Variational Monte Carlo algorithm for lattice gauge theories with continuous gauge groups: A study of $(2+1)$-dimensional compact QED with dynamical fermions at finite density

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 $(2+1)$-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 $\text{U}\left(1\right)$ 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.

Figure 1

Naming conventions on the periodic lattice: the gauge-field degrees of freedom ${\theta }_{\mathbf{x},i},E_{\mathbf{x},i}$ (blue) reside on the links $\mathbf{x},i$ while the fermionic degrees of freedom ${\psi }_{\mathbf{x},\alpha }^{†}$ (red), which can come in several species $\alpha$, are located on the sites $\mathbf{x}$. The circular arrows on the plaquettes denote the plaquette variables ${\theta }_{\mathbf{p}}$. The global loops ${\theta }_i$ wind around the axis given by ${\mathbf{e}}_i$ and are illustrated by blue lines.

Figure 2

Scheme of variational Monte Carlo procedure: the ansatz is formulated in the full gauge-field basis denoted by $|\theta \rangle$, in our case the $\text{U}\left(1\right)$ gauge group, consisting of a pure gauge part ${\mathrm{\Psi }}_G\left(\theta \right)$ and gauge-field-dependent fermionic Gaussian states $|{\mathrm{\Psi }}_F\left(\theta \right)\rangle$. Expectation values of observables $O$ can be carried out analytically with respect to the fermionic part (which involves the eigendecomposition of the gauge-matter interactions for fixed $\theta$). The resulting expressions $O_{\text{loc}}\left(\theta \right)$ are diagonal in $\theta$ and sampled with Monte Carlo techniques according to a probability distribution $p\left(\theta \right)$ in which only ${\mathrm{\Psi }}_G\left(\theta \right)$ appears since the gauge-field-dependent fermionic Gaussian states are normalized. The variational parameters are adapted according to stochastic reconfiguration.

Figure 3

Gauss law violation $\langle G_{\mathbf{x}}\rangle -q_{\mathbf{x}}$ for a $12\times 12$ lattice at $g^2=0.25$, $t=1$, and $g_{\text{mag}}=-1$ with a random choice of variational parameters and a sampling size of $N=10$. The Gauss law violation is of the order of machine precision even for a small sampling size, demonstrating that the ansatz is inherently gauge-invariant.

Figure 4

The flux ${\theta }_{\mathbf{p}}$ per plaquette is shown for the variational ground state at $g^2=0.0$, $t=1$ and $g_{\text{mag}}=-1$ for a $12\times 12$ lattice. The average deviation of ${\theta }_{\mathbf{p}}$ from $\pi$ is on the order of ${10}^{-4}$. The global loops ${\theta }_1$ and ${\theta }_2$ (winding around the axes of the lattice) also acquire a $\pi$-flux with a similar deviation as the plaquette fluxes.

Figure 5

Benchmark for pure gauge compact ${\text{QED}}_3$ of our ansatz (denoted by FG) against the variational method in Ref. [48] based on complex periodic Gaussian states (denoted by CPG): we compare the ground-state energy $E_0$ for an $8\times 8$ lattice over the whole coupling region and compute the relative error (see inset).

Figure 6

Benchmark for compact ${\text{QED}}_3$ coupled to $N_f=2$ species of dynamical fermions at half-filling: the shown averaged flux energy per plaquette $cos\left({\theta }_{\mathbf{p}}\right)$ in the variational ground state is to be compared with results obtained in an Euclidean Monte Carlo study shown in Fig. 13 in Ref. [53]. The data agree over the whole coupling region, showing no evidence of a discontinuous phase transition.

Figure 7

Benchmark for compact ${\text{QED}}_3$ coupled to $N_f=2$ species of dynamical fermions at half-filling: we compute the AFM correlation ratio $r_{\text{AFM}}$ in the variational ground state for lattice size up to $16\times 16$ (left). The AFM correlation ratio is computed from the spin correlations and quantifies the strength of antiferromagnetic order. The crossing points are extracted and extrapolated to the thermodynamic limit, resulting in $g_{c,\infty }^2=0.15\left(2\right)$ (right). This is to be compared with the Euclidean Monte Carlo study in Ref. [53] where also a nonzero coupling was extrapolated but at a higher value of $g_{c,\infty ,\text{EMC}}^2=0.40\left(5\right)$.

Figure 8

Benchmark for compact ${\text{QED}}_3$ coupled to $N_f=2$ species of dynamical fermions at half-filling: we compute the decay of spin correlations (both for the full correlation function and the connected correlation function) in the variational ground state for a $16\times 16$ lattice at weak coupling ($g^2=0.1$) and at stronger coupling ($g^2=0.85$). Note that we only use odd distances in $r$ to avoid oscillations. At strong coupling (where one expects behavior similar to the Heisenberg model) the full correlations decay to a constant while the connected part decays exponentially. At weak coupling the decay is rather algebraically similar to the decay shown in the Euclidean Monte Carlo study in Fig. 4 in Ref. [53].

Figure 9

Finite chemical potential study for compact ${\text{QED}}_3$ coupled to $N_f=2$ species of dynamical fermions: we compute the variational ground-state energies (corrected by an overall constant $\mu N/2$) on an $8\times 8$ lattice for different isospin numbers $\mathrm{\Delta }N$ depending on the chemical potential difference $\mu$ between the two species. We study both the case of massless and massive fermions [see top row, (a) and (c)]. By computing the crossing points between the ground-state energies we can extract the phase transitions between neighboring $\mathrm{\Delta }N$ phases [see bottom row, (b) and (d)].

Figure 10

The acceptance probability in the variational ground states of the $N_f=2$ model. The acceptance probability is on a high level down to the lowest nonzero coupling and only for $g^2=0.0$ it sharply drops. This is expected since the ground state there is diagonal in $\theta$ with all plaquettes ${\theta }_{\mathbf{p}}$ having $\pi$ flux so that our variational probability, while approaching the delta distribution, gets lower and lower in acceptance probability.