The interplay between pairing and shape deformations at finite temperatures is studied with reference to an exactly soluble SU(2)\ifmmode\times\else\texttimes\fi{}SU(2) model which is able to suitably mock up both superconductivity and the effects of long-range residual forces. Thermodynamic and ground state phase transitions are studied, both in the thermodynamic limit and in the case of a finite number of particles. Several independent quasiparticle approximations are analyzed with reference to the corresponding exact results.
The interplay between pairing and shape deformations at finite temperatures is studied with reference to an exactly soluble SU(2)\ifmmode\times\else\texttimes\fi{}SU(2) model which is able to suitably mock up both superconductivity and the effects of long-range residual forces. Thermodynamic and ground state phase transitions are studied, both in the thermodynamic limit and in the case of a finite number of particles. Several independent quasiparticle approximations are analyzed with reference to the corresponding exact results.
The thermal and the time-dependent versions of the Hartree-Fock-Bogoliubov approximation are used to describe the effects of the pairing interaction in a simple model. Similarities and differences between the two approaches are discussed. In the one-shell case analytic results are obtained which clearly show that as the degeneracy increases the exact energies tend to the limits provided by the approximate methods. A relation between the quasiparticle occupation probability and the seniority is established. In the two-shell case the time-dependent approach approximates all the families of exact states labeled by the shell seniority and gives information on the two different phase transitions present in the model. The finite-temperature method, instead, only approximates the exact energy for the lowest state of each family.
Abstract We analyze the thermodynamics and the critical behavior of the supersymmetric t - J model with long-range interactions. Using the transfer matrix formalism, we obtain a closed-form expression for the free energy per site both for a finite number of sites and in the thermodynamic limit. Our approach, which is different from the usual ones based on the asymptotic Bethe ansatz and generalized exclusion statistics, can in fact be applied to a large class of models whose spectrum is described in terms of supersymmetric Young tableaux and their associated Haldane motifs. In the simplest and most interesting case, we identify the five ground state phases of the model and derive the complete low-temperature asymptotic series of the free energy per site, the magnetization and charge densities, and their susceptibilities. We verify the model’s characteristic spin-charge separation at low temperatures, and show that it holds to all orders in the asymptotic expansion. Using the low-temperature asymptotic expansions of the free energy, we also analyze the critical behavior of the model in each of its ground state phases. While the standard phase is described by two independent conformal field theories (CFTs) with central charge c = 1 in correspondence with the spin and charge sectors, we find that the low-energy behavior of the and phases is that of a single c = 1 CFT. We show that the model exhibits an even richer behavior on the boundary between zero-temperature phases, where it can be non-critical but gapless, critical in the spin sector but not in the charge one, or critical with central charge c = 3/2.
Pairing and topological phases in cold atoms with long range interactions Chapter 1 Introduction The systematic study and classification of phase transitions has become a popular research area in physics over the past decades. The most familiar phenomena associated to the concept of phase transition are the ones giving rise to macroscopic changes in the properties of strongly correlated many-body systems due to thermal fluctuations. The most prominent examples of this kind of behaviour are the ferromagnetic-paramagnetic transition in lattice spin models and the liquid-gas transition. On the other hand, when dealing with quantum-mechanical systems changes of matter from one state to another can arise as the result of the variation of parameters other than temperature. In particular, when a given many-body quantum problem is studied at zero temperature, thereby suppressing thermal fluctuations, it is the interplay between kinetic energy and interactions that can cause the system to jump into different phases as the model parameters are varied. Phase transitions occurring at zero temperature are widely known as quantum phase transitions. In the framework of the physics of phase transitions, the role of dimensionality is a noticeable one. The major challenge in this perspective is the treatment of low dimensional systems, where standard mean-field treatments and ordinary perturbation theory are known to fail due to the enhancement of fluctuations. The goal of describing collective behaviours in reduced dimensionality is usually pursued in one-dimensional (1D) systems, which turn out to be simple enough to be studied in an effective way through both numerical and analytical techniques, while exhibiting a rich and deeply fascinating phenomenology. As a matter of fact, since particles confined to one dimension cannot avoid interactions with each other, the typical low energy excitations of the system are represented by collective density waves, which substitute nearly free quasiparticle excitations characterizing the behaviour of three-dimensional Fermi liquids in the description of the effects of the interplay between kinetic fluctuations and interactions. On the other hand, the tools employed in the activity of tackling 1D manybody problems are efficient and well developed. The analytical techniques aremainly based on the bosonization technique, which allows for the reformulation of strongly correlated 1D systems onto free bosonic theories, which can then be analyzed by means of standard path integral techniques. Meanwhile, the density matrix renormalization group (DMRG) algorithm represents the state-of-theart numerical machinery in the study of 1D physics, since it allows to efficiently extract the low energy properties of the model Hamiltonian and consequently characterize the corresponding phase diagram by means of the behaviour of properly chosen order parameters. As a final remark pointing out the relevance of the research effort in the direction of a better understanding of 1D many-body quantum systems, it is worth noticing that 1D systems do not represent only a playground for theorists, but have been experimentally realized by means of setups with cold atoms trapped in optical lattices, where the experimentalist is able to tune the model parameters by varying the characteristic features of the trapping laser beams and meanwhile has a comfortable access to the most relevant observables.
The dynamical effects of ground state correlations for excitation energies and transition strengths near the superfluid phase transition are studied in the soluble two level pairing model, in the context of the particle-particle self-consistent random phase approximation (SCRPA). Exact results are well reproduced across the transition region, beyond the collapse of the standard particle-particle random phase approximation. The effects of two-body correlation in the SCRPA are displayed explicitly.
We employ two different Lipkin-like, exactly solvable models so as to display features of the competition between different fermion–fermion quantum interactions (at finite temperatures). One of our two interactions mimics the pairing interaction responsible for superconductivity. The other interaction is a monopole one that resembles the so-called quadrupole one, much used in nuclear physics as a residual interaction. The pairing versus monopole effects here observed afford for some interesting insights into the intricacies of the quantum many body problem, in particular with regards to so-called quantum phase transitions (strictly, level crossings).
The effective-pseudospin model of the local-electron-pair superconductor, being equivalent to a hard-core charged Bose gas on a lattice, is analyzed for the case of anisotropic lattice structures and arbitrary electron concentration with the use of self-consistent random-phase approximation. The evolution of the ground-state phase diagrams, the low-temperature thermodynamic properties, and the superconducting critical temperature (Tc), as a function of the lattice anisotropy and electron concentration (n), are determined. We also discuss the superfluid properties within the framework of the Landau quasiparticle model. In the quasi-two-dimensional case and for low concentrations, the changeover from Tc\ensuremath{\sim}n2/3 to Tc\ensuremath{\sim}n is found to occur for ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$/${\mathit{J}}_{\mathrm{\ensuremath{\parallel}}}$\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}2}$, with ${\mathit{J}}_{\mathrm{\ensuremath{\parallel}}}$ and ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ being the intraplanar and interplanar pair hopping amplitude parameters, respectively. It turns out that the increasing lattice anisotropy (i.e., reduction of interplanar or interchain couplings) extends the range of stability of the homogeneous superconducting phase and suppresses the charge-density-wave state. The results for the ground-state superconducting order parameter are in good agreement with recent Monte Carlo and exact-diagonalization studies for the two-dimensional square lattice. The relationship between the pseudospin and bosonic descriptions of local-electron-pair superconductor in the dilute limit is also further analyzed.