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Theory of the quantized hall effect
147
Surface Science 142 (1984) 147-154 North-Holland, Amsterdam
THEORY OF THE QUANTIZED
HALL EFFECT
D.J. THOULESS Deportment of Physics FM - 15, Unioerslty of Washington, Seattle, Washington md Cauendtsh Laboratory, Madingley Road, Cambridge CB3 OHE, UK Received
6 July 1983; accepted
for publication
6 September
98195, USA
1983
A review is given of the current state of the theory of the quantum Hall effect. The integer values of the Hall conductance seen in systems with moderate disorder and predicted for systems with periodic modulation are fairly well understood. Charge density wave and Fermi liquid theories of the fractional quantization observed in systems with low disorder are briefly described.
1. Integer quantization The discovery by von Klitzing, Dorda and Pepper [l] that the Hall conductance of a two-dimensional electron system can be, with very high precision, an integer multiple of e2/h was a triumph of experimental physics. In most comparable cases, such as the quantization of flux in superconductivity or the quantization of circulation for superfluid helium, there have been previous theoretical suggestions of the existence of the effect, even if there were unexpected features in the experimental result. In this case there was no more than approximate quantization suggested [2], and so there was no reason for the experimentalists to examine the transverse voltage in their device with the precision which they used. Once the discovery had been made, we theorists rushed in to show why the result had been obvious all the time. There were a large number of papers, of which I list only a few [3-61, which used bulk perturbative arguments to show that the disorder in the substrate potential gave no correction to the simple quantized result which can be obtained from full Landau levels in an ideal two-dimensional system with no disorder. These arguments are sufficient to show that corrections due to disorder or electron-electron interactions vanish to all orders in perturbation theory. One such argument says that the Lorentz transformation allows us to move to a frame of reference moving with velocity -E/B, in which there is no electric field, and in this frame the substrate potential makes a local perturbation of the electron density, but does not carry 003%6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
148
D.J. Thoultxs / Theory of quantized Hall effect
any electron charge with it when the Fermi energy lies in a gap because the perturbation of the electron density falls off exponentially with distance from the perturbing potential. In this frame the current is entirely carried by the positively charged substrate (in an n-type layer) moving with velocity E/B. These arguments also show that localized states, even if they are shifted from the unperturbed Landau level enough to pass through the Fermi surface, do not affect the quantization and do not change the value of the quantized current. A more profound approach to the problem was introduced by Laughlin [7]. He used gauge invariance to show that, for a system with the topology of an annulus, a change in the flux threading the annulus by an integer amount would lead to the transfer of an integer number of electrons from one edge to the other. Since the change in flux, by Faraday’s law, is given by the time integral of the EMF, and the transfer of charge is given by the time integral of the current, it follows that the ratio of current in one direction to voltage in the perpendicular direction is an integer multiple of e divided by the quantum of flux. This powerful argument has the great advantage that it can be applied in circumstances in which the perturbative arguments are not applicable, such as, for example, when a Landau level is split into sub-bands by a periodic potential; in this case, as I shall discuss in the next section, each sub-band must carry a multiple of the quantized current carried by the entire level. It has the disadvantage that it makes the Hall current appear to be an edge effect, and therefore possibly sensitive to boundary conditions, while the perturbative arguments make it clear that it is a bulk effect. One knows, however, that two-dimensional bulk effects and edge effects can be related by some version of Stokes’ theorem. All these arguments assume that the electric field is weak. In the perturbative arguments the response linear in the electric field is calculated, while in Laughlin’s argument it is assumed that the rate of change of the vector potential is slow enough that the electrons can follow it adiabatically. In a strong field we can be sure that the quantization is not exact, since there will be a tunnelling current parallel to the electric field due to electrons tunnelling from a full Landau level to an empty one, and this longitudinal current must in turn affect the transverse current. Such corrections must depend exponentially on the inverse of the field strength, so it is reasonable to ask if there are any corrections which are algebraic in the field strength. It seems likely, both from the precision of the experimental results and on theoretical grounds, that there are no such algebraic terms, but I do not know of any general proof of this result. It is generally thought that in the absence of a magnetic field all states in a two-dimensional disordered system are localized [8-lo]. The existence of the quantum Hall effect shows that this is not the case in the presence of a magnetic field, since some nonlocalized states are needed to carry the Hall
