Edge waves in the quantum Hall effect

Edge waves in the quantum Hall effect

ANNALS OF PHYSICS 207, 38-52 (1991) Edge Waves in the Quantum Hall Effect MICHAEL STONE University of Illinois at Urbana Champaign, 1110 West G...

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207, 38-52


Edge Waves in the Quantum

Hall Effect


of Illinois at Urbana Champaign, 1110 West Green Street, Urbana, Received


30, 1990; revised

Department of Physics, Illinois 61801 August

14, 1990

I show that the bosonized form of the low energy edge excitations at the surface of a droplet of two-dimensional quantum Hall liquid can be understood in terms of coherent “ripplon” deformations in the shape of the droplet. When interacting with an electromagnetic field the resulting chiral boson theory has anomalies that are cancelled, via the Callan-Harvey mechanism, by contributions from the bulk. I also consider non-Abelian generalization of the bosonization that may be useful for chiral spin states. ?’ 1991 Academic Press, Inc.

1. INTRODUCTION The flow of a globule of incompressible and irrotational fluid is determined entirely by the changing shape of its free surface. Considering the fluid as a dynamical system we would say that all the degrees of freedom reside on the boundary. A Chern-Simons field theory defined in a simply connected two-dimensinal region shares this property [ 1, 21 and this is no coincidence: a droplet of two-dimensional electron gas (2DEG) in the quantum Hall regime is effectively incompressible and its motion irrotational at energies below the magnetophonon gap [4, 51, and the Hall liquid responds to electromagnetic fields via a Chern-Simons effective action containing these (non-dynamical) external fields. Recent work by Wen [S-S] uses this observation to deduce that the only low-energy excitations, the edge states, form representations of chiral Kac-Moody algebras [9]. In this paper I expand on the results of Wen and show that these algebras may be represented by chiral boson fields [lo] describing “ripplon” fluctuations in the shape of the 2DEG droplet. The ripplons may be regarded as the edge degrees of freedom of a Chern-Simons action with dynamical fields [2]. The connection between the edge states and the bulk 2DEG provides an example of the Callan-Harvey anomaly cancellation mechanism

Clll. The picture that emerges should be of interest to both solid state and particle physicists. In the quantum Hall system many of the mathematical constructions of string theory are available for experimental investigation: The central charges of the Kac-Moody algebras are related to the electric charge of the edge excitations [IS], the central charges of the intertwining Virasoro algebras determine the 2DEG 38 0003-4916/91 Copyright All rights


0 1991 by Academic Press, Inc. of reproduction in any form reserved.





contributions to the specific heat [6], while the uni-directional motion of the edge waves provides a concrete example of the key concept in the heterotic string. It is also possible that the technology developed by string theorists may have other applications in these systems. In the fractional quantum Hall effect (FQHE) there are many branches of gapless edge excitations [12] but in this paper I will, for simplicity, concentrate on the the case of a single branch. This means that most of my results apply, in the first instance, only to the integer effect or to the lowest level of the FQHE hierarchy. I hope, however, that the technique and language will be useful in the more general cases. In Section 2 we will examine in detail the case of the integer quantum Hall effect. We will see that there is an essentially exact description of the dynamics of this state in terms of chiral edge states and that these states can be interpreted as bosonic “ripplon” fluctuations in the droplet shape. In Section 3 we will review the bosonization of general chiral excitations. In Section 4 we examine the “anomalies” that appear in the edge system when it is coupled to external electromagnetic fields and how they are removed when we consider the system as a whole. In an appendix we will discuss the analogy with the Callan-Harvey effect.


