Abstract
It is widely expected that systems which fully thermalize are chaotic in the sense of exhibiting random-matrix statistics of their energy level spacings, whereas integrable systems exhibit Poissonian statistics. In this paper, we investigate a third class: spin glasses. These systems are partially chaotic but do not achieve full thermalization due to large free energy barriers. We examine the level spacing statistics of a canonical infinite-range quantum spin glass, the quantum p-spherical model, using an analytic path integral approach. We find statistics consistent with a direct sum of independent random matrices, and show that the number of such matrices is equal to the number of distinct metastable configurations — the exponential of the spin glass “complexity” as obtained from the quantum Thouless-Anderson-Palmer equations. We also consider the statistical properties of the complexity itself and identify a set of contributions to the path integral which suggest a Poissonian distribution for the number of metastable configurations. Our results show that level spacing statistics can probe the ergodicity-breaking in quantum spin glasses and provide a way to generalize the notion of spin glass complexity beyond models with a semi-classical limit.
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References
J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.
M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.
M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452 (2008) 854.
H. Liu and P. Glorioso, Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics, PoS TASI2017 (2018) 008 [arXiv:1805.09331] [INSPIRE].
M. Crossley, P. Glorioso and H. Liu, Effective field theory of dissipative fluids, JHEP 09 (2017) 095 [arXiv:1511.03646] [INSPIRE].
S. Grozdanov and J. Polonyi, Viscosity and dissipative hydrodynamics from effective field theory, Phys. Rev. D 91 (2015) 105031 [arXiv:1305.3670] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Effective Action for Relativistic Hydrodynamics: Fluctuations, Dissipation, and Entropy Inflow, JHEP 10 (2018) 194 [arXiv:1803.11155] [INSPIRE].
K. Jensen, N. Pinzani-Fokeeva and A. Yarom, Dissipative hydrodynamics in superspace, JHEP 09 (2018) 127 [arXiv:1701.07436] [INSPIRE].
O. Bohigas, M.J. Giannoni and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52 (1984) 1 [INSPIRE].
F.J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3 (1962) 140 [INSPIRE].
M. Mehta, Random Matrices, Elsevier Science, Amsterdam, The Netherlands (2004).
T. Guhr, A. Müller-Groeling and H.A. Weidenmüller, Random matrix theories in quantum physics: Common concepts, Phys. Rept. 299 (1998) 189 [cond-mat/9707301] [INSPIRE].
L. D’Alessio, Y. Kafri, A. Polkovnikov and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65 (2016) 239 [arXiv:1509.06411] [INSPIRE].
L.F. Santos and M. Rigol, Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization, Phys. Rev. E 81 (2010) 036206.
A. Chan, A. De Luca and J.T. Chalker, Spectral statistics in spatially extended chaotic quantum many-body systems, Phys. Rev. Lett. 121 (2018) 060601 [arXiv:1803.03841] [INSPIRE].
S. Moudgalya, A. Prem, D.A. Huse and A. Chan, Spectral statistics in constrained many-body quantum chaotic systems, Phys. Rev. Res. 3 (2021) 023176.
M. Schiulaz, E.J. Torres-Herrera and L.F. Santos, Thouless and relaxation time scales in many-body quantum systems, Phys. Rev. B 99 (2019) 174313 [arXiv:1807.07577] [INSPIRE].
D. Roy and T. Prosen, Random Matrix Spectral Form Factor in Kicked Interacting Fermionic Chains, Phys. Rev. E 102 (2020) 060202 [arXiv:2005.10489] [INSPIRE].
M. Winer and B. Swingle, Hydrodynamic Theory of the Connected Spectral form Factor, Phys. Rev. X 12 (2022) 021009 [arXiv:2012.01436] [INSPIRE].
M. Winer and B. Swingle, Spontaneous symmetry breaking, spectral statistics, and the ramp, Phys. Rev. B 105 (2022) 104509 [arXiv:2106.07674] [INSPIRE].
D. Roy, D. Mishra and T. Prosen, Spectral form factor in a minimal bosonic model of many-body quantum chaos, arXiv:2203.05439 [INSPIRE].
K. Binder and A.P. Young, Spin glasses: Experimental facts, theoretical concepts, and open questions, Rev. Mod. Phys. 58 (1986) 801 [INSPIRE].
