Non-abelian bosonization in two and three spatial dimensions and applications

ucb,lbl]Yen-Ta Huang

ucb,lbl]Dung-Hai Lee

[ucb]Department of Physics, University of California, Berkeley, California 94720, USA
[lbl]Materials Sciences Division, Lawrence Berkeley National Laboratories, Berkeley, California 94720, USA

In this paper, we generalize Witten's non-abelian bosonization in $(1+1)$-D to two and three spatial dimensions. Our theory applies to fermions with relativistic dispersion. The bosonized theories are non-linear sigma models with level-1 Wess-Zumino-Witten terms. We apply the bosonization results to the $SU(2)$ gauge theory of the $\pi$-flux phase, critical spin liquids in 1,2,3 spatial dimensions, and twisted bilayer graphene.

Bosonization in $(1+1)$-D has been a very useful theoretical tool.  It allows one to map a theory, where the fundamental degrees of freedom are fermionic, to a theory with bosonic degrees of freedom.  Often, things that can be seen easily in one picture are difficult to see in the other. The best-known bosonization is the abelian bosonization [1, 2, 3, 4], where fermions are  solitons in the Bose field. A shortcoming of the abelian bosonization, when fermions have flavor (e.g., spin) degrees of freedom, is that the flavor symmetries are hidden. This problem was solved by Witten's non-abelian bosonization [5]. In this paper we generalize Witten's non-abelian  bosonization to $(2+1)$ and $(3+1)$ space-time dimensions.

The limitation of our theory is that it only applies to fermions with relativistic dispersion. (However, we do not restrict the Fermi velocity to be  the speed of light.) In the absence of a mass gap, such theories have Dirac-like dispersion relation. In one space dimension, massless fermions are generically relativistic at low energies. In two and three space dimensions,  relativistic massless fermions have  been discovered in many experimental condensed matter systems. Examples include graphene and twisted bilayer graphene, Dirac and Weyl semi-metal,...etc. Moreover,  relativistic massless fermions can appear in the mean-field theory of strongly correlated systems. Such theory serves as the starting point of a more rigorous treatment. For example,  the “spinon $\pi$-flux phase” mean-field theory sets the stage for a gauge theory description of the Mott insulating state of the half-filled Hubbard model.

Another important area where relativistic massless fermions appear is at the boundary of topological insulators or superconductors, which are simple examples of symmetry-protected topological (SPT) phases. The classification of topological insulator/superconductor[6, 7] can be viewed as asking how many copies of the massless fermion theories on the boundary are required to couple together before a symmetry-allowed mass term emerges.

The paper contains two major parts: I. bosonization and II. applications. Each of them contains several sections, namely, 14 sections in part I and 3 sections in part II.  In each section of part I, we discuss an important step or input of the bosonization. We shall illustrate the relevant concepts with examples in the lowest spatial dimension where it first appears. For higher spatial dimensions, we simply present the result while leaving the details to the appendices. Together, the 14 sections in part I provide the readers with the idea and technical details of the bosonization. In part II there are 3 sections, each gives an example of how this bosonization can be applied.  The topics include the $SU(2)$ gauge theory of the $\pi$-flux phase of half-filled Hubbard model, the critical spin liquid of “bipartite-Mott insulators” in spatial dimensions 1, 2, 3, and the twisted bilayer graphene. Finally,  the 11 appendices provide the details omitted in the main text.

§ THE IDEA

In this paper, by “Bosonization”, we mean to construct bosonic theories that are equivalent to theories of massless relativistic fermions. In the rest of the paper, unless otherwise stated, “massless fermion” always refers to massless relativistic fermion. Here we stress again  that “relativistic massless fermion” does not imply the Fermi velocity is the speed of light. As mentioned in the introduction, in several $(2+1)$ and $(3+1)$ dimensional condensed matter systems, relativistic massless fermions have been encountered.

We look at the massless fermion theories from two points of view. On one hand, as $d$ (spatial) dimensional theories, the massless fermion theories have  emergent symmetries and symmetry anomalies. On the other hand, the massless fermions can be realized on the boundary of $d+1$ topological insulators/superconductors where the emergent symmetries are the protection symmetries.

If the emergent symmetries are to be respected, the (massless) fermions can not develop an energy gap. But what if we introduce mass terms (or the bosonic order parameters), at the expense of breaking the emergent symmetries, then fluctuate the order parameters smoothly (in both space and time) until the symmetries are restored ? Since the order parameter fluctuations are smooth, we expect the fermion gap to remain intact. Under such conditions, we can integrate out the fermions to yield  bosonic non-linear sigma models governing the dynamics of the order parameters. From the perspective of the boundary of topological insulators/superconductors, after integrating out the fermions, what's left are fluctuating order parameters and the non-linear sigma models. Because the protection symmetries are restored by the order parameter fluctuations, the non-linear sigma models are either gapless or possess topological order.  It turns out that the non-linear sigma models have a special type of topological term: the level-one Wess-Zumino-Witten (WZW) term. Such term encodes the 't Hooft anomaly[The 't Hooft anomaly refers to the obstruction in gauging the continuous part of the emergent global symmetries. Under such conditions, once gauge field is introduced, the partition function fails to be gauge invariant.] associated with the boundary of topological insulators/superconductors.
Due to the WZW term, the non-linear sigma models are gapless, hence potentially can be equivalent to the massless fermion theories. This equivalence is supported by the fact that the fermion and boson theories have (1) the same symmetries, (2) the same anomalies, and (3) the boson theories have fermionic solitons.

As to the question of why do we bother to bosonize? One reason is it allows us to determine the low energy physics of a non-trivial boson theory by solving the theory of free massless fermions, and often what is subtle in one picture can become clearer in the other. Of course, we will not stop at the massless free fermion theories, the goal of bosonization is to enable one to go further. This will become clear in the applications.

§ EMERGENT SYMMETRIES OF THE MASSLESS FERMION THEORY

A necessary condition for two theories to be equivalent is that they have the same symmetry.  Thus it is important to determine the symmetry of massless fermion theories.    It turns out the symmetries of such theories are rather rich. Because the massless fermion theories are low energy effective theories,  we shall refer to their symmetries as the emergent symmetries.

In the following, we shall consider massless $n$-flavor Dirac (or Majorana) fermion theories in spatial dimensions 1, 2, and 3. Such theories can be split into two main categories, namely, complex class and real class.  A theory in complex classes can be solely written in terms of complex Dirac fermion fields. Moreover, in the presence of a cutoff, its Hilbert space is the eigenspace of certain “charge” operator $Q$. In the following we shall focus on the $Q=0$ eigenspace, i.e.,  the “charge neutral point” in condensed matter physics. The charge operator is the generator of a (continuous) global U(1) symmetry. In contrast, a theory in the real class is expressed in terms of Majorana fermion fields. In this class, there is no requirement for a conserved charge operator.

§.§ Complex class

Now, as an example, let's determine the emergent symmetry group of a one dimensional massless fermion theory. To this end let's first consider a complex class, $n$-flavor, massless Dirac fermion theory described by the following action

\begin{align}
\label{S01C}
&S_0 = \int dx^0 dx^1 \psi^\dagger (\partial_0  - i \Gamma_1 \partial_1 )\psi  ~~{\rm where}\\
& \Gamma_1 = Z I_n \notag
\end{align}

Here $I_n$ denotes $n\times n$ identity matrix. In the following we shall use the shorthand $I,X,Y,Z,E$ to denote the Pauli matrix $\s_{0,x,y,z},i\s_y$, and when two matrix symbols stand next to each other, e.g., $Z I_n$, it means tensor product $Z \otimes I_n$.
For complex fermion field $\psi$, the possible unitary transformations include
ψU ·ψ      ψC ·(ψ^†)^T
where $U$ and $C$ are unitary matrices. Note that as a discrete transformation (the second line of the above equations), the charge conjugation transformation  does leave the $Q=0$ eigenspace invariant [However, we do not allow the charge conjugation operator to generate continuous transformations, since under such transformations $\psi$ will go into the superposition of $\psi$ and $\psi^\dagger$. This violates the requirement that the Hilbert space is the eigenspace of the charge operator.].

One can easily show that the full emergent symmetries of the action in S01C are
Chiral $U(n)$ symmetry:        U(n)_+ ×U(n)_-:ψ(P_+ ⊗g_+ + P_- ⊗g_-)ψ where  g_±∈U(n)       Charge conjugation  symmetry:         C:ψ→( Z ⊗I_n ) (ψ^†)^T       Time reversal  symmetry (anti-unitary):         T:ψ→( X ⊗I_n ) ψHere

\begin{align}
\label{Ppm}
P_\pm := \frac{I \pm Z}{2}
\end{align}

are the projection operators with the subscript $\pm$ denoting the “right/left” moving fermions, respectively. Note that any other anti-unitary symmetry can be written in terms of the composition of a unitary symmetry and the time reversal transformation above.

§.§ Real class

Next, we consider the one-dimensional massless theory in the real class. In this case, we write the action in terms of the n-component Majorana fermion field

\begin{align}
\label{S01R}
& S_0 = \int dx^0 dx^1 \, \chi^T \left[\partial_0 - i \Gamma_1 \partial_1 \right]\chi~~{\rm where} \\
&\Gamma_1 := Z I_n \notag
\end{align}

For Majorana fermion field, the possible unitary transformations are of the form

$$\chi\ra O \cdot\chi $$

where $O$ is an orthogonal matrix. The full emergent symmetries of the action in S01R are
Chiral $O(n)$ symmetry:      O(n)_+ ×O(n)_-:χ(P_+ ⊗g_+ + P_- ⊗g_-)χ
where  g_±∈O(n)       Time reversal symmetry (anti-unitary):      T: χ→(X⊗I_n) χ.

In $D=d+1$ space-time dimension, the massless fermion actions are

\begin{align}
\text{Complex class:} ~~&S_0 = \int d^D x \, \psi^\dagger \left[\partial_0 - i \sum\limits_{i=1}^d \Gamma_i \partial_i \right]\psi \notag\\
\text{Real class:} ~~&S_0 = \int d^D x\, \chi^T\left[\partial_0 - i \sum\limits_{i=1}^d \Gamma_i \partial_i \right]\chi
\label{s0}
\end{align}

where $\psi$ and $\chi$ are complex and Majorana fermion fields, respectively. In table <ref> we summarize the emergent symmetries of massless fermion theories in 1,2 and 3 dimensions. See the detailed derivation in appendix <ref>. Here the discrete symmetries, such as charge conjugation or time-reversal,  should be viewed as the generators of more general charge conjugation and time-reversal transformations. For example, compounding the time-reversal transformation with an arbitrary unitary symmetry yields another anti-unitary symmetry. The reason for the particular choice of the discrete symmetry generators in table <ref> will be discussed in subsection <ref> of appendix<ref>.

$(1+1)$-D 		   	Real class	    Complex class

$\Gamma_i$		     	$Z\otimes I_n $ 	   	$Z \otimes I_n$

Emergent symmetries
$T = X \otimes I_n$

$O_+(n) \times O_-(n): P_+ \otimes g_+ + P_- \otimes g_-$

where $g_+ \in O_+(n)$ and $g_- \in O_-(n)$
$T = X \otimes I_n$
$C=Z \otimes I_n$

$U_+(n) \times U_-(n): P_+ \otimes g_+ + P_- \otimes g_-$

where $g_+ \in U_+(n)$ and $g_- \in U_-(n)$

$(2+1)$-D 		   	Real class	    Complex class

$\Gamma_i$		     	$Z \otimes I_n$	, $X \otimes I_n$ 	   	$Z \otimes I_n$	, $X \otimes I_n$

Emergent symmetries
$T = E \otimes I_n$

$O(n) : I \otimes g$

where $g \in O(n)$
$T = Y \otimes I_n$

$C=I \otimes I_n$

$U(n) : I \otimes g$

where $g \in U(n)$

$(3+1)$-D 		   	Real class	    Complex class

$\Gamma_i$		     	$ZI \otimes I_n$	, $XI \otimes I_n$, $YY \otimes I_n$ 	   	$ZI \otimes I_n$	, $XI \otimes I_n$, $YZ \otimes I_n$

Emergent symmetries
$T = EZ \otimes I_n$

$U(n) : II \otimes g_1 - IE \otimes g_2$

where $u=g_1 + i g_2 \in U(n)$
$T = YZ \otimes I_n$

$C=IX \otimes I_n$

$U_+(n)\times U_-(n) : IP_+ \otimes g_+ + IP_- \otimes g_-$

where $g_+ \in U_+(n)$ and $g_- \in U_-(n)$

A summary of the emergent symmetries of massless fermions in  $(1+1)$-D, $(2+1)$-D, and $(3+1)$-D. Here $P_\pm:=(I\pm Z)/2$ as in Ppm.

§ MASS TERMS AND MASS MANIFOLDS

Mass terms, or order parameters, are fermion bilinears, namely,        ψ^†Mψ,  or      χ^T Mχ,which opens an energy gap when added to s0. To achieve that, the hermitian mass matrix $M$ must anti-commute with all the gamma matrices, i.e.,

\begin{align}
\label{mass}
&\{M,\G_i\}=0~{\rm for}~i=1,...,d
\end{align}

We will further require that the gap is flavor independent by imposing

\begin{align}
\label{constantGap}
M^2 = m^2 \cdot 1
\end{align}

Here $1$ means the identity matrix of appropriate size.

The mass matrices satisfying mass and constantGap form a topological space – the
mass manifold.  In the simplest case, it can be a k-dimensional sphere. In general, it is a closed $k$-dimensional manifold. If, in addition to mass, the mass terms are required to be invariant under certain unitary or anti-unitary transformations, the mass manifold will be affected. In the classification of the free fermion SPTs, it is important to know what is the homotopy group of the mass manifold [6].

In the following we give two examples in one spatial dimension, to let the readers get a feeling of what's involved in figuring out the mass manifold.

§.§ Complex class

Let the U(1) symmetry transforms the field field according to

$$\psi\ra e^{i\theta}\psi.$$
Then all mass terms in the form

\begin{align*}
\psi^\dagger M^{\mathbb{C}} \psi,
\end{align*}

are invariant under U(1).  Here the superscript $\mathbb{C}$ is to remind us that this is  a mass matrix in the complex fermion class.
$M^{\mathbb{C}}$ is a $2n\times 2n$ ($2n$ is the number of component of $\psi$) satisfying

\begin{align*}
&M^{\mathbb{C}} = \left( M^{\mathbb{C}}\right)^\dagger\\
&\{ M^{\mathbb{C}}, \Gamma_i \} = 0 \\
&\left( M^{\mathbb{C}} \right)^2 = m^2  I_{2n}
\end{align*}

Here $I_{2n}$ is the $2n\times 2n$ identity matrix. Associated with the massless fermion action given in S01C, the first two conditions require $M^{\mathbb{C}}$ to be of the form

M^ℂ = m (X ⊗H_1 + Y ⊗H_2)

where $H_1$ and $H_2$ are $n\times n$ hermitian matrices. If we define
Q^ℂ:= H_1+ i H_2, it can be easily shown that the third condition requires
$$Q^{\mathbb{C}} \cdot \left( Q^{\mathbb{C}}\right)^\dagger = I_n.$$
Therefore the mass manifold for one dimension, in complex class, is the topological space formed by $n\times n$ unitary matrices.

§.§ Real class

In this case, the mass term is the Majorana fermion bilinear

\begin{align*}
\chi^T M^{\mathbb{R}} \chi
\end{align*}

where the matrix $M^{\mathbb{R}}$ is an anti-symmetric matrix satisfying

\begin{align*}
&M^{\mathbb{R}} = \left( M^{\mathbb{R}}\right)^\dagger\\
&\{ M^{\mathbb{R}}, \Gamma_i \} = 0 \\
&\left( M^{\mathbb{R}} \right)^2 = m^2  I_{2n}
\end{align*}

The first two conditions require

\begin{align*}
M^{\mathbb{R}} = m \left(Y\otimes S + X\otimes (i A) \right)
\end{align*}

where $S$ and $A$  are real symmetric and anti-symmetric matrix, respectively. If we define
$$Q^{\mathbb{R}}:= S+A$$

the last condition requires
$$Q^{\mathbb{R}} \cdot \left( Q^{\mathbb{R}}\right)^T = I_n.$$
Thus, the mass manifold is the space of $n\times n$ orthogonal matrices.

In table <ref> we summarize the mass manifolds for 1,2 and 3 dimensions. The detailed derivations are left in appendix <ref>.

$(1+1)$-D 		   	Real class     Complex class

$\Gamma_i$		     	$Z\otimes I_n $ 	   	$Z \otimes I_n$

Mass manifold
$M=Y \otimes S + X \otimes (iA)$

where $Q^{\mathbb{R}} = S+A \in O(n)$
$M=X \otimes H_1 + Y \otimes H_2$

where $Q^{\mathbb{C}} = H_1+i H_2 \in U(n)$

$(2+1)$-D 		   	Real class     Complex class

$\Gamma_i$		     	$Z \otimes I_n$	, $X \otimes I_n$ 	   	$Z \otimes I_n$	, $X \otimes I_n$

Mass manifold
$M= Y \otimes S $

where $Q^{\mathbb{R}} = S \in \bigcup_{l=0}^n\frac{O(n)}{O(l) \times O(n-l)}$
$M= Y \otimes H $

where $Q^{\mathbb{C}} = H \in \bigcup_{l=0}^n\frac{U(n)}{U(l) \times U(n-l)}$

$(3+1)$-D 		   	Real class     Complex class

$\Gamma_i$		     	$ZI \otimes I_n$	, $XI \otimes I_n$, $YY \otimes I_n$ 	   	$ZI \otimes I_n$	, $XI \otimes I_n$, $YZ \otimes I_n$

Mass manifold
$M= YX \otimes S_1 + YZ \otimes S_2 $

where $Q^{\mathbb{R}} = S_1 + i S_2 \in \frac{U(n)}{O(n)}$
$M= YX \otimes H_1 + YY \otimes H_2 $

where $Q^{\mathbb{C}} = H_1 + i H_2 \in U(n)$

A summary of the mass manifolds for the real and complex class fermions in $(1+1)$-D, $(2+1)$-D, and $(3+1)$-D.

§ THE SYMMETRY ANOMALIES OF THE FERMIONIC THEORIES

Emergent symmetries of a low-energy effective theory can be broken when a cutoff is imposed.
In this section, we review the symmetry anomalies of the massless fermion theories.

§.§ The 't Hooft anomaly of continuous symmetry

The emergent symmetries discussed in the section <ref> can suffer the “'t Hooft anomaly”. A theory is said to have the 't Hooft anomaly with respect to global  symmetry group $G$ if there are obstructions against gauging $G$ [8]. In the following we shall use the $(1+1)$-D complex class to illustrate the ideas.

