Nonperturbative renormalization group treatment of amplitude fluctuations for $|\varphi|^4$ topological phase transitions

TL;DR

This paper develops a nonperturbative renormalization group method in amplitude-phase representation to analyze topological phase transitions in $| ext{phi}|^4$ models, accurately capturing fixed points and thermodynamics.

Contribution

It introduces a novel functional renormalization group scheme in AP representation, effectively handling amplitude fluctuations and unifying lattice XY and continuum $| ext{phi}|^4$ models.

Findings

Correctly reproduces fixed point line and universal thermodynamics.

Provides estimates for universal and nonuniversal quantities matching Monte Carlo results.

Flexible approach applicable to experimentally relevant parameter ranges.

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.

Equations

$$\beta H_{XY}=-K\sum _{⟨ij⟩}\left[\cos \left({\theta }_i-{\theta }_j\right)-1\right]$$

$$S[\phi ]=\int d^{2}x\left\{\frac{1}{2m}{\partial }_{\mu }\phi {\partial }_{\mu }{\phi }^{\ast }-\mu |\phi {|}^2+\frac{U}{2}|\phi {|}^4\right\}.$$

$$\beta H_{\text{sw}}=\frac{K}{2}\int {\left(\nabla \theta \right)}^2{d}^{2}x.$$

$$M_L\propto {\left(\frac{a}{L}\right)}^{\frac{1}{2\pi K}},G(x)\propto {\left(\frac{a\pi }{x}\right)}^{\frac{1}{2\pi K}},$$

$$\eta (T)=\frac{T}{2\pi J}.$$

$$T_{\text{BKT}}\approx \frac{\pi J}{2}.$$

$${\oint }_C\nabla \theta \cdot d\mathbf{\ell }=2\pi m_i$$

$$S_{SG}[\Phi ]=\int d^{2}x\left(\frac{1}{2}{\partial }_{\mu }\Phi {\partial }_{\mu }\Phi -u\cos \left(\beta \Phi \right)\right),$$

$$U(\rho )=\frac{{\lambda }_k}{2}(\rho -{\kappa }_k{)}^2$$

$${\partial }_t{\tilde{\lambda }}_k=(2-2{\eta }_k){\tilde{\lambda }}_k-{\tilde{\lambda }}_k^2\frac{\left(4-{\eta }_k\right)}{8\pi }\left(N-1+\frac{1}{(1+2{\tilde{\kappa }}_k{\tilde{\lambda }}_k{)}^3}\right),$$

$${\partial }_t{\tilde{\kappa }}_k={\eta }_k{\tilde{\kappa }}_k-\frac{\left(4-{\eta }_k\right)}{16\pi }\left(N-1+\frac{1}{(1+2{\tilde{\kappa }}_k{\tilde{\lambda }}_k{)}^2}\right),$$

$${\eta }_k=\frac{1}{\pi }\frac{{\tilde{\kappa }}_k{\tilde{\lambda }}_k^2}{(1+2{\tilde{\kappa }}_k{\tilde{\lambda }}_k{)}^2}.$$

$$H_{XY}=-J\sum _{⟨ij⟩}\left(s_{x,i}{s}_{x,j}+s_{y,i}{s}_{y,j}\right),$$

$$Z(\beta )=\int \mathcal{D}s{e}^{\beta J{\sum }_{⟨ij⟩}\left(s_{x,i}{s}_{x,j}+s_{y,i}{s}_{y,j}\right)}{\Pi }_j\delta \left({\mathbf{s}}_j^2-1\right)$$

$$Z(\beta )=\int \mathcal{D}s{e}^{\mathbf{s}\cdot \frac{K^{\prime }}{2}\cdot \mathbf{s}}{\Pi }_j\delta \left({\mathbf{s}}_j^2-1\right)$$

$$e^{\mathbf{s}\cdot \frac{K^{\prime }}{2}\cdot \mathbf{s}}={\left[(2\pi {)}^N\sqrt{\det K^{\prime }}\right]}^{-1}\int \mathcal{D}\varphi e^{-\mathbf{\varphi }\cdot \frac{K^{\prime -1}}{2}\cdot \mathbf{\varphi }-\mathbf{s}\cdot \mathbf{\varphi }}$$

$$K=K^{\prime }+2\beta \mu \mathbb{I}$$

$$Z(\beta )={\left[(2\pi {)}^N\sqrt{\det K}\right]}^{-1}\int \mathcal{D}\varphi e^{-\mathbf{\varphi }\cdot \frac{K^{-1}}{2}\cdot \mathbf{\varphi }+{\sum }_{j}U({\mathbf{\varphi }}_j)},$$

$$e^{U({\mathbf{\varphi }}_j)}$$

$$U(\mathbf{\varphi })=\log \left(\pi I_0\left(\sqrt{\rho }\right)\right)$$

$$K(q)=2\beta (\mu +J{\epsilon }_0(q))$$

$${\epsilon }_0(q)=\sum _{\nu =1}^d\cos (q_{\nu }a)$$

$$S_{\text{kin}}[\varphi ]=\frac{1}{2}\sum _q{\mathbf{\varphi }}_q\left(\frac{1}{K(q)}-\frac{1}{K(0)}\right){\mathbf{\varphi }}_{-q}.$$

$$\mathbf{\varphi }\to 2\sqrt{\frac{\beta }{J}}(Jd+\mu )\mathbf{\phi }$$

$$S_{\text{kin}}[\phi ]=\frac{1}{2}\sum _q{\mathbf{\phi }}_q\epsilon (q){\mathbf{\phi }}_{-q}$$

$$\epsilon (q)=2(Jd+\mu )\frac{d-{\epsilon }_0(q)}{J{\epsilon }_0(q)+\mu }.$$

$${\epsilon }_{\text{cl}}(q)=q^2$$

$$S_{\text{pot}}[\phi ]=\int d^{d}x\left[-U\left(2\sqrt{\frac{\beta }{J}}(Jd+\mu )|\mathbf{\phi }|\right)+\frac{Jd+\mu }{J}|\mathbf{\phi }{|}^2\right].$$

$$\phi (x)=\sqrt{\rho (x)}{e}^{i\theta (x)}.$$

$$S[\phi ]=\int d^{d}x\left\{\frac{1}{8\rho }{\partial }_{\mu }\rho {\partial }_{\mu }\rho +\frac{\rho }{2}{\partial }_{\mu }\theta {\partial }_{\mu }\theta +U(\rho )\right\}.$$

Full text

I Introduction

The study of topological phase transitions plays a major role in modern physics, both for the importance of having non-local order parameters in absence of conventional spontaneous symmetry breaking and for their occurrence in a wide variety of low-dimensional systems, including superfluid Bishop1978 and superconducting films Epstein1981 , two-dimensional ($2d$) superconducting arrays Resnick1981 ; Martinoli2000 ; Fazio2001 , granular superconductors Simanek1994 , $2d$ cold atomic systems Hadzibabic2006 ; Schweikhard2007 ; Murthy2015 and one-dimensional ($1d$) quantum models Giamarchi2004 .

The standard understanding of the main properties of phase transitions in $2d$ interacting systems is based on the role of topological defects Nelson2002 as relevant excitations of these models. In $2d$ systems with continuous symmetry the unbinding of vortex excitations drives the system out of the superfluid state above a finite critical temperature $T_{\text{BKT}}$. The mechanism for this topological phase transition in $2d$ with continuous symmetry—in which there is no local order parameter according to the Mermin-Wagner (MW) theoremMermin1966 ; Hohenberg1967 —was first explained by Berezinskii, Kosterlitz and ThoulessBerezinskii1972 ; Kosterlitz1973 ; Kosterlitz1974 and lead to the paradigm of the BKT critical behaviorKosterlitz2016 .

The importance of the BKT mechanism can hardly be overestimated. On the one hand, it explained $2d$ superfluidity at finite temperature despite the lack of off-diagonal long-range orderPenrose1956 , which manifests itself in the absence of magnetization in $2d$ magnetic models such as the $XY$ modelMermin1966 and in a vanishing condensate fraction at finite temperature in $2d$ bosonic modelsStringari1995 . Nevertheless, because of the power-law decay of correlation functions in the low-temperature phase Berezinskii1972 one can still have superfluid/superconducting behaviorNelson1977 . The physical consequences have been studied in very different $2d$ systems, with applications ranging from soft matterNelson2002 and magnetic systemsBramwell1993 to layered and high-$T_{c}$ superconductorsTinkham1996 , where the strong anisotropyLawrence1971 may induce BKT behaviorTinkham1996 ; Benfatto2007 ; for an overview of the relevant literature we refer the reader to the recent review Kosterlitz2016 . At the same time, despite extensive work the effects of disorder, spatial anisotropy and more complex or long-range interactions in real systems require the development of advanced theoretical tools to extend our understanding of BKT topological phase transitions to these cases.

