Nonequilibrium Kosterlitz-Thouless transition in a three-dimensional driven disordered system

We demonstrate a three-dimensional Kosterlitz-Thouless (KT) transition in the random field $XY$ model driven out of thermal equilibrium. By employing the spin-wave approximation and functional renormalization group approach, in the weak disorder regime, the three-dimensional driven random field $XY$ model is found to exhibit a quasi-long-range order phase, wherein the correlation function shows power-law decay with a nonuniversal exponent that depends on the disorder strength. This result is consistent with that reported in a previous numerical study. We further develop a phenomenological theory of the three-dimensional KT transition by taking into account the effect of vortices. The point of this theory is that the cross-section of the system with respect to a plane perpendicular to the driving direction is essentially identical to the two-dimensional pure $XY$ model.

Figure 1

(a) RG evolution of $\delta \left(u\right)$ for the RFXYM at $D=4$ calculated by Eq. (62). The values of the renormalization scale are $l=0.0,1.0,2.0,3.0,4.0$ from the bottom to the top at $u=\pi$. The bare cumulant is given by the cosine function. $\delta \left(u\right)$ flows to zero in the limit $l\to \infty$. (b) RG evolution of $\delta \left(u\right)$ for the DRFXYM at $D=3$ calculated by Eq. (71). $\delta \left(u\right)$ flows to a constant $\delta \left(u\right)={\delta }_{\mathrm{B}}\left(0\right)\ne 0$ in the limit $l\to \infty$.

Figure 2

RG trajectories for the spin-wave model with the correction of the vortices. The horizontal axis is $\delta \left(0\right)$ and the vertical axis is $y$. The thin dashed curve is the bare value of the vortex fugacity Eq. (80). The thick line on the horizontal axis is the fixed line characterizing the QLRO. The inset represents the trajectories near the endpoint of the fixed line.

Figure 3

Schematic picture of the vortex dissociation process for the 3D-DRFXYM. The lines represent vortex lines. The transverse section corresponds to a snapshot of the 2D pure $XY$ model. (a) QLRO phase. The vortex lines form into small loops, whose volume density is low enough at weak disorder. (b) Disordered phase. The space is filled with tangled long vortex lines.

Figure 4

Graphical representation of the flow equation for ${\mathrm{\Gamma }}_1$.

Figure 5

Graphical representation of the flow equation for ${\mathrm{\Gamma }}_2$.

Figure 6

Graphical representation of the flow equation for ${\mathrm{\Gamma }}_3$.