NUCLEAR P H YS I C S B
Nuclear Physics B396 (1993) 465—490 NorthHolland
________________
Infinite symmetry in the quantum Hall effect Andrea Cappelli
~,
Carlo A. Trugenberger and Guillermo R. Zemba
Theory Division, CERN, 121] Geneva 23, Switzerland Received 12 June 1992 (Revised 19 August 1992) Accepted for publication 2 December 1992
Free planar electrons in a uniform magnetic field are shown to possess the dynamical symmetry of areapreserving diffeomorphisms (Winfinity algebra). Intuitively, this is a consequence of gauge invariance, which forces dynamics to depend only on the flux. The infinity of generators of this symmetry act within each Landau level, which is infinite dimensional in the thermodynamic limit. The incompressible ground states corresponding to completely filled Landau levels (integer quantum Hall effect) possess a dynamical symmetry, since they are left invariant by an infinite subset of generators. This geometrical characterization of incompressibility also holds for fractional fillings of the lowest level (simplest fractional Hall effect) in the presence of Haldane’s effective twobody interactions. Although these modify the symmetry algebra, the corresponding incompressible ground states proposed by Laughlin are again symmetric with respect to the modified infinite algebra.
1. Introduction The quantum Hall system [1] is a fascinating example of a quantum fluid. Twodimensional electrons in an external uniform magnetic field occupy highly degenerate Landau levels. At low temperatures and large fields, they have strong quantum correlations which lead to collective motion and macroscopical quantum effects. The macroscopical quantum states are few, “simple” and universal, and they find their experimental evidence in a discrete series of quantized values for the Hall conductivity: =
4,4,4,
p=1(±1O8), 4(±1O~),
4,...,~y
4,2,
4,...,
(1.1)
where r’ is interpreted as the filling fraction, the ratio between the number of electrons and the degeneracy of the Landau levels. The observed universality clearly calls for an approach based on an effective field theory and renormalization group ideas [2,31. However, the most striking On leave of absence from I.N.F.N., Firenze, Italy.
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Elsevier Science Publishers B.V. All rights reserved
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feature is the extraordinary precision of these rational values for the conductivity. Exact numbers usually indicate that dynamics is dominated by symmetry and topology, as is familiar from infinite conformal symmetry in two dimensions [41, and from topological quantization conditions of gauge fields [5]. Due to these expectations, the quantum Hall effect has attracted the interest of theoretical physicists beyond the border of the solidstate community. In this paper, we show that an infinitedimensional symmetry is indeed present in the Hall system at integer and (the simplest) fractional fillings. This yields a strong geometrical characterization of incompressibility of the ground state and it may be a key idea to understand the exactness of the collective behavior of electrons. Before specifying the content of this paper, let us briefly review the phenomenology of the problem and previous approaches based on symmetry principles. The established phenomenology is based on the seminal work of Laughlin, Haldane, Halperin and others [1,6—81.It comprises a body of theoretical arguments substantiated by numerical simulations, which account for the exactness and stability of the ground state at the plateaus of the Hall conductivity. The main idea is the existence of an incompressible quantum fluid at specific rational values of the filling. Repelling electrons assume the most “symmetric” configuration compatible with their density, such that the ground state has a gap and such that there are no phonons. This picture clearly applies to the case of completely filled Landau levels (integer Hall effect). Compressions would excite electrons to higher levels, but these are suppressed by the large cyclotron energy. The Hall conduction is due to an overall rigid motion of the droplet of quantum fluid, which yields (1.1) for integer r~.In this case, Coulomb repulsion can be neglected since its typical scale is much smaller than the gap. Moreover, topological arguments have been developed to show that the freeelectron picture is robust in the presence of impurities [9]. At filling fractions r’ 1/rn, with m an odd integer, Laughlin’s wave function =
1P’(x1)
m exp ~ =
J](z~—z~) i
1
(1.2)
1’
(z 3 xj + iy~is the position of the jth electron and 12 %/2hc/eB is the magnetic length) exhibits a similar incompressibility [10]. It correctly approximates the numerical ground state for a large class of repulsive interactions, and excitations have a finite gap [11]. Laughlin developed the incompressibility picture, as well as the properties of excitations, by interpreting 2 as a classical probability distribution for a twodimensional Coulomb gas of charges, the plasma analogy [10]. For p 1/rn (m ~ 70), this plasma is a liquid, the charges are completely screened and stability amounts to local electrical neutrality. A very interesting consequence of this theory is that the charged excitations are =
=
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Infinite symmetry in the quantum Hall effect
