The fermion sign problem in Gauss law sectors of quantum link models with dynamical matter

TL;DR

This work analyzes the fermion sign problem in Abelian lattice gauge theories with dynamical fermions by studying Gauss Law sectors in spin-$\tfrac{1}{2}$ quantum-link models. It identifies GL sectors that are free from the sign problem, notably the $(d,-d)$ sector and its shift partner, and shows that the conventional zero-charge sector suffers from the sign problem. Using large-scale exact diagonalization and meron-cluster Monte Carlo, it characterizes the ground-state structure and phase behavior across sectors, and demonstrates how magnetic energy can drive sector transitions. The findings have implications for simulating truncated Kogut-Susskind theories and benchmarking quantum simulators, with future work aimed at extending meron methods to additional GL sectors and exploring QED in higher dimensions.

Abstract

The fermion sign problem poses a formidable challenge to the use of Monte Carlo methods for lattice gauge theories with dynamical fermionic matter fields. A meron cluster algorithm recently formulated for gauge fields represented as spin-$\frac{1}{2}$ quantum links coupled to a single flavour of staggered fermions samples only two of the exponentially many Gauss law (GL) sectors at low temperatures, making it possible to simulate either of those two GL sectors at zero temperature in polynomial time. In this article, we analytically identify GL sectors which can be simulated without encountering the fermion sign problem in arbitrary spatial dimensions. Using large-scale exact diagonalization and cluster Monte Carlo methods, we further explore the nature of phases in the GL sectors dominating at zero temperature. The vacuum states lie in sectors which satisfy a staggered Gauss law, in contrast to the zero GL sector familiar in particle physics. Moreover, we prove that while the ground state GL sectors do not suffer from the fermion sign problem, the usual zero-charge GL sector (often considered the physical sector) does. We outline the role of the magnetic energy in causing transitions between GL sectors. We expect our results to be valid for truncated Kogut-Susskind gauge theories, beyond quantum link models.

Figures (12)

• Figure 1: (Top): The fermion hop from left to right is accompanied by a $\sigma^- (U^\dagger)$ operator on the link, causing the spin-$1/2$ electric flux to flip its orientation, while the right to left hop is accompanied by a $\sigma^+ (U)$ operator on the link and is thus constrained by the state of the flux on the link. (Bottom): Examples of electric flux configurations belonging to two different Gauss Law sectors. The direction of the arrows on the links indicates the flux to be $E_{x,\hat{i}} = \pm \frac{1}{2}$, while the filled (empty) sites indicate the site to be occupied (empty).
• Figure 2: The orientation of the gauge links in the sector $(3,-3)$ in $d=3$ do not allow the movement of fermions beyond one lattice spacing. Thus, positions of $f_1$ and $f_2$ cannot be switched by the action of the Hamiltonian.
• Figure 3: (Left): The GL constraints in the sector $(2,-2)$ in $d=3$ are relaxed enough to allow fermions $f_1$ and $f_2$ to exchange positions with each other following the general prescription described in the text. (Right) The different GL sectors that are sampled by the QMC algorithm at different $\beta$. For low temperature (large $\beta$) only the GL $(d,-d)$ and its shifted partner arises.
• Figure 4: Ground state energy difference between the GL sectors $(2,-2)$ and $(0,0)$ for bosons (open symbols) and fermions (solid lines) respectively in $d=2$. While the fermionic and bosonic results are the same for the sector $(2,-2)$, there is a difference in the $(0,0)$ sector for $V/t \sim 0$, leading to the deviation between the two sets of data in that region.
• Figure 5: Top two figures show $\braket{\bar{\psi} \psi}$ vs $V/t$ in the two GL sectors in $d=2$. The $\mathbb{Z}_2$ chiral symmetry breaks for $V/t \gg 0$ in both sectors, while phase separation occurs for $V/t \ll 0$. The shaded band for $V/t \sim 0$ indicates a region where a liquid phase is suspected. The bottom two panels show $\epsilon_{x,y}$ for each GL sector in $d=2$. In $(2,-2)$ the symmetric ordering of the flux causes both $\epsilon_{x,y}$ to reach the same value for $V/t >0$ while due to phase separation at $V/t < 0$, $\epsilon_x$ and $\epsilon_y$ have different values, also depending on whether it is a ladder or a square geometry. In GL sector $(0,0)$, the ordering of the flux requires a larger unit cell, and is thus sensitive to a multiple of 2 vs 4.