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Physical review. D/Physical review. D.

Tridiagonalizing random matrices

Vijay Balasubramanian, Javier M. Magán, Qingyue Wu

Published Jun 6, 2023• 60 citations
Publisher

Abstract

The Hungarian physicist Eugene Wigner introduced random matrix models in physics to describe the energy spectra of atomic nuclei. As such, the main goal of random matrix theory (RMT) has been to derive the eigenvalue statistics of matrices drawn from a given distribution. The Wigner approach gives powerful insights into the properties of complex, chaotic systems in thermal equilibrium. Another Hungarian, Cornelius Lanczos, suggested a method of reducing the dynamics of any quantum system to a one-dimensional chain by tridiagonalizing the Hamiltonian relative to a given initial state. In the resulting matrix, the diagonal and off-diagonal Lanczos coefficients control transition amplitudes between elements of a distinguished basis of states. We connect these two approaches to the quantum mechanics of complex systems by deriving analytical formulas relating the potential defining a general RMT, or, equivalently, its density of states, to the Lanczos coefficients and their correlations. In particular, we derive an integral relation between the average Lanczos coefficients and the density of states, and, for polynomial potentials, algebraic equations that determine the Lanczos coefficients from the potential. We obtain these results for generic initial states in the thermodynamic limit. As an application, we compute the time-dependent ``spread complexity'' in thermofield double states and the spectral form factor for Gaussian and non-Gaussian RMTs.

