\partial_{\nu}w_{\mu}$
(36)
$\frac{1}{2}\hat{n}^{a}(\partial_{\mu}^{\phantom{x}}J_{\nu}^{a}-\partial_{\nu}^{\phantom{x}}J_{\mu}^{a})=-F_{\mu\nu}^{\phantom{x}}+\epsilon_{\mu\nu\alpha\beta}^{\phantom{x}}\partial^{\alpha}w^{\beta}$

relate the spin currents and chirality to the internal electromagnetic field
$F_{\mu\nu}$, or equivalently to the external electromagnetic field
$f_{\mu\nu}$ by the virtue of (32). Evidently, $w_{\mu}$ represents the
independent spin wave content. The dynamics of spin waves is governed in the
field equation (31). After some algebra, one finds that spin waves in the
absence of charge currents propagate according to:

$\displaystyle
S\epsilon^{abc}\hat{n}^{b}\partial_{0}\hat{n}^{c}-(\delta^{ab}-\hat{n}^{a}\hat{n}^{b})(\kappa_{s}^{\phantom{x}}\partial_{\mu}\partial^{\mu}\hat{n}^{b}-\mu
b^{b})=$
$\displaystyle\qquad=-3CF^{\mu\nu}\left(\partial_{\mu}^{\phantom{x}}J_{\nu}^{a}-\frac{1}{2}\hat{n}^{a}\epsilon_{\mu\nu\alpha\beta}J^{\alpha\beta}\right)\
.$ (37)

The term $\epsilon_{\mu\nu\alpha\beta}F^{\mu\nu}J^{\alpha\beta}$ on the right-
hand side is the chiral interaction in disguise,

$\displaystyle\frac{1}{4}\epsilon_{\mu\nu\alpha\beta}F^{\mu\nu}J^{\alpha\beta}$
$\displaystyle=$ $\displaystyle
B^{i}\frac{1}{2}\epsilon_{ijk}^{\phantom{x}}\epsilon^{abc}\hat{n}^{a}(\partial_{j}^{\phantom{x}}\hat{n}^{b})(\partial_{k}^{\phantom{x}}\hat{n}^{c})$
$\displaystyle-E^{i}\epsilon^{abc}\hat{n}^{a}(\partial_{0}^{\phantom{x}}\hat{n}^{b})(\partial_{i}^{\phantom{x}}\hat{n}^{c})\
,$

where $E^{i}$ and $B^{i}$ are the spin-chirality electric and magnetic fields
respectively, correlated with the external electromagnetic field by (32).
Consequently, the spin chirality can be induced by an external magnetic field
and moved around by an electric field. This affects the dispersion of spin
waves in the complicated manner captured by (III.2). Qualitatively, the
propagation of spin waves can mimic the features of particle propagation in
magnetic fields. The presence of charge density and currents further
complicates the equations by introducing sources to $W_{\mu\nu}$ in (34).

In order to explore the effects of hopfion dynamics in the field equations, we
need additional real-time Lagrangian density terms related to hopfions. When
topological order is possible, a Lagrangian term of the form
$j_{\mu}\mathcal{J}^{\mu}$ implements the conservation
$\partial_{\mu}\mathcal{J}^{\mu}=0$ of the topological defect current at the
level of field equations. The topological conservation law formally arises due
to abundant phase $\theta$ fluctuations in the particle current
$j_{\mu}\sim\partial_{\mu}\theta+a_{\mu}$, with a remnant
$a_{\mu}\mathcal{J}^{\mu}$ given by the Chern-Simons or generalized
background-field coupling in the context of $\pi_{d-1}(S^{d-1})$ homotopy
groups. A consequence of such a coupling is the attachment of charge to
topological defects, characteristic for topologically ordered phases. A
singular topological defect of a spin vector field $\hat{\bf n}$ in $d=3$
dimensions is a hedgehog, and indeed topologically ordered quantum liquids of
hedgehogs are possible. However, hopfions are $d=3$ classical topological
defects without singularities, similar to skyrmions in $d=2$. A topological
current of hopfion singularities can be constructed only in $d=4$ spatial
dimensions, i.e. $D=d+1=5$ dimensional space-time:

$\mathcal{J}^{\mu}\sim\epsilon^{\mu\nu\alpha\beta\gamma}\partial_{\nu}(A_{\alpha}\partial_{\beta}A_{\gamma})\equiv\partial_{\nu}\mathcal{I}^{\mu\nu}\xrightarrow{D=4}{}\partial_{\nu}\mathcal{I}^{\nu}\
.$ (38)

Its projection to the $D=4$ real-world, with one index stripped, is the
divergence $\partial_{\nu}\mathcal{I}^{\nu}$ of the Hopf index current density
(26). We cannot construct a topological Lagrangian term
$j_{\mu}\mathcal{J}^{\mu}$ in $D=4$, but

$\mathcal{L}_{i}^{\phantom{x}}=K_{i}^{\textrm{c}}\,(\partial_{\mu}^{\phantom{x}}\mathcal{I}^{\mu}_{\textrm{c}})^{2}+K_{i}^{\textrm{s}}\,(\partial_{\mu}^{\phantom{x}}\mathcal{I}^{\mu}_{\textrm{s}})^{2}$
(39)

is permissible and shapes the dynamics of instantons. Hopf index conservation
amounts to $\partial_{\mu}\mathcal{I}^{\mu}=0$, so $\mathcal{L}_{i}$ regulates
the amount of fluctuations in which the topological charge conservation is
locally violated.

Unfortunately, the instanton Lagrangian has no impact on the field equations.
The $A_{\mu}$ variations of $\mathcal{L}_{i}$ cannot introduce any correction
to (30) because $\partial_{\mu}\mathcal{I}^{\mu}$ is a real scalar lacking the
structure that could support topological singularities. For example, in the
spin sector with $a_{\mu}=0$:

$\displaystyle\frac{\delta\mathcal{L}_{i}}{\delta A_{\sigma}}$
$\displaystyle=$
$\displaystyle\frac{2K_{i}}{\phi_{0}^{4}}(\partial_{\mu}\mathcal{I}^{\mu})\frac{\delta}{\delta
A_{\sigma}}\epsilon^{\alpha\beta\gamma\delta}\Bigl{(}\partial_{\alpha}(A_{\beta}\partial_{\gamma}A_{\delta})\Bigr{)}$
(40) $\displaystyle\to$
$\displaystyle\frac{4K_{i}}{\phi_{0}^{4}}\epsilon^{\sigma\alpha\beta\gamma}A_{\alpha}(\partial_{\beta}\partial_{\gamma}\partial_{\mu}\mathcal{I}^{\mu})=0$

(note that integrations-by-parts were carried out assuming the absence of
monopoles,
$\epsilon^{\mu\alpha\beta\gamma}\partial_{\alpha}\partial_{\beta}A_{\gamma}=0$).
While instantons apparently do not influence the classical dynamics, they do
leave a signature in the quantum noise.

Is there any other hopfion mechanism for the phenomenology of topological
order? A method for analyzing the topological ground state degeneracy,
presented in Section IV.3, finds that no topological order of hopfions is
possible in $d=3$ spatial dimensions. Nevertheless, we can attempt to obtain
some form of charge-hopfion attachment in the $d=3$ field equations by
considering another addition to the Lagrangian density,

$\mathcal{L}_{t}^{\phantom{x}}=-K_{t}^{\textrm{s}}\rho\,j_{\mu}^{\phantom{x}}\mathcal{I}^{\mu}_{\textrm{s}}-K_{t}^{\textrm{c}}\rho(\partial_{\mu}^{\phantom{x}}\chi+qA_{\mu}^{\phantom{x}})\mathcal{I}^{\mu}_{\textrm{c}}\
.$

This introduces various complications in the field equations, and a notable
correction to (29) which takes a simple form in the limit $a_{\mu}\to 0$:

$j^{\mu}=\frac{K_{t}^{\textrm{s}}\rho}{\kappa_{c}}\mathcal{I}^{\mu}_{\textrm{s}}\
.$ (41)

This is problematic for several reasons. The charge density $j^{0}$ is
effectively bound to the Hopf index density $\mathcal{I}_{0}$ and spread-out
over an extended region in space where $\mathcal{I}_{0}\neq 0$. There is no
singular point to which the charge can attach. Since fluctuations can smoothly
vary the spatial distribution of $\mathcal{I}_{0}$, no local quasiparticles
can carry a quantized fractional amount of charge despite the quantization of
the total Hopf index. Hence, we cannot expect topological order from
$\mathcal{L}_{t}$. We also do not expect the appearance of $\mathcal{L}_{t}$
in the effective theories of realistic systems. Topological terms are normally
generated by a microscopic Berry curvature, which inserts the static
topological current $\mathcal{J}_{\mu}$ into the Hamiltonian instead of
$\mathcal{I}_{\mu}$. If, alternatively, a certain two-particle interaction
were able to generate $\mathcal{L}_{t}$, then the amount $\nu$ of attached
charge per topological flux quantum would scale in proportion to the particle
density $\rho$; this is also incompatible with topological order because the
co-mobility of particles and their topological defects requires an
incompressible state with a constant and quantized filling factor $\nu$.

###  III.3 Thermodynamic signatures and the chiral anomaly

The confined phase of hopfion instantons can be physically identified despite
lacking obvious signatures in classical observables. The key to this is
quantum noise. Instanton confinement is sharply characterized by a
generalization of the Wilson loop operator

$\mathcal{C}(S^{3})=\oint\limits_{S^{3}}d^{3}x\,\hat{\eta}_{\mu}\mathcal{I}^{\mu}\
,$ (42)

which is defined on a 3-sphere $S^{3}$ embedded in $D=4$ space-time;
$\hat{\eta}_{\mu}$ is a unit-vector locally perpendicular to the sphere.
According to the definition of the Hopf index (17) and its current density
$\mathcal{I}_{\mu}$ (26), this counts the Hopf-instanton charge, i.e. the
number of instantons minus the number of anti-instantons inside the space-time
volume $B^{4}$ bounded by the closed “surface” $S^{3}$. The variance of the
random outcomes in the quantum measurements of $\mathcal{C}(S^{3})$ is a
thermodynamic characterization of the instanton phase Nikolic (2023):

$\textrm{Var}\,\mathcal{C}(S^{3})\xrightarrow{S^{3}\to\infty}{}\begin{cases}aS^{3}&,\quad\textrm{confined
phase}\\\ bB^{4}&,\quad\textrm{deconfined phase}\end{cases}\ .$ (43)

If the instantons are confined, then they contribute to (42) only within a
finite distance from the “surface” $S^{3}$ given by their confinement length
$\lambda$; every instanton is compensated further away, so that
$\textrm{Var}\,\mathcal{C}(S^{3})$ grows in proportion to the “area” of the
boundary $S^{3}$. Otherwise, topological instanton fluctuations matter in the
entire space-time and $\textrm{Var}\,\mathcal{C}(S^{3})$ scales in proportion
to the bounded volume $B^{4}$.

