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arXiv:cond-mat/0311043 · Physical Review A

Self-energy and critical temperature of weakly interacting bosons

Sascha Ledowski, N. Hasselmann, Peter Kopietz

Published Jun 17, 2004• 51 citations
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Abstract

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 ${T}_{c}$ of Bose-Einstein condensation in dimensions $3\ensuremath{\leqslant}D<4$. We obtain the complete crossover function interpolating between the critical regime $k⪡{k}_{c}$, where $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2\ensuremath{-}\ensuremath{\eta}}$, and the short-wavelength regime $k⪢{k}_{c}$, where $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2(D\ensuremath{-}3)}$ in $D>3$ and $\ensuremath{\Sigma}(k)\ensuremath{\propto}\mathrm{ln}(k∕{k}_{c})$ in $D=3$. Our approach yields the crossover scale ${k}_{c}$ on the same footing with a reasonable estimate for the critical exponent $\ensuremath{\eta}$ in $D=3$. From our $\ensuremath{\Sigma}(k)$ we find for the interaction-induced shift of ${T}_{c}$ in three dimensions $\ensuremath{\Delta}{T}_{c}∕{T}_{c}\ensuremath{\approx}1.23a{n}^{1∕3}$, where $a$ is the $s$-wave scattering length and $n$ is the density.

DOI
10.1103/physreva.69.061601

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Using the exact renormalization group we calculate 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. We obtain the complete crossover function interpolating between the critical regime k⪡kck⪡{k}_{c}k⪡kc​, where \ensuremathΣ(k)\ensuremath∝k2\ensuremath−\ensuremathη\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2\ensuremath{-}\ensuremath{\eta}}\ensuremathΣ(k)\ensuremath∝k2\ensuremath−\ensuremathη, and the short-wavelength regime k⪢kck⪢{k}_{c}k⪢kc​, where \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 and \ensuremathΣ(k)\ensuremath∝ln(k∕kc)\ensuremath{\Sigma}(k)\ensuremath{\propto}\mathrm{ln}(k∕{k}_{c})\ensuremathΣ(k)\ensuremath∝ln(k∕kc​) 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 \ensuremathη\ensuremath{\eta}\ensuremathη in D=3D=3D=3. From our \ensuremathΣ(k)\ensuremath{\Sigma}(k)\ensuremathΣ(k) we find for the interaction-induced shift of Tc{T}_{c}Tc​ in three dimensions \ensuremathΔTc∕Tc\ensuremath≈1.23an1∕3\ensuremath{\Delta}{T}_{c}∕{T}_{c}\ensuremath{\approx}1.23a{n}^{1∕3}\ensuremathΔTc​∕Tc​\ensuremath≈1.23an1∕3, where aaa is the sss-wave scattering length and nnn is the density.
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 Tc{T}_{c}Tc​ 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⪡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 $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2\ensuremath{-}\ensuremath{\eta}}$, and in the short-wavelength regime the behavior is $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2(D\ensuremath{-}3)}$ in D>3D>3D>3. In D=3D=3D=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 kc{k}_{c}kc​ as well as a reasonable estimate for the critical exponent $\ensuremath{\eta}$ in D=3D=3D=3. From our scaling function we find for the interaction-induced shift in Tc{T}_{c}Tc​ in three dimensions, $\ensuremath{\Delta}{T}_{c}∕{T}_{c}=1.23a{n}^{1∕3}$, 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 $\ensuremath{\eta}\ensuremath{\approx}0.0513$. 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.
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.
We investigate Bose-Einstein condensation for interacting bosons at zero and nonzero temperature. Functional renormalization provides us with a consistent method to compute the effect of fluctuations beyond the Bogoliubov approximation. For three-dimensional dilute gases, we find an upper bound on the scattering length aaa which is of the order of the microphysical scale---typically the range of the van der Waals interaction. In contrast to fermions near the unitary bound, no strong interactions occur for bosons with approximately pointlike interactions, thus explaining the high quantitative reliability of perturbation theory for most quantities. For zero temperature we compute the quantum phase diagram for bosonic quasiparticles with a general dispersion relation, corresponding to an inverse microphysical propagator with terms linear and quadratic in the frequency. We compute the temperature dependence of the condensate and particle density nnn, and find for the critical temperature Tc{T}_{c}Tc​ a deviation from the free theory, $\ensuremath{\Delta}{T}_{c}/{T}_{c}=2.1a{n}^{1/3}$. For the sound velocity at zero temperature we find very good agreement with the Bogoliubov result, such that it may be used to determine the particle density accurately.
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.
The critical behavior of dense and dilute Bose systems is considered within the same framework in terms of an additional parameter y introduced in the renormalization-group transformation. The dense case is obtained when y=0, and the formalism becomes identical to the one previously devised by Singh which gives a two-thirds correlation-length critical exponent. The dilute case corresponds to the condition y=d>2, where the renormalization procedure leads to Bose-Einstein condensation in the vicinity of the critical point. As a consequence, the appearance of a crossover exponent results in a condensate correlation-length critical exponent equal to unity. The critical exponents in both cases are in agreement with an experiment done recently by Crooker et al., where a crossover from a two-thirds to a linear power law is observed in the temperature dependence of the superfluid mass with a systematic reduction in density. The critical masses of the condensate and noncondensate are found to be zero and infinite, respectively.
This article reviews recent advances in the theory of the three-dimensional dilute homogeneous Bose gas at zero and finite temperature. Effective-field-theory methods are used to formulate a systematic perturbative framework that can be used to calculate the properties of a system at T=0.T=0.T=0. The perturbative expansion of these properties is essentially an expansion in the gas parameter na3,\sqrt{{\mathrm{na}}^{3}},na3​, where aaa is the sss-wave scattering length and nnn is the number density. In particular, the leading quantum corrections to the ground-state energy density, the condensate depletion, and long-wavelength collective excitations are rederived in an efficient and economical manner. Nonuniversal effects are also discussed. These effects are higher-order corrections that depend on properties of the interatomic potential other than the scattering length, such as the effective range. The article critically examines various approaches to the dilute Bose gas in equilibrium at finite temperature. These include the Bogoliubov approximation, the Popov approximation, the Hartree-Fock-Bogoliubov approximation, the \ensuremath{\Phi}-derivable approach, optimized perturbation theory, and renormalization-group techniques. The article ends with a look at recent calculations of the critical temperature of the dilute Bose gas, which include 1/N1/N1/N techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.