DOI
10.1103/physrevd.107.126001

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The Hungarian physicist Eugene Wigner introduced random matrix models in physics to describe the energy spectra of atomic nuclei. As such, the main goal of random matrix theory (RMT) has been to derive the eigenvalue statistics of matrices drawn from a given distribution. The Wigner approach gives powerful insights into the properties of complex, chaotic systems in thermal equilibrium. Another Hungarian, Cornelius Lanczos, suggested a method of reducing the dynamics of any quantum system to a one-dimensional chain by tridiagonalizing the Hamiltonian relative to a given initial state. In the resulting matrix, the diagonal and off-diagonal Lanczos coefficients control transition amplitudes between elements of a distinguished basis of states. We connect these two approaches to the quantum mechanics of complex systems by deriving analytical formulas relating the potential defining a general RMT, or, equivalently, its density of states, to the Lanczos coefficients and their correlations. In particular, we derive an integral relation between the average Lanczos coefficients and the density of states, and, for polynomial potentials, algebraic equations that determine the Lanczos coefficients from the potential. We obtain these results for generic initial states in the thermodynamic limit. As an application, we compute the time-dependent ``spread complexity'' in thermofield double states and the spectral form factor for Gaussian and non-Gaussian RMTs.
The Hungarian physicist Eugene Wigner introduced random matrix models in physics to describe the energy spectra of atomic nuclei. As such, the main goal of Random Matrix Theory (RMT) has been to derive the eigenvalue statistics of matrices drawn from a given distribution. The Wigner approach gives powerful insights into the properties of complex, chaotic systems in thermal equilibrium. Another Hungarian, Cornelius Lanczos, suggested a method of reducing the dynamics of any quantum system to a one-dimensional chain by tridiagonalizing the Hamiltonian relative to a given initial state. In the resulting matrix, the diagonal and off-diagonal Lanczos coefficients control transition amplitudes between elements of a distinguished basis of states. We connect these two approaches to the quantum mechanics of complex systems by deriving analytical formulae relating the potential defining a general RMT, or, equivalently, its density of states, to the Lanczos coefficients and their correlations. In particular, we derive an integral relation between the average Lanczos coefficients and the density of states, and, for polynomial potentials, algebraic equations that determine the Lanczos coefficients from the potential. We obtain these results for generic initial states in the thermodynamic limit. As an application, we compute the time-dependent ``spread complexity'' in Thermo-Field Double states and the spectral form factor for Gaussian and Non-Gaussian RMTs.
Quantum chaotic systems are conjectured to display a spectrum whose fine-grained features (gaps and correlations) are well described by random matrix theory (RMT). We propose and develop a complementary version of this conjecture: quantum chaotic systems display a Lanczos spectrum whose local means and covariances are well described by RMT. To support this proposal, we first demonstrate its validity in examples of chaotic and integrable systems. We then show that for Haar-random initial states in RMTs the mean and covariance of the Lanczos spectrum suffice to produce the full long-time behavior of general survival probabilities including the spectral form factor, as well as the spread complexity. In addition, for initial states with continuous overlap with energy eigenstates, we analytically find the long-time averages of the probabilities of Krylov basis elements in terms of the mean Lanczos spectrum. This analysis suggests a notion of eigenstate complexity, the statistics of which differentiate integrable systems and classes of quantum chaos. Finally, we clarify the relation between spread complexity and the universality classes of RMT by exploring various values of the Dyson index and Poisson distributed spectra.
We consider the problem of measuring the spread of quantum states and justify the use of the Krylov basis to measure the spread of quantum states as a Hamiltonian system evolves, resulting in a measure of "spread complexity" akin to Krylov operator complexity but applied to states. In this basis, Hamiltonians become tridiagonal. The time-evolved thermofield double states in chaotic systems shows four regimes of this spread complexity: a linear ramp up to a peak that is exponential in the entropy, followed by a slope down to a plateau. We then derive analytical approximations for the mean of the tridiagonal matrix coefficients from the density of states in an general quantum system, as well as for both the mean and covariance for random matrices with a single-trace potential. Other quantities in the long time limit are shown to be computable as well. This is then applied in two ways: We show in examples that the covariance of the tridiagonal coefficients distinguish integrable and chaotic systems. Essentially, we provide a new set of tools capable of probing the detailed spectral statistics of quantum systems. We also find an application of the mean to infer the late time behavior of double scaled SYK. Lastly, as a somewhat separate topic, we shine a spotlight on the differences between the properties of eigenbasis chaos (e.g. the Eignestate Themalization Hypothesis) and spectral chaos (e.g. the eigenvalue repulsion found in chaotic systems). We show that without any constraints on the system or with a exponentially small tolerance in the constraints, these two properties are unrelated. We then provide a numerically-constructed example that shows that even with exact k-locality it may be possible to construct systems where the two properties do not agree. This suggests that careful considerations of typicality, restrictions, and/or the order of limits, may be needed to fully connect eigenbasis and spectral chaos.
Krylov methods have reappeared recently, connecting physically sensible notions of complexity with quantum chaos and quantum gravity. In these developments, the Hamiltonian and the Liouvillian are tridiagonalized so that Schrodinger/Heisenberg time evolution is expressed in the Krylov basis. In the context of Schrodinger evolution, this tridiagonalization has been carried out in Random Matrix Theory. We extend these developments to Heisenberg time evolution, describing how the Liouvillian can be tridiagonalized as well until the end of Krylov space. We numerically verify the analytical formulas both for Gaussian and non-Gaussian matrix models.
We establish a direct correspondence between the Lanczos approach and the orthogonal polynomials approach in random matrix theory. In the large-NNN and continuum limits, the average Lanczos coefficients and the recursion coefficients become equivalent, with the precise mapping b(1−x)=R(x)b(1-x)=\sqrt{R(x)}b(1−x)=R(x)​ and a(1−x)=S(x)a(1-x)=S(x)a(1−x)=S(x). As a result, the two formalisms yield identical expressions for the leading density of states. We further analyze the Krylov dynamics associated with the recursion coefficients and show that the orthogonal polynomials admit a natural interpretation as Krylov polynomials. This picture is realized explicitly in the Gaussian Unitary Ensemble, where all quantities can be computed analytically.
We study the spread complexity of the infinite-temperature thermofield double state in large-NNN random matrix theory and investigate the physical meaning of its Lanczos coefficients bnb_nbn​. The leading density of states defines a family of orthogonal polynomials whose recursion coefficients coincide with the Lanczos coefficients of the associated Krylov dynamics. Starting from the Coulomb-gas equation, we derive a pair of nonlinear relations for these coefficients, which we call the Lanczos equations. For normalizable one-cut ensembles, the Lanczos coefficients approach constants at large Krylov index, rendering the asymptotic Krylov chain translationally invariant and the late-time growth of spread complexity linear, with the growth rate given by the spectral average of the group velocity obtained from a Bloch-wave analysis. For an even potential, we further show that the difference between consecutive squared Lanczos coefficients, bn+12−bn2b_{n+1}^2-b_n^2bn+12​−bn2​, is the generating function for two-legged maps whose marked legs are separated by the exact geodesic distance nnn. Consequently, the spread complexity acceleration operator equals twice the geodesic generating operator. We illustrate these results in the quartic matrix model, where the expansion in the quartic coupling counts tetravalent planar maps, and in the double-scaled Sachdev--Ye--Kitaev model, where the auxiliary Hilbert space can be identified with the physical chord Hilbert space.