The Hopf index is conserved in the confined instanton phase. This means in
practice that the noise spectrum of the Hopf index fluctuations is depleted at
low frequencies Nikolic (2023). Quantum noise quickly fills up the low-
frequency spectrum upon entering the deconfined phase. The main challenge for
the detection of this spectrum change is whether one can indirectly measure
the Hopf index of the system. In the worst case scenario, we don’t have any
other probe but the heat capacity. The onset of low-frequency noise is naively
expected to boost the heat capacity across the critical temperature for
instanton deconfinement Nikolic (2023). In some cases, it might be possible to
indirectly probe the Hopf index fluctuations with quantum noise measurements.
Such experiments would be certainly difficult for a variety of reasons, but
they could, for example, exploit the chiral quantum anomaly in materials with
a Dirac and Weyl electron spectrum. There is also a general possibility that a
confined instanton phase could be identified by unconventional transport
properties or observable local fluctuations, but it is not presently clear how
and it could be system-specific.

The chiral anomaly of quantum electrodynamics is, actually, closely related to
the Hopf index physics. It relates the chiral current non-conservation to the
electromagnetic field fluctuations:

$\partial_{\mu}j^{5\mu}=\frac{1}{16\pi^{2}}\,\epsilon^{\mu\nu\alpha\beta}f_{\mu\nu}f_{\alpha\beta}$
(44)

when the Dirac quasiparticle mass $m$ vanishes Adler (1969); Bell and Jackiw
(1969); Fujikawa (1980); Nielsen and Ninomiya (1981, 1983); Peskin and
Schroeder (1995); Zinn-Justin (2005). This result was originally obtained in
the perturbation theory by introducing gauge-invariant regularizations of
three-vertex one-loop Feynman diagrams Adler (1969); Bell and Jackiw (1969).
Different regularization schemes, each complicated in its own way, point to
the same conclusion. The physics of this is peculiar because a massless Dirac
theory possesses the chiral symmetry and conserves the chiral current
$j^{5\mu}$ at the level of field equations. It is the quantum fluctuations
which necessarily violate the chiral symmetry, for the reasons which are not
made very transparent in the perturbation theory. For our purposes, we
immediately recognize the hopfion current

$\mathcal{I}^{\mu}=\frac{1}{(2\pi)^{2}}\,\epsilon^{\mu\nu\alpha\beta}a_{\nu}\partial_{\alpha}a_{\beta}$
(45)

of the electromagnetic field in the charge sector ($\phi_{0}=2\pi$) on the
right-hand side of (44):

$\partial_{\mu}j^{5\mu}=\frac{1}{16\pi^{2}}\,\epsilon^{\mu\nu\alpha\beta}f_{\mu\nu}f_{\alpha\beta}=\partial_{\mu}\mathcal{I}^{\mu}\
.$ (46)

This provides new insight: the local events which violate the chiral
conservation law are bound to the Hopf instantons. While particles are
attached to topological defects in topologically ordered phases, here we have
attachment of particle creation/annihilation events to the analogous events
for topological defects. Many interesting questions arise from this similarity
between the topological order and quantum anomaly. Are both of them
manifestations of correlations caused by instanton confinement? If yes, then
can a quantum anomaly be fractionalized? Is the chiral anomaly a relativistic
variant of the non-relativistic topological order?

We can gain valuable insight into these questions from the following
transparent physical picture of the chiral anomaly. For simplicity, and in
order to relate to the traditional context, let us scrutinize quantum
electrodynamics whose real-time Lagrangian density is

$\mathcal{L}=\psi^{\dagger}(i\mathcal{D}-\gamma^{0}m)\psi-\frac{1}{4}f_{\mu\nu}f^{\mu\nu}\
.$ (47)

$\gamma^{\mu}$ are the standard Dirac matrices. We must regularize this theory
on a lattice in order to provide flux quantization for the gauge field
$a_{\mu}$, a prerequisite for the Hopf index quantization. We first introduce
a regulator which breaks the Lorentz invariance at high energies,

$\mathcal{D}=D_{0}+\frac{i\gamma^{0}}{2M}(D_{i}D^{i}+\mathcal{A}_{i}\mathcal{A}^{i})\quad,\quad
D_{\mu}=\partial_{\mu}+ia_{\mu}+i\mathcal{A}_{\mu}$ (48)

featuring a large mass $M$ and a static background SU(2) gauge field

$\mathcal{A}_{0}=0\quad,\quad\mathcal{A}_{i}=-M\gamma_{i}\ .$ (49)

The usual relativistic Dirac theory
$\mathcal{L}=\bar{\psi}(i\cancel{D}-m)\psi$ with
$\bar{\psi}=\psi^{\dagger}\gamma^{0}$ and
$\cancel{D}=\gamma^{\mu}(\partial_{\mu}+ia_{\mu})$ is recovered in the
$M\to\infty$ limit. This regularization is physically transparent and converts
the QED into the continuum limit description of a condensed matter system with
a Berry curvature. It is now easy to construct a lattice Hamiltonian

$H=-t\sum_{\bf r}\sum_{\mu}\psi_{\bf r}^{\dagger}e^{i(a_{{\bf
r},\mu}+\mathcal{A}_{{\bf r},\mu})}\psi_{{\bf
r}+\hat{\boldsymbol{\mu}}}^{\phantom{\dagger}}+h.c.+\Delta H\ ,$ (50)

where ${\bf r}$ are lattice sites and $\mu$ are the directions to nearest-
neighbor sites. The continuum limit of $H$ reproduces the Hamiltonian of the
theory (47). Some spurious two-fold degenerate Dirac nodes might arise at the
corners of the first Brillouin zone due to time-reversal and lattice
symmetries; these are easily gapped-out to high energies by appropriate
chirality-altering hopping terms $\Delta H$, without disturbing the Dirac node
at the $\Gamma$ point.

In the pristine QED with $M\to\infty$, the structure:

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\gamma^{0}=\gamma^{0\dagger}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)=\tau^{z}\otimes 1$ (53) $\displaystyle\alpha^{k}$
$\displaystyle=$
$\displaystyle\alpha^{k\dagger}=\gamma^{0}\gamma^{k}=\left(\begin{array}[]{cc}0&\sigma^{k}\\\
\sigma^{k}&0\end{array}\right)=\tau^{x}\otimes\sigma^{k}$ (56)
$\gamma^{5}=\gamma^{5\dagger}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)=\tau^{x}\otimes 1$

expressed in terms of Pauli matrices $\sigma^{a}$ and $\tau^{a}$ produces the
Hamiltonian

$H=-i\alpha^{k}(\partial_{k}+ia_{k})+\gamma^{0}m\;\xrightarrow[a_{\mu}\to
0]{m\to 0}\;\alpha^{k}p_{k}\ .$ (57)

Massless $m=0$ Dirac electrons have the spectrum

$E_{\tau\sigma{\bf p}}=\tau\sigma|{\bf p}|\ .$ (58)

Since the matrices $\sigma^{a}$ couple to spin, the eigenvalue $\sigma=\pm 1$
of $\boldsymbol{\sigma}\hat{{\bf p}}$ is helicity (alignment or anti-alignment
between the electron’s spin and momentum). The $\tau=\pm 1$ eigenvalue of
$\tau^{x}$ (as well as $\gamma^{5}$) is chirality. The band index
$s=\tau\sigma$ is the product of helicity and chirality. The continuum-limit
regularization (48) only adds

$\delta H=\gamma^{0}\frac{(p_{i}+a_{i})^{2}}{2M}$ (59)

to the Dirac Hamiltonian (57). This preserves the particle-hole symmetry, but
explicitly breaks the chiral symmetry of QED at high energies. Photon
absorption and emission that violates chiral current conservation is also made
possible, with an amplitude proportional to $M^{-1}$. Note that removing the
factor of $\gamma^{0}$ to restore the chiral symmetry is not permissible
because, without it, the spectrum would possess a large Fermi surface that
coexists with the Dirac node.

The regularization of QED plays an important role in the emergence of the
chiral anomaly, and simultaneously reveals the origin of the anomaly in
condensed matter systems. However, the physical picture which we will now
construct invokes the regularization only in very subtle ways. The chiral
anomaly is a correlation between the changes of chirality and the topological
Hopf index. The classical field equations are derived from small and smooth
variations of the fields, which by definition cannot capture the changes of a
topological invariant. This is why the field equations fail to reveal the
quantum anomaly. We will show next that as long as there is some microscopic
mechanism (provided by the regularization) for the electromagnetic field
fluctuations to alter electrons’ chirality, the topological considerations
alone link the chirality fluctuations to the Hopf index.

Figure 4: Two-photon absorption behind the chiral quantum anomaly, analogous
(after time reversal) to the neutral pion decay $\pi^{0}\to\gamma+\gamma$. A
massless Dirac electron in the occupied valence band has a well-defined
chirality, say $\tau=-1$. This fixes its helicity to $\sigma=1$. By absorbing
a photon ($\gamma$) with right-handed circular polarization, the electron
flips its spin and gains some energy, leaving a hole in the valence band. This
removal of $\tau=-1$ chirality from the valence band is equivalent to the
injection of $\tau=1$. The absorption of a second right-handed photon, moving
in the opposite direction from the first photon, flips the electron’s spin
again. The final particle excitation has chirality $\tau=1$ because it has the
same helicity $\sigma=1$ (same spin and momentum) as the original electron but
lives in a different band ($\sigma\tau$ is the band index, see Eq.58).
Therefore, the particle-hole pair, created by absorbing a spinless photon
pair, has no net angular momentum and carries the net chirality $\tau=2$. The
pristine massless QED forbids the above second photon process in which the
electron’s chirality changes. However, the required physical regularization
necessarily introduces a chirality-changing photon absorption in order to
protect the gauge invariance.

Particle-hole excitations which carry a neutral chiral current

$j^{\mu}=\bar{\psi}\gamma^{\mu}\psi=0\quad,\quad
j^{5\mu}=\bar{\psi}\gamma^{5}\gamma^{\mu}\psi\neq 0$ (60)

necessarily have a zero net spin due to the spin-momentum locking in the
conduction and valence bands. Therefore, the conservation of charge, momentum
and angular momentum permits only zero-spin photon pairs to create such
excitations, assuming that the chirality need not be conserved, see Fig.4. It
turns out that zero-spin photon fluctuations also change the Hopf index of the
electromagnetic field. This is easy to show by expressing the gauge field
operator ${\bf a}$ in the Coulomb gauge using the creation $a_{{\bf
k},p}^{\dagger}$ and annihilation $a_{{\bf k},p}^{\phantom{\dagger}}$
operators of photons with momentum ${\bf k}$ and linear polarization $p=\pm$:

${\bf a}({\bf
r},t)=\sum_{p}\\!\int\\!\frac{d^{3}k}{(2\pi)^{3}}\frac{\hat{\boldsymbol{\epsilon}}_{1,{\bf
k}}^{p}}{\sqrt{2\omega_{k}}}\Bigl{[}a_{{\bf k},p}^{\phantom{x}}e^{i({\bf
k}{\bf r}-\omega_{k}t)}+h.c.\Bigr{]}$ (61)

The two orthogonal polarization vectors $\hat{\epsilon}_{1,{\bf k}}^{p}$ and
$\hat{\epsilon}_{2,{\bf k}}^{p}$ satisfy $\hat{\boldsymbol{\epsilon}}_{1,{\bf
k}}^{p}\times\hat{\boldsymbol{\epsilon}}_{2,{\bf k}}^{p}=\hat{{\bf k}}$ and
$\hat{\boldsymbol{\epsilon}}_{i,{\bf
k}}^{+}\hat{\boldsymbol{\epsilon}}_{i,{\bf k}}^{-}=0$, and the photon
dispersion is $\omega_{k}=|{\bf k}|$. Substituting into (44) gives us:

