A controlled coarse-grained theory of classical dimers and the quantum dimer phase diagram on the Ammann-Beenker quasiperiodic tiling

The hard-core dimer (perfect-matching) model on the Ammann-Beenker (AB) quasiperiodic tiling admits perfect matchings whose statistics are controlled by a hierarchy of 'pseudomembrane' constraints that reproduce the dimer constraint at every inflation scale, producing long-ranged, anisotropic, site-dependent correlations consistent with log-periodically modulated power laws, together with unusual monomer behavior (radial anti-confinement and annular deconfinement). The source paper (Lloyd, Biswas, Simon, Parameswaran, and Flicker, Phys. Rev. B 106, 094202 (2022)) leaves as its central future direction the construction of a controlled, quantitative coarse-grained description of these statistics - a height-like action, possibly nonlocal or scale-hierarchical - and, conditioned on it, the fate of the quantum dimer model on the same tiling: whether its Rokhsar-Kivelson-like point is a fine-tuned multicritical point, as on periodic bipartite lattices, or whether quasiperiodicity stabilizes robust long-wavelength quantum dimer correlations. Later work by two of the same authors confirmed an exact real-space renormalization group and a discrete-scale-invariant critical point numerically, and machine-learning coarse graining independently identified hard-core dimers as the effective degrees of freedom; however, no analytic theory exists, the critical data remain fitted rather than derived, and the quantum dimer model on the AB tiling is entirely unaddressed in the literature. The problem remains open.

Background

A classical dimer model places dimers (hard rods) on the edges of a graph subject to a close-packing constraint: in a perfect matching, every vertex is touched by exactly one dimer. On periodic bipartite planar lattices such as the square and honeycomb lattices, this combinatorial problem maps onto a height model: an integer-valued height is defined on the faces of the lattice so that crossing an edge shifts the height by a rule fixed by the dimer occupancy, and the close-packing constraint becomes single-valuedness of the height. The coarse-grained limit of the height ensemble is a Gaussian free field (the height action, an entropic analogue of electrostatics), whose logarithmically correlated fluctuations imply algebraically decaying connected dimer correlations - a Coulomb phase - and confinement of test monomers, meaning that two vertices left uncovered cost a free energy that grows with their separation. The quantum dimer model introduced by Rokhsar and Kivelson promotes this ensemble to a quantum Hamiltonian with a potential term $V$ and a kinetic term $t$ that resonates dimers around plaquettes; at the Rokhsar-Kivelson (RK) point $t=V$ the ground state is the equal-amplitude superposition of all classical coverings, so its equal-time dimer correlations coincide with the classical ones. On periodic bipartite lattices, the height description in 2+1 spacetime dimensions combined with Polyakov's argument that height vortices (instantons) proliferate in compact U(1) gauge theory in three spacetime dimensions forces algebraic quantum correlations to occur only at the fine-tuned RK multicritical point, never as a stable phase.