On the other hand, $1d$ quantum systems at zero temperature can be mapped via the quantum-to-classical correspondence to $2d$ classical models at finite temperature and share the same universal properties. This motivated extensive study of BKT properties in $1+1$ dimensional models and field theories, in particular the sine-Gordon (SG) modelColeman1975 ; Rajaraman1987 . The $XY$ model can be linked to SG theory in two steps: first, the Villain approximationVillain1975 to the $XY$ model preserves the periodicity of the phase variable but approximates the cosine angular dependence with a harmonic one, and can be mapped exactly onto the $2d$ Coulomb gasJose1977 ; Minnhagen1987 ; Gulacsi1998 . It was shown rigorouslyFrohlich1982 that both the Coulomb gas and the Villain model exhibit a BKT transition. In a second step, by neglecting irrelevant higher vorticities/charges, the Coulomb gas is mapped onto the (single-frequency) SG model, which also exhibits a BKT transition Coleman1975 ; Amit1980 ; Nandori2001 . In the Villain model, vortex and spin-wave degrees of freedom are decoupled, unlike in the $XY$ model LeBellac1991 . Note that in general a strong spin-vortex coupling can destroy the BKT transition.

The BKT scenario clearly applies when one can define local phases and explicitly detect and study vortices, such as in the $2d$ $XY$ model on a lattice, but it also applies in cases where it is difficult to define the phase or detect vortices. In all cases, the BKT transition separates a low-temperature phase with power-law decaying correlations from a high-temperature phase with exponentially decaying correlations, without an explicit need to monitor vortex configurationsHadzibabic2006 . Even then, one cannot disregard the periodic (compact) nature of the phase variables, which is necessary to obtain the BKT transition and is correctly taken into account in the SG model. The subtlety of the compact phase is the reason why the results from the SG and $|\varphi|^{4}$ models are not easily related: the SG model provides an excellent description of the RG flow near the critical point and is the natural formalism to include the compact phase, but the inclusion of fluctuations apart from vortices is not straightforward. In contrast, within the $O(2)$ symmetric $|\varphi|^{4}$ theory one readily incorporates amplitude fluctuations but it is difficult to access the properties related to the periodicity of the phase and to recover the BKT transition in the thermodynamic limit (see below).

In this work we discuss the role of the phase variable in $|\varphi|^{4}$ theory and show that the amplitude-phase (AP) Madelung representation of the field $\varphi=\sqrt{\rho}\,e^{i\theta}$ leads to a consistent and efficient treatment combining the advantages of the SG and $|\varphi|^{4}$ approaches. With this tool we can treat on equal footing both the $XY$ lattice model, where amplitude is fixed by construction, and the $|\varphi|^{4}$ model, for which we show without a priori assumptions that amplitude (density) fluctuations are gapped at the critical point, at least at leading order in the truncation. For this purpose we employ an exact mapping from the $XY$ model to an appropriate $|\varphi|^{4}$ theory. Our approach with a periodic phase variable recovers the universal properties of the BKT transition including the line of fixed points, essential scaling and the equation of state in the fluctuation regime; this would be lost without phase periodicity. More importantly, we can also study the contribution of amplitude and longitudinal spin fluctuations to non-universal quantities such as the critical temperature, which is useful for BKT studies of $2d$ superconductorsSchneider2000 ; Benfatto2012 and other materials.

We perform our study in the framework of the functional renormalization group (FRG), which generalizes the idea of Wilson renormalization to the full functional form of the Landau-Ginzburg free energy. Since its introductionWetterich1993 , FRG has been able to recover and expand most of the traditional RG results and provides a systematic approach for the investigation of high-energyBerges2002 , condensed matterKopietz2010 ; Metzner2012 ; Boettcher2012 and statistical physicsDelamotte2012 . An advantage of FRG is particularly evident when considering the universal critical exponents of $O(N)$ field theories as a function of the spatial dimension $d$ and the field component number $N$. The FRG approach combined with lowest order derivative expansion Morris1997 gives numerical results for the anomalous dimension $\eta$ and correlation length exponent $\nu$, which reproduce the expected behavior in the limiting cases $N\to\infty$, $d\to 4$ and $d\to 2$ Codello2013 ; Codello2015 ; also $O(N)$ models with long-range interaction have been studied Defenu2015 ; Defenu2016 .

Since several works already addressed the BKT transition using FRG Grater1995 ; Gersdorff2001 ; Krahl2007 ; Nagy2009 ; Machado2010 ; Jakubczyk2014 ; Jakubczyk2017 , we think it is useful to explain here in detail our motivation to study $2d$ systems in an FRG framework using the AP parameterization. FRG reproduces for $d\to 2$ the exact behavior required by the MW theorem Mermin1966 ; Hohenberg1967 . Moreover, it is possible to recover the MW theorem already at the lowest order of the derivative expansion, i.e., in the local potential approximation Defenu2015-2 . The compatibility of FRG results with the MW theorem also leads in the $N>2$ case to an exact agreement of numerical critical exponents with the lowest order $4-\varepsilon$ LeGuillou1980 and $2+\tilde{\varepsilon}$ expansion Brezin1976 for the $O(N)$ nonlinear $\sigma$ models. Furthermore, for the anomalous dimension $\eta$ in general $d$ one finds $\eta\to 0$ for $d>2$ and $N\geq 2$ in the limit $d\to 2$ Codello2013 ; Codello2015 . However, in the BKT case $d=2$ and $N=2$, the application of FRG is much less straightforward.

The field theoretical and FRG approaches to the $d=2$, $N=2$ case in general use a two-component, complex $|\varphi|^{4}$ theory in the continuum. The field $\varphi$ entering the partition function can be parametrized in the following ways:

• (i)

the field and its complex conjugate, $\varphi$ and $\varphi^{\ast}$;

• (ii)

the real and imaginary parts of $\varphi$, i.e., $\operatorname{Re}{\varphi}$ and $\operatorname{Im}{\varphi}$;

• (iii)

the amplitude $\rho$ and phase $\theta$ of the field $\varphi=\sqrt{\rho}\,e^{i\theta}$.

In the paper Grater1995 the $|\varphi|^{4}$ model in $d=2$ is studied within FRG by the derivative expansion formalism using the parametrization (i), where the phase periodicity is implicitly implemented. Proceeding in this way, one can show that there is a line of (pseudo)-fixed points, which is a hallmark of BKT, and $\eta$ can be estimated in good agreement with the BKT prediction, even though it is not possible to unambiguously locate the critical point. Indeed, in order to locate the critical point it is necessary to terminate the FRG flow at a finite scale, corresponding to a reasonable (but arbitrary) size of the system as also used inKrahl2007 . The $\beta$ function for the interaction coupling $\lambda$ obtained in this FRG scheme agrees with the one of the nonlinear $\sigma$ model only at first order in the temperature $T$. This discrepancy leads to a rather different behavior: in the loop expansion of the non-linear $\sigma$ model the flow of the interaction $\lambda$ is trivial since all loop contributions vanish, and the model remains always in its low-temperature phase. On the other hand, the FRG treatmentGrater1995 gives a nontrivial flow for the $\lambda$ coupling with a line of pseudo-fixed points appearing at low temperature and a high-temperature phase where the system renormalizes to a symmetric state with $\lambda\to 0$; this is interpreted as a hint of BKT behavior. However, the low-temperature pseudo-fixed points in the FRG flow are unstable and the system is always driven to the high-temperature state in the thermodynamic limit, in contradiction to the BKT picture. This instability remains also in FRG with higher-order truncationsGersdorff2001 ; Jakubczyk2017 .

Parametrization (ii)Jakubczyk2014 ; Jakubczyk2017 has the advantage that the transverse mode ($\operatorname{Im}\varphi$) alone reproduces the BKT scenario, if one disregards the massive longitudinal ($\operatorname{Re}\varphi$) mode. However, in view of more complex cases in which the existence of the BKT transition is not a priori known, it is important to study also the effect of the massive longitudinal mode (in which case the flow equations derived in parametrization (ii) become equivalent to (i)). It turns out that the interplay of massless transverse modes with the massive longitudinal mode makes the line of fixed points unstable and drives the RG flow to the high temperature phase for all initial conditions Jakubczyk2017 . Instead, with a temperature dependent regulator that is optimized (fine-tuned) for each initial condition of the RG flow, a line of true fixed points is found with very good results for the anomalous dimension and the jump of the stiffness at $T_{\text{BKT}}$ Jakubczyk2014 .

In this paper we argue that FRG in the AP parametrization (iii) overcomes possible ambiguities in the other parametrizations and achieves two goals: first, it recovers the BKT transition in the $XY$ and $|\varphi|^{4}$ models without any ad hoc assumption on its existence and validity; and second, it quantifies the effect of amplitude fluctuations on the superfluid stiffness and nonuniversal properties of both models. The paper is structured as follows: Section II defines the models and recapitulates previous FRG results; Section III explains the mapping from the lattice $XY$ model to the continuum $|\varphi|^{4}$ model so that we can treat both on equal footing. Section IV introduces our new FRG approach, which proceeds in two stages: first, we formulate the FRG in the AP parametrization to integrate over amplitude and longitudinal phase fluctuations; at the end of this flow we obtain an effective SG model with a renormalized superfluid stiffness. Subsequently, transverse phase (vortex) excitations in the SG model with compact phase drive the BKT transition and yield a line of true fixed points. In Sec. V we present our results for the $|\varphi|^{4}$ model, where we recover the universality of thermodynamic functions in the fluctuation regime Popov1983 ; Fisher1988 ; Baym1999 ; Prokofev2001 ; Prokofev2002 ; Pilati2008 ; Rancon2012 , as well as for the $XY$ model where we discuss the temperature dependent renormalization of the superfluid stiffness. Finally, we conclude in Sec. VI.

