Non-reciprocal Ising gauge theory

Abstract

Non-reciprocity and geometric frustration enable many-body systems to avoid crystalline order and instead exhibit complex, liquid-like behavior. Here we show that their interplay is richer than the sum of its parts, leading to surprising structural and dynamical phenomena. In our minimal model, two copies of Ising gauge theory are non-reciprocally coupled in a way that crucially preserves a local $\mathbb{Z}_2$ symmetry. We discover that the combined Wilson loop observable of the two copies exhibits linear asymptotic scaling, with a quasiparticle-pair confinement length tuned by the strength of the non-reciprocal coupling. Key dynamical features are revealed in the behavior of individual deconfined excitations due to strong interactions induced by the non-reciprocity, leading to motion on a critical percolation cluster that follows a self-avoiding trail. Mapping from this quasiparticle dynamics onto the magnetic noise spectrum, we discover that non-reciprocity tunes topological logarithmic contributions and causes long-lived metastable states due to quasiparticle trapping. Our work opens the way for broader investigations of geometrically frustrated non-reciprocity.

Figures (8)

• Figure 1: Non-reciprocal Ising gauge theory. (a) Two species ($A - \text{red}, B - \text{blue}$) of Ising variables ($\sigma^{A,B}_l$) live on the links $l$ of a square lattice, subject to a reciprocal intra-species plaquette interaction of strength $J$ and a nearest-neighbor inter-species interaction that has both reciprocal ($K_p$) and non-reciprocal ($K_m$) components. (b) The selfish energies are invariant under flipping all spins on the links connected to a pair of nearest-neighbor $A$ and $B$ sites. This extends to a general invariance of the simultaneous flip of all spins ($A,B$) along a closed loop on the dual lattice (black dashed lines).
• Figure 2: (a) Snapshots of $A$ (red) and $B$ (blue) spins, and their products $\sigma^A_i \sigma^B_i$ (black), with $-1$ values in bold, for $J/T = 3$, $K_p/T = 0.6$, $K_m/T = 0.35$ and $L = 50$. Quasiparticles are highlighted by filled circles. (b) Time and history averages ($\langle \ldots \rangle$) of the WW observables. We see that $\langle W_{A,B} \rangle$ show the expected quadratic scaling for all distances (the quantitative difference is due to the different selfish inter-species couplings: $K_p \pm K_m$), whereas the average of their product $\langle W_{AB} \rangle$ displays a crossover from quadratic to linear scaling. (c) Dependence of $\langle W_{AB}(l_c) \rangle$ on $K_m$, with vertical dashed lines identifying the corresponding confinement length $l_{\rm cross}$. All WW curves are normalised by their value at $l_c = 50$, for ease of comparison. The data in (b) and (c) are for $J/T = 4$, $K_p/T = 0.6$ and $L = 160$ and in (b) $K_m = 0.25$.
• Figure 3: Quasiparticle dynamics in the low density regime ($J-K_m \gg T$). (a) A snapshot of the two species in one of the fluctuating windows, with two $A$ (red) quasiparticles and none in $B$ (blue). (b) Local snapshot showing the spins of both species near a single $A$ quasiparticle which hops upon flipping one of the $A$ spins around the excited plaquette. Strong non-reciprocal coupling biases such dynamics towards selfish-energy-lowering spin flips, resulting in the quasiparticles performing a self-avoiding trail (SAT) on a diluted lattice. (c) When $J-K_m \gg T$, finite size systems exhibit recurring time windows without excitations ($W_{A,B}=1, \, \forall l_c$ ) separated by fluctuating windows when quasiparticle pairs are created, propagate (deconfined, as evident from the quadratic length dependent scaling), and eventually annihilate within a species ($W_B$ is shifted by $0.05$ for visual clarity). Data shown for $L = 100$, $J/T = 6, K_m/T = 3.2$. (d) Mean squared displacement of a quasiparticle in the two-quasiparticle model for different values of $K_m/T$ with $L = 200$, averaged over $100$ trajectories. The data contrast conventional RW behavior at small coupling with SAT behavior on a diluted lattice at strong coupling (see text and EM Fig. \ref{['fig5']}).
• Figure 4: (a) Second moment of $M(t)$ for the two-quasiparticle model and different values of $K_m/T$ (weak and strong coupling). The solid line is a guide to the eye showing linear $t$ scaling. The dashed line corresponds to Eq. \ref{['logmag']} (no fitting parameters). Inset: $\langle [ M(t) - M(0) ]^2 \rangle / t$ plotted in linear-log scale, to highlight the topological logarithmic contribution at weak coupling. (b) Power spectral density of the magnetization averaged over the time windows when the WW of one species is fluctuating and that of the other is saturated in Fig. \ref{['fig3']}(c). We considered $J/T = 5$, $K_m/T = 0.3$ and $L = 120$; and $J/T = 6$, $K_m/T = 3.2$ and $L = 100$. We multiplied the $K_m/T = 0.3$ curve by $3.5$ for a better visual comparison. The departure from inverse square scaling at weak coupling is due to topological logarithmic contributions that are lifted by the SAT behavior at strong coupling (see text).
• Figure 5: The confinement length scale (linear to quadratic crossover) from the $W_{AB}$ curves in Fig. \ref{['fig2']}(c), compared to the diameter of gyration of the antiferromagnetic clusters of $\sigma^A_l \sigma^B_l = -1$, for different values of $K_m/T$.