Theoretical Aspects of the Quantum Hall Effect S

Proc. Nati. Acad. Sci. USA Vol. 84, pp. 4698-4700, July 1987 Symposium Paper

This paper was presented at a symposium "Interfaces and Thin Films," organized by John Armstrong, Dean E. Eastman, and George M. Whitesides, held March 23 and 24, 1987, at the National Academy of Sciences, Washington, D.C. Theoretical aspects of the quantum Hall effect S. M. GIRVIN Surface Science Division, National Bureau of Standards, Gaithersburg, MD 20899 This talk will focus on the fractional quantum Hall effect, Here z x + iy is a complex number representing the posi- which is a remarkable many-body phenomenon occurring in tion of a particle in the plane [in units of the magnetic length I the two-dimensional electron gas at high magnetic fields and (Ic/eB)½], and m is an odd integer (to satisfy the Pauli low temperatures. The Hall conductance of a real, macro- principle). The energy is very nearly optimized because the scopic device is quantized in the form: a- = ve2/h, where v zeros of the polynomial factor in T are attached to the parti- is a rational fractional quantum number. Associated with cles (9) so that T vanishes rapidly if two particles approach this are vortex-like excitations that have fractional charge each other, thereby reducing the potential energy. Laughlin and other bizarre features. There are deep connections be- also proposed variational quasi-electron and quasi-hole tween this phenomenon and superfluidity and analogies with states that have a finite excitation gap above the ground state models of current interest in high-energy physics. The es- and demonstrated that these quasi-particles carry fractional sence of the effect is that electrons in a magnetic field can charge e* +ve.P turn into bosons by attaching themselves to flux tubes. It turns out that there are deep connections between this The discovery of the quantum Hall effect (for a recent re- problem of fermions in a magnetic field and superfluidity in view, see ref. 1) has revolutionized our understanding of boson systems (10-13). The first hint that this is so comes transport in disordered systems in high magnetic fields in from the fact that application of Feynman's theory of the two dimensions and has important implications for other ar- collective excitations in 4He yields excellent results in this eas of physics as well as metrology. The integer quantum problem as well (10-12). Feynman argues on quite general Hall effect is a one-body phenomenon associated with the grounds (14) that, because of Bose statistics, the only low- gap between states of different kinetic energy (Landau lev- lying collective excitations are density waves that are well- els) in a magnetic field. The basic experimental observation described by: is that under special conditions the excitation gap causes the system to become dissipationless, and the Hall conductance Tk = Pk(F; Pk Z e1 rj [3] takes on a universal value: j where 1 is the ground state and pk is the Fourier transform of e2 the density. This leads to a simple expession for the excita- any= Ve [1] tion energy: h2)2 1 t where v is an integer quantum number. It turns out that dis- A(k) = -i--; s(k) =((DIpk PkI(D½, [4] order and Anderson localization of the wave functions are of paramount importance to this effect and that the Hall con- where s(k) is the static structure factor of the ground state. ductance is quantized because it can be expressed as a topo- This result can be shown to be equivalent to the assumption logical invariant (a winding number) (2, 3). This beautiful that all of the available oscillator strength is absorbed by a idea also makes its appearance in a field-theoretic descrip- single mode. This works well in Bose systems but normally tion of localization in a magnetic field (4), which turns out to fails in Fermi systems because of the continuum of single- yield a "6-vacuum" model of the sort of current interest to particle excitations. Here, however, the magnetic field field theorists. comes to the rescue and destroys the Fermi surface by The fractional quantum Hall effect (FQHE) is a rather dif- quenching the kinetic energy into discrete Landau levels. ferent and even more surprising phenomenon (5, 6). In low- Projection of Pk in Eq. 3 onto the lowest Landau level yields disorder samples at still higher magnetic fields, one finds an excellent description of the collective mode, which agrees quantization of ocry according to Eq. 1 but with the quantum well (see Fig. 1) with first-principle numerical calculations number v being a rational fraction. The lowest Landau level (15-19). Notice that like helium there is a "roton minimum" is only fractionally filled, and the excitation gap is due not to at a finite wave vector, but unlike helium the Goldstone the kinetic energy but rather to collective correlations arising mode at a small wave vector has acquired a large' gap from the Coulomb interaction. ("mass") associated with the incompressibility of the sys- The unusual nature of the correlations that can occur in a tem. Experimental attempts are underway to directly mea- partially filled Landau level was first delineated in a seminal sure the collective-mode dispersion, but it will be very diffi- paper by Laughlin (7, 8) who developed a remarkably accu- cult for a number of technical reasons. rate variational wave function to describe the ground state at A further helium analogy is that Laughlin's quasi-particles filling factor v = 1/m: are precisely quantized vortices (9, 12). To see this, consider the state describing a quasi-hole at the origin (7, 8): 'Im(Zl, *- -* ZN) = 17 (z-Zi)mexp( ! EIZkj [2] i