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anyons, i.e., particles with fractional statistics O/~ 1/rn [12], and fractional charge e/rn [13]. Moreover, Haldane [7] and Halperin [8] constructed a whole hierarchy of new incompressible quantum fluids for other filling fractions xi 4, 4,..., by extending Laughlin’s arguments on electrons to anyons. The hierarchical structure of plateaus and excitations has been observed experimentally and strongly supports the entire picture. As we already mentioned, we believe that this picture should be supplemented with exact statements coming from symmetry. We have in mind the exact fractional critical exponents that follow from the infinite conformal symmetry of twodimensional critical phenomena [4]. Actually, a formal analogy between the quantum Hall effect and twodimensional conformal field theory was observed by Fubini [14], and Stone [15], and further developed in ref. [16]. This analogy is based on the similarity between the (holomorphic part of the) Laughlin wave function (1.2) and the correlators of vertex operators in a conformal theory with central charge c 1. Further extensions of this analogy have produced new results [17]. A new type of incompressible fluid was suggested at xi which has been recently observed experimentally in doublelayer samples [16]. Moreover, the possibility that the excitations in this system possess nonabelian statistics is currently being debated [17,181. These interesting developments call for an explanation of this analogy from a direct analysis of the microscopic physics of the quantum Hall effect. At first sight, this looks rather odd, given that conformal symmetry acts in (1 + 1)dimensional relativistic field theory, whereas the problem at hand is (2 + 1)dimensional and nonrelativistic. A first positive indication is that the Hilbert space of electrons restricted to a given Landau level is the Bargmann space of analytic functions [19,20]. This projection produces a dimensional reduction of the particle phase space from four to two dimensions. In sect. 2, we shall indeed show that there exists an infinitedimensional symmetry in the free electron theory of Landau levels. However, the relevant algebra is not the Virasoro algebra but a quantum version of the algebra of areapreserving diffeomorphisms. These types of quantum algebras are called Wc~ and are currently the subject of active investigations in twodimensional gravity [211. In sect. 3, the incompressibility of completely filled Landau levels is related to the infinite symmetry. Namely, the ground state is shown to be annihilated by infinitely many (in the thermodynamic limit) symmetry generators. This is analogous to the invariance of the vacuum in conformal field theory, i.e., the highestweight conditions. In sect. 4, the symmetry characterization of incompressibility is extended to the Laughlin ground state (1.2) at xi 1/rn. It is convenient and customary in this case to limit the theory to the lowest Landau level, and to introduce an effective shortrange interaction for which the Laughlin wave function is an exact incom=
=
=
=
=
4,
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pressible eigenstate [7,11]. This interaction deforms the infinite symmetry algebra of the free case. While the full properties of this theory, and of the deformed algebra, are not discussed in this paper, we prove that the Laughlin wave function for xi 1/rn satisfies again the highestweight conditions as in the xi n case. Finally, in the conclusions we discuss excitations and a geometrical description of the fluid. =
=
2. Infinite dynamical symmetry in the noninteracting theory
2.1. LANDAU LEVELS
Let us begin by reviewing some basic facts about noninteracting electrons in an external uniform magnetic field B orthogonal to the plane [19,20]. We first discuss the onebody problem and shall deal later with the manybody situation and second quantization. The action S, and the hamiltonian H, of an electron of charge e and mass m are
S=fdt (4mv2+~v.A),
H
=
4rnv2
=
(p
~
—
~A)2,
(2.1)
where we choose the symmetric gauge A 4B(—y, x). At the semiclassical level (exact in this case), the classical circular motion of cyclotron frequency w eB/mc gives rise to quantization of the (kinetic) energy E horn and thus to quantized radii r~.The corresponding orbits enclose a flux multiple of the quantum unit of flux D~ hc/e, i.e., 7rr~B n1 0. The center of the orbit is instead free, thus excitations of arbitrary angular momentum cost no energy and are degenerate. The flux quantization implies that the magnetic field has an associated unit of length, the magnetic length 8 ~~/2hc/eB. From now on we choose units h c 8= 1, so that eB 2 and D0 2~r/e. Moreover, for convenience we take m e 1 in these units, thus w 2. At the quantum level, the canonical momentum is realized as p —iV acting on functions of the coordinate x (x, y). The hamiltonian and (canonical) angular *
=
=
=
=
=
=
=
= =
=
=
=
=
=
=
*
Our results will naturally be gauge independent. We shall comment later on the formulas for other gauges.
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momentum can be written in terms of a pair of independent harmonic oscillators:
H
2ata
=
+
1,
J=btb_ata. In complex notation z
=
x
+
iy, ~
x
=
—
iy, d
=
3/dz,
(2.2)
a
=
9/82, these operators are
a=4z+ã,
at=42_a,
b=42+a,
bt=4z_3,
(2.3)
[b,bt]=1,
(2.4)
and satisfy the commutation relations [a,at]=1,
with all other commutators vanishing. Starting from the vacuum (1/V~) exp( 41 z 2) satisfying alit0 bli’~~0, one indeed finds energy (at) excitations, the Landau levels, which are infinitely degenerate with respect to the angular momentum (bt). The general wave function of energy n and angular momentum 1 is given by =
—
=
=
2)exp(41z12), =
~(1+n)!
~~~(x)
=
V~l±n)!z’L~(lzI (2.5)
where L~(x)are the generalized Laguerre polynomials and n ~ 0, 1 + n ~ 0. For example, the wave functions of the lowest Landau level satisfy alit 01 0, and read =
z’ =
1
7=—~=exp(—4Iz12).
(2.6)
These functions are peaked at circular orbits, such that the degenerate levels have an onionlike structure. The degeneracy Nd of levels occupying an area d is equal to the flux through it in quantum units, Nd ‘~/~ Bd/2ir, thus the degeneracy is finite in a finite sample. The manybody problem of N electrons will clearly be described in first quantization by taking N copies of the previous operators labeled by an index i: a, b a1, b,. In this case, the parameter B acts as an external pressure, because it controls the number of states and thus the density of electrons per state. Actually, the latter is the correct quantum measure of electron density, the fillingfraction =
=
—*
xi,
N Nd
a
1 —
B
(finite sample).
(2.7)
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In the infinite plane, and within the lowest Landau level, the commonly used definition is N(N1) 2J
(infinite plane).
