A quantum Monte Carlo algorithm for arbitrary high-spin Hamiltonians

TL;DR

This work introduces PMR-QMC, a universal quantum Monte Carlo method for simulating arbitrary high-spin Hamiltonians by casting the Hamiltonian in Permutation Matrix Representation and sampling closed-walk configurations. It extends previous spin-$\tfrac{1}{2}$ PMR-QMC to spin-$s>\tfrac{1}{2}$, providing model-agnostic updates, a cycle-based ergodicity framework, and a sign-problem analysis via vanishing geometric phases. Demonstrations on spin-$1$ and spin-$\tfrac{3}{2}$ Heisenberg models and random high-spin Hamiltonians show the method’s versatility and scalability, including mixed-species extensions with fermions and bosons. The approach yields a single, versatile code path applicable to complex, nonlocal, and mixed-spin systems, with parallel scalability and transparent observables. The authors release their code on GitHub, enabling broad adoption and exploration in quantum magnetism and related many-body phenomena.

Abstract

We present a universal quantum Monte Carlo algorithm for simulating arbitrary high-spin (spin greater than 1/2) Hamiltonians, based on the recently developed permutation matrix representation (PMR) framework. Our approach extends a previously developed PMR-QMC method for spin-1/2 Hamiltonians [Phys. Rev. Research 6, 013281 (2024)]. Because it does not rely on a local bond decomposition, the method applies equally well to models with arbitrary connectivities, long-range and multi-spin interactions, and its closed-walk formulation allows a natural analysis of sign-problem conditions in terms of cycle weights. To demonstrate its applicability and versatility, we apply our method to spin-1 and spin-3/2 quantum Heisenberg models on the square lattice, as well as to randomly generated high-spin Hamiltonians. Additionally, we show how the approach naturally extends to general Hamiltonians involving mixtures of particle species, including bosons and fermions. We have made our program code freely accessible on GitHub.

Figures (6)

• Figure 1: Calculations of the spin-$1$ quantum Heisenberg model on a square $L\times L$ lattice. Top: Specific heat as a function of inverse-temperature $\beta$. Bottom: magnetic susceptibility as a function of $\beta$. The standard errors are no larger than the size of a marker.
• Figure 2: Calculations of the spin-$3/2$ quantum Heisenberg model on a square $L\times L$ lattice. Top: Specific heat as a function of inverse-temperature $\beta$. Bottom: magnetic susceptibility as a function of $\beta$. The standard errors are no larger than the size of a marker.
• Figure 3: Top: Average energy $\langle E\rangle$ over $200$ randomly generated spin-$1$ Hamiltonian instances as a function of $m$ for random $k$-local $40$-spin Hamiltonians for $k=3$, $k=5$, and $k=8$ at $\beta=1$. Bottom: A similar plot for $\langle\textrm{sgn}\rangle$, averaged over the $200$ Hamiltonian instances.
• Figure 4: Top: Average energy $\langle E\rangle$ over $200$ randomly generated spin-$1$ Hamiltonian instances as a function of $m$ for random $k$-local $40$-spin Hamiltonians for $k=3$, $k=5$, and $k=8$ at $\beta=5$. Bottom: A similar plot for $\langle\textrm{sgn}\rangle$, averaged over the $200$ Hamiltonian instances.
• Figure 5: Top: Average energy $\langle E\rangle$ over $200$ randomly generated spin-$3/2$ Hamiltonian instances as a function of $m$ for random $k$-local $40$-spin Hamiltonians for $k=3$, $k=5$, and $k=8$ at $\beta=1$. Bottom: A similar plot for $\langle\textrm{sgn}\rangle$, averaged over the $200$ Hamiltonian instances.