D.J. Thouless / Theory of quantized Hall effect
149
current. The question naturally arises whether in the presence of disorder the Landau level is broadened into a band whose tail states are localized but whose central region consists of extended states, as some numerical calculations have suggested [ll], or if all the current is carried at some singular energy, as a simple semiclassical argument suggests [12,13]. If the potential varies slowly over a magnetic length, the electrons should be confined to constant energy curves on the potential surface. For a random potential these curves will either be electron-like, surrounding a minimum, or hole-like, surrounding a maximum. It is only at the singular energy dividing the two that there can be an open orbit capable of varying a current. In the presence of an electric field this open orbit is broadened into a band of open orbits whose width depends on the field strength. In this model, which seems to give a fairly good account of the experimental situation, the steps between Hall plateaus, and the corresponding regions of nonzero longitudinal resistance, should get sharper and sharper as the electric field strength is reduced. The problem of the existence of extended states in the presence of a magnetic field has recently been discussed on the basis of the nonlinear sigma model [14], but the question of whether the extended states form a broad band or lie at a singular energy is still open.
2. Periodic potentials It has already been mentioned that a periodic modulation of the substrate breaks the degeneracy of the Landau level in the ideal system and breaks it up into a set of sub-bands separated by gaps. The number of sub-bands produced is p, where the number of flux quanta per unit cell is p/q, so the structure of the energy bands is highly sensitive to whether the number of flux quanta per unit cell is rational or irrational. This can be seen most clearly in the energy level diagrams calculated for p up to about 50 by Hofstadter [15] for a modulation with square symmetry, and by Claro and Wannier [16] for a modulation with hexagonal symmetry. The quantum number associated with the Hall current gives a useful way of classifying the energy gaps in this sort of diagram, and the study of periodic modulations sheds some light on the problem of a random substrate. There are not yet any experiments to which this theory is applicable, but it is likely that some experiments on two-dimensionally modulated layers will be done soon. Studies of this problem have been made by various authors [17-191. We found that from the Bloch wave function +t,+, the Hall conductance of a sub-band is given by the integral over the Brillouin zone
(2.1) This is a topological
invariant
which can be shown to be an integer
by using
D.J. Thouless / Theory of quantized Hull
150
effect
Stokes’s theorem to reduce it to an integral round the perimeter of the Brillouin zone, and then the periodicity of the Hamiltonian in the Brillouin zone can then be used to show that the integrand is just the derivative of a phase round the perimeter. A nonzero value of the Hall current implies that the wave function cannot be written as a continuous single valued function satisfying periodic boundary conditions in the unit cell, and therefore well-behaved Wannier functions do not exist for a sub-band with nonzero Hall current. We derived, with some difficulty, the general relation pt + qs =
(2.2)
r
for the Hall current 1 in the (r - 1)th gap; all these quantities is equivalent to Streda’s [18] expression te/h
= dn/d
B,
are integers.
This
(2.3)
where n is the electron density. For an incommensurate flux density, for which would have to be replaced by an irrational number, the equation equivalent to (2.2) has a unique solution, but for a rational flux density this equation obviously gives a value of t which is only determined modulo q. For the modulation with simple rectangular symmetry we argued that s should be as small as possible, and this determines the solution uniquely, and apparently correctly. In the case of a modulation with hexagonal symmetry, with its minima at points of sixfold symmetry, there is an additional restriction that s and t cannot both be odd. It turns out, however, that s does not always have the lowest possible magnitude, as can been seen from an inspection of the energy level diagram in Claro and Wannier [16]. The gap for p = 5, q = 1, r = 4 clearly corresponds to t = 0, s = 4, not to t = 1, s = - 1, for example. In an indirect fashion this model can shed some light on the idea that in a random potential the Hall current is all carried at one singular energy. A slowly varying potential is one in which p is much larger than q. For q = 1 it is certainly true that t is zero below the center of the Landau level and unity above the centre, for the square lattice, so that all the Hall current is carried by the central sub-band or pair of sub-bands. If q is larger than unity each of the sub-bands carries a nonzero current, but it is still true that the sub-bands fall into approximately p/q groups, with each group other than the central group carrying no total Hall current. The energy gaps within groups are very small compared with the energy gaps between groups. For more rapid modulations, so that p and q are of comparable magnitude, the Hall current varies in a rather exotic manner from gap to gap. It is generally true that the larger gaps correspond to smaller Hall currents, but there is no obvious sign that the current is carried in one particular part of the spectrum.
p/q
D.J. Thouless / Theory of quantized Hall effect 3.
