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Physical Review Letters

Exact description of spectral correlators by a quantum one-dimensional model with inverse-square interaction

Benjamin D. Simons, Patrick A. Lee, B. L. Altshuler

Published Jun 28, 1993• 123 citations
Publisher

Abstract

The universal correlations which describe the response of energy levels of a disordered metallic grain to an arbitrary perturbation are shown to be equivalent to time-dependent correlations of a one-dimensional quantum Hamiltonian with inverse-square interaction. These resutls establish a direct connection between a strongly interacting quantum Hamiltonian, the nonlinear \ensuremath{\sigma} model of disordered electronic systems, and quantum chaotic spectra. As a consequence we have an expression for the correlation function of the quantum Hamiltonian which we believe to be exact for all space and time.

DOI
10.1103/physrevlett.70.4122

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The universal correlations which describe the response of energy levels of a disordered metallic grain to an arbitrary perturbation are shown to be equivalent to time-dependent correlations of a one-dimensional quantum Hamiltonian with inverse-square interaction. These resutls establish a direct connection between a strongly interacting quantum Hamiltonian, the nonlinear \ensuremath{\sigma} model of disordered electronic systems, and quantum chaotic spectra. As a consequence we have an expression for the correlation function of the quantum Hamiltonian which we believe to be exact for all space and time.
The spectra of nonintegrable or quantum chaotic systems with unitary or orthogonal symmetry were recently shown to display universality in the response of energy levels to arbitrary perturbations. We show that this type of universality extends to symplectic ensembles. In doing so, we obtain an exact expression for the autocorrelator of particle density fluctuations of a system of one-dimensional particles interacting through a repulsive 1/r2{\mathit{r}}^{2}r2 pairwise potential.
We study chaotic many-body quantum dynamics in a minimal model with spatial structure and local interactions. It has a time-independent Hamiltonian, in contrast to quantum circuits and Brownian models, and is simple at the single-site level, in contrast to Sachdev-Ye-Kitaev chains. It is analytically tractable for large local Hilbert space dimension and weak intersite coupling. In this limit, we show that energy dynamics is described by a classical master equation and is diffusive. We also show that the spectral form factor can be expressed exactly in terms of the solution to this master equation. For a two-site system, we obtain closed-form expressions for both the two-point correlator of energy density and the spectral form factor, in essentially perfect agreement with numerical simulations. For an L -site system we show at late times how a linear ramp emerges in the spectral form factor, as universally expected from level repulsion in chaotic quantum systems. Conversely, at earlier times we identify two distinct mechanisms for an increase of the spectral form factor above its ramp value. One of these is associated with energy diffusion and is effective until the Thouless time, which varies as L 2 . The other involves contributions like those that would appear if the system were composed of many uncoupled subsystems: They generate a large enhancement of the spectral form factor and are suppressed on a timescale varying as ( ln L ) 2 . Besides being exact for the limit considered, we believe our approach provides the natural approximation even for small local Hilbert space dimension and strong intersite coupling. We present a numerical study of a spin-half chain, finding an early-time enhancement of the spectral form factor which is qualitatively similar to that in our solvable model.
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.
This work demonstrates that sparse long-range random bonds on a one-dimensional lattice alone can generate quantum-chaotic spectral correlations and also drive a localization transition in a noninteracting single-particle Hamiltonian. The model is a one-dimensional ring in which each pair of sites is connected independently with a probability pij=dij−(1+σ)p_{ij}= d_{ij}^{-(1+σ)}pij​=dij−(1+σ)​. Each bond carries identical unit hopping and on-site disorder is absent. Despite the absence of on-site disorder and interaction, the model displays quantum chaotic spectra with Gaussian orthogonal ensemble (GOE) level statistics at small σσσ and localized eigenstates with Poisson statistics at larger σσσ. The transition occurs in the range $ 0.80 \lesssim σ_c \lesssim 0.85,farabovethesummabilitythresholdofthemeanhoppingprofile(, far above the summability threshold of the mean hopping profile (,farabovethesummabilitythresholdofthemeanhoppingprofile(σ=0$). A Gaussian field theory retaining only the mean and variance of the Bernoulli bonds instead predicts a threshold at σ=1σ=1σ=1, suggesting that higher cumulants are infrared-relevant. Our findings hint towards a universality class that is distinct from both the power-law random banded matrix model and the standard Anderson transition.
We determine the exact time-dependent non-idempotent one-particle reduced density matrix and its spectral decomposition for a harmonically confined two-particle correlated one-dimensional system when the interaction terms in the Schrödinger Hamiltonian are changed abruptly. Based on this matrix in coordinate space we derivea precise condition for the equivalence of the purity and the overlap-square of the correlated and non-correlated wave functions as the system evolves in time. This equivalence holds only if the interparticle interactions are affected, while the confinement terms are unaffected within the stability range of the system. Under this condition we also analyze various time-dependent measures of entanglement and demonstrate that, depending on the magnitude of the changes made in the Schrödinger Hamiltonian, periodic, logarithmically incresing or constant value behavior of the von Neumann entropy can occur.