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. Neither is guaranteed for a generic bipar- resulting height functions typically do not lead to ana- tite graph. lytically tractable coarse-grained continuum free energy 3

FIG. 1. Clockwise: (a) A finite patch of the Ammann-Beenker (AB) tiling. Composed of copies of square and rhombus tiles, the (infinite) tiling covers the plane in an ordered fashion, yet never repeats periodically. (b) A maximum matching (dimer covering) of a patch of the AB tiling. The tiling can be perfectly matched in the thermodynamic limit; on finite patches, an O(1) number of unmatched vertices (monomers) generally appear, which can be moved to the boundary. Thick black lines indicate links which comprise overlapping pseudomembranes, each of which collectively host one dimer. Yellow edges indicate a dimer on a pseudomembrane. (c) A patch of the AB∗ tiling, obtained from AB by removing all 8-connected vertices. (d) A maximum matching of a patch of the AB∗ tiling. Pseudomembranes become membranes, hosting zero dimers and leading to a decoupling of the partition function. functionals. This is because there is no longer a simple making it challenging to define a crisp notion of monomer relationship between the local configuration of the height confinement. These facts challenge the goal of a precise field and the statistical weight of the global dimer cover- characterization of long-wavelength properties of dimer ing. In any case, due to a sizeable density of monomers in models in quasiperiodic environments. the cases studied in Refs. 58 and 63, connected correla- In this work, we meet this challenge in the setting tion functions of dimers are short-ranged and more or less of a distinct quasiperiodic dimer problem for which we unremarkable. In addition, monomer correlation func- can make a series of exact, and asymptotic, statements. tions are non-monotonic and strongly site-dependent, Specifically, we study classical dimers on the Ammann- 4

ples into perfectly matched one-dimensional regions that we call ladders, separated by membranes. These mem- branes, like those in Refs. 58 and 63, are comprised of edges between different ladders and do not host a dimer in any maximum matching. However, unlike the monomer-confining regions of Refs. 58 and 63, they sepa- rate perfectly matched regions. This structure allows us to exactly compute the partition function of the dimer model on the AB∗ tiling. We show that membranes in the AB∗ tiling can be decomposed into non-intersecting but overlapping pseudomembranes in the full AB tiling when the 8-vertices are reintroduced: the edges belonging to every pseudomembrane now collectively host exactly one dimer. Each 8-vertex is surrounded by at least one pseu- domembrane. Loosely speaking, a double deflation maps this 8-vertex and its surrounding pseudomembrane- bounded region into a single vertex at the next scale. Intuitively, since each pseudomembrane is pierced by ex- actly one dimer, this procedure preserves the dimer con- straint at each successive scale. This remarkable property provides a heuristic picture of how power law connected correlations emerge. It also suggests that the AB tiling is an intriguing example of a deterministic lattice-level (rather than continuum) coarse-graining of a constrained system that faithfully imposes the constraint at each suc- cessive decimation scale 1. FIG. 2. The e -dependence of the connected dimer corre- j We show that the discrete scale invariance exhibited by lations, C(e , e ) (Eq. (33)). The source edge e , indicated 0 j 0 perfect matchings leaves its imprint in connected dimer by the green triangle, connects an 82-unit to a nearby lad- der. The resulting correlations are typical of slowly decaying correlations, as exhibited in Fig. 2. Investigating dimer examples, and resemble a power law. (See Sec. VI) correlations numerically we show that, while they are strongly anisotropic and site-dependent, the asymptotic behaviour of connected correlations of dimers are consis- tent with power law scaling. This is particularly striking Beenker (AB) tiling, shown in Fig. 1. This tiling has given the absence of the continuum height description been the topic of much recent attention, with investi- which mandates power law correlations in periodic bi- gations involving its magnetism [69–71], superconductiv- partite lattices [47, 51]. We also investigate the phase ity [72], critical eigenstates [73] and protected Majorana diagram of dimer models on both the AB tiling and modes [74]. Like the Penrose tilings, AB tilings exhibit AB∗ tiling in the presence of a classical aligning inter- discrete scale invariance: deflations (vertex decimations action resembling the Rokhsar-Kivelson potential term, followed by rescaling lengths by the irrational silver ra- as a step on the road towards a study of quasiperiodic tio) map an AB tiling to another AB tiling. We prove quantum dimer models. that these tilings host a perfect matching in the thermo- The remainder of this paper is organized as follows. dynamic limit, in contrast to the Penrose tilings investi- We introduce the necessary background on dimer covers gated in Ref. [58]. Our proof makes use of the discrete and AB tilings in Sec. II. In Sec. III we prove that per- scale invariance characteristic of quasicrystals: we find fect matchings exist for AB tilings in the thermodynamic that at any given coarse-graining scale, special vertices limit. The construction leads to the introduction of the that are left invariant by double deflations retain strong auxiliary AB∗ tiling, which we also prove to be perfectly dimer-dimer correlations at the next scale. This sug- matched. In Sec. IV A we prove the existence of mem- gests that certain regions associated with these vertices branes in the AB∗ tiling; in Sec. IV B we demonstrate retain mutual dimer correlations. The preserved vertices how these become pseudomembranes in the AB tiling. have edge-co-ordination eight and we refer to them as In Sec. V we present an exact calculation of the partition ‘8-vertices’. function and free energy of the classical dimer model on The success of this iterative construction of dimer cov- the AB∗ tiling. Turning to numerical results, we consider erings motivates us to consider an auxiliary problem on a related graph that we dub the AB∗ tiling, Fig. 1c. This is obtained from the AB tiling by removing all the 8- vertices. The AB∗ tiling is also perfectly matched and 1 We emphasize the deterministic nature, since real-space decima- does not host monomers, and we show that it decou- tions can have an especially simple structure in random systems. 5 the full AB tiling, where, after outlining our choice of samples and boundary conditions in Sec. VI A, we iden- tify connected correlations consistent with power laws in Sec. VI B. In Sec. VI C we include a classical aligning interaction to both the AB tiling and AB∗ tiling. We provide concluding remarks in Sec. VII.

II. BACKGROUND

A. Dimer models and graph theory

In this section we introduce the necessary terminol- ogy to discuss dimer coverings on graphs. The graphs of interest in this paper have two important properties. First, they are bipartite, meaning the vertices can be par- titioned into two mutually exclusive subsets, and , so that every vertex in ( ) only has edges toU verticesV in ( ). If two verticesU V belong to the same subset, we willV U say they have equal (bipartite) charge; otherwise, they are oppositely charged. Second, the graphs admit planar embeddings. We keep the geometry of the tiling, although strictly speaking only the graph topology is rel- evant to the matching problem. A matching of a graph is a subset of edges such that no vertex is incident with more than one edge in the subset [75]. A matching is equivalent to a dimer con- figuration, as edges in the matching can be covered by dimers, with no vertex touching more than one dimer. A vertex connected to an edge in the matching is said to be matched; an unmatched vertex is called a monomer. FIG. 3. (a) A maximum matching which is not perfect. Un- matched vertices are marked with a blue circle. (b) A perfect A perfect matching is a matching with every vertex matching. (c) An alternating cycle; augmenting this cycle — matched or, equivalently, a monomer-free dimer cover- swapping covered and uncovered edges — yields a new match- ing. Not every graph admits a perfect matching — a ing with the same number of dimers. (d) An even-length simple counterexample is any graph with an odd number alternating path terminating at a monomer; upon augment- of vertices. A maximum matching is a matching with ing this path, the monomer is transported to the other end the maximum number of dimers (minimum number of of the path. (e) Augmenting an odd-length alternating path monomers): Fig. 3a-b shows examples of maximum and between monomers, or augmenting path, annihilates the two perfect matchings. If a graph admits a perfect match- monomers in favour of a dimer. ing, then any maximum matching is necessarily perfect. Usually a graph will have multiple maximum or perfect computation is usually a formidable task. Detailed dis- matchings. From a statistical mechanics perspective, our cussions on classical dimer coverings and their applica- interest is in understanding the complete space of such tion to physics can be found in e.g. Refs. 68, 76, and matchings. Of particular interest is the partition function 77. Z, which is a sum over all possible dimer configurations Much of the recent interest in dimers originated with weighted according to the details of the configuration the quantum dimer model introduced on the square lat- andC the model under consideration, w( ): C tice by Rokhsar and Kivelson [2]. Defining a plaquette X Z = w( ). (1) to be the four edges of a single square tile, the Rokhsar- C C Kivelson Hamiltonian is defined to be For example, we may be interested in penalising monomers, or favouring certain alignments of dimers. In Sec. V, we consider the partition function of an equally weighted sum over perfectly matched configurations of (2) the AB∗ tiling, i.e. w( ) = 1 only if is a perfect match- C C ing, otherwise w( ) = 0. Monomer-monomer, dimer- where the sum is taken over all plaquettes . The nota- dimer, and monomer-dimerC correlations are then typi- tion indicates that the objects of interest are flippable pla- cally the observables of interest, although their analytic quettes: plaquettes which host a pair of parallel dimers. 6

Such a configuration can always be ‘flipped’ to the other dynamic limit) admits a perfect matching is in general pair of dimers without changing the rest of the graph. non-trivial. Quasiperiodic Penrose tilings cannot be per- The kinetic term t takes advantage of this local move fectly matched: though the infinite tiling is charge neu- to introduce a natural form of local quantum dynamics, tral, monomers are confined within regions with an ex- quantum ‘resonances’ between the two perfectly matched cess of one or the other type of the bipartite charge. configurations of the plaquette. These resonances give Such confinement emerges as a direct consequence of resonating valence bond states their name [7]. In this the quasiperiodic geometry: monomers cannot cross paper we will instead exclusively consider the classical monomer membranes to annihilate those of the oppo- problem with t = 0. The V terms, which are essen- site charge. Monomer membranes are closed loops of tially classical, favour (disfavour) alignment of dimers on edges which cannot be covered by dimers in any maxi- plaquettes for V < 0 (V > 0). We return to these in mum matching; equivalently, no augmenting paths exist Sec. VI C. The square plaquette notation remains rele- between monomers on opposite sides of a membrane. In vant in our case since all the tiles we consider have four Sec. IV we will demonstrate that similar membrane struc- edges. tures exist on the AB tiling, but in this case they do not An alternating path is a connected subset of edges frustrate the perfect matching. along which edges are alternately covered and uncov- ered by dimers. Starting from any matching, switching which edges are covered/uncovered in a closed alternat- B. Ammann-Beenker tilings ing path (alternating cycle) results in another matching with the same number of dimers (see Fig. 3c). All alter- Although we will use the language of graph theory in nating cycles are of even length (on a bipartite graph this our discussion of AB, the mathematical theory of qua- statement is trivial since all cycles are of even length). sicrystals has traditionally been developed in the lan- This process of switching covered and uncovered edges is guage of tilings. A tiling is a filling of space with con- known as augmenting the path. The simplest such aug- gruent shapes without overlaps or gaps [61]. For clar- mentation, of the smallest alternating cycle, is the flip of ity we will use tiling to refer to an infinite tiling of the a flippable plaquette. plane; finite sections of a tiling we refer to as patches. Augmenting an odd-length alternating path changes Simple tilings can be formed by periodic repetition of a the number of dimers on the matching. At the most single tile, such as a square or hexagon. Conventional basic level, two monomers can be created by remov- crystals can be seen as regular tilings of space by an ing one dimer. On a bipartite graph, these monomers atomic unit cell; the underlying symmetries can be de- have opposite charge, and so the process has a physical duced from diffraction experiments, and yield the 230 analogy in the excitation of a particle-antiparticle or de- crystallographic space groups [80]. However, the only ro- fect pair above the vacuum state (viewed as one where tational symmetries compatible with such periodic long- all monomers are paired into neutral dimers). These range order (in 2D or 3D) are 2-, 3-, 4-, and 6-fold. When monomers are then able to ‘move’ through the dimer diffraction experiments on AlMn-alloys revealed struc- vacuum: taking an even-length alternating path with one tures compatible with 5-fold rotational symmetry [81], end terminating on one of the monomers, augmenting the crystallographic theory had to be extended to include path results in the monomer being translated to the op- quasicrystals — alloys with long-range atomic order but posite end of the path, while conserving the monomer’s no translation symmetry. Quasicrystals can feature finite charge (see Fig. 3d). Given a maximum matching, all patches of 5-, 8-, 10- or 12-fold rotational symmetry [82– other maximum matchings can be generated by combi- 84]. nations of two basic ‘moves’: (i) augmenting alternating Quasiperiodic tilings are a class of planar tilings with- cycles; and (ii) transporting monomers along alternating out translation symmetry, capable of accounting for the paths of even length. symmetries observed in quasicrystals which are forbidden By reversing the above logic, starting from a non- in periodic tilings [60, 61, 85]. They are distinguished perfect matching, if an odd-length alternating path can from the more general set of aperiodic tilings by the fact be found with end points terminating on two monomers that their diffraction patterns can be indexed by a finite (an augmenting path), then this path can be augmented number of wavevectors [61]. The most famous family to annihilate the two monomers in favour of a dimer (see of quasiperiodic tilings is due to Penrose [59]. Penrose Fig. 3e). If augmenting paths can be sequentially found tilings have local patches of 5-fold symmetry (and can between all remaining monomers of a matching, so that have at most one true 5-fold rotational centre). The no monomer appears in more than one path, then the Ammann-Beenker tilings [60, 86] similarly have 8-fold augmentation of these paths results in a perfect match- symmetry, and together with the decagonal (10-fold) and ing. We will use these facts to prove the AB tiling can dodecagonal (12-fold) tilings, they make up the ‘Penrose- be perfectly matched. like’ tilings [87]. The remarkable fact that the four sym- Efficient algorithms exist to determine the maximum metries displayed by these tilings account for the sym- matching of finite graphs [78, 79]. However, showing metries of all physical quasicrystals (i.e. those observ- that an infinite graph (the relevant case in the thermo- able in experiment) was explained by Levitov based on 7

