We study a general class of easy-axis spin models on a lattice of corner sharing even-sided polygons with all-to-all interactions within a plaquette. The low-energy description corresponds to a quantum dimer model on a dual lattice of even coordination number with a multidimer constraint. At an appropriately constructed frustration-free Rokhsar-Kivelson (RK) point, the ground-state wavefunction can be exactly mapped onto a classical vertex model on the dual lattice. When the dual lattice is bipartite, the vertex models are bonded and are self-dual under Wegner's duality, with the self-dual point corresponding to the RK point of the original multidimer model. We argue that the self-dual point is a critical point based on known exact solutions to some of the vertex models. When the dual lattice is nonbipartite, the vertex model is arrowed, and we use numerical methods to argue that there is no phase transition as a function of the vertex weights. Motivated by these wavefunction dualities, we construct two other distinct families of frustration-free Hamiltonians whose ground states can be mapped onto these vertex models. Many of these RK Hamiltonians provably host ${\mathbb{Z}}_{2}$ topologically ordered phases.
We study a general class of easy-axis spin models on a lattice of corner sharing even-sided polygons with all-to-all interactions within a plaquette. The low-energy description corresponds to a quantum dimer model on a dual lattice of even coordination number with a multidimer constraint. At an appropriately constructed frustration-free Rokhsar-Kivelson (RK) point, the ground-state wavefunction can be exactly mapped onto a classical vertex model on the dual lattice. When the dual lattice is bipartite, the vertex models are bonded and are self-dual under Wegner's duality, with the self-dual point corresponding to the RK point of the original multidimer model. We argue that the self-dual point is a critical point based on known exact solutions to some of the vertex models. When the dual lattice is nonbipartite, the vertex model is arrowed, and we use numerical methods to argue that there is no phase transition as a function of the vertex weights. Motivated by these wavefunction dualities, we construct two other distinct families of frustration-free Hamiltonians whose ground states can be mapped onto these vertex models. Many of these RK Hamiltonians provably host Z2 topologically ordered phases.
We construct a broad class of frustration-free quantum vertex models in 3+1 dimensions (3+1D) whose ground states are weighted superpositions of classical 3D vertex model configurations. Our results are illustrated for diamond, cubic, and BCC lattices, but hold in general for 3D lattices with even coordination number. The corresponding classical vertex models have a Z2 gauge constraint enriched with a Z2 global symmetry. We study the interplay between these symmetries by exploiting exact wave function dualities and effective field theories. We find an exact gapless point which by duality is related to the Rokhsar-Kivelson (RK) point of U(1) spin liquids. At this point, both the symmetry breaking and deconfinement order parameters exhibit long range order. The gapless point is additionally a self-dual point of a second duality that maps the Z2 deconfined and Z2 symmetry-broken phases to one another. For the BCC lattice vertex model, we find that gapless point is proximate to an unusual intermediate phase where symmetry breaking and deconfinement coexist.
We consider a family of generalized Rokhsar-Kivelson (RK) Hamiltonians, which are reverse-engineered to have an arbitrary edge-weighted superposition of dimer coverings as their exact ground state at the RK point. We focus on a quantum dimer model on the triangular lattice, with doubly periodic edge weights. For simplicity we consider a 2×1 periodic model in which all weights are set to one except for a tunable horizontal edge weight labeled α. We analytically show that the model exhibits a continuous quantum phase transition at α=3, changing from a topological Z2 quantum spin liquid (α<3) to a columnar ordered state (α>3). The dimer-dimer correlator decays exponentially on both sides of the transition with the correlation length ξ∝1/∣α−3∣ and as a power-law at criticality. The vison correlator exhibits an exponential decay in the spin liquid phase, but becomes a constant in the ordered phase, which we explain in terms of loop statistics of the double-dimer model. Using finite-size scaling of the vison correlator, we extract critical exponents consistent with the 2D Ising universality class. Additionally, we analytically show that the topological Rényi entropy of order ∞ (topological min-entropy) changes from log2 for the quantum spin liquid phase α<3, to 0 for the ordered phase α>3, thereby analytically confirming the topological nature of the phase transition.
