Density matrix quantum Monte Carlo

TL;DR

This paper introduces a quantum Monte Carlo method that samples the full density matrix of many-particle systems at finite temperatures, enabling the calculation of complex observables and entanglement measures across all temperatures.

Contribution

The method extends quantum Monte Carlo techniques to directly sample the density matrix, allowing comprehensive finite-temperature analysis and entanglement calculations in many-body systems.

Findings

Successfully computed energy and magnetization of Heisenberg models

Calculated entanglement measures like concurrence and Renyi entropy

Demonstrated the method's ability to study temperature dependence

Abstract

We present a quantum Monte Carlo method capable of sampling the full density matrix of a many-particle system at finite temperature. This allows arbitrary reduced density matrix elements and expectation values of complicated non-local observables to be evaluated easily. The method resembles full configuration interaction quantum Monte Carlo but works in the space of many-particle operators instead of the space of many-particle wave functions. One simulation provides the density matrix at all temperatures simultaneously, from $T=\infty$ to $T=0$, allowing the temperature dependence of expectation values to be studied. The direct sampling of the density matrix also allows the calculation of some previously inaccessible entanglement measures. We explain the theory underlying the method, describe the algorithm, and introduce an importance-sampling procedure to improve the stochastic efficiency. To demonstrate the potential of our approach, the energy and staggered magnetization of the isotropic antiferromagnetic Heisenberg model on small lattices, the concurrence of one-dimensional spin rings, and the Renyi $S_2$ entanglement entropy of various sublattices of the $6 \times 6$ Heisenberg model are calculated. The nature of the sign problem in the method is also investigated.