of two steps: decomposition, where every tile is divided into smaller tiles as shown in Fig. 4, followed by rescal- ing, where the decomposed tiling is scaled so that the new tiling is formed from exact copies of the original tiles. The inflation of the square tile breaks the square rotation symmetry down to a single mirror symmetry. Therefore we must keep track of the orientation of the square, as captured by the triangular motif shown. When this in- formation is not pertinent to the discussion, we will gen- erally leave the markings absent in future figures. The scaling factor is the silver ratio δS, defined by FIG. 4. The inflation rule σ, in which a tile is decomposed into smaller tiles, followed by a rescaling of all lengths by 2 √ δS = 2δS + 1 (3) a factor of the silver ratio δS = 1 + 2 (which we do not −1 show here). The inverse σ , deflation, is uniquely specified. and equal to An arbitrarily large AB tiling can be generated from a small patch by repeated inflations. Note the triangular motif on the δS = 1 + √2. (4) square tile, to indicate the orientation of the inflation. Note that it is the length (rather than area) of the tile edges that scales by δS under decomposition: this can be arguments of energetic stability [88]. The Penrose-like seen geometrically in Fig. 4, with each edge divided into a tilings have therefore seen extensive study in connection rhombus edge and the square diagonal. The area of each 2 to physics. tile scales by δS under decomposition. In the following, if Classical dimers on Penrose tilings were studied in pre- we say one tiling is larger than another by some power of vious work [58]; here we focus on the Ammann-Beenker the silver ratio, we are always referring to the respective (AB) tiling, a patch of which is shown in Fig. 1. The lengths of their tile edges. The Penrose tiling√ instead has AB tiling is constructed from copies of two inequivalent 1+ 5 as its scale factor the golden ratio ϕ = 2 . tiles: a square, and a rhombus with angles π/4 and 3π/4. Patches of arbitrarily large size can be generated by Both tiles have edges of unit length. No shifted copy the inflation process; the tiling is recovered after an of this tiling can be exactly overlaid with the original. infinite number of inflations. TheT inverse of the inflation Many mathematical properties of AB tilings are well un- rule, σ−1, deflation, is also uniquely specified on a tiling, derstood and can be found in e.g. [61]. Here we briefly consisting of composition — reconstruction of the larger mention those relevant to the following discussion. tiles from smaller tiles — and rescaling by the inverse Formally, one should really refer to Ammann-Beenker of the silver ratio. Acting with σ or σ−1 on returns tilings (plural), the uncountable set of equivalence classes another tiling in the same LI class. T of LI (locally indistinguishable or locally isomorphic) Every possible configuration of tiles around a vertex in tilings. Two tilings are LI if any finite patch of one can AB is identical to one of the seven vertex configurations be found in the other. In this way, all AB tilings ‘look the shown in Fig. 5 (up to rotations). The 5-vertices (5A same’ from a local perspective, distinguished only in their and 5B) yield two distinct results upon inflation depend- global structure. In fact, every finite patch recurs with ing on the orientation of the adjacent square plaquettes. positive frequency, in all tilings, a property that accounts All other vertices can be uniquely identified by their co- for the long-range order of the quasiperiodic tilings. This ordination number (valence, in graph theory nomencla- structure can be compared to that of the periodic tilings, ture). Under σ, each vertex configuration is mapped to where the long-range order arises from the repeated unit a different configuration, e.g. 4 6, with the exception cell. Results that we discuss in the following sections of the 8-vertex, which inflates to→ another 8-vertex. These apply in generality to the entire set of AB tilings. inflations are also shown in Fig. 5. Any AB tiling can be generated by several meth- The scale symmetry of the Ammann-Beenker tilings T ods [60, 61, 89]. Square and rhombus tiles can have their arises from this inflation structure, and specifically from edges decorated by matching rules which force the tiles the invertibility of σ. One expression of this symmetry is to fit together quasiperiodically; an AB tiling can be cre- that all 8-vertices of are positioned at the vertices of ated as a slice through a higher dimensional periodic lat- the tiling obtained byT twice-composing i.e. the twice- −2 T 2 tice (the cut-and-project technique); or the tilings can be deflated tiling −2 σ ( ) with lengths rescaled by δS. generated via an inflation procedure. This follows fromT noticing≡ T that every vertex is mapped This inflation method allows us to discuss the scale to an 8-vertex under two inflations, as can be checked symmetry of the tilings, and is our primary workhorse in from the vertex inflations in Fig. 5. By the inverse, de- this paper. Starting from a finite seed patch (e.g. a flating twice maps the 8-vertices of to the vertices of T0 T single square tile) an inflation rule σ is repeatedly ap- −2. This scale symmetry is more clearly shown in Fig. 6, n T plied to grow the patch as n = σ ( 0), with the number which shows a patch of overlaid with the scaled −2. of tiles growing exponentiallyT underT inflation. σ consists The 8-vertices of are colouredT by their bipartite charge,T T 8

FIG. 5. The seven vertex configurations, and their action under inflation (Fig. 4). All vertices map to 8-vertices under at most two inflations.

FIG. 7. The σ2 (in/de)flation rules, which map vertices di- rectly to 8-vertices in the completed tiling. Here we leave off triangular motifs in the inflation for clarity.

venient to define an ordering to the 8-vertices. An 8- vertex is order-zero if it maps to a 7-, 5A-, or 6-vertex under a single deflation; order-one if it deflates to an order-zero 8-vertex; and so on. This flow of vertices un- der (in/de)flation is represented in Fig. 8. We denote an order-n 8-vertex an 8n-vertex. An 8n-vertex is therefore th FIG. 6. The scale symmetry of the AB tiling: the 8-vertices the n inflation of an order-0 8-vertex, 80. We find that of an AB tiling are positioned to sit at the vertices of another 8n-vertices of sit at the vertices of an AB tiling, LI to T locally indistinguishable AB tiling, with lengths increased by (n+2) √ , with lengths scaled by a factor of δS . Since all 2 T a factor of δS , where δS = 1+ 2 is the silver ratio. 8-vertices vertices of are mapped to order-n or higher 8-vertices are coloured by their bipartite charge. under σ(n+2)T , 8-vertices of order-n and higher can also be viewed as those vertices of that are preserved un- der n + 2 deflations. We will denoteT the n-times deflated and are seen to coincide with the vertices of the scaled tiling of as −n. T T − . We will rely on this symmetry in the rest of the T 2 To conclude this section, we discuss the 8-fold sym- paper. Anticipating this 8-vertex vertex mapping, we metry that defines the Ammann-Beenker tilings. First, 2 ↔ introduce σ inflation tiles, shown in Fig. 7. Compared as mentioned previously, the Bragg spectrum (Fourier 2 to the basic inflation tiles (Fig. 4), the σ tiles have the transform of the lattice) has a discrete 8-fold rotational property that their vertices become 8-vertices in the in- symmetry (D8 in Sch¨onfliesnotation) [90]. Second, an 8- flated tiling. fold rotation of an AB tiling returns another AB tiling in This scale hierarchy continues: under σ2, the 8-vertices the same LI class. Third, and most importantly to us, the of inflate to 8-vertices, and these twice-inflated 8- symmetry shows up in the structure of tiles surrounding verticesT are positioned to sit at the vertices of a tiling an 8-vertex. Every vertex configuration has a set of tiles 4 with lengths scaled by δS. With this in mind, it is con- which always appears around the vertex wherever it is 9

where the tiling covers the infinite plane and boundary effects can be ignored. Taking the thermodynamic limit is a delicate procedure in quasiperiodic systems since one must necessarily work with open boundary conditions and bound their contribution as the system size increases. For a first pass at the problem, and to underscore its nontrivial nature, we note that the matching problem on finite patches of the AB tiling depends strongly on the exact patch considered. Trivially, a patch with a net imbalance of bipartite charge never admits a perfect FIG. 8. The flow of vertices under successive inflations. The matching. Less trivially, it is also possible to find charge- ordering of 8-vertices is defined so that an order-zero 8-vertex, neutral patches with unpaired monomers. An example 80, deflates once to a 5A-, 6-, or 7-vertex, and an order-n 8- is the patch generated by inflating the basis square four vertex, 8n, deflates to 8n−1. times: exactly two monomers remain in the maximum matching, with opposite charges. In contrast to the Pen- rose case, where the presence of membranes leads to a monomer density that approaches 10%, the observed number of unpaired monomers on∼ AB remains of order unity as the tiling is grown. This suggests that monomers on AB are artefacts of finite tilings, representing defects that ‘migrate in’ from the boundary, rather than a finite bulk monomer density persisting in the thermodynamic limit. We have verified these statements using standard graph-theoretic methods [78, 79] to compute maximum matchings on a large sample of patches generated via inflation. We will prove these statements regarding fi- nite patches, and that the planar tilings can be perfectly matched. The strategy of our proof is as follows. Con- sider the perfect matching problem on an AB tiling . T FIG. 9. The local empire of an 8 -vertex, the simply con- We first match up all vertices of except for the 8- 0 T nected set of tiles that always appear around an 80-vertex in vertices (recall that these are the vertices which remain the tiling. The local empire has D8 symmetry. under two deflations). We then show that the problem of matching the remaining 8-vertices maps to a perfect matching problem on the twice-deflated tiling −2. On placed in the tiling. Such a motif is known as the vertex this tiling, we match all but the 8-vertices againT (this empire [60, 91]. Vertex empires are generally not simply matches all 80- and 81-vertices on the original ). We connected; the set of empire tiles simply connected to a then use the scale symmetry of to iterate thisT proce- vertex is called the local empire of the vertex. The local dure and obtain an upper bound onT the monomer density empire of the 80-vertex is shown in Fig. 9, and has D8 of the AB tiling after n deflations. This density vanishes symmetry. The inflation σ preserves this symmetry, and exponentially as n , corresponding to taking the → ∞ so the radius of symmetry of an 8n-vertex is a factor of thermodynamic limit. δS larger than for an 8n−1-vertex. The local empire of an Our motivation for separating out the 8-vertices comes 8n-vertex in fact includes that of an 8n−1-vertex within from the requirement for a bipartite graph to be charge- it. According to the definition of the LI-class of tilings, neutral in order to admit a perfect matching. This is true every finite region can be found with positive frequency of the AB tiling in the thermodynamic limit, because across the tiling. It follows that D8-symmetric 8-empires the average vertex connectivity is the same for the two of arbitrarily large size can be found across the tiling, bipartite subsets and . Note, however, that any D8 with a frequency corresponding to the frequency of the local empire has anU excessV charge of at least one. This respective 8-vertices. This heavily constrains the struc- follows from the fact that every D8 local empire has an ture of dimer configurations on the AB tilings. 8-vertex at its centre (by definition), and its eight-fold rotational symmetry mandates that the total number of vertices within the region must be 8m+1, for some integer III. EXISTENCE OF PERFECT MATCHINGS m. Since this number is odd, there must be an excess of ON THE AMMANN-BEENKER TILINGS bipartite charge. In fact, the same argument shows that any symmetric We now show that Ammann-Beenker tilings can be region centred on 8-vertices cannot be perfectly matched perfectly matched. We work in the thermodynamic limit, without placing at least one dimer outside of the region. 10

