Critical behavior of su(1|1) supersymmetric spin chains with long-range interactions

TL;DR

This paper introduces a class of su(1|1) supersymmetric spin chains with long-range interactions, analyzing their quantum criticality, entanglement properties, and phase transitions, revealing they exhibit conformal field theory behavior with central charge 1.

Contribution

The study extends understanding of su(1|1) spin chains by analyzing their critical behavior and entanglement, including specific models like elliptic and Haldane-Shastry chains, using algebraic fermionization techniques.

Findings

Models are quantum critical with central charge c=1 when chemical potential is in the critical interval.

Entanglement entropies scale logarithmically, characteristic of (1+1)-D CFT.

Quantum phase transitions occur at the ends of the critical interval.

Abstract

We introduce a general class of su$(1|1)$ supersymmetric spin chains with long-range interactions which includes as particular cases the su$(1|1)$ Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su$(1|1)$ permutation operator, and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low energy excitations and the low temperature behavior of the free energy, which coincides with that of a $(1+1)$-dimensional conformal field theory (CFT) with central charge $c=1$ when the chemical potential lies in the critical interval $(0,\mathcal E(\pi))$, $\mathcal E(p)$ being the dispersion relation. We also analyze the von Neumann and R\'enyi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson $(1+1)$-dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge $c=1$. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su$(1|1)$ elliptic chain.