We show how multi-level BCS Hamiltonians of finite systems in the strong pairing interaction regime can be accurately approximated using multi-dimensional shifted harmonic oscillator Hamiltonians. In the Shifted Harmonic Approximation (SHA), discrete quantum state variables are approximated as continuous ones and algebraic Hamiltonians are replaced by differential operators. Using the SHA, the results of the BCS theory, such as the gap equations, can be easily derived without the BCS approximation. In addition, the SHA preserves the symmetries associated with the BCS Hamiltonians. Lastly, for all interaction strengths, the SHA can be used to identify the most important basis states -- allowing accurate computation of low-lying eigenstates by diagonalizing BCS Hamiltonians in small subspaces of what may otherwise be vastly larger Hilbert spaces.
In this thesis, we will show how certain classes of quantum many-body Hamiltonians with
su21oplussu22oplusldotsoplussu2k spectrum generating algebras can be approximated by multi-dimensional shifted harmonic oscillator Hamiltonians. The dimensions of the Hilbert spaces of such Hamiltonians usually depend exponentially on
k. This can make obtaining eigenvalues by diagonalization computationally challenging. The Shifted Harmonic Approximation (SHA) developed here gives good predictions of properties such as ground state energies, excitation energies and the most probable states in the lowest eigenstates. This is achieved by solving only a system of
k equations and diagonalizing
ktimesk matrices. The SHA gives accurate approximations over wide domains of parameters and in many cases even across phase transitions.\n\nThe SHA is first illustrated using the Lipkin-Meshkov-Glick (LMG) model and the Canonical Josephson Hamiltonian (CJH) which have
su2 spectrum generating algebras. Next, we extend the technique to the non-compact
su1,1 algebra, using the five-dimensional quartic oscillator (5DQO) as an example. Finally, the SHA is applied to a
k-level Bardeen-Cooper-Shrieffer (BCS) pairing Hamiltonian with fixed particle number. The BCS model has a
su21oplussu22oplusldotsoplussu2k spectrum generating algebra.\n\nAn attractive feature of the SHA is that it also provides information to construct basis states which yield very accurate eigenvalues for low-lying states by diagonalizing Hamiltonians in small subspaces of huge Hilbert spaces. For Hamiltonians that involve a smaller number of operators, accurate eigenvalues can be obtained using another technique developed in this thesis: the generalized Rowe-Rosensteel-Kerman-Klein equations-of-motion method (RRKK). The RRKK is illustrated using the LMG and the 5DQO. In RRKK, solving unknowns in a set of
10times10 matrices typically gives estimates of the lowest few eigenvalues to an accuracy of at least eight significant figures. The RRKK involves optimization routines which require initial guesses of the matrix representations of the operators. In many cases, very good initial guesses can be obtained using the SHA. \n\nThe thesis concludes by exploring possible future developments of the SHA.
Exact eigenstates of the pairing-force Hamiltonian have been calculated for
2N particles in
2N equispaced, doubly degenerate single-particle levels for
N=4,8,12,and16. The states calculated are the ground state and the first two excited seniority-zero and seniority-two states. These states were calculated for values of the interaction strength that give pairing energies that are typical of nuclear systems. Numerical results are given for the energies and occupation probabilities of these states and are compared with the corresponding results of the Bardeen-Cooper-Schrieffer (BCS) theory. It is shown that the BCS theory is a worse approximation for the larger values of
N than it is for the smaller values, with errors as high as 140% (20%) for the excitation energies of the seniority-zero (-two) states. The methods used in the calculation of these states are outlined.
The dynamics of BCS (Bardeen-Cooper-Schrieffer) superconductors is fairly well understood due to the availability of a mean-field solution for the pairing Hamiltonian, a solution which gives the quantum state of a superconductor as a state of almost-free fermions interacting only with a condensate. As a result, transition probabilities may be computed, and expressed in terms of matrix elements of electron creation and annihilation operators between approximate eigenstates. These matrix elements are also called ``coherence factors.'' Mean-field theory is however not sufficient to describe all eigenstates of a superconductor, a deficiency which is hardly important in (or very close to) equilibrium, but one that becomes relevant in certain out-of-equilibrium situations. We report here on a computation of matrix elements (coherence factors) for the pairing Hamiltonian between any ``two-arc'' eigenstates in the thermodynamic limit.
The spectrum and eigenstates of Bardeen-Cooper-Schrieffer's reduced Hamiltonian in the theory of superconductivity are investigated in a mathematically rigorous manner. The Hamiltonian is defined first for a finite number of electrons contained in a finite volume. The interaction Hamiltonian is assumed to take a simple form. The Hamiltonian after the Bogoliubov transformation (transformed Hamiltonian) has a limit in the weak topology of Hilbert space when the volume tends to infinity with the number density of electrons kept constant. In the limit the interaction part vanishes and the free part gives a continuous excitation spectrum of Bogolons (transformed electrons). It should be noticed that the operator norm of the former does not tend to zero and has no strong limit as well as the total Hamilton. An equivalent boson which consists of two Bogolons with opposite momenta and spins is introduced. In the Fock space for the bosons the Hamiltonian has a strong limit for infinite volume. The limit operator has the same excitation spectrum of Bogolons as the free Hamiltonian and the existence of the lower limit of the spectrum is shown for some special class of the interactions. For large volume the limit operator is a good approximation to be transformed Hamiltonian. The ground state of the free part of the transformed Hamiltonian (i.e. the ground state of Bardeen-Cooper-Schrieffer's calculation) is not the exact eigenstate of the transformed total Hamiltonian and the difference does not vanish when the volume tends to infinity.
We present a derivation of a previously announced result for matrix elements between exact eigenstates of the pairing Hamiltonian. Our results, which generalize the well-known Bardeen–Cooper–Schrieffer (BCS) (Bardeen et al 1957 Phys. Rev. 108 1175; 1957 Phys. Rev. 106 162) expressions for what are known as ‘coherence factors’, are derived based on the Slavnov (1989 Theor. Math. Phys. 79 502) formula for overlaps between Bethe-ansatz states, thus making use of the known connection between the exact diagonalization of the BCS Hamiltonian, due to Richardson (1963 Phys. Lett. 3 277; 1964 Nucl. Phys. A 52 221), and the algebraic Bethe ansatz. The resulting formula has a compact form after a suitable parameterization of the energy plane. Although we apply our method here to the pairing Hamiltonian, it may be adjusted to study what is termed the ‘Sutherland limit’ (Sutherland 1995 Phys. Rev. Lett. 74 816) for exactly solvable models, namely where a macroscopic number of rapidities form a large string.