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.
E.P. Wigner, Gruppentheorie (in German), Vieweg, Berlin Germany (1931) [Group Theory, Academic Press Inc., New York U.S.A. (1959)].
M.V. Lawson, Inverse Semigroups — The Theory of Partial Symmetries, World Scientific, Singapore (1998).
J. Kellendonk and M.V. Lawson, Tiling Semigroups, J. Algebra 224 (2000) 140.
D.P. Di Vincenzo and P.J. Steinhardt, Quasicrystals: The State of the Art, World Scientific, Singapore (1991).
C. Janot, Quasicrystals — A Primer, Clarendon Press, Oxford U.K. (1992).
M. Senechal, Quasicrystals and Geometry, Cambridge University Press, Cambridge U.K. (1995).
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.
R. Exel, D. Goncalves and C. Starling, The tiling C ∗ -algebra viewed as a tight inverse semigroup algebra, arXiv:1106.4535.
J. Kellendonk, Topological equivalence of tilings, J. Math. Phys. 38 (1997) 1823 [cond-mat/9609254].
D. Damanik, A. Gorodetski and W. Yessen, The Fibonacci Hamiltonian, arXiv:1403.7823.
J. Bellissard, A. Bovier and J.-M. Ghez, Gap Labelling Theorems for One Dimensional Discrete Schrodinger Operators, Rev. Math. Phys. 4 (1992) 1.
V.V. Wagner, The theory of generalised heaps and generalised groups, Mat. Sb. (N.S.) 32 (1953) 545.
G.B. Preston, Representations of inverse semi-groups, J. London Math. Soc. 29 (1954) 411.
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].
P. Ramond, Dual Theory for Free Fermions, Phys. Rev. D 3 (1971) 2415 [INSPIRE].
A. Neveu and J.H. Schwarz, Factorizable dual model of pions, Nucl. Phys. B 31 (1971) 86 [INSPIRE].
D.V. Volkov and V.P. Akulov, Is the Neutrino a Goldstone Particle?, Phys. Lett. B 46 (1973) 109 [INSPIRE].
J. Wess and B. Zumino, Supergauge Transformations in Four-Dimensions, Nucl. Phys. B 70 (1974) 39 [INSPIRE].
M.F. Sohnius, Introducing Supersymmetry, Phys. Rept. 128 (1985) 39 [INSPIRE].
E. Witten, Dynamical Breaking of Supersymmetry, Nucl. Phys. B 188 (1981) 513 [INSPIRE].
F. Cooper and B. Freedman, Aspects of Supersymmetric Quantum Mechanics, Annals Phys. 146 (1983) 262 [INSPIRE].
E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].
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].
M.F. Atiyah and I.M. Singer, The index of elliptic operators on compact manifolds, Bull. Am. Math. Soc. 69 (1963) 422.
M. Atiyah, R. Bott and V.K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973) 279.
R. Melrose, The Atiyah-Patodi-Singer Index Theorem, Taylor and Francis, London U.K. (1993).
F. Gesztesy and B. Simon, Topological Invariance of the Witten Index, J. Funct. Anal. 79 (1988) 91.
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].
G. Junker, Supersymmetric Methods in Quantum and Statistical Physics, Springer-Verlag, Heidelberg Germany (1996).
H. Nicolai, Supersymmetry and Spin Systems, J. Phys. A 9 (1976) 1497 [INSPIRE].
H. Moriya, On Supersymmetric Fermion Lattice Systems, Ann. Henri Poincaré 17 (2016) 2199.
P.H. Dondi and H. Nicolai, Lattice Supersymmetry, Nuovo Cim. A 41 (1977) 1 [INSPIRE].
M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.
P.W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109 (1958) 1492 [INSPIRE].
A. Larkin and Y.N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Sov. JETP 28 (1969) 1200.
J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.
H. Tasaki, From Quantum Dynamics to the Canonical Distribution: General Picture and a Rigorous Example, Phys. Rev. Lett. 80 (1998) 1373 [cond-mat/9707253].
M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452 (2008) 854 [arXiv:0708.1324].
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].
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].
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].
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].
A. Das, Supersymmetry and Finite Temperature, Physics A 158 (1989) 1.
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].
S. Nag and A. Garg, Many-body mobility edge in a quasi periodic system, arXiv:1701.00236.
S. Aubry and G. Andrè, Analyticity Breaking and Anderson Localization in incommensurate lattices, Ann. Israel Phys. Soc. 3 (1980) 133.
A. Kitaev, A simple model of quantum holography, talks at the KITP 2015, Santa Barbara U.S.A. (2015).
V.A. Rubakov and V.P. Spiridonov, Parasupersymmetric Quantum Mechanics, Mod. Phys. Lett. A 3 (1988) 1337 [INSPIRE].
J. Beckers and N. Debergh, On parasupersymmetry and remarkable Lie structures, J. Phys . A 23 (1990) L751S.
A Khare, Parasupersymmetric quantum mechanics of arbitrary order, J. Phys. A 25 (1992) L749.
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].
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].
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Padmanabhan, P., Rey, SJ., Teixeira, D. et al. Supersymmetric many-body systems from partial symmetries — integrability, localization and scrambling. J. High Energ. Phys. 2017, 136 (2017). https://doi.org/10.1007/JHEP05(2017)136