# Instanton confinement-deconfinement transitions: The stability of pseudogap
phases and topological order

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

We explore the stability of certain many-body quantum states which may exist
at zero or finite temperatures, may lack long-range order and even topological
order, and still are thermodynamically distinct from uncorrelated disordered
phases. We sharply characterize such states by the conservation of topological
charge, or equivalently confinement of instantons, using a generalization of
the Wilson loop and the correlation length of an emergent gauge field. Our
main conclusions are: (i) topological orders can exist at finite temperatures,
(ii) relativistic liquids of topological defects can also exist as stable
phases at finite temperatures, and (iii) there are two universality classes of
instanton suppression. We also relate the instanton dynamics to the problem of
the pseudogap state in underdoped cuprates. A universal experimental signature
of the instanton deconfinement transition is a change of the quantum noise
spectrum, which can perhaps be measured in some situations, for example via a
quantum anomaly, or indirectly detected with a specific heat jump. The method
of analysis is a functional renormalization group that generalizes the Coulomb
gas treatment of Kosterlitz and Thouless to arbitrary interactions and
dimensions. In particular, we construct an exact non-perturbative technique
for confining interactions between instantons, which introduce irreparable
infra-red divergences in the standard perturbative approaches.

##  I Introduction

Strongly correlated states of matter pose a great challenge to our universal
understanding of materials, especially when interactions between particles are
combined with quantum entanglement, thermal effects and non-trivial topology.
Topological order is a form of many-body quantum entanglement which cannot be
unambiguously detected by correlations between any local properties, and yet
produces ground state degeneracy on topologically non-trivial manifolds Wen
(2004). Fractional quantum Hall states are the only experimentally confirmed
realizations of topological order up to date. However, the theory of
topological orders has advanced very far. A complete classification of
topological orders in two spatial dimensions may have been achieved Wen
(2017), and at least a partial classification in three dimensions is emerging
Levin and Stern (2009); Maciejko  _et al._ (2010); Swingle  _et al._ (2011);
Walker and Wang (2012); von Keyserlingk  _et al._ (2013); Jian and Qi (2014);
Wang and Levin (2014); Ye and Gu (2015, 2016); Putrov  _et al._ (2017); Fuji
and Furusaki (2019); Lan  _et al._ (2018); Lan and Wen (2019). This major gap
between theory and experiment is actually also manifest in the theory itself
since it seems unclear how to take a random case from the available
classification schemes and construct a realistic microscopic model (beyond a
specialized exactly solvable model) which realizes the chosen topological
order in its ground state. This is in contrast to the phenomenon of symmetry
breaking and conventional long-range order: the moment we choose a group for
symmetry transformations, we have an idea about what degrees of freedom to use
and how to couple them within a physical Hamiltonian, and then we have many
techniques at our disposal to analyze the dynamics. The present study attempts
to approach the problem of topological order from a similar physical point of
view, still quite universal but focused on the issues of feasibility and
stability instead of classification.

Here, we explore certain general conditions for the stability of
unconventional quantum phases which exhibit strong correlations and no
spontaneous symmetry breaking. We are interested in the phases shaped by
delocalized but uncondensed topological defects. A topological protection
mechanism that conserves topological charge, and hence allows particles to be
delocalized as well, is engaged through the confinement of instantons. Thus,
we determine, using renormalization group, which types of instanton
interactions in $D=d+1$ space-time dimensions lead to instanton confinement at
zero and finite temperatures. While much is already known about the issues of
confinement Berezinsky (1972); Kosterlitz and Thouless (1973); Kosterlitz
(1974); Wilson (1974); Polyakov (1975); Belavin  _et al._ (1975); ’t Hooft
(1976); Kosterlitz (1977); Polyakov (1977, 1978); Osterwalder and Seiler
(1978); Guth (1980); Lautrup and Nauenberg (1980); Svetitsky and Yaffe (1982);
Nagaosa (1993); Schäfer and Shuryak (1998); Kondo (1998); Zheng and Sachdev
(1989); Read and Sachdev (1990); Kondo (1999); ’t Hooft and Bruckmann (2000);
Nagaosa and Lee (2000); Yoshida (2001); Sachdev and Park (2002); Kleinert  _et
al._ (2002); Herbut  _et al._ (2003); Case  _et al._ (2004); Hermele  _et al._
(2004); Kragset  _et al._ (2004); Børkje  _et al._ (2005); Engelhardt (2005);
Wang (2005); Nogueira and Kleinert (2005, 2008); Yamamoto (2008); Ogilvie
(2012); Anber (2014), the general scope and various technical aspects of this
study make it non-trivial. Also, the instanton context of confinement leads to
several important and surprising conclusions: (i) topological order is
possible at finite temperatures, (ii) relativistic correlated quantum liquids
of topological defects and anti-defects are also possible at finite
temperatures, with or without topological order, and (iii) there is more than
one universality class of instanton deconfinement.

We will show that stable thermodynamic phases of confined instantons can exist
independently of spontaneous symmetry breaking in generic systems which
support topological defects. An important prerequisite is the presence of an
intrinsic length scale distinct from the ultra-violet cut-off and unrelated to
temperature or disorder. The new length scale is required in order to
determine a finite coherence length $\xi$ of the matter field. If the
instantons are neutralized within distances $\lambda<\xi$, then topological
defects are well-defined and conserved for all practical purposes. The
fractional quantum Hall state is an example: it lacks long-range order because
the introduced vortices are mobile, but its filling factor $\nu$, the number
of particles per vortex, is rationally quantized. Since charge is conserved,
this is possible only if topological charge, or vorticity, is conserved as
well.

This paper is organized as follows. Section II introduces instantons and
explains a thermodynamic characterization of their confinement with a
generalized Wilson operator and a correlation length for an emergent gauge
field. Then, Section II.1 begins with a survey of the literature and facts
about confinement-deconfinement transitions in the contexts related to this
study. Here we also present the general idea behind our analysis, and the
manner in which it connects to topological orders and pseudogap states in
cuprates. Technical presentation of the instanton gas renormalization group is
given in Section III. After a brief introduction, we consider non-confining
potentials in Section III.1, logarithmic potentials in Section III.2, general
confining potentials in Section III.3, and finite-temperature transitions in
Section III.4. At the end, the critical scaling of the correlation length with
temperature is derived for two universality classes of instanton
deconfinement. Then, Section IV explores possible experimental signatures of
the confined-instanton phases, hidden in the quantum noise. Conclusions and
final discussions are summarized in Section V.

Throughput 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 Thermodynamic distinction between confined and deconfined phases

Consider a generic quantum system in the continuum limit whose local degrees
of freedom are accessed by a field operator $\psi({\bf x})$. If the system
supports topological defects, it is also possible to construct an operator
$\mathcal{J}_{0}[\psi]$ which probes their density. The topological index or
“charge” of all defects inside a $d$-dimensional volume $B^{d}$ is

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

No smooth field deformations can change the topological index. We
mathematically represent this fact by a conservation law
$\partial_{\mu}\mathcal{J}_{\mu}=0$, where $\mathcal{J}_{i}$ is the current
density of topological defects. The solution of the conservation equation can
be expressed using an antisymmetric tensor gauge field
$\mathcal{A}_{\mu_{1}\cdots\mu_{d-1}}$,

$\mathcal{J}_{\mu}=\epsilon_{\mu\nu\lambda_{1}\cdots\lambda_{d-1}}\partial_{\nu}\mathcal{A}_{\lambda_{1}\cdots\lambda_{d-1}}\
.$ (2)

Explicit constructions of $\mathcal{J}_{\mu}$ have been obtained for spinor
and vector fields whose topological defects (vortices, monopoles and
hedgehogs) are classified by the $\pi_{n}(S^{n})$ homotopy groups Nikolić
(2020a). Using Stokes-Cartan theorem, we find that the topological invariant
is the gauge flux through the boundary $S^{d-1}$ of $B^{d}$:

$\displaystyle N$ $\displaystyle=$
$\displaystyle\int\limits_{B^{d}}d^{d}x\,\epsilon_{0ij_{1}\cdots
j_{d-1}}\partial_{i}\mathcal{A}_{j_{1}\cdots j_{d-1}}$ $\displaystyle=$
$\displaystyle\oint\limits_{S^{d-1}}d^{d-1}x\,\epsilon_{j_{1}\cdots
j_{d-1}}\mathcal{A}_{j_{1}\cdots j_{d-1}}\ .$

In our notation, the Levi-Civita tensor $\epsilon$ either carries all space-
time indices, or displays only the indices which live on the integration
manifold as in the last line.

In order to study dynamics at any temperature, we construct the imaginary-time
path integral for the partition function

$\mathcal{Z}=\textrm{tr}(e^{-\beta
H[\psi]})=\int\mathcal{D}\psi\,e^{-S[\psi]}\ .$ (4)

Here, $H$ is the Hamiltonian operator, $S$ is the imaginary-time action, and
$\beta=1/T$ is the inverse temperature. The action is an integral of the
Lagrangian density $\mathcal{L}$ over the infinite $d$-dimensional space and a
finite imaginary time extent $\tau\in(0,\beta)$ with periodic boundary
conditions. Let us introduce the instanton density operator

$\mathcal{I}=\partial_{\mu}\mathcal{J}_{\mu}\ ,$ (5)

which vanishes if the topological charge is conserved. For our purposes,
instantons are the lowest-action events of quantum tunneling between
topologically distinct classical configurations, i.e. events in which
topological defects are created or annihilated. Therefore, instantons require
singularities in the field configuration $\psi(x_{\mu})$. We can apply Gauss’
theorem to a space-time volume $B^{d+1}$ and express the number of instantons
in it as the flux of $\mathcal{J}_{\mu}$ through the boundary $S^{d}$:

$\mathcal{N}=\int\limits_{B^{d+1}}d^{d+1}x\,\mathcal{I}=\oint\limits_{S^{d}}d^{d}x\,\hat{\eta}_{\mu}\mathcal{J}_{\mu}\
,$ (6)

where $\hat{\eta}_{\mu}$ is the unit-vector locally perpendicular to the
oriented surface $S^{d}$.

By simple symmetry considerations, we could naively capture the essential
dynamics of topological defects with a gradient term
$\mathcal{J}_{\mu}\mathcal{J}_{\mu}$ in the Lagrangian density, which is the
Maxwell term of the gauge field $\mathcal{A}_{\mu_{1}\cdots\mu_{d-1}}$. If no
other Lagrangian term is relevant for instanton dynamics, then the action cost
of $\mathcal{N}\neq 0$ will be minimized by evenly distributing the flux of
$\mathcal{J}_{\mu}$ over the entire $S^{d}$ manifold. This produces a Coulomb
attractive interaction potential between opposite-charge instantons separated
by $r$ in the $D=d+1$ dimensional space-time:

$V_{\textrm{C}}(r)\sim 1/r^{D-2}$ (7)

This estimate comes from the $\mathcal{N}=1$ solution of (6),
$\mathcal{J}_{\mu}(r)\propto\hat{r}_{\mu}/r^{d}$, and the ensuing action cost
$\int d^{d+1}x\,(\mathcal{J}_{\mu})^{2}\propto 1/r^{d-1}$ cut-off by the
distance between the two instantons. It is known that the Coulomb interaction
in $D>2$ is not able to confine its charged particle sources into neutral
dipoles Kosterlitz (1977). Consequently, instantons with this interaction
would be free to roam and destroy the conservation of topological charge at
macroscopic length and time scales.

However, the above picture is inconsistent with the smooth low-energy
configurations of the matter field $\psi$; it requires a continuous
distribution of singularities, which physically corresponds to a Mott
insulator. Instead, a coherent $\psi$ field, governed by a
$(\partial_{\mu}\psi)^{2}$ Lagrangian density, focuses its non-zero
topological currents into lower-dimensional singular domains. The flux of
$\mathcal{J}_{\mu}$ satisfying (6) then stretches through a microscopically
narrow tube, as a string connecting the two opposite-charge instantons. The
string acquires tension from the depletion of the coherent $\psi$ background
inside the tube, leading to a linear interaction between instantons:

$V_{\textrm{conf.}}(r)\sim r\ .$ (8)

An example with vortices is shown in Fig.1. This kind of interaction is
confining. Instantons of opposite charge are asymptotically free at short
distances but cannot drift away from each other. Their dynamics is, however,
complicated by the instanton dipole fluctuations. If one tries to separate a
pair of opposite-charge test instantons to an arbitrarily large distance, one
eventually supplies enough action to generate another dipole of instantons.
The generated instantons can move to completely neutralize the original
instantons, and allow one to continue pulling them apart without further
resistance. Therefore, screening is inevitable despite a finite action cost or
“mass” of an instanton. The resulting finite screening length $\lambda$ may be
further reduced by virtual instanton fluctuations.

Figure 1: An illustration of instantons for vortices in $D=2+1$ space-time
dimensions. (a) Worldlines of a vortex and antivortex (blue) are terminated by
a pair of opposite-charge instantons. In this example, the two instantons
occur at the same time $t=t_{0}$, but this is not generally required. The
phase gradients $\boldsymbol{\nabla}\theta$ of $\psi\sim e^{i\theta}$, which
build the action cost, are indicated with colored clouds. Far away from the
instantons, the phase gradients are minimal (green), and typically produce a
Coulomb-like interaction between the topological defects (screened to a finite
range in the presence of gauge fields). But, the gradients must smoothly
deform near the instantons in order to minimize the impact of a sudden change
across $t=t_{0}$. In the best case scenario, there is a large action cost
$V(r)$ on the temporal space-time lattice links (due to $\partial_{t}\theta$)
at the time $t=t_{0}$ and within a region of space which grows in proportion
to the distance $r$ between the instantons. The resulting $V(r)\propto r$
confines the instantons and suppresses the shown vortex creation/annihilation
events. However, the underlying assumption was that discontinuous changes of
$\theta$ are very costly. Instead, phase fluctuations on a lattice can be
abundant enough to uncorrelate $\theta$ even across distances comparable to
the lattice constant. This situation characterizes Mott insulators and removes
the linear confinement of instantons (i.e., $V(r)$ becomes short-ranged). (b)
The case of confined instantons: vortex flux lines cannot terminate. In this
example, a vortex-antivortex pair annihilates, but the horizontal connecting
flux line $\boldsymbol{\nabla}\times{\bf A}\neq 0$ corresponds to the electric
field ${\bf E}=\partial_{t}{\bf A}$ which is generated via Faraday’s law. The
Maxwell action cost of this electric field suppresses vortex
creation/annihilation process across large distances.