$\displaystyle\frac{1}{16\pi^{2}}\int
d^{3}r\,\epsilon^{\mu\nu\alpha\beta}f_{\mu\nu}f_{\alpha\beta}=\frac{\partial
N_{\textrm{Hopf}}}{\partial t}$ (62)
$\displaystyle~{}\to-\frac{1}{2\pi^{2}}\\!\sum_{p,p^{\prime}}\\!\int\\!\\!\frac{d^{3}k}{(2\pi)^{3}}\frac{\omega_{k}}{2}\hat{\boldsymbol{\epsilon}}_{1,{\bf
k}}^{p}\hat{\boldsymbol{\epsilon}}_{2,-{\bf k}}^{p^{\prime}}a_{{\bf
k},p}^{\phantom{x}}a_{-{\bf
k},p^{\prime}}^{\phantom{x}}e^{-2i\omega_{k}t}\\!+\\!h.c.$
$\displaystyle~{}=\frac{1}{8\pi^{2}}\frac{\partial}{\partial
t}\sum_{\sigma}\\!\int\\!\\!\frac{d^{3}k}{(2\pi)^{3}}\,\sigma\,b_{{\bf
k},\sigma}^{\phantom{x}}b_{-{\bf
k},\sigma}^{\phantom{x}}e^{-2i\omega_{k}t}\\!+\\!h.c.$
$\displaystyle~{}\propto\frac{\partial N^{5}}{\partial t}\ .$

Behind the arrow, we retained only the terms which describe the photon
absorption and emission. At the end, we switched to the circular polarization
field operators $b_{{\bf k},\sigma}$, where $\sigma$ is photon’s helicity:
$\sigma=+1$ indicates that photon’s spin is aligned with its momentum ${\bf
k}$, and $\sigma=-1$ means anti-alignment. The final proportionality follows
from the fact that whenever two photons of zero net momentum and spin are
annihilated with $b_{{\bf k},\sigma}^{\phantom{x}}b_{-{\bf
k},\sigma}^{\phantom{x}}$, i.e. successfully absorbed, their energy
$2\omega_{k}$ is used to create an electron-hole pair with net spin zero,
which carries a non-zero chirality in the massless Dirac spectrum. Therefore,
the processes of quantum electrodynamics make it impossible to change the
electron chirality without changing the Hopf index of the gauge field.

Figure 5: The Hopf index and chiral anomaly. (a) Create two unlinked flux-
quantum loops A and B (black lines, $\boldsymbol{\nabla}\times{\bf a}$) and
accompany each by a $2\pi$ phase twist. Let the twists of A and B carry
particle (blue) and hole (red) current respectively governed by the Lagrangian
(III.3); both currents have the same chirality. Dashed and dotted segments are
out of sight, e.g. at infinity. The dotted colored loops represent far-away
vortex singularities of $\theta^{+}$ (blue) and $\theta^{-}$ (red); they are
interlinked with one flux loop each. In this configuration, each twisted flux
loop contributes 1 to the Hopf index, totaling the change $\Delta
N_{\textrm{Hopf}}=2$ according to (67). When the straight flux segments are
smoothly deformed and pushed to overlap, their gauge flux quanta and charge
currents cancel out. What remains is two units of the chiral current,
$\boldsymbol{\nabla}\phi=\boldsymbol{\nabla}(\theta^{+}+\theta^{-})\sim
2\times 2\pi$. Therefore, $\Delta N^{5}=\Delta N_{\textrm{Hopf}}=2$. (b) An
equivalent local process. Creating a pair of interlinked flux loops gives
$\Delta N_{\textrm{Hopf}}=2$. A chiral current $\Delta N^{5}=2$ is left behind
at the end of the process.

In order to find the exact proportionality constant in (62), consider a
qualitative description of massless Dirac particles coupled to a gauge field,

$\displaystyle\mathcal{L}_{\textrm{QED}_{0}}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{2}(\partial_{\mu}\theta^{+}+a_{\mu})^{2}+\frac{\kappa}{2}(\partial_{\mu}\theta^{-}-a_{\mu})^{2}$
$\displaystyle=$
$\displaystyle\frac{\kappa}{4}(\partial_{\mu}\theta+2a_{\mu})^{2}+\frac{\kappa}{4}(\partial_{\mu}\phi)^{2}\
.$

It is not crucial to capture the fermionic statistics of charged matter fields
$\theta^{\pm}$, we only must consider them uncondensed. By construction,
$\theta^{\pm}$ carry opposite charge $\pm 1$ with respect to the U(1) gauge
field $a_{\mu}$, and will represent the massless Dirac particles and holes
which carry the same helicity $\tau$. By particle-hole symmetry,
$\theta^{\pm}$ simultaneously represent particle (charge 1) currents of
opposite helicity. Defining

$\theta=\theta^{+}-\theta^{-}\quad,\quad\phi=\theta^{+}+\theta^{-}$ (64)

allows us to separately track the charge $j^{\mu}$ and chiral $j^{5\mu}$
currents

$j_{\mu}^{\phantom{x}}\sim\partial_{\mu}^{\phantom{x}}\theta+2a_{\mu}^{\phantom{x}}\quad,\quad
j^{5}_{\mu}\sim\partial_{\mu}^{\phantom{x}}\phi\ .$ (65)

This theory has only one gauge transformation

$\theta^{\pm}\to\theta\pm\lambda\quad,\quad a_{\mu}\to
a_{\mu}-\partial_{\mu}\lambda\ ,$ (66)

so we expect only one Hopf index:

$N_{\textrm{Hopf}}(t)=\frac{1}{(2\pi)^{2}}\oint\limits_{S^{3}}d^{3}x\,\epsilon^{ijk}(\partial_{i}\theta+2a_{i})\partial_{j}a_{k}\
.$ (67)

The normalization is fixed by the requirement that the $2\pi$ winding of
either $\theta^{+}$ or $\theta^{-}$ along a flux loop give a unit Hopf index.
According to (12), the inter-linking of flux loops gives only even values of
the Hopf index. Therefore, changing $N_{\textrm{Hopf}}$ by $\pm 1$ necessarily
excites charge (twist) currents. A neutral chiral current
($j^{\phantom{x}}_{\mu}=0$, $j^{5}_{\mu}\neq 0$) carries at least $N^{5}=2$
helicity which has to be matched to some $N_{\textrm{Hopf}}\neq 0$; it turns
out this is $N_{\textrm{Hopf}}=2$. Fig.5 explains the basic process of
creating a particle-hole excitation characterized by $\Delta N^{5}=\Delta
N_{\textrm{Hopf}}=2$, while all other processes can be derived from it by
adding low-energy Hopf-preserving fluctuations. This reproduces the chiral
anomaly

$\Delta N^{5}=\Delta N_{\textrm{Hopf}}\ .$ (68)

From the current perspective, chiral anomaly requires the conservation of the
combined Hopf index (17) of the gauge and twist (matter) fields. This is
putatively realized below the instanton deconfinement temperature
$T_{\textrm{c}2}$. The correlation (68) is lost at $T>T_{\textrm{c}2}$, but
reflects via quantum noise the deconfinement of pure gauge field instantons at
a lower critical temperature $T_{\textrm{c}1}$. Hence, the Hopf index (9) of
the gauge field alone is conserved at $T<T_{\textrm{c}1}$, and its
fluctuations are mirrored by chiral currents at
$T_{\textrm{c}1}<T<T_{\textrm{c}2}$. The latter can be further modified if
interactions produce additional effective gauge fields Rylands  _et al._
(2021) which carry a Hopf index. Note that fermionic matter cannot screen out
the magnetic flux loops, but it can affect the action cost of instantons by
suppressing their electric field fluctuations at sufficiently low frequencies.
The regime $T<T_{\textrm{c}1}$ can be probably accessed with a sufficiently
low fermion density (i.e. chemical potential close to the Dirac node) because
fermions can maintain phase coherence across the mean inter-particle distance
(as in quantum Hall states).

Figure 6: Particle-hole excitations with and without net spin are
distinguished by different energy/momentum transfers from photons when the
spin degeneracy (left) is lifted (right) by a magnetic field or spin-orbit
coupling. (a) The case of Dirac or Weyl spectrum: A spin-flip particle-hole
excitation with the minimum threshold energy $2\epsilon_{\textrm{f}}$ on the
Fermi sea (shaded) occurs with no momentum transfer, while the chiral
excitations created from the quantum anomaly fluctuations require momentum
transfer equal to the momentum separation $\Delta{\bf Q}$ between the
intrinsic or generated Weyl nodes. The relativistic spin-momentum-locked
spectrum makes $\Delta{\bf Q}$ independent of the Fermi energy
$\epsilon_{\textrm{f}}$, but sensitive to a magnetic field. (b) The
conventional bands without spin-momentum locking: The same-spin threshold
transitions still require momentum transfers, but the transferred momentum
also depends on the energy transfer.

In principle, the correlation between currents and the Hopf index can be
exploited to detect instanton deconfinement. The chiral anomaly is just a
specific manifestation of this correlation by Dirac electrons, least difficult
to detect, but converting chiral to charge currents requires finite momentum
transfers between photons and electrons. Since the total momentum of the
electron gas is not conserved, the measurements of charge currents will
exhibit quantum randomness which is correlated with the gauge field dynamics.
We are interested in the resulting quantum noise spectrum. The noise
contributed by virtual zero-spin particle-hole pairs is suppressed at low
frequencies in a confined instanton phase. Fig.6 illustrates that zero-spin
particle-hole fluctuations can be distinguished from the non-zero-spin
fluctuations by momentum and energy transfer when the magnetic field or spin-
orbit coupling lifts the spin degeneracy of the electron spectrum. This is, of
course, a very difficult measurement for several reasons. It requires enough
sensitivity and a subtraction of other well-understood sources of noise. And,
the probability of photon-electron momentum transfer is suppressed by a
negative power of the speed of light. Natural photons carry very small
momenta, so it is really an emergent gauge field (with a speed of light
comparable to a typical electron speed) that would give this method a better
chance. One could devise a dielectric constant superlattice to impart umklapp
scattering on the (emergent) photons that matches the needed momentum
transfers. A layered Weyl heterostructure Burkov and Balents (2011) might just
serve this purpose, and tuning the magnetic field can perhaps bring the
momentum transfer into resonance. While measuring the noise spectrum would be
the least ambiguous detection method, there are other options. An instanton
deconfinement transition should theoretically boost the specific heat Nikolic
(2023), and it may also be accompanied by other observable properties
(transport, for example).

