Deforming the Fredkin spin chain away from its frustration-free point

TL;DR

This paper investigates the stability and phase transitions of the Fredkin spin chain when deformed away from its exact frustration-free point, revealing multiple phases and the fragility of its ground state.

Contribution

It introduces a generalized spin chain model interpolating between Fredkin and Heisenberg models, providing exact and numerical analysis of its phase diagram.

Findings

Fredkin ground state is unstable to infinitesimal antiferromagnetic frustration.

Multiple phase transitions and unexpected ordered phases are identified.

Existence of an 'anti-Fredkin' point with opposite spin configuration structure.

Abstract

The Fredkin model describes a spin-half chain segment subject to three-body, correlated-exchange interactions and twisted boundary conditions. The model is frustration-free, and its ground state wave function is known exactly. Its low-energy physics is that of a strong xy ferromagnet with gapless excitations and an unusually large dynamical exponent. We study a generalized spin chain model that includes the Fredkin model as a special tuning point and otherwise interpolates between the conventional ferromagnetic and antiferromagnetic quantum Heisenberg models. We solve for the low-lying states, using exact diagonalization and density-matrix renormalization group calculations, in order to track the properties of the system as it is tuned away from the Fredkin point; we also present exact analytical results that hold right at the Fredkin point. We identify a zero-temperature phase diagram with multiple transitions and unexpected ordered phases. The Fredkin ground state turns out to be particularly brittle, unstable to even infinitesimal antiferromagnetic frustration. We remark on the existence of an "anti-Fredkin" point at which all the contributing spin configurations have a spin structure exactly opposite to those in the Fredkin ground state.