Edge modes in the fractional quantum Hall effect without extra edge fermions

TL;DR

This paper demonstrates that the Chern-Simons-Landau-Ginsburg theory for the quantum Hall effect inherently maintains gauge invariance on bounded samples, eliminating the need for additional edge fermions.

Contribution

It shows that the quantum Hall effect's theoretical description is anomaly-free without extra boundary fermions, challenging previous assumptions.

Findings

Chern-Simons-Landau-Ginsburg theory is anomaly free on bounded samples

No extra edge fermions are needed for gauge invariance

Simplifies understanding of edge modes in quantum Hall systems

Abstract

We show that the Chern-Simons-Landau-Ginsburg theory that describes the quantum Hall effect on a bounded sample is anomaly free and thus does not require the addition of extra chiral fermions on the boundary to restore local gauge invariance.

Equations

$$H_{\mathrm{F}}=\frac{1}{2m}\sum _r{\left[{\mathbf{p}}_r-\frac{e}{c}\mathbf{A}({\mathbf{x}}_r)\right]}^2+\sum _{r}e{A}_0({\mathbf{x}}_r)$$

$$+\sum _{r<s}V({\mathbf{x}}_r-{\mathbf{x}}_s)+\sum _r{V}_c({\mathbf{x}}_r).$$

$$U=\exp \left(-i\sum _{r<s}\frac{\theta }{\pi }{\alpha }_{rs}\right),$$

$$U^{-1}\left({\mathbf{p}}_r-\frac{e}{c}\mathbf{A}({\mathbf{x}}_r)\right)U={\mathbf{p}}_r-\frac{e}{c}\mathbf{A}({\mathbf{x}}_r)-\hslash \frac{\theta }{\pi }\sum _{r\ne s}\nabla {\alpha }_{rs}.$$

$${\partial }_i{\alpha }_{rs}\equiv {\partial }_i\alpha ({\mathbf{x}}_r{-\mathbf{x}}_s)=-{\epsilon }_{ij}\frac{x_r^j-x_s^j}{|{\mathbf{x}}_r-{\mathbf{x}}_s{|}^2}.$$

$${\mathbf{a}(\mathbf{x}}_r)\equiv \frac{{\varphi }_0}{2\pi }\frac{\theta }{\pi }\sum _{s\ne r}\nabla {\alpha }_{rs},$$

$$H_{\mathrm{B}}=\frac{1}{2m}\sum _r{\left({\mathbf{p}}_r-\frac{e}{c}[\mathbf{A}({\mathbf{x}}_r)+\mathbf{a}({\mathbf{x}}_r)]\right)}^2$$

$$+\sum _{r}e{A}_0({\mathbf{x}}_r)+\sum _{r<s}V({\mathbf{x}}_r-{\mathbf{x}}_s)+\sum _r{V}_c({\mathbf{x}}_r).$$

$$H_{\mathrm{M}}={\int }_{\Gamma }{d}^{2}x\left\{\frac{{\hslash }^2}{2m}{\left(\mathbf{D}\varphi \left(\mathbf{x}\right)\right)}^{†}\cdot \mathbf{D}\varphi \left(\mathbf{x}\right)\right\}$$

$$+{\int }_{\Gamma }{d}^{2}x\left\{e{A}_0\left(\mathbf{x}\right){\varphi }^{†}\left(\mathbf{x}\right)\varphi \left(\mathbf{x}\right)\right\}$$

$$+\frac{1}{2}{\int }_{\Gamma }{d}^{2}x{d}^{2}y\delta \rho \left(\mathbf{x}\right)V(\mathbf{x}-\mathbf{y})\delta \rho \left(\mathbf{y}\right).$$

$$D^k={\partial }^k+\frac{ie}{\hslash c}\left(A^k(\mathbf{x})+a^k(\mathbf{x})\right),$$

$$S_{\mathrm{M}}=\int d^{3}x\left\{\frac{i\hslash c}{2}\Theta \left(\mathbf{x}\right){\varphi }^{†}\left(x\right)D_0\varphi (x)\right\}$$