In an interesting series of papers [bS] Wen has focused attention on the edges of the quantum hall sample and, under very general assumptions, has shown that these edges support operators with the structure of Kac-Moody algebras. Since these algebras are rather abstract objects, it is worth considering the simplest case in some detail and trying to extract the physical content from the mathematics. The simplest example is provided by the edge of the lowest Landau-level in the n = v = 1 integer Hall effect state. The application of a large magnetic field supresses many of the degrees of freedom of a quantum mechanical particle. In two dimensions restricting the motion to the lowest Landau level reduces the original four-dimensional phasespace to one with only two canonically conjugate variables, the coordinates X, y. In addition, for fermions, the only low energy degrees of freedom are transitions from an occupied state to an unoccupied state in the same level, excitations to the next level costing energy o, = eB/m. In a quantum Hall device the fermions form a twodimensional droplet conlined by a potential well and, for the v = 1 QHE state, they have uniform density B/27-c.Moving an electron from within the droplet to an unoccupied state outside is stongly reminscent of exciting an electron-hole pair from a Dirac, or Fermi, sea. The purpose of this section is to argue that this analogy is, in essence, exact: the v = 1 droplet is a Fermi sea and its surface is its Fermi surface. We will take the particles to have mass m = 1 and charge e = 1. Using the linear




gauge, A,. = Bx, and keeping only states from the lowest Landau level, the secondquantizied field operator is (2.1)

For convenience in counting densities of states we have taken the system to be periodic in the y direction so the allowed momenta are k, = 2xn/L,,. The wave functions have been normalised in this volume and the annihilation and creation operators obey {a,, al;!> = 6,,. Since the physical droplet has a boundary with the topology of a circle these boundary conditions are sensible-indeed, since the edge excitations will turn out to be chiral, there will be no suitable self-adjoint definition for the edge state Hamiltonian unless we take periodic boundary conditions [ 131. In the absence of an electric field the states created by u: all have the same energy and carry no net current-but once an electrochemical potential gradient V(x) = Ex is added to the Hamiltonian, the energy of the state with canonical momentum k becomes E(k) = + Ek/B and each state carries a Hall curent. The group and phase velocities in the y direction coincide with the Hall current drift velocity E/B. Such a potential occurs natually at the edge of any Hall effect device c141. Since we are measuring the energy from the Fermi surface, the negative energy states will be occupied by electrons. With V= Ex the physical surface of the 2DEG will then be at x = 0, the states to the left being full and those to the right empty. This physical edge of the droplet in x space can be identified, through the relation x, = k,/B between the location of the centres of the Gaussian wavefunctions and their momentum k,, with the Fermi surface at k = 0, the “speed of light” or “Fermi velocity,” vr, being the drift velocity E/B. Since all the states move in the same direction the droplet is the Dirac sea of a chid, or Weyl, fermion. To probe the excitations near the Fermi surface we define the y dependent “surface charge” operator, j(y), as the charge density integrated over an x interval [ -A, A] containing the edge. (2.2)

We must take the cutoff A larger than the amplitude of any expected ripple in the surface but the Fourier components with non-zero momentum will be independent of the particular choice of A We can write j( y) = C eeiknyj,, where k, = 2nnlL,,, and

j,= +f f~jn+~u~e-~~~. m= -cc


By focusing on excitations whose wavelengths are long compared to the magnetic









length, the exponential cutoff can be ignored and j, is precisely the density operator of a “relativistic” right-going chiral fermion with hamiltonian,

The total charge operator & =j, has an unimportant, but ,4 dependent, c-number contribution from states far inside the droplet. This is removed in the usual manner by normal ordering the operator with regard to the k = 0 Fermi surface. It has been shown by many authors, begining with Jordan [lS] that the commutator of the j, operators has a Schwinger term, or central charge, whose magnitude in the case of a normal Fermi sea is determined by the f sum-rule [16]. (A particularly clear demonstration is provided by Manton [17].) The commutator is




and the j, form the simplest example of a Kac-Moody algebra, i.e., U( 1) at level one [9]. It is worth stressing that the Schwinger term is a Berry phase [lS] and is thus a property of the states to which the operator is applied, rather than a property of the operator stemming from the normal ordering (which only affects j,): if the positive k states were filled, rather than the negative k ones, the central charge has opposite sign [19]. The commutator (2.5) is the key ingredient in the solution of the Tomonaga [20] and Luttinger [21] models as well as the physics of a number of other one-dimensional electron systems [16]. In terms of the coordinate along the edge of the droplet the commutator reads

CAyLAy’) =z ~,JLv-y’).