The simplest example is the chiral anomaly associated with the $(1+1)$-D complex class theory defined in s0. This theory has emergent global $U_+(n) \times U_-(n)$ symmetry. However, when one tries to gauge this symmetry, an anomaly is encountered. Namely, in the presence of gauge field with non-zero curvature, the theory can not be made to conserve the  Noether's current associated with the full   $U_+(n) \times U_-(n)$ symmetry.

Starting from the massless fermion theory, we can introduce the $U_+(n) \times U_-(n)$ gauge field (i.e., “gauging” $U_+(n) \times U_-(n)$) via minimal coupling. Moreover, we can define the effective gauge action after integrating out fermions,

\begin{align}
&W[A_+, A_-] = -\ln \left[ \int D\psi \, D\bar{\psi} e^{-S[\psi,\bar{\psi},A_+,A_-]} \right], ~~	\text{where }\notag \\
&S[\psi,\bar{\psi},A_+,A_-] = \int d^2 x~ \, \bar{\psi} \, \left[ i \gamma^\mu \left( \partial_\mu + i  P_+\otimes  A_{+,\mu}  + i P_-  \otimes A_{-,\mu}   \right) \right] \psi.
\label{weffA}\end{align}

Here $A_\pm$ are the $n\times n$ matrix value gauge fields associated with $U_\pm(n)$, and $P_\pm$ are the projection operators selecting the chiral fermion modes defined in Ppm. Adler[9], Bell, and Jackiw[10] first showed that in the presence of a diagonal (i.e., $A_+=A_-$) $U(1)$ gauge field, the axial current is not conserved. Shortly after, this was generalized by Bardeen [11] who showed that under infinitesimal gauge transformation, $W$ in weffA is not gauge invariant, namely,
δW := W[A_+ + dϵ_+ , A_- + dϵ_-] - W[A_+, A_-]       =  -i/4π ∫_ℳ  tr [ A_+  dϵ_+ - A_- dϵ_-   ].

This is the 't Hooft anomaly.

This phenomenon is also connected to the physics of SPT. In odd space dimension, this connection constitutes the so-called “anomaly inflow picture” [12]. In fact, each of the emergent symmetry groups in table <ref> protects a $D+1$ dimensional $\mathbb{Z}$-classified free fermion SPT. The $D$-dimensional massless free fermion theories in s0 describe the boundary of these SPTs.  We shall discuss this point further in appendix <ref>.

The most familiar anomaly inflow example is for $n=1$ in 1D. In this case, we can view the 1D (non-chiral) massless fermions  as the edge modes of two Chern insulators stacked together, with each Chern insulator having Hall conductivity $\sigma_{xy}=\pm 1$ (see DCI). In the presence of a time-dependent flux associated with the diagonal gauge field, there will be  the electric fields in the azimuthal direction. This induces a Hall current causing the charge to flow from the outer to the inner edge on one layer, and from the inner to the outer edge on the other layer.  Viewing from the edge (one-dimensional world), the chiral current $J_+-J_-$ is not conserved. This manifests the chiral anomaly, namely gauging the diagonal U(1) symmetry breaks axial U(1) symmetry - an example of the 't Hooft anomaly.

Two layers of annulus shape Chern insulators with $\sigma_{xy}=\pm 1$ stacked together. The outer edge harbors the 1D $n=1$ non-chiral massless fermion modes. The green and red arrows represent the opposite chiralities. When a time-dependent diagonal $U(1)$ flux pierces the inner hole, the induced electric field in the azimuthal direction causes a Hall current (dashed arrows) flowing from inner to outer boundary in the top layer and from outer to inner boundary in the bottom layer. As the result,  the chiral current $J_+-J_-$ is not conserved viewed from the outer edge alone. This system is realized as the “spin Hall insulator” experimentally.

Although the $U_+(n) \times U_-(n)$ anomaly makes it impossible to gauge the whole group consistently, it's possible to gauge a subgroup of it. For example, if we only gauge the diagonal subgroup $U(n) $ within $U_+(n) \times U_-(n)$, i.e., if

\begin{align*}
&A:=A_+ = A_-\\
&\epsilon := \epsilon_+ = \epsilon_-
\end{align*}

then the two terms in Bardeen1c cancel out, hence the theory is anomaly free with respect to  diagonal $U(n)$ subgroup.

§.§ A heuristic way to determine the 't Hooft anomaly

The discussions presented above require rather involved field theory calculations. However, there is a heuristic way to get the correct answer. The basis of this heuristic argument is the fact that if a theory can be defined on a lattice with all its (continuous) symmetry, then these symmetries can be gauged without anomaly.  Again, the above statement is suggested by the SPT physics, namely, the boundary modes (which has 't Hooft anomaly) of an SPT can not be defined on a lattice in the dimension of the boundary. In the following we shall again use the $(1+1)$-D complex class to illustrate the ideas.

Under Wilson's regularization[13](see later), whether a theory with global symmetry group $G$ can be defined on a lattice, is determined by whether there is a mass term that respects $G$
[Using Wilson's regularization method [13], the existence of such a mass term is a sufficient condition for the theory to be regularizable on a lattice. However, it is more involved to show that it is the necessary condition [14].
Thus, a theory with the $U_+(n) \times U_-(n)$ anomaly, means no mass term is $U_+(n) \times U_-(n)$ symmetric. Again, this is the condition that the gaplessness of the boundary modes
is symmetry protected.

First, we show that no mass term is allowed if  $U_+(n) \times U_-(n)$ symmetry is to be respected. Under $U_+(n) \times U_-(n)$ the fermion field transform as

\begin{align*}
\psi \rightarrow  \left( P_+ \otimes g_+ + P_- \otimes g_- \right) \psi~~{\rm where}~~P_\pm={I\pm Z\over 2}.
\end{align*}

Under such transformation, there is, e.g., no mass term preserving the axial $U_A(1)$ generated by $ZI_n$. This is because according to table <ref> the mass terms have the form
$$\psi^\dagger \left( X\otimes H_1 + Y \otimes H_2\right) \psi.$$
In fact, the anomaly is not only in the axial $U_A(1)$ part. To see that, let's consider $n>1$. The diagonal $U(n)$ symmetry requires that both $H_1$ and $H_2$ be proportional to the identity matrix. However, such mass term would break $U_+(n)$.

Now we show that if we relax the condition to only demanding the diagonal  $U(n)$ symmetry, there is a mass term. For example,

\begin{align*}
M_{\rm reg} = X \otimes I_n.
\end{align*}

This means that we can then write down a lattice model in momentum space using Wilson's regularization[13]

\begin{align*}
{\rm \hat{H}} = \sum_{k\in{\rm  BZ}} \psi_k^\dagger \left[ \sin k \, \Gamma_1 + (1- \cos k ) M_{\rm reg}\right] \psi_k
\end{align*}

where “BZ” stands for the Brillouin zone. We can Fourier transform the above hamiltonian back to the real space which gives us a lattice tight-binding model.  The  diagonal $U(n)$ gauge field can then be introduced via Peierls' substitution

\begin{align*}
\psi_j^\dagger \psi_i \rightarrow \psi_j^\dagger e^{i A_{i,j}} \psi_i
\end{align*}

for two adjacent sites $i,j$. Here $A_{i,j}$ is the gauge connection from site $i$ to $j$.

§.§ Discrete global symmetry anomaly

A (global) discrete symmetry in a fermion theory can also be broken by regularization. In this subsection, we shall review the simplest example – the “parity anomaly"[15, 16] of the $(2+1)$-D Dirac fermions in the complex class.

When the anomaly-free $U(n)$  symmetry is gauged,
the low energy fermion action  is given by

\begin{align}
\label{actGauged2C}
S =& \int d\tau \, d^2{\v x} \, \psi^\dagger \left[ \left(\partial_0 + i\, I \otimes A_0 \right) - i \Gamma_i \left(\partial_i + i \, I\otimes A_i\right)\right] \psi \\
&\text{ where } \Gamma_1 = ZI_n ,~~~ \Gamma_2=XI_n \notag
\end{align}

Here $A_\mu$ is the $n\times n$ matrix-valued  $U(n)$ gauge field. Under the global emergent symmetries listed in table <ref>, the gauged field transforms as

\begin{align}
\label{gaugeTrans2C}
&\text{$U(n)$:~} 	A_\mu \rightarrow g \cdot A_\mu \cdot g^\dagger\notag\\
&\text{Time reversal:~} A_\mu \rightarrow  -\left(A_\mu\right)^* \notag\\
&\text{Charge conjugation:~} 	A_\mu \rightarrow  -\left(A_\mu\right)^T
\end{align}

It's easy to check that the low energy action actGauged2C is invariant under the combined transformation of the fermion and the gauge field.

As we saw in the preceding subsection, the condition for a symmetry to be anomaly-free is the theory can be regularized while preserving the symmetry. In the present case, to preserve  $U(n)$  we need to choose a regularization that is $U(n)$ invariant. In Wilson's regularization[13] this amounts to choose a $U(n)$ invariant regularization mass. The most general mass term is given by
$$M=m \, Y\otimes H,$$
where $H$ is an $n\times n$ hermitian matrix with $H^2=I_n$. When acted upon by the global $U(n)$,
$$M\ra (I\times g)^\dagger \cdot M \cdot (I\times g)$$
(see table <ref>). Requiring it to be invariant forces us to choose M_reg= m  Y ⊗I_n.Under Wilson's regularization the momentum space Hamiltonian of the massless Dirac fermion (without gauge field) read,

\begin{align}
{\rm \hat{H}} = \sum_{{\v k} \in BZ} \psi^\dagger_{\v k} \left[ \sin k_1 \Gamma_1 + \sin k_2 \Gamma_2 + \left(2 - \cos k_1 - \cos k_2 \right)M_{\rm reg} \right]\psi_{\v k}
\end{align}

To incorporate the gauge field, we Fourier transform the above equation back to real space and introduce the gauge field by Peierls' substitution. This is all good as far as regularizing actGauged2C is concerned.

Under the action of the discrete symmetries, however

\begin{align*}
\text{Charge conjugation: }& M_{\rm reg} \rightarrow -\left( I\otimes I_n \right) \cdot M_{\rm reg}^T \cdot \left( I\otimes I_n \right) = M_{\rm reg} \\
\text{Time reversal: }& M_{\rm reg} \rightarrow \left( Y\otimes I_n \right) \cdot M_{\rm reg}^* \cdot \left( Y\otimes I_n \right) = -M_{\rm reg}
\end{align*}

Therefore charge conjugation is respected by the regularization, however, time-reversal symmetry is not.

It was first shown by Redlich [15, 16] that one can detect the time-reversal anomaly through the effective $U(n)$ gauge action after integrating out the fermions. We reproduce his argument in the following. In momentum space (the Brillouin zone) we have four low energy Dirac fermions, each around a time-reversal invariant ${\v k}$ points:

\begin{align}
\label{DiracNodes2C}
{\v k} = (0,0) + {\v q}:  ~~~ & {\rm \hat{H}}_{(0,0)} \approx \sum_{\text{small }{\v q}} \psi^\dagger_{(0,0)+ {\v q}} \left[ q_1 \, \Gamma_1 + q_2 \, \Gamma_2  \right]\psi_{(0,0)+ {\v q}}\\
{\v k} = (\pi,0) + {\v q}:  ~~~ & {\rm \hat{H}}_{(\pi,0)} \approx \sum_{\text{small }{\v q}} \psi^\dagger_{(\pi,0)+ {\v q}} \left[ -q_1 \, \Gamma_1 + q_2 \, \Gamma_2  + 2m \, M_{\rm reg} \right]\psi_{(\pi,0)+ {\v q}}\notag\\
{\v k} = (0,\pi) + {\v q}:  ~~~ & {\rm \hat{H}}_{(0,\pi)} \approx \sum_{\text{small }{\v q}} \psi^\dagger_{(0,\pi)+ {\v q}} \left[ q_1 \, \Gamma_1 - q_2 \, \Gamma_2  + 2m \, M_{\rm reg} \right]\psi_{(0,\pi)+ {\v q}}\notag\\
{\v k} = (\pi,\pi) + {\v q}:  ~~~ & {\rm \hat{H}}_{(\pi,\pi)} \approx \sum_{\text{small }{\v q}} \psi^\dagger_{(\pi,\pi)+ {\v q}} \left[ -q_1 \, \Gamma_1 - q_2 \, \Gamma_2  + 4m \, M_{\rm reg} \right]\psi_{(\pi,\pi)+ {\v q}}\notag
\end{align}

Among the four, the first is massless and preserves the time-reversal symmetry. The remaining three, however, acquire a large regularization mass, which is time-reversal breaking. In the presence of the $U(n)$ gauge field, these massive Dirac fermions would each contribute a Chern-Simons effective gauge action after the fermions are integrated out[16]. In particular, for each massive fermion the effective gauge action is
$$\frac{1}{2}\times (\pm 1)\times {i\over 4\pi}\int  AdA,$$
where the sign depends on the product of the signs in front of $q_1 \Gamma_1$, $q_2 \Gamma_2$, and $M_{\rm reg}$. Combing them, the massive fermions contribute the following breaking effective action
( -1/2-1/2+1/2 ) m/|m| i/4π ∫[A   dA + 2i/3A^3 ] = -i/8π∫[A   dA + 2i/3A^3 ].
This is time-reversal odd, as can be explicitly shown by replacing $A_\mu \rightarrow  -\left(A_\mu\right)^*$ and complex conjugating the action. As to the massless fermions near $\v k=(0,0)$, based on the fact that the first line of DiracNodes2C is time reversal invariant so should their effective gauge action. Thus after regularization, the time-reversal symmetry of actGauged2C is broken!
As expected, charge conjugation is not broken by the regularization. Since $T$ is broken while $C$ is not, based on the $CPT$ invariance, the parity should also be broken
In two space dimension, the "parity" transformation $P$ is realized by spatial reflection. Take the reflection in $x$-direction as an example, the fermion field transforms according to
$$\psi(\tau,x,y) \xrightarrow{P} XI_n \cdot \psi(\tau,-x,y).$$
It is easy to see the that the regularization mass defined in mregg changes sign under $P$. However, the combined $CPT$ transformation leaves it invariant.
Thus, there is no $CPT$ anomaly. The same conclusion can be drawn by looking at the parity transformation of the effective gauge action. Under $P$ the gauge field transforms as

\begin{align*}
A_\tau(\tau,x,y) \xrightarrow{P}& A_\tau(\tau,-x,y)  \\
A_x(\tau,x,y) \xrightarrow{P}& -A_x(\tau,-x,y)  \\
A_y(\tau,x,y) \xrightarrow{P}& A_y(\tau,-x,y)
\end{align*}

Again, panom changes sign under $P$, but is invariant under $CPT$.

In table <ref>, we summarize the maximal anomaly-free continuous symmetry and the discrete symmetry  that is broken after regularization. The only discrete symmetry which possesses anomaly occurs in $(2+1)$-D for the time-reversal symmetry. More detailed discussions are left to appendix <ref>.

$(1+1)$-D 		   	Real class     Complex class

Global Symmetry
Anti-unitary: $T^2=+1$

Continuous unitary
Chiral $O(n) \times O(n)$
Anti-unitary: $T^2=+1$

Unitary: $C^2=+1$
Continuous unitary
Chiral $U(n) \times U(n)$

Aanomaly free part
Diagonal $O(n)$, $T$     Diagonal $U(n)$, $T$, $C$

$(2+1)$-D 		   	Real class     Complex class

Global Symmetry
Anti-unitary: $T^2=-1$
Continuous unitary
Anti-unitary: $T^2=-1$

Unitary: $C^2=+1$
Continuous unitary

Anomaly free part
$O(n)$     $U(n)$, $C$

$(3+1)$-D 		   	Real class     Complex class

Global Symmetry
Anti-unitary: $T^2=-1$
Continuous unitary
Anti-unitary: $T^2=-1$

Unitary: $C^2=+1$
Continuous unitary
Chiral $U(n)\times U(n)$

Anomaly free part
$O(n)$, $T$     Diagonal $U(n)$, $T$, $C$

The summary of the global symmetry groups and the anomaly-free parts of the symmetry groups of the massless fermions (and the bosonized non-linear sigma models) in $(1+1)$-D,$(2+1)$-D, and $(3+1)$-D.

§ BREAKING THE EMERGENT SYMMETRY BY THE MASS TERMS

The mass terms discussed in the last section necessarily break some of the emergent symmetries in table <ref>.  This is because so long as the full emergent symmetries remain unbroken, the fermions will remain massless. In the rest of this section, we use one-dimensional examples to illustrate this.

§.§ Complex class

The mass terms for the complex class in $(1+1)$-D can be written as

\begin{align*}
\psi^\dagger \left( X\otimes H_1 + Y \otimes H_2\right) \psi
= \psi^\dagger \begin{bmatrix} 0 & \left( Q^{\mathbb{C}} \right)^\dagger \\ Q^{\mathbb{C}} & 0 \end{bmatrix} \psi.
\end{align*}

When acted upon by the emergent symmetries in symm1C,  $Q^{\mathbb{C}}$ transforms as

\begin{align}
\label{symmTransform1c}
&U_+(n) \times U_-(n): Q^{\mathbb{C}}\ra 	g_-^\dagger \cdot Q^{\mathbb{C}} \cdot g_+ \\
&\text{Charge conjugation}: Q^{\mathbb{C}}\ra							 \left(Q^{\mathbb{C}}\right)^* 	\notag\\
&\text{Time reversal}: Q^{\mathbb{C}}\ra						 \left(Q^{\mathbb{C}}\right)^T. \notag
\end{align}

Thus a space-time constant $Q^{\mathbb{C}}$ breaks the emergent symmetry because both $g_+$ and $g_-$ can be arbitrary unitary matrices.

§.§ Real class

For the real class in $(1+1)$-D, the mass term  can be written as

\begin{align}
\chi^\dagger \left[ Y\otimes S + X \otimes (iA)\right] \chi
= \chi^T \begin{bmatrix} 0 & -i\left( Q^{\mathbb{R}} \right)^T \\ i Q^{\mathbb{R}} & 0 \end{bmatrix} \chi.
\label{mass1R}
\end{align}

When the emergent symmetries in symm1R acts on it $Q^{\mathbb{R}}$ transforms as

\begin{align*}
&O_+(n) \times O_-(n): Q^{\mathbb{R}}\ra 	g_-^T \cdot Q^{\mathbb{R}} \cdot g_+ \\
&\text{Time reversal}: Q^{\mathbb{R}}\ra						 \left(Q^{\mathbb{R}}\right)^T.\\
\end{align*}

Therefore a space-time non-zero $Q^{\mathbb{R}}$  breaks the emergent symmetry because both $g_+$ and $g_-$ can be arbitrary orthogonal matrices.

§ RESTORING THE EMERGENT SYMMETRIES

So far we have seen that space-time constant  $Q^{\mathbb{C}}$ or $Q^{\mathbb{R}}$ breaks the emergent symmetry. But what if $Q^{\mathbb{C}}$ and $Q^{\mathbb{R}}$ fluctuates in space-time? As in statistical mechanics, when the order parameters fluctuate, the broken  symmetry can be restored. Likewise, if we fluctuate $Q^{\mathbb{C}}$ and $Q^{\mathbb{R}}$ over the appropriate mass manifold we expect the emergent symmetry to be restored.

