Interplay of Gauss Law and the fermion sign problem in quantum link models with dynamical matter

Abstract

Quantum Link Models with dynamical matter coupled to spin-$\frac{1}{2} \ \rm U(1)$ gauge fields in $d=2+1 $ and $3+1$ can potentially give rise to the Coulomb phase expected in quantum electrodynamics (QED) and other confining phases. Using exact diagonalization techniques, we show that the ground state in a class of models without the magnetic field always lies in the sector which satisfies $(G_e,G_o) = (d,\ -d)$, where $d$ is the spatial dimension and $e$ and $o$ are even and odd sites. It can be analytically proven that this sector is free of the fermion sign problem. We also demonstrate that a meron cluster algorithm for the problem naturally samples the ground states of the Hamiltonian in the aforementioned Gauss Law sector.

Figures (5)

• Figure 1: 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.
• Figure 2: (Left): 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. (Right) The GL constraints in the sector $(0,0)$ 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.
• Figure 3: Breakup and corresponding weights for the spin-$\frac{1}{2}$ U(1) gauge links coupled with matter Hamiltonian in \ref{['eq:H_U1']}.
• Figure 4: (Left) 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. (Right) Transition between the $(2,-2)$ and $(0,0)$ GL sectors as the magnetic coupling is increased.
• Figure 5: The different GL sectors that are sampled by the QMC algorithm at different $\beta$ in both $d=2$ and $d=3$ spatial dimensions. For low temperature (large $\beta$) only the GL $(d,\ -d)$ and its shifted partner arises.