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  • Letter
  • Open Access

Correlated Chern insulators in two-dimensional Raman lattices: A cold-atom regularization of strongly coupled four-Fermi field theories

L. Ziegler1, E. Tirrito2,3, M. Lewenstein3,4, S. Hands5, and A. Bermudez1,6

  • 1Departamento de Física Teórica, Universidad Complutense, 28040 Madrid, Spain
  • 2International School for Advanced Studies (SISSA), via Bonomea 265, 34136 Trieste, Italy
  • 3ICFO - Institut de Ciencies Fotoniques, Barcelona Institute of Science and Technology, Avenida Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
  • 4ICREA, Lluis Companys 23, 08010 Barcelona, Spain
  • 5Department of Physics, Faculty of Science and Engineering, Swansea University, Singleton Park, Swansea SA28PP, United Kingdom
  • 6Instituto de Física Teórica, UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain

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Phys. Rev. Research 4, L042012 – Published 20 October, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.L042012

Abstract

We show that synthetic spin-orbit coupling for ultracold atoms in optical Raman potentials can be exploited to build versatile quantum simulators of correlated Chern insulators connected to strongly coupled four-Fermi field theories similar to the Gross-Neveu model in (2+1) dimensions. Exploiting this multidisciplinary perspective, we identify a large-N quantum anomalous Hall (QAH) effect in absence of any external magnetic field, and use it to delimit regions in parameter space where these correlated topological phases appear, the boundaries of which are controlled by strongly coupled fixed points of these four-Fermi relativistic field theories. We further show how, for strong interactions, the QAH effect gives way to magnetic phases described by a two-dimensional quantum compass model in a transverse field. We present a detailed description of the phase diagram using the large-N effective potential, and variational techniques such as projected entangled pairs.

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References (110)

  1. L. H. Thomas, The motion of the spinning electron, Nature 117, 514 (1926).
  2. P. A. M. Dirac, The quantum theory of the electron, Proc. R. Soc. London, Ser. A 117, 610 (1928).
  3. P. A. M. Dirac, A theory of electrons and protons, Proc. R. Soc. London, Ser. A 126, 360 (1930).
  4. S. S. Schweber, QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga (Princeton University Press, Princeton, 1994).
  5. J. Schwinger, On quantum-electrodynamics and the magnetic moment of the electron, Phys. Rev. 73, 416 (1948).
  6. T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, Tenth-Order QED Contribution to the Electron g−2 and an Improved Value of the Fine Structure Constant, Phys. Rev. Lett. 109, 111807 (2012).
  7. D. Hanneke, S. Fogwell, and G. Gabrielse, New Measurement of the Electron Magnetic Moment and the Fine Structure Constant, Phys. Rev. Lett. 100, 120801 (2008).
  8. R. H. Parker, C. Yu, W. Zhong, B. Estey, and H. Müller, Measurement of the fine-structure constant as a test of the standard model, Science 360, 191 (2018).
  9. Y. A. Bychkov and E. I. Rashba, Oscillatory effects and the magnetic susceptibility of carriers in inversion layers, J. Phys. C 17, 6039 (1984).
  10. G. Dresselhaus, Spin-orbit coupling effects in zinc blende structures, Phys. Rev. 100, 580 (1955).
  11. C. L. Kane and E. J. Mele, Z2 Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett. 95, 146802 (2005).
  12. C. L. Kane and E. J. Mele, Quantum Spin Hall Effect in Graphene, Phys. Rev. Lett. 95, 226801 (2005).
  13. B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314, 1757 (2006).
  14. M. König, S. Wiedmann, C. Brune, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Quantum spin Hall insulator state in HgTe quantum wells, Science 318, 766 (2007).
  15. D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, A topological dirac insulator in a quantum spin Hall phase, Nature (London) 452, 970 (2008).
  16. M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  17. X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  18. C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
  19. F. D. M. Haldane, Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the “Parity Anomaly”, Phys. Rev. Lett. 61, 2015 (1988).
  20. X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors, Phys. Rev. B 74, 085308 (2006).
  21. X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B 78, 195424 (2008).
  22. M. Nakahara, Geometry, Topology and Physics (CRC Press, Bristol and Philadelphia, 2018).
  23. S.-S. Chern, Characteristic classes of Hermitian manifolds, Ann. Math. 47, 85 (1946).
  24. F. D. M. Haldane, Nobel lecture: Topological quantum matter, Rev. Mod. Phys. 89, 040502 (2017).
  25. C.-X. Liu, S.-C. Zhang, and X.-L. Qi, The quantum anomalous hall effect: Theory and experiment, Annu. Rev. Condens. Matter Phys. 7, 301 (2016).
  26. A. Soumyanarayanan, N. Reyren, A. Fert, and C. Panagopoulos, Emergent phenomena induced by spin–orbit coupling at surfaces and interfaces, Nature (London) 539, 509 (2016).
  27. K. G. Wilson, in New Phenomena in Subnuclear Physics (Springer, New York, 1977), pp. 69–142.
  28. S. Hands, A. Kocic, and J. B. Kogut, Four-Fermi theories in fewer than four dimensions, Ann. Phys. 224, 29 (1993).
  29. S. Hands, Fixed point four Fermi theories, arXiv:hep-lat/9706018.
  30. J. Braun, Fermion interactions and universal behavior in strongly interacting theories, J. Phys. G: Nucl. Part. Phys. 39, 033001 (2012).
  31. E. Fermi, Versuch einer Theorie der β-Strahlen. i, Z. Phys. 88, 161 (1934).
  32. Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. i, Phys. Rev. 122, 345 (1961).
  33. D. J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D 10, 3235 (1974).
  34. J. Hubbard, Electron correlations in narrow energy bands, Proc. R. Soc. London A 276, 238 (1963).
  35. K. G. Wilson, Confinement of quarks, Phys. Rev. D 10, 2445 (1974).
  36. C. Gattringer and C. Lang, Quantum Chromodynamics on the Lattice: An Introductory Presentation (Springer Science & Business Media, Berlin, Heidelberg, 2009).
  37. V. Galitski and I. B. Spielman, Spin–orbit coupling in quantum gases, Nature (London) 494, 49 (2013).
  38. L. Zhang and X.-J. Liu, in Synthetic Spin-Orbit Coupling in Cold Atoms (World Scientific, Singapore, 2018), pp. 1–87.
  39. R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).
  40. M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, Ultracold atomic gases in optical lattices: Mimicking condensed matter physics and beyond, Adv. Phys. 56, 243 (2007).
  41. I. Bloch, J. Dalibard, and S. Nascimbene, Quantum simulations with ultracold quantum gases, Nat. Phys. 8, 267 (2012).
  42. J. Kogut and L. Susskind, Hamiltonian formulation of Wilson's lattice gauge theories, Phys. Rev. D 11, 395 (1975).
  43. H. B. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice: (i). Proof by homotopy theory, Nucl. Phys. B 185, 20 (1981).
  44. H. B. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice: (ii). Intuitive topological proof, Nucl. Phys. B 193, 173 (1981).
  45. Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Realization of two-dimensional spin-orbit coupling for Bose-Einstein condensates, Science 354, 83 (2016).
  46. W. Sun, B.-Z. Wang, X.-T. Xu, C.-R. Yi, L. Zhang, Z. Wu, Y. Deng, X.-J. Liu, S. Chen, and J.-W. Pan, Highly Controllable and Robust 2D Spin-Orbit Coupling for Quantum Gases, Phys. Rev. Lett. 121, 150401 (2018).
  47. B. Song, L. Zhang, C. He, T. F. J. Poon, E. Hajiyev, S. Zhang, X.-J. Liu, and G.-B. Jo, Observation of symmetry-protected topological band with ultracold fermions, Sci. Adv. 4, eaao4748 (2018).
  48. X.-J. Liu, K. T. Law, and T. K. Ng, Realization of 2D Spin-Orbit Interaction and Exotic Topological Orders in Cold Atoms, Phys. Rev. Lett. 112, 086401 (2014).
  49. X.-J. Liu, K. T. Law, and T. K. Ng, Erratum: Realization of 2D Spin-Orbit Interaction and Exotic Topological Orders in Cold Atoms, Phys. Rev. Lett. 113, 059901(E) (2014).
  50. N. Q. Burdick, Y. Tang, and B. L. Lev, Long-Lived Spin-Orbit-Coupled Degenerate Dipolar Fermi Gas, Phys. Rev. X 6, 031022 (2016).
  51. B. Song, C. He, S. Zhang, E. Hajiyev, W. Huang, X.-J. Liu, and G.-B. Jo, Spin-orbit-coupled two-electron Fermi gases of ytterbium atoms, Phys. Rev. A 94, 061604(R) (2016).
  52. L. F. Livi, G. Cappellini, M. Diem, L. Franchi, C. Clivati, M. Frittelli, F. Levi, D. Calonico, J. Catani, M. Inguscio, and L. Fallani, Synthetic Dimensions and Spin-Orbit Coupling with an Optical Clock Transition, Phys. Rev. Lett. 117, 220401 (2016).
  53. S. Kolkowitz, S. Bromley, T. Bothwell, M. Wall, G. Marti, A. Koller, X. Zhang, A. Rey, and J. Ye, Spin–orbit-coupled fermions in an optical lattice clock, Nature (London) 542, 66 (2017).
  54. S. Bromley, S. Kolkowitz, T. Bothwell, D. Kedar, A. Safavi-Naini, M. Wall, C. Salomon, A. Rey, and J. Ye, Dynamics of interacting fermions under spin–orbit coupling in an optical lattice clock, Nat. Phys. 14, 399 (2018).
  55. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Cold Bosonic Atoms in Optical Lattices, Phys. Rev. Lett. 81, 3108 (1998).
  56. W. Hofstetter, J. I. Cirac, P. Zoller, E. Demler, and M. Lukin, High-Temperature Superfluidity of Fermionic Atoms in Optical Lattices, Phys. Rev. Lett. 89, 220407 (2002).
  57. I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  58. G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Experimental realization of the topological haldane model with ultracold fermions, Nature (London) 515, 237 (2014).
  59. C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang et al., Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator, Science 340, 167 (2013).
  60. C.-Z. Chang, W. Zhao, D. Y. Kim, H. Zhang, B. A. Assaf, D. Heiman, S.-C. Zhang, C. Liu, M. H. Chan, and J. S. Moodera, High-precision realization of robust quantum anomalous Hall state in a hard ferromagnetic topological insulator, Nat. Mater. 14, 473 (2015).
  61. A. Bermudez, L. Mazza, M. Rizzi, N. Goldman, M. Lewenstein, and M. A. Martin-Delgado, Wilson Fermions and Axion Electrodynamics in Optical Lattices, Phys. Rev. Lett. 105, 190404 (2010).
  62. D. B. Kaplan and S. Sun, Spacetime as a Topological Insulator: Mechanism for the Origin of the Fermion Generations, Phys. Rev. Lett. 108, 181807 (2012).
  63. J. Jünemann, A. Piga, S.-J. Ran, M. Lewenstein, M. Rizzi, and A. Bermúdez, Exploring Interacting Topological Insulators with Ultracold Atoms: The Synthetic Creutz-Hubbard Model, Phys. Rev. X 7, 031057 (2017).
  64. A. Bermudez, E. Tirrito, M. Rizzi, M. Lewenstein, and S. Hands, Gross–Neveu–Wilson model and correlated symmetry-protected topological phases, Ann. Phys. 399, 149 (2018).
  65. Y. Kuno, Phase structure of the interacting Su-Schrieffer-Heeger model and the relationship with the Gross-Neveu model on lattice, Phys. Rev. B 99, 064105 (2019).
  66. E. Tirrito, M. Rizzi, G. Sierra, M. Lewenstein, and A. Bermudez, Renormalization group flows for Wilson-Hubbard matter and the topological Hamiltonian, Phys. Rev. B 99, 125106 (2019).
  67. G. Roose, N. Bultinck, L. Vanderstraeten, F. Verstraete, K. Van Acoleyen, and J. Haegeman, Lattice regularisation and entanglement structure of the Gross-Neveu model, J. High Energy Phys. 07 (2021) 207.
  68. S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. Ludwig, Topological insulators and superconductors: Tenfold way and dimensional hierarchy, New J. Phys. 12, 065010 (2010).
  69. D. B. Kaplan, A method for simulating chiral fermions on the lattice, Phys. Lett. B 288, 342 (1992).
  70. H. So, Induced Chern-Simons class with lattice fermions, Prog. Theor. Phys. 73, 528 (1985).
  71. M. F. Golterman, K. Jansen, and D. B. Kaplan, Chern-simons currents and chiral fermions on the lattice, Phys. Lett. B 301, 219 (1993).
  72. S. Sen, Chern insulator transitions with Wilson fermions on a hyperrectangular lattice, Phys. Rev. D 102, 094520 (2020).
  73. S. Aoki, New phase structure for lattice QCD with Wilson fermions, Phys. Rev. D 30, 2653 (1984).
  74. S. Sharpe, R. Singleton et al., Spontaneous flavor and parity breaking with Wilson fermions, Phys. Rev. D 58, 074501 (1998).
  75. S. Coleman, Aspects of Symmetry: Selected Erice Lectures (Cambridge University Press, Cambridge, 1988).
  76. S. Coleman, R. Jackiw, and H. Politzer, Spontaneous symmetry breaking in the O (N) model for large N, Phys. Rev. D 10, 2491 (1974).
  77. S. Coleman and E. Weinberg, Radiative corrections as the origin of spontaneous symmetry breaking, Phys. Rev. D 7, 1888 (1973).
  78. In Euclidean anisotropic lattices, dimensionless fields have a symmetric rescaling Ψ(x)=Ψ̃(x)/a0a1+a1a2+a2a0. This leads to the following dimensionless bare mass m̃=ma1/(1+ξ1+ξ2) with ξ1=a1/a0 and ξ2=a1/a2, and coupling strength g̃2=(g2/a1)ξ1ξ2/(1+ξ1+ξ2)2. Additionally, due to the presence of additional doublers in the time direction, there is an additive renormalization m̃→m̃−12g̃2(1+ξ1+ξ2)2/ξ1. The condensates become dimensionless through Σ̃0=Σ0a1/(1+ξ1+ξ2), and Π̃=Πa1/(1+ξ1+ξ2).
  79. Only in the isotropic case, a1=a2=:a, and along the symmetric line (−ma,g2)=(2,g2), does the scalar condensate vanish for any coupling strength Σ0(g2)=0, which signals a cancellation of the additive mass renormalizations arising from the two fermion doublers at k∈{(0,π/a),(π/a,0)}. The lack of additive renormalization in this so-called central Wilson branch is not a large-N artifact, which has interesting implications for Monte Carlo studies of other lattice gauge theories [106, 107].
  80. Z. Wang and S.-C. Zhang, Simplified Topological Invariants for Interacting Insulators, Phys. Rev. X 2, 031008 (2012).
  81. Z. Wang and S.-C. Zhang, Strongly correlated topological superconductors and topological phase transitions via Green's function, Phys. Rev. B 86, 165116 (2012).
  82. Z. Wang and B. Yan, Topological Hamiltonian as an exact tool for topological invariants, J. Phys.: Condens. Matter 25, 155601 (2013).
  83. W. Chen, Weakly interacting topological insulators: Quantum criticality and the renormalization group approach, Phys. Rev. B 97, 115130 (2018).
  84. Z. Wang, X.-L. Qi, and S.-C. Zhang, Topological Order Parameters for Interacting Topological Insulators, Phys. Rev. Lett. 105, 256803 (2010).
  85. V. Gurarie, Single-particle Green's functions and interacting topological insulators, Phys. Rev. B 83, 085426 (2011).
  86. D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett. 49, 405 (1982).
  87. At the isotropic point a1=a2=:a, there is an additional O(2) symmetry, such that the parity-broken phase will spontaneously select the direction of the ferromagnetic order in the xy plane. In this limit, one can also envisage the appearance of a Chern-Simons term. However, by its very nature, it can only influence dynamics while our large-N solution focuses on static properties. The dynamical consequences of this term will be studied in the future.
  88. P. W. Anderson, Antiferromagnetism. Theory of superexchange interaction, Phys. Rev. 79, 350 (1950).
  89. P. W. Anderson, in Solid State Physics, Vol. 14 (Elsevier, New York, 1963), pp. 99–214.
  90. W. Heisenberg, Zur Theorie des Ferromagnetismus, Z. Phys. 49, 619 (1928).
  91. Z. Nussinov and J. van den Brink, Compass models: Theory and physical motivations, Rev. Mod. Phys. 87, 1 (2015).
  92. A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006).
  93. B. Douçot, M. V. Feigel'man, L. B. Ioffe, and A. S. Ioselevich, Protected qubits and chern-simons theories in josephson junction arrays, Phys. Rev. B 71, 024505 (2005).
  94. D. Bacon, Operator quantum error-correcting subsystems for self-correcting quantum memories, Phys. Rev. A 73, 012340 (2006).
  95. M. Li, D. Miller, M. Newman, Y. Wu, and K. R. Brown, 2D Compass Codes, Phys. Rev. X 9, 021041 (2019).
  96. J. Dorier, F. Becca, and F. Mila, Quantum compass model on the square lattice, Phys. Rev. B 72, 024448 (2005).
  97. H.-D. Chen, C. Fang, J. Hu, and H. Yao, Quantum phase transition in the quantum compass model, Phys. Rev. B 75, 144401 (2007).
  98. R. Orús, A. C. Doherty, and G. Vidal, First Order Phase Transition in the Anisotropic Quantum Orbital Compass Model, Phys. Rev. Lett. 102, 077203 (2009).
  99. F. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, arXiv:cond-mat/0407066.
  100. J. Jordan, R. Orús, G. Vidal, F. Verstraete, and J. I. Cirac, Classical Simulation of Infinite-Size Quantum Lattice Systems in Two Spatial Dimensions, Phys. Rev. Lett. 101, 250602 (2008).
  101. H.-C. Jiang, Z.-Y. Weng, and T. Xiang, Accurate Determination of Tensor Network State of Quantum Lattice Models in Two Dimensions, Phys. Rev. Lett. 101, 090603 (2008).
  102. We have implemented the variational iPEPS algorithms described in [108, 109, 110], the accuracy of which relies on a refinement parameter D related to the maximum entanglement content of the Ansatz. In practice, increasing D leads to better descriptions of the ground state, and D=2 is the minimum value that captures the effect of quantum correlations.
  103. D. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Fractional quantum Hall effect in the absence of Landau levels, Nat. Commun. 2, 389 (2011).
  104. T. Neupert, L. Santos, C. Chamon, and C. Mudry, Fractional Quantum Hall States at Zero Magnetic Field, Phys. Rev. Lett. 106, 236804 (2011).
  105. D. B. Kaplan and S. Sen, Fractional Quantum Hall Effect in a Relativistic Field Theory, Phys. Rev. Lett. 124, 131601 (2020).
  106. T. Misumi and Y. Tanizaki, Lattice gauge theory for the Haldane conjecture and central-branch Wilson fermion, Prog. Theor. Exp. Phys. 2020, 033B03 (2020).
  107. T. Misumi and J. Yumoto, Varieties and properties of central-branch Wilson fermions, Phys. Rev. D 102, 034516 (2020).
  108. R. Orús and G. Vidal, Simulation of two-dimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction, Phys. Rev. B 80, 094403 (2009).
  109. H. N. Phien, J. A. Bengua, H. D. Tuan, P. Corboz, and R. Orús, Infinite projected entangled pair states algorithm improved: Fast full update and gauge fixing, Phys. Rev. B 92, 035142 (2015).
  110. P. Corboz, Variational optimization with infinite projected entangled-pair states, Phys. Rev. B 94, 035133 (2016).

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Phys. Rev. Research 4, L042012– Published 20 October, 2022
  • Received 22 December 2020
  • Revised 8 July 2022
  • Accepted 25 July 2022
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