The representation of this algebra by bosonic operators has been described in one language by Wen in Ref. [7]. Here I will give a somewhat different view of the same physics. Denote the state of the undisturbed droplet by 10). We can produce coherent excitations of the droplet by using these j(y) operators: define the family of states

remember that (0 (j( y)l 0 ) = 0, and use ,-ij

H(y’)j(y’)d.v’ j(Y)





+ &





=j(y) + & ~,,W)








to see that the state 113) has charge


These states are seen to correspond to ripples in the surface density. For long wavelength the effect of the excitation operator is to change the local value of k to k + a,8 in the Landau-level eigenstates. Because the new state is a still a linear superposition of lowest Landau-level states, and because for these states there is a connection between k and the location of the state along the x axis, one knows that the surface density fluctuations correspond to changes in the droplet shape-as they should since the 2DEG is incompressible. Further, because all states near the edge propagate at the same speed, the ripples should move at this speed as well. To see this formally we commute the Hamiltonian fi through exp(i J O(y’)j( y’) &j(y)). The Hamiltonian Z? comes from the potential gradient and the dependence of the position of the state on its momentum; (2.10) so we find eand, as claimed, unchanged:



the ripples move along at the drift velocity with their shape e -‘“‘le(y))=Ie(y-qt)).

The description of the fermionic edge states in an example of chiral bosonization. It is by no ) O(x)) states exhaust all possible excitations-but may be demonstrated by examining the partition origin from which we measure the energy so that and the allowed energies become

(2.12) terms of coherent boson states is means obvious that the bosonic the plausibility of this assertion function: For simplicity move the it lies midway between two levels

s,=32n-1). .L’ Write q= exp( -/?(E/B)(rr/L,)). Then, remembering that we must count both particle states and hole states, the partition function is .Y=Tr(e

[email protected]@fi}=



[email protected])(1




(Strictly speaking, we should include a Casimir energy factor e+2”B”2L~. This factor







plays a vital role when one is interested in the the modular transformation properties of b-but it plays no role in the present, purely combinatorial, argument.) The Jacobi triple-product formula [22] ,,e, (1 - q2”)(






1) =




then shows that 9 can be written as (2.16) which is the partition energies

function for neutral bosonic excitations (the ripplons)


(2.17) together with a sum over charged sectors, each created by first inserting (or removing) n extra fermions in the lowest available states above the sea, and then creating the ripplons as distortions of the new state. The total energy of these n extra fermions is proportional to I2 + 32 + 5’ + . . + 2n - 1= rr2 and accounts for the qnZ. In the bosonized language the charged states are composed of solitons where the U( 1) field winds once for each extra unit of charge. (This is another way of saying that the U( 1) group is disconnected and an irreducible representation of the whole group contains many copies of the representation of the Lie algebra.) In the next sections we will generalize these coherent excitations and examine their connection to the bulk flow of the Hall liquid.




In the previous section we have discussed, in operator language, the edge states and their bosonization. It useful to have a path-integral description of the chiral bosons but the chiral constraint serves to complicate matters [23-251. It is actually easiest to understand the physics by using a non-Abelian generalization of the chiral bosons invented by Sonnenschein [26] and then specialising to the Abelian case after understanding the broader setting. This general machinary of chiral bosonisation is useful in its own right when we have more than one “flavour” of fermions-as occurs when we wish to include spin excitations or, for example, in the chiral spin states of Wen, Wilczek, and Zee [27, 63. The simplest route to chiral bosonization [lo] is via the quantum mechanical concept of coherent-state path-integrals [28]. These begin with an irreducible representation D(g) of some continuous group G and define a collection of




generalized coherent states [29], labeled by elements ge G, analogous labeling of the states 10) by the function 13(x), [email protected]> =Wg)lO). Because of the irreducibility, completeness relation

to the (3.1)

Schur’s lemma shows that these states satisfy an over-



where d[g] is the invariant measure on G. We can use the I g) to give a path-integral persistence amplitude,



for the vacuum



(The trace is taken only over the states in the representation D(g).) By repeatedly inserting the resolution of the identity, (3.2) into the trace, we obtain the path integral

where d[g] is the path measure made from taking the invariant measure at each time step and the integral is over all paths in G taking time t. This construction applies in a natural way to the loop group [30] generated by the Kac-Moody algebra [S], CJi(XL


= iffjJk(X) 6(X - JJ) - $ tr(&Aj)


