Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Topological Phase Transitions of Interacting Fermions in the Presence of a Commensurate Magnetic Flux (2403.04622v2)

Published 7 Mar 2024 in cond-mat.mes-hall and cond-mat.quant-gas

Abstract: Motivated by recently reported magnetic-field induced topological phases in ultracold atoms and correlated Moir\'e materials, we investigate topological phase transitions in a minimal model consisting of interacting spinless fermions described by the Hofstadter model on a square lattice. For interacting lattice Hamiltonians in the presence of a commensurate magnetic flux it has been demonstrated that the quantized Hall conductivity is constrained by a Lieb-Schultz-Mattis (LSM)-type theorem due to magnetic translation symmetry. In this work, we revisit the validity of the theorem for such models and establish that a topological phase transition from a topological to a trivial insulating phase can be realized but must be accompanied by spontaneous magnetic translation symmetry breaking caused by charge ordering of the spinless fermions. To support our findings, the topological phase diagram for varying interaction strength is mapped out numerically with exact diagonalization for different flux quantum ratios and band fillings using symmetry indicators. We discuss our results in the context of the LSM-type theorem.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (49)
  1. F. D. M. Haldane, Model for a quantum hall effect without landau levels: Condensed-matter realization of the “parity anomaly”, Physical Review Letters 61, 2015–2018 (1988).
  2. C. L. Kane and E. J. Mele, Quantum spin hall effect in graphene, Physical Review Letters 95, 10.1103/physrevlett.95.226801 (2005).
  3. K. v. Klitzing, G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance, Physical Review Letters 45, 494 (1980).
  4. R. B. Laughlin, Quantized hall conductivity in two dimensions, Physical Review B 23, 5632 (1981).
  5. I. Dana, Y. Avron, and J. Zak, Quantised hall conductance in a perfect crystal, Journal of Physics C: Solid State Physics 18, L679 (1985).
  6. E. Brown, Bloch electrons in a uniform magnetic field, Physical Review 133, A1038 (1964).
  7. J. Zak, Magnetic translation group, Physical Review 134, A1602 (1964).
  8. D. R. Hofstadter, Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields, Physical Review B 14, 2239 (1976).
  9. G. H. Wannier, A result not dependent on rationality for bloch electrons in a magnetic field, physica status solidi (b) 88, 757–765 (1978).
  10. Y. Fang and J. Cano, Symmetry indicators in commensurate magnetic flux, Physical Review B 107, 10.1103/physrevb.107.245108 (2023).
  11. Y. Guan, A. Bouhon, and O. V. Yazyev, Landau levels of the euler class topology, Physical Review Research 4, 10.1103/physrevresearch.4.023188 (2022).
  12. N. Goldman, J. C. Budich, and P. Zoller, Topological quantum matter with ultracold gases in optical lattices, Nature Physics 12, 639–645 (2016).
  13. Q. Niu, D. J. Thouless, and Y.-S. Wu, Quantized hall conductance as a topological invariant, Physical Review B 31, 3372 (1985).
  14. E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Annals of Physics 16, 407–466 (1961).
  15. M. B. Hastings, Lieb-schultz-mattis in higher dimensions, Physical Review B 69, 10.1103/physrevb.69.104431 (2004).
  16. M. Oshikawa, Topological approach to luttinger’s theorem and the fermi surface of a kondo lattice, Physical Review Letters 84, 3370–3373 (2000b).
  17. Y.-M. Lu, Y. Ran, and M. Oshikawa, Filling-enforced constraint on the quantized hall conductivity on a periodic lattice, Annals of Physics 413, 168060 (2020).
  18. J. E. Avron and L. G. Yaffe, Diophantine equation for the hall conductance of interacting electrons on a torus, Physical Review Letters 56, 2084 (1986).
  19. A. Kol and N. Read, Fractional quantum hall effect in a periodic potential, Physical Review B 48, 8890–8898 (1993).
  20. J. M. Luttinger, The effect of a magnetic field on electrons in a periodic potential, Physical Review 84, 814 (1951).
  21. D. H. Kobe, Gauge invariance in second quantization: Applications to hartree-fock and generalized random-phase approximations, Physical Review A 19, 1876 (1979).
  22. A. Fünfhaus, T. Kopp, and E. Lettl, Winding vectors of topological defects: multiband chern numbers, Journal of Physics A: Mathematical and Theoretical 55, 405202 (2022).
  23. B. A. Bernevig and N. Regnault, Emergent many-body translational symmetries of abelian and non-abelian fractionally filled topological insulators, Physical Review B 85, 10.1103/physrevb.85.075128 (2012).
