Supersymmetric many-body systems from partial symmetries — integrability, localization and scrambling

  • Pramod Padmanabhan
  • Soo-Jong Rey
  • Daniel Teixeira
  • Diego Trancanelli
Open Access
Regular Article - Theoretical Physics


Partial symmetries are described by generalized group structures known as symmetric inverse semigroups. We use the algebras arising from these structures to realize supersymmetry in (0+1) dimensions and to build many-body quantum systems on a chain. This construction consists in associating appropriate supercharges to chain sites, in analogy to what is done in spin chains. For simple enough choices of supercharges, we show that the resulting states have a finite non-zero Witten index, which is invariant under perturbations, therefore defining supersymmetric phases of matter protected by the index. The Hamiltonians we obtain are integrable and display a spectrum containing both product and entangled states. By introducing disorder and studying the out-of-time-ordered correlators (OTOC), we find that these systems are in the many-body localized phase and do not thermalize. Finally, we reformulate a theorem relating the growth of the second Rényi entropy to the OTOC on a thermal state in terms of partial symmetries.


Discrete Symmetries Extended Supersymmetry Lattice Integrable Models Random Systems 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    E.P. Wigner, Gruppentheorie (in German), Vieweg, Berlin Germany (1931) [Group Theory, Academic Press Inc., New York U.S.A. (1959)].Google Scholar
  2. [2]
    M.V. Lawson, Inverse Semigroups — The Theory of Partial Symmetries, World Scientific, Singapore (1998).Google Scholar
  3. [3]
    J. Kellendonk and M.V. Lawson, Tiling Semigroups, J. Algebra 224 (2000) 140.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    D.P. Di Vincenzo and P.J. Steinhardt, Quasicrystals: The State of the Art, World Scientific, Singapore (1991).Google Scholar
  5. [5]
    C. Janot, Quasicrystals — A Primer, Clarendon Press, Oxford U.K. (1992).Google Scholar
  6. [6]
    M. Senechal, Quasicrystals and Geometry, Cambridge University Press, Cambridge U.K. (1995).Google Scholar
  7. [7]
    B. Unal et al., Nucleation and growth of Ag islands on fivefold Al-Pd-Mn quasicrystal surfaces: Dependence of island density on temperature and flux, Phys. Rev. B 75 (2007) 064205.ADSCrossRefGoogle Scholar
  8. [8]
    R. Exel, D. Goncalves and C. Starling, The tiling C -algebra viewed as a tight inverse semigroup algebra, arXiv:1106.4535.
  9. [9]
    J. Kellendonk, The Local structure of tilings and their integer group of coinvariants, Commun. Math. Phys. 187 (1997) 115 [cond-mat/9508010] [INSPIRE].
  10. [10]
    J. Kellendonk, Topological equivalence of tilings, J. Math. Phys. 38 (1997) 1823 [cond-mat/9609254].
  11. [11]
    D. Damanik, A. Gorodetski and W. Yessen, The Fibonacci Hamiltonian, arXiv:1403.7823.
  12. [12]
    J. Bellissard, A. Bovier and J.-M. Ghez, Gap Labelling Theorems for One Dimensional Discrete Schrodinger Operators, Rev. Math. Phys. 4 (1992) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    J. Kellendonk, Non Commutative Geometry of Tilings and Gap Labelling, cond-mat/9403065 [INSPIRE].
  14. [14]
    V.V. Wagner, The theory of generalised heaps and generalised groups, Mat. Sb. (N.S.) 32 (1953) 545.Google Scholar
  15. [15]
    G.B. Preston, Representations of inverse semi-groups, J. London Math. Soc. 29 (1954) 411.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Yu. A. Golfand and E.P. Likhtman, Extension of the Algebra of Poincaré Group Generators and Violation of p Invariance, JETP Lett. 13 (1971) 323 [INSPIRE].ADSGoogle Scholar
  17. [17]
    P. Ramond, Dual Theory for Free Fermions, Phys. Rev. D 3 (1971) 2415 [INSPIRE].ADSMathSciNetGoogle Scholar
  18. [18]
    A. Neveu and J.H. Schwarz, Factorizable dual model of pions, Nucl. Phys. B 31 (1971) 86 [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    D.V. Volkov and V.P. Akulov, Is the Neutrino a Goldstone Particle?, Phys. Lett. B 46 (1973) 109 [INSPIRE].
  20. [20]
    J. Wess and B. Zumino, Supergauge Transformations in Four-Dimensions, Nucl. Phys. B 70 (1974) 39 [INSPIRE].
  21. [21]
    M.F. Sohnius, Introducing Supersymmetry, Phys. Rept. 128 (1985) 39 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    E. Witten, Dynamical Breaking of Supersymmetry, Nucl. Phys. B 188 (1981) 513 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  23. [23]
    F. Cooper and B. Freedman, Aspects of Supersymmetric Quantum Mechanics, Annals Phys. 146 (1983) 262 [INSPIRE].
  24. [24]
    F. Cooper, A. Khare and U. Sukhatme, Supersymmetry and quantum mechanics, Phys. Rept. 251 (1995) 267 [hep-th/9405029] [INSPIRE].