Although other dimer inflation tiles could have been chosen, all tile choices that consistently match along boundaries are equivalent up to augmenting cycles in the bulk, and/or reflections such that black and white ver- tices are exchanged. It is not possible to devise inflation tiles that perfectly match the tiling. This is on account of the previously considered 8-fold symmetry: for example, the local empire of an 82-vertex can be created by attach- ing eight copies of the rhombus dimer inflation tile — any dimer added to the tile to match the central monomer would be repeated eight times, resulting in a vertex with an 8-fold dimer cover, violating the dimer constraint. The unpaired 8-vertices already result in an upper FIG. 10. (Left) The dimer inflation tiles: deflating with bound on the monomer density of , of T the squared tiles in Fig. 7, T → T−2, and then re-inflating (0) with these dimer-decorated tiles places dimers on T . This ρ ν8v, (5) matches all vertices of T except the 8-vertices (marked with ≤ −4 red and blue circles denoting bipartite charge). Unmatched where ν8v = δS 0.03 is the density of 8-vertices on vertices on tile boundaries (black) overlap with matched ver- the Ammann-Beenker∼ tiling [61]. tices (white) in the completed tiling. (Right) Augmenting Next, we allow these monomers to move around the paths always exist between neighbouring 8-vertices (the other tiling by augmenting the alternating paths which termi- choice of charge-neutral pairing follows from the symmetry of nate on them. Moving a monomer along an alternat- the dimers). ing path cannot create additional monomers, so Eq. (5) remains a good upper bound on the monomer density. The bound will be reduced if we can identify augment- Since such regions exist at all scales across the tiling, ing paths connecting two oppositely charged monomers we can always find arbitrarily large regions that can- — augmentation then annihilates the monomers to give not be perfectly matched internally. This structure of a dimer, reducing the bound on number of monomers by coupled D8 regions complicates the problem. By first two. removing the 8-vertices at the centres of these regions, We define 8-vertices (all of which host monomers at we can match the rest of the tiling systematically before this stage) to be deflate neighbours on if they become progressively reintroducing and matching the remaining true nearest-neighbours under two deflations,T i.e. on monomers. −2. On the dimer-decorated , deflate-neighbouring To isolate the 8-vertices from the matching, we use the Tmonomers sit on two of the cornerT vertices of one of the fact that under two deflations (σ−2) the 8-vertices of dimer-decorated tiles in Fig. 10, connected along one of T are preserved, mapping to the vertices of another tiling the large tile edges (the monomers have opposite charge). 2 −2. Replacing the tiles of −2 with the σ inflation tiles, As is clear from the right of Fig. 10, an augmenting T T and rescaling, brings us back to , with the 8-vertices path always exists between two monomers on deflate- T sitting at the corners of the inflation tiles. Replacing neighbouring 8-vertices. In addition, augmenting paths the tiles of −2 instead with the dimer decorated tiles between disjoint pairs of deflate-neighbouring monomers T on the left of Fig. 10, and rescaling, returns a copy of can be chosen so as not to intersect. This means that an- T decorated with dimers. We call these the dimer inflation nihilating two deflate-neighbouring monomers does not tiles. affect the possibility of further annihilations, except in Two observations are pertinent regarding this process. the obvious way that the annihilated monomers are no First, the 8-vertices of remain unmatched. This fol- longer available for pairing. The paths shown in Fig. 10 lows straightforwardlyT since no dimers touch the cor- are non-intersecting in this way. ner vertices of the decorated tiles (coloured red and Finding a complete set of augmenting paths be- blue representing bipartite charge), which map to the tween deflate-neighbouring monomers is therefore equiv- 8-vertices. Second, the remaining vertices in the bulk of alent to finding a perfect matching of the twice- are matched. This may be directly verified by check- deflated tiling −2. This follows from our definition of ingT that all vertices in the interior of the decorated tiles deflate-neighbouringT 8-vertices and the existence of non- are matched, and that the boundaries of the decorated intersecting augmenting paths between them. We re- tiles overlap consistently to match all boundary vertices fer the reader back to Fig. 6 to see that the identifica- (excluding the corner 8-vertices) with no double-covered tion of augmenting paths between deflate-neighbouring vertices. While each decorated tile has four unpaired 8-vertices indeed has the structure of a matching prob- monomers on its boundary (large, black), in the bulk of lem at the next scale. −2 all tiles will be completely surrounded, so that all A perfect matching of −2 is evidently no easier to ob- unpairedT large black vertices overlap with paired large tain than for . However,T we can find a partial matching white vertices on adjacent tile boundaries. with the sameT method used previously: by performing 11 two deflations of −2 to obtain the tiling −4, and re- inflating with theT dimer inflation tiles. ThisT matches all vertices of − apart from the 8-vertices, and by iden- T 2 tifying dimers on − with augmenting paths on , an- T 2 T nihilates all monomers on 80- and 81-vertices of (the 8-vertices that deflate to non-8-vertices under twoT defla- tions). The remaining unmatched vertices correspond to the 8n>1-vertices of , and the bound on the monomer density on is decreasedT by another factor of ν : T 8v (2) 2 ρ ν8v. (6) FIG. 11. The ladder tiles: dimer inflation tiles of Fig. 10 with ≤ 8-vertices of the smaller tiles removed along with the edges To complete the proof of perfect matching, we apply this incident on them. The locations of the removed 8-vertices procedure to all scales of the AB tiling. From now on are marked with solid blue and red circles, the colours de- we will refer to a monomer on an 8n-vertex as an order-n noting the two bipartite charges. The yellow and blue shaded monomer. Finding augmenting paths between all order-n segments join up to form ladders which host dimers in a max- and order-(n + 1) monomers of corresponds to finding imum matching of AB∗. The edges in the unshaded regions augmenting paths between all order-(T n 2) and order- comprise membranes, which never host a dimer in any maxi- − mum matching of AB∗. The green shaded region forms stars (n 1) monomers of the −2 tiling, and by induction, to a matching− of the non-8-verticesT of the tiling (here of 16 edges around the absent 8-vertices, and can be perfectly −(n+2) matched in the bulk of AB∗. we identify order-n for n < 0 with theT non-8-vertices ac- cording to Fig. 8). At each matching of the non-8-vertices of −(n+2) via the dimer inflation tiles, the density bound namic limit. However, we also observed that finite re- onT monomers is reduced by a factor of ν . Since can 8v T gions admit monomers. Indeed, we argued that there be deflated an arbitrary number of times and ν8v < 1, must exist at least a single monomer on any finite patch the density of monomers must tend to zero in the ther- with 8-fold symmetry. A key step in our proof of perfect modynamic limit: matchings on the full tiling was to pair up monomers (2n) 2n (2n) on adjacent patches by augmenting the path connecting ρ ν8v , lim ρ 0, (7) ≤ n→∞ −→ them. Now, we argue that this protocol is linked to the and the tiling admits a perfect matching . existence of ‘pseudomembranes’ — finite closed loops of Physically, this hierarchy of perfect matchings arises edges that collectively host exactly one dimer in any per- from the fact that higher-order monomers are placed at fect matching. the centres of increasingly large D8 regions by the dimer Refs. 58 and 63 identified exact membranes hosting inflation, and, due to the symmetry, have to travel out- zero dimers separating unmatched regions with an ex- side of their region to be paired. D8 regions exist at all cess of bipartite charge of one sign. In contrast, these scales within the AB tiling, and so naturally monomers pseudomembranes are each comprised of edges which of equal orders have to be paired together at each step. collectively host a single dimer. However, like mem- This picture of monomers pairing at all orders has an branes, pseudomembranes capture how certain aspects interesting consequence. According to our proof, all finite of the quasiperiodic graph structure are encoded in the order monomers are matched after enough iterations of set of perfect matchings. The properties of membranes the dimer inflation: any 8-vertices that, for all successive and pseudomembranes can be formally understood in deflations, deflate to an 8-vertex (an ‘order- ’ vertex in terms of the Dulmage-Mendelsohn decomposition of the the terminology of Fig. 8), will never be matched∞ by this graph and its ‘fine’ generalization as discussed in Ap- process. A vertex that deflates to an 8-vertex, for all pendix A. Here, we provide a more intuitive picture based successive deflations, must be the 8-fold centre of an in- on first introducing an auxiliary ‘AB∗ tiling’, on which finitely large symmetric empire. However, their existence the existence of exact membranes and perfect match- does not spoil our proof: such monomers occur with zero ings can be seen clearly. We then demonstrate that the frequency on the infinite tiling, and can be moved arbi- exact membranes become pseudomembranes on the full trarily far from the centre of the region, as follows from AB tiling. Note that in contrast to Refs. 58 and 63 the matching process given above for any finite-order 8- where membranes confine monomers, the membranes in vertex. The monomer can therefore be moved ‘to the our AB∗ tiling separate perfectly matched regions. boundary’ of the infinite tiling, and the tiling can be said to be perfectly matched. A. Membranes in the AB∗ tiling

IV. MEMBRANES AND PSEUDOMEMBRANES In proving the existence of perfect matchings on the AB tiling, at each iteration we first matched all vertices As we have shown in the preceding section, the except those 8-vertices preserved by two deflations. In AB tiling admits perfect matchings in the thermody- completing the proof, we matched a fraction of these 12

FIG. 12. All vertex configurations of the ladder-decorated tiles of Fig. 11. This shows all possible ways in which ladder segments can join up to form ladders in the AB∗ tiling. The segments shaded with yellow and light blue in Fig. 11 match up to form ladders, whereas the green segments match up to form stars in the bulk of the AB∗ tiling. remaining vertices by associating augmenting paths be- formation of stars from the green regions, is apparent. tween 8-vertices with dimers on the twice-deflated tiling. We now argue that in every perfect matching, the stars A key aspect of this procedure is that only such aug- and ladders of the AB∗ tiling form perfectly matched re- menting paths between two 8-vertices survive as a dimer gions, separated by exact membranes which do not host at the next double deflation. Crucially, all information any dimers. Fig. 11 has dimers residing only on edges that correlates different inflation scales involves the 8- that belong to the stars and ladders. Now, we show that vertices. This observation naturally leads us to consider all vertices in each star (ladder) must be matched to ver- a simplified dimer problem on an auxiliary tiling that we tices in the same star (ladder) in all perfect matchings. ∗ dub the AB tiling, obtained from the AB tiling by re- Therefore, any edges external to the stars and ladders moving all 8-vertices. The proof of perfect matchings on (thick white edges in Fig. 11 and Fig. 12) are never cov- the AB tiling (Sec. III) automatically generates a perfect ered by dimers in any perfect matching and hence form ∗ matching of the AB tiling using the dimer inflation tiles exact membranes. of Fig. 10. The proof follows from two observations. First, stars We can construct a set of dimer-decorated tiles for the are closed loops of 16 vertices, with eight vertices in each ∗ bipartite subset (we denote the two subsets and ) AB tiling by deleting 8-vertices and their incident edges U V from the dimer inflation tiles for the AB tiling in Fig. 10. and no vertices in the interior (due to the removal of the ∗ In Fig. 11, the locations of 8-vertices are now marked 8-vertices in constructing AB ). For each star, all ‘exte- with solid blue and red circles to emphasize the absence rior’ vertices with edges to the rest of the graph (i.e. the of vertices, and the colours indicate bipartite charge as ‘points’ of the star) are of the same charge. The ‘inte- before. We have coloured plaquettes by their participa- rior’ vertices alternate with these and therefore are all tion in structures that emerge when the tiles are placed of the opposite charge. Since they cannot match with on the full tiling. The light-blue- and yellow-shaded pla- any other vertex, we must match each interior vertex quettes join to form regions which we call ladders (the with an adjacent exterior vertex. Therefore each star colour codes are made clear when we discuss ladders in hosts eight dimers forming one of two possible alternating detail in Sec. V). The green shaded regions form closed paths around the star (see Fig. 16a), leaving no exterior loops, which we call stars, of 16 edges and 16 vertices vertices unmatched. around the missing 8-vertices. To investigate all the dif- Second, ladder segments in the tiles of Fig. 11 always ferent ways in which the ladder segments match up to contain the same number of vertices of each charge, a form ladders in the graph, it is useful to refer to the ver- property that is therefore inherited by any section of a tex configurations of such dimer-decorated tiles for the ladder built up from these segments. Fig. 12 shows the AB∗ tiling, and they are shown in Fig. 12 (compare Fig. 5 vertex configurations for the ladder-decorated tiles, al- for the undecorated tiles). The continuation of light blue lowing us to read off all the ways in which different lad- and yellow regions into ladder segments, as well as the der segments can match up to form ladders. For each 13 ladder section in Fig. 12, all vertices with edges to other ladder segments are of the same charge, say of . While -vertices do have edges connecting them to stars,U those edgesV can never host dimers in a perfect matching, as we have already shown that the vertices in a star are always matched to vertices within the star. Since there are an equal number of vertices of each charge in any ladder, this immediately implies that all -vertices in a ladder must match to -vertices in the sameU ladder in a perfect matching to avoidV a contradiction. Thus, edges outside ladders and stars constitute membranes which never host a dimer in any perfect matching of the AB∗ tiling. These results imply that the partition function of the dimer problem on the AB∗ tiling decomposes into a prod- uct of partition functions of ladders of different lengths, and stars. This enables us to make several exact state- ments about the dimer problem on the AB∗ tiling in Sec. V. Ladder regions and membranes in any perfectly matched patch of the AB∗ tiling can be determined algo- rithmically by using the Dulmage-Mendelsohn decompo- sition described in Appendix A. Fig. 15 shows the ladders and stars for an 8-fold symmetric patch of the AB∗ tiling. All the ladders are closed loops with 8-fold symmetry. Note that a generic finite patch of the AB∗ tiling can have ladders that span the patch, terminating at the bound- aries. However, these ladders will generally close upon FIG. 13. Membranes in perfectly matched punctured 8 - completion of the boundary: infinite system-spanning 2n empires, which separate perfectly matched regions demar- ladders occur with zero frequency on the tiling. cated with different colours. An 8n-empire is the local em- pire of an 8n-vertex. A punctured 8n-empire is an 8n-empire without the central 8-vertex. On adding the 8-vertex back, B. Pseudomembranes in the AB tiling each membrane becomes a pseudomembrane which can host a single dimer on its edges. Top: membranes in a punctured ∗ 82-empire. Bottom: membranes in a punctured 84-empire, Armed with our results on AB , we now return to containing the 8 -empire. the full AB tiling, and show that the restoration of the 2 deleted 8-vertices transforms the membranes of AB∗ to pseudomembranes on AB. As we defined earlier, a pseu- domembrane is a connected set of edges which collec- perfectly matched subregions i are separated by mem- H tively host a maximum of one dimer between them in branes which are concentric with the deleted 8-vertex. any maximum matching. Pseudomembranes satisfy two This can be shown using arguments similar to ones used additional properties: (i) deleting all the edges in a pseu- in Sec. III and Sec. IV A. We have relegated the proof of domembrane disconnects the graph into two components; the statment to Appendix B. Here, we display the mem- (ii) pseudomembranes close with D8 symmetry, and are branes and perfectly matched regions for punctured 82 centred on 8-vertices. and 84-empires in Fig. 13. Anticipating the second point, we consider local em- If the central 8-vertex is reinstated to this perfect pires of 8n-vertices, which we refer to as 8n-empires. An matching, it will host a monomer. Moving this monomer 8n-empire is generated by inflating the local empire of out of any region enclosed by a membrane of the punc- an 80-vertex (Fig. 9) n times. We first remove the cen- tured empire converts the membranes to pseudomem- tral 8n-vertex from a 8n-empire, yielding what we term branes. This follows since moving the monomer requires a punctured 8n-empire. Certain D8-symmetric annular the augmentation of alternating paths of even lengths subregions , concentric with the 8-vertex of the punc- which terminate on the monomer. Since such paths tra- Hi tured 8n-empire (to be specified in Appendix B), ad- verse a membrane that originally enclosed the monomer, mit perfect matchings. If we choose boundary condi- they will place a single dimer across the former mem- tions to exclude vertices which lie out outside the outer- brane. Since only a single monomer was added to the most annular subregion n, then the region so obtained perfect matching, it follows that there can be no more also hosts perfect matchings.H In these perfect match- than one such dimer in any pseudomembrane. One might ings, vertices in each annular subregion i are perfectly wonder if a monomer can recross the membrane in the op- matched to other vertices within the subregionH . The posite direction and thereby place another dimer across Hi 14 the membrane; this is ruled out by some simple obser- V. EXACT RESULTS ON THE AB∗ TILING vations about the perfectly matched regions. Any two perfectly matched regions separated by a membrane have So far, we have only considered the single perfect the property that all edges in the membrane connect ver- matching that arises from our chosen decoration of the tices of the same bipartite charge in one region to vertices dimer inflation tiles, either from repeated application 2 of the opposite bipartite charge in the other region . If of the rule specified in Fig. 10 (in the case of the full the central 8-vertex is an -vertex, this implies that each U AB tiling), or single application of the rule specified in perfectly matched component of a punctured 8n-empire Fig. 11 (for the AB∗ tiling). However, in both cases the has only -vertices on its inner boundary and -vertices V U complete space of dimer configurations is exponentially on its outer boundary. When a monomer is introduced large in the number of vertices. The most general parti- by reinstating the central 8-vertex, the monomer recross- tion function specified in Eq. (1) is a weighted sum over ing a membrane would imply the existence of a perfectly all dimer configurations. Here we consider a statistical matched region such that its inner boundary is crossed ensemble initially restricted to perfect matchings of the twice by the monomer and its outer boundary is not AB∗ tiling, with all such configurations having equal sta- crossed by the monomer. This matches 2 -vertices of V tistical weight. In this way the partition function simply that region with vertices outside the region, leaving an counts perfect matchings. excess of 2 -vertices in the rest of the region, and is U The nontrivial physics of the dimer problem stems from therefore not possible in a maximum matching. the matching constraint, which makes the problem of enumerating all configurations on any graph notoriously In general, an 8n-empire hosts n+2 pseudomembranes concentric with the 8-vertex at the empire’s centre. The difficult. The degrees of freedom (dimers) are no longer independent as they are, for example, in spin models. 8n-empire also hosts additional pseudomembranes con- centric with other 8-vertices within the empire. These For planar graphs, there is an in-principle exact solu- 8-vertices are necessarily of order m < n. The edges tion [1, 29, 30, 37, 76]. However, this calculation, requir- comprising the pseudomembranes are those that form ing the evaluation of a certain Pfaffian, is often compu- membranes in the AB∗ tiling, as described in Sec. IV A. tationally demanding. On some periodic lattices, as we have noted, effective long-wavelength ‘height’ representa- Note that the results of the previous paragraph also tions allow the use of field-theoretic techniques to com- hold for 8n-empires with generic boundary conditions. pute monomer and dimer correlations. While it is pos- Punctured 8n empires admit a perfect matching only sible to construct a height representation for the dimer when specific boundary conditions are considered, other- problem on the AB tiling, there is no clear physical prin- wise they typically host a few monomers at their bound- ciple (comparable to ‘maximize flippable loops’ for many aries depending on the specific choice of boundary con- periodic dimer problems) that would allow us to infer ditions. While such monomers have a vanishing density a tractable local free energy density on which to base a in the thermodynamic limit, our construction of pseu- systematic height field theory. domembranes outlined above does not carry over. How- Accepting these difficulties, the AB∗ tiling introduced ever, when such an 8n-empire is embedded in a larger in the preceding section provides a powerful simplifica- system, the monomers at the boundaries are annihilated tion. The constraint that no dimers can sit on mem- with other monomers in a maximum matching (recall branes dramatically simplifies the enumeration of dimer that AB tilings host perfect matchings in the thermo- coverings, since the dimer partition function on the full dynamic limit), and pseudomembranes appear. 2D tiling factorizes into a product of partition functions for lower-dimensional quasiperiodic dimer models. Sim- As noted earlier, an 8n-empire has smaller 8m-empires ply put, there can be no correlation across the mem- (m < n) within it corresponding to 8-vertices of different branes. We can leverage this property to obtain an orders, both concentric and otherwise. All these empires asymptotically exact result for the free energy of dimers host their own pseudomembranes. The full AB tiling con- on AB∗. While this does not immediately yield an exact sequently exhibits a rich hierarchical structure of nested result for the AB tiling (since the latter lacks the exact pseudomembranes. Further, each region bounded by a factoring property) it provides important clues to aid our pseudomembrane acts as an effective unit which is con- numerical investigation of AB in Sec. VI. nected by a single dimer to the rest of the system. This hints at a possible scale invariance in the dimer problem, which we probe numerically in Sec. VI. A. Stars and ladders

