A web of exact mappings from RK models to spin chains

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.

💡 Research Summary

The authors investigate a broad class of Rokhsar‑Kivelson (RK) dimer and spin‑ice models that realize U(1) lattice gauge theories, focusing on quasi‑one‑dimensional (quasi‑1D) geometries where the low‑energy degrees of freedom are closed electric‑field lines (“strings”). By representing a string as a continuous path of links with σᶻ = −1, they show that gauge constraints force these strings to be closed and to propagate from left to right without back‑tracking. This string picture makes the underlying U(1) structure explicit and enables a systematic mapping of the original two‑dimensional RK Hamiltonians onto three distinct one‑dimensional quantum chains:

1. Spin‑½ XXZ chain – the simplest mapping where string hopping corresponds to spin‑flip terms.
2. Spin‑1 chain with global U(1) symmetry – here the average magnetization (⟨Sᶻ⟩) plays the role of a “string magnetization”, and the chain’s stiffness varies continuously with this quantity.
3. Kinetically constrained fermion (tile) chain – configurations are encoded as tilings of a rectangular strip; the kinetic constraints reflect the hard‑core repulsion of strings.

A key insight is that the twist φ of the boundary conditions in the 1D chains maps exactly onto the transverse momentum k⊥ of the closed string, while the Drude weight D of the chain equals the inverse of the string’s effective mass (D = 1/m). Thus transport properties of the 1D model provide direct information about the dynamics of the string in the original 2D gauge theory.

Using density‑matrix renormalization group (DMRG) simulations, the authors explore the phase diagrams for v = V/t < 1 (the regime where kinetic and potential terms compete). In all but one model they find two generic phases:

• A gapped solid phase where strings are locked into a regular pattern.
• An XY (Luttinger‑liquid) phase with quasi‑long‑range order, interpreted as the one‑dimensional descendant of the resonating‑plaquette phase of the full 2D RK model.

The spin‑1 chain exhibits a remarkable feature at the RK point (v = 1): an entire Bloch‑sphere‑like manifold of exact zero‑energy ground states, despite the absence of any microscopic SU(2) symmetry. Consequently, the low‑energy stiffness of the gapless mode changes continuously with the chain’s magnetization, implying that the string’s effective mass can be tuned by adjusting its “magnetization”.

The most novel results arise for the tile chain, which corresponds to a compactified six‑vertex model. Here destructive interference of plaquette resonances generates an extensive set of local conserved quantities, fragmenting the Hilbert space into exponentially many disconnected sectors. The largest sector, which hosts the physical ground‑state manifold, occupies an exponentially small fraction of the full Hilbert space. Moreover, the authors identify a Landau‑forbidden line of quantum critical points separating an antiferromagnetic (AFM) phase, a resonating‑plaquette (RP) phase, and the XY phase. Bosonization and numerical evidence suggest that this critical line is a stable, non‑Landau criticality characterized by a Luttinger‑liquid with unconventional exponents, a phenomenon not present in the full 2D RK models.

Overall, the paper demonstrates that “string‑to‑chain” mappings provide a powerful analytical and numerical toolbox for probing constrained gauge theories. By reducing the problem to well‑studied 1D spin chains, the authors uncover rich physics: multi‑string interactions, emergent continuous degeneracies, exotic criticality beyond Landau’s paradigm, and Hilbert‑space fragmentation due to extensive local integrals of motion. These insights open avenues for experimental realization in platforms such as Rydberg atom arrays, superconducting qubit lattices, or other quantum simulators where U(1) gauge constraints and tunable boundary twists can be engineered. Future work may extend these mappings to non‑bipartite lattices, higher‑dimensional compactifications, or coupling to external fields, further exploiting the deep connection between lattice gauge theories and low‑dimensional quantum spin systems.