$$+\int d^{3}x\left\{-\frac{i\hslash c}{2}\Theta \left(\mathbf{x}\right){\left(D_0\varphi \left(x\right)\right)}^{†}\varphi (x)\right\}$$

$$+\int d^{3}x\left\{-\frac{{\hslash }^2}{2m}\Theta \left(\mathbf{x}\right){\left(\mathbf{D}\varphi \left(x\right)\right)}^{†}\cdot \mathbf{D}\varphi (x)\right\}$$

$$-\frac{1}{2}\int d^{3}x{d}^{3}y\Theta \left(\mathbf{x}\right)\Theta \left(\mathbf{y}\right)\delta \rho \left(x\right)V(\mathbf{x}-\mathbf{y})\delta \rho \left(y\right),$$

$$D_0=\frac{1}{c}{\partial }_t+\frac{ie}{\hslash c}{A}_0\left(\mathbf{x}\right)$$

$$\equiv {\partial }_0+\frac{ie}{\hslash c}{A}_0\left(\mathbf{x}\right),$$

$$\Theta\left(\mathbf{x}\right)=\left\{\begin{array}[c]{c}1,\;\;\mbox{if $\mathbf{x}\in{\Gamma}$},\\ 0,\;\;\mbox{if $\mathbf{x}\notin{\Gamma}$}.\end{array}\right.$$

$${\partial }_{\mu }{j}_{\mathrm{M},\Gamma }^{\mu }={\partial }_0{j}_{\mathrm{M},\Gamma }^0+{\partial }_i{j}_{\mathrm{M},\Gamma }^i\left(x\right)=0,$$

$$j_{\mathrm{M},\Gamma }^0=\Theta \left(\mathbf{x}\right){\varphi }^{†}\left(x\right)\varphi \left(x\right)=\Theta \left(\mathbf{x}\right)\rho \left(x\right),$$

$${\mathbf{j}}_{\mathrm{M},\Gamma }=\Theta \left(\mathbf{x}\right){\mathbf{j}}_{\mathrm{M}}\left(x\right),$$

$${\mathbf{j}}_{\mathrm{M}}\left(x\right)=\frac{i\hslash }{2m}\left\{{\varphi }^{†}\left(x\right)\mathbf{D}\varphi \left(x\right)-{\left(\mathbf{D}\varphi \left(x\right)\right)}^{†}\varphi \left(x\right)\right\}.$$

$$a_i(x)=-\frac{{\varphi }_0}{2\pi }\frac{\theta }{\pi }{\epsilon }_{ij}{\int }_{\Gamma }{d}^{2}y\frac{x_j-y_j}{|\mathbf{x}-\mathbf{y}{|}^2}\rho \left(y\right)$$

$$=-\frac{{\varphi }_0}{2\pi }\frac{\theta }{\pi }{\epsilon }_{ij}\int d^{2}y\frac{x_j-y_j}{|\mathbf{x}-\mathbf{y}{|}^2}\Theta \left(\mathbf{y}\right)\rho \left(y\right).$$

$${\epsilon }_{ij}{\partial }_i{a}_j(x)={\varphi }_0\frac{\theta }{\pi }\Theta \left(\mathbf{x}\right)\rho (x),$$

$${\partial }_i{a}_i=0.$$

$${\epsilon }_{ij}{\partial }_0{a}_j(x)=-{\varphi }_0\frac{\theta }{\pi }\Theta \left(\mathbf{x}\right)j_{\mathrm{M}}^i.$$

$$S=S_{\mathrm{M}}\left(a_0\right)+S_{\mathrm{CS}},$$

$$S_{\mathrm{CS}}\equiv \frac{{\sigma }_{xy}}{2}\int d^{3}x{\epsilon }^{\mu \nu \rho }{a}_{\mu }{\partial }_{\nu }{a}_{\rho },$$

$${\sigma }_{xy}\equiv \frac{\pi }{\theta }\frac{1}{{\varphi }_0}.$$

$$S_{\mathrm{CS}}[a_{\mu }+{\partial }_{\mu }\alpha ]=S_{\mathrm{CS}}[a_{\mu }]+\frac{{\sigma }_{xy}}{2}\int d^{3}x{\epsilon }^{\mu \nu \rho }\left({\partial }_{\mu }\alpha \right){\partial }_{\nu }{a}_{\rho }$$

