Relationship of Edge States to Anomaly and Construction of Dual Systems in Quantum Hall Systems

Paul Bracken

Department of Mathematics, University of Texas, Edinburg, TX, USA.

DOI: 10.4236/jmp.2024.156037                

Abstract

The hierarchy of bulk actions is developed which are associated with Chern-Simons theories. The connection between the bulk and edge arising from the requirement there is a cancelation of an anomaly which arises in the theory. A duality transformation is studied for the Chern-Simons example. The idea that is used has been employed to describe duality in a scalar theory. The link between the edge theory with the Chern-Simons theory in the bulk then suggests that similar transformations can be implemented in the bulk Chern-Simons theory as well.

Keywords

Quantum, Algebra, Hall, Conductivity, Forms, Chiral, Fermion

1. Introduction

The quantum Hall effect has received a lot of attention in no small part due to its wide ranging and deep mathematical properties [1] [2] . The fractional Hall states are new states of matter which appear due to the interactions of electrons in 2 dimensional layers under strong magnetic fields and very low temperatures $T\to 0$ .

When the electronic degrees of freedom are integrated out, the effective action for the electromagnetic vector potential has a Chern-Simons (CS) term [3] . Mathematically the CS form is related to a topological density in 2n dimensions known as a characteristic class $C_{2n}$ . Since the forms are local $\left(2k+1\right)$ -forms, they are integrated without much difficulty over, for example, homology spheres without additional structures. The metric of the embedding space is irrelevant, a consequence of the topological origin of the CS forms. In fact, the function involves explicitly the connection A and cannot be expressed as an integral of a gauge invariant local function [4] . The coefficient of the term that is left upon integration is proportional to the Hall conductivity. To see why this arises, assume the effective action is given apart from the Maxwell term by

$S_e\left[A\right]=-\frac{1}{2}{\sigma }_{xy}{\displaystyle \int d^{3}x{\epsilon }^{\mu \nu \lambda }{A}_{\mu }{\partial }_{\nu }{A}_{\lambda }}.$ (1.1)

The expectation value of the current is readily calculated

$\langle j_{em}^{\mu }\rangle =-\frac{\delta }{\delta A_{\mu }}{S}_e\left[A\right]={\sigma }_{xy}{\epsilon }^{\mu \nu \lambda }{\partial }_{\nu }{A}_{\mu }\mathrm{.}$ (1.2)

It is apparent that there is a current in the x-direction when there is an electric field in the y-direction giving rise to the Hall effect, with conductivity ${\sigma }_{xy}$ [5] [6] [7] [8] .

It is possible CS theories which involve several vector potentials can be considered, one of which is the electromagnetic field [9] . These include statistical gauge fields and fields describing excitations in the bulk. The former is introduced for the purpose of changing the statistics of the excitation fields in the action, while the latter describes collective degrees of freedom, such as vortices or other quasiparticles. They can describe the case of either bosons or fermions. Experimentally, it is found that the Hall conductivity is expressed in terms of certain definite fractions corresponding, respectively, to integer and fractional effects [10] [11] .

The CS action is not gauge invariant on a boundary like an annulus, or such a manifold having realistic geometry for a Hall system. This means nontrivial dynamical degrees of freedom have to be included at the edge to restore gauge invariance. This leads to the production of edge states with completely different corresponding to the existence of chiral edge state currents in the Hall sample [12] . One way to describe them is by means of a conformal field theory of a set of massless chiral scalar fields taking values on a torus. The most general action for such scalar fields is made up of a symmetric matrix $G_{ij}$ and an antisymmetric matrix $B_{ij}$ . The actual Hall conductivity depends on $G_{ij}$ . It is possible to show how the anomaly cancellation argument enables one to relate this matrix to a matrix that is relevant to the CS theory.

The objective at this point is to see how hierarchies emerge using such CS theories and to examine the connection between bulk and edge states. It is also possible to construct dual theories starting from CS based actions. It will be seen how to implement duality in CS theories. As in string theories there exist duality transformations of the edge theory that leave the spectrum invariant [13] . Then $B_{ij}$ and $G_{ij}$ are changed in well defined ways under these transformations, and consequently also change the Hall conductance as well. That a connection exists between the edge theory by means of CS theory in bulk suggests that similar transformations can be implemented in the bulk as well. This conjecture seems to be at least partially feasible in that it can be implemented by a type of duality transformation given in the bulk [14] .

2. A Brief Introduction to Chern-Simons Quantum Field Theory

Generally the field variables of the $U\left(1\right)$ CS theory over a manifold M are described by a one-form $A\in {\Omega }^1\left(M\right)$ with components $A=A_{\mu }\left(x\right)d{x}^{\mu }$ as

$S\left[A\right]=2\pi k{\displaystyle {\int }_M{d}^{3}x{\epsilon }^{\mu \nu \rho }{A}_{\mu }{\partial }_{\nu }{A}_{\rho }}=2\pi k{\displaystyle {\int }_{M}A\wedge dA}.$ (2.1)

where k not zero denotes the real coupling constant of the model. The action is invariant under gauge transformations $A_{\mu }\left(x\right)\to A_{\mu }\left(x\right)+{\partial }_{\mu }\zeta \left(x\right)$ . This means the action can be understood as a function of the gauge orbits.