151
Fractional quantization
The discovery of fractional quantization [20] served to undermine the complacency of theorists even further. Recent work seems to show clearly that the Hall conductance of high mobility inversion layers at very low temperatures is precisely l/3 [21], and can probably be other exact fractions, of e2/h [22]. So clean an experimental result demands a clear theoretical explanation, and promises to lead to a lot of new and unexpected physics, but, in my opinion, there is so far no theory which matches the elegance of the experimental results. There are a few things that are generally agreed. In the first place, since the integer Hall quantization is observed in lower mobility devices and the fractional effect is observed in very high mobility devices, and since integer quantization appears to be an almost inevitable consequence of a theory of noninteracting electrons in a weak substrate potential, it seems reasonable to suppose that fractional quantization is a result of the dominance of electron-electron interactions over the effects due to disorder. It is a natural extension of this observation that a model in which the degeneracy of the Landau levels is broken by the Coulomb interaction between electrons, but the substrate potential and transitions to other Landau levels are ignored, should be sufficient to show fractional quantization. It is also clear that any mechanism which makes rational fraction occupation of the Landau level energetically favorable will go a long way towards explaining fractional quantization. The argument from the Lorentz transformation shows that, provided the electrons are not locked to the substrate, a one third full Landau level will carry one third of a quantum of Hall current. Eq. (2.3) also shows that if the occupancy of the Landau level is what determines the energy, then it will also give an appropriate fractional Hall current. Two types of theories which are in accord with the principles outlined in the previous paragraph have been studied. There are models in which there is some sort of the charge density wave, and those in which there is no charge density wave. It is natural to expect a charge density wave in such a system, since at very high magnetic fields or very low electron densities the system behaves classically and forms a hexagonal lattice. Yoshioka and Fukuyama [23] and Yoshioka and Lee [24] are among the many authors who have studied this problem recently. There are two major difficulties with this approach, however. The first is that a charge density wave is likely to be pinned by substrate inhomogeneities, as it is in other systems, and should require some minimum electric field to free it from the substrate [25]. The second is that there does not seem to be any particular stability associated with simple fractional occupation of the Landau level, or, equivalently, with the commensurability of the electron lattice and the flux quantum. It can be seen why this is from the type of energy band diagram shown in Claro and Wannier [16]. If there is one electron per
152
D.J. Thouless / Theory of quantized Hall effect
unit cell the Fermi energy lies in a gap that extends from the bottom left to the top right of the diagram, and it is easy to show that if the Fermi energy is in a gap the energy is an analytic function of the magnetic field, and so there can be no commensuration energy in a mean field theory. There are gaps that close at simple rational values of the flux density, but it is hard to see how these could lead to stable enough fractional occupations to explain the observed results, or how pinning could be avoided with such a mechanism. A rather different version of the charge density wave mechanism has been proposed by Haldane [26]. In this model the lattice is locked to commensurate values by the reduced energy of defects such as vacancies and interstitials which occurs when the lattice spacing is commensurate with the flux. The lattice is rigid, and free to carry the Hall current by sliding over the substrate, and it is only the defects that get pinned to the substrate. Yoshioka, Halperin and Lee [27] have carried out a calculation for up to six electrons in a single Landau level