# Jordan-Wigner Composite-fermion liquids in 2D quantum spin-ice

Leonardo Goller  SISSA, Via Bonomea 265, 34136 Trieste, Italy  Inti Sodemann
Villadiego  Institut für Theoretische Physik, Universität Leipzig,
Brüderstraße 16, 04103, Leipzig, Germany  Max-Planck-Institut für Physik
komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany

###### Abstract

The Jordan-Wigner map in 2D is as an exact lattice regularization of the
$2\pi$-flux attachment to a hard-core boson (or spin-$1/2$) leading to a
composite-fermion particle. When the spin-$1/2$ model obeys ice rules this map
preserves locality, namely, local Rohkshar-Kivelson models of spins are mapped
onto local models of Jordan-Wigner/composite-fermions. Using this composite-
fermion dual representation of RK models, we construct spin-liquid states by
projecting Slater determinants onto the subspaces of the ice rules.
Interestingly, we find that these composite-fermions behave as “dipolar”
partons for which the projective implementations of symmetries are very
different from standard “point-like” partons. We construct interesting
examples of these composite-fermi liquid states that respect all microscopic
symmetries of the RK model. In the six-vertex subspace, we constructed a time-
reversal and particle-hole-invariant state featuring two massless Dirac nodes,
which is a composite-fermion counterpart to the classic $\pi$-flux state of
Abrikosov-Schwinger fermions in the square lattice. This state is a good
ground state candidate for a modified RK-like Hamiltonian of quantum spin-ice.
In the dimer subspace, we construct a state fearturing a composite Fermi
surface but with nesting instabilities towards ordered phases such as the
columnar state. We have also analyzed the low energy emergent gauge structure.
If one ignores confinement, the system would feature a $U(1)\times U(1)$ low
energy gauge structure with two associated gapless photon modes, but with the
composite fermion carrying gauge charge only for one photon and behaving as a
gauge neutral dipole under the other. These states are examples of pseudo-
scalar $U(1)$ spin liquids where mirror and time-reversal symmetries act as
composite-fermion particle-hole conjugations, and the emergent magnetic fields
are even under such time-reversal or lattice mirror symmetries.

###### Contents

  1. I Equivalence of Jordan-Wigner Transformation and flux attachment in 2D
  2. II Jordan Wigner/Composite fermions as extended partons in 2D quantum spin-ice
    1. II.1 2+1D Quantum spin-ice and its Jordan-Wigner Composite Fermion representation
    2. II.2 Review of Abrikosov-Schwinger parton states
    3. II.3 Extended parton states for quantum spin-ice
      1. II.3.1 Symmetry implementation on JW composite fermions: general considerations
      2. II.3.2 A specific implementation of symmetries of 2D quantum spin-ice on JW composite fermions.
      3. II.3.3 Connection to pseudo-scalar spin liquids.
      4. II.3.4 Dirac and Fermi surface mean-field states for the six-vertex model and quantum dimer models
  3. III Gauge field fluctuations and effective low energy continuum field theory
    1. III.1 Review of Gauge field fluctuations for $U(1)$ spin liquids from standard parton constructions
    2. III.2 Gauge field fluctuations for $U(1)$ spin liquids from extended parton constructions in 2D quantum spin-ice
    3. III.3 Gauge field and matter couplings, low energy effective field theory and dipolar nature of composite fermions
      1. III.3.1 $\textbf{p}=(0,0)$ scattering terms
      2. III.3.2 $\textbf{p}=(\pi,\pi)$ scattering terms
    4. III.4 $U(1)\times U(1)$ Gauge structure
  4. IV Summary and discussion
  5. A $2\pi$ Flux attachment equivalence
  6. B Projective symmetry implementations
    1. B.1 $\frac{\pi}{2}$ rotation enforcement
    2. B.2 Time-reversal enforcement
    3. B.3 Reflection symmetries enforcement
    4. B.4 Particle-Hole enforcement
  7. C Low energy effective theory derivation
    1. C.0.1 $\textbf{q}=(0,0)$ scattering processes analysis
    2. C.0.2 $\textbf{q}=(\pi,\pi)$ momentum scattering processes

## Introduction

The appearance of fermionic particles in systems whose microscopic building
blocks are spins or bosons is a remarkable example of the emergence of non-
local excitations in quantum states of matter. An exact and powerful map that
allows to understand this phenomenon in one-dimension is the Jordan-Wigner
transformation, which provides a re-writing of one-dimensional
spin-$\frac{1}{2}$ models in terms of fermions. In two-dimensions the Jordan-
Wigner transformation is closely related to another celebrated statistical
transmutation procedure known as flux attachment [1, 2, 3, 4, 5, 6]. More
specifically, as we will review in detail in Sec. I, a standard Jordan-Wigner
Fermion constructed by ordering spin-$\frac{1}{2}$ operators in a 2D lattice
is exactly equivalent to a hard-core boson carrying a fictitious solenoid of
$2\pi$-flux. In this sense, the 2D Jordan-Wigner Fermion is an exact lattice
regularized version of the composite Fermion particle that is commonly used to
understand certain quantum Hall states emerging from microscopic bosons, such
as those making the bosonic composite Fermi liquid state at filling $\nu=1$
[7, 8, 9].

This 2D Jordan-Wigner/flux-attachment has been exploited in several studies of
non-trivial quantum disordered states (“spin liquids”) and their competition
with traditional ordered phases [4, 10, 11, 12, 13, 14, 15, 16, 17, 18]. One
of the central challenges with the models investigated in these previous
studies is that the Jordan-Wigner/flux-attachment map in 2D does not preserve
space locality, in the sense that not all local spin-$\frac{1}{2}$ operators
appearing in the Hamiltonian are mapped onto local fermionic operators. This
sharply contrasts with the situation in 1D, where simply imposing a global
parity symmetry guarantees that local spin Hamiltonians map onto local
fermionic Hamiltonians. In most of the 2D studies this difficulty is dealt
with in a non-systematic manner by adding background magnetic fields that
account for the relation between particle density and flux in an average
fashion, similarly to how it is done in mean-field treatments of composite
fermions in quantum Hall states [19, 20, 21, 22].

Recently, however, it has been emphasized that another kind of exact Jordan-
Wigner-like maps in 2D are possible which in some sense preserve space
locality [23, 24, 25, 26]. This is achieved by imposing local conservation of
certain $\mathbb{Z}_{2}$-valued operators and thus endowing the Hilbert space
of spin-$\frac{1}{2}$ with a $\mathbb{Z}_{2}$ lattice gauge theory structure.
The gauge invariant spin operators (namely those commuting with the
$\mathbb{Z}_{2}$ conservation laws) can then be mapped exactly into bilinears
of fermion operators. The single fermion creation operator remains non-local
and can be explicitly constructed as a Jordan-Wigner-like string operator in
2D [23, 24, 25, 26]. This construction realizes an exact lattice
regularization of a different kind of flux attachment, namely the one
associated with a mutual Chern-Simons theory comprised of two U(1) gauge
fields and the following $K$-matrix:

$K=\begin{pmatrix}0&2\\\ 2&0\end{pmatrix},$

which is the Chern-Simons description of the topological order associated with
Kitaev’s Toric code model [23, 24, 25, 26], and the Jordan-Wigner fermions are
the $\epsilon$ particles, while the operator associated with the local
$\mathbb{Z}_{2}$ conservation law measures the parity of one of the other
self-bosonic anyons, e.g. the $e$ or $m$ particle. For related constructions
see also [27, 28, 29, 30, 31, 32, 33].

Motivated by these precedents, our current work investigates systems with a
different kind of local conservation law that allows to preserve the locality
of the usual Jordan-Wigner map in 2D associated with attaching $2\pi$-flux to
a hard-core boson. Our local conservation laws will be associated with a two
dimensional spin “ice rule” with a correspondingly conserved local operator
that generates a U(1) gauge group, and the models of interest will be the
classic 2D Rokshar-Kivelson-like (RK) Hamiltonians [34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46]111See Ref.[47] for a recently proposed interesting
variant.. Similar to the situation in 1D, these models will remain local under
Jordan-Wigner maps, however the price we will have to pay for this is that the
resulting fermionic model will be necessarily interacting and endowed with
local conservation laws. Despite this, the advantage of re-writing the RK
models in terms fermionic variables is that they are much more flexible
degrees of freedom to construct non-trivial quantum disordered states than the
original spin degrees of freedom. For example, simple fermionic Slater
determinant states would serve already as a mean-field approximation to
describe quantum spin liquid states. There is however a crucial caveat to this
mean field approach, which is that generically free fermion Slater determinant
states will not obey the local spin-ice rules. In other words, the naive free
fermion states would break the local gauge invariance and violate Elitzur’s
theorem [48]. To remedy this, we will project the free-fermion states onto the
susbpace of the Hilbert space satisfying the ice rules, in an analogous
fashion to how the Gutzwiller projection is employed in parton constructions
of spin liquid states [49]. Despite the similarity of spirit, we have
encountered that these states differ in crucial aspects from the standard
parton constructions, such as the Abrikosov-Schwinger fermions [50, 51, 52].

One of the key distinctions, is that the Abrikosov-Schwinger fermion of
standard parton constructions behaves as a point-like object under the parton
gauge transformations, whereas the Jordan-Wigner fermion behaves as a dipole-
like object under the local $U(1)$ gauge transformations of the RK models.
This is because the Abrikosov-Schwinger fermion operator at a given lattice
site only transforms non-trivially under the parton gauge transformations
acting on its site, but it transforms trivially under gauge transformations of
different sites. In other words, the creation of a single Abrikosov-Schwinger
fermion would violate the constraint defining the physical Hilbert space only
at a single lattice site, and in this sense it is point-like. The Jordan-
Wigner fermion violates the spin-ice rule of two neighboring vertices, in such
a way so as to create a dipole under of the $U(1)$ Gauss’ law of the RK type
Hamiltonians.

Because of the above, we will refer to our construction of spin-ice projected
Slater determinants of the Jordan-Wigner fermions as an “extended parton
construction”. These differences between extended vs point-like partons lead
to crucial physical differences between for the states constructed from them.
Some of these differences will manifest as unconventional implementations of
lattice symmetries. For example, we will show that $\pi/2$ rotation
symmetries222These are $\pi/2$ rotations that will be denoted by
$R_{\frac{\pi}{2}}$. do not admit an ordinary fermion implementation for the
Jordan-Wigner fermions, but need to be dressed by a unitary transformation
that is not part of the $U(1)$ lattice gauge group.

But the most remarkable difference we have found between the extended partons
and the point-like partons, is the nature of the gauge fluctuations around
their mean field Slater determinant states. According to the principles of the
projective symmetry group constructions for ordinary point-like partons, like
Abrikosov-Schwinger fermions, a Slater determinant state which conserves the
global particle number fermions will describe a $U(1)$ spin liquid state,
whenever it is stable against gauge confinement. The deconfined state has
therefore an emergent $U(1)$ photon gauge field, and the fermionic parton will
carry charge under this field. As we will see, however, the a Slater
determinant of the Jordan-Wigner extended partons will feature a $U(1)\times
U(1)$ gauge structure, namely two distinct gapless photons. The Jordan-Wigner
fermion will carry a net gauge charge under one of these two photons, but it
will be gauge neutral under the other photon, for which it will only carry a
gauge dipole.

Moreover despite the fact that the Jordan-Wigner fermion is a composite
fermion that can be viewed as a boson attached to $2\pi$ flux, we will see
that the expected action of the two $U(1)\times U(1)$ gauge fields is an
ordinary Maxwell-like action with no Chern-Simons terms, as a result of the
enforcement of time reversal and microscopic mirror symmetries of the models
in question. This is interesting because it demonstrates an explicit instance
of the existence of a composite Fermi liquid-like states arising from flux
attachment, for which the emergent gauge structure does not feature an
explicit Chern-Simons term. This feature is somewhat reminiscent of the Dirac
composite fermion theories of the half-filled Landau level [53, 54, 55], and
of some of the more sophisticated composite Fermi liquid theories of bosons at
$\nu=1$ [7, 8, 9], which contrast from the more traditional explicit forms of
flux attachment in the HLR description of composite fermions [56].

We will also construct interesting explicit examples of mean-field spin-liquid
states for RK-like 2D quantum spin-ice Hamiltonians. As we will see, the
sectors defined by different values of the spin-ice rules will correspond to
different fillings of the Jordan-Wigner/composite-fermion bands. For example
the sector with zero spin, which maps to the quantum six-vertex model [38,
39], will correspond to half-filling of a two-band model. We will construct an
explicit mean-field state that satisfies all the space symmetries of the
lattice, time-reversal and particle-hole transformations, and that features
two gapless linearly dispersing Dirac fermion modes at low energy, which can
be pictures as a composite fermion counterpart to the classic $\pi$-flux state
of Abrikosov-Schwinger fermions [51, 57]. Ignoring compactification-driven
instabilities (see below however), this would be therefore a specific kind of
Dirac composite Fermi liquid state, with an emergent low energy $U(1)\times
U(1)$ gauge structure with two massless photons, with the fermions carrying
charge under only one $U(1)$ photon and neutral under the other $U(1)$ photon.

Field theories of masless Dirac fermions coupled to a single $U(1)$ compact
gauge field are known to remain deconfined at low energies in the limit of
large-$N$ number of Dirac fermions flavors [58, 59] and to also avoid
spontaneous chiral symmetry breaking [60, 61]. However, understanding the
ultimate infrared fate of these field theories at finite $N$ has remained
challenging [62, 63, 64]. In our case we have $N=2$ Dirac fermions and two
photons (with the fermions carrying charge under only one of these photons).
We will not address systematically the impact of gauge compactification, but
we expect that at least the photon under which the fermions are neutral will
undergo Polyakov-like confinement [65, 66], which will remove it from low
energies, leaving possibly only two masless Dirac fermions coupled to a single
$U(1)$ photon at low energies, analogously to $\text{QED}_{3}$ with $N=2$ (to
the extend that this theory avoids confinement and other instabilities at low
energies).

On the other hand, we will see that for the subspace of spin-ice that maps
onto the quantum dimer model [38], the band structure of the Jordan-
Wigner/composite-fermions will be at quarter-filling leading to the appearance
of a composite Fermi-surface state. Moreover, the state arising when the
composite fermions only hop between nearest neighbor sites will display a
perfectly nested Fermi surface. This nesting is accidental in the sense that
it can be removed by adding symmetry-allowed further-neighbor hopping terms.
Nevertheless, such strong tendency for perfect nesting can be viewed as
related to the tendency of the quantum dimer model systems to have ordinary
gauge confined ground states (such as the resonant plaquette or the columnar
phases [67, 68, 69, 70, 71, 72]), which would appear if the Fermi surface is
fully gapped via a composite fermion particle-hole pair condensation. This
nested state could be therefore useful as a mother state to understand the
descending competing broken symmetry states of the quantum dimer model and
perhaps help understand the strong tendency towards the columnar phase of the
classic RK model, which has been advocated in recent studies to take over the
complete phase diagram on the side of the RK point where a unique ground state
exists ($v/t<1$) [71, 72].

Table 1: Dictionary spin-$\frac{1}{2}$ to hard-core-boson language. Spin-$\frac{1}{2}$ | Boson
---|---
$\ket{\uparrow}$ | $\ket{0}$
$\ket{\downarrow}$ | $\ket{1}$
$\sigma^{+}$ | $b$
$\sigma^{-}$ | $b^{{\dagger}}$
$\frac{1-\sigma^{z}}{2}$ | $n$

Our manuscript is organized as follows: Chapter I reviews the one-dimensional
Jordan-Wigner transformation and its interpretation as flux-attachment in the
2D square lattice. Chapter II applies this construction to 2D quantum spin-ice
models, and introduces the general extended parton construction of mean-field
states. Then we apply this to the specific cases of Quantum Six Vertex and
Quantum Dimer Models, and construct the mean-field states with two Dirac cones
and a Fermi Surface for each of these models respectively. Chapter III
develops a description of the gauge field fluctuations, and discusses the
derivation of the effective low energy theories for these states,
demonstrating the appearance of two U(1) gauge fields with two associated
gapless photons, with the fermions being charged under only one of the U(1)
fields. We close then with a summary and discussion where we also comment on
future research directions.

##  I Equivalence of Jordan-Wigner Transformation and flux attachment in 2D

Let us consider a two-dimensional square lattice with a spin-$\frac{1}{2}$
degree of freedom residing in each site denoted by ${\bf r}$, as depicted in
Fig.1. These spin-$\frac{1}{2}$ degrees of freedom can also be viewed as hard-
core bosons, according to the convention of Table 1. By choosing a convention
for the ordering of sites, we can write the standard Jordan-Wigner fermion
creation operators as follows (see Fig.1):

${f}^{{\dagger}}(\textbf{r})={b}^{{\dagger}}(\textbf{r})\prod_{1\leq\textbf{r}^{\prime}<\textbf{r}}\sigma^{z}(\textbf{r}^{\prime}).$
(1)

We will order the sites using “western typing” convention, as depicted in
Fig.1. Since, $\sigma^{z}(\textbf{r})=\exp(i\pi n(\textbf{r}))$, it follows
that for any pair of sites $\textbf{r},\textbf{r}^{\prime}$ the boson hopping
operators can be written as:

${b}^{{\dagger}}(\textbf{r}){b}(\textbf{r}^{\prime})={f}^{{\dagger}}(\textbf{r})e^{i\pi\sum_{{\textbf{r}^{\prime}}\leq\textbf{r}^{\prime\prime}<{\textbf{r}}}{n}(\textbf{r}^{\prime\prime})}{f}(\textbf{r}^{\prime}).$
(2)

Figure 1: Physical spin-$\frac{1}{2}$ degrees of freedom reside at the blue
sites of a 2D square lattice labeled by r. The light blue shaded region
denotes the membrane operator made from the product of all the
$\sigma^{z}(\textbf{r}^{\prime})$ spin operators in such region, associated
with the Jordan-Wigner fermion creation operator at site r (see Eq.(1)). The
directed arrows illustrate our “western typing” convention for ordering the 2D
lattice sites.

