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

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

Published Sep 28, 2017• 0 citations
OpenAlex

Abstract

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.

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

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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.
Nonperturbative RG treatment of amplitude fluctuations for ∣φ∣4\boldsymbol{|\varphi|^4}∣φ∣4 topological phase transitions
The study of the Berezinskii-Kosterlitz-Thouless transition in two-dimensional ∣φ∣4|\varphi|^4∣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 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 ∣φ∣4|\varphi|^4∣φ∣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.
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 XYXYXY 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.
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.
Proliferation of defects is a mechanism that allows for topological phase transitions. Such a phase transition is found in two dimensions for the XY model, which lies in the Berezinskii-Kosterlitz-Thouless (BKT) universality class. The transition point can be found using renormalization group analysis. We apply renormalization group arguments to determine the nature of BKT transitions for the three-dimensional plaquette-dimer model, which is a model that exhibits fractonic mobility constraints. We show that an important part of this analysis demands a modified dimensional analysis that changes the interpretation of scaling dimensions upon coarse-graining. Using this modified dimensional analysis, we compute the beta functions of the model and predict a finite critical value above which the fractonic phase melts, proliferating dipoles. Importantly, the transition point is found through a renormalization group analysis that accounts for the phenomenon of UV/IR mixing, characteristic of fractonic models.
It is argued that two-dimensional U(N) spin models for any N undergo a Berezinskii-Kosterlitz-Thouless (BKT)-like phase transition, similarly to the famous XY model. This conclusion follows from the Berezinskii-like calculation of the two-point correlation function in U(N) models, approximate renormalization group analysis, and numerical investigations of the U(2) model. It is shown, via Monte Carlo simulations, that the universality class of the U(2) model coincides with that of the XY model. Moreover, preliminary numerical results point out that two-dimensional SU(N) spin models with the fundamental and adjoint terms and N>4 exhibit two phase transitions of BKT type, similarly to Z(N) vector models.
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.