Abstract
The Spectral Form Factor (SFF) measures the fluctuations in the density of states of a Hamiltonian. We consider a generalization of the SFF called the Loschmidt Spectral Form Factor, \( \textrm{tr}\left[{e}^{i{H}_1T}\right]\textrm{tr}\left[{e}^{-i{H}_2T}\right] \), for H1 − H2 small. If the ensemble average of the SFF is the variance of the density fluctuations for a single Hamiltonian drawn from the ensemble, the averaged Loschmidt SFF is the covariance for two Hamiltonians drawn from a correlated ensemble. This object is a time-domain version of the parametric correlations studied in the quantum chaos and random matrix literatures. We show analytically that the averaged Loschmidt SFF is proportional to eiλT T for a complex rate λ with a positive imaginary part, showing in a quantitative way that the long-time details of the spectrum are exponentially more sensitive to perturbations than the short-time properties. We calculate λ in a number of cases, including random matrix theory, theories with a single localized defect, and hydrodynamic theories.
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References
F. Haake, Quantum signatures of chaos, Springer, Berlin, Heidelberg, Germany (2010).
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].
M. Mehta, Random matrices, Elsevier Science (2004).
M. Winer and B. Swingle, Hydrodynamic theory of the connected spectral form factor, Phys. Rev. X 12 (2022) 021009 [arXiv:2012.01436] [INSPIRE].
A. Chenu, I.L. Egusquiza, J. Molina-Vilaplana and A. del Campo, Quantum work statistics, Loschmidt echo and information scrambling, Sci. Rep. 8 (2018) 12634 [arXiv:1711.01277] [INSPIRE].
A. Chenu, J. Molina-Vilaplana and A. del Campo, Work statistics, Loschmidt echo and information scrambling in chaotic quantum systems, Quantum 3 (2019) 127.
B.D. Simons and B.L. Altshuler, Universal velocity correlations in disordered and chaotic systems, Phys. Rev. Lett. 70 (1993) 4063 [INSPIRE].
H.A. Weidenmüller, Parametric level correlations in random-matrix models, J. Phys. Cond. Matter 17 (2005) S1881 [math-ph/0412057].
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].
J. Cotler and K. Jensen, A precision test of averaging in AdS/CFT, arXiv:2205.12968 [INSPIRE].
M.V. Berry, Semiclassical theory of spectral rigidity, Proc. Roy. Soc. Lond. A 400 (1985) 229.
M. Sieber and K. Richter, Correlations between periodic orbits and their rôle in spectral statistics, Phys. Scr. T90 (2001) 128.
P. Saad, S.H. Shenker and D. Stanford, A semiclassical ramp in SYK and in gravity, arXiv:1806.06840 [INSPIRE].
A. Chan, A. De Luca and J.T. Chalker, Spectral Lyapunov exponents in chaotic and localized many-body quantum systems, Phys. Rev. Res. 3 (2021) 023118 [arXiv:2012.05295] [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 [arXiv:2009.11863] [INSPIRE].
A.J. Friedman, A. Chan, A. De Luca and J.T. Chalker, Spectral statistics and many-body quantum chaos with conserved charge, Phys. Rev. Lett. 123 (2019) 210603 [arXiv:1906.07736] [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, R. Barney, C.L. Baldwin, V. Galitski and B. Swingle, Spectral form factor of a quantum spin glass, JHEP 09 (2022) 032 [arXiv:2203.12753] [INSPIRE].
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
A. Kitaev, A simple model of quantum holography (part 1), in KITP progr. Entanglement strongly-correlated quantum matter, https://online.kitp.ucsb.edu/online/entangled15/kitaev/, University of California, Santa Barbara, CA, U.S.A., 7 April 2015.
A. Kitaev, A simple model of quantum holography (part 2), in KITP progr. Entanglement strongly-correlated quantum matter, https://online.kitp.ucsb.edu/online/entangled15/kitaev2/, University of California, Santa Barbara, CA, U.S.A., 27 May 2015.