- y),

for k= 1. This is discussed for k= 1 in [lo] and for k> 1 in [31]. The loop group, LG, is the “gauge” group made by taking an element of an ordinary Lie group, G, at each point of the one-dimensional space. Multiplication is done point-by-point and seems trivial-but all non-trivial representations are projectiue, i.e. representations up to a phase. They are spanned (for level k = 1) by coherent deformations, made by application of group elements g(x, t) which are exponentials of integrals of the generating currents Ji(x), of a chiral Dirac sea. These coherent states are labeled by the G-valued fields I g(x, t)) =2(x, t)[O). Setting ur= 1, we calculate the ingredients for (3.4) and find that the action contains a Wess-Zumino term. To write it in a simple form we have to extend the functions g(x, t) to g(x, t, r) defined in the interior of the region bounded by the two-dimensional space time. In terms of these extended fields the action in the path integral is found to be [lo] S(g)=









Despite the extended domain of definition for the fields, the variation of the action depends only on values of g at physical points. We find

so the clasical equation of motion is



with solutions

.dx, t) =g,(x-



where g2(f) is arbitrary x independent factor. The solutions look like right-going waves-but, in addition, there is a hidden gauge invariance which shows itself in the factor g*(t). The reason for the gauge invariance is that any x independent right factor in the 2 leaves the Dirac sea invariant up to a phase so the coherent states are really elements of the coset space LG/G [ 10,301. The physical currents have bosonized form

$+(x1Aill/ = & W&a, gg-‘)


which is insensitive to this arbitrary factor. (There are also “unphysical” currents which are sensitive to gz(t) and they can be used for sewing two chiral bosons together to make one Wess-Zumino-Witten bosonized Dirac fermion [32, 331. These may be useful when two edges are in proximity [8].) The simplest Hall effect case considered in the previous section uses the Abelian, U(l), group where g(x, t) = e iB(-‘.r). In this case the action becomes (3.11) with equation of motion a,(a, + a,) e = 0


and solution qx, t) = exhibiting

the undetermined



t) + e,(t)

gauge degree of freedom in e,(t).





The momentum field conjugate to 0(x, t) is n(x, t) = commutation relations for the rc(x) field are [7r(x), 71(x’)] = - & a,6(x-xr).

(1/27c) i3Y8 and the


Writing (j(.y) = q. - $ pox + i 1 : e2nin-r/L R#O


and thus 7r(x) =i


p. + C a,e2nin”iL . > n+O


We obtain (3.14) by taking

Cqoypal= i.


If the 0 field contains a classical soliton twist, where 6’ -+ 6 + 27~2 as we circle the boundary, we would expect the eigenvalue of p. to equal -n. This looks innocuous until we realise that q. contains the unphysical gauge degree of freedom and the gauge transformations are generated by po. Demanding that the wave function be independent of the q. requires p. = 0 and thus there are no solitions. Although it is customary to require the wave function to be gauge invariant, it is is not strictly necessary: the wave function may be a section of libre bundle over the configuration space of the system; i.e., it changes by a prescribed phase as the unphysical degrees of freedom vary. The phase must, however, form a representation of the global U( 1) charge group and this selects the allowed eigenvalues of p. to be integers-consistent with the soliton requirement. These phases are natural here as the twisted states have charges and therefore change by a phase under global rotations of the Dirac sea [lo]. The Hamiltonian for the chiral boson is’ a sum of free oscillators and one rotator (3.18) from which we may read off the allowed energies and find the partition function to be identical to that found in (2.16). Wen has also discussed [S] the bosonization of the v = 1/(2n + 1) FQHE states. These are still level-one algebras (although with a differential normalization of the generators) and the only significant effect is on the compactification radius of the U( 1) group where we now have to wind 2n + 1 times to create a state with charge e= 1. ’ I am ignoring

the extra



term which

yields e,(f).