Our approach is conceptually similar to that in Ref.[17, 18] where, on the surface of the topological insulator, the fluctuating superconducting order parameters restore the symmetries of the massless fermions. The important difference is that the required order parameter fluctuation in Ref.[17, 18] is not smooth, because it involves the proliferation of superconducting vortices. Since the structure of vortex cores is important in that approach, and such structure depends on the short-distance physics, this approach is constrained to the surface of SPTs where regularization is not an issue. In contrast, our goal is to bosonize the low energy effective theory, where the emergent symmetry is necessarily broken at short distances (due to anomaly).  As the result, we restrict our order parameter to be smooth in space and time, so that they act on the low energy theory only.

But what does “appropriate mass manifold” mean? For complex class in $(1+1)$-D,  $Q^{\mathbb{C}}$  needs to fluctuate over the space formed by  $n\times n$ unitary matrices, or $U(n)$. Such a space is connected and has a single component. On the other hand for the real class in 1D, $Q^{\mathbb{R}}$ needs to fluctuate in the space formed by $n\times n$ orthogonal matrices, or $O(n)$.  This space has two disconnected  components, corresponding to $\det [ Q^{\mathbb{R}}] = \pm 1$. It's only when $Q^{\mathbb{R}}$ fluctuates in both components with the equal statistical weight we can restore the emergent symmetry.

In $(3+1)$-D the mass manifold consists of a single component, in which $Q^{\mathbb{C,R}}$ fluctuate.  However, in $(2+1)$-D the mass manifold in complex class is $\cup_{l=0}^n\frac{U(n)}{U(l) \times U(n-l)}$ which contains $n+1$ disconnected components. Here $Q^{\mathbb{C}}$ needs to fluctuate in the component $l=n/2$ in order to restore the time reversal symmetry[Of course this requires $n$ to be even.]. In real class, the mass manifold in two space dimension is $\cup_{l=0}^n\frac{O(n)}{O(l) \times O(n-l)}$, and  $Q^{\mathbb{R}}$ needs to fluctuate in the $l=n/2$ component in order to restore the time reversal symmetry. We summarize the results for higher dimensions in table <ref> and leave the detail in appendix <ref>.

$(1+1)$-D 		   	Real class     Complex class

Symmetry transformations
of $Q^{\mathbb{C},\mathbb{R}}$
$T: Q^{\mathbb{R}} \rightarrow \left( Q^{\mathbb{R}} \right)^T $

$O_+(n) \times O_-(n): Q^{\mathbb{R}} \rightarrow g_-^T \cdot Q^{\mathbb{R}} \cdot g_+$
$ T: Q^{\mathbb{C}} \rightarrow \left( Q^{\mathbb{C}} \right)^T $

$C: Q^{\mathbb{C}} \rightarrow \left( Q^{\mathbb{C}} \right)^* $

$U_+(n) \times U_-(n): Q^{\mathbb{C}} \rightarrow g_-^\dagger \cdot Q^{\mathbb{C}} \cdot g_+$

The mass manifold required to
restore the full emergent symmetries

$(2+1)$-D 		   	Real class     Complex class

Symmetry transformations
of $Q^{\mathbb{C},\mathbb{R}}$
$T: Q^{\mathbb{R}} \rightarrow - Q^{\mathbb{R}}  $

$O(n) : Q^{\mathbb{R}} \rightarrow g^T \cdot Q^{\mathbb{R}} \cdot g$
$T: Q^{\mathbb{C}} \rightarrow -\left( Q^{\mathbb{C}} \right)^* $

$C: Q^{\mathbb{C}} \rightarrow \left( Q^{\mathbb{C}} \right)^T $

$U(n) : Q^{\mathbb{C}} \rightarrow g^\dagger \cdot Q^{\mathbb{C}} \cdot g$

The mass manifold required to
restore the full emergent symmetries
$\frac{O(n)}{O(n/2) \times O(n/2)}$
for $n \in $ even
$\frac{U(n)}{U(n/2) \times U(n/2)}$
for $n \in $ even

$(3+1)$-D 		   	Real class     Complex class

Symmetry transformations
of $Q^{\mathbb{C},\mathbb{R}}$
$T: Q^{\mathbb{R}} \rightarrow \left( Q^{\mathbb{R}} \right)^* $

$U(n): Q^{\mathbb{R}} \rightarrow u^T \cdot Q^{\mathbb{R}} \cdot u$
$T: Q^{\mathbb{C}} \rightarrow \left( Q^{\mathbb{C}} \right)^* $

$C: Q^{\mathbb{C}} \rightarrow \left( Q^{\mathbb{C}} \right)^T $

$U_+(n) \times U_-(n) : Q^{\mathbb{C}} \rightarrow g_-^\dagger \cdot Q^{\mathbb{C}} \cdot g_+$

The mass manifold required to
restore the full emergent symmetries

The summary of the symmetry transformations of $Q^{\mathbb{R},\mathbb{C}}$, and the mass manifolds in which the $Q^{\mathbb{R},\mathbb{C}}$ fluctuations can restore the full emergent symmetries.

§ THE CONDITIONS FOR THE EFFECTIVE THEORY BEING BOSONIC

In order to achieve bosonization, the fermions in S01C and S01R must not appear in the low energy theory.
To ensure that, we need to impose some conditions on the space-time dependence of $Q^{\mathbb{C}}$ and $Q^{\mathbb{R}}$. Namely, as functions of $\v x$ and $\t$,  $Q^{\mathbb{C}}(\t,\v x)$ and $Q^{\mathbb{R}}(\t,\v x)$ needs to fluctuate smoothly (comparing with the length and time scale set by $m$).  Under such conditions, the original fermions can be integrated out, yielding a non-linear sigma model for the order parameters. The idea is similar to that encountered in magnetism, where electrons  form local moments. After integrating out the electrons we arrive at an effective theory – a non-linear sigma model describing the fluctuations of the local moments in space and time.

§ FERMION INTEGRATION

In this section, using $(1+1)$-D as an example, we shall describe how to integrate out the fermions. In higher spatial dimensions we shall present the results while leaving the details in appendix <ref>.

§.§ Complex class

The fermion action with a space-time dependent mass term reads
S =    ∫dτ  dx̌   ψ^†[ ∂_0 - iΓ_1 ∂_1 + m M̂(τ, x̌) ] ψ
where and $\G_1=ZI_n$, and  {_1,M̂(t,x̌)}=0,  and  M̂(,x̌)^2=I_2n.The $\hat{M}(\t,\v x)$ that satisfies mcondit is given by

\hat{M}(\t,\v x) = m \left[X \otimes H_1 (\t, \v x) + Y \otimes H_2 (\t,\v x)\right],

For smooth order parameter configurations $\hat{M}(\t,\v x)$, the fermion integration can be done via gradient expansion (See [19] for example. We shall convert the action to a Lorentz invariant form and present the general formalism applicable for all spatial dimensions in appendix <ref>). The resulting effective action consists of two types of terms: the non-topological and topological terms. For the non-topological term (the stiffness term) we shall keep the one with the smallest number of space-time derivatives  (they are the most relevant in the renormalization group sense). The topological term is dimensionless. In $(1+1)$-D, explicit fermion integration yields (see appendix <ref> for details)

\begin{align}
W[Q^{\mathbb{C}}]  = \frac{1}{8\pi}   \int\limits_{\mathcal{M}} d^2x \, \tr\left[\p_\mu Q^{\mathbb{C}\dagger}  \partial^\mu Q^{\mathbb{C}}\right] -  \frac{2\pi i}{24\pi^2}    \int\limits_{\mathcal{B}}   \, {\rm tr} \Big[   \left( \tilde{Q}^{\mathbb{C}\dagger}   d \tilde{Q}^{\mathbb{C}}  \right)^3    \Big],  \label{wzw1C}
\end{align}

where $Q^{\mathbb{C}} $ is given in mass1C and qcdef. The first term in wzw1C is the stiffness term and the second is the Wess-Zumino-Witten (WZW) topological term. wzw1C reproduces the  level-1 $U(n)$ (abbreviated as $U(n)_1$) WZW model in Witten's non-abelian bosonization [5]. Note that the symbol “$\tr$” means tracing over the $n\times n$ portion of the matrix. (In doing fermion integration, we have already traced out the matrix part involving $\g^\mu$'s). In wzw1C $\mathcal{M}$ is the space-time manifold, and $\mathcal{B}$ is the extension of the space-time manifold $\mathcal{M}$ so that
$$\partial \mathcal{B} = \mathcal{M}.$$
In addition, $\tilde{Q}^{\mathbb{C}}(u,x)$ is an extension field of $Q^{\mathbb{C}}(x)$ so that       Q̃^ℂ(u=1,x)= Q^ℂ(x)  and      Q̃^ℂ(u=0,x)= constantIn the equation above,  “constant” means a space-time independent matrix.

For simplicity we shall focus on the space-time manifold $\mathcal{M}=S^D$ so that $\mathcal{B}$ is a
$D+1$-dimensional disk. The reason for this choice is to ensure the extension $\tilde{Q}^{\mathbb{C}}(u,x)$ exists. Because we require a smooth evolution from $Q^{\mathbb{C}}(u=0,x)$ to $\tilde{Q}^{\mathbb{C}}(u=1,x) $ ($x$ denotes $(\t,\v x)$), it means the mapping
$$Q^{\mathbb{C}}: (u=1,x)\ra \text{mass manifold}$$
is homotopically equivalent to the mapping
$$Q^{\mathbb{C}}: (u=0,x)\ra \text{mass manifold}.$$
Since $\tilde{Q}^{\mathbb{C}}(u=0,x)= {\rm constant}$ is homotopically trivial, a necessary condition for the smooth extension to exist is

$$ \pi_D(\text{mass manifold})=0,$$
i.e., all smooth mappings from the space-time manifold to the mass manifold are homotopically trivial. It turns out this condition is met for sufficiently large $n$ in all spatial dimensions. We shall return to this point in appendix <ref>, <ref>, and <ref>. For $(1+1)$-D, $ \pi_2(U(n))=0$ for any $n$.

For the WZW term to be well defined, it had better not depend on the extension. When there are two different extensions on the $D+1$ dimensional disk, say one defined by $\tilde{Q}_1^{\mathbb{C}}$ on $\mathcal{B}_1$ and the other by $\tilde{Q}_2^{\mathbb{C}}$ on $\mathcal{B}_2$, the difference in the WZW term associated with these two extensions is given by

ΔW_WZW[Q̃^ℂ] = -2πi/24π^2    ∫_ℬ_1∪(-ℬ_2)      tr [   ( Q̃^ℂ†   d Q̃^ℂ  )^3    ]

where $-\mathcal{B}_2$ is the mirror reflection of $\mathcal{B}_2$. Since $\mathcal{B}_1 \cup (-\mathcal{B}_2)=S^{D+1}$, removing the factor $2\pi i$, wzwdiff is the topological invariant associated with $\pi_{2+1}(\text{mass manifold})$. It turns out that for all relevant cases, $\pi_{D+1}(\text{mass manifold})$ $=$$\mathbb{Z}$ (see appendix <ref>). In $(1+1)$-D, $\pi_3(U(n))=\mathbb{Z}$ for $n\ge 2$ ($n=1$ corresponds to flavorless or spinless fermion where the bosonization is abelian.). The coefficient of the  WZW term renders $\Delta W_{WZW}=2\pi i \times \text{integer}$. The fact that the WZW term is $2\pi i$ times the topological invariant implies the level ($k$) is 1.  After the exponentiation, the phase factor associated with the WZW term is well-defined.

§.§ Real class

The 1+1-D Majorana fermion action with a space-time dependent mass read

\begin{align}
S =& \int d\tau \, d{\v x} \, \chi^T \left[ \partial_0 - i\Gamma_1 \partial_1 +m \hat{M}(\tau, \v x) \right] \chi
\label{S1R}
\end{align}

\begin{align*}
\Gamma_1 = ZI_n \text{ and } \hat{M}(\t,\v x)=	\left[Y\otimes S + X\otimes (i A) \right].
\end{align*}

Following the same steps discussed in the last subsection, fermion integration yields the following effective action (see appendix <ref>)

\begin{align}
W[Q^{\mathbb{R}}] &=  \frac{1}{16\pi}   \int\limits_{\mathcal{M}} d^2x \, {\rm tr}\left[ \p_\mu Q^{\mathbb{R}T} \partial^\mu Q^{\mathbb{R}} \right] -\frac{2\pi i}{48\pi^2}    \int\limits_{\mathcal{B}}   {\rm tr} \Big[   \left(\tilde{Q}^{\mathbb{R}T}   d \tilde{Q}^{\mathbb{R}}\right)^3\Big].
\label{wzw1R}
\end{align}

wzw1R is the $O(n)_{k=1}$ WZW model. Again, $\tilde{Q}^{\mathbb{R}}(u,x)$ is extension field of $Q^{\mathbb{R}}(x)$, which exists if $\pi_D(\text{mass manifold})=0$. In $(1+1)$-D, $\pi_2(O(n))=0$ for $n\ge 3$.  Here the difference in the WZW term associated with two different extension is the topological invariant associated with $\pi_3(O(n))=\mathbb{Z}$ for relevant $n$ (see appendix <ref>). The coefficient of the WZW term renders $\Delta W_{WZW}=2\pi i\times\text{integer}$ hence yields the same phase factor upon exponentiation. Again,  the fact that the WZW term is $2\pi i$ times the topological invariant implies the level ($k$) is 1.

Thus, for both complex and real classes, the bosonization of massless fermion is the non-linear sigma model with WZW term. This reproduces  Witten's non-abelian bosonization results, which was obtained using a totally different method (the current algebra).

The above bosonization scheme can be straightforwardly   generalized to higher dimensions. One thing that needs some care is the fact that the homotopy group of the mass manifold depends on $n$. For $n$ exceeds certain value $\pi_{D+1}(\text{mass manifold}) $ $=$ $\mathbb{Z}$. In that case fermion integration does lead to a nonlinear sigma model with $k=1$ WZW term. However, for small $n$ (before the “homotopy stabilization”) sometimes, e.g., $\pi_{D+1}(\text{mass manifold})=0$. We shall discuss one such instance in appendix <ref>. Fortunately, for the vast majority of applications $n$ is sufficiently big so that $\pi_{D+1}(\text{mass manifold})=\mathbb{Z}$.

§ NON-LINEAR SIGMA MODELS IN $(2+1)$-D AND $(3+1)$-D

As mentioned, the bosonization strategy described in the preceding section can be applied to two and three spatial dimensions. To facilitate later discussions, including the applications in $(2+1)$-D and $(3+1)$-D, the explicit form of the nonlinear sigma models in table <ref> are given here. For briefness, we shall only include the results for sufficiently large $n$ so that $\pi_{D+1}({\rm mass manifold})=\mathbb Z$. As discussed earlier, under such conditions the non-linear sigma model possesses a WZW term.

§.§ Complex class in $(2+1)$-D

For Dirac fermions with $n$ flavors in the complex class, after bosonization the sigma model matrix field (or the order parameter)  lives in the space of complex Grassmannian, namely,
$$Q^{\mathbb{C}}(x)\in\frac{U(n)}{U(n/2) \times U(n/2)}.$$
This means that at any space-time point $x$, $Q^{\mathbb{C}}(x)$ is an $n \times n$ hermitian matrix with half of the eigenvalues  $+1$, and the other half $-1$. One can specify $Q^{\mathbb{C}}(x)$ by the unitary matrix, $C(x)$, which renders $Q^{\mathbb{C}}(x)$ diagonalized upon similarity transformation, i.e.,

$$Q^{\mathbb{C}}(x)=C(x) \cdot \begin{pmatrix}I_{n/2}&0\\0 &-I_{n/2}\end{pmatrix} \cdot C^\dagger(x).$$

Obviously two different $C(x)$s related by

\begin{align*}
C^\prime(x) =
C(x) \cdot \begin{pmatrix} g_1(x) & 0 \\ 0 & g_2(x) \end{pmatrix},
\end{align*}

where $g_1(x),g_2(x)\in U(n/2)$, will lead to identical $Q^{\mathbb{C}}(x)$. Due to this redundancy, the order parameter lives in the quotient space $\frac{U(n)}{U(n/2) \times U(n/2)}$.

Explicit fermion integration yields the following non-linear sigma model

\begin{align}
W[Q^{\mathbb{C}}] ={1\over 2\lambda_3} \int\limits_{\mathcal{M}} d^3 x \, \tr\Big[\Big(\p_\mu Q^{\mathbb{C}}\Big)^2\Big] -  \frac{ 2 \pi i }{256 \pi^2}      \int\limits_{\mathcal{B}} \, \tr \Big[\tilde{Q}^{\mathbb{C}}    \,\left( d \tilde{Q}^{\mathbb{C}} \right)^4 \Big],
\label{wzw2C}
\end{align}

where $\lambda_3$ is a parameter having the dimension of length. In the limit where the short distance cutoff is zero, λ_3=8πmwhere $m$ is the fermion energy gap.

The first term in wzw2C is the stiffness term and the second is the level-1 ($k=1$) Wess-Zumino-Witten term. $\tilde{Q}^{\mathbb{C}}(x,u)$ is the extended field of $Q^{\mathbb{C}}(x)$, which exist because $\pi_3(\frac{U(n)}{U(n/2) \times U(n/2)})=0$ for $n\geq 4$. The difference in the WZW term associated with two different extensions is $2\pi i$ times the topological invariant associated with $\pi_4(\frac{U(n)}{U(n/2) \times U(n/2)})=\mathbb{Z}$. Consequently upon exponentiation, different extensions yield the same phase factor. (To recapitulate the explanation, the readers are referred to subsection <ref>.)

§.§ Real class in $(2+1)$-D

For massless $n$-flavor Majorana fermions in the real class,  the fluctuating order parameters $Q^{\mathbb{R}}(x)$  lives in the space of real Grassmannian, namely,
$$Q^{\mathbb{R}}(x)\in \frac{O(n)}{O(n/2) \times O(n/2)}.$$
This means that at any space-time point $x$, $Q^{\mathbb{R}}(x)$ is an $n \times n$ real symmetric matrix, with half of the eigenvalues  $+1$, and the other half $-1$. One can specify $Q^{\mathbb{R}}(x)$ by  the orthogonal matrix, $R(x)$, required to render $Q^{\mathbb{R}}(x)$ diagonalized, namely,
$$Q^{\mathbb{R}}(x)=R(x) \cdot  \begin{pmatrix}I_{n/2}&0\\0 &-I_{n/2}\end{pmatrix} \cdot R^T(x).$$
Two different $R(x)$s related by

\begin{align*}
R(x) \cdot \begin{pmatrix} g_1(x) & 0 \\ 0 & g_2(x) \end{pmatrix},
\end{align*}

where $g_1(x),g_2(x)\in O(n/2)$, will lead to identical $Q^{\mathbb{R}}(x)$. Due to this redundancy, the order parameter lives in the quotient space $\frac{O(n)}{O(n/2) \times O(n/2)}$.