(2.8)
Indeed, the area occupied by the electrons is proportional to the total angular momentum J i1, whose minimal value is N(N 1)/2, by Fermi statistics. To be more precise, the filling fraction xi has to be understood as the thermodynamic expectation value, and in this sense it appears in the conductivity, eq. (1.1). Therefore, eq. (2.8) indicates that 1/xi is proportional to the average value of J. In order to control J in the infinite plane with an external parameter, we introduce a central harmonic potential which confines the particles, =
—
2,
(2.9)
H~~=H+4ALlz~l where is the Gibbs free energy. The harmonic potential gives the simplest realization of the boundary effects in a finite sample [37]. It can be written in terms of a’s and b’s and, therefore, of J and H, ~‘
(1 +ct)H+aJ,
.~‘=
(2.10)
where A a(a + 2). Thus, a is the external parameter conjugate to H + J, i.e., to 1/v. Since (J + H) ~ 0 measures the area occupied by the electrons, a is a confining pressure (infinite plane) and it is equivalent to 1/B (finite sample). Finally, note that the value of xi will be dominated by the angular momentum of the ground state and fluctuations in J of 0(1, N) will describe the characteristic excitations. In second quantization, the field operator is written as =
~‘(x, t)
=
~
F,~’i’flk_fl(x)
exp[—i(2n
+
1)t],
(2.11)
n,k=O
in terms of the wave functions (2.5), the fermionic Fock annihilators ~ creators F~)t,with ~~
k
‘
F(m)tl / J
=
nm .s k,l~
and
~2 12
The Nbody wave function is given by the Fock average ~E(xI,..,xN)_(0l~(xl,0)..(xN,0)lN,
E>,
(2.13)
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where 0) is the Fock vacuum, and I N, E) the corresponding Fock state. Second quantization corresponds to the canonical quantization of the nonrelativistic Schrodinger field theory whose action is
s=
d2x
fdt
4(D~)t.(D~)),
~
(2.14)
where D V—iA. The commutation relations (2.12) correspond to the canonical commutator =
{~P(x,t), ~
(2.15)
t)) =~2~(x—y),
and the conserved hamiltonian H and number operator N are given by H= fd2x 4(D~)t.(D~P),
N= fd2x
~
(2.16)
2.2. INFINITE QUASILOCAL DYNAMICAL SYMMETRY TRANSFORMATIONS
We now return to discuss the meaning of the a and boscillators. The operators a, at are covariant derivatives, as can be seen by restoring their Bdependence, a=3+4Bz,
at= —3+4B~,
(2.17)
[a,at]=4B.
The operators b and bt are generators of rnagnetic translations, i.e., of translations combined with gauge transformations. In a uniform magnetic field, there is clearly translation invariance. Therefore, there should be two derivative operators commuting with the hamiltonian, [b, H] [bt, H] 0. However, the choice of gauge apparently breaks translation invariance. Therefore, a translation should be accompanied by a compensating gauge transformation Actually, by exponentiation, the finite magnetic translations T~ give rise to a projective representation of the translation group =
=
~.
=
exp[4(X —aA)]1~± (2.18)
5~÷~.
The fact that the two correct translation operators commute with H, but not among themselves ([b, bt] 1), accounts for the infinite degeneracy of the Landau levels on the unbounded plane. It also gives other interesting effects in finite geometries [16]. =
*
A general discussion of symmetry in the presence of a background is given in ref.
1231.
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It is worth stressing that the magnetic translation algebra is gauge invariant, even if the explicit form of b and bt in terms of 3~and x depends on the gauge. As discussed in refs. [14,24], in a general gauge they are defined by b, lJ~+ iA1, =
where A,(x) ar at most linear in the coordinates and are determined by the conditions [b,, p~—A’] 0, i, I 1, 2. These imply [b1, b2] [p1 —A1, p2 —A2] —iB, which establishes manifest gauge invariance. Thus far, we discussed wellestablished matters. We would now like to make a new, yet simple, remark: there are more operators which commute with the harniltonian. They are given by polynomials of the noncommuting operators b and bt, defined by =
=
=
=
n, m ~ —1,
H]
[5~n,m’
=
0.
(2.19)
These generate quantum areapreserving diffeomorphisms (up to gauge transformations). Indeed, from the definitions (2.3) and (2.4) we deduce that 2), (2.20) [~i,m,~k,1]((m+1)(1)(n+1)(1))~÷k,m±/+0(P1 where we resealed b and bt to [b, bt] h to compare with ref. [21]. In eq. (2.20), the missing terms correspond to contractions of more derivatives, and hence have higher powers of h The 0(h) term, the classical algebra, identifies our algebra as that of areapreserving diffeomorphisms or isometries. This is called w~in the context of twodimensional gravity [21]. Quantum deformations of this algebra are called, collectively, W~algebras. There are many such deformations, which differ in the higherorder terms in h they can be obtained by modifing the ordering of b, bt in the definition (2.19). The 5~°n,m generically involve powers of derivatives higher than one, and therefore are not generators of local coordinate transformations on the wave function; they are quasilocal operators. The generating function of ~m is actually the finite magnetic translation discussed before eq. (2.18). The only local coordinate transformations are the translations 2’~—~ ~ and rotations ~ ata) with which we started. For N particles, the firstquantized form of the generators 5~m is =
~.
**;
.~‘_
—
~n,mm
~
n, m~ —1.
(2.21)
Note that in the thermodynamic limit they become an infinite set of independent charges. * **
Note that the full quantum algebra closes, as is clear from eq. (2.31) below. The classical limit is further discussed in ref. [38].
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The secondquantized formulae might help to convince the reader that the 2,m are bonafide conserved charges arising from conserved (quasi)local currents. Their secondquantized expression is
~n,m
=
fd2x
~t(bt)~l(b)m±h1[f
(2.22)
where ~ are the field operators introduced in eq. (2.11). Since [(bt)~l(b)m±t, H] 0, the 5~mare conserved charges. This can be cast in the usual form of current conservation, =
~O~(n,m)
+ 1~i~(nm) =
0,
(2.23)
~t(btY±1bm÷1DJ~].
(2.24)
where ~1(n,m)(X)
1~t(bt)~I(b)m+
4j[(DJ~)t(bt)bm+1~
—
Current conservation follows from the operator equation of motion of the hamiltonian (2.16), and the mentioned fact that b, bt commute with the covariant derivatives. It is worthwhile to point out that, strictly speaking, the observables have to be real, and they correspond to the eigenvalues of Re(~,m)and Im(~m). 2.3. CLASSICAL SYMMETRY
In order to gain some geometrical intuition, let us clarify the classical origin of the symmetry in our problem. In relativistic field theory we usually look for symmetries as those coordinate transformations that leave the action invariant. However, from a hamiltonian point of view, symmetries are canonical transformations that leave the hamiltonian invariant. Besides coordinate transformations, there are more general transformations which change momenta independently of the coordinates. These are usually disregarded in field theory, since they act nonlocally in the Hubert space, and therefore are difficult to implement at the quantum level As we now show, the symmetry in our problem is the quantum version of a canonical symmetry of the classical hamiltonian (2.1). To begin, we briefly recall some basic facts about canonical transformations in the simplest example of a twodimensional phase space (q, p). Canonical transformations are diffeomorphisms of the phase space which preserve the symplectic “.
