Anomalous Landau levels and quantum oscillation in rotation-invariant insulators
Source abstract
Abstract
Landau levels in certain models are known to protrude into the zero-field energy gap. These are known as anomalous Landau levels (ALLs). We study whether ALLs can lead to Fermi-surface like quantum oscillation in the absence of a zero-field Fermi surface. Focusing on two-dimensional multi-band low-energy models of electrons with continuous rotation symmetry, we show that an effective-band description, akin to the semiclassical treatment of Landau level problems in metals, can be used to predict the Landau level spectrum, including possible ALLs. This description describes ALL induced quantum oscillation for certain insulating models, which we demonstrate through numerical calculations. We demonstrate that these results could guide the search for ALL in 2D topological materials with effective Hamiltonian similar to that of magnetically doped (Bi, Sb)2Te3 ultrathin films.
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OpenAlex topics: Topological Materials and Phenomena · Quantum and electron transport phenomena · Quantum Mechanics and Non-Hermitian Physics
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (100+)
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1. Shoenberg, D. Magnetic Oscillations in Metals. Cambridge Monographs on Physics (Cambridge University Press, 1984).10.1017/CBO9780511897870 ↗
2. Landau, L. Diamagnetismus der Metalle. Z. Phys. 64, 629 (1930).10.1007/BF01397213 ↗
3. Onsager, L. Interpretation of the de Haas-van Alphen effect. Lond. Edinburgh Dublin Philos. Mag. J. Sci. 43, 1006–1008 (1952).10.1080/14786440908521019 ↗
4. Lifshitz, I. M. & Kosevich, L. M. On the theory of the Shubnikov-de Haas effect. Sov. Phys. JETP 6, 67 (1958).
5. Hartnoll, S. A. & Hofman, D. M. Generalized Lifshitz-Kosevich scaling at quantum criticality from the holographic correspondence. Phys. Rev. B 81, 155125 (2010).10.1103/PhysRevB.81.155125 ↗
6. Alexandradinata, A. & Glazman, L. Semiclassical theory of Landau levels and magnetic breakdown in topological metals. Phys. Rev. B 97, 144422 (2018).10.1103/PhysRevB.97.144422 ↗
7. Zhang, W. et al. Tunable non-Lifshitz-Kosevich temperature dependence of Shubnikov-de Haas oscillation amplitudes in SMSB. npj Quantum Mater. 8, 55 (2023).10.1038/s41535-023-00588-1 ↗
8. Knolle, J. & Cooper, N. R. Quantum oscillations without a Fermi surface and the anomalous de Haas–van Alphen effect. Phys. Rev. Lett. 115, 146401 (2015).10.1103/PhysRevLett.115.146401 ↗
9. Wu, S. The detection of unconventional quantum oscillations in insulating 2d materials. 2D Mater. 11, 033004 (2024).10.1088/2053-1583/ad4b35 ↗
10. Tan, B. S. et al. Unconventional Fermi surface in an insulating state. Science 349, 287–290 (2015).10.1126/science.aaa7974 ↗
11. Li, G. et al. Two-dimensional Fermi surfaces in Kondo insulator SmB6. Science 346, 1208–1212 (2014).10.1126/science.1250366 ↗
12. Thomas, S. M. et al. Quantum oscillations in flux-grown SmB6 with embedded aluminum. Phys. Rev. Lett. 122, 166401 (2019).10.1103/PhysRevLett.122.166401 ↗
13. Xiang, Z. et al. Quantum oscillations of electrical resistivity in an insulator. Science 362, 65–69 (2018).10.1126/science.aap9607 ↗
14. Mizzi, C. A. et al. The reverse quantum limit and its implications for unconventional quantum oscillations in YbB12. Nat. Commun. 15, 1607 (2024).10.1038/s41467-024-45801-2 ↗
15. Sato, Y. et al. Unconventional thermal metallic state of charge-neutral fermions in an insulator. Nat. Phys. 15, 954 (2019).10.1038/s41567-019-0552-2 ↗
16. Wang, H. et al. Discovery of log-periodic oscillations in ultraquantum topological materials. Sci. Adv. 4, 5096 (2018).10.1126/sciadv.aau5096 ↗
