Nonperturbative renormalization group treatment of amplitude fluctuations for ${\left|\mathbf{\phi }\right|}^{\mathit{4}}$ topological phase transitions

The study of the Berezinskii-Kosterlitz-Thouless transition in two-dimensional ${\left|\phi \right|}^4$ models can be performed in several representations, and the amplitude-phase (AP) Madelung parametrization is a natural way to study the contribution of density fluctuations to nonuniversal quantities. We introduce a functional renormalization group scheme in AP representation where amplitude fluctuations are integrated first to yield an effective sine-Gordon model with renormalized superfluid stiffness. By a mapping between the lattice $XY$ and continuum ${\left|\phi \right|}^4$ models, our method applies to both on equal footing. Our approach correctly reproduces the existence of a line of fixed points and of universal thermodynamics and it allows to estimate universal and nonuniversal quantities of the two models, finding good agreement with available Monte Carlo results. The presented approach is flexible enough to treat parameter ranges of experimental relevance.

Figure 1

Flow diagram of the $\text{O}\left(2\right)$ symmetric ${\left|\phi \right|}^4$ theory with flowing effective potential (9) in parametrization (i). The flow is first attracted toward a line of pseudofixed points at large ${\tilde{\kappa }}_k$; then the flow proceeds very slowly along this line toward $({\tilde{\kappa }}_k,{\tilde{\lambda }}_k)=(0,0)$, which corresponds to the high-temperature phase.

Figure 2

The anomalous dimension $\eta$ in the $\text{O}\left(2\right)$ model (blue solid line) represents the power-law decay of the two-point correlation function. Its value is found from Eq. (12) along the pseudofixed line in Fig. 1. The star represents the choice of $\eta$ in [48, 53]. The lower, red solid curve gives the values of ${\partial }_t{\tilde{\kappa }}_k$ along the same pseudofixed line, while a line of true fixed points would have ${\partial }_t{\tilde{\kappa }}_k=0$, as one has in [53] by a temperature-dependent choice of the cutoff function.

Figure 3

Flow diagram for the rescaled superfluid stiffness ${\tilde{\kappa }}_k$ and interaction ${\tilde{\lambda }}_k$ due to amplitude fluctuations in $d=2$. For large enough ${\tilde{\lambda }}_{\mathrm{\Lambda }}$, the flow always proceeds towards an infinitely interacting ${\tilde{\lambda }}_{k\simeq 0}\simeq +\infty$ fixed line where the expectation value ${\tilde{\rho }}_0={\kappa }_k$ is effectively frozen. (a) The naive AP flow is (incorrectly) attracted for ${\tilde{\lambda }}_{\mathrm{\Lambda }}\ll 1$ toward the free theory. (b) The modified AP flow with the Gaussian contributions subtracted reproduces the expected flow diagram.

Figure 4

Superfluid density ${\rho }_s$ as a function of chemical potential. (a) ${\rho }_s/mT$ vs $\mu /U$ for seven different values of $mU=0.6...,0.02$ from top to bottom. (b) ${\rho }_s/mT$ vs dimensionless chemical potential $X$. Inset: critical chemical potential ${\mu }_c/U$ vs $U$.

Figure 5

Superfluid scaling function $f\left(X\right)=\pi {\rho }_s/2mT$ (black line) as a function of the chemical potential variable $X$ (average and variance over 30 sets of data for different interaction values $U)$, the standard deviation is shown as a red shadow. Black dots are the MC data from [62].

Figure 6

Superfluid stiffness $J_s$ in units of $J$ as a function of the temperature for the $XY$ model. The purple lines represent the ${\left|\phi \right|}^4$ model with initial condition (28) for the potential and $\mu =Jd$, as detailed in the case a of the main text. The blue lines are the case b, the gray lines the case c, and the green lines the case d. Solid and dashed lines represent, respectively, the results without and with the inclusion of vortex excitations.