When $\textbf{r},\textbf{r}^{\prime}$ are nearby, the above operator is
clearly local in its physical bosonic representation, however is it is
generally non-local in its dual fermion representation as illustrated in
Fig.2.

Figure 2: Solid directed arrows represent the local boson hopping operator
between sites from site $\textbf{r}^{\prime}$ towards site r from Eq.(2). Blue
lines represent the Jordan Wigner strings associated with the fermion
representation of these same operators. We see that for our convention (see
Fig.1), the horizontal boson hoppings remain local in the fermion
representation, whereas the vertical hoppings have a non-local fermion
representation.

Let us now demonstrate the equivalence of Eq.(2) to $2\pi$-flux attachment.
Consider spin-less fermions, ${f}^{{\dagger}}(\textbf{r})$ located at the
sites r of the square lattice. We attach a thin solenoid to each of these
fermions which we view as located in the center of the plaquette that is
north-east to the site r (see Fig.3). The solenoid carries a $2\pi$-flux and
we choose a gauge that concentrates its vector potential,
$\textbf{A}(\textbf{x})$, into two strings, depicted as dotted lines in Fig.3.
This gauge is chosen so that the flux attachment exactly matches our specific
choice of “western typing” ordering convention of the Jordan-Wigner
transformation, and different ordering conventions lead to different gauge
choices for the flux-attachment (see e.g. [1, 2, 3, 4, 5, 6]). Here x can be
viewed as a coordinate on the ambient 2D space in which the lattice is
embedded. Each one of these strings is chosen so that the line integral of the
vector potential across a path that intersects the strings is exactly $\pi$.
Therefore, when another fermion hops across a bond that intersects one of
these strings, its hopping amplitude will have an extra minus sign, relative
to the hopping it has when the string is not present (namely each string acts
as a “branch-cut” that dresses the fermion hopping phase by $\pi$).

Figure 3: Flux attachment is performed by binding to each boson (located in
the sites marked by blue dots) a thin solenoid depicted by the star which is
located in the plaquette northeast from the boson site. This thin solenoid
carries $2\pi$ flux, whose vector potential is chosen to be concentrated in
the two dotted lines connected to the star. The hopping operators (depicted by
solid black lines) that intersect such dotted lines are multiplied by $-1$
when the solenoid is present, namely there is an extra $\pi$ phase for hopping
across the dotted lines.

Therefore, the above convention fixes the vector potential
$\textbf{A}(\textbf{x})$ to be a unique operator which is a function of all
the fermion occupations operators,
${n}(\textbf{r})={f}^{{\dagger}}(\textbf{r}){f}(\textbf{r})$. Therefore,
establishing the equivalence of the above flux attachment procedure to the
Jordan-Wigner transformation reduces to demonstrating that the following
operator identity holds:

$\exp\left(i\pi\sum_{{\textbf{r}^{\prime}}\leq\textbf{r}^{\prime\prime}<{\textbf{r}}}{n}(\textbf{r}^{\prime\prime})\right)=\exp\left(i\int_{\textbf{r}^{\prime}}^{\textbf{r}}\textbf{A}(\textbf{x})\cdot
d\textbf{x}\right).$ (3)

To demonstrate the above relation, let us first consider the line integral of
$\textbf{A}(\textbf{x})$ when $\textbf{r},\textbf{r}^{\prime}$ are nearest
neighbor sites. From Fig.4, we can see that the following holds for the
horizontal and vertical nearest neighbor hoppings:

$\displaystyle\frac{1}{\pi}\int_{\textbf{r}}^{\textbf{r}+\textbf{e}_{x}}\textbf{A}(\textbf{x})\cdot
d\textbf{x}={n}(\textbf{r}),$ (4)
$\displaystyle\frac{1}{\pi}\int_{\textbf{r}}^{\textbf{r}-\textbf{e}_{y}}\textbf{A}(\textbf{x})\cdot
d\textbf{x}=\sum_{x\leq
x^{\prime}}{n}(x^{\prime},y)+\sum_{x^{\prime}<x}{n}(x^{\prime},y-1),$

where $\textbf{r}=(x,y)$ are the coordinates of the lattice sites measured in
units of lattice constant, and the integration path is chosen respectively to
be the bonds $\\{\textbf{r},\textbf{r}+\textbf{e}_{x}\\}$ and
$\\{\textbf{r},\textbf{r}-\textbf{e}_{y}\\}$ (see Fig.4). The relations in Eq.
(4) are the same expected from Eq. (3).

Figure 4: A local boson hopping operator (depicted by the directed black solid
line), can be equivalently represented as a fermion hopping operator with its
hoppings dressed by the vector potentials that capture the $2\pi$-flux
attachment, according to the rules depicted in Fig.3 (see Eq.(4)).

Let us now show that the line integral of $\textbf{A}(\textbf{x})$ in Eq.(3)
is independent of the specific path that connects the points
$\textbf{r},\textbf{r}^{\prime}$, modulo $2\pi$. Let us consider two paths
$\gamma_{1}$ and $\gamma_{2}$ connecting $\textbf{r},\textbf{r}^{\prime}$.
These two paths define a closed path $\gamma$ which is the boundary of a
region $\Omega$ (see Fig.5). From Stokes’ theorem it follows that:

$\oint_{\gamma}\textbf{A}(\textbf{x})\cdot
d\textbf{x}=\iint_{\Omega}(\nabla\times\textbf{A})(\textbf{x})\cdot
d\sigma=2\pi\sum_{\textbf{r}\in\Omega}n(\textbf{r}).$ (5)

The sum over $\textbf{r}\in\Omega$ in the above expression is perfomed over
those sites r for which the solenoid is strictly in the interior of $\Omega$
(see Fig.5). Therefore since the fermion number operators $n(\textbf{r})$ are
integer valued, it follows from Eq. (5), that:

$\int_{\gamma_{1}}\textbf{A}(\textbf{x})\cdot
d\textbf{x}=\int_{\gamma_{2}}\textbf{A}(\textbf{x})\cdot d\textbf{x}\ \ {\rm
mod}(2\pi).$ (6)

Figure 5: Illustration of the paths used to derive Eqs.(5) and (6).

The above equation demonstrates that there is no ambiguity in the line
integrals in the right hand side of Eq.(3). A detailed derivation that Eq. (3)
holds for any pair $\textbf{r},\textbf{r}^{\prime}$ is presented in Appendix
A.

Therefore, we see that there is a precise equivalence between the notion of
the statistical transmutation of a hard-core boson and a “composite fermion”
carrying a solenoid of $2\pi$ flux, and the statistical transmutation of
spin-$\frac{1}{2}$ degrees of freedom onto Jordan-Wigner fermions in 2D
lattices. The non-locality of the Jordan-Wigner transformation in 2D should
not be viewed as “bug” but rather as a “feature” that secretly encodes the
natural non-locality associated with flux attachment. This equivalence could
also be useful to understand the precise lattice versions of transformations
discussed within the web of dualities [73].

##  II Jordan Wigner/Composite fermions as extended partons in 2D quantum
spin-ice

Quantum spin-ice in the 2D square lattice is a classic example of a lattice
gauge theory, namely, a model with a set of local conservation laws [20, 21,
34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]. For different values for
these local conservation laws the Hilbert space can be reduced to that of the
Quantum Six Vertex Model (Q6VM) or the celebrated Quantum Dimer Model (QDM)
introduced by Rokhsar and Kivelson [34]. In this chapter, we will develop a
dual representation of these models in terms of Jordan-Wigner/composite
fermions, and exploit the fact that these models of spins remain local in
terms of their dual Jordan-Wigner/composite fermions. We will show that these
Jordan-Wigner/composite fermions behave in certain sense like partons, such as
Abrikosov-Scwhinger fermions [49, 74], but with crucial qualitative
differences arising from the fact that they carry not only lattice gauge
charge, but also a lattice gauge dipole moment.

###  II.1 2+1D Quantum spin-ice and its Jordan-Wigner Composite Fermion
representation

To describe quantum spin-ice models it is convenient to introduce a different
lattice convention relative to that of the previous section. We first divide
the plaquettes of the 2D square lattice into two sublattices, that we will now
call “vertices” and “plaquettes”, so that the spin-$\frac{1}{2}$ degrees of
freedom are viewed as residing in the “links” connecting such vertices (see
Fig.6). These links therefore form another square lattice which is rotated
$45^{\circ}$ relative to the orginal square lattice. The Bravais lattice of
the spin-ice models is spanned by two vectors
$\textbf{R}=n_{1}\textbf{R}_{1}+n_{2}\textbf{R}_{2}$, $n_{1,2}\in\mathbb{Z}$,
which can be viewed as the position of vertices (see Fig.6). Therefore, the
unit cell has a basis with two spin-$\frac{1}{2}$ degrees of freedom, which we
will distinguish by subscripts $a,b$. For example,
$\sigma_{a}^{i}(\textbf{R})$ will denote the $i$-th Pauli matrix associated
with the site $(\textbf{R},a)$ (see Fig.6) 333We will also continue to label
the spin sites with lower-case letter r when there is no need to specify its
detailed Bravais lattice label, namely r is also understood to be the physical
coordinate of the spin site with Bravais lattice label $(\textbf{R},i)$ with
$i=a,b$.. From here on, we will assume that the original square lattice has an
even number of spin sites both in the x- and y-directions, because this is
needed in order to make spin-ice lattice periodic in a torus (see fig. 6) .

Figure 6: The original square lattice of spins is spanned by vectors
$\hat{\textbf{e}}_{x}$ and $\hat{\textbf{e}}_{y}$, whose sites (blue and red
dots) are denoted by r. The “spin-ice” lattice is the Bravais lattice spanned
by vectors $\textbf{R}_{1}$ and $\textbf{R}_{2}$, and with a basis of two spin
sites: the “a” sites (blue dots) and “b” sites (red dots). The plaquettes of
the original square lattice are now separated into “vertices” located at R and
“plaquettes” (e.g. the square shaded in gray) of the “spin-ice lattice”. The
plaquette resonance operator of the RK model, $L_{\textbf{R}}$ from Eq.(10),
is also illustrated.

For every vertex, we define an “ice charge operator” as the sum of the
$z$-components of the spins in its four links:

$\small
Q_{\text{ice}}(\textbf{R})\doteq\sigma^{z}_{a}(\textbf{R})+\sigma^{z}_{b}(\textbf{R})+\sigma^{z}_{a}(\textbf{R}-\textbf{R}_{1})+\sigma^{z}_{b}(\textbf{R}-\textbf{R}_{2}).$
(7)

The ice charge operators are the locally conserved quantities, and they are
the generators of the following “UV lattice gauge group” of unitary
transformations:

$G[\\{\theta(\textbf{R})\\}]=\exp\bigg{(}i\sum_{\textbf{R}}\theta(\textbf{R})Q_{\text{ice}}(\textbf{R})\bigg{)}.$
(8)

where $\theta(\textbf{R})$ are arbitrary real numbers. The lattice gauge
theory structure is imposed by demanding that the Hamiltonian, $H$, is
invariant under the UV lattice gauge group, or equivalently, that it commutes
with all the ice charge operators:

$[{Q}_{\text{ice}}({\textbf{R}}),{H}]=0,\,\,\,\forall\textbf{R}.$ (9)

The subspace with ${Q}_{\text{ice}}({\textbf{R}})=2$ at every vertex is
equivalent to that of the quantum dimer model (QDM), whereas the subspace with
${Q}_{\text{ice}}({\textbf{R}})=0$ is equivalent to the quantum six-vertex
model (Q6VM) (see Figs. 7 and 8 for illustration of the allowed
configurations). Gauge invariant operators include spin diagonal operators
such as $\sigma^{z}_{a}(\textbf{R})$ (boson number), and products of
spin/raising lowering operators (boson creation/ annihilation) over a sequence
of links forming a closed loop, the smallest of which is the “plaquette
flipping” operator:

$L_{\textbf{R}}=\sigma^{+}_{a}(\textbf{R})\sigma^{-}_{b}(\textbf{R}-\textbf{R}_{2})\sigma^{+}_{a}(\textbf{R}-\textbf{R}_{2})\sigma^{-}_{b}(\textbf{R}+\textbf{R}_{1}-\textbf{R}_{2}).$
(10)

The above operator can be viewed as centered around the plaquette that is
neighboring to the right the vertex located at R as shown in Fig.6. A classic
gauge invariant Hamiltonian is the Rokhsar-Kivelson model:

$H=-t\sum_{\textbf{R}}L_{\textbf{R}}+L^{{\dagger}}_{\textbf{R}}+v\sum_{\textbf{R}}L_{\textbf{R}}L^{{\dagger}}_{\textbf{R}}+L^{{\dagger}}_{\textbf{R}}L_{\textbf{R}}.$
(11)

Additionally, when placed in a 2D torus each gauge invariant subspace splits
into “winding” sectors, due to the existence of two conserved t’Hooft loop
operators, one for each direction of the torus, defined as:

$\ell_{x}\doteq\sum_{\textbf{r}\in L_{x}}\sigma^{z}(\textbf{r}),\
\ell_{y}\doteq\sum_{\textbf{r}\in L_{y}}\sigma^{z}(\textbf{r}).$ (12)

Where $\textbf{r}\in L_{x,y}$ denotes a sum over the sites in the non-
contractible loops of the torus depicted in Fig.9.

Figure 7: The four configurations of $\sigma^{z}$ on the links allowed by
${Q}_{\text{ice}}(\textbf{R})=2$. This sector corresponds to the quantum dimer
model, and the dimers are located at links where spins are reversed relative
to the state with all spins pointing up. Therefore, the dimers also mark the
location of JW/composite-fermions. Figure 8: The six configurations of
$\sigma^{z}$ on the links allowed by ${Q}_{\text{ice}}(\textbf{R})=0$. This is
the sector of to the six vertex model. Figure 9: Non-contractible loops
$L_{x,y}$ for the definition t’Hooft operators in Eq.(12) for a lattice with
periodic boundary conditions. Notice that in order to place the spin-ice
lattice on a torus there needs to be an even number of spin along the $x$ and
$y$ directions.

One of the remarkable properties of the lattice gauge structure of quantum
spin-ice models, is that any local gauge invariant operator remains local in
its dual Jordan-Wigner/composite-fermion representation. For example, the
elementary plaquette flipping operator from Eq.(10), after using the the
Jordan-Wigner transformation described in Sec. I, can be written as:

$L_{\textbf{R}}=f_{a}(\textbf{R})f^{\dagger}_{b}(\textbf{R}+\textbf{R}_{1})f_{a}(\textbf{R}+\textbf{R}_{2})f^{\dagger}_{b}(\textbf{R}).$
(13)

Therefore we see that the RK model can be equivalently represented as a local
model of interacting Jordan-Wigner/composite fermions. For larger gauge
invariant loop operators (e.g. those enclosing two adjacent plaquettes), the
dual fermion operators would also include the products of the fermion parities
for the links inside the loop, but in general any local gauge invariant
operator of spins maps onto a local fermion operator without any left-over
trace of the long-range part of the Jordan-Wigner strings 444This follows from
the fact that plaquette operators, $L_{\bf R},L^{\dagger}_{\bf R}$, and
$\sigma^{z}_{\bf r}$, form a complete algebraic basis from which any local
gauge invariant operator can be obtained by addition and multiplication of
these. Since this basis operators are mapped into local fermion operators via
the JW transformation, it follows that any local gauge invariant operator
remains local in its dual JW/composite-fermion representation..

The ice-charge operators are represented in terms of Jordan-Wigner/composite
fermions as follows:

$\small
Q_{\text{ice}}(\textbf{R})=4-2\,n_{\text{ice}}(\textbf{R})=4-2\sum_{\textbf{r}\in\textbf{R}}f^{\dagger}(\textbf{r})f(\textbf{r}),$
(14)

Where $\textbf{r}\in\textbf{R}$ denotes the spin sites, r, in the four links
connected to vertex R, and $n_{\text{ice}}(\textbf{R})$ is the total number of
fermions in such links. From the above we see that the subspaces obeying with
different values of ice charge correspond to different lattice fillings of the
Jordan-Wigner/composite-fermions. The QDM and Q6VM spaces have $\frac{1}{4}$
and $\frac{1}{2}$ filling of the fermion sites respectively. Some
representative configurations illustrating these fillings are shown in Fig.11.

In this work we will be interested in constructing spin-liquid states that are
relevant not only for the RK model, but for the universality class that the RK
Hamiltonian defines. This universality class is defined as the set of local
spin Hamiltonians555The locality is defined with respect to the tensor product
structure of the Hilbert space of underlying microscopic spin degrees of
freedom., with the same spin-ice local conservation laws, and the same global
symmetries of the RK model. Some of these global symmetries of the RK model
are listed in Table 2, and the notation for some of its space symmetries is
also depicted in figure 10. The particle-hole symmetry can only be enforced
for the filling associated with the subspace of the Q6VM (see Table 2).