*
~,,
Nevertheless, we quote the example of the Bäcklund transformation in the canonical quantization of Liouville theory [25].
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twoform u dq A dp, i.e., the area form [26]. The infinitesimal transformations have a generating function F F(p, q), =
=
Q=q+5q
3F 6q={q,F}pB=—,
P=p+~p,
~p={p,F}pB=———,
aF
(2.25)
where { }PB are the Poisson brackets, {q, p) 1. The “point” transformations, 5q =f(q), are given by F =pf(q), under which p transforms as a derivative. A complete basis of generators is given by the full power series expansion of F(p, q), Fnm q” lpm +1. The Poisson brackets of two such generators give a representation of the classical w,~algebra [21]. Let us now explain how this algebra can arise in our problem, which has a fourdimensional phase space (x, p~, y, pr). The classical hamiltonian (2.1) can be written as =
=
—
+
H=4((p~+y)2+(p~x)2)=4(v~+v~).
(2.26)
The most general canonical transformations which leave it invariant are generated by ~
p)
=
(bt)~’bm+1,
=
{H,
~~}PB
b
=
4((p~+x)
=
+
i(p~—y)),
0,
(2.27)
as is clear from the classical limit of the discussion in subsect. 2.2, 5~m and [ I ih{ ‘1PB + 0(h2). This also implies a crucial property of these transformations, namely, that they leave invariant the two combinations v~ Im a, v~ Re a, where a 4(—(p~ x) + i(p~+ y)), i.e., ,
—~
=
—
=
=
—
~(p~+y)=0,
~(p~—x)=0.
(2.28)
Therefore, the 5~(~act nontrivially only on a twodimensional subspace of the fourdimensional phase space, characterized by constant energy. It is a peculiar property of this problem that a twodimensional subspace admits a symplectic structure in terms of b and bt. Thus, the infinite canonical transformations on this subspace are infinite symmetries and satisfy the w~algebra,
~
~‘
P}PB
=
—i((m + 1)(k + 1)
which is the classical limit of (2.20).
—
(n
+
1)(l + 1))~2~~m+t, (2.29)
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Upon quantization, something peculiar happens: within each Landau level, the previous twodimensional subspace is identified with the actual coordinate space of the electrons. As emphasized in ref. [27], this phasespace reduction can be obtained by taking the limit m —s 0 of the action (2.1): S=limS=~ m—20
dtA=P=fdt(I’x_~y),
(2.30)
orbit
where ~Pdenotes the flux piercing the area encircled by the orbit. The action S (“Chern—Simons mechanics”) describes classically the residual degree of freedom within each Landau level (the constant energy of the level is renormalized to zero). Note that ~ in (2.30) is of first order in time derivatives, thus the symplectic structure is immediately evident and implies that only one of the original coordinates remains a coordinate in the hamiltonian sense the other becomes the conjugate momentum. This is the advertised phasespace reduction induced by the external magnetic field. Since H 0, the symmetries of the hamiltonian are all canonical transformations of the reduced phase space. Equivalently, the symmetries of the action S are the areapreserving diffeomorphisms of the coordinate space, which leave cP invariant. —
=
2.4. THE FULL W,, ALGEBRA
We now present some additional properties of the full quantum algebra and discuss its relation to the standard notation [21]. The full commutator (2.20) follows from the Leibnitz differentiation rule (m+1)!(k+1)! (rn s) ‘(k —s) (s + 1)
Min(m,k)
[~,m’
~k,1I
—
—
—(m
I, n
~
k).
~
(2.31)
Special cases are given by: (i) (ii)
[5~,~’ ~k,oI (k
—
[~?‘~, ~‘O,kI
=
—
(iii) (iv)
~ [~,
=
=
(n
k)~)fl+m;
~,kI = 0
~
(Cartan subalgebra); (n m)5~m.
(2.32)
The first two are Virasoro algebras, but the negative (n < 1) modes are missing. Later, we shall show that 2~ E~z111 1d1, when acting on analytic functions of the lowest level (2.6): these look like the conformal transformations which were —
+
=
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Infinite symmetry in the quantum Hall effect
assumed in ref. [16]; however, there are important differences. The missing negative modes correspond to singular transformations at the origin, which are not allowed for our twodimensional Hilbert space ((bt)~~’~ is not defined). On the other hand, Virasoro transformations are reparametrizations of the circle and can be extended to singular transformations of the plane. Moreover, the incompleteness of the algebra prevents us from considering possible central extensions. In the Virasoro algebra L~ L_k, but in our case ~ ~ and these do not satisfy a Virasoro algebra with the The standard notation of W. algebra stresses its relation to the Virasoro algebra. Indeed, we can express our operators as =
9~’kO~
“?
