Strong randomness criticality in the scratched XY model

We study the finite-temperature superfluid transition in a modified two-dimensional (2D) XY model with power-law-distributed “scratch”-like bond disorder. As its exponent decreases, the disorder grows stronger and the mechanism driving the superfluid transition changes from conventional vortex-pair unbinding to a strong randomness criticality (termed scratched XY criticality) characterized by a nonuniversal jump of the superfluid stiffness. The existence of the scratched XY criticality at finite temperature and its description by an asymptotically exact semi-renormalization group theory, previously developed for the superfluid-insulator transition in one-dimensional disordered quantum systems, is numerically proven by designing a model with minimal finite-size effects. Possible experimental implementations are discussed.

Figure 1

The log-log plot of the distribution of renormalized strengths $t\left(\ell \right)$ at $\mathrm{\Gamma }=0.7$ for $L=8$, 16, and 32 from $8\times {10}^5, 1.2\times {10}^6$, and $2.0\times {10}^6$ disorder realizations (in contrast to a few thousands in Ref. [20]). The slope of the tail keeps changing until small enough $t\left(\ell \right)$ (requiring a sufficient number of disorder realizations) is reached. The tail part on the log-log scale is perfectly fitted by a linear line with a slope of 0.43(2) which agrees with the exponent $1/\mathrm{\Gamma }-1$ within error bars. The fitted values extracted from the last four points of the tail part are 0.414 for $L=8$, 0.448 for $L=16$, and 0.441 for $L=32$. Therefore, the dominant barriers joining adjacent superfluid regions are formed by the individual deepest scratches.

Figure 2

A plot of the disorder averaged $K(lnL)$ for $L=16,32,\cdots ,512$. The brown dashed line is a plot of the critical $1/\zeta$ line. We find a critical value of ${\mathrm{\Gamma }}_c=0.764\left(2\right)$ (vertical grid line) with a nonuniversal value of $K_c=4.24\left(4\right)$ (upper horizontal grid line) at the transition (cf. Sec. 4). The nonuniversal value of $K_c$ is larger than that in the BKT case where $K_c=2$ (lower horizontal grid line).

Figure 3

Shown is the data collapse of $\left(K\left(\ell \right)-{\zeta }_c^{-1}\right)\left(ln\left(L\right)+C\right)$ over $({\zeta }_c-\zeta ){(lnL+C)}^2$. With ${\zeta }_c=0.236\left(4\right)$ and $C=3.86\left(5\right)$, all the finite-size data collapse onto a single line satisfying the constraint $F\left(0\right)=2(1-{\zeta }_c)/{\zeta }_c^2$. The critical $K_c$ is given by $K_c=1/{\zeta }_c=4.24\left(7\right)>2$ as predicted by the strong randomness criticality.

Figure 4

The Weber-Minnhagen [25] root-mean-square-error $\sigma$ by fitting the flow of $K(lnL)$ for $L=16$, 32, 64, 128, 256, and 512 to the critical flow of the sXY criticality (red dots, lower and left axis) and to the BKT criticality (blue squares, upper and right axis). While $\sigma$ displays a sharp minimum at ${\mathrm{\Gamma }}_c=0.764\left(2\right)$ for the sXY RG, there is no such minimum for the BKT RG.