J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
A. Wisniacki, Loschmidt echo, Scholarpedia 7 (2012) 11687.
H. Kohler and C. Recher, Fidelity and level correlations in the transition from regularity to chaos, EPL (Europhys. Lett.) 98 (2012) 10005 [arXiv:1204.1747].
T. Gorin, T. Prosen, T.H. Seligman and M. Žnidarič, Dynamics of Loschmidt echoes and fidelity decay, Phys. Rept. 435 (2006) 33 [quant-ph/0607050].
A. Wisniacki, Loschmidt echo, Scholarpedia 7 (2012) 11687.
F.J. Dyson, A brownian-motion model for the eigenvalues of a random matrix, J. Math. Phys. 3 (1962) 1191 [INSPIRE].
C.H. Joyner and U. Smilansky, Dyson’s brownian-motion model for random matrix theory — revisited. With an appendix by Don Zagier, arXiv:1503.06417.
M. Winer and B. Swingle, Spontaneous symmetry breaking, spectral statistics, and the ramp, Phys. Rev. B 105 (2022) 104509 [arXiv:2106.07674] [INSPIRE].
L.V. Keldysh, Diagram technique for nonequilibrium processes, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 [Sov. Phys. JETP 20 (1965) 1018] [INSPIRE].
A. Kamenev and A. Levchenko, Keldysh technique and nonlinear sigma-model: basic principles and applications, Adv. Phys. 58 (2009) 197 [arXiv:0901.3586] [INSPIRE].
A. Kamenev, Field theory of non-equilibrium systems, Cambridge University Press (2011).
K.-C. Chou, Z.-B. Su, B.-L. Hao and L. Yu, Equilibrium and nonequilibrium formalisms made unified, Phys. Rept. 118 (1985) 1 [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Schwinger-Keldysh formalism. Part I. BRST symmetries and superspace, JHEP 06 (2017) 069 [arXiv:1610.01940] [INSPIRE].
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.
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.
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
V. Rosenhaus, An introduction to the SYK model, J. Phys. A 52 (2019) 323001 [arXiv:1807.03334] [INSPIRE].
R.A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen and S. Sachdev, Thermoelectric transport in disordered metals without quasiparticles: the Sachdev-Ye-Kitaev models and holography, Phys. Rev. B 95 (2017) 155131 [arXiv:1612.00849] [INSPIRE].
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].
M. Crossley, P. Glorioso and H. Liu, Effective field theory of dissipative fluids, JHEP 09 (2017) 095 [arXiv:1511.03646] [INSPIRE].
P. Glorioso, M. Crossley and H. Liu, Effective field theory of dissipative fluids. Part II. Classical limit, dynamical KMS symmetry and entropy current, JHEP 09 (2017) 096 [arXiv:1701.07817] [INSPIRE].
H. Liu and P. Glorioso, Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics, PoS TASI2017 (2018) 008 [arXiv:1805.09331] [INSPIRE].
S. Grozdanov and J. Polonyi, Viscosity and dissipative hydrodynamics from effective field theory, Phys. Rev. D 91 (2015) 105031 [arXiv:1305.3670] [INSPIRE].
P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, J. Phys. A 45 (2012) 473001 [arXiv:1205.5040] [INSPIRE].
S. Dubovsky, L. Hui, A. Nicolis and D.T. Son, Effective field theory for hydrodynamics: thermodynamics, and the derivative expansion, Phys. Rev. D 85 (2012) 085029 [arXiv:1107.0731] [INSPIRE].
S. Endlich, A. Nicolis, R.A. Porto and J. Wang, Dissipation in the effective field theory for hydrodynamics: first order effects, Phys. Rev. D 88 (2013) 105001 [arXiv:1211.6461] [INSPIRE].
J. Cotler and K. Jensen, A precision test of averaging in AdS/CFT, arXiv:2205.12968 [INSPIRE].
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Winer, M., Swingle, B. The Loschmidt spectral form factor. J. High Energ. Phys. 2022, 137 (2022). https://doi.org/10.1007/JHEP10(2022)137
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DOI: https://doi.org/10.1007/JHEP10(2022)137