Figure 10: Illustration of the point group symmetry operations of the quantum spin-ice model centered on a spin-ice plaquette (corresponding to Dihedral Group $D_{8}$). The model also has a similar set of symmetry operations centered around the vertices, which we also consider for constructing states. Symmetries | Symbol |  | Q6VM | QDM | Linear | Antilinear | Action on $b(\textbf{r})$ | Action on $Q_{\text{ice}}(\textbf{R})$
---|---|---|---|---|---|---|---|---
Time Reversal | $\Theta$ |  |  |  |  |  | $b(\textbf{r})$ | $Q_{\text{ice}}(\textbf{R})$
Spatial transformations |  | $R_{\frac{\pi}{2}}$ |  |  |  |  |  |
| $S_{x}$ |  |  |  |  |  |
$U_{d}$ | $S_{y}$ |  |  |  |  | $b\big{(}U_{d}(\textbf{r})\big{)}$ | $Q_{\text{ice}}(U_{d}(\textbf{R}))$
| $S_{1}$ |  |  |  |  |  |
| $S_{2}$ |  |  |  |  |  |
Particle-Hole | $X$ |  |  |  |  |  | $b^{\dagger}(\textbf{r})$ | $-Q_{\text{ice}}(\textbf{R})$
Table 2: Table of symmetries of the Rokhsar-Kivelson Hamiltonian of quantum
spin-ice. $U_{d}(\textbf{r})$ and $U_{d}(\textbf{R})$ denote the image of the
site r and the vertex R under the corresponding spatial transformation. See
Fig.10 for a definition of the spatial transformations and Fig.13 for a
depiction of the action of $R_{\frac{\pi}{2}}$. Figure 11: Left: depiction of
a configuration with half of spins reversed (half-filling of JW/composite-
fermions) relative to fully polarized state (denoted by solid bars) belonging
to six-vertex subspace. Right: depiction of a configuration with one quarter
of spins reversed (quarter-filling of JW/composite-fermions) relative to fully
polarized state (denoted by solid bars) belonging to quantum dimer subspace.

###  II.2 Review of Abrikosov-Schwinger parton states

Before introducing the extended parton construction of states for Jordan-
Wigner/composite-fermions in quantum spin-ice, we would like to review some of
the key ideas of the more traditional construction of states for Abrikosov-
Schwinger fermions, which we will sometimes refer to as “point-like” partons
(for more detailed discussions see e.g. Refs.[75, 76, 49, 74]). The same
previously discussed physical spin-$\frac{1}{2}$ degrees of freedom at the
lattice site r can be alternatively represented in terms of spinful Abrikosov-
Schwinger fermions $\psi^{{\dagger}}_{s}(\textbf{r})$
($s=\,\uparrow,\downarrow$):

$\sigma^{i}(\textbf{r})=\sigma^{i}_{ss^{\prime}}\psi^{{\dagger}}_{s}(\textbf{r})\psi_{s^{\prime}}(\textbf{r}).$
(15)

where $\sigma^{i}_{ss^{\prime}}$ is the $ss^{\prime}$ element of the $i$-th
Pauli Matrix. The above representation enlarges the physical Hilbert space
from a two dimensional $\\{\ket{\uparrow},\ket{\downarrow}\\}$ into a four
dimensional
$\\{\ket{0},\ket{\uparrow},\ket{\downarrow},\ket{\uparrow\downarrow}\\}$. In
this case, the “UV lattice gauge group” is generated by the fermion number at
each site:

$n(\textbf{r})=\sum_{s}f^{{\dagger}}_{s}(\textbf{r})f_{s}(\textbf{r}).$ (16)

The above operator is the counterpart of the spin-ice charge for this lattice
gauge structure. Gauge invariant operators are defined as those commuting with
$n(\textbf{r})$, and in this case they are are the spin operators themselves,
$\sigma^{i}(\textbf{r})$. The physical subspace is a gauge invariant subspace
satisfying:

$n(\textbf{r})\ket{\psi}=\ket{\psi}.$ (17)

Therefore, in this parton construction physical states are restrited to have
$\frac{1}{2}$ fermion filling of the lattice, which is already a crucial
difference with respect to the Jordan-Wigner/composite-fermions.

When restricted to the physical subspace, $n(\textbf{r})=\mathbb{1}$, there is
actually a $SU(2)$ group that leaves all gauge invariant, and which is larger
than the $U(1)$ UV lattice gauge group generated by (17). Such larger group of
operations that leave the gauge invariant operators invariant, is called the
parton Gauge group (PGG). To construct spin-liquid states it is convenient to
introduce an auxiliary mean-field Hamiltonian that parametrizes a Slater of
fermions:

$H_{\rm
MF}=\sum_{ss^{\prime}}\sum_{\textbf{r},\textbf{r}^{\prime}}t_{ss^{\prime}}(\textbf{r},\textbf{r}^{\prime})f^{{\dagger}}_{s}(\textbf{r})f_{s^{\prime}}(\textbf{r}^{\prime}).$
(18)

The hopping elements $t_{ss^{\prime}}(\textbf{r},\textbf{r}^{\prime})$ in the
mean-field Hamiltonian can be viewed as variational parameters of its Slater
determinant ground state, that we will denote by
$\ket{\Omega_{0}[t_{ss^{\prime}}(\textbf{r},\textbf{r}^{\prime})]}$. The above
mean field Hamiltonian conserves the total fermion number, and, therefore, it
is invariant under a global $U(1)$ subgroup of the PGG. More generally, the
group that leaves the mean-field Hamiltonian invariant is called invariant
gauge group (IGG). The importance of the IGG is that it determines the
expected true low energy emergent gauge group of the spin liquid state [49,
74] (assuming it does not suffer from instabilities such as gauge
confinement). For the above mean-field Hamiltonian with a $U(1)$ IGG we expect
then to have $U(1)$ spin liquid [49, 74]666But had we chosen a BCS-like mean
field state with a $\mathbb{Z}_{2}$ IGG, we would expect a $\mathbb{Z}_{2}$
spin liquid. For concretenes in this work we will be focusing on spin liquids
with low energy emergent $U(1)$ gauge groups.

The ground state,
$\ket{\Omega_{0}[t_{ss^{\prime}}(\textbf{r},\textbf{r}^{\prime})]}$, of the
above mean field Hamiltonian generically is not invariant under the UV gauge
group and violates the constraint of Eq.(17). The correct physical mean-field
state is obtained by projecting this state onto the physical gauge invariant
subspace (Gutzwiller projection), as follows:

$\ket{\Omega[t_{ss^{\prime}}(\textbf{r},\textbf{r}^{\prime})]}=\prod_{\textbf{r}}\bigg{(}\frac{1-(-1)^{n(\textbf{r})}}{2}\bigg{)}\ket{\Omega_{0}[t_{ss^{\prime}}(\textbf{r},\textbf{r}^{\prime})]},$
(19)

The Gutzwiller projection is a nontrivial operation that generally makes
difficult the calculation gauge invariant operators. It is possible, however,
to develop a precise understanding of the symmetry properties of the
Gutzwiller projected physical state. To illustrate this, let us imagine that
there is some global physical symmetry operation acting on the spins, denoted
by $S$ (e.g. a lattice translation or a mirror symmetry). We say that two
operations $S_{1}$ and $S_{2}$ defined by their action on the parton fermions
represent the same physical symmetry, if they have the same action on all
gauge invariant operators. However, if $S_{1}$ and $S_{2}$ differ by an
element of the parton gauge group, their enforcement on $\ket{\Omega_{0}}$ can
lead to two distinct physical states $\ket{\Omega}$. In this case, then
$S_{1}$ and $S_{2}$ are said to be two distinct projective symmetry group
(PSG) implementations on the partons of the same underlying physical symmetry
(for a recent discussion illustrating this, see e.g.[77]).

###  II.3 Extended parton states for quantum spin-ice

We are now ready to present our extended parton construction of mean field
states for composite fermions obtained from the JW transformation applied to
the quantum spin-ice Hamiltonians. The idea is to parallel the construction
for Abrikosov-Schwinger fermions, but for the UV gauge structure defined by
the ice charge operators from Eq.(7). We begin by introducing an auxiliary
mean-field Hamiltonian of composite fermions:

$H_{\text{MF}}=\sum_{\textbf{r},\textbf{r}^{\prime}}t(\textbf{r},\textbf{r}^{\prime})f^{{\dagger}}(\textbf{r})f(\textbf{r}^{\prime}).$
(20)

here $f^{{\dagger}}(\textbf{r})$ is the creation operator of the spinless
Jordan-Wigner fermion at the spin site r. The hopping amplitudes,
$t(\textbf{r},\textbf{r}^{\prime})$, are again viewed as parametrizing the
Slater determinant ground state of the mean field Hamiltonian, denoted by
$\ket{\Phi_{0}[t(\textbf{r},\textbf{r}^{\prime})]}$. The physical spin
orientation is encoded in the composite fermion occupation at each site, and
therefore there is no enlargement of the full spin Hilbert space.
Nevertheless, the composite fermion hopping bilinears in the above mean-field
Hamiltonian generically do not commute with the generators of the UV lattice
gauge transformations, and therefore its ground state, $\ket{\Phi_{0}}$
violates the ice rules. However, this is forbidden by Elitzur’s theorem: local
gauge symmetries cannot be spontaneously broken. As a consequence, the naive
ground state of the above mean-field Hamiltonian is not a satisfactory
approximation to the true gauge invariant ground states of quantum spin-ice
models. However, this deficiency can be cured in an analogous way as in the
case of Abrikosov-Schwinger partons, by projecting $\ket{\Phi_{0}}$ onto the
Gauge invariant subspaces. Therefore, in analogy to the Gutzwiller projection,
we introduce a projector into a Gauge invariant subspace, specified by the
local ice charges $Q_{\text{ice}}(\textbf{R})$ and the t’Hooft operators
$(\ell_{x},\ell_{y})$ (for the case of a torus), given by:

$\displaystyle P(\\{Q_{\text{ice}}(\textbf{R}),\ell_{x},\ell_{y}\\})\doteq
P(\ell_{x})P(\ell_{y})\prod_{\textbf{R}}P(Q_{\text{ice}}(\textbf{R})).$ (21)
$\displaystyle\ket{\Phi[t(\textbf{r},\textbf{r}^{\prime})]}=P(\\{Q_{\text{ice}}(\textbf{R}),\ell_{x},\ell_{y}\\})\ket{\Phi_{0}[t(\textbf{r},\textbf{r}^{\prime})]}.$

The above projected state is also parametrized by the hoppings,
$t(\textbf{r},\textbf{r}^{\prime})$, that could be in principle optimized as
variational parameters to minimize the energy of RK-like Hamiltonians.

| Abrikosov-Schwinger Fermions | Jordan-Wigner Composite-Fermions
---|---|---
Local Hilbert Space Enlargement | YES | NO
Internal degrees of freedom | Spin $\frac{1}{2}$ | spinless
UV Gauge Transformations generators | $n(\textbf{r})=\sum_{s}f^{{\dagger}}_{s}(\textbf{r})f_{s}(\textbf{r})$ | $Q_{\text{ice}}(\textbf{R})=4-2\sum_{\textbf{r}\in\textbf{R}}f^{\dagger}(\textbf{r})f(\textbf{r})$
Physical lattice fillings | $n(\textbf{r})=1$ | Any (e.g. $\langle f^{\dagger}(\textbf{r})f(\textbf{r})\rangle=\frac{1}{4}$ for QDM)
Table 3: Comparison between traditional point-like Abrikosov-Schwinger fermion
partons and the extended Jordan-Wigner/composite-fermion parton constructions
for 2D quantum spin-ice. Here $s\in\\{\uparrow,\downarrow\\}$, ${\bf r}$
denotes spin sites, ${\bf r}\in{\bf R}$ denotes the four spins adjacent to a
quantum spin-ice vertex located at ${\bf R}$. See Eq.(14) and Fig.6 for
definitions and depictions.

####  II.3.1 Symmetry implementation on JW composite fermions: general
considerations

Let us now consider the implementation of symmetries on these mean field
states of Jordan-Wigner/composite-fermions. As in the case of Abrikosov-
Schwinger fermions, the key idea is that the task of enforcing symmetries in
the physical projected states is traded by the easier task of enforcing
symmetries in the un-projected mean-field Hamiltonians. However, one needs to
develop a set of consistency criteria for these implementations because there
are multiple ways in which one given symmetry can be implemented in the un-
projected state, leading to the rich structure of projective symmetry group
implementations [49, 74].

At first glance it might appear as if there was no freedom on how to implement
symmetries on the Jordan-Wigner fermions, because any prescription on how
physical symmetries act on the underlying spin-$\frac{1}{2}$ degrees of
freedom would fix a unique symmetry action of the Jordan-Wigner/composite-
fermion operators. We will refer to this underlying symmetry implementation as
the “bare” symmetry action. However, this bare symmetry implementation cannot
be suitably enforced in the mean field Hamiltonians from Eq.(20). This is
because the specific choice for implementing the Jordan-Wigner ordering of the
2D lattice (e.g. the western typing convention of Sec. I) does not manifestly
preserve the symmetries of the lattice, and thus, for example, the bare action
of the bare implementation of a $\pi/2$ lattice rotation would map the fermion
bilinear mean field Hamiltonian from Eq.(20) onto a complex operator which is
no longer fermion bilinear Hamiltonian and does not appear local in its dual
fermion representation. Our goal in this subsection will be therefore to
develop a precise but more flexible notion of symmetry implementations on the
Jordan-Wigner/composite-fermions that is amenable to enforcement on mean-field
Hamiltonians.

Some of these difficulties of bare symmetry actions are not peculiar to the 2D
Jordan-Wigner transformation but are also reminiscent of those appearing in
the 1D Jordan-Wigner transformation, e.g. in the anomalous implementation of
lattice translations, which we will now discuss in order to motivate the 2D
construction. For example, consider a standard 1D finite lattice with periodic
boundary conditions and a standard translational symmetry implemented on the
microscopic spin operators located at site $r$ as follows:

$T\sigma^{i}({r})T^{{\dagger}}=\sigma^{i}(r+1)$ (22)

However, when this “bare” symmetry is implemented on the JW fermions it does
not act like a standard fermionic lattice translation, which we denote by
$\tau$, defined as:

$\tau f^{{\dagger}}(r)\tau^{{\dagger}}=f^{{\dagger}}(r+1)\neq
Tf^{{\dagger}}(r)T^{{\dagger}}$ (23)

The above arises because the JW string becomes translated by $T$ and therefore
it does not follow the initial JW convention (it does not start at spin “1”
any more)777For a recent discussion of the connection between lattice
translational symmetries, anomalies and dualities associated with the 1D JW
transformation see Ref. [78].. However while $T$ and $\tau$ are different
operations when acting on the single fermion operator, they would act
identically on fermion bilinear operators supported in the interior of the 1D
chain:

$\tau
f^{{\dagger}}(r)f(r)\tau^{{\dagger}}=Tf^{{\dagger}}(r)f(r)T^{{\dagger}}=f^{{\dagger}}(r+1)f(r+1)$
(24)

Therefore we can say that when the symmetry operations $T$ and $\tau$ are
restricted to act on parity even operators, they are essentially the same
symmetry888Up to corrections associated with boundary terms, but in this work
we will focus on implementations of symmetry in the bulk.. The parity
restriction in 1D plays an analogous role to the spin gauge structure in 2D,
in the sense that local spin operators that are invariant under the UV lattice
gauge symmetry, remain local in their dual fermion representation after the JW
map. In other words, after a quantum spin-ice model is mapped onto fermions
via the JW map, it appears to be a bona fide local fermionic model, similar to
how a parity even spin Hamiltonian looks like an ordinary fermionic model
after 1D JW map.

Therefore we define a generalized notion of equivalence among symmetries of
the 2D quantum spin-ice model when these are implemented on JW transformation,
as follows:

For a quantum spin-ice model, we say that two operators $S_{1}$ and $S_{2}$
that implement a symmetry action are equivalent, when they have the same
action on all local operators that are invariant under the UV lattice gauge
transformations defined in Eq.(8) (see Fig.12).

The usefulness of this notion of equivalent symmetries is that instead of
enforcing the non-trivial “bare” action of a symmetry, $S_{1}$, we can enforce
instead a simpler but equivalent symmetry implementation, $S_{2}$, which maps
fermion bilinear Hamiltonians onto fermion bilinear Hamiltonians. By enforcing
$S_{2}$ on the fermion bilinear mean-field Hamiltonian from Eq.(20), then the
expectation value of any gauge invariant operator computed from its
corresponding Gutzwiller projected state from Eq.(21), will obey the same
symmetry constraints as if we had enforced the bare symmetry action $S_{1}$.
In particular, if $G$ is a lattice UV gauge transformation from Eq.(8), then
$S$ and $GS$ are equivalent implementations of a symmetry. Interestingly, as
we will see, enforcing symmetries that differ by such a pure UV lattice gauge
transformation, $G$, on the mean-field Hamiltonian of Eq.(20), can lead to
physically distinct states after the generalized Gutzwiller projection of
Eq.(21). This situation is analogous to that of PSG implementations of
symmetry on the Abrikosov-Schwinger partons (see discussion following
Eq.(19)). However several interesting qualitative differences will appear
between these two cases, and this is partly why we call our construction an
extended projective symmetry group implementation (see Table 3).

