May 20, 2002 - fractional quantum Hall effect FQHE but, like the gauge theory, stops .... quantum Hall effect is defined by ... While the definition 7 works well.
PHYSICAL REVIEW B, VOLUME 65, 205325
Mean-field theory of fractional quantum Hall effect Igor Dzyaloshinskii* Max Planck Institut—CNRS, Laboratoire des Champs Magnetiques Intenses, 25 Avenue des Martyrs, 38049 Grenoble Cedex 9, France; Institute Max von Lane-Paul Langevin, 6 rue Jules Horovitz, Boite Postale 38042, Grenoble Cedex 9, France; and Department of Physics and Astronomy, University of California, Irvine, California 92697 共Received 20 November 2001; published 20 May 2002兲 A phenomenological theory based on the Bloch-Zak waves representation of electron motion in constant magnetic field reduces all calculations to the standard operations of the theory of electron spectra in a crystalline field in the pseudopotential approximation. The theory automatically predicts the Laughlin odd fraction fractional quantum Hall effect 共FQHE兲 but, like the gauge theory, stops short of forbidding even fractions. Physical arguments involving peculiarity of time-reversal symmetry breaking and the well-known Mott criterion for stability of insulating states make the absence of even fraction FQHE plausible. Clear physical conditions ensure that the pseudopotentials that formally describe effects of periodic density waves 共BravaisZak lattices兲 on electrons motion actually represent phenomena in a genuine liquid 共or a liquid crystal兲. DOI: 10.1103/PhysRevB.65.205325
PACS number共s兲: 73.43.⫺f
I. INTRODUCTION
The phenomenal breakthrough by Laughlin demystified fractional quantum Hall effect 共FQHE兲 共see Ref. 1兲. It revealed the particle density-magnetic flux commensurability as the source of stable incompressible plasmalike FQHE states, provided a variational description of the odd fraction states, and explicitly forbade the even fraction states, indeed only rarely observed. Finally, Laughlin constructively introduced one-particle excitations across the gap and found their charge to be fractional. The second major development was based on a new concept: a gauge field acting on electrons alongside a magnetic field 共see Ref. 1兲. This field is a special feature of strictly two-dimensional geometry and was previously considered in the context of statistics transmutation 共see Ref. 2 for detailed bibliography兲. The gauge-field scheme is more lenient than that of Laughlin and does not proscribe outright even fractions excluding only 1/2 filling. However, this relatively weak constraint apparently contradicts some experimental data3 and it required serious efforts like taking the gauge field fluctuations into account2 to open a gap at 1/2. However, the very beauty of both theories that do not contain any parameters but magnetic field B, particularly density e and the quantum magnetic flux ⌽ 0 ⫽2 បc/e, I would say its closedness, makes it hard to see the real players behind the scene: Coulomb force, other Landau levels, and so on. Here a less ambitious phenomenological scheme is presented. It operates directly with Landau levels splitting and broadening by Coulomb force. Formally, the approach is nothing but the band-structure theory in the standard pseudopotential approximation. FQHE liquid is treated as if it were a crystal. The fact that Bravais lattices and Bloch waves may be used meaningfully in the liquid state as an alternative to wave functions with definite angular momentum 共used by Laughlin in his determinants兲 was discovered by Zak4 and is guaranteed by the enormous macroscopic degeneracy of Landau levels. The properties of Bloch-Zak functions and 0163-1829/2002/65共20兲/205325共7兲/$20.00
corresponding Bravais lattices and Wannier representations are discussed in Sec. II and in the Appendix. The crucial feature here is that the lattices and waves themselves may be introduced only under the Zak quantization condition: the magnetic flux through the unit cell ⌽ must be an integer N⫽1, 2 . . . of the unit quantum flux ⌽ 0 :⌽⫽⫾N⌽ 0 . This immediately establishes a commensurability relation between unit cell area, electron density, and magnetic field. Coulomb force splits and broadens Landau bands into Zak subbands separated by gaps. The number of states in a subband is the rational fracture L /N of the number of states in a filled Landau level: a clear picture of FQHE. The major problem here is that the gaps in Landau bands imply real density waves generated by Coulomb force. We must have a criterion telling which of the pseudopotentials represents a real crystal and which leaves the electrons liquid. I could not formulate the criterion in a workable form. Instead some natural properties of liquid state and their compatibility with the pseudopotential are discussed in Sec. V. Besides, in Sec. IV a skeleton pseudopotential is discussed, which apparently splits the Landau level but leaves the system liquid. At first glance our mean-field approach does not discriminate at all between even and odd fractions, and the problem is how to explain why evens are observed so rarely. It seems there is a physical mechanism based upon a peculiarity of time-reversal symmetry breaking by homogeneous magnetic fields, which discriminates between even and odd denominators 共Sec. IV兲. The Coulomb problem in a homogeneous field is invariant under the product of time-reversal T and twodimensional space inversion P 2 (x→x,y→⫺y):T P 2 共Sec. III兲. The T-invariant part of the mean field removes the Landau degeneracy at odd fillings completely, but leaves even fillings doubly degenerate and the T-breaking part is actually needed. In Sec. IV I argue that this circumstance, combined with the Mott criterion of stability of an insulating state, provides a plausible explanation of the event’s rarity.