whose degeneracy is broken by the Coulomb interaction. They found some signs of the l/3 quantization even in such a small system, but no charge density wave. The theory of Laughlin [28] does not involve a charge density wave. In this theory the wave function for the lowest Landau level is written in the form (3.1) where z = x - iy and p is odd, so that the level has a l/p occupation. This gives a specially low energy for commensurate occupation, but has no variations in charge density to provide a pinning mechanism, so it should give a l/p Hall quantization. It is not immediately obvious how this can be generalized to give fractional occupation with denominator greater than unity, but Anderson [29] has shown how this might be done. The state described by eq. (3.1) has a broken symmetry, and there are p - 1 equivalent states that can be generated from this state by gauge transformations. States with fractional occupation q/p can be produced by taking the product of q of these equivalent states. Haldane [30] has set up the problem in spherical geometry to construct a generalization of Laughlin’s wave function for these cases with 4 > 1. Our own approach to this problem [31,32] has some features in common with the work of Laughlin [28], although our approach is quite different. We have tried to use conventional many-body techniques to deal with this problem in which the complete degeneracy of the partially occupied Landau level is broken by the Coulomb interaction. The difficulty in comparison with the more usual sort of theory is that the splitting between the energy levels is itself produced by the perturbation, so that is no variable parameter that gives the ratio of the perturbation to the unperturbed energy denominators. On the other hand there is a high degree of symmetry that can be exploited to simplify calculations. In our work so far we have used the Landau gauge, but the
D.J. Thouless / Theory
of quantized Hall gffect
153
symmetric gauge could be used without it making any essential difference, and in that gauge it is easier to compare our results with Laughlin’s. Alternatively Laughlin’s wave function could be written in the Landau gauge as 2nr,/L. -
Lye
e2wL)P
exp - f:
i
$ .
I
(3.2)
We assume that there is some suitable unperturbed ground state in which some fraction of the one-particle states in the Landau level are occupied and the rest are unoccupied. We then use many-body techniques to calculate the self-energies of the various one-particle states. If the occupied states are chosen in a regular manner, so that, for example, one state is occupied and the next p - 1 states are unoccupied through the whole array of one-particle states allowed by the boundary conditions, we find an energy gap between the hole and the particle states which stabilizes the l/p occupation and an enhanced correlation energy. This, like Laughlin’s state, has a broken symmetry with p equivalent ground states. Unfortunately in the present state of the theory we find even denominators as well as the odd denominators found both by the experimentalists and by Laughlin. Tao [32] has generalized the work to cases in which the denominator is greater than unity. If the techniques of many-body theory can be adapted successfully to deal with this problem it will be much easier to calculate the effects of disorder and nonzero temperature.
Acknowledgements I am grateful to Dr. R. Tao for many useful discussions. I am also grateful to the many people who have sent me preprints of their work. This work was supported in part by the National Science Foundation under grant number DMR79-20795.
References [l] [2] [3] [4] [5] [6] [7] [S]
K. van Klitzing, G. Dorda and M. Pepper, Phys. Rev. Letters 45 (1980) 494. ‘T. Ando, Y. Matsumoto and Y. Uemura, J. Phys. Sot. Japan 39 (1973) 279. H. Aoki and T. Ando, Solid State Commun. 38 (1981) 1079. R.E. Prange, Phys. Rev. B23 (1981) 4802. D.J. Thouless, J. Phys. Cl4 (1981) 3475. R.E. Prange and R. Joynt, Phys. Rev. B25 (1982) 2943. R.B. Laughlin, Phys. Rev. B23 (1981) 5632. E. Abrahams, P.W. Anderson, D.C. Licciardello and T.V. Ramakrishnan, Phys. Rev. Letters 42 (1979) 673. [9] F. Wegner, Z. Physik B36 (1980) 209. [lo] S. Hikami, Phys. Rev. B24 (1981) 2671.
