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International Symposium on Capture Gamma-Ray Spectroscopy and Related Topics (CGS)

THE TAMM-DANCOFF APPROXIMATION AS THE CONTRACTION LIMIT OF THE RICHARDSON-GAUDIN EQUATIONS

Stijn De Baerdemacker

Published Apr 28, 2013• 0 citations
Publisher

Abstract

A connection is made between the exact eigen states of the level-independent Bardeen-Cooper-Schrieffer (BCS) Hamiltonian and its Tamm-Dancoff Approximation (TDA). This is done by means of a deformation of the quasi-spin algebra, which connects the Bethe Ansatz states with a unique multi-phonon mode of the TDA. The procedure is illustrated with a model describing neutron superfluidity in 56 Fe.

DOI
10.1142/9789814383646_0091

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Background: The reduced, level-independent, Bardeen-Cooper-Schrieffer Hamiltonian is exactly diagonalizable by means of a Bethe ansatz wave function, provided the free variables in the ansatz are the solutions of the set of Richardson-Gaudin equations. On the one side, the Bethe ansatz is a simple product state of generalized pair operators. On the other hand, the Richardson-Gaudin equations are strongly coupled in a nonlinear way, making them prone to singularities. Unfortunately, it is nontrivial to give a clear physical interpretation to the Richardson-Gaudin variables because no physical operator is directly related to the individual variables.Purpose: The purpose of this paper is to shed more light on the singular behavior of the Richardson-Gaudin equations, and how this is related to the product wave structure of the Bethe ansatz.Method: A pseudodeformation of the quasispin algebra is introduced, leading towards a Heisenberg-Weyl algebra in the contraction limit of the deformation parameter. This enables an adiabatic connection of the exact Bethe ansatz eigenstates with pure bosonic multiphonon states. The physical interpretation of this approach is an adiabatic suppression of the Pauli exclusion principle.Results: The method is applied to a so-called ``picket-fence'' model for the BCS Hamiltonian, displaying a typical critical behavior in the Richardson-Gaudin variables. It was observed that the associated bosonic multiphonon states change collective nature at the critical interaction strengths of the Richardson-Gaudin equations.Conclusions: The Pauli exclusion principle is the main responsible for the critical behavior of the Richardson-Gaudin equations, which can be suppressed by means of a pseudodeformation of the quasispin algebra.

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A connection is made between the exact eigen states of the level-independent Bardeen-Cooper-Schrieffer (BCS) Hamiltonian and its Tamm-Dancoff Approximation (TDA). This is done by means of a deformation of the quasi-spin algebra, which connects the Bethe Ansatz states with a unique multi-phonon mode of the TDA. The procedure is illustrated with a model describing neutron superfluidity in 56 Fe.
The present contribution discusses a connection between the exact Bethe Ansatz eigenstates of the reduced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian and the multi-phonon states of the Tamm-Dancoff Approximation (TDA). The connection is made on the algebraic level, by means of a deformed quasi-spin algebra with a bosonic Heisenberg-Weyl algebra in the contraction limit of the deformation parameter. Each exact Bethe Ansatz eigenstate is mapped on a unique TDA multi-phonon state, shedding light on the physics behind the Bethe Ansatz structure of the exact wave function. The procedure is illustrated with a model describing neutron pairing in 56 Fe.
A connection is made between the exact eigen states of the BCS Hamiltonian and the predictions made by the Tamm-Dancoff Approximation. This connection is made by means of a parametrised algebra, which gives the exact quasi-spin algebra in one limit of the parameter and the Heisenberg-Weyl algebra in the other. Using this algebra to construct the Bethe Ansatz solution of the BCS Hamiltonian, we obtain parametrised Richardson-Gaudin equations, leading to the secular equation of the Tamm-Dancoff Approximation in the bosonic limit. An example is discussed in depth.
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.
Background: The reduced, level-independent, Bardeen-Cooper-Schrieffer Hamiltonian is exactly diagonalizable by means of a Bethe ansatz wave function, provided the free variables in the ansatz are the solutions of the set of Richardson-Gaudin equations. On the one side, the Bethe ansatz is a simple product state of generalized pair operators. On the other hand, the Richardson-Gaudin equations are strongly coupled in a nonlinear way, making them prone to singularities. Unfortunately, it is nontrivial to give a clear physical interpretation to the Richardson-Gaudin variables because no physical operator is directly related to the individual variables.Purpose: The purpose of this paper is to shed more light on the singular behavior of the Richardson-Gaudin equations, and how this is related to the product wave structure of the Bethe ansatz.Method: A pseudodeformation of the quasispin algebra is introduced, leading towards a Heisenberg-Weyl algebra in the contraction limit of the deformation parameter. This enables an adiabatic connection of the exact Bethe ansatz eigenstates with pure bosonic multiphonon states. The physical interpretation of this approach is an adiabatic suppression of the Pauli exclusion principle.Results: The method is applied to a so-called ``picket-fence'' model for the BCS Hamiltonian, displaying a typical critical behavior in the Richardson-Gaudin variables. It was observed that the associated bosonic multiphonon states change collective nature at the critical interaction strengths of the Richardson-Gaudin equations.Conclusions: The Pauli exclusion principle is the main responsible for the critical behavior of the Richardson-Gaudin equations, which can be suppressed by means of a pseudodeformation of the quasispin algebra.
The Dicke model is derived in the contraction limit of a pseudo-deformation of the quasispin algebra in the su (2)-based Richardson-Gaudin models. Likewise, the integrability of the Dicke model is established by constructing the full set of conserved charges, the form of the Bethe Ansatz state, and the associated Richardson-Gaudin equations. Thanks to the formulation in terms of the pseudo-deformation, the connection from the su (2)-based Richardson-Gaudin model towards the Dicke model can be performed adiabatically.