As explained in Sec. IV, the structure of AB∗ is com- prised of stars, ladders and membranes. As the stars and ladders form a subset of the AB tiling, we expect these structures to obey the inflation symmetry. Fig. 14 2 This follows from the properties of Dulmage-Mendelsohn decom- shows how the ladder tiles of Fig. 11 inflate under σ position reviewed in Appendix A. (technically, to perform inflation on AB∗, the inflation 15

whole segment: (B1/2)2 = B. We are free to shift our inflation rule by B1/2 to the right, obtaining the more convenient form ξ : P A, A AB, B A4B. (9) → → → A 1D inflation rule ξ over the three-letter3 ‘alphabet’ (P, A, B) is then obtained if we further define ξ(MN) = ξ(M)ξ(N) (10) for any ‘words’ M,N. Due to the invertibility of σ, the inverse of ξ must also exist: we denote it ξ−1. The 2D inflation σ is therefore reduced to an effective 1D inflation ξ on the AB∗ tiling. The P segments always close into isolated stars with 8 D8 symmetry, i.e. P (with periodic boundary condi- tions). Each ladder either closes in a loop or spans the entire tiling — this follows by noting from the config- urations in Fig. 12 that ladders cannot branch or ter- minate in the tiling. Additionally, if a ladder closes, it does so with D8 symmetry. To see this, observe that FIG. 14. The inflation of the ladder tiles. We distinguish the inflation of the ladder tiles (Fig. 14) preserves the three distinct units that build the stars and ladders of the symmetry of the segments. Then, given a closed ladder AB∗ tiling: P , A, and B. White space separating segments loop, repeated deflation must eventually result in a loop represents impermeable membranes. Denoting the three dif- containing P segments, since ξ−1 only destroys P s. It ferent tile environments that appear in any segment as S, R, follows that ladders can only close with D8 symmetry. and R0, and noting that R0 is topologically equivalent to S for The possibility of a system-spanning ladder on the infi- 2 2 2 the purpose of matching, we have A = SRS, B = S R S . nite tiling is evident by considering the case of repeated The segments are marked on the tiles by coloured arcs of in- inflation of any single segment; the infinite ladder is ob- ternal angles π , π , π for segments P , A, B respectively, which 4 4 2 tained in the limit. The possibility of multiple system retain the symmetry of the original segment. spanning ladders is more interesting. Similar curves to those in Fig. 14 can be used to decorate the tiles of the Penrose tiling; in that case, Penrose and Conway have is performed on the AB tiling and then the 8-vertices independently shown that at most two system-spanning are once again removed, mapping back to AB∗). Rather curves can exist in a tiling [92]. We expect that a similar than work at the level of the tiles, we distinguish three statement holds for the Ammann-Beenker tiling. Such repeating units: the basic 2-edge unit of the star, which infinite ladders are very rare in any case — from the LI we label P (shaded green); a ladder segment with the property of AB, any finite patch of tiling can be taken to structure SRS (square-rhombus-square), which we label lie inside a closed ladder of large enough extent, wherein as A (shaded light blue); and a ladder segment with the all ladders must close. The existence of these rare lad- structure SR0RRR0S, which we label B (shaded yellow). ders will have negligible effect on the matching problem, Each unit connects at its two end-points with other units, and from now on we assume all ladders are closed with forming longer segments. We note the basic rhombus tile D symmetry. can appear in two distinct environments (R and R0): the 8 To streamline the remaining discussion, we define the R rhombi appear in the A segment or in the centre of star, P 8, to be the order-0 ladder, set A P , and hence- the B segment, and have a single vertex which only has 0 forth phrase our discussion in terms of≡ ladders of order edges to two other vertices in the ladder; the R0 rhombi n 0 according to the inflation hierarchy. With the D appear sandwiched between S and R in B. In the graph 8 symmetry≥ constraint, and assuming periodic boundary topology of the ladder, the R0 rhombi are equivalent to conditions, we obtain the structure of the order-n ladder the S. Therefore, for the purposes of matching we may as identify R0 S and only discuss S and R, so that the ∼ 2 2 2 n 8 8 structure of B is equivalent to S R S . We denote a tile Ln = (ξ (A0)) An. (11) or segment T repeated consecutively n times as T n. ≡ Thus the order-1 ladder corresponds to A8, the order-2 From Fig. 14 we can read off the inflation of the seg- ladder to (AB)8, the order-3 ladder to (ABA4B)8, and ments: so on. P A, A B1/2AB1/2,B B1/2A4B1/2. (8) → → → Although a one-to-one identification creates fractional- 3 Since P never occurs after the first inflation the alphabet is ef- power B segments, two of these always join to form a fectively two-letter. 16

with it, and centred on other 8-vertices of order m < n. These properties follow from the fact that all 80-vertices are enclosed by stars, and an order-n ladder inflates from a star. Consequently, the number of ladders at a given order n is in one-to-one correspondence with the num- ber of 8-vertices of order m n. This in turn can be computed by counting the number≥ of all 8-vertices on the tiling obtained by deflating n times. Given a tiling with N vertices, for N large, the n-fold deflated tiling has 2n 4 N/δS vertices, a fraction 1/δS of which are 8-vertices. ∼These statements become exact as N . Combin- → ∞ ing these results, we see that the number n of order-n ladders on an N-site tiling is given by N 1 n νnN with νn 4+2n as N . (14) N ≡ → δS → ∞

Here, νn is the frequency of order-n ladders, equal to the sum of frequencies of 8m-vertices for m n (so ν0 νv8 in the notation of section III). Note that≥ N refers≡ here to the number of vertices on the ‘parent’ AB tiling; the number of vertices on the AB∗ tiling is given by elimi- nating the N/δ4 8-vertices, with FIG. 15. A finite patch of the AB∗ tiling, in terms of stars S and ladders. An 8n-vertex is the centre of concentric ladders −4 lim N∗ = N(1 δS ). (15) of orders 0 ≤ m ≤ n, which close with D8 symmetry. N→∞ −