We will show that instanton confinement is a thermodynamic phase transition
which can occur at finite temperatures independently of symmetry breaking. A
sharp distinction between the confined and deconfined phases is given by a
generalization of the Wilson loop Wilson (1974) operator defined on the closed
space-time surface $S^{d}$. This operator counts the total instanton charge
enclosed by $S^{d}$, and we can set it up in imaginary time using (6) at zero
temperature:

$\mathcal{C}(S^{d})\equiv\mathcal{N}\ .$ (9)

This is a random variable and we are interested in its variance. If instantons
are deconfined, then there is a finite probability of an uncompensated
instanton appearing at any point in space-time, uncorrelated with the
locations of other instantons. Such random fluctuations make the variance
$\textrm{Var}\,\mathcal{C}(S^{d})\propto B^{d+1}$ scale as the volume
$B^{d+1}$ of the space-time enclosed by $S^{d}$. In contrast, confined
instantons are always compensated within a finite radius $\lambda$, so their
number inside $S^{d}$ can be affected only by small dipole fluctuations near
$S^{d}$, giving rise to an “area-law” behavior
$\textrm{Var}\,\mathcal{C}(S^{d})\propto S^{d}$. This distinction is made in
the limit of infinite surface $S^{d}\to\infty$, so it is impossible to
smoothly deform the “volume-law” into the “area-law” behavior. Ultimately,
$\textrm{Var}\,\mathcal{C}(S^{d})$ serves the role of an “order parameter” for
the thermodynamic phase transition between the confined and deconfined states
of instantons:

$\textrm{Var}\,\mathcal{C}(S^{d})\xrightarrow{S^{d}\to\infty}{}\begin{cases}aS^{d}&,\quad\textrm{confined
phase}\\\ bB^{d+1}&,\quad\textrm{deconfined phase}\end{cases}\ .$ (10)

The most general definition of the operator $\mathcal{C}(S^{d})$ is obtained
in real time by exploiting the formal correspondence between the imaginary and
real time path integrals at zero temperature. We just interpret (6) as a
quantum mechanical operator expressed in terms of the field
(creation/annihilation) operators $\psi$, and replace the imaginary time
coordinate $x_{0}\equiv\tau$ with real time $x_{0}\to it$:

$\mathcal{C}(S^{d})=\oint\limits_{S^{d}}d^{d}x\,\hat{n}_{\mu}\mathcal{J}_{\mu}[\psi]\
.$ (11)

Note that a factor of $i$ inherited from the integral measure was absorbed
into the appropriate (re)definition of $\mathcal{J}_{\mu}$ as a Hermitian
operator. The time dependence of field operators is handled as usual with
$\psi(t)=e^{iHt}\psi(0)e^{-iHt}$ in the Heisenberg picture. Measurements of
$\mathcal{C}(S^{d})$ in the ground state $|0\rangle$ yield random outcomes,
and their variance follows (10) by the same topological considerations as
before. Having a proper quantum operator prescribes measurements at any
temperature as

$\textrm{Var}\,\mathcal{C}\equiv\bigl{\langle}(\mathcal{C}-\langle\mathcal{C}\rangle)^{2}\bigr{\rangle}\quad,\quad\langle\cdots\rangle=\textrm{tr}(\cdots
e^{-\beta H})\ ,$ (12)

even when the exact real-imaginary time correspondence is lost due to
$\beta<\infty$. This enables a sharp distinction between confined and
deconfined phases at $T>0$; the infinite extent of real time allows taking the
$S^{d}\to\infty$ limit.

Another important characterization of the topological defect dynamics is the
correlation length $\zeta$ of the effective gauge field
$\mathcal{A}_{\lambda_{1}\cdots\lambda_{d-1}}$. When instantons fluctuations
are suppressed, i.e. $\partial_{\mu}\mathcal{J}_{\mu}=0$, then the Maxwell
Lagrangian density $\frac{1}{2e^{2}}\mathcal{J}_{\mu}\mathcal{J}_{\mu}$ with
(2) yields a gauge-invariant correlation function in momentum space

$\displaystyle\left\langle\mathcal{A}_{\mu_{1}\cdots\mu_{d-1}}({\bf
q})\mathcal{A}_{\nu_{1}\cdots\nu_{d-1}}({\bf q}^{\prime})\right\rangle\sim$
(13)
$\displaystyle\qquad\sim\frac{e^{2}}{q^{2}}\frac{q_{\alpha}q_{\beta}}{q^{2}}\epsilon_{\lambda\alpha\mu_{1}\cdots\mu_{d-1}}\epsilon_{\lambda\beta\nu_{1}\cdots\nu_{d-1}}\,\delta({\bf
q}+{\bf q}^{\prime})\ .$

This algebraic form signifies the presence of gapless photon modes and a
formally infinite correlation length that characterizes the gauge field
fluctuations. Note that a topological Lagrangian term which arises from a
Berry curvature, or the Higgs mechanism, can open a gap in the photon
spectrum, but this may also modify the instanton interaction potential. If
uncompensated instantons are present and separated by a mean space-time
distance $\zeta$, then the photons can propagate coherently only across the
length and time scales of $\zeta$. This effectively gives the photons a mass
$m\sim\zeta^{-1}$. We will find that the correlation length $\zeta$ diverges
as a power of $T-T_{c}$ when the temperature $T$ approaches from above its
critical value $T_{c}$ for the instanton confinement transition.

###  II.1 Cases of confinement and deconfinement

Most studies of confinement are specialized to a particular gauge theory, and
ask if the gauge field fluctuations produce a potential energy $U(r)$ between
static test charges which grows indefinitely with the distance $r$. The common
Abelian electrodynamics ($D=3+1$) does not confine charge, but its compact
variety, whose lattice action

$S=-\frac{1}{g^{2}}\sum_{\square}\cos\left(\sum_{\mu\in\square}A_{\mu}\right)$
(14)

evaluates the gauge flux on lattice plaquettes $\square$, causes confinement
in the strong coupling $g\gg 1$ regime Wilson (1974); Polyakov (1975);
Osterwalder and Seiler (1978); Guth (1980); Lautrup and Nauenberg (1980).
Confinement-deconfinement transitions are found in all $D\geq 4$ compact U(1)
gauge theories Svetitsky and Yaffe (1982), while the $D=3$ case is special and
always confined in the absence of a dynamical matter field Polyakov (1975).
Here is a naive explanation. The weak coupling regime $g\ll 1$ justifies
taking the continuum limit

$S\to\frac{1}{2\gamma^{2}}\int
d^{D}x\,(\epsilon_{\cdots\mu\nu}\partial_{\mu}A_{\nu})^{2}\ .$ (15)

This non-compact Maxwell form suppresses monopoles, i.e. terminations of the
vortex (world)lines, and hence mediates plain Coulomb interactions between
charges. It just happens that the Coulomb potential $U(r)\propto 1/r^{d-2}$
between particles separated by $r$ vanishes as $r\to\infty$ in $d\geq 3$, but
becomes confining $U(r)\propto\log(r/a)$ in $d=2$ ($D=3$), making only neutral
bound states possible. The opposite $g\gg 1$ limit of (14) has to be
considered on the lattice. It admits free flux-quantum fluctuations through
lattice plaquettes and hence allows monopoles. Abundant monopoles that cannot
be removed by a gauge transformation strongly frustrate the dynamics of
uncompensated charges via quantum interference, until only neutral objects can
exist as low-energy excitations. The mechanism is similar to the particle
localization in Mott insulators when viewed from a dual theory Fisher  _et
al._ (1989); Fisher (1990). Since gauge symmetry breaking is not possible
Elitzur (1975), the phase transitions in pure gauge theories have topological
character.

The deconfinement of particles in the presence of dynamical matter fields has
been intensely studied in condensed matter physics Fradkin and Shenker (1979);
Nagaosa and Lee (2000), especially in relation to the stability of spin
liquids Kleinert  _et al._ (2002); Herbut  _et al._ (2003); Case  _et al._
(2004); Kragset  _et al._ (2004); Hermele  _et al._ (2004); Nogueira and
Kleinert (2008). Massless particles suppress gauge field fluctuations and
facilitate charge deconfinement. In $d=2$ spatial dimensions, where this
matters most, massless Dirac fermions coupled to a U(1) gauge field can be
deconfined if they exist in sufficiently many flavors Hermele  _et al._
(2004); Nogueira and Kleinert (2008). Fermi liquids expel electric fields and
superconductors expel all gauge fields, so their particles are self-
consistently deconfined. Everything becomes more complicated with non-Abelian
gauge fields. A crucial property of Yang-Mills gauge theory is the existence
of instanton configurations which cost a finite action even in the continuum
limit Belavin  _et al._ (1975); ’t Hooft (1976); Schäfer and Shuryak (1998);
’t Hooft and Bruckmann (2000); Ogilvie (2012). Similar to skyrmions in $d=2$,
these instantons are characterized by an arbitrary length-scale which
determines their size, so they are relevant macroscopically (in the infra-red
limit). Their proliferation is responsible for the confinement of quarks.

Our interest in this paper is the confinement of instantons instead of
particles. The fate of instantons is most directly decided by their
interaction potential. We only tackle a small part of this problem by asking
which instanton interactions lead to confinement, and what kinds of
deconfinement transitions occur in any $D$ space-time dimensions at zero or
finite temperatures. We answer these questions by an “instanton gas
renormalization group” (IGRG), building on the Coulomb gas renormalization
group of Kosterlitz and Thouless Kosterlitz and Thouless (1973); Kosterlitz
(1974, 1977). A significant part of quantum dynamics are instanton number
fluctuations; analogous processes are indirectly incorporated in the original
approach of Kosterlitz and Thouless, but curbed in most studies of the Coulomb
gas which aim to address soft condensed matter systems Caillol and Levesque
(1986); Minnhagen (1987); Levin  _et al._ (1994); Fisher (1994); Gama
Goicochea and Nussinov (2023).

The main RG extension in this work is an exact functional treatment of
confining potentials. We will show that potentials which are asymptotically
free and more confining than the logarithm generally induce instanton
confinement below a finite critical temperature. Confining potentials which
are singular at the origin, such as the logarithm, will be considered not
asymptotically free. They stabilize a different fixed point, with deconfined
but suppressed instantons at zero temperature and no finite temperature
transition. By contrast, potentials which vanish at large distances are
regarded as naively non-confining. We find that the Coulomb Kosterlitz (1977)
and weaker potentials cannot confine the instantons in any $D$ dimensions at
any temperature. The employed perturbative RG method breaks down for stronger
non-confining potentials, but indicates their functional flow into confining
forms.

In bosonic systems, a superfluid phase gaps out any gauge field and thus
suppress the instantons. As explained earlier, the interaction between
instantons is then confining, in fact linear $V(r)\propto r$ if the instantons
create and destroy singular point defects with a $\pi_{d-1}(S^{d-1})$ homotopy
invariant (see Appendix A for a general derivation). Instantons associated
with the Hopf index interact at least logarithmically, which is still
confining (shown in Appendix B), but instantons associated with finite loops
and other contractible structures tend to have short-range interactions. If
the potential is confining, every instanton will be compensated by its anti-
instanton within a finite screening radius $\lambda$. Even when fluctuations
destabilize the superfluid state, the dynamics resembles that of a superfluid
below the length and time scales given by the correlation length $\xi$. If
$\lambda<\xi$, we may compute the confining instanton interaction potential as
if we were in the superfluid phase and then let the renormalization group
confirm confinement from the macroscopic point of view by showing the flow
$\lambda\to 0$. We, therefore, find instanton confinement in the absence of
spontaneous symmetry breaking, and sharply distinguish it from the opposite
regime $\lambda\gg\xi$ by the generalized Wilson correlation (10).

This fits the description of known topologically ordered phases. A
prerequisite for topological order is the conservation of topological charge,
i.e. instanton confinement. It was observed recently that monopoles and
hedgehogs in higher dimensions $d>2$ can seemingly exist in topologically
ordered phases at finite temperatures Nikolić (2020a), because their
degenerate ground states on topologically non-trivial manifolds are separated
by (free) energy barriers that grow as a positive power of the system size.
This is similar to the barriers between the symmetry-related degenerate ground
states which enable spontaneous symmetry breaking in macroscopic systems. The
present analysis supports finite-temperature topological order by
demonstrating instanton confinement at finite temperatures.

Instanton confinement by itself does not imply topological order. Several
interesting systems of particles with strong spin-orbit coupling in $d=2$ and
$d=3$ spatial dimensions exhibit periodic arrays of topological defects and
anti-defects in their mean-field ground states Nikolić  _et al._ (2013);
Nikolić (2014, 2016, 2020b). The spin-orbit coupling provides a “magnetic”
length that determines the separation between the defects. One imagines that
quantum fluctuations could melt such a defect lattice. If that happens, some
defects and anti-defects could annihilate, but a finite density of both would
still be energetically protected, even if the defects became delocalized. It
is not obvious if such a quantum liquid would possess topological order
because the average numbers of defects and anti-defects could be the same and
finite. However, the present study shows that this quantum liquid can have a
zero _conserved_ topological charge, and thus be distinct from a Mott
insulator.

Pseudogap states have a long history in the context of high-temperature
superconductors, and also can be considered in cold atom gases. A pseudogap
state can be smoothly connected to a disordered high-temperature phase. A
simple example is the crossover from a band-insulator of weakly interacting
fermions to a Mott insulator of tightly bound $s$-wave Cooper pairs Nikolić
and Tesanović (2011). Such a crossover parallels the BCS-BEC crossover in cold
atom gases; there is no phase transition between the two types of insulators,
but the Mott insulator is special because its lowest energy excitations are
bosonic and its transition to the superconducting phase is in a different
universality class than the analogous transition from the band-insulator Moon
_et al._ (2007); Nikolić (2011). More intricate possibility is a correlated
pseudogap phase, thermodynamically distinct from conventional disordered
states. Any state of confined instantons is a candidate because it is sharply
characterized at least by its instanton confinement via (10). Many experiments
have revealed distinctive short-range correlations in the underdoped pseudogap
states of cuprates Vershinin  _et al._ (2004); Fang  _et al._ (2004); Steiner
_et al._ (2005); Valla  _et al._ (2006); Crane  _et al._ (2007); Boyer  _et
al._ (2007); Kohsaka  _et al._ (2007); Gomes  _et al._ (2007); Kohsaka  _et
al._ (2008); Ghiringhelli  _et al._ (2012); Torchinsky  _et al._ (2013), above
the superconducting critical temperature $T_{c}$ and below the doping-
dependent “pseudogap” temperature $T^{*}$. Notably, however, Nernst effect
measurements Wang  _et al._ (2001, 2006); Li  _et al._ (2010) find a plausible
interpretation in the picture of protected and mobile vortices which drift due
to a temperature gradient and produce a voltage drop in the direction
perpendicular to their drift. One may ask how vortices can survive when the
superconducting order parameter loses long-range coherence. The present study
provides a rigorous explanation, and (re)opens the possibility that a phase
transition associated with instanton deconfinement happens at the $T^{*}$
temperature.

Of course, the interesting confined-instanton quantum liquids may be routinely
precluded by conventional phase transitions out of the coherent (e.g. Higgs)
phase of the matter field. At least a dedicated length scale must be provided
in the Hamiltonian for the finite correlation length $\xi$ if the confined-
instanton phase is to be possible. If, instead, the model defines only the
lattice constant, or some equivalent ultra-violet cut-off length, then the
matter field cannot be short-range coherent across any relevant finite
distances $\xi>\lambda$. This, in fact, keeps the phase diagram of the compact
XY model simple and without “pseudogap” phases. If the XY model is gauged and
the external magnetic field specifies a magnetic length, then quantum Hall
liquids become possible.

So far we have emphasized the role of bosons in the instanton confinement, but
fermions also have a role to play. Due to the Pauli exclusion, fermions cannot
maintain a macroscopic phase coherence in their charge currents. Nevertheless,
their quantum coherence is still evident at zero temperature in the simple
phenomena like the sharp Fermi surface in interacting Fermi liquids and zero
conductivity in band insulators. These features characterize the thermodynamic
limit, but degrade at any finite temperature: the Fermi-Dirac distribution of
occupation numbers becomes a continuous function of energy, and the
conductivity of band-insulators becomes thermally activated. A more
complicated phenomenon is the electronic quantum Hall liquid at zero
temperature, which can be fractionalized. This type of a state still requires
instanton confinement for the quantization of the filling factor $\nu$. How is
this possible? We find an answer by observing that delocalized fermions at
zero temperature still possess quantum coherence up to the mean inter-particle
separation distance $l$, limited by the Pauli exclusion. If $\xi\sim l$ is
sufficiently larger than the lattice constant, and the topological defects are
separated by comparable distances, we have the basic condition for instanton
confinement just as in the case of short-range coherent bosons.