In order to define the expectation value $\langle A_{\mu }\left(x\right)A_{\nu }\left(y\right)\cdots A_{\lambda }\left(z\right)\rangle$ of the products of fields , one needs to introduce a gauge-fixing procedure because the gauge field $A_{\mu }\left(x\right)$ is not gauge invariant but if one is interested in the correlation function $\langle F_{\mu \nu }\left(x\right)F_{\rho \sigma }\left(y\right)\cdots F_{\lambda \tau }\rangle$ of the curvature $F_{\mu \nu }\left(x\right)={\partial }_{\mu }{A}_{\nu }\left(x\right)-{\partial }_{\nu }{A}_{\mu }\left(x\right)$ , gauge fixing is not required. If a one-form $B=B_{\mu }\left(x\right)d{x}^{\mu }$ is a classical external source, the integral satisfies

${\displaystyle {\int }_{M}dA\wedge B}={\displaystyle {\int }_{M}A\wedge dB}.$ (2.2)

It is invariant under gauge transformations acting on A because the curvature $F=dA$ is gauge invariant. The generating functional $G\left[B\right]$ for the correlation function of the curvature is defined by

$G\left[B\right]=\langle e^{2\pi i{\displaystyle \int A\wedge dB}}\rangle \mathrm{<mo>\; =\; </mo>}\frac{{\displaystyle \int DA{e}^{2\pi ik{\displaystyle \int A\wedge dA}}{e}^{2\pi i{\displaystyle \int A\wedge dB}}}}{{\displaystyle \int DA{e}^{2\pi ik{\displaystyle \int A\wedge dA}}}}\mathrm{.}$ (2.3)

The coefficients of the Taylor expansion of $G\left[B\right]$ in powers of B coincide with correlation functions of the curvature. Any $A_{\mu }\left(x\right)$ can be written as

$A_{\mu }\left(x\right)=-\frac{1}{2k}{B}_{\mu }\left(x\right)+{\omega }_{\mu }\left(x\right)\mathrm{,}$ (2.4)

where $B_{\mu }\left(x\right)$ is fixed and ${\omega }_{\mu }\left(x\right)$ may vary. Since

$k{\displaystyle {\int }_{M}A\wedge dA}={\displaystyle {\int }_{M}A\wedge dB}+k{\displaystyle {\int }_M\omega \wedge d\omega }-\frac{1}{4k}{\displaystyle {\int }_{M}B\wedge dB}$ (2.5)

The functional integration is invariant under translation and $DA=D\omega$ , so

$\langle e^{2\pi i{\displaystyle \int A\wedge dB}}\rangle =e^{-\left(2\pi i/4k\right){\displaystyle \int B\wedge dB}}\frac{{\displaystyle \int D\omega e^{2\pi ki{\displaystyle \int \omega \wedge d\omega }}}}{{\displaystyle \int DA{e}^{2\pi ki{\displaystyle \int A\wedge dA}}}}=e^{-\left(2i\pi /4k\right){\displaystyle \int B\wedge dB}}\mathrm{.}$

Without the introduction of gauge fixing, and without any metric in M, the Feynman path integral gives

$G\left[B\right]=e^{i{G}_c\left[B\right]}\mathrm{<mo>\; =\; </mo>}\exp \left(-\frac{2\pi i}{4k}{\displaystyle {\int }_{M}B\wedge dB}\right)\mathrm{.}$ (2.6)

The generating function of the connected correlation functions of the curvature formally coincides with the Chern-Simons action under the replacement $k\to -1/k$ .

3. Appearance of Hierarchies

Following Zee [5] , the electron can be described by a scalar field coupled to a statistical gauge field $a_{\mu }$ . Furthermore, if the bosonic order parameter develops an expectation value, a massless Goldstone boson γ results, the phase of the original scalar field. It satisfies an equation, ${\partial }_{\mu }{\partial }^{\mu }\gamma =0$ since it has no mass. The field equation for γ turns into an identity in the dual representation. A minimal coupling to the external electromagnetic potential $A_{\mu }$ can also be implemented. The action for the system so far is

$S={\displaystyle {\int }_M{d}^{3}x}\left[-e{J}^{\mu }\left(A_{\mu }-a_{\mu }\right)-\frac{e^2}{4\pi }{\epsilon }^{\mu \nu \lambda }{a}_{\mu }{\partial }_{\nu }{a}_{\lambda }\right],$ (3.1)

$J^{\mu }={\epsilon }^{\mu \nu \lambda }{\partial }_{\nu }{\alpha }_{\lambda }\mathrm{,}$

where M is the manifold $D\backslash H\times ℝ$ and $D\backslash H$ is a disk D with a hole H due to the removal of a piece, which reduces to an annulus. The time variable is accounted for by the factor $ℝ$ . The last term in (3.1) is an abelian CS term pertaining to the statistical gauge field $a_{\mu }$ . The coefficient has been chosen to ensure that it converts the boson to a fermion as may be seen in the following way. Varying (3.1) with respect to ${\alpha }_{\mu }$ we obtain the equation,

${\epsilon }^{\mu \nu \lambda }{\partial }_{\nu }{A}_{\lambda }={\epsilon }^{\mu \nu \lambda }{\partial }_{\nu }{\alpha }_{\lambda }\mathrm{.}$ (3.2)

On varying (3.1) with respect to $a_{\mu }$ we have,

${\epsilon }^{\mu \nu \lambda }{\partial }_{\nu }{\alpha }_{\lambda }=\frac{e}{2\pi }{\epsilon }^{\mu \nu \lambda }{\partial }_{\nu }{\alpha }_{\lambda }\mathrm{.}$ (3.3)

Equations (3.2) and (3.3) imply that

${\epsilon }^{\mu \nu \lambda }{\partial }_{\nu }{\alpha }_{\lambda }=\frac{e}{2\pi }{\epsilon }^{\mu \nu \lambda }{\partial }_{\nu }{A}_{\lambda }\mathrm{.}$ (3.4)

Now (3.4) yields a relation between the number density $J^0=N_e$ of electrons to the number density $N_{\varphi }=eB/2\pi$ of flux quanta 2π/e. It states in fact that $N_e=N_{\varphi }$ , or there is one flux quantum per electron such that the electron is converted to a boson. This can be regarded as the first equation in the rung of a hierarchy. The filling factor $\nu$ is 1 by (3.2) since $N_{\varphi }=N_e$ , so it describes the integer quantum Hall effect.

