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arXiv:cond-mat/0409167 · Physical Review A

Critical behavior of weakly interacting bosons: A functional renormalization-group approach

N. Hasselmann, Sascha Ledowski, Peter Kopietz

Published Dec 30, 2004• 34 citations
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Abstract

We present a detailed investigation of the momentum-dependent self-energy $\ensuremath{\Sigma}(k)$ at zero frequency of weakly interacting bosons at the critical temperature ${T}_{c}$ of Bose-Einstein condensation in dimensions $3\ensuremath{\leqslant}D<4$. Applying the functional renormalization group, we calculate the universal scaling function for the self-energy at zero frequency but at all wave vectors within an approximation which truncates the flow equations of the irreducible vertices at the four-point level. The self-energy interpolates between the critical regime $k⪡{k}_{c}$ and the short-wavelength regime $k⪢{k}_{c}$, where ${k}_{c}$ is the crossover scale. In the critical regime, the self-energy correctly approaches the asymptotic behavior $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2\ensuremath{-}\ensuremath{\eta}}$, and in the short-wavelength regime the behavior is $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2(D\ensuremath{-}3)}$ in $D>3$. In $D=3$, we recover the logarithmic divergence $\ensuremath{\Sigma}(k)\ensuremath{\propto}\mathrm{ln}(k∕{k}_{c})$ encountered in perturbation theory. Our approach yields the crossover scale ${k}_{c}$ as well as a reasonable estimate for the critical exponent $\ensuremath{\eta}$ in $D=3$. From our scaling function we find for the interaction-induced shift in ${T}_{c}$ in three dimensions, $\ensuremath{\Delta}{T}_{c}∕{T}_{c}=1.23a{n}^{1∕3}$, where $a$ is the $s$-wave scattering length and $n$ is the density, in excellent agreement with other approaches. We also discuss the flow of marginal parameters in $D=3$ and extend our truncation scheme of the renormalization group equations by including the six- and eight-point vertex, which yields an improved estimate for the anomalous dimension $\ensuremath{\eta}\ensuremath{\approx}0.0513$. We further calculate the constant ${\mathrm{lim}}_{k\ensuremath{\rightarrow}0}\phantom{\rule{0.2em}{0ex}}\ensuremath{\Sigma}(k)∕{k}^{2\ensuremath{-}\ensuremath{\eta}}$ and find good agreement with recent Monte Carlo data.

DOI
10.1103/physreva.70.063621

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We present a detailed investigation of the momentum-dependent self-energy \ensuremathΣ(k)\ensuremath{\Sigma}(k)\ensuremathΣ(k) at zero frequency of weakly interacting bosons at the critical temperature Tc{T}_{c}Tc​ of Bose-Einstein condensation in dimensions 3\ensuremath⩽D<43\ensuremath{\leqslant}D<43\ensuremath⩽D<4. Applying the functional renormalization group, we calculate the universal scaling function for the self-energy at zero frequency but at all wave vectors within an approximation which truncates the flow equations of the irreducible vertices at the four-point level. The self-energy interpolates between the critical regime k⪡kck⪡{k}_{c}k⪡kc​ and the short-wavelength regime k⪢kck⪢{k}_{c}k⪢kc​, where kc{k}_{c}kc​ is the crossover scale. In the critical regime, the self-energy correctly approaches the asymptotic behavior \ensuremathΣ(k)\ensuremath∝k2\ensuremath−\ensuremathη\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2\ensuremath{-}\ensuremath{\eta}}\ensuremathΣ(k)\ensuremath∝k2\ensuremath−\ensuremathη, and in the short-wavelength regime the behavior is \ensuremathΣ(k)\ensuremath∝k2(D\ensuremath−3)\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2(D\ensuremath{-}3)}\ensuremathΣ(k)\ensuremath∝k2(D\ensuremath−3) in D>3D>3D>3. In D=3D=3D=3, we recover the logarithmic divergence \ensuremathΣ(k)\ensuremath∝ln(k∕kc)\ensuremath{\Sigma}(k)\ensuremath{\propto}\mathrm{ln}(k∕{k}_{c})\ensuremathΣ(k)\ensuremath∝ln(k∕kc​) encountered in perturbation theory. Our approach yields the crossover scale kc{k}_{c}kc​ as well as a reasonable estimate for the critical exponent \ensuremathη\ensuremath{\eta}\ensuremathη in D=3D=3D=3. From our scaling function we find for the interaction-induced shift in Tc{T}_{c}Tc​ in three dimensions, \ensuremathΔTc∕Tc=1.23an1∕3\ensuremath{\Delta}{T}_{c}∕{T}_{c}=1.23a{n}^{1∕3}\ensuremathΔTc​∕Tc​=1.23an1∕3, where aaa is the sss-wave scattering length and nnn is the density, in excellent agreement with other approaches. We also discuss the flow of marginal parameters in D=3D=3D=3 and extend our truncation scheme of the renormalization group equations by including the six- and eight-point vertex, which yields an improved estimate for the anomalous dimension \ensuremathη\ensuremath≈0.0513\ensuremath{\eta}\ensuremath{\approx}0.0513\ensuremathη\ensuremath≈0.0513. We further calculate the constant limk\ensuremath→0\ensuremathΣ(k)∕k2\ensuremath−\ensuremathη{\mathrm{lim}}_{k\ensuremath{\rightarrow}0}\phantom{\rule{0.2em}{0ex}}\ensuremath{\Sigma}(k)∕{k}^{2\ensuremath{-}\ensuremath{\eta}}limk\ensuremath→0​\ensuremathΣ(k)∕k2\ensuremath−\ensuremathη and find good agreement with recent Monte Carlo data.
Using the exact renormalization group we calculate the momentum-dependent self-energy $\ensuremath{\Sigma}(k)$ at zero frequency of weakly interacting bosons at the critical temperature Tc{T}_{c}Tc​ of Bose-Einstein condensation in dimensions $3\ensuremath{\leqslant}D<4$. We obtain the complete crossover function interpolating between the critical regime k⪡kck⪡{k}_{c}k⪡kc​, where $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2\ensuremath{-}\ensuremath{\eta}}$, and the short-wavelength regime k⪢kck⪢{k}_{c}k⪢kc​, where $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2(D\ensuremath{-}3)}$ in D>3D>3D>3 and $\ensuremath{\Sigma}(k)\ensuremath{\propto}\mathrm{ln}(k∕{k}_{c})$ in D=3D=3D=3. Our approach yields the crossover scale kc{k}_{c}kc​ on the same footing with a reasonable estimate for the critical exponent $\ensuremath{\eta}$ in D=3D=3D=3. From our $\ensuremath{\Sigma}(k)$ we find for the interaction-induced shift of Tc{T}_{c}Tc​ in three dimensions $\ensuremath{\Delta}{T}_{c}∕{T}_{c}\ensuremath{\approx}1.23a{n}^{1∕3}$, where aaa is the sss-wave scattering length and nnn is the density.
We calculate the temperature-dependent condensate density rho0(T) of interacting bosons in three dimensions using the functional renormalization group (FRG). From the numerical solution of suitably truncated FRG flow equations for the irreducible vertices we obtain rho0(T) for arbitrary temperatures. We carefully extrapolate our numerical results to the critical point and determine the order parameter exponent beta approximately 0.32 in reasonable agreement with the expected value 0.345 associated with the XY -universality class. We also calculate the condensate density in two dimensions at zero temperature using a truncation of the FRG flow equations based on the derivative expansion including cubic and quartic terms in the expansion of the effective potential in powers of the density. As compared with the widely used quadratic approximation for the effective potential, the coupling constants associated with the cubic and quartic terms lead to small corrections of the condensate density.