Explicit fermion integration leads to the following non-linear sigma model

\begin{align}
W[Q^{\mathbb{R}}] ={1\over 4\lambda_3} \int\limits_{\mathcal{M}} d^3 x \, \tr\Big[\Big(\p_\mu Q^{\mathbb{R}}\Big)^2\Big] -  \frac{ 2 \pi i }{512 \pi^2}      \int\limits_{\mathcal{B}} \tr \Big[\tilde{Q}^{\mathbb{R}}    \,\left( d \tilde{Q}^{\mathbb{R}} \right)^4 \Big] .
\label{wzw2R}
\end{align}

Again, $\lambda_3$ has the dimension of length, and in the limit where the short-distance cutoff is zero $\lambda_3$ is given by lambda2D.

The first term in wzw2R is the stiffness term and the second is the Wess-Zumino-Witten topological term of level $k=1$. $\tilde{Q}^{\mathbb{R}}(x,u)$ is the extended field of $Q^{\mathbb{R}}(x)$, which exist because $\pi_3(\frac{O(n)}{O(n/2) \times O(n/2)})=0$ for $n\geq 6$.  The difference in the WZW term associated with two different extensions is $2\pi i$ times the topological invariant associated with $\pi_4(\frac{O(n)}{O(n/2) \times O(n/2)})=\mathbb{Z}$. Consequently upon exponentiation
different extensions yield the same phase factor. (Again, to recapitulate the explanation, the readers are referred to subsection <ref>.)

§.§ Complex class in $(3+1)$-D

For the $n$-flavor massless Dirac fermions in the complex class,
the fluctuating order parameters $Q^{\mathbb{C}}(x)$ lives in the space of $n\times n$ unitary matrices, namely,

$$Q^{\mathbb{C}}(x)\in U(n).$$

Explicit fermion integration leads to the following non-linear sigma model

\begin{align}
W[Q^{\mathbb{C}}] = &  \frac{1}{2\lambda_4^2}   \int_\mathcal{M} d^4 x \, \tr\left[  \partial_{\mu} Q^{\mathbb{C}} \partial^{\mu}Q^{\mathbb{C}\dagger}\right]
-\frac{2\pi}{480\pi^3}    \int\limits_{\mathcal{B}}    \, \tr \Big[\left(\tilde{Q}^{\mathbb{C}\dagger} d\tilde{Q}^{\mathbb{C}} \right)^5 \Big],
\label{wzw3C}
\end{align}

where $\lambda_4$ has the dimension of length. Using dimensional regularization  $\lambda_4$ is given by
1λ_4= [ Γ(0^+)m^2/8 π^2 ]^1/2,signifying that $\lambda_4$ is cutoff-dependent.  Here $\Gamma(0^+)$ is the gamma function evaluated at $0^+$ from dimensional regularization (see appendix <ref> for the details).

The first term in wzw2C is the stiffness term and the second is the level $k=1$ Wess-Zumino-Witten term. $\tilde{Q}^{\mathbb{C}}(x,u)$ is the extended field of $Q^{\mathbb{C}}(x)$, which exist because $\pi_4(U(n))=0$ for $n\geq 3$. The difference in the WZW term associated with two different extensions is $2\pi i$ times the topological invariant associated with $\pi_5(U(n))=\mathbb{Z}$. Consequently upon exponentiation different extensions yield the same phase factor. (Again, to recapitulate the explanation, the readers are referred to subsection <ref>.)

§.§ Real class in $(3+1)$-D

For the $n$-flavor massless Majorana fermions in the complex class, the fluctuating order parameters  $Q^{\mathbb{R}}(x)$ lives in the space of “real Lagrangian Grassmannian”, namely,
$$Q^{\mathbb{R}}(x)\in \frac{U(n)}{O(n)}.$$
This means that at any space-time point $x$, $Q^{\mathbb{R}}(x)$ is an $n \times n$ symmetric unitary matrix. According to the Autonne decomposition (e.g., corollary 2.6.6 of [20]), any symmetric unitary matrix can be decompose into
$$Q^{\mathbb{R}}(x)= W(x) \cdot W^T(x),$$
where $W(x)$ is unitary. Hence, two different $W(x)$s related by

\begin{align*}
W^\prime(x)= W(x) \cdot g(x),
\end{align*}

where $g(x)\in O(n)$, will lead to identical $Q^{\mathbb{R}}(x)$. Due to this redundancy, the order parameter lives in the quotient space $\frac{U(n)}{O(n)}$.

Explicit fermion integration yields the following non-linear sigma model

\begin{align}
W[Q^{\mathbb{R}}] = &  \frac{1}{4\lambda_4^2}  \int_\mathcal{M} d^4 x \, \tr\left[  \partial_{\mu} Q^{\mathbb{R}} \partial^{\mu}Q^{\mathbb{R}\dagger}\right]
-\frac{2\pi}{960\pi^3}    \int\limits_{\mathcal{B}}   \, \tr \Big[\left(\tilde{Q}^{\mathbb{R}\dagger} d\tilde{Q}^{\mathbb{R}} \right)^5 \Big].
\label{wzw3R}
\end{align}

The first term in wzw2R is the stiffness term and the second is the  level $k=1$ Wess-Zumino-Witten topological term. $\tilde{Q}^{\mathbb{R}}(x,u)$ is the extended field of $Q^{\mathbb{R}}(x)$, which exist because $\pi_4(U(n)/O(n))=0$ for $n\geq 5$. The difference in the WZW term associated with two different extensions is $2\pi i$ times the topological invariant associated with $\pi_5(U(n)/O(n))=\mathbb{Z}$.  Consequently upon exponentiation different extensions yield the same phase factor. (Again, to recapitulate the explanation, the readers are referred to subsection <ref>.)

In table <ref> we summarize the $n$ values above which $\pi_{D+1}(\text{mass manifold})$ is stabilized. We shall discuss some of the small $n$ cases which are relevant to our applications in appendix <ref>.

Real class     complex class

$O(n)_1$ WZW term
stabilized for $n\geq 3$
$U(n)_1$ WZW term
stabilized for $n\geq 2$

$\left[\frac{O(n)}{O(n/2)\times O(n/2)}\right]_1$ WZW term
stabilized for $n\geq 6$
$\left[\frac{U(n)}{U(n/2)\times U(n/2)}\right]_1$ WZW term
stabilized for $n\geq 4$

$\left[U(n)/O(n)\right]_1$ WZW term
stabilized for $n\geq 5$
$U(n)_1$ WZW term
stabilized for $n\geq 3$

The $n$ values above which the $\pi_{D+1}(\text{mass manifold})$ is stabilized.

§.§ The value of the stiffness constant and the phases of non-linear sigma models

Unlike in $(1+1)$-D, the stiffness constants of the non-linear sigma models in $(2+1)$-D and $(3+1)$-D are dimensionful parameters. A natural question then arises, how does the values of these parameters determine the phase of the non-linear sigma models ?  For small $\lambda_3$ and $\lambda_4^2$ the action costs of space-time varying $Q^{\mathbb{R}, \mathbb{C}}$ is large, hence we expect spontaneous symmetry breaking to occur.  Quantum disorder sets in for large $\lambda_3$ and $\lambda_4^2$. In the presence of the WZW term, the quantum disordered phase is gapless. It is in the latter phase do the non-linear sigma models represent the massless free fermions.

§ NON-LINEAR SIGMA MODELS AS THE EFFECTIVE THEORIES OF INTERACTING FERMION MODELS

As we have seen in section <ref>, while the coefficient in front of the stiffness term in the non-linear sigma model is dimensionless in $(1+1)$-D, those in $(2+1)$-D and $(3+1)$-D are dimensionful parameters. This begs the question of what are these parameters? and for what values of these parameters are the non-linear sigma models equivalent to the massless fermion theories? In addition, for $D=2+1$  the mass manifold consists of more than one connected components. What kind of model can realize phases correspond to different components of the mass manifold? In the following, we answer these questions by focusing on the complex class. It is straightforward to generalize the result to the real class.

As listed in table <ref>, the mass terms correspond to  $Q^{\mathbb{C}}$ are given by
(1+1)-D:  M[Q^ℂ]= X⊗12[Q^ℂ+(Q^ℂ)^†]+ Y⊗12i[Q^ℂ-(Q^ℂ)^†]       (2+1)-D:  M[Q^ℂ]=Y⊗Q^ℂ       (3+1)-D:  M[Q^ℂ]=YX⊗12[Q^ℂ+(Q^ℂ)^†]+YY⊗12i[Q^ℂ-(Q^ℂ)^†]

Let's consider the four-fermion interacting
generated by the  following inverse Hubbard-Stratonovich transformation,

\begin{align}
\label{invhs}
&\exp\left\{-S_{\rm I}\left[\psi^\dagger,\psi\right]\right\}:=\notag\\&\int D\left[\mathcal{Q}(x)\right] \, \exp\left\{-\int d^Dx \left[ \psi^\dagger~M[\mathcal{Q}(x)] ~\psi + \frac{1}{2\lambda_I} \tr\left[\mathcal{Q}(x)^\dagger \mathcal{Q}(x) \right] \right]\right\}
\end{align}

where $\mathcal{Q}(x)$ is an $n \times n$  matrix-valued function of space-time. We note that the strength of the four fermion interaction in invhs is proportional to $\lambda_I$.

The emergent global symmetries transform $\mathcal{Q}(x)$ in exactly the same way as $Q^{\mathbb{C}}$ (see table <ref>). This is because $\mathcal{Q}(x)$ and $Q^{\mathbb{C}}$ couple to the same fermion bi-linears. Such transformation can be absorbed by the redefinition of the integration variable $\mathcal{Q}(x)$. Therefore as long as the integration measure in invhs is symmetric under the symmetry transformations, $S_I$ is invariant under the action of emergent symmetries.

When $\lambda_I$ is sufficiently large, it is energetically favorable for
$$\tr \left[\Avg{\mathcal{Q}^\dagger (x) \mathcal{Q}(x)}\right]$$
to acquire a non-zero expectation value. Assuming such expectation value doesn't spontaneously break the continuous symmetry[The possible symmetry breaking phases are captured by the non-zero expectation value $\Avg{\mathcal{Q}(x)}$.] it must satisfy

\begin{align*}
\Avg{\mathcal{Q}^\dagger (x) \mathcal{Q}(x)} \rightarrow g^\dagger \cdot \Avg{\mathcal{Q}^\dagger (x) \mathcal{Q}(x)} \cdot g = \Avg{\mathcal{Q}^\dagger (x) \mathcal{Q}(x)}
\end{align*}

for all $g \in U(n)$ (for $(1+1)$-D and $(3+1)$-D $g\in U_+(n)$). This requires the expectation value of $\mathcal{Q}^\dagger (x) \mathcal{Q}(x)$ to be proportional identity matrix,

\begin{align*}
\Avg{\mathcal{Q}^\dagger (x) \mathcal{Q}(x)} = \kappa^2 I_n
\end{align*}

where  $\kappa^2$ should grow monotonically with $\lambda_I$. At low energy and long wavelength, the dynamics of $\mathcal{Q}$ is governed by the Goldstone modes $Q^\mathbb{C}(x)$, where

$$\mathcal{Q}(x) \rightarrow \kappa Q^\mathbb{C}(x),~~{\rm and}~~\left(Q^\mathbb{C}\right)^\dagger Q^\mathbb{C} = I_n.$$
The manifold in which $Q^\mathbb{C}(x)$ fluctuates is exactly  the mass manifold given in table <ref>.

The effective action governing the fluctuations of $Q^\mathbb{C}(x)$ is given by the results of section <ref>, where the stiffness term coefficients ${1\over 2\lambda_3}$ and ${1\over 2\lambda_4^2}$ should grow with $\kappa^2$ which, in turn, monotonically increases with $\lambda_I$. As the result, strong four-fermion interaction implies small $\lambda_3$ and $\lambda_4^2$, while weak four fermion-interaction implies large   $\lambda_3$ and $\lambda_4^2$. Thus, we obtain a duality-like relation, namely, strong coupling fermion theory corresponds to weak coupling non-linear sigma model, and weak coupling fermion theory corresponds to strong coupling non-linear sigma model. Since, by dimension counting, local four-fermion interaction is an irrelevant perturbation to the massless theory in $(2+1)$- and $(3+1)$-D, we expect there is a range of large $\lambda_3$ and $\lambda_4^2$ where the non-linear sigma model is massless.

Now we come to $(2+1)$-D, where according to table <ref> , the mass manifold has $n+1$ components, namely,
$$Q^\mathbb{C} \in \bigcup_{l=0}^n \frac{U(n)}{U(l) \times U(n-l) }.$$
(Here $l$ corresponds to the number positive eigenvalues of $Q^\mathbb{C}(x)$, the readers are referred to appendix <ref> for details.) The condition that the order parameter is a smooth function of space-time confines $Q^\mathbb{C}(x)$ to fluctuate in one of the mass manifold components. If such fluctuation is to restore the time-reversal symmetry, it further restricts $l=n/2$ (we focus on $n=$ even). However, if we allow the possibility of spontaneous time-reversal symmetry breaking, then  $Q^\mathbb{C}(x)$ can fluctuate in the $l\ne n/2$ mass manifold. It is  interesting whether the order parameter fluctuation in the $l\ne n/2$ mass manifolds can restore the unitary part of the emergent symmetry, and if it does can the resulting phase be gapless.

§ GLOBAL SYMMETRIES OF THE NON-LINEAR SIGMA MODELS

Up to this point, we have derived the non-linear sigma model. The bosonic partition function is given by

\mathcal{Z}=\int D[Q^{\mathbb{C,R}}] ~e^{-S_{\rm{NL}\s}\left[Q^{\mathbb{C,R}}\right]}.$$

Here $Q^{\mathbb{C,R}}\in \text{mass manifolds}$, and the integration measure is defined so that at every space-time point
$Q^{\mathbb{C,R}}$ and the symmetry transformed $Q^{\mathbb{C,R}}$ (see table <ref>) have the same weight.

Now, using the  complex class in $(1+1)$-D as an example, we demonstrate that the non-linear sigma model in wzw1C respects the emergent symmetries of the massless free fermion theory.
Under the  action of the global emergent symmetries, a configuration $Q^{\mathbb{C}}(\tau, \bf x)$  transforms by  symmTransform1c, namely,

\begin{align*}
&U_+(n) \times U_-(n): Q^{\mathbb{C}}(\tau, {\bf x})\ra  g_-^\dagger \cdot Q^{\mathbb{C}}(\tau, {\bf x}) \cdot g_+ \\
&\text{Charge conjugation}: Q^{\mathbb{C}}(\tau, {\bf x})\ra  	\left(Q^{\mathbb{C}}(\tau, {\bf x})\right)^* 	\\
&\text{Time reversal}: Q^{\mathbb{C}}(\tau, {\bf x})\ra		 	 \left(Q^{\mathbb{C}}(\tau, {\bf x})\right)^T.
\end{align*}

Under the action of $U_+(n) \times U_-(n)$
Q^ℂ†∂_μ Q^ℂg_+^†·( Q^ℂ†∂_μ Q^ℂ) ·g_+
Due to the cyclic invariance  of trace, the similarity transformations cancel out and the action wzw1C is invariant.

Under  charge conjugation, the stiffness term transforms as

\begin{align*}
& -\frac{1}{8\pi}   \int\limits_{\mathcal{M}} d^2x \, \tr\left[  \left(Q^{\mathbb{C}\dagger}  \partial^\mu Q^{\mathbb{C}}\right)^2 \right]
\rightarrow  -\frac{1}{8\pi}   \int\limits_{\mathcal{M}} d^2x \, \tr\left[  \left(Q^{\mathbb{C}T}  \partial^\mu Q^{\mathbb{C}*}\right)^2 \right] \\
=& -\frac{1}{8\pi}   \int\limits_{\mathcal{M}} d^2x \, \tr\left[  \left(  \partial^\mu Q^{\mathbb{C}\dagger}Q^{\mathbb{C}}\right)   \left(  \partial^\mu Q^{\mathbb{C}\dagger}Q^{\mathbb{C}}\right)\right]=-\frac{1}{8\pi}   \int\limits_{\mathcal{M}} d^2x \, \tr\left[  \left(  Q^{\mathbb{C}\dagger} \partial^\mu Q^{\mathbb{C}}\right)   \left(   Q^{\mathbb{C}\dagger}\partial^\mu Q^{\mathbb{C}}\right)\right]
\end{align*}

hence is invariant. Here the first equality in the second line is due to the invariance of trace under transposing, and the second equality is due to $ \partial^\mu Q^{\mathbb{C}\dagger}Q^{\mathbb{C}}=-Q^{\mathbb{C}\dagger}\partial^\mu Q^{\mathbb{C}}.$ A similar argument applies to the WZW term,

\begin{align*}
&\frac{2\pi i}{24\pi^2}    \int\limits_{\mathcal{B}}   \tr \Big[   \left( \tilde{Q}^{\mathbb{C}\dagger}   d \tilde{Q}^{\mathbb{C}}  \right)^3    \Big]
\rightarrow  \frac{2\pi i}{24\pi^2}    \int\limits_{\mathcal{B}}   \tr \Big[   \left( \tilde{Q}^{\mathbb{C}T}   d \tilde{Q}^{\mathbb{C}*}  \right)^3    \Big]\\
= & -\frac{2\pi i}{24\pi^2}    \int\limits_{\mathcal{B}}   \, {\rm tr} \Big[   \left(d \tilde{Q}^{\mathbb{C}\dagger}  \tilde{Q}^{\mathbb{C}}     \right)^3    \Big]
= \frac{2\pi i}{24\pi^2}    \int\limits_{\mathcal{B}}   \,{\rm tr} \Big[   \left( \tilde{Q}^{\mathbb{C}\dagger}   d \tilde{Q}^{\mathbb{C}}  \right)^3    \Big].
\end{align*}

The extra minus sign in the second line is because transposing causes an odd number of crossing of the  differential 1-forms. This negative sign is canceled out in the last term due to the odd number of negative signs arising from $d Q^{\mathbb{C}\dagger}Q^{\mathbb{C}}=-Q^{\mathbb{C}\dagger}d Q^{\mathbb{C}}.$  Therefore wzw1C is invariant under charge conjugation.