The fields α and a can be eliminated to get an effective action which depends only on the electromagnetic gauge field. The electromagnetic current $-e{J}^{\mu }$ of in (3.1) is equal to

$-\frac{e^2}{2\pi }{\epsilon }^{\mu \nu \lambda }{\partial }_{\nu }{A}_{\lambda }\mathrm{.}$ (3.5)

by (3.4). It is reproduced by the action,

$S=-\frac{e^2}{4\pi }{\displaystyle {\int }_M{d}^{3}x{\epsilon }^{\mu \nu \lambda }{A}_{\mu }{\partial }_{\nu }{A}_{\lambda }}\mathrm{.}$ (3.6)

This is the electromagnetic CS term, and moreover, a signature of the Hall effect for the Hall conductivity, ${\sigma }_H=e^2/2\pi$ .

One can immediately generalize (3.1) to obtain more of the hierarchy and the Laughlin functions by changing the coefficient $e^2/4\pi$ to $e^2/4\pi m$ with m odd so $m\in 2\mathbb{Z}+1$ ,

$S={\displaystyle {\int }_M{d}^{3}x}\left[-e{J}^{\mu }\left(A_{\mu }-a_{\mu }\right)-\frac{e^2}{4\pi m}{\epsilon }^{\mu \nu \lambda }{a}_{\mu }{\partial }_{\nu }{a}_{\lambda }\right]\mathrm{.}$ (3.7)

Proceeding as done already, variation leads to the following pair

${\epsilon }^{\mu \nu \lambda }{\partial }_{\nu }{\alpha }_{\lambda }=\frac{e}{2\pi m}{\epsilon }^{\mu \nu \lambda }{\partial }_{\nu }{a}_{\lambda }\mathrm{,}{\epsilon }^{\mu \nu \lambda }{\partial }_{\nu }{a}_{\lambda }=\frac{e}{2\pi m}{\epsilon }^{\mu \nu \lambda }{\partial }_{\nu }{A}_{\lambda }\mathrm{.}$ (3.8)

The first equation of (3.8) has the implication that

$N_e=\frac{1}{m}{N}_{\varphi }.$ (3.9)

Since m is odd, this is the same as

$N_{\varphi }=m{N}_e+N^{\left(1\right)}\mathrm{,}$ (3.10)

and implies the composite is bosonic, as it should be so the description of the electron is consistent. The filling fraction is now

$\nu =\frac{1}{m}\mathrm{,}$ (3.11)

The action (3.6) now becomes,

$S^m=-\frac{e^2}{4\pi m}{\displaystyle {\int }_M{d}^{3}x{\epsilon }^{\mu \nu \lambda }{A}_{\mu }{\partial }_{\nu }{A}_{\lambda }}\mathrm{.}$ (3.12)

This is the form of the CS action which gives the next level of the hierarchy.

The next step is to modify (3.7) by including a coupling of the quasiparticles current $J^{\left(1\right)\nu }$ to the gauge field $a_{\mu }$ to get the action

${\displaystyle {\int }_M{d}^{3}x}\left[-e{J}^{\mu }\left(A_{\mu }-a_{\mu }\right)-\frac{e^2}{4\pi m}{\epsilon }^{\mu \nu \lambda }{a}_{\mu }{\partial }_{\nu }{a}_{\lambda }+2\pi J^{\left(1\right)\nu }{a}_{\mu }\right]\mathrm{.}$ (3.13)

To motivate the choice of the coefficient 2π in the last term of (3.13) suppose there is a vortex localized at z so that

$J^{\left(1\right)0}\left(z\right)={\delta }^2\left(z-x\right)\mathrm{,}$

while the electron density $\rho$ is some smooth function. Since the equations of motion imply,

$\rho =\frac{e}{2\pi m}{\epsilon }^{0ij}{\partial }_i{a}_j\mathrm{.}$ (3.14)

So it follows that ${\epsilon }^{0ij}{\partial }_i{a}_j$ is also smooth. Variation of $\alpha$ produces the equation of motion

$\frac{2\pi }{e}{J}^{\left(1\right)0}={\epsilon }^{0ij}\left({\partial }_i{A}_j+{\partial }_i{a}_j\right)\mathrm{,}$ (3.15)

The magnetic flux attached to the vortex is the flux quantum, 2π/e. This is a unit of magnetic flux to be attached to the vortex and the factor of 2π is correct.

Suppose the quasiparticles condense so we can write $J^{\left(1\right)\mu }={\partial }^{\mu }{\gamma }^1$ is the Goldstone boson phase degree of freedom. As before the ${\gamma }^1$ field is massless and so it has to satisfy ${\partial }_{\mu }{\partial }^{\mu }{\gamma }^1=0$ . A dual version of the current can be given by defining a field ${\beta }_{\mu }$ as

$J^{\left(1\right)\mu }={\partial }^{\mu }{\gamma }^1={\epsilon }^{\mu \nu \lambda }{\partial }_{\nu }{\beta }_{\lambda }\mathrm{.}$ (3.16)

A statistical gauge field $b_{\mu }$ can be included such that flux tubes of b are attached to the quasi-particle. The quasiparticles correspond to vortices, which are assumed to be bosonic, hence an even number of elementary b flux tubes are attached to each to preserve their bosonic nature.

Based on this more CS terms can be added to (3.13) to obtain

$S^{\left(1\right)}={\displaystyle {\int }_M}\left(-e\left(A-a\right)d\alpha -\frac{e^2}{4\pi m}ada+2\pi \alpha d\beta -ebd\beta -\frac{e^2}{4\pi \left(2m_1\right)}bdb\right)\mathrm{.}$ (3.17)

The wedge is omitted and the one-forms are simply written as $\eta =A,\alpha ,\beta ,a,b$ meaning $\eta ={\eta }_{\mu }d{x}^{\mu }$ . As indicated already, the equations of motion obtained from the action (3.17) are

$\frac{e}{2\pi }dA=\frac{e}{2\pi }da+d\beta ,md\alpha =\frac{e}{2\pi }da,d\alpha =\frac{e}{2\pi }db,d\beta =-\frac{e}{2\pi \left(2m_1\right)}db.$ (3.18)