We end this discussion by envisioning a mechanism for the fractional
quantization of the chiral anomaly,

$\Delta N^{5}=\nu\,\Delta N_{\textrm{Hopf}}\quad,\quad\nu=\frac{p}{q}\ .$ (69)

This requires a fractionalization of chirality without spoiling its classical
conservation. Since massless Dirac electrons carry both charge and chirality,
charge fractionalization will generally imply some form of chirality
fractionalization. Charge fractionalization in $d=3$ spatial dimensions
manifestly occurs in nuclear matter (irrespective of the microscopic
mechanism), and can occur by the topological order involving hedgehogs or
monopoles Nikolić (2020a). An effective theory that captures the dynamics of
fractional quasiparticles (partons) always requires an emergent gauge field
$\widetilde{\bf a}$. If $\widetilde{\bf a}$ experiences extremely strong
fluctuations, it necessarily recombines the fractional quasiparticles into
electrons and stabilizes a conventional phase of matter. But, a Coulomb phase
of $\widetilde{\bf a}$ allows the partons to be deconfined because they carry
unit-charge relative to $\widetilde{\bf a}$ (note that the physical U(1) gauge
field may be gapped in order to accommodate monopoles for the topological
order). Since the Coulomb phase hosts flux loops characterized by a conserved
Hopf index of $\widetilde{\bf a}$, the fluctuations of this Hopf index are
correlated with the fluctuations of the _fractionalized_ quasiparticle’s
chirality. A rational quantization of the “filling factor” $\nu$ is inherited
from the parent topological order. Disregarding questions about the
feasibility of $d=3$ fractionalization in materials, the main theoretical
issue is whether the gap of the topologically ordered phase can be
disconnected from the usual chirality-mixing mechanism that gives Dirac
electrons a mass. Naively, the classical chirality conservation should be able
to survive because the energy gap can be produced purely from the destructive
quantum interference caused by mobile topological defects.

##  IV Hopfions in four dimensions: Topological order

Interlinked loops of quantized flux can form topological defects with point
singularities only in $d=4$ spatial dimensions. The topological charge of such
a singularity is given by the Hopf index on a 3-sphere which encloses the
singularity. Neither smooth transformations nor local quantum tunneling can
change the Hopf topological charge of the field configuration. The latter
protects Hopf singularities from quantum fluctuations and enables topological
order with fractionalized charge and braiding statistics. Here we explore the
Hopf topological order for the purposes of adding the $\pi_{3}(S^{2})$
homotopy group to the general topological order classification. Higher order
homotopy groups $\pi_{2n-1}(S^{n})$ which generalize the Hopf singularities to
$d=2n$ spatial dimensions (for even $n$) are also able to support topological
order, but instead of analyzing this in detail we only construct the
corresponding homotopy invariants in Appendices B and C for spin and charge
sectors respectively.

###  IV.1 Charge and angular momentum fractionalization

A method for analyzing the topologically ordered states of spinor fields was
discussed in Ref.Nikolić (2020a). We first need to construct a “singularity
gauge field” whose flux is computed as the Hopf index on an $S^{3}$ manifold
surrounding a Hopf singularity. Then, we can formulate a field theory which
captures the dynamics of this gauge field even when its flux diffuses as a
result of quantum fluctuations. The topological protection mechanism will be
implemented with a topological Lagrangian density term involving the
singularity gauge field.

We already have the basic setup. The fundamental fields carry charge
($\theta$) and magnetic moments ($\hat{\bf n}$) in $d=4$ dimensional space.
Both degrees of freedom can be packed into an SU(2) spinor $\psi({\bf x})$
whose magnitude $|\psi|^{2}=\rho\neq 0$ is fixed:

$\psi=\psi_{0}e^{i\theta}\quad,\quad\hat{\bf
n}=\frac{1}{\rho}\psi^{\dagger}\boldsymbol{\sigma}\psi\ .$ (70)

There are two relevant gauge fields, one $a_{\mu}=\partial_{\mu}\theta$
extracted from the U(1) phase winding of $\psi$ and another $A_{\mu}$
representing the topological Hall effect from the skyrmion textures of
$\hat{\bf n}$. To be concrete, we will focus on the spin sector
($\phi_{0}=4\pi$) and construct the Hopf index on a closed $S^{3}$ spatial
manifold which encloses a volume $B^{4}$,

$N=\frac{1}{\phi_{0}^{2}}\oint\limits_{S^{3}}d^{3}x\,\epsilon_{ijk}A_{i}\partial_{j}A_{k}=\frac{1}{\phi_{0}^{2}}\oint\limits_{S^{3}}d^{3}x\,\epsilon_{ijk}\mathcal{A}_{ijk}\
.$ (71)

The antisymmetric singularity gauge field can be expressed as

$\displaystyle\mathcal{A}_{ijk}$ $\displaystyle=$
$\displaystyle\frac{1}{6}(A_{i}\partial_{j}A_{k}+A_{j}\partial_{k}A_{i}+A_{k}\partial_{i}A_{j}$
$\displaystyle-
A_{k}\partial_{j}A_{i}-A_{j}\partial_{i}A_{k}-A_{i}\partial_{k}A_{j})\ .$

The density $\mathcal{J}_{0}$ of Hopf singularities is revealed by
representing the Hopf index as a $B^{4}$ volume integral using Stokes-Cartan
theorem:

$N=\frac{1}{\phi_{0}^{2}}\int\limits_{B^{4}}d^{4}x\,\mathcal{J}_{0}\ .$ (73)

From this, we can identify the current density of Hopf singularities

$\displaystyle\mathcal{J}^{\mu}$ $\displaystyle=$
$\displaystyle\epsilon^{\mu\nu\alpha\beta\gamma}\partial_{\nu}\mathcal{A}_{\alpha\beta\gamma}=\epsilon^{\mu\nu\alpha\beta\gamma}\partial_{\nu}(A_{\alpha}\partial_{\beta}A_{\gamma})$
$\displaystyle=$
$\displaystyle\epsilon^{\mu\alpha\beta\gamma\delta}(\partial_{\alpha}A_{\beta})(\partial_{\gamma}A_{\delta})-\epsilon^{\mu\alpha\beta\gamma\delta}A_{\alpha}\partial_{\beta}\partial_{\gamma}A_{\delta}\
.$

The last term containing an antisymmetrized second derivative responds only to
the monopole singularities of the gauge field $A_{\mu}$, which are line
defects in $d=4$. We can omit this term if the monopoles have decidedly higher
energy than other excitations. For our purposes, the presence of both charge
and spin degrees of freedom introduces two gauge fields and two gauge
transformations. We will show in Section IV.4 that the gauge invariant
topological current density has an expanded form

$\mathcal{J}^{\mu}=\frac{1}{q^{2}}\epsilon^{\mu\alpha\beta\gamma\delta}\Bigl{(}\partial_{\alpha}(a_{\beta}+qA_{\beta})\Bigr{)}\Bigl{(}\partial_{\gamma}(a_{\delta}+qA_{\delta})\Bigr{)}\
.$ (75)

The topological Lagrangian density term has the purpose to implement the
conservation of topological charge despite the defect delocalization. We can
construct it in real time by coupling the charge current $j_{\mu}$ to the Hopf
singularity current

$\mathcal{L}_{t}=-K_{t}\rho\,j_{\mu}\mathcal{J}^{\mu}\ .$ (76)

Substituting $j_{\mu}=\partial_{\mu}\theta+a_{\mu}+qA_{\mu}$ with
$q\phi_{0}=2\pi$ and integrating out the phase fluctuations $\theta$ gives us
the topological conservation law $\partial_{\mu}\mathcal{J}^{\mu}=0$ (see
Appendix A). What remains is a generalization of the Chern-Simons coupling to
the $\pi_{3}(S^{2})$ homotopy group,

$\displaystyle\mathcal{L}_{t}$ $\displaystyle=$
$\displaystyle-\frac{K_{t}\rho}{q^{2}}\,\epsilon^{\mu\alpha\beta\gamma\delta}(a_{\mu}+qA_{\mu})$
$\displaystyle\quad\times\Bigl{(}\partial_{\alpha}(a_{\beta}+qA_{\beta})\Bigr{)}\Bigl{(}\partial_{\gamma}(a_{\delta}+qA_{\delta})\Bigr{)}\
.$

We can alternatively implement the topological current conservation with the
twist field $\chi$ minimally coupled to $A_{\mu}$; this allows the $a_{\mu}$
and $A_{\mu}$ gauge fields to have independent couplings in $\mathcal{L}_{t}$.

The rest of the real-time Lagrangian density is a generalization of (III.2) to
$d=4$:

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{\kappa_{c}}{2}\left(\partial_{\mu}\theta\\!+\\!a_{\mu}+qA_{\mu}\right)^{2}+\frac{\kappa_{s}}{2}(\partial_{\mu}\hat{n}^{a})^{2}+\mu
b_{a}\hat{n}^{a}+\mathcal{L}_{\textrm{B}}$
$\displaystyle+\frac{\kappa_{t}}{2}\left(\partial_{\mu}\chi\\!+\\!qA_{\mu}\right)^{2}+\frac{1}{12e^{2}}(\epsilon^{\mu\nu\lambda\alpha\beta}\partial_{\alpha}a_{\beta})^{2}$
$\displaystyle+\frac{C}{6}\left(\epsilon^{\mu\nu\lambda\alpha\beta}\partial_{\alpha}A_{\beta}-J^{\mu\nu\lambda}\right)^{2}+\mathcal{L}_{t}\
.$

The featured currents are (IV.1) and

$\displaystyle j_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\theta+a_{\mu}+qA_{\mu}$ (79) $\displaystyle
J_{\mu}^{a}$ $\displaystyle=$
$\displaystyle\epsilon^{abc}\hat{n}^{b}\partial_{\mu}^{\phantom{x}}\hat{n}^{c}\
.$

The spin chirality is a rank-3 tensor in $D=d+1=5$ space-time,

$J_{\mu\nu\lambda}^{\phantom{x}}=\frac{1}{2}\epsilon_{\mu\nu\lambda\alpha\beta}^{\phantom{x}}\epsilon^{abc}\hat{n}^{a}(\partial_{\alpha}^{\phantom{x}}\hat{n}^{b})(\partial_{\beta}^{\phantom{x}}\hat{n}^{c})=\frac{1}{2}\epsilon_{\mu\nu\lambda\alpha\beta}^{\phantom{x}}\hat{n}^{a}\partial_{\alpha}^{\phantom{x}}J_{\beta}^{a}$

The field equations reproduce the same kinds of phenomena as in $d=3$, but
with a new important feature attributed to the topological term. Quantized
charge becomes attached to quantized Hopf singularities,

$\kappa_{c}j^{\mu}-\frac{1}{e^{2}}\partial_{\nu}f^{\mu\nu}-3K_{t}\rho\mathcal{J}^{\mu}=0\
.$ (80)

The system can respond to the presence of charge currents by binding
topological currents to them

$j_{\mu}=\frac{\nu}{\phi_{0}^{2}}\mathcal{J}_{\mu}\quad;\quad\frac{\nu}{\phi_{0}^{2}}=\frac{3K_{t}\rho}{\kappa_{c}}$
(81)

instead of generating the electromagnetic field $f_{\mu\nu}$. The
proportionality constant, or the “filling factor”, is rationally quantized
$\nu=p/q$ in any incompressible quantum liquid. Otherwise, fluctuations can
push some frustrated defects through the space occupied by particles (or vice
versa) and hence localize the particles (or defects respectively) by
destructive quantum interference.