Quasicrystals are materials with long-range order - sharp diffraction peaks - but no translational periodicity, and are modeled by quasiperiodic tilings. The Ammann-Beenker (AB) tiling is the two-dimensional octagonal tiling built from squares and 45-degree rhombi, self-similar under inflation by the silver mean $1+\sqrt{2}$. Dimer physics on quasiperiodic graphs differs sharply from the periodic case. On Penrose tilings there are no perfect matchings at all, and monomers are trapped in charge-imbalanced regions separated by membranes, which are sets of edges that carry no dimer in any maximum matching (Flicker, Simon, and Parameswaran, 2020); Bhola et al. (2022) identified such monomer-confining structures with components of the Dulmage-Mendelsohn decomposition, a canonical decomposition of a bipartite graph into perfectly matched parts and monomer-confining parts. Lloyd et al. (Phys. Rev. B 106, 094202 (2022)) showed that the AB tiling, in contrast, admits perfect matchings. Deleting the 8-fold-coordinated vertices yields an auxiliary quasiperiodic tiling AB* in whose perfect matchings the dimers lie along disjoint one-dimensional loops and ladders separated by exact membranes, so the dimer partition function factorizes; reintroducing the 8-fold vertices turns the membranes into pseudomembranes, sets of edges that collectively host exactly one dimer in any perfect matching. Pseudomembrane-bounded regions then act as effective vertices that are matched exactly like elementary vertices, so the dimer constraint is reproduced at every inflation scale - a discrete scale invariance, meaning invariance only under rescalings by powers of $(1+\sqrt{2})^2$ rather than under arbitrary rescalings. Monte Carlo simulations found connected dimer correlations consistent with power laws (anisotropic, site-dependent, and modulated on the inflation scales) and remarkable monomer behavior: radial anti-confinement, where a monomer pair's free energy is lowest at separations of order the system size, together with annular deconfinement, where a monomer can move on annuli around the 8-fold vertices. The paper argues that these correlations make a local, isotropic, translation-invariant continuum height action unlikely, and leaves open whether any nonlocal or scale-hierarchical quantitative description exists; it also anticipates a quantum RK-like point on AB between columnar-like and staggered-like ordered analogues, noting that without a height action the instanton reasoning is inapplicable, so whether that point is fine-tuned or part of a robust phase is unknown. As a related observation, the paper connects monomers of the matching problem to zero-energy modes of tight-binding models on the same graph: the monomer density equals the density of strong zero modes (modes surviving arbitrary hopping disorder), whereas the finite density of zero modes found on AB by Koga (2020) is fragile, moving away from zero energy under arbitrarily weak disorder. The machine-learning coarse-graining study of Gökmen et al. (2024) and the constructive height-field derivation of Powell (2026) for the square and honeycomb lattices delimit the current state of the art: the height framework remains available, in analytic form, only for periodic bipartite lattices.

Problem Statement

The problem is to replace the periodic-lattice height framework with a controlled quantitative theory of dimer statistics on the Ammann-Beenker quasiperiodic tiling, and to use it to settle the quantum dimer phase diagram on the same tiling. It has two linked deliverables.

(1) Classical characterization. Determine whether there exists a tractable coarse-grained description of the ensemble of perfect matchings on the AB tiling - the source paper's 'maximally probable dimer configurations' - whether a height-like action, local or nonlocal, possibly organized by the inflation hierarchy, or an equivalent analytic characterization of the fixed point of the exact inflation renormalization group implied by the pseudomembrane structure. Here 'tractable' means analytic - a description expressible in closed form, or derived by a controlled approximation, rather than obtainable only by exhaustive enumeration or numerical fitting. The description must quantitatively reproduce the observed multiscale connected dimer correlations (power-law decay with log-periodic modulation, including their anisotropy and site dependence) and the monomer radial anti-confinement and annular deconfinement, and the associated critical data - the correlation-decay exponent and the fractal dimension of double-dimer overlap loops (formed by superposing two independent perfect matchings of the same patch: the symmetric difference decomposes into tightly-packed self-avoiding loops with dimers alternating between the two covers, while edges occupied in both count as length-2 loops) - must follow analytically rather than being fitted. A complementary admissible outcome is a rigorous nonexistence result answering the question as posed: a proof that none of the tractable descriptions listed above can capture these correlations, settling both the conjectured absence of a local, isotropic, translationally invariant action of coarse-grained fields and the fate of the nonlocal and scale-hierarchical alternatives the question asks about. A nonexistence result confined to a narrower, solver-chosen class does not complete part (1) unless it also resolves whether nonlocal or scale-hierarchical descriptions can capture these correlations.

(2) Quantum phase structure. For the quantum dimer (Rokhsar-Kivelson) model on the AB tiling - plaquette-flip kinetic term and dimer-number potential, approached in the thermodynamic limit through the sequence of inflations with the open boundary conditions that quasiperiodicity forces - determine the phase diagram as a function of the ratio $V/t$ (and of any alignment potential distinguishing columnar-like from staggered-like order): in particular, whether the point at which equal-time dimer correlations reproduce the classical discrete-scale-invariant correlations is a fine-tuned multicritical point, as on periodic bipartite lattices, or whether quasiperiodicity and the pseudomembrane hierarchy stabilize long-wavelength quantum dimer correlations as a robust phase.

The two parts form one scientific objective: the controlled classical description of part (1) is the natural instrument for deciding part (2), because the height-instanton reasoning that settles the periodic-bipartite case has no known analogue on the AB tiling. Should part (1) instead terminate in a rigorous nonexistence result, that result would be the input constraining how part (2) is decided; in either case part (2) remains to be determined in its own right.

References

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