The equations in (3.18) for α and β correspond to hierarchy equations on eliminating a and b of the form

$N^{\left(e\right)}=2\left|m_1{N}^{\left(1\right)}\right|+N^{\left(2\right)},N^{\left(1\right)}=2\left|m_2{N}^{\left(2\right)}\right|+N^{\left(3\right)},N^{\left(2\right)}=0.$ (3.19)

The equations for α and β can be reproduced by means of an action

$\begin{array}{c}{S}^{\left(1\right)}={\displaystyle {\int }_M}-eAd\alpha +\pi m\alpha d\beta +\pi \left(2m_1\right)\beta d\beta \\ ={\displaystyle {\int }_M}\left[-eAd\alpha +\pi \left(\alpha \beta \right)\left(\begin{array}{cc}{m}_1& 1\\ 1& 2m_1\end{array}\right)\left(\begin{array}{c}d\alpha \\ d\beta \end{array}\right)\right]\mathrm{.}\end{array}$ (3.20)

This result suggests a generalization to higher levels. Introduce q vector fields denoted ${\alpha }_I$ where index $I=\mathrm{1,}\cdots \mathrm{,}q$ . In the case above, $q=2$ , and the ${\alpha }_i$ are ${\alpha }_1=\alpha$ x and ${\alpha }_2=\beta$ . Consider the Lagrangian form given by

$\mathcal{L}=-eAd{\alpha }_1+\pi {\alpha }_I{N}^{IJ}d{\alpha }_J,{\alpha }_i={\alpha }_{I\mu }d{x}^{\mu }.$ (3.21)

In (3.21), $N^{IJ}$ is a matrix with main diagonal $\left(m\mathrm{,2}{m}_1\mathrm{,}\cdots \right)$ and the full matrix is given by

$K^{IJ}=\left(\begin{array}{cccccc}m& 1& 0& 0& 0& \cdots \\ 1& 2m_1& 1& 0& 0& \cdots \\ 0& 1& 2m_2& 1& 0& \cdots \\ 0& 0& 1& 2m_3& 1& \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮& \end{array}\right)$

The Lagrangian in (3.21) gives rise to its own set of equations of motion. The equation for ${\alpha }_1$ obtained from (3.21) is

$edA=2\pi N^{1J}d{\alpha }_J\mathrm{.}$ (3.22)

The equations of motion for the remaining ${\alpha }_I$ for $I\ne 1$ are given by

$N^{IJ}d{\alpha }_J=0.$ (3.23)

Clearly a hierarchy of equations has been constructed this way. From (3.22), solve for the forms $d{\alpha }_j$ as

$d{\alpha }_I=\frac{e}{2\pi }{\left(N^{-1}\right)}_{I1}dA\mathrm{.}$ (3.24)

Solution (3.24) can be replaced back into $\mathcal{L}$ to generate the following form

$\begin{array}{c}\mathcal{L}=-eAd{\alpha }_1+\pi {\alpha }_i{N}^{IJ}\frac{e}{2\pi }{\left(N^{-1}\right)}_{IJ}dA\\ =-\frac{e}{2\pi }{\left(N^{-1}\right)}_{11}dA+\frac{e^2}{4\pi }A{\left(K^{-1}\right)}_{I1}{K}^{IJ}{\left(K^{-1}\right)}_{J1}dA\\ =-\frac{e^2}{4\pi }A{\left(K^{-1}\right)}_{11}dA.\end{array}$ (3.25)

This is the CS form of Lagrangian that gives rise to a complete hierarchy and the filling fraction is $\nu ={\left(N^{-1}\right)}_{11}$ .

It has been seen the fermion electron field can be re-expressed in terms of a bosonic field and a statistical gauge field, and these ideas generalize to get filling fractions. Let us summarize this section in physical terms. As the magnetic field is altered it is magnetically favorable for the excess of deficit magnetic field to organize itself as flux tubes threading vortices in the condensate so that quasi-particles which are one of these vortices are formed. At a certain moment a large number of these quasiparticles form and condense so there is a finite number density of quasiparticles and a new ground state is formed.

If the quasiparticles or vortices are thought of as carrying a new form of charge, then the gauge field to which they couple are actually the duals of Goldstone phase mode of the condensate. This is illustrated by noting that if the electron current is represented in a dual representation by using a one form, then the electric field associated to the one form has behavior, outside a vortex, identical to that of an electric field outside an ordinary electric charge. The flux quanta in this dual representation are the electrons themselves. The bosonic nature is supported provided an even number $2m_1$ of dual flux quanta electrons get attached to each of these quasiparticles. These statements are summarized mathematically in the form of (3.18).

4. Anomaly and Bulk-Edge Coupling

Gauge invariance forces the matrix $N^{IJ}$ to be the same as the inverse of the target space metric $G_{IJ}$ which pertains to the scalar theory for the edge excitations. To understand this, start with a CS action with no electromagnetic coupling

$S=\frac{1}{2}{\displaystyle {\int }_M\alpha d\alpha }\mathrm{.}$ (4.1)

Although the arguments apply to any manifold, M is given coordinates $\left(r\mathrm{,}\theta \right)$ , so to picture what is going on, it can be supposed the fields are valued in a circle with $r=R$ on the boundary. If the manifold has a closed, compact and boundaryless spatial slice, there is gauge invariance of the action under the transformation

$\alpha \to \alpha +d\Lambda \mathrm{.}$ (4.2)

If on the other hand, $\bar{M}$ is a manifold such as M where the spatial slice Σ is an annulus $D\backslash H$ and has a boundary, then gauge variation results in a surface term. For the manifold $M=D\backslash H\times ℝ$ , the variation of the action is

$\delta S=\frac{1}{2}{\displaystyle {\int }_{\partial D\times ℝ}}\Lambda d\alpha -\frac{1}{2}{\displaystyle {\int }_{\partial M}}\Lambda d\alpha \mathrm{.}$ (4.3)