65 205325-1
©2002 The American Physical Society
IGOR DZYALOSHINSKII
PHYSICAL REVIEW B 65 205325
II. BLOCH AND WANNIER WAVE FUNCTIONS
In what follows, we keep the so-called magnetic length ⫽(បc/eB) 1/2 unity. In terms of the Landau energy ⑀ L ⫽ប 2 /m 2 , the Coulomb energy ⑀ c ⫽e 2 /. Besides
One recognizes in Eq. 共5兲 the conditions of the BohmAronov constructive or distractive interference when a particle moves along the closed loop a→b→⫺a→⫺b. The commutativity of T a and T b permits to construct Bloch-Zak 共BZ兲 waves BZ requiring
2 B⫽⌽ 0 /2 , where the quantum flux ⌽ 0 ⫽2 បc/e. The crucial ratio between Landau levels and their Coulomb broadening and splitting
冉 冊
⑀ c me 2 m m B0 ⫽ ⫽ ⫽ ⑀L ប me rB me B
冉 冊 m me
,
v K⫽
1 e2 ⑀ L⫹ f 共 e 2 兲, 2
where e is the actual electron density. The Schro¨dinger equation for a free particle written in the symmetric gauge A⫽ 12 (B⫻r) is ⫺i
1 ⫹ ⑀ x x ␣ 2 ␣ 
冊
This reduction signifies that there are N states for each value of k. Consequently, each Bloch-Zak wave is labeled by three numbers: n⫽0, 1, . . . for Landau level, k and the projection at ‘‘isotop spin’’ ⫽ 21 (N⫺1),
BZ ⫽ nk . The actual BZ may be found in the Appendix where we closely follow the work by Dubrovin and Novikov.5 Alongside BZ waves we may define their natural Wannier-Zak 共WZ兲 counterparts. WZ function W R (x) localized around a lattice point R⫽La⫹M b (L, M are integers兲 are the integrals over the reciprocal unit cell
2
n ⫽ 共 2n⫹1 兲 n ,
共1兲
where ␣ ⫽x,y and the unit antisymmetric tensor ⑀ xy ⫽⫺ ⑀ yx ⫽1. Due to physical homogeneity of the problem the Hamiltonian in Eq. 共1兲 commutes with two operators of infinitesimal translation
1 ˆ ␣ ⫽ ⫺i ⑀ ␣ x  x␣ 2
共2兲
and with translation by a finite distance a ␣ ,
冉
冊
1 T a 共 x 兲 ⫽exp i ⑀ ␣ x ␣ a  共 x⫹a 兲 . 2
共3兲
The quantum observability of vector potential leads to noncommutativity of ˆ x , and ˆ y and T a , T b . However, in 1964 Zak4 made a crucial observation: two finite translations T a , T b commute if the area v of the cell defined by a and b is quantized, v ⫽ ⑀ ␣ a ␣ b  ⫽⫾2 N, N⫽1, 2, 3, . . . .
共4兲
W R共 x 兲 ⫽
冕
vK
d 2 ke i(kR) k 共 x 兲 .
共7兲
Obviously T a W R ⫽W R⫹a , meaning that starting at the point R⫽0 we may build the complete Wannier set. However, some caution should be exercised in using the Wannier representation. While the definition 共7兲 works well for free electrons in magnetic field the Coulomb force specifically under QHE conditions can make Eq. 共7兲 problematic.5 In what follows, Wannier functions are used in the oversimplified model at the beginning of Sec. IV where electrons eigenfunctions remain that of free particles. III. TP 2 AND CHARGE CONJUGATION C SYMMETRIES
The Hamiltonian of free motion 共1兲 and the corresponding problem of Coulomb repulsion are invariant under combined time-reversal T and two-dimensional space inversion P 2 (x →x,y→⫺y),
In normal units Eq. 共4兲 expresses quantization of the magnetic flux ⌽ a,b through the cell ⌽ a,b ⫽⫾N⌽ 0 .
42 2 ⫽ N v
N ⫽ L /N. B0 .