The equation (81) with a rational $\nu$ describes charge fractionalization
because topological defects carry an integer-quantized Hopf index. This is a
hallmark of topological order, seen in all $\pi_{d-1}(S^{d-1})$ homotopy
groups and now in $\pi_{3}(S^{2})$ as well. The flux-loop structure of the
Hopf topological defects provides another mechanism for fractionalization
which is not present in the $\pi_{d-1}(S^{d-1})$ homotopies. Consider a
hopfion made from U(1) flux loops in the charge sector. The fractional charge
$\nu$ attached to the topological defect experiences a system of inter-linked
or self-linked flux loops on the $S^{3}$ manifolds that enclose the
singularity. The relevant manifolds are those with a radius smaller than the
particle-defect binding length. Within such an $S^{3}$ manifold, quantum
fluctuations of the topologically ordered state explore many locations and
orientations of the flux loops. This frustrates the motion of the attached
charge $\nu$ and localizes it to a flux loop, but gives it freedom to move
along the loop (parallel to the magnetic field). Other inter-linked loops are
seen as the source of Aharonov-Bohm flux by this charge. Consequently, the
charge is encouraged to move around its loop and produce a circulating charge
current with non-zero orbital angular momentum and magnetic moment. For a
circular loop $C$ of radius $r$ and perimeter $l=2\pi r$, interlinked with $n$
quantized flux loops (including self), the resulting orbital magnetic moment
amplitude is (in the units $c=1$)

$\mu=IS=\nu e\frac{2\pi n}{2\pi r}\pi r^{2}=\frac{\nu en}{2}l\ .$ (82)

The magnetic moment is itself an antisymmetric rank-2 tensor in $d=4$
dimensions. In this derivation, we interpreted the gauge field $a_{i}$ as the
background charge current $I/\nu e\equiv j_{i}=a_{i}$ on the loop, normalized
to the unit particle number and then divided $\oint dx_{i}a_{i}=2\pi n$ with
the loop perimeter to extract the average current along the loop. The orbital
angular momentum is

$L=\nu m|2\pi rj_{i}|r=\nu m\cdot 2\pi r^{2}\cdot\frac{2\pi n}{2\pi r}=\nu
mnl\ ,$ (83)

(note that the quasiparticle mass $\nu m$ is also fractionalized), so that the
Bohr magneton

$\mu_{\textrm{B}}=\frac{\mu}{L}=\frac{e}{2m}$ (84)

still has the standard non-fractionalized value (in the $\hbar=1$ units)
associated with orbital motion in quantum mechanics. Nevertheless, the
magnetic moment and angular momentum are both fractionalized according to the
filling factor $\nu$. We ought to recover the proper angular momentum units by
comparing with a non-fractionalized particle ($\nu=1$) whose angular momentum
quantization $L=n\hbar\to n$ is clear. The fractionalization by hopfion loops
effectively reduces the microscopic angular momentum quantum $\hbar$ to a
fractional value $\nu\hbar$, and the magnetic moment to

$\mu=\nu\mu_{\textrm{B}}\times\textrm{integer}\ .$ (85)

It should be noted that the total fractional angular momentum is conserved in
the presence of rotational symmetry, and restricted to small magnitudes due to
the short length-scale of particle-defect binding (the flux loops in this
analysis live on microscopic $S^{3}$ manifolds). But, there are further
complications which we will not attempt to analyze. Even though the fractional
charge $\nu$ is indivisible, it can tunnel from one linked loop to another and
form superposition states which are still deeply quantized due to the
microscopic binding to the topological defect. This drives the twist-exchange
processes shown in Fig.3. Other Hopf index preserving processes which link or
unlink flux loops also tend to preserve the Aharonov-Bohm flux seen by the
fractional charge.

###  IV.2 Fractional braiding statistics

The topological order of Hopf singularities is equipped with a topologically
protected braiding operation. This cannot be a particle exchange like in the
fractional quantum Hall state because no topological invariant can be defined
on contractible loop-paths. Instead, the braiding operation must be carried
out on a 3-sphere path that can serve as a manifold for the computation of the
Hopf index. A fractional quasiparticle is a point-like object which combines a
fractional charge $\nu$ with a unit Hopf singularity. The mentioned 3-sphere
manifold has to enclose the quasiparticle and reflect on its “surface” the
Hopf charge $N=1$ of the singularity. The lower-dimensional object we braid
should traverse the path comprised of this 3-sphere manifold. Therefore, the
braided object will be a 2-sphere, and the braiding operation will be a $d=4$
analogue of the process depicted in Fig.7. After a braiding operation, the
many-body wavefunction of the system acquires an Abelian fractionally
quantized “Aharonov-Bohm phase”. No other braiding operations have
topologically protected outcomes. Specifically, the braiding of flux loops
does not enjoy topological protection because loops are volatile under the
Hopf index preserving dynamics (see Fig.3).

Figure 7: A three-dimensional counterpart of the four-dimensional braiding.
Set up a sphere around a fractional quasiparticle that contains a topological
defect (a monopole in this example). Create an infinitesimal loop at the south
pole of the sphere, then stretch and sweep it across the sphere, and finally
collapse it into a point to annihilate at the north pole. All entities in this
example acquire one additional dimension for the braiding with a Hopf
singularity.

For simplicity, let us neglect any Hopf structure in the charge sector. A
hopfion singularity is surrounded by the gauge field
$\mathcal{A}_{\mu\nu\lambda}$ defined in (IV.1). The topological charge of the
singularity determines the quantized flux (71) of
$\mathcal{A}_{\mu\nu\lambda}$ on any 3-sphere $S^{3}$ that encloses the
singularity. We wish to assign to this an Aharonov-Bohm “phase”

$\phi=\frac{q_{3}}{\phi_{0}^{2}}\oint d^{3}x\,\epsilon_{ijk}A_{ijk}=q_{3}N$
(86)

accumulated by a 2-sphere that we create at the “south” pole point of $S^{3}$,
continuously stretch and sweep across the $S^{3}$ manifold, then finally
shrink into a point and consume at the “north” pole. Alternatively, we can
consider an $S^{2}\times S^{1}$ manifold to capture a braiding operation
between a 2-sphere and a quasiparticle in which a quasiparticle is pushed
through a 2-sphere on a closed loop. The current $J_{\mu\nu\lambda}$ of the
2-sphere objects minimally couples to $\mathcal{A}_{\mu\nu\lambda}$ and
generates these braiding operations via the operator
$\exp(i\,d^{3}x\,\epsilon_{ijk}J_{ijk})$. The bare current is the
topologically trivial “pure gauge” content of $\mathcal{A}_{\mu\nu\lambda}$,
which integrates out to $N=0$:

$J_{\mu\nu\lambda}=\partial_{\mu}\theta_{\nu\lambda}-\partial_{\nu}\theta_{\mu\lambda}+\partial_{\lambda}\theta_{\mu\nu}\
.$ (87)

The matter field $\theta_{\mu\nu}$ introduced here is an antisymmetric tensor.
The full gauge-invariant current we need is

$J_{\mu\nu\lambda}=\partial_{\mu}\theta_{\nu\lambda}-\partial_{\nu}\theta_{\mu\lambda}+\partial_{\lambda}\theta_{\mu\nu}+\mathcal{A}_{\mu\nu\lambda}\
.$ (88)

We must construct the braiding operator $\mathcal{B}$ using the gauge-
invariant generator $\mathcal{J}_{\mu\nu\lambda}$, but the matter fields in
its definition immediately drop out because they are integrated over a
manifold without a boundary. Hence,

$\mathcal{B}=\exp\left(iq_{3}\oint\limits_{S^{3}}d^{3}x\,\epsilon_{ijk}\mathcal{A}_{ijk}\right)=e^{iq_{3}^{\phantom{x}}\phi_{0}^{2}N}$
(89)

should create a 2-sphere excitation $J_{0ij}$ at the “south” pole of $S^{3}$,
drive it around a stationary fractionalized quasiparticle at the origin, and
then annihilate it at the “north” pole of $S^{3}$. If the quasiparticle at the
origin carried $N$ hopfion quanta, then $iq_{3}^{\phantom{x}}\phi_{0}^{2}N$
would be the Aharonov-Bohm phase acquired in the braiding operation. There is
a problem, however. The charge $q_{3}$ associated with the current
$J_{\mu\nu\lambda}$ should be an integer in any conventional state of matter,
yet the acquired phase proportional to $\phi_{0}^{2}$ would be non-trivial due
to not being an integer multiple of $2\pi$. Analogous constructions of
braiding operations for vortices and monopoles are not plagued by this
problem. We must renormalize the generating current operator to fix this
problem.

Consider the analogy with particle braiding, where $\exp(\oint dx_{i}A_{i})$
creates a unit dipole, drives the particle of the dipole around a loop and
then annihilates it with the antiparticle. The dipole creation step,

$e^{dx_{i}a_{i}}=e^{dx_{i}\partial_{i}\theta}=e^{i\theta({\bf x}+{\bf
dx})}e^{-i\theta({\bf x})}\ ,$ (90)

manifestly involves the particle creation $e^{i\theta}$ and annihilation
$e^{-i\theta}$ operators. The phase operator $\theta$ is canonically conjugate
to the number operator $n$. Charge quantization implies integer eigenvalues
for $n$, making $\theta$ an angle from a $2\pi$ interval. This determines the
normalization of the current operator $j_{\mu}=\partial_{\mu}\theta+a_{\mu}$.
Going back to our problem, the initial creation
$\exp(-iJ^{\mu\nu\lambda}\epsilon_{\mu\nu\lambda}d^{3}x)$ of a braided
2-sphere ought to utilize the properly quantized “phase” operators
$\theta_{\mu\nu}$ within (87). We can reveal this quantization by constructing
everything from the singular vortex-loop configurations of the twist field
$\chi$. We begin by applying a singular gauge transformation to extract
vorticity from $\chi$ into the gauge field

$A_{\mu}=\frac{1}{q}\partial_{\mu}\chi\ .$ (91)

Then, by comparing (87) and (IV.1) through a similar singular gauge
transformation $\mathcal{J}_{\mu\nu\lambda}\to\mathcal{A}_{\mu\nu\lambda}$, we
identify

$\theta_{\mu\nu}=\frac{1}{3q^{2}}\chi\partial_{\mu}\partial_{\nu}\chi$ (92)

with an implicit index antisymmetrization. An underlying assumption is that
the $\chi$ configuration does not contain monopoles, i.e.
$\epsilon_{\cdots\mu\nu\lambda}\partial_{\mu}\partial_{\nu}\partial_{\lambda}\chi=0$.
The rank-3 current currently has the form

$J_{\mu\nu\lambda}=\frac{1}{3q^{2}}\Bigl{[}\partial_{\mu}(\chi\partial_{\nu}\partial_{\lambda}\chi)-\partial_{\nu}(\chi\partial_{\mu}\partial_{\lambda}\chi)+\partial_{\lambda}(\chi\partial_{\mu}\partial_{\nu}\chi)\Bigr{]}$

where the antisymmetrization of indices is implicitly understood. The $\chi$
field carries $2\pi$-quantized vorticity, so that

$\frac{1}{q^{2}}\partial_{\mu}(\chi\partial_{\nu}\partial_{\lambda}\chi)\,d^{3}x\to\frac{2\pi}{q^{2}}(\partial_{\mu}\chi)dx^{\mu}\
.$ (93)

The omitted index antisymmetrization keeps the $\mu$ direction orthogonal to
the directions $\nu,\lambda$. Then, the $\partial_{\mu}$ derivative acts along
the singular flux line and sees
$\epsilon_{\cdots\mu\nu\lambda}\partial_{\nu}\partial_{\lambda}\chi$ as a
constant. By definition, the twist field $\chi$ is a proper angle, so
$\partial_{\mu}\chi$ will integrate to an integer multiple of $2\pi$.
Evidently, the extra factor $2\pi/q^{2}$ must be divided out in order to get
an adequately normalized rank-3 current:

$\widetilde{J}_{\mu\nu\lambda}=\frac{q^{2}}{2\pi}\Bigl{(}\partial_{\mu}\theta_{\nu\lambda}-\partial_{\nu}\theta_{\mu\lambda}+\partial_{\lambda}\theta_{\mu\nu}+\mathcal{A}_{\mu\nu\lambda}\Bigr{)}\
.$ (94)

Now,

$\widetilde{\mathcal{B}}=\exp\left(iq_{3}\oint\limits_{S^{3}}d^{3}x\,\epsilon_{ijk}\widetilde{J}_{ijk}\right)=e^{iq_{3}^{\phantom{x}}(q^{2}\phi_{0}^{2}/2\pi)N}$
(95)

correctly braids a quantized 2-sphere around a point Hopf singularity because
$q\phi_{0}=2\pi$ in both charge and spin sectors.

