Renormalization-group flow and asymptotic behaviors at the Berezinskii-Kosterlitz-Thouless transitions

We investigate the general features of the renormalization-group flow at the Berezinskii-Kosterlitz-Thouless (BKT) transition, providing a thorough quantitative description of the asymptotc critical behavior, including the multiplicative and subleading logarithmic corrections. For this purpose, we consider the RG flow of the sine-Gordon model around the renormalizable point which describes the BKT transition. We reduce the corresponding $\beta$ functions to a universal canonical form, valid to all perturbative orders. Then we determine the asymptotic solutions of the RG equations in various critical regimes: the infinite-volume critical behavior in the disordered phase, the finite-size scaling limit for homogeneous systems of finite size, and the trap-size scaling limit occurring in two-dimensional bosonic particle systems trapped by an external space-dependent potential.

Figure 1

Sketch of the RG flow at the BKT transition. The dashed curve in the region $C$ ($v>0$) shows the approach to criticality ($Q=0$ line) of the 2D $\mathit{XY}$ model from the HT phase. The correlation length is singular along the $Q=0$ line in the region $v>0$, while it is analytical along the $Q=0$ for $v<0$.

Figure 2

FSS plot of the two-point function $G(\mathbf{0},\mathbf{x})$ at $T_c$ for $\mathbf{x}=(r,0)$, $0\le r\le l_t=(L-1)/2$, for a homogeneous $\mathit{XY}$ model on a $L^2$ lattice with open boundary conditions (OBCs).

Figure 3

TSS plot of the two-point function $G(\mathbf{0},\mathbf{x})$ at $T_c$ for a harmonic potential $V={(r/l_t)}^2$. We set $\kappa =0$ (top) and $\kappa =1$ (bottom). To check finite-size effects, we report two data sets: $L\approx 4l_t$ and $L\approx 8l_t$ in the two cases, respectively.

Figure 4

TSS plot of the two-point function $G(\mathbf{0},\mathbf{x})$ at $T_c$ for a trap with potential $V={(r/l_t)}^4$. We set $\kappa =0$ (top) and $\kappa =1/2$ (bottom). To check finite-size effects, we report data for $L\approx 4l_t$ and $L\approx 2l_t$.

Figure 5

Plot of $G_r\left(\mathbf{x}\right)\equiv l_t^{1/4}{\left(\ln l_t\right)}^{-1/8-\kappa /4}G(\mathbf{0},\mathbf{x})$ vs $r_r\equiv r{\left(\ln l_t\right)}^{\kappa }/l_t$ for $p=2$, $p=4$, and for the homogeneous system with OBCs (it corresponds to $p\to \infty$), using the data for the largest available trap sizes. We use $\kappa =1,1/2,0$ for $p=2,4$ and the OBC case, respectively.