The order-n ladders are fractal structures in the limit B. Analytic calculation of the free energy of dimers n , and their fractal dimension can be derived from on the AB∗ tiling the→ preceding ∞ inflation rules. Denoting the number of A (B) segments in an order-n ladder as αn (βn), the growth We now compute the free energy of the dimer model under ξ is specified by the inflation matrix, on the AB∗ tiling, building on the properties of stars and ladders just described. Starting from a reference per- α  1 4 α  n+1 = n . (12) fectly matched configuration, all perfect matchings can βn+1 1 1 βn be enumerated by a sequence of moves which augment alternating cycles, each leading to a new perfect match- The largest eigenvalue of this matrix is 3, and so the ing. An alternating cycle cannot intersect a membrane, number of both A and B segments in a ladder grows by because augmenting the cycle would lead to the place- 3 under inflation, in the above limit. It would appear ment of two additional dimers on the membrane, which to follow that the length of a ladder likewise increases is impossible in a perfect matching. Therefore, the to- by a factor of 3 under inflation. However, there is a tal number of dimer configurations across the tiling can subtlety here in the fact that the orientations of A and be obtained as the product over the number of configu- B segments are reversed under a single inflation; this is rations on each ladder. That is, the partition function easily resolved by considering the action under ξ2. Then factorizes over ladders, with each ladder defining an in- the length of a ladder increases by a factor of 9, whereas dependent 1D dimer problem. Since all ladders of a given all edge lengths are scaled by the silver ratio δ2 . The S order n have the identical partition function Z , the par- (box-counting) fractal dimension d of the ladders is then n F tition function in the thermodynamic limit, Z , is simply 2 dF ∗ found by setting (δS) = 9, from which ∞ Y Nn 1 Z∗ = Z . (16) dF = 1.246 ... (13) n log3 δS ≈ n=0 This is similar to the case on the Penrose tiling, where While the partition function naturally diverges in the the membranes were found to have fractal dimension thermodynamic limit, the free energy density is directly 1/ log2 ϕ [58], where ϕ is the golden ratio. derivable from the partition function, and is bounded. We show a finite patch of the AB∗ stars and ladders The edges are the natural degrees of freedom on the in Fig. 15. Observe that an 8n-vertex is the centre of tiling. The average edge connectivity on the AB tiling concentric ladders of orders 0 m n. An order-n lad- is 4 as all tiles have 4 edges. This is not true for the der also encloses other ladders≤ which≤ are not concentric AB∗ tiling, with its 16-edge star tiles. However, in the 17 thermodynamic limit we have the number of edges of the difference in the number of dimers on the inner and ∗ −4 AB expressed in terms of N, as NE∗ = N(4 8δS )/2. outer legs of this square [93]. While the choice of square Thus, the free energy per edge of the AB∗ tiling− in the is arbitrary, the same choice must be made consistently thermodynamic limit is in comparing two configurations. There are then three possible sectors, w = 1, 0, and the problem reduces to ∞ ± ln Z∗ 1 X ln Zn counting the configurations in each sector on the ladders f∗ = lim = 2n , (17) defined by Eq. (11). We show example configurations for N→∞ − NE∗ −6(1 + 4δS) δ n=0 S the different winding sectors in Fig. 16. We start by noting that with w = 1, i.e. a square where we have used Eq. (14) and Eq. (15), and simpli- with a dimer on exactly one of its leg± edges, the entire fied powers of δ using Eq. (3). The factorization of S ladder configuration is forced, as can be seen in Fig. 16. the partition function, so that the free energy density in Thus there are exactly two ‘staggered’ states for each Eq. (17) can be expressed as a sum over contributions ladder, providing the analogy to the two star states. In of ‘free ladders’, is a remarkable property of AB∗. Note the w = 0 sector, for an n-th order ladder, let us take as that this decomposition of the partition function is non- (0) trivial: even after the 8-vertex deletions, the AB∗ tiling a reference configuration Cn the dimer arrangement of retains a single connected component. The free ladder the ladder tiles in Fig. 11. This state has the maximum decomposition emerges from the interplay of the dimer number of flippable plaquettes, maximising dimers on the constraint and the quasiperiodic geometry, which discon- rungs of the ladder. Such maximally flippable states are nects the configuration space of dimers. often referred to as ‘columnar’ states [5] in the litera- ture 4. Owing to the symmetry of each segment about The task of computing f∗ now reduces to computing the rhombi, the ladder has 2kA 3kB degenerate columnar the partition function Zn of the order-n ladder. The lad- ders and stars are effectively one dimensional, and this states, where kA (kB) is the number of A (B) segments. (0) suggests that the ladder partition function can be effi- Starting from Cn , all possible configurations within the ciently enumerated in terms of transfer matrices. To w = 0 sector can be found using all possible combinations construct such transfer matrices, we start with a brief of plaquette flips. discussion of the dimer configurations of the perfectly To outline the transfer matrix approach we take on the matched ladders. Given a perfect matching of any bipar- AB∗ ladders, we will first consider the simpler problem tite graph, all other perfect matchings can be obtained of enumerating coverings of a periodic ladder consisting by augmenting alternating cycles (Sec. II A). The aug- only of square tiles. In the w = 0 columnar state with mentation of some, but not all, alternating cycles can dimers on every rung, every square plaquette is flippable. be achieved by sequentially augmenting alternating cy- We can associate each plaquette in this initial configura- cles on the elementary flippable plaquettes. Recall from tion with a particle in the ground state, and a flipped pla- Eq. (2) that a plaquette is the set of four edges of a single quette (with dimers on the legs) with the excited state of tile, and a plaquette is flippable if it hosts dimers on par- the particle. Two neighbouring plaquettes cannot then allel edges. These local plaquette flips naturally divide be simultaneously flipped into an ‘excited’ state (with the configuration space into sectors, such that configu- respect to the reference configuration). In other words, rations within each sector are reachable from each other two excitations cannot neighbour one other. The same by such plaquette flips. For periodic lattices, such sec- constraint appears in chains of Rydberg atoms [97, 98]. tors are topological in origin, and transitions between The solution to this problem can be expressed in terms sectors are associated with alternating cycles which wind of Fibonacci numbers. The transfer matrix that enforces around one of the directions with periodic boundary con- the neighbour-exclusion constraint is ditions. Specializing to ladders, the partition function Z n 1 1 enumerates configurations that can be reached via alter- = , (18) 1 0 nating cycles remaining within the ladder. The smallest F move augments (flips) a single flippable plaquette. Dis- where the 0 (1) element represents the possibility for a tinct sectors are connected by loops which wind around particle to exist in the ground (excited) state. Then, for the whole ladder. We label them by a winding number a closed chain consisting of M plaquettes, the (w = 0) w. partition function is The 0-ladders (stars) are a special case since they lack ˜(0) M plaquettes. They admit only two possible configurations, ZM = Tr( ) = FM+1 + FM−1, (19) the two alternating cycles connected by augmentation of F the whole loop. Single plaquette flips cannot connect the two configurations. Thus, Z = 2 and we label the two 0 4 sectors by w = 1. For the higher-order ladders, let When dealing with dimers on periodic ladders, the equivalent ± states to our columnar states are also called ‘rung-dimer’ states us introduce the terminology ‘rung’ for an edge lying in in the literature, with ‘columnar’ reserved to describe dimers the ladder interior, and ‘leg’ for an edge on the inner or lining up along the ladder legs [94–96]. More generally, columnar outer boundary. Choosing any square plaquette on the is used to describe the maximally flippable states, which is the ladder as a reference, we define the winding number to be terminology we adopt here. 18

matrix, , multiplied by the Pauli matrix σ , F x 1 1 = σ = . (20) W F· X 0 1

(0) Note that in the Cn state, the transfer matrix from a square to a rhombus is still . We therefore have two transfer matrices: followingF a square plaquette, and following a rhombusF plaquette. With our definition of AWand B ladder segments, we will work instead with the two matrices 3 1 9 7 = , 2 2 2 = . A1 ≡ FWF 2 1 B1 ≡ F W F 5 4 (21) The inflation rule ξ then generalises straightforwardly to the generalized transfer matrices n and n via FIG. 16. (a) The two staggered configurations of the star. A B One can be obtained from the other by augmenting the alter- 4 n+1 = n n, n+1 = n n. (22) nating cycle. (b) The two staggered configurations of a ladder A A B B A B (only a segment of the ladder is shown). The winding number The dimer partition function of the order-n ladder in the w is defined to be the difference in the number of dimers on the w = 0 topological sector is inner and outer legs of a chosen square plaquette (here marked (0) (0)  8  with black cross). (c) Top: the reference configuration Cn Zn = Tr n . (23) for an n-th order ladder in the w = 0 columnar sector (only a A segment of the ladder is shown). We associate each plaquette To see the nontrivial role played by the rhombi, it is use- (0) in Cn with a particle in its ‘ground’ state (black circles). A ful to compare the partition function of the n = 1 ladder plaquette is ‘excited’ by augmenting the minimum 4-edge cy- to the partition function Z˜ obtained for the periodic lad- cle around the plaquette (‘flipping’ the flippable plaquette): der obtained by replacing the eight rhombi with squares. plaquettes that cannot be excited, e.g. the S2 plaquette, we From Eq. (19) the partition function Z˜ is simply show with a red outline. Bottom: exciting plaquettes (teal) results in a new perfect matching, and constrains a new set of Z˜(0) = Tr( 24), (24) plaquettes as unexcitable. For example, exciting the central M=24 F R means S2 can now be excited, but S1 cannot. whence we find that

Z(0)/Z˜(0) 0.363. (25) 1 M=24 ≈ with FM the M-th Fibonacci number, and F1 = F2 = 1. In other words, the rhombi remove a large fraction of The ladders in AB∗ are not periodic. Their sequences configurations available to the square-only ladder. The of rhombi and squares contain no repeating part other total free energy is now determined via the multiplication than that implied by their D8 symmetry; the sequences of the transfer matrices. Including the two additional tend to quasiperiodicity in the thermodynamic limit. staggered configurations contributed to each ladder by Consequently Eq. (19) does not hold, but will still be the w = 1 sectors, we find the total free energy density the transfer matrix between two neighbouringF square pla- from Eq.± (17) to be quettes (recall that two of the rhombi in the B segment " ∞  
# are treated as squares for this purpose, via the identifica- 1 X ln Tr 8 + 2 f = ln 2 + An . (26) tion R0 S). The rhombi in the ladder segments impose ∗ 2n −6(1 + 4δS) δS a constraint,∼ distinct from neighbour exclusion, which n=1 modifies the exclusion condition. This is easiest to see When working with transfer matrices in periodic sys- in the A segment, where a single rhombus sits between tems, translational invariance permits a closed-form ex- (0) two squares. Our choice of Cn breaks the symmetry pression for Z in terms of the transfer matrix eigenvalues. about the rhombus. One of the square plaquettes (S1) is In our case, apart from the eight-fold repetition required flippable in this configuration, while the other (S2) is not by the symmetry of the tiling, there is no periodically re- (see Fig. 16). Flipping the rhombus causes S to become peating unit smaller than . The two matrices and 1 An F unflippable, and makes S2 flippable. If we again associate (and therefore 1 and 1) do not commute, so can- (0) W A B every plaquette in the Cn configuration with the ground not be simultaneously diagonalised. Consequently, each (0) state of a particle, the neighbour-exclusion constraint is Zn calculation requires multiplication of an ever-longer reversed for plaquette S2: only when the preceding rhom- (exponentially growing) string of matrices which tends bus is in the excited state can S2 also be excited. The to quasiperiodicity in the thermodynamic limit. How- rhombus transfer matrix is therefore the square transfer ever, by iteratively computing in terms of Eq. (22), the 19 traces can be computed efficiently (approximately lin- Upon converting AB∗ back to AB by reinstating the early in n) to arbitrary finite order. Further, the infinite deleted 8-vertices, the exact membranes become pseu- series in Eq. (26) converges exponentially to its limit, on domembranes, and the exact factorization property is account of the exponential drop-off in the frequency at lost. Nevertheless, the existence of pseudomembranes which higher-order ladders occur. suggests that dimer correlations on the full AB tiling To prove this, first note that and (and hence could continue to have a rich structure. In this sec- A1 B1 An and n for n > 1) are unimodular matrices. Labeling the tion, we explore these correlations via a numerical study B + − maximum and minimum eigenvalues of n as an and an of the classical dimer model on finite patches of the + A− respectively, and those of n as bn and bn , it follows that AB tiling using the directed-loop algorithm. Originally in- − + − B+ + + an = 1/an and bn = 1/bn . The eigenvalues an , bn are troduced to efficiently sample space-time configurations bounded according to Eq. (22) as using quantum Monte Carlo algorithms [10, 99, 100],

+ + + + + 4 + this method has been adapted to sample the config- an < an−1bn−1, bn < (an−1) bn−1. (27) uration space of classical dimer models and to access Taking logarithms of Eq. (27) we have the matrix equa- their monomer correlations [35, 101]. The algorithm in- tion volves introducing two monomer defects into a maximum matching, and transporting one of the monomers around  +    +  ln an 1 1 ln an−1 an alternating cycle until annihilating it with its part- + < + . (28) ln bn 4 1 · ln bn−1 ner monomer (or with any other monomer, in graphs not admitting perfect matchings). Since detailed balance is We then write satisfied for these intermediate configurations with two + monomers, the corresponding partition function can be ln Zn < 8 ln an + ln 2 (29) sampled. This gives us access to monomer correlations where we take the ln 2 bound for simplicity. The max- without additional computational effort. For complete- imum eigenvalue of the matrix in Eq. (28) is 3, and we ness, we review the algorithm in Appendix C. + n 2 have asymptotically ln an 3 ; with 3 < δS, Eq. (26) is bounded by a convergent∼ geometric series. We can use Eq. (29) to bound the total error incurred by truncating A. Choice of samples and boundary conditions the ladder series at finite order M. Denoting the error M on the free energy by δf∗ we have As we have noted previously, since quasiperiodic sys- ∞ M 1 1 X ln Zn+M tems are not translationally invariant and do not admit δf∗ = 2M 2n . (30) periodic boundary conditions, some care must be taken −δ 6(1 + 4δS) δ S n=1 S in choosing appropriate finite patches for our numerical Diagonalising Eq. (28), and using Eq. (29), we have studies of the AB tiling. We consider finite patches with exact D8 symmetry. We anticipate that understanding n−1 + +
ln Zn < 3 4 ln a + 2 ln b matching problems on such patches can yield results rep- 1 1 − n + +
resentative of an arbitrary finite patch of AB tilings, since ( 1) 4 ln a1 2 ln b1 + ln 2, (31) − − those are characterized by an effective matching problem from which of D8 empires. Additionally, every finite patch of tiling is a part of a larger D8 empire of the infinite tiling. M 1
δf∗ < ΓM + ζ , 24(1 + 4δ )δ2M+1 For our largest simulations we consider an 84-empire. S S We find a small number of vertices near the bound- M + + ΓM = 3 (1 + δS)(4 ln a1 + 2 ln b1 ), aries of this 84-empire to belong to perfectly matched + + components of the Dulmage-Mendelsohn decomposition ζ = (δS 1)(4 ln a1 2 ln b1 ) + 2 ln 2. (32) − − of Appendix A. The central 8-vertex has 6 concentric + 4 + 2 (here we used (a1 ) > (b1 ) as can be readily pseudomembranes around it. We remove the parts of checked from Eq. (21)). Taking for example the first the tiling outside the largest pseudomembrane as they 40 terms in the ladder summation, we find f∗ = essentially generate boundary effects. As explained in 0.06884471896847(17). The quasiperiodic geometry of Sec. IV B, in any perfect matching of the AB tiling, pre- −AB∗ significantly lowers the number of available dimer cisely one dimer straddles the pseudomembrane to cor- states. relate the dimer configurations inside and outside the re- gion it encircles. Hence, it is reasonable to expect that calculations of observables defined entirely inside a re- VI. NUMERICAL RESULTS ON THE gion enclosed by a pseudomembrane will not be severely AB TILING affected by the rest of the tiling even in finite patches. With these modifications, the largest AB patch we con- In Sec. V we used the existence of membranes on the sider contains 15473 vertices and 30136 edges. In some AB∗ tiling to factorize the tiling’s dimer partition func- cases, we consider a smaller patch consisting of the region tion into contributions from ladders of different orders. inside the 4th pseudomembrane, which can be derived 20