$$=S_{\mathrm{CS}}[a_{\mu }]-\frac{{\sigma }_{xy}}{2}\int d^{3}x{\partial }_{\mu }\left({\epsilon }^{\mu \nu \rho }\alpha {\partial }_{\nu }{a}_{\rho }\right)$$

$$=S_{\mathrm{CS}}[a_{\mu }].$$

$$S_{\mathrm{CS}}^b\equiv \frac{{\sigma }_{xy}}{2}\int d^{3}x\Theta \left(\mathbf{x}\right){\epsilon }^{\mu \nu \rho }{a}_{\mu }{\partial }_{\nu }{a}_{\rho },$$

$${\epsilon }_{ij}{\partial }_i{a}_j(x)={\varphi }_0\frac{\theta }{\pi }\rho (x),$$

$${\epsilon }_{ij}{\partial }_0{a}_j(x)=-{\varphi }_0\frac{\theta }{\pi }{j}^i,$$

Full text

quantum Hall effects; gauge field theories; chiral fermions

I Introduction

From a field theoretical point of view, the fractional quantum Hall effect (FQHE) is well understood since 1989, when it was shown that it could be described by a Chern-Simons-Landau-Ginsburg (CSLG) effective theory zhang . Two years later, the effect of boundaries was analysed wen and a breakdown of gauge invariance was found, due to a bounded Chern-Simons (CS) action. Noticing the fact that the microscopic theory is gauge invariant, several authors wen ; maeda concluded that an anomaly appeared, that would have to be cancelled in order to recover gauge invariance. This was done by the addition of extra degrees of freedom, in the form of chiral fermions circulating at the edge, whose well known gauge anomaly restored gauge invariance of the complete theory. These extra degrees of freedom were, since then, usually identified with chiral edge modes, which are fundamental in the wave function approach to the FQHE girvin .

In this letter, we briefly review the steps conducting from the microscopic Hamiltonian to the CSLG theory in the presence of boundaries. We show that, by carefully considering the influence of the boundary in the dynamics of the CS field, one finds no such anomaly and the resulting theory is gauge invariant. Thus, there is no need to introduce extra degrees of freedom such as chiral edge fermions. The edge modes appear naturally from the dynamics of the CS field, which is determined by its coupling to the Noether current of the CSLG theory. Thus, they have nothing to do with the chiral edge fermions, introduced in previous approaches.

Edge states were found experimentally wang-goldman and appear as an essential ingredient in several recent works. We can quote some of them which deal with various subjects such as edge states in graphene yao ; hatsugai , descriptions of chiral Luttinger liquids boyarsky ; chang and relations between edge electrons and Berry’s phase basu . Accordingly, the association of these modes with the original degrees of freedom of the CSLG effective theory is very important both for present and future applications.

II FQHE on a bounded surface

We briefly review the derivation of the CSLG model for the FQHE zhang , modifying the procedure when necessary to take into account the finiteness of the surface. We start from the microscopic Hamiltonian of a two-dimensional system of polarized electrons interacting with an external electromagnetic field**.**

[TABLE]

In (1), $V_{c}(\mathbf{x}_{r})$ is an electrostatic potential which is responsible for confining the electrons into the bounded region $\Gamma$. It is essentially zero in the bulk of $\Gamma$ and very large as one approaches the boundary. As the electrons are polarized, the wave function is completely antisymmetric. It is possible to map this fermionic problem into a bosonic one. This can be done by means of an unitary transformation

[TABLE]

were $\alpha_{rs}$ is the angle between $\mathbf{x}_{r}-\mathbf{x}_{s}$ with an arbitrary direction that may be chosen as the $x$-axis and $\theta=(2k+1)\pi$ with $k$ being an integer. Under this choice, one can easily verify that an antisymmetric wave function $\psi$ is mapped into a symmetric (bosonic) one $\phi\equiv U^{-1}\psi$. It is easy to check that