##  III Instanton gas renormalization group

In order to study instanton confinement, we introduce instantons at space-time
positions ${\bf x}_{i}$ as boundary conditions on the $\psi$ field in the
partition function (4). Then we integrate out the smooth fluctuations of
$\psi$ in order to obtain the instanton path integral:

$\mathcal{Z}=\sum_{N=0}^{\infty}\frac{y_{0}^{N}}{R_{0}^{DN}}\frac{1}{N_{+}!N_{-}!}\int\prod_{i=1}^{N}d^{D}x_{i}\,e^{-S_{N}({\bf
x}_{1},\dots,{\bf x}_{N})}\ .$ (16)

The configurations of $N_{\pm}$ identical instantons of charge $\pm$ are still
summed over, with $N_{+}+N_{-}=N$. The action $S_{0}$ of an instanton core is
consumed into the fugacity $y_{0}=e^{-S_{0}}$, and an ultra-violet cut-off
length scale $R_{0}$ is included in order to keep $y_{0}$ dimensionless. We
will not attempt to derive the effective instanton action $S_{N}$ from any
microscopic model. Instead, we will consider all possible pair-wise
interaction potentials and attempt to classify all possible scenarios that
lead to confinement or deconfinement. We will neglect three-body and other
many-body interactions with an assumption that interactions over large
distances are the ones most important for the macroscopic behavior of the
system.

The usual picture of non-exclusive pair-wise interactions

$S_{N}=\frac{1}{2}\sum_{i\neq j}^{1\dots N}q_{i}q_{j}V({\bf x}_{i}-{\bf
x}_{j})$ (17)

works well for non-confining potentials that vanish at large distances,
$V(r\to\infty)\to 0$. Here, $q_{i}=\pm 1$ is the instanton charge at position
${\bf x}_{i}$. However, this breaks down for confining potentials
$V(r\to\infty)\to\infty$ which allow only bound states. We will show that
sustainable long-distance confining interactions must be exclusive and act
only between opposite-charge instantons:

$S_{N}=\sum_{i=1}^{N/2}V({\bf x}_{i}^{\phantom{x}}-{\bf
x}^{\prime}_{\mathcal{P}(i)})\ .$ (18)

Here, instantons are first grouped into $N/2$ dipoles with $+$ charges at
${\bf x}_{i}^{\phantom{x}}$ and $-$ charges at ${\bf x}^{\prime}_{i}$. Then,
the $+$ charge from each dipole $i$ engages in an exclusive interaction with
only one $-$ charge from the same or any other dipole $\mathcal{P}(i)$. All
possible interaction pairings are captured by the permutations $\mathcal{P}$
and need to be summed over in (16). We can visualize this as flux-string
connections between opposite-charges, with a restriction that only one string
emanates from each charge. While natural for $V(r)\propto r$, this character
of interactions is required of any confining potential.

Section III.1 analyzes the non-exclusive interactions (17) with a functional
extension of the Coulomb gas renormalization group Kosterlitz (1977). We
reproduce the well-known result that Coulomb and weaker potentials cannot
confine the instantons, but also discover an indication that stronger
algebraic potentials renormalize into confining forms. We show in Section
III.2 that logarithmic potentials in $D>2$ dimensions also become confining at
macroscopic scales. Then, Section III.3 develops a non-perturbative functional
renormalization group for the exclusive confining interactions (18).
Considering also finite temperatures in Section III.4, we compile a picture of
two universality classes for instanton confinement.

###  III.1 Non-confining interactions

Here we consider the attractive interaction potential

$C(\delta{\bf x})\equiv-V(\delta{\bf x})=-\sum_{n}\frac{K_{n}}{\delta
x^{n}}\quad,\quad K_{n}>0$ (19)

which vanishes at large distances $\delta x\equiv|\delta{\bf x}|\to\infty$ and
has a singularity at $\delta x=0$. This is the Laurent expansion of an
arbitrary non-confining potential about $\delta x=0$. The screened potential
$C^{\prime}(\delta x)\equiv C^{\prime}(|{\bf x}-{\bf x}^{\prime}|)$ can be
determined by inserting a “test” instanton dipole at ${\bf x},{\bf
x}^{\prime}$ and computing

$\displaystyle e^{-C^{\prime}(\delta x)}$ $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{Z}}\sum_{N=0}^{\infty}\frac{y_{0}^{N}}{R_{0}^{DN}}\frac{1}{N_{+}!N_{-}!}\int\prod_{i=1}^{N}d^{D}x_{i}\,\exp\biggl{\\{}$
$\displaystyle\quad\frac{1}{2}\sum_{i\neq j}q_{i}q_{j}C(|{\bf x}_{i}-{\bf
x}_{j}|)-C(|{\bf x}-{\bf x}^{\prime}|)$
$\displaystyle\quad+\sum_{i}q_{i}C(|{\bf x}_{i}-{\bf x}|)-\sum_{i}q_{i}C(|{\bf
x}_{i}-{\bf x}^{\prime}|)\biggr{\\}}$

in the path integral with the instanton action (17). We use the fugacity as an
expansion parameter, assuming $y_{0}\ll 1$, and restrict to globally neutral
instanton configurations, i.e. $N=2k$ and $\sum q_{i}=0$. Substituting (16),
we find to the lowest order in fugacity:

$\displaystyle e^{-C^{\prime}(\delta x)}e^{C(\delta
x)}=1+\frac{y_{0}^{2}}{R_{0}^{2D}}\int d^{D}R\,d^{D}r\,e^{-C(r)}$ (21)
$\displaystyle\;\quad\times\left[e^{C\left(\left|{\bf R}\\!+\\!\frac{{\bf
r}}{2}\\!-\\!{\bf x}\right|\right)\\!-\\!C\left(\left|{\bf R}\\!-\\!\frac{{\bf
r}}{2}\\!-\\!{\bf x}\right|\right)\\!-\\!C\left(\left|{\bf R}\\!+\\!\frac{{\bf
r}}{2}\\!-\\!{\bf x}^{\prime}\right|\right)\\!+\\!C\left(\left|{\bf
R}\\!-\\!\frac{{\bf r}}{2}\\!-\\!{\bf x}^{\prime}\right|\right)}-1\right]$

Here, only one fluctuating dipole with size ${\bf r}={\bf x}_{1}-{\bf x}_{2}$
and the center of mass ${\bf R}=\frac{1}{2}({\bf x}_{1}+{\bf x}_{2})$ screens
the test dipole. We anticipate that only small fluctuating dipoles make a
significant impact, but this must be verified by scrutinizing the $r\to\infty$
limit inside the integral:

$\displaystyle e^{-C^{\prime}(\delta x)}e^{C(\delta
x)}\xrightarrow{r\to\infty}{}1+\frac{y_{0}^{2}}{R_{0}^{2D}}\int
d^{D}R\,d^{D}r$ (22)
$\displaystyle\;\quad\times\biggl{\\{}-\sum_{n}\frac{2^{2+n}nK_{n}}{r^{1+n}}\,\hat{\bf
r}\,\delta{\bf
x}+\frac{1}{2}\biggl{[}\sum_{n}\frac{2^{2+n}nK_{n}}{r^{1+n}}\,\hat{\bf
r}\,\delta{\bf x}\biggr{]}^{2}$ $\displaystyle\qquad+\cdots\biggr{\\}}\ .$

Only the quadratic term remains after integrating out the fluctuating dipole
orientations $\hat{\bf r}$,

$e^{-C^{\prime}(\delta x)}e^{C(\delta
x)}\xrightarrow{r\to\infty}{}1+\frac{y_{0}^{2}}{R_{0}^{2D}}\int
dr\,r^{d}\sum_{n}\frac{a_{n}K_{n}\delta x^{2}}{r^{2+2n}}+\cdots$

We are not concerned with the $R$ integral because we assumed $|{\bf
r}|\gg|{\bf R}|$. But, there is a lower bound

$n>\frac{d-1}{2}$ (23)

for the acceptable terms in the expansion (19) which do not cause an infra-red
divergence of (III.1). A well-behaved integral is indeed dominated by small
$r$, so we estimate its value by Taylor-expanding (21) about $r=0$:

$\displaystyle e^{-C^{\prime}(\delta x)}e^{C(\delta
x)}=1+\frac{y_{0}^{2}}{2DR_{0}^{2D}}\int d^{D}R\,d^{D}r\,e^{-C(r)}$ (24)
$\displaystyle\;\quad\times\left\\{r^{2}\Bigl{[}\boldsymbol{\nabla}_{{\bf
R}}C(|{\bf R}-{\bf x}|)-\boldsymbol{\nabla}_{{\bf R}}C(|{\bf R}-{\bf
x}^{\prime}|)\Bigr{]}^{2}+\mathcal{O}(r^{4})\right\\}$

Again, we only keep the $\hat{\bf r}$-independent terms 111Sufficiently high
order terms in this expansion create infra-red divergence, but this is an
artifact of the expansion. The exact integral is convergent and approximated
by its well-behaved lowest order term.. Next, we turn our attention to the $R$
integral and find

$\displaystyle\int d^{D}R\,\Bigl{[}\boldsymbol{\nabla}_{{\bf R}}C(|{\bf
R}-{\bf x}|)-\boldsymbol{\nabla}_{{\bf R}}C(|{\bf R}-{\bf
x}^{\prime}|)\Bigr{]}^{2}=$
$\displaystyle\qquad\times\sum_{n,m}nmK_{n}K_{m}\beta_{n,m}\,\delta
x^{D-2-(n+m)}\ .$ (25)

Defining the area of a unit $n$-sphere

$S_{n}=\frac{2\pi^{(n+1)/2}}{\Gamma\left(\frac{n+1}{2}\right)}\ ,$ (26)

the positive dimensionless coefficients $\beta_{n,m}$ are given by

$\displaystyle\beta_{n,m}=\frac{S_{D-2}}{2^{D-2-(n+m)}}\int\limits_{0}^{\infty}d\xi\,\xi^{-D+1+(n+m)}\int\limits_{0}^{\pi}d\theta\,\sin^{D-2}\theta$
(27)
$\displaystyle\quad\times\\!\left(\frac{1}{(1+\xi^{2}-2\xi\cos\theta)^{1+(n+m)/2}}\\!+\\!\frac{1}{(1+\xi^{2}+2\xi\cos\theta)^{1+(n+m)/2}}\\!-\\!\frac{2(1-\xi^{2})}{(1+\xi^{2}+2\xi\cos\theta)^{1+n/2}(1+\xi^{2}-2\xi\cos\theta)^{1+m/2}}\right)$

(note $\xi=\delta x/2R$). The infra-red behavior of $\beta_{n,m}$ obtains by
taking the limit $\xi\to 0$ (i.e. $R\to\infty$) inside its integrand:

$\displaystyle\beta_{n,m}\xrightarrow{\xi\to
0}\textrm{const.}\times\int\limits_{0}^{\infty}d\xi\,\xi^{-D+1+(n+m)}\xi^{2}\propto\xi^{4-D+(n+m)}$

This imposes another requirement on the acceptable terms in the expansion
(19), stricter than (23):

$n>\frac{D}{2}-2=\frac{d-3}{2}\ .$ (28)

The integral is well-behaved in the $\xi\to\infty$ limit ($D>0$), but has an
ultra-violet singularity for $n+m>D-2$ at $\xi=1$, $\theta\in\\{0,\pi\\}$.
Substituting $\xi=1+\delta\xi$ and $\theta=\delta\theta$ or
$\theta=\pi-\delta\theta$, then expanding to the lowest orders in
$\delta\xi,\delta\theta$ and integrating out
$\delta\xi\in(\delta\xi_{0},\infty)$, $\delta\theta\in(0,\infty)$ reveals:

$\displaystyle\beta_{n,m}\xrightarrow[\sin\theta\to 0]{\xi\to
1}\frac{S_{D-2}\Gamma\left(\frac{D-1}{2}\right)}{2^{D-2-(n+m)}}\,\frac{\Gamma\left(\frac{n+m-D+3}{2}\right)}{\Gamma\left(1+\frac{n+m}{2}\right)}$
(29)
$\displaystyle\qquad\qquad\qquad\times\frac{\left|\delta\xi_{0}\right|^{D-2-(n+m)}}{|D-2-(n+m)|}-\textrm{const}$

The ultra-violet singularity corresponds to the overlaps between the positions
of the virtual and test instantons. Overlapping instantons are physically
indistinguishable below the ultra-violet cut-off length scale $R_{0}$, so the
integral over $R$ should be regularized by cutting out the volumes of the
order of $R_{0}^{D}$ around ${\bf R}\pm\delta{\bf x}/2$. In other words,

$\xi=\frac{\delta x}{2R}\xrightarrow{R\to\delta
x/2}1+\delta\xi_{0}\sim\frac{\delta x}{\delta x\mp
R_{0}}=1\pm\frac{R_{0}}{\delta x}\ .$ (30)

So, substituting $\delta\xi_{0}\sim R_{0}/\delta x$ in (29) and (III.1) gives
us a qualitatively accurate renormalization (24) of the instanton potential to
the order of $y_{0}^{2}$:

$\displaystyle e^{-C^{\prime}(\delta x)}e^{C(\delta
x)}=1\\!+\\!\frac{y_{0}^{2}}{2DR_{0}^{2D}}\sum_{n,m}nmK_{n}K_{m}\\!\int\\!d^{D}r\,e^{-C(r)}$
$\displaystyle\qquad\quad\times
r^{2}\Bigl{(}\gamma^{\phantom{x}}_{n,m}R_{0}^{D-2-(n+m)}-\beta^{\prime}_{n,m}\,\delta
x^{D-2-(n+m)}\Bigr{)}\ .$

The coefficients $\beta^{\prime}_{n,m}$ and $\gamma^{\phantom{x}}_{n,m}$ are
positive.