FIG. 17. Left: Dimer occupation densities on an 84-unit, a finite D8-symmetric patch of AB tiling. The region is bounded by the 6-th pseudomembrane. The lighter colours indicate a rich nested structure of pseudomembranes associated with different 8-vertices and their local empires. The black lines emphasize the scale symmetry of the lattice (and consequently maximum matchings): they denote sections of the larger AB-tilings (T−2 and T−4) composed of the 82 and 80-units. Each 8n-unit is surrounded by a pseudomembrane, and hence connected to the rest of the graph by at most one dimer. They sit at the vertices of a larger AB-tiling (T−n), and act as effective units. Right: An 80-unit (top) and an 82-unit (bottom), with edges coloured to indicate dimer-occupation densities.

from an 82-empire. This sample contains 481 vertices branes concentric with it (counting the eight edges of and 872 edges. For reasons outlined below, we call the the octagon around the 8-vertex as a pseudomembrane). larger sample an 84-unit, and the smaller sample an 82- The pseudomembrane structure of the AB tiling can be unit. Note that all maximum matchings of this patch described as follows. Each pseudomembrane bounds a host a single monomer with the same bipartite charge as region (set of edges and vertices) which acts as an effec- the central 8-vertex. If the finite patch were extended to tive unit, such that only one dimer connects the region the infinite tiling, this monomer would annihilate with an to the rest of the graph. Motivated by this, we define th oppositely charged monomer in another region, creating an 8n-unit to be the region bounded by the n pseu- the single dimer crossing the pseudomembrane. Unless domembrane from the centre of the local empire of any otherwise specified, when we discuss ‘correlations on the 8n-vertex. An 8n-unit contains n + 2 pseudomembranes. AB tiling’ we mean those obtained on the finite patches The smallest membrane, the octagon around an 8-vertex, just described. and the eight edges within it, together form the (small- est) 8−2-unit. From the discussion of scale invariance of the AB tilings in Sec. II B, every 80-unit forms another B. Monomer and dimer correlations AB patch, larger by the square of the silver ratio. How- ever, these 80-units are mediated by the ladders (formed First we calculate the dimer occupation densities, with of 8−2-units) of Sec. V. This means that each 80-unit results displayed in Fig. 17. We clearly see that the pseu- matches to a ladder with a single dimer (living on a pseu- domembranes of Fig. 13 appear as rings of edges with domembrane). The edges of this ladder in turn match near-zero dimer density, concentric with the centres of lo- to other 80-units. Similarly, 8n-units form AB tilings mediated by ladders formed of 8 -units, for all even cal D8 symmetry. Every 8n-vertex has n+2 pseudomem- n−2 21 n. This effective description in terms of matching up e0 = a 8n-units through ladders suggests the possibility of non- e0 = b 1 e0 = c trivial long-range dimer correlations. )) − e0 = d , x

With this motivation, we investigate the connected cor- 0 2 e = e e 0

( − relations C(ei, ej) of dimers on edges e1 and e2, defined e0 = f to be max 3 e0 = g

C − ( 10 C(ei, ej) = n(ei)n(ej) n(ei) n(ej) , (33) 4

h i − h ih i log −

5 where n(ei) = 1 if the edge ei hosts a dimer, and − n(e ) = 0 otherwise. For the system sizes under con- 0.5 1.0 1.5 2.0 i log (x) sideration we find a striking result: connected dimer 10 correlations do not decay exponentially, as they would for either long-range ordered or disordered dimer cov- FIG. 18. Power law connected correlations of dimers for a ers. Characterizing C(ei, ej) is complicated owing to its D8-symmetric sample (an 84-unit). For several edges e0, lack of translational invariance, and its high degree of Cmax(e0, x), the maximum absolute value of dimer correla- inhomogeneity. We have already displayed, in Fig. 2, tions at a graph distance of x edges from e0 has a slow decay the dimer-correlation function C(e0, ej) for an edge e0 consistent a power law. The apparent dip at log10(x) ∼ 2 which connects a 4-unit to a ladder (formed of 2-units). corresponds to effects of the sample boundary. We have dis- While this clearly indicates slowly decaying dimer corre- played several choices of e0 in this figure: a, b and c are edges lations, we take up a more careful investigation of dimer- between 80-units and L2 ladders; d is a edge between an 80- correlations here. To characterize the decay of correla- unit and an L3 ladder; e is a edge on an L1 ladder; f connects an 82-unit to L4; g is a edge between an 82-unit and an L1 tions, we calculate Cmax(e0, x): the maximum value of ladder. A large-scale plot of the tiling showing all the different C(e , e ) such that the edge e has a graph distance of | 0 j | j source edges for this plot is provided in Appendix D. x edges from e0. We display this quantity, computed for seven different choices of e0, in Fig. 18. We see a slow decay consistent with power law asymptotic behaviour. 0 e0 = a We conjecture that this slow power law is a manifesta- e = b )) 0 tion of the description in terms of the effective matching 1 e = c , x − 0 0 problems, at all scales, in terms of 8n-units mediated by e e0 = d ( 2 e0 = e ladders formed of 8n−2-units as described above. We em- − phasize that these power-law-like correlations are neither max e0 = f C

( e = g homogeneous nor translationally invariant, and are un- 3 0 10 − related to the familiar power laws appearing in bipartite log 4 lattices with a continuum Gaussian action. However, not − all edges have such power law dimer correlations. For ex- 0.5 1.0 1.5 2.0 ample, inside an 82-unit, the edges connecting 80-units to log (x) L1 ladders do not have significant connected correlations 10 outside the 82-unit. Such bounded correlations are dis- played in Fig. 19. FIG. 19. Short-ranged connected correlations of dimers for a Next we turn to monomer correlations. For dimer D8-symmetric sample (an 84-unit). In contrast to Fig. 18, for problems on bipartite graphs with perfect matchings, the several edges e0, Cmax(e0, x), the maximum absolute value of monomer correlation functions can be obtained as the dimer-correlations at a graph distance of x-edges from e0, is partition function Zmm(r1, r2) of two monomers with op- sizeable only inside the 82-unit in which e0 lies. In this figure, posite bipartite charge situated on vertices r1 and r2. we have chosen e0 to be edges a − h, all connecting an 80-unit Our finite patches always have one extra vertex in the to an L1 ladder, inside a larger 82-unit. A large-scale plot of bipartite subset of the central 8-vertex. Without loss of the tiling showing all the different source edges for this plot is provided in Appendix D. generality we denote this the -subset. We define an UUVU auxiliary partition function Zmmm(r1, r2, r3) with three monomers on vertices (r1, r2, r3), where r3 belongs to the -subset, while r1 and r2 belong to . We now de- fineV monomer correlations for this gasU of 3 monomers. UV X UUV M (r1, r2) = Z (ri, rj, r2)(δri,r1 + δrj ,r1 ). (34) Correlations between the two -monomers are domi- ri,rj nated by the density of the singleU -monomer, which U UV our D8-symmetric samples host in the maximum match- Here, M (r1, r2) corresponds to a -monomer at r1 and U UV ing. To probe the physics associated with the creation of a -monomer at r2. We find that M (r1, r2) is non- monomer defects in the maximum matching, we measure monotonicV as a function of the graph distance between UV the correlation function r1 and r2. Irrespective of the location of r1, M (r1, r2) 22

V = 1.0 V = 1.0 − 0.04 m = 1 ) m m = 2 v C 6 m = 3 0.03 m = 4 Max(

1.2 m N m m 4

1.0 /N 0.02 /N m m v v

0.8 C C 2 0.6 0.01 0.4 0.00 0 0.2 0 1 2 0.8 1.0 1.2 1.4 0.0 T T

m m th FIG. 21. The intensive specific heat, Cv /N , of an m - order ladder with aligning interactions, as a function of the temperature T . Lines denote transfer matrix calculations, whereas the points represent data obtained from Monte Carlo FIG. 20. Density plot of the monomer correlation function simulations. Left: At V = 1.0, the ladders exhibit a sharp M UV (r , r ), Eq. (34), for a sample obtained from an 8 - 1 2 2 feature in the intensive specific heat, which scales with the empire. M UV (r , r ) is displayed as a function of r , with r 1 2 2 1 number of edges in a ladder (shown in inset). Right: At fixed at a vertex marked by a solid red circle. Non-monotonic V = −1.0, the intensive specific heat has a broad feature behaviour of the monomer correlation function is clearly vis- independent of the ladder size, associated with a crossover to ible, with larger values near the central 8-vertex. columnar states.

V = 1.0 V = 1.0 is strongly peaked for r2 within the first few pseudomem- − branes near the central 8-vertex. A typical profile of 82-unit

UV 0.03 84-unit /N 0.08 M (r1, r2) for a fixed (r1) is displayed in Fig. 20. We f N found similar non-monotonic monomer correlations on σ T Penrose tilings. We conjecture that such correlations 1.0 1.5 0.06

/N 0.02 /N are generic to systems with membranes and pseudomem- v v branes. Such non-monotonicities make it tricky to define C 0.04 C a sharp notion of monomer confinement. This can be 0.01 understood in terms of dynamics of the loop algorithm, 0.02 whose intermediate configurations sample this auxiliary partition function with two monomer defects. The loop 1 2 1.0 1.5 spends a lot of time trapped in the first few pseudomem- T T branes near the central 8-vertex. Once a loop enters the region bounded by a membrane, it can only exit through the same edge through which it entered the re- FIG. 22. The intensive specific heat, Cv/N of finite AB patches with aligning interactions, as a function of the gion. This also holds for pseudomembranes, save when temperature T . Left: At V = 1.0, the sharp peak seen in the loop enters the region through the single edge on the the case of the ladders in Fig. 21 is significantly broadened pseudomembrane which hosts a dimer. here. Inset: σNf /N (Eq. (37)), indicating fluctuations in the number of flippable plaquettes, has a broad feature which in- dicates a crossover to staggered states on the ladders. Right: C. Aligning Interactions At V = −1.0, the crossover to columnar states on the ladders is associated with a broad feature which does not scale with Finally, we briefly discuss the role of aligning interac- the size of the tiling. tions. To do so, we include an energy function which either favours or disfavours the presence of flippable pla- ings. The partition function is modified to quettes. Specifically, we consider the classical energy function which results when the Rokhsar-Kivelson hamil- X − C Z = e VNF ( )/T , (35) tonian of Eq. (2) has its kinetic energy t set to zero. We C maintain the variable alignment potential V which mim- ics the diagonal terms in quantum dimer models [2, 5]. where NF ( ) counts the number of flippable plaquettes Taking V positive (negative) penalizes (favours) configu- in a dimerC configuration . T is a temperature that we rations with a large number of flippable plaquettes. We introduce for convenience.C Only the combination V/T work entirely within the manifold of maximum match- has physical significance. Regardless of sign, a non-zero 23

value stabilizes ordered phases which break lattice sym- We calculate the standard deviation σNF of the number metries. The motivation of our investigation of this clas- of flippable plaquettes NF as sical model is to look for possible ordered phases, and to 2 2 investigate the form of the the associated transitions, in σNF = ( NF NF ). (37) the absence of periodic lattice symmetries. h i − h i To orient the discussion, first consider the effect of The results are displayed in the inset of Fig. 22. We aligning interactions on the ladders which make up the see that σNF /N exhibits a broad feature at T 1, which AB∗ tiling. Negative V favours the maximally flippable suggests that the dimers settle into configurations≈ resem- columnar states on the ladders, while positive V favours bling staggered states on the ladders without undergoing the two staggered states, which are in topologically dis- a phase transition. tinct sectors. That is, each staggered state cannot be reached from any other state by a sequence of local pla- VRK = 3,T = 1 quette flips. These ladders can be treated exactly using − transfer matrices. However, to account for aligning in- teractions, we must replace Eq. (18) and Eq. (20) with 0 1 1 1 0 0 ˆ = 0 κ 1 ˆ = κ 0 0 , (36) F κ 0 0 W 0 κ 1 where κ = e−V/T . Note that the limit of no interactions (κ 1) yields 3 3 transfer matrices for the problem considered→ in Sec.× V (compared to the 2 2 matrices presented there, these use a basis which× differentiates between flippable and non-flippable plaquettes), and re- covers the results presented there. First, we calculate the intensive specific heat for an mth-order ladder with N edges (this is obtained by tak- m VRK = +3,T = 1 ing derivatives of the free energy in the usual way). We show the result in Fig. 21. For V = 1.0, the intensive specific heat exhibits a broad peak which− does not scale with the size of the ladder. This feature corresponds to the loss of entropy incurred in the crossover to the colum- nar state. Each ladder has a macroscopically degenerate number of columnar ground states — in the language of Sec. V, each A segment contributes two states while each B segment contributes three. When V = 1.0, the intensive specific heat has a sharp feature at T 1.0, sig- nalling the onset of staggered states in the ladders.≈ As m shown in the inset of Fig. 21, CV /Nm Nm. This con- ventionally signals a first-order phase transition,∼ though the latter term should be used cautiously since one can only take a particular sequence of Nm. Note, however, not that this does drive a first-order transition in the 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 AB∗ tiling, since any thermodynamic singularity is sup- pressed by the exponential decay in the density of ladders FIG. 23. Dimer densities on an AB patch (82-unit) with align- with their order m. ing interactions. Left: at V = −3.0, the dimer densities re- We now compute the intensive specific heat cv = Cv/N veal a strong tendency to align along the legs of the ladders, of AB patches with N edges (Fig. 22) using Monte Carlo indicating that staggered configurations on the ladders are simulations. As in the case of the AB∗ ladders, for favoured. Right: at V = 3.0, the dimers tend to align along V = 1.0, cv has a broad feature at T 0.5 which the rungs of the ladders, indicating that columnar states on corresponds− to the onset of columnar-like configurations≈ the ladders are favoured. on the ladders. On AB, dimer correlations are no longer restricted to ladders, but since the ladders host most of Fig. 23 shows the dimer densities of an AB patch in the flippable plaquettes, tuning V enhances the role of the presence of aligning interactions of both signs. We the ladders even in this case. However, when V = 1.0, see that at large negative values of V the dimers align the sharp feature associated with the transition to the along the rungs of the ladders, while at large positive staggered states at T 1.0 in the AB∗ ladders (Fig. 21) values of V the dimers align along the legs of the ladders. is no longer present in≈ the AB patches. This suggests that typical dimer configurations of the full 24

