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arXiv:1901.10797 · SciPost Physics

On the size of the space spanned by a nonequilibrium state in a quantum spin lattice system

Maurizio Fagotti

Published May 15, 2019• 6 citations
arXivHTMLPDF

Abstract

We consider the time evolution of a state in an isolated quantum spin lattice system with energy cumulants proportional to the number of the sites L^d L d . We compute the distribution of the eigenvalues of the time averaged state over a time window [t_0,t_0+t] [ t 0 , t 0 + t ] in the limit of large L. This allows us to infer the size of a subspace that captures time evolution in [t_0,t_0+t] [ t 0 , t 0 + t ] with an accuracy 1-\epsilon 1 − ϵ . We estimate the size to be \frac{\sqrt{2{\mathfrak e}_2}}{\pi}erf^{-1}(1-\epsilon) L^{\frac{d}{2}}t 2 𝔢 2 π e r f − 1 ( 1 − ϵ ) L d 2 t , where {\mathfrak e}_2 𝔢 2 is the energy variance per site, and erf^{-1} e r f − 1 is the inverse error function.

DOI
10.21468/scipostphys.6.5.059

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We consider the time evolution of a state in an isolated quantum spin lattice system with energy cumulants proportional to the number of the sites L^d L d . We compute the distribution of the eigenvalues of the time averaged state over a time window [t_0,t_0+t] [ t 0 , t 0 + t ] in the limit of large L. This allows us to infer the size of a subspace that captures time evolution in [t_0,t_0+t] [ t 0 , t 0 + t ] with an accuracy 1-\epsilon 1 − ϵ . We estimate the size to be \frac{\sqrt{2{\mathfrak e}_2}}{\pi}erf^{-1}(1-\epsilon) L^{\frac{d}{2}}t 2 𝔢 2 π e r f − 1 ( 1 − ϵ ) L d 2 t , where {\mathfrak e}_2 𝔢 2 is the energy variance per site, and erf^{-1} e r f − 1 is the inverse error function.
The evolution of an isolated quantum system inevitably exhibits recurrence: the state returns to the vicinity of its initial condition after finite time. Despite its fundamental nature, a rigorous quantitative understanding of recurrence has been lacking. We establish upper bounds on the recurrence time, t_{rec}≲t_{exit}(ε)(1/ε)^{d}, where d is the Hilbert-space dimension, ε the neighborhood size, and t_{exit}(ε) the escape time from this neighborhood. For pure states evolving under a Hamiltonian H, estimating t_{exit} is equivalent to an inverse quantum speed limit problem: finding upper bounds on the time a time-evolved state ψ_{t} needs to depart from the ε vicinity of the initial state ψ_{0}. We provide a partial solution, showing that under mild assumptions t_{exit}(ε)≈ε/sqrt[Δ(H^{2})], with Δ(H^{2}) the Hamiltonian variance in ψ_{0}. We show that our upper bound on t_{rec} is generically saturated for random Hamiltonians. Finally, we analyze the impact of coherence of the initial state in the eigenbasis of H on recurrence behavior.
We report an attempt to calculate energy eigenvalues of large quantum systems by the diagonalization of an effectively truncated Hamiltonian matrix. For this purpose we employ a specific way to systematically make a set of orthogonal states from a trial wavefunction and the Hamiltonian. In comparison with the Lanczos method, which is quite powerful if the size of the system is within the memory capacity of computers, our method requires much less memory resources at the cost of the extreme accuracy. In this paper we demonstrate that our method works well in the systems of one-dimensional frustrated spins up to 48 sites, of bosons on a chain up to 32 sites and of fermions on a ladder up to 28 sites. We will see this method enables us to study eigenvalues of these quantum systems within reasonable accuracy.
We report an attempt to calculate energy eigenvalues of large quantum systems by the diagonalization of an effectively truncated Hamiltonian matrix. For this purpose we employ a specific way to systematically make a set of orthogonal states from a trial wavefunction and the Hamiltonian. In comparison with the Lanczos method, which is quite powerful if the size of the system is within the memory capacity of computers, our method requires much less memory resources at the cost of the extreme accuracy. In this paper we demonstrate that our method works well in the systems of one-dimensional frustrated spins up to 48 sites, of bosons on a chain up to 32 sites and of fermions on a ladder up to 28 sites. We will see this method enables us to study eigenvalues of these quantum systems within reasonable accuracy.
Using a Krylov-subspace time evolution algorithm, we simulate the real-time dynamics of translation invariant non-integrable finite spin rings to quite long times with high accuracy. We systematically study the finite-size deviation between the resulting equilibrium state and the thermal state, and we highlight the importance of the energy variance on the deviations. We find that the deviations are well described by the eigenstate thermalization hypothesis, and that the von Neumann entropy correction scaling is the square of the local operator scaling. We reveal also an area law contribution to the relaxed von Neumann entropy, which we connect to the mutual information between the considered subsystem and its immediate environment. We also find that local observables relax towards equilibrium exponentially with a relaxation time scale that grows linearly with system length and is somewhat independent of the local operator, but depends strongly on the energy of the initial state, with the fastest relaxation times found towards one end of the overall energy spectrum. To contrast this behaviour we also study domain wall initial states, which exhibit clear diffusive behaviour, with a Thouless time scaling quadratically with the systems size, leading to a rather precise estimate for the diffusion constant for states in the vicinity of the middle of the energy spectrum.
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In this paper we propose the recursive stochastic state selection method, an extension of the recently developed stochastic state selection method in Monte Carlo calculations for quantum spin systems. In this recursive method we use intermediate states to define probability functions for stochastic state selections. Then we can diminish variances of samplings when we calculate expectation values of the powers of the Hamiltonian. In order to show the improvement we perform numerical calculations of the spin-1/2 anti-ferromagnetic Heisenberg model on the triangular lattice. Examining results on the ground state of the 21-site system we confide this method in its effectiveness. We also calculate the lowest and the excited energy eigenvalues as well as the static structure factor for the 36-site system. The maximum number of basis states kept in a computer memory for this system is about 3.6 x 10**7. Employing a translationally invariant initial trial state, we evaluate the lowest energy eigenvalue within 0.5 % of the statistical errors.