Now we can derive the renormalization group beta functions for the interaction
couplings $K_{n}$. We first coarse-grain the view of dynamics by integrating
out fluctuations of small dipoles, $r\in(R_{0},e^{\delta l}R_{0})$, where
$\delta l\ll 1$:

$\displaystyle e^{-C^{\prime}(\delta x)}e^{C(\delta
x)}=1\\!+\\!\frac{y_{0}^{2}}{2DR_{0}^{2D}}\sum_{n,m}nmK_{n}^{\phantom{x}}K_{m}^{\phantom{x}}S_{D-1}^{\phantom{x}}R_{0}^{D+2}\delta
l$ $\displaystyle\qquad\times
e^{-C(R_{0})}\Bigl{(}\gamma^{\phantom{x}}_{n,m}R_{0}^{D-2-(n+m)}-\beta^{\prime}_{n,m}\,\delta
x^{D-2-(n+m)}\Bigr{)}$

Then, to the lowest order in $y_{0}^{2}$,

$\displaystyle C^{\prime}(\delta x)=C(\delta
x)-\frac{y_{0}^{2}}{2DR_{0}^{2D}}\sum_{n,m}nmK_{n}^{\phantom{x}}K_{m}^{\phantom{x}}S_{D-1}^{\phantom{x}}R_{0}^{D+2}\delta
l$ $\displaystyle\qquad\times
e^{-C(R_{0})}\Bigl{(}\gamma^{\phantom{x}}_{n,m}R_{0}^{D-2-(n+m)}-\beta^{\prime}_{n,m}\,\delta
x^{D-2-(n+m)}\Bigr{)}$

It can be formally seen from (21) that adding a constant to the potential has
no physical effect. Thus, we are free to interpret the entire bracket content
in the last expression as the non-confining interaction with power
$D-2-(n+m)<0$ (shifted by a constant). Then, substituting the expansion (19)
and matching powers gives us:

$K^{\prime}_{n}=K^{\phantom{x}}_{n}-\frac{y_{0}^{2}}{R_{0}^{D-2}}\sum_{m}\beta^{\prime\prime}_{D-2+n-m,m}K_{D-2+n-m}^{\phantom{x}}K_{m}^{\phantom{x}}\delta
l$

with positive coefficients $\beta^{\prime\prime}_{n,m}$ for $n,m>0$ limited by
(28). This yields the contribution of coarse-graining to the beta function
$(K^{\prime}_{n}-K_{n})/\delta l\to\delta K_{n}/\delta l$. After coarse-
graining, we rescale the coordinates to restore the original cut-off,
$e^{\delta l}R_{0}\to R_{0}$:

$C^{\prime}(r)=C(e^{\delta
l}r)\quad\Rightarrow\quad\frac{K^{\prime}_{n}-K_{n}}{\delta l}=\frac{\delta
K_{n}}{\delta l}=-nK_{n}\ .$ (31)

Combining the coarse-graining and rescaling gives us the full beta function:

$\frac{\delta K_{n}}{\delta
l}=-\frac{y_{0}^{2}}{R_{0}^{D-2}}\sum_{m}\beta^{\prime\prime}_{D-2+n-m,m}K_{D-2+n-m}^{\phantom{x}}K_{m}^{\phantom{x}}-nK_{n}\
.$ (32)

The renormalization group flow of fugacity $y_{0}=e^{-S_{0}}$ is obtained by
fixing the partition function under the rescaling of the ultra-violet cut-off
$R_{0}$. The action of a single instanton

$\mathcal{S}_{1}=\mathcal{S}_{0}+C(\lambda;R_{0})\ ,$ (33)

contains the core part $S_{0}$ and the “coherent” part estimated by the
(attractive) instanton interaction potential $C(r)$ at an infra-red cut-off
distance $\lambda$. The potential can generally depend on the ultra-violet
cut-off $R_{0}$. The partition function of a single instanton stays invariant
under the change of scale $l\to l+\delta l$,

$\displaystyle\mathcal{Z}_{1}\sim\left(\frac{\lambda}{R_{0}}\right)^{D}e^{-\mathcal{S}_{1}}=y_{0}(R_{0})\left(\frac{\lambda}{R_{0}}\right)^{D}e^{-C(\lambda,R_{0})}$
$\displaystyle\qquad\quad=y_{0}(e^{\delta
l}R_{0})\left(\frac{\lambda}{e^{\delta
l}R_{0}}\right)^{D}e^{-C(\lambda,e^{\delta l}R_{0})}$ (34)

so that

$\displaystyle y_{0}(R_{0}+R_{0}\delta l)=y_{0}(R_{0})+\delta y_{0}$ (35)
$\displaystyle\qquad\quad=y_{0}(R_{0})\,\left(1+D\delta l+\frac{\partial
C(\lambda)}{\partial R_{0}}R_{0}\delta l\right)\ .$

This yields the beta function for fugacity

$\frac{\delta y_{0}}{\delta l}=y_{0}\left(D+\frac{\partial
C(\lambda)}{\partial R_{0}}R_{0}\right)\ .$ (36)

The non-confining potential (19) must be cut-off in the ultra-violet limit
$r\to 0$. The proper way to do it is to shift it by a constant that achieves
$C(R_{0})=0$:

$C(\delta{\bf x})\to\sum_{n}K_{n}\left(\frac{1}{R_{0}^{n}}-\frac{1}{\delta
x^{n}}\right)\ .$ (37)

The implication for the fugacity is

$\frac{\delta y_{0}}{\delta
l}=y_{0}\left(D-\sum_{n}\frac{nK_{n}}{R_{0}^{n}}\right)\ .$ (38)

The full set of renormalization group equations for non-confining instanton
interactions is:

$\displaystyle\frac{\delta K_{n}}{\delta l}$ $\displaystyle=$
$\displaystyle-\frac{y_{0}^{2}}{R_{0}^{D-2}}\sum_{m}\beta^{\prime\prime}_{D-2+n-m,m}K_{D-2+n-m}^{\phantom{x}}K_{m}^{\phantom{x}}-nK_{n}$
$\displaystyle\frac{\delta y_{0}}{\delta l}$ $\displaystyle=$ $\displaystyle
y_{0}\left(D-\sum_{n}\frac{nK_{n}}{R_{0}^{n}}\right)\ .$ (39)

Since $n>0$ and $\beta^{\prime\prime}_{n,m}>0$, attractive interaction
couplings $K_{n}>0$ generally decrease with the scale parameter $l$,
irrespective of the value of fugacity $y_{0}$. Eventually, sufficiently small
$K_{n}$ cannot prevent the growth of $y_{0}$. There is only one basin of
attraction, $K_{n}\to 0$, $y_{0}\to 1(\infty)$. The fixed point corresponds to
prolific non-interacting instantons with a vanishing core action $S_{0}$, and
hence represents the deconfined phase.

Even though the last conclusion looks simple, the generation of new
interaction channels from the microscopic ones holds important further
insight. If interaction channels with $K_{n}\neq 0$ are limited to
$n\in[M,N]$, then the first equation in (III.1) generates new interactions in
channels $n^{\prime}\in[2M+2-D,2N+2-D]$. Only the generalized Coulomb
interaction $n=D-2$ yields a closed set of interactions channels, which can be
handled in a more straight-forward manner Kosterlitz (1977). Weaker-than-
Coulomb potentials $n>D-2$ generate an endless set of even weaker non-
confining potentials whose main effect is to alter the interactions at short
distances. Therefore, the concluded absence of instanton confinement holds
consistently for the Coulomb and less confining potentials.

Note that a microscopic attractive interaction in channel $n>D-2$ can formally
induce a first-generation repulsive interaction between opposite-charge
instantons in a higher channel $n^{\prime}>n$. Then, the attractive $n$ and
repulsive $n^{\prime}$ channels together generate an attractive second-
generation potential in the channel
$n^{\prime\prime}=n+n^{\prime}-(D-2)>n^{\prime}$. As the new interaction
channels are introduced, the net interaction remains attractive at large
distances as dictated by the original lowest channel $n$. The generation of
formally repulsive channels is innocuous for two reasons: (i) the net
interaction is renormalized only at short distances, and (ii) all the
couplings $K_{n}\to 0$ are irrelevant in the renormalization group sense in
all channels.

The most interesting aspect of the interaction renormalization is that
stronger-than-Coulomb $n<D-2$ potentials generate an open set of interaction
channels $n^{\prime}<n$ which progressively become more rather than less
important at large distances. This breaks down the current approach when the
channels violating (28), or even confining channels $n<0$, are reached.
Attempting to still take the equations (III.1) seriously reveals a
transformation of the $n<0$ interaction couplings $K_{n}$ into relevant
operators which flow away from zero while $y_{0}\ll 1$. We interpret all these
features as an indication that stronger-than-Coulomb interactions produce new
physics ultimately controlled by confining potentials.

###  III.2 Logarithmic interactions

A special case of interactions between instantons is the logarithmic potential

$C(r)=-V(r)=K\log\left(\frac{r}{R_{0}}\right)\ .$ (40)

This is a Coulomb potential in $D=2$ space-time dimensions which leads to the
Kosterlitz-Thouless transition Kosterlitz (1974) between confined and
deconfined phases of instantons. In higher dimensions $D>2$, we must check if
an infra-red divergence arises in (III.1):

$\displaystyle\int d^{D}R\,\Bigl{[}\boldsymbol{\nabla}_{{\bf R}}C(|{\bf
R}-{\bf x}|)-\boldsymbol{\nabla}_{{\bf R}}C(|{\bf R}-{\bf
x}^{\prime}|)\Bigr{]}^{2}=$ $\displaystyle\quad=K^{2}\delta
x^{2}\int\frac{d^{D}R}{\left(R^{2}+\frac{1}{4}\delta
x^{2}\right)^{2}-\left({\bf R}\delta{\bf x}\right)^{2}}\ .$ (41)

It is evident that divergences from $R\to\infty$ occur in $D\geq 4$.
Specifically, in $D=4$ we have the weakest logarithmic divergence:

$\displaystyle\delta x^{2}\int\frac{d^{4}R}{\left(R^{2}+\frac{1}{4}\delta
x^{2}\right)^{2}-\left({\bf R}\delta{\bf x}\right)^{2}}=$
$\displaystyle\quad=4\pi^{2}\int\limits_{0}^{\infty}dR\,R\left(1-\frac{\left|R^{2}-\frac{1}{4}\delta
x^{2}\right|}{R^{2}+\frac{1}{4}\delta x^{2}}\right)$
$\displaystyle\quad\xrightarrow{R\to\infty}{}-2\pi^{2}\delta
x^{2}\log\left(\frac{\delta x}{R}\right)+\pi^{2}\delta
x^{2}+\mathcal{O}\left(\frac{\delta x^{2}}{R^{2}}\right)\ .$

If the naive renormalization of the microscopic potential appears infra-red
divergent, then a different microscopic mechanism must be engaged in the
generation and renormalization of interactions. We will discuss this mechanism
in the next section. Logarithmic interactions in all $D\geq 4$ space-time
dimensions must be treated as confining.

The direct renormalization of a logarithmic interaction in $D=3$ is both
infra-red and ultra-violet convergent:

$\delta x^{2}\int\frac{d^{3}R}{\left(R^{2}+\frac{1}{4}\delta
x^{2}\right)^{2}-\left({\bf R}\delta{\bf x}\right)^{2}}=\pi^{3}\delta x\ .$
(43)

However, coarse-graining generates a new confining potential
$C^{\prime}(\delta x)\propto\delta x$. This cannot be handled perturbatively,
and hints instead that the logarithmic interaction renormalizes into a
confining one at macroscopic scales. The emergent confining potential requires
a treatment which we develop in the next section. Earlier studies have
suggested that the $D=3$ logarithmic interaction can drive a confinement-
deconfinement phase transition Kleinert  _et al._ (2002); Kragset  _et al._
(2004), but did not consider this important issue.

###  III.3 Confining interactions

We previously found that instanton interactions which do not decrease fast
enough with the distance cause an infra-red divergence of the screening
(III.1). This includes algebraic potentials (19) that violate (28), and
logarithmic potentials in $D\geq 4$ space-time dimensions. The infra-red
divergence comes from the wandering of the virtual instanton dipoles’ center-
of-mass ${\bf R}$ throughout space-time. If the largest possible values of
$|{\bf R}|$ produce the most statistically-important instanton configurations
for screening, then the virtual dipole fluctuations are effectively suppressed
in the thermodynamic limit since they are unlikely to occur anywhere near the
test dipole.

A physical system can always find its way out of this trouble. If instanton
fluctuations are to take place, their interactions must be restructured by the
appropriate dynamics of the system’s matter field $\psi$. Instanton
interactions are subject to the boundary condition (6). One way to satisfy
this condition is to evenly spread the flux of the topological current
$\mathcal{J}_{\mu}$ over the spherical manifolds $S^{d}$ which enclose an
instanton. If this leads to infra-red divergences, then the flux must spread
differently. The alternative extreme option is for the flux to focus into
strings. Preserving (6) then leads to the exclusive confining two-body
interactions (18) with a linear form $C(r)\propto r$ at large distances. Any
non-linear form is either not exclusive, or exhibits some additional bias
among the interacting instantons which requires a more complex dynamics.
Interestingly, it is possible to calculate the renormalization of exclusive
potentials exactly. We will show that the exclusive interaction given by (18)
always resolves the problem of infra-red divergence.

Consider $+$ instantons at locations ${\bf x}_{i}^{\phantom{x}}$ and an equal
number of $-$ instantons at locations ${\bf x}^{\prime}_{i}$. We visualize
exclusive interactions by connecting every instanton with exactly one other
instanton of opposite charge. A connection diagram is specified by a
permutation $\mathcal{P}$: the $+$ instanton $i$ is connected to the $-$
instanton $\mathcal{P}(i)$. The instanton partition function with exclusive
interactions is:

$\displaystyle\mathcal{Z}=\sum_{N=0}^{\infty}\left(\frac{1}{N!}\right)^{2}\frac{y_{0}^{2N}}{R_{0}^{2ND}}$
(44)
$\displaystyle\qquad\times\sum_{\mathcal{P}[N]}\int\prod_{i=1}^{N}d^{D}x_{i}^{\phantom{x}}\,d^{D}x^{\prime}_{i}\,\exp\left[-\sum_{i=1}^{N}C({\bf
x}_{i}^{\phantom{x}}-{\bf x}^{\prime}_{\mathcal{P}(i)})\right]\ .$

When a test dipole is inserted at $({\bf x}_{0}^{\phantom{x}},{\bf
x}^{\prime}_{0})$, its interaction potential acquires the following
renormalization by virtual instanton fluctuations:

$\displaystyle e^{-C^{\prime}({\bf x}_{0}^{\phantom{x}}-{\bf
x}^{\prime}_{0})}=\frac{1}{\mathcal{Z}}\sum_{N=0}^{\infty}\left(\frac{1}{N!}\right)^{2}\frac{y_{0}^{2N}}{R_{0}^{2ND}}$
(45)
$\displaystyle\quad\times\sum_{\mathcal{P}[N+1]}\int\prod_{i=1}^{N}d^{D}x_{i}^{\phantom{x}}\,d^{D}x^{\prime}_{i}\,\exp\left[-\sum_{i=0}^{N}C({\bf
x}_{i}^{\phantom{x}}-{\bf x}^{\prime}_{\mathcal{P}(i)})\right]$

If the instantons of the test dipole are not connected to any of the virtual
instantons, we will say that the connection diagram is disconnected. Let us
first demonstrate that only the connected diagrams contribute to the
interaction renormalization. Consider a generic diagram of $N$ virtual
dipoles, $i=1,\dots N$ in which only $k$ dipoles make a connected cluster with
the test dipole $i=0$. There are $\binom{N}{k}$ ways to chose $k$ out of $N$
available instantons of each charge to create the connected sub-diagram.
Combining with the $1/N!$ factors gives us the sum over connected clusters
only ($N=k+m$):

$\displaystyle e^{-C^{\prime}({\bf x}_{0}^{\phantom{x}}-{\bf
x}^{\prime}_{0})}=\frac{1}{\mathcal{Z}}\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}\left(\frac{1}{(k+m)!}\right)^{2}\binom{k+m}{k}^{2}$
$\displaystyle~{}\,\times\\!\frac{y_{0}^{2k}}{R_{0}^{2kD}}\\!\\!\\!\sum_{\mathcal{P}[k+1]}^{\textrm{conn.}}\int\\!\prod_{i=1}^{k}d^{D}x_{i}^{\phantom{x}}d^{D}x^{\prime}_{i}\,\exp\\!\left[-\\!\sum_{i=0}^{k}C({\bf
x}_{i}^{\phantom{x}}\\!\\!-\\!{\bf x}^{\prime}_{\mathcal{P}(i)})\right]\\!$
$\displaystyle~{}\,\times\\!\frac{y_{0}^{2m}}{R_{0}^{2mD}}\\!\\!\sum_{\mathcal{P}[m]}\int\\!\prod_{j=1}^{m}d^{D}x_{j}^{\phantom{x}}d^{D}x^{\prime}_{j}\,\exp\\!\left[-\\!\sum_{j=1}^{m}C({\bf
x}_{j}^{\phantom{x}}\\!\\!-\\!{\bf x}^{\prime}_{\mathcal{P}(j)})\right]\\!$
$\displaystyle\,=\sum_{k=0}^{\infty}\left(\frac{1}{k!}\right)^{2}\frac{y_{0}^{2k}}{R_{0}^{2kD}}\sum_{\mathcal{P}[k+1]}^{\textrm{conn.}}$
$\displaystyle\quad\times\int\prod_{i=1}^{k}d^{D}x_{i}^{\phantom{x}}d^{D}x^{\prime}_{i}\,\exp\left[-\sum_{i=0}^{k}C({\bf
x}_{i}^{\phantom{x}}-{\bf x}^{\prime}_{\mathcal{P}(i)})\right]\ .$ (46)