  25. [25]
    E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    O. Buerschaper, J.M. Mombelli, M. Christandl and M. Aguado, A hierarchy of topological tensor network states, J. Math. Phys. 54 (2013) 012201 [arXiv:1007.5283].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    X. Chen, Z.-C. Gu, Z.-X. Liu and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87 (2013) 155114 [arXiv:1106.4772] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    M.J.B. Ferreira, P. Padmanabhan and P. Teotonio-Sobrinho, 2D Quantum Double Models From a 3D Perspective, J. Phys. A 47 (2014) 375204 [arXiv:1310.8483] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  29. [29]
    M.F. Atiyah and I.M. Singer, The index of elliptic operators on compact manifolds, Bull. Am. Math. Soc. 69 (1963) 422.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    M. Atiyah, R. Bott and V.K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973) 279.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    R. Melrose, The Atiyah-Patodi-Singer Index Theorem, Taylor and Francis, London U.K. (1993).Google Scholar
  32. [32]
    F. Gesztesy and B. Simon, Topological Invariance of the Witten Index, J. Funct. Anal. 79 (1988) 91.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    K. Aghababaei Samani and A. Mostafazadeh, Quantum mechanical symmetries and topological invariants, Nucl. Phys. B 595 (2001) 467 [hep-th/0007008] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    S.M. Girvin, A.H. MacDonald, M.P.A. Fisher, S.-J. Rey and J.P. Sethna, Exactly soluble model of fractional statistics, Phys. Rev. Lett. 65 (1990) 1671 [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    G. Junker, Supersymmetric Methods in Quantum and Statistical Physics, Springer-Verlag, Heidelberg Germany (1996).Google Scholar
  36. [36]
    H. Nicolai, Supersymmetry and Spin Systems, J. Phys. A 9 (1976) 1497 [INSPIRE].ADSMathSciNetGoogle Scholar
  37. [37]
    H. Moriya, On Supersymmetric Fermion Lattice Systems, Ann. Henri Poincaré 17 (2016) 2199.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    P.H. Dondi and H. Nicolai, Lattice Supersymmetry, Nuovo Cim. A 41 (1977) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    C. Hagendorf, Spin chains with dynamical lattice supersymmetry, J. Stat. Phys. 150 (2013) 609 [arXiv:1207.0357] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    N. Ilieva, H. Narnhofer and W.E. Thirring, Supersymmetric models for fermions on a lattice, Fortsch. Phys. 54 (2006) 124 [quant-ph/0502100] [INSPIRE].
  41. [41]
    J. de Gier, G.Z. Feher, B. Nienhuis and M. Rusaczonek, Integrable supersymmetric chain without particle conservation, J. Stat. Mech. 1602 (2016) 023104 [arXiv:1510.02520] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  42. [42]
    H. Saleur and N.P. Warner, Lattice models and N = 2 supersymmetry, hep-th/9311138 [INSPIRE].
  43. [43]
    P. Fendley, K. Schoutens and B. Nienhuis, Lattice fermion models with supersymmetry, J. Phys. A 36 (2003) 12399 [cond-mat/0307338] [INSPIRE].
  44. [44]
    L. Huijse and B. Swingle, Area law violations in a supersymmetric model, Phys. Rev. B 87 (2013) 035108 [arXiv:1202.2367] [INSPIRE].
  45. [45]
    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP 02 (2016) 004 [arXiv:1511.04021] [INSPIRE].
  47. [47]
    Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.ADSGoogle Scholar
  51. [51]
    P.W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109 (1958) 1492 [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    R. Nandkishore and D.A. Huse, Many body localization and thermalization in quantum statistical mechanics, Ann. Rev. Condensed Matter Phys. 6 (2015) 15 [arXiv:1404.0686] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    B. Swingle and D. Chowdhury, Slow scrambling in disordered quantum systems, Phys. Rev. B 95 (2017) 060201 [arXiv:1608.03280] [INSPIRE].
  54. [54]
    R. Fan, P. Zhang, H. Shen and H. Zhai, Out-of-Time-Order Correlation for Many-Body Localization, arXiv:1608.01914 [INSPIRE].
  55. [55]
    Y. Huang, Y.-L. Zhang and X. Chen, Out-of-Time-Ordered Correlator in Many-Body Localized Systems, arXiv:1608.01091 [INSPIRE].
  56. [56]
    Y. Chen, Quantum Logarithmic Butterfly in Many Body Localization, arXiv:1608.02765 [INSPIRE].
  57. [57]
    X. Chen, T. Zhou, D.A. Huse and E. Fradkin, Out-of-time-order correlations in many-body localized and thermal phases, arXiv:1610.00220 [INSPIRE].
  58. [58]
    A. Larkin and Y.N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Sov. JETP 28 (1969) 1200.ADSGoogle Scholar
  59. [59]
    J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.ADSCrossRefGoogle Scholar
  60. [60]
    H. Tasaki, From Quantum Dynamics to the Canonical Distribution: General Picture and a Rigorous Example, Phys. Rev. Lett. 80 (1998) 1373 [cond-mat/9707253].