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(111 T. Ando, Surface Sci. 113 (1982) 182. 1121 S. Luryi and R.F. Kazarinov, Phys. Rev. B27 (1983) 1386. [13] S.A. Trugman, Phys. Rev. B27 (1983) 7539. [14] [15] [16] [17] [18] [19] (201 [21] [22] [23] [24] [25] 126) [27] [28] [29] [30] [31] [32]
H. Levine, S.B. Libby and A.M.M. Pruisken. Phys. Rev. Letters 51 (1983) 1915. D.R. Hofstadter, Phys. Rev. B14 (1976) 2239. F.H. Clara and G.H. Wannier, Phys. Rev. B19 (1979) 6068. D.J. Thouless, M. Kohmoto, M.P. Nightingale and M. den NiJs. Phys. Rev. Letters 49 (1982) 405. P. Streda, J. Phys. Cl5 (1982) L717. D. Yoshioka, Phys. Rev. B27 (1983) 3637. D.C. Tsui, H.L. Stlirmer and A.C. Gossard, Phys. Rev. Letters 48 (1982) 1559. D.C. Tsui, H.L. StGrmer, J.C.M. Hwang, J.S. Brooks and M.J. Naughton, Phys. Rev. B28 (1983) 2274. H.L. StGrmer, A. Chang, D.C. Tsui, J.C.M. Hwang, A.C. Gossard and W. Wiegmann, Phys. Rev. Letters 50 (1983) 1953. D. Yoshioka and H. Fukuyama, J. Phys. Sot. Japan 47 (1979) 394. D. Yoshioka and P.M. Lee, Phys. Rev. B27 (1983) 4986. H. Fukuyama and P.M. Platzman, Phys. Rev. B25 (1982) 2934. F.D.R. Haldane, unpublished. D. Yoshioka, B.I. Halperin and P.A. Lee, Phys. Rev. Letters 50 (1983) 1219. R.B. Laughlin, Phys. Rev. Letters 50 (1983) 1395. P.W. Anderson. Phys. Rev. B28 (1983) 2264. F.D.R. Haldane, Phys. Rev. Letters 51 (1983) 605. R. Tao and D.J. Thouless, Phys. Rev. B28 (1983) 1142. R. Tao, Phys. Rev. B29 (1984) 636.
Surface Science 142 (1984) 147-154 North-Holland, Amsterdam
THEORY OF THE QUANTIZED
HALL EFFECT
D.J. THOULESS Deportment of Physics FM - 15, Unioerslty of Washington, Seattle, Washington md Cauendtsh Laboratory, Madingley Road, Cambridge CB3 OHE, UK Received
6 July 1983; accepted
for publication
6 September
98195, USA
1983
A review is given of the current state of the theory of the quantum Hall effect. The integer values of the Hall conductance seen in systems with moderate disorder and predicted for systems with periodic modulation are fairly well understood. Charge density wave and Fermi liquid theories of the fractional quantization observed in systems with low disorder are briefly described.
1. Integer quantization The discovery by von Klitzing, Dorda and Pepper [l] that the Hall conductance of a two-dimensional electron system can be, with very high precision, an integer multiple of e2/h was a triumph of experimental physics. In most comparable cases, such as the quantization of flux in superconductivity or the quantization of circulation for superfluid helium, there have been previous theoretical suggestions of the existence of the effect, even if there were unexpected features in the experimental result. In this case there was no more than approximate quantization suggested [2], and so there was no reason for the experimentalists to examine the transverse voltage in their device with the precision which they used. Once the discovery had been made, we theorists rushed in to show why the result had been obvious all the time. There were a large number of papers, of which I list only a few [3-61, which used bulk perturbative arguments to show that the disorder in the substrate potential gave no correction to the simple quantized result which can be obtained from full Landau levels in an ideal two-dimensional system with no disorder. These arguments are sufficient to show that corrections due to disorder or electron-electron interactions vanish to all orders in perturbation theory. One such argument says that the Lorentz transformation allows us to move to a frame of reference moving with velocity -E/B, in which there is no electric field, and in this frame the substrate potential makes a local perturbation of the electron density, but does not carry 003%6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
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D.J. Thoultxs / Theory of quantized Hall effect
any electron charge with it when the Fermi energy lies in a gap because the perturbation of the electron density falls off exponentially with distance from the perturbing potential. In this frame the current is entirely carried by the positively charged substrate (in an n-type layer) moving with velocity E/B. These arguments also show that localized states, even if they are shifted from the unperturbed Landau level enough to pass through the Fermi surface, do not affect the quantization and do not change the value of the quantized current. A more profound approach to the problem was introduced by Laughlin [7]. He used gauge invariance to show that, for a system with the topology of an annulus, a change in the flux threading the annulus by an integer amount would lead to the transfer of an integer number of electrons from one edge to the other. Since the change in flux, by Faraday’s law, is given by the time integral of the EMF, and the transfer of charge is given by the time integral of the current, it follows that the ratio of current in one direction to voltage in the perpendicular direction is an integer multiple of e divided by the quantum of flux. This powerful argument has the great advantage that it can be applied in circumstances in which the perturbative arguments are not applicable, such as, for example, when a Landau level is split into sub-bands by a periodic potential; in this case, as