Coupled Fredkin and Motzkin chains from quantum six- and nineteen-vertex models

TL;DR

This paper introduces two-dimensional quantum vertex models extending Fredkin and Motzkin chains, revealing a phase transition in entanglement entropy and analyzing the spectral gap with variational methods.

Contribution

It generalizes Fredkin and Motzkin models to 2D with correlated interactions, uncovering entanglement phase transitions and providing bounds on the spectral gap.

Findings

Entanglement entropy exhibits a phase transition between area- and volume-law.

Critical point shows entanglement scaling as L log L.

Spectral gap upper bound scales as q^{-L^3/8}.

Abstract

We generalize the area-law violating models of Fredkin and Motzkin spin chains into two dimensions by building quantum six- and nineteen-vertex models with correlated interactions. The Hamiltonian is frustration free, and its projectors generate ergodic dynamics within the subspace of height configuration that are non negative. The ground state is a volume- and color-weighted superposition of classical bi-color vertex configurations with non-negative heights in the bulk and zero height on the boundary. The entanglement entropy between subsystems has a phase transition as the $q$-deformation parameter is tuned, which is shown to be robust in the presence of an external field acting on the color degree of freedom. The ground state undergoes a quantum phase transition between area- and volume-law entanglement phases with a critical point where entanglement entropy scales as a function $L\log L$ of the linear system size $L$. Intermediate power law scalings between $L\log L$ and $L^2$ can be achieved with an inhomogeneous deformation parameter that approaches 1 at different rates in the thermodynamic limit. For the $q>1$ phase, we construct a variational wave function that establishes an upper bound on the spectral gap that scales as $q^{-L^3/8}$.