Statistical mechanics of dimers on quasiperiodic tilings Jerome Lloyd,1, 2, 3, ∗ Sounak Biswas,1, ∗ Steven H. Simon,1 S. A. Parameswaran,1 and Felix Flicker1, 4 1Rudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford OX1 3PU, United Kingdom 2School of Physics and Astronomy, University of Birmingham, Edgbaston Park Road, Birmingham, B15 2TT, United Kingdom 3Max Planck Institute for the Physics of Complex Systems, N¨othnitzerStraße, Dresden, 01187, Germany 4School of Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA, United Kingdom We study classical dimers on two-dimensional quasiperiodic Ammann-Beenker (AB) tilings. De- spite the lack of periodicity we prove that each infinite tiling admits `perfect matchings' in which every vertex is touched by one dimer. We introduce an auxiliary `AB∗' tiling obtained from the AB tiling by deleting all 8-fold coordinated vertices. The AB∗ tiling is again two-dimensional, in- finite, and quasiperiodic. The AB∗ tiling has a single connected component, which admits perfect matchings. We find that in all perfect matchings, dimers on the AB∗ tiling lie along disjoint one- dimensional loops and ladders, separated by `membranes', sets of edges where dimers are absent. As a result, the dimer partition function of the AB∗ tiling factorizes into the product of dimer partition functions along these structures. We compute the partition function and free energy per edge on the AB∗ tiling using an analytic transfer matrix approach. Returning to the AB tiling, we find that membranes in the AB∗ tiling become `pseudomembranes', sets of edges which collectively host at most one dimer. This leads to a remarkable discrete scale-invariance in the matching problem. The structure suggests that the AB tiling should exhibit highly inhomogenous and slowly decaying connected dimer correlations. Using Monte Carlo simulations, we find evidence supporting this sup- position in the form of connected dimer correlations consistent with power law behaviour. Within the set of perfect matchings we find quasiperiodic analogues to the staggered and columnar phases observed in periodic systems. CONTENTS A. Membranes in general bipartite graphs and the Dulmage-Mendelsohn decomposition 25 I. Introduction 1 B. Membranes in punctured 8n-empires 27 II. Background 5 A. Dimer models and graph theory 5 C. The directed loop algorithm 28 B. Ammann-Beenker tilings 6 D. Source edges for dimer correlations 29 III. Existence of perfect matchings on the Ammann-Beenker tilings 9 References 29 IV. Membranes and pseudomembranes 11 A. Membranes in the AB∗ tiling 11 B. Pseudomembranes in the AB tiling 13 I. INTRODUCTION V. Exact results on the AB∗ tiling 14 Dimer models have long attracted interest as ele- A. Stars and ladders 14 gant routes to capture the interplay of local constraints B. Analytic calculation of the free energy of and lattice geometry [1]. Much of their study has dimers on the AB∗ tiling 16 been spurred by the rich phase structure of the quan- tum dimer models originally introduced by Rokhsar and VI. Numerical results on the AB tiling 19 Kivelson [2{5] as effective descriptions of short-range res- A. Choice of samples and boundary conditions 19 onating valence bond physics in high-temperature super- arXiv:2103.01235v1 [cond-mat.stat-mech] 1 Mar 2021 B. Monomer and dimer correlations 20 conductors [6, 7]. Quantum dimer models have since C. Aligning Interactions 22 outgrown this original motivation and now rank among the paradigmatic models of quantum statistical mechan- VII. Conclusions 24 ics. They are known to host a rich variety of phases and phase transitions [8{14], including both gapped and Acknowledgments 25 gapless quantum spin liquids [4, 15{20] and deconfined quantum critical points [21{23], whose emergent gauge structure and fractionalized excitations have particularly intuitive descriptions in terms of dimers [5]. More re- ∗ These authors contributed equally. cently, their local constraints have been proposed as a 2 route to glassy quantum dynamics and slow thermaliza- Classical dimers have also been studied in settings with tion [24{27]. disorder, such as random regular graphs and Erd}os-Renyi Here we consider classical dimer models on bipartite random graphs [52{55], using approaches that are asymp- graphs [1, 19, 28{44]. A graph is a set of vertices con- totically exact in the thermodynamic limit [56, 57]. How- nected by edges. It is bipartite if its vertices can be par- ever, the absence of locality in these ensembles rules out titioned into two mutually exclusive sets such that there any simple generalization of the notions of dimer corre- are no edges between vertices belonging to the same set lations and monomer confinement. (same bipartite `charge'). Dimers are placed on the edges Recent work explored the problem of classical dimer such that each vertex connects to zero or one dimers models on Penrose tilings [58]. These are infinite tilings (a hard-core constraint). This defines a dimer covering, of the plane constructed from two types of tile. The or matching. An unmatched