[TABLE]

The gradient of the angle between the vector $\mathbf{x}_{r}-\mathbf{x}_{s}$ and the $x$-axis is given by

[TABLE]

Now, a statistical field is defined

[TABLE]

where $\phi_{0}=hc/e$ is the quantum of flux. The bosonized Hamiltonian is $H_{\mathrm{B}}=U^{-1}H_{\mathrm{F}}U$ or

[TABLE]

The second quantized Hamiltonian is obtained through the introduction of a bosonic field $\phi\left(\mathbf{x}\right)$ ($\mathbf{x}$ denotes $\left(x_{1},x_{2}\right)$) satisfying$\left[\phi\left(\mathbf{x}\right),{\phi}^{\dagger}{(\mathbf{y})}\right]=\delta^{\left(2\right)}\left(\mathbf{x}-\mathbf{y}\right),$ and generalizing $H_{\mathrm{B}}$ to the (hermitian) matter Hamiltonian

[TABLE]

Notice that we incorporated the effect of the confining potential $V_{c}$ by restricting the integration domain over the region $\Gamma$. An alternative approach would be to explicitly take into account the confining potential by adding a Gaussian term $\int d^{2}x\left\{V_{c}\left(\mathbf{x}\right)\phi^{\dagger}\left(\mathbf{x}\right)\phi\left(\mathbf{x}\right)\right\}$ and not restricting integration. We will follow the first approach to make it easier to compare our results with the literature. In (7) the covariant derivative was defined as

[TABLE]

$\rho\left(\mathbf{x}\right)={\phi}^{\dagger}{(\mathbf{x})\phi(\mathbf{x})}$ and $\delta\rho\left(\mathbf{x}\right)=\rho\left(\mathbf{x}\right)-\bar{\rho}$ (the average density $\bar{\rho}$ is included here to avoid problems in the case of a long range potential zhang ). Taking all operators in the Heisenberg picture, they become functions of time. The action below generates the correct Heisenberg equations

[TABLE]

with

[TABLE]

and the integration being effectively over the surface of the sample, which is obtained by the use of a step function $\Theta$ defined as

[TABLE]

Global phase invariance of the action (9) under the transformations $\phi^{\prime}\left(x\right)=e^{i\alpha}\phi\left(x\right)$ implies the continuity equation

[TABLE]

where the components of the matter current are given by

[TABLE]

with $\mathbf{j}_{\mathrm{M}}$ given by

[TABLE]

The field $\mathbf{a}\left(x\right)$ is completely determined in terms of the density operator $\rho\left(x\right)$, and is given by the second quantized version of equation (5),

[TABLE]

The field $\mathbf{a}\left(x\right)$ shown above can be seen as an auxiliary field. It is the solution of the following pair of equations

[TABLE]

Using the continuity equation (13) we can derive a third equation for the field $a_{i}\left(x\right)$ involving a time derivative

[TABLE]

Equations (15) and (16) may be viewed as the equations of motion of a new dynamical field, if we make the substitution $D_{0}=\partial_{0}+\frac{ie}{\hbar c}A_{0}\left(\mathbf{x}\right)\rightarrow\partial_{0}+\frac{ie}{\hbar c}\left(A_{0}\left(\mathbf{x}\right)+a_{0}\left(x\right)\right)$ (which means that now $S_{\mathrm{M}}=S_{\mathrm{M}}\left(a_{0}\right)$), and replace $S_{\mathrm{M}}$ by the action

[TABLE]

with

[TABLE]

where $S_{\mathrm{CS}}$ is known as the Chern-Simons (CS) action and we define the Hall conductivity as

[TABLE]

The additional field $a_{0}\left(x\right)$ introduced in (18) and in $S_{\mathrm{M}}$ can be eliminated by requiring the condition $a_{0}\left(x\right)=0$, what is legitimate in the context of a gauge field theory. Thus, gauge invariance is crucial for the correct introduction of the statistical CS field $a_{\mu}\left(x\right)$. Without it, the equations of motion obeyed by the field are not enough to eliminate this extra component and the identification of the dynamics of $a_{\mu}\left(x\right)$ with that of a CS field cannot be made.

