Monopole-controlled phase structure of exactly solvable 3D quantum vertex models away from the Rokhsar–Kivelson manifold

A family of three-dimensional quantum vertex (quantum dimer) models on bipartite lattices — diamond (coordination $V_p=4$), cubic ($V_p=6$), and body-centered cubic ($V_p=8$), generalizing to all even coordination — admits exactly solvable Rokhsar–Kivelson (RK) ground-state wavefunctions, exact Wegner self-duality maps of the vertex weight, and a height-field description in which monopole (instanton) operators decide whether emergent U(1) or $\mathbb{Z}_2$ phases survive. Mean-field theory and Monte Carlo locate, on the body-centered cubic lattice, a phase in which $\mathbb{Z}_2$ topological order and dimer symmetry breaking coexist. The open problem — stated explicitly in the published version of the construction — is the full phase structure away from the solvable RK manifold once the monopole terms and a $\delta(\nabla\theta)^2$ perturbation are turned on: whether the coexistence phase survives monopole tunneling and what becomes of the $z_{\mathrm{dyn}}=1$ gapless line; the coexistence phases anticipated at coordination $V_p>8$ analogously to the BCC lattice; and the distinct phase diagrams of arrowed versus bond formulations on non-bipartite 3D lattices. No resolution of any component is known in the literature to date.

Background

Quantum dimer models (QDMs) place hard-core dimers on the links of a lattice under a close-packing constraint, with kinetic (dimer-flip) and potential (dimer-interaction) terms. At the Rokhsar–Kivelson (RK) point, where the potential-to-kinetic ratio takes a special value, the ground-state wavefunction is exactly the equal-amplitude superposition of all allowed dimer coverings, and equal-time correlators reduce to classical combinatorics (Rokhsar and Kivelson, PRL 61, 2376 (1988)). In three dimensions, RK states on bipartite lattices realize either a U(1) Coulomb phase — a gapless phase with algebraic correlations described by a coarse-grained height field $\theta$ whose transverse fluctuations act as an emergent photon, as in Hermele–Fisher–Balents' U(1) spin liquid — or $\mathbb{Z}_2$ topological order, i.e. the deconfined phase of a $\mathbb{Z}_2$ lattice gauge theory (the 3D toric code) with pointlike electric charges $e$ and looplike magnetic fluxes $m$. Because the height field is compact, it admits monopole (instanton) operators $\cos(4n\theta)$; when these are relevant in $3+0$ dimensions they gap the photon, so the fate of U(1) phases in 3D dimer models is controlled by monopole scaling.

Balasubramanian, Bulmash, Galitski, and Vishwanath (arXiv:2201.08856, published as PRB 110, L180401 (2024)) constructed a family of 3D quantum vertex models on the diamond ($V_p=4$), cubic ($V_p=6$), and body-centered-cubic ($V_p=8$) lattices, generalizing to arbitrary even coordination $V_p$, that (i) are exactly solvable on their RK manifold, (ii) possess exact Wegner self-duality transformations of the vertex weight $u$ — the 3D analogue of Kramers–Wannier duality, mapping the phase diagram onto itself; for example $u\mapsto(15+u)/(u-1)$ on the cubic lattice (self-dual at $u=5$), and a two-parameter map on the BCC lattice whose self-dual line is $u-4v=5$ — and (iii) admit decorated Wegner dualities on the diamond and BCC lattices that map the ice-rule point $u=v=0$ (on diamond/BCC, the constraint that the number of dimers equal the number of empty links at each site) to a gapless point. The RK manifold is described by a quantum Lifshitz theory (the field theory of the height mode with dynamical exponent $z=2$), extended by perturbations:

$S=\int d^3x\,d\tau\left[\frac{1}{2}(\partial_\tau\theta)^2+\frac{\kappa}{2}(\nabla^2\theta)^2+\delta(\nabla\theta)^2+g_4\cos(4\theta)+g_8\cos(8\theta)\right]$

with potential $V(\theta)=\alpha\cos(4\theta)+\beta\cos(8\theta)$. At $\delta>0$ the $(\nabla^2\theta)^2$ term is irrelevant, giving a $z_{\mathrm{dyn}}=1$ gapless line where $g_4=g_8=0$; along it, $\cos(2\theta)$ and $\sin(2\theta)$ correlators remain long-range ordered, signaling coexistence of $\mathbb{Z}_2$ deconfinement and dimer symmetry breaking. Mean-field theory plus Monte Carlo on the BCC model locate this coexistence phase between a first-order self-dual line ($\alpha=0$, $\beta>0$) and two second-order lines in the 3D Ising universality class ($\alpha/(4\beta)=\pm1$, $\beta<0$); the self-dual gapless point $(u,v)=(35/3,5/3)$ is apparently not tricritical. On non-bipartite lattices the arrow and bond (dimer) formulations differ, and only two-dimensional numerics (the companion paper, PRB 106, 195127) bear on their expected distinct behavior. What is not known is the phase structure once $\delta>0$ and the monopole fugacities are nonzero — the subject of this problem.

Problem Statement

Determine the phase structure of this family of exactly solvable 3D quantum vertex models away from the Rokhsar–Kivelson manifold — that is, for positive perturbation $\delta(\nabla\theta)^2$ and nonzero monopole fugacities — on the lattices addressed by the three components below, taking the microscopic vertex weights and the exact RK-manifold results of Balasubramanian, Bulmash, Galitski, and Vishwanath as the solvable starting point. Together these three components constitute the task and jointly serve one objective, the phase diagram(s) of the family beyond the solvable manifold:

1. (Monopole stability of coexistence) Establish whether the phase of the body-centered-cubic ($V_p=8$) model in which $\mathbb{Z}_2$ topological order and dimer symmetry breaking coexist remains stable once the monopole terms $g_4\cos(4\theta)+g_8\cos(8\theta)$ are turned on at $\delta>0$; determine the resulting phase boundaries and critical behavior across the $(u,v)$ plane, including the fate of the $z_{\mathrm{dyn}}=1$ line where $g_4=g_8=0$ and whether any multicritical structure (e.g. at the self-dual gapless point $(u,v)=(35/3,5/3)$) survives.

2. (Higher coordination) Determine the phase diagram for coordination $V_p>8$, confirming or refuting the coexistence phases anticipated in the source analogously to the BCC lattice — an anticipation drawn from the mean-field analysis of $V(\theta)=\alpha\cos(4\theta)+\beta\cos(8\theta)$ at the gapless point, with allowed monopole operators $\cos(4n\theta)$, $4n\le V_p$.

3. (Non-bipartite lattices) Determine the phase diagrams of the arrowed and bond formulations on non-bipartite 3D lattices (e.g. the hyperkagome lattice, the 3D network of corner-sharing triangles), where the two formulations are expected to differ (bond: a generic gapless point; arrowed: a single $\mathbb{Z}_2$ topologically ordered phase).

References

1. Shankar Balasubramanian, Daniel Bulmash, Victor Galitski, Ashvin Vishwanath, 'Interplay of symmetry breaking and deconfinement in three-dimensional quantum vertex models', Physical Review B 110, L180401 (2024), DOI 10.1103/PhysRevB.110.L180401, arXiv:2201.08856, https://arxiv.org/abs/2201.08856
2. Shankar Balasubramanian, Victor Galitski, Ashvin Vishwanath, 'Classical vertex model dualities in a family of two-dimensional frustrated quantum antiferromagnets', Physical Review B 106, 195127 (2022), DOI 10.1103/PhysRevB.106.195127, https://doi.org/10.1103/PhysRevB.106.195127
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