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arXiv:1706.00618 · Physical review. B./Physical review. B

Nonperturbative renormalization group treatment of amplitude fluctuations for |φ|4 topological phase transitions

Nicolò Defenu, Andrea Trombettoni, I. Nándori, Tilman Enss

Published Nov 13, 2017• 29 citations
arXivHTMLPDF

Abstract

The study of the Berezinskii-Kosterlitz-Thouless transition in two-dimensional ${|\ensuremath{\varphi}|}^{4}$ 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 ${|\ensuremath{\varphi}|}^{4}$ 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.

DOI
10.1103/physrevb.96.174505

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The study of the Berezinskii-Kosterlitz-Thouless (BKT) transition in two-dimensional |ϕ| 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 non-universal quantities.We show how one can obtain a consistent phase diagram in the AP representation using the functional renormalization group scheme.Constructing the mapping between |ϕ| 4 and the XY models allows us to treat these models on equal footing.We estimate universal and non-universal quantities of the two models and find good agreement with available Monte Carlo results.The presented approach is flexible enough to treat parameter ranges of experimental relevance.
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)PLEEE81539-375510.1103/PhysRevE.82.041128] 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.
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We analyze the interplay of thermal amplitude and phase fluctuations in a U(1)U(1)U(1) symmetric two-dimensional ϕ4\phi^4ϕ4-theory. To this end, we derive coupled renormalization group equations for both types of fluctuations. Discarding the amplitude fluctuations, the expected Berezinskii-Kosterlitz-Thouless (BKT) phase characterized by a finite phase stiffness and an algebraic decay of order parameter correlations is recovered at low temperatures. However, in contrast to the widespread expectation, amplitude fluctuations are not innocuous, since their mass vanishes due to a strong renormalization by phase fluctuations. Even at low temperatures the amplitude fluctuations lead to a logarithmic renormalization group flow of the phase stiffness, which ultimately vanishes. Hence, the BKT phase is strictly speaking replaced by a symmetric phase with a finite correlation length, which is however exponentially large at low temperatures. The vortex-driven BKT transition is then rounded to a crossover, which may be practically indistinguishable from a true phase transition in real systems such as 4^44He-films.