AB tiling in these limits can be described in terms of the the other bipartite charge alongside perfectly matched cartoons of columnar and staggered states on the ladders. regions. The division between different regions is always formed of sets of edges which cannot host dimers in max- imum matchings. These cases, while still quasiperiodic VII. CONCLUSIONS and long-range ordered, begin to approach the generic results found in random or disordered bipartite graphs with two-dimensional embeddings, as characterised us- We have demonstrated that classical dimers on the ing the Dulmage-Mendelsohn decomposition [63–65]. It AB tiling admit perfect matchings in the thermodynamic is in this context that the results in the present study limit, with a rich structure linked to the interplay of truly stand out. The behaviour of the long-range dimer constraints with quasiperiodicity. A crucial feature of correlations we have identified in these graphs appears to our analysis is the identification of collections of edges be related to the discrete scale invariance implied by the whose dimer content can be strictly bounded from above quasiperiodic inflation rules of Fig. 4; quantifying this in any maximum matching. Previous work on the Pen- connection more precisely is a worthwhile objective. rose tiling identified exact membranes, sets of edges that host zero dimers in maximum matchings, separating re- Turning to the more thorny issue of quantum fluctu- gions hosting monomers (unmatched vertices) with dis- ations, the existence of columnar and staggered states tinct bipartite charge [58]. The present work extends for opposing signs of the alignment potential suggests this analysis in two ways. First, we identified exact mem- that upon including dimer resonance moves (t > 0 in branes on the auxiliary quasiperiodic AB∗ tiling obtained Eq. (2)), the phase diagram of quantum dimers on qua- by deleting 8-vertices from the AB tiling. These mem- sicrystals will resemble the Rokshar-Kivelson picture for branes can be understood in terms of the fine Dulmage- periodic lattices [2, 5]. Therefore we anticipate the exis- Mendelsohn decomposition of a bipartite graph, reviewed tence of at least a point in the quantum dimer phase dia- in Appendix A. Second, returning to the full AB tiling gram, between the columnar and staggered limits, where by replacing the 8-vertices, we found that membranes equal-time dimer correlations resemble those of the clas- become pseudomembranes, sets of edges that collectively sical models studied in this work. However, whether this host precisely one dimer in any perfect matching. point requires fine tuning, or instead represents the prop- The coexistence of perfect matchings and exact mem- erties of a robust phase of matter, is a much more del- branes allowed us to exactly compute the partition func- icate question. Historically, studies of dimer models on tion of the AB∗ tiling as the product of partition func- periodic bipartite lattices have made use of the height tions for disjoint dimer covers on ‘ladders’, sets of edges representation [21, 102]; in the quantum case, the height on which the dimers are not fully constrained. The pseu- action must be supplemented by instanton contributions domembrane structure of AB implies that there are ef- linked to the integer-valued nature of the height field. fective ‘units’ which exist at all scales which are con- Instantons destroy the long-range quantum dimer cor- nected by at most one dimer to the rest of the graph, relations in two spatial dimensions, rendering them ex- leading to long-range connected dimer correlations that ponentially short-ranged: in essence, this is Polyakov’s can be understood in terms of the discrete scale invari- argument for the absence of a deconfined phase of U(1) ance of the quasiperiodic tiling. Using classical Monte quantum lattice gauge theories in three spacetime dimen- Carlo simulations we found evidence for long-range and sions [103]. This means that the power law correlations highly heterogeneous connected dimer correlations. Fi- of the RK model require the fine-tuning characteristic nally, we demonstrated that the introduction of an align- of a multicritical point rather than the stability of a ro- ing interaction can favour various subsets of perfectly bust phase. The situation is less clear in the present case matched dimer configurations. This allowed us to iden- since, as we have noted above, there does not seem to be tify the quasiperiodic analogue of ‘staggered’ and ‘colum- an obvious local height action that characterizes max- nar’ states, linked to the dominant correlations along imally probable dimer configurations on quasicrystals. those sets of edges on AB that form ladders on AB∗. Consequently there is a possibility that long-wavelength The existence of perfect dimer covers on the AB tiling dimer correlations persist in the quantum problem. Ver- in the thermodynamic limit is in striking contrast with ifying this is challenging, since in order to make precise the situation in other two-dimensional quasicrystals [58]. statements one must approach the thermodynamic limit There are six minimal quasicrystals in two dimensions as via a discrete sequence of inflations, and we are forced identified in the classification scheme of Ref. 87. All ver- by quasiperiodicity to work with open boundary condi- tices of the Penrose tiling belong to regions of one or the tions. This means that the relevant computational cost other excess bipartite charge. The oppositely charged re- likely becomes prohibitive before finite-size effects have gions are separated by sets of edges which cannot host been suppressed. In light of this, further study of the dimers in maximum matchings, the prototypes for the classical problem to determine if there is a convenient, AB membranes in the present study. In some sense, how- possibly non-local, characterization of maximally proba- ever, it is the lack of perfectly matched regions which ble dimer configurations seem warranted, as this might makes the Penrose tiling unique. The remaining four 2D open a route to an analytical treatment. A more numer- quasicrystals contain regions with an excess of one or ically tractable direction is to explore dimer models on 25 the D8-symmetric ladders introduced in the context of the other closes. However, a key feature of mobile fracton the AB∗ tiling in Sec. V. Recall that each of the eight pairs is that their separation is fixed, while the monomer symmetry-related segments of the ladder forms a system pair has no such constraint. Very recent work [118] has which tends to quasiperiodicity in the thermodynamic extended the duality between fractons and elasticity the- limit in its own right. This presents an interesting av- ory [119] to quasicrystalline systems, but as yet it is un- enue for investigating quasiperiodic quantum dimer lad- clear whether this has direct implications for the results ders. Periodic ladders, and closely associated frustrated presented here. In type-II fracton phases the excitations spin ladders, have long been studied to shed light on the can only move along fractal subsets of the full system. phases and transitions of quantum dimers [94–96, 104– Loosely speaking, they are an instance where a compli- 106]. We expect studies of quasiperiodic ladders to be cated set of gauge constraints on a simple lattice leads similarly fruitful. Monomers in bipartite graph matching to emergent low-energy behaviour in which the natural problems have been mapped to exact zero-energy modes gauge-charged objects are fractals. Contrast this with of the hopping problem on the same graph [63, 107]. the present example, where a conventional Gauss-law- Here, the number of monomers hosted in a maximum like structure (imposed by the dimer constraint) leads to matching is associated with the number of zero modes, a setting where ‘gauge lines’ are themselves subject to while the monomer-confining regions are associated with fractal — and fractally distributed — barriers. It seems wave functions whose support is confined within a com- therefore that the study of dimer models on quasicrys- pact subgraph. The computed monomer densities and tals presents an intriguing counterpoint to fractons. In geometry of monomer-confining regions on the Penrose future, it will be interesting to explore whether proper- tiling [58] are consistent with density of zero modes and ties such as unusual topological robustness at finite tem- the nature of confined states obtained from investiga- perature and glassy dynamics characteristic of fractonic tions of hopping problems on the Penrose tiling [108– phases also emerge in the quasiperiodic dimer setting. 110]. Ref. [71] has recently computed a finite density of confined zero modes on the AB tiling; naively, this appears to be in conflict with our results that demon- strate that the AB tiling can be perfectly matched with ACKNOWLEDGMENTS vanishing monomer density in the thermodynamic limit. This apparent contradiction may be resolved by noting We thank Kedar Damle, Paul Fendley, and Shivaji that the zero modes obtained in Ref. [71] are what we Sondhi for insightful discussions. We acknowledge sup- term ‘fragile’: they move away from E = 0 on intro- port from the European Research Council under the ducing arbitrarily weak disorder in the hopping matrix European Union Horizon 2020 Research and Innova- elements. In contrast, the Penrose tiling hosts ‘strong’ tion Programme via Grant Agreement No. 804213- zero modes that survive to any disorder strength. For- TMCS (S.A.P., S.B.), and EPSRC Grant EP/S020527/1 mally, the monomer density computed in the dimer cover (S.A.P., S.H.S.). F. F. acknowledges support from the problem exactly equals the density of strong zero modes, Astor Junior Research Fellowship of New College, Ox- but is not linked to the density of fragile zero modes. The ford. J. L. acknowledges support from the Ogden Trust. fragility of the AB zero modes may be explicitly verified Statement of compliance with EPSRC policy framework by computing the spectrum of the random-hopping prob- on research data: This publication is theoretical work lem: for any nonzero randomness, exactly one zero mode that does not require supporting research data. from Ref. [71] survives (the strong mode associated with the unavoidable central monomer on 8-fold symmetric patches), with all the remaining modes moving to finite energy. Appendix A: Membranes in general bipartite graphs and the Dulmage-Mendelsohn decomposition Finally, we comment on a relationship between the ideas explored in this paper and fracton phases of mat- ter [111–117]. The latter are usually defined on transla- We provide a graph-theoretic picture of monomer- tionally invariant lattices, and blend topological features confining regions and their associated membranes in with sensitivity to geometry. Type-I fracton phases host maximum matchings. These regions turn out to be quasiparticle excitations which cannot move individually, components of the Dulmage-Mendelsohn decomposition, but which can combine into pairs or quadruplets to move which we will briefly review here. along lines or planes. There is a passing resemblance to A bipartite graph can be represented by a matrix G the membranes in AB∗, in which the minimum excita- with rows denoting one bipartite subset , and columns tion out of a perfect matching would be the deletion of denoting the other subset . An edgeU between an - a single dimer, creating a monomer-antimonomer pair. vertex i and a -vertex j correspondsV to a nonzero valueU V Membranes restrict each individual monomer to move on of the element Gij. Readers may be more familiar with a subset of vertices of the same bipartite charge. But the the graph adjacency matrix , a square matrix labelled pair together is free to move anywhere, with one or other by vertices of the graph, andA whose nonzero entries cor- monomer ‘opening a door’ through a membrane which respond to edges. In terms of G, the adjacency matrix 26