17. Xiao, D., Liu, C.-X., Samarth, N. & Hu, L.-H. Anomalous quantum oscillations of interacting electron-hole gases in inverted type-ii InAs/GaSb quantum wells. Phys. Rev. Lett. 122, 186802 (2019).10.1103/PhysRevLett.122.186802 ↗
18. Han, Z., Li, T., Zhang, L., Sullivan, G. & Du, R.-R. Anomalous conductance oscillations in the hybridization gap of InAs/GaSb quantum wells. Phys. Rev. Lett. 123, 126803 (2019).10.1103/PhysRevLett.123.126803 ↗
19. Czajka, P. et al. Oscillations of the thermal conductivity in the spin-liquid state of α-RuCl3. Nat. Phys. 17, 915 (2021).10.1038/s41567-021-01243-x ↗
20. Leeb, V. et al. Anomalous quantum oscillations in a heterostructure of graphene on a proximate quantum spin liquid. Phys. Rev. Lett. 126, 097201 (2021).10.1103/PhysRevLett.126.097201 ↗
21. Lee, P. A. Quantum oscillations in the activated conductivity in excitonic insulators: possible application to monolayer WTe2. Phys. Rev. B 103, 041101 (2021).10.1103/PhysRevB.103.L041101 ↗
22. He, W.-Y. & Lee, P. A. Quantum oscillation of thermally activated conductivity in a monolayer WTe2-like excitonic insulator. Phys. Rev. B 104, 041110 (2021).10.1103/PhysRevB.104.L041110 ↗
23. Erten, O., Ghaemi, P. & Coleman, P. Kondo breakdown and quantum oscillations in SmB6. Phys. Rev. Lett. 116, 046403 (2016).10.1103/PhysRevLett.116.046403 ↗
24. Sodemann, I., Chowdhury, D. & Senthil, T. Quantum oscillations in insulators with neutral fermi surfaces. Phys. Rev. B 97, 045152 (2018).10.1103/PhysRevB.97.045152 ↗
25. Shen, H. & Fu, L. Quantum oscillation from in-gap states and a non-Hermitian Landau level problem. Phys. Rev. Lett. 121, 026403 (2018).10.1103/PhysRevLett.121.026403 ↗
26. Xiao, Y., Pelletier, V., Chaikin, P. M. & Huse, D. A. Landau levels in the case of two degenerate coupled bands: Kagomé lattice tight-binding spectrum. Phys. Rev. B 67, 104505 (2003).10.1103/PhysRevB.67.104505 ↗
27. Rhim, J.-W., Kim, K. & Yang, B. Quantum distance and anomalous Landau levels of flat bands. Nature 584, 59–63 (2020).10.1038/s41586-020-2540-1 ↗
28. Lian, B., Xie, F. & Bernevig, B. A. Landau level of fragile topology. Phys. Rev. B 102, 041402 (2020).10.1103/PhysRevB.102.041402 ↗
29. Devakul, T., Kwan, Y. H., Sondhi, S. L. & Parameswaran, S. A. Quantum oscillations in the zeroth Landau level: Serpentine Landau fan and the chiral anomaly. Phys. Rev. Lett. 127, 116602 (2021).10.1103/PhysRevLett.127.116602 ↗
30. Roth, L. M. Semiclassical theory of magnetic energy levels and magnetic susceptibility of Bloch electrons. Phys. Rev. 145, 434–448 (1966).10.1103/PhysRev.145.434 ↗
31. Chang, M.-C. & Niu, Q. Berry phase, hyperorbits, and the Hofstadter spectrum. Phys. Rev. Lett. 75, 1348–1351 (1995).10.1103/PhysRevLett.75.1348 ↗
32. Chang, M.-C. & Niu, Q. Berry phase, hyperorbits, and the Hofstadter spectrum: semiclassical dynamics in magnetic Bloch bands. Phys. Rev. B 53, 7010–7023 (1996).10.1103/PhysRevB.53.7010 ↗
33. Chang, M.-C. & Niu, Q. Berry curvature, orbital moment, and effective quantum theory of electrons in electromagnetic fields. J. Phys.: Condens. Matter 20, 193202 (2008).
34. Koshino, M. & Ando, T. Anomalous orbital magnetism in Dirac-electron systems: role of pseudospin paramagnetism. Phys. Rev. B 81, 195431 (2010).10.1103/PhysRevB.81.195431 ↗
35. Gao, Y. & Niu, Q. Zero-field magnetic response functions in Landau levels. Proc. Natl. Acad. Sci. 114, 7295–7300 (2017).10.1073/pnas.1702595114 ↗
36. Fuchs, J.-N., Piéchon, F. & Montambaux, G. Landau levels, response functions and magnetic oscillations from a generalized Onsager relation. SciPost Phys. 4, 024 (2018).10.21468/SciPostPhys.4.5.024 ↗
37. Rhonald Burgos Atencia, A. A. & Culcer, D. Orbital angular momentum of Bloch electrons: equilibrium formulation, magneto-electric phenomena, and the orbital Hall effect. Adv. Phys.: X 9, 2371972 (2024).