It is assumed that Λ vanishes in the infinite past and future. To restore gauge invariance at the boundary, the following two-dimensional action is added to S, and $\mu =\mathrm{0,1,2}$ . It implements a new scalar field $\phi$ ,

$S^{\left(1\right)}=-\frac{1}{2}{\displaystyle {\int }_{\partial M}}d\phi \wedge \alpha +\frac{1}{4}{\displaystyle {\int }_{\partial M}}{d}^{2}x\left(D_{\mu }\phi \right)\left(D^{\mu }\phi \right)\mathrm{.}$ (4.4)

This means field $\phi$ gauge transforms as $\phi \to \phi -\Lambda$ , so the covariant derivative is $D_{\mu }\phi ={\partial }_{\mu }\phi +{\alpha }_{\mu }$ . The edge current is supposed to be chiral and this fixes the factor 1/4 outside the kinetic energy term in (4.4). This means the following condition can be imposed consistently with the equations of motion,

$D_{-}\phi =\left(D_0-D_{\theta }\right)\phi =0.$ (4.5)

The combined action $S+S^{\left(1\right)}$ is gauge invariant.

The operator that generates (4.2) at a fixed time with ${\Lambda |}_{\partial D}\ne 0$ and ${\Lambda |}_{\partial H}=0$ is

$Q\left(\Lambda \right)={\displaystyle {\int }_{D\backslash H}}d\Lambda \alpha \mathrm{.}$ (4.6)

Here Λ is a function on the annulus $D\backslash H$ and ${\Lambda |}_{\partial H}=0$ for simplicity. The algebra generated by the operators is specified by

$\left[Q\left(\Lambda \right)\mathrm{,}Q\left({\Lambda }^{\prime }\right)\right]=-i{\displaystyle {\int }_{\partial D}}\Lambda d{\Lambda }^{\prime }\mathrm{.}$ (4.7)

Imposing the condition of gauge invariance condition $Q\left(\Lambda \right)|p\rangle =0$ on physical states $|p\rangle$ leads to a contradiction as the commutator of two Q’s acting on a physical state would also have to vanish. However, (4.7) specifies the value of this commutator to be a non-zero c-number.

If the action is expanded by including term (4.4) which accounts for new degrees of freedom at the boundary, the generators of edge gauge transformations are modified. The modification is by means of the terms

$q\left(\Lambda \right)={\displaystyle {\int }_{\partial D}}\Lambda \left({\Pi }_{\phi }-\frac{1}{2}{\phi }^{\prime }\right)\mathrm{.}$ (4.8)

In (4.8), we have

${\Pi }_{\phi }=\frac{1}{2}\left(D_0\phi +A_{\theta }\right)$

is the conical momentum conjugate to $\phi$ and satisfies the usual commutation relations, so $q\left(\Lambda \right)$ generates the transformations

$\phi \to \phi -\Lambda \mathrm{,}{\Pi }_{\phi }\to {\Pi }_{\phi }+\frac{1}{2}{\partial }_{\theta }\Lambda \mathrm{.}$ (4.9)

Thus the algebra is generated by the $q\left(\Lambda \right)$ and is given by

$\left[q\left(\Lambda \right)\mathrm{,}q\left({\Lambda }^{\prime }\right)\right]=i{\displaystyle {\int }_{\partial D}}\Lambda d{\Lambda }^{\prime }\mathrm{.}$ (4.10)

The new generators

$\bar{Q}\left(\Lambda \right)=Q\left(\Lambda \right)+q\left(\Lambda \right)\mathrm{,}$ (4.11)

now commute between themselves and can be chosen to annihilate the physical states.

To now attempt to couple electromagnetism to S in (4.1) if A is a background electromagnetic field and $\ast d\alpha$ represents a current so that the most obvious coupling is

$S^{\prime }=-q{\displaystyle {\int }_M}Ad\alpha \mathrm{.}$ (4.12)

When the equations of motion implied by the action $S+S^2$ are examined, a problem is encountered. The equation in the bulk which is obtained by varying α is

$d\alpha =qdA\mathrm{.}$ (4.13)

The integrand of (4.1) can be written as $\left(1/2\right)d{\alpha }^2$ , so using Stokes theorem, on the boundary, it is

$\frac{1}{2}\alpha =qA\mathrm{.}$ (4.14)

Clearly, equations (4.13) and (4.14) are incompatible and moreover (4.13) implies a relation between the values of the field strengths of α and A on the boundary that differs by the factor of two from that implied by (4.13) in bulk, whereas by continuity, they should be equal.

There is a simple modification of the minimal coupling (4.11) given a consistent set of equations. To see this consider the modified action

$S^2=-\frac{1}{2}{\displaystyle \int q}\left(Ad\alpha +\alpha dA\right)\mathrm{.}$ (4.15)

Based on this boundary equation (4.14) is modified to

$\alpha =qA.$ (4.16)

With (4.13) and (4.16), together they imply the result $\alpha =qA$ everywhere classically up to gauge transformations that vanish on the boundary. Gauge transformations that do not vanish on the boundary which are consistent with the equations of motion have the form

$\alpha \to \alpha +qd\Lambda ,A\to A+d\Lambda \mathrm{.}$

The equations of motion in bulk and edge are now consistent, however, the action $S+S^2$ given by (4.1) and (4.15) is no longer gauge invariant under $\alpha \to \alpha +qd\Lambda$ and $A\to A+d\Lambda$ . This is similar to what happens at the edge where gauge invariance and chirality are found to be incompatible with the equations of motion. Here the solution is to couple to degrees of freedom at the boundary. A scalar field $\phi$ needs to be introduced with a boundary action of the form

$S^3=\frac{q}{2}{\displaystyle {\int }_{\partial M}}d\phi A+\frac{1}{4}{\displaystyle {\int }_{\partial M}}{d}^{2}x{\left(D_{\mu }\phi \right)}^2\mathrm{,}$ (4.17)

