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Physics and Chemistry of Liquids

Three-dimensional dilute Bose liquid at finite temperature: a Renormalization Group approach

M. Crişan, Marina Bianca Trif, I. Grosu, I. Ţifrea

Published Feb 1, 2007• 0 citations
Publisher

Abstract

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.

DOI
10.1080/00319100600941722

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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 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 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.
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.
We use the functional renormalization group (FRG) to derive analytical expressions for thermodynamic observables (density, pressure, entropy, and compressibility) as well as for single-particle properties (wave-function renormalization and effective mass) of interacting bosons in two dimensions as a function of temperature TTT and chemical potential $\ensuremath{\mu}$. We focus on the quantum disordered and the quantum critical regime close to the dilute Bose gas quantum critical point. Our approach is based on a truncated vertex expansion of the hierarchy of FRG flow equations and the decoupling of the two-body contact interaction in the particle-particle channel using a suitable Hubbard-Stratonovich transformation. Our analytic FRG results extend previous analytical renormalization-group calculations for thermodynamic observables at $\ensuremath{\mu}=0$ to finite values of $\ensuremath{\mu}$. To confirm the validity of our FRG approach, we have also performed quantum Monte Carlo simulations to obtain the magnetization, susceptibility, and correlation length of the two-dimensional spin-1/21/21/2 quantum XYXYXY model with coupling JJJ in a regime where its quantum critical behavior is controlled by the dilute Bose gas quantum critical point. We find that our analytical results describe the Monte Carlo data for $\ensuremath{\mu}\ensuremath{\le}0$ rather accurately up to relatively high temperatures $T\ensuremath{\lesssim}0.1J$.
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.