I shall discuss in the next section, each sub-band must carry a multiple of the quantized current carried by the entire level. It has the disadvantage that it makes the Hall current appear to be an edge effect, and therefore possibly sensitive to boundary conditions, while the perturbative arguments make it clear that it is a bulk effect. One knows, however, that two-dimensional bulk effects and edge effects can be related by some version of Stokes’ theorem. All these arguments assume that the electric field is weak. In the perturbative arguments the response linear in the electric field is calculated, while in Laughlin’s argument it is assumed that the rate of change of the vector potential is slow enough that the electrons can follow it adiabatically. In a strong field we can be sure that the quantization is not exact, since there will be a tunnelling current parallel to the electric field due to electrons tunnelling from a full Landau level to an empty one, and this longitudinal current must in turn affect the transverse current. Such corrections must depend exponentially on the inverse of the field strength, so it is reasonable to ask if there are any corrections which are algebraic in the field strength. It seems likely, both from the precision of the experimental results and on theoretical grounds, that there are no such algebraic terms, but I do not know of any general proof of this result. It is generally thought that in the absence of a magnetic field all states in a two-dimensional disordered system are localized [8-lo]. The existence of the quantum Hall effect shows that this is not the case in the presence of a magnetic field, since some nonlocalized states are needed to carry the Hall
D.J. Thouless / Theory of quantized Hall effect
149
current. The question naturally arises whether in the presence of disorder the Landau level is broadened into a band whose tail states are localized but whose central region consists of extended states, as some numerical calculations have suggested [ll], or if all the current is carried at some singular energy, as a simple semiclassical argument suggests [12,13]. If the potential varies slowly over a magnetic length, the electrons should be confined to constant energy curves on the potential surface. For a random potential these curves will either be electron-like, surrounding a minimum, or hole-like, surrounding a maximum. It is only at the singular energy dividing the two that there can be an open orbit capable of varying a current. In the presence of an electric field this open orbit is broadened into a band of open orbits whose width depends on the field strength. In this model, which seems to give a fairly good account of the experimental situation, the steps between Hall plateaus, and the corresponding regions of nonzero longitudinal resistance, should get sharper and sharper as the electric field strength is reduced. The problem of the existence of extended states in the presence of a magnetic field has recently been discussed on the basis of the nonlinear sigma model [14], but the question of whether the extended states form a broad band or lie at a singular energy is still open.
2. Periodic potentials It has already been mentioned that a periodic modulation of the substrate breaks the degeneracy of the Landau level in the ideal system and breaks it up into a set of sub-bands separated by gaps. The number of sub-bands produced is p, where the number of flux quanta per unit cell is p/q, so the structure of the energy bands is highly sensitive to whether the number of flux quanta per unit cell is rational or irrational. This can be seen most clearly in the energy level diagrams calculated for p up to about 50 by Hofstadter [15] for a modulation with square symmetry, and by Claro and Wannier [16] for a modulation with hexagonal symmetry. The quantum number associated with the Hall current gives a useful way of classifying the energy gaps in this sort of diagram, and the study of periodic modulations sheds some light on the problem of a random substrate. There are not yet any experiments to which this theory is applicable, but it is likely that some experiments on two-dimensionally modulated layers will be done soon. Studies of this problem have been made by various authors [17-191. We found that from the Bloch wave function +t,+, the Hall conductance of a sub-band is given by the integral over the Brillouin zone
(2.1) This is a topological
invariant
which can be shown to be an integer
by using
D.J. Thouless / Theory of quantized Hull
150
effect
Stokes’s theorem to reduce it to an integral round the perimeter of the Brillouin zone, and then the periodicity of the Hamiltonian in the Brillouin zone can then be used to show that the integrand is just the derivative of a phase round the perimeter. A nonzero value of the Hall current implies that the wave function cannot be written as a continuous single valued function satisfying periodic boundary conditions in the unit cell, and therefore well-behaved Wannier functions do not exist for a sub-band with nonzero Hall current. We derived, with some difficulty, the general relation pt + qs =
(2.2)
r
for the Hall current 1 in the (r - 1)th gap; all these quantities is equivalent to Streda’s [18] expression te/h
= dn/d
B,
are integers.
