arXiv:hep-th/0111114

Integrability and Conformal Symmetry in the BCS model

G. Sierra

Published Nov 13, 2001• 3 citations

Abstract

The exactly solvable BCS Hamiltonian of superconductivity is considered from several viewpoints: Richardson's ansatz, conformal field theory, integrable inhomogenous vertex models and Chern-Simons theory.

DOI: 10.48550/arxiv.hep-th/0111114

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arXiv:hep-th/0111114

Integrability and Conformal Symmetry in the BCS model

G. Sierra

Published Nov 13, 2001• 3 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.hep-th/0111114

Related papers

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J. Dukelsky, B. Errea, Sergio Lerma-Hernández +1 more

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Conformal Field Theory and the Exact Solution of the BCS Hamiltonian

Sierra, G

We propose a connection between conformal field theory (CFT) and the exact solution and integrability of the reduced BCS model of superconductivity. The relevant CFT is given by the $SU(2)_k$ -WZW model in the singular limit when the level k goes to -2. This theory has to be perturbed by an operator proportional to the inverse of the BCS coupling constant. Using the free field realization of this perturbed Wess-Zumino-Witten model, we derive the exact Richardson's wave function and the integrals of motion of the reduced BCS model in the saddle point approximation. The construction is reminiscent of the CFT approach to the Fractional Quantum Hall effect.We propose a connection between conformal field theory (CFT) and the exact solution and integrability of the reduced BCS model of superconductivity. The relevant CFT is given by the $SU(2)_k$-WZW model in the singular limit when the level k goes to -2. This theory has to be perturbed by an operator proportional to the inverse of the BCS coupling constant. Using the free field realization of this perturbed Wess-Zumino-Witten model, we derive the exact Richardson's wave function and the integrals of motion of the reduced BCS model in the saddle point approximation. The construction is reminiscent of the CFT approach to the Fractional Quantum Hall effect.

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In many exactly solvable models of quantum statistical mechanics, an interaction Hamiltonian is equal to an integral of a product of operators of annihilation and creation and a potential divided by a volume of a system raised to a certain power. For example, the interaction Hamiltonian of the BCS model of superconductivity has the form ${H_{I,M}} = \frac{g}{{2V}}\iint {\iint {\Phi ({x_1} - {x_2},{{x'}_1} - {{x'}_2}){\psi ^ + }({x_2})\psi ({{x'}_1})\psi ({{x'}_2}})d{x_1}d{x_2}d{{x'}_1}d{{x'}_2},}$ (1) where integration is carried out over the whole three-dimensional Euclidean space ℝ3, V is the volume of ℝ3, ψ+ and ψ- are operators of creation and annihilation of Fermi particles, and Φ is a potential that satisfies the conditions $\Phi ({x_1} - {x_2},{x'_1} - {x'_2}) = \Phi (\overline {{x_1} - {x_2},{{x'}_1} - {{x'}_2}} ) = \Phi ({x'_2} - {x'_1},{x_2} - {x_2}),\Phi ({x_2} - {x_1},{x'_1} - {x'_2}) = \Phi ({x_2} - {x_1},{x'_2} - {x'_1})$ (2) and is a test function with respect to x 1-x 2 and ${x'_1} - {x'_2}$ . The dependence on spin is omitted here.In many exactly solvable models of quantum statistical mechanics, an interaction Hamiltonian is equal to an integral of a product of operators of annihilation and creation and a potential divided by a volume of a system raised to a certain power. For example, the interaction Hamiltonian of the BCS model of superconductivity has the form $$ {H_{I,M}} = \frac{g}{{2V}}\iint {\iint {\Phi ({x_1} - {x_2},{{x'}_1} - {{x'}_2}){\psi ^ + }({x_2})\psi ({{x'}_1})\psi ({{x'}_2}})d{x_1}d{x_2}d{{x'}_1}d{{x'}_2},} $$ (1) where integration is carried out over the whole three-dimensional Euclidean space ℝ3, V is the volume of ℝ3, ψ+ and ψ- are operators of creation and annihilation of Fermi particles, and Φ is a potential that satisfies the conditions $$ \Phi ({x_1} - {x_2},{x'_1} - {x'_2}) = \Phi (\overline {{x_1} - {x_2},{{x'}_1} - {{x'}_2}} ) = \Phi ({x'_2} - {x'_1},{x_2} - {x_2}),\Phi ({x_2} - {x_1},{x'_1} - {x'_2}) = \Phi ({x_2} - {x_1},{x'_2} - {x'_1}) $$ (2) and is a test function with respect to x 1-x 2 and $ {x'_1} - {x'_2} $. The dependence on spin is omitted here.

