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International Symposium on Lattice Field Theory (LATTICE)

Ground state search of interacting boson–fermion systems using variational Monte Carlo

Pranay Naredi, Stefan Kühn, Enrico Brehm

Published May 30, 2026• 0 citations
Publisher

Abstract

We study lattice QED in $2+1$ dimensions coupled to Wilson fermions in the Hamiltonian formulation and develop a sign-problem-free variational Monte Carlo algorithm. Using a continuous basis to describe the gauge degrees of freedom and fermionic Gaussian states for the matter fields, we construct a variational ansatz for the ground state. The parameters of the ansatz are optimized using gradient descent and/or imaginary time evolution based on the time-dependent variational principle. For a lattice with $N$ matter sites, the computational complexity for sampling a gauge configuration is $\mathcal{O}(N^2\log N)$, and the complexity for computing the energy per gauge configuration is $\mathcal{O}(N^4)$. Comparing our method to results from exact diagonalization on small lattices, we demonstrate good agreement. In addition, we present preliminary results for the ground state for two fermion flavors, $N_f=2$, in the presence of a non-zero chemical potential, a regime in which the conventional action-based Monte Carlo methods suffer from the sign problem.

DOI
10.22323/1.518.0049

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Lattice gauge theories coupled to fermionic matter account for many interesting phenomena in both high-energy physics and condensed-matter physics. Certain regimes, e.g., at finite fermion density, are difficult to simulate with traditional Monte Carlo algorithms due to the so-called sign problem. We present a variational, sign-problem-free Monte Carlo method for lattice gauge theories with continuous gauge groups and apply it to (2+1)(2+1)(2+1)-dimensional compact QED with dynamical fermions at finite density. The variational ansatz is formulated in the full gauge-field basis, i.e., without having to resort to truncation schemes for the U(1)\text{U}(1)U(1) gauge-field Hilbert space. The ansatz consists of two parts: first, a pure gauge part based on Jastrow-type ansatz states (which can be connected to certain neural-network ansatz states) and, second, a fermionic part based on gauge-field-dependent fermionic Gaussian states. These are designed in such a way that the gauge-field integral over all fermionic Gaussian states is gauge-invariant and at the same time still efficiently tractable. To ensure the validity of the method we benchmark the pure gauge part of the ansatz against another variational method and the full ansatz against an existing Monte Carlo simulation where the sign problem is absent. Moreover, in limiting cases where the exact ground state is known we show that our ansatz is able to capture this behavior. Finally, we study a sign-problem affected regime by probing density-induced phase transitions.
Gauge theories describe fundamental interactions in the standard model of particle physics and build effective theories in condensed matter physics. Being notoriously hard to simulate, these theories are commonly regularized as lattice gauge theories which can be evaluated with high precision through Monte Carlo algorithms or variational methods. However, Monte Carlo algorithms are not applicable in all regimes due to the sign problem; while variational methods, quantum and classical alike, only give upper-bounds on the ground-state energy. Their precision crucially depends on the chosen ansatz. Here, we present a framework based on a hierarchy of semidefinite programs to yield increasingly good lower-bounds on the ground-state energy and propagate these bounds to extended observables like Wilson loops and mesonic strings. We demonstrate the numerical capabilities by applying the algorithm to a (1+1)-dimensional Z2\mathbb{Z}_2Z2​ theory with dynamic fermionic matter and its pure-gauge version in (2+1)-dimensions. In both cases, we obtain certified intervals for the ground-state energy. For a one-dimensional system with 129 sites, the certified interval has a relative spread of 0.04%0.04\%0.04%. For a two-dimensional system with 97 sites (corresponding to a 7×87 \times 87×8 lattice), the relative spread is 2.08%2.08\%2.08%. Beyond energy estimates, the proposed method provides rigorous bounds on both short- and long-range observables such as Wilson loops over multiple plaquettes and the magnitude of the mesonic string. The method can be both used as a computational microscope into many-body physics and a way to certify results of quantum simulators.
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Lattice gauge theories (LGTs) provide a powerful framework for studying non-perturbative phenomena in gauge theories. However, conventional approaches such as Monte Carlo (MC) simulations in imaginary time are limited, as they do not allow real time evolution and suffer from a sign problem in many important cases. Using Gauged Gaussian fermionic projected entangled pair states (GGFPEPS) as a variational ground state ansatz offers an alternative for studying LGTs through a sign-problem-free variational MC. As this method is extended to larger and more complex systems, understanding its numerical behavior becomes essential. While conventional action based MC has been extensively studied, the performance and characteristics of non-action-based MC within the GGFPEPS framework are far less explored. In this work, we investigate these algorithmic aspects, identifying an optimal update size for GGFPEPS-based MC simulations for Z2\mathbb{Z}_2Z2​ in 2+12+12+1 dimensions. We show that gauge fixing generally slows convergence, and demonstrate that not exploiting the translation-invariance can, in some cases, improve the computational time scaling of error convergence. We expect that these improvements will allow advancing the simulation to larger and more complex systems.
We propose a new ansatz for the ground-state wave function of quantum many-body systems on a lattice. The key idea is to cover the lattice with plaquettes and obtain a state whose configurational weights can be optimized by means of a variational Monte Carlo algorithm. Such a scheme applies to any dimension, without any 'sign' instability. We show results for various two-dimensional spin models (including frustrated ones). A detailed comparison with available exact results, as well as with variational methods based on different ansatze, is offered. In particular, our numerical estimates are in quite good agreement with exact ones for unfrustrated systems, and compare favorably to other methods for frustrated ones.
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