冉 冊
冉
共6兲
so the number of states available is N times smaller than the Landau degeneracy L ⫽1/2 ,
2
In the extreme quantum limit ⑀ c Ⰶ ⑀ L we neglect the interference of higher 共empty兲 and lower 共filled兲 Landau bands and the problem becomes purely numerical. The energies in a single Landau level, E n ⫽ n⫹
T b BZ ⫽e i(kb) BZ ,
where k are quasimomenta. The reciprocal lattice K a ,K b is defined in the conventional way. The area of the reciprocal cell
1/2
where m e is electron mass, r B is Bohr radius, and B 0 is the atomic magnetic field of the order 103 T. The domain of quantum Hall effect is defined by
⑀ c ⱗ ⑀ L , Bⲏ
T a BZ ⫽e i(ka) BZ ,
共5兲 205325-2
T:
ˆ →H ˆ *, H P2 :
⌿ 共 x 1 ,x 2 , . . . 兲 →⌿ * 共 x 1 ,x 2 , . . . 兲 ,
ˆ →H ˆ 共 P 2 x 兲 ⬅H ˆ * ,⌿ 共 x 兲 →⌿ 共 P 2 x 兲 , H
A MEAN-FIELD THEORY OF FRACTIONAL QUANTUM . . .
ˆ 共 x 兲 →H ˆ 共 x 兲, H
T P2 :
⌿ 共 x 兲 →⌿ * 共 P 2 x 兲 .
共8兲
If the ground state is not degenerate then T P 2 ⌿⫽C⌿, 兩 C 兩 2 ⫽1 and the same is true for the density matrix (x 1 ,x 2 ), T P 2 共 x 1 ,x 2 兲 ⫽ * 共 P 2 x 1 , P 2 x 2 兲 ⫽ 共 x 1 ,x 2 兲 .
PHYSICAL REVIEW B 65 205325
The lattice P 2 a, P 2 b obviously has the unit cell of the same shape as a,b and consequently matrix elements of isotropic and homogeneous interactions are identical for both lattices. This allows for transformation
共9兲
ˆ⫽ H
Besides ˆ is a Hermitian operator,
共 x 1 ,x 2 兲 ⫽ * 共 x 2 ,x 1 兲 .
共10兲
In a liquid state its density matrix is invariant under magnetic translations ˆ in Eq. 共2兲,
冉
冊
1 ⫹ ⫺i ⑀ ␣ 共 x 1  ⫺x 2  兲 l ⫽0. x 1␣ x 2␣ 2
The phase transformation
冉
冊
1 l ⫽exp i ⑀ ␣ 共 x 1 ␣ ⫹x 2 ␣ 兲共 x 1  ⫺x 2  兲 ¯ l 共 x 1 ,x 2 兲 . 4
共11兲
gives immediately ¯ l ⫽¯ l (x 1 ⫺x 2 ). In the nondegenerate T P 2 -invariant state, ¯ l* 共 x 兲 ⫽¯ l 共 ⫺x 兲 ,
¯ * ¯ l 共 x 兲 ⫽ l共 P 2x 兲 .
共12兲
Equation 共12兲 allows both isotropic liquid and T P 2 -invariant liquid crystal. T P 2 violation opens the way to a different variety of liquid crystals. Important is that an isotropic liquid is T P 2 invariant. In the extreme quantum case only a single Landau level is relevant, which gives grounds to particle-hole symmetry: charge conjugation invariance C. To ascertain this we have to look more closely at BZ waves 共Appendix兲. BZ waves and Bravais lattices are not T P 2 invariant due to the Landau degeneracy and dependence at BZ on k and lattice periods should be considered explicitly. According to Eqs. 共A11兲 and 共A12兲 BZ depend on k only via two angles 1 , 2 , ˙ 共 kb 兲 , 1 ⫽ 共 ka 兲 , 2 ⫽
⫺ ⭐ 1,2⭐ .
共13兲
Under T P 2 , T P 2 共 x 兩 a,b 兲 ⫽ ˜ 共 x 兩 P 2 a, P 2 b 兲 ,
⫽共 1 ,2兲,
˜ ⫽共 ⫺1 ,2兲.
共14兲
Strictly speaking, BZ and T P 2 BZ belong to different Bravais lattices, viz., a,b and P 2 a, P 2 b. Let us define the hole creation operator ˆ h (x)⬅C ˆ (x) as
ˆ h 共 x 兲 ⫽C ˆ 共 x 兲 ⫽ P 2 ˆ ⫹ 共 x 兲 . In terms of operators cˆ ˜ in BZ representation
ˆ h 共 x 兲 ⫽ 兺 ˜ 共 x 兩 P 2 a, P 2 b 兲 cˆ ⫹ ⫽ 兺 h 共 x 兩 a,b 兲 cˆ h ,
cˆ h ⫽ 共 cˆ ˜ 兲 ⫹ ,
h 共 x 兩 a,b 兲 ⫽ ˜ 共 x 兩 P 2 a, P 2 b 兲 .