38. Rhim, J.-W. & Yang, B.-J. Classification of flat bands according to the band-crossing singularity of Bloch wave functions. Phys. Rev. B 99, 045107 (2019).10.1103/PhysRevB.99.045107 ↗
39. Hwang, Y., Rhim, J.-W. & Yang, B. Geometric characterization of anomalous Landau levels of isolated flat bands. Nat. Commun. 12, 6433 (2021).10.1038/s41467-021-26765-z ↗
40. Jung, J., Lim, H. & Yang, B.-J. Quantum geometry and Landau levels of quadratic band crossings. Phys. Rev. B 109, 035134 (2024).10.1103/PhysRevB.109.035134 ↗
41. Oh, C., Rhim, J.-W. & Yang, B.-J. Revisiting the magnetic responses of bilayer graphene from the perspective of quantum distance. Phys. Rev. B 110, 155412 (2024).10.1103/PhysRevB.110.155412 ↗
42. Yu, J., Bernevig, B. A., Queiroz, R., Rossi, E., Törmä, P. & Yang, B.-J. Quantum geometry in quantum materials. npj Quantum Mater. 10, 101 (2025).10.1038/s41535-025-00801-3 ↗
43. Kohn, W. Theory of Bloch electrons in a magnetic field: the effective Hamiltonian. Phys. Rev. 115, 1460–1478 (1959).10.1103/PhysRev.115.1460 ↗
44. Blount, E. I. Bloch electrons in a magnetic field. Phys. Rev. 126, 1636–1653 (1962).10.1103/PhysRev.126.1636 ↗
45. Zak, J. Dynamics of electrons in solids in external fields. Phys. Rev. 168, 686–695 (1968).10.1103/PhysRev.168.686 ↗
46. Nenciu, G. Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective hamiltonians. Rev. Mod. Phys. 63, 91–127 (1991).10.1103/RevModPhys.63.91 ↗
47. Zhang, Y. et al. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201 (2005).10.1038/nature04235 ↗
48. Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009).10.1103/RevModPhys.81.109 ↗
49. Knolle, J. & Cooper, N. R. Anomalous de Haas–van Alphen effect in InAs/GaSb quantum wells. Phys. Rev. Lett. 118, 176801 (2017).10.1103/PhysRevLett.118.176801 ↗
50. Yu, R. et al. Quantized anomalous Hall effect in magnetic topological insulators. Science 329, 61–64 (2010).10.1126/science.1187485 ↗
51. Zhang, Y. et al. Crossover of the three-dimensional topological insulator Bi2Se3 to the two-dimensional limit. Nat. Phys. 6, 584–588 (2010).10.1038/nphys1689 ↗
52. Lu, H.-Z., Zhao, A. & Shen, S.-Q. Quantum transport in magnetic topological insulator thin films. Phys. Rev. Lett. 111, 146802 (2013).10.1103/PhysRevLett.111.146802 ↗
53. Chang, C.-Z., Liu, C.-X. & MacDonald, A. H. Colloquium: quantum anomalous hall effect. Rev. Mod. Phys. 95, 011002 (2023).10.1103/RevModPhys.95.011002 ↗
54. Chang, C.-Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).10.1126/science.1234414 ↗
55. Guo, H.-M. & Franz, M. Topological insulator on the kagome lattice. Phys. Rev. B 80, 113102 (2009).10.1103/PhysRevB.80.113102 ↗
56. Weeks, C. & Franz, M. Topological insulators on the Lieb and perovskite lattices. Phys. Rev. B 82, 085310 (2010).10.1103/PhysRevB.82.085310 ↗
57. Dalal, A., Ruhman, J. & Venderbos, J. W. Flat band physics in the charge-density wave state of 1 T-TaS2 and 1 T-TaSe2. npj Quantum Mater. 10, 31 (2025).10.1038/s41535-025-00747-6 ↗
58. Green, D., Santos, L. & Chamon, C. Isolated flat bands and spin-1 conical bands in two-dimensional lattices. Phys. Rev. B 82, 075104 (2010).10.1103/PhysRevB.82.075104 ↗
59. Slot, M. R. et al. Experimental realization and characterization of an electronic Lieb lattice. Nat. Phys. 13, 672–676 (2017).10.1038/nphys4105 ↗
60. Whittaker, C. E. et al. Exciton polaritons in a two-dimensional Lieb lattice with spin-orbit coupling. Phys. Rev. Lett. 120, 097401 (2018).10.1103/PhysRevLett.120.097401 ↗
61. Chiu, C.-K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).10.1103/RevModPhys.88.035005 ↗
62. Taut, M., Eschrig, H. & Richter, M. Effective hamiltonian approach for magnetic band structure and additional oscillations in the magnetization originating from the fermi surface at finite magnetic fields. Phys. Rev. B 72, 165304 (2005).10.1103/PhysRevB.72.165304 ↗
63. Zhang, L., Song, X.-Y. & Wang, F. Quantum oscillation in narrow-gap topological insulators. Phys. Rev. Lett. 116, 046404 (2016).10.1103/PhysRevLett.116.046404 ↗