II The models and discussion of previous FRG results

In this section we introduce the $XY$ and $|\varphi|^{4}$ models studied in this work and recapitulate basic properties of the BKT phase transition. We then discuss previous FRG work before presenting our results in Secs. IV–V.

II.1 The $\bm{XY}$ and $\bm{|\varphi|^{4}}$ models in $\bm{2d}$

The Hamiltonian of the XY or plane rotor model reads

[TABLE]

where $K=\beta J>0$ denotes the spin coupling in units of temperature and as usual $\beta=1/k_{B}T$. The angles $\theta_{i}$ are defined at the sites $i$ of a $2d$ lattice; in the following we consider a square lattice. The ground state is fully magnetized with all spins pointing in the same direction, $\theta_{i}=\theta_{0}\,\forall\,i$, and is infinitely degenerate. At any $T>0$ symmetry breaking is forbidden in $2d$ by the MW theorem. Nevertheless, finite systems can have a nonzero magnetization, which is used to detect the BKT transition Bramwell1994 . In ultracold atomic gases the counterpart of the magnetization is the $\bm{k}=0$ component of the momentum distribution and the central peak of the atomic density profile sharply decreases around $T_{\text{BKT}}$ Trombettoni2005 .

The action for the $|\varphi|^{4}$ model reads

[TABLE]

Note that Eq. (2) has been written for a classical field $\varphi$, but in the following it will be applied also to the interacting boson case. There, $\mu$ would represent the chemical potential, $U$ the local interaction and $m$ the boson mass. In the following we will use unit mass $m=1$ but restore it when convenient. We shall use units in which $\hbar=k_{B}=1$.

Continuous $O(N)$ field theories, with the action (2) corresponding to $N=2$, have been studied extensively and provide important examples of the field theoretical treatment of phase transitions. The nonperturbative FRG has produced a comprehensive picture of the universality classes of such theories for every real dimension $d$ and number of field components $N$ Codello2013 ; Codello2015 . In Section III we discuss how to map the lattice $XY$ model (1) onto the continuum $|\varphi|^{4}$ theory (2).

To fix the notation and state results used later, we briefly recapitulate basic results of the BKT universality class Pelissetto2002 referring to the $XY$ model. A discussion of BKT theory in the $|\varphi|^{4}$ model can be found, e.g., in Prokofev2002 . Within a spin-wave analysis of the $XY$ model (1), we can expand around the symmetry broken state for small phase displacements $\theta_{i}-\theta_{j}\ll 1$, which in the continuum limit leads to

[TABLE]

When the phase $\theta$ is treated as periodic the latter model is equivalent to the Villain modelVillain1975 . Neglecting the compactness of phase variable $\theta$, one readily finds

[TABLE]

where $M_{L}$ is the magnetization of a finite system of size $L$ and lattice spacing $a$, and $G(x)$ denotes the two-point correlation function between two spins at distance $x$ in the thermodynamic limit (see App. A for a derivation).

The magnetization $M_{L}$ decays as a power law of the system size, and in the thermodynamic limit the system has no finite order parameter at finite temperature, in agreement with the MW theorem. On the other hand, the two-point correlation $G(x)$ displays algebraic behavior with temperature dependent anomalous dimension

[TABLE]

This result is generally valid also at higher order in the low-temperature expansion of the system.

The spin-wave analysis suggests that the ordered phase is stable at all temperatures and the correlation functions have power-law behavior even for small $K$ values. However, this is inconsistent with an intuitive argument Kosterlitz1973 based on the free energy $F=(\pi J-2T)\log\left(\frac{L}{a}\right)$ of a macroscopic vortex configuration (see, e.g., LeBellac1991 ). Accordingly, vortex configurations of the spin should become favorable for temperatures larger than

[TABLE]

For $T>T_{\text{BKT}}$, one expects vortex excitations to proliferate and destroy the long-range order found in the spin-wave analysis. Monte Carlo simulations have established $T_{\text{BKT}}\simeq 0.893J$ Gupta1988 ; Schultka1994 ; Olsson1995 ; Hasenbusch2005 ; Komura2012 . A review of the critical properties of the Villain model is provided in Kleinert1989 , and for comparison its critical temperature is $\simeq 1.330J$ Janke1993 .

The continuous field theory for the spin-wave approximation is, however, not suited to account for vortex configurations, which are characterized by

[TABLE]

when integrating over a closed contour $C$. The single-valued complex field $\varphi$ allows for differences in the phase field $\theta$ by multiples of $2\pi$, and thereby imposes the condition $m_{i}\in\mathbb{Z}$ for the winding number of the vortex configurations. Instead, the path integral formulation with a single-valued field $\theta$ does not include vortex configurations.

It is possible to take exact account of the vortex configurations by means of a dual transformation Jose1977 . One can extract the contribution from the multivalued configuration by means of the decomposition $\theta(x)=\theta^{\prime}(x)+\tilde{\theta}(x)$, where $\oint_{C}\nabla\theta^{\prime}=0$ and $\oint_{C}\nabla\tilde{\theta}=2\pi m_{i}\neq 0$. Substituting this into Eq. (3), one can show that the vortex part of the XY Hamiltonian in $2d$ is equivalent to a Coulomb gas Minnhagen1987 with charges playing the role of vortices. More precisely, it is the Villain model that can be exactly mapped onto the Coulomb gas, and spin-wave–vortex interactions give rise to additional contributions that can be computed. In absence of a magnetic field, the mapping leads to a neutral Coulomb gas with $\sum_{i}m_{i}=0$. The Coulomb gas formalism allows for a sensible low temperature expansion, indeed for $T\leq T_{\text{BKT}}$ we expect only singly charged vortices to be relevant and we thus include only $m_{i}=\pm 1$ configurations. The latter give rise to an additional cosine potential in the spin-wave Hamiltonian, and the duality transformation maps this to the SG model in the dual phase field $\Phi$,

[TABLE]

with dimensional coupling $u$ and dimensionless SG coupling $\beta$ (not to be confused with the symbol $\beta=1/k_{B}T$). Also the $XY$ model can be mapped onto the SG model (8) via the Coulomb gas Jose1977 ; this can be intuitively understood because the compact nature of the variable $\theta$ allows only perturbations in the form of a periodic operator. Thus, from the RG point of view the theory space of a periodic field $\theta$ is naturally described, at least at lowest order, by the SG model ZinnJustin1996 . Note, however, that the original compact phase $\theta$ is replaced by the dual phase $\Phi$ in the SG model. We also observe that the mapping between the $XY$ and the SG model Jose1977 ; Kleinert1989 ; Benfatto2012 has the advantage of giving an explicit form for the bare coupling of the SG model.

A key point, which we will use in the following, is that the spin-wave Hamiltonian (3) with a compact variable $\theta$ is equivalent to the Villain model, which, once vortices with $|m_{i}|>1$ are neglected, is dual to the SG model (8) with $\beta^{2}=4\pi^{2}K$, which becomes critical at $\beta^{2}=8\pi$.

The SG model has also been studied extensively in the FRG framework, which provides a nonperturbative generalization of the original Kosterlitz-Thouless RG equationsNandori2001 ; Jentschura2006 ; Nagy2008 ; Nagy2009 ; Nandori2009 ; Nandori2010 ; Bacso2015 . In the following, after briefly reviewing in Section II.2 previous FRG work for the $O(2)$ model, we will combine the AP parametrization of the FRG with the SG results into a comprehensive FRG treatment including amplitude, spin-wave and vortex excitations.

II.2 FRG results for the $\bm{O(2)}$ model in $\bm{2d}$

In this section we review and discuss previous FRG results for the $O(N=2)$ field theory in $d=2$. One can write the quartic potential with $\rho=|\varphi|^{2}$ as

[TABLE]

and derive FRG equations for the flow of the scale-dependent couplings. In the LPA*′* approximation Berges2002 one has

[TABLE]

where $k\propto L^{-1}$ is an infrared momentum cutoff, $\tilde{\lambda}_{k}=k^{-2}\lambda_{k}$ and $t=-\log\left(ka\right)=0\dotsc\infty$ is the RG “time”. The flow equation for $\tilde{\kappa}_{k}$ reads

[TABLE]

with $\tilde{\kappa}_{k}=Z_{k}\kappa_{k}$. The anomalous dimension at scale $k$ is given by

[TABLE]

These flow equations for $\tilde{\lambda}_{k}$ and $\tilde{\kappa}_{k}$ may easily be integrated numerically Grater1995 , and the resulting phase diagram is shown in Fig. 1. For initial conditions with sufficiently large $\tilde{\kappa}_{k}$ the flow is rapidly attracted to a line of pseudo-fixed points at an almost constant value of $\tilde{\lambda}_{k}$. Once this line is reached, the flow slows down substantially and leaves the system in its symmetry broken phase for intermediate RG times. For larger $t\to\infty$ the flow eventually escapes the low-temperature phase and reaches the high-temperature phase with $\tilde{\kappa}_{k}=0$ at a finite time $t<\infty$.