1 2
ˆ h⫽ ⫺H
冕
dx 1 dx 2 ˆ ⫹ 共 x 1 兲 ˆ ⫹ 共 x 2 兲
冕
ˆ⫹ dx 1 dx 2 ˆ ⫹ h 共 x1兲h 共 x2兲
1 2
e2 ˆ 共 x 2 兲 ˆ 共 x 1 兲 , 兩 x 1 ⫺x 2 兩 e2 ˆ 共 x 兲 ˆ 共 x 兲 . 兩 x 1 ⫺x 2 兩 h 2 h 1
Finally, a remark with the group-theory undertone. The macroscopic Landau degeneracy permits to construct a new set of wave functions generated by BZ waves. Let us transform the set based upon a,b to another one based on a new lattice a ⬘ ,b ⬘ . The transformation a,b→a ⬘ ,b ⬘ preserving the area is the well-known two-dimensional 共2D兲 affine group SL共2兲. Acting upon complex waves it becomes SU共2兲: the unitary group of rotations of 3D ‘‘isotopic’’ space. Its irreducible representations are given by the value of ‘‘isotop spin’’ ⫽0, 21 ,1, 32 , . . . . They may be easily identified if one remembers that actually BZ waves depend on quasimomentum k only via angles 关see Eq. 共13兲兴 that are not sensitive to the lattice choice. Now at fixed the group SL共2兲 acts only upon a,b in (x 兩 a,b). There are just N linearly independent waves at each , which permits to identify the subspace of N waves BZ with the irreducible subspace of the fixed isotop spin ⫽ 21 (N⫺1),
BZ →⌿ . We will not discuss ⌿ any further. IV. THE PSEUDOPOTENTIAL APPROACH
To begin, we consider the simplest possible mean field in the Wannier representation corresponding to a specially chosen density wave in real space. In Wannier picture with N⫽1 particles are localized at the sites of Bravais-Zak lattice, exactly one particle per site being permitted. When all sites are occupied the Landau level is completely filled. Now let us move to a new cell N times larger 共it contains N old sites 1, 2, . . . ,N) and assign different new energies ⑀ 1 ⬍ ⑀ 2 ⬍••• ⑀ N to each of the N sites in the new cell. The Landau level is thus split into N subbands having e ⫽ L /N states each. When the subbands are either completely filled or completely empty a FQHE state with filling factor e ⫽ p L /N;p⭐N is realized. Moreover, the state is a liquid because its total energy
共15兲
E⫽
1 共 ⑀ ⫹ ⑀ ⫹••• ⑀ p 兲 L N 1 2
apparently does not depend on the shape of the unit cell used, that is, on deformations. Amusingly, this liquid is not isotropic. Indeed, a quick inspection using formulas in the Appendix shows that written in real space variable the density waves involved violate T P 2 invariance that according to Sec. III signifies a liquid crystal. 205325-3
IGOR DZYALOSHINSKII
PHYSICAL REVIEW B 65 205325
To ascertain T P 2 conservation we have to work with general k-dependent mean fields: pseudopotentials. A T P 2 -invariant pseudopotential kernel V(x 1 ,x 2 ) satisfies the conditions 共9兲, V * 共 x 1 ,x 2 兲 ⫽V 共 P 2 x 1 , P 2 x 2 兲 .
共16兲
In Bloch-Zak representation with N fluxes per unit cell and wave functions BZ (x)⬅ n , where n denotes a Landau level, is the angle 共13兲 and is a component of isospin; ⫽ 12 (N⫺1) the pseudopotential becomes a matrix in n, : Vˆ ( ). Neglecting the influence of higher Landau levels we may expand the matrix Vˆ in powers of isospin operators ˆ i ,i⫽1, 2, 3,
冉
1 Vˆ 共 兲 ⫽V 0 共 兲 ⫹m i 共 兲 ˆ i ⫹ Q i j 共 兲 ˆ i ˆ j ⫹ ˆ j ˆ i 2
冊
2 ⫺ ␦ i j 共 ⫹1 兲 ⫹•••, 3
共17兲
with real functions V 0 ( ),m i ( )•••. T P 2 invariance 共16兲 means 关see Eq. 共14兲兴 Vˆ * 共 1 , 2 兲 ⫽Vˆ 共 ⫺ 1 , 2 兲 兩 ˆ →⫺ ˆ or V 0 共 1 , 2 兲 ⫽V 0 共 ⫺ 1 , 2 兲 ,Q i j 共 1 , 2 兲 ⫽Q i j 共 ⫺ 1 , 2 兲 , . . .
共18兲
for even-order coefficients and m i 共 1 , 2 兲 ⫽⫺m i 共 ⫺ 1 , 2 兲 , . . .