There are only two connected clusters with sizes $k=0$ and $k=1$, because the
test dipole has only two terminals which can be either connected to each other
($k=0$) or to the terminals of another dipole ($k=1$). Therefore,

$\displaystyle e^{-C^{\prime}({\bf x}-{\bf x}^{\prime})}$ $\displaystyle=$
$\displaystyle e^{-C({\bf x}-{\bf x}^{\prime})}$
$\displaystyle+\frac{y_{0}^{2}}{R_{0}^{2D}}\int
d^{D}y\,d^{D}y^{\prime}\,e^{-C({\bf y}-{\bf x}^{\prime})-C({\bf
y}^{\prime}-{\bf x})}$

yields the exact renormalized potential $C^{\prime}(r)$ to all orders in
$y_{0}$:

$\displaystyle C^{\prime}(r)$ $\displaystyle=$
$\displaystyle-\log\left(e^{-C(r)}+\alpha^{2}y_{0}^{2}\right)$
$\displaystyle\to$
$\displaystyle\begin{cases}C(r)-\alpha^{2}y_{0}^{2}e^{C(r)}&,\quad\alpha^{2}y_{0}^{2}\ll
e^{-C(r)}\\\
-\log(\alpha^{2}y_{0}^{2})-\frac{e^{-C(r)}}{\alpha^{2}y_{0}^{2}}&,\quad\alpha^{2}y_{0}^{2}\gg
e^{-C(r)}\end{cases}$

where

$\alpha[C]=\frac{1}{R_{0}^{D}}\int d^{D}y\,e^{-C(y)}\ .$ (49)

We may consider various forms of the unscreened potential $C(r)$ in order to
understand the present approach better. Confining attractive potentials
$C(r)\propto r^{n}$, $n>0$ always keep this procedure infra-red convergent,
while the logarithmic attractive potential $C(r)=K\log(r/R_{0})$ needs a large
enough coupling constant $K>D$ to keep $\alpha$ finite. The renormalized
potential $C^{\prime}$ is not confining even though the microscopic one is.
This makes sense only when $C^{\prime}$ is correctly interpreted as the
effective action $C^{\prime}(r)\equiv S_{\textrm{eff}}(r)$ for an inserted
instanton dipole of size $r$. If the dipole starts off small and we gradually
increase its size, then for a while we keep paying more and more confinement
potential “energy” according to $C^{\prime}(r)\approx C(r)$. The eventual
flattening of $C^{\prime}(r)\to\textrm{const.}$ at sufficiently large
$r>\lambda$ indicates that the inserted test instantons become screened. The
cross-over from the confining to the flattened regime reveals the screening
length $\lambda$:

$e^{-C(\lambda)}=\alpha^{2}y_{0}^{2}\quad\Rightarrow\quad
C(\lambda)=-\log(\alpha^{2}y_{0}^{2})\ .$ (50)

We can also define the bare confinement length $\lambda_{0}$ as the dipole
size which provides enough unscreened potential “energy” for a new
compensating dipole,

$C(\lambda_{0}^{\phantom{x}})=2S_{0}^{\phantom{x}}=-\log(y_{0}^{2})\ .$ (51)

In comparison to $\lambda_{0}$, the screening length $\lambda$ is affected by
a certain renormalization that enters through the factor of $\alpha^{2}$ in
(50); generally, $\alpha>1$ implies $\lambda<\lambda_{0}$. While $\lambda_{0}$
unambiguously increases when the confining interaction $C(r)$ becomes weaker,
$\lambda$ can exhibit the opposite behavior for sufficiently weak interactions
because $\alpha$ increases at the same time according to (49). This effect is
due to entropy. A strong confining interaction places the compensating
instantons very close to the original test instantons. In contrast, a weak
confining interaction (more asymptotically free) is much less selective; even
though the corresponding $\lambda_{0}$ is larger, virtual instanton
fluctuations have an enhanced probability and contribute significantly to
screening, thereby reducing $\lambda$. Note, however, that the present
approach cannot be trusted to determine a small $\lambda$ in the weak-
interaction limit since the underlying assumption of exclusive interactions
need not apply to very short distances.

An important feature of the renormalized potential (III.3) is that screening
occurs inevitably at a finite length scale $\lambda$ regardless of whether the
unscreened potential $C(r)$ is truly confining in the $r\to\infty$ limit.
Exclusive confining interactions between instantons are formed beyond some
short length-scale, and may extend to a finite distance $\xi$, given for
example by the coherence length of the fundamental field $\psi$, as long as
$\xi>\lambda$. If circumstances reduce $\xi$ below $\lambda$, instanton
confinement will be lost. Similarly, thermal deconfinement requires a finite
temperature scale in the presence of asymptotic freedom.

Our main goal is to identify the conditions for confinement phase transitions
and explore their universal properties. The usual tool for this purpose is
renormalization group, but here we have to slightly depart from the tradition
and construct a stripped-down scaling argument. The scaling $\lambda\to 0$ of
the screening length will serve as an indicator of confinement, and computing
$\lambda$ with (50) requires the unscreened potential $C(r)$, not the
renormalized one. At the same time, the fugacity $y_{0}$ is determined by the
full partition function, so it implicitly depends on the renormalized
potential. The general beta function for fugacity (36) involves the scale-
dependent renormalized potential only through its singular dependence on the
ultra-violet cut-off length $R_{0}$. The key observation for our purposes is
that all screening renormalizations alter only the large-distance behavior of
the interaction potential $C(r)$, so the renormalized potential has the same
short-distance singularities as the unscreened one. Therefore, we may use the
unscreened potential $C(r)$ in the beta function (36) for fugacity.

We will track the unscreened potential and fugacity as functions of the
observation scale parametrized by $l\geq 0$. Coordinate rescaling modifies the
potential according to

$C^{\prime}({\bf r})=C({\bf r})+\delta C({\bf r})=C(e^{\delta l}{\bf
r})=C({\bf r})+{\bf r}\boldsymbol{\nabla}C({\bf r})\,\delta l$

which amounts to $\delta C({\bf r})/\delta l={\bf r}\boldsymbol{\nabla}C({\bf
r})$. No further renormalization of the unscreened potential takes place under
coarse-graining. For completeness, the full set of scaling equations for
isotropic potentials is:

$\frac{\delta C(r)}{\delta l}=r\frac{\partial C(r)}{\partial
r}\quad,\quad\frac{\delta y_{0}}{\delta l}=y_{0}\left(D+\frac{\partial
C(\lambda)}{\partial R_{0}}R_{0}\right)\ .$ (52)

These equations are accurate to all orders of $y_{0}$, so we can reliably
track the flow of fugacity to large values. They also apply to arbitrary
confining functions $C(r;l;R_{0})$ which do not create infra-red divergences
in the present approach. The exact solution for the potential scaling is

$C(r;l)=C(re^{l})\ .$ (53)

It turns out that identifying fixed points is not particularly useful because
they lie outside of the limits in which the scaling equations are valid.
Formally, $C(r)=\textrm{const.}$ is a fixed point potential, but it breaks
down this procedure by introducing an infra-red divergence in (49). Fugacity
always has $y_{0}=0$ and $y_{0}\to 1(\infty)$ fixed points. The logarithmic
potential $C(r)=K\log(r/R_{0})$ scales in a manner
$C(r;l)=K\log(e^{l}r/R_{0})$ which keeps the interaction strength $K$ fixed
and allows a non trivial fixed point for fugacity at $K=D$, but this and
smaller couplings also make (49) divergent.

Instead of considering fixed points and their stability, we can analyze the
evolution of concrete potentials. It is easy to see that a generic
asymptotically-free power-law potential becomes only more confining at larger
length scales

$C(r)=\sum_{n>0}A_{n}r^{n}\quad\Rightarrow\quad
C(r;l)=\sum_{n>0}A_{n}e^{nl}r^{n}\ ,$ (54)

without jeopardizing asymptotic freedom. Since this interaction has no ultra-
violet singularities, the fugacity scales as

$\frac{dy_{0}}{dl}=Dy_{0}\quad\Rightarrow\quad y_{0}(l)=y_{0}(0)\,e^{Dl}$ (55)

This is unbounded flow toward infinity, but we should stop it at $y_{0}=1$
because fugacity $y_{0}=e^{-S_{0}}$ is given by the action $S_{0}$ of an
instanton core, so $y_{0}\to 1$ corresponds to the vanishing cost of a core.
If we focus on a single channel $n$, i.e. $C(r)=A_{n}r^{n}$, we can also
easily compute (49)

$\alpha(l)=\frac{S_{d}\,\Gamma\\!\left(\frac{D}{n}\right)}{nR_{0}^{D}A_{n}^{D/n}}\,e^{-Dl}$
(56)

and solve (50) to find the running screening length

$\lambda(l)=\lambda(0)\,e^{-l}$ (57)

in terms of the microscopic screening length

$\lambda(0)=\begin{cases}\left[\frac{2}{A_{n}}\left|\log\bigl{[}y_{0}(0)\,\alpha(0)]\right|\right]^{1/n}&,\quad
y_{0}(0)\,\alpha(0)<1\\\ \;R_{0}\to 0&,\quad y_{0}(0)\,\alpha(0)>1\end{cases}$
(58)

Recall that the formal saturation $\lambda\to R_{0}$ in the $\alpha y_{0}>1$
regime cannot be trusted because it pertains to weak interactions which need
not assume an exclusive form at short distances. The main insight for now is
that instantons are necessarily confined at $T=0$ according to $\lambda(l)\to
0$ as $l\to\infty$.

We can repeat this exercise for logarithmic potentials

$C(r;l)=K\log\left(\frac{r}{R_{0}}e^{l}\right)\ .$ (59)

The new ingredient is an ultra-violet singularity from $R_{0}\to 0$ that
enables a non-trivial flow of fugacity:

$\frac{dy_{0}}{dl}=(D-K)y_{0}\quad\Rightarrow\quad
y_{0}(l)=y_{0}(0)\,e^{(D-K)l}\ .$ (60)

Only the cases $K>D$ are accessible without infra-red divergences in the
present approach. The fugacity then flows into the $y_{0}\to 0$ fixed point
and pushes the screening length $\lambda$ toward infinity,

$\lambda(l)=e^{-l}R_{0}\left(e^{(2K-D)l}\frac{K-D}{S_{d}\,y_{0}(0)}\right)^{2/K}\xrightarrow{l\to\infty}\infty\
.$ (61)

Instantons are formally deconfined, but their fluctuations are suppressed by
$y_{0}\to 0$ and the photons of the gauge field (13) have an infinite
correlation length at $T=0$. At finite temperatures, the absence of
confinement makes the correlation length finite. We will find indications that
the state with $K<D$ is deconfined and short-range correlated even at zero
temperature in the next section.

An exclusive logarithmic potential considered here is not realistic. However,
the last analysis reveals a general possibility of a non-trivial $y_{0}\to 0$
fixed point shaped by the short-distance modifications of a physical confining
potential. Note that a crossover to a non-exclusive interaction at short
distances does not by itself jeopardize this argument because (36) holds for
non-exclusive potentials as well. It is not obvious what microscopic dynamics
may pull the fugacity toward zero. Nevertheless, instantons’ positional
fluctuations can alter the topological constraint (6) at short distances from
the instanton, and accommodate the potential. Overall, the behavior of
fugacity determines two different universality classes of instanton
suppression at $T=0$, characterized by the fixed points $y_{0}\to 0$ and
$y_{0}\to 1(\infty)$.

###  III.4 Instanton confinement at finite temperatures

Here we analyze the fate of instanton confinement at finite temperatures $T$
by developing a two-stage renormalization group. In stage 1, we start from a
$D=d+1$ dimensional space-time which hosts the imaginary-time path integral
(4) of a $d$-dimensional quantum system at a finite temperature $T$. The time-
direction has periodic boundary conditions and a finite extent
$\tau\in(0,\beta)$, where $\beta=T^{-1}$ is the inverse temperature. The
system remains infinite in all $d$ spatial directions, and the Lagrangian
density of the system is the same as in the $T=0$ case. The renormalization
group proceeds exactly as in the previous section, but the span of the
imaginary time coordinate changes under scaling as:

$\frac{dT}{dl}=lT\quad\Rightarrow\quad
T(l)=T(0)\,e^{l}\quad,\quad\beta(l)=\beta(0)\,e^{-l}\ .$ (62)

We must stop the stage 1 at a finite scale $l=l_{0}$ when $\beta\to R_{0}$
becomes comparable to the ultra-violet cut-off length and the system
effectively flattens to its $d$-dimensional spatial manifold. The running
coupling constants have picked by then the essential quantum renormalizations,
while the pure scaling of the energy couplings $w(l)=w(0)e^{l}$ has imparted
on them a factor of $\beta$. Then, we embark on the “classical” stage 2 of
renormalization group, utilizing the last $d$-dimensional action of
$\beta=R_{0}$ which has a Landau-Ginzburg form. The final fixed point at
$l\to\infty$ signifies the phase in which the microscopic theory lives. The
entire procedure is depicted in Fig.2.

Figure 2: (a) Stage 1 of the renormalization group: the temporal (vertical)
extent of the world $\beta=1/T$ flows toward zero. While $\beta$ is large
enough, the configurations of topological defects and instantons can
comfortably fit into the world view. In particular, the field configuration
(e.g. the shown phase gradient $\boldsymbol{\nabla}\theta$) can maintain a
confining action potential $V(r)\propto r$ between the instantons. Such
confined instantons are macroscopically irrelevant if their confinement length
$\lambda$ scales down to zero during this stage. (b) Stage 2 of the
renormalization group: $\beta$ has scaled down to the cut-off length, and the
world has lost its imaginary time dimension. If the instantons have not been
confined under $\lambda\to 0$ yet, they cannot be properly defined any more.
Any surviving spatial gradients, which build topological defects, can take
their minimum-action form. This typically produces an algebraic $U(r)\propto
r^{-\alpha}$ potential energy between the defects ($\alpha>0$). One could
interpret $U(r)$ as the interaction potential between instanton remnants, but
this is not enough for confinement. An exception is $U(r)\propto\log(r)$ in
$d=2$ spatial dimensions, which permits a Kosterlitz-Thouless transition.

The following discussion assumes that the coherence length $\xi$ of the
microscopic field $\psi$ remains larger than the instanton screening length
$\lambda$ at all temperatures. During stage 1, the confining interaction
potential $C(r)$ and fugacity $y_{0}$ flow according to (52). Let us first
consider the asymptotically free regime $y_{0}\to 1(\infty)$. The confinement
length $\lambda$ and the inverse temperature $\beta$ both decrease with the
scale parameter and race toward the cut-off $R_{0}$. If $\lambda$ reaches the
cut-off first, the instantons become strictly confined and neutralized at
microscopic scales during the stage 1. Further scale evolution of dynamics
features only the fluctuations of neutral dipole excitations, since the
running theory cannot keep any information about the microscopic details below
the cut-off length $R_{0}$. Dipoles are all that’s left when the stage 2
begins and we merely expect that their positive mass drives the remaining
renormalization group flow as a relevant operator. The final fixed point is
confined.