Following Grater1995 , one can identify the unstable pseudo-fixed line at finite $\tilde{\lambda}_{k}$ with the low-temperature phase of the BKT transition. In this symmetry broken phase, the complex field can be decomposed into radial and transverse modes. The radial (or massive) mode $\rho$ is effectively frozen by its finite mass $m_{m}\propto 2\tilde{\lambda}_{k}\tilde{\kappa}_{k}$, while the remaining massless Goldstone mode ($m_{g}=0$) is effectively described by the spin-wave Hamiltonian (3) and has algebraic correlations. On the other hand, for initial conditions in the small $\tilde{\kappa}_{k}$ region, the flow is rapidly attracted to the point $\tilde{\kappa}_{k}=\tilde{\lambda}_{k}=0$ and enters a high-temperature $U(1)$ symmetric phase with exponential correlations, which is identified with the disordered, high-temperature phase of the BKT transition.

It is remarkable that the FRG treatment of the $O(2)$ model is able to recover the high-temperature phase without explicitly considering vortex configurations. Indeed, the complex field parametrization (i) implicitly includes the compact phase variable responsible for vortex excitations, in contrast to the spin-wave action (3) when the phase is considered non-periodic.

On the other hand, in the thermodynamic limit $k\to 0$ there is only one regime in Fig. 1, showing that spin-wave excitations are always massive in this approximation. More precisely, the FRG flow presented in Fig. 1 does not exhibit a sharp BKT transition but rather a smooth crossover. Indeed, for large enough length scales $k^{-1}\gg a$ the flow always reaches the symmetric phase and algebraic correlations disappear in the thermodynamic limit for any $T>0$. This is the result of vortex unbinding, hence the FRG calculation Grater1995 overestimates the effect of vortex configurations which appear to be relevant at any finite temperature.

Note that a similar behavior was already found in the Migdal approach to the $XY$ model Jose1977 . There, the RG equations are written in terms of the periodic potential $V(\theta)$ between the phases of two neighboring spins. Even in that scheme an unstable pseudo-fixed line is found with a phase potential very similar to the one of the Villain model Villain1975 . On the other hand, for small enough values of $k$ the interaction potential always reaches a high-temperature fixed point.

The failure of the Migdal approximation to reproduce the expected low-energy physics of the $2d$ $XY$ model has been attributed to an insufficient representation of vortex correlations, which leads to a systematic overestimation of the vortex contribution in the long-wavelength limit Jose1977 . A similar effect may be responsible for the picture found in the lowest-order FRG truncation. Indeed, neglecting higher derivative terms in the $|\varphi|^{4}$ action may overestimate the effect of vortex degrees of freedom in the thermodynamic limit. Nevertheless, it is an open question whether fully including higher derivatives reproduces vortex-vortex correlations with the correct power-law decay to stabilize the low-temperature phase.

One way to extract an anomalous dimension from the LPA*′* FRG treatment with complex field parametrization (i) is to effectively discard the finite flow along the pseudo-fixed line. The condition $\partial_{t}\tilde{\lambda}_{k}=0$ is evaluated numerically to obtain a curve $\tilde{\lambda}_{k}=f(\tilde{\kappa}_{k})$ in $(\tilde{\kappa}_{k},\tilde{\lambda}_{k})$ space. Once the finite flow $\partial_{t}\tilde{\kappa}_{k}$ is discarded along this line, one may compute the power-law exponent $\eta$ of the correlation function in the thermodynamic limit using Eq. (12).

The result for $\eta$ along the line of pseudo-fixed points is depicted as a blue line in Fig. 2. The red curve below shows the residual flow $\partial_{t}\tilde{\kappa}_{k}$ along the pseudo-fixed line. This flow should vanish for a line of true fixed points, while as it can be seen in in Fig. 2 it vanishes only in the limit $\tilde{\kappa}_{k}\to\infty$, remaining finite for smaller $\tilde{\kappa}_{k}$. Nevertheless, it is possible to identify a point where $|\partial_{t}\tilde{\kappa}_{k}|$ starts to increase sharply and drives the system to the disorder phase for small scales $k$. The anomalous dimension at the turning point is surprisingly close to the expected value $1/4$. In the Sections IV–V below we show how a line of true fixed points and gapped amplitude excitations are found with the AP parametrization. As a basis for this, we first discuss the mapping between the $XY$ and $|\varphi|^{4}$ models in Section III.

III Mapping of the models

In this section we derive the explicit mapping of the $XY$ model into a suitable $|\varphi|^{4}$ theory via a Hubbard-Stratonovich transformation Simanek1994 ; Machado2010 ; Nishimori2011 . While this mapping is well known, we present it briefly in order to demonstrate how the XY model of unitary spins is equivalent to a complex field $\varphi$ with density fluctuations. Via the mapping, our subsequent FRG analysis of the $|\varphi|^{4}$ model applies also to the XY model.

Our starting point is the $XY$ model (1), which can be written—apart from a constant energy—as

[TABLE]

where $s_{x,i}\equiv\cos\theta_{i}$, $s_{y,i}\equiv\sin\theta_{i}$ can be combined into a vector $\bm{s}_{i}=\left(s_{x,i},s_{y,i}\right)$ with $\bm{s}_{i}^{2}=1$. The partition function is then given by

[TABLE]

with $Ds=\Pi_{i}ds_{x,i}ds_{y,i}\equiv\Pi_{i}d\bm{s}_{i}$. One can rewrite the partition function in the form

[TABLE]

where $\bm{s}=(s_{x,1},s_{y,1},\cdots,s_{x,N},s_{y,N})$ is a $2N$-dimensional vector and the matrix $K^{\prime}$ has elements $2\beta J$ on the neighboring upper and lower diagonals. To perform the Hubbard-Stratonovich transformation we use the Gaussian identity

[TABLE]

where $\bm{\phi}$ is a $2N$ vector composed of $N$ two-component vectors $\bm{\phi}_{j}$. Since $K^{\prime}$ is not positive definite, we replace it by a shifted interaction

[TABLE]

that is positive definite for an appropriately chosen constant $\mu$; this amounts to a redefinition of the zero point energy of the system. We then obtain

[TABLE]

where the potential $U$ is defined by

[TABLE]

$U$ can depend only on the quadratic invariant $\rho_{j}=\phi_{x,j}^{2}+\phi_{y,j}^{2}$, and we obtain

[TABLE]

in terms of the modified Bessel function $I_{0}$. The matrix $K^{-1}$ is diagonal in Fourier space with entries

[TABLE]

where

[TABLE]

is the dispersion relation on a $d$-dimensional cubic lattice for momentum components $q_{\nu}$ and lattice spacing $a$ (in our case $d=2$). It will be convenient to shift the kinetic term as

[TABLE]

After a field rescaling

[TABLE]

one obtains the kinetic term

[TABLE]

with dispersion relationMachado2010

[TABLE]

In the continuum limit $a\to 0$ we recover

[TABLE]

to lowest order in $q$, where the subscript “cl” stands for continuum limit. The potential term in the rescaled field reads

[TABLE]

With this mapping of the $XY$ model into a $|\varphi|^{4}$ theory, we can subsequently use our functional RG equations for both models; only the initial conditions, i.e., the functional forms of the dispersion $\varepsilon(q)$ and the potential $U(\rho)$, are different and discriminate between the microscopic $XY$ and $|\varphi|^{4}$ models. We finally observe that in the $XY$ model, amplitude fluctuations are absent by construction, but there are spin-wave excitations which interact with vortex fluctuations to modify the effective phase stiffness in the thermodynamic limit. In the $O(2)$ equivalent (28) of the XY model, the finite renormalization of the stiffness is partly due to the gapped amplitude fluctuations, and partly due to longitudinal phase fluctuations, too. Hence, amplitude fluctuations originate from the re-parametrization of the interaction between vortex and spin-wave fluctuations and, as it will be shown in the following, represent a large part of the renormalization of the superfluid stiffness.

IV The Amplitude-Phase parametrization

The complex field $\varphi$ in (2), which is equivalent to the two-component field $\bm{\varphi}$ in (28), can be parametrized in terms of real amplitude $\rho$ and phase $\theta$ according to

[TABLE]

In this AP parametrization (iii) the $|\varphi|^{4}$ action (2) reads

[TABLE]

When applied to the $XY$ model with the mapping (28), the field expectation value is related to the $XY$ magnetization by $\langle\varphi\rangle=\sqrt{\beta J}m$.

It is worth noting that the derivation of action (30) is a direct consequence of Leibniz rule for continuous spatial derivatives and it is not straightforwardly applicable to lattice models where the kinetic contribution to the action cannot be expressed in terms of continuous spatial derivatives. In any case, lattice effects can be introduced by integrating the model in the $\varphi$ parametrization till an effective scale where only quadratic momentum terms dominate, as it should always happen due to universality.