Boson Gauge-Invariant Operator Boson Gauge-Invariant Operator Fermion Gauge-
Invariant Operator Fermion Gauge-Invariant Operator Jordan-Wigner Tf.
$\scriptstyle{\text{adj}_{X}(\cdot)}$ Jordan-Wigner Tf.
$\scriptstyle{\text{adj}_{\Xi}(\cdot)}$ Figure 12: Illustration of the notion
of equivalence of symmetry actions of operators. Two operators $X$ and $\Xi$
are equivalent implementations of a symmetry, if their action is identical on
all the operators that are invariant under the spin-ice UV lattice gauge
transformations (defined in Eq.(8)). This notion allows us to trade the
possibly complicated “bare” action of the microscopic symmetry, $X$, on the
JW/composite-fermions, by a simpler but equivalent symmetry implementation,
$\Xi$, which maps JW fermion bilinears onto JW fermion bilinears. This is a
natural extension of the notion of symmetry equivalence in standard parton
constructions Abrikosov-Schwinger fermions (see e.g. Refs.[49, 74]).

####  II.3.2 A specific implementation of symmetries of 2D quantum spin-ice on
JW composite fermions.

We will now construct a concrete example of extended projective symmetry
implementation for the symmetries of the quantum spin-ice model (see Table 2).
Our objective is to illustrate the general ideas by constructing interesting
and perhaps even energetically competitive spin liquid states (although we
will not compute explicitly their energy). It is clear, in analogy to ordinary
parton constructions [49, 74], that there is a large landscape of possible
extended projective symmetry implementations beyond the ones we will illstrate
concretely. We leave to future work the development of a more global
understanding and classification of the large and colorful landscape of
extended projective symmetry group implementations.

Let us begin by considering a $\pi/2$ spatial rotation centered on a plaquette
(see Fig.13), denoted by $R_{\frac{\pi}{2}}$. We define its action on the
microscopic degrees of freedom from an implementation that is natural when
viewed as spinless bosons, namely as follows:

$\displaystyle R_{\frac{\pi}{2}}b^{{\dagger}}({\bf
r})R_{\frac{\pi}{2}}^{\dagger}$
$\displaystyle=b^{{\dagger}}(R_{\frac{\pi}{2}}{\bf r}).$ (25)

Here $b^{{\dagger}}({\bf r})$ is the hard-core boson equivalent of the spin
lowering operator (see Table 1), and $R_{\frac{\pi}{2}}{\bf r}$ is the image
of site ${\bf r}$ under the rotation. The action of $R_{\frac{\pi}{2}}$ on the
gauge invariant plaquette operator from Eq.(10) is thus simply:

$\displaystyle
R_{\frac{\pi}{2}}(b_{2}^{{\dagger}}b_{1}b^{{\dagger}}_{4}b_{3})R_{\frac{\pi}{2}}^{\dagger}$
$\displaystyle=b^{{\dagger}}_{4^{\prime}}b_{1^{\prime}}b_{2^{\prime}}^{{\dagger}}b_{3^{\prime}},$
(26)

where $1,2,3,4$ denote the sites in the plaquette from figure 13 and
$1^{\prime},2^{\prime},3^{\prime},4^{\prime}$ their images after the
$\frac{\pi}{2}$ rotation. As discussed in Eq.(13), this same plaquette
operator can be alternatively written as product of JW/composite-fermion
operators. However, while the action of $R_{\frac{\pi}{2}}$ is simple on this
four fermion operator, it is complex and cumbersome on JW/composite-fermion
operators themselves, as it involves a $\pi/2$ rotation of the JW-strings.
More importantly $R_{\frac{\pi}{2}}$ does not map fermion bilinear operators
onto fermion bilinears, because, for example, it maps a fermion horizontal
hopping into a fermion vertical hopping dressed by JW-strings (see Fig.2).
Therefore, we would like to find an alternative but gauge equivalent
implementation of $R_{\frac{\pi}{2}}$ to overcomes this difficulty.

To do so, we define a collection of auxiliary operators associated with each
of the microscopic symmetries listed in Tables 2 and 4, whose action is
defined by replacing the boson operator, $b^{{\dagger}}({\bf r})$ with the
fermion operator $b^{{\dagger}}({\bf r})$ in Table 2. For example, for the
microscopic symmetry $R_{\frac{\pi}{2}}$, we associate the auxiliary fermion
operator $P_{\frac{\pi}{2}}$, whose action is obtained from Eq.(25) by
replacing $b^{{\dagger}}({\bf r})\rightarrow f^{{\dagger}}({\bf r})$, leading
to:

$P_{\frac{\pi}{2}}f^{{\dagger}}(\textbf{r})P_{\frac{\pi}{2}}^{\dagger}=f^{{\dagger}}(R_{\frac{\pi}{2}}\textbf{r}).$
(27)

Thus the idea is that these auxiliary fermion operators are intuitive and
natural symmetry implementations on fermions, but they are not necessarily
equivalent implementations of the microscopic symmetries on gauge invariant
operators, as we now explain. This auxiliary fermion rotation acts on the same
plaquette operator from Eq.(26), which can be equivalently represented with
fermions using Eq.(13), as follows:

$P_{\frac{\pi}{2}}(f_{2}^{{\dagger}}f_{1}f^{{\dagger}}_{4}f_{3})P_{\frac{\pi}{2}}^{\dagger}=-f^{{\dagger}}_{4^{\prime}}f_{1^{\prime}}f_{2^{\prime}}^{{\dagger}}f_{3^{\prime}}$
(28)

Figure 13: Action of a 90° rotation (denoted by $R_{\frac{\pi}{2}}$) centered
on the plaquette where the dashed dotted lines intersect, acting on the
plaquette resonace operators marked by thick squares (according to convention
in fig. 6).

Therefore the fermion rotation, $P_{\frac{\pi}{2}}$, is not an equivalent
implementation of the the underlying physical symmetry, $R_{\frac{\pi}{2}}$,
because it additionally multiplies the gauge invariant plaquette operator by a
global minus sign. The extra minus sign can be removed by dressing
$P_{\frac{\pi}{2}}$ with a staggered $U(1)$ transformation that we call
$U_{\frac{1}{4}}$, which rotates the phase of bosons with opposite signs in
the $a$ and $b$ sublattices (see Fig.14 $(a)$) as follows:

$\displaystyle
U_{\frac{1}{4}}b^{{\dagger}}_{a}(\textbf{R})U^{{\dagger}}_{\frac{1}{4}}$
$\displaystyle=e^{i\frac{\pi}{4}}b^{{\dagger}}_{a}(\textbf{R}),$
$\displaystyle
U_{\frac{1}{4}}b^{{\dagger}}_{b}(\textbf{R})U^{{\dagger}}_{\frac{1}{4}}$
$\displaystyle=e^{-i\frac{\pi}{4}}b^{{\dagger}}_{b}(\textbf{R}).$

where we are using the Bravais labels of the sites of the model (see Sec. II.1
for the convention). Notice that the action of $U_{\frac{1}{4}}$ on boson and
JW fermion operators is the same. Therefore, its action on the plaquette
operator is:

$U_{\frac{1}{4}}(f_{2}^{{\dagger}}f_{1}f^{{\dagger}}_{4}f_{3})U_{\frac{1}{4}}^{{\dagger}}=-f_{2}^{{\dagger}}f_{1}f^{{\dagger}}_{4}f_{3}.$
(29)

Figure 14: Phases gained by creation operators $f^{\dagger}(\textbf{r})$ or
$b^{\dagger}(\textbf{r})$ under the action of the site dependent $U(1)$
transformations: (a) $U_{\frac{1}{4}}$ (from Eq.(29)), (b) $V_{\frac{1}{2}}$
(from Eq.(40)). Figure 15: Right: Depiction of the allowed nearest neighbour
JW/composite-fermion hoppings associated with $\Theta_{x}$ and $\Theta_{y}$
extended projective symmetry groups from Table 5. Here the amplitude $t$ is
real for $\Theta_{x}$ and purely imaginary for $\Theta_{y}$. Left: The phase
of fermion hopping around a vertex or plaquette of the original square lattice
(dotted line) is $\pi$. This $\pi$ phase is behind the Dirac spectrum of
JW/composite-fermions for these state (see Fig.17).

Therefore the action of $R_{\frac{\pi}{2}}$ and
$U_{\frac{1}{4}}P_{\frac{\pi}{2}}$ on plaquette operators is identical.
Moreover, their action is also identical on $\sigma^{z}(\textbf{r})$ (or
equivalently the on-site fermion number operator). Since these operators
together with the plaquette operators form a complete algebraic basis for all
local gauge invariant operators, it follows that
$U_{\frac{1}{4}}P_{\frac{\pi}{2}}$ is an equivalent implementation of the
underlying physical symmetry $R_{\frac{\pi}{2}}$ on gauge invariant operators:

$R_{\frac{\pi}{2}}\equiv U_{\frac{1}{4}}P_{\frac{\pi}{2}}.$ (30)

Table 4 presents a list of the microscopic symmetries of the quantum spin-ice
model and a corresponding equivalent symmetry operation acting on the JW-
fermions. We see that in addition to the rotations, the natural fermionic
implemention of diagonal mirrors $S_{1}$ and $S_{2}$ (see Fig.10) also need to
be dressed by $U_{\frac{1}{4}}$ in order to make them equivalent to the
underlying microscopic symmetries. We will also enforce Bravais lattice
translational symmetries, which are understood to act identically on bosons
and fermions (up to boundary terms) and thus are not listed explicitly in
Table 4. Details of the derivations for these additional symmetries can be
found in Appendix B.

Symmetry | Bare microscopic boson | Auxiliary fermion transformation | Equivalent JW fermion symmetry
---|---|---|---
Time Reversal | $\Theta$ | $\Theta$ | $\Theta$
Spatial | $R_{\frac{\pi}{2}}$ | $P_{\frac{\pi}{2}}$ | $U_{\frac{1}{4}}P_{\frac{\pi}{2}}$
$S_{x}$ | $\Sigma_{x}$ | $\Sigma_{x}$
$S_{y}$ | $\Sigma_{y}$ | $\Sigma_{y}$
$S_{1}$ | $\Sigma_{1}$ | $U_{\frac{1}{4}}\Sigma_{1}$
$S_{2}$ | $\Sigma_{2}$ | $U_{\frac{1}{4}}\Sigma_{2}$
Particle-Hole | $X$ | $\Xi$ | $\Xi$
Table 4: Summary of RK Hamiltonian symmetries and their implementation on
bosons (spins) and JW fermions. The operations listed under the column “bare
microscopic boson” are the underlying bare microscopic symmetries implemented
on the boson creation operators, for example the 90° rotation
$R_{\frac{\pi}{2}}$ acts as defined in Eq.(25). For each of these we introduce
an “auxiliary fermion transformation” which acts in a simple way as expected
for spinless fermions, such as the fermion rotation $P_{\frac{\pi}{2}}$
defined in Eq.(27). However, this auxiliary fermion transformation is not
always equivalent to the “bare microscopic boson” symmetry (see Fig.12 for
notion of equivalence), and might need to be dressed by an extra site
dependent $U(1)$ gauge transformation to make it equivalent, as listed under
“equivalent JW fermion symmetry” (see Eq.(30) as an example for the
$R_{\frac{\pi}{2}}$ rotation and Fig.14 for a definition of
$U_{\frac{1}{4}}$). The above “equivalent JW fermion symmetries” define only
one possible extended projective symmetry group implementation on the
JW/composite-fermions. Two other examples, that are the focus of this work,
are described Table 5. In all the examples we implement the translations by
Bravais vectors ${\bf R}_{1},{\bf R}_{2}$ in the standard trivial non-
projective way for bosons and fermions without dressing the auxiliary fermion
transformations by gauge transformations.

This set of equivalent symmetries listed under the Jordan-Wigner fermion
column of Table 4 maps fermion bilinears onto fermion bilinears. Therefore,
any such equivalent symmetry implementation, denoted by $S$, can be used to
enforce the symmetry on the fermion mean-field Hamiltonian $H_{\text{MF}}$ of
Eq.(20), by determining the hoppings that are satisfy by the following
relation:

$SH_{\text{MF}}S^{-1}=H_{\text{MF}}.$ (31)

Interestingly one can show that after enforcing all the equivalent symmetries
from Table 4 and Bravais lattice translations, there are no allowed nearest
neighbor fermion hoppings in the lattice. For typical RK models, one expects
that short distance correlations determine a sizable portion of the energy-
density of the state and one would therefore expect that this projective
symmetry implementation from Table 4 is not a very energetically favorable
choice for reasonably simple microscopic Hamiltonians. However, as mentioned
before, the Fermionic symmetries listed in Table 4 are only one choice among a
large set of possibilities.

It is therefore interesting to consider the following question: can we
construct an alternative equivalent symmetry implementations that impose all
the symmetries of the RK spin-ice model but which allow for the nearest
neighbor hoppings to be non-zero? We have found two modified symmetry
implementations that allow for non-zero nearest neighbor hoppings, which we
shall denote as $\Theta_{x}$ and $\Theta_{y}$ implementations, and on which we
focus for the remainder of the paper. These projective symmetry
implementations are obtained by dressing the implementations of Table 4 with
the operations listed in Table 5, which are obtained after composition with
the following UV lattice gauge group elements $G_{x}$ and $G_{y}$:

$\displaystyle G_{x}b^{{\dagger}}(\textbf{r})G^{\dagger}_{x}$
$\displaystyle=(-1)^{x}b^{{\dagger}}(\textbf{r}).$ (32) $\displaystyle
G_{y}b^{{\dagger}}(\textbf{r})G^{\dagger}_{y}$
$\displaystyle=(-1)^{y}b^{{\dagger}}(\textbf{r}).$

Here we write the sites as $\textbf{r}=(x,y)$, where $x,y$ are understood to
be integers. These two transformations can be viewed as generated by the UV
lattice gauge transformations from Eq. (8), by choosing $\theta(\textbf{r})$
so that it takes the values depicted respectively in Fig.16 $(a)$ and 16
$(b)$.

Figure 16: Phases gained by creation operators $f^{\dagger}(\textbf{r})$ or $b^{\dagger}(\textbf{r})$ under the action of the UV gauge transformations: (a) $G_{x}$, (b) $G_{y}$. The transformations are obtained by choosing $\theta({\bf r})=\frac{\pi}{2}$ in Eq.(8) over the vertices contained in the gray regions and zero in the remainder. Symmetry |  $\Theta_{x}$ (extended PSG) |  $\Theta_{y}$ (extended PSG)
---|---|---
Time Reversal | $G_{x}\Theta$ | $G_{y}\Theta$
Spatial | $U_{\frac{1}{4}}P_{\frac{\pi}{2}}$ | $U_{\frac{1}{4}}P_{\frac{\pi}{2}}$
$G_{x}\Sigma_{x}$ | $G_{y}\Sigma_{x}$
$G_{x}\Sigma_{y}$ | $G_{y}\Sigma_{y}$
$G_{y}U_{\frac{1}{4}}\Sigma_{1}$ | $G_{x}U_{\frac{1}{4}}\Sigma_{1}$
$G_{y}U_{\frac{1}{4}}\Sigma_{2}$ | $G_{x}U_{\frac{1}{4}}\Sigma_{2}$
Particle-Hole | $G_{y}\Xi$ | $G_{x}\Xi$
Table 5: The two distinct extended projective symmetry group implementations
on the JW/composite-fermions that are the focus of this work. These symmetries
are all equivalent to the microscopic symmetries listed in Tables 2 and 4 (see
Fig-12 for summary of notion of equivalence).

####  II.3.3 Connection to pseudo-scalar spin liquids.

So far we have used an implementation of microscopic symmetries which is more
natural when we view the microscopic degrees of freedom as hard-core bosons,
but which is not necessarily natural when we view them as spin-$\frac{1}{2}$.
However, thanks to the large set of microscopic symmetries of the RK model of
spin-ice, we are implicitly also enforcing symmetries whose action is the
natural one when we view the microscopic degrees of freedom as spins.

For example the time-reveral operator $\Theta$, defined in Table 2, acts as
complex conjugation in the standard choice of Pauli matrices where only
$\sigma^{y}$ is imaginary, and $\sigma^{x,z}$ are real. Therefore it does not
square to $-1$. The more standard time-reversal operator of
spin-$\frac{1}{2}$, would act on the spin at site ${\bf r}$, as
$\mathcal{T}=i\sigma^{y}({\bf r})\Theta$. However, the operator
$i\sigma^{y}({\bf r})$ is equivalent to composition $U(\pi)\sigma^{x}({\bf
r})$, where $U(\pi)$ is a $\pi$ spin rotation around the z-axis, which acts on
the fermions as:
$U(\pi)f^{\dagger}(\textbf{r})U^{\dagger}(\pi)=-f^{\dagger}(\textbf{r})$.
Therefore $i\sigma^{y}({\bf r})$ is equivalent to a composition of the
particle-hole $X$, implemented $\sigma^{x}({\bf r})$, and a boson global U(1)
symmetry operation, implemented by $U(\pi)$, which we are already enforcing,
namely, we have:

$\mathcal{T}=U(\pi)X\Theta.$ (33)

Therefore, we are also implicitly enforcing such standard time-reversal
action, $\mathcal{T}$, on spin-$\frac{1}{2}$, and one can similarly understand
other natural spin symmetries of the RK model, as products of the natural
boson symmetries that we are already enforcing.