  24. N. B. Backhouse and C. Bradley, Projective representations of abelian groups, Proceedings of the American Mathematical Society 36, 260 (1972).
  25. N. B. BACKHOUSE and C. J. BRADLEY, PROJECTIVE REPRESENTATIONS OF SPACE GROUPS, i: TRANSLATION GROUPS, The Quarterly Journal of Mathematics 21, 203 (1970).
  26. J. Yang and Z.-X. Liu, Irreducible projective representations and their physical applications, Journal of Physics A: Mathematical and Theoretical 51, 025207 (2017).
  27. P. Goddard and D. I. Olive, Magnetic monopoles in gauge field theories, Reports on Progress in Physics 41, 1357–1437 (1978).
  28. M. Kupczyński, B. Jaworowski, and A. Wójs, Interaction-driven transition between the wigner crystal and the fractional chern insulator in topological flat bands, Physical Review B 104, 10.1103/physrevb.104.085107 (2021).
  29. H. Pan, F. Wu, and S. D. Sarma, Quantum phase diagram of a moiré-hubbard model, Physical Review B 102, 10.1103/physrevb.102.201104 (2020).
  30. Y. Hatsugai and M. Kohmoto, Energy spectrum and the quantum hall effect on the square lattice with next-nearest-neighbor hopping, Physical Review B 42, 8282 (1990).
  31. A. Mishra, S. R. Hassan, and R. Shankar, Effects of interaction in the hofstadter regime of the honeycomb lattice, Physical Review B 93, 10.1103/physrevb.93.125134 (2016).
  32. N. Goldenfeld, Lectures on phase transitions and the renormalization group (CRC Press, 2018).
  33. In contrast, a long-range structure of period q𝑞qitalic_q could be provided for example by a stripe pattern of impurities that would only break translation symmetry in one direction and would in fact result in a nondegenerate and topologically trivial ground state if the stripe pattern is commensurable with the given cluster.
  34. This does not rule out the possibility of a mean-field ansatz in terms of an effective quasiparticle Hamiltonian [75].
  35. X. G. Wen and Q. Niu, Ground-state degeneracy of the fractional quantum hall states in the presence of a random potential and on high-genus riemann surfaces, Physical Review B 41, 9377–9396 (1990).
  36. F. D. M. Haldane, Many-particle translational symmetries of two-dimensional electrons at rational landau-level filling, Physical Review Letters 55, 2095–2098 (1985).
  37. A. Paramekanti and A. Vishwanath, Extending luttinger’s theorem to z2subscript𝑧2z_{2}italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT fractionalized phases of matter, Physical Review B 70, 10.1103/physrevb.70.245118 (2004).
  38. Z. Tešanović, F. Axel, and B. I. Halperin, ‘“hall crystal”’ versus wigner crystal, Physical Review B 39, 8525–8551 (1989).
  39. R. Côté and H. A. Fertig, Collective modes of quantum hall stripes, Physical Review B 62, 1993–2007 (2000).
  40. M. M. Fogler, A. A. Koulakov, and B. I. Shklovskii, Ground state of a two-dimensional electron liquid in a weak magnetic field, Physical Review B 54, 1853–1871 (1996).
  41. T. Banks and B. Zhang, On the low density regime of homogeneous electron gas, Annals of Physics 412, 168019 (2020).
  42. B. I. Halperin, On the hohenberg–mermin–wagner theorem and its limitations, Journal of Statistical Physics 175, 521–529 (2018).
  43. S. A. Trugman and S. Kivelson, Exact results for the fractional quantum hall effect with general interactions, Physical Review B 31, 5280–5284 (1985).
  44. R. Narevich, G. Murthy, and H. A. Fertig, Hamiltonian theory of the composite-fermion wigner crystal, Physical Review B 64, 10.1103/physrevb.64.245326 (2001).
  45. H. Yi and H. A. Fertig, Laughlin-jastrow-correlated wigner crystal in a strong magnetic field, Physical Review B 58, 4019–4027 (1998).
  46. X. Zhu, P. B. Littlewood, and A. J. Millis, Sliding motion of a two-dimensional wigner crystal in a strong magnetic field, Physical Review B 50, 4600–4621 (1994).
  47. J. Wu, T.-L. Ho, and Y.-M. Lu, Symmetry-enforced quantum spin hall insulators in π𝜋\piitalic_π-flux models (2017).
  48. S. Jiang, D. J. Scalapino, and S. R. White, Ground-state phase diagram of the t−t′−j𝑡superscript𝑡′𝑗t-t^{\prime}-jitalic_t - italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_j model, Proceedings of the National Academy of Sciences 118, 10.1073/pnas.2109978118 (2021).
  49. M. S. Sánchez and T. Stauber, Correlated phases and topological phase transition in twisted bilayer graphene at one quantum of magnetic flux (2024).

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now