# Free fermions in disguise

## TL;DR

This paper constructs free-fermion operators for a complex quantum chain with local four-fermion interactions, revealing its spectrum, symmetries, and phase transitions, including gapless and Ising-like behavior.

## Contribution

It introduces a method to solve a four-fermion Hamiltonian using non-local free-fermion operators, uncovering its spectrum and symmetry structure.

## Key findings

- The system is gapless with dynamical critical exponent z=3/2.
- Staggered couplings lead to an Ising transition.
- The model exhibits supersymmetry and large degeneracies.

## Abstract

I solve a quantum chain whose Hamiltonian is comprised solely of local four-fermi operators by constructing free-fermion raising and lowering operators. The free-fermion operators are both non-local and highly non-linear in the local fermions. This construction yields the complete spectrum of the Hamiltonian and an associated classical transfer matrix. The spatially uniform system is gapless with dynamical critical exponent z=3/2, while staggering the couplings gives a more conventional free-fermion model with an Ising transition. The Hamiltonian is equivalent to that of a spin-1/2 chain with next-nearest-neighbour interactions, and has a supersymmetry generated by a sum of fermion trilinears. The supercharges are part of a large non-abelian symmetry algebra that results in exponentially large degeneracies. The model is integrable for either open or periodic boundary conditions but the free-fermion construction only works for the former, while for the latter the extended symmetry is broken and the degeneracies split.