To determine the explicit action of $\mathcal{T}$ on JW/composite-fermions,
let us first describe the action of $\Theta$. On spin raising/lowering
operators this acts as a trivial anti-unitary operator (complex conjugation):

$\Theta\sigma^{+}({\bf r})\Theta^{-1}=\sigma^{+}({\bf r}).$

Therefore this operator acts similarly on JW/composite-fermions:

$\Theta f^{\dagger}(\textbf{r})\Theta^{-1}=f^{\dagger}(\textbf{r}).$ (34)

Let us now describe the implementation of the particle-hole conjugation of
hard-core bosons, denoted by $X=\prod_{\bf r}\sigma^{x}({\bf r})$. From the
action of this operator on spin operators, $X\sigma^{+}({\bf
r})X^{\dagger}=\sigma^{-}({\bf r})$, one obtains the action on the
JW/composite-fermions:

$Xf^{\dagger}(\textbf{r})X^{\dagger}=(-1)^{L_{s}(\textbf{r})}f(\textbf{r}).$
(35)

Where $L_{s}(\textbf{r})$ is the length of the JW string. The factor
$(-1)^{L_{s}(\textbf{r})}$ can be viewed as pure UV gauge transformation, and
therefore $X$ is gauge equivalent to the natural JW/composite-fermion
particle-hole conjugation, denoted by $\Xi$ (see Table 4), and defined as:

$\Xi f^{\dagger}(\textbf{r})\Xi^{\dagger}=f(\textbf{r}).$ (36)

The spin time-reversal symmetry, $\mathcal{T}$, reverses the direction of all
the spin components, and in particular the z-direction:
$\mathcal{T}\sigma^{z}(\textbf{r})\mathcal{T}^{-1}=-\sigma^{z}(\textbf{r})$.
Therefore, since $\sigma^{z}(\textbf{r})$ encodes the information of the
JW/composite-fermion, it is clear that $\mathcal{T}$ maps a fermion particle
into a hole and viceversa. Therefore, we see that $\mathcal{T}$ is therefore a
type of anti-unitary particle hole conjugation on the JW/composite-fermion
operator, which explicitly reads as:

$\mathcal{T}f^{\dagger}(\textbf{r})\mathcal{T}^{-1}=(-1)^{L_{s}(\textbf{r})+1}f(\textbf{r}).$
(37)

Where $L_{s}(\textbf{r})$ is the length of the JW string, and the factor
$(-1)^{L_{s}(\textbf{r})+1}$ is a pure UV gauge transformation identical to
$G_{x}$ defined in Eq.(32)999Assuming the lattice has an even number of sites
in the x-direction, which is natural for quantum spin-ice in a torus (see
Fig.9).. Because of the above we see the JW/composite-fermion behaves under
$\mathcal{T}$ as a pseudoscalar spinon, in the sense defined in Ref. [77].

We have introduced other space symmetries in their natural boson
representation in Tables 2 and 4 that also would act as particle-hole
conjugations on the JW/composite-fermions when implemented as standard
spin-$\frac{1}{2}$ symmetries. For example, for the space mirror operations
$S_{x},S_{y},S_{1},S_{2}$ $\sigma^{z}(\textbf{r})$ transform as a scalar,
e.g.: $S_{y}\sigma^{z}(\textbf{r})S_{y}^{-1}=\sigma^{z}(S_{y}\textbf{r})$.
However, its spin spin version would include an additional boson particle-hole
conjugation, leading to the standard action of spins which are pseudo-vectors
under mirrors, and which would reverse $\sigma^{z}(\textbf{r})$ because it is
parallel to these mirror planes. Therefore, these mirrors act as unitary
particle-hole conjugations on the JW/composite-fermions, and the spin liquid
states that we will be discussing in this paper can be viewed as pseudo-scalar
spin liquids with respect to spin implementations of time-reversal and space
mirror symmetries in the sense defined in Ref.[77].

####  II.3.4 Dirac and Fermi surface mean-field states for the six-vertex
model and quantum dimer models

The procedure described in the previous section allows us to fix the nearest
neighbour hopping amplitudes in Eq.(20). The resulting pattern of hoppings is
illustrated in fig. 15, and the corresponding mean-field hamiltonian reads as:

$\displaystyle H_{\text{MF}}=$
$\displaystyle\sum_{\textbf{R}}it^{*}\big{(}f^{{\dagger}}_{a}(\textbf{R}-\textbf{R}_{1}+\textbf{R}_{2})f_{b}(\textbf{R})+f^{{\dagger}}_{a}(\textbf{R})f_{b}(\textbf{R})\big{)}+t\big{(}f^{{\dagger}}_{a}(\textbf{R}-\textbf{R}_{1})f_{b}(\textbf{R})+f^{{\dagger}}_{a}(\textbf{R}+\textbf{R}_{2})f_{b}(\textbf{R})\big{)}+h.c.$
(38)

Which in crystal momentum basis can be re-expressed as:

$H_{\text{MF}}=\sum_{\textbf{q}\in\text{BZ}}\begin{pmatrix}f^{{\dagger}}_{a}(\textbf{q})&f^{{\dagger}}_{b}(\textbf{q})\end{pmatrix}\begin{pmatrix}0&h_{ab}(\textbf{q})\\\
h_{ab}^{*}(\textbf{q})&0\end{pmatrix}\begin{pmatrix}f_{a}(\textbf{q})\\\
f_{b}(\textbf{q})\end{pmatrix},$

where we are using the crystal momentum basis
$f^{{\dagger}}_{a}(\textbf{R})=N_{\Lambda}^{-1/2}\sum_{\textbf{q}\in\text{BZ}}e^{-i\textbf{q}\cdot\textbf{R}}f^{{\dagger}}_{a}(\textbf{q})$,
and the matrix entry is:

$h_{ab}(\textbf{q})=2e^{\frac{i}{2}(q_{1}-q_{2})}\bigg{[}it^{*}\,\cos\big{(}\frac{q_{1}-q_{2}}{2}\big{)}+t\,\cos\big{(}\frac{q_{1}+q_{2}}{2}\big{)}\bigg{]}.$

Where $q_{i}={\bf q}\cdot{\bf R_{i}}$, $i=1,2$. The associated band energy
dispersions is:

$\epsilon_{\pm}(\textbf{q})=\pm
2|t|\sqrt{\cos\big{(}\frac{q_{1}-q_{2}}{2}\big{)}^{2}+\cos\big{(}\frac{q_{1}+q_{2}}{2}\big{)}^{2}}.$
(39)

These bands are illustrated in Fig.17. The two extended projective symmetry
implementations $\Theta_{x}$ and $\Theta_{y}$ (see table 5) impose different
constraints on the hopping amplitude $t$ be either purely real or purely
imaginary:

$\begin{cases}t=t^{*}\,\,\,\,\,\,\,\,\,\text{for the}\,\,\Theta_{x}\\\
t=-t^{*}\,\,\,\,\text{for the}\,\,\Theta_{y}\end{cases}$

Details on how the above follows from implementing the symmetries from Table 5
are shown in Appendix B.

Despite their similarity, the extended projective symmetry group
implementations $\Theta_{x}$ and $\Theta_{y}$ (see Table 5) are inequivalent.
This can be seen by considering the action of a particular unitary
transformation, denoted by $V_{\frac{1}{2}}$, which acts on the fermion
operator, $f^{\dagger}(\textbf{r})$, as a local site dependent U(1)
transformation multiplying it by the specific phases shown in Fig.14(b). It
turns out that $V_{\frac{1}{2}}$ maps the $\Theta_{x}$ mean-field Hamiltonian
onto the $\Theta_{y}$ mean-field Hamiltonian, as can be seen from its action
of the following fermion bilinears (the $1,2,3,4$ sub-indices below are the
sites shown in Fig.15):

$\displaystyle tf_{3}^{{\dagger}}f_{1}$ $\displaystyle\to
itf_{3}^{{\dagger}}f_{1}$ (40) $\displaystyle-itf_{1}^{{\dagger}}f_{2}$
$\displaystyle\to tf_{1}^{{\dagger}}f_{2}$ $\displaystyle
tf_{2}^{{\dagger}}f_{4}$ $\displaystyle\to itf_{2}^{{\dagger}}f_{4}$
$\displaystyle-itf_{4}^{{\dagger}}f_{3}$ $\displaystyle\to
tf_{4}^{{\dagger}}f_{3}$

On the other hand, the operators $L_{\textbf{R}}$ and
$L_{\textbf{R}}^{{\dagger}}$ that enter in the microscopic RK Hamiltonian (see
Eq.(10)) can be shown to be odd under the action $V_{\frac{1}{2}}$. Since
$L_{\textbf{R}}$ is a invariant under the UV Lattice Gauge group, it follows
that $V_{\frac{1}{2}}$ is not a pure gauge transformation but a transformation
with non-trivial action within the gauge invariant subspaces, and therefore
the $\Theta_{x}$ and $\Theta_{y}$ mean-field Hamiltonians are not gauge
equivalent, but rather realizing to two physically distinct generalized
projective symmetry group implementations. This implies that only one them
will have lower energy as a trial ground state for a specific microscopic RK
Hamiltonian. Since the plaquette resonance term $L_{\textbf{R}}$ is odd under
$V_{\frac{1}{2}}$, the one that is more energetically favorable will be
determined by the sign of the plaquette resonance term in the microscopic
Hamiltonian101010Notice that $(V_{\frac{1}{2}})^{2}$ would map both the
$\Theta_{x}$ and $\Theta_{y}$ mean-field Hamiltonians into minus themselves.
However, $(V_{\frac{1}{2}})^{2}$ leaves all the gauge invariant operators
unchanged, and it is therefore an element of UV gauge group. Therefore we see
that changing the global sign of $t$ in either the $\Theta_{x}$ or
$\Theta_{y}$ mean-field Hamiltonians leads to the same physical state..

Figure 17: Dispersions of the JW/composite-fermions associated with the
extended projective symmetry groups $\Theta_{x}$, $\Theta_{y}$ (see Table 5
for their definitions, Fig.15 for the hoppings, and Eq.(39) for dispersion).
There are two Dirac cones at $(0,\pi)$ and $(\pi,0)$ and a Fermi surface at
$\frac{1}{4}$-filling which is perfectly nested, and correspons to the
straight lines separating blue from red regions in the equal energy contours
shown in the right panel. The crystal momenta is defined as $q_{1,2}={\bf
q}\cdot{\bf R}_{1,2}$ and therefore measure along the ${\bf R}_{1,2}$
directions (see Fig.15).

As described in Sec.II.1, for the cases of the $\mathcal{H}_{\text{Q6VM}}$ and
$\mathcal{H}_{\text{QDM}}$, the system is respectively at half-filling and
quarter filling, therefore, as depicted in Fig.17, these systems have a mean
field dispersion featuring two massless Dirac cones and a Fermi surface,
respectively. The Dirac points are located at ${\bf
q}_{0}=(q_{1},q_{2})=(\pi,0)$ and ${\bf q}_{0}=(q_{1},q_{2})=(0,\pi)$. By
writing ${\bf q}={\bf q}_{0}+{\bf p}$ and expanding the mean-field Hamiltonian
to linear order in the momentum ${\bf p}$, we obtain the following effective
Dirac Hamiltonian for the $\Theta_{x}$ extended PSG ($t\in\mathbb{R}$) is:

$\displaystyle h({\bf q}_{0}+{\bf p})\simeq
v\begin{cases}p_{x}\tau^{x}+p_{y}\tau^{y}\,\,\,\,\,\text{for}\,\,{\bf
q}_{0}=(\pi,0)\\\ p_{x}\tau^{x}-p_{y}\tau^{y}\,\,\,\,\,\text{for}\,\,{\bf
q}_{0}=(0,\pi)\\\ \end{cases}$ (41)

where $\tau^{x,y}$ are Pauli matrices in the $a/b$ sublattice space,
$v=\sqrt{2}t|{\bf R}_{1}|$, and $p_{x}=({\bf p}\cdot\hat{{\bf R}}_{1}-{\bf
p}\cdot\hat{{\bf R}}_{2})/\sqrt{2}|{\bf R}_{1}|,p_{y}=({\bf p}\cdot\hat{{\bf
R}}_{1}+{\bf p}\cdot\hat{{\bf R}}_{2})/\sqrt{2}|{\bf R}_{1}|$. On the other
hand, for the $\Theta_{y}$ extended PSG ($t\in i\mathbb{R}$) the linearized
Hamiltonian is:

$\displaystyle h({\bf q}_{0}+{\bf p})\simeq
v\begin{cases}p_{x}\tau^{y}+p_{y}\tau^{x}\,\,\,\,\,\text{for}\,\,{\bf
q}_{0}=(\pi,0)\\\ p_{x}\tau^{y}-p_{y}\tau^{x}\,\,\,\,\,\text{for}\,\,{\bf
q}_{0}=(0,\pi)\\\ \end{cases}$ (42)

where $v=-i\sqrt{2}t|{\bf R}_{1}|$.

Figure 18: Further neighbour hoppings considered to deform the Fermi surface
at $\frac{1}{4}$-filling and remove its perfect nesting. $t^{\prime}$ is
forbidden by the $\Theta_{x}$ and $\Theta_{y}$ extended projective symmetry
groups. $t_{2}$ and $t_{4}$ are both allowed, but $t_{2}$ alone does not
remove the perfect nesting (see Eq.(43)). Such resilience of the nesting is
indicative of the fragility of the JW/composite-fermi liquid state at
$\frac{1}{4}$-filling, and thus it might be a useful parent to understand the
competing confined broken symmetry states of the RK-like quantum dimer models
(e.g. columnar and resonant plaquette states).

Now for the case of the subspace of the QDM model which corresponds to a
quarter filling of the bands by the JW/composite fermions, there is a Ferrmi
surface that consists of straight lines that are perfectly nested by $(\pi,0)$
and $(0,\pi)$ vectors (see Figs. 17,19). This indicates that such putative
composite Fermi liquid state would be highly unstable towards forming a state
which spontaneously breaks the lattice translational symmetry and gaps the
Fermi surface. This perfect nesting occurs only for the strict nearest
neighbor mean-field Hamiltonian, and therefore can be removed by adding longer
range hoppings which are allowed by the extended projective symmetry
implementations under consideration ($\Theta_{x},\Theta_{y}$ from Table 5). To
illustrate this, we consider the further neighbor hoppings depicted in Fig.18.
One can show that the second neighbor hopping, denoted by $t^{\prime}$ and
depicted by blue arrows in Fig.18, vanishes for the $\Theta_{x},\Theta_{y}$
symmetry implementations. The further neighbor hopping denoted by $t_{2}$ and
depicted by green arrows in Fig.18 is allowed by $\Theta_{x},\Theta_{y}$
symmetry implementations, and it leads to the following sublattice-diagonal
entries to the mean-field Hamiltonian:

${\Delta h_{2}}({\bf q})_{ab}\sim
t_{2}[\cos(q_{1}+q_{2})+\cos(q_{1}-q_{2})]\delta_{ab}.$ (43)

However, since $\cos(q+\pi/2)+\cos(q-\pi/2)=0$, the above correction vanishes
exactly along the lines that define the nested Fermi surface (see Figs.17 and
19), and therefore does not remove the perfect nesting. Nevertheless there are
symmetry-allowed hoppings that lifts the nesting. One of them is denoted by
$t_{4}$ and depicted in Fig.18 by the red arrows. This hopping adds the
following sublattice-diagonal entries to the mean-field Hamiltonian:

$\Delta h_{4}({\bf q})_{ab}\sim
t_{4}[\cos(2q_{1}+2q_{2})+\cos(2q_{1}-2q_{2})]\delta_{ab}.$

Figure 19 illustrates how the perfect nesting is destroyed as $t_{4}$
increases relative to $t$, leading to a composite Fermi liquid state with two
Fermi surfaces centered around $(\pi,0)$ and $(0,\pi)$. The above illustrates
that a composite fermi liquid state at $\frac{1}{4}$-filling could be in
principle a true stable spin liquid ground state. However the strong
resilience of the nesting to near neighbor hopping corrections is indicative
that the Fermi surface has a strong tendencies to be gapped out and destroyed
via instabilities of composite-fermion particle-hole pair condensation with
finite crystal momentum, leading to ordinary confined states with
spontaneously broken lattice translational symmetries, such as the columnar,
staggered and resonant plaquette phases that are believed to be typically
realized for RK Hamiltonians of quantum dimer models.

Figure 19: Fermi Surfaces at quarter-filling for various values of $t_{4}/t$,
illustrating the lifting the perfect nesting of the JW/composite-fermi surface
associated with the QDM model, by adding this further-neighbor hopping.

We would like to close this subsection by noting that our mean-field states
associated with the $\Theta_{x},\Theta_{y}$ extended projective symmetry
groups have a resemblance to the classic $\pi$-flux state of standard
Abrikosov-Schwinger fermions introduced in Refs. [51, 57]. In fact from
Fig.15, we see that the fermions are hopping around every plaquette of the
original square lattice (which are now subdivided into vertices and plaquettes
of the “spin-ice” lattice) accumulating a phase $\pi$ over the closed loop.
There are, however, several crucial physical differences with the classic
$\pi$-flux state of Abrikosov-Schwinger fermions. First, the classic
$\pi$-flux state is a spin singlet in which each spin specie of Abrikosov-
Schwinger fermions has the same hoppings in the square lattice, whereas in our
construction the JW/composite-fermions are spin-less with only one fermion
specie hopping around the plaquette, in a state that is not a spin
singlet111111The RK Hamiltonian of quantum spin-ice is highly anisotropic in
spin space and far from having SU(2) symmetry.. More fundamentally, an onsite
U(1) transformation such as the $V_{\frac{1}{2}}$ transformation defined in
Fig.14, would be an element of the UV parton gauge of Abrikosov-Schwinger
fermions and therefore, the analogue of the $\Theta_{x},\Theta_{y}$ projective
symmetry groups would be two physically equivalent states for the classic
$\pi$-flux state of Abrikosov-Schwinger fermions. Generally speaking
symmetries are much more constraining for the JW/composite-fermions relative
to Abrikosov-Schwinger fermions, as they fix the phases of hopping and
different phase might lead to physically distinct states.

