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arXiv:2305.07505 · Journal of High Energy Physics

Sparse random matrices and Gaussian ensembles with varying randomness

Takanori Anegawa, Norihiro Iizuka, Arkaprava Mukherjee, Sunil Kumar Sake, Sandip P. Trivedi

Published Nov 30, 2023• 9 citations
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Abstract

Abstract We study a system of N qubits with a random Hamiltonian obtained by drawing coupling constants from Gaussian distributions in various ways. This results in a rich class of systems which include the GUE and the fixed q SYK theories. Our motivation is to understand the system at large N. In practice most of our calculations are carried out using exact diagonalisation techniques (up to N = 24). Starting with the GUE, we study the resulting behaviour as the randomness is decreased. While in general the system goes from being chaotic to being more ordered as the randomness is decreased, the changes in various properties, including the density of states, the spectral form factor, the level statistics and out-of-time-ordered correlators, reveal interesting patterns. Subject to the limitations of our analysis which is mainly numerical, we find some evidence that the behaviour changes in an abrupt manner when the number of non-zero independent terms in the Hamiltonian is exponentially large in N. We also study the opposite limit of much reduced randomness obtained in a local version of the SYK model where the number of couplings scales linearly in N, and characterise its behaviour. Our investigation suggests that a more complete theoretical analysis of this class of systems will prove quite worthwhile.

DOI
10.1007/jhep11(2023)234

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Abstract We study a system of N qubits with a random Hamiltonian obtained by drawing coupling constants from Gaussian distributions in various ways. This results in a rich class of systems which include the GUE and the fixed q SYK theories. Our motivation is to understand the system at large N. In practice most of our calculations are carried out using exact diagonalisation techniques (up to N = 24). Starting with the GUE, we study the resulting behaviour as the randomness is decreased. While in general the system goes from being chaotic to being more ordered as the randomness is decreased, the changes in various properties, including the density of states, the spectral form factor, the level statistics and out-of-time-ordered correlators, reveal interesting patterns. Subject to the limitations of our analysis which is mainly numerical, we find some evidence that the behaviour changes in an abrupt manner when the number of non-zero independent terms in the Hamiltonian is exponentially large in N. We also study the opposite limit of much reduced randomness obtained in a local version of the SYK model where the number of couplings scales linearly in N, and characterise its behaviour. Our investigation suggests that a more complete theoretical analysis of this class of systems will prove quite worthwhile.
In this work, we consider a model of a subsystem interacting with a reservoir and study dynamics of entanglement assuming that the overall time-evolution is governed by non-integrable Hamiltonians. We also compare to an ensemble of Integrable Hamiltonians. To do this, we make use of unitary invariant ensembles of random matrices with either Wigner-Dyson or Poissonian distributions of energy. Using the theory of Weingarten functions, we derive universal average time evolution of the reduced density matrix and the purity and compare these results with several physical Hamiltonians: randomized versions of the transverse field Ising and XXZ models, Spin Glass and, Central Spin and SYK model. The theory excels at describing the latter two. Along the way, we find general expressions for exponential n-point correlation functions in the gas of GUE eigenvalues.
Abstract We use random matrix theory to explore late-time chaos in supersymmetric quantum mechanical systems. Motivated by the recent study of supersymmetric SYK models and their random matrix classification, we consider the Wishart-Laguerre unitary ensemble and compute the spectral form factors and frame potentials to quantify chaos and randomness. Compared to the Gaussian ensembles, we observe the absence of a dip regime in the form factor and a slower approach to Haar-random dynamics. We find agreement between our random matrix analysis and predictions from the supersymmetric SYK model, and discuss the implications for supersymmetric chaotic systems.
We use random matrix theory to explore late-time chaos in supersymmetric quantum mechanical systems. Motivated by the recent study of supersymmetric SYK models and their random matrix classification, we consider the Wishart-Laguerre unitary ensemble and compute the spectral form factors and frame potentials to quantify chaos and randomness. Compared to the Gaussian ensembles, we observe the absence of a dip regime in the form factor and a slower approach to Haar-random dynamics. Here we find agreement between our random matrix analysis and predictions from the supersymmetric SYK model, and discuss the implications for supersymmetric chaotic systems.
We study the Wishart-Sachdev-Ye-Kitaev (WSYK) model consisting of two $\stackrel{^}{q}$-body Sachdev-Ye-Kitaev (SYK) models with general complex couplings, one the Hermitian conjugate of the other, living in off-diagonal blocks of a larger WSYK Hamiltonian. The spectrum is positive with a hard edge at zero energy. We employ diagrammatic and combinatorial techniques to compute analytically the low-order moments of the Hamiltonian. In the limit of large number NNN of Majoranas, we have found striking similarities with the moments of the weight function of the Al-Salam-Chihara QQQ-Laguerre polynomials. For $\stackrel{^}{q}=3$, 4, the QQQ-Laguerre prediction, with $Q=Q(\stackrel{^}{q},N)$ also computed analytically, agrees well with exact diagonalization results for $30<N\ensuremath{\le}34$ while we observe some deviations for $\stackrel{^}{q}=2$. The most salient feature of the spectral density is that, for odd $\stackrel{^}{q}$, low-energy excitations grow as a stretched exponential, with a functional form different from that of the supersymmetric SYK model. For $\stackrel{^}{q}=4$, a detailed analysis of level statistics reveals quantum chaotic dynamics even for time scales substantially shorter than the Heisenberg time. More specifically, the spacing ratios in the bulk of the spectrum and the microscopic spectral density and the number variance close to the hard edge are very well approximated by that of an ensemble of random matrices that, depending on NNN, belong to the chiral or superconducting universality classes. In particular, we report the first realization of level statistics belonging to the chiral Gaussian Unitary Ensemble universality class, which completes the tenfold-way classification in the SYK model.
The SYK model involves N Majorana fermions in 1+0 dimensions with quenched Gaussian dis- order and proves to be exactly solvable in the large N limit at strong coupling. It has been initially proposed by Sachdev and Ye as a model of condensed matter and later gained some interest as a toy model of AdS/CFT correspondence, thanks to the work of Kitaev. On the other side, random tensors are generalisations of random matrices to objets that carry more than two indices. It turns out that the SYK model and random tensors involve a special class of Feynman graphs known as "melons". We will briefly review both constructions. Then, we will show how non Gaussian disorder can be reduced to a Gaussian one, treating the coupling as a random tensor, thanks to Gurau’s Gaussian universality result.
We show that non-Hermitian Ginibre random matrix behaviors emerge in spatially extended many-body quantum chaotic systems in the space direction, just as Hermitian random matrix behaviors emerge in chaotic systems in the time direction. Starting with translational invariant models, which can be associated with dual transfer matrices with complex-valued spectra, we show that the linear ramp of the spectral form factor necessitates that the dual spectra have nontrivial correlations, which in fact fall under the universality class of the Ginibre ensemble, demonstrated by computing the level spacing distribution and the dissipative spectral form factor. As a result of this connection, the exact spectral form factor for the Ginibre ensemble can be used to universally describe the spectral form factor for translational invariant many-body quantum chaotic systems in the scaling limit where t and L are large, while the ratio between L and L_{Th}, the many-body Thouless length is fixed. With appropriate variations of Ginibre models, we analytically demonstrate that our claim generalizes to models without translational invariance as well. The emergence of the Ginibre ensemble is a genuine consequence of the strongly interacting and spatially extended nature of the quantum chaotic systems we consider, unlike the traditional emergence of Hermitian random matrix ensembles.