An alternative to this outcome is the scenario in which $\beta$ reaches
$R_{0}$ sooner than $\lambda$. If that happens, then individual instanton
charges, or more accurately their projections, are still discernible in the
$d$-dimensional world view at the end of stage 1. However, these instanton
projections cannot interact with confining potentials in the resulting $d=D-1$
dimensional world. Confining interactions were derived in the first place from
the action cost of the stiff matter field $\psi$. Let us assume that $\psi$ is
coherent across some length scale $\xi$. Consider a pair of opposite-charge
topological defects coexisting at the same instant of time $\tau$ a spatial
distance $\delta r<\xi$ apart. They can be simultaneously removed by a pair of
instantons at time $\tau_{0}$. In order to minimize action, a coherent $\psi$
field configuration corresponding to this removal must involve a deformation
of $\psi({\bf r},\tau)$ from the (locally) optimal spatial configuration at
$\tau<\tau_{0}-\delta\tau$ to a costly spatial configuration at time
$\tau=\tau_{0}$ which is uniform everywhere except inside a narrow flux tube
that connects the defect and antidefect. The instanton pair abruptly removes
the tube and replaces it with the surrounding uniform $\psi$ at times
$\tau>\tau_{0}$. This was illustrated in Fig.1(a). The action cost
proportional to $\delta r$ comes from the rapid temporal variations
$(\partial_{0}\psi)^{2}$ along the tube, but the required gradient terms are
expelled from the purely spatial $d$-dimensional action of the stage 2. This
obviously cannot be repaired by non-simultaneous instantons, so no mechanism
is left to maintain confining interactions between instantons. Ultimately,
instantons are not well-defined in stage 2. Their projections assume at best a
dynamics analogous to that of spatial topological defects; the strength of
interactions is limited by the Coulomb law $C(r)\sim r^{2-d}$ (at least in the
case of monopoles and hedgehogs Nikolić (2020a)), which is not able to confine
its charges (see Section III.1). The final fixed point is deconfined.

Since two different fixed points of renormalization group can be reached at
macroscopic $l\to\infty$ scales, a thermodynamics phase transition marks the
resulting sharp distinction between the confined and deconfined phases of
instantons. Asymptotically free confining interactions $C(r)=A_{n}r^{n}$ in
the $y_{0}\to 1(\infty)$ universality class produce a deconfinement phase
transition at a critical temperature $T_{c}>0$. Given (57), (62) and the above
discussion, the critical temperature is

$T_{c}=\frac{1}{\lambda(0)}=\left[\frac{2}{A_{n}}\left|\,\log\left(y_{0}(0)\,\frac{S_{d}\,\Gamma\\!\left(\frac{D}{n}\right)}{nR_{0}^{D}A_{n}^{D/n}}\right)\right|\right]^{-1/n}\
.$ (63)

The critical temperature is generally finite, but goes to zero if $y_{0}\to 0$
at any fixed $A_{n}$. It increases with the interaction strength $A_{n}$ when
$A_{n}$ is large enough. Formally, $T_{c}$ has a local minimum at a small
value of $A_{n}=\widetilde{A}_{n}$ and also grows with decreasing
$A_{n}<\widetilde{A}_{n}$, but again, this weak-interaction regime pushes the
theory out of its applicability limits. We can also define a critical
screening length at temperature $T$

$\lambda_{c}(T)=\frac{1}{T}\ ,$ (64)

such that $\lambda(0)<\lambda_{c}(T)$ yields a confined phase.

In the absence of a Berry curvature, the photon correlation length $\zeta$
introduced in Section II exhibits a critical behavior near the confinement
phase transition of the $y_{0}\to 1(\infty)$ universality class. The exact
screened potential (III.3) and the stage 1 renormalization group show that
instanton dipoles are statistically relevant in $T=0$ confined phases only if
their sizes are limited by the screening length $\lambda$, i.e. $r<\lambda$.
Conversely, the dipoles with sizes $r>\lambda$ are statistically irrelevant.
At finite temperatures, we find from the renormalization group that the
dipoles of microscopic sizes $r<\lambda_{c}=1/T$ remain confined in the
macroscopic point of view, while the microscopically large $r>\lambda$ dipoles
continue to be statistically irrelevant because they are still neutralized in
spatial directions by the confining potential of the microscopic theory, i.e.
they are confined in the microscopic point of view. Therefore, the
statistically important dipoles which contribute to the volume-law behavior of
the correlation function (10) are only those whose microscopic sizes are
$\lambda_{c}<r<\lambda$. Consider an instanton placed at the origin, and let
$P(r)$ be the microscopic cumulative probability that this instanton is
neutralized by an anti-instanton within the radius $r$ as a result of the
dynamics at $T=0$. We know that $P(0)=0$ and $P(\infty)=1$. Continuity and the
absence of other length scales in the problem also ensure that $P(r)$ has no
extrema on $0<r<\infty$. At finite temperatures, an anti-instanton placed at
$\lambda>r>\lambda_{c}$ will end up being deconfined from the instanton. The
probability of this occurrence when $\Delta T=T-T_{c}\ll T_{c}$ is:

$P(\lambda)-P(\lambda_{c})\approx
P^{\prime}(\lambda_{c})\,(\lambda-\lambda_{c})\approx\lambda_{c}P^{\prime}(\lambda_{c})\,\frac{\Delta
T}{T_{c}}\ .$ (65)

This is a conditional probability, assuming that an instanton was present at
the origin in the first place (a compensating anti-instanton is then
required). The probability density of satisfying the condition can be
estimated as $R_{0}^{-D}y_{0}^{\phantom{x}}$. Therefore, the probability per
unit space-time volume of finding a free deconfined instanton is roughly:

$p\approx
R_{0}^{-D}y_{0}^{\phantom{x}}\lambda_{c}^{\phantom{x}}P^{\prime}(\lambda_{c}^{\phantom{x}})\,\frac{\Delta
T}{T_{c}}\ .$ (66)

We may interpret this as the inverse volume $p\sim\zeta^{-D}$ which contains
only one free instanton. Then, the mean distance between uncompensated
instantons is

$\zeta\sim m^{-1}\sim
R_{0}^{\phantom{x}}\left[y_{0}^{\phantom{x}}\lambda_{c}^{\phantom{x}}P^{\prime}_{\lambda}(\lambda_{c}^{\phantom{x}})\,\frac{\Delta
T}{T_{c}}\right]^{-1/D}\ .$ (67)

The photons of the effective gauge field can propagate coherently only across
distances where they do not encounter uncompensated instantons. Therefore, the
last equation estimates the photons’ correlation length $\zeta$ and their
effective mass $m$. We see that the correlation length diverges as a power-law
on the approach to the critical temperature. It should be noted, however, that
a phase transition with different critical properties can take place if the
coherence length $\xi$ of the fundamental field limits the screening length
$\lambda$ due to a temperature increase. Analyzing such a scenario requires a
more comprehensive theory, so we will not attempt it here.

The alternative $y_{0}\to 0$ universality class is qualitatively different. We
have seen that this regime sends the confining length $\lambda$ toward
infinity at $T=0$. Consequently, $\lambda$ never reaches the cut-off $R_{0}$
and $\beta$ trivially wins the race in the stage 1 of renormalization group.
Only a deconfined phase exists any finite temperature. Since the fugacity
reaches a finite value $y_{0}>0$ at the end of stage 1, thermal fluctuations
are always able to generate some instantons. The scale parameter $l$ at which
the stage 1 ends determines the length scale beyond which instantons
effectively lose their confinement. By relating this to temperature, we can
find the crossover temperature $T^{*}$ below which the suppression of
instantons approximately conserves the topological charge. Let us first
compute the photon correlation length $\zeta$. If the fugacity scales toward
zero as $y_{0}(l)=y_{0}(0)e^{\kappa l}$, then

$y_{0}(l)=y_{0}(0)\,\left(R_{0}T\right)^{\kappa}\ .$ (68)

at the end of stage 1 where $\beta(l)=R_{0}$. The space-time volume that
typically contains a single instanton is proportional to the inverse
probability $y_{0}=e^{-S_{0}}$ of a bare instanton occurrence. Since photons
propagate freely only until they encounter an instanton, this volume limits
the correlation length $\zeta$ of photons,

$\zeta^{D}\sim\frac{R_{0}^{D}}{y_{0}(l)}\quad\Rightarrow\quad\zeta\sim\frac{R_{0}}{y_{0}^{1/D}\,\left(R_{0}T\right)^{\kappa/D}}\
.$ (69)

The scale $\zeta$ computed this way pertains to the beginning of the second
renormalization group stage; all quantum effects have been taken into account,
and any further renormalizations would be calculated as in the standard
classical renormalization group in $d$ spatial dimensions. The photon mass
affected by the quantum processes is $m\sim\zeta^{-1}$.

In the case of a logarithmic potential $C(r)=K\log(r/R_{0})$, we have
$\kappa=K-D$. The correlation length diverges for $K>D$ when the temperature
approaches zero. Supposing that this expression could be trusted for $K<D$ as
well, its prediction $\zeta\to 0$ on the approach to zero temperature is
consistent with the deconfined instanton plasma. The low-temperature regime
with an approximate topological charge conservation is bounded by a crossover
temperature $T^{*}$. Setting $\zeta(T^{*})=xR_{0}$, we find

$T^{*}\sim\frac{1}{R_{0}}\Bigl{(}x\,y_{0}^{1/D}\Bigr{)}^{-D/\kappa}\propto
R_{0}^{-1}\,y_{0}^{-1/(K-D)}\ .$ (70)

At $T<T^{*}$, photon coherence is still discernible at least at some finite
short length scales. We must pick a value for $x$ such that $xy_{0}^{1/D}>1$
in order to also get $T^{*}=0$ when $K=D$. Ultimately, $T^{*}$ can be made
large by a large instanton interaction coupling $K>D$, together with a
naturally small ultra-violet cut-off length $R_{0}$. In this sense, the low-
temperature dynamics may be significantly influenced by the quantum phase
transition between the states of conserved and unconserved topological charge.

##  IV Experimental signatures: noise correlations

The phase transition between instanton confined and deconfined phases need not
involve any latent heat or symmetry breaking. The question is, then, if it is
possible to experimentally detect this transition. Here we argue that noise
correlations, measurable in the specific heat, provide a probe for instanton
confinement.

The practical feasibility of detailed experimental detection rests upon the
means to indirectly observe the fluctuations of topological charge in the
system. For example, since vortices are attached to particles in quantum Hall
liquids, one can gain insight about vortex dynamics by measuring electron
density fluctuations. A relativistic example is given by the chiral quantum
anomaly of quantum electrodynamics: the relevant instantons are the creations
and annihilations of inter-linked flux loops that carry the Hopf invariant of
the $\pi_{3}(S^{2})$ homotopy group, but it is their link to the electron
chiral currents that provides the means for detection. The fluctuations driven
by instanton dynamics should be visible in the specific heat as well, although
with much less detail.

Let us assume for simplicity that we have some indirect way of measuring the
total topological charge $N(t)$ inside the entire system at any time $t$. In
order to theoretically characterize the fluctuations of $N(t)$, we want to
consider the generalized Wilson correlation function (10) on the boundary
$S^{d}$ of a slab which stretches infinitely in all spatial dimensions but has
a finite temporal extent within the interval $t\in(t_{1},t_{2})$. The noise
spectrum $\textrm{Var}\,\mathcal{C}(\omega)$ of this correlation function can
distinguish between the confined and deconfined instanton phases. Let

$\bar{t}=\frac{t_{2}+t_{1}}{2}\quad,\quad\delta t=t_{2}-t_{1}\ .$ (71)

Then,

$\displaystyle\delta\mathcal{C}(\delta t)$ $\displaystyle=$ $\displaystyle
N(t_{2})-N(t_{1})=\int d\omega\,(e^{i\omega t_{2}}-e^{i\omega
t_{1}})N(\omega)=2i\int
d\omega\,e^{i\omega\bar{t}}\sin\left(\frac{\omega\delta t}{2}\right)N(\omega)$
(72)

relates the correlation function to the noise spectrum $N(\omega)$ of the
topological charge in the system. The variance of these random fluctuations
is:

$\displaystyle\textrm{Var}\,\mathcal{C}(\delta
t)=\langle\delta\mathcal{C}^{2}(\delta t)\rangle=\left\langle\left|\int
d\omega\,(e^{i\omega t_{2}}-e^{i\omega
t_{1}})N(\omega)\right|^{2}\right\rangle-\left|\left\langle\int
d\omega\,(e^{i\omega t_{2}}-e^{i\omega
t_{1}})N(\omega)\right\rangle\right|^{2}$ (73) $\displaystyle\quad=2\int
d\omega_{1}d\omega_{2}\,e^{i(\omega_{1}-\omega_{2})\bar{t}}\left[\cos\left(\frac{(\omega_{2}-\omega_{1})\delta
t}{2}\right)-\cos\left(\frac{(\omega_{2}+\omega_{1})\delta
t}{2}\right)\right]\Bigl{[}\left\langle
N(\omega_{1})N^{*}(\omega_{2})\right\rangle-\left\langle
N(\omega_{1})\right\rangle\left\langle N^{*}(\omega_{2})\right\rangle\Bigr{]}\
.$

The noise spectrum should not depend on the overall times $\bar{t}$ at which
the experiment is conducted, so we may take the average over $\bar{t}$ without
loss of information:

$\frac{1}{\Delta\bar{t}}\int d\bar{t}\,\textrm{Var}\,\mathcal{C}(\delta
t)=\frac{8\pi}{\Delta\bar{t}}\int d\omega\,\sin^{2}\left(\frac{\omega\delta
t}{2}\right)\,\textrm{Var}\Bigl{\\{}|N(\omega)|\Bigr{\\}}$

The remaining dependence on $\delta t$ is different in the confined “area-law”
and deconfined “volume-law” regimes. The slab volume enclosed by the $S^{d}$
manifold, on which we calculate $\mathcal{C}$, is linearly proportional to
$\delta t$. In contrast, the area of $S^{d}$ has a negligible dependence on
$\delta t$ in an infinite system. These behaviors restrict the frequency
dependence of the variance of $N(\omega)$:

$\textrm{Var}\Bigl{\\{}|N(\omega)|\Bigr{\\}}=\begin{cases}A\,\delta(\omega)+f(\omega)&,\quad\textrm{confined}\\\
B\,|\omega|^{-2}+g(\omega)&,\quad\textrm{deconfined}\end{cases}$ (74)

where

$f(\omega),g(\omega)=\begin{cases}0&|\omega|<\delta t_{\textrm{min}}^{-1}\\\
\neq 0&\textrm{otherwise}\end{cases}\ .$ (75)

The arbitrary functions $f,g$ specify the high-frequency spectrum and affect
the noise correlations only on short time intervals $\delta t<\delta
t_{\textrm{min}}$. They have no bearing on the thermodynamic properties
observed in the $S^{d}\to\infty$, i.e. $\delta t\to\infty$ limit. Note that
the “constants” $A$ and $B$ can depend on temperature and other parameters.
Additional sources of noise in the indirect measurements of $N(t)$ can blur
the distinction between these frequency behaviors, but the depletion of low-
frequency noise remains a characteristic signature of the confined instanton
phase.

