Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Acta Physica Polonica A

Thermodynamics of the Two-Level Richardson Model

Anna Ciechan, Karol Izydor Wysokiński

Published Apr 1, 2007• 1 citation
Publisher

Abstract

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.

DOI
10.12693/aphyspola.111.467

Related papers

Finite Size Corrections in the Two-Level BCS Model

Anna Ciechan, Karol Izydor Wysokiński

Superconducting properties of small metallic grains modelled by highly degenerate two-level spectrum have been studied.We have solved numerically Richardson's exact equations for the system of 2N electrons in two levels.Characterising the size of the grain by the level degeneracy we study the finite size corrections to the thermodynamic limit of the ground and lowest excited state energy.The interparticle distance ∝ N -1/3 seems to be the expansion parameter.The obtained results have been compared with those of other authors.

Acta Physica Polonica A•Apr 1, 2006•1 citation

Pairing and thermodynamics properties of finite-systems with fixed number of particles

D. Gambacurta, Denis Lacroix A

A canonical description of the thermodynamical pairing properties of small systems is achieved by using the Variation After Projection approach at finite temperature. The minimization of the free energy is made by a direct evaluation of the energy and full diagonalization of the entropy. We use the Richardson – pairing model whose exact solution allows to study the reliability of different approaches. We show that the Projection After Variation approach, that is usually performed at zero temperature with rather good success, provides a quite poor description at finite temperature. On the contrary, the Variation After Projection applied at finite temperature provides a perfect reproduction of the exact canonical properties of odd or even systems from very low to high temperature.

EPJ Web of Conferences•Jan 1, 2012•0 citations

Exactly Solvable Many-Boson Model

R. W. Richardson

A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.

Journal of Mathematical Physics•Sep 1, 1968•82 citations

The many levels pairing Hamiltonian for two pairs

M. B. Barbaro, R. Cenni, Alfredo Molinari +1 more

We address the problem of two pairs of fermions living on an arbitrary number of single particle levels of a potential well (mean field) and interacting through a pairing force. The associated solutions of the Richardson’s equations are classified in terms of a number vl, which reduces to the seniority v in the limit of large values of the pairing strength G and yields the number of pairs not developing a collective behaviour, their energy remaining finite in the G → ∞ limit. We express analytically, through the moments of the single particle levels distribution, the collective mode energy and the two critical values G + cr and G − cr of the coupling which can exist on a single particle level with no pair degeneracy. Notably G + cr and G − cr merge when the number of single particle levels goes to infinity, where they coincide with the Gcr (when it exists) of a one pair system, not envisioned by the Richardson theory. In correspondence of Gcr the system undergoes a transition from a mean field to a pairing dominated regime. We finally explore the behaviour of the excitation energies, wave functions and pair transfer amplitudes finding out that the former, for G> G − cr, come close to the BCS predictions, whereas the latter display a divergence at Gcr, signaling the onset of a long range off-diagonal order in the system. PACS numbers: 21.60.-n, 21.30 Fe, 24.10.Cn 1 1

•Jan 1, 2004•2 citations

A new method in the many-body problem

Igor Krasovsky, V. I. Peresada

A new method is proposed for exact calculation of the energy levels of Hamiltonians H and the quantities Sp(f(H+W)-f(H)), where W is a finite-dimensional perturbation, in the case when H and H+W may be represented by a certain wide class of (2n+1)-diagonal matrices. As an example, two problems for a linear chain of spins with nearest- and next-nearest neighbour interactions described by the Heisenberg Hamiltonian are considered.

Journal of Physics A Mathematical and General•Mar 23, 1995•4 citations

Exact thermodynamics and Luttinger liquid properties of the integrable t-J model

Guido Jüttner, Andreas Klümper, Junji Suzuki +1 more

A Trotter-Suzuki mapping is used to calculate the finite-temperature properties of the one-dimensional supersymmetric t − J model. This approach allows for the exact calculation of various thermodynamical properties by means of the quantum transfer matrix (QTM). The free energy and other interesting quantities are obtained such as the specific heat and compressibility. For the largest eigenvalue of the QTM leading to the free energy a set of just two non-linear integral equations is presented. These equations are studied analytically and numerically for different particle densities and temperatures. The structure of the specific heat is discussed in terms of the elementary charge as well as spin excitations. Special emphasis is placed on the study of the low-temperature behavior confirming scaling predictions by conformal field theory and Luttinger liquid theory. To our knowledge this is the first complete investigation of a strongly correlated electron system on a lattice at finite temperature.

•Jan 1, 1997•35 citations
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.
Superconducting properties of small metallic grains modelled by highly degenerate two-level spectrum have been studied.We have solved numerically Richardson's exact equations for the system of 2N electrons in two levels.Characterising the size of the grain by the level degeneracy we study the finite size corrections to the thermodynamic limit of the ground and lowest excited state energy.The interparticle distance ∝ N -1/3 seems to be the expansion parameter.The obtained results have been compared with those of other authors.
A canonical description of the thermodynamical pairing properties of small systems is achieved by using the Variation After Projection approach at finite temperature. The minimization of the free energy is made by a direct evaluation of the energy and full diagonalization of the entropy. We use the Richardson – pairing model whose exact solution allows to study the reliability of different approaches. We show that the Projection After Variation approach, that is usually performed at zero temperature with rather good success, provides a quite poor description at finite temperature. On the contrary, the Variation After Projection applied at finite temperature provides a perfect reproduction of the exact canonical properties of odd or even systems from very low to high temperature.
A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.
We address the problem of two pairs of fermions living on an arbitrary number of single particle levels of a potential well (mean field) and interacting through a pairing force. The associated solutions of the Richardson’s equations are classified in terms of a number vl, which reduces to the seniority v in the limit of large values of the pairing strength G and yields the number of pairs not developing a collective behaviour, their energy remaining finite in the G → ∞ limit. We express analytically, through the moments of the single particle levels distribution, the collective mode energy and the two critical values G + cr and G − cr of the coupling which can exist on a single particle level with no pair degeneracy. Notably G + cr and G − cr merge when the number of single particle levels goes to infinity, where they coincide with the Gcr (when it exists) of a one pair system, not envisioned by the Richardson theory. In correspondence of Gcr the system undergoes a transition from a mean field to a pairing dominated regime. We finally explore the behaviour of the excitation energies, wave functions and pair transfer amplitudes finding out that the former, for G> G − cr, come close to the BCS predictions, whereas the latter display a divergence at Gcr, signaling the onset of a long range off-diagonal order in the system. PACS numbers: 21.60.-n, 21.30 Fe, 24.10.Cn 1 1
A Trotter-Suzuki mapping is used to calculate the finite-temperature properties of the one-dimensional supersymmetric t − J model. This approach allows for the exact calculation of various thermodynamical properties by means of the quantum transfer matrix (QTM). The free energy and other interesting quantities are obtained such as the specific heat and compressibility. For the largest eigenvalue of the QTM leading to the free energy a set of just two non-linear integral equations is presented. These equations are studied analytically and numerically for different particle densities and temperatures. The structure of the specific heat is discussed in terms of the elementary charge as well as spin excitations. Special emphasis is placed on the study of the low-temperature behavior confirming scaling predictions by conformal field theory and Luttinger liquid theory. To our knowledge this is the first complete investigation of a strongly correlated electron system on a lattice at finite temperature.