v is The boundary vertices of Gi belong to the -subgraph, and the edges connecting the boundary verticesV to the  0 G  = T . (A1) rest of the graph constitute membranes. A G 0 Membranes separating perfectly matched regions cor- respond to a finer decomposition of the perfectly matched The Dulmage-Mendelsohn decomposition can be suc- s s cinctly expressed in terms of a block-triangular factorisa- region G . G can be further block-triangularised as  s s s  tion (BTF) of the matrix G, achieved by a permutation G1 M12 ...M1n s s of its rows and columns. In general, such a BTF is of the  0 G2 ...M2n Gs =   . (A4) form  . . .. .   . . . .   h hs hv s G M M 0 0 0 Gn s sv G =  0 G M  . (A2) s v Each block Gi corresponds to a region of the graph such 0 0 G s that all vertices in Gi are perfectly matched to vertices within Gs in all maximum matchings. This implies that The matrix Gh has dimensions h h with h < h i U × V U V edges in the off-diagonal blocks M s are never matched in and the matrix Gv has dimensions v v with v > v. ij U ×V U V any maximum matching, and correspond to membranes Gh is a square matrix with dimensions s s, with U × V separating perfectly matched regions. s = s. The subscripts h, v, s denote horizontal, vertical andU squareV respectively. Having identified different kinds of membranes and monomer-confining regions with components of the In a maximum matching, all the -vertices in Gh are U Dulmage-Mendelsohn decomposition, we outline the pro- matched to -vertices in Gh, leaving behind h h V V − U cedure to compute the decomposition. Given any max- monomers. Similarly all the -vertices in Gv are matched to -vertices in G , leaving behindV monomers. - imum matching, find all vertices reachable from the - v v v v U verticesU in G are perfectly matchedU to−V -vertices in GU . monomers by alternating paths. G is the subgraph in- s s h This implies that none of the edges inV the off-diagonal duced by these vertices. Similarly, G is the subgraph blocks M hv,M hs,M sv are matched in any maximum induced by the vertices reachable from -monomers by s V matching: these edges constitute membranes. Note that alternating paths. G is the subgraph induced by ver- for a particular bipartite graph, some of these blocks tices unreachable from monomers by alternating paths. h v might not appear in the BTF: a graph with a perfect The further decomposition of G and G into monomer- matching would only have the block Gs, while the Pen- confining regions (Eq. (A3)) corresponds to computing rose tiling (which decomposes into monomer-confining re- their connected components. s gions) would only have the matrices Gh, Gv and M hv The decomposition of the perfectly matched G in its BTF. This is the content of the coarse Dulmage- (Eq. (A4)) requires a bit more work. Given two - s U Mendelsohn decomposition. vertices in G , being a part of a closed alternating path Monomer-confining regions (such as those studied defines an equivalence relation. Let the corresponding U U Refs. [58, 63]) as well as perfectly matched regions sep- equivalence classes be C1 ...Cn . Further, we denote the U V arated by membranes in our problem are components set of -vertices matched to -vertices in Ci by Ci . The V U U V of the fine Dulmage-Mendelsohn decomposition [64–67]. subgraph induced by Ci and Ci , for each i, correspond s s This involves a further decomposition of the matri- to the blocks Gi appearing in the decomposition of G (Eq. (A4)). These are perfectly matched regions sepa- ces Gh,Gv and Gs by row and column permutations. Monomer regions correspond to a block-diagonalisation rated by membranes (sets of edges which do not host dimers in any maximum matching). These equivalence of Gh and Gv. classes can be determined by first forming a directed  h  s G1 0 0 graph Gd corresponding to the perfectly matched sub- h s s  0 G2 0  graph G . To construct Gd, given a perfect matching of Gh =   . (A3) s   G , we first direct all unmatched edges from -vertices ··· h U 0 0 Gn to the -vertices. Then all matched edges are collapsed to a singleV vertex, labelled by the -site. This speci- h s U Each rectangular block Gi corresponds to a monomer- fies the directed graph Gd, with vertices labelled by - confining region with more vertices than vertices, vertices. The strongly connected components of this di-U V U U and hosting a corresponding number of monomers on - rected graph give us the equivalence classes Ci . Strongly vertices. All vertices on the boundary of such regionsV connected components of a directed graph can be effi- belong to the -subgraph, and all edges connecting such ciently determined using Tarjan’s algorithm. [120] This U s boundary vertices to the rest of the graph constitute allows an efficient determination of the components Gi membranes which never host a dimer. of the perfectly matched subgraph Gs— all -vertices in v v U G can be similarly block diagonalized into Gi . These each component are perfectly matched to -vertices in blocks correspond to regions confining monomers on - the same component, and the components areV separated vertices. Each block now has an excess of -vertices andU by membranes, which never host a dimer in any maxi- host a corresponding number of monomersU on -vertices. mum matching. U 27

vertices of the punctured 8n-empire belong to one of the i for some i. In an 8n-empire, the cen- tral 8-vertexH is surrounded by a star (we define H0 to be the star, 0 0), which in turn is sur- rounded by ,H and≡ a region L is surrounded by H1 Hi the region i+1. The inflation rule λ can be eas- ily read offH from the inflation of the ladder tiles in Fig. 14. Vertices in a ladder are no longer entirely matched to vertices within the same ladder (dimers are placed on the membranes). Howerver, it is con- venient to describe the inflation rules in terms of the ladder segments. A region i is comprised of certain closed ladders that followH from the inflation, as well as the links connecting them, which can be read off from Fig. 12 (previously membranes on the AB∗ tiling). We present the λ inflation rule in Fig. 24. From λ, all regions can be constructed Hi starting from 1. The first two regions constructed using the inflationH rules of Fig. 24 are displayed in Fig. 25.

• Second, each component i generated by the in- flation rule λ can be perfectlyH matched. Such a perfect matching can be constructed following the arguments of Sec. III used to construct a perfect matching of the AB tiling: contains ladder seg- Hi FIG. 24. Starting from H1, the order-1 ladder, an infla- ments which can be perfectly matched, using the tion rule λ generates concentric, D8-symmetric regions Hi = i dimer-decorated ladder tiles displayed in Fig. 11. λ (H1) which can be perfectly matched, such that all sites in The 8-vertices (located at the centres of green cir- an 8 -empire belong to one of the H for some i. We have n i cles in Figs. 24 and 25, not shown) now lie at presented the inflation rules in terms of ladder segments of −2 Fig. 11. As before, the ladder segments are represented by the vertices of the component i−2 = λ ( i), H 2 H coloured arcs of internal angles π/4,π/4,π/2 for the segments with edge-lengths larger by a factor of δs . Each P , A and B respectively. Each region H is comprised of edge of the larger i−2 component implies an odd- i H closed ladders as well as the links connecting those ladders. length alternating path between the corresponding For clarity, we have suppressed both the 8-vertices located at 8-vertices in i, and can be augmented to match the centre of the green circles (stars) and the links between the 8-vertices.H This follows directly from applying the ladder segments. the augmenting paths for the dimer-inflation tiles, Fig. 10, to connect up the (now reintroduced) 8- vertices in the ladder tiles (Fig. 11). Since both the Appendix B: Membranes in punctured 8n-empires star and order-1 ladder can be perfectly matched, all i can be perfectly matched. If boundary con- ∗ H In Sec. IV A, we showed that the AB tiling hosts a ditions are imposed on an 8n-empire such that all perfect matching, and membranes separate stars and lad- vertices outside the largest component n are ex- ders, which host dimers in the perfect matching. Here, cluded, then the bounded empire hostsH a perfect we consider punctured 8n-empires, obtained by remov- matching. Now we argue the existence of mem- ing the central 8n-vertex from the 8n-empire, and prove branes within the perfect matching. that when such 8n-empires are terminated with certain • Each component i has the property that if the D8-symmetric boundary conditions they host a perfect H matching. Further, concentric membranes around the central 8-vertex is (say) a -vertex, all vertices on the inner boundary of U(towards the central absent central 8-vertex separate perfectly matched com- Hi ponents, which we label as . These statements can be 8-verex) are -vertices while those on the outer Hi boundary areV -vertices. This can be seen by first proven as follows: U noting that this is true for 1. If i has ladder segments with -vertices atH a boundary,H inflations • Starting with 1, which we take as the first closed H of Fig. 24 resultU in segments with -vertices at the ladder (L1, in the language of Sec. V A) which sur- U rounds the absent 8-vertex and the star around it, boundary. there exists an inflation rule λ such that i+1 = In a perfect matching, vertices in the smallest i H λ ( 1) are mutually exclusive D8-symmetric re- component 0 (the star surrounding the absent gions,H concentric with the central 8-vertex, and all 8-vertex) mustH be perfectly matched to vertices 28

FIG. 25. Using the inflation rules of Fig. 24, we construct the first two inflated regions H2 and H3, starting from the order-1 ladder H1 ≡ L1. Ladder segments P , A, and B are represented by coloured arcs as in Fig. 14. The H regions are comprised of closed ladders as well as the links connecting those ladders. Each Hi region admits a perfect matching.

within 0. 0 has an equal number (8) of - matchings. Note that this approach works both for max- and -vertices,H H with only -vertices having edgesU imum as well as perfect matchings, so we develop the connectingV them to the restU of the graph. In a discussion without specializing to the perfect-matching perfect matching, this constrains the -vertices to case. Given a maximum matching, the algorithm gener- U match to -vertices on the interior of 0, lest the ates new maximum matchings as follows: -verticesV remain unmatched. For theH component V 1. Start with a maximum matching. Randomly pick 1, the outer boundary has only -vertices having H U a matched vertex s . Let s be matched to s . Set edges connecting them to vertices in 2 . At the 0 0 1 inner boundary, only -vertices have edgesH connect- si = s0 and sj = s1. ing them to vertices inV , but these edges cannot 0 2. Pivot: Randomly choose a neighbour of s , say s . be matched as verticesH in are always matched j k 0 (The probability governing this choice is described to vertices within . ThisH implies that vertices 0 below.) Remove the dimer on the edge (s , s ) and in are matchedH to other vertices in in all i j 1 1 place a dimer on the edge (s , s ). This is the el- perfectH matchings. H j k ementary step of the update, where a dimer ‘piv- Extending this argument implies that for all com- ots’ over the vertex sj from the edge (si, sj) to the ponents i, vertices of i are matched to vertices edge (s , s ). Note that in the first step, the dimer- H H j k within i. i are the perfectly matched compo- pivoting move leaves an extra monomer at s0. Hs H nents Gi of the Dulmage-Mendelsohn decomposi- tion reviewed in Appendix A. 3. Stop if sk hosted a monomer before the pivot.

We have shown that if we choose boundary conditions 4. Grow: if sk hosts a dimer (sk, sm) with sm = sj, the intermediate configuration has a monomer6 at of the 8n-empire which exclude vertices lying outside the the starting vertex s0, and an antimonomer (two outermost component n, then the 8n-empire hosts a perfect matching with nHconcentric membranes. dimers touching a site) at the vertex sk. Set si = sk, sj = sm, and go to Step 2 (Pivot). Intuitively, each step of the algorithm creates a monomer- Appendix C: The directed loop algorithm antimonomer defect-pair and moves the antimonomer around until it annihilates with another monomer. The In this appendix, we summarize the directed-loop value of sk when the procedure terminates determines algorithm for sampling dimer configurations/maximum which of two possible updates have been implemented: 29 sk = s0 corresponds to flipping the dimer-occupancies in maximum matching, but with one additional monomer a closed alternating path of edges (a loop update), while and one additional antimonomer. The loop update sam- s = s corresponds to transporting a monomer from s ples this partition function with the correct weights. This k 6 0 k to s0 (string update). is very nearly the quantity we are interested in when The utility of the algorithm lies in the fact that any understanding questions of confinement, though there two maximum matchings can be connected by a sequence one usually considers a closely related partition function of loop or string updates, and so it can sample the whole Zmm, which involves two more monomers than the maxi- configuration space of maximum matchings. mum matching. In fact, the loop-construction procedure To ensure detailed balance in the space of maximum outlined above could equivalently be described as creat- matchings generated by the update, we implement it in ing two monomers in a maximum matching, and prop- the enlarged configuration space that includes all max- agating one of the monomers until it annihilates with imum matchings as well as the intermediate configu- another monomer to give a new maximum matching. To rations generated by the algorithm. The latter corre- be precise, a step where the dimer pivots on the vertex sj spond to configurations which have an extra monomer- from the edge (si, sj) to the edge (sj, sk) can be equiv- antimonomer pair relative to a maximum matching. Im- alently described in terms of a monomer hopping from posing detailed balance at each step of the update yields the vertex sj to the vertex sm (which is matched to sk), the transition probabilities Pij;jk for a dimer to pivot on while another monomer is fixed at the starting vertex a vertex j from the edge (si, sj) to the edge (sj, sk): s0. The dimer on the edge (sk, sm) is moved to the edge (sj, sk) during this hop. However, the detailed balance equations are satisfied with respect to the partition func- wijPij;jk = wjkPjk;ij (C1) X tion Zma instead of Zmm. To sample from Zmm correctly Pij;jk = 1. (C2) we weight each intermediate configuration generated in (jk) the loop update (with monomers at sj and s0) with a P −1 factor of ( wjk) . Measurement of Zmm closely corre- wij is the weight contributed to the partition function by sponds to the monomer correlations as discussed in the a dimer on the edge (si, sj). In general, the system of main text. equations (C2) subject to the constraints (C1) is under- determined. It is necessary to look for solutions which minimize the probabilities Pij;ij of the loop retracing it- Appendix D: Source edges for dimer correlations self. Such solutions can be found using linear program- ming techniques [121]. If the dimers are non-interacting, To investigate connected correlations of dimers, we in- as is the case in most of this paper, the backtracking vestigated the quantity CMax(e0, x), the maximum abso- probabilities can be set to zero. In this case, for an n- lute value of the dimer correlation function at a distance coordinated vertex sj, Pij;jk = 1/(n 1) for all edges of x edges from e , in Sec. VI B. Fig. 18 shows slow de- − 0 (sj, sk) = (si, sj). cay of C (e , x), consistent with power laws, for many 6 Max 0 Since the algorithm respects detailed balance in different source edges e0. Fig. 19 shows that for some the extended configuration space with the monomer- other choices of source edges, the connected correlations antimonomer pair created while making the loop, it af- are bounded within pseudomembranes. For a 6-unit con- fords access to the partition function Zma. This involves sidered in Sec. VI B, we label the source edges considered configurations with the same number of dimers as the in Fig. 18 and Fig. 19 in Fig. 26.

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FIG. 26. Top: The set of source edges e0 considered in Fig. 18, which are representative of edges for which connected correlations of dimers decay as power laws. Bottom: The set of source edges e0 considered in Fig. 19, for which connected correlations of dimers are bounded within pseudomembranes. The colour of the edges indicate dimer occupation density to reveal the structure of pseudomembranes. We use the D8-symmetry to choose (and display) source edges within a wedge— eight of these wedges make up the whole sample.

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