$D_{\mu }\phi ={\partial }_{\mu }\phi -a{A}_{\mu }\mathrm{.}$

This maintains invariance under the transformations $\alpha \to \alpha +qd\Lambda$ , $A\to A+d\Lambda$ , the electromagnetic gauge transformations and $\phi$ transforms in the following way under these transformations

$\phi \to \phi +q\Lambda \mathrm{.}$ (4.18)

Under (4.18), the generators of the edge gauge transformations can be required to annihilate the states. The total action $\mathcal{S}=S+S^2+S^3$ is then gauge invariant under the class of electromagnetic gauge transformations, the bulk and boundary equations are compatible. The total action $\mathcal{S}$ is

$\mathcal{S}={\displaystyle {\int }_{DH\times R}}\left(\frac{1}{2}\alpha d\alpha -\frac{q}{2}\left(Ad\alpha +\alpha dA\right)\right)+\frac{q}{2}{\displaystyle {\int }_{\partial D\times R}}Ad\phi +\frac{1}{4}{\displaystyle {\int }_{\partial D\times R}}{d}^{2}x{\left(D_{\mu }\phi \right)}^2\mathrm{.}$ (4.19)

To generalize this to the case in which there are m CS fields, introduce the action

$S=\pi N^{IJ}{\displaystyle {\int }_M}{\alpha }_{I}d{\alpha }_J\mathrm{,}$ (4.20)

where $K^{IJ}$ is the matrix introduced before. This theory has a $U\left(1\right)$ gauge invariance denoted

${\alpha }_I\to {\alpha }_I+d{\Lambda }_I\mathrm{.}$

Introduce m background gauge fields $A^I$ such that one of them represents the physical electromagnetic field and the rest are fictitious. After the quasiparticles are integrated out, the action which results depends on the gauge fields. Functional differentiation with respect to these fields then gives the correlators of the currents or connected Greens functions. Introducing edge scalar fields to restore gauge invariance, the final action assumes the following form

$\begin{array}{c}\mathcal{S}={\displaystyle {\int }_M}\left(-\frac{1}{2}\left(A^{I}d{\alpha }_I+{\alpha }_{I}d{A}^I\right)+\pi N^{IJ}{\alpha }_{I}d{\alpha }_J\right)+{\displaystyle {\int }_{\partial M}}\frac{1}{4\pi }{\left(N^{-1}\right)}_{IJ}{\phi }^I{A}^J\\ +\frac{1}{8\pi }{\displaystyle {\int }_{\partial M}}{\left(N^{-1}\right)}_{IJ}{D}_{\mu }{\phi }^I{D}^{\mu }{\phi }^J\mathrm{.}\end{array}$ (4.21)

As usual, the coefficient of the kinetic term is fixed by requiring consistency between the chirality of the edge currents and the equations of motion.

5. Construction of Related Dual Theories

To demonstrate exactly how dual theories can be developed from CS theories, start with the action

$S=\frac{k}{2\pi }{\displaystyle {\int }_M}\alpha d\alpha \mathrm{.}$ (5.1)

In (5.1) M is an oriented three manifold with an annulus as its spatial slice and time compactified to a circle. This means fields at $t=\pm \infty$ take the same values, so the path integral leads to a transition amplitude between states when displaced around a closed loop in configuration space.

To forbid arbitrary rescaling of the field α an extra condition on α is needed. If this is not done, constant k can be changed to ${\lambda }^{2}k$ by mapping α according to $\alpha \to \alpha \lambda$ where $\lambda \in ℝ$ . The imposed condition is

${\displaystyle {\int }_{C\in \partial M}}\alpha \in 2\pi \mathbb{Z}$ (5.2)

and C any closed loop on the boundary of M.

By means of the transformation $\alpha \to \alpha +\omega$ on α and ω is a closed one-form so $d\omega =0$ . The Lagrangian three form is not invariant as it changes by an exact three form

$\alpha d\alpha \to \alpha d\alpha -d\left(\omega \alpha \right)\mathrm{.}$ (5.3)

Suppose a connection one form A is introduced which transforms according to the rule $A\to A-\omega$ . Gauging S we find it satisfies

$S^{\prime }=\frac{k}{2\pi }{\displaystyle {\int }_M}\left(\alpha +\omega \right)d\alpha +\frac{k}{2\pi }{\displaystyle {\int }_M}\left(A-\omega \right)d\omega =\frac{k}{2\pi }{\displaystyle {\int }_M}\alpha d\alpha +\frac{k}{2\pi }{\displaystyle {\int }_M}Ad\alpha \mathrm{.}$ (5.4)

However S' is not equivalent to S because the equations of motion are different. At this point, a Lagrange multiplier λ can be introduced which acts to constrain A through the equations

$dA=\mathrm{0,}{\displaystyle {\int }_{C\in \partial M}}A\in 2\pi \mathbb{Z}\mathrm{.}$ (5.5)

When α satisfies these equations, α can be chosen using $\alpha \to \alpha +\omega$ and get (5.1) and (5.2) back. Then we write

$S^{\prime }=\frac{k}{2\pi }{\displaystyle {\int }_M}\alpha d\alpha +\frac{k}{2\pi }{\displaystyle {\int }_M}Ad\alpha +\frac{1}{2\pi }{\displaystyle {\int }_M}d\lambda A\mathrm{,}$ (5.6)

${\displaystyle {\int }_{C\in \partial M}}\lambda \in 2\pi \mathbb{Z}\mathrm{.}$

A path integral which integrates λ out is

$Z_{\lambda }={\displaystyle \int d\lambda }\exp \left(\frac{i}{2\pi }{\displaystyle \int d\lambda A}\right)\mathrm{.}$ (5.7)

Let each connected component of the boundary $\partial M$ denoted as ${\left(\partial M\right)}_a$ contain $p_a$ cycles $C_a$ which can serve to define the generators of the first homology group. As well there exist $p_a$ one forms ${\omega }_{ai}$ , for each a on $\partial M$ such that