64. Kittel, C., Fong, C.-y. Quantum Theory of Solids (Wiley New York, 1963).
65. Khait, I., Bhattacharyya, S., Samanta, A. & Auerbach, A. Hall anomalies of the doped Mott insulator. npj Quantum Mater. 8, 75 (2023).10.1038/s41535-023-00611-5 ↗
66. Tisserond, E. et al. Aperiodic quantum oscillations of particle-hole asymmetric Dirac cones. Europhys. Lett. 119, 67001 (2017).10.1209/0295-5075/119/67001 ↗
67. Kumari, A. et al. Observation of the magnetic field induced Fermi surface expansion in aperiodic quantum oscillations. Adv. Functional Mater. 2422986, https://doi.org/10.1002/adfm.202422986 (2025).10.1002/adfm.202422986 ↗
68. Rubi, K. et al. Aperiodic quantum oscillations in the two-dimensional electron gas at the LaAlO3/SrTiO3 interface. npj Quantum Mater. 5, 9 (2020).10.1038/s41535-020-0210-z ↗
69. Rubi, K. et al. Unconventional quantum oscillations and evidence of nonparabolic electronic states in quasi-two-dimensional electron system at complex oxide interfaces. Phys. Rev. Res. 6, 043231 (2024).10.1103/PhysRevResearch.6.043231 ↗
70. Jiang, Y. et al. Landau quantization and the thickness limit of topological insulator thin films of Sb2Te3. Phys. Rev. Lett. 108, 016401 (2012).10.1103/PhysRevLett.108.016401 ↗
71. Chang, C.-Z. et al. Thin films of magnetically doped topological insulator with carrier-independent long-range ferromagnetic order. Adv. Mater. 25, 1065–1070 (2013).10.1002/adma.201203493 ↗
72. Checkelsky, J. et al. Trajectory of the anomalous Hall effect towards the quantized state in a ferromagnetic topological insulator. Nat. Phys. 10, 731–736 (2014).10.1038/nphys3053 ↗
73. Kou, X. et al. Scale-invariant quantum anomalous Hall effect in magnetic topological insulators beyond the two-dimensional limit. Phys. Rev. Lett. 113, 137201 (2014).10.1103/PhysRevLett.113.137201 ↗
74. Bestwick, A. J. et al. Precise quantization of the anomalous Hall effect near zero magnetic field. Phys. Rev. Lett. 114, 187201 (2015).10.1103/PhysRevLett.114.187201 ↗
75. Lachman, E. O. et al. Visualization of superparamagnetic dynamics in magnetic topological insulators. Sci. Adv. 1, 1500740 (2015).10.1126/sciadv.1500740 ↗
76. Chang, C.-Z. et al. High-precision realization of robust quantum anomalous hall state in a hard ferromagnetic topological insulator. Nat. Mater. 14, 473–477 (2015).10.1038/nmat4204 ↗
77. Chang, C.-Z. et al. Zero-field dissipationless chiral edge transport and the nature of dissipation in the quantum anomalous Hall state. Phys. Rev. Lett. 115, 057206 (2015).10.1103/PhysRevLett.115.057206 ↗
78. Bansil, A., Lin, H. & Das, T. Colloquium: topological band theory. Rev. Mod. Phys. 88, 021004 (2016).10.1103/RevModPhys.88.021004 ↗
79. He, K., Wang, Y. & Xue, Q.-K. Topological materials: quantum anomalous Hall system. Annu. Rev. Condens. Matter Phys. 9, 329–344 (2018).10.1146/annurev-conmatphys-033117-054144 ↗
80. Li, Y.-Y. et al. Intrinsic topological insulator Bi2Te3 thin films on Si and their thickness limit. Adv. Mater. 22, 4002–4007 (2010).10.1002/adma.201000368 ↗
81. Feng, X. et al. Thickness dependence of the quantum anomalous Hall effect in magnetic topological insulator films. Adv. Mater. 28, 6386–6390 (2016).10.1002/adma.201600919 ↗
82. Qu, D.-X., Hor, Y. S., Xiong, J., Cava, R. J. & Ong, N. P. Quantum oscillations and Hall anomaly of surface states in the topological insulator Bi2Te3. Science 329, 821–824 (2010).10.1126/science.1189792 ↗