M. Mézard, G. Parisi and M.A. Virasoro, Spin Glass Theory and Beyond, World Scientific, Singapore (1987).
K.H. Fischer and J.A. Hertz, Spin Glasses, Cambridge University Press, Cambridge, U.K. (1991).
H. Nishimori, Statistical Physics of Spin Glasses and Information Processing, Oxford University Press, Oxford, U.K. (2001).
T. Castellani and A. Cavagna, Spin-glass theory for pedestrians, J. Stat. Mech. 2005 (2005) P05012.
M. Mézard and A. Montanari, Information, Physics, and Computation, Oxford University Press, Oxford, U.K. (2009).
D.L. Stein and C.M. Newman, Spin Glasses and Complexity, Princeton University Press, Princeton, U.S.A. (2013).
P. Saad, S.H. Shenker and D. Stanford, A semiclassical ramp in SYK and in gravity, arXiv:1806.06840 [INSPIRE].
P. Saad, Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity, arXiv:1910.10311 [INSPIRE].
M. Winer, S.-K. Jian and B. Swingle, An exponential ramp in the quadratic Sachdev-Ye-Kitaev model, Phys. Rev. Lett. 125 (2020) 250602 [arXiv:2006.15152] [INSPIRE].
S. Müller, S. Heusler, P. Braun, F. Haake and A. Altland, Periodic-orbit theory of universality in quantum chaos, Phys. Rev. E 72 (2005) 046207.
Y. Chen, Spectral form factor for free large N gauge theory and strings, JHEP 06 (2022) 137 [arXiv:2202.04741] [INSPIRE].
J.S. Cotler et al., Black Holes and Random Matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
E. Brézin and S. Hikami, Spectral form factor in a random matrix theory, Phys. Rev. E 55 (1997) 4067.
K. Papadodimas and S. Raju, Local operators in the eternal black hole, Phys. Rev. Lett. 115 (2015) 211601.
A. Altland and M.R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55 (1997) 1142 [cond-mat/9602137] [INSPIRE].
T. Tao, Topics in Random Matrix Theory, Graduate studies in mathematics, American Mathematical Society, Providence, U.S.A. (2012).
E. Wigner and J. Griffin, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Pure and applied Physics, Academic Press, Cambridge, U.S.A. (1959).
T. Anous and F.M. Haehl, The quantum p-spin glass model: a user manual for holographers, J. Stat. Mech. 2111 (2021) 113101 [arXiv:2106.03838] [INSPIRE].
T.R. Kirkpatrick and D. Thirumalai, Dynamics of the structural glass transition and the p-spin interaction spin-glass model, Phys. Rev. Lett. 58 (1987) 2091.
A. Crisanti, H. Horner and H.J. Sommers, The spherical p-spin interaction spin-glass model, Z. Phys. B 92 (1993) 257.
L.F. Cugliandolo and J. Kurchan, Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model, Phys. Rev. Lett. 71 (1993) 173.
A. Barrat, R. Burioni and M. Mézard, Dynamics within metastable states in a mean-field spin glass, J. Phys. A 29 (1996) L81.
B. Altshuler, H. Krovi and J. Roland, Anderson localization makes adiabatic quantum optimization fail, Proc. Nat. Acad. Sci. 107 (2010) 12446.
V. Bapst, L. Foini, F. Krzakala, G. Semerjian and F. Zamponi, The quantum adiabatic algorithm applied to random optimization problems: The quantum spin glass perspective, Phys. Rept. 523 (2013) 127.
B. Zhao, M.C. Kerridge and D.A. Huse, Three species of Schrödinger cat states in an infinite-range spin model, Phys. Rev. E 90 (2014) 022104.
C.L. Baldwin and C.R. Laumann, Quantum algorithm for energy matching in hard optimization problems, Phys. Rev. B 97 (2018) 224201.
V.N. Smelyanskiy, K. Kechedzhi, S. Boixo, S.V. Isakov, H. Neven and B. Altshuler, Nonergodic delocalized states for efficient population transfer within a narrow band of the energy landscape, Phys. Rev. X 10 (2020) 011017.
D.J. Thouless, P.W. Anderson and R.G. Palmer, Solution of ‘solvable model of a spin glass’, Phil. Mag. 35 (1977) 593.
H. Ishii and T. Yamamoto, Effect of a transverse field on the spin glass freezing in the Sherrington-Kirkpatrick model, J. Phys. C 18 (1985) 6225.
