A web of exact mappings from RK models to spin chains

TL;DR

This paper maps RK dimer and spin ice models to various quantum chains, revealing new phases, critical points, and conserved quantities, thus providing insights into lattice gauge theories and their low-energy physics.

Contribution

It introduces novel mappings of RK models to spin chains and fermion chains, uncovering new phases, critical points, and conserved quantities in quasi-one-dimensional gauge theories.

Findings

Mapped RK models to three quantum chains: XXZ, spin-1, and fermion chains.

Discovered a continuous degenerate ground state at the RK point in the spin-1 chain.

Identified a stable Landau-forbidden gapless critical point using bosonization and numerics.

Abstract

We study Rokhsar-Kivelson (RK) dimer and spin ice models realizing $U(1)$-lattice gauge theories in a wide class of quasi-one-dimensional settings, which define a setup for the study of few quantum strings (closed electric field lines) interacting with themselves and each other. We discover a large collection of mappings of these models onto three quantum chains: the spin-1/2 XXZ chain, a spin-1 chain, and a kinetically constrained fermion chain whose configurations are best described in terms of tilings of a rectangular strip. We show that the twist of boundary conditions in the chains maps onto the transverse momentum of the electric field string, and their Drude weight to the inverse of the string mass per unit length. We numerically determine the phase diagrams for these spin chains, employing DMRG simulations and find global similarities but also many interesting new features in comparison to the full 2D problems. For example, the spin-1 chain we obtain features a continuous family of degenerate ground states at its RK point analogous to a Bloch sphere, but without an underlying microscopic global $SU(2)$ symmetry. We also argue for the existence of a (stable) Landau-forbidden gapless critical point away from the RK point in one of the models we study using bosonization and numerics. This is surprising given that the full 2D problem is generically gapped away from the RK point. The same model also displays extensively many local conserved quantities which fragment the Hilbert space, arising as a consequence of destructive resonances between the electric field lines. Our findings highlight spin-chain mappings as a potent technique for the exploration of unusual dynamics, exotic criticality, and low-energy physics in lattice gauge theories.