Under the action of time-reversal transformation, the stiffness term transforms as

\begin{align*}
& \frac{1}{8\pi}   \int\limits_{\mathcal{M}} d^2x \,{\rm  tr}\left[\p_\mu Q^{\mathbb{C}\dagger}  \partial^\mu Q^{\mathbb{C}}\right]
\rightarrow  \frac{1}{8\pi}  \int\limits_{\mathcal{M}} d^2x \,{\rm  tr}\left[\p_\mu Q^{\mathbb{C}*}  \partial^\mu Q^{\mathbb{C}T}\right] \\
=&\frac{1}{8\pi}  \int\limits_{\mathcal{M}} d^2x \,{\rm  tr}\left[ \partial^\mu Q^{\mathbb{C}}\p_\mu Q^{\mathbb{C}\dagger} \right]
=  \frac{1}{8\pi}   \int\limits_{\mathcal{M}} d^2x \,{\rm  tr}\left[\p_\mu Q^{\mathbb{C}\dagger}  \partial^\mu Q^{\mathbb{C}}\right]
\end{align*}

As for the WZW term, note that the $i$ in front becomes $-i$ due to the complex conjugation involved in the time-reversal transformation [The time-reversal symmetry in Euclidean space-time requires a complex conjugation on the Boltzmann weight. It is important to check whether a term is real or complex before deciding how time-reversal transformation acts.]. Thus the WZW term transforms as

2πi/24π^2    ∫_ℬ    tr [   ( Q̃^ℂ†   d Q̃^ℂ  )^3    ]
→-2πi/24π^2    ∫_ℬ     tr [   ( Q̃^ℂ*   d Q̃^ℂT  )^3    ]      =  2πi/24π^2    ∫_ℬ     tr [   (d Q̃^ℂ  Q̃^ℂ†     ) (d Q̃^ℂ  Q̃^ℂ†     ) (d Q̃^ℂ  Q̃^ℂ†     )    ]      =  2πi/24π^2    ∫_ℬ    tr [   ( Q̃^ℂ†   d Q̃^ℂ  )^3    ]

The disappearance of the minus sign in the second line is because transposing causes an odd number of crossings of  differential 1-forms. The passing to the third line follows from the cyclic invariance of trace.

In summary, the non-linear sigma model is invariant under the action of the global emergent symmetries. The same conclusion applies to the real and complex classes nonlinear sigma models in other space-time dimensions.
The details is left in appendix <ref>.

§ THE SYMMETRY ANOMALIES OF THE NONLINEAR SIGMA MODELS

A necessary condition for the bosonized non-linear sigma model to be equivalent to the massless fermion theory is that the former  has the same symmetry anomalies as the original massless fermion theories. In this section, we will show this is indeed the case.

§.§ Gauging the non-linear sigma models and the 't Hooft anomalies

In table <ref>  we see that in $(1+1)$-D and $(3+1)$-D, the massless free fermion theories have the 't Hooft anomalies (with respect to the continuous symmetries). In this subsection, we first gauge the non-linear sigma models and then determine their 't Hooft anomalies.

Again, taking the complex class $(1+1)$-D example, under an infinitesimal $U_+(n) \times U_-(n)$ transformation, $Q^{\mathbb{C}}$ and gauge fields transformed as
Q^ℂ →e^-i ϵ_- Q^ℂ e^i ϵ_+      A_±→A_±+ dϵ_±+ i [A_±, ϵ_±]
where we let $g_\pm = e^{i\epsilon_\pm}$ in the symmetry transformation. For the stiffness term, the usual minimal coupling guarantees the gauge invariance

\begin{align*}
W_{\rm stiff}[Q^{\mathbb{C}},A_{+},A_{-}] = -\frac{1}{8\pi} \int\limits_{\mathcal{M}} d^2 x \, \tr\left[\left(Q^{\mathbb{C} \dagger} \left( \partial_\mu Q^{\mathbb{C}} - i Q^{\mathbb{C}} A_{+,\mu} + i A_{-,\mu} Q^{\mathbb{C}} \right) \right)^2 \right].
\end{align*}

However, it is less clear how to gauge the WZW term.
Here we follow Witten's “trial-and-error” method [21], which we shall explain in the following.

First, we determine the variation of the WZW term when $Q^{\mathbb{C}}$ undergoes  space-time dependent transformation given by the first line of gtr

\begin{align*}
&\delta \Big[ - \frac{i}{12\pi} \int\limits_{\mathcal{B}} {\rm tr} \left[\left(Q^{\mathbb{C} \dagger} d Q^{\mathbb{C}} \right)^3 \right]\Big] \\
=& \frac{1}{4\pi} \int\limits_{\mathcal{M}}  {\rm tr} \left[d\epsilon_+ \left( Q^{\mathbb{C} \dagger} d Q^{\mathbb{C} } \right) + d\epsilon_- \left(d Q^{\mathbb{C} } Q^{\mathbb{C} \dagger}  \right) \right]
\end{align*}

Here we remark that although writing down the action requires the extended space-time manifold $\mathcal{B}$, the variation of the action can be expressed solely in the space-time manifold $\mathcal{M}$, which is $(1+1)$-D in the example.

In an attempt to make the theory gauge invariant, we subtract a term with $d\epsilon_\pm $ replaced by $A_\pm$. Together, the gauge variant part becomes

\begin{align*}
&\delta \Big[ - \frac{i}{12\pi} \int\limits_{\mathcal{B}} {\rm tr} \left[\left(Q^{\mathbb{C} \dagger} d Q^{\mathbb{C}} \right)^3 \right] - \frac{1}{4\pi} \int\limits_{\mathcal{M}} {\rm tr} \left[ A_+ \left( Q^{\mathbb{C} \dagger} d Q^{\mathbb{C} } \right) + A_- \left(d Q^{\mathbb{C} } Q^{\mathbb{C} \dagger}  \right)\right]\Big] \\
=&-\frac{i}{4\pi} \int\limits_{\mathcal{M}} {\rm tr} \left[ A_+ \left( d\epsilon_+ - Q^{\mathbb{C} \dagger} d\epsilon_- Q^{\mathbb{C} } \right) + A_- \left( -d\epsilon_- + Q^{\mathbb{C} } d\epsilon_+ Q^{\mathbb{C} \dagger}   \right)   \right]
\end{align*}

Last, we repeat the previous step by adding another term with $d\epsilon_\pm $ in the above equation replaced by $A_\pm$. After some work we obtain

\begin{align*}
&\delta \Big[  -\frac{i}{12\pi} \int\limits_{\mathcal{B}} {\rm tr} \left[\left(Q^{\mathbb{C} \dagger} d Q^{\mathbb{C}} \right)^3 \right] - \frac{1}{4\pi} \int\limits_{\mathcal{M}} {\rm tr} \left[ A_+ \left( Q^{\mathbb{C} \dagger} d Q^{\mathbb{C} } \right) + A_- \left(d Q^{\mathbb{C} } Q^{\mathbb{C} \dagger}  \right)+ i A_+ Q^{\mathbb{C} \dagger}  A_- Q^{\mathbb{C} } \right]\Big] \\
=&-\frac{i}{4\pi} \int\limits_{\mathcal{M}} {\rm tr} \left[ A_+  d\epsilon_+ - A_- d\epsilon_-   \right]
\end{align*}

Now the gauge variant part contains no $Q^{\mathbb{C}}$ anymore. Hence we cannot find any term to cancel the remaining non-gauge-invariance. This result reproduces Bardeen's result in Bardeen1c.

In summary, the gauged WZW model is given by

\begin{align*}
W[Q^{\mathbb{C}},A_+,A_-]=&-\frac{1}{8\pi} \int\limits_{\mathcal{M}} d^2 x \, \tr\left[\left(Q^{\mathbb{C} \dagger} \left( \partial_\mu Q^{\mathbb{C}} - i Q^{\mathbb{C}} A_{+,\mu} + i A_{-,\mu} Q^{\mathbb{C}} \right) \right)^2 \right]\\
-&\frac{i}{12\pi} \int\limits_{\mathcal{B}} {\rm tr} \left[\left(Q^{\mathbb{C} \dagger} d Q^{\mathbb{C}} \right)^3 \right] \\
-& \frac{1}{4\pi} \int\limits_{\mathcal{M}} {\rm tr} \left[ A_+ \left( Q^{\mathbb{C} \dagger} d Q^{\mathbb{C} } \right) + A_- \left(d Q^{\mathbb{C} } Q^{\mathbb{C} \dagger}  \right)+ i A_+ Q^{\mathbb{C} \dagger}  A_- Q^{\mathbb{C} } \right]\Big] .
\end{align*}

Moreover, we have shown that it has the same 't Hooft anomaly for the continuous symmetry as the original massless fermion. In appendix <ref>  we summarize the gauged non-linear sigma model in $d=1,2,3$.

§.§ Discrete symmetry anomalies

In section <ref>, we saw that massless fermion theory has a time-reversal anomaly for the complex class in $(2+1)$-D.  This anomaly originates from the massive Dirac fermion at time reversal invariant $\v k$ points other than $\v k=(0,0)$ where the mass breaks time-reversal.  We would like to see the same phenomenon in the nonlinear sigma model.

In the following, we focus on the complex class in $(2+1)$-D. First, let's focus on the vicinity of $\v k = 0$ (under Wilson's regularization). The bosonized model is given by wzw2C. Following Witten's trial-and-error method discussed in the preceding subsection (see appendix <ref> for the detail), we obtain  the following gauged nonlinear sigma model,

\begin{align}
\label{MaingaugedWZW2C}
W[Q^{\mathbb{C}}, A] =&{1\over 2\lambda_3} \int\limits_{\mathcal{M}} d^3 x \, \tr\Big[\Big(\p_\mu Q^{\mathbb{C}}+i [A_\mu,Q^{\mathbb{C}}]\Big)^2\Big]  \\
-&  \frac{ 2 \pi i }{256 \pi^2}   \Big\{   \int\limits_{\mathcal{B}} \tr \Big[\tilde{Q}^{\mathbb{C}}    \,\left( d \tilde{Q}^{\mathbb{C}} \right)^4 \Big] \notag\\
+& 8 \int\limits_{\mathcal{M}} \tr \Big[ i A Q^{\mathbb{C}} (dQ^{\mathbb{C}})^2 - (A Q^{\mathbb{C}})^2 dQ^{\mathbb{C}} \notag\\
-& \frac{i}{3} (A Q^{\mathbb{C}})^3 + i A^3 Q^{\mathbb{C}} -AQ^{\mathbb{C}}F - AFQ^{\mathbb{C}} \Big]
\Big\} \notag
\end{align}

This action  is invariant under global symmetry transformations where the gauge field and $Q^{\mathbb{C}}$ are transformed according to gaugeTrans2C and table <ref>. This is expected, given the low energy fermion theory near $\v k=0$ respects these symmetries.

For the Dirac fermions near $\v k = (\pi,0), (0, \pi)$, and $(\pi,\pi)$, there are time reversal breaking masses, namely,  $M = 2m \, Y\otimes I_n$ for $\v k=(\pi,0), (0,\pi)$ and $M=4m \, Y\otimes I_n$ for $(\pi,\pi)$. The non-linear sigma model describes these massive fermions is again given by
MaingaugedWZW2C except that now $l=n$ or $0$. Due to the signs in front of $q_1\G_1$ and $q_2\G_2$ at $\v k = (\pi,0), (0, \pi)$, and $(\pi,\pi)$ the effective  mass sign for these massive fermions are given by
$$\eta_{\v k}:=\text {sign of~} (q_1\G_1)\times\text {sign of~} (q_2\G_2)\times\text {sign of~}(m).$$
Consequently the $Q^{\mathbb{C}}$ associated with the massive fermions obeys

We can thus use the gauged nonlinear sigma model in appendix <ref> to predict the Chern-Simons term due to the massive fermions at $\v k = (\pi,0), (0, \pi)$, and $(\pi,\pi)$ by plug in regf.
For these space-time constant $Q^{\mathbb{C}}$ we can drop all the terms with derivatives on $Q^{\mathbb{C}}$. The remaining can be combined into the Chern-Simons term. Summing the contribution from ${\v k}$ around $(\pi,0)$, $(0,\pi)$, and $(\pi,\pi)$, we get

\begin{align*}
W_{(\pi,0)}+W_{(0,\pi)}+W_{(\pi,\pi)} =&\left( -\frac{1}{2}-\frac{1}{2}+\frac{1}{2} \right) \frac{m}{|m|} \frac{i}{4\pi} \int \tr\left[A \, dA + \frac{2i}{3}A^3 \right] \\
=& -\frac{i}{8\pi}\int \tr\left[A \, dA + \frac{2i}{3}A^3 \right]
\end{align*}

which agrees with panom.

As for other discrete symmetry anomalies, with the input of how $Q^\mathbb{C,R}$ and the gauge field transform under discrete symmetries, it's simple to show that in $(1+1)$-D and $(3+1)$-D, there is no discrete-symmetry-anomaly after gauging the anomaly-free part of the continuous symmetries. In $(2+1)$-D, gauging the continuous symmetry breaks the time-reversal symmetry as discussed in subsection <ref>.

In appendix <ref>, we show that all the symmetry anomalies of massless fermions in table <ref> are reproduced by the corresponding gauged nonlinear sigma models. This lends strong support to the idea that the nonlinear sigma models are equivalent to the original massless fermion theories.

§ SOLITON OF THE NON-LINEAR SIGMA MODELS AND THE WESS-ZUMINO-WITTEN TERMS

In order for the bosonization to hold, somehow the bosonic non-linear sigma model must possess fermion degrees of freedom.  In this section, we show that due to the WZW term, the solitons of the non-linear sigma model are fermions.

§.§ Soliton classification

Soliton is a spatial texture of the “order parameter” ($Q^{\mathbb{C,R}}$). Such texture represents a non-trivial mapping from the spatial space to the mass manifold, i.e., the space where the order parameter lives.  In $d$ spatial dimension,  solitons are classified by the $d$-th homotopy group of the mass manifold, namely,

\begin{align*}
\pi_d \left( \text{mass manifold} \right).
\end{align*}

In appendix <ref>, we list the relevant homotopy groups. Since exchange  statistics only make sense for spatial dimension greater than one,  in the following we shall focus on $d\ge 2$. For the nonlinear sigma models considered in section <ref>, when $n$ is sufficiently large so that there is a WZW term, the soliton classifications are $\mathbb{Z}$ for the complex classes, and are $\mathbb{Z}_2$ for the real classes, namely,

\begin{align*}
&\pi_2 \Big( \frac{U(n)}{U(n/2) \times U(n/2)} \Big) = \mathbb{Z} &\text{  for } n\geq 4 \\
&\pi_2 \Big( \frac{O(n)}{O(n/2) \times O(n/2)} \Big) = \mathbb{Z}_2 &\text{  for } n\geq 6 \\
&\pi_3 \Big( U(n) \Big) = \mathbb{Z} &\text{  for } n\geq 3 \\
&\pi_3 \Big( \frac{U(n)}{O(n)} \Big) = \mathbb{Z}_2 &\text{  for } n\geq 5 \\
\end{align*}

This means that for the complex classes, we can define a topological quantum number, namely, the “soliton charge” $Q_{\rm sol}$. When we fuse two solitons of different charges, $Q_{\rm sol}$ adds; for the real classes, on the other hand, this soliton charge is defined $mod~ 2$  so that two solitons with unit soliton charges can fuse into zero soliton charge.

§.§ Soliton charge and the conserved $U(1)$ charge $Q$

For the complex classes, it is natural to ask what is the relation between the soliton charge $Q_{\rm sol}$ and the conserved charge $Q$. The conserved charge $Q$ is associated with a global U(1) symmetry. In $(3+1)$-D such U(1) symmetry belongs to a diagonal subgroup of  $U_+(n) \times U_-(n)$. As shown in table <ref>, it is anomaly-free. For $(2+1)$-D the $U(1)$ symmetry is a subgroup of the global symmetry group $U(n)$, which is also anomaly-free according to table <ref>.

In appendix <ref> we present the gauged non-linear sigma model. In particular, by focusing on the term linear in the gauge field (associated with the anomaly-free $U(1)$ subgroup) derived from the WZW term, we can extract the $U(1)$ current. The answer is [The same result can be derived by fermion integration.]

\begin{align}
\label{solitonCurrent}
&\text{$(2+1)$-D}:~ J^\mu = -\frac{i}{16 \pi} \epsilon^{\mu\nu\rho}\tr\left[Q^\mathbb{C} \partial_\nu Q^\mathbb{C} \partial_\rho Q^\mathbb{C}  \right] \\
&\text{$(3+1)$-D}:~J^\mu = -\frac{1}{24 \pi^2} \epsilon^{\mu\nu\rho\sigma}\tr\left[\left( Q^{\mathbb{C}\dagger} \partial_\nu Q^\mathbb{C}\right) \left( Q^{\mathbb{C}\dagger} \partial_\rho Q^\mathbb{C}\right)\left( Q^{\mathbb{C}\dagger} \partial_\sigma Q^\mathbb{C}\right)  \right]. \notag
\end{align}

Thus the $U(1)$ charge given by

\begin{align*}
&\text{$(2+1)$-D}:~Q =  -\frac{i}{16 \pi} \int d^2 {\v x} \, \epsilon^{ij}\tr\left[Q^\mathbb{C} \partial_i Q^\mathbb{C} \partial_j Q^\mathbb{C}  \right] \\
&\text{$(3+1)$-D}:~Q = -\frac{1}{24 \pi^2} \int d^3 {\v x} \, \epsilon^{ijk}\tr\left[\left( Q^{\mathbb{C}\dagger} \partial_i Q^\mathbb{C}\right) \left( Q^{\mathbb{C}\dagger} \partial_j Q^\mathbb{C}\right)\left( Q^{\mathbb{C}\dagger} \partial_k Q^\mathbb{C}\right)  \right].\\
\end{align*}

These are, in fact, exactly the same expression as the topological invariant corresponding to $\pi_2(\frac{U(n)}{U(n/2) \times U(n/2)})=\mathbb{Z}$ in $(2+1)$-D and $\pi_3(U(n))=\mathbb{Z}$ in $(3+1)$-D (see appendix <ref> for the details). Thus, for both cases

\begin{align}
\label{solitonU(1)charge}
Q=Q_{\rm sol}.
\end{align}

§.§ Statistics of soliton

One way to derive the statistics of soliton is to calculate the topological spin by comparing Berry's phase difference between the following two processes. In the first process, we have a static soliton. In the second process, the spatial soliton configuration is adiabatically rotated by $2\pi$ in time. Following Witten [22], we show in appendix <ref> that such Berry's phase difference is $e^{-i k \pi}$, where $k$ is the level of the WZW term (see appendix <ref> for the details).  Since all nonlinear sigma models in section <ref> have $k=1$ WZW term, their solitons are fermion.

§ A SUMMARY OF PART I

So far, we have established the fact that the fermion and boson theories have the same global symmetries and anomalies. In addition, we have shown that the solitons of the bosonic theories are fermions.  All these support the equivalence between the fermion and boson theories. Now we present a brief summary of part I.

