Regression Relation for Pure Quantum States and Its Implications for Efficient Computing

Abstract

We obtain a modified version of the Onsager regression relation for the expectation values of quantum-mechanical operators in pure quantum states of isolated many-body quantum systems.We use the insights gained from this relation to show that high-temperature time correlation functions in many-body quantum systems can be controllably computed without complete diagonalization of the Hamiltonians, using instead the direct integration of the Schroedinger equation for randomly sampled pure states. This method is also applicable to quantum quenches and other situations describable by time-dependent many-body Hamiltonians. The method implies exponential reduction of the computer memory requirement in comparison with the complete diagonalization.We illustrate the method by numerically computing infinite-temperature correlation functions for translationally invariant Heisenberg chains of up to 29 spins 1/2. Thereby, we also test the spin diffusion hypothesis and find it in a satisfactory agreement with the numerical results. Both the derivation of the modified regression relation and the justification of the computational method are based on the notion of quantum typicality.

Finite-temperature charge transport in the one-dimensional Hubbard model

“…(18) is not the vanishing mean errorǭ = 0 but the knowledge about the standard deviation of errors Σ(ǫ). This standard deviation is bounded from above by [33,68,69,77],…”

“…These differential equations can be solved by the use of straightforward iterative methods such as, e.g., RungeKutta [33,47,77]. We use a massively parallel implementation of a Suzuki-Trotter product formula or Chebyshev polynomial algorithm [78,79], allowing us to study quantum systems with as many as 2L = 36 lattice sites (L = 18 in the fermionic language), where the Hilbertspace dimension is d = O(10 11 ).…”

“…We incorporate finite temperatures β = 0 by introducing |ψ N (β) = exp(−βH/2) |ψ N and rewriting the current autocorrelation function in Eq. (10) in the form [33,47,68,69,76,77] (skipping the index σ for clarity)…”

Real-time broadening of nonequilibrium density profiles and the role of the specific initial-state realization

“…While this question has a long and fertile history, it has experienced an upsurge of interest in recent years [1] due to the advent of cold atomic gases [2] as well as due to the discovery of new states of matter such as many-body localized phases [3]. In particular, the theoretical understanding has seen substantial progress by the fascinating concepts of eigenstate thermalization [4][5][6] and typicality of pure quantum states [7][8][9][10][11][12][13][14] as well as by the invention of powerful numerical methods such as density-matrix renormalization group [15]. Much less is known on the route to equilibrium as such [16] and still the derivation of the conventional laws of (exponential) relaxation and (diffusive) transport on the basis of truly microscopic principles is a challenge to theory [17].…”

Interaction-induced weakening of localization in few-particle disordered Heisenberg chains

“…First, we prepare a pure initial state of densely packed particles (also known as bound states 47,48 ), where all particles (↑-spins) are concentrated at adjacent sites and holes (↓-spins) are located on the other sites. Then, we calculate the evolution of the particle distribution in real time and real space, using a Runge-Kutta integration of the time-dependent Schrödinger equation 36,37,49,50 . While the time dependence of the distribution width allows us to study the type of dynamics in general, a convergence of this width to a constant value in the long-time limit (which may or may not exist) also allows us to extract a finite local-ization length.…”