Hamiltonian Monte Carlo enhanced by Exact Diagonalization

Abstract

Strongly correlated fermionic systems are of great interest in condensed matter physics and numerical methods are indispensable tools for their study. However, existing approaches such as exact diagonalization (ED) and stochastic quantum Monte Carlo methods each suffer from fundamental limitations: ED is hindered by exponential scaling in system size, while Monte Carlo methods are plagued by sign problems and long autocorrelation times. These limitations restrict the accessible parameter space and developing algorithms that efficiently alleviate them remains a central challenge in computational physics. In this work, we propose a hybrid algorithm that combines ED and Hamiltonian Monte Carlo (HMC) to simulate 2D arrays of coupled quantum wires, modeled as interacting fermionic Hubbard chains. We demonstrate how our hybrid implementation of HMC, which we dub H$^2$MC, outperforms either method alone across several key simulation facets. When compared to pure ED, H$^2$MC has a much more favorable computational scaling, which allows us to push simulations to much larger 2D arrays. H$^2$MC also greatly alleviates the sign problem and reduces autocorrelation times when compared to pure HMC formulations utilizing either real or imaginary auxiliary fields. Our formalism demonstrates how complementary strengths of seemingly disparate methods can be leveraged to enable feasible simulations in an extended parameter space.

Figures (3)

• Figure 1: Benchmark of the H$^2$MC framework against full ED on a $2\times 2$ lattice at $U=3$, $V=1$, $\beta=4$, and $t=1$ around half-filling, corresponding to $\mu =-3.5$. Simulations recorded $N_{\mathrm{cfg}}=10^4$ configurations for $N_t=32$ (blue circles) and $N_t=40$ (orange cross) and are compared to exact continuum results obtained from full ED (black dashed line). Left: Average charge density $\langle q\rangle$ () as a function of chemical potential $\mu$. Near half-filling both simulations agree well with the exact result, while deviations grow at large $|\mu|$ for the coarse discretization, reflecting the increasing Suzuki-Trotter error. Right: On-site imaginary-time correlator $C(\tau)$ () at half-filling, compared to the exact continuum result. Both values correctly reproduce the correlator accurately across the full imaginary-time extent.
• Figure 2: On-site imaginary time correlator $C(\tau)$ () at half-filling for a $4\times 6$ lattice at $\beta = 1$, $U = 3$, $V = 0.2$, and $t=1$, with chemical potential tuned to half-filling $\mu = -1.9$. Simulations compare the H$^2$MC formulation to the real-field $(\chi)$ and imaginary-field $(i\chi)$ pure HMC formulations at $N_t=32$. H$^2$MC recorded $N_{\mathrm{cfg}}=10^4$ configurations, while pure HMC generated $N_{\mathrm{cfg}}=10^5$ configurations each. All three methods agree within statistical uncertainties, while the imaginary field-pure HMC exhibits visibly enlarged error bars due to a substantial sign problem.
• Figure 3: Sampling properties of the H$^2$MC , real-field pure HMC ($\chi$), and imaginary-field pure HMC ($i\chi$) formulations as a function of the on-site interaction strength $U$, for a $4\times6$ lattice at half-filling with $\beta = 1$, $V = 0.2$, and $t=1$. Top: Maximum integrated autocorrelation time $\tau_{\mathrm{int},C}$ () over all spatial correlator and imaginary-time slices. Bottom: Average phase $\Sigma$ () quantifying the sign problem of the respective simulations. The two pure HMC formulations exhibit complementary limitations with increasing $U$; We observe growing autocorrelation times for the real-field and deteriorating average phase for the imaginary-field formulation, while H$^2$MC maintains both a stable short autocorrelation time and an average phase indicating the absence of a sign problem across the full range of $U$ shown.