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arXiv:2508.16298 · SciPost Physics

Scalable hybrid quantum Monte Carlo simulation of $U(1)$ gauge field coupled to fermions on GPU

Kexin Feng, Chuang Chen, Zi Yang Meng

Published Feb 25, 2026• 4 citations
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Abstract

The problem of a U(1) U ( 1 ) gauge field coupled to fermions in (2+1) dimensions is of fundamental importance, as it is believed to give rise to U(1) U ( 1 ) Dirac spin liquid (DSL) state in quantum magnets and the strongly coupled conformal field theory in quantum electrodynamics. However, numerical progress has long been hampered by the steep computational cost of traditional determinant quantum Monte Carlo (QMC). Here, we develop a GPU-accelerated hybrid QMC algorithm assisted with several novel technical improvements, which significantly reduces the complexity to be linear with respect to space-time volume, and scales the simulation up to an unprecedented size. At this scale, we observe asymptotic convergence of fermion bilinear correlator and conserved current correlator, which was unclear previously, and find their scaling dimensions in good agreement with field theory, which supports the conformal nature of the Dirac spin liquid. Our advances establish a scalable framework to study DSL physics and its transitions at larger scales.

DOI
10.21468/scipostphys.20.2.060

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Scalable hybrid quantum Monte Carlo simulation of U(1)U(1)U(1) gauge field coupled to fermions on GPU
The problem of a U(1) U ( 1 ) gauge field coupled to fermions in (2+1) dimensions is of fundamental importance, as it is believed to give rise to U(1) U ( 1 ) Dirac spin liquid (DSL) state in quantum magnets and the strongly coupled conformal field theory in quantum electrodynamics. However, numerical progress has long been hampered by the steep computational cost of traditional determinant quantum Monte Carlo (QMC). Here, we develop a GPU-accelerated hybrid QMC algorithm assisted with several novel technical improvements, which significantly reduces the complexity to be linear with respect to space-time volume, and scales the simulation up to an unprecedented size. At this scale, we observe asymptotic convergence of fermion bilinear correlator and conserved current correlator, which was unclear previously, and find their scaling dimensions in good agreement with field theory, which supports the conformal nature of the Dirac spin liquid. Our advances establish a scalable framework to study DSL physics and its transitions at larger scales.
We study a compact U(1) lattice gauge theory using a self-built Hybrid Monte Carlo (HMC) program. The starting point is the parton construction of the Heisenberg spin chain where the spins are fractionalized into spinons coupled to an emergent compact U(1) gauge field. Here we do not impose Gauss's law, such that the Hamiltonian describes an unconstrained lattice gauge theory. We make theoretical calculations of the ground-state phase diagram and identify an emergent SO(4) symmetry in the strong-coupling limit. Furthermore, the weak coupling limit corresponds to frozen gauge fields which results in a free fermionic phase that is unstable in one dimension. The phase diagram is supported by carrying out HMC simulations which also allow us to study the dynamical generation of Gauss's law at low energy scales. We find that the theory accurately predicts the numerical findings and establish that HMC works as a method for simulating unconstrained gauge theories in (1+1)-dimensions.
Thanks to the development in quantum Monte Carlo technique, the compact U(1) lattice gauge theory coupled to fermionic matter at (2+12+12+1)D is now accessible with large-scale numerical simulations, and the ground state phase diagram as a function of fermion flavor (Nf{N}_{f}Nf​) and the strength of gauge fluctuations is mapped out. Here we focus on the large fermion flavor case (Nf=8{N}_{f}=8Nf​=8) to investigate the dynamic properties across the deconfinement-to-confinement phase transition. In the deconfined phase, fermions coupled to the fluctuating gauge field to form U(1) spin liquid with continua in both spin and dimer spectral functions, and in the confined phase fermions are gapped out into valence bond solid phase with translational symmetry breaking and gapped spectra. The dynamical behaviors provide supporting evidence for the existence of the U(1) deconfined phase and could shine light on the nature of the U(1)-to-VBS phase transition which is of the QED3{\mathrm{QED}}_{3}QED3​-Gross-Neveu chiral O(2) universality whose properties are still largely unknown.
