Dual lattice functional renormalization group for the Berezinskii-Kosterlitz-Thouless transition: Irrelevance of amplitude and out-of-plane fluctuations

We develop a functional renormalization group (FRG) approach for the two-dimensional $XY$ model by combining the lattice FRG proposed by Machado and Dupuis [Phys. Rev. E 82, 041128 (2010)] with a duality transformation that explicitly introduces vortices via an integer-valued field. We show that the hierarchy of FRG flow equations for the infinite set of relevant and marginal couplings of the model can be reduced to the well-known Kosterlitz-Thouless renormalization group equations for the renormalized temperature and the vortex fugacity. Within our approach it is straightforward to include weak amplitude as well as out-of-plane fluctuations of the spins, which lead to additional interactions between the vortices that do not spoil the Berezinskii-Kosterlitz-Thouless transition. This demonstrates that previous failures to obtain a line of true fixed points within the FRG are a mathematical artifact of insufficient truncation schemes.

Figure 1

Schematic representation of the physical lattice (blue dots), the dual lattice (green dots), and of the related fields. The red arrows denote the currents $p_{i\mu }$ flowing into and out of the site $i$ of the physical lattice in direction $\mu =x,y$. The $m$ field introduced in Eqs. (2.15a)–(2.15d) is defined on the sites of the dual lattice.

Figure 2

Graphical representation of the exact FRG flow equations for the irreducible vertices with two and four legs, see Eqs. (3.20) and (3.22), as well as for the irreducible six-point vertex [47]. Solid lines represent the cutoff-dependent propagator $G_{\lambda }\left(\mathit{q}\right)$, while slashed solid lines denote the corresponding single-scale propagator ${\dot{G}}_{\lambda }\left(\mathit{q}\right)$. The dot over the vertex on the left-hand side represents the derivative with respect to the cutoff parameter $\lambda$. We have labeled the external momenta by integers, omitting them where there is no danger of ambiguity.

Figure 3

Graphical representation of the approximate flow equations for the irreducible vertices ${\mathrm{\Gamma }}_{\mathrm{\Lambda }}^{\left(2\right)}\left(\mathit{k}\right)$, ${\mathrm{\Gamma }}_{\mathrm{\Lambda }}^{\left(4\right)}\left(\mathit{k}\right)$, and ${\mathrm{\Gamma }}_{\mathrm{\Lambda }}^{\left(6\right)}\left(\mathit{k}\right)$ as given in Eqs. (4.14), (4.16), and (4.17). Here external legs without labels carry vanishing momentum.

Figure 4

Flow diagram for the $XY$ model in the vicinity of the BKT transition at ${\tau }_{*}=T_c/J=\pi /2$, which was obtained by numerically solving the flow Eqs. (4.57) and (4.58). The separatrix (red line) separates the high-temperature regime on the right from the low-temperature regime on the left. In the disordered high-temperature regime ${\tilde{y}}_l$ and ${\tau }_l$ flow to infinity, while in the quasi-long-range ordered low-temperature regime the flow terminates at the Gaussian fixed point manifold (green line) at $\tilde{y}=0$ and $\tau \le \pi /2$.

Figure 5

Representative plot of the RG flow of the irrelevant coupling ${\tilde{g}}_l$ generated by amplitude fluctuations as a function of the logarithmic scale parameter $l=ln({\mathrm{\Lambda }}_0/\mathrm{\Lambda })$. The curves are obtained from the numerical solution of the RG Eqs. (4.57), (4.58), and (5.48), with initial conditions ${\tilde{y}}_0=0.01$ and $\tau =0.99{\tau }_{*}=0.99\pi /2$. The upper curve (red) corresponds to a finite initial value of ${\tilde{g}}_0=0.01$ (corresponding to a model with finite amplitude fluctuations), while the lower curve (blue) corresponds to ${\tilde{g}}_0=0$ as is the case for the $XY$ model. We find that for sufficiently large $l$ both curves merge and eventually approach zero for $l\to \infty$, so that the large $l$ behavior of ${\tilde{g}}_l$ is independent of the initial strength of the amplitude fluctuations.