Topological-sector fluctuations and ergodicity breaking at the Berezinskii-Kosterlitz-Thouless transition

The Berezinskii-Kosterlitz-Thouless (BKT) phase transition drives the unbinding of topological defects in many two-dimensional systems. In the two-dimensional Coulomb gas, it corresponds to an insulator-conductor transition driven by charge deconfinement. We investigate the global topological properties of this transition, both analytically and by numerical simulation, using a lattice-field description of the two-dimensional Coulomb gas on a torus. The BKT transition is shown to be an ergodicity breaking between the topological sectors of the electric field, which implies a definition of topological order in terms of broken ergodicity. The breakdown of local topological order at the BKT transition leads to the excitation of global topological defects in the electric field, corresponding to different topological sectors. The quantized nature of these classical excitations, and their strict suppression by ergodicity breaking in the low-temperature phase, afford striking global signatures of topological-sector fluctuations at the BKT transition. We discuss how these signatures could be detected in experiments on, for example, magnetic films and cold-atom systems.

Figure 1

Topological-sector fluctuations of lattice electric fields in the two-dimensional Coulomb gas on a torus. Shown is the $x$ component of the normalized total harmonic mode $L{\overline{E}}_x/2\pi$ and winding field $L{\overline{E}}_{\mathrm{w},x}/2\pi$ vs Monte Carlo time for an $L\times L$ system of linear size $L=16$ at $T=1.34$ (top) and $T=2.0$ (bottom). The system was simulated using the MR algorithm with local moves only. At the lower temperature (top), harmonic-mode fluctuations are finite (black) but there is no winding-field component (blue), while at the higher temperature (bottom), the winding-field component becomes finite, indicating topological-sector fluctuations.

Figure 2

The susceptibility quotient ${\chi }_{\mathrm{w}}^{\text{local}}/{\chi }_{\mathrm{w}}^{\text{all}}$ vs temperature for an $L\times L$ Coulomb gas of linear size $L=64$. In the region $T<1.2$, the quotient is zero, while for $T>1.6$, the quotient approaches unity. This divergence between the results of the local-update and the all-updates simulations, accompanied by striking fluctuations in the intermediate region, signals an ergodicity breaking as the system is cooled through the BKT transition. The line is a guide to the eye.

Figure 3

The winding-field susceptibility ${\chi }_{\mathrm{w}}$ as a function of temperature for $L\times L$ Coulomb gases of linear size $L=8,16,32$, and 64. The curves intersect at low and high temperature. Inset: An expanded plot of the data in the region of the low-temperature intersections. The indicated crossover temperatures are given by $T_{\text{Cross}}(L=16)=1.45, T_{\text{Cross}}(L=32)=1.40$, and $T_{\text{Cross}}(L=64)=1.37$ (to within estimated error), based on a data fit.

Figure 4

The crossover temperature $T_{\text{Cross}}$ (black data) and crossover susceptibility ${\chi }_{\mathrm{w}}^{\mathrm{Cross}}$ (red data) as functions of inverse linear system size $1/L$. Lines are linear-regression fits to each data set, from which the $y$ intercept ($L\to \infty$) was calculated. $T_{\mathrm{Cross}}(L\to \infty )=1.350\left(2\right)$, equal to the BKT transition temperature $T_{\mathrm{BKT}}$ [46]. The crossover susceptibility ${\chi }_{\mathrm{w}}^{\mathrm{Cross}}(L\to \infty )=4\times {10}^{-4}$ with estimated error of the same order: there is no measurable difference between this quantity and the winding-field susceptibility due to global updates only at $T=1.35$.

Figure 5

A charge-hop update in the positive $x$ direction: The $s_{ij}$ variable (red arrow) has its value decreased by an amount $q$. The value of the electric-field flux $E_{\alpha \beta }$ (black arrow) flowing from site $\alpha$ to site $\beta$ then decreases by $q/{\epsilon }_0$, corresponding to a charge-hop update. Red circles represent positive charges; white circles represent empty charge sites.

Figure 6

An update of the rotational degrees of freedom of the electric field: The value of the $\phi$ variable at the center of a randomly chosen lattice plaquette decreases by an amount $\Delta$. This rotates the electric flux by an amount $\Delta /{\epsilon }_0$ around the plaquette, leaving Gauss' law satisfied. Red arrows represent $\phi$ variables, black arrows represent the electric field, dashed red lines represent the conjugate lattice $D^{\prime }$, the blue arrow represents the direction of the field rotation, and gray circles represent sites of arbitrary charge.