Energy filter methods in combination with quantum simulation can efficiently access the properties of quantum many-body systems at finite energy densities [S. Lu , ]. Classically simulating this algorithm with tensor networks can be used to investigate the microcanonical properties of large spin chains, as recently shown by Y. Yang []. Here we extend this strategy to explore the properties of off-diagonal matrix elements of observables in the energy eigenbasis, fundamentally connected to the thermalization behavior and the eigenstate thermalization hypothesis. We test the method on integrable and nonintegrable spin chains of up to 60 sites, much larger than accessible with exact diagonalization. Our results allow us to explore the scaling of the off-diagonal functions with the size and energy difference, and to establish quantitative differences between integrable and nonintegrable cases. Published by the American Physical Society 2024
Energy filter methods in combination with quantum simulation can efficiently access the properties of quantum many-body systems at finite energy densities [S. Lu , ]. Classically simulating this algorithm with tensor networks can be used to investigate the microcanonical properties of large spin chains, as recently shown by Y. Yang []. Here we extend this strategy to explore the properties of off-diagonal matrix elements of observables in the energy eigenbasis, fundamentally connected to the thermalization behavior and the eigenstate thermalization hypothesis. We test the method on integrable and nonintegrable spin chains of up to 60 sites, much larger than accessible with exact diagonalization. Our results allow us to explore the scaling of the off-diagonal functions with the size and energy difference, and to establish quantitative differences between integrable and nonintegrable cases. Published by the American Physical Society 2024
The thermalization phenomenon and many-body quantum statistical properties are studied on the example of several observables in isolated spin-chain systems, both integrable and generic nonintegrable. While diagonal matrix elements for nonintegrable models comply with the eigenstate thermalization hypothesis, the integrable systems show evident deviations and similarity to properties of noninteracting many-fermion models. The finite-size scaling reveals that the crossover between the two regimes is given by a scale closely related to the scattering length. Low-frequency off-diagonal matrix elements related to dc transport quantities also follow in a generic system a behavior analogous to the eigenstate thermalization hypothesis, however, unrelated to one of the diagonal matrix elements.
The eigenstate thermalization hypothesis explains the emergence of the thermodynamic equilibrium in isolated quantum many-body systems by assuming a particular structure of the observable's matrix elements in the energy eigenbasis. Schematically, it postulates that off-diagonal matrix elements are random numbers and the observables can be described by random matrix theory (RMT). To what extent a RMT description applies, more precisely at which energy scale matrix elements of physical operators become truly uncorrelated, is, however, not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matrix elements for Hilbert-space dimensions beyond those accessible by exact diagonalization. Our analysis is based on the evaluation of higher moments of operator submatrices, defined within energy windows of varying width. Considering nonintegrable quantum spin chains, we observe that matrix elements remain correlated even for narrow energy windows corresponding to timescales of the order of thermalization time of the respective observables. We also demonstrate that such residual correlations between matrix elements are reflected in the dynamics of out-of-time-ordered correlation functions.
Using numerical exact diagonalization, we study matrix elements of a local spin operator in the eigenbasis of two different nonintegrable quantum spin chains. Our emphasis is on the question to what extent local operators can be represented as random matrices and, in particular, to what extent matrix elements can be considered as uncorrelated. As a main result, we show that the eigenvalue distribution of band submatrices at a fixed energy density is a sensitive probe of the correlations between matrix elements. We find that, on the scales where the matrix elements are in a good agreement with all standard indicators of the eigenstate thermalization hypothesis, the eigenvalue distribution still exhibits clear signatures of the original operator, implying correlations between matrix elements. Moreover, we demonstrate that at much smaller energy scales, the eigenvalue distribution approximately assumes the universal semicircle shape, indicating transition to the random-matrix behavior, and in particular that matrix elements become uncorrelated.
We investigate off-diagonal matrix elements of local operators in integrable spin chains, focusing on the isotropic spin- 1 / 2 Heisenberg chain ( X X X chain). We employ state-of-the-art algebraic Bethe ansatz results, which allow us to efficiently compute matrix elements of operators with support up to two sites between generic energy eigenstates. We consider both matrix elements between eigenstates that are in the same thermodynamic macrostate, as well as eigenstates that belong to different macrostates. In the former case, focusing on thermal states we numerically show that matrix elements are compatible with the exponential decay as exp ( − L | M i j O | ) . The probability distribution functions of M i j O depend on the observable and on the macrostate, and are well described by Gumbel distributions. On the other hand, matrix elements between eigenstates in different macrostates decay faster as exp ( − | M ′ i j O | L 2 ) , with M ′ i j O , again, compatible with a Gumbel distribution.
For an isolated generic quantum system out of equilibrium, the long time average of observables is given by the diagonal ensemble, i.e., the mixed state with the same probability for energy eigenstates as the initial state but without coherences between different energies. In this work we present a method to approximate the diagonal ensemble using tensor networks. Instead of simulating the real time evolution, we adapt a filtering scheme introduced earlier [M. C. Ba\~nuls, D. A. Huse, and J. I. Cirac, Phys. Rev. B 101, 144305 (2020)] to this problem. We analyze the performance of the method on a nonintegrable spin chain, for which we observe that local observables converge towards thermal values polynomially with the inverse width of the filter.
We study the statistics of matrix elements of local operators in the basis of energy eigenstates in a paradigmatic, integrable, many-particle quantum theory, the Lieb-Liniger model of bosons with repulsive delta-function interactions. Using methods of quantum integrability, we determine the scaling of matrix elements with system size. As a consequence of the extensive number of conservation laws, the structure of matrix elements is fundamentally different from, and much more intricate than, the predictions of the eigenstate thermalization hypothesis for generic models. We uncover an interesting connection between this structure for local operators in interacting integrable models and the one for local operators that are not local with respect to the elementary excitations in free theories. We find that typical off-diagonal matrix elements ⟨ μ | O | λ ⟩ in the same macrostate scale as exp ( − c O L ln ( L ) − L M μ , λ O ) , where the probability distribution function for M μ , λ O is well described by Fréchet distributions and c O depends only on macrostate information. In contrast, typical off-diagonal matrix elements between two different macrostates scale as exp ( − d O L 2 ) , where d O depends only on macrostate information. Diagonal matrix elements depend only on macrostate information up to finite-size corrections. Published by the American Physical Society 2024