D. Thirumalai, Q. Li and T.R. Kirkpatrick, Infinite-range Ising spin glass in a transverse field, J. Phys. A 22 (1989) 3339.
Y.Y. Goldschmidt, Solvable model of the quantum spin glass in a transverse field, Phys. Rev. B 41 (1990) 4858.
G. Büttner and K.D. Usadel, Replica-symmetry breaking for the Ising spin glass in a transverse field, Phys. Rev. B 42 (1990) 6385.
L.F. Cugliandolo and G. Lozano, Real-time nonequilibrium dynamics of quantum glassy systems, Phys. Rev. B 59 (1999) 915.
L.F. Cugliandolo, D.R. Grempel and C.A. da Silva Santos, Imaginary-time replica formalism study of a quantum spherical p-spin-glass model, Phys. Rev. B 64 (2001) 144031.
C.R. Laumann, A. Pal and A. Scardicchio, Many-body mobility edge in a mean-field quantum spin glass, Phys. Rev. Lett. 113 (2014) 200405.
C.L. Baldwin, C.R. Laumann, A. Pal and A. Scardicchio, Clustering of nonergodic eigenstates in quantum spin glasses, Phys. Rev. Lett. 118 (2017) 127201.
G. Biroli, D. Facoetti, M. Schiró, M. Tarzia and P. Vivo, Out-of-equilibrium phase diagram of the quantum random energy model, Phys. Rev. B 103 (2021) 014204.
L.F. Cugliandolo and G. Lozano, Quantum aging in mean-field models, Phys. Rev. Lett. 80 (1998) 4979.
A. Crisanti and H.J. Sommers, The spherical p-spin interaction spin glass model: the statics, Z. Phys. B 87 (1992) 341.
E. Gardner, Spin glasses with p-spin interactions, Nucl. Phys. B 257 (1985) 747 [INSPIRE].
T.M. Nieuwenhuizen and F. Ritort, Quantum phase transition in spin glasses with multi-spin interactions, Physica A 250 (1998) 8.
V. Dobrosavljevic and D. Thirumalai, 1/p expansion for a p-spin interaction spin-glass model in a transverse field, J. Phys. A 23 (1990) L767.
L. De Cesare, K. Lukierska-Walasek, I. Rabuffo and K. Walasek, Replica-symmetry breaking and quantum fluctuation effects in the p-spin interaction spin-glass model with a transverse field, J. Phys. A 29 (1996) 1605.
S. Thomson, P. Urbani and M. Schiró, Quantum quenches in isolated quantum glasses out of equilibrium, Phys. Rev. Lett. 125 (2020) 120602.
G. Biroli and L.F. Cugliandolo, Quantum Thouless-Anderson-Palmer equations for glassy systems, Phys. Rev. B 64 (2001) 014206.
A. Crisanti and H. Sommers, Thouless-Anderson-Palmer approach to the spherical p-spin spin glass model, J. Phys., I 5 (1995) 805.
R.L. Graham, D.E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley Publishing Company (1994).
G. Biroli and J. Kurchan, Metastable states in glassy systems, Phys. Rev. E 64 (2001) 016101.
D. Facoetti, G. Biroli, J. Kurchan and D.R. Reichman, Classical Glasses, Black Holes, and Strange Quantum Liquids, Phys. Rev. B 100 (2019) 205108 [arXiv:1906.09228] [INSPIRE].
L.F. Cugliandolo, G.S. Lozano, N. Nessi, M. Picco and A. Tartaglia, Quenched dynamics of classical isolated systems: the spherical spin model with two-body random interactions or the Neumann integrable model, J. Stat. Mech. 2018 (2018) 063206.
Y. Liao, A. Vikram and V. Galitski, Many-body level statistics of single-particle quantum chaos, Phys. Rev. Lett. 125 (2020) 250601 [arXiv:2005.08991] [INSPIRE].
T. Plefka, Convergence condition of the TAP equation for the infinite-ranged Ising spin glass model, J. Phys. A 15 (1982) 1971.
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Winer, M., Barney, R., Baldwin, C.L. et al. Spectral form factor of a quantum spin glass. J. High Energ. Phys. 2022, 32 (2022). https://doi.org/10.1007/JHEP09(2022)032
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DOI: https://doi.org/10.1007/JHEP09(2022)032