Perturbative arguments suggest that the amplitude mode is always gapped and does not influence the critical behaviorPopov1983 ; Jakubczyk2017 . Instead, the critical behavior is dominated by massless phase fluctuations. Indeed, in $d=2$ only single vertex diagrams are relevant Rajaraman1987 , and since the perturbative expansion for the phase correlation function does not contain any single vertex diagram, we expect only a finite renormalization of the superfluid stiffness in the action (30). In the following we will explicitly treat amplitude fluctuation effects to show how, even in the nonperturbative picture, they remain gapped at criticality.

Consistent with these arguments, we propose a RG investigation which treats amplitude and phase fluctuations separately in two steps. The overall RG flow is thus effectively separated into two scales: first, at high momenta, only non-critical fluctuations are considered. They turn out to be irrelevant and remain always gapped in the long-wavelength limit. At the end of the AP flow the minimum $\kappa_{k}$ freezes, and we obtain an effective SG model with a renormalized superfluid stiffness. The effective couplings resulting from the amplitude stage of the flow are then considered as initial conditions for the traditional BKT flow Kosterlitz1973 of the SG model, which describe low-energy vortex excitations and produce the universal long-wavelength behavior. The validity of our technique relies on the separation of the scales for amplitude and vortex phase fluctuations, and on the smallness of the interaction between these excitations. Traditional perturbative arguments as well as the consistency of our results suggest that these assumptions are well justified.

As a preliminary step, we first discuss uncoupled amplitude and phase fluctuations. In this case, the superfluid stiffness $\rho=\kappa_{k}$ in the phase kinetic term remains fixed at the minimum $\rho_{0}$ of the potential $U(\rho)$. The total action (30) then decouples into a sum of two actions

[TABLE]

where

[TABLE]

The phase action (33) is equivalent to spin-wave model (3) with $K=\kappa_{k}$, while for the $XY$ model it is $K=\kappa_{k}\beta J$. If one considers the phase variable $\theta$ in (33) as noncompact, the correlation function $\langle e^{i[\theta(x)-\theta(y)]}\rangle$ is algebraicBerezinskii1972 and no regularization is necessary to obtain this behaviorJakubczyk2017 .

The treatment within the AP parametrization shows that the low-temperature expansion of the $|\varphi|^{4}$ and $XY$ models must coincide, at least as long as perturbative arguments are correct and amplitude fluctuations do not influence the thermodynamic behavior. However, it is worth noting that this analysis still does not yield a conclusive picture. Indeed, while the previous FRG analysis based on the $|\varphi|^{4}$ action (2) leads to a finite correlation length at any temperature and reproduces the BKT behavior only as a crossover, the amplitude and phase scheme is equivalent to the spin-wave approximation of the $XY$ model and yields algebraic correlation at any temperature, $T_{\text{BKT}}=\infty$.

To complete the picture, it is therefore necessary to introduce vortex configurations. The spin-wave analysis in Appendix A does not include discontinuous configurations of the field $\theta$ and perturbative arguments cannot account for topological excitations. These can be included using the dual mapping described in Jose1977 ; Kleinert1989 or by explicitly introducing singular phase configurations Mudry2014 ; Benfatto2012 . The total partition function of the system is then given by

[TABLE]

where we used the decomposition in Eq. (31).

In the case of frozen amplitude fluctuations, this model becomes a pure phase SG model with a line of fixed points and is described by the BKT flow equations

[TABLE]

where $K$ is the superfluid (phase) stiffness and $g_{k}$ is the vortex fugacity. The fugacity $g$ is related to the SG parameter as $u=g/\pi$.

At the bare level, $K$ and $g$ assume the values

[TABLE]

for the $|\varphi|^{4}$ model, and

[TABLE]

for the $XY$ model. In order to derive Eqs. (35)–(36) one has to assume a UV regularization, which traditionally relies in considering the Coulomb gas charges as hard disks of finite radiusKosterlitz1973 .

It should be also noted that Eqs. (37)–(40) have been obtained in the case of a purely quadratic kinetic phase term, as in the Villain model. In the $O(2)$ model the absence of higher gradient terms in the phase is the result of the decoupling in equation (31). In principle, one expects amplitude fluctuations to generate also higher gradient terms and therefore action (33) represent only the lowest-order approximation in derivative expansion.

In the small vortex fugacity limit $g_{k}\ll 1$ the BKT flow Eqs. (35) and (36) reproduce the BKT temperature in Eq. (6), while for larger values of the initial condition $g_{\Lambda}$ the BKT flow introduces multi-vortex corrections which lower the BKT temperature. For a discussion of these effects and of vortex core energies we refer to Prokofev2000 ; Hasenbusch2005 ; the prediction for the jump of the superfluid stiffness, $\frac{2mT}{\pi}\left(1-16\pi e^{-4\pi}\right)$ with a correction of $0.02\%$ with respect to the Nelson-Kosterlitz prediction $2mT/\pi$, has been tested in extensive Monte Carlo simulations Hasenbusch2005 ; Komura2012 .

V Results

Although the universal behavior of the BKT transition is completely driven by topological excitations, in the $|\varphi|^{4}$ and $XY$ models the contribution of, respectively, longitudinal and amplitude fluctuations to non-universal quantities may be different. Due to the mapping discussed in Section III, it is possible to build a $|\varphi|^{4}$ model which exactly reproduces the $XY$ model and where amplitude fluctuations play the role of longitudinal spin excitations. It is then convenient to study the BKT transition first in the $|\varphi|^{4}$ formalism and then transfer the results to the $XY$ model, which we do subsequently in the two next Sections V.1 and V.2.

V.1 $\bm{|\varphi|^{4}}$ model

In this section we apply the FRG to the $|\varphi|^{4}$ action in the AP parametrization (30). As discussed in the previous section, at the perturbative level the amplitude mode $\rho$ remains gapped while the phase fluctuations $\theta$ produce power-law correlations at any finite temperature, so the high-temperature phase of the BKT transition is not reproduced. In this section we revisit this issue at the nonpertubative level.

Our FRG procedure is based on two steps: (a) we first perform the FRG flow for the amplitude part $S_{A}$ of the action (32), which yields a renormalized superfluid stiffness; (b) we then insert this stiffness into the phase part $S_{P}$ of the action (33), which for a compact phase is equivalent to the SG model (8) so we can use the BKT flow Eqs. (35)–(36).

In the FRG approach for the amplitude part we introduce as infrared regulator a momentum dependent mass term for the amplitude fluctuations. As the cutoff scale is lowered, the effective action flows from the model-dependent initial condition (32) to the full effective action. For the flowing effective action we choose the ansatz

[TABLE]

and with the regulator (73) we obtain the flow Eq. (77) for the effective potential of amplitude and phase fluctuations, for details see Appendix B.

The flow equation is solved numerically for the full potential $U_{k}(\rho)$. In order to draw a flow diagram, we Taylor expand the potential $U_{k}=\lambda_{k}(\rho-\kappa_{k})^{2}/2$ around its minimum $\rho=\kappa_{k}$ for every $k$ and trace the flow in $(\kappa_{k},\lambda_{k})$ space. The resulting flow diagram is shown in the left Fig. 3(a) in terms of the rescaled “dimensionless” couplings $\tilde{\lambda}_{k}$ and $\tilde{\kappa}_{k}$. This first naive attempt at the AP flow is not yet correct: indeed in the lower left corner of the phase diagram the $\lambda_{k}$ coupling becomes irrelevant and the flow runs toward a region of gapless amplitude fluctuations; although this effect is not as severe as in previous parametrization, since it arises only for small values of the bare coupling $\lambda_{\Lambda}$, it is not in agreement with the expectation of irrelevant amplitude fluctuations in the thermodynamic limit. This inconsistency arises from an IR divergent term in the standard formulation of the Wetterich equation. Indeed, already the flow of the free Gaussian model in the AP parametrization has the same divergence because the phase kinetic term depends on the field $\rho$.

This spurious contribution originates from the different ways the Gaussian theory is represented in the amplitude and phase parametrization. In the path integral formulation the Gaussian $O(2)$ model, (2) with $U=0$, can be exactly integrated, yielding an effective action with the same functional form of the microscopic action, in agreement with mean field approximation being exact for Gaussian theories. Thus also in the FRG formalism the flow of Gaussian theories should vanish and the bare action should be equivalent to the fully renormalized one. Nevertheless when one uses the Wetterich equation to calculate the flow of a Gaussian ansatz in the traditional $O(N)$ formalism one gets a constant flow for the effective action Krippa2009 ; Gies2012 . Thus, the functional form of a Gaussian action is preserved by the FRG formalism apart for a field independent term, which is diverging in the long-wavelength limit. Such a term is unnecessary in the computation of most of the system properties and it is usually neglected.

In free energy calculations the constant term in the effective action is essential and should be kept finite. The most common regularization procedure is to subtract the noninteracting ($\lambda_{k}=0$) component from the right-hand side of the Wetterich equation under study Krippa2009 ; Gies2012 . The Gaussian theory in the AP parametrization has a linear field potential term and a nonanalytic kinetic term, as in Eq. (41) with $U_{k}(\rho)=\mu\rho$. The nonanalytic term prevents the exact integration in the AP representation and makes it appear to be not exactly solvable; in fact, the AP representation is singular in the $\rho\to 0$ limit. In our application the amplitude fluctuations remain gapped, so the AP parametrization is always valid.