4698 Downloaded by guest on October 2, 2021 Symposium Paper: Girvin Proc. NatL. Acad. Sci. USA 84 (1987) 4699

' bous in two dimensions (21-25) and that fermions are equivalent to bosons in a singular gauge in which a fake-gauge flux tube is attached to each particle. The singular-gauge density matrix for Laughlin's ground state exhibits power-law ODLRO: N- 0.10 cmJ a) P(Z, Z') _ IZ - Z'l-m/2 [8] and represents in a certain sense (20) a "condensation" of composite objects consisting of electrons and gauge flux tubes analogous to the phenomenon of "oblique confine- ment" (26, 27). More precisely, power-law behavior in p represents the binding of the zeros of the wave function to the particles, which was discussed above. Within the single- mode approximation (10-12), ODLRO can be shown to be a ki necessary and sufficient condition for the existence' of an FIG. 1. Comparison of predicted collective-mode energy (solid excitation gap and hence the FQHE. We can now justify the lines) with exact small-system numerical results. x, (N = 7, v = 1/3) Landau-Ginsburg theory discussed above by identifying the spherical system; A, (N = 6, v = 1/3) hexagonal unit cell (15, 16); *, order parameter if as the coherent state ofthe composite par- (N = 9, v = 1/3) and (N = 7, v = 1/5) spherical system calculations ticles (since they are bosons), and Eq. 7 follows from the fact of Fano et al. (17). Arrows at the top show the magnitude of the that these composite particles carry flux tubes. Recent work reciprocal lattice vector of the Wigner crystal at the indicated filling by Read (28) addresses this issue. One can also view the factors. fake-gauge field discussed here as arising from "Berry phase" effects associated with adiabatic transport of parti- Notice that if any one of the particles circles the origin, the cles or quasi-particles around closed loops (refs. 29 and 30; phase of NPqh winds up by an additional 2'ir relative to the R. B. Laughlin, unpublished data). Finally, it should be not- ground state. This is precisely what defines the vortex state ed that there is a nice connection between this picture and in a Bose system (14). The phase gradient leads to current the theory of Kivelson et al. (31), which focuses on the exis- circulating around the vortex center, which in helium films tence of infinite ring exchanges. This connection follows produces a logarithmically divergent total energy. In the from the fact that in the path-integral representation of the FQHE the vortex energy is finite. This can be viewed as partition function for helium, ODLRO appears in the form of being due to screening of the current by the fractional charge infinite ring exchanges (14). that accumulates on the vortex (12, 13). The nondivergent In summary, the quantum Hall effect is a remarkable and energy of the vortices precludes the possibility of a Koster- surprising phenomenon that is rich in connections with other litz-Thouless transition at finite temperature and explains areas of physics. the finite though exponentially small dissipation at low temperatures. The effects just described are very reminiscent of the An- 1. Prange, R. E. & Girvin, S. M., eds. (1987) The Quantum Hall derson-Higgs mechanism in superconductors in which the Effect (Springer, New York). coupling of the ord&r parameter to the vector potential leads 2. Laughlin, R. B. (1981) Phys. Rev. B 23, 5632-5633. 3. Thouless, D. J. (1987) in The Quantum Hall Effect, eds. to a collective-mode gap (the plasmon energy) and to quan- Prange, R. E. & Girvin, S. M. (Springer, New York), pp. 101- tized fluxoids with finite energy due to screening of the cir- 116. culating current by the trapped flux. By using these ideas, a 4. Pruisken, A. M. M. (1987) in The Quantum Hall Effect, eds. very simple phenomenological Landau-Ginsburg theory has Prange, R. E. & Girvin, S. M. (Springer, New York), pp. 117- been developed (13) for the FQHE. The action is: 173. 5. Tsui, D. C., Stormer, H. L. & Gossard, A. C. (1982) Phys. Rev. Lett. 48, 1559-1562. S = fd2rl(-lV + A)0l2 [6] 6. Chang, A. M. (1987) in The Quantum Hall Effect, eds. Prange, R. E. & Girvin, S. M. (Springer, New York), pp. 175-232. 7. Laughlin, R. B. (1983) Phys. Rev. Lett. 50, 1395-1398. with 8. Laughlin, R. B. (1987) in The Quantum Hall Effect, eds. Prange, R. E. & Girvin, S. M. (Springer, New York), pp. 233- 301. - 1 i2V x A = [7] 9. Halperin, B. I. (1983) Helv. Phys. Acta 56, 75-102. 10. Girvin, S. M., MacDonald, A. H. & Platzman, P. M. (1986) Phys. Rev. Lett. 54, 581-583. where the "vacuum angle" 2irz' in Eq. 7 relates the flux 11. Girvin, S. M., Mact;onald, A. H. & Platzman, P. M. (1986) density to the local charge density. Here A is not the true Phys. Rev. B 33, 2481-2494. vector potential (whose fluctuations are negligible) but rath- 12. Girvin, S. M. (1987) in The Quantum Hall Effect, eds. Prange, er a fake gauge field representing the "frustration" associat- R. E. & Girvin, S. M. (Springer, New York), pp. 353-380. ed with density fluctuations (13). The action in Eq. 6 exhibits 13. Girvin, S. M. (1987) in The Quantum Hall Effect, eds. Prange, the usual flux quantization familiar from superconductivity R. E. & Girvin, S. M. (Springer, New York), pp. 381-399. but which from Eq. 7 implies that each vortex has finite en- 14. Feynman, R. P. (1972) Statistical Mechanics (Benjamin, Read- ergy and carries fractional charge e* = Pey, where y is the ing, MA), pp. 312-350.- 15. Haldane, F. D. M. & Rezayi, E. H. (1985) Phys. Rev. Lett. 54, winding humber of the phase around the vortex. 237-240. It is remarkable that this picture, which assumes that the 16. Haldane, F. D. M. & Rezayi, E. H. (1985) Phys. Rev. B 31, electrons are behaving like bosons, works so well. The rea- 2529-2532. sons for the success of this approach have been clarified re- 17. Fano, G., Ortolani, F. & Colombo, E. (1986) Phys. Rev. B 34 cently by the demonstration of the existence of a peculiar 2670-2680. type of off-diagonal long-range order (ODLRO) in this sys- 18. Zhang, F. C., Vulovic, V. Z., Guo, Y. & Das Sarma, S. (1985) tem (20). It is known that the statistics ofparticles are ambig- Phys. Rev. B 32, 6920-6923. Downloaded by guest on October 2, 2021 4700 Symposium Paper: Girvin Proc. Natl. Acad. Sci. USA 84 (1987)