a BZ wave doubly degenerate for a given angle with energies ⑀ ↑ ( )⬅ ⑀ ↓ ( ). T-breaking 共odd order兲 terms remove this degeneracy but due to T P 2 invariance expressed by Eqs. 共18兲 and 共19兲 the relation
共19兲
for odd-order coefficients. The pseudopotential 共17兲 splits the Landau level in N subbands with subband widths defined by dependence of coefficients in 共17兲. Actually T P 2 invariance requirements 共18兲, 共19兲 lead to differences in subbands for even and odd N 共see below兲. If the width of subbands W is less than their splitting ⌬:W⬍⌬, the integer N—the number of subbands, viz., the number of fluxes becomes an independent observable. There is no one-to-one correspondence between N and the actual filling e . The latter may be achieved in many ways: N ⫽r/ e , where r is an integer, provided the width at given N is still less than splitting, W N ⬍⌬ N . Exceedingly large N→⬁ may be rejected outright. The overall scale of the Landau level broadening is just Coulomb energy ⑀ c ⬃e 2 1/2 e . The gap ⌬ N is asymptotically proportional to 1/N: ⌬ N ⬃ ⑀ c /N, but apparently the width W N does not decay that fast, if at all. The general pseudopotential 共17兲 removes N-fold degeneracy of BZ waves completely: now there is exactly one wave for each angle , for both odd 共integer isospin ) and even N 共half-integer isospin ). The difference between odd and even becomes evident when one calculates density of states. For integer spins already T-invariant 共even order兲 terms in Eq. 共17兲 remove the N-fold degeneracy completely and densities of states of different subbands are generally speaking unrelated. For half integers T-invariant terms leave
˜ 兲, ˜ ⫽共 ⫺1 ,2兲 ⑀ ↑共 兲 ⫽ ⑀ ↓共
共20兲
holds. The spectra 共20兲 reminds that of a typical antiferromagnetic metal with ‘‘doubly-degenerate density of states.’’ Thus if we are concerned only with energetics: subbands fillings, total energy, and so on, the subbands for even N’s remain virtually doubly degenerate. Immediately it becomes possible to have both Fermiliquid and FQHE states at 21 filling. At N⫽2 the Landau level broadens into two ‘‘antiferromagnetic’’ subbands, viz., one doubly-degenerate subband 共in energy variable兲 with L /2 states in each. A clear picture of FL with ‘‘antiferromagnetic’’ FL bands emerges. Picking larger N⫽4, 8, . . . we split the Landau level into, viz., 2, 4, . . . doubly degenerate subbands with L /4, L /8, . . . states in each and a clear picture of a FQHE liquid at e ⫽ 21 emerges. Why are even fractions so rarely observed? Within the framework of the mean-field phenomenology the only difference between odd and even N 共integer and half integer ) is the double degeneracy of subbands in the latter case, and as will be clear shortly, a new physical assumption is needed. Here we assume that T-breaking terms 共odd order terms兲 are much weaker than T-conserving terms in the T P 2 preserving pseudopotential 共17兲. Under this condition, even order terms completely define band splitting ⌬ ⫹ ⬇⌬,W ⫹ ⬇W:W ⫺ ⰆW. However, as we have seen for even N, T-invariant terms alone leave the states doubly degenerate: there are two waves for each value at angle and energy ⑀ ( ). There are some reasons justifying this assumption. I mention here the most transparent: the mean-field approximation of the gauge-field theory2 corresponds to T-invariant pseudopotential 共17兲. Indeed, in this case a particle moves in a homogeneous effective gauge field and obviously all subbands have zero width. All coefficients in Eq. 共17兲 become -independent constants, and due to Eq. 共19兲, the odd order (T-breaking兲 terms are identically zero,
冉
冊
1 2 Vˆ ⫽Vˆ 0 ⫹ Q i j ˆ i ˆ j ⫹ ˆ j ˆ i ⫺ 共 ⫹1 兲 ␦ i j ⫹ ˆ 4 ⫹•••. 2 3 共21兲 The resulting double 共Kramers兲 degeneracy at half integer 共even N) leads to singular behavior of BZ waves as functions of angle . The waves are not periodic functions of ,
⫹2 , → , ⬘ ,
⫽⬘
共22兲
共see detailed analysis in Ref. 6兲. In their turn singularities of BZ waves in quasimomentum space have as a consequence that corresponding excitation across the gap are quasiparticles poorly defined in real space x and time t. While in the case of odd N wave packets decay exponentially outside their average space-time localization areas, excitations for even N are poorly localized: exponential decay switches to a power
205325-4
A MEAN-FIELD THEORY OF FRACTIONAL QUANTUM . . .
law 共e.g., 1/兩 x 兩 if BZ is discontinuous in ). To see this let us take a look at the excitations propagator in our model, g ex 共 x,t 兲 ⫽
兺
冕
* 共 0,0兲 d ⑀ d 2 共 x,t 兲
1 , ⑀ ⫺ ⑀ 共 兲 共23兲
where summation in goes over empty subbands for particles and over filled subbands for holes. The theorems of the theory of Fourier transformation immediately lead to the above conclusions. Poor localization of wave packets of particles and holes results in an abnormally large effective Coulomb matrix element in the Mott criterion of stability of an insulating state, which effectively precludes FQHE at even fractions. Indeed, even at odd fraction FQHE, when excitations are welldefined particles in space and time, the Mott criterion is not easy to satisfy. In the standard form it puts a constraint on the gap ⌬ and the effective mass M of excitations, ⌬⬎
M e4 ប2
.