We begin in section <ref> by presenting the essential idea underlying the present work. Prior to performing the fermion integration, we first identify the emergent symmetries in section <ref>, and the mass manifolds in section <ref>. For a given massless fermion theory, the mass manifold is the topological space formed by all mass terms that can fully gap out the fermions. We then work out the anomalies with respect to the emergent symmetries in section <ref>. Afterward, we introduce mass terms at the expense of breaking the emergent symmetries in section <ref> and fluctuate the mass terms smoothly to regain the emergent symmetries in section <ref>. As discussed in section <ref>, the smoothness of the mass fluctuations is to ensure that the original fermions remain gapped, hence can be integrated out to yield non-linear sigma models in section <ref> and section <ref>.[The procedure can be easily applied to higher dimensions, though we shall not pursue it in the present paper.]. The level-1 WZW term  resulting from the fermion integration is checked against the prediction of homotopy groups in the appendix, which is referred to in sections <ref> and <ref>. In section <ref>, we present local interacting fermion theories that have duality-like relationships with the bosonized non-linear sigma models. In section  <ref>, we analyze the symmetries of the non-linear sigma models. A comparison with the results obtained in section <ref> leads to the conclusion that the fermion and boson theories have the same symmetry. Using the method of reference [21] we determine the anomalies of the non-linear sigma models in section <ref>. A comparison with the results obtained in section <ref> leads to the conclusion that the fermion and boson theories have the same anomalies. Finally, in section <ref>, we show the bosonized theories have fermionic degrees of freedom, namely the solitons of the non-linear sigma models.

§ THE SU(2) GAUGE THEORY OF THE $\PI$-FLUX PHASE OF THE HALF-FILLED HUBBARD MODEL

§.§ The “spinon” representation of the spin operator

The paradigmatic model describing a Mott insulator is the Hubbard model in the large $U$ limit. At half-filling, every site is occupied by one electron. Below the Mott-Hubbard gap, the active degrees of freedom are those of spins. Through Anderson's super-exchange [23], the dynamics of the spins is governed by the anti-ferromagnetic Heisenberg interaction

\begin{align*}
\hat{H} =& \sum_{\Avg{ij }} J_{ij} \vec{S}_i \cdot \vec{S}_j.
\end{align*}

In the “spinon” treatment [24, 25] one decomposes a spin-$1/2$ operator into auxiliary fermion (spinon) operators
S_i^a =1/2 f_i^†σ^a_ f_i,

and supplement it with the single occupation constraints       f_i^†f_i+f_i^†f_i=1      f^†_if^†_i=0      f_if_i=0.
In the following we shall refer to the above constraints as the “Mott constraint”.
The decomposition in stspn, where one separates the physical spin degrees of freedom into the auxiliary “spinon” degrees of freedom, is an example of the so-called “slave particle” approach.

In terms of the spinon operators the Heisenberg Hamiltonian read

\begin{align*}
\hat{H} =&\frac{1}{4} \sum_{\Avg{ij }} J_{ij}\left( f_{i\alpha}^\dagger \sigma^a_{\alpha \beta} f_{i\beta} \right) \left(  f_{j\gamma}^\dagger \sigma^a_{\gamma \delta} f_{j\delta} \right) \\
=&\frac{1}{4} \sum_{\Avg{ij }} J_{ij}\left( -f_{i\alpha}^\dagger f_{i\alpha} f_{j\beta}^\dagger f_{j\beta} - 2 f_{i \alpha}^\dagger f_{j \alpha} f_{j \beta}^\dagger f_{i \beta}  \right)  \\
= &-\frac{1}{2} \sum_{\Avg{ij }} J_{ij}\left(\frac{1}{2}f_{i\alpha}^\dagger f_{i\alpha} f_{j\beta}^\dagger f_{j\beta} +f_{i \alpha}^\dagger f_{j \alpha} f_{j \beta}^\dagger f_{i \beta}  \right)
\end{align*}

Upon Hubbard-Stratonovich transformation, we express

\begin{align}
&\exp\Big\{-\int_0^\b d\t \Big[\sum_i f^\dagger_{i\a}\p_0f_{i\a}+H\Big]\Big\}=\nn
&\int D[U]\exp\Big\{-\int_0^\b d\t \Big[\sum_i\ \psi^\dagger_{i}\p_0 \psi_{i}+\sum_{\Avg{ij}} \frac{3}{8} J_{ij}\Big( - \left(\psi^\dagger_{i}U_{ij} \psi_{j} + h.c.\right) + \frac{1}{2}\Tr \left[U_{ij}^\dagger U_{ij} \right] \Big)\Big]\Big\}.\nn
\end{align}

\begin{align}
\psi_i=\begin{pmatrix}f_{i\ua}\\f^\dagger_{i\da}\end{pmatrix}, U_{ij}=
\begin{bmatrix}
\chi_{ij}^*		&	\Delta_{ij}	\\
\Delta_{ij}^*	&	-\chi_{ij}
\end{bmatrix}.\label{mfan}
\end{align}

For later convenience, we  rewrite the spinon operator in terms of Majorana fermions

\begin{align*}
f_{i \alpha} := F_{i ,1 \alpha} + i F_{i, 2 \alpha},
\end{align*}

in terms of which, the  spin operators are represented as
S_i^a =1/2 F_i^†Σ^a F_i,  where      Σ^a = (YX, IY, YZ ).

In the last line, the first and second Pauli matrices carry  the Majorana and spin indices, respectively.

The spin operators in sop are invariant under the following local “charge-SU(2)”

$$F_i\ra W_i F_i$$

where $W_i$ is generated by

$$T^b =( XY, YI, ZY).$$

In terms of The Majorana fermion operators, the Mott constraint in constraint becomes
f^†_i α f_i α -1=  F^T_i ( YI ) F_i =F^T_i T^2 F_i=0       ϵ^αβ ( f_i α f_i β +f^†_iβ f^†_i α  )=  F^T_i ( XY ) F_i =F^T_i T^1 F_i=0       iϵ^αβ (  f_i α f_i β - f^†_iβ f^†_i α  )=  F^T_i ( ZY ) F_i =F^T_i T^3 F_i=0
These constraints are implemented via the Lagrange multipliers in the  path integral

Z=\int D[F] D[U]D[a_0] \exp{\left(-S\right)}

\begin{align}
S=&\int_0^\b d\t \Big\{\sum_i F^T_{i}\p_0 F_{i}+ \sum_{\Avg{ij}} \frac{3}{8} J_{ij} \Big[F_i^T\Big(  Re[\chi_{ij} ] YI + i \, Im[\chi_{ij}] II  + Re[\Delta_{ij}] XY\notag\\&- \, Im[\Delta_{ij}] ZY \Big) F_j
+|\chi_{ij}|^2 + |\eta_{ij}|^2 \Big] + i\sum_i   a^b_{i0}  \left( F^T_i T^b F_i\right)\Big\}.
\label{spap}
\end{align}

§.§ The $\pi$-flux phase mean-field theory and the $SU(2)$ gauge fluctuations

In treating the path integral, spap, one often starts from a mean-field theory where $U_{ij}$ and $a^b_{i0}$ are assumed to be space-time independent. To see the many possible mean-field ansatzes we refer the readers to, e.g., Ref.[25]. In the following, we shall focus on the so-called “$\pi$-flux phase mean-field theory”[26] for the nearest neighbor Heisenberg model.

The $\pi$-flux mean field theory assumes the following mean-field $\bar{U}_{ij}$ and $\bar{a}^b_{i0}$
Δ̅_ij=0,  a̅^b_i0=0,      χ̅_i,i+x̂ = i χ,       χ̅_i,j+ŷ = i (-1)^i_x χ
where $\chi$ is a real parameter (see piflux).

The $\pi$-flux mean-field theory. Here the black bonds represent hopping amplitude $i\chi$ in the positive x- or y-direction and the green bonds represent $-i\chi$. The unit cell is enclosed by the orange rectangle.

This leads to the
following fermion mean-field Hamiltonian,

\begin{align*}
\hat{H}_{\rm MF}= -\frac{3}{4} J \sum_{i} \Big\{ i \, \chi \left[F_{i+\hat{x}}^T \left(II\right) F_{i }\right]+ i \, (-1)^{i_x} \chi \left[F_{i+\hat{y}}^T \left(II\right) F_{i }\right]+ h.c.  \Big\}
\end{align*}

Because the Pauli matrices are identity in both the Majorana and spin spaces, this mean-field Hamiltonian enjoys both global spin- and charge-SU(2) symmetries generated by
Spin-SU(2) generators:  Σ^a = (YX, IY, YZ )      Charge-SU(2) generators:  T^a = (XY, YI, ZY ).

Using the eigenvalues $\pm 1$ of the “sub-lattice Pauli matrix” $Z$ to label the blue and red sub-lattices in piflux,  and performing Fourier transform we obtain the following momentum-space mean-field Hamiltonian

Ĥ_MF = - 3/4J χ  ∑_k F_-k^T   [ II ⊗
i ( e^i k_2 - e^-i k_2 ) 	   	-i + i e^2i k_1

i - i e^-2ik_1								   	-i( e^ik_2 - e^-ik_2 )
] F_k      =- 3/4J χ  ∑_k F_-k^T   [ II ⊗( - sin2k_1   X + ( 1-cos2k_1 )Y -2 sink_2   Z )] F_k.
In the above equation the tensor product of Pauli matrices are ordered according to

$$\text{Majorana}\otimes \text{spin}\otimes \text{sub-lattice}.$$
In mjmf the (halfed) Brillouin zone is
$$-\pi/2\le k_1< \pi/2, ~~-\pi\le k_2<\pi$$
and the Dirac nodes are situated at ${\bf k}_0= (0, 0)$ and $(0, \pi)$, which are referred to as two “valleys” in the following.

Expand $\bf k = \v k_0 + \v q$ around these two Dirac nodes, and Fourier transform (w.r.t. $\v q$) back to the real space, we obtain the following low energy mean-field Hamiltonian

\begin{align*}
\hat{H}_{\rm MF} = \int d^2 x  \, \tilde{F}^T \left( -i \Gamma_i \partial_i  \right) \tilde{F},
\end{align*}

Γ_1 = IIXI       Γ_2 = IIZZ.

The tensor product of four Pauli matrices in gmmt are arranged according to
$$\text{Majorana}\otimes \text{spin}\otimes \text{sub-lattice}\otimes \text{valley}.$$
Including the sub-lattice and valley Pauli matrices the generators of the charge and spin SU(2) transformations are given by
Spin-SU(2) generators:  Σ^a = (YXII, IYII, YZII )      Charge-SU(2) generators:  T^a = (XYII, YIII, ZYII ).

Because the local charge-SU(2) gauge degrees of freedom is a redundancy in the original half-filled Mott insulator, we expect the field theory in spap to have the local charge-SU(2) symmetry. This motivates one to think the low energy theory, including fluctuations in $U_{ij}$ and $a_{i0}^b$, is a charge-SU(2) gauge theory with

$$U_{ij}=\bar{U}_{ij}e^{i a_{ij}}$$

where $a_{ij}=a_{ij}^b T^b$ is the spatial component of the charge-SU(2) gauge field. According to Ref.[25, 27], because the mean-field $\bar{U}_{ij}$ commutes with the global charge-SU(2) transformations, the low theory is a charge-SU(2) gauge theory, with  $a_0$ and $a_{ij}$ playing the roles of the time and spatial components of the gauge field, respectively.

The partition function of the charge-SU(2) gauge theory reads
𝒵=∫D[F̃] D[a_μ] e^-S[F̃,a_μ]      S=∫d^3 x {F̃^T[( ∂_0 + i a_0^a T^a )-i ∑_i=1^2 Γ^i(∂_i + i a_i^a T^a)]F̃+ 12g f_μν^2}.

In SU2Higgs the ${1\over 2g} f_{\mu\nu}^2$ is generated by integrating out the higher energy fermion degrees of freedom.
The theory in SU2Higgs describes the $n=8$ real class fermion theory coupled to a dynamic charge-SU(2) gauge field.

According to the bosonization in section <ref>, the bosonized theory is a gauged $\frac{O(8)}{O(4) \times O(4)}$ nonlinear sigma model with the $k=1$ WZW term [Note that although for $n=8$, the homotopy group are not yet stabilized, fermion integration still gives a WZW term. When $\mathcal{B}$ is a closed manifold, and after division by $2\pi i$, the WZW term is the topological invariant of one of the $\mathbb{Z}$ factor of the $\pi_4$.]. Here the charge-SU(2) subgroup  of the fermion (emergent) global symmetry group $O(8)$ is gauged.

In the following let's assume that the effect of the SU(2) gauge field is to cause confinement (note, however, we are not implying the deconfined phase does not exist)
Note that unlike the compact U(1) gauge field, here the confinement can be not caused by the proliferation of monopoles. This is based on the following homotopy argument. The $SU(2)$ gauge configurations on the space-time surface $S^2$ surrounding the location of the monopole are classified by the mapping  classes of $S^2\ra BSU(2)$, where $BSU(2)$ is the classifying space of $SU(2)$. Using the following identity in algebraic topology,

\begin{align*}
[S^2, BSU(2) ]_* = [\Sigma S^1, BSU(2)]_* = [S^1,SU(2)]_* = \pi_1(SU(2))=0,
\end{align*}

it follows that there is no topologically non-trivial gauge field configuration on $S^2$, hence there is no monopole. Here $\Sigma$ denotes “reduced suspension”, and $[X_1,X_2]_*$ is the homotopy class of base-point-preserving maps $X_1\rightarrow X_2$. Physically speaking, assuming the  $SU(2)$ monopole exists, we can take the northern and southern hemispheres as the patches to define the gauge connection so that on each patch, the gauge field configuration is non-singular. On the equator, $S^1$, where the two patches overlap, a gauge transformation must relate the gauge fields originated from the two patches. At each point of $S^1$ the gauge transformation is an element in $SU(2)$.
Therefore the monopole classification is given by the homotopy class of gauge transformation on the $S^1$, i.e., $\pi_1(SU(2))$.
Under such condition, the fermion-antifermion pair oder parameter (analogous to mesons in QCD) must be a charge-SU(2) singlet.
Since $Q^\mathbb{R}$ is precisely the “meson” field, it follows that in the charge-SU(2) confined phase the finite energy $Q^\mathbb{R}$ are restricted to a sub-manifold of $\frac{O(8)}{O(4)\times O(4)}$ which are invariant under the charge-SU(2) transformation [In addition to restricting $Q^\mathbb{R}$ to be invariant under charge-SU(2) transformations, the charge-SU(2) gauge fluctuations can also generate four-fermion interactions, the effects of which are not studied in the current work.][Our result is analogous to Witten's non-linear sigma model description of QCD in the color SU(3) confined phase[22].]. This sub-manifold is the $S^4$ spanned by the following $5$ mutually anti-commuting masses,
S^4={∑_i=1^5 n_i M_i;   ∑_i=1^5 n_i^2=1},   where      M_i = YXZX, IYZX, YZZX, IIYI, IIZY.

In order to match the gamma matrices and mass matrices convention in table <ref> and <ref> (based on which the non-linear sigma models in subsection <ref> and appendix <ref> and <ref> are derived), we will make the following change the basis. We first exchange the order of the third and the fourth (i.e., sub-lattice and valley) Pauli matrices, followed by the orthogonal transformation,

\begin{align*}
II\otimes \begin{bmatrix} I & 0 \\ 0 & X \end{bmatrix}
\end{align*}

In the new basis the gamma matrices and the mass terms become
Γ̃_1 	=IIIX       Γ̃_2 	=IIIZ       M̃_̃ĩ			=YXYY, IYYY, YZYY, IIXY, IIZY
These are consistent with the matrices shown in table table <ref> and table <ref>, except a trivial exchange of the first and the last Pauli matrices. In this basis, the order parameter $Q^\mathbb{R}$ is defined by $\tilde{M} =m\, Q^\mathbb{R} \otimes Y$.

§.§ Antiferromagnet, Valence bond solid, and the “deconfined” quantum crtical point

For the mass manifold in NV, we expect the non-linear sigma model to have a WZW term because $\pi_4(S^4)=\mathbb{Z}$.
Q^ℝ=n_i N_i where      N_i =(YXY, IYY, YZY, IIX, IIZ )
into the non-linear sigma model given by wzw2R in subsection <ref>
we obtain
W[n̂] =2λ_3∫_ℳ d^3 x   ( ∂_μn_i )^2 - 2πi /64 π^2 ∫_ℬ ϵ^ijklm ñ_i   dñ_j    dñ_k   dñ_l   dñ_m.
This model has O(5) global symmetry generated by the pair-wise product of the matrices in $\tilde{M_i}$, which are also the generators of $O(8)$ that commutes with the charge-SU(2) generators. Hence o5 is often referred to as the “O(5)” non-linear sigma model in the literature [28, 29, 30, 31] (a recent related work can be found in [32]).

Now we address the physical meaning of the five masses given in NV (or equivalently the physical meaning of  $\tilde{M}_i$ in fmss).
The first three of the masses in NV correspond to the Néel order parameters, while the last two to the valence bond solid (VBS) orders. To see this, we first note that the first three masses rotate into each other under spin-SU(2),

\begin{align*}
\Sigma^a = \left( YXII, IYII, YZII \right)
\end{align*}

while the last two are invariant.

We can also deduce the effect of translation by one lattice constant on these mass terms. In writing down the mean-field Hamiltonian we have chosen a particular charge-SU(2) gauge that explicitly breaks the symmetry associated with x-translation by one-lattice spacing. However, this is an artifact of gauge choice. The compounded transformation where the x-translation is followed by the  gauge transformation  which multiplies the fermion operator located on the orange rows in PSG by $-1$
( F_ℐ,1, F_ℐ,2 )  (-1)^ℐ_y ×( F_ℐ,2 , F_ℐ+x̂,1 )       ( F_ℐ,1, F_ℐ,2 )   ( F_ℐ+ŷ,1 , F_ℐ+ŷ,2 ),

leaves the mean-field ansatz invariant. This is an example of “projective transformation”. In tsg1 $\mathcal{I}$ label the unit cell in piflux
, and we have omitted the Majorana and spin indices because they are unaffected by the translation.

Translation by one lattice constant in the x-direction compounded with the gauge transformation which multiplies the fermion operators on sites in the orange rows by -1 leaves the mean-field Hamiltonian invariant.

In the following, we derive the effects of the “projective translation” on the fermion operator $\tilde{F}$ which is related to $F$ via

\begin{align*}
F_{\mathcal{I},l}=\sum\limits_{\text{small } \v q} \left( \tilde{F}_{q,(l,1)} e^{i {\v q} \cdot \mathcal{I}} +\tilde{F}_{q,(l,2)} e^{i ((0,\pi)+{\v q}) \cdot \mathcal{I}}\right)
\end{align*}

where the $(l,v)$  are the indices for sub-lattice and valleys respectively. Doing the inverse Fourier transform, the above projective translation transforms $\tilde{F}$ according to

$$\tilde{F}\ra T_{\hat{x},\hat{y}}\tilde{F}$$

T_x̂ = IIXX      T_ŷ = IIIZ.
Here we have put back the Majorana and spin  (i.e., the first two) Pauli matrices.