19. Haldane, F. D. M. (1987) in The Quantum Hall Effect, eds. 25. Halperin, B. I. (1984) Phys. Rev. Lett. 52, 1583-1586. Prange, R. E. & Girvin, S. M. (Springer, New York), pp. 303- 26. Cardy, J. L. & Rabinovici, E. (1982) Nucl. Phys. B 205, 1-16. 352. 27. Cardy, J. L. (1982) Nucl. Phys. B 205, 17-26. 20. Girvin, S. M. & MacDonald, A. H. (1987) Phys. Rev. Lett. 58, 28. Read, N. (1987) Bull. Am. Phys. Soc. 32, 923. 1252-1255. 29. Arovas, D., Schrieffer, J. R. & Wilczek, F. (1984) Phys. Rev. 21. Wilczek, F. (1982) Phys. Rev. Lett. 49, 957-962. Lett. 53, 722-723. 22. Arovas, D. P., Schrieffer, J. R., Wilczek, F. & Zee, A. (1985) 30. Semenoff, G. & Sodano, P. (1986) Phys. Rev. Lett. 57, 1195- Nucl. Phys. B 251, 117-126. 1198. 23. Wu, Y. S. (1983) Phys. Rev. Lett. 52, 2103-2106. 31. Kivelson, S., Kallin, C., Arovas, D. P. & Schrieffer, J. R. 24. Wu, Y. S. (1984) Phys. Rev. Lett. 53, 111-114. (1986) Phys. Rev. Lett. 56, 873-876. Downloaded by guest on October 2, 2021