Nevertheless, the fact that our mean-field hamiltonians of JW/composite-
fermions can be viewed as states with $\pi$-flux in each plaquette of the
original square lattice is useful for understanding the properties of the
mean-field states. For example, for a $\pi$-flux mean-field state there exists
an intra-unit cell magnetic translation that is not part of the spin-ice
Bravais lattice, which can be taken to be a translation by
$(\textbf{R}_{1}+\textbf{R}_{2})/2$ (see Fig.15). This magnetic translation
would anti-commute with the ordinary elementary translations along either of
the two basis vectors of the Bravais lattice $\textbf{R}_{1},\textbf{R}_{2}$,
because the parallelogram spanned by $\textbf{R}_{1}$ and
$(\textbf{R}_{1}+\textbf{R}_{2})/2$ encloses $\pi$-flux, and similarly for the
parallelogram spanned by $\textbf{R}_{2}$ and
$(\textbf{R}_{1}+\textbf{R}_{2})/2$. As a consequence this magnetic
translation boosts the standard crystal momentum by
$(q_{1},q_{2})\rightarrow(q_{1}+\pi,q_{2}+\pi)$, and this explains why the
mean-field fermion dispersions that we have found display this translational
symmetry in momentum space (see Figs.17,18). However, while this magnetic
translation by $(\textbf{R}_{1}+\textbf{R}_{2})/2$ is a symmetry of the
unprojected mean-field Hamiltonian, this cannot be a symmetry of the
microscopic RK-model of quantum spin-ice, because a translation by
$(\textbf{R}_{1}+\textbf{R}_{2})/2$ would map spin-ice vertices onto spin-ice
plaquettes, which are clearly distinct in the RK model, and in any typical
model with the same spin-ice rules, since the ice rules themselves are
incompatible with a symmetry that would exchange vertices and plaquettes
(except for trivial models without quantum fluctuations). However this
symmetry of the bare-unprojected mean-field state would not be present for the
full physical trial state obtained after the spin-ice Gutzwiller projection,
because the Gutzwiller projection operator from Eq.(21) is not invariant
unders such translation by $(\textbf{R}_{1}+\textbf{R}_{2})/2$, since it is
defined by projecting onto the spin-ice rules associated with the vertices. As
we will see the effective Hamiltonian capturing the gauge field fluctuations
that we will discuss in the next section, in fact does not have any associated
translational symmetry by $(\textbf{R}_{1}+\textbf{R}_{2})/2$, and thus this
symmetry of the bare mean-field state will be lifted by gauge fluctuations.

##  III Gauge field fluctuations and effective low energy continuum field
theory

The Gutzwiller projection is a non-trivial operation that substantially
changes the character of the un-projected mean-field state. Computing
analytically the properties of the projected state is a however a highly non-
trivial task. In a sense, this projection can be viewed as giving rise to the
appearance of strong gauge field fluctuations around the mean-field state [21,
49, 74], and, accounting for such fluctuations is necessary to capture, even
qualitatively, the correct behavior of the phase of matter in question at low
energies.

The mean-field description introduced in the previous section still conceals
the emergence of low-energy dynamical gauge fields which can be viewed as
arising from fluctuations of the hopping amplitudes of the mean-field state in
Eq. (20). While a desciption of these gauge field fluctuations is often
performed by enforcing constraints and performing saddle point expansions in a
path integral representation (see e.g. Ref.[79, 80]), we will device here a
more phenomenological approach to infer the field content and emergent gauge
structure of the low energy field theory that describes the phase of matter
for the itinerant liquids of JW/composite-Fermions associated with the mean
field states constructed in the previous section.

We will include only the fluctuations of the phases of the complex hoppings
$t(\textbf{r},\textbf{r}^{\prime})$ of the mean field Hamiltonian (see
Eq.(20)) but not the fluctuations their amplitudes, because we assume that the
latter can be viewed as being gapped and thus not important at low energies.
To capture the fluctuations of such phases, we introduce additional bosonic
degrees of freedom associated with the non-zero hopping,
$t(\textbf{r},\textbf{r}^{\prime})$, of the mean field state from Eq.(20),
that connect a pair of fermion lattice sites ${\bf r},\textbf{r}^{\prime}$. We
denote the deviation of the phase from its mean field value by
$A(\bf{r},\bf{r}^{\prime})$, and we promote the mean field Hamiltonian to a
new Hamiltonian including this phase fluctuation variables as follows:

$\displaystyle H[t]$ $\displaystyle\mapsto H[t,A],$ (44) $\displaystyle
t(\textbf{r},\textbf{r}^{\prime})f^{{\dagger}}(\textbf{r})f(\textbf{r}^{\prime})$
$\displaystyle\to
t(\textbf{r},\textbf{r}^{\prime})f^{{\dagger}}(\textbf{r})e^{iA(\textbf{r},\textbf{r}^{\prime})}f(\textbf{r}^{\prime}).$

The scalar phase $A(\textbf{r},\textbf{r}^{\prime})$ can be interpreted as a
lattice version of $\int_{\textbf{r}^{\prime}}^{\textbf{r}}\textbf{A}\cdot
d\textbf{x}$. Notice that hermiticity demands that
$A(\textbf{r},\textbf{r}^{\prime})=-A(\textbf{r},\textbf{r}^{\prime})$ and
$t(\textbf{r},\textbf{r}^{\prime})=t^{*}(\textbf{r}^{\prime},\textbf{r})$. The
above Hamiltonian describes the coupling of the matter fields to the gauge
fields, and therefore we need to provide another Hamiltonian for the “pure”
gauge field sector. This Hamiltonian can be otained by demanding invariance
under a generalized version of lattice UV gauge structure and simple symmetry
considerations. In the next section III.1, we will review this construction
first for the case of usual Abrikosov-Schwinger partons and subsequently in
section III.1 we will apply it to the case of the extended parton
constructions for quantum spin-ice.

###  III.1 Review of Gauge field fluctuations for $U(1)$ spin liquids from
standard parton constructions

In this section we will derive the effective field theory governing a U(1)
spin liquid associated with the standard Abrikosov-Schwinger parton mean field
states (see section 51). The conclusion in this section will be simple and
well established before, namely, when the spin-liquid state associated with a
given mean-field parton state is stable, the low-energy de-confined gauge
structure will be given by the invariant gauge group (IGG) [49, 74]. We will
illustrate this for a mean-field parton state with a global U(1) particle-
conservation symmetry, and thus a U(1) IGG, leading, therefore, to a low-
energy U(1) gauge group minimally coupled to the parton fermions (i.e. an
standard U(1) spin-liquid). We wish, however, here to rederive these results
in what is hopefully a more conceptually intuitive construction, so that we
can use it as a template of reasoning for deriving the new results of the
emergent low energy gauge structure of our extended parton constructions of
JW/composite-fermion states in the next section.

We begin by promoting the phases of the hoppings into dynamical degrees of
freedom and the mean field Hamiltonian from Eq.(18), into the following
Hamiltonian capturing the field matter coupling:

$H[t,A]\doteq\sum_{s,s^{\prime}}\sum_{\textbf{r},\textbf{r}^{\prime}}t_{ss^{\prime}}(\textbf{r},\textbf{r}^{\prime})e^{iA(\textbf{r},\textbf{r}^{\prime})}f^{{\dagger}}_{s}(\textbf{r})f_{s^{\prime}}(\textbf{r}^{\prime}),$
(45)

Here $A(\bf{r},\bf{r}^{\prime})$ is viewed as a dynamical compact periodic
phase taking values $A(\textbf{r},\textbf{r}^{\prime})\in[0,2\pi)$. We would
like now to define an extension of the local UV parton gauge symmetry, but
which acts not only on the fermions but also on the dynamical gauge fields
$A(\bf{r},\bf{r}^{\prime})$. As discussed in section 51, the local U(1)
transformations of the parton gauge group are generated by the local fermion
occupations $n(\textbf{r})$, which transform the fermion bilinears as (see
Eq.(18)):

$f^{{\dagger}}_{s}(\textbf{r})f_{s}(\textbf{r}^{\prime})\xrightarrow[]{\text{Gauge}}e^{-i[\theta(\textbf{r})-\theta(\textbf{r}^{\prime})]}f^{{\dagger}}_{s}(\textbf{r})f_{s}(\textbf{r}^{\prime})$
(46)

Therefore, in order to leave the Hamiltonian from Eq. (45) invariant, we
demand that these transformations act on the dynamical phase gauge degrees of
freedom as follows:

$A(\textbf{r},\textbf{r}^{\prime})\xrightarrow[]{\text{Gauge}}A(\textbf{r},\textbf{r}^{\prime})+\theta(\textbf{r})-\theta(\textbf{r}^{\prime})$
(47)

For simplicity, from now on we will assume that the hoppings only connect
nearest neighbour sites r and $\textbf{r}^{\prime}=\textbf{r}+\textbf{e}_{i}$
(with $i=x,y$) and we will label the bond connecting them by $(\textbf{r},i)$,
and the gauge fields by $A({\bf r},i)$. To implement the transformation from
Eq.(47) quantum-mechanically, we introduce a canonically conjugate variable to
the vector potentials denoted by $E({\bf r},i)$, and take these variables to
satisfy the following commutation relations:

$\begin{cases}[A(\textbf{x},i),E(\textbf{y},j)]&=-i\delta_{\textbf{x},\textbf{y}}\delta_{ij}\\\
[E(\textbf{x},i),E(\textbf{y},j)]&=0\\\
[A(\textbf{x},i),A(\textbf{y},j)]&=0\\\ \end{cases}$ (48)

Since $A(\textbf{x},i)$ is an angle, $E(\textbf{x},i)$ is an angular momentum
with integer-valued spectrum. It is then easy to show, that the generalized UV
gauge transformations acting on matter and dynamical phase gauge fields are
generated by exponentials of the following operators:

$\displaystyle G(\textbf{r})$ $\displaystyle=n(\textbf{r})-\nabla\cdot
E(\textbf{r}).$ (49)

where:

$\nabla\cdot
E(\textbf{r})=E(\textbf{r},x)+E(\textbf{r},y)-E(\textbf{r}-\textbf{e}_{x},x)-E(\textbf{r}-\textbf{e}_{y},y)$
(50)

We will demand that the combined effective Hamiltonian of matter and gauge
fields is invariant under the above local gauge group, and we will interpret
then the values of $G(\textbf{r})$ as a constraint that can be consistently
imposed on the states in order to represent the subspace of physical interest.
The subspace of physical interest will be that for which $G(\textbf{r})=0$ for
all r, and therefore this constraint can be viewed as a lattice version of
Gauss’s law (see Fig.20 for a depiction).

Figure 20: Depiction of generator of generalized gauge transformations (see
Eq.(49)) acting on the matter (residing on blue sites) and vector potentials
(residing on links), relevant for the emergent lattice U(1) gauge theory of
standard Abrikosov-Schwinger partons.

Let us now determine the simplest operators that are made only from gauge
fields which commute with every $G(\textbf{r})$. It is easy to verify that one
of them is the magnetic field operator associated with the curl of $A$ around
a plaquette:

$B(\textbf{r})\doteq
A(\textbf{r},x)+A(\textbf{r}+\textbf{e}_{x},y)-A(\textbf{r}+\textbf{e}_{y},x)-A(\textbf{r},y)$
(51)

Here we view the plaquette of interest as being northeast from lattice site r,
and thus we are using this as a label of the plaquette as well. The
canonically conjugated variable to $B(\textbf{r})$ can be shown to be the
lattice curl of $E$:

$\nabla\times E(\textbf{r})\doteq
E(\textbf{r},x)+E(\textbf{r}+\textbf{e}_{x},y)-E(\textbf{r}+\textbf{e}_{y},x)-E(\textbf{r},y)$

Figure 21: The solid black lines depict the sum of vector potentials that
enter the definition of the magnetic flux operator
$B(\textbf{r}-\textbf{e}_{x})$ (from Eq.(51)). The orange lines depict the
sums of electric fields that enter the divergence operator (from Eq.(50)). The
fact that these two operators commute can be visualized by noting that the
number of segments in which parallel black and orange arrows overlap equals
the number of segments in which anti-parallel arrows overlap.

Notice that at any site r, $\nabla\times E(\textbf{r})$ and $(\nabla\cdot
E)(\textbf{r})$ are two indipendent degrees of freedom. Following an analogous
reasoning to the one we did to define the action of Gauge transformations on
gauge fields, we extend the symmetries in 4 onto gauge fields by requiring
that the interaction hamiltonian (46) remains invariant. Importantly, the
action of symmetries on gauge fields is independent of the specific extended
projective symmetry group implementation for the fermionic matter, because the
“projective” factors are already fully taken into account in the fermion
transformation rules and the choice of mean-field hopping amplitudes.
Moreover, under space transformations, the vector potential $A(\textbf{r},i)$
transforms as a vector directed along the bond $(\textbf{r},i)$. Its
transformation under time reversal, $\Theta$, can be fixed by demanding that
the exponent in Eq.(45) is left invariant:

$e^{iA(\textbf{r},i)}=\Theta e^{iA(\textbf{r},i)}\Theta^{-1}=e^{-i\Theta
A(\textbf{r},i)\Theta^{-1}},$ (52)

thus, $\Theta A(\textbf{r},i)\Theta^{-1}=-A(\textbf{r},i)$.

So far we have kept track of the compactification of the $A$ field. When the
low energy phase is deconfined, it is appropriate to simplify the description
by neglecting the compactification and view the fields $A$ as taking values on
the real axis. With this simplification and after enforcing the symmetries it
is easy to see that the simplest Hamiltonian that is bilinear in the local
gauge-invariant fields, $E$ and $B$, is the standard Maxwell Hamiltonian in
the lattice, given by:

$H_{\text{Gauge}}=\frac{\epsilon}{2}\sum_{\textbf{r}}(E^{2}(\textbf{r},x)+E^{2}(\textbf{r},y))+\frac{1}{2\mu}\sum_{\textbf{r}}B^{2}(\textbf{r}).$
(53)

Where $\epsilon$ and $\mu$ are constants. The above Hamiltonian can be
diagonalized in terms of “normal modes” of the pure-gauge in the absence of
coupling to fermionic matter. Since we have two independent scalar degrees of
freedom per unit cell, associated with $A(\textbf{r},x)$ and
$A(\textbf{r},y)$, but one non-dynamical constraint per unit cell (since
$\nabla\cdot E(\textbf{r})$ commutes with $H$), there is only one truly
dynamical Harmonic oscillator degree of freedom per unit cell, associated with
the magnetic field $B(\textbf{r})$. Its equations of motion can be determined
easily from the Hamiltonian using the commutators from eq.(48), and read as
follows:

$\displaystyle\frac{dB(\textbf{r})}{dt}$ $\displaystyle=-\nabla\times
E(\textbf{r})$ (54) $\displaystyle\frac{d}{dt}\nabla\times E(\textbf{r})$
$\displaystyle=\frac{4}{\mu\epsilon}B(\textbf{r})-\frac{1}{\mu\epsilon}\sum_{\bm{\xi}=\pm\textbf{e}_{x}\atop\bm{\xi}=\pm\textbf{e}_{y}}B(\textbf{r}+\bm{\xi}).$

The above can be solved by expanding fields in crystal momentum basis (Fourier
transform), to obtain:

$\frac{d^{2}B(\textbf{q})}{dt^{2}}+\frac{1}{\mu\epsilon}[4-2\cos(q_{x})-2\cos(q_{y})]B(\textbf{q})=0,$
(55)

and therefore the dispersion of the normal modes is (illustrated in Fig.22):

$\omega^{2}(\textbf{q})=\frac{1}{\epsilon\mu}[4-2\cos(q_{x})-2\cos(q_{y})].$
(56)

Figure 22: Left: Dispersion relation of the standard emergent photon of a U(1)
spin liquid of Abrikosov-Schwinger fermions, from Eq.(56). Right: Cut of the
dispersion relations along $q_{y}=0$, with the dashed line illustrating the
linearized photon dispersion near $(q_{x},q_{y})=(0,0)$.

The above dispersion features a linearly dispersing photon-like mode centered
at momentum $(q_{x},q_{y})=(0,0)$ with a speed of (see Fig.22):

$v=\frac{1}{\sqrt{\mu\epsilon}}.$

This photon would be minimally coupled to the fermionic matter through
Eq.(44). We see, therefore, that our phenomelogical procedure is able to
describe the low energy field content expected at low energies for a $U(1)$
spin liquid associated with the standard Abrikosov-Schwinger parton
construction [49, 74]. Let us pause to consider what protects the gaplessness
of this photon mode?. Once deconfinement is presumed, so that it is valid to
replace vector potentials by continuum real-valued variables, the lattice
Faraday law from Eq.(54), can be re-interpreted as a continuity equation:

$\frac{\partial B}{\partial t}(\textbf{r},t)+\nabla\cdot\varepsilon=0,$ (57)

where $\varepsilon$ is a dual electric field. It is a rotated version of the
previously defined electric field, so that its lattice divergence is centered
on the plaquettes, and is defined as:

$\varepsilon_{i}=\epsilon_{ij}E_{j},$

where $\epsilon_{ij}$ is the 2D Levi-civita symbol. The photon can be viewed
as a Goldstone mode of a spontaneously broken global U(1) 1-form symmetry
associated with the conservation of magnetic flux, as it is usually discussed
in boson-vortex dualities in 2+1D [81, 82, 83]. In the absence of gapless
fermionic matter and due compact nature of the gauge fields, the above photon
would ultimately become gapped at low energies due to Polyakov confinement
[84], because the global conservation law of magnetic flux would be explicitly
broken by fluctuations associated with local magnetic flux creation and
destruction events.

