Abstract
We argue that in a large class of disordered quantum many-body systems, the late time dynamics of time-dependent correlation functions is captured by random matrix theory, specifically the energy eigenvalue statistics of the corresponding ensemble of disordered Hamiltonians. We find that late time correlation functions approximately factorize into a time-dependent piece, which only depends on spectral statistics of the Hamiltonian ensemble, and a time-independent piece, which only depends on the data of the constituent operators of the correlation function. We call this phenomenon “spectral decoupling”, which signifies a dynamical onset of random matrix theory in correlation functions. A key diagnostic of spectral decoupling is k-invariance, which we refine and study in detail. Particular emphasis is placed on the role of symmetries, and connections between k-invariance, scrambling, and OTOCs. Disordered Pauli spin systems, as well as the SYK model and its variants, provide a rich source of disordered quantum many-body systems with varied symmetries, and we study k-invariance in these models with a combination of analytics and numerics.
Highlights
New connections between many-body quantum chaos and random matrix theory (RMT) have emerged over the last several years [1,2,3,4,5,6,7,8,9,10,11,12]
We argue that in a large class of disordered quantum many-body systems, the late time dynamics of time-dependent correlation functions is captured by random matrix theory, the energy eigenvalue statistics of the corresponding ensemble of disordered Hamiltonians
We find that late time correlation functions approximately factorize into a time-dependent piece, which only depends on spectral statistics of the Hamiltonian ensemble, and a time-independent piece, which only depends on the data of the constituent operators of the correlation function
Summary
New connections between many-body quantum chaos and random matrix theory (RMT) have emerged over the last several years [1,2,3,4,5,6,7,8,9,10,11,12]. We can predict the time dependence of disorder-averaged, highly non-local correlation functions by calculating energy eigenvalue correlations of the corresponding Hamiltonian ensemble. We can calculate energy eigenvalue correlations of a Hamiltonian ensemble by studying the dynamics of disorder-averaged, highly non-local correlation functions. It is possible that correlation functions containing O(t) will satisfy an equation like eq (1.1) after sufficiently long times In this way, we can imagine relating RMT energy eigenvalue statistics to the late time dynamics of correlation functions of initially local operators. In this paper we make the above idea precise, and study how late time dynamics of correlation functions of disordered systems are captured by RMT energy eigenvalue statistics via spectral decoupling. The appendices contain various formulas and derivations, including the first two moments of the Haar unitary, Haar orthogonal, and Haar symplectic ensembles
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