共24兲
The mass M here is to be expressed through the subband width W using the relations W⬃
ប2 M 2
,
e2 ⌬⬃ ,
where is the magnetic length, ⌬⬎
⌬2 共 ! 兲. W
level ⑀ ⫽0. You may just as well diagonalize corresponding ˆ: G ˆ ⫺1 ⫽ ⑀ ⫺⌺ˆ . In this way the ground-state self-energy ⌺ pseudopotential is naturally defined as ˆ 共 , ⑀ ⫽0 兲 . Vˆ gr 共 兲 ⬅⌺
Vˆ p 共 兲 ⫽⌺ˆ 共 , ⑀ ⫽⌬ p 兲 , Vˆ h 共 兲 ⫽⌺ˆ 共 , ⑀ ⫽⫺⌬ h 兲 ,
共27兲
where ⌬ p is the lowest particle energy and ⫺⌬ h is the highest hole energy, the gap ⌬⫽⌬ p ⫹⌬ h . Within the fieldtheoretical approach in Eq. 共27兲 only zero-order terms in small ⑀ ⫺⌬ p or ⑀ ⫹⌬ h are kept, thus neglecting quasiparticle decay. Equation 共27兲 describes thermodynamics of FQHE liquid, however, to calculate longitudinal currents in the pseudopotential phenomenology the quasiparticle charge q is to be brought from outside. The very fact that states of quasiparticles are BZ waves in a Bravais-Zak lattice with the flux ⌽⫽N⌽ 0 implies that q must satisfy the Zak condition q q ⌽⬅ N⌽ 0 ⫽2 r, បc បc
共28兲
r⫽1, 2, . . . .
Equation 共28兲 is the quantization rule 共5兲 written for a particle with charge q. Hence we have
共25兲
To realize the state even for odd fractions we must assume 共not to say prove兲 that C is small for some purely numerical reasons. For even fractions as we have just argued, C is definitely much larger than for the odds, making even fractions much less probable. The nonzero T-breaking terms remove the double degeneracy, but due to our assuming T breaking to be weak, W ⫺ ⰆW, the resulting effective space-time size of the wave packet remains large ⬃W/W ⫺ thus enhancing the value of C in Eq. 共25兲. Finally, let us consider the excitation charge q within the mean-field phenomenology. Formally here the charge of an excitation q and the charge of a bare particle 共electron兲 e should be equal. In reality, however, the ground state and the excitations must be described by different pseudopotentials, Vˆ gr and Vˆ ex . Both are connected with the exact 共renormalˆ ( , ⑀ ) written in BZ representation. Folized兲 propagator G lowing the Luttinger paper7 共the site of the famous Luttinger theorem兲 the bands defining the ground state, and their filling and corresponding wave functions are to be calculated by ˆ at the chemical potential diagonalization of the propagator G
共26兲
Equation 共26兲 gives the filling factor of bare particles e and the Hall conductance H ⫽e 2 e /ប with the bare charge e. In a Fermi-liquid Vˆ ex obviously coincides with Eq. 共26兲. In an insulating case chemical potential lies inside the gap. We still should use Eq. 共26兲 in analyzing the ground state. But in the Luttinger spirit to construct the meaningful pseudopotentials for particles Vˆ p and holes Vˆ h we have to use the following relations:
共24⬘兲
Combining Eq. (24⬘ ) with obvious condition: W⬍⌬ we finally have ⌬⬎W⬎C⌬.
PHYSICAL REVIEW B 65 205325
q⫽
r e, N
or
e q min ⫽ , N
共29兲
q⫽rq min ,
the quasiparticle charge is an integer r of the minimal charge q min defined by the flux number N. To find the actual value of r one has to move beyond the phenomenology and to look at its field-theoretical source. Apparently there are some field-theoretical theorems appending the Ward identities in the case of quantizing magnetic fields, which connect the integer r with the properties of ground state, its filling factor e , and so on. This problem will be addressed elsewhere. V. LIQUID OR CRYSTAL?
The general criteria of an isotropic liquid state are expressed by the relation 共11兲 with ¯ l (x 1 ,x 2 )⬅¯ l ( 兩 x 1 ⫺x 2 兩 ) for density matrix and the analogous formula for the renormalized propagator in terms of space x and energy ⑀ variables,
205325-5
冉
1 G 共 x 1 ,x 2 , ⑀ 兲 ⫽exp i ⑀ ␣ 共 x 1 ␣ ⫹x 2 ␣ 兲共 x 1  ⫺x 2  兲 4 ⫻g 共 兩 x 1 ⫺x 2 兩 , ⑀ 兲 .
冊 共30兲
IGOR DZYALOSHINSKII
PHYSICAL REVIEW B 65 205325
In BZ representations n (x) it defines the general form of ˆ ( , ⑀ ) for a liquid, matrix G ˆ 共 , ⑀ 兲兴 n ,n ⫽ 关G ⬘ ⬘
冕
dx 21 dx 22 *n 共 x 1 兲 n ⬘ ⬘ 共 x 2 兲
冉
VI. CONCLUSION
1 ⫻exp i ⑀ ␣ 共 x 1 ␣ ⫹x 2 ␣ 兲 4
冊
⫻ 共 x 1  ⫺x 2  兲 g 共 兩 x 1 ⫺x 2 兩 , ⑀ 兲 .