The last task is to determine the “charge” $q_{3}$ associated with the current
$J_{\mu\nu\lambda}$. The U(1) flux quantum carried by $F_{\mu\nu}$ remains
$\phi_{0}$ in the fractionalized phases, but the charge unit carried by
$J_{\mu}$ becomes a rational number $\nu$. We do not want to embed a factor of
$\nu$ into the operator $\widetilde{J}_{\mu\nu\lambda}$, which ought to create
unit “charges”. Instead, we explicitly provide a factor of $q_{3}=\nu$ in the
braiding generator, to be associated with the $J_{\mu}$ operators within (94).
In this sense, the braiding operator will braid the smallest possible bundle
of charge and flux comprising $\widetilde{J}_{\mu\nu\lambda}$ that the
incompressible quantum liquid can support. The resulting topologically-
protected hopfion braiding statistics (phase) between a 2-sphere and a point
fractional quasiparticle made of $N$ Hopf quanta is:

$\widetilde{\mathcal{B}}=e^{2\pi i\nu N}\ .$ (96)

Note that further induced field corrections to braiding have not been
considered here.

###  IV.3 Topological ground state degeneracy

The incompressible quantum liquid of Hopf singularities exhibits topological
order, i.e. a ground state degeneracy on topologically non-trivial manifolds.
We will demonstrate this on the four-dimensional spatial manifold
$\mathcal{M}=S^{1}\otimes M^{3}$ with $M^{3}=S^{1}\otimes\mathbb{R}$. Quantum
fluctuations on such manifolds, which one might call vacuum instantons, can
convert a classical ground state from one topological sector with a given Hopf
index to another. Our goal here is to demonstrate that vacuum instantons lift
the classical ground state degeneracy down to a finite residual degeneracy. We
will consider only a simple subset of the full problem in order to exhibit
topological order at zero temperature. Vacuum instantons should not be
confused with defect instantons which were mentioned in earlier sections.
Defect instantons cost more action, and create or annihilate local topological
defect excitations. Their confinement, measured with a separate long-range
correlation function, is a prerequisite for topological order.

For our purposes, a classical ground state is characterized by two integer
topological invariants, a Hopf index $N$ on $M^{3}$ and a vortex winding
number $N^{\prime}$ on the complement $S^{1}$ submanifold of $\mathcal{M}$.
The gauge field configurations in these states are

$\displaystyle a_{i}({\bf x})$ $\displaystyle=$ $\displaystyle
2\pi\frac{N^{\prime}}{L}\delta_{i,4}$ (97) $\displaystyle A_{i}({\bf x})$
$\displaystyle=$
$\displaystyle\frac{1}{q}\left(\frac{x_{2}\delta_{i,1}-x_{1}\delta_{i,2}}{x_{1}^{2}+x_{2}^{2}}n+\frac{2\pi
N}{L}\delta_{i,3}\right)\ ,$

where $x_{1},x_{2}\in\mathbb{R}^{2}$, $x_{3}\in S^{1}\subset M$, $x_{4}\in
S^{1}\subset\mathcal{M}$, and both $S^{1}$ submanifolds have the perimeter
$L$. A vortex line

$\epsilon_{ijk}\partial_{j}A_{k}\to\frac{2\pi
n}{q}\delta(x_{1})\delta(x_{2})\delta_{i,3}$ (98)

is threaded along the $x_{3}$ axis through the origin of $\mathbb{R}$; this
singularity will need to be regularized later on. We mention in passing that
the quantization of $N$ and $N^{\prime}$ is enforced by the presence of matter
fields that couple to $a_{i}$ and $A_{i}$; for example, the integer winding of
$\theta$ over $S^{1}$ can be compensated without a high gradient
$(\partial_{\mu}\theta+a_{\mu})^{2}$ price only by quantizing the appropriate
gauge field components. The configurations (97) are classically degenerate in
the $L\to\infty$ limit. We implicitly integrate out all other configurations
which cost more energy, and explore how the remaining macroscopic degeneracy
can be lifted by quantum fluctuations. The relevant effective Lagrangian
density has the form

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{12e^{2}}(\epsilon_{\mu\nu\lambda\alpha\beta}\partial_{\alpha}a_{\beta})^{2}+\frac{C}{6}(\epsilon_{\mu\nu\lambda\alpha\beta}\partial_{\alpha}A_{\beta})^{2}$
(99)
$\displaystyle-\frac{q^{2}\nu}{(2\pi)^{2}}\epsilon^{\mu\nu\alpha\beta\gamma}a_{\mu}\partial_{\nu}(A_{\alpha}\partial_{\beta}A_{\gamma})\
,$

where we have retained only a part of the full topological term for the
demonstration of topological order. Choosing Coulomb gauge $a_{0}=0$,
$A_{0}=0$ and allowing time variations of the topological invariants, we
substitute (97) into the Lagrangian density and get

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{(2\pi)^{2}}{2e^{2}L^{2}}(\partial_{0}N^{\prime})^{2}+\frac{(2\pi)^{2}C}{q^{2}}\left[\delta(x_{1})\delta(x_{2})\right]^{2}n^{2}$
(100)
$\displaystyle+\frac{C}{q^{2}}\left[\frac{(\partial_{0}n)^{2}}{x_{1}^{2}+x_{2}^{2}}+\frac{(2\pi)^{2}(\partial_{0}N)^{2}}{L^{2}}\right]$
$\displaystyle-4\pi\nu\,\frac{n}{L^{2}}\delta(x_{1})\delta(x_{2})N^{\prime}\partial_{0}N\
.$

The Lagrangian $L_{i}$ of vacuum instantons obtains by integrating out ${\bf
x}\in\mathcal{M}$. We are forced to regulate the infra-red and ultra-violet
divergence by reducing the $\mathbb{R}^{2}$ submanifold to a ring with inner
radius $r\to 0$ and outer radius $R\to\infty$:

$\displaystyle L_{i}$ $\displaystyle=$
$\displaystyle\frac{(2\pi)^{2}}{2e^{2}}\pi
R^{2}(\partial_{0}N^{\prime})^{2}+\frac{(2\pi)^{2}C}{q^{2}}\frac{L^{2}}{\pi
r^{2}}\,n^{2}$ (101) $\displaystyle+\frac{C}{q^{2}}\left[2\pi
L^{2}\log\left(\frac{R}{r}\right)(\partial_{0}n)^{2}+(2\pi)^{2}\pi
R^{2}(\partial_{0}N)^{2}\right]$ $\displaystyle-4\pi
n\nu\,N^{\prime}\partial_{0}N\ .$

The ultraviolet cut-off distance $r$ appears only in the terms that contain
the vorticity $n$, and these terms are relevant only if the vorticity can
fluctuate. We may neglect such fluctuations here because their energy cost is
logarithmically enhanced in comparison to the cost of all other fluctuations.
So, if we assume that the fluctuations of $n$ are frozen, i.e.
$\partial_{0}n\to 0$, the vacuum instanton Lagrangian becomes

$\displaystyle L_{i}$ $\displaystyle=$
$\displaystyle\frac{(2\pi)^{2}}{2e^{2}}\pi
R^{2}(\partial_{0}N^{\prime})^{2}+\frac{(2\pi)^{2}C}{q^{2}}\pi
R^{2}(\partial_{0}N)^{2}$ (102) $\displaystyle-4\pi
n\nu\,N^{\prime}\partial_{0}N\ .$

up to a constant. We can now treat $N$ and $N^{\prime}$ as canonical
coordinates for instanton fluctuations. The corresponding canonical momenta
are

$P=\frac{\delta L_{i}}{\delta\partial_{0}N}=\frac{2(2\pi)^{2}C}{q^{2}}\pi
R^{2}\,\partial_{0}N-\frac{4\pi n\nu}{m}\,N^{\prime}$ (103)
$P^{\prime}=\frac{\delta
L_{i}}{\delta\partial_{0}N^{\prime}}=\frac{(2\pi)^{2}}{e^{2}}\pi
R^{2}\partial_{0}N^{\prime}\ .$ (104)

The instanton Hamiltonian is then

$\displaystyle H_{i}$ $\displaystyle=$ $\displaystyle
P\partial_{0}N+P^{\prime}\partial_{0}N^{\prime}-L_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2M}\left(P+4\pi n\nu
N^{\prime}\right)^{2}+\frac{1}{2M^{\prime}}P^{\prime 2}\ ,$

where the masses

$M=\frac{(2\pi)^{3}R^{2}C}{q^{2}}\quad,\quad
M^{\prime}=\frac{(2\pi)^{3}R^{2}}{2e^{2}}$ (106)

are macroscopically large due to $R\to\infty$.

As in the treatment of fractional quantum Hall states, we imagine that
$N,N^{\prime}\in\mathbb{Z}$ are integer-valued coordinates of sites on a
fictitious square lattice. Their canonical conjugates $P,P^{\prime}$ must be
angles, so it is necessary to regularize the instanton Hamiltonian into a
compact form

$H_{i}\to-t\cos(P+4\pi n\nu N^{\prime})-t^{\prime}\cos(P^{\prime})\ .$ (107)

This is Hofstadter model expressed in Landau gauge, with $2n\nu$ flux quanta
per lattice plaquette. Its ground states capture the fate of the classical
degeneracy after taking vacuum instantons into account. If the filling factor
$\nu$ is rational so that $2n\nu=p/q$ with mutually prime integers $p,q$
($n\in\mathbb{Z}$), then the ground state degeneracy is $q$ for any finite
$t,t^{\prime}$. Under current assumptions, $t,t^{\prime}\propto R^{-2}\to 0$
both vanish in the thermodynamic $R\to\infty$ limit, and hence preserve the
infinite degeneracy of the classical topological sectors. However, this is a
consequence of suppressing the local flux-changing events in our formalism.
The U(1) flux lines which determine $N$ and $N^{\prime}$ do not live in the
physical space $\mathcal{M}$, so no energy cost is associated with their
presence or absence. One can imagine these flux lines protruding through
$\mathcal{M}$ due to occasional fluctuations, and temporarily creating visible
magnetic dipoles of any size. This dynamics is governed only by the local
energy of the visible dipoles, but may result with changes of the total flux
$N$ and $N^{\prime}$ threaded through the openings of $\mathcal{M}$. As a
consequence, we should indeed assume that the hopping integrals
$t,t^{\prime}>0$ are finite.