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First, we reformulate the BCS-Bogoliubov theory of superconductivity from the viewpoint of linear algebra. We define the BCS Hamiltonian on $\mathbb{C}^{2^{2M}}$ , where $M$ is a positive integer. We discuss selfadjointness and symmetry of the BCS Hamiltonian as well as spontaneous symmetry breaking. Beginning with the gap equation, we give the well-known expression for the BCS state and find the existence of an energy gap. We also show that the BCS state has a lower energy than the normal state. Second, we introduce a new superconducting state explicitly and show from the viewpoint of linear algebra that this new state has a lower energy than the BCS state. Third, beginning with our new gap equation, we show from the viewpoint of linear algebra that we arrive at the results similar to those in the BCS-Bogoliubov theory.First, we reformulate the BCS-Bogoliubov theory of superconductivity from the viewpoint of linear algebra. We define the BCS Hamiltonian on $\mathbb{C}^{2^{2M}}$, where $M$ is a positive integer. We discuss selfadjointness and symmetry of the BCS Hamiltonian as well as spontaneous symmetry breaking. Beginning with the gap equation, we give the well-known expression for the BCS state and find the existence of an energy gap. We also show that the BCS state has a lower energy than the normal state. Second, we introduce a new superconducting state explicitly and show from the viewpoint of linear algebra that this new state has a lower energy than the BCS state. Third, beginning with our new gap equation, we show from the viewpoint of linear algebra that we arrive at the results similar to those in the BCS-Bogoliubov theory.

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We propose a connection between conformal field theory (CFT) and the exact solution and integrability of the reduced BCS model of superconductivity. The relevant CFT is given by the

SU(2)_k

-WZW model in the singular limit when the level k goes to -2. This theory has to be perturbed by an operator proportional to the inverse of the BCS coupling constant. Using the free field realization of this perturbed Wess-Zumino-Witten model, we derive the exact Richardson's wave function and the integrals of motion of the reduced BCS model in the saddle point approximation. The construction is reminiscent of the CFT approach to the Fractional Quantum Hall effect.

{H_{I,M}} = \frac{g}{{2V}}\iint {\iint {\Phi ({x_1} - {x_2},{{x'}_1} - {{x'}_2}){\psi ^ + }({x_2})\psi ({{x'}_1})\psi ({{x'}_2}})d{x_1}d{x_2}d{{x'}_1}d{{x'}_2},}

(1) where integration is carried out over the whole three-dimensional Euclidean space ℝ3, V is the volume of ℝ3, ψ+ and ψ- are operators of creation and annihilation of Fermi particles, and Φ is a potential that satisfies the conditions

\Phi ({x_1} - {x_2},{x'_1} - {x'_2}) = \Phi (\overline {{x_1} - {x_2},{{x'}_1} - {{x'}_2}} ) = \Phi ({x'_2} - {x'_1},{x_2} - {x_2}),\Phi ({x_2} - {x_1},{x'_1} - {x'_2}) = \Phi ({x_2} - {x_1},{x'_2} - {x'_1})

(2) and is a test function with respect to x 1-x 2 and

{x'_1} - {x'_2}

. The dependence on spin is omitted here.

First, we reformulate the BCS-Bogoliubov theory of superconductivity from the viewpoint of linear algebra. We define the BCS Hamiltonian on

\mathbb{C}^{2^{2M}}

, where

M

is a positive integer. We discuss selfadjointness and symmetry of the BCS Hamiltonian as well as spontaneous symmetry breaking. Beginning with the gap equation, we give the well-known expression for the BCS state and find the existence of an energy gap. We also show that the BCS state has a lower energy than the normal state. Second, we introduce a new superconducting state explicitly and show from the viewpoint of linear algebra that this new state has a lower energy than the BCS state. Third, beginning with our new gap equation, we show from the viewpoint of linear algebra that we arrive at the results similar to those in the BCS-Bogoliubov theory.