###  III.2 Gauge field fluctuations for $U(1)$ spin liquids from extended
parton constructions in 2D quantum spin-ice

Let us now generalize the previous construction to try to elucidate the low
energy emergent gauge structure associated with the extended parton composite
fermi liquid states of quantum spin-ice models discussed in Sec.II.3. Just as
we did for the Abrikosov-Schwinger fermions, we begin by writing the mean
field Hamiltonian of the composite fermions and introduce a real-valued
variable that captures the fluctuations of the phase of the hoping amplitude
connecting a pair of fermion sites $(\bf{r},\bf{r}^{\prime})$ and denote it by
$A(\bf{r},\bf{r}^{\prime})$. For concreteness we will focus on the
fluctuations of the mean-field states described in Sec. II.3.4 which had non-
zero hoppings only for $(\textbf{r},\textbf{r}^{\prime})$ being nearest
neighbour sites, so that the reasulting mean field Hamiltonian, analogously to
(45), reads as:

$\displaystyle H(t,A)=\sum_{\textbf{R}}$ $\displaystyle
it^{*}e^{-iA_{1}(\textbf{R}+\textbf{R}_{2})}f^{{\dagger}}_{a}(\textbf{R}-\textbf{R}_{1}+\textbf{R}_{2})f_{b}(\textbf{R})+it^{*}e^{iA_{3}(\textbf{R})}f^{{\dagger}}_{a}(\textbf{R})f_{b}(\textbf{R})$
(58) $\displaystyle+$ $\displaystyle
te^{-iA_{4}(\textbf{R})}f^{{\dagger}}_{a}(\textbf{R}-{\textbf{R}_{1}})f_{b}(\textbf{R})+te^{iA_{2}(\textbf{R}+\textbf{R}_{2})}f^{{\dagger}}_{a}(\textbf{R}+{\textbf{R}_{2}})f_{b}(\textbf{R})+h.c.$

where the convention for the labelling of Gauge fields is depicted in Fig.23.
As before we promote the above phases into angular quantum-rotor bosonic
degrees of freedom, with an associated canonically conjugate degree of freedom
denoted by $E(\textbf{r},\textbf{r}^{\prime})$, with the same commutation
relations described in Eqs.(48). However, the first crucial difference that
appears for the extended partons is that the UV $U(1)$ gauge transformations
are not acting as in Eq.(46) for the Abrikosov-Schwinger fermions. Instead the
UV $U(1)$ gauge transformation are generated by the spin-ice charge operators
from Eqs.(7),(14), or equivalently by the total number of fermions in the
links connected to vertex ${\bf R}$, denoted by $n_{\text{ice}}(\textbf{R})$,
and which in Bravais lattice notation reads as (see Fig.23):

$n_{\text{ice}}(\textbf{R})=n_{a}(\textbf{R})+n_{b}(\textbf{R})+n_{a}(\textbf{R}-\textbf{R}_{2})+n_{b}(\textbf{R}-\textbf{R}_{1}).$
(59)

Therefor, the generator of the generalized lattice gauge transformations
analogous to the one from Eq.(49), that also acts on the dynamical phase
degrees of freedom, is a sum of the corresponding four generators from
Eq.(49), and is given by:

$\ G_{\text{ice}}(\textbf{R})\doteq n_{\text{ice}}(\textbf{R})-\sum_{{\bf
r}\in\textbf{R}}(\nabla\cdot E)(\textbf{r}),$ (60)

where $\textbf{r}\in\textbf{R}$ denotes the four spin sites that contribute to
the ice-rule associated to the vertex R, as depicted in Fig.23, and the
lattice divergence $(\nabla\cdot E)(\textbf{r})$ is defined in the same way as
in Eq.(50).

Figure 23: Bottom left: convention for labeling gauge fields. These reside at
the links (blue lines) that connect the JW/composite fermion sites (located at
the solid dots). Top right: depiction of the operators entering in the
generator of generalized spin-ice gauge transformations,
$n_{\text{ice}}(\textbf{R})$ from Eq.(59), which is ceneterd at the spin-ice
vertices.

As before we demand that $G_{\text{ice}}$ commutes with every term in the
Hamiltonian and interpret the physical Hilbert space as the one satisfying the
constraint $G_{\text{ice}}(\textbf{R})=0$ for every R, which can be re-written
as a Gauss law of the form:

$(\nabla\cdot E)_{\text{ice}}(\textbf{R})=n_{\text{ice}}(\textbf{R}),$ (61)

where $(\nabla\cdot E)_{\text{ice}}$ is given by (see Fig.24):

$(\nabla\cdot E)_{\text{ice}}(\textbf{R})=\sum_{{\bf
r}\in\textbf{R}}(\nabla\cdot E)(\textbf{r}).$ (62)

We can also write a canonically conjugate partner to the above gauge
constraint operator, given by:

$(\nabla\cdot A)_{\text{ice}}(\textbf{R})=\sum_{{\bf
r}\in\textbf{R}}(\nabla\cdot A)(\textbf{r}).$ (63)

Let us now construct the analogue of the Maxwell Hamiltonian from Eq.(53). To
do so, we need to find all the linearly independent gauge field operators that
commute with the gauge field part of the constraint operator
$G_{\text{ice}}(\textbf{R})$ from Eq.(60), namely with $(\nabla\cdot
E)_{\text{ice}}(\textbf{R})$ and its canonical partner $(\nabla\cdot
A)_{\text{ice}}(\textbf{R})$. Since the Bravais unit cell contains four scalar
vector potential degrees of freedom (see Fig.23), but there is one Gauss law
constraint per unit cell, we expect three independent harmonic oscillator
modes per cell and therefore three dynamical gauge field bands. To find a
basis for such modes, we notice that since $(\nabla\cdot
E)_{\text{ice}}(\textbf{R})$ is a sum of divergences from the previous section
on Abrikosov-Schiwnger fermions (see Eq.(50)), the gauge invariant operators
we discussed in the previous section, would also be gauge invariant in the new
spin-ice construction. These include the magnetic operators $B(\textbf{r})$
from Eq.(51), but now there are two such operators per spin-ice Bravais unit
cell, one associated with the spin-ice vertex and one with the spin-ice
plaquette, which we denote respectively by $B_{V}(\textbf{R})$,
$B_{P}(\textbf{R})$, and together with their canonically conjugate partners
are given by are explicitly given by (see Fig.23):

$\displaystyle B_{V}(\textbf{R})=$ $\displaystyle
A_{1}(\textbf{R})+A_{2}(\textbf{R})-A_{3}(\textbf{R})-A_{4}(\textbf{R}),$ (64)
$\displaystyle(\nabla\times E)_{V}(\textbf{R})=$ $\displaystyle
E_{1}(\textbf{R})+E_{2}(\textbf{R})-E_{3}(\textbf{R})-E_{4}(\textbf{R}),$
$\displaystyle B_{P}(\textbf{R})=$ $\displaystyle
A_{3}(\textbf{R}-\textbf{R}_{2})+A_{4}(\textbf{R}+\textbf{R}_{1}-\textbf{R}_{2})\cdots$
$\displaystyle-A_{1}(\textbf{R}+\textbf{R}_{1})-A_{2}(\textbf{R}),$
$\displaystyle(\nabla\times E)_{P}(\textbf{R})=$ $\displaystyle
E_{3}(\textbf{R}-\textbf{R}_{2})+E_{4}(\textbf{R}+\textbf{R}_{1}-\textbf{R}_{2})\cdots$
$\displaystyle-E_{1}(\textbf{R}+\textbf{R}_{1})-E_{2}(\textbf{R}),$

Figure 24: Left: depiction of generalized spin-ice electric field divergence
operator, $(\nabla\cdot E)_{\text{ice}}=\sum_{{\bf
r}\in\textbf{R}}(\nabla\cdot E)(\textbf{r})$ from Eqs.(60),(62). The red
arrows depict the convention for adding electric fields. Right: depiction of
spin-ice magnetic field operator $B_{\text{ice}}(\textbf{R}-\textbf{R}_{1})$
from Eq.(66). The blue arrows depict the convention for adding vector
potentials. The blue crosses mark the location of the spin-ice vertices.

where $B_{V}(\textbf{R})$ can viewed as a lattice curl centered around vertex
R and $B_{P}(\textbf{R})$ as a curl centered around the plaquette which is
neighboring to the right the vertex R (see Fig.25). However, there are certain
additional operators containing only gauge fields, that commute with every
$(\nabla\cdot E)_{\text{ice}}(\textbf{R})$, but which would not be gauge
invariant under the convention of previous section, namely they would not
commute with all the divergences of electric fields defined in Eq.(50). These
operators and their canonically conjugate partners (see Fig.25) can be taken
to be:

$\displaystyle B_{x}(\textbf{R})$
$\displaystyle=A_{3}(\textbf{R}-\textbf{R}_{2})-A_{1}(\textbf{R}+\textbf{R}_{1}),$
(65) $\displaystyle E_{x}(\textbf{R})$
$\displaystyle=E_{3}(\textbf{R}-\textbf{R}_{2})-E_{1}(\textbf{R}+\textbf{R}_{1}),$
$\displaystyle B_{y}(\textbf{R})$
$\displaystyle=A_{4}(\textbf{R}+\textbf{R}_{1}-\textbf{R}_{2})-A_{2}(\textbf{R}),$
$\displaystyle E_{y}(\textbf{R})$
$\displaystyle=E_{4}(\textbf{R}+\textbf{R}_{1}-\textbf{R}_{2})-E_{2}(\textbf{R}),$

Figure 25: Left: depiction of $B_{x}$ operators (from Eq.(65)), as pairs of
blue arrows. Notice that whenever the blue arrows (vector potentials) of a
$B_{x}$ operator overlap with the red arrows (electric fields) of a
$(\nabla\cdot E)_{\text{ice}}$ operator, there is always an equal number of
parallel and antiparallel arrows, illustrating that these operators commute.
Right: analgous depictions for the $B_{y}$ operators (from Eq.(65)). The blue
crosses mark the location of the spin-ice vertices.

where the $B_{x}(\textbf{R}),B_{y}(\textbf{R})$ fields can viewed as centered
around the plaquette of the spin-ice model which is neighboring to the right
the vertex R (see Fig.25). Notice that
$B_{P}(\textbf{R})=B_{x}(\textbf{R})+B_{y}(\textbf{R})$. Therefore, the set of
linearly independent dynamical fields could in principle be chosen to be
$B_{x}(\textbf{R}),B_{y}(\textbf{R}),B_{V}(\textbf{R})$. There is however a
much better choice of local gauge invariant fields that will highly simplify
the dynamics and the final physical picture. The idea is that instead of
$B_{V}(\textbf{R})$, we would like to construct a local magnetic field
strength that fits more naturally within the spin-ice gauge structure, which
we will denote by $B_{\text{ice}}(\textbf{R})$. This quantity and its
canonical partner can be chosen as follows:

$\displaystyle B_{\text{ice}}(\textbf{R})$
$\displaystyle=B_{x}(\textbf{R})+B_{x}(\textbf{R}+\textbf{R}_{1}+\textbf{R}_{2})+B_{y}(\textbf{R}+\textbf{R}_{2})+B_{y}(\textbf{R}+\textbf{R}_{1})+2B_{V}(\textbf{R}+\textbf{R}_{1}),$
(66) $\displaystyle(\nabla\times E)_{\text{ice}}(\textbf{R})$
$\displaystyle=E_{x}(\textbf{R})+E_{x}(\textbf{R}+\textbf{R}_{1}+\textbf{R}_{2})+E_{y}(\textbf{R}+\textbf{R}_{2})+E_{y}(\textbf{R}+\textbf{R}_{1})+2(\nabla\times
E)_{V}(\textbf{R}+\textbf{R}_{1}).$

Figure 24 illustrates the terms that enter into $B_{\text{ice}}(\textbf{R})$
making more clear why it has a natural interpretation of a spin-ice lattice
curl. Notice that $B_{\text{ice}}(\textbf{R})$ is naturally viewed as centered
around the vertex $\textbf{R}+\textbf{R}_{1}$ (see Fig.24), but it will be
convenient to keep its position label as R as we will see later on. The three
fields $B_{x}(\textbf{R}),B_{y}(\textbf{R}),B_{\text{ice}}(\textbf{R})$ and
their canonical conjugate partners
$E_{x}(\textbf{R}),E_{y}(\textbf{R}),(\nabla\times
E)_{\text{ice}}(\textbf{R})$, commute with the gauge constraint field,
$(\nabla\cdot E)_{\text{ice}}(\textbf{R})$, and its canonical partner,
$(\nabla\cdot A)_{\text{ice}}(\textbf{R})$, and thus form a basis for the
three independent modes of physical gauge fluctuations. The advantage of this
basis over the $B_{x}(\textbf{R}),B_{y}(\textbf{R}),B_{V}(\textbf{R})$ basis,
is that these fields form a sets of decoupled canonical coordinates, namely
their mutual commutators vanish:

$\displaystyle=[B_{x}(\textbf{R}),(\nabla\times
E)_{\text{ice}}(\textbf{R}^{\prime})]=0,$
$\displaystyle[B_{y}(\textbf{R}),E_{x}(\textbf{R}^{\prime})]$
$\displaystyle=[B_{y}(\textbf{R}),(\nabla\times
E)_{\text{ice}}(\textbf{R}^{\prime})]=0,$
$\displaystyle[B_{\text{ice}}(\textbf{R}),E_{x}(\textbf{R}^{\prime})]$
$\displaystyle=[B_{\text{ice}}(\textbf{R}),E_{y}(\textbf{R}^{\prime})]=0.$

The action of the microscopic lattice space symmetries on these fields is the
same as in the case of Abrikosov-Schwinger fermions, and the additional pure
gauge group transformation that enter into the extended projective symmetry
group implementation on the fermions do not affect the gauge fields, therefore
the fields $A_{i}(\textbf{R})$ transform as ordinary vectors according to the
directions specified by the sites they connect, which is depicted in Fig.23.
From this the transformations of dynamical fields under space symmetries
follow easily. The action of time reversal ($\Theta$ in Table 2) can be also
inferred analogously to Eq.(52), and one concludes that:

$\displaystyle\Theta A_{i}(\textbf{R})\Theta^{-1}$
$\displaystyle=-A_{i}(\textbf{R})$ (67) $\displaystyle\Theta
E_{i}(\textbf{R})\Theta^{-1}$ $\displaystyle=E_{i}(\textbf{R})$

where $i=1,2,3,4$ are the components depicted in Fig.23, and the
transformations of $E_{i}(\textbf{R})$ can be inferred from its canonical
commutator with $A_{i}(\textbf{R})$. Let us now consider the action of the
microscopic particle-hole conjugation of hard-core bosons, denoted by $X$ (see
Table 2). From its action on JW/composite-fermions (see Eq.(35)) we obtain
that the phases dressing the mean-field Hamiltonian should transform as:

$\displaystyle Xe^{iA(\textbf{r},\textbf{r}^{\prime})}X^{\dagger}$
$\displaystyle=e^{iA(\textbf{r}^{\prime},\textbf{r})}=e^{-iA(\textbf{r},\textbf{r}^{\prime})}$
(68)

where we used that
$A(\textbf{r}^{\prime},\textbf{r})=-A(\textbf{r},\textbf{r}^{\prime})$
(hermiticity). Therefore the fields transform as:

$\displaystyle XA_{i}(\textbf{R})X^{\dagger}$
$\displaystyle=-A_{i}(\textbf{R})$ (69) $\displaystyle
XE_{i}(\textbf{R})X^{\dagger}$ $\displaystyle=-E_{i}(\textbf{R})$

It is interesting to note that under the natural microscopic time-reversal
symmetry of spin-$\frac{1}{2}$ denoted by $\mathcal{T}$ (see Sec.II.3.3), it
follows from Eq.(33) and Eqs.(67),(70) that the gauge fields transform as:

$\displaystyle\mathcal{T}A_{i}(\textbf{R})\mathcal{T}^{-1}$
$\displaystyle=A_{i}(\textbf{R})$ (70)
$\displaystyle\mathcal{T}E_{i}(\textbf{R})\mathcal{T}^{-1}$
$\displaystyle=-E_{i}(\textbf{R})$

and therefore, interestingly, all the magnetic fields,
$B_{x}(\textbf{R}),B_{y}(\textbf{R}),B_{\text{ice}}(\textbf{R})$, are even and
the electric fields are odd under this time-reversal, which is opposite to the
standard situation in QED. This is a manifestation of the psedo-scalar
transformation of the JW/composite-fermions under this symmetry, as discussed
in Sec.II.3.3 and Ref.[77]. Similar considerations also apply to other space
symmetries such as mirrors, which in order to be implemented as natural
spin-$\frac{1}{2}$ symmetries need to be dressed by the hard-core boson
particle-hole conjugation $X$, which would lead to transformations on gauge
fields opposite to those of ordinary QED (e.g. the electric field transforming
as a pseudo-vector under mirrors).