As stated above, the FRG flow of Gaussian theories does not completely vanish, and in the amplitude and phase representation the remaining contribution also pick up a spurious field dependence, which is the counterpart of the nonanalytic kinetic term preventing an exact integration in the path integral formalism. Since one knows that the functional form of quadratic theories remains the same at bare and renormalized level and this property must remain valid regardless of the chosen representation, one can safely subtract the equivalent Gaussian contribution from the flow equations and force them to be zero for quadratic theories, as it is done in the traditional case for free energy calculations.

With this modification the potential flow equation (77) becomes

[TABLE]

The parameter $\alpha=\alpha_{\rho}$ characterizes the regulator (73); Fig. 3 has been plotted with $\alpha_{\rho}=2$, but below we set $\alpha=(4\kappa_{*})^{-1}$ self-consistently with the value of $\kappa$ at the end of the flow.

The modified Eq. (42) now produces the correct flow diagram shown in the right Fig. 3(b). The flow equations for the dimensionless minimum $\tilde{\kappa}_{k}$ and interaction $\tilde{\lambda}_{k}$ read

[TABLE]

As expected, the mass term of amplitude fluctuations does not vanish. The dimensionless $\tilde{\lambda}_{k}$ keeps growing because the action (41) with a noncompact phase has no fixed point. Indeed, $\tilde{\kappa}_{k}$ is marginal in $2d$, and after an initial renormalization by amplitude fluctuations at finite $\tilde{\lambda}_{k}$, it remains frozen up to infinite length scales ($k\to 0$).

The results of Fig. 3(b) are in agreement with the expectation from perturbation theory and show that amplitude fluctuations are irrelevant in the RG sense and only lead to a finite renormalization of the stiffness. Note that this irrelevance has been proven in the truncation scheme described by the ansatz (41), where only the lowest-order coupling between amplitude and phase is present. Indeed, also phase fluctuations drive the flow of the effective potential $U_{k}(\rho)$ in the first term of the original flow equation (77), but this contribution is canceled when subtracting the Gaussian part to obtain (42). In a more general truncation scheme we do not expect this cancellation to persist, but we are confident that the remaining phase-amplitude terms will be irrelevant.

It is useful to compare the results of Figs. 1 and 3(a) with those of Fig. 3(b): both represent the theory space of a $2d$ two-component field theory where the order parameter has $U(1)$ symmetry. However, differences arise in the treatment of the kinetic term: in Fig. 1 the flow for the couplings has been obtained including the full $|\varphi|^{4}$ invariant kinetic term, which incorporates both amplitude and phase degrees of freedom. There, for $\lambda_{\Lambda}$ large enough, the flow is attracted to a pseudo-fixed line and the IR theory appears to have finite $\tilde{\kappa}_{k}$, finite $\tilde{\lambda}_{k}$ and massive amplitude fluctuations at finite $k$. At the same time, a fixed $\tilde{\lambda}_{k}$ produces a vanishing dimensionful $\lambda_{k}=k^{2}\tilde{\lambda}_{k}$ in the thermodynamic limit $k\to 0$. Hence, it is not surprising that for $k$ small enough the superfluid density $\kappa_{k}$ tends to vanish because of the increasing relevance of amplitude fluctuations. In contrast, the modified flow in the right Fig. 3(b) is consistent with fully gapped amplitude fluctuation and frozen amplitude (superfluid stiffness) $\kappa_{k}\equiv\kappa_{*}$. Indeed, for every finite $\tilde{\lambda}_{\Lambda}>0$ the flow is attracted by a stream line at fixed $\tilde{\kappa}_{k}$ and $\tilde{\lambda}_{k}\propto k^{-2}\lambda_{*}$, yielding $\lambda_{k}\simeq\lambda_{*}$ for $k\ll\Lambda$.

Having shown that the modified AP flow agrees with the perturbative results and the BKT scenario, we are in a position to verify that our approach reproduces the expected universality of the thermodynamics of the $2d$ Bose gasPopov1983 ; Fisher1988 ; Baym1999 ; Prokofev2001 ; Prokofev2002 ; Rancon2012 and to quantify the agreement with Monte-Carlo results. In particular, starting the flow from the initial conditions (37) we can compute:

1. a.

the superfluid density $\rho_{s}$, which is equal to the coupling $\kappa_{*}$; and 2. b.

the critical chemical potential $\mu_{c}$ as a function of the bare interaction $U$.

To achieve this result we perform the renormalization group procedure described above with the initial condition

[TABLE]

where $U$ is the effective interaction and $\mu=U\kappa_{\Lambda}$ is the chemical potential of the classical $2d$ $|\varphi|^{4}$ model we are studying.

The known results for the $2d$ quantum Bose gas with which we want to compare are the following Prokofev2001 :

the thermodynamic quantities have to collapse once expressed in terms of the dimensionless variable

[TABLE]

which measures the distance from the critical point.

• 2)

The superfluid density defines a function $f(X)$ via the relation

[TABLE]

Note that the predicted jump of the superfluid stiffness $\rho_{s}=2mT/\pi$ at criticality Nelson1977 implies that $f(X)$ jumps from 0 to 1 at $X=0$. The collapse of the superfluidity function using the variable $X$ is shown in Figs. 4(a)–4(b).

• 3)

for small $X>0$ one has

[TABLE]

with coefficientProkofev2002

[TABLE]

• 4)

For $2d$ quantum systems in the continuum, one has the following results in the weakly interacting limit for the critical density $\rho_{c}$ and the critical chemical potential $\mu_{c}$ (respectively in the canonical and grand-canonical ensembles):

[TABLE]

[TABLE]

The parameters $\xi$, $\xi_{\mu}$, extracted from Monte Carlo simulations in a classical lattice $|\varphi|^{4}$ model and via a careful analysis of the mapping between the simulated lattice model and the continuum limits, have been estimated to be $\xi=380\pm 3$ and $\xi_{\mu}=13.2\pm 0.4$ Prokofev2001 ; Prokofev2002 . An earlier FRG approach, once the transition point has been empirically fixed, yields $\xi_{\mu}=9.48$ in good agreement with MC simulationsRancon2012 . The logarithm of their ratio,

[TABLE]

is a non-trivial universal number, determined to be Prokofev2002

[TABLE]

We now present our results for these non-universal and universal properties of the $|\varphi|^{4}$ model. In Fig. 4(a) we report our results for the superfluid fraction $\rho_{s}$ for different values of $U$. In Fig. 4(b) we plot the same curves vs the dimensionless variable $X$. We find that they collapse almost perfectly even for a wide range of interactions $mU=0.02,\dotsc,0.6$. Note that the spreading between the curves increases for large $X$, as expected, since the universality should hold only in the fluctuation regime up to $X\approx 1/mU$ and we use also rather large values of $U$. To quantitatively determine the function $f(X)$ we perform an interpolation of the curves for $\rho_{s}(X)$, some of them shown in Fig. 4(b), and compute their average and variance, which are reported in Fig. 5. The average has been computed over a total number of $30$ curves obtained for $30$ different values of the interaction logarithmically spaced in the interval $mU\in[0,1]$, the curve $f(X)$ can be trusted also for large $X$ since the statistical weight of large interaction $U>0.5$ is small. Agreement with Monte Carlo dataProkofev2002 is rather good, also considering that we are using the lowest order perturbative SG results (37)–(38).

Our findings for $\mu_{c}$ as a function of $U$ are given in the inset of Fig. 4(b). Logarithmic corrections to the relation $\mu_{c}\propto U$ are found, in agreement with Eq. (51). The coefficient $\xi_{\mu}$ entering such logarithmic corrections is not reported since the fitting procedure employed was not robust enough and the result strongly depends on the range of interactions considered, even for $U\leq 0.3$ which should be within the range of validity of Eq. (51) Hadzibabic2011 .

The $\rho_{s}(X)$ in Fig. 4(b) determines the function $f(X)=\pi\rho_{s}(X)/2mT$ reported in Fig. 5. From $f(X)$ we can obtain estimates for the universal quantities $\kappa^{\prime}$ and $\theta_{0}$. Fitting with expression Eq. (48), the data in Fig. 5 yield

[TABLE]

in reasonably good agreement with the Monte Carlo result (49). The latter result has been obtained from a linear fit of the curves in Fig. 4(a) and averaging $\kappa^{\prime}$ over the values obtained for different interactions. The average is consistent with (54) while the error is partly due to difficulties in fitting procedure close to the transition point and partially to non-perfect universality of the curves in Fig. 4(a).

Regarding $\theta_{0}$, we observe that for relatively large $X$ one has $f(X)\approx(\pi/2)\theta(X)-1/4$ in terms of the universal equation of state $\theta(X)$ Prokofev2002 . It should be noted that in order to evaluate $\theta_{0}=\theta(X=0)$ from $f(X)$ one shall extrapolate the value of a curve obtained for large $X$ to the point $X=0$. Such extrapolation has been done assuming polynomial behavior of $\theta(X)$. A polynomial fit of the $\rho_{s}$ curves of different interactions at high values of $X$ yields

[TABLE]

again in fairly good agreement with the Monte Carlo result (53).