4. ATTACHING THE SURFACE TO THE BULK Suppose the two-dimensional electron liquid occupies a region Q. We know that the bulk of the liquid responds to an external electromagnetic field by producing currents whose maginitude and direction are precisely quantified by the integer Hall effect. If the filling factor is v = k this means that the effective action for the bulk is a Chern-Simons action



where A, is the electromagnetic potential. The current derived from this action, J,, = SSjSA, = (k/271)E~~=F~~,is conserved within the region, but not at the edge, X2. To make a gauge invariant theory we need to add terms describing the motion of the edge, or, if the walls present expansion, the accumulation of charge there. I have already described the edge motion of a single Landau level in terms of a chiral boson field and it seems reasonable to do the same for the higher (integer) v cases. We need to see how to consistently sew together the bulk action and a chiral boson field. So as to have a concrete picture in mind while examining the general case, let me describe the physical origin of the gauge anomaly in the edge states: A chiral fermion interacting with an external electromagnetic field has Hamiltonian 8=

‘~~~‘(-i(?x-A,-Al)~. sil


I have taken the particle to live on a circle of circumference L. The fields occur in the combination A, + A, because $?I++ is both the charge density and the current density. Temporarily set A, = 0 and look at the single particle eigenstates at fixed A,r. They are simply ,bn(x) = pin-w,

27c &,=-n-A L



When A, is made vary with time as A, = -Et, the particles gain energy from the electric field as they should-but this means that states are crossing the fermi surface at rate E/27cper unit length, and, counting from the fermi surface, the particle density is changing. The naive conservation law (8, + a,) tit+ = 0 must be replaced by the anomalous version

If these fermions were the whole story the anomaly would prevent us from producing a consistent, gauge invariant, theory-as is the case in the the Standard 595/207/M




Model of Weak Interactions where, complete quark-lepton “generations” of chirally coupled particles are needed for satisfactory physics. In the Hall droplet system an everywhere tangential E field must come from a changing magnetic flux, and the anomaly is simply the expansion or contraction of the droplet as the B field is varied. It is another statement of the Hall effect. The non-conservation of charge implied by the bulk CS effective action will match, and cancel, the non-conservation of charge for the edge states. For demonstrating the general structure, and, in fact, for case of calculation (the non-abelian group structure provides a structure that guides the calculation), it is again easiest to consider the non-Abelian extension of the problem. In the case of a non-Abelian group the (2 + I)-dimensional Chern-Simons term is, in the compact notation of differential forms,

For the non-Abelian have

case, k must be an integer. Under a variation A + A + 6A, we (4.6)

F is the field strength 2-form F= ~54 + A. Under a gauge transformation

A +hAh-‘-dhh-‘,


the field strength changes by F + hFh ~ ’ and, for infinitessimal

h = 1 + E, (4.8 1

6,A = [E, A] - d&.


(4.8) into (4.6) gives

6,&-= --$ja~xRTr(E(F-A2))_ -&jaaxR*dEdAh


and the CS action is not gauge invariant on its own. At the edge of the droplet we attach a chiral boson for the level k algebra, d3x Tr(g-‘a, So(g)= -&s,,,, TrW1dxg)‘+&jQxR whose variation




we have already found,


dx dt Tr(g-‘6g

k =tii




g +g-‘a,

g)) (4.11)







Here 8 + = 8, + a,. We wish to couple the external gauge fields A,, A, to the boson so the resulting theory is gauge invariant under the combined variation E= 6gg- ‘, A -+ A + 6,A. Since, as mentioned earlier, the 8, gg-’ curent represents both the charge density and the current density we expect it to couple to the combination A + = A, + A,. After some exploration one sees that

S(g,A)=&-&I,,,d~~drTr((A,+A,)d,gg~‘) k -- 4n dQxRdxdfTr((A,+At)A.J) s


has as its gauge variation, Tr(.s(F- A2)),


which is equal and opposite to the variation of the bulk Chern-Simons effective action. (The actual form of the variation is that of the so-called “consistent anomaly.“) The combined action Scs( A) + S( g, A) is gauge invariant and describes the entire system in a unified manner. This cancellation of a gauge variation in one system by contributions from a system of different dimensionality is an example of the “Callan-Harvey” effect [ 111 which is reviewed in the Appendix. Notice that there is ome mixing between the A+ and A, that might not have been expected. This mixing is familiar from string theory where it is known as the “holomorphy” anomaly. DISCUSSION