~
i>_ —1,
n> —i—i,
(2.33)
where the index i + 2 corresponds to the conformal spin. For i 0, the Virasoro subalgebra is satisfied by the Fourier modes of the spintwo stress tensor. For 1, one has the Zamolodchikov spinthree current, and similarly for higher spins. The index n is the conformal dimension, namely the eigenvalue of .2’~.In our case, the Fourier modes l’~’ are bounded from below (n ~ —i 1), while in conformal field theory they are unbounded {1’~’,n e 7L, i ~ 0 in CFT). Therefore, our algebra corresponds to the socalled “wedge” WA {J’~,I n I ~ i + 1), plus the positive modes n > i + 1. In the standard notation of W 2~,it reads (restoring h for comparison with ref. [21]) 1~+ ~ + h3y’~~2+ ... +ht+ ~ (2.34) [I’~I” l’~] hA V,1 where q Min(Max(i m, j), Max(i, j n)). In eq. (2.20), we omitted the structure constants, which differ from those of ref. [21], apart from the classical one. As anticipated, their form depends on the choice of quantum ordering and it is largely arbitrary, provided the Jacobi identity is satisfied. Therefore, the notion of W 0,, really stands for a large class of quantum algebras. As in the case ofminimal the Virasoro 813~n+mocj(n). In its form subalgebra, W,,~admits in general a central term [21], c,(n) vanish inside the wedge, thus it cannot be added to our algebra. =
=
—
=
=
=
—
—
3. Infinite symmetry and incompressibility of the ground state at integer fillings In the previous section, we revealed an infinitedimensional algebra in the noninteracting theory; we now investigate its relevance for the integer Hall effect. Up to now, we have been concerned primarily with the commutators of the generators with the hamiltonian. Before we can assert that there is a symmetry in the quantum theory, we should discuss the action of the generators on the states, in particular the ground state I (~). We shall find an implementation of the
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symmetry analogous to conformal field theory. The ground state satisfies highestweight conditions, i.e., it is annihilated by an infinite subset of “lowering” operators, while the others create excitations. In conformal field theory, this condition is L~IfTl)=0,
n> —1.
(3.1)
Let us first understand the action of the ~ in Fock space. Their secondquantized expression is obtained by inserting the field operator (2.11) into definition (2.22)
=
k~=O
li’~(x)~
~I~fk©I~(t)fd2x
t)’~lbm+1
~
=
r=0
Q9(r)
n,m
=
k~m+1
~
—
+n —m)! F(~~)t F(r) ~ k+n—m k mij.
3 2
The ~ split into identical copies 2~ acting independently within each level (r). The bilinears F,t~+SFk show that ~ for fixed s, displaces electrons by increasing (s > 0) or lowering (s <0) their angular momentum. They act as raising and lowering operators within each level. Clearly, the bilinears F~+SFkgive the simplest bosonic description of the Hilbert space at each level, because any fermionic state can be connected to any other by their repeated action. Therefore, they achieve a “bosonization” of the nonrelativistic fermion theory in each level. Moreover, the ~ similarly span the singlelevel Hilbert space, given that (F~F~~} and ~ are equivalent sets. Actually, eq. (2.21) can be inverted within the rth level as follows: 1 ( 1)k+1 F(~~tF(~~) a b V~T~T k=—1 (k+1)! — —
3 3
~r)a±k,b+k~
5’~m is a “spectrumgenerating algebra” for As consequence, the algebra of the the aangular momentum. Next, we discuss some physical issues of the integer Hall effect. The noninteracting electron theory considered so far is generically unphysical. The Coulomb interaction lifts the degeneracy of the Landau levels such that the properties of the ground state depend on the value of the filling fraction However, for integer filling xi 1, 2,..., the noninteracting theory makes physical sense [9]. Under xi.
=
*
*
A more precise statement based on the renormalization group will be made in sect. 4.
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the experimental conditions of large magnetic fields, the energy gap between Landau levels is larger than the typical energy scale of the Coulomb interaction: e2 xc8
eB mc
(3.4)
where K is the dielectric constant. Therefore, for integer xi, the Landau levels retain their character, the ground state is unique and has a gap: this corresponds to an incompressible quantum fluid [10]. Transitions of one or more electrons to higher levels would reduce the angular momentum, i.e., would compress the fluid; however, these are forbidden by the large gap. Moreover, angularmomentum excitations, i.e., decompressions, are controlled in our geometry by the confining potential (2.10) discussed in subsect. 2.1. Their energy is given by the Gibbs energy aJ a.2’ 00 (within each Landau level), where the value of a a 1/B allows for completely filled Landau levels, say the first one. Angularmomentum excitations become energy excitations, and we achieve a very close analogy with conformal field theory, where similarly H L0. Indeed, the incompressible ground state at xi 1 satisfies highestweight conditions analogous to (3.1), as we now show. The ground state is given by ~‘
=
=
=
=
Ifl)= IN, xi=1)=F~l)tF~©...F~)2.tiI0)
(3.5)
in second quantization, and by ~‘~1(x1,...,xN)
=
fl(z~—z1) exP(_4 ~ IzjI2) 1<)
(3.6)
i—I
in first quantization. The highestweight conditions read m~1’~=10,
for —1
5~m~Indeed, ~m
(3.7)
when acting on the and follow thethey nature of the angular momentum for n
Note that the condition of magnetic translation invariance,
~
—s5~
~,~‘I’= 0, has been previously recog
nized as a necessary condition for incompressibility of the ground state 120,221. The magnetic translation group has been also used in the field theoretical approach of ref. 1281.
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Infinite symmetry in the quantum Hall effect
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More precisely, let us compare both conditions in the customary W0, notation (see subsect. 2.4): V~IQ>0,
i=0,1,...,—i—1~n<0
(Halleffect).
(3.8)
These conditions have to be compared with those corresponding to the \\~~symmetric vacuum in conformal theory: J’~IQ)=0,
i=0,1,...,—i—1
(CFT).
(3.9)
Note that all conditions in (3.8) are (3.9). In the case, one finds 2)contained nontrivialin conditions forHall N electrons, such by inspection that there are 0(Nsymmetric in the thermodynamic limit. that the ground state is infinitely The same incompressibility conditions (3.7) are easily extended to k completely filled Landau levels. They read for —1
mIN,u’o>O,
Indeed, using second quantization, for N
=
k=1,2
(3.10)
kNL electrons:
IN, xi=k)= fl(F)tF)t...F~
1)Io),
(3.11)
thus, tFt...F~0)=0, ~
(3.12)
IN, xi=k)= ~.~ç~
by commuting operators for different levels. In summary, we have shown that in the case of the integer Hall effect, the incompressibility of the ground state corresponds to an infinite symmetry under the Vv~, 0 algebra.