V.2 $\bm{XY}$ model

As we discussed in Sections III–IV, one can treat the $XY$ model as a $|\varphi|^{4}$ model, provided that one uses the appropriate initial condition for the RG flow, as extracted from the mapping of Section III, and that one rescales the field by $\sqrt{\beta J}$ to have a magnetization with absolute value smaller than one.

The $XY$ model has been the subject of intense investigations from different perspectives and several quantities have been studied in detail, which we can now study with the FRG approach presented in this paper. Here, to test the validity of our approach, we focus on the renormalized phase (superfluid) stiffness $J_{s}(T)$ and quantify the effect of amplitude fluctuations on it. We proceed by computing $\kappa_{*}$ as discussed in the previous Section V.1, then the stiffness is given by

[TABLE]

where $J_{s}(T)$ indicates the effective bare superfluid stiffness without inclusion of vortex configurations. In the following, the same notation will be used also to indicate the fully renormalized spin stiffness in presence of vortex excitations, since the two definitions can be simply regarded as two levels of approximation for the same quantity.

All the physical quantities should be independent of the mapping parameter $\mu$ Machado2010 . However, in the following we are going to discard lattice effects, effectively replacing the lattice dispersion (26) with the continuum dispersion (28). Such an approximation introduces a $\mu$ dependence in the physical quantities, which we may fix either from mean-field or low-temperature results.

To clarify the different approximations which we are going to consider for the FRG computation of $J_{s}(T)$, let us recapitulate the logic followed so far. Starting from the action of the $|\varphi|^{4}$ model in the continuum limit, we introduced the AP parametrization (29) and we decoupled the phase and amplitude degrees of freedom by substituting $\rho=\kappa_{k}$ into the phase kinetic term. The phase action (33) is then equivalent to the low-temperature expression of the $XY$ Hamiltonian (3) and we can apply the usual BKT flow Eqs. (35) and (36). The amplitude fluctuations then encode all fluctuations except for vortices, which are encoded at perturbative level in the BKT flow equations.

It is instructive to consider first the mean-field approximation. A first step is to completely discard amplitude fluctuation and simply set $\kappa_{k}=\text{const}$, which can be re-absorbed into the definition of $J$. A further step is to consider only a saddle point approximation for the amplitude fluctuations. Their expectation value is given by $\kappa_{MF}=\rho_{\mathrm{MF}}(T)$ such that

[TABLE]

where $S_{\mathrm{pot}}[\varphi]$ is defined in Eq. (28) and the additional $\sqrt{\beta J}$ factor in the argument is needed to reproduce the $K\equiv\beta J$ factor in Eq. (3). Thus, at first order in our treatment we find

[TABLE]

For small $T$, longitudinal fluctuation are practically frozen and $\lim_{T\to 0}J_{s}(T)=J$. At larger temperatures $\kappa_{\mathrm{MF}}(T)$ decreases since longitudinal fluctuations reduce the stiffness. Finally, $J_{s}(T)$ vanishes at a finite temperature value $T_{\mathrm{MF}}>T_{\text{BKT}}$. The mean-field critical temperature $T_{\mathrm{MF}}$ is given by $T_{\mathrm{MF}}=2J$ ($T_{\mathrm{MF}}=dJ$ for a hypercubic lattice in $d$ dimensions). To obtain this value of $T_{\mathrm{MF}}$ one has to fix $\mu=0$. This choice turns out to be a reasonable one, since one finds for small $T$

[TABLE]

in agreement with the results of the self-consistent harmonic approximation Pires1994 , which predicts $J_{s}(T)/J=1-T/zJ$ for a model with $z$ nearest neighbors at small $T$. In $1d$ this agrees also with the exact low-temperature result Stanley1969 ; see as well the discussion in Benfatto2012 on the low temperature behaviour of $J_{s}(T)/J$. We also mention that Monte Carlo simulations private_comm confirm that for low temperature one has that the slope $\partial J_{s}/\partial T$ for $T\to 0$ is $1/4$, as given in (59).

In order to go beyond the saddle-point approximation, it is necessary to explicitly solve the flow Eq. (42). Then the expectation value $\kappa_{*}$ for the field $\rho$ is defined by the minimum

[TABLE]

and the phase stiffness is given by (56).

Our results are summarized in Fig. 6 which shows the temperature dependence of the spin stiffness $J_{s}(T)$. In this figure the solid lines correspond to the results generated by amplitude fluctuations using Eq. (56), but without considering the vortex fluctuations. The different lines correspond to different approximations discussed in the following. The dashed lines represent the vortex renormalized stiffness and are obtained by considering the effect of vortex fluctuations via the perturbative SG Eqs. (35) and (36) with initial conditions (39) and (40) after performing the RG for the amplitude modes. Without vortex fluctuations the BKT temperature is simply obtained by the intersection of $J_{s}(T)$ with $\frac{2T}{\pi}$. From top to bottom of Fig. 6 we have:

1. a.

FRG, initial condition (28), $\mu=Jd$ (purple lines):

$T_{\text{BKT}}/J=1.19\pm 0.02$. 2. b.

Low-temperature expansion (59) (blue lines):

$T_{\text{BKT}}/J=1.00\pm 0.02$. 3. c.

Mean-field estimate (61) of $J_{s}(T)$, $\mu=0$, (gray lines):

$T_{\text{BKT}}/J=0.96\pm 0.02$. 4. d.

FRG, initial condition (28), $\mu=0$ (green lines):

$T_{\text{BKT}}/J=0.94\pm 0.02$.

In the figure we also plot for comparison the Monte Carlo result $T_{\text{BKT}}/J=0.893$ (red star).

A remark is in order here: when mapping the $XY$ model onto the two-component $|\varphi|^{4}$ lattice field theory in section III, we underlined that the mapping is exact and the results should be $\mu$ independent as long as $\mu\geq Jd$. However, as discussed above, our FRG flow equation for the action (30) is applied to the $XY$ model by modifying only the initial condition for the bare potential. This procedure is incomplete since the lattice field theory equivalent of the $XY$ model has the lattice dispersion (26) rather than the continuous one (27). Therefore, the application of the FRG flow with continuous dispersion (27) and $\mu=Jd$ (purple line in Fig. 6) is a rather crude approximation and does not agree with the low-temperature expansion (blue line).

Moreover, approximating the lattice dispersion with a continuous dispersion introduces a $\mu$ dependence in our result. We can exploit this and fix $\mu=0$ to approach the exact low-temperature asymptotics. While such a value of $\mu$ would not be allowed in the lattice theory with dispersion (26), it is permitted in the continuous case. The resulting green solid line in Fig. 6 shows a consistent improvement over the low-temperature expansion (blue line).

Since the effect of the amplitude fluctutations in the continuous $|\varphi|^{4}$ model with effective potential (28) is rather small, we expect that analytic results for the superfluid stiffness obtained from the saddle point solution follow very closely the exact results in all the range of the temperature between zero and $T_{\text{BKT}}$. This can be made quantitative by observing that one could obtain very good results (plotted as gray lines in Fig. 6) by solving the following mean-field equation for the superfluid stiffness $J_{s}(T)/J$:

[TABLE]

(which is the solid gray line), and then use it as initial condition in the perturbative SG Eqs. (35)–(36). The procedure gives the dashed gray line and $T_{\text{BKT}}=0.96\pm 0.02$, worse than the value (62) we find using $\mu=0$, but again reasonably good.

In conclusion, our most accurate results come from the nonperturbative evaluation of the FRG flow for the amplitude mode combined with the perturbative SG flow for the phase:

[TABLE]

in good agreement with the expected result for the XY model $T_{\text{BKT}}\simeq 0.893J$ obtained by MC simulations Gupta1988 ; Schultka1994 ; Olsson1995 ; Hasenbusch2005 ; Komura2012 . Note that this very good agreement for the critical temperature has been obtained by matching with the appropriate choice of $\mu$ the low-temperature behaviour of the superfluid stiffness.

VI Conclusions

The topological phase transition in two-dimensional spin models with continuous symmetry as explained by the Berezinskii, Kosterlitz and Thouless (BKT) theory is a celebrated result. Our aim in this paper has been to set up and implement a renormalization group framework for the BKT universality class to quantitatively determine nonuniversal properties such as the temperature dependence of the superfluid fraction, the critical chemical potential and the transition temperature, given their relevance in $2d$ realizations of BKT physics and in current experiments.

After discussing the role of the parametrization of the field in functional RG approaches to $2d$ BKT phase transitions, we argue that the amplitude-phase (AP) Madelung representation of the field is the natural choice to study the contribution of longitudinal spin fluctuations to nonuniversal quantities and we show that amplitude fluctuations are gapped at the critical point. With the AP parametrization we have been able to study the RG flow directly in the relevant degrees of freedom: amplitude (density) fluctuations, longitudinal spin waves, and vortex excitations, and we discuss their mutual interplay.