This
(2.3)
where n is the electron density. For an incommensurate flux density, for which would have to be replaced by an irrational number, the equation equivalent to (2.2) has a unique solution, but for a rational flux density this equation obviously gives a value of t which is only determined modulo q. For the modulation with simple rectangular symmetry we argued that s should be as small as possible, and this determines the solution uniquely, and apparently correctly. In the case of a modulation with hexagonal symmetry, with its minima at points of sixfold symmetry, there is an additional restriction that s and t cannot both be odd. It turns out, however, that s does not always have the lowest possible magnitude, as can been seen from an inspection of the energy level diagram in Claro and Wannier [16]. The gap for p = 5, q = 1, r = 4 clearly corresponds to t = 0, s = 4, not to t = 1, s = - 1, for example. In an indirect fashion this model can shed some light on the idea that in a random potential the Hall current is all carried at one singular energy. A slowly varying potential is one in which p is much larger than q. For q = 1 it is certainly true that t is zero below the center of the Landau level and unity above the centre, for the square lattice, so that all the Hall current is carried by the central sub-band or pair of sub-bands. If q is larger than unity each of the sub-bands carries a nonzero current, but it is still true that the sub-bands fall into approximately p/q groups, with each group other than the central group carrying no total Hall current. The energy gaps within groups are very small compared with the energy gaps between groups. For more rapid modulations, so that p and q are of comparable magnitude, the Hall current varies in a rather exotic manner from gap to gap. It is generally true that the larger gaps correspond to smaller Hall currents, but there is no obvious sign that the current is carried in one particular part of the spectrum.
p/q
D.J. Thouless / Theory of quantized Hall effect 3.
151
Fractional quantization
The discovery of fractional quantization [20] served to undermine the complacency of theorists even further. Recent work seems to show clearly that the Hall conductance of high mobility inversion layers at very low temperatures is precisely l/3 [21], and can probably be other exact fractions, of e2/h [22]. So clean an experimental result demands a clear theoretical explanation, and promises to lead to a lot of new and unexpected physics, but, in my opinion, there is so far no theory which matches the elegance of the experimental results. There are a few things that are generally agreed. In the first place, since the integer Hall quantization is observed in lower mobility devices and the fractional effect is observed in very high mobility devices, and since integer quantization appears to be an almost inevitable consequence of a theory of noninteracting electrons in a weak substrate potential, it seems reasonable to suppose that fractional quantization is a result of the dominance of electron-electron interactions over the effects due to disorder. It is a natural extension of this observation that a model in which the degeneracy of the Landau levels is broken by the Coulomb interaction between electrons, but the substrate potential and transitions to other Landau levels are ignored, should be sufficient to show fractional quantization. It is also clear that any mechanism which makes rational fraction occupation of the Landau level energetically favorable will go a long way towards explaining fractional quantization. The argument from the Lorentz transformation shows that, provided the electrons are not locked to the substrate, a one third full Landau level will carry one third of a quantum of Hall current. Eq. (2.3) also shows that if the occupancy of the Landau level is what determines the energy, then it will also give an appropriate fractional Hall current. Two types of theories which are in accord with the principles outlined in the previous paragraph have been studied. There are models in which there is some sort of the charge density wave, and those in which there is no charge density wave. It is natural to expect a charge density wave in such a system, since at very high magnetic fields or very low electron densities the system behaves classically and forms a hexagonal lattice. Yoshioka and Fukuyama [23] and Yoshioka and Lee [24] are among the many authors who have studied this problem recently. There are two major difficulties with this approach, however. The first is that a charge density wave is likely to be pinned by substrate inhomogeneities, as it is in other systems, and should require some minimum electric field to free it from the substrate [25]. The second is that there does not seem to be any particular stability associated with simple fractional occupation of the Landau level, or, equivalently, with the commensurability of the electron lattice and the flux quantum. It can be seen why this is from the type of energy band diagram shown in Claro and Wannier [16]. If there is one electron per
