Projected Density Matrix Sampling for Lattice Hamiltonians

TL;DR

This paper introduces a continuous-time path-integral Monte Carlo method that efficiently computes low-energy spectra of lattice quantum Hamiltonians, overcoming some limitations of traditional quantum Monte Carlo techniques.

Contribution

The authors develop a sign-problem-tolerant Monte Carlo approach that projects the density matrix onto a chosen subspace, enabling spectral analysis of complex quantum systems.

Findings

Successfully computed low-energy levels of the 1D Ising model, illustrating the flow to the $E_8$ quantum field theory.

Benchmarked the method against exact diagonalization for the Shastry-Sutherland model.

Extended analysis to larger systems beyond the reach of exact methods, despite sign problems.

Abstract

Quantum Monte Carlo methods are powerful tools for studying quantum many-body systems but face difficulties in accessing excited states and in treating sign problems. We present a continuous-time path-integral Monte Carlo method for computing the low-lying spectrum of generic quantum Hamiltonians within a projection subspace. The method projects the thermal density matrix onto a subspace spanned by a chosen set of linearly independent states. It is free of Trotter discretization errors and systematically converges to the low-energy states which have finite overlap with the projection subspace as the $\beta$ parameter increases. While most effective for systems without a sign problem, the method also yields information about low-energy spectra when sign problems are present. We illustrate the approach on two problems. For the sign-free case, we compute the first four low-energy levels in the scaling limit of the one-dimensional Ising model with both transverse and longitudinal fields, demonstrating the flow from the conformal limit to the massive $E_8$ quantum field theory. For the sign-problem case, we apply the method to the frustrated Shastry-Sutherland model and benchmark it against exact diagonalization on small lattices. We also present results for larger systems beyond the lattice sizes accessible to exact diagonalization, while limited to small $\beta$ where sign problems occur. Our method provides a general route toward quantum Monte Carlo spectroscopy for lattice Hamiltonians.