Following the ideas of Wen [S-S] I have shown that the edge of a droplet of quantum Hall liquid can be described in terms of either the fermionic excitation of a “Fermi surface” or as bosonic ripples. In the present paper I have considered only the case of a single branch of excitations where the mapping is straightforward. I have also considered a non-Abelian generalization, which needs additional degrees of freedom, such as unpolarised spins, to be physically realised. These degrees of freedom do occur in chiral spin states so the extension is not entirely accademic. The fractional Hall effect has been discussed by Wen but the representations were infered from general principles and not constructed explicitly from the underlying electron wavefunction. A concrete construction of the fractional edge states in terms of deformed Laughlin states would be very interesting. It does not seem possible to have non-Abelian analogues of these states as the level of the non-Abelian algebras would be equal to v but have to be integers [9]. The chiral boson formalism presented here is not as unified as it might be since the component of the electric field providing the “speed of light” for the edge states



is treated differently from the other component and from the magnetic field. It may be that using the technology of [2] to express (4.12) entirely in terms of Chern-Simons fields will help here. In addition an alternative formulation of the QHE droplet should exist: one in which the geometry of the droplet shape enters the formalism explicitly-instead of appearing implicitly in the surface charge. Such a formalism would be be able to describe large deformations in the droplet shape and would provide a physical application for the machinary of string theory.


In Ref. [ 1 l] Callan and Harvey demonstrated the cancellation of anomalies between two systems of differing dimensionality. This cancellation has had previous application to condensed matter systems [34361. The original Callan-Harvey effect was driven by a slightly different mechanism than the Hall effect but one that has many features in common. As an illustration of their machinary, consider a (2 + 1 )-dimensional massive Fermi system with Dirac-Hamiltonian H= -ia,a,-ia,a,.+m(x)a,.

(Al 1

The mass m(x) is taken to be asymptotically constant, is independent of y, and changes sign as we pass through x = 0. The vanishing of m creates a one-dimensional “string” or domain boundary. We would expect there to be states near x = 0 which are bound because they have energies in the asymptotic mass gap-but one of these states is different in that it musf exist for topological reasons: we look for eigenstates of H of the form @(x, y) = eikqo(x);


then io=u,exp(Elbrn(xf~~xf), is a normalizable



eigenstate with

H$ = 8k.v+,


E= - f[sgn(m( 00)) - sgn(m( - co)].



These states move in only one direction despite their originating from fermions with both chiralities. A person made out of these particles would think of herself as living in a one-dimensional universe with a net anomaly: On applying an electric field Ey along the string, the k, momenta are replaced by k, + eE, t. There is a steady flow of states across E = 0 at a rate Eei?,/2n per unit length and since these



states are charged, the one-dimensional observer sees charge being created out of the vacuum at a rate of dE,/27r, violating charge conservation in accord with the (1 + 1)-dimensional anomalous conservation law for particles with chirality E: t.46)

Being able to see the larger picture, we know that the extra charge is not really coming out of the depths of an infinite one-dimensional Dirac sea but is actually flowing in from outside the universe: Each momentum k,, looks like a one-dimensional fermion theory and can easily be seen to contribute an amount (A71 to the current leading to a total flux of sg 1 j.Sx) = 2nm I 2n I + k2/m2 = & sgn(m) e2E,.. This flux is exactly what is needed to account for the extra charge seen by our anomalous observer. For a general applied field, electric or magnetic, we would have j, = g sgn(m)


The motion of the charge perpendicular to the applied field described by (A9) is analogous to the Hall effect but distinct from it. The flow has to be dissipationless as there is a gap, but the gap is created by a conventional mass term rather than by a magnetic field. It is known as the parity “anomaly” because, at first sight, the Levi-Civita symbol suggests that we have a parity-even expression equal to a parity-odd expression. The resolution of this paradoxical dependence on the choice of what to call left and right needs the observation that m, and hence sgn(m), is parity-odd in an even number of space dimensions: In 2 + 1 dimensions, parity is defined by $(x, y) + G*$(x, -y) (9(x, y) + $(--x, -y) is merely a rotation) and m&b changes sign under this transformation.

ACKNOWLEDGMENTS This work was suported by NSF-DMR-88-18713 and by NSF-DMR-86-12860. Fradkin for drawing my attention to Refs. [f%S] and for much useful discussion.

I thank Eduardo



1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

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