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unit cell the Fermi energy lies in a gap that extends from the bottom left to the top right of the diagram, and it is easy to show that if the Fermi energy is in a gap the energy is an analytic function of the magnetic field, and so there can be no commensuration energy in a mean field theory. There are gaps that close at simple rational values of the flux density, but it is hard to see how these could lead to stable enough fractional occupations to explain the observed results, or how pinning could be avoided with such a mechanism. A rather different version of the charge density wave mechanism has been proposed by Haldane [26]. In this model the lattice is locked to commensurate values by the reduced energy of defects such as vacancies and interstitials which occurs when the lattice spacing is commensurate with the flux. The lattice is rigid, and free to carry the Hall current by sliding over the substrate, and it is only the defects that get pinned to the substrate. Yoshioka, Halperin and Lee [27] have carried out a calculation for up to six electrons in a single Landau level whose degeneracy is broken by the Coulomb interaction. They found some signs of the l/3 quantization even in such a small system, but no charge density wave. The theory of Laughlin [28] does not involve a charge density wave. In this theory the wave function for the lowest Landau level is written in the form (3.1) where z = x - iy and p is odd, so that the level has a l/p occupation. This gives a specially low energy for commensurate occupation, but has no variations in charge density to provide a pinning mechanism, so it should give a l/p Hall quantization. It is not immediately obvious how this can be generalized to give fractional occupation with denominator greater than unity, but Anderson [29] has shown how this might be done. The state described by eq. (3.1) has a broken symmetry, and there are p - 1 equivalent states that can be generated from this state by gauge transformations. States with fractional occupation q/p can be produced by taking the product of q of these equivalent states. Haldane [30] has set up the problem in spherical geometry to construct a generalization of Laughlin’s wave function for these cases with 4 > 1. Our own approach to this problem [31,32] has some features in common with the work of Laughlin [28], although our approach is quite different. We have tried to use conventional many-body techniques to deal with this problem in which the complete degeneracy of the partially occupied Landau level is broken by the Coulomb interaction. The difficulty in comparison with the more usual sort of theory is that the splitting between the energy levels is itself produced by the perturbation, so that is no variable parameter that gives the ratio of the perturbation to the unperturbed energy denominators. On the other hand there is a high degree of symmetry that can be exploited to simplify calculations. In our work so far we have used the Landau gauge, but the
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of quantized Hall gffect
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symmetric gauge could be used without it making any essential difference, and in that gauge it is easier to compare our results with Laughlin’s. Alternatively Laughlin’s wave function could be written in the Landau gauge as 2nr,/L. -
Lye
e2wL)P
exp - f:
i
$ .
I
(3.2)
We assume that there is some suitable unperturbed ground state in which some fraction of the one-particle states in the Landau level are occupied and the rest are unoccupied. We then use many-body techniques to calculate the self-energies of the various one-particle states. If the occupied states are chosen in a regular manner, so that, for example, one state is occupied and the next p - 1 states are unoccupied through the whole array of one-particle states allowed by the boundary conditions, we find an energy gap between the hole and the particle states which stabilizes the l/p occupation and an enhanced correlation energy. This, like Laughlin’s state, has a broken symmetry with p equivalent ground states. Unfortunately in the present state of the theory we find even denominators as well as the odd denominators found both by the experimentalists and by Laughlin. Tao [32] has generalized the work to cases in which the denominator is greater than unity. If the techniques of many-body theory can be adapted successfully to deal with this problem it will be much easier to calculate the effects of disorder and nonzero temperature.
Acknowledgements I am grateful to Dr. R. Tao for many useful discussions. I am also grateful to the many people who have sent me preprints of their work. This work was supported in part by the National Science Foundation under grant number DMR79-20795.
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