Second Quantized Reduced Bloch Equations and the Exact Solutions for Pairing Hamiltonian

TL;DR

This paper develops hierarchy Bloch equations for reduced density operators in quantum statistical mechanics and applies them to solve for the pairing Hamiltonian, including the ground state of a superconductor.

Contribution

It introduces a new set of hierarchy Bloch equations for reduced density operators and demonstrates their application to exactly solve the pairing Hamiltonian.

Findings

Exact solutions for the pairing Hamiltonian

Reduced density matrices for the superconductor ground state

Utilization of symplectic group symmetry

Abstract

In this article, we present a set of hierarchy Bloch equations for the reduced density operators in either canonical or grand canonical ensembles in the occupation number representation. They provide a convenient tool for studying the equilibrium quantum statistical mechanics for some model systems. As an example of their applications, we solve the equations for the model system with a pairing Hamiltonian. With the aid of its symplectic group symmetry, we obtain the statistical reduced density matrices with different orders. As a special instance for the solutions, we also get the reduced density matrices of the ground state for a superconductor.