We investigate the single-particle spectral density of interacting bosons within the nonperturbative functional renormalization group technique. The flow equations for a Bose gas are derived in a scheme which treats the two-particle density-density correlations exactly but neglects irreducible correlations among three and more particles. These flow equations are solved within a truncation which allows to extract the complete frequency and momentum structure of the normal and anomalous self-energies. Both the asymptotic small momentum regime, where the perturbation regime fails, as well as the perturbative regime at larger momenta are well described within a single unified approach. The self-energies do not exhibit any infrared divergences, satisfy the U(1) symmetry constraints, and are in accordance with the Nepomnyashchy relation, which states that the anomalous self-energy vanishes at zero momentum and zero frequency. From the self-energies we extract the single-particle spectral density of the two-dimensional Bose gas. The dispersion is found to be of the Bogoliubov form and shows the crossover from linear Goldstone modes to the quadratic behavior of quasifree bosons. The damping of the quasiparticles is found to be in accordance with the standard Beliaev damping. We furthermore recover the exact asymptotic limit of the propagators derived by Gavoret and Nozi\`eres and discuss the nature of the nonanalyticities of the self-energies in the very small momentum regime.
The behavior of a dilute Bose superfluid at low temperatures, $T<<{T}_{\ensuremath{\lambda}}$, is considered with emphasis on the superfluid density, ${\ensuremath{\rho}}_{s}(T)$. Many of the predictions of the Bogoliubov model are exact in this regime and are shown to lead to a scaling description of the crossover from ideal to interacting Bose gas behavior as the density is reduced. Combined with previous work on scaling near criticality, $T\ensuremath{\simeq}{T}_{\ensuremath{\lambda}}$, these results complete the crossover description of the dilute superfluid over the full temperature range. Diagrammatic perturbation theory in the superfluid phase is outlined and previous results from the helium literature are summarized. The various divergences encountered have previously been shown to arise from nonanalyticities in the self-energy functions at zero momentum, and these are related to the spin-wave nature of the superfluid phase. A full renormalization-group fixed-point picture is presented summarizing the zero-temperature and finite-temperature regimes and the flows connecting them.
The topic of this thesis is the functional renormalization group. We discuss some approximations schemes. Thereafter we apply these approximations to study different fields of condensed matter physics. Generally we have to evaluate an infinite set of vertex functions describing the scattering of particles. These vertex functions get renormalized away from their bare values governed by an infinite hierarchy of flow equations. We cannot expect to actually solve these equations but have to apply a couple of approximations. The aim is to somehow separate relevant contributions from irrelevant ones. One possible scheme opens up if we rescale fields and vertices. Here "relevance" is used in a quantitative way to describe the scaling behaviour of vertices close to a fixed point of the RG. One disadvantage of describing the system in terms of infinitely many vertices is that the majority of these vertices we have to evaluate are not of interest to us. In most cases we are just looking for the self-energy or the two-particle effective interaction. However there might be contributions to the flow of these vertices that are generated by irrelevant vertices. We generally assume that we can express irrelevant vertices in terms of the relevant and marginal ones. Then in turn it should be possible to write the contributions of these irrelevant vertices to the flow of relevant and marginal ones in terms of relevant and marginal vertices as well. We show how this can be achieved by what we term the adiabatic approximation. We now consider weakly interacting bosons at the critical point of Bose-Einstein condensation. As the transition takes place at a finite temperature this temperature defines an effective ultraviolet cut-off. For the investigation of physical properties that depend on momenta smaller than this cut-off it is therefore sufficient to describe the system by a classical field theory. Our central topic here is the self-energy of the bosons and we are able to evaluate it with the full momentum dependence. For small momenta it approaches a scaling form and as the momentum is gradually increased we observe a crossover to the perturbative regime. As a test for the reliability