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Physica Scripta

Thermodynamic properties of condensates with spin- f atoms ( f ≥ 1 ) in the neighborhood of zero temperature

Y. Z. He, Yimin Liu, Bao Cheng-guang, Zhibing Li

Published Dec 18, 2024• 0 citations
Publisher

Abstract

Abstract When the Hamiltonian of a condensate containing spin- f bosons is dominated by attractive singlet pairing interaction with the strength − ∣G 0∣ and when the temperature T is in the neighborhood of zero, the many-body Schrödinger equation can be exactly solved. Based on the complete and exact spectra the thermodynamic properties of these systems different in particle number N and f have been studied. The variation of the function ∂ C ∂ x against x is firstly studied, where C is the capacity, x ≡ lg(k B T/∣G 0∣), and k B is the Boltzmann constant. It was found that the variation can be divided into four stages. In the very narrow first stage, ∂ C ∂ x appears as a sharp peak. The second stage is broad where ∂ C ∂ x = 0 . The third stage is also very narrow where ∂ C ∂ x appears as a sharp anti-peak. In the fourth stage, ∂ C ∂ x becomes zero again. All the thermodynamic functions: the internal energy, the capacity, and the entropy vary accordingly. How these functions are affected by N, f , and ∣G 0∣ in each stage has been clarified. The spin-thermodynamics would replace the traditional thermodynamics in the neighborhood of zero temperature, where the spin degrees of freedom play an essential role.

DOI
10.1088/1402-4896/ada117

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Abstract When the Hamiltonian of a condensate containing spin- f bosons is dominated by attractive singlet pairing interaction with the strength − ∣G 0∣ and when the temperature T is in the neighborhood of zero, the many-body Schrödinger equation can be exactly solved. Based on the complete and exact spectra the thermodynamic properties of these systems different in particle number N and f have been studied. The variation of the function ∂ C ∂ x against x is firstly studied, where C is the capacity, x ≡ lg(k B T/∣G 0∣), and k B is the Boltzmann constant. It was found that the variation can be divided into four stages. In the very narrow first stage, ∂ C ∂ x appears as a sharp peak. The second stage is broad where ∂ C ∂ x = 0 . The third stage is also very narrow where ∂ C ∂ x appears as a sharp anti-peak. In the fourth stage, ∂ C ∂ x becomes zero again. All the thermodynamic functions: the internal energy, the capacity, and the entropy vary accordingly. How these functions are affected by N, f , and ∣G 0∣ in each stage has been clarified. The spin-thermodynamics would replace the traditional thermodynamics in the neighborhood of zero temperature, where the spin degrees of freedom play an essential role.
Abstract The spectra of N -boson systems with arbitrary nonzero spin f have been studied. Firstly, only the singlet pairing interaction is considered, a set of exact eigenstates together with the eigenenergies are analytically obtained. The completeness of this set is proved. The analytical expression allows us to see clearly the spin textures of various states different in N and/or f , and to find out the similarity and relationship lying among them. Secondly, the effect of other interactions is evaluated via exact numerical calculations on the systems with a smaller N . Some features and notable phenomena that might emerge in high- f systems, say, the ground band might have extremely high level density, have been discussed.
Thermodynamical properties of an interacting boson system at finite temperatures and zero chemical potential are studied within the framework of the Skyrme-like mean-field toy model. It is assumed that the mean field contains both attractive and repulsive terms. Self-consistency relations between the mean field and thermodynamic functions are derived. It is shown that, for sufficiently strong attractive interactions, this system develops a first-order phase transition via the formation of a Bose condensate. An interesting prediction of the model is that the condensed phase is characterized by a constant total density of particles. It is shown that the energy density exhibits a jump at the critical temperature.
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.
Bose system of zero spin particles is considered in the presence of the Bose–Einstein condensate in the vicinity of the phase transition point. The system is investigated in the framework of the Bogolyubov model with the separated condensate. In this model an effective Hamiltonian of the system is introduced by replacing condensate creation and annihilation operators in system Hamiltonian by n01/2 where n0 is occupation number of the condensate state. According to Bogolyubov, the grand canonical thermodynamic potential related to the effective Hamiltonian is considered as nonequilibrium thermodynamic potential. In the present paper this potential is investigated as a function of the small variable n0. With the help of the thermodynamic perturbation theory it is shown that it is expanded in a series over integer powers of n0. This corresponds to the basic idea of the Landau theory of the phase transitions of the second kind. Coefficients at terms of the first and second orders in n0 in the expansion are calculated for Bose gas in the main approximation in small interaction. Calculation of the coefficients at terms of the third and fourth orders needs accounting contributions of the thermodynamic perturbation theory at least of the 4th order and will be done elsewhere. It is established that the results obtained for Bose gas do not fit into the Landau theory of phase transitions of the second kind. Some comments that discuss the situation are given.
The ground states of Bose-Einstein condensates (BECs) of spin-2 bosons are classified into three distinct phases (ferromagnetic, ``antiferromagnetic,'' and cyclic) depending on the s-wave scattering lengths of binary collisions for total-spin 0, 2, and 4 channels. Many-body spin correlations and magnetic response of the condensate in each of these phases are studied in a mesoscopic regime, while low-lying excitation spectra are investigated in the thermodynamic regime. In the mesoscopic regime, where the system is so tightly confined that the spatial degrees of freedom are frozen, the exact, many-body ground state for each phase is found to be expressed in terms of the creation operators of pair or trio bosons having spin correlations. These pairwise and trio-wise units are shown to bring about some unique features of spin-2 BECs such as a huge jump in magnetization from minimum to maximum possible values and the robustness of the minimum-magnetization state against an applied magnetic field. In the thermodynamic regime, where the system is spatially uniform, low-lying excitation spectra in the presence of magnetic field are obtained analytically using the Bogoliubov approximation. In the ferromagnetic phase, the excitation spectrum consists of one Goldstone mode and four single-particle modes. In the antiferromagnetic phase, where spin-singlet ``pairs'' undergo Bose-Einstein condensation, the spectrum consists of two Goldstone modes and three massive ones, all of which become massless when magnetic field vanishes. In the cyclic phase, where boson ``trios'' condense into a spin-singlet state, the spectrum is characterized by two Goldstone modes, one single-particle mode having a magnetic-field-independent energy gap, and a gapless single-particle mode that becomes massless in the absence of magnetic field.
Abstract We obtain and justify a many-body Hamiltonian of pairwise interacting spin-1 atoms, which includes eight generators of the SU(3) group associated with spin and quadrupole degrees of freedom. It is shown that this Hamiltonian is valid for non-local interaction potential, whereas for local interaction specified by s -wave scattering length, the Hamiltonian should be bilinear in spin operators only (of the Heisenberg type). We apply the obtained Hamiltonian to study the ground-state properties and single-particle excitations of a weakly interacting gas of spin-1 atoms with Bose–Einstein condensate (BEC) taking into account the quadrupole degrees of freedom. It is shown that the system under consideration can be in ferromagnetic, quadrupole, and paramagnetic phases. The corresponding phase diagram is constructed and discussed. The main characteristics such as the density of the grand thermodynamic potential, condensate density, and single-particle excitation spectra modified by quadrupole degrees of freedom are determined in different phases.