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

2004; American Physical Society; Volume: 70; Issue: 6 Linguagem: Inglês

10.1103/physreva.70.063621

1538-4446

N. Hasselmann, Sascha Ledowski, Peter Kopietz,

Physics of Superconductivity and Magnetism

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.