4. Infinite symmetry and incompressibility at fractional fillings In this section, we show that the incompressibility of the Laughlin wave function at fractional fillings xi 1/rn, m odd, is again characterized by a set of highestweight conditions, similar to the integer case. =
4.1. PROJECTION ONTO THE LOWEST LANDAU LEVEL
For fractional fillings, noninteracting electrons have access to a large reservoir of degenerate states. One has to understand how the repulsive interaction man
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ages to arrange the electrons in a collective “symmetric” state, such that the ground state is unique and has a gap in the thermodynamic limit. We already remarked that under the experimental conditions, the Landau level gap is larger than the typical scale of the Coulomb interaction, eq. (3.4). A better estimate of the interaction scale is given by the size of the gap for the lowlying excitations above the xi 1/rn ground state, which numerical simulations and Laughlin’s theory predict to be a few percent of the Landau level gap [10,111. Therefore, it is probably sufficient to limit the theory to the lowest level. The Laughlin wave functions (1.2) belong indeed to this level, i.e., they are of the general form =
li’(xl,...,xN)
=exp(_4L IzjI2)~(zl,...,zN),
where ~ is an entire analytic function of
(4.1)
z
(see subsect. 2.1). The projection onto the lowest level is largely accepted in the literature for xi 1/rn, while it is being debated in the case of hierarchical constructions at fillings =p/q [29]. The projection leads us to the Bargmann space of coherent states for the b, bt oscillators. Let us recall some of its basic features [19,201.The exponential factor in (4.1) is absorbed into the measure and operators act only on the analytic part of the wave function ~‘(Z), 1,. .., ZN
=
xi
dzd2
(~Ic)=f
li’=exp(4z2)q~(Z),
2~i exp(Z2)q5(Z)~(z).
(4.2)
From eqs. (2.3), b and bt are indeed represented by Z and 8, respectively. The orthogonality and completeness conditions for coherent states are (see subsect. 2.1) t) I ~ ~7(Z) Kz I ~>, (4.3) I Z) exp(2b I ~ exp(Z~) (identity kernel), (4.4) =
=
(Z
=
if
dZ
d2
2~ñ exp(—Z2)IZXZI.
(4.5)
Notice that 2 acts on the left and is the adjoint of 8, because one finds by partial integration that <4. I8~) . This confirms that the twodimensional plane (z, 2) is a phase space for the onelevel theory, in agreement with the classical analysis of subsect. 2.3. Any operator A is represented by an integral kernel A(Z, ~) and acts as follows: =
(Aq~)(z)
=
I dij2~id~exp(—1~)A(Z,~)c(~i).
(4.6)
/
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Infinite symmetry in the quantum Hall effect
481
In particular, the kernel A(z, ~) of a (quasi)local operator A is the same operator applied to the identity kernel
~) =Zk(~)
A~(z) =Zk8~(Z) ~A(z,
exp(Z~)=Z~
exp(z~). (4.7)
The projector P0 onto the lowest level acts on a generic wave function li’, setting to zero its higherlevel components. This is achieved by using the identity kernel li’(Z,
(P0cP)(Z)
=
f
2) =exp(—4Z2)cP(Z, 2),
exp(—~) exp(z~)cP(s~, ~).
.
2iri
(4.8)
Indeed, if D (at)4~, ~) (~ 8,~)4,it vanishes inside the integral by partial integration. Similarly, the projection of local operators V(2, Z) is obtained by moving all 2 powers to the left of the z’s (normal ordering) and by replacing 2 —s 8 [19]: =
=
—
P0V(2,
Z)P0=
: V(a,
z):
(4.9)
.
2P
For example, P0 I z I 0 8Z 1 + Z8. The same result is obtained by applying P0 in the integral form (4.8) on both sides [20]. The generators of areapreserving diffeomorphisms (2.21) take the form =
=
N ~n,m
=
m+1
~
~
~—
(4.10)
.
i=1
As anticipated in subsect. 2.4, the ~5?~ actually generates conformal transformations on analytic wave functions. Note that, as a result of the projection, the subset of local transformations of our algebra is enlarged to include this infinite subset. 4.2. EFFECTIVE INTERACTION FOR v
=
1/m
A convenient parametrization of projected twobody interactions is given by a “partialwave” analysis or, equivalently, by a shortrange expansion [19,20]. The *
*
This is also reminiscent of the operator product expansion in conformal field theory [30].
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previous rule of projection (4.8) tells us that the kernel of a projected twobody operator, V(I z1 z2 I), is given by —
(P0VP0)(z1,
Z2 i~, ?7~)
2~exp(p~ =
exp(~
+Z 2~)V(
e~
~
Zl+Z2
=
~)

fd2p d
exp
~
exp
Ip
Z1—Z2
~ I) exp(p~1+

~11~12
n
[
n=~0
,
(4.11)
where the amplitudes en of the partial waves are 2r2nV(1~r). e~
(4.12)
e_r
~fdr2
=
The terms in the expansion of the kernel can be rewritten as P 0V(1, 2)P0= ~ e~(1, 2), n
=
J12=n>
17=
(4.13)
t)
Here, J/~ is the projection on the relative angular momentum J12 when acting on a twoparticle wave function, it gives
~) ( ~) ( ~) ) Z1—Z2
~
Z2)
=
(~)(z~,
Z2)
=
~
Z1+Z2
,
m
~
n. Indeed,
m
~Cnm(
ECnm(
=
(4.14)
(4.15)
,
i.e., it selects the term of O((z1 —z2)~) in the expansion of ~(Z1, Z2) around (Z1 + Z2)/2. By antisymmetry, even powers never appear; thus, we should only consider V2k±l,k 0, 1, 2 An equivalent notation [22], valid only within expectation values, is =
2~(x fr~(1,2)
~‘~
1 —x2),
(li’
I
I li’)
=
fd2x
lift(x)~lit(x),
(4.16)
where ~ is the laplacian. For example, the partial waves of the Coulomb interaction are H~(1, 2)
1 =
1x1—x21
~~e~=
F(n+’) 2
n!%/~
I ~i:~~~ ~.
