Abstract: The one-loop effective potential for non-relativistic bosons with a delta function repulsive potential is calculated for a given chemical potential using functional methods. After renormalization and at zero temperature it reproduces the standard ground state energy and pressure as function of the particle density. At finite temperatures it is found necessary to include ring corrections to the one-loop result in order to satisfy the Goldstone theorem. It is natural to introduce an effective chemical potential directly related to the order parameter and which uniformly decreases with increasing temperatures. This is in contrast to the the ordinary chemical potential which peaks at the critical temperature. The resulting thermodynamics in the condensed phase at very low temperatures is found to be the same as in the Bogoliubov approximation where the degrees of freedom are given by the Goldstone bosons. At higher temperatures the ring corrections dominate and result in a critical temperature unaffected by the interaction. 1
In this thesis, we investigate thermodynamic properties, magnetic phase transitions and correlation functions of integrable quantum many-body systems in the field of ultracold atomic gases. The scope of our investigation encompasses three models in one-dimension that are integrable by means of the Bethe ansatz method. They are the multi-component 5-interacting Fermi gas, the spin-l Bose gas with antiferromagnetic spin interaction and the SU(2) spinor Bose gas with finite-range Gaussian interaction. We derive the thermodynamic Bethe ansatz equations for the general SU(k) Fermi gas. We illustrate the root patterns and give a quantitative description of the ground state properties for the strongly and weakly interacting regimes. The thermodynamics and magnetic phase transitions are described quantitatively by means of the thermodynamic Bethe ansatz equations. We find that all phase transitions in the attractive regime are of second order with a linear field dependent magnetization in the vicinities of critical points. The SU(2) Fermi gas is studied in more detail. The low temperature thermodynamics for the repulsive case is derived by means of the Wiener-Hopf technique. Finite-size corrections and correlation functions for both the repulsive and attractive cases are derived from the nested Bethe ansatz equations and conformal field theory. In the attractive case, we show that spatial oscillations of the leading terms in the pair correlation function and spin correlation function solely depend on the mismatch between Fermi surfaces of spin-up and spin-down fermions. Such spatial modulations are characteristic of a Fulde- Ferrell-Larkin-Ovchinnikov (FFLO) state. For the spin-l Bose gas with antiferromagnetic spin interaction, we find that at zero temperature, the system exhibits three quantum phases: a phase with singlet pairs of bosons, a fully-polarized Tonks-Girardeau gas phase of unpaired ms = 1 bosons, and a mixed phase of singlet pairs and unpaired ms = 1 bosons. Phase transitions in the vicinities of the critical fields are also of second order with a linear field dependent magnetization. Finally, we use the asymptotic Bethe ansatz method to derive the quasimomenta distribution and ground state energy for the SU(2) spinor Bose gas with finite-range Gaussian interaction. We show that finite range potentials are more likely to lead to quasi Bose-Einstein condensation than zero range potentials by studying the spatial density profiles as the interaction width increases.
The crossover from ideal-Bose-gas behavior to nonclassical, XY-like criticality in a dilute, interacting Bose fluid is considered in detail with emphasis on the superfluid density, ${\ensuremath{\rho}}_{s}$(T), below the transition. A general discussion of crossover scaling in constrained systems shows that ${\ensuremath{\rho}}_{s}$ and other temperature-dependent thermodynamic functions at controlled overall density can be expressed in terms of universal scaling functions. The finite-temperature Green's function formalism is used to calculate the associated linear scaling fields and the crossover exponent exactly for all dimensionalities d>2, scaling predictions, including logarithmic singularities just at d=3, being checked via second-order computations. A mapping of the Bose problem onto the classical
s4 spin model is achieved which is asymptotically exact in the critical region. In this way, previously known renormalization-group results are used to calculate the scaling functions for ${\ensuremath{\rho}}_{s}$, for the off-diagonal susceptibility, etc., to first order in \ensuremath{\epsilon}=4-d. Good agreement is found with the experimental measurements by Reppy and co-workers of ${\ensuremath{\rho}}_{s}$(T) for helium absorbed at low coverages in Vycor glass (of pore size 50--80 A\r{}). An effective-mass ratio
m∗/m\ensuremath{\simeq}1.5, and an effective-scattering-length ratio
a∗/a\ensuremath{\simeq}1.3 are indicated for mobile helium in the Vycor (although the irregular character of Vycor has not, at this point, been fully allowed for in the theory). Other approaches in the literature to computing the critical behavior of quantum-mechanical systems are reviewed and related to the present analysis.
One-dimensional spinless Bose and Fermi gases with contact interactions have the close interrelation via Girardeau's Bose-Fermi mapping, leading to the correspondences in their energy spectra and thermodynamics. However, correlation functions are in general not identical between these systems. We derive in both systems the exact universal relations for correlation functions, which hold for any energy eigenstate and any statistical ensemble of the eigenstates with or without a trapping potential. These relations include the large-momentum behaviors of static structure factors and of momentum distributions as well as energy relations, which connect the sums of kinetic and interaction energies to the momentum distributions. The relations involve two- and three-body contacts, which are the integrals of local pair and triad correlations, respectively. We clarify how the relations for bosons and fermions differ and are connected with each other. In particular, we find that the three-body contact makes no contribution to the bosonic energy relation, but it plays a crucial role in the fermionic one. In addition, we compute the exact momentum distribution for any number of fermions in the unitary limit.
Inverse Fourier transforms of dynamic correlation functions of order-parameter fluctuations in the singlet-spin-pairing and equal-spin-pairing states of an interacting Fermi system are calculated in a mean-field approximation. The behavior of these functions in the corresponding critical regions, i.e., in the neighborhood of a critical point for an anisotropic, singlet superfluid (and for a superconductor) and for an equal-spin-pairing-type, triplet superfluid, are examined. It is found that these functions do not conform to certain generalized homogeneity assumptions. The static counterparts of these functions are found to be nonsingular at corresponding critical points. A comparison of these findings with corresponding findings for a Bose system, reported in a previous work, highlights the role of Fermi statistics in the critical regions under consideration.
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
T 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-
1/2 quantum
XY model with coupling
J 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 $T\ensuremath{\lesssim}0.1J$.