${\displaystyle {\int }_{C_{aj}}}{\omega }_{ba}=2\pi {\delta }_{ij}{\delta }_{ab},i,j=1,2,\cdots ,p_a.$ (5.8)

If M is assumed to be a compact manifold, then ${\left(\partial M\right)}_a$ is compact and boundaryless, and M oriented means $\partial M$ is as well. So each connected piece ${\left(\partial M\right)}_a$ is a sphere with handles and its homology group has an even number of generators. Hence $p_a$ has to be even and the ${\omega }_{ai}$ can be ordered in such a way that

${\displaystyle {\int }_{\partial M}}{\omega }_{a\mathrm{,2}l-1}{\omega }_{bj}=4{\pi }^2{\delta }_{2l\mathrm{,}j}{\delta }_{a\mathrm{,}b}\mathrm{,}l=\mathrm{1,2,}\cdots \mathrm{,}{p}_a/2\mathrm{.}$ (5.9)

To any such ω on $\partial M$ can be associated an ω on M by requiring

${\nabla }^2\omega =0.$ (5.10)

Some Euclidean metric on M is needed to define the operator in (5.10), and the Laplacian operator on M. Other operators on M can be defined by using the same Euclidean metric which is also used to define an inner product. The pullback of ω to $\partial M$ must of course agree with the ω established there.

Based on (5.6) specifying the integral of λ, the following expansion holds

$\lambda ={\lambda }^{\left(0\right)}+{\displaystyle \underset{a\mathrm{,}i}{\sum }}{n}_{a\mathrm{,}i}{\omega }_{a\mathrm{,}i}\mathrm{,}{n}_{a\mathrm{,}i}\in \mathbb{Z}\mathrm{.}$ (5.11)

On the right side of (5.11), ${\lambda }^{\left(0\right)}$ is a one form on M which satisfies

${\displaystyle {\int }_{C_{a,j}}}{\lambda }^{\left(0\right)}=0.$ (5.12)

Also there is a boundary condition, namely, pullback of γ to $\partial M$ is ${\gamma |}_{\partial M}=0$ , and so (5.11) for λ becomes

$\lambda ={\displaystyle \underset{n}{\sum }}{\beta }_n{\gamma }_n+{\displaystyle \underset{a\mathrm{,}i}{\sum }}{n}_{a\mathrm{,}i}{\omega }_{ai}\mathrm{,}{{\gamma }_a|}_{\partial M}=0.$ (5.13)

Using these facts, we can integrate

$\begin{array}{c}{Z}_{\lambda }={\displaystyle \underset{n_{a,i}}{\sum }{\displaystyle \int {\displaystyle \underset{n}{\prod }d{\beta }_n}}}\exp \left(\frac{i}{2\pi }\left({\displaystyle \underset{n}{\sum }{\beta }_n}{\displaystyle {\int }_{M}d{\gamma }_{n}A}+{\displaystyle \underset{n_{a,i}}{\sum }{n}_{a,i}}{\displaystyle {\int }_{M}d{\omega }_{ai}A}\right)\right)\\ =\mathcal{N}{\displaystyle \underset{n}{\prod }\delta \left({\displaystyle \int d{\gamma }_{n}A}\right)}{\displaystyle \underset{n_{a,i}}{\sum }\exp \left(\frac{i}{2\pi }{\displaystyle \underset{n_{a,i}}{\sum }{n}_{a,i}{\displaystyle {\int }_{M}d{\omega }_{a,i}A}}\right)}\\ =\mathcal{N}\delta \left(dA\right){\displaystyle \underset{n_{a,i}}{\sum }\exp }\left(\frac{i}{2\pi }{\displaystyle \underset{n_{a,i}}{\sum }{n}_{a,i}{\displaystyle {\int }_{\partial M}{\omega }_{ai}A}}\right).\end{array}$ (5.14)

To arrive at this, two partial integrations are done, using completeness of the ${\gamma }_a$ in first step, and neglecting the bulk term in the following two lines. Also $\mathcal{N}$ is a generic constant developed along the way. Using the fact that

${\displaystyle \underset{n}{\sum }}{e}^{in\zeta }=2\pi {\displaystyle \underset{m}{\sum }}\delta \left(\zeta -2\pi m\right)\mathrm{,}$

then (5.14) becomes for $m_{a\mathrm{,}i}\in \mathbb{Z}$

$Z_{\lambda }=\mathcal{N}\delta \left(dA\right){\displaystyle \underset{a\mathrm{,}i}{\prod }}\left({\displaystyle \underset{m_{a\mathrm{,}i}}{\sum }}\delta \left({\displaystyle {\int }_{\partial M}}\frac{{\omega }_{ai}}{2\pi }A-2\pi m_{a\mathrm{,}i}\right)\right)\mathrm{.}$ (5.15)

The delta function implies that A is a closed one-form and A admits an expansion on the boundary

${A|}_{\partial M}=d\xi +{\displaystyle \underset{a\mathrm{,}i}{\sum }}{r}_{a\mathrm{,}i}{\omega }_{ai}\mathrm{,}$ (5.16)

where $\xi$ is a function on the boundary and $r_{ai}$ are real valued. Since

${\displaystyle {\int }_{\partial M}}{\omega }_{ai}d\xi =\mathrm{0,}$

as the ${\omega }_{ai}$ are closed at the boundary, substituting ${A|}_{\partial M}$ ,

$Z_{\lambda }=\mathcal{N}\delta \left(dA\right){\displaystyle \underset{a\mathrm{,}i}{\prod }}\left({\displaystyle \underset{m_{a\mathrm{,}i}}{\sum }}\delta \left(r_{a\mathrm{,}i}-m_{a\mathrm{,}i}\right)\right)\mathrm{.}$ (5.17)

The result of integrating λ gives the Equations (5.5) that we seek and this implies that the action S' is equivalent to the original action (5.1).

