Collaborators

1. Study of Integer and Fractional Quantum Hall Effects using Topological Quantum Numbers* *Supported by NSF-MRSEC at Princeton University Ravin Bhatt, Princeton University Collaborators Yan Huo Princeton University Xin Wan Zhejiang Inst.of Modern Physics Duncan Haldane Princeton University Ed Rezayi California State University D.N. Sheng California State University Kun Yang NHMFL,Florida State Univ.

2. Outline • Introduction: Integer quantum Hall effect (IQHE) • Topological quantum number • Critical behavior of integer quantum Hall transition • Fractional quantum Hall effect (FQHE) and the difficulties of a topological description • Topological quantum numbers in FQHE • Topological ground state degeneracies • Effects of disorder: Mobility gap • Effects of layer thickness and correlated potential

3. Integer Quantum Hall Effect I RH (h/e2) VH B R (a.u.) VL von Klitzing, Dorda, Pepper (80) classical B (T) Quantum numbers !?

4. A Tale of Two Quantum Numbers • Quantum numbers related to symmetry • Example: angular momentum  rotational symmetry • Degeneracy from the algebra of the group generators • Structure destroyed by symmetry breaking • Quantum numbers determined by topology • Examples: • Quantized circulation in superfluid He • Flux quantization in superconductors • Related to the winding number of a condensate wave function • Survive relatively strong perturbation • Which category does quantized Hall conductance fit?

5. Laughlin’s Gedanken Experiment "By gauge invariance, adding Φ0 maps the system back to itself, … [which results in] the transfer of n electrons.“ – R. B. Laughlin Why the average transferred charge has to be an integral multiple of e?

6. Edge States - Halperin (1982) B EF Chiral, gapless edge excitation

20. Hall Conductance as Curvature • Hall conductance: Kubo formula derived from the linear-response theory: • Geometric interpretation of Hall conductance: local curvature in the boundary condition space or the flux space

21. Geometry and Topology • Hall conductance averaged over a torus of boundary conditions  a quantized integral (Thouless et al.) • Berry’s phase around the magnetic Brillouin zone • First Chern class of a principal U(1) fiber bundle over a torus  Chern number (named after Chern Shiing-shen) • Topological: small perturbation  no change in Chern number • Transitions between Chern numbers: level crossing (curvature diverges)

22. Significance of Chern Numbers • Chern numbers, being topological invariant, characterize the topological properties of wave functions. (Thouless et al., ’84) • Topological properties of the eigenstates can be used to distinguish between localized and extended states. (Arovas et al., ’88) • First Chern number: sensitivity of nodes of the wave function to changes in boundary conditions • Current carrying state can vanish at any position in real space by a proper choice of boundary conditions

23. Plateau Transitions - Critical behavior at strong B?

24. Finite-Size Scaling DE: width of extended states DE DE

25. Fractional Quantum Hall Effect • Discovered by Tsui, Stormer, Gossard (82) • Laughlin sequence • Jain sequence

26. How to understand the fractional quantum Hall conductance in terms of topological (integral) quantum numbers?

27. Disorder: Roles from IQHE to FQHE • How to define mobility gap and mobility edge? • Can we distinguish insulating and current carrying states? • Are mobility gap and spectral gap the same? • How to understand topological nature of FQHE? • A clean system has q-fold degeneracy for n = p/q on a torus; • Does weak disorder lift degeneracy? • How to describe transitions from FQHE to insulator? • FQHE is destroyed by strong disorder in a dirty sample; • What is the critical disorder? • Comparison with experiments quantitatively Sheng, XW, Rezayi, Yang, Bhatt & Haldane, PRL 90, 256802 (2003) XW, Sheng, Rezayi, Yang, Bhatt & Haldane, PRB 72, 075325 (2005)

28. Mobility Dependence of Activation Gap • Thermally excited quasiparticles • Resistivity ~ density of QP Arrhenius type G. S. Boebinger et al., PRB 36, 7919 (1987)