This establishes topological order at zero temperature, with ground state
degeneracy on $\mathcal{M}$ determined by the filling factor $\nu$. The vacuum
instanton Hamiltonian (107) is ultimately analogous to that of a fractional
quantum Hall liquid in two dimensions, although the ground state degeneracy is
slightly different for the same filling factor. This topological order does
not survive at finite temperatures. However, the defect instanton events which
create and annihilate the Hopf singularities in the $D=d+1=5$ dimensional
space-time are strongly confined by a linear potential in the action, so a
confined instanton phase survives at finite low temperatures. When defect
instantons are confined, the topological charge of Hopf singularities is
conserved regardless of whether topological order is present. This opens a
possibility of other correlated phases at low temperatures.

It is interesting to point out that the presented method can be also applied
to the more physical $d=3$ models of hopfion dynamics. The result is, however,
trivial. The ground state of a putative incompressible quantum liquid of
conserved hopfions is non-degenerate in three spatial dimensions. The vacuum
instanton Hamiltonian equivalent to (107) would be one-dimensional because a
$d=3$ spatial manifold leaves no room for the $N^{\prime}$ variable.

###  IV.4 Spinor representation of the topological term

Electrons are naturally represented by a spinor field $\psi$ that carries
information about charge and spin. The charge content is handled by the phase
factor $\psi\propto e^{i\varphi}$ and the spin content $\hat{{\bf
n}}(\theta,\phi)$ is given by the spinor structure in some representation of
the Spin(3)=SU(2) group:

$\psi(\hat{{\bf n}})=e^{-iJ_{3}\phi}e^{-iJ_{2}\theta}e^{i\varphi}\psi_{0}\ ,$
(108)

where $J^{a}$ are the spin projection operators. The reference spinor
$\psi_{0}$ is constant and all dynamics is expressed with the space-time
variations of $\varphi,\theta,\phi$. Here we construct the topological
Lagrangian density directly in terms of the spinor field, and study its
properties. This provides the most accurate formulation of the topological
field theory which resolves some issues with the previous more effective
approach.

In the previous discussions, we normalized the charge current
$j_{\mu}\sim\partial_{\mu}\varphi+a_{\mu}+qA_{\mu}$ in a manner that
emphasizes its relationship to the phase gradient $\partial_{\mu}\varphi$ of
the matter field. We now wish to use the standard normalization of the current
extracted from a spinor,

$\rho=\psi^{\dagger}\psi\quad,\quad
j_{\mu}=-\frac{i}{2}\Bigl{(}\psi^{\dagger}(\partial_{\mu}\psi)-(\partial_{\mu}\psi^{\dagger})\psi\Bigr{)}\
.$ (109)

The corresponding imaginary-time topological Lagrangian density with the same
coupling constant $K_{t}$ is

$\mathcal{L}_{t}=iK_{t}\,j_{\mu}\mathcal{J}_{\mu}\ .$ (110)

Our goal is to reproduce this form and the Hopf singularity current

$\mathcal{J}_{\mu}=\epsilon_{\mu\nu\alpha\beta\gamma}\partial_{\nu}(A_{\alpha}\partial_{\beta}A_{\gamma})$
(111)

by applying a certain singular derivative operator
$\psi^{\dagger}\mathcal{D}\psi$ to the spinor field $\psi$. This operator will
contain antisymmetrized derivatives which automatically give zero at every
point $x_{\mu}$ where $\psi(x_{\mu})$ is analytic; we want $\mathcal{D}$ to
pick only the singularities of $\psi$, and for that purpose a mathematical
rule for taking derivatives of singular functions will be implicitly provided.
The structure of the Hopf index suggests that we should explore the Lagrangian
density

$\mathcal{L}_{t}\propto\epsilon_{\mu\nu\alpha\beta\gamma}\psi^{\dagger}\partial_{\mu}\partial_{\nu}\partial_{\alpha}\partial_{\beta}\partial_{\gamma}\psi\
,$ (112)

with understanding that the correct type of topological defects will be
detected when present. The gradient of (108) is

$\partial_{\mu}\psi=i\left[\partial_{\mu}\varphi-
J_{3}\partial_{\mu}\phi-e^{-iJ_{3}\phi}J_{2}e^{+iJ_{3}\phi}\partial_{\mu}\theta\right]\psi\
.$ (113)

We will at first concentrate on the charge sector and consider
$\theta,\phi\to\textrm{const}$. After applying a singular gauge transformation
to transfer the gauge field $a_{\mu}$ into $\partial_{\mu}\varphi$, we get

$\theta,\phi\to\textrm{const}\quad\Rightarrow\quad\partial_{\mu}\psi\to
ia_{\mu}\psi\ .$ (114)

Then, the formal application of the product rule for derivatives gives us

$\displaystyle\epsilon_{\mu\nu\alpha\beta\gamma}\partial_{\mu}\partial_{\nu}\partial_{\alpha}\partial_{\beta}\partial_{\gamma}\psi=i\epsilon_{\mu\nu\alpha\beta\gamma}\partial_{\mu}\partial_{\nu}\partial_{\alpha}\partial_{\beta}a_{\gamma}\psi$
(115)
$\displaystyle\qquad=iq\epsilon_{\mu\nu\alpha\beta\gamma}\partial_{\mu}\partial_{\nu}\partial_{\alpha}(\partial_{\beta}a_{\gamma})\psi-\epsilon_{\mu\nu\alpha\beta\gamma}\partial_{\mu}\partial_{\nu}\partial_{\alpha}a_{\gamma}a_{\beta}\psi$
$\displaystyle\qquad=iq\epsilon_{\mu\nu\alpha\beta\gamma}\partial_{\mu}\partial_{\nu}(\partial_{\alpha}\partial_{\beta}a_{\gamma})\psi-\epsilon_{\mu\nu\alpha\beta\gamma}\partial_{\mu}\partial_{\nu}a_{\alpha}(\partial_{\beta}a_{\gamma})\psi$
$\displaystyle\qquad=iq\epsilon_{\mu\nu\alpha\beta\gamma}\partial_{\mu}(\partial_{\nu}\partial_{\alpha}\partial_{\beta}a_{\gamma})\psi-\epsilon_{\mu\nu\alpha\beta\gamma}\partial_{\mu}a_{\nu}(\partial_{\alpha}\partial_{\beta}a_{\gamma})\psi$
$\displaystyle\qquad\quad-\partial_{\mu}^{\phantom{x}}\mathcal{J}_{\mu}^{\textrm{c}}\psi\
,$

where

$\mathcal{J}_{\mu}^{\textrm{c}}=\epsilon_{\mu\nu\alpha\beta\gamma}^{\phantom{x}}\partial_{\nu}^{\phantom{x}}(a_{\alpha}^{\phantom{x}}\partial_{\beta}^{\phantom{x}}a_{\gamma}^{\phantom{x}})$
(116)

is the Hopf index current density in the charge sector. The first term in the
final expression describes $d=4$ monopoles; we will drop this term because
monopoles are homotopically distinct topological defects from hopfions. The
second term similarly captures the line singularities of $a_{\mu}$ in $d=4$,
which would be point-like monopoles in $d=3$; hopfions are not equipped with
such singularities. So, for the purposes of hopfions in the charge sector, the
topological Lagrangian density (110) should be

$\mathcal{L}_{t}=K_{t}\,\psi^{\dagger}\partial_{\mu}\mathcal{J}_{\mu}\psi=-K_{t}\,\psi^{\dagger}\epsilon_{\mu\nu\alpha\beta\gamma}\partial_{\mu}\partial_{\nu}\partial_{\alpha}\partial_{\beta}\partial_{\gamma}\psi\
.$ (117)

Antisymmetrized derivatives were retained in various steps of this derivation.
The implicit rule for calculating such derivatives of singular functions is
given by the Stokes-Cartan theorem and similar topological relationships. For
example, a complex field $\psi=\psi_{0}e^{i\varphi}$ with non-zero magnitude
$\rho=|\psi_{0}|^{2}$ in two dimensions may have a vortex singularity at the
origin; then, the “singularity gauge field” $a_{i}=\partial_{i}\varphi$ has a
singular flux
$\epsilon_{ijk}\partial_{j}a_{k}\equiv-i\rho^{-1}\psi^{\dagger}\epsilon_{ijk}\partial_{j}\partial_{k}\psi=2\pi\delta_{i,0}\delta({\bf
x})$ because $\oint dx_{\mu}a_{\mu}=2\pi$, has a fixed well-defined value on
any loop that encloses the origin. This specifies the basic rule for taking
antisymmetric derivatives, and its extension to higher orders can be
constructed recursively Nikolić (2020a).

We will show next that the same antisymmetric derivative extracts the Hopf
singularity in the spin sector as well. The $\phi\in(0,2\pi)$ angle in (108)
microscopically provides both the twist field $\chi\sim\phi$ and the spin
chirality gauge field $qA_{\mu}\sim\partial_{\mu}\phi$. Consider the following
integral over a spatial volume $B^{4}$ bounded by $S^{3}$:

$\displaystyle I_{4}$ $\displaystyle=$
$\displaystyle\int\limits_{B^{4}}d^{4}x\,\psi^{\dagger}\epsilon_{ijkl}\partial_{i}\partial_{j}\partial_{k}\partial_{l}\psi$
$\displaystyle=$ $\displaystyle
I_{3}-\int\limits_{B^{4}}d^{4}x\,\epsilon_{ijkl}(\partial_{i}\psi^{\dagger})(\partial_{j}\partial_{k}\partial_{l}\psi)$
$\displaystyle=$ $\displaystyle
I^{\phantom{x}}_{3}+I^{\prime}_{3}+\int\limits_{B^{4}}d^{4}x\,\epsilon_{ijkl}(\partial_{j}\partial_{i}\psi^{\dagger})(\partial_{k}\partial_{l}\psi)\
.$

Two integrations by parts and Stokes-Cartan theorem produce the boundary
integrals

$\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\oint\limits_{S^{3}}d^{3}x\,\psi^{\dagger}\epsilon_{ijk}\partial_{i}\partial_{j}\partial_{k}\psi$
(119) $\displaystyle I^{\prime}_{3}$ $\displaystyle=$
$\displaystyle\oint\limits_{S^{3}}d^{3}x\,\epsilon_{ijk}(\partial_{i}\psi^{\dagger})(\partial_{j}\partial_{k}\psi)$

which add up to zero because $S^{3}$ itself has no boundary:

$I^{\phantom{x}}_{3}+I^{\prime}_{3}=\oint\limits_{S^{3}}d^{3}x\,\epsilon_{ijk}\partial_{i}(\psi^{\dagger}\partial_{j}\partial_{k}\psi)=0\
.$ (120)

The remaining volume integral still responds only to the singularities of
$\psi({\bf x})$ which arise in (108) due to the spin texture $\hat{\bf n}$ or
charge current vortices. The spin-rotation generators $J_{i}$ place the spin
in the unique direction $\hat{{\bf n}}(\theta,\phi)$, but also introduce a
sign-changing branch-cut in the spinor for fermionic particles due to having
eigenvalues equal to odd integer halves. This is formally corrected by
providing a U(1) rotation:

