Dedicated to Professor Józef Spa lek on the occasion of his 60th birthdayThe two-level version of the Richardson model presents a unique possibility to calculate numerically exactly thermodynamic properties of the system it describes.The point is that all energies and the degeneracies of the many-body system can be easily calculated.The energies are given by the eigenvalues of the small (of order of N × N , where 2N is a number of electrons in the system) tridiagonal matrices.Here we numerically obtain a complete spectrum of the interacting two-level model and calculate the specific heat and the pairing energy of the small system at finite temperatures.
The ground-state energy and energy gap to the first excited state are calculated for the attractive Hubbard model in one dimension using both the Bethe ansatz equations and the variational BCS wave function. Comparisons are provided as a function of coupling strength and electron density. While the ground-state energies are always in very good agreement, the BCS energy gap is sometimes incorrect by an order of magnitude, particularly at half-filling. Finite-size effects are also briefly discussed for cases where an exact solution in the thermodynamic limit is not possible. In general, the BCS result for the energy gap is poor compared to the exact result.
The exact ground state of the reduced BCS Hamiltonian is investigated numerically for large system sizes and compared with the BCS ansatz. A ``canonical'' order parameter is found to be equal to the largest eigenvalue of Yang's reduced density matrix in the thermodynamic limit. Moreover, the limiting values of the exact analysis agree with those obtained for the BCS ground state. Exact results for the ground-state energy, level occupations, and a pseudospin-pseudospin correlation function are also found to converge to the BCS values already for relatively small system sizes. However, discrepancies persist for a pair-pair correlation function, for interlevel correlations of occupancies and for the fidelity susceptibility, even for large system sizes where these quantities have visibly converged to well-defined limits. Our results indicate that there exist nonperturbative corrections to the BCS predictions in the thermodynamic limit.
The two-dimensional Hubbard model is studied for small values of the interaction strength (
U of the order of the hopping amplitude
t), using a variational ansatz well suited for this regime. The wave function, a refined Gutzwiller ansatz, has a BCS mean-field state with
d-wave symmetry as its reference state. Superconducting order is found for densities
n<1 (but not for
n=1). This resolves a discrepancy between results obtained with the functional renormalization group, which do predict superconducting order for small values of
U, and numerical simulations, which did not find superconductivity for $U\ensuremath{\lesssim}4t$. Both the gap parameter and the order parameter have a domelike shape as a function of
n with a maximum for $n\ensuremath{\approx}0.8$. Expectation values for the energy, the particle number, and the superconducting order parameter are calculated using a linked-cluster expansion up to second order in
U. In this way large systems (millions of sites) can be readily treated and well converged results are obtained. A big size is indeed required to see that the gap becomes very small at half filling and probably tends to zero in the thermodynamic limit, whereas away from half filling a finite asymptotic limit is reached. For a lattice of a given size the order parameter becomes finite only above a minimal coupling strength
Uc. This threshold value decreases steadily with increasing system size, which indicates that superconductivity exists for arbitrarily small
U for an infinite system. For moderately large systems the size dependence is quite irregular, due to variations in level spacings at the Fermi energy. The fluctuations die out if the gap parameter spans several level spacings.
The exact solution of the reduced pairing Hamiltonian is compared to the BCS mean field approximation for low particle numbers in finite systems. A combination of two different methods for solving the Richardson equations is presented. One of the methods originally intended to be solved analytically is instead adapted to a numerical scheme. The comparison includes the pairing gap parameter, as approximated by the three-point formula in the exact solution, and a microcanonical investigation of systems with levels of varying degeneracies. The gaps as calculated within the two approaches is found to deviate more and more from each other with increasing pairing strength. However testing the three-point-formula with the BCS approximation suggests that this might be due to the formula itself. Furthermore the BCS approximation, when re-evaluating the gap for each excitation, does a very good job at reproducing the energy spectra as obtained for the Richardson solution of the highly excited states where the pairing effects are small. However for the low lying excitations the assessment is not so good. When keeping the gap constant for all excitations the excitation spectra of the BCS approximation broaden a lot more than that of the exact method. This procedure gives unreasonable results for the systems and low particle numbers considered.
We consider an exactly solvable inhomogeneous Dicke model which describes an interaction between a disordered ensemble of two-level systems with single mode boson field. The existing method for evaluation of Richardson–Gaudin equations in the thermodynamical limit is extended to the case of Bethe equations in Dicke model. Using this extension, we present expressions both for the ground state and lowest excited states energies as well as leading-order finite-size corrections to these quantities for an arbitrary distribution of individual spin energies. We then evaluate these quantities for an equally-spaced distribution (constant density of states). In particular, we study evolution of the spectral gap and other related quantities. We also reveal regions on the phase diagram, where finite-size corrections are of particular importance.