Lowest energy states of an $O(N)$ fermionic chain

TL;DR

This paper characterizes the ground states of finite-size fermionic chains with $O(N)$ symmetry, revealing their structure, degeneracies, and quantum numbers, including extensions to long-range interactions and translation invariance.

Contribution

It provides a rigorous description of the ground state multiplets and quantum numbers for $O(N)$ symmetric fermionic chains with various boundary conditions and interactions.

Findings

Ground state in invariant sector is a single rank-$m$ antisymmetric multiplet.

Particle-hole quantum number relates to the parity of $m$ for even-length chains.

Odd-length chains exhibit twofold degeneracy due to particle-hole symmetry.

Abstract

A quite general finite-size chain of fermions with $N$ internal degrees of freedom (flavors) and $O(N)$ symmetry is considered. In the case of the free boundary condition, we prove that the ground state in the invariant sector having exactly $m$ flavors with an odd particle number is represented by a single rank-$m$ antisymmetric multiplet. For the even-length chains, its particle-hole quantum number (if it's a good one) is given by the parity of the $m$. For the odd-length chains, the particle-hole symmetry leads to the twofold degeneracy among the conjugate multiplets. Similar statements are proven for the $O(N)$ mixed-spin chains in antisymmetric representations. The results are extended to the long-range interacting fermions and (partially) to the translation invariant chains.