$\psi(\hat{{\bf
n}})=e^{-iJ_{3}\phi}e^{-iJ_{2}\theta}e^{i(\varphi+s\phi/2)}\psi_{0}\ ,$ (121)

and amounts to the shift $J_{3}\to J_{3}-s/2$ of the spin projection operator.
The value of $s$ must be an odd integer for fermions and an even integer for
bosons, unless $\phi$ winds by an integer multiple of $4\pi$. Looking at the
two equivalent minimal representations $\psi_{\pm}=e^{\pm i\phi/2}\psi$ of
(108) with $\psi_{0}=|\uparrow\rangle$,

$\psi_{+}=\left(\begin{array}[]{c}\cos\left(\frac{\theta}{2}\right)\\\
e^{i\phi}\sin\left(\frac{\theta}{2}\right)\end{array}\right)\quad,\quad\psi_{-}=\left(\begin{array}[]{c}e^{-i\phi}\cos\left(\frac{\theta}{2}\right)\\\
\sin\left(\frac{\theta}{2}\right)\end{array}\right)\quad,$ (122)

we see that the choices $s=1$ and $s=-1$ minimally regularize the vortex
singularity at $\theta=0$ and $\theta=\pi$ respectively. This generalizes to
$s=2S$ and $s=-2S$ respectively in higher spin representations. The values
$s=2(S+n)$ at $\theta=0$ and $s=2(-S+n)$ at $\theta=\pi$ leave a $2\pi n$
vortex singularity of $\phi$ in the spinor $\psi$. We will pick $s=2(S+1)$ in
order to capture the $2\pi$ vorticity of $\phi$ at “skyrmion centers”,

$S^{\prime}=S-\frac{s}{2}\xrightarrow{s\to 2(S+1)}-1\ .$ (123)

This amounts to the choice of an arbitrary spin quantization axis $\hat{\bf
z}$ such that $\hat{\bf n}=\hat{\bf z}$ ($\theta=0$) at the skyrmion centers
and $\hat{\bf n}\to-\hat{\bf z}$ ($\theta=\pi$) in the far-away zero-vorticity
ferromagnetic regions. A skyrmion center defined hereby will coincide with the
symmetry center when the spin texture is rotationally symmetric, but generally
cannot be unambiguously identified because skyrmions are not singular spin
configurations. Since we pinned the skyrmion centers to $\theta=0$, we always
have $J_{3}\psi=S\psi$ there given that (121) is an eigenvector of $J_{3}$ at
$\theta=0$. So, starting from (121), we find

$\partial_{j}\psi=i\left[\partial_{j}\varphi-\left(J_{3}-\frac{s}{2}\right)\partial_{j}\phi-e^{-iJ_{3}\phi}J_{2}e^{iJ_{3}\phi}\partial_{j}\theta\right]\psi\
,$

and after some algebra

$\epsilon_{\cdots ij}\partial_{i}\partial_{j}\psi=i(\epsilon_{\cdots
ij}\partial_{i}\partial_{j}\varphi)\psi-i(\epsilon_{\cdots
ij}\partial_{i}\partial_{j}\phi)\left(J_{3}-\frac{s}{2}\right)\psi\ .$ (124)

The derivation discards $\epsilon_{\cdots ij}\partial_{i}\partial_{j}\theta$
because $\theta\in[0,\pi]$ cannot wind and support a vortex singularity. We
point out in passing that the statistics correction $s$ has no impact on the
dynamics of spin-hedgehog topological defects Nikolić (2020a) because
hedgehogs introduce $\phi$ vorticity at both north $\theta=0$ and south
$\theta=\pi$ poles, and the effects of $s$ at both poles cancel out.

Let us simplify the further analysis by temporarily neglecting charge
currents, i.e setting $\varphi=\textrm{const}$. When we substitute (124) in
(IV.4) with $\varphi=\textrm{const.}$, we get

$\displaystyle I_{4}$ $\displaystyle=$
$\displaystyle-\int\limits_{B^{4}}d^{4}x\,\epsilon_{ijkl}(\partial_{i}\partial_{j}\phi)(\partial_{k}\partial_{l}\phi)\,\psi^{\dagger}\left(J_{3}-\frac{s}{2}\right)^{2}\psi$
(125) $\displaystyle=$ $\displaystyle-S^{\prime
2}|\psi|^{2}\int\limits_{B^{4}}d^{4}x\,\epsilon_{ijkl}(\partial_{i}\partial_{j}\phi)(\partial_{k}\partial_{l}\phi)\
.$

If we introduce a parameter $\xi\in\mathbb{R}$ which specifies the points
${\bf r}^{(n)}(\xi)$ on the $n^{\textrm{th}}$ skyrmion’s center, then we can
express the singular vorticity of $\phi$ in a $d=3$ manifold as

$\displaystyle\epsilon_{ijk}^{\phantom{x}}\partial_{j}^{\phantom{x}}\partial_{k}^{\phantom{x}}\phi$
$\displaystyle=$ $\displaystyle\sum_{n}\int
dr_{i}^{(n)}\,2\pi\delta\bigl{(}{\bf r}-{\bf r}^{(n)}(\xi)\bigr{)}$
$\displaystyle=$ $\displaystyle\int d\xi\,\frac{\partial
r_{i}^{(n)}}{\partial\xi}\,2\pi\delta\bigl{(}{\bf r}-{\bf
r}^{(n)}(\xi)\bigr{)}\ ,$

where the $\delta$-function is 3-dimensional. The parametrization of a
$\phi$-vortex in $d=4$ must take the form of a rank-2 tensor because a
skyrmion center is a sheet of points ${\bf r}^{(n)}(\xi_{1},\xi_{2})$. Summing
over all skyrmions ($n$), we obtain

$\displaystyle\epsilon_{ijkl}^{\phantom{x}}\partial_{k}^{\phantom{x}}\partial_{l}^{\phantom{x}}\phi=\sum_{n}\int
d\xi_{1}^{\phantom{x}}d\xi_{2}^{\phantom{x}}\left(\frac{\partial
r_{i}^{(n)}}{\partial\xi_{1}}\frac{\partial
r_{j}^{(n)}}{\partial\xi_{2}}-\frac{\partial
r_{i}^{(n)}}{\partial\xi_{2}}\frac{\partial
r_{j}^{(n)}}{\partial\xi_{1}}\right)$ $\displaystyle\qquad\times
2\pi\,\delta\bigl{(}{\bf r}-{\bf
r}^{(n)}(\xi_{1}^{\phantom{x}},\xi_{2}^{\phantom{x}})\bigr{)}\ .$ (127)

The $\delta$-function is four-dimensional here. Substituting the last
expression into (125) and using

$\epsilon_{pqij}\epsilon_{pqkl}=2(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})$
(128)

yields

$\displaystyle I_{4}$ $\displaystyle=$ $\displaystyle-(2\pi
S^{\prime})^{2}|\psi|^{2}\sum_{n,m}\,\int\limits_{B^{4}}d^{4}r\,\int
d\xi_{1}^{(m)}d\xi_{2}^{(m)}d\xi_{1}^{(n)}d\xi_{2}^{(n)}$
$\displaystyle\times\epsilon_{ijkl}\,\frac{\partial
r_{i}^{(m)}}{\partial\xi_{1}^{(m)}}\frac{\partial
r_{j}^{(m)}}{\partial\xi_{2}^{(m)}}\frac{\partial
r_{k}^{(n)}}{\partial\xi_{1}^{(n)}}\frac{\partial
r_{l}^{(n)}}{\partial\xi_{2}^{(n)}}$ $\displaystyle\times\delta\bigl{(}{\bf
r}-{\bf r}^{(n)}(\xi_{1}^{(n)},\xi_{2}^{(n)})\bigr{)}\,\delta\bigl{(}{\bf
r}-{\bf r}^{(m)}(\xi_{1}^{(m)},\xi_{2}^{(m)})\bigr{)}\ .$

Let us assume that a Hopf singularity of spins is at the origin. We can
enclose it by concentric $S^{3}$ 3-spheres that each carry the same Hopf
index. The inter-linked skyrmions which materialize the Hopf index are loops
in each $S^{3}$ manifold, so every pair of coordinates $r_{i}$ and $r_{j}$ in
the last integral lives on a cone-shaped skyrmion sheet embedded in the $d=4$
space; one coordinate (parametrized by $\xi_{1}$) is bound to the skyrmion
loop on the 3-sphere and the other (parametrized by $\xi_{2}$) to the “radial”
direction which emanates from the Hopf singularity. All skyrmions need to meet
at the singularity in order for $I_{4}$ to capture them; the skyrmion lines do
not intersect on any $S^{3}$ away from the singularity, so only the
singularity itself can contribute to $I_{4}$. We will extract the skyrmion
linking number from $I_{4}$. The first step is to integrate out ${\bf r}$:

$\displaystyle I_{4}$ $\displaystyle=$ $\displaystyle-(2\pi
S^{\prime})^{2}|\psi|^{2}\sum_{n,m}\int
d\xi_{1}^{(m)}d\xi_{2}^{(m)}d\xi_{1}^{(n)}d\xi_{2}^{(n)}$
$\displaystyle\quad\times\epsilon_{ijkl}\,\frac{\partial
r_{i}^{(m)}}{\partial\xi_{1}^{(m)}}\frac{\partial
r_{j}^{(m)}}{\partial\xi_{2}^{(m)}}\frac{\partial
r_{k}^{(n)}}{\partial\xi_{1}^{(n)}}\frac{\partial
r_{l}^{(n)}}{\partial\xi_{2}^{(n)}}$
$\displaystyle\quad\times\delta^{4}\bigl{(}{\bf
r}^{(m)}(\xi_{1}^{(m)},\xi_{2}^{(m)})-{\bf
r}^{(n)}(\xi_{1}^{(n)},\xi_{2}^{(n)})\bigr{)}$

We now distinguish the loop $\xi_{1}$ and radial $\xi_{2}$ parameters for each
pair $n,m$ of skyrmion cones. The loop parameter is integrated over a one-
dimensional manifold without a boundary (the perimeter of the 2-cone base),
and the radial parameter is integrated out on a line such that ${\bf
r}(\xi_{1},\xi_{2})$ is carried from the singularity to the final $S^{3}$
shell of some radius $R$ (the boundary of $B^{4}$). Then, the four-dimensional
Dirac function can be broken up into the radial and $S^{3}$ factors

$\delta^{4}(\Delta{\bf r})=\delta(\Delta r_{r})\delta^{3}(\Delta{\bf
r}_{S^{3}})$ (131)

where $\Delta r_{r}$ and $\Delta{\bf r}_{S^{3}}$ are the projections of
$\Delta{\bf r}={\bf r}^{(m)}-{\bf r}^{(n)}$ onto the radial direction and the
$S^{3}$ manifold orthogonal to it respectively. We immediately observe that,
by construction, $\Delta{\bf r}_{S^{3}}^{\phantom{x}}=\Delta{\bf
r}_{S^{3}}^{\phantom{x}}(\xi_{1}^{(m)},\xi_{1}^{(n)})\propto R$ is
parametrized only by the loop coordinates $\xi_{1}$ of the two skyrmions,
provided that both live on the same shell, $\xi_{2}^{(m)}=\xi_{2}^{(n)}$, of
radius $R$. The latter condition is enforced by the first $\delta$-function
factor. Now we can integrate out one of the radial $\xi_{2}$ coordinates and
rename the other one into $\eta$: