Fractional quantum Hall physics in topological flat bands

Siddharth A. Parameswaran; Rahul Roy; Shivaji L. Sondhi

doi:10.1016/j.crhy.2013.04.003

Topological insulators/Isolants topologiques

Fractional quantum Hall physics in topological flat bands
[Physique de lʼeffet Hall quantique fractionnaire dans des bandes plates topologiques]

Siddharth A. Parameswaran ¹ ; Rahul Roy ² ; Shivaji L. Sondhi ³

¹ Department of Physics, University of California, Berkeley, CA 94720, United States of America
² Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547, United States of America
³ Department of Physics, Princeton University, Princeton, NJ 08544, United States of America

Comptes Rendus. Physique, Volume 14 (2013) no. 9-10, pp. 816-839.

Résumé
Abstract

Nous présentons une revue didactique sur la physique des isolants de Chern, qui se concentre plus particulièrement sur lʼeffet Hall quantique fractionnaire. Habituellement, cet effet apparaît typiquement dans des hétérostructures semiconductrices à basse température et sous champ magnétique fort. En revanche, les isolants de Chern peuvent abriter des phases topologiques aux propriétés similaires, mais stabilisées à lʼéchelle du paramètre de réseau, ce qui peut conduire un ordre topologique à haute température. Nous décrivons la construction des modèles avec bande(s) plate(s), passons en revue les résultats numériques et établissons une comparaison entre les isolants de Chern sur réseau et le problème de Landau défini dans le continuum. Nous discutons alors brièvement les aspects de la physique des isolants de Chern qui nont pas dʼanalogues dans le continuum, avant de passer aux possibles réalisations expérimentales. Nous concluons par une liste de perspectives et de problèmes encore ouverts dans ce domaine, ainsi que par une discussion des extensions de ces idées à des dimensions supérieures et à dʼautres phases topologiques.

We present a pedagogical review of the physics of fractional Chern insulators with a particular focus on the connection to the fractional quantum Hall effect. While the latter conventionally arises in semiconductor heterostructures at low temperatures and in high magnetic fields, interacting Chern insulators at fractional band filling may host phases with the same topological properties, but stabilized at the lattice scale, potentially leading to high-temperature topological order. We discuss the construction of topological flat band models, provide a survey of numerical results, and establish the connection between the Chern band and the continuum Landau problem. We then briefly summarize various aspects of Chern band physics that have no natural continuum analogs, before turning to a discussion of possible experimental realizations. We close with a survey of future directions and open problems, as well as a discussion of extensions of these ideas to higher dimensions and to other topological phases.

Publié le : 2013-06-04

DOI : 10.1016/j.crhy.2013.04.003

Keywords: Chern insulators, Flat bands, Fractional quantum Hall effect, Topological order
Mot clés : Isolants de Chern, Bandes plates, Effet Hall fractionnaire, Ordre topologique

Affiliations des auteurs :

Siddharth A. Parameswaran ¹ ; Rahul Roy ² ; Shivaji L. Sondhi ³

¹ Department of Physics, University of California, Berkeley, CA 94720, United States of America
² Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547, United States of America
³ Department of Physics, Princeton University, Princeton, NJ 08544, United States of America

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     title = {Fractional quantum {Hall} physics in topological flat bands},
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Siddharth A. Parameswaran; Rahul Roy; Shivaji L. Sondhi. Fractional quantum Hall physics in topological flat bands. Comptes Rendus. Physique, Volume 14 (2013) no. 9-10, pp. 816-839. doi : 10.1016/j.crhy.2013.04.003. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2013.04.003/

Bibliographie
Cité par

[1] K. von Klitzing; G. Dorda; M. Pepper New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett., Volume 45 (1980) no. 6, p. 494 | DOI

[2] D.C. Tsui; H.L. Störmer; A.C. Gossard Two-dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett., Volume 48 (1982) no. 22, p. 1559 | DOI

[3] R.B. Laughlin Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett., Volume 50 (1983) no. 18, p. 1395 | DOI

[4] X.-G. Wen Topological order in rigid states, Int. J. Mod. Phys. B, Volume 4 (1990), p. 239 | DOI

[5] F.D.M. Haldane Fractional quantization of the Hall effect: A hierarchy of incompressible quantum fluid states, Phys. Rev. Lett., Volume 51 (1983) no. 7, p. 605 | DOI

[6] B.I. Halperin Statistics of quasiparticles and the hierarchy of fractional quantized Hall states, Phys. Rev. Lett., Volume 52 (1984) no. 18, p. 1583 | DOI

[7] G. Moore; N. Read Nonabelions in the fractional quantum Hall effect, Nucl. Phys. B, Volume 360 (1991) no. 2–3, p. 362

[8] S.-C. Zhang; T.H. Hansson; S.A. Kivelson Effective field theory model for the fractional quantum Hall effect, Phys. Rev. Lett., Volume 62 (1989) no. 1, p. 82

[9] S.C. Zhang The Chern–Simons–Landau–Ginzburg theory of the fractional quantum Hall effect, Int. J. Mod. Phys. B, Volume 6 (1992) no. 1, pp. 43-77

[10] D.J. Thouless; M. Kohmoto; M.P. Nightingale; M. den Nijs Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett., Volume 49 (1982) no. 6, p. 405 | DOI

[11] F.D.M. Haldane Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”, Phys. Rev. Lett., Volume 61 (1988) no. 18, pp. 2015-2018 | DOI

[12] E. Tang; J.-W. Mei; X.-G. Wen High-temperature fractional quantum hall states, Phys. Rev. Lett., Volume 106 (2011) no. 23, p. 236802 | DOI

[13] K. Sun; Z.-C. Gu; H. Katsura; S. Das Sarma Nearly flatbands with nontrivial topology, Phys. Rev. Lett., Volume 106 (2011) no. 23, p. 236803 | DOI

[14] T. Neupert; L. Santos; C. Chamon; C. Mudry Fractional quantum Hall states at zero magnetic field, Phys. Rev. Lett., Volume 106 (2011) no. 23, p. 236804 | DOI

[15] D. Sheng; Z. Gu; K. Sun; L. Sheng Fractional quantum Hall effect in the absence of Landau levels, Nat. Commun., Volume 2 (2011), p. 389

[16] Y.-F. Wang; Z.-C. Gu; C.-D. Gong; D.N. Sheng Fractional quantum Hall effect of hard-core bosons in topological flat bands, Phys. Rev. Lett., Volume 107 (2011), p. 146803 | DOI

[17] N. Regnault; B. Bernevig Fractional Chern insulator, Phys. Rev. X, Volume 1 (2011) no. 2, p. 021014

[18] X.-L. Qi Generic wave-function description of fractional quantum anomalous Hall states and fractional topological insulators, Phys. Rev. Lett., Volume 107 (2011), p. 126803 | DOI

[19] S.A. Parameswaran; R. Roy; S.L. Sondhi Fractional Chern insulators and the $w_{\infty}$ algebra, Phys. Rev. B, Volume 85 (2012) no. 24, p. 241308

[20] R. Roy Band geometry of fractional topological insulators | arXiv

[21] G. Murthy; R. Shankar Composite fermions for fractionally filled Chern bands | arXiv

[22] G. Murthy; R. Shankar Hamiltonian theory of fractionally filled Chern bands | arXiv

[23] A.S. Sørensen; E. Demler; M.D. Lukin Fractional quantum Hall states of atoms in optical lattices, Phys. Rev. Lett., Volume 94 (2005) no. 8, p. 086803 | DOI

[24] S.M. Girvin; A.H. MacDonald; P.M. Platzman Magneto-roton theory of collective excitations in the fractional quantum Hall effect, Phys. Rev. B, Volume 33 (1986) no. 4, pp. 2481-2494 | DOI

[25] M. Barkeshli; J. McGreevy Continuous transitions between composite Fermi liquid and Landau Fermi liquid: A route to fractionalized mott insulators, Phys. Rev. B, Volume 86 (2012), p. 075136 | DOI

[26] M. Barkeshli; J. McGreevy | arXiv

[27] Y. Zhang; A. Vishwanath Establishing non-Abelian topological order in Gutzwiller projected Chern insulators via entanglement entropy and modular S-matrix | arXiv

[28] G. Möller; N.R. Cooper Composite fermion theory for bosonic quantum Hall states on lattices, Phys. Rev. Lett., Volume 103 (2009) no. 10, p. 105303 | DOI

[29] D. Xiao; W. Zhu; Y. Ran; N. Nagaosa; S. Okamoto Interface engineering of quantum Hall effects in digital transition metal oxide heterostructures, Nat. Commun., Volume 2 (2011) | DOI

[30] N.R. Cooper Optical flux lattices for ultracold atomic gases, Phys. Rev. Lett., Volume 106 (2011), p. 175301 | DOI

[31] N.Y. Yao; A.V. Gorshkov; C.R. Laumann; A.M. Läuchli; J. Ye; M.D. Lukin Realizing fractional Chern insulators with dipolar spins | arXiv

[32] N.Y. Yao; C.R. Laumann; A.V. Gorshkov; S.D. Bennett; E. Demler; P. Zoller; M.D. Lukin Topological flat bands from dipolar spin systems, Phys. Rev. Lett., Volume 109 (2012), p. 266804 | DOI

[33] X. Hu; M. Kargarian; G.A. Fiete Topological insulators and fractional quantum Hall effect on the ruby lattice, Phys. Rev. B, Volume 84 (2011) no. 15, p. 155116

[34] F.D.M. Haldane Many-particle translational symmetries of two-dimensional electrons at rational Landau-level filling, Phys. Rev. Lett., Volume 55 (1985), pp. 2095-2098 | DOI

[35] A. Kol; N. Read Fractional quantum Hall effect in a periodic potential, Phys. Rev. B, Volume 48 (1993), pp. 8890-8898 | DOI

[36] F.D.M. Haldane “Fractional statistics” in arbitrary dimensions: A generalization of the Pauli principle, Phys. Rev. Lett., Volume 67 (1991), pp. 937-940 | DOI

[37] B.A. Bernevig; N. Regnault Emergent many-body translational symmetries of abelian and non-abelian fractionally filled topological insulators, Phys. Rev. B, Volume 85 (2012), p. 075128 | DOI

[38] A. Kitaev; J. Preskill Topological entanglement entropy, Phys. Rev. Lett., Volume 96 (2006), p. 110404 | DOI

[39] M. Levin; X.-G. Wen Detecting topological order in a ground state wave function, Phys. Rev. Lett., Volume 96 (2006), p. 110405 | DOI

[40] H. Li; F.D.M. Haldane Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum Hall effect states, Phys. Rev. Lett., Volume 101 (2008) no. 1, p. 010504 | DOI

[41] M. Srednicki Entropy and area, Phys. Rev. Lett., Volume 71 (1993), pp. 666-669 | DOI

[42] A. Sterdyniak; N. Regnault; B.A. Bernevig Extracting excitations from model state entanglement, Phys. Rev. Lett., Volume 106 (2011), p. 100405 | DOI

[43] Y. Zhang; T. Grover; A. Turner; M. Oshikawa; A. Vishwanath Quasiparticle statistics and braiding from ground-state entanglement, Phys. Rev. B, Volume 85 (2012), p. 235151 | DOI

[44] C.H. Lee; R. Thomale; X.-L. Qi Pseudopotential formalism for fractional Chern insulators | arXiv

[45] Z. Liu; E.J. Bergholtz From fractional Chern insulators to abelian and non-abelian fractional quantum Hall states: Adiabatic continuity and orbital entanglement spectrum, Phys. Rev. B, Volume 87 (2013), p. 035306 | DOI

[46] T. Scaffidi; G. Möller Adiabatic continuation of fractional Chern insulators to fractional quantum Hall states, Phys. Rev. Lett., Volume 109 (2012), p. 246805 | DOI

[47] Y.-H. Wu; J.K. Jain; K. Sun Adiabatic continuity between Hofstadter and Chern insulator states, Phys. Rev. B, Volume 86 (2012), p. 165129 | DOI

[48] S.A. Trugman Localization, percolation, and the quantum Hall effect, Phys. Rev. B, Volume 27 (1983) no. 12, p. 7539 | DOI

[49] Y.-L. Wu; N. Regnault; B.A. Bernevig Gauge-fixed Wannier wave functions for fractional topological insulators, Phys. Rev. B, Volume 86 (2012), p. 085129 | DOI

[50] E. Brown Bloch electrons in a uniform magnetic field, Phys. Rev. A, Volume 133 (1964) no. 4, pp. 1038-1044 | DOI

[51] R. Roy, 2011, unpublished.

[52] N. Regnault, B.A. Bernevig, 2011, personal communication.

[53] E. Kapit; E. Mueller Exact parent Hamiltonian for the quantum Hall states in a lattice, Phys. Rev. Lett., Volume 105 (2010), p. 215303 | DOI

[54] E. Kapit; P. Ginsparg; E. Mueller Non-abelian braiding of lattice bosons, Phys. Rev. Lett., Volume 108 (2012), p. 066802 | DOI

[55] S. Boldyrev; V. Gurarie The integer quantum Hall transition and random su(N) rotation, J. Phys. Condens. Matter, Volume 15 (2003), p. L125-L132 | DOI

[56] D. Page Geometrical description of Berryʼs phase, Phys. Rev. A, Volume 36 (1987), pp. 3479-3481

[57] J. Anandan; Y. Aharonov Geometry of quantum evolution, Phys. Rev. Lett., Volume 65 (1990), pp. 1697-1700 | DOI

[58] S. Kobayashi; K. Nomizu Foundations of Differential Geometry, vol. 2, Interscience Publishers, New York, 1969

[59] A.K. Pati Relation between “phases” and “distance” in quantum evolution, Phys. Lett. A, Volume 159 (1991) no. 3, pp. 105-112 | DOI

[60] R. Resta The insulating state of matter: A geometrical theory, Eur. Phys. J. B, Volume 79 (2011), pp. 121-137 | arXiv | DOI

[61] A.G. Grushin; T. Neupert; C. Chamon; C. Mudry Enhancing the stability of a fractional Chern insulator against competing phases, Phys. Rev. B, Volume 86 (2012), p. 205125 | DOI

[62] T. Neupert; L. Santos; S. Ryu; C. Chamon; C. Mudry Noncommutative geometry for three-dimensional topological insulators, Phys. Rev. B, Volume 86 (2012), p. 035125 | DOI

[63] C. Chamon; C. Mudry Magnetic translation algebra with or without magnetic field in the continuum or on arbitrary Bravais lattices in any dimension, Phys. Rev. B, Volume 86 (2012), p. 195125 | DOI

[64] B. Estienne; N. Regnault; B.A. Bernevig d-algebra structure of topological insulators, Phys. Rev. B, Volume 86 (2012), p. 241104 | DOI

[65] Y.M. Lu; Y. Ran Symmetry-protected fractional Chern insulators and fractional topological insulators, Phys. Rev. B, Volume 85 (2012), p. 165134 | DOI

[66] X.-G. Wen Quantum orders and symmetric spin liquids, Phys. Rev. B, Volume 65 (2002), p. 165113 | DOI

[67] J. McGreevy; B. Swingle; K.-A. Tran Wave functions for fractional Chern insulators, Phys. Rev. B, Volume 85 (2012), p. 125105 | DOI

[68] Y. Zhang; T. Grover; A. Vishwanath Topological entanglement entropy of $Z_{2}$ spin liquids and lattice Laughlin states, Phys. Rev. B, Volume 84 (2011), p. 075128 | DOI

[69] Y.-F. Wang; H. Yao; Z.-C. Gu; C.-D. Gong; D.N. Sheng Non-abelian quantum Hall effect in topological flat bands, Phys. Rev. Lett., Volume 108 (2012), p. 126805 | DOI

[70] Y.-F. Wang; H. Yao; C.-D. Gong; D.N. Sheng Fractional quantum Hall effect in topological flat bands with Chern number two, Phys. Rev. B, Volume 86 (2012), p. 201101 | DOI

[71] S. Yang; Z.-C. Gu; K. Sun; S. Das Sarma Topological flat band models with arbitrary Chern numbers, Phys. Rev. B, Volume 86 (2012), p. 241112 | DOI

[72] F. Wang; Y. Ran Nearly flat band with Chern number $c = 2$ on the dice lattice, Phys. Rev. B, Volume 84 (2011) no. 24, p. 241103

[73] M. Trescher; E.J. Bergholtz Flat bands with higher Chern number in pyrochlore slabs, Phys. Rev. B, Volume 86 (2012) no. 24, p. 241111

[74] Z. Liu; E.J. Bergholtz; H. Fan; A.M. Läuchli Fractional Chern insulators in topological flat bands with higher Chern number, Phys. Rev. Lett., Volume 109 (2012), p. 186805 | DOI

[75] A. Sterdyniak; C. Repellin; B.A. Bernevig; N. Regnault Series of Abelian and non-Abelian states in $C > 1$ fractional Chern insulators | arXiv

[76] M. Barkeshli; X.-L. Qi Topological nematic states and non-abelian lattice dislocations, Phys. Rev. X, Volume 2 (2012), p. 031013 | DOI

[77] Y.-L. Wu; N. Regnault; B.A. Bernevig Bloch model wavefunctions and pseudopotentials for all fractional Chern insulators | arXiv

[78] B.I. Halperin Theory of the quantized Hall conductance, Helv. Phys. Acta, Volume 56 (1983), p. 75

[79] M. Barkeshli; C.-M. Jian; X.-L. Qi Genons, twist defects, and projective non-Abelian braiding statistics | arXiv

[80] S. Kourtis; J.W.F. Venderbos; M. Daghofer Fractional Chern insulator on a triangular lattice of strongly correlated $t_{2 g}$ electrons, Phys. Rev. B, Volume 86 (2012), p. 235118 | DOI

[81] J.W.F. Venderbos; M. Daghofer; J. van den Brink Narrowing of topological bands due to electronic orbital degrees of freedom, Phys. Rev. Lett., Volume 107 (2011), p. 116401 | DOI

[82] J.W.F. Venderbos; S. Kourtis; J. van den Brink; M. Daghofer Fractional quantum-Hall liquid spontaneously generated by strongly correlated $t_{2 g}$ electrons, Phys. Rev. Lett., Volume 108 (2012), p. 126405 | DOI

[83] N.R. Cooper; R. Moessner Designing topological bands in reciprocal space, Phys. Rev. Lett., Volume 109 (2012), p. 215302 | DOI

[84] N.R. Cooper; J. Dalibard Reaching fractional quantum Hall states with optical flux lattices | arXiv

[85] M. Onoda; N. Nagaosa Quantized anomalous Hall effect in two-dimensional ferromagnets: Quantum Hall effect in metals, Phys. Rev. Lett., Volume 90 (2003) no. 20, p. 206601 | DOI

[86] S.A. Kivelson; D.-H. Lee; S.-C. Zhang Global phase diagram in the quantum Hall effect, Phys. Rev. B, Volume 46 (1992) no. 4, p. 2223

[87] S.L. Sondhi; S.A. Kivelson Long-range interactions and the quantum Hall effect, Phys. Rev. B, Volume 46 (1992) no. 20, p. 13319

[88] T. Liu; C. Repellin; B.A. Bernevig; N. Regnault Fractional Chern insulators beyond Laughlin states | arXiv

[89] A.M. Läuchli; Z. Liu; E.J. Bergholtz; R. Moessner Hierarchy of fractional Chern insulators and competing compressible states | arXiv

[90] X.-G. Wen Chiral Luttinger liquid and the edge excitations in the fractional quantum Hall states, Phys. Rev. B, Volume 41 (1990) no. 18, p. 12838 | DOI

[91] D.A. Abanin; T. Kitagawa; I. Bloch; E. Demler Interferometric approach to measuring band topology in 2D optical lattices | arXiv

[92] H.M. Price; N.R. Cooper Mapping the Berry curvature from semiclassical dynamics in optical lattices, Phys. Rev. A, Volume 85 (2012), p. 033620 | DOI

[93] M. Atala; M. Aidelsburger; J.T. Barreiro; D. Abanin; T. Kitagawa; E. Demler; I. Bloch Direct measurement of the Zak phase in topological Bloch bands | arXiv

[94] M. Fannes; B. Nachtergaele; R.F. Werner Finitely correlated states on quantum spin chains, Commun. Math. Phys., Volume 144 (1992), pp. 443-490 | DOI

[95] S.R. White Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett., Volume 69 (1992), pp. 2863-2866 | DOI

[96] S. Östlund; S. Rommer Thermodynamic limit of density matrix renormalization, Phys. Rev. Lett., Volume 75 (1995), pp. 3537-3540 | DOI

[97] F. Verstraete; J.I. Cirac Renormalization algorithms for quantum-many body systems in two and higher dimensions | arXiv

[98] M.P. Zaletel; R.S.K. Mong Exact matrix product states for quantum Hall wave functions, Phys. Rev. B, Volume 86 (2012), p. 245305 | DOI

[99] B. Estienne; Z. Papic; N. Regnault; B.A. Bernevig Matrix product states and the fractional quantum Hall effect | arXiv

[100] B. Béri; N.R. Cooper Local tensor network for strongly correlated projective states, Phys. Rev. Lett., Volume 106 (2011), p. 156401 | DOI

[101] M. Freedman; C. Nayak; K. Shtengel; K. Walker; Z. Wang A class of $P, T$ -invariant topological phases of interacting electrons, Ann. Phys., Volume 310 (2004), pp. 428-492 | arXiv | DOI

[102] M. Levin; A. Stern Fractional topological insulators, Phys. Rev. Lett., Volume 103 (2009), p. 196803 | DOI

[103] J. Maciejko; X.-L. Qi; A. Karch; S.-C. Zhang Fractional topological insulators in three dimensions, Phys. Rev. Lett., Volume 105 (2010), p. 246809 | DOI

[104] B. Swingle; M. Barkeshli; J. McGreevy; T. Senthil Correlated topological insulators and the fractional magnetoelectric effect, Phys. Rev. B, Volume 83 (2011), p. 195139 | DOI

[105] T. Neupert; L. Santos; S. Ryu; C. Chamon; C. Mudry Fractional topological liquids with time-reversal symmetry and their lattice realization, Phys. Rev. B, Volume 84 (2011), p. 165107 | DOI

[106] L. Santos; T. Neupert; S. Ryu; C. Chamon; C. Mudry Time-reversal symmetric hierarchy of fractional incompressible liquids, Phys. Rev. B, Volume 84 (2011), p. 165138 | DOI

[107] M. Levin; F.J. Burnell; M. Koch-Janusz; A. Stern Exactly soluble models for fractional topological insulators in two and three dimensions, Phys. Rev. B, Volume 84 (2011), p. 235145 | DOI

[108] X. Chen; Z.-C. Gu; Z.-X. Liu; X.-G. Wen Symmetry protected topological orders in interacting bosonic systems | arXiv

[109] F. Pollmann; A.M. Turner; E. Berg; M. Oshikawa Entanglement spectrum of a topological phase in one dimension, Phys. Rev. B, Volume 81 (2010), p. 064439 | DOI

[110] F. Pollmann; E. Berg; A.M. Turner; M. Oshikawa Symmetry protection of topological phases in one-dimensional quantum spin systems, Phys. Rev. B, Volume 85 (2012), p. 075125 | DOI

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