We are now in a position to write a simple bilinear Maxwell-like model
Hamiltonian for the pure gauge field part invariant under all microscopic
symmetries of the RK model, which we write as:

$\displaystyle H_{\rm
Gauge}=\frac{\epsilon}{2}\sum_{\textbf{R}}\sum_{i=1}^{4}E_{i}^{2}(\textbf{R})+\frac{\chi_{B}}{2}\sum_{\textbf{R}}B_{\text{ice}}^{2}(\textbf{R})+\frac{\chi_{P}}{2}\sum_{\textbf{R}}(B_{x}^{2}(\textbf{R})+B_{y}^{2}(\textbf{R}))$
(71)

Here we have ignored again for simplicity the compactification of gauge
fields, and $\epsilon,\chi_{B},\chi_{P}$ are phenomenological coupling
constants. The equations of motion for the Hamiltonian from Eq.(71) are:

$\displaystyle\frac{d^{2}B_{x}}{dt^{2}}(\textbf{R})=$
$\displaystyle-2\frac{\chi_{P}}{\epsilon}B_{x}(\textbf{R}),$ (72)
$\displaystyle\frac{d^{2}B_{y}}{dt^{2}}(\textbf{R})=$
$\displaystyle-2\frac{\chi_{P}}{\epsilon}B_{y}(\textbf{R}),$
$\displaystyle\frac{d^{2}B_{\text{ice}}}{dt^{2}}(\textbf{R})=$
$\displaystyle-\frac{2\chi_{B}}{\epsilon}\big{[}4B_{\text{ice}}(\textbf{R})-\sum_{\bm{\xi}=\pm\textbf{R}_{1}\atop\bm{\mu}=\pm\textbf{R}_{2}}B_{\text{ice}}\big{(}\textbf{R}+\bm{\xi}+\bm{\mu}\big{)}\big{]}.$

Which in crystal momentum basis reduce to:

$\displaystyle\frac{d^{2}B_{x}}{dt^{2}}(\textbf{q})$
$\displaystyle=-\frac{2\chi_{P}}{\epsilon}B_{x}(\textbf{q}),$ (73)
$\displaystyle\frac{d^{2}B_{y}}{dt^{2}}(\textbf{q})$
$\displaystyle=-\frac{2\chi_{P}}{\epsilon}B_{y}(\textbf{q}),$
$\displaystyle\frac{d^{2}B_{\text{ice}}}{dt^{2}}(\textbf{q})$
$\displaystyle=-\omega^{2}(\textbf{q})B_{\text{ice}}(\textbf{q}),$

where:

$\omega^{2}(\textbf{q})\doteq\frac{4\chi_{B}}{\epsilon}\big{[}2-\cos\big{(}{q}_{1}+{q}_{2}\big{)}-\cos\big{(}{q}_{1}-{q}_{2}\big{)}\big{]}$
(74)

and $q_{i}\doteq\textbf{q}\cdot\textbf{R}_{i}$ $i=1,2$. The dispersion
relations show features that are crucially different with respect to the case
of usual lattice QED and the emergent gauge fields discussed in Sec.III.1 in
the context of Abrikosov-Schwinger fermions. The $B_{x/y}$ modes display a
fully gapped and dispersive-less flat band with energy
$\sqrt{2\chi_{P}/\epsilon}$, as depicted in Fig. 26. While the exact flatness
is a consequence of our simple model, the fact that these modes are gapped is
a generic feature. Therefore the fluctuations associated with these modes are
expected to be irrelevant at low energies, and they can be safely neglected
from the low energy effective theory. On the other hand the $B_{\text{ice}}$
mode displays two distinct gapless photon-like (74) linearly dispersing modes
centered around $\textbf{q}=(0,0)$ and $\textbf{q}=(\pi,\pi)$ as depicted in
Fig.26.

To close this subsection, we would like to remark that the Maxwell-like
Hamiltonian of Eq.(71) does not have the half-translational symmetry by
$(\textbf{R}_{1}+\textbf{R}_{2})/2$, that we encountered in the bare mean-
field fermion Hamiltonian. This can be easily seen by noting that this
symmetry would exchange the spin-ice vertices with the spin-ice plaquettes.
However the $B_{x}(\textbf{R}),B_{y}^{2}(\textbf{R})$ fields are only centered
around the spin-ice plaquettes, whereas the fields
$B_{\text{ice}}(\textbf{R})$ are centered only around vertices, and therefore
clearly the Hamiltonian of Eq.(71) has no symmetry relating spin-ice vertices
and spin-ice plaquettes. The apparent translational symmetry in momentum space
by $(\pi,\pi)$ of the pure gauge field modes that we see in Fig.26, is a
result of fine tunning of the model, which we have done for simplicity. For
example gauge invariant terms can be easily added to the Maxwell-Hamiltonian
that would delocalize the $B_{x}(\textbf{R}),B_{y}(\textbf{R})$ modes and make
them itinerant and with dispersions that would have different energies near
$(0,0)$ vs near $(\pi,\pi)$. Similarly it is possible to add gauge invariant
terms to the Hamiltonian that would make the photons originating from
fluctuations of $B_{\text{ice}}(\textbf{R})$ to have different speeds near
$(0,0)$ vs near $(\pi,\pi)$. Therefore, the full theory of fermions coupled to
gauge field fluctuations does not have the translational symmetry by
$(\textbf{R}_{1}+\textbf{R}_{2})/2$ that we saw in the bare mean-field fermion
Hamiltonian, reflecting the fact that this is not a true microscopic symmetry
of the underlying RK Hamiltonian of spin-ice.

Figure 26: Dispersion relations of the gauge fields obtained from the pure
Maxwell-like Hamiltonian (i.e. ignoring coupling to matter fermions), along a
cut with $q_{1}=q_{2}$. There are two linearly dispersing photon-like modes
near $(q_{1},q_{2})=(0,0)$ and $(q_{1},q_{2})=(\pi,\pi)$, with their
dispersion shown in blue obtained from Eq.(74) (the dashed lines are
linearized approximations of the photon dispersions). There are also two fully
gapped gauge fluctuation modes associated with the $B_{x},B_{y}$ fields (red
lines), which are localized modes and hence have strictly flat dispersions for
the ideal Maxwell-like Hamiltonian from Eq.(71).

###  III.3 Gauge field and matter couplings, low energy effective field theory
and dipolar nature of composite fermions

Let us now determine the matter coupling to the low energy gauge fields and
the low-energy effective field theory. For concreteness we will focus on the
case of gapless Dirac fermions obtained for the six-vertex subspace, but
similar considerations would apply to the Fermi surface state of the quantum
dimer model. Ignoring compactification and expanding Eq. (58) up to linear
order on the vector potentials, we obtain the following:

$\displaystyle H(t,A)$
$\displaystyle=\sum_{\textbf{R}}it^{*}(1-iA_{1}(\textbf{R}+\textbf{R}_{2}))f^{{\dagger}}_{a}(\textbf{R}-\textbf{R}_{1}+\textbf{R}_{2})f_{b}(\textbf{R})+it^{*}(1+iA_{3}(\textbf{R}))f^{{\dagger}}_{a}(\textbf{R})f_{b}(\textbf{R})$
(75)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+t(1-iA_{4}(\textbf{R}))f^{{\dagger}}_{a}(\textbf{R}-{\textbf{R}_{1}})f_{b}(\textbf{R})+t(1+iA_{2}(\textbf{R}+\textbf{R}_{2}))f^{{\dagger}}_{a}(\textbf{R}+{\textbf{R}_{2}})f_{b}(\textbf{R})+h.c+O(A^{2})$

As discussed in Sec. II.3.4, the fermions have gapless Dirac nodes at the two
valleys $(\pi,0)$ and $(0,\pi)$ while the gauge field has gapless photon-like
modes at $(0,0)$ and $(\pi,\pi)$. Therefore we expect that the dominant
effects at low energy include:

  1. 1.

Intra-valley scattering within each Dirac cone mediated by exchange of long-
wavelength gauge field fluctuations with momenta near $(0,0)$.

  2. 2.

Inter-valley scattering process connecting the two Dirac cones mediated by the
exchange of gauge fluctuations with momenta near $(\pi,\pi)$.

These two kind of processes are depicted in Fig.27. Therefore, in the spirit
of $k\cdot p$ theory, we define the following fields by expanding the fermion
and gauge fields around their different respective gapless points:

$\displaystyle\Psi(\textbf{q})$
$\displaystyle\doteq\begin{pmatrix}f_{a}(\textbf{p}+(\pi,0))\\\
f_{b}(\textbf{p}+(\pi,0))\\\ f_{a}(\textbf{p}+(0,\pi))\\\
f_{b}(\textbf{p}+(0,\pi))\end{pmatrix}$ $\displaystyle A^{0}_{j}(\textbf{p})$
$\displaystyle\doteq
A_{j}(\textbf{p})\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,j\in\\{1,2,3,4\\}$
$\displaystyle A^{\pi}_{j}(\textbf{p})$ $\displaystyle\doteq
A_{j}(\textbf{p}+(\pi,\pi))\,\,\,\,\,j\in\\{1,2,3,4\\}$

where p is understood to be “small” with respect to the size of the Brillouin
zone, so that we can expand the hamiltonian (75) to the first order (for the
convention sublattice indices see Fig.23). Details of the derivations of the
small momentum expansion can be found in appendix C, we will here summarize
the final results next.

$(-\pi,0)$$(0,\pi)$$q_{1}$$q_{2}$$(0,0)$ Photon$(0,0)$ Photon$(\pi,\pi)$
Photon$(\pi,\pi)$ Photon Figure 27: Illustration of the two type of fermion
scattering processes arising from their coupling to gauge fields. The gauge
modes near $q=(0,0)$ mediate “intra-valley” fermion scattering processes
(depicted by orange circles), and the gauge modes near $q=(\pi,\pi)$ mediate
“inter-valley” scattering processes (depicted by red straight arrows).

####  III.3.1 $\textbf{p}=(0,0)$ scattering terms

The hamiltonian density describing processes of the first type for the state
with $\Theta_{x}$ extended PSG ($t\in\mathbb{R}$) is:

$H=v\Psi^{\dagger}({\bf x})\left[(p_{x}-A_{x}^{0}({\bf
x}))\tau^{x}+(p_{y}-A_{y}^{0}({\bf x}))\tau^{y}\rho^{z}\right]\Psi({\bf x}),$

and for the state with $\Theta_{y}$ extended PSG ($t\in i\mathbb{R}$) is:

$H=v\Psi^{\dagger}({\bf x})\left[(p_{x}-A_{x}^{0}({\bf
x}))\tau^{y}+(p_{y}-A_{y}^{0}({\bf x}))\tau^{x}\rho^{z}\right]\Psi({\bf x}),$

where the convention of momenta is the same as in Eq.(41), and $\tau^{i}$,
$\rho^{i}$ denote Pauli matrices in $\\{a,b\\}$ sub-lattice and on
$\\{(\pi,0),(0,\pi)\\}$ valley spaces respectively, and we have defined
continuum vector potential fields as follows:

$\displaystyle A^{0}_{x}({\bf x})$ $\displaystyle\doteq\frac{A^{0}_{1}({\bf
x})+A^{0}_{3}({\bf x})}{\sqrt{2}|{\bf R}_{1}|},$ (76) $\displaystyle
A^{0}_{y}({\bf x})$ $\displaystyle\doteq\frac{A^{0}_{2}({\bf
x})+A^{0}_{4}({\bf x})}{\sqrt{2}|{\bf R}_{1}|}.$

Therefore, we see that the long-wavelength fluctuations gauge fluctuations
that are gapless near $(0,0)$, simply behave as the standard minimal coupling
of a photon-like mode to the matter fields (compare with mean field
Hamiltonian from Eq.(41)).

####  III.3.2 $\textbf{p}=(\pi,\pi)$ scattering terms

The contribution to the hamiltonian density accounting processes of the second
type, for the state with $\Theta_{x}$ extended PSG ($t\in\mathbb{R}$) is:

$\delta H=-v\Psi^{\dagger}({\bf x})\left[B^{\pi}_{x}({\bf
x}){\tau^{x}}\rho^{1}+B^{\pi}_{y}({\bf x})\tau^{x}\rho^{2}\right]\Psi({\bf
x}),$ (77)

and for the state with $\Theta_{y}$ extended PSG ($t\in i\mathbb{R}$) is:

$\delta H=-v\Psi^{\dagger}({\bf x})\left[B^{\pi}_{x}({\bf
x}){\tau^{y}}\rho^{1}-B^{\pi}_{y}({\bf x})\tau^{y}\rho^{2}\right]\Psi({\bf
x}),$ (78)

where the continuum vector potential fields are defined as follows:

$\displaystyle B^{\pi}_{x}(\textbf{x})$
$\displaystyle=\frac{A^{\pi}_{3}(\textbf{x})-A^{\pi}_{1}(\textbf{x})}{\sqrt{2}|\textbf{R}_{1}|},$
(79) $\displaystyle B^{\pi}_{y}(\textbf{x})$
$\displaystyle=\frac{A^{\pi}_{4}(\textbf{x})-A^{\pi}_{2}(\textbf{x})}{\sqrt{2}|\textbf{R}_{1}|}.$

Notice that the fields $B^{\pi}_{x},B^{\pi}_{y}$ are the continuum limits of
the fields defined in Eq. (65) expanded around momentum $(\pi,\pi)$.
Therefore, remarkably, what we are finding here is that to linear order in
vector potentials, there is no coupling to the linearly dispersing gapless
photon modes near $(\pi,\pi)$, but instead the inter-valley scattering process
are only mediated by gauge fields associated with the $B_{x}$ and $B_{y}$
modes, which are fully gapped throughout the entire Brillouin. Therefore, at
low energies compared to the gap of the $B_{x},B_{y}$ modes and the band-width
of the photon modes, we have two emergent massless photon modes, and two
massless Dirac fermions. But the fermions are only carry gauge charge under
the $(0,0)$ photon but appear as gauge neutral under the $(\pi,\pi)$ photon.

Figure 28: Depiction of the expected infrared effective low energy theory,
which is a U(1) compact QED in 2+1 dimensions minimally coupled to two
massless Dirac fermions. The Dirac fermions are centered at $(0,\pi)$ and
$(\pi,0)$, and they are minimally coupled to the single U(1) photon gapless at
$(0,0)$. The photon at $(\pi,\pi)$ likely undergoes Polyakov style
confinement, hence disappearing at low energies.

###  III.4 $U(1)\times U(1)$ Gauge structure

In this section we will explain why the occurrence of two gapless photon modes
and the gauge coupling of the Dirac composite fermions to only one of them,
that we encountered in sections III.2 and III.2, is not accidental. We will
show there are two emergent $U(1)$ gauge structures with independent local
Gauss laws and two global flux conservation, as if we had two copies of
ordinary lattice QED.

To see this it is convenient to split the Bravais lattice of vertices of the
spin-ice model, which are located at vectors ${\bf R}$, into two sublattices
denoted by $\Lambda_{A}$ and $\Lambda_{B}$, as depicted in Fig. 29. The
sublattice $\Lambda_{B}$ can be obtained by displacing the $\Lambda_{A}$ by
either the Bravais vector ${\bf R}_{1}$ or ${\bf R}_{2}$, and vice-versa.
Therefore, the Bravais unit vectors of the lattice $\Lambda_{A}$ can be taken
to be $\\{{\bf R}_{1}-{\bf R}_{2},{\bf R}_{1}+{\bf R}_{2}\\}$, and similarly
for $\Lambda_{B}$ (see Fig. 29). Notice that the operators that measure the
divergence of the dynamical emergent fields, $(\nabla\cdot
E)_{\text{ice}}(\textbf{R})$ defined in Eq. (62) and illustrated in Fig.24,
behave as two independent divergences obeying separate Gauss’ laws. Namely
when we sum $(\nabla\cdot E)_{\text{ice}}(\textbf{R})$ over R restricted to
region of points residing only on sublattice $\Lambda_{A}$, we will get a sum
of electric fields residing only at the boundary of such region and normal to
the boundary, as expected for a lattice divergence, and similarly for regions
of points contained only in sublattice $\Lambda_{B}$. Moreover, we can also
restrict the operators $G_{\text{ice}}(\textbf{R})$ to reside over either of
the sublattices and in this way we can view the $U(1)$ gauge group as a
product $U(1)_{A}\times U(1)_{B}$. We can assign a pair charges
$(q_{A},q_{B})$ with $q_{A,B}\in\mathbb{Z}$ to matter operators (namely those
constructed as products of fermion creation/destruction operators) under this
$U(1)_{A}\times U(1)_{B}$ gauge group. In particular the JW/composite-fermion
creation operator, would transforms as a charge $(q_{A},q_{B})=(1,1)$ under
such sublattice gauge groups.

There are also two independent global flux conservation symmetries (when
ignoring gauge field compactification), one associated with sublattice
$\Lambda_{A}$ and the other with sublattice $\Lambda_{B}$, which are
responsible for the gaplessness of the two photons. This can be seen by adding
the operators $dB_{\text{ice}}(\textbf{R})/dt$, defined in Eq.(66) and
illustrated in Fig.24, over some region of R that only contains points in the
sublattice $\Lambda_{A}$, resulting into a boundary operator, that can be
viewed as a line integral of the operator $(\nabla\times