If neither direct nor indirect measurements of $N(t)$ can be performed, it is
still possible to see the change of noise correlations in the heat capacity.
Generally, heat capacity at temperature $T$ is related to the internal energy
$\mathcal{E}$ fluctuations:

$C(T)=\frac{\textrm{Var}\\{\mathcal{E}\\}}{k_{B}T^{2}}\ .$ (76)

Creating a topological defect in an instanton event, perhaps together with an
attached particle excitation, always costs some energy $\epsilon$. Therefore,
the relevant energy fluctuations should be proportionally derived from the
fluctuations of the topological charge $N(t)$. The total internal energy
$\mathcal{E}$ inherits such topological fluctuations from a range of low
frequencies in the spectrum:

$\displaystyle\textrm{Var}\\{\mathcal{E}\\}$ $\displaystyle\sim$
$\displaystyle\frac{\epsilon^{2}}{\Delta\bar{t}}\\!\int\limits_{\omega_{0}}^{\delta
t_{\textrm{min}}^{-1}}\\!d\omega\,\textrm{Var}\\{|N(\omega)|\\}$
$\displaystyle\sim$
$\displaystyle\frac{\epsilon^{2}}{\Delta\bar{t}}\begin{cases}A(T)&,\quad\textrm{confined}\\\
\frac{B(T)}{\omega_{0}}&,\quad\textrm{deconfined}\end{cases}$

A lower frequency bound $\omega_{0}$ is formally necessary for the deconfined
phase in order to avoid infra-red divergence (we naively set $\omega_{0}=0$
for the confined phase). The high-temperature behavior
$C(T)\propto(\epsilon/T)^{2}$ expected for uncorrelated gap-$\epsilon$ degrees
of freedom means that $\omega_{0}$ cannot be limited by temperature. Also, the
fluctuations of $N$ are governed by an action instead of a Hamiltonian, so
there is no clear energy scale either that could determine $\omega_{0}$. In
fact, the only available physical source for $\omega_{0}$ is the (long) time
interval during which the noise statistics is measured. This experiment
duration limits both $\bar{t}$ and $\delta t$ in (71), so we expect
$\omega_{0}\Delta\bar{t}\sim 1$. Since the noise statistics is accurately
collected only in the $\Delta\bar{t}\to\infty$ limit, we conclude that
instanton deconfinement produces a heat capacity jump at the critical
temperature $T_{c}$:

$C_{\textrm{inst.}}(T)\propto\begin{cases}0&,\quad
T<T_{c}\;\;\textrm{(confined)}\\\ (\epsilon/T)^{2}&,\quad
T>T_{c}\;\;\textrm{(deconfined)}\end{cases}$ (78)

##  V Conclusions

The main finding of this analysis is that correlated quantum phases of
confined instantons can exist at low finite temperatures without spontaneous
symmetry breaking. Instanton confinement is a synonym for the conservation of
topological charge, i.e. the invariant which characterizes topological
defects. Long range coherence of the matter field is not a prerequisite for
these phases. The presented renormalization group mathematically reveals that
coherence needs to extend only up to the instanton screening distance
$\lambda$. Therefore, confined phases can be stabilized with bosonic and
fermionic particles alike. In the case of fermions at zero temperature,
$\lambda$ must be smaller than the mean distance between particles.

Instanton deconfinement transitions can occur at finite temperatures and
exhibit critical properties of second order transitions even though
spontaneous symmetry breaking does not take place. This universality class
characterizes asymptotically free instantons. We computed the critical scaling
of the emergent gauge field’s correlation length above the critical
temperature, assuming that particles remain coherent across sufficiently long
distances. At the same time, we found that an alternative universality class
characterizes instantons which are not asymptotically free: topological charge
can be conserved only at zero temperature, but the influence of this non-
trivial quantum dynamics can be seen below a finite crossover temperature.

A prominent non-relativistic realization of instanton confinement is
topological order. A topologically ordered ground state features a finite
density of topological defects with definite topological charge, which are
attached to particles of the corresponding physical charge. The number of
defects and the number of particles are both conserved, so neither are
condensed. The presented renormalization group confirms the possibility of
stabilizing topological orders at finite temperatures. A deeper analysis of
quantum tunneling between classical vacuum states Nikolić (2020a) shows that
finite temperature topological orders featuring monopoles or hedgehogs of
spinor fields also require $d\geq 3$ spatial dimensions. Fractional quantum
Hall states are realized in $d=2$ dimensions and their topological order
survives only at zero temperature.

A non-zero density of topological charge is also not a prerequisite for
instanton confinement. Therefore, correlated condensed phases with conserved
topological charge are also possible in relativistic systems. Such phases can
be relativistic analogues of topological orders, although it is not clear yet
how to precisely characterize them. One context in which instanton confinement
could be relevant is the pseudogap state of high temperature superconductors.
Various studies have found evidence of charge coherence outside of the
superconducting phase in cuprates Corson  _et al._ (1999); Crane  _et al._
(2007), most notably the Nernst effect measurements which can be
phenomenologically ascribed to the dynamics of well-defined vortices Wang  _et
al._ (2001, 2006); Li  _et al._ (2010). Often invoked qualitative picture of
these phenomena is that Cooper pairs are still present, in the mean-field
sense, above the critical temperature for superconductivity, but lack long-
range coherence. Such a vague picture does not provide any reason to regard
the pseudogap state as a thermodynamic phase separated from the high-
temperature disordered phase. Here, we demonstrated a generic physical
mechanism which can stabilize a thermodynamic pseudogap phase at finite
temperatures, even without symmetry breaking and possibly also without a non-
relativistic topological order.

Still, there is no guarantee that a non-condensed confined instanton phase is
a particular unconventional phase in a particular physical system. To
illustrate this point, we can think about bosons living on a lattice and
interacting with a U(1) gauge field. Interactions can stabilize a Mott
insulator, whereby the boson field $\psi$ becomes incoherent. A dual picture
of a Mott insulator is a vortex or monopole superconductor, so the appropriate
topological charge is not conserved. States with instanton confinement and the
topological charge conservation which accompanies it, therefore, are not Mott
insulators. Any low temperature phase in which the bosons are not localized is
a state of confined instantons. This statement can be extended to fermions at
zero temperature as well, i.e. the conventional states such as Fermi liquids,
integer quantum Hall states and ordinary band insulators. So, the question is
if there is anything interesting about instanton confinement in weakly
correlated systems.

For example, it is possible that instanton confinement can be related to
quantum anomalies in relativistic systems. The chiral quantum anomaly of
quantum electrodynamics in $D=4$ space-time dimensions links the non-
conservation of the chiral current to the Hopf $\pi_{3}(S^{2})$ topological
invariant of the electromagnetic field. This invariant, actually,
characterizes instantons because the spatial field configurations can only
realize singularity-free topological objects with a Hopf index in $d=3$
spatial dimensions, analogous to skyrmions in $d=2$ dimensions. Even though
local quantum tunneling can readily change the Hopf index of these
configurations, instanton confinement corresponds to the dynamics which
effectively preserves the topological index of the ground state. So, perhaps,
the chiral anomaly is a correlation effect which replaces topological order in
the non-relativistic systems of massless Dirac particles. After all, it does
require interactions via a gauge field. But then, the main question is if
sufficient correlations can be achieved at low temperatures in some Dirac
systems to sharply distinguish a non-trivial quantum-anomalous phase from the
completely uncorrelated high-temperature phase.

##  VI Acknowledgements

This research was partly supported by the Department of Energy, Basic Energy
Sciences, Materials Sciences and Engineering Award DE-SC0019331. Support was
also provided by the Quantum Science and Engineering Center at George Mason
University.

##  Appendix A Instantons for the $\pi_{n}(S^{n})$ homotopy groups

Spinor fields $\psi$ which transform in a representation of the Spin($d$)
group in $d$ spatial dimensions generally admit singular configurations that
can be characterized by $\pi_{n}(S^{n})$ homotopy groups Nikolić (2020a).
Separate singularities exist in charge and spin sectors. At the lowest rank,
the $\pi_{1}(S^{1})$ singularities are vortices in the charge sector or
skyrmions in the spin sector; these are point topological defects in $d=2$,
lines or loops in $d=3$, two-dimensional sheets in $d=4$, etc. At the highest
rank, the $\pi_{d-1}(S^{d-1})$ singularities are always topologically
protected point-like monopoles in the charge sector and hedgehogs in the spin
sector. At any rank $n<d$, the dynamics of charge-sector singularities can be
captured by the Lagrangian density with gradient ($\kappa_{n}$) and Maxwell
($e_{n}$) terms

$\displaystyle\mathcal{L}_{n}$ $\displaystyle=$
$\displaystyle\frac{\kappa_{n}}{2}\left(\sum_{i=1}^{n}(-1)^{i-1}\partial_{\mu_{i}}A_{\mu_{1}\cdots\mu_{i-1}\mu_{i+1}\cdots\mu_{n}}+A_{\mu_{1}\cdots\mu_{n}}\right)^{2}$
(79)
$\displaystyle+\frac{1}{2e_{n}^{2}}(\epsilon_{\mu_{1}\cdots\mu_{d-n}\nu\lambda_{1}\cdots\lambda_{n}}\partial_{\nu}A_{\lambda_{1}\cdots\lambda_{n}})^{2}\
,$

where $A_{\mu_{1}\cdots\mu_{n}}$ is an antisymmetric rank-$n$ tensor gauge
field whose flux evaluated on closed $n$-manifolds is the quantized
topological charge of the enclosed $\pi_{n}(S^{n})$ singularities. The gauge
field at rank $n-1$ serves as a matter field minimally coupled to the rank-$n$
gauge field. Analogous hierarchy exists in the spin sector, but the gauge
fields at lower ranks have non-Abelian form.

Only monopoles and hedgehogs at the highest rank $d-1$ can enjoy topological
protection. Consider the Higgs phase of the matter field at rank $d-1$. The
suppressed fluctuations of the matter field quantize the rank $d-1$ flux

$\frac{1}{q}\oint\limits_{S^{d-1}}d^{d-1}x\,\epsilon_{\mu_{1}\cdots\mu_{d-1}}A_{\mu_{1}\cdots\mu_{d-1}}\in\mathbb{Z}$
(80)

which emanates from the monopole or hedgehog singularity, where $q$ is the
flux quantum ($q=2\pi$ in the charge sector, and $q=S_{n}$ is the area of a
unit-radius $n$-sphere in the spin sector). The gauge field, then, depends on
the distance $r$ from the singularity as $A_{\mu_{1}\cdots\mu_{d-1}}\sim
1/r^{d-1}$. The Maxwell term of (79) vanishes everywhere except at the
singularity, so its cost is local. The gradient term can be compensated via
the Higgs mechanism if the matter field embeds the singularity in its
configuration, otherwise the energy cost

$\frac{\kappa_{d-1}}{2}\\!\\!\int\limits_{|{\bf
x}<R}\\!\\!d^{d}x\left(\frac{1}{r^{d-1}}\right)^{2}\propto\frac{1}{R^{d-2}}$
(81)

with an infra-red cut-off $R$ would impart the Coulomb interaction between
static monopoles separated by the distance $R$, with potential energy
$U(R)\sim 1/R^{d-2}$. Note that a static monopole is a quantized worldline
stretching in the temporal direction of the $D=d+1$ dimensional space-time.
Now consider instantons which create or annihilate a point topological defect
at time $t=0$. Their action potential $V(r)$ is largest when the matter field
at rank $d-1$ is in the Higgs phase. Since the matter field must compensate a
singularity, an instanton needs to change its $\pi_{d-1}(S^{d-1})$ topological
invariant. This cannot be done with a smooth transformation. The matter field
will be discontinuous across $t=0$ and the question is only what time-
dependent configuration minimizes the discontinuity action to form an
instanton. As explained in the introduction (Section II), the best option is
to focus the flux into singular worldlines. The annihilation of a monopole is
just a deflection of its world line from the temporal into a spatial
direction, which generalizes the Faraday’s law of electrodynamics. The gauge
flux is ultimately conserved and the monopoles/hedgehogs are topologically
protected. The interaction between instantons is linear in the distance
between them, $V(r)\propto r$, because it obtains from the string tension
(Maxwell term) associated with the flux worldline that connects them. Note
that we can trust this mean-field type of conclusion without much concern in
$d>2$ dimensions because we did assume a Higgs phase.

None of the finite-sized singular structures at lower ranks are topologically
protected. A singular manifold at any rank $n<d-1$ must be without a boundary
if it is to have a finite size. An example is a vortex loop in $d=3$ spatial
dimensions, which becomes a vortex 2-sphere in $d=4$, etc. This disturbs the
gauge fields at all ranks only locally. One can, actually, carry out a smooth
deformation of the fields to collect all energy density into a finite volume
around the singularity. Specifically, the rank-$n$ gauge field
$A_{\mu_{1}\cdots\mu_{n}}$ has zero flux on generic $n$-manifolds which
“encircle” the singularity, because they are not interlinked with the manifold
of the singularity (only the rank $d-1$ is special since there the
“encircling” manifold for the flux calculation is by definition interlinked
with the point-manifold of the singularity). Therefore, the most dangerous
rank-$n$ gauge field can be made to vanish in all directions a finite distance
away from the singularity. Since all energy cost can be consumed into a finite
volume, an instanton which removes the finite boundary-less singularity also
costs a finite action. The action interaction potential between instantons can
only be short-ranged, so the instantons cannot be confined and the number of
lower-rank singularities cannot be conserved. It should be emphasized that
this concerns quantum fluctuations; instantons are quantum tunneling
processes. It doesn’t help to interlink a singular structure with something
else – this creates an energy barrier for the removal of the singularity, but
instanton quantum tunneling still occurs with a finite probability. The only
possible protection against tunneling at ranks $n<d-1$ is to make the singular
domains infinite.

The instanton deconfinement at lower ranks $n<d-1$ is even more pronounced in
the spin sector where the gauge fields at these ranks are non-Abelian. See the
next Appendix for a brief example from the Yang-Mills gauge theory.

##  Appendix B Action interaction potential for skyrmion and hopfion
instantons

Certain topological defects do not have a singularity. This makes them more
vulnerable to quantum tunneling. The best known example are skyrmions. In
$d=2$ spatial dimensions, skyrmions are topological defects of the unit-
magnitude vector spin configuration $\hat{\bf
n}=(\hat{n}^{x},\hat{n}^{y},\hat{n}^{z})$ in the continuum limit. If one
defines a gauge field from the spin chirality

$\epsilon_{\mu\nu\lambda}\partial_{\nu}A_{\lambda}=\frac{1}{2}\epsilon_{\mu\nu\lambda}\epsilon^{abc}\hat{n}^{a}(\partial_{\nu}\hat{n}^{b})(\partial_{\lambda}\hat{n}^{c})\
,$ (82)

then the topological invariant of the spin configuration is the total flux

$\frac{1}{4\pi}\oint\limits_{S^{2}}d^{2}x\,\hat{\bf
z}(\boldsymbol{\nabla}\times{\bf A})\in\mathbb{Z}$ (83)

of the gauge field over the entire space. This number counts the total
skyrmion charge (number) in the system. The spatial manifold must, actually,
be a sphere $S^{2}$ in order to obtain quantization, but this is in practice
arranged on the infinite open plane by insisting that $\hat{\bf n}$ be uniform
and ferromagnetic in far-away regions. In physical terms, $A_{\mu}$ is the
gauge field which imparts “topological” Hall effect on electrons that move in
the topologically non-trivial magnetic background. A skyrmion does not have a
singularity, but it does have a center where all gradients of the
magnetization field can be focused by smooth transformations. Once this is
done, a tunneling event which removes the skyrmion costs only a finite action.
Finite-action instantons interact via short-range action potentials, so they
cannot be confined. Skyrmion charge conservation is violated by quantum
tunneling events, and hence skyrmions cannot produce a truly quantized
“topological” Hall effect.

Most generally, an instanton with creates or annihilates a skyrmion is a
hedgehog of $\hat{\bf n}$ in space-time. This can be seen by taking the
difference between the skyrmion charge $N(t)$ computed from (83) at two
different times $t_{1}\neq t_{2}$. The space-time boundary at times $t_{1}$
and $t_{2}$ can be continuously deformed into a 2-sphere $S^{2}$ embedded in
space-time, and then (83) becomes the hedgehog topological index from the
$\pi_{2}(S^{2})$ homotopy group. In the worst-case scenario, the interaction
between two hedgehogs a space-time distance $r$ apart is given by the Coulomb
potential $V(r)\sim 1/r$ in $D=d+1=3$. This is also unable to stabilize a
confined phase of instantons Kosterlitz (1977).