The action $S$ is gauge invariant, because the CS part $S_{\mathrm{CS}}$ (equation 18) is not being integrated over a finite surface. Then, under a gauge transformation,

[TABLE]

The restriction of the domain of integration to the area of the sample is the origin of the destruction of gauge invariance wen ; maeda . This problem is absent here. Thus, gauge invariance is a consequence of unbounded integration in equation (18).

If one insisted to consider the CS action on a surface with a boundary,

[TABLE]

the resulting equations of motion for the CS field would be (setting $a_{0}=0$ artificially, as this is only allowed if the theory is gauge invariant)

[TABLE]

and these equations would result in the solution

[TABLE]

which does not contain the restriction of the integration to the surface of the sample and, so, does not coincide with (14). This would result in a second quantized action which is not equivalent to the original problem. Therefore another reason why the CS action must be unbounded is to provide the correct solution for the statistical field.

Although the Chern-Simons part of the action makes no reference to the boundary of the sample, we have to remember that the CS field is minimally coupled to the matter fields. This coupling involves the Noether current and forces the equations of motion for the CS field to depend on the boundary, as can be explicitly seen in (15) and (16). This has to be so, if one wants to recover (14).

III Compatibility with known results

We can follow zhang and seek for a mean field solution in the presence of a magnetic field $\Theta\left(\mathbf{x}\right)B=-\varepsilon_{ij}\partial_{i}A_{j}$. Again the proposed form is:

[TABLE]

where $\bar{\rho}$ is the average particle density. The current takes the form $j^{\mu}=\left(\Theta\left(\mathbf{x}\right)\bar{\rho},\mathbf{0}\right)$. The equations of motion in the CS sector are

[TABLE]

Identifying the density of magnetic flux as $\rho_{{\Gamma}}=B/\phi_{0}$, one obtains the condition for the validity of the mean field approximation

[TABLE]

This is exactly the same result obtained in zhang for the filling factors.

Concerning the expression of the current in terms of the CS field, we can see, using the CS equations of motion

[TABLE]

So, $j_{\mathrm{M,}{\Gamma}}^{\mu}=$ $-\delta S_{\mathrm{CS}}/\delta a_{\mu}$. Computing this last quantity, we obtain

[TABLE]

This is precisely the final current obtained in maeda , after the effect of the chiral edge fermions has been taken into account. In this paper, the author integrates over chiral $(1+1)$-dimensional edge fermions to obtain an effective action which, considered along with a bounded Chern-Simons action (not gauge invariant), results in a gauge invariant theory. The anomalous term in the current is cancelled and it remains only the contribution equal to that of a non-bounded Chern-Simons action (which is gauge invariant). Our approach leads directly to this result.

It must also be emphasised that the result (27), contained in wen ; maeda , means that the edge modes can not be identified with the extra edge fermions. This is explicitly said by the author of ref. maeda , which follows the approach proposed in ref. wen . Our approach gives the same current without the need of introduction of any extra degrees of freedom.

All that remains is to see what happens at the edge using the equation $\partial_{\mu}j_{\mathrm{M},{\Gamma}}^{\mu}=0$. This equation gives

[TABLE]

Equation (28) implies two separated equations: one for the bulk and other for the edge of the surface of the sample. We obtain,

[TABLE]

and

[TABLE]

where $n_{i}$ is a vector field that is normal to the boundary. Condition (30) says that the current at the edge is only tangential, and this completes the identification of the edge modes relevant to the FQHE with the matter current, as expected.

IV Conclusion

Close inspection on the calculation done in references zhang ; wen ; maeda shows that one obtains, starting from $S$, given by equation (17) and gauge invariant, the same value for the Hall conductance and edge current that were obtained previously, starting with a theory which was not gauge invariant and adding extra degrees of freedom in the form of chiral edge fermions. So, there is no reason for the introduction of extra one-dimensional chiral fermions circulating on the boundary.

Acknowledgements.

G. L. S. Lima has been financially supported by CAPES (Brazil) during the realization of this work.