of our expression for the selfenergy we investigate the interaction induced shift of the critical. Our results compare quite satisfactory to the best available estimates for this shift. For the anomalous dimension our approach predicts the correct order of magnitude however with a considerable error. As an improvement we include more vertices into our calculations. Here we observe that our fixed point estimates indeed approach the best known results but this convergence is quite weak. We turn toward systems of interacting fermions. The formulation of the functional renormalization group implicitly requires knowledge of the true Fermi surface of the full interacting system. In general however we can just calculate it a-posteriori from the self-energy. The requirement to flow into a fixed point can be translated into a fine-tuning of the frequency/momentum independent part r_0 of the rescaled 2-point function. We show how this bare value is related to the momentum dependent effective interaction along the complete trajectory of the RG. On the other hand r_0 expresses the difference between the bare and the true Fermi surface. Putting both equations together results into an exact selfconsistency equation for the Fermi surface. We apply our self-consistency equation above to tackle the problem of finding the true Fermi surface of interacting fermions in low dimensions. The most simple non-trivial model with an inhomogeneous Fermi surface is a system of two coupled metallic chains. The process of interband backward scattering leads to a smoothing of the Fermi surface. Of special interest is if the Fermi momenta of the two bands collapse into just one value. We propose the term confinement transition for this behaviour. We bosonize the interband backward scattering by means of a Hubbard-Stratonovich transformation and treat our system as a single channel problem. This bosonization together with the adiabatic approximation allows us to investigate the system even at strong coupling. Within a simple one-loop treatment our method predicts a confinement transition at strong coupling. However taken vertex renormalizations into account we observe that this confinement is destroyed by fluctuations beyond one-loop. Actually we observe how the confined phase can be stabilized by the inclusion of interband umklapp scattering. Thereafter we consider the physically more relevant case of a two-dimensional system of infinitely many coupled metallic chains. Here the Fermi surface consists of two disconnected weakly curved sheets. We are able to repeat the calculations we have performed for our toy model. Within a self-consistent 2-loop calculation indeed signs for a confinement transition at finite coupling strength emerge.
In a Bose superfluid, the coupling between transverse (phase) and longitudinal fluctuations leads to a divergence of the longitudinal correlation function, which is responsible for the occurrence of infrared divergences in the perturbation theory and the breakdown of the Bogoliubov approximation. We report a nonperturbative renormalization-group calculation of the one-particle Green's function of an interacting boson system at zero temperature. We find two regimes separated by a characteristic momentum scale kG{k}_{G}kG​ (``Ginzburg'' scale). While the Bogoliubov approximation is valid at large momenta and energies, $|\mathbf{p}|,\text{ }|\ensuremath{\omega}|/c⪢{k}_{G}$ (with ccc as the velocity of the Bogoliubov sound mode), in the infrared (hydrodynamic) regime, $|\mathbf{p}|,\text{ }|\ensuremath{\omega}|/c⪡{k}_{G}$, the normal and anomalous self-energies exhibit singularities reflecting the divergence of the longitudinal correlation function. In particular, we find that the anomalous self-energy agrees with the Bogoliubov result ${\ensuremath{\Sigma}}_{\text{an}}(\mathbf{p},\ensuremath{\omega})\ensuremath{\simeq}\text{const}$ at high energies and behaves as ${\ensuremath{\Sigma}}_{\text{an}}(\mathbf{p},\ensuremath{\omega})\ensuremath{\sim}{({c}^{2}{\mathbf{p}}^{2}\ensuremath{-}{\ensuremath{\omega}}^{2})}^{(d\ensuremath{-}3)/2}$ in the infrared regime (with ddd as the space dimension), in agreement with the Nepomnyashchii identity ${\ensuremath{\Sigma}}_{\text{an}}(0,0)=0$ and the predictions of Popov's hydrodynamic theory. We argue that the hydrodynamic limit of the one-particle Green's function is fully determined by the knowledge of the exponent $3\ensuremath{-}d$ characterizing the divergence of the longitudinal susceptibility and the Ward identities associated to gauge and Galilean invariances. The infrared singularity of ${\ensuremath{\Sigma}}_{\text{an}}(\mathbf{p},\ensuremath{\omega})$ leads to a continuum of excitations (coexisting with the sound mode) which shows up in the one-particle spectral function.