Berry phase, entanglement entropy, and algebraic properties of ground states of BCS and BEC superfluids

TL;DR

This paper analyzes the algebraic structures, Berry phases, and entanglement properties of BCS and BEC superfluid ground states using Bogoliubov transformations, revealing their quantized geometric phases and underlying algebraic classifications.

Contribution

It introduces a detailed algebraic and geometric analysis of superfluid ground states, highlighting their quantized Berry phases and classification via q-deformed Hopf algebras, which is novel.

Findings

Ground states are generalized squeezed coherent states.

Berry phases depend on particle number and are quantized.

Ground states exhibit entanglement and are classified by q-deformed algebras.

Abstract

By using Bogoliubov transformations to construct the ground states of fermionic Bardeen-Cooper-Schrieffer (BCS) superfluids and weakly-interacting Bose gases supporting Bose Einstein Condensation (BEC), their algebraic structures and implications can be analyzed in detail. Both ground states are generalized squeezed coherent states saturating a generalized Heisenberg uncertainty relation, and they acquire quantized Berry phases when the corresponding systems are transported along a closed path in their parameter spaces. While the Berry phase of the BCS ground state depends on the total particle number, the Berry phase of the BEC ground state depends only on the particles outside the BEC. The Berry phases are associated with magnetic monopoles in the parameter spaces and we found that the Dirac quantization condition is satisfied. Moreover, both ground states are entangled states of the fermion or boson quanta and we found the entanglement entropy quantifying the internal correlations. A fixed particle-number approach of fermionic superfluids does not saturate the generalized uncertainty relation, exhibits internal entanglement, and gives corresponding Berry phase. In addition, the algebraic structures of the ground states can be classified by the $q$-deformed Hopf algebra, $\bigoplus_{\mathbf{k}}h_{q_{\mathbf{k}}}(1)$ for bosons and $q$-deformed Hopf superalgebra $\bigoplus_{\mathbf{k}}h_{q_{\mathbf{k}}}(1|1)$ for fermions, respectively.