共31兲
ˆ depends explicitly on parameters of BravaisThe matrix G Zak lattice via BZ . The criterion 共31兲 is practically useless, however, it is relatively easy to formulate some physical requirements the eigenvalues of Vˆ gr and Vˆ ex must satisfy. The lattice dependence of Vˆ gr eigenvalues is virtually irrelevant provided the subbands are completely filled or empty. The only weak constraint is the ground-state being energy independent of the lattice parameters. Expending energy spectrum of a filled subband in the Fourier series in ,
⑀ gr 共 兲 ⫽ ⑀ 0 ⫹ 兺 ⑀ n e i(n ) , n⫽0
E tot ⫽
42 N
兺
filled bands
⑀0 .
The lattice independence of thermodynamics imposes a really strong 共but obvious兲 constraint on excitation spectra: density of states D(E) lattice independence. Writing D(E) as D共 E 兲 ⫽
冕
unit cell
1 N
冕 冕
⫺
⫺
d 2 ␦ „E⫺ ⑀ ex 共 兲 ….
⑀ ex ␦ ⫹ 共 ˆ ␣ 兲 ⬘ , k ␣ ⬘
The Schro¨dinger equation 共1兲 may be rewritten in complex variable z⫽x⫹iy for a new function n (z,z * ),
冉
共A1兲
1 1 n ⫽⫺ n n . ⫺ z* z 2 2 z*
共A2兲
冊
For the zero Landau level n⫽0, Eq. 共A2兲 gives
0 共 z,z * 兲 ⬅ f 共 z 兲 ,
共A3兲
where f is an arbitrary analytic function of z. The arbitrariness obviously is another manifestation of the Landau degeneracy. It is easy to see that for higher Landau levels
共33兲
where the nondiagonal matrix accounts for transitions between N different subbands: a new phenomenological feature 共alongside V ex , relaxation rates, . . . 兲 in our mean-field theory. The abundance of adjustable parameters leaves prac-
冉 冊
1 zz * n 共 z,z * 兲 , 4
n 共 x,y 兲 ⫽exp
共32兲
Obviously excitation energies ⑀ ex expressed in terms of angles cannot depend on lattice parameters. Actually kinetics 共its lattice independence兲 imposes new constraints only on new physical quantities: relaxation time and so on. Besides, quantum calculations of, say, longitudinal conductivity introduce a new physical quantity: velocity operator of excitations, 关vˆ ␣ 共 k 兲兴 ⬘ ⫽
What are the benefits of the mean-field approach based on Bravais-Zak lattices and Bloch-Zak waves? 共1兲 It offers greater flexibility in classifying a FQHE state not only by the fractional filling factor e but by another observable integer: N the number of subbands into which the Landau level is split. 共2兲 It uses different meanfields for ground state and excitations. 共3兲 It connects the number of subbands N to the minimal possible excitations charge: q min ⫽e/N. 共4兲 It uses fully the fundamental symmetry of the quantum problem of charges moving in a homogeneous magnetic field: T P 2 invariance. This invariance and the method flexibility explain why we may have either Fermi liquid or FQHE liquid at the same 1/2 filling: the former is realized only at N⫽2 while liquids with larger even N⫽4, 8, . . . exhibit FQHE. 共5兲 The method permits to incorporate routinely transitions between different Landau levels. 共6兲 Finally, basing on the Mott criterion for stability of insulating state and T P 2 invariance and assuming 共quite plausibly兲 that the part of the mean field responsible for T violation is weak the method suggests a qualitative explanation of why even-fraction FQHE is so rarely observed. Of course, despite all its benefits the method remains heavily phenomenological. It definitely lacks the beauty of Laughlins and gauge-field microscopic approaches. APPENDIX BLOCH-ZAK WAVE FUNCTIONS
d 2 k ␦ „E⫺ ⑀ ex 共 k 兲 …
and moving here from quasimomenta k to angle variables 共the corresponding Jacobian is just the area of unit cell N) we have D共 E 兲 ⫽
tically no doubts that Bravais-Zak lattices and Bloch-Zak waves may be safely used to describe isotropic liquid and liquid crystal states.
n 共 z,z * 兲 ⫽
冉
1 ⫺ z* z 2
冊
n
f 共 z 兲.
共A4兲
Now BZ waves may be routinely built starting from n ⫽0. Here we follow the paper by Dubrovin and Novikov.6
205325-6
A MEAN-FIELD THEORY OF FRACTIONAL QUANTUM . . .