As a preliminary step, we have derived an explicit mapping from the $2d$ lattice $XY$ model to a continuum $|\varphi|^{4}$ field theory. While in three and higher dimensions this continuum limit is straightforward, in two dimensions the mapping depends, qualitatively and quantitatively, on nonuniversal ultraviolet details of the initial model. As a result, we have mapped the original $XY$ coupling $J$ to the initial superfluid stiffness $\rho$ and interaction $\lambda$ at cutoff scale $\Lambda$ of the corresponding $|\varphi|^{4}$ model. Therefore, the RG equations are the same and only the initial conditions differ to characterize the $XY$ and $|\varphi|^{4}$ models, so that they can be treated within the same formalism on equal footing.

We then proceeded to write the action in the amplitude and phase degrees of freedom and we have shown that amplitude excitations are gapped, such that the BKT behavior is correctly recovered as a transition and not as a crossover at large distances. This result is based on the explicit subtraction in the functional RG equations of the Gaussian energy. While this is mainly a technical point, we think it is an interesting one since (i) in many other applications such contributions do not have any physical effect in the determination of the critical properties of $O(N)$ models, and (ii) the AP representation provides a straightforward way to show this effect.

Our FRG procedure is then based on two steps: we first perform FRG on the amplitude part $S_{A}$ of the action (32). We then insert the obtained stiffness into the phase part of the action, which is given by the spin-wave action (33) with the phase crucially considered as a periodic variable. This allows us to correctly take into account the compact nature of the phase variable and to use the results of the sine-Gordon model.

The combination of the nonperturbative functional RG analysis of the amplitude part of the action with the perturbative flow for the sine-Gordon model is already sufficient to give rather good results for nonuniversal and universal quantities. In particular, we determined the critical chemical potential for the $|\varphi|^{4}$ model and the nontrivial universal parameters $\kappa^{\prime}$ and $\theta_{0}$ defined in Eqs. (49) and (53). Our results for these two parameters are $\kappa_{\text{(FRG)}}^{\prime}=0.67\pm 0.07$ and $\theta_{0\text{(FRG)}}=1.033\pm 0.032$, which should be compared with the Monte Carlo results $\kappa^{\prime}=0.61\pm 0.01$ and $\theta_{0}=1.068\pm 0.01$ Prokofev2002 . For the $XY$ model we obtained the temperature dependence of the stiffness $J_{s}(T)$, which receives nonuniversal corrections from amplitude fluctuations. It reproduces the exact low-temperature limit and predicts the critical temperature with an error of $\approx 5\%$.

In conclusion, our findings confirm that amplitude fluctuations only result in a finite renormalization of the stiffness and do not completely deplete the superfluid fraction. We also find, without a priori assumptions, that amplitude fluctuations are frozen for the $|\varphi|^{4}$ model and yield effectively a phase-only model of spin-wave and vortex excitations. Finally, we proved that the combined use of the functional RG for the amplitude modes and of perturbative results for the sine-Gordon model allows one to quantify the effect of vortex excitations at finite temperature, which depends on the value of the vortex core energy and yields a further lowering of $T_{c}$ Benfatto2012 . Results for several universal and nonuniversal quantities are presented, with a very good agreement with known results.

To further improve the results obtained for both the $|\varphi|^{4}$ and the $XY$ models, one can include nonperturbative effects in the sine-Gordon part of the RG flow. To this end, one should compute the anomalous dimension $\eta$ in the nonperturbative RG flow of the sine-Gordon model. Moreover, for the $XY$ model, one should include lattice effects Machado2010 which are beyond the scope of this paper. The study of lattice effects leads in a natural way to generalized sine-Gordon models, which we think is promising for future work. Although the obtained results are rather good, we think that the nonperturbative treatment of the SG part of the action, and of the lattice effects for the $XY$ model, may lead to further improvements that are worthwhile to estimate.

This work can provide a basis for future efforts to derive a generalized sine-Gordon model which comprehensively includes amplitude fluctuations on equal footing with phase fluctuations, and not as an initial condition from a previous RG step, as we did in this paper. In this way one should be able to describe also the feedback of vortex excitations onto amplitude fluctuations. We think that it would be interesting to extend the results of this work to $2d$ quantum systems in order to quantitatively determine $T_{c}$ as a function of interaction strength in ultracold Bose Hadzibabic2006 ; Schweikhard2007 and Fermi gases Bauer2014 ; Ries2015 ; Murthy2015 ; Madeira2017 ; Murthy2017 and for out-of-equilibrium situations Chu2001 ; He2017 ; Schweigler2017 .

Acknowledgements. The authors wish to thank L. Benfatto, C. Castellani, N. Dupuis, G. Gori, Z. Gulacsi, H. Knörrer, J. Lorenzana, and I. Maccari for insightful discussions. We also thank M. Hasenbusch and N. Prokofev for useful correspondence. This work is part of and supported by the DFG Collaborative Research Centre “SFB 1225 (ISOQUANT)”. T.E. thanks the Erwin-Schrödinger Institute in Vienna for hospitality during the initial stages of this work. This work was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. A.T. and I.N. acknowledge support from Progetto Premiale ABNANOTECH. N.D. and A.T. thank the Institut Henri Poincaré–Centre Emile Borel for hospitality, where the final part of this work was done during the trimester “Stochastic Dynamics Out of Equilibrium”.

Note added. After the submission of this paper, an FRG treatment of the $XY$ model by J. Krieg and P. Kopietz Krieg2017 appeared on arXiv. Within the Coulomb gas representation, these authors include amplitude fluctuations perturbatively and find a true line of fixed points, confirming the importance of explicitly using vortex degrees of freedom. The main difference is that our approach treats amplitude fluctuations nonperturbatively, while Krieg2017 includes lattice effects explicitly Machado2010 .

Appendix A Spin-wave approximation

The expression for the magnetization is given by

[TABLE]

while the expression of the spin-spin correlation function on the lattice is

[TABLE]

Both are conveniently rewritten in continuous notation as

[TABLE]

with $J(x^{\prime})=i\delta(x^{\prime})$ and $J(x^{\prime})=i\delta(x-x^{\prime})-i\delta(x^{\prime})$, where the two expressions are valid respectively for the magnetization and the two-point correlation function. The integral in latter expression yields

[TABLE]

where

[TABLE]

(the $x=0$ case must be evaluated separately in a finite volume and in the thermodynamic limit). In a finite system of size $L$ we obtain in $d=2$

[TABLE]

leading to a vanishing magnetization in the $2d$ system in the thermodynamic limit.

In order to evaluate $\mathcal{G}(x)$ it is convenient to perform the computation directly in the thermodynamic limit. We first consider a general dimension $d$ and then compute the $d\to 2$ limit. One has then

[TABLE]

where $s_{d}$ is the surface of the $d$-dimensional unit sphere divided by $(2\pi)^{d}$. The finite $x$ expression can be obtained in the continuum limit $a\to 0$ as

[TABLE]

and one obtains

[TABLE]

Appendix B Flow equations for the amplitude and phase scheme

In order to derive the FRG flow equations, we project the Wetterich equationWetterich1993 onto the theory space defined by the effective action ansatz (41) to obtain Berges2002

[TABLE]

We choose both amplitude and phase regulators $R^{(\ell)}$, with $\ell=\rho,\theta$, of the form

[TABLE]

where $\alpha_{\ell}$ is a dimensional coefficient necessary to have the correct scaling dimension of the regulator terms. The scale derivative of the regulator is then

[TABLE]

These $\theta$ functions in the numerator of the integral in Eq. (72) constrain the momenta to $q^{2}\in[0,k^{2}]$, where the regulator $\theta$ functions in the denominators are always unity. We are then left with the calculation of two integrals of the type (in $d=2$)

[TABLE]

where $a$ and $b$ are two $q$-independent constants. It is convenient to define the rescaled variable $x=q^{2}/k^{2}$ leading to

[TABLE]

Substituing $a$ and $b$ with the coefficients of the integrals in Eq. (72), one obtains the full potential flow equation

[TABLE]

where we have used the fact that $\partial_{t}\alpha_{\ell}=0$ in two dimensions. The flow for Gaussian theories $U^{\prime\prime}_{k}(\rho)=0$ is simply

[TABLE]

According to the discussion in the text the flow for Gaussian theories must vanish, thus, in order to enforce this condition, we simply subtract the r.h.s. of Eq. (78) from the r.h.s. of Eq. (77). The latter procedure finally produces Eq. (42) in the text.

This equation is solved numerically for the full potential function to produce the numerical results shown in Section V. Nevertheless, in order to gain a qualitative understanding of the flow, it is useful to employ a second-order Taylor expansion around the running potential minimum

[TABLE]

which leads to the following flowing RG couplings:

[TABLE]

The general flow equation (77) contains two free parameters $\alpha_{\theta,\rho}$, which are dimensionless in $d=2$. The phase diagram in Fig. 3 has been obtained with $\alpha_{\theta}=\kappa_{k}$ and $\alpha_{\rho}=1/\left(4\kappa_{k}\right)$ in order to simplify the flow equations, but different choices of these parameters give equivalent results.