Disordered XY model: Effective medium theory and beyond

We study the effect of uncorrelated random disorder on the temperature dependence of the superfluid stiffness in the two-dimensional classical XY model. By means of a perturbative expansion in the disorder potential, equivalent to the $T$-matrix approximation, we provide an extension of the effective-medium-theory result able to describe the low-temperature stiffness and its separate diamagnetic and paramagnetic contributions. These analytical results provide an excellent description of the Monte Carlo simulations for two prototype examples of uncorrelated disorder. Our findings offer an interesting perspective on the effects of quenched disorder on longitudinal phase fluctuations in two-dimensional superfluid systems.

Figure 1

Top: comparison between the Monte Carlo results and the low-temperature analytical expressions for the three response functions $J_s$, $J_d$, and $J_p$ based on Eqs. (8) and (9). Bottom: vortex density of the system as a function of the temperature. The linear size of the system simulated is $L=256$.

Figure 2

Diagrammatic representation of the local self-energy corrections to the bare Green's function at $T=0$. Here the solid line represents the bare Green's function $G_0$, the double line is the dressed one $G$, and each cross accounts for a single-impurity scattering, contributing with a factor proportional to $\delta J_i$.

Figure 3

Diagrammatic representation of the local self-energy corrections to the bare Green's function up to the linear terms in temperature. In the diagrammatic expansion, each arrow stands for a bare Green's function, while the hatched square accounts for two different impurity-scattering contributions. As shown in the second line, indeed, it corresponds to the sum of the zero-temperature correction, proportional to $\delta J_i$, and the linear temperature one, proportional to $J_i$, arising from the four-leg vertex in $H_i^{\left(4\right)}$. The closed loop appearing in the second line accounts self-consistently for all the local self-energy corrections, as shown in the last line. As a matter of fact, however, since we are considering the self-energy corrections up to the linear terms in temperature, in our calculation we will consider the closed loop corrected by only the zero-temperature contribution; that is, we will replace the hatched square with the bare circle $\delta J_i$.

Figure 4

Monte Carlo results on a disordered XY model with Gaussian distributed random couplings. The three response functions are plotted: (a) $J_s$, (b) $J_d$, and (c) $J_p$ as a function of the temperature. The dashed lines correspond to the analytical results (53, 54, 55) obtained with the EMA and reproduce perfectly the numerical simulations.

Figure 5

Monte Carlo results for a disordered XY model with random diluted couplings. The three response functions are plotted: (a) $J_s$, (b) $J_d$, and (c) $J_p$ as a function of the temperature. The dashed lines correspond to the analytical results (59, 60, 61) obtained with the EMA. Even close to the percolation threshold at $p=0.5$, above which no superfluidity is possible, the analytical formulas fit very well the numerical data.