(4.17)
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Infinite symmetry in the quantum Hall effect
483
Note that the e~are positive and monotonically decreasing. Their overall dimensionful factor e2/8 is set to one in our units. Let us now discuss an effective shortrange interaction which has been introduced for the Hall systems at filling fractions xi 1/rn, say 1/3 (Haldane’s “pseudopotentials” [11,7]). We would like to motivate it by using a renormalizationgroup picture. Indeed, the phenomenological and numerical results in the literature on the Laughlin wave function strongly suggest such a picture. We have in mind the renormalizationgroup flow in the space of effective interactions (4.13) and we would like to associate a fixed point to each xi I/rn plateau, with a corresponding effective interaction. The main supporting facts are: (i) There is universality, namely the Laughlin wave function =
=
zN)= fl(z~—z
m,
xi=
1/rn,
(4.18)
1)
i
is an extremely good approximation to the numerical ground state for a large class of repulsive interactions at xi 1/rn [10]. The approximation improves as one approaches the thermodynamic limit [31] ~. (ii) The Laughlin wave function is the exact incompressible ground state for the shortrange potential [7,22], =
m—2
~
(4.19)
ekVk,
k=1
4 odd
where e 1,. . . , e,~,_2are positive couplings (this statement will be proven later). (iii) There is a tunable mass scale A in the fieldtheoretical sense, 1
a
—
finite sample ,
(4.20)
which controls the density xi a I/J a, by eqs. (2.8) and (2.10), and allows one to move among plateaus. Indeed, the typical interelectron distance (in correct quantum units) is 1/ V~, which plays the role of correlation length. (iv) The incompressibility of the xi 1/rn ground states should be of the same nature as in the xi I case. These facts suggest the following renormalizationgroup scenario. The Coulomb interaction (including possible lattice impurities) corresponds to the “bare”, or classical, interaction, whose couplings are given by (4.17). At a given value of a, the electrons test the Coulomb interaction at all scales smaller than 1/v~. The bare couplings ek, originally monotonically decreasing for k ~, have a renormalizationgroup flow. =
=
—~
*
Compare with finitesize effects at the critical point in field theory.
484
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For (a such that) point,
xi
(e~°( 1/v
/
Infinite symmetry in the quantum Hall effect
1/m, we assume that this flow is attracted to the fixed
=
m)}
=
{e1( m),
.. .
,em _2(m),
0, 0,..
. },
(4.21)
characterized by nonvanishing shortrange effective couplings (ek, k
=
=
=
—
—~
4.3. EXACTNESS AND INFINITE SYMMETRY OF LAUGHLIN’S WAVE FUNCTION
This renormalizationgroup scenario suggests that the original Coulomb interaction can be replaced by Haldane’s effective interaction for describing the Hall system at the fractional filling xi 1/rn. As we now show these effective fixedpoint theories are characterized by an infinite symmetry, as in the case of integer filling. They describe electrons constrained to the lowest Landau level and interacting via the twobody repulsive potential, =
m—2
~ ek)Vk(i, i)~
Vk(i,
I)
=
IJ~ 1=k)(J1~=kI,
k=1 k odd 17k
where
(4.22)
i
is the shortrange potential defined previously. A more explicit form of
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Infinite symmetry in the quantum Hall effect
485
its action on the holomorphic wave function is given by
(~) a
1 =
(Z~ _ZJ)k~
z.+Z.+X
k
Z.+ZX
~
~
I
,...,
. ., ZN)~,
(4.23)
which generalizes the twobody case (4.15). By antisymmetry, the analytic wave function ~ in (4.1) has the general form (4.24)
97= fl(Z~—ZJ)P(zl,...,ZN), i
where Ii (Z, is the Vandermonde determinant and P is a completely symmetry homogeneous polynomial. For the first nontrivial case, xi 1/3, the hamiltonian is — Z)
=
Hh3~=eiLVi(i,j),
e 1>0.
(4.25)
i
We now discuss its spectrum. The E 0 eigenspace is highly degenerate and contains wave functions which go to zero at coincidence points faster than (z, z~) for any pair i, j 1, . . . , N. The E> 0 eigenstates those which for vanish 3~ can be contain easily diagonalized N as 2, O(Z1 z.), i.e., minimally. For example, H~ by using eq. (4.23), =
—
=
=
—
E
=
0,
~o
=
(~ Z
3P(Z
—
2)
E
=
e1,
4~
=
(~
— Z2)Q(Z1
1, Z2), + Z2),
@0
=
1
—
V1,
@~ ~
(4.26)
=
where @~,@~are the projectors on the corresponding eigenspaces, 3k9 [email protected] 1 @~+ @~, @O~9~ 0, H~ 0 0, H~ 1 [email protected] (4.27) =
=
=
=
=
We now find the ground state of the interacting theory and later discuss its incompressibility. We prove the following statement: the E 0 eigenspace of the hamiltonian (4.25) consists of wave functions which contain the third power of the Vandermonde as a factor, 3P(Zl,...,ZN) ~‘H~3~ c= fl(Z~—ZJ) 970=0, (4.28) =
i
where P is a symmetric homogeneous polynomial. Clearly wave functions of this form have vanishing energy, by have eq. (4.23). Conversely, let value, us findthus thewe solutions to 3~çb 0. Note that projectors positive expectation can write H~ ~ Vi,j. (4.29) =
i
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Infinite symmetry in the quantum Hall effect
There are as many independent conditions as the number of pairs. Again, by eq. (4.23), 4. should vanish faster than (z1 z1) for any pair; thus, by antisymmetry, it should have a least a thirdorder zero for every pair. Moreover, 1j is an analytic polynomial (i.e., entire) function of z1 ZN, which is completely determined by its zeroes (whose number is given by J). Therefore, 4 should have a third power of the Vandermonde determinant as a factor, q.e.d. ~. A corollary to this statement is that the Laughlin wave function (4.18) is the exact nondegenerate ground state at 1/3, as firstly observed by Haldane [7,11]. On the plane, the value xi 1/3 corresponds to angular momentum J 3N(N 1)/2. Actually, J ).~ z181 gives the degree of homogeneity, so that P has degree zero in this case, P 1; thus, 97ü at xi 1/3 is the Laughlin wave function (4.18). This ground state is unique and incompressible, because it has the minimal value of J in the E 0 eigenspace. Indeed, in eq. (4.28) we proved that E 0 implies J 3N(N 1)/2. Conversely, any state of smaller J should necessarily have E > 0, i.e., it has an energy gap. On the other hand, decompressions, J> 3N(N 1)/2, also develop a gap after inclusion of the confining potential (2.10), 3~ H~3~ + a..~’ H~ 00, (4.30) —
xi
=
=
=
=
—
=
=
=
~
=
—
—
~
=
in complete analogy with the integer filling case discussed in sect. 3. We shall now discuss the geometrical conditions expressing the incompressibility of the Laughlin wave function. To this end, we introduce the projector @() on the JJ(3) 0 subspace 13~9 I~ I, H 0 0, (4.31) =
=
=
t~o,~1 is an orthogonal set of 97~’5of eq. (4.26). where Next, we modify the generators .S~mof the noninteracting symmetry algebra (2.19) as follows =
Ez~’a~’,~
(4.32)
@[email protected]
The projected generators commute with H~3~ and produce compressions and decompressions in the E 0 subspace described before (4.28). They satisfy a modified algebra, due to the projection @~.Nevertheless, their angular momentum eigenvalue is unchanged because 5f’~ and [2’~~, ~,,m] (n rn)S~m in the E 0 subspace. =
~
=
—
=
*
The structure of inclusion of the Hubert space for
v
=
1,
~,
ky...
