Projected Density Matrix Sampling for Lattice Hamiltonians

TL;DR

This work develops a continuous-time Monte Carlo method that projects the thermal density matrix onto a d_proj-dimensional subspace to recover the low-energy spectrum of generic lattice Hamiltonians, circumventing Trotter errors and mitigating sign problems when sampling within carefully chosen projection states. By constructing Z and E matrices within the projection subspace and evaluating e_a(β) = eigenvalues of E Z^{-1}, the approach yields the projected low-energy levels as β → ∞, with convergence governed by the overlap structure of the subspace and spectral gaps. The authors demonstrate the method on the transverse-field Ising model (recovering E8 mass ratios in the scaling regime) and the Shastry–Sutherland model (validating results against exact diagonalization on small lattices and exploring larger systems at finite β), highlighting how projection subspace design controls convergence and sign-problem effects. Overall, the framework provides a versatile, nonperturbative route for quantum Monte Carlo spectroscopy, with potential extensions to frustrated magnets, lattice gauge theories, and phase-informed projection bases. The work suggests practical guidelines for constructing near-ideal projection subspaces and points to analytic connections to finite-temperature effective theories as avenues for extrapolations from high to low temperature.

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 $β$ 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 $β$ where sign problems occur. Our method provides a general route toward quantum Monte Carlo spectroscopy for lattice Hamiltonians.

Figures (13)

• Figure 1: Illustration of a continuous-time path-integral configuration $[\ell]$, introduced in \ref{['eq:rhoth']}, for a nearest-neighbor Hamiltonian in one spatial dimension. The configuration shown contains $N_\ell=4$ bond operators $(-\varepsilon H_{\ell_i})$ at times $t_i$ ($i=1,2,3,4$). Each operator is accompanied by a factor $\varepsilon$, representing the continuous-time measure $dt$, which cancels naturally during updates enforcing detailed balance. The full configuration $[\ell,s]$ is obtained by inserting complete sets of spin states, as explained in \ref{['eq:rhoth-1']}.
• Figure 2: The loop-cluster configuration $[\ell,s]$ associated with the continuous-time path-integral configuration $[\ell]$ shown in \ref{['fig:CTconf']}. The two colors represent opposite spins.
• Figure 3: A 2-domain wall state of length $\ell=4$ for a chain of length $L=12$. States like these make up our projection subspace.
• Figure 4: The lowest four projected effective thermal energies obtained using the method for $(L = 12,\, h = 0.001)$ (left), $(L = 12,\, h = 0.2)$ (center), and $(L = 96,\, h = 0.01)$ (right). The solid lines passing through the Monte Carlo data in the left and center panels represent exact diagonalization results. The dashed lines in these panels indicate the projected low-energy spectrum to which $e_a(\beta)$ is expected to converge in the $\beta \rightarrow \infty$ limit. The fourth level in the left and center panels begins to exhibit larger errors around $\beta \approx 3$, which we attribute to its proximity to the two-particle threshold, as discussed in the text. The right panel shows that the convergence of the energy levels is relatively slow due to the small gaps in the energy spectrum. Because the energies are extensive quantities, they scale with $L$, whereas the energy gaps do not. Consequently, obtaining accurate estimates of the gaps requires high statistics.
• Figure 5: Monte Carlo estimate of the mass ratio $m_{2}/m_{1}$ (left) and $m_{3}/m_{1}$ (right) as a function of $\mu$. The dashed horizontal line shows the expected $E_8$ result of $m_{2}/m_{1}=1.618$ and $m_{3}/m_{1}=1.989$. The inset shows our estimate at some of the largest values of $\mu$ we have computed. The light dots give estimates from exact diagonalization results at various small values of $L$ and $h$.