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.
We study spin-1/2 Heisenberg antiferromagnets on a diamond-like-decorated square lattice. The diamond-like-decorated square lattice is a lattice in which the bonds in a square lattice are replaced with diamond units. The diamond unit has two types of antiferromagnetic exchange interactions, and the ratio λ of the diagonal bond strength to that of the other four edges controls the frustration strength. For 0.974 < λ < 2, the present system has a nontrivial macroscopic degeneracy, which is called the macroscopically degenerated tetramer-dimer (MDTD) states. The MDTD states are identical to the Hilbert space of the Rokhsar–Kivelson (RK) quantum dimer model (QDM). By introducing further neighbor couplings in the MDTD states, we calculate the second-order effective Hamiltonian, which is exactly the same as the square-lattice QDM with a finite hopping amplitude t and dimer-dimer interaction v. Furthermore, we calculate v/|t| as a function of the ratio λ in the Heisenberg model and examine which phases of the square-lattice QDM appear in our obtained states. Our obtained QDM has a region where λ exhibits a finite hopping amplitude (|t| > 0) and repulsive interaction between dimers (v > 0). This suggests the possibility of realizing the resonating valence bond (RVB) state because the RVB state is obtained at v = |t|, which is known as the RK point.
Quantum phases of two-dimensional Z2 gauge theory coupled to spinless fermion matter
We investigate the rich quantum phase diagram of Wegner's theory of discrete Ising gauge fields interacting with U(1) symmetric single-component fermion matter hopping on a two-dimensional square lattice. In particular limits the model reduces to (i) pure Z2 even and odd gauge theories, (ii) free fermions in a static background of deconfined Z2 gauge fields, (iii) the kinetic Rokhsar-Kivelson quantum dimer model at a generic dimer filling. We develop a local transformation that maps the lattice gauge theory onto a model of Z2 gauge-invariant spin 1/2 degrees of freedom. Using the mapping, we perform numerical density matrix renormalization group calculations that corroborate our understanding of the limits identified above. Moreover, in the absence of the magnetic plaquette term, we reveal signatures of topologically ordered Dirac semimetal and staggered Mott insulator phases at half-filling. At strong coupling, the lattice gauge theory displays fracton phenomenology with isolated fermions being completely frozen and dimers exhibiting restricted mobility. In that limit, we predict that in the ground state dimers form compact clusters, whose hopping is suppressed exponentially in their size. We determine the band structure of the smallest clusters numerically using exact diagonalization.
Recent work [Kundu, Damle, Phys. Rev. X 15, 011018 (2025)] has demonstrated that a class of S=1 kagome lattice antiferromagnets with competing anisotropies can exhibit unusual low-temperature behavior that is best described in terms of the statistical mechanics of an interesting interpolation between the fully-packed dimer model and the fully-packed loop model on the bipartite honeycomb lattice. On this and other bipartite lattices in two and three dimension, such fully-packed dimer-loop models generically have two distinct power-law correlated Coulomb phases, both of which admit a low-energy description in terms of a fluctuating divergence-free polarization vector field [Phys. Rev. X 15, 011018 (2025); arXiv:2510.11607]. The transition between these phases corresponds to a confinement transition for the Z2 part of the corresponding U(1) flux [Phys. Rev. X 15, 011018 (2025); arXiv:2510.11607]. Here we demonstrate that a variant of such a topological confinement transition can also exist in dimer-loop models on non-bipartite lattices such as the kagome and triangular lattices although the low-energy description does not feature any conserved U(1) flux. In these non-bipartite dimer-loop models, both phases are short-range correlated liquids, and the topological distinction between them is in terms of a Z2-valued winding number instead of the flux of a divergence-free polarization field. In spite of this key difference, this transition appears to be in the same Ising universality class as the analogous transition of the bipartite dimer-loop models. These kagome and triangular lattice dimer-loop models correspond respectively to the low-energy limit of a class of S=1 antiferromagnets with competing anisotropies on the ruby lattice comprising a corner-sharing arrangement of fully-frustrated squares and a lattice of corner-sharing fully-frustrated hexagons.