resembles the projections in rational
conformal field theories, like the projection of the free bosonic Fock space carried on by the Coulombgas technique
133].
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Infinite symmetry in the quantum Hall effect
487
Acting on the Laughlin wave function (4.18), we find ~m~l/i_0O~,.m0OfT(ZiZj)_0,
for n
I
(4.33)
1 (z, ‘~,m lowers the angular momentum, Proof. Consider ~ ~i,[email protected] thus cP has J 3N(N 1)/2 (m n) <3N(N 1)/2. We already proved that functions with such values of J are necessarily a superposition of E > 0 eigenstates only. Therefore, @~i 0, by eq. (4.31), q.e.d. — Z
=
—
—
—
3.
1)
—
=
Therefore, eqs. (4.33) are the infinite incornpressibility conditions satisfied by the Laughlin wave function at xi 1/rn. We achieve a description of incompressibility of the quantum fluid analogous to the integer xi case. Among the transitions generated by the ~m’ we neglected compressions to E > 0 states, which are forbidden by the macroscopic gap. The remaining ones (the ~ n
=
5. Summary and perspectives In summary, we have exposed the existence of an infinite symmetry in the manybody quantum problem of planar electrons in a uniform magnetic field. In the absence of twobody interactions, the symmetry generators satisfy an algebra of
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Infinite symmetry in the quantum Hall effect
(quantum) areapreserving diffeomorphisms. Its classical origin are canonical transformations of the fourdimensional phase space that leave invariant the hamiltonian. Upon quantization, the external magnetic field quenches the kinetic energy and the phase space is consequently reduced from four to two dimensions. At the quantum level, these generators have been used to describe excitations within any Landau level. The incompressibility of completely filled levels was expressed by a set of highestweight conditions (3.7), (3.12), of the type which appear in conformal field theory. In the presence of Haldane’s effective twobody interactions, the basic picture does not change: there is still an infinite algebra and the Laughlin ground states (1.2) are annihilated by an infinite subset of generators. In this case, the explicit expressions of the generators and their algebra are not fully understood at the moment. The exact nature of the latter and the eventual emergence of a full Virasoro algebra constitute important unsettled issues which are under current investigation [37]. Another interesting problem is to further elucidate the geometrical properties of excited states. For example, let us consider the (secondquantized) Fouriertransformed density operator p(k): p(k)
=
fd2x elkxp(x).
(5.1)
When projected onto the first Landau level this operator can be written —kk p(k) =exP(_~_)
~
1
iknjkm
Ei,~~(~)
~zi,mp
(5.2)
Due to the character of the ~m’ it creates intralevel bosonic excitations labelled by a wave vector k: indeed, these are the magnetophonons and magnetorotons neutral excitations studied in ref. [34]. More relevant for Laughlin’s theory are the quasihole and quasiparticle charged excitations. We expect these to fall into representations of the projected algebra of the fractional case. Hopefully, this will provide the correct characterization of anyons as “vertex operators”. As a last point, we mention the geometrical treatment of the Fermi fluid in terms of distribution functions. The expectation value of the density operator p(x, t) in any quantum state gives a corresponding quantum distribution function w(x, t) of the fluid. This expectation value is also expressed in terms of
w(x,
t)
=
1 —exp(
1 Z2)
m
~ .J~Z~Z n,m=0 n.m.
~
k=
‘ 1
(k
K2~+k,m+k)
.‘
+
1).
(5.3)
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Infinite symmetry in the quantum Hall effect
489
Thus, the space of all distribution functions w carries a representation of ~ The semiclassical approximation consists in considering distribution functions which carry a representation of w, which is a more tractable algebra. Using this framework, we hope to understand better its geometrical meaning in terms of deformations of the droplet of fluid and the edge states [35,37]. This semiclassical construction was recently discussed in a “string inspired” geometrical treatment of (1 + 1)dimensional Fermi fluids [36]. Note, however, that the Fermi fluid considered here is (2 + 1)dimensional and that the reduction to a twodimensional phase space is produced by the external magnetic field.
We gratefully acknowledge helpful discussions with Luis AlvarezGaumé and Andy Lütken. We also thank Gerald Dunne for useful comments on the manuscript. The work of G.R.Z. is partially supported by the World laboratory. References 11] RE. Prange and SM. Girvin, ed., The quantum Hall effect (Springer, Berlin, 1990)
12]
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