We consider relativistic U(1) gauge theories in 2+12+12+1 dimensions, with Nb{N}_{b}Nb​ species of complex bosons and Nf{N}_{f}Nf​ species of Dirac fermions at finite temperature. The quantum phase transition between the Higgs and Coulomb phases is described by a conformal field theory (CFT). At large Nb{N}_{b}Nb​ and Nf{N}_{f}Nf​, but for arbitrary values of the ratio Nb∕Nf{N}_{b}∕{N}_{f}Nb​∕Nf​, we present computations of various critical exponents and universal amplitudes for these CFTs. We make contact with the different spin liquids, charge liquids, and deconfined critical points of quantum magnets that these field theories describe. We compute physical observables that may be measured in experiments or numerical simulations of insulating and doped quantum magnets.
Monte Carlo simulations of strongly interacting fermionic systems are plagued by the fermion sign problem, making the nonperturbative study of many interesting regimes of dense quantum matter, or of theories of odd numbers of fermion flavors, challenging. Moreover, typical fermion algorithms require the computation (or sampling) of the fermion determinant. We focus instead on the meron cluster algorithm, which can solve the fermion sign problem in a class of models without involving the determinant. We develop and benchmark new meron algorithms to simulate fermions coupled to Z2 and U(1) gauge fields in the presence of appropriate four-fermi interactions. Such algorithms can be used to uncover potential exotic properties of matter, particularly relevant for quantum simulator experiments. We demonstrate the emergence of the Gauss’ law at low temperatures for a U(1) model in (1+1)D . Published by the American Physical Society 2024
An emergent gauge field can control boundary critical behavior without changing the bulk quantum spin liquid. We investigate the boundary criticality in (2+1)(2+1)(2+1)-dimensional U(1)U(1)U(1) Dirac spin liquid, whose low-energy physics is described by massless quantum electrodynamics. Using a perturbative renormalization-group analysis in a half-space, we show that Neumann and Dirichlet boundary conditions for the emergent gauge field lead to distinct boundary universality classes. We determine the boundary scaling dimensions of the fundamental fields and gauge-invariant operators that provide observable signatures. We further propose a fermion-gauge lattice model with tunable boundary interactions as a microscopic setting for realizing and probing these boundary universality classes.
We design a lattice model of a “mixed” U(1) gauge field coupled to fermions with a flavor chemical potential and solve it with large-scale determinant quantum Monte Carlo simulations. In this mixed construction, dynamical magnetic monopole events are suppressed in the temporal-continuum limit. For zero flavor chemical potential, the model realizes three-dimensional quantum electrodynamics ( QED 3 ) which has been argued to describe the ground state and low-energy excitations of the Dirac spin liquid phase of quantum antiferromagnets. At finite flavor chemical potential, which corresponds to a Zeeman field perturbing the Dirac spin liquid, we find a “chiral flux” phase which is characterized by the generation of a finite mean emergent gauge flux and, accordingly, the formation of relativistic Landau levels for the Dirac fermions. We use field-theory arguments to argue that in this state a U ( 1 ) m magnetic symmetry is spontaneously broken, leading to a gapless photon mode. Because of spin-flux attachment this mode would contribute to the low-energy longitudinal spin response. We numerically compute longitudinal and transverse spin structure factors which match our continuum and lattice mean-field theory predictions. In a different region of the phase diagram, strong fluctuations of the emergent gauge field give rise to an antiferromagnetically ordered state with gapped Dirac fermions coexisting with a deconfined gauge field. We also find an interesting intermediate phase where the chiral flux phase and the antiferromagnetic phase coexist. We argue that our results pave the way to testable predictions for magnetized Dirac spin liquids in frustrated quantum antiferromagnets.