Under $T_{\hat{x},\hat{y}}$ the mean-field Hamiltonian is invariant, but the first three mass terms change sign under $\hat{T}_{\hat{x}}$ and  $\hat{T}_{\hat{y}}$ (as should the Néel order parameter) while the remaining two masses each breaks $\hat{T}_{\hat{x}}$ or $\hat{T}_{\hat{y}}$. These are the expected transformation properties of the VBS order parameters.

In appendix <ref> we show that the order parameters in fmsg completely decouple from the charge-SU(2) gauge field.  Thus even in the presence of such gauge field the non-linear sigma model preserves the form in o5.
Before moving on, there is one additional thing worth mentioning, namely,
Hence there is no soliton in the order parameter associated with o5.

In summary, we have found that after the charge-SU(2) confinement  o5 describes the critical point between the AFM and VBS phases the so-called “deconfined quantum critical point”[28, 29, 33, 34]. It is important to note that in the treatment so far, we have assumed that the sole effect of the charge-SU(2) confinement is to restrict $Q^\mathbb{R}$ to the appropriate submanifold of $\frac{O(8)}{O(4)\times O(4)}$. However, the SU(2) gauge field fluctuations can also induce four-fermion interactions, the effects of which are not studied in the current work.
It is very satisfying that aside from the Néel order parameter, fmsg captures the best-known VBS order parameter once the Néel is destabilized by quantum fluctuation[35, 36, 37, 38, 39].

§ THE CRITICAL SPIN LIQUID OF “BIPARTITE MOTT INSULATORS” IN  $D=1+1,2+1$ AND $3+1$.

The idea explained in the preceding section can be generalized to the insulating phase of “bipartite Mott insulators”.

A bipartite Mott insulator is a Mott insulator whose lattice consists of two sub-lattices, and hoppings only occur between different sub-lattices. The nearest-neighbor spin-$1/2$ antiferromagnetic Heisenberg model in one spatial dimension  describes the dynamics of spin degrees of freedom in a one-dimensional bipartite Mott insulator. It realizes the $SU(2)_1$ WZW non-linear sigma model, where the emergent symmetries are realized in a non-onsite (e.g., translation) fashion. This model serves as a paradigm of, e.g., quantum number fractionalization, and has profoundly influenced theoretical physics. It is natural to ask what is the generalization of this non-linear sigma model in the Mott insulating phase of higher dimensions. In the present section, we answer this question.

In a Mott insulating phase, the low energy degrees of freedom are the spins.  Since the spin operators are invariant under the charge-SU(2) transformation, there are lots of choices in fractionalizing the spin into spinons. Different choices are related by the spinon charge-SU(2) gauge transformation. The spin-spin interaction is generated by Anderson's super-exchange,  the spinon mean-field theory amounts to choosing a spinon tight-binding model which reproduces the spin-spin interaction after super-exchange. Since the spin-spin interaction is independent of which charge-SU(2) gauge we choose, we shall choose the gauge so that the hoppings are purely imaginary in the following. The reason for doing so is because in such a gauge, the mean-field spinon Hamiltonian is charge-SU(2) invariant. This gauge choice exists when the Mott insulator is bipartite.

In section <ref>, we saw that the Mott insulating condition is imposed by the constraint that the order parameter  $Q^{\mathbb{R}}$ is a charge-SU(2) singlet.
In this section we show that imposing such constraints allows us to derive the spin effective theory in  bipartite Mott insulators in spatial dimensions 1,2 and 3 [Although we will not pursue it in the present paper, the discussion in the following can be generalized to the cases with larger flavor number or in higher dimensions.].

§.§ (1+1)-D

§.§.§ The analog of the $\pi$-flux phase

For the nearest neighbor tight-binding model with real hopping in 1D, one can break the lattice into $A,B$ sub-lattice and do the transformation $(c^A_j ,c^B_{ j }) \rightarrow ( c^A_j, i \,c^B_{ j })$ to make the hopping purely imaginary (see figure <ref>). This leads to the lattice model

\begin{align*}
\hat{H} = &
t \sum_k c_k^\dagger\left[  I \otimes \begin{pmatrix}
0	&	i + i e^{-ik} 	\\
-i-ie^{+ik}	&	0		\\
\end{pmatrix} \right] c_k \\
=& -t  \sum_k c_k^\dagger \left[ -(\sin k) IX +(1+\cos k) IY  \right] c_k
\end{align*}

Here identity matrix $I$ part acts on the spin. After linearizing around $k_F=\pi$,  the low energy effective theory in Majorana fermion basis reads

H=\int dx ~\chi^T(x)~ \left[ -i \G_1 \partial_1 \right]~\chi(x)$$

\begin{align}
\Gamma_1 = I I X.
\end{align}

To comply with the gamma matrix notation in table <ref>, we perform a basis transformation $\chi \rightarrow e^{-i\frac{\pi}{4}IIY} \chi$, so that the gamma matrix becomes

\begin{align}
\label{ggmm1}
\Gamma_1 = I I Z.
\end{align}

Here the tensor product of Pauli matrices are arranged according to

$$\text{Majorana}\otimes \text{spin}\otimes \text{sub-lattice}.$$
In the presence of Hubbard $U$, there is the global charge-SU(2)
symmetry at half-filling. In the low energy theory, the charge-SU(2) transformation is generated by

\begin{align*}
\text{Charge-$SU(2)$ generators}: T^a = (XYI, YII, ZYI)
\end{align*}

On the other hand, the spin-SU(2) transformations are generated by the following charge-SU(2) invariant matrices,

\begin{align*}
\text{Spin-$SU(2)$ generators}: \Sigma^a = (YXI, IYI, YZI)
\end{align*}

Real hopping

Imaginary hopping

(a) The usual 1D nearest neighbor tight-binding with real hopping. (b) Upon the gauge transformation $(c^A_j ,c^B_{ j }) \rightarrow ( c^A_j, i \,c^B_{ j })$, hoppings become purely imaginary with alternating sign. The arrows point in the direction where the hopping is $+it$. The hopping Hamiltonian in panel (b) is charge-SU(2) invariant.

§.§.§ The charge-SU(2) confinement

Following the discussion in section <ref>, the massless free fermion theory is equivalent to the $O(4)$ level-$1$ WZW model. Gauging the charge-SU(2) symmetry of the sigma model, and integrating over the gauge field, amounts to imposing the Mott insulating constraint.
Assume the system is in the charge-SU(2) confined phase, only charge-SU(2) singlet order parameters (mass terms) can exist at low energies. These mass terms satisfy

\begin{align*}
&\{ \Gamma_1 , M \} = 0 \\
&[T^a , M ] =0 \\
&M^2 = I_{8}
\end{align*}

The most general mass $M$ satisfying the first two lines has the form

\begin{align}
\label{mass1H}
M =	n_0 \, IIY+ n_1 \, YXX+n_2 \, IYX  + n_3  \, YZX
\end{align}

Among the mass terms

\begin{align*}
IIY, YXX, IYX, YZX,
\end{align*}

the last three rotate into each other under the action of spin-SU(2) transformations, and the first one is invariant. They corresponds to the dimer and Néel order parameters respectively.
The condition that $M^2=I_8$ gives
$$\sum_{i=0}^3 n_i^2=1.$$

The non-linear sigma model describing the fluctuations of $\hat{n}$ has a WZW term because $\pi_3(S^3)=\mathbb{Z}$, namely,

\begin{align}
W[\hat{n}] = \frac{1}{4\pi} \int\limits_{\mathcal{M}} d^2 x \, \left( \partial_\mu \hat{n} \right)^2 - \frac{2\pi i }{12 \pi^2} \int\limits_{\mathcal{B}} \epsilon^{ijkl} \tilde{n}_i \, d\tilde{n}_j  \, d\tilde{n}_k \, d\tilde{n}_l .
\label{o4}
\end{align}

This is the $SU(2)_1$ non-linear sigma model, known to be the effective theory of the Heisenberg spin chain [40].

§.§ $(2+1)$-D

§.§.§ The analog of the  $\pi$-flux phase

Real hopping

Imaginary hopping

(a) The usual nearest neighbor tight-binding model on the honeycomb lattice with real hopping. (b) Upon the gauge transformation $(c^A_j ,c^B_{ j }) \rightarrow ( c^A_j, i \,c^B_{ j })$, hoppings become purely imaginary with alternating sign. The arrows point in the direction where the hopping is $+it$. The tight-binding Hamiltonian in panel (b) is charge-SU(2) invariant.

In $(2+1)$-D we use the honeycomb lattice to write down the tight-binding model. The lattice vectors in the real and momentum space are

\begin{align*}
{\v a}_1 = \sqrt{3} a \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right), ~~~
{\v a}_2 = \sqrt{3} a \left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right)
\end{align*}

\begin{align}\label{reci}
\v b_1 = \frac{4 \pi}{3a} \left(\frac{\sqrt{3}}{2}, \frac{1}{2} \right), ~~~
\v b_2 = \frac{4 \pi}{3a} \left(-\frac{\sqrt{3}}{2}, \frac{1}{2} \right),
\end{align}

In the following we perform the gauge transformation
$$(c^A_j ,c^B_{ j }) \rightarrow ( c^A_j, i \,c^B_{ j })$$
on the two sub-lattices, so that the nearest-neighbor hopping becomes purely imaginary (see figure <ref>).
The tight-binding Hamiltonian reads

\begin{align*}
\hat{H} = &
t \sum_{\v k} c_{\v k}^\dagger\left[  I \otimes \begin{pmatrix}
0	&	i + i e^{i {\v k} \cdot {\v a_1}} + i e^{i {\v k} \cdot {\v a_2}}	\\
-i - i e^{-i {\v k} \cdot {\v a_1}}  - i e^{-i {\v k} \cdot {\v a_2}}	&	0		\\
\end{pmatrix} \right] c_{\v k} \\
t \sum_{\v k} c_{\v k}^\dagger \Big[
-\left(\sin (\v k \cdot \v a_1) + \sin (\v k \cdot \v a_2) \right) IX
-\left( 1 +\cos (\v k \cdot \v a_1) + \cos (\v k \cdot \v a_2) \right) IY
\Big] c_{\v k}
\end{align*}

Here the Pauli matrices are arranged according to

$$ {\rm spin}\otimes \text{ sub-lattice}.$$

In the Majorana fermion basis,
Ĥ=t ∑_ǩ χ_ǩ^†[
-(sin(ǩ ·ǎ_1) + sin(ǩ ·ǎ_2) ) IIX
-( 1 +cos(ǩ ·ǎ_1) + cos(ǩ ·ǎ_2) ) IIY
] χ_ǩ.
Here the Pauli matrices are arranged according to

In the presence of repulsive Hubbard $U$, there is charge-SU(2) symmetry at half-filling. In the low energy theory the charge-SU(2) transformation is generated by

\begin{align*}
\text{Charge-SU(2) generators}: T^a = (XYI, YII, ZYI)
\end{align*}

On the other hand, the spin-SU(2) transformations are generated by the following  matrices,

\begin{align*}
\text{Spin-SU(2) generators}: \Sigma^a = (YXI, IYI, YZI)
\end{align*}

latticeHoneycomb is invariant under both the charge- and spin-SU(2). In momentum space
the Dirac points are located at $K$ and $\hat{K}$ points, i.e., $\pm k_0$ where $k_0 :=  \frac{1}{3} \left( \v b_1 - \v b_2 \right)$ (see reci). Note that in the Majorana fermion basis, the contribution of Hamtiltonian from $\v k$ and $-\v k$ are the same due to the constraint $\chi_{- \v k}^T = \chi^\dagger_{\v k}$. This means that one can take the fermion $\chi_{\v k_0 + \v q}$ around $k_0$ as the Fourier modes of complex fermion $\tilde{c}_{\v q}$ and discard the other node. We then break this complex fermion $\tilde{c}$ into real fermion by $\tilde{c} = \tilde{\chi}_1 + i \tilde{\chi}_2$ (in the following we shall refer to this 1 and 2 as the “valley” indices). In this final Majorana representation, the low energy Hamiltonian reads

\begin{align*}
\hat{H}=\int d\v x ~\tilde{\chi}^T(\v x)~ \left[ -i \G_1 \partial_1 - i \G_2 \partial_2 \right]~\tilde{\chi}(\v x)
\end{align*}

where $\G_1 = IIIX$ and $\G_2 = IIYY$ [This is related to the gamma matrices in table <ref> by a basis transformation]. Here the Pauli matrices are arranged according to

In this basis, the symmetry generators are

\begin{align*}
&\text{Charge $SU(2)$ generators}: T^a = (XYII, YIII, ZYII)\\
&\text{Spin $SU(2)$ generators}: \Sigma^a = (YXII, IYII, YZII).
\end{align*}

§.§.§ The charge-SU(2) confinement

Following the discussions in section <ref>, the massless fermion theory is equivalent to the $\frac{O(8)}{O(4) \times O(4)}$ level-$1$ WZW model. Notice that the low energy fermion theory is identical to the $\pi$ flux phase spinon mean-field theory discussed in section <ref>.
Imposing the Mott constraint  constraints the mass manifold. Specifically it requires the mass terms to commute with the charge-SU(2) generators. Under conditions the allowed mass terms satisfy

\begin{align*}
&\{ \Gamma_i , M \} = 0 \\
&[T^a , M ] =0 \\
&M^2 = I_{16}
\end{align*}

The most general mass, $M \in \frac{O(8)}{O(4) \times O(4)}$, satisfying the first two equations has the form

\begin{align*}
M =	n_1 \, YXIZ + n_2 \, IYIZ + n_3 \, YZIZ + n_4 \, IIXY + n_5 \, IIZY
\end{align*}

Similar to the discussion in section <ref>, the first three of the masses in NV correspond to the Néel order parameters, while the last two to the valence bond solid (VBS) order parameters. The order parameter space forms an $S^4$. Plugging it into the $\frac{O(8)}{O(4) \times O(4)}$ level-$1$ WZW model, we arrive at the $O(5)$ WZW theory

W[n_i] = \frac{2}{\lambda_3} \int\limits_{\mathcal{M}} d^3 x \, \left( \partial_\mu n_i \right)^2 - \frac{2\pi i }{64 \pi^2} \int\limits_{\mathcal{B}} \epsilon^{ijklm} \tilde{n}_i \, d\tilde{n}_j  \, d\tilde{n}_k \, d\tilde{n}_l \, d\tilde{n}_m.
Here we note that because $\pi_2(S^4)=0$ there is no soliton.

§.§ $(3+1)$-D

§.§.§ The analog of the $\pi$-flux phase

Real hopping

Imaginary hopping

(a) The tight-binding model on a stacked honeycomb lattice with real-valued nearest-neighbor hopping. Blue/red (A/B) mark the two sub-lattices of the honeycomb lattice respectively. The positive hoppings are drawn in black, while the negative hoppings in white.(b) After the gauge transformation $(c^{A_1}_j , c^{B_1}_j, c^{A_2}_j , c^{B_2}_j) \rightarrow (c^{A_1}_j , i c^{B_1}_j, i c^{A_2}_j , c^{B_2}_j)$,  a unit cell contains four sites. This is marked by blue/red/green/orange and labeled as $A_1/B_1/A_2/B_2$  respectively. The hoppings become purely imaginary. The arrows point in the direction where the hopping is $+it$. The tight-bonding Hamiltonian in panel (b) has charge-SU(2) symmetry.

As a model for bipartite Mott insulator, we begin with a 3-dimensional tight-binding model consists of stacked honeycomb lattice.  Here the lattice sites of each layer are stacked on top of those in the layer beneath. Within each layer, we have real hopping between the nearest-neighbor sites described in section <ref>. Between layers, the (real) hopping have the opposite sign for the $A$ and $B$ sub-lattice (see figure <ref>). In order to make the hopping terms global charge-SU(2) invariant, we first enlarge the unit cell by grouping two adjacent layers to form $A_1$, $B_1$, $A_2$, $B_2$ sub-lattices as shown in figure <ref>. We then perform the following gauge transformation,

\begin{align*}
(c^{A_1}_j , c^{B_1}_j, c^{A_2}_j , c^{B_2}_j) \rightarrow (c^{A_1}_j , i c^{B_1}_j, i c^{A_2}_j , c^{B_2}_j).
\end{align*}

Here the lattice vectors in the real and momentum spaces are

\begin{align*}
{\v a}_1 = \sqrt{3}  \left( \frac{1}{2}, \frac{\sqrt{3}}{2} ,0\right), ~~~
{\v a}_2 = \sqrt{3}  \left( -\frac{1}{2}, \frac{\sqrt{3}}{2} ,0\right), ~~~
{\v a}_3 = 3  \left( 0,0,1 \right)
\end{align*}

\begin{align}
\v b_1 = \frac{4 \pi}{3} \left(\frac{\sqrt{3}}{2}, \frac{1}{2} ,0\right), ~~~
\v b_2 = \frac{4 \pi}{3} \left(-\frac{\sqrt{3}}{2}, \frac{1}{2} ,0 \right),~~~
\v b_3 = \frac{2 \pi}{3} \left( 0 , 0 , 1 \right)
\label{reci3D}
\end{align}

respectively. In the above we have assumed the magnitude of the hopping in the z-direction is the same as those within each layer. Moreover, we have tuned the lattice constant in the z-direction so that the Dirac cone is isotropic. The resulting tight-binding model reads
Ĥ = i   t ∑_ǩ c^†_ǩ  ·I ⊗
0														   	S_xy(ǩ)	    S_z(ǩ)	   	0

-S_xy^*(ǩ)	   	0	   	0	   S_z(ǩ)

-S_z^*(ǩ)	   	0	   	0	   	-S_xy(ǩ)

0	   -S_z^*(ǩ)	   	S_xy^*(ǩ)	   	0
c_ǩ      =t ∑_ǩ c^†_ǩ I⊗{

-[ sin(ǩ ·a_1 )+ sin(ǩ ·a_2)  ] ZX  -[ 1+ cos(ǩ ·a_1 )+ cos(ǩ ·a_2)  ] ZY

+ sin( ǩ ·a_3 ) XI - [1+cos(ǩ ·ǎ_3) YI].

} c_ǩwhere the $S_{xy}$ and $S_z$ in cpxhop are defined as
S_xy(ǩ)= 1 + e^i ǩ ·ǎ_1 +  e^i ǩ ·ǎ_2      S_z(ǩ)=	1+ e^-i ǩ ·ǎ_3,and the Pauli matrices are arranged according to

$$ {\rm spin}\otimes \text{ sub-lattice } (4 \times 4).$$
It is simple to show that in the momentum space the Dirac points are located at $\pm \v k_0$, where $\v k_0 :=  \frac{1}{3} \left( \v b_1 - \v b_2 \right)$.