  61. [61]
    M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452 (2008) 854 [arXiv:0708.1324].ADSCrossRefGoogle Scholar
  62. [62]
    H. Kim, T.N. Ikeda, D.A. Huse, Testing whether all eigenstates obey the Eigenstate Thermalization Hypothesis, Phys. Rev. E 90 (2014) 052105 [arXiv:1408.0535].ADSGoogle Scholar
  63. [63]
    E. Altman and R. Vosk, Universal dynamics and renormalization in many body localized systems, Ann. Rev. Condens. Matter Phys. 6 (2015) 383 [arXiv:1408.2834].ADSCrossRefGoogle Scholar
  64. [64]
    M. Serbyn, Z. Papić and D.A. Abanin, Local conservation laws and the structure of the many-body localized states, Phys. Rev. Lett. 111 (2013) 127201 [arXiv:1305.5554].ADSCrossRefGoogle Scholar
  65. [65]
    D.A. Huse, R. Nandkishore and V. Oganesyan, Phenomenology of fully many-body-localized systems, Phys. Rev. B 90 (2014) 174202 [arXiv:1408.4297] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    R. Vosk and E. Altman, Many-body localization in one dimension as a dynamical renormalization group fixed point, Phys. Rev. Lett. 110 (2013) 067204 [arXiv:1205.0026].ADSCrossRefGoogle Scholar
  67. [67]
    A. Das, Supersymmetry and Finite Temperature, Physics A 158 (1989) 1.ADSGoogle Scholar
  68. [68]
    S. Iyer, V. Oganesyan, G. Refael and D.A. Huse, Many-Body Localization in a Quasiperiodic System, Phys. Rev. B 87 (2013) 134202 [arXiv:1212.4159].ADSCrossRefGoogle Scholar
  69. [69]
    S. Nag and A. Garg, Many-body mobility edge in a quasi periodic system, arXiv:1701.00236.
  70. [70]
    S. Aubry and G. Andrè, Analyticity Breaking and Anderson Localization in incommensurate lattices, Ann. Israel Phys. Soc. 3 (1980) 133.MathSciNetzbMATHGoogle Scholar
  71. [71]
    A. Kitaev, A simple model of quantum holography, talks at the KITP 2015, Santa Barbara U.S.A. (2015).Google Scholar
  72. [72]
    W. Fu, D. Gaiotto, J. Maldacena and S. Sachdev, Supersymmetric Sachdev-Ye-Kitaev models, Phys. Rev. D 95 (2017) 026009 [arXiv:1610.08917] [INSPIRE].ADSGoogle Scholar
  73. [73]
    T. Li, J. Liu, Y. Xin and Y. Zhou, Supersymmetric SYK model and random matrix theory, arXiv:1702.01738 [INSPIRE].
  74. [74]
    L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, hep-th/9605187 [INSPIRE].
  75. [75]
    H. Moriya, Breakdown of ergodicity induced by infinitely many local kinematical supercharges for the Nicolai supersymmetric fermion lattice model, arXiv:1610.09142 [INSPIRE].
  76. [76]
    P. Fendley, K. Schoutens and J. de Boer, Lattice models with N = 2 supersymmetry, Phys. Rev. Lett. 90 (2003) 120402 [hep-th/0210161] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    V.A. Rubakov and V.P. Spiridonov, Parasupersymmetric Quantum Mechanics, Mod. Phys. Lett. A 3 (1988) 1337 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  78. [78]
    J. Beckers and N. Debergh, On parasupersymmetry and remarkable Lie structures, J. Phys . A 23 (1990) L751S.
  79. [79]
    A Khare, Parasupersymmetric quantum mechanics of arbitrary order, J. Phys. A 25 (1992) L749.ADSMathSciNetzbMATHGoogle Scholar
  80. [80]
    M. Stosic and R. Picken, Parasupersymmetric Quantum Mechanics of Order 3 and a Generalized Witten Index, Mod. Phys. Lett. A 20 (2005) 1395 [math-ph/0407019].
  81. [81]
    A. Mostafazadeh, Parasupersymmetric quantum mechanics and indices of Fredholm operators, Int. J. Mod. Phys. A 12 (1997) 2725 [hep-th/9603163] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  82. [82]
    A. Mostafazadeh, Spectrum degeneracy of general (p = 2) parasupersymmetric quantum mechanics and parasupersymmetric topological invariants, Int. J. Mod. Phys. A 11 (1996) 1057 [hep-th/9410180] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Pramod Padmanabhan
    • 1
  • Soo-Jong Rey
    • 1
    • 2
    • 3
  • Daniel Teixeira
    • 4
  • Diego Trancanelli
    • 4
  1. 1.Fields, Gravity & Strings, CTPU, Institute for Basic ScienceDaejeonKorea
  2. 2.School of Physics and Astronomy & Center for Theoretical PhysicsSeoul National UniversitySeoulKorea
  3. 3.Department of Basic SciencesUniversity of Science and TechnologyDaejeonKorea
  4. 4.Institute of PhysicsUniversity of São PauloSão PauloBrazil

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