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Nonperturbative RG treatment of amplitude fluctuations for $\boldsymbol{|\varphi|^4}$ topological phase transitions

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

Published Jun 2, 2017• 0 citations
OpenAlex

Abstract

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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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.
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.
We show that four-dimensional systems may exhibit a topological phase transition analogous to the well-known Berezinskii-Kosterlitz-Thouless vortex unbinding transition in two-dimensional systems. We study a suitable generalization of the sine-Gordon model in four dimensions and the renormalization group flow equation of its couplings, showing that the critical value of the frequency is the square of the corresponding value in 2D. The value of the anomalous dimension at the critical point is determined (η=1/32) and a conjecture for the universal jump of the superfluid stiffness (4/π2) presented.
We reexamine the two-dimensional linear O(2)\mathrm{O}(2)O(2) model (${\ensuremath{\varphi}}^{4}$ theory) in the framework of the nonperturbative renormalization-group. From the flow equations obtained in the derivative expansion to second order and with optimization of the infrared regulator, we find a transition between a high-temperature (disordered) phase and a low-temperature phase displaying a line of fixed points and algebraic order. We obtain a picture in agreement with the standard theory of the Kosterlitz-Thouless (KT) transition and reproduce the universal features of the transition. In particular, we find the anomalous dimension $\ensuremath{\eta}({T}_{\mathrm{KT}})\ensuremath{\simeq}0.24$ and the stiffness jump ${\ensuremath{\rho}}_{s}({T}_{\mathrm{KT}}^{\ensuremath{-}})\ensuremath{\simeq}0.64$ at the transition temperature TKT{T}_{\mathrm{KT}}TKT​, in very good agreement with the exact results $\ensuremath{\eta}({T}_{\mathrm{KT}})=1/4$ and ${\ensuremath{\rho}}_{s}({T}_{\mathrm{KT}}^{\ensuremath{-}})=2/\ensuremath{\pi}$, as well as an essential singularity of the correlation length in the high-temperature phase as $T\ensuremath{\rightarrow}{T}_{\mathrm{KT}}$.
The scheme dependence of the renormalization group (RG) flow has been investigated in the local potential approximation for two-dimensional periodic, sine-Gordon type field-theoretic models discussing the applicability of various functional RG methods in detail. It was shown that scheme-independent determination of such physical parameters is possible as the critical frequency (temperature) at which Kosterlitz-Thouless-Berezinskii type phase transition takes place in the sine-Gordon and the layered sine-Gordon models, and the critical ratio characterizing the Ising-type phase transition of the massive sine-Gordon model. For the latter case, the Maxwell construction represents a strong constraint on the RG flow, which results in a scheme-independent infrared value for the critical ratio. For the massive sine-Gordon model also the shrinking of the domain of the phase with spontaneously broken periodicity is shown to take place due to the quantum fluctuations.