Converting cpxhop into the Majorana fermion basis, the Hamiltonian reads

\begin{align}
\hat{H}  =t \sum\limits_{\v k} \chi^T_{-\v k} II\otimes \left[
\thead{
-\left( \sin (\v k \cdot a_1 )+ \sin (\v k \cdot a_2)  \right)ZX  -\left( 1+ \cos(\v k \cdot a_1 )+ \cos (\v k \cdot a_2)  \right)ZY \\
+ \sin ( \v k \cdot a_3 ) XI - \left(1+\cos(\v k \cdot \v a_3) YI\right)
\right] \chi_{\v k}
\label{Mott3D_lattice}
\end{align}

where the first Pauli matrix $I$ acts in the Majorana space. The Hamiltonian in Mott3D_lattice is invariant under the global charge and spin $SU(2)$ transformations generated by

\begin{align}
&\text{Charge-SU(2): } T^a = (XYII, ~YIII, ~ZYII)\notag\\
&\text{Spin-SU(2): ~~~~} \Sigma^a = (YXII, ~IYII, ~YZII)
\label{tsu2g}
\end{align}

When  performing the mode expansion near $\pm \v k_0$, because $\chi_{- \v k}^T = \chi^\dagger_{\v k}$, one can keep the complex fermion operator  $\tilde{c}_{\v q}=\chi_{\v k_0 + \v q}$ while disregard the mode expansion near $-\v k_0$. We subsequently break $\tilde{c}$ into real fermion operators $\tilde{c} = \tilde{\chi}_1 + i \tilde{\chi}_2$ (in the following we shall refer to this 1 and 2 as the “valley” indices). Omitting the tilde, in this final Majorana representation, the low energy theory of the Hamiltonian Mott3D_lattice is given by the $n=8$ real class massless fermion Hamiltonian
Ĥ_eff = ∫d^3 x  χ^T [ - i  ∑_i=1^3  Γ_i ∂_i ]χ      where  Γ_1 = IIZXI,   Γ_2 = IIZYY,   Γ_3 = IIXII,

and tsu2g is given by

\begin{align}
&\text{Charge-SU(2): } T^a = (XYIII, ~YIIII, ~ZYIII)\notag\\
&\text{Spin-SU(2): ~~~~} \Sigma^a = (YXIII, ~IYIII, ~YZIII)
\label{tsu2g2}\end{align}

In this basis, the Pauli matrices correspond to

$$ \text{Majorana} \otimes \text{spin}\otimes \text{ sub-lattice } (4 \times 4) \otimes \text{valley}.$$

For the gamma matrices to be in the standard basis used in table <ref>, we can do the transformation
$$\chi \rightarrow e^{i \frac{\pi}{4} IIZYI} \cdot e^{i \frac{\pi}{4} IIIXY}  \chi,$$
and then switch between the third and the fifth Pauli matrices. In the new basis, 3Dlow becomes
Ĥ_eff = ∫d^3 x  χ^T [ - i  ∑_i=1^3  Γ_i ∂_i ]χ      where  Γ_1 = IIIZI,   Γ_2 = IIIXI,   Γ_3 = IIIYY,

while the symmetry generators in tsu2g2 remain unchanged.  Upon bosonization, 3Dlow1 is equivalent to the $U(8)/O(8)$ nonlinear sigma model in wzw3R.

§.§.§ The charge-SU(2) confinement

Following the discussion in section <ref>, the Mott  constraint  can be imposed by demanding the order parameter to be charge $SU(2)$ singlet.

It is straightforward (but lengthy) to show that the following $Q^\mathbb{R}$ satisfies the charge-SU(2) singlet requirement

\begin{align}
\label{U(1)_S5}
Q^\mathbb{R}(x) = e^{i \theta(x)} \left[ n_0(x) \, N_0 + i \sum_{i=1}^5 n_i(x) \, N_i   \right]:=e^{i \theta(x)}\cG_S(x),
\end{align}

\begin{align*}
&N_0 = III, ~N_1=IIZ, ~N_2=IIX, ~N_3=IYY, ~N_4=YZY, ~N_5=YXY\\
&\text{and}~\sum_{i=0}^5 n_i^2 = 1, \text{i.e.}, (n_0,n_1,n_2,n_3,n_4,n_5)\in S^5.
\end{align*}

In addition, in U(1)_S5 $\cG_S$ is a symmetric special unitary $8\times 8$ matrix, namely,

$$\cG_S(x)\in {SU(8)\over O(8)}.$$

Substituting U(1)_S5 into the bosonized nonlinear sigma model wzw3R and noting that

$$\frac{1}{i}Q^{\mathbb{R}\dagger}\partial_\mu Q^{\mathbb{R}} = \frac{1}{i} \cG_S^\dagger\partial_\mu \cG_S + \partial_\mu \theta,$$

the stiffness term becomes
1/4λ_4^2   ∫_ℳ d^4 x   [  ∂_μ Q^ℝ ∂^μQ^ℝ†]       =8/4λ_4^2   ∫_ℳ d^4 x   [  _μθ^μθ] +1/4λ_4^2  ∫_ℳ d^4 x   [  ∂_μ _S ∂^μ_S^†]       =2/λ_4^2   ∫_ℳ d^4 x   [  _μθ^μθ] +2/λ_4^2   ∫_ℳ d^4 x   ∑_i=0^5 (∂_μ n_i )^2

The cross term vanishes because 1/i[_S^†∂_μ_S]=0.notrace is due to the fact that
$\cG_S$ is a symmetric special unitary matrix hence $\in SU(n)$. As a result, the matrix part of $\frac{1}{i}\cG_S^\dagger \partial_\mu \cG_S$ can be decomposed into the generators $\{ t^a\}$ of $su(n)$, which are traceless.

As to the WZW term it's can be shown that
-2π/960π^3    ∫_ℬ      [(Q̃^ℝ† dQ̃^ℝ )^5 ]=-2π/960π^3    ∫_ℬ      [(_̃S̃^† d_̃S̃)^5 ]      =-2πi/120π^3    ∫_ℬ    ϵ^i_1i_2i_3i_4i_5i_6 ñ_i_1  dñ_i_2  dñ_i_3  dñ_i_4  dñ_i_5 dñ_i_6.(We shall prove this relation in appendix <ref>.)

Putting together stftinto and wzwtinto, the non-linear sigma model action is given by
W[θ,β]       = 2/λ_4^2   ∫_ℳ d^4 x   [  _μθ^μθ] +2/λ_4^2  ∫_ℳ d^4 x   ∑_i=0^5 (∂_μ n_i )^2        -2πi/120π^3    ∫_ℬ    ϵ^i_1i_2i_3i_4i_5i_6 ñ_i_1  dñ_i_2  dñ_i_3  dñ_i_4  dñ_i_5 dñ_i_6.Therefore unlike $(1+1)$- and $(2+1)$-D, the spin effective theory for $(3+1)$-D bipartite Mott insulator has an extra $U(1)$ mode!

§.§.§ Gapping out the $U(1)$ mode

In this subsection we show that there is a charge-SU(2) singlet fermion interaction term that gaps out the $\theta$ degree of freedom. For convenience, we use the basis in 3Dlow1.

The emergent symmetry is $U(n)$ which includes a subgroup $U(1)$ (not to be confused with the extra $U(1)$ mode discussed earlier)  generated by
$$Q_{U(1)} =  I_n \otimes IY.$$
We can use this $Q_{U(1)}$ to complexify the Majorana fermion
Note that although we have complexified the Majorana fermion using the emergent $U(1)$, this is different from the complex class because we allow the mass term to break this $U(1)$.
, namely,
ψ_i^α:= 1/√(2) ( χ_αi1 + i  χ_αi2 ).
Here the Majorana field $\chi_{\alpha i  a  }$ carries three indices: $\alpha=1,2,...,n$ is the flavor  index;
$i$ indexes the second Pauli matrix in 3Dlow1 and $a=1,2$ indexes the last Pauli matrix.  In terms of the complexified  fermion operators the mass term reads (see table<ref>)

χ^T [    S_1 ⊗YX  +  S_2 ⊗YZ   ] χ      = [    ψ_i^α  ( i   Y_ij )( S_1+ i   S_2)_αβ  ψ_j^β+ h.c. ]      = [    ψ_i^α  E_ij  Q^ℝ_αβ  ψ_j^β+ h.c. ]

where $S_1$ and $S_2$ are symmetric real matrices.

Now we are ready to construct the desired interaction term to gap out the $U(1)$ mode in U(1)_S5

\begin{align}
\hat{H}_{\rm int}=- \frac{U_\theta}{2} \int d^4 x \Big[&E_{i_1 j_1} E_{i_2 j_2 }... E_{i_n j_n }\Big( \epsilon_{\alpha_1 \alpha_2 ... \alpha_n} \psi^{ \alpha_1}_{i_1 } \psi^{\alpha_2}_{i_2 } ... \psi^{ \alpha_n}_{i_n } \Big)\nn
&\times \Big( \epsilon_{\beta_1 \beta_2 ... \beta_n} \psi_{j_1 }^{\beta_1} \psi_{j_2 }^{\beta_2} ... \psi_{j_n }^{\beta_n} \Big) + {\rm h.c.} \Big].
\label{hitr}
\end{align}

First we note that hitr is a charge-SU(2) singlet, hence is unaffected by the Mott constraint. The proof goes as follows. When acted upon by the charge-SU(2) transformation, the fermion operator  in cpf transforms according to
$$ \psi^{ \alpha}_{i}\ra u^{\alpha}_{\beta} \psi^{ \beta}_{i },$$
where $u^{\alpha}_{\beta}$ is the charge-SU(2) transformation matrix. Under such transformation, the term in each parenthesis of hitr transforms according to
\begin{align*}
&\epsilon_{\alpha_1 \alpha_2 ... \alpha_n} \psi^{ \alpha_1}_{i_1 } \psi^{\alpha_2}_{i_2 } ... \psi^{ \alpha_n}_{i_n } \ra
\epsilon_{\alpha_1 \alpha_2 ... \alpha_n} u^{\alpha_1}_{\beta_1} u^{\alpha_2}_{\beta_2}...u^{\alpha_n}_{\beta_n} \psi^{ \beta_1}_{i_1 } \psi^{\beta_2}_{i_2 } ... \psi^{ \beta_n}_{i_n }\\&=\left(\det{u}\right) \epsilon_{\beta_1 \beta_2 ... \beta_n}\psi^{ \beta_1}_{i_1 } \psi^{\beta_2}_{i_2 } ... \psi^{ \beta_n}_{i_n }= \epsilon_{\alpha_1 \alpha_2 ... \alpha_n} \psi^{ \alpha_1}_{i_1 } \psi^{\alpha_2}_{i_2 } ... \psi^{ \alpha_n}_{i_n } .
\end{align*}

Therefore hitr is charge-SU(2) invariant.

Next, we note, upon bosonization

\begin{align*}
E_{ij} \psi_i^\alpha \psi_j^\beta  \rightarrow Q^{\mathbb{R}}_{\alpha \beta} = \left( S_1 + i S_2\right)_{\alpha \beta},
\end{align*}

$$Q^{\mathbb{R}}\in{U(8)\over O(8)}$$
is the order parameter of the nonlinear sigma model in wzw3R.
As the result, the action corresponds to $\hat{H}_{\rm int}$ is S_int=- U_θ/2 ∫d^4 x {[Q^ℝ]+c.c}.Substituting U(1)_S5 into  stheta1 we obtain
S_int= - U_θ ∫d^4 x  cos(8θ).

Naively, it might appear that the $\theta$-vacuum is $8$-fold degenerate, corresponding to
$$\theta = \frac{2\pi l}{8} ~\text{with~} l=0,1,...7.$$
However, this is due to a redundancy in the splitting $U(8)/O(8) \ra  U(1) \times SU(8)/O(8)$. The transformation

\begin{align*}
e^{i \theta} \rightarrow&  e^{i (\theta + \frac{2\pi}{8}) }
\end{align*}

can be absorbed by the following transformation of $\cG_S$

\begin{align*}
\cG_S \rightarrow& e^{i \frac{2 \pi}{8}} \cG_S.
\end{align*}

Because $\left(e^{i \frac{2 \pi}{8}} \right)^8=1$, the transformed $\cG_S$ still belongs to $SU(8)/O(8)$. As a result,  the $8$ different $\theta$ vacua should be counted as one, as long as there is no spontaneous symmetry breaking in $\cG_S$ (i.e., when $\cG_S(x)$ fluctuates over all possible configurations in $SU(8)/O(8)$).

In the phase that the $\theta$ degrees of freedom are gapped out, we have

\begin{align}
\label{S5}
Q^\mathbb{R}(x) =  \left[ n_0(x) \, N_0 + i \sum_{i=1}^5 n_{i}(x) \, N_{i}   \right].
\end{align}

Among the six order parameters, the first three are spin-SU(2) singlet and the last three are spin-SU(2) triplet. The latter can be interpreted as the anti-ferromagnetic order parameters. As to the first three, they break the lattice rotation symmetry, and can be identified as the VBS order parameters. The non-linear sigma model governing the $n_i$ degrees of freedom read
W[n_i] = 2/λ_4^2   ∫_ℳ d^4 x   ∑_i=0^5 (∂_μ n_i )^2 -2πi/120π^3    ∫_ℬ    ϵ^i_1i_2i_3i_4i_5i_6 ñ_i_1  dñ_i_2  dñ_i_3  dñ_i_4  dñ_i_5 dñ_i_6,

which is the $S^5$ (or $O(6)$) nonlinear sigma model with $k=1$ WZW term. Note that since $\pi_3(S^5)=0$, there is no soliton. This model is a natural generalization of the spin effective theory in $(1+1)$- and $(2+1)$-D.

Finally, the cautionary remarks in the summary of section <ref>, concerning the four-fermion interactions, induced by the charge-SU(2) gauge field fluctuations, also apply here.

§ TWISTED BI-LAYER GRAPHENE

Another 2D system where relativistic electron dispersion comes into play is the twisted bilayer graphene (TBLG). When the twisting angle is close to the “magic” value, the relevant bands become very flat [41], which suggests strong correlation. Under that condition, as a function of band filling $\nu$, a rich phase diagram emerges. This includes various insulating phases near integer filling and superconductivity when $\nu$ deviates from integer [42, 43, 44, 45, 46, 47]. In the following, we shall hold the point of view that the essence of the TBLG physics is the fact that the interaction energy overwhelms the bandwidth, which does not require the bandwidth to be zero. Therefore we  restrict ourselves to twisting angles close but not exactly equal to the magic values.

In the non-interacting picture, the Fermi energy ($E_F$) only intersects the Dirac nodes at the charge neutral point $\nu=0$. However, by measuring the electronic compressibility, it is recently suggested that the coincidence of $E_F$ and Dirac nodes  reappears  at all integer filling factors [48, 49].  Such “Dirac revivals” is interpreted as the evidence of the unequal filling of bands induced by the polarization of the flavor (including valley and spin) degrees of freedom. Therefore the relativistic massless fermions and bosonized non-linear sigma models discussed in Part I are good starting points to address the physics of TBLG.

The real space structure of the TBLG is shown in fig:TBLGa for a certain small but commensurate twisting angle.  In fig:TBLGb we show the associated momentum space structure. The large blue and the red hexagons are the original graphene Brillouin zones for the two layers.
The small hexagons colored orange are the Brillouin zone of the Moiré superlattice. In fig:TBLGc we blow up one of the Moiré Brillouin zones. Here $K_M$ and $K_M^\prime$ labels the two valleys in the Moiré Brillouin zone, while the blue/red $K$ and $K^\prime$ labels the valleys of the graphene Brillouin zone. Note that each valley of the Moiré Brillouin zone consists of two opposite valleys of the graphene Brillouin zone,

§.§ Charge neutral point $\nu=0$

In the presence of inter-layer hybridization, there are eight “active”  graphene-like bands. We can label these eight “flavors” by the flavor index which represents
$$\text{graphene valley, Moir\'e valley,  spin}$$
degrees of freedom. At the charge neutral point, the Fermi level crosses the Dirac points at $K_M$ and $K_M^\prime$.

In the momentum space we expand the band dispersion around $K_M$ and $K_M^\prime$, the resulting low energy Dirac-like band structure is described by the following continuum real-space Hamiltonian
Ĥ=∫d^2 x ψ^†(x̌)(-i_1_x-i_2_y)ψ(x̌),
where $\psi$ is an eight-component complex fermion field, and

\begin{align}
\G_1=XZII,~~\G_2=YIII.
\label{GammasTBLG}
\end{align}

Here the tensor product of Pauli matrices is arranged according to
$$ \text{ sub-lattice}\otimes {\rm graphene~valley}\otimes \text{Moir\'e valley}\otimes {\rm spin}.$$

Real space

Momentum space

Moiré Brillouin zone

(a) A real space picture of twisted bilayer graphene. (b) Blue and red color the Brillouin zones of the first and second layer graphene. Orange colors the Brillouin zone of the Moiré superlattice.  (c) At $K_M$ there are the $K_1$ of the first layer and $K_2'$ of the first layer. At $K'_M$ there are the $K_1'$ of the first layer and $K_2$ of the second layer.

The reason we use the complex fermion (rather than Majorana) representation in drc is that at integer band fillings there is no evidence of superconductivity [44]. Therefore drc belong to the complex class.

The massless free fermion Hamiltonian in drc has emergent $U(8)$ symmetry. After performing the  the basis transformation
$$\psi \rightarrow  e^{i \frac{\pi}{4} XIII} \cdot \left[ \begin{pmatrix} I &  0 \\ 0 & Z \end{pmatrix} \otimes II \right] \psi$$
to cast the gamma matrices into the form used in table <ref>, namely,

$$ \G_1=XIII, ~~\G_2=ZIII,$$
we can use our bosonization result (see appendix <ref>).

In the presence of the electromagnetic  ($U(1)$) gauge field $A$, the massless fermion theory in drc is equivalent to the following gauged non-linear sigma model
W[Q^ℂ, A] =12λ_3  ∫_ℳ d^3 x   [(_μQ^ℂ)^2]-   2 πi /256 π^2   {   ∫_ℬ [Q̃^ℂ     ( d Q̃^ℂ )^4 ]       + 8 ∫_ℳ  [ i A Q^ℂ (dQ^ℂ)^2  - 2 AFQ^ℂ ]

As discussed in section <ref>, there exists a local interacting fermion model which respects all emergent symmetries
and the phases (which might spontaneously break the continuous or discrete symmetries) are described by the effective theories given by gwzw but with

\begin{align*}
Q^{\mathbb{C}}\in ~\bigcup_{l=0}^8~{U(8)\over U(l)\times U(8-l)}
\end{align*}

Among the last two terms of gwzw, the term  linear in $A_{\mu}$ measures the soliton current

J^\mu=\frac{ i }{16 \pi} \e^{\mu\nu\rho} \tr \Big[Q^{\mathbb{C}}\p_\nu Q^{\mathbb{C}} \p_\rho Q^{\mathbb{C}}  \Big].

The term proportional to $AF$ gives rise to a Chern-Simons term

$$- \frac{i }{8 \pi}     \int\limits_{\mathcal{M}} \tr[Q^{\mathbb{C}}] AF,$$
with the corresponding Hall conductance