29. Model • Projected into lowest Landau level (LLL) • Generalized periodic boundary conditions (torus) • Gaussian white noise -- short-range scatterers • Up to 8 electrons, n = 1/3 • Hilbert space of size 735,471 • Exact diagonalization with Lanczos algorithm

30. GS Degeneracy in Clean Sample • Disorder-free: symmetry of magnetic translations • (By large gauge transformations) generalized to cases with random potential, spatial-dependent magnetic field, and other perturbations, as long as all quasiparticle excitations have finite-energy gaps – in the thermodynamic limit

31. Evolution of Ground State Manifold Thermodynamic limit L, with disorder Finite system, with disorder Finite system, clean Degeneracy preserved in the thermodynamic limit (Wen & Niu, 90)

32. Em Es Chern Numbers and Mobility Gap • Ground state manifold: • Shared by 3 states, Hall conductance 1/3, FQHE! • Topological – preserved when gap exists • Mobility edge: Fluctuation of Chern numbers mobility gap Em  spectral gap ES

33. Disorder dependence of Em • Mobility m vs. disorder • Dominated by short-range scatterers • Born approximation • Empirical m ~ nb, b ~ 1.5 • Blue dot: creation energy for quasiparticle-quasihole pair at large separation • Green curve in (b): converted from experimental data

34. Willett et al., 1988 Comparison with Experiments • Finite layer thickness • Landau level mixing • Correlated potential Boebinger et al., 1987

35. Effects of Layer Thickness • Fang-Howard wave function • softens the bare Coulomb interaction between electrons by • so that Wan, Sheng, Rezayi, Yang, Bhatt & Haldane, PRB 72, 075325 (2005); Virt. J. Nano. Sci. Tech. 12, Issue 8 (2005).

36. Layer Thickness Effect on E0 • Ground state energy decreases with increasing layer thickness • Quantitatively agrees with the results found by MacDonald & Aers (‘84), and Chakraborty (’86) • e-e interaction softened.

37. Layer Thickness Effect on Es • Spectral gap Es (extrapolated to N) decreases with increasing layer thickness (as well as with increasing disorder).

38. Layer Thickness Effect on Em • Mobility gap differs from spectral gap in magnitude • The two gaps disappear at about the same disorder strength

39. Effects of Correlated Potential • Introduce Gaussian correlation • Equivalently, • Mobility of the 2DEG enhanced (Born approximation) • I0: the zeroth-order modified Bessel function of the first kind • L0:the zeroth-order modified Struve function

40. Effects of Correlated Potential on Em • Effect of the range of potential absorbed into sample mobility; comparisons between samples meaningful.

41. Breakdown of T-orders: FQHE-I-T • FQHE unstable for large disorder W • Need clean samples, high mobility to observe FQHE • Mobility gap Em to close • Near WC, 0C (~sH) drops rapidly with increasing W • Peak in dsH/dW  WC • Em survives deviation of n from 1/3  plateau

42. Other Theoretical Development • Bilayer quantum Hall effect Sheng et al., Phys. Rev. Lett. 91, 116802 (2003) • Spin quantum Hall effect Cui, XW & Yang, Phys. Rev. B 70, 094506 (2004)

43. Experimental Data Then and Now Boebinger, 1987 Pan, unpublished

44. Summary • Calculation of topological quantum number is a powerful method to study of quantum Hall effects (& related effects). • Integer quantum Hall effect and related problems: • Critical behavior of plateau transitions at strong B, and random B • Understanding phase transitions in electrons with periodic and random potential (Hofstadter problem with disorder) • Fractional quantum Hall effect: • Ground state manifold topologically degenerate,characterized by topological quantum numbers(both robust against disorder) fractional plateaus • Mobility gap and spectral gap: quantitatively different • Excellent agreement with experimentally measured activation gap: effects of disorder, thickness of 2DEG, and range of potential.