We consider the influence of temperature on the critical behavior of the weakly interacting three-dimensional Bose system. Using the flow equations of the Renormalization group and a Φ4 model with dynamical critical exponent z = 2, we calculated the critical exponent ν, and the thermodynamic parameters near the critical temperature, taking into consideration the quantum effects. The quantum effects considered in this method gives ν the value 0.75, which is close to the value 0.73 obtained by the polynomial method. The critical temperature shift ΔTc was obtained, in the lowest order approximation, proportional to the scattering length.
We consider the influence of temperature on the critical behavior of the weakly interacting three-dimensional Bose system. Using the flow equations of the Renormalization group and a Φ4 model with dynamical critical exponent z = 2, we calculated the critical exponent ν, and the thermodynamic parameters near the critical temperature, taking into consideration the quantum effects. The quantum effects considered in this method gives ν the value 0.75, which is close to the value 0.73 obtained by the polynomial method. The critical temperature shift ΔTc was obtained, in the lowest order approximation, proportional to the scattering length.
We consider the influence of temperature on the critical behavior of a weakly interacting three dimensional Bose system. Using the flow equations of the Renormalization Group and a Φ4 model with dynamical critical exponent z = 2, we calculated the critical exponent ν, and the thermodynamical parameters near the critical temperature, taking in consideration the quantum effects. The quantum effects considered using this method gives for ν the value 0.75, which is close to the value 0.73 obtained by the polynomial method. The critical temperature shift ΔT c was obtained, in the lowest-order approximation, and turns out to be proportional to the scattering length.
We applied the Renormalization Group method at finite temperature to reconsider the two-dimensional dilute Bose gas. The general flow equations are obtained for the case of arbitrary dimensions, and by considering the two-dimensional limit, we estimate the value of the critical temperature, coherence length and specific heat. The value of the critical temperature is in agreement with previous calculations performed using the t-matrix method. The coherence length and the specific heat present a non-universal behavior, a logarithmic temperature dependence in the critical region being identified.
The critical phenomenon of the zero-temperature superfluid--Bose-glass phase transition for hard-core bosons on a three-dimensional disordered lattice is studied using a quantum real-space renormalization-group method. The correlation-length exponent \ensuremath{\nu} and the dynamic exponent z are computed. The critical exponent z is found to be 2.5 for compressible states and 1.3 for incompressible states. The exponent \ensuremath{\nu} is shown to be insensitive to z as that in the two-dimensional case, and has a value roughly equal to 1.
Within a canonical ensemble treatment, we study the finite-size scaling behaviour of the specific heat near the critical region for the weakly interacting Bose gases, and obtain the specific heat exponent α and correlation length exponent ν which agree well with experimental data and previous theoretical predictions. For the small and fixed interaction parameter an 1/3 with a being the scattering length and n the particle number density, we report for the first time that the dimensionless specific heats per particle c ( t , N ) for various system sizes ( L = ( N / n ) 1/3 in a square box potential) and reduced temperatures ( t = T / T c − 1) collapse onto a single form, and the borders of universality validity are obtained numerically. From scaling arguments, we study the interaction-induced shift of the transition temperature (Δ T c = T c − T 0 c ) and find that Δ T c / T 0 c = ban 1/3 with b = 0.42 ± 0.05, where T 0 c is the ideal-gas thermodynamic critical temperature. We also compare our findings with experimental and theoretical results for in cubic lattices, showing the very different behaviours for the two systems.
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. The perturbative expansion of these properties is essentially an expansion in the gas parameter na3, where a is the s-wave scattering length and n 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/N techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.
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 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⪡kc and the short-wavelength regime k⪢kc, where 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>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 kc 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 Tc 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.