Application of the functional renormalization group to Bose gases: From linear to hydrodynamic fluctuations

Abstract

We study weakly interacting Bose gases using the functional renormalization group with a hydrodynamic effective action. We use a scale-dependent parametrization of the boson fields that interpolates between a Cartesian representation at high momenta and an amplitude-phase one for low momenta. We apply this to Bose gases in two and three dimensions near the superfluid phase transition where they can be described by statistical O(2) models. We are able to give consistent physical descriptions of the infrared regime in both two and three dimensions. In particular, and in contrast to previous studies using the functional renormalization group, we find a stable superfluid phase at finite temperatures in two dimensions. We compare our results for the superfluid and boson densities with Monte-Carlo simulations, and we find they are in reasonable agreement. arXiv:1806.10373v2 [cond-mat.quant-gas] 21 Sep 2018 in the physical limit will depend on the choices made in setting up the flow equations. These include the form of the regulator, and ways of optimizing this choice have been developed [29,30].There are many versions of the FRG; in this work we use the one introduced by Wetterich [26, 27] for the average effective action, which is based on the Legendre transform of the logarithm of the partition function. The exact flow equation for this effective action has a one-loop structure. However, in practice, a truncation scheme needs to be used in order to solve for the RG flow. In this work we employ a derivative expansion; other choices can be found in the literature [31][32][33]. Truncations based on derivative expansions have been widely used in applications of the FRG and, despite the approximations introduced, they have been successful in describing phase transitions and critical exponents in a variety of systems [27,28].In particular, with fields in the Cartesian representation, the FRG has had some success in describing the weaklyinteracting Bose gas and related O(N ) models in three dimensions. In the critical regime, it yields results that are in good agreement with Monte-Carlo simulations. Away from that region, it has also been used to investigate bulk thermodynamic properties [18,27,31,[34][35][36][37][38]. As shown by Dupuis [18,39], one appealing aspect of this FRG approach is that, despite the potential IR divergences of loop diagrams as the physical limit is approached, it does respect the relevant Ward identity [17], giving a vanishing anomalous self-energy.This self-energy, and the resulting IR-divergent propagator for the longitudinal mode, arise from a momentumindependent interaction that vanishes in the physical limit. This treatment thus fails to capture the interaction between the amplitude fluctuations of the condensate that is present in Popov's effective theory. It also omits the leading interaction between the Goldstone modes, which is of second order in momenta. These limitations may be remedied by the inclusion of higher-order terms in the derivative expansion of the action. However, in current ...

Thermodynamics of Bose gases from functional renormalization with a hydrodynamic low-energy effective action

“…For each case we first compare the flows obtained for the interpolating and Cartesian representations. From now on, we will refer to the mass renormalization Z m as Z ϑ , to emphasize that it describes the renormalization of the Goldstone (phase) mode [43]. Subsequently, we present results for thermodynamic properties and compare them with known results.…”

“…Ref. [43], and a detailed discussion can be found there. Here, we focus on features that are particular to dynamical Bose gases.…”

“…In this work, we have studied the superfluid phases of weakly-interacting Bose gases in one, two and three dimensions using the functional renormalization group. The boson fields are parametrized using the interpolating representation developed in our previous work [43]. We also have developed a rather natural approach for calculating the thermodynamic properties of Bose gases.…”

“…As argued in Ref. [43], the Cartesian representation must be used in UV regime where both longitudinal and Goldstone fluctuations are important, that is, where the contribution 2u 2 ρ 0 is small compared to the kinetic term. In that regime the path integral over fluctuations is approximately Gaussian.…”

“…Such problems are caused by the fact that just as the Cartesian representation is not the best choice in the IR, the AP is in general a poor choice in the UV regime. In a previous paper [43], we implemented a scale-dependent parametrization of the boson fields that interpolates between the Cartesian representation in the UV and the AP representation in the IR. This "interpolating representation" enables us to treat correctly both Gaussian and Goldstone fluctuations in the UV and IR, respectively.…”