83. Zhao, W. et al. Quantum oscillations of robust topological surface states up to 50 k in thick bulk-insulating topological insulator. npj Quantum Mater. 4, 56 (2019).10.1038/s41535-019-0195-7 ↗
84. Ren, Z., Taskin, A. A., Sasaki, S., Segawa, K. & Ando, Y. Large bulk resistivity and surface quantum oscillations in the topological insulator Bi2Te2Se. Phys. Rev. B 82, 241306 (2010).10.1103/PhysRevB.82.241306 ↗
85. Huang, S.-M. et al. Shubnikov-de Haas oscillation of Bi2Te3 topological insulators with cm-scale uniformity. J. Phys. D Appl. Phys. 49, 255303 (2016).10.1088/0022-3727/49/25/255303 ↗
86. Rogacheva, E., Budnik, A., Sipatov, A.Y., Nashchekina, O., & Dresselhaus, M. Thickness dependent quantum oscillations of transport properties in topological insulator Bi2Te3 thin films. Appl. Phys. Lett. 106(5) https://doi.org/10.1063/1.4907319 (2015).10.1063/1.4907319 ↗
87. Barua, S. & Rajeev, K. P. Robust quantum oscillations with non-zero Berry phase in Bi2Te3. Solid State Commun. 248, 68–76 (2016).10.1016/j.ssc.2016.09.014 ↗
88. Zhang, H. et al. Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nat. Phys. 5, 438–442 (2009).10.1038/nphys1270 ↗
89. Lu, H.-Z. & Shen, S.-Q. Weak localization of bulk channels in topological insulator thin films. Phys. Rev. B 84, 125138 (2011).10.1103/PhysRevB.84.125138 ↗
90. Zhang, F., Zhou, J., Xiao, D. & Yao, Y. Tunable intrinsic plasmons due to band inversion in topological materials. Phys. Rev. Lett. 119, 266804 (2017).10.1103/PhysRevLett.119.266804 ↗
91. Shan, W.-Y., Lu, H.-Z. & Shen, S.-Q. Effective continuous model for surface states and thin films of three-dimensional topological insulators. N. J. Phys. 12, 043048 (2010).10.1088/1367-2630/12/4/043048 ↗
92. Island, J. et al. Spin–orbit-driven band inversion in bilayer graphene by the van der Waals proximity effect. Nature 571, 85–89 (2019).10.1038/s41586-019-1304-2 ↗
93. Jung, J. & MacDonald, A. H. Accurate tight-binding models for the π bands of bilayer graphene. Phys. Rev. B 89, 035405 (2014).10.1103/PhysRevB.89.035405 ↗
94. McCann, E. & Koshino, M. The electronic properties of bilayer graphene. Rep. Prog. Phys. 76, 056503 (2013).10.1088/0034-4885/76/5/056503 ↗
95. Qiao, Z. et al. Quantum anomalous Hall effect in graphene proximity coupled to an antiferromagnetic insulator. Phys. Rev. Lett. 112, 116404 (2014).10.1103/PhysRevLett.112.116404 ↗
96. Wang, Z. F., Liu, Z. & Liu, F. Quantum anomalous Hall effect in 2d organic topological insulators. Phys. Rev. Lett. 110, 196801 (2013).10.1103/PhysRevLett.110.196801 ↗
97. Liu, Z., Wang, Z.-F., Mei, J.-W., Wu, Y.-S. & Liu, F. Flat Chern band in a two-dimensional organometallic framework. Phys. Rev. Lett. 110, 106804 (2013).10.1103/PhysRevLett.110.106804 ↗
98. Li, J. et al. Intrinsic magnetic topological insulators in van der Waals layered MnBi2Te4-family materials. Sci. Adv. 5, 5685 (2019).10.1126/sciadv.aaw5685 ↗
99. Zhang, D. et al. Topological axion states in the magnetic insulator MnBi2Te4 with the quantized magnetoelectric effect. Phys. Rev. Lett. 122, 206401 (2019).10.1103/PhysRevLett.122.206401 ↗
100. Deng, Y. et al. Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi2Te4. Science 367, 895–900 (2020).10.1126/science.aax8156 ↗