PHYSICAL REVIEW B 65 205325
Let us choose the Bravais-Zak lattice 共basic vectors a 1 and a 2 ) with N quantum fluxes through its unit cell⫹ and write f (z) in Eq. 共A3兲 in the following way: f 共 z 兲 ⫽exp共 ␣ z⫹ ␥ z 兲 共 z⫺  1 兲 ••• 共 z⫺  N 兲 , 2
兿 共 z⫺⍀ 兲 exp ⍀⫽0
冉
z2 2⍀
2
⫹
冊
z , ⍀
共A6兲
⍀⫽2M 1 1 ⫹2M 2 2 , M 1 ,M 2 integers, 2 1 ⫽a 1x ⫹ia 2y ,
e i 1,2 BZ 共 z,z * 兲 ⫽exp
共A5兲
where ␣ , ␥ ,  1 , . . . ,  N are complex constants what are to be adjusted. (z) is the so-called function.8 It is expressed in the form of infinite product over the complex lattice sites,
共 z 兲 ⫽z
T 2 1,2 BZ ⫽e i 1,2 BZ ,
共 z⫹2 1,2兲 ⫽⫺exp共 2 1,2z⫹2 1,2 1,2兲 共 z 兲
1 1 * 1 2 * 1 2 ␥ ⫽⫺ N ⫹ ⬅⫺ N ⫹ 2 1 41 2 2 42
␣⫽ ¯ ⫽
共A8兲
1 and 2 are not independent,
共A9兲
Finally, the area of unit cell given by Eq. 共4兲 is now expressed as
1 2* ⫺ 2 1* ⫽⫺i N.
1 1 共  ⫹•••⫹  N 兲 ⫽ 共 1 ⌽ 2 ⫺ 2 ⌽ 1 兲 . N 1
共A12兲
⌽ 1,2⫽ 1,2⫹N ⫹2 L 1,2 ,
8
1 2 1⫺ 1 2⫽ i . 2
1 共 ⌽ ⫺ 2⌽ 1 兲, 1 2
New angles ⌽ 1 and ⌽ 2 are
共 z⫹2 1,2兲 ⫽2 1,2⫹ 共 z 兲 ,
1,2⫽ 共 1,2兲 .
共A11兲
and
the -dependent constants 1,2 here are connected with socalled functions8
兺
冊
The exponential here is the corresponding exponential in the definition of magnetic translations 共3兲 written in complex variables. Using the whole set of formulas 共A1兲 through 共A10兲 we may express the constants ␣ , ␥ ,  1 , . . . ,  N via angles 1 and 2 and the lattice periods 2 1 and 2 2 ,
共A7兲
1 1 共 z 兲⫽ ⫹ , z ⍀⫽0 z⫺⍀
1 * z兲 共 z * ⫺ 1,2 2 1,2
* 兲. ⫻ BZ 共 z⫹2 1,2 ,z * ⫹2 1,2
2 2 ⫽a 2x ⫹ia 2y .
(z) acquires a z dependent new phase under lattice translations 2 1 ,2 2 共Ref. 8兲,
冉
共A10兲
A BZ wave with quasimomentum k or, better still, two angles 1 , 2 共13兲 is defined by the conditions 共6兲 that now takes the form
where integers L 1,2 are to be chosen as to keep  1 , . . . ,  N inside the original unit cell. The fact that Eq. 共A12兲 fixes only the sum ¯ of  1 , . . . ,  N indicates the aforementioned degeneracy of each BZ state. The detailed analysis in Ref. 5 shows that there are exactly N linearly independent waves for each angle . It is easy to see that if N solutions f 1 (z), . . . , f N (z) in Eq. 共A3兲 for the zero Landau levels are known the corresponding N solutions for a higher level n1 (z,z * ), . . . , nN (q,q * ) are given by n differentiations in Eq. 共A4兲. Finally applying time reversal T: → * and space inver← z * )兴 and using relevant sion P 2 关in complex form P 2 is z → formulas above we arrive at the results listed in Sec. III.
J. Zak, Phys. Rev. A 136, 776 共1964兲. D. Thouless, J. Phys. C 17, L325 共1984兲. 6 B.A. Dubrovin and S.R. Novikov, JETP 52, 511 共1980兲. 7 J.M. Luttinger, Phys. Rev. 119, 1153 共1960兲. 8 Higher Transcendental Functions 共McGraw-Hill, New York, 1953兲, Vol. 2.
*Permanent address: University of California, Irvine, CA 92697.
4
1
5
The Quantum Hall Effect, 2nd ed., edited by R. E. Prange and S. M. Girvin 共Springer Verlag, New York, 1990兲. 2 B.I. Halperin et al., Phys. Rev. B 47, 7312 共1993兲. 3 R.L. Willett et al., Phys. Rev. Lett. 59, 1776 共1987兲; W. Pan et al., Solid State Commun. 119, 641 共2001兲.
205325-7