vertex not connected to a tiles fit together without gaps or defects, in such a way dimer is termed a monomer. A monomer-free configura- that no patch can be tessellated periodically to reproduce tion, if one exists, is called a perfect matching. Classi- the pattern [59{61]. Despite lacking the discrete trans- cal dimer models and their associated matching problems lational symmetries of crystal lattices, they nevertheless have attracted sustained interest from the physics, math- feature a great deal of order. For example, their Fourier ematics, and computer science communities over the last transforms, which are tenfold rotationally symmetric, century owing to their ubiquity in problems of constraint feature sharp Bragg peaks which can be labelled by a fi- satisfaction, optimization, and combinatorics [5, 45{47]. nite number of wave vectors. This latter condition defines The perfect matchings of graphs admitting planar em- the Penrose tiling to be quasiperiodic [61]. Penrose tilings beddings can be counted exactly using Pfaffian tech- came to prominence in the physics community with the niques [1, 29, 30]. However, while mathematically rig- discovery of quasicrystals, real materials whose atoms are orous, these do not provide especially transparent phys- arranged quasiperiodically [62]. Considering the edges ical insight, and are often computationally demanding. and vertices of the tiles as those of a bipartite graph, A more intuitive perspective is afforded by the height Penrose tilings do not admit perfect matchings despite representation [48{51], particularly when it is applied to having no net imbalance in their bipartite charge [58]. In- periodic bipartite lattices that admit perfect matchings. stead, they have a finite density of monomers in the ther- On these lattices, the statistical mechanics of dimer con- modynamic limit. The maximum matchings on Penrose figurations can be understood by mapping dimer cover- tilings, which contain the maximum number of dimers, ings to configurations of an integer-valued `height' field have an unusually rich underlying structure, quite dis- on edges of the dual lattice. The hard-core constraint be- tinct from either periodic or random systems. In general comes a zero-divergence condition on this field | i.e. a maximum matchings, monomers can be thought of as Gauss law | allowing it to be re-expressed as the lattice moving via dimer re-arrangements. On Penrose tilings, curl of a scalar (vector) height variable in 2D (3D). The monomers are always confined within regions bound by height mapping is most useful when entropically favoured nested loops, or membranes. These membranes are com- dimer coverings correspond to locally flat height config- prised of edges which do not host a dimer in any max- urations. This allows us to deduce dimer correlations imum matching. Each such region has an excess of using a coarse-grained free energy density for the height vertices belonging to one or the other bipartite charge, field, taking a local Gaussian form at long wavelengths. and hosts a corresponding number of monomers. Ad- Monomers appear as vortex defects of the height field. In jacent regions have monomers of the opposite bipartite 2D, if the microscopic parameters correspond to the vor- charge. The properties of these membranes and the re- ticity being irrelevant (as on the square and honeycomb gions that they enclose follow directly from the dimer lattices), the height model is in a rough phase, implying constraint and the underlying symmetry of the tiling, critical (power law) dimer correlations and logarithmic and can hence be precisely determined. Ref. 63 identified confinement of monomers (i.e. the free energy cost of a similar monomer-confining regions, separated either by pair of test monomers in an otherwise-perfect matching membranes or perfectly matched regions, as components diverges logarithmically with their separation). When of the Dulmage-Mendelsohn decomposition of generic bi- the vorticity is relevant (as on the square lattice with in- partite graphs [64{67]. This was used to investigate teractions [35]), the height model is in its flat phase, with phase transitions of such monomer-confining regions in exponentially decaying connected correlations of dimers ensembles of periodic lattices with random vertex dilu- and linearly confined monomers. Since vortex defects tion, such as those used to model vacancy disorder in are never relevant in 3D, dimer correlations are always quantum magnets. algebraic, and monomers are deconfined, i.e. a test pair While both Refs. 58 and 63 consider bipartite dimer can be separated to arbitrary distance with finite free models, the usual mapping to height models does not energy cost. However, these arguments rely on (i) the apply to quasiperiodic graphs, or graphs where vertices existence of perfect matchings; and (ii) the identification can have different co-ordination numbers. While a more of locally flat height configurations with high-probability general height mapping is possible in principle [68], the configurations.