One might naively suspect that topological defects without a singularity
cannot be protected against quantum fluctuations. However, this is not the
case. The vector field $\hat{\bf n}$ supports another type of topological
defects in $d=3$ spatial dimensions, usually called hopfions. The
corresponding topological invariant can again be expressed using the gauge
field from (82),

$\frac{1}{16\pi^{2}}\int\limits_{S^{3}}d^{3}x\,{\bf
A}(\boldsymbol{\nabla}\times{\bf A})\in\mathbb{Z}\ .$ (84)

This is the Hopf index of the $\pi_{3}(S^{2})$ homotopy group. Geometrically,
hopfions are interlinked loops of skyrmions, which themselves are lines in
$d=3$, so they do not contain a singularity. Now, set up a 3-sphere $S^{3}$ in
space-time and evaluate (84) on it. If $S^{3}$ is smoothly deformed into a
slab between times $t_{1}$ and $t_{2}$, then the computed integer is the
number of hopfions that were created (or destroyed) in the time interval
$(t_{1},t_{2})$. Thus, (84) gives us the instanton charge. By dimensional
analysis, we see that a local non-zero instanton charge generates a gauge
field $A_{\mu}\sim 1/r$ a (large) distance away from the source. This
disturbance cannot be compressed into a finite volume because we need both a
non-zero gauge field and its magnetic flux at any distance $r$. So, in the
best case scenario, the Maxwell energy cost of a single Hopf instanton scales
as

$\mathcal{L}\propto(\epsilon_{ijk}\partial_{j}A_{k})^{2}+(\partial_{0}A_{i})^{2}\sim\frac{1}{r^{4}}$
(85)
$S(R)=\mathcal{S}_{0}+\int\limits_{R_{0}}^{R}dr\\!\\!\int\limits_{S^{3}(r)}\\!\\!d^{3}x\,\mathcal{L}\sim
S_{0}+\log\left(\frac{R}{R_{0}}\right)$ (86)

with the system radius $R$. Therefore, Hopf instantons interact at least via a
logarithmic potential $\log(r/R_{0})$. We found in Section III.2 that this
interaction cannot apply to every pair of instantons in $D=4$ space-time
dimensions. Instead, it evolves into an exclusive interaction at large
distances, linear in the distance between the instantons. Consequently, Hopf
index is globally conserved at sufficiently low temperatures despite the
quantum tunneling fluctuations.

This conclusion holds for the Abelian gauge field $A_{\mu}$. However, the Hopf
index can be also introduced for non-Abelian gauge fields
$A_{\mu}^{\phantom{x}}=A_{\mu}^{a}\gamma^{a}$, where $\gamma^{a}$ are the
gauge group generators. Defining

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-i[A_{\mu},A_{\nu}]\quad,\quad\widetilde{F}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}F_{\alpha\beta}$

(the square brackets are a commutator), we evaluate the Hopf index on the
closed “surface” $S^{3}$ by integrating over the space-time volume $B^{4}$
bounded by it Zinn-Justin (2005):

$\frac{1}{32\pi^{2}}\int\limits_{B^{4}}d^{4}x\;F_{\mu\nu}^{a}\widetilde{F}_{\mu\nu}^{a}\in\mathbb{Z}\
.$ (87)

Even though smooth transformations of the gauge field protect this integer,
there are singular instanton configurations which alter it with only a finite
action cost – at least in the $D=4$ Yang-Mills gauge theory Belavin  _et al._
(1975); ’t Hooft (1976); Schäfer and Shuryak (1998); ’t Hooft and Bruckmann
(2000); Ogilvie (2012). Their interaction potential is short-ranged in the
best case scenario, and the ensuing instanton deconfinement is responsible for
quark confinement in QCD.

## References

  * Wen (2004) X.-G. Wen,  _Quantum Field Theory of Many-Body Systems_ (Oxford University Press, New York, 2004).
  * Wen (2017) X.-G. Wen, Reviews of Modern Physics 89, 041004 (2017).
  * Levin and Stern (2009) M. Levin and A. Stern, Physical Review Letters 103, 196803 (2009).
  * Maciejko  _et al._ (2010) J. Maciejko, X.-L. Qi, A. Karch, and S.-C. Zhang, Physical Review Letters 105, 246809 (2010).
  * Swingle  _et al._ (2011) B. Swingle, M. Barkeshli, J. McGreevy, and T. Senthil, Physical Review B 83, 195139 (2011).
  * Walker and Wang (2012) K. Walker and Z. Wang, Frontiers of Physics 7, 150 (2012).
  * von Keyserlingk  _et al._ (2013) C. W. von Keyserlingk, F. J. Burnell, and S. H. Simon, Physical Review B 87, 045107 (2013).
  * Jian and Qi (2014) C.-M. Jian and X.-L. Qi, Physical Review X 4, 041043 (2014).
  * Wang and Levin (2014) C. Wang and M. Levin, Physical Review Letters 113, 080403 (2014).
  * Ye and Gu (2015) P. Ye and Z.-C. Gu, Physical Review X 5, 021029 (2015).
  * Ye and Gu (2016) P. Ye and Z.-C. Gu, Physical Review B 93, 205157 (2016).
  * Putrov  _et al._ (2017) P. Putrov, J. Wang, and S.-T. Yau, Annals of Physics 384, 254 (2017).
  * Fuji and Furusaki (2019) Y. Fuji and A. Furusaki, Physical Review B 99, 241107 (2019).
  * Lan  _et al._ (2018) T. Lan, L. Kong, and X.-G. Wen, Physical Review X 8, 021074 (2018).
  * Lan and Wen (2019) T. Lan and X.-G. Wen, Physical Review X 9, 021005 (2019).
  * Berezinsky (1972) V. L. Berezinsky, Soviet Physics JETP 34, 610 (1972), zh. Eksp. Teor, Fiz. 61, 1144 (1972).
  * Kosterlitz and Thouless (1973) J. M. Kosterlitz and D. J. Thouless, Journal of Physics C: Solid State Physics 6, 1181 (1973).
  * Kosterlitz (1974) J. M. Kosterlitz, Journal of Physics C: Solid State Physics 7, 1046 (1974).
  * Wilson (1974) K. G. Wilson, Physical Review D 10, 2445 (1974).
  * Polyakov (1975) A. M. Polyakov, Physics Letters B 59, 82 (1975).
  * Belavin  _et al._ (1975) A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Y. S. Tyupkin, Physics Letters B 59, 85 (1975).
  * ’t Hooft (1976) G. ’t Hooft, Physical Review Letters 37, 8 (1976).
  * Kosterlitz (1977) J. M. Kosterlitz, Journal of Physics C: Solid State Physics 10, 3753 (1977).
  * Polyakov (1977) A. M. Polyakov, Nuclear Physics B 120, 429 (1977).
  * Polyakov (1978) A. M. Polyakov, Physics Letters B 72, 477 (1978).
  * Osterwalder and Seiler (1978) K. Osterwalder and E. Seiler, Annals of Physics 110, 440 (1978).
  * Guth (1980) A. H. Guth, Physical Review D 21, 2291 (1980).
  * Lautrup and Nauenberg (1980) B. Lautrup and M. Nauenberg, Physics Letters B 95, 63 (1980).
  * Svetitsky and Yaffe (1982) B. Svetitsky and L. G. Yaffe, Nuclear Physics B 210, 423 (1982).
  * Nagaosa (1993) N. Nagaosa, Physical Review Letters 71, 4210 (1993).
  * Schäfer and Shuryak (1998) T. Schäfer and E. V. Shuryak, Reviews of Modern Physics 70, 323 (1998).
  * Kondo (1998) K.-I. Kondo, Physical Review D 58, 085013 (1998).
  * Zheng and Sachdev (1989) W. Zheng and S. Sachdev, Physical Review B 40, 2704 (1989).
  * Read and Sachdev (1990) N. Read and S. Sachdev, Physical Review B 42, 4568 (1990).
  * Kondo (1999) K.-I. Kondo, Physics Letters B 455, 251 (1999).
  * ’t Hooft and Bruckmann (2000) G. ’t Hooft and F. Bruckmann, (2000), arXiv:hep-th/0010225.
  * Nagaosa and Lee (2000) N. Nagaosa and P. A. Lee, Physical Review B 61, 9166 (2000).
  * Yoshida (2001) K. Yoshida, Journal of High Energy Physics 04, 030 (2001).
  * Sachdev and Park (2002) S. Sachdev and K. Park, Annals of Physics 298, 58 (2002).
  * Kleinert  _et al._ (2002) H. Kleinert, F. S. Nogueira, and A. Sudbø, Physical Review Letters 88, 232001 (2002).
  * Herbut  _et al._ (2003) I. F. Herbut, B. H. Seradjeh, S. Sachdev, and G. Murthy, Physical Review B 68, 195110 (2003).
  * Case  _et al._ (2004) M. J. Case, B. H. Seradjeh, and I. F. Herbut, Nuclear Physics B 676, 572 (2004).
  * Hermele  _et al._ (2004) M. Hermele, T. Senthil, M. P. A. Fisher, P. A. Lee, N. Nagaosa, and X. G. Wen, Physical Review B 70, 214437 (2004).
  * Kragset  _et al._ (2004) S. Kragset, A. Sudbø, and F. S. Nogueira, Physical Review Letters 92, 186403 (2004).
  * Børkje  _et al._ (2005) K. Børkje, S. Kragset, and A. Sudbø, Physical Review B 71, 085112 (2005).
  * Engelhardt (2005) M. Engelhardt, Nuclear Physics B - Proceedings Supplements 140, 92 (2005).
  * Wang (2005) Z. Wang, Physical Review Letters 94, 176804 (2005).
  * Nogueira and Kleinert (2005) F. S. Nogueira and H. Kleinert, Physical Review Letters 95, 176406 (2005).
  * Nogueira and Kleinert (2008) F. S. Nogueira and H. Kleinert, Physical Review B 77, 045107 (2008).
  * Yamamoto (2008) N. Yamamoto, Journal of High Energy Physics 12, 060 (2008).
  * Ogilvie (2012) M. C. Ogilvie, Journal of Physics A: Mathematical and Theoretical 45, 483001 (2012).
  * Anber (2014) M. M. Anber, Annals of Physics 341, 21 (2014).
  * Nikolić (2020a) P. Nikolić, Physical Review B 101, 115144 (2020a).
  * Fisher  _et al._ (1989) M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Physical Review B 40, 546 (1989).
  * Fisher (1990) M. P. A. Fisher, Physical Review Letters 65, 923 (1990).
  * Elitzur (1975) S. Elitzur, Physical Review D 12, 3978 (1975).
  * Fradkin and Shenker (1979) E. Fradkin and S. H. Shenker, Physical Review D 19, 3682 (1979).
  * Caillol and Levesque (1986) J. M. Caillol and D. Levesque, Physical Review B 33, 499 (1986).
  * Minnhagen (1987) P. Minnhagen, Reviews of Modern Physics 59, 1001 (1987).
  * Levin  _et al._ (1994) Y. Levin, X. jun Li, and M. E. Fisher, Physical Review Letters 73, 2716 (1994).
  * Fisher (1994) M. E. Fisher, Journal of Statistical Physics 75, 1 (1994).
  * Gama Goicochea and Nussinov (2023) A. Gama Goicochea and Z. Nussinov, Physical Review E 107, 014104 (2023).
  * Nikolić  _et al._ (2013) P. Nikolić, T. Durić, and Z. Tesanović, Physical Review Letters 110, 176804 (2013).
  * Nikolić (2014) P. Nikolić, Physical Review A 90, 023623 (2014).
  * Nikolić (2016) P. Nikolić, Physical Review B 94, 064523 (2016).
  * Nikolić (2020b) P. Nikolić, Physical Review B 102, 075131 (2020b).
  * Nikolić and Tesanović (2011) P. Nikolić and Z. Tesanović, Physical Review B 83, 064501 (2011).
  * Moon  _et al._ (2007) E. G. Moon, P. Nikolić, and S. Sachdev, Physical Review Letters 99, 230403 (2007).
  * Nikolić (2011) P. Nikolić, Physical Review B 83, 064523 (2011).
  * Vershinin  _et al._ (2004) M. Vershinin, S. Misra, S. Ono, Y. Abe, Y. Ando, and A. Yazdani, Science 303, 1995 (2004).
  * Fang  _et al._ (2004) A. Fang, C. Howald, N. Kaneko, M. Greven, and A. Kapitulnik, Physical Review B 70, 214514 (2004).
  * Steiner  _et al._ (2005) M. A. Steiner, G. Boebinger, and A. Kapitulnik, Physical Review Letters 94, 107008 (2005).
  * Valla  _et al._ (2006) T. Valla, A. V. Fedorov, J. Lee, J. C. Davis, and G. D. Gu, Science 314, 1914 (2006).
  * Crane  _et al._ (2007) R. W. Crane, N. P. Armitage, A. Johansson, G. Sambandamurthy, D. Shahar, and G. Gruner, Physical Review B 75, 184530 (2007).
  * Boyer  _et al._ (2007) M. C. Boyer, W. D. Wise, K. Chatterjee, M. Yi, T. Kondo, T. Takeuchi, H. Ikuta, and E. W. Hudson, Nature Physics 3, 802 (2007).
  * Kohsaka  _et al._ (2007) Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007).
  * Gomes  _et al._ (2007) K. K. Gomes, A. N. Pasupathy, A. Pushp, S. Ono, Y. Ando, and A. Yazdani, Nature 447, 569 (2007).
  * Kohsaka  _et al._ (2008) Y. Kohsaka, C. Taylor, Wahl, A. Schmidt, J. Lee, K. Fujita, J. W. Alldredge, K. McElroy, J. Lee, H. Eisaki, Uchida, D.-H. Lee, and J. C. Davis, Nature 454, 1072 (2008).
  * Ghiringhelli  _et al._ (2012) G. Ghiringhelli, M. L. Tacon, M. Minola, S. Blanco-Canosa, C. Mazzoli, N. B. Brookes, G. M. D. Luca, A. Frano, D. G. Hawthorn, F. He, T. Loew, M. M. Sala, D. C. Peets, M. Salluzzo, E. Schierle, R. Sutarto, G. A. Sawatzky, E. Weschke, B. Keimer, and L. Braicovich, Science 337, 821 (2012).
  * Torchinsky  _et al._ (2013) D. H. Torchinsky, F. Mahmood, A. T. Bollinger, I. Bozovic, and N. Gedik, Nature Materials 12, 387 (2013).
  * Wang  _et al._ (2001) Y. Wang, Z. A. Xu, T. Kakeshita, S. Uchida, S. Ono, Y. Ando, and N. P. Ong, Physical Review B 64, 224519 (2001).
  * Wang  _et al._ (2006) Y. Wang, L. Li, and N. P. Ong, Physical Review B 73, 024510 (2006).
  * Li  _et al._ (2010) L. Li, Y. Wang, S. Komiya, S. Ono, Y. Ando, G. D. Gu, and N. P. Ong, Physical Review B 81, 054510 (2010).
  * Note (1) Sufficiently high order terms in this expansion create infra-red divergence, but this is an artifact of the expansion. The exact integral is convergent and approximated by its well-behaved lowest order term.
  * Corson  _et al._ (1999) J. Corson, R. Mallozzi, J. Orenstein, J. N. Eckstein, and I. Bozovic, Nature 398, 221 (1999).
  * Zinn-Justin (2005) J. Zinn-Justin, Lecture Notes in Physics 659, 167 (2005), topology and Geometry in Physics, Ed. Eike Bick and Frank Daniel Steffen.