If the intention is to integrate A out of action S', the result is

$\begin{array}{c}{Z}_{\lambda }={\displaystyle \int DA\exp }\left(i\left(\frac{k}{2\pi }{\displaystyle \int \alpha d\alpha }+\frac{k}{2\pi }{\displaystyle \int Ad\alpha }+\frac{1}{2\pi }{\displaystyle \int d\lambda A}\right)\right)\\ =\mathcal{N}\delta \left(\frac{k}{2\pi }d\alpha -\frac{1}{2\pi }d\lambda \right)\exp \left(i\frac{k}{2\pi }{\displaystyle \int \alpha d\alpha }\right).\end{array}$ (5.18)

The delta in (5.18) implies the following equation holds

$d\alpha =\frac{1}{k}d\lambda \mathrm{.}$ (5.19)

This implies that α differs from λ by a closed one form which we call ${\omega }^{\left(1\right)}$ as

$\alpha =\frac{1}{k}\lambda +{\omega }^{\left(1\right)}\mathrm{.}$ (5.20)

Substituting α into $Z_{\lambda }$ , the following simple expression is obtained

$Z_{\lambda }=\mathcal{N}\delta \left(\frac{k}{2\pi }d\alpha -\frac{1}{2\pi }d\lambda \right)\exp \left(\frac{i}{2\pi k}{\displaystyle {\int }_M}\lambda d\lambda +\frac{i}{2\pi }{\displaystyle {\int }_M}d\left({\omega }^{\left(1\right)}\lambda \right)\right)\mathrm{.}$ (5.21)

The last term in this exponential is clearly a surface term since ${\omega }^{\left(1\right)}$ is closed, a fact which has been used to write it in this form. The dual action we get from this result by integrating A out is

$S_D=\frac{1}{2\pi k}{\displaystyle {\int }_M}\lambda d\lambda -\frac{1}{2\pi }{\displaystyle {\int }_{\partial M}}{\omega }^{\left(1\right)}\lambda \mathrm{.}$ (5.22)

The integral of λ is subject to the constraint condition where C is any cycle on the boundary

${\displaystyle {\int }_{C\in \partial M}}\lambda \in 2\pi \mathbb{Z}\mathrm{.}$ (5.23)

The second term in (5.22) is a surface term and it does not influence the equations of motion or contribute. Using the equations of motion $d\lambda =0$ , from the first term, it may be concluded that the second term must vanish. These considerations generalize to the case of several scalar fields coupled to a matrix and the case with many CS fields coupled together by means of a matrix $K^{IJ}$ as seen already.

6. Conclusion

The quantum Hall effect continues to advance especially on the experimental front [15] [16] [17] . The fractional Hall effect is observed in two-dimensional layers and arises under strong magnetic fields (~30 Tesla) with temperatures approaching zero. Since at very low temperatures the interactions of electrons are strong, the fractional effect represents a strongly correlated system. It has been shown that using a sequence of CS theories, a hierarchy can be established. This is actually what is known as the Haldane hierarchy. It has been indicated that there is a connection between these CS theories to chiral scalar field theories at the edge. Both give rise to the same Hall conductivity in the bulk. The CS theory when gauged gives rise to an effective CS theory for the electromagnetic potential, but is not gauge invariant. A surface action is required to restore gauge invariance. It is the gauged chiral scalar field theory at the boundary that functions as this surface action. It has become apparent recently that attaching more flux tubes can alter statistics. So an electron with 2 flux tubes attached may be regarded as a boson and the $m=0$ solution in [12] admitted giving even numbers in the associated fractions, as in the 5/2 state now known to exist.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

[1]	Fradkin, E. (1991) Field Theories of Condensed Matter Systems. Addison-Wesley, Reading.
[2]	Nagaosa, N. (1999) Quantum Field Theory of Condensed Matter Physics. Springer, Berlin. https://doi.org/10.1007/978-3-662-03774-4
[3]	Oh, P. (1996) Nuclear Physics B, 462, 551-570. https://doi.org/10.1016/0550-3213(95)00669-9
[4]	Balachandran, A.P., Chander, L. and Sathiapalen, B. (1995) Nuclear Physics B, 443, 465-500. https://doi.org/10.1016/0550-3213(95)00122-9
[5]	Zee, A. (1992) Progress of Theoretical Physics, Supplement, 107, 77-100. https://doi.org/10.1143/PTPS.107.77
[6]	Fröhlich, J. and Keller, T. (1991) Nuclear Physics B, 354, 369-417. https://doi.org/10.1016/0550-3213(91)90360-A
[7]	Karabali, D. and Nair, V.P. (2006) Journal of Physics A: Mathematical and General, 39, 12735-12763. https://doi.org/10.1088/0305-4470/39/41/S05
[8]	Watson, G. (1996) Contemporary Physics, 37, 127-143. https://doi.org/10.1080/00107519608230340
[9]	Laughlin, R.B. (1983) Physical Review B, 27, 3383-3389. https://doi.org/10.1103/PhysRevB.27.3383
[10]	Stone, M. (1991) Annals of Physics, 207, 38-52. https://doi.org/10.1016/0003-4916(91)90177-A
[11]	Girvin, S. and Jach, T. (1984) Physical Review B, 29, 5617-5625. https://doi.org/10.1103/PhysRevB.29.5617
[12]	Bracken, P. (2001) Canadian Journal of Physics, 79, 121-131. https://doi.org/10.1139/p01-073
[13]	Lütken, C.A. and Ross, G.G. (1992) Physical Review B, 45, 11837. https://doi.org/10.1103/PhysRevB.45.11837
[14]	Balatsky, A. and Fradkin, E. (1991) Physical Review B, 43, 10622-10634.
[15]	Karabali, D. and Nair, V. (2016) Physical Review D, 94, 064057. https://doi.org/10.1103/PhysRevD.94.064057
[16]	Jotzu, G., Messer, M., Desbuquois, R., Lebrat, M., Uehlinger, T., Grief, D. and Esslinger, T. (2014) Nature, 515, 237-240. https://doi.org/10.1038/nature13915
[17]	Mandel, S. and Basu, S. (2023) Physical Review B, 107, 035421.