Numerical approaches to isolated many-body quantum systems

Abstract

Ultracold atoms have revolutionized atomic and condensed matter physics. In addition to having clean, controllable Hamiltonians, ultracold atoms are near-perfect realizations of isolated quantum systems, in which weak environmental coupling can be neglected on experimental time scales. This opens new opportunities to explore these systems not just in thermal equilibrium, but out of equilibrium as well.

In this dissertation, we investigate some properties of closed quantum systems, utilizing a combination of numerical and analytical techniques. We begin by applying full configuration-interaction quantum Monte Carlo (FCIQMC) to the Fermi polaron,

which we use as a test bed to improve the algorithm. In addition to adapting standard QMC techniques, we introduce novel controlled approximations that allow mitigation of the sign problem and simulation directly in the thermodynamic limit. We also contrast the sign problem of FCIQMC with that of more standard techniques, focusing on FCIQMC's capacity to work in a second quantized determinant space.

Next, we discuss nonequilibrium dynamics near a quantum critical point, focusing on the one-dimensional transverse-field Ising (TFI) chain. We show that the TFI dynamics exhibit critical scaling, within which the spin correlations

exhibit qualitatively athermal behavior. We provide strong numerical evidence for the universality of dynamic scaling by utilizing time-dependent matrix product states to simulate a non-integrable model in the same equilibrium universality class. As this non-integrable model has been realized experimentally, we investigate the robustness of our predictions against the presence of open boundary conditions and disorder. We find that the qualitatively athermal correlations remain visible, although other phenomena such as even/odd effects become relevant within the finite size scaling theory.

Finally, we investigate the properties of the integrable TFI model upon varying the strength of a non-integrable perturbation, attempting to understand the finite-size scaling behavior of the eigenstate thermalization crossover. We numerically estimate the crossover scale of this perturbation strength using exact diagonalization, finding that the crossover scale decreases strongly with system size, and analytically provide a lower bound on the finite-size scaling of this crossover. However, we are unable to solve for the thermodynamic limit of this crossover; doing so will require larger systems and more comprehensive theory.