# Discrete scale invariant fixed point in a quasiperiodic classical dimer
model

Sounak Biswas  Institut für Theoretische Physik und Astrophysik, Universiät
Würzburg, 97074 Würzburg, Germany  Rudolf Peierls Centre for Theoretical
Physics, Parks Road, Oxford OX1 3PU, United Kingdom  S. A. Parameswaran
Rudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford OX1 3PU,
United Kingdom

###### Abstract

We study close-packed dimers on the quasiperiodic Ammann-Beenker (AB) graph,
that was recently shown to have the unusual feature that hard-core dimer
constraints are exactly reproduced at successive discrete length scales. This
observation led to a conjecture that it would be possible to construct an
exact real-space decimation scheme where each iteration preserves both the
quasiperiodic tiling structure and the constraint. Here, we confirm this
conjecture by explicitly constructing the corresponding renormalization group
transformation and show, using large-scale Monte Carlo simulations, that the
dimer distributions flow to a fixed point with non-zero dimer potentials. We
use the fixed-point Hamiltonian to demonstrate the existence of slowly
decaying dimer correlations. We thus identify a remarkable example of a
classical statistical mechanical model whose properties are controlled by the
fixed point of an exact renormalization group procedure exhibiting discrete
scale invariance but lacking translational and continuous rotational
symmetries.

##  I Introduction

Figure 1: A finite patch of the quasiperiodic Ammann-Beenker tiling (or
graph). The tiling is built out of two kinds of plaquettes: a square and a
rhomb. The plaquettes have the same edge length; the rhomb has internal angles
$3\pi/4$ and $\pi/4$. The tiling has no symmetries of translation. The grey
shaded region shows the local empire of an 8-vertex: the set of tiles which
always simply connected to an 8-vertex in the tiling

Scale invariance is a striking phenomenon that appears in various guises
across a range of problems in physics and beyond [1, 2]. Perhaps its most
familiar setting is in the context of critical phenomena, where it is one of
the hallmarks of continuous phase transitions, both classical and quantum [3,
4, 5]. In the standard lore, critical fluctuations are correlated on all
length scales, making their long-wavelength asymptotic properties — that
average over many correlated degrees of freedom — insensitive to microscopic
details. This leads to striking universality across superficially very
different problems. Similar arguments also apply to stable critical phases
that lack long-range order, as in the two-dimensional XY model. This intuitive
link between scale invariance and universality can be made more quantitative
via the renormalization group (RG) [6, 4, 1]. The RG describes a class of
iterative procedures for increasing the length scale over which a system is
studied, while discarding all but the essential data about smaller scales.
Fixed points of such rescaling procedures that describe the universal
behaviour of critical points or stable critical phases usually exhibit
continuous scale invariance, signalled by the divergence of the correlation
length, i.e. the distance beyond which correlations become statistically
independent.

An implicit assumption in nearly all RG approaches and in the preceding
discussion is that as a lattice model is coarse-grained, its microscopic
details cease to matter and therefore the emergent scale invariance is
equivalent to that of a continuum field theory. Indeed, often one simply
asserts that a continuum description of the fixed point exists, and then
justifies the neglect of lattice-scale effects a posteriori by showing that,
in RG parlance, they are irrelevant perturbations, i.e. shrink under
successive iterations of the RG. Long-wavelength properties controlled by such
a continuum fixed point exhibit continuous rotational and translational
symmetry, and are invariant under arbitrary rescalings of the coordinates.
Often these combine to generate invariance under a larger group of conformal
transformations, with particular significance in two dimensions [7, 8, 9].

In this work we identify a rare exception to these general rules: namely, a
statistical mechanical problem that admits an exact RG fixed point that lacks
translational symmetry, and is only invariant under discrete point-group
rotations. Most strikingly, however, the fixed point only has discrete scale
invariance (DSI), under rescaling distances by an even power $\delta^{2n}$ of
the silver mean $\delta=1+\sqrt{2}$. This is in striking contrast to the
behaviour of RG fixed points of continuum field theories.

The new fixed point combines two ingredients: quasiperiodicity and
constraints. We study a model of close-packed dimers [10, 11] on the
quasiperiodic Ammann-Beenker (AB) tiling [12, 13], (Fig. 1) with the usual
constraint that exactly one dimer touches each site111On finite geometries the
constraint may be modified to allow a single unmatched site, as we clarify
below., defining a so-called maximum dimer covering. (In the graph theory
literature, dimer coverings are often termed [vertex] matchings [14] since
each dimer uniquely matches a pair of vertices; we will use these terms
interchangeably below.) Constrained models resist a naive application of the
RG, since hard constraints usually soften on coarse-graining. Hence such
problems are usually tackled by reformulating them in terms of new variables
that automatically satisfy the constraint. For example, two-dimensional dimer
models on periodic lattices can be mapped to “height models” that can then be
studied by taking a continuum limit [15, 16, 17, 18]. Similarly,
quasiperiodicity of the lattice is often an irrelevant perturbation to a
continuum fixed point [19]: intuitively, the distinction between periodic and
quasiperiodic structures is averaged out at the longest length scales. (This
can be formalized by considering the ‘wandering exponent’ of averages over
quasiperiodic regions.) Surprisingly, however, the combination of
quasiperiodicity and constraints resists such simplifications. The
quasiperiodic structure forces the height description to have a complicated
nonlocal free energy. As we show below, the constraint in turn forces the RG
to retain information about the inflation properties of the underlying
quasiperiodic tiling, with DSI as a natural consequence.

The fixed point we identify in this paper cannot be readily accessed from a
continuum limit, since such theories usually have continuous scale and
translational symmetries. Hence the RG scheme must work directly with lattice
variables. Consequently, our approach is rooted in an older yet arguably more
physically transparent RG procedure: the “block spin” decimation scheme
originally proposed by Kadanoff and Migdal in the analysis of local spin
models [20, 21, 22]. Similarly to the block-spin scheme, we work directly in
real space, and in each iteration of the RG identify subsets of degrees of
freedom (dimers) that define new degrees of freedom at the next scale. The
choice of scale transformation $x\to\delta^{2}x$ in each step is dictated by
the discrete scale symmetry encoded in the “inflation rule” that generates the
AB tiling222The tiling is self-similar under a pair of inflations..
Remarkably, our RG transformation preserves the single-dimer-per site
constraint exactly at each successive scale, leading to a direct RG map
between dimer models at successive length scales. The well-known failure of
the block-spin approach to eliminate lattice-scale information now emerges as
a feature, since this enables the retention of quasiperiodic structure
inherent to the fixed point. We circumvent the technical challenges usually
encountered in implementing real-space blocking transformations by using
large-scale efficient Monte Carlo sampling of dimer correlations to implement
the RG transformation explicitly. This strategy leads to a huge variety of
algorithms varying in the specifics of the choice and implementation of the RG
transformation, and are often collectively referred to as the Monte Carlo
Renormalisation Group (MCRG) methods [23, 24, 25].

The self-similar fractal properties of quasiperiodic systems are well-known:
for instance, the inflation procedure at the root of our RG decimation is one
example of such self-similarity. However, the self-similarity of the dimer
model defined on the quasiperiodic AB tiling lies beyond this: requiring that
the dimer partition function reproduce itself, including constraints, under
discrete scale transformations is highly nontrivial. DSI at the level of the
partition function does not emerge in other problems: for example,
implementing a real-space RG transformation on Ising and Potts ferromagnets on
quasiperiodic tilings do not lead to a DSI fixed point [26, 27]. Lattice
models where exact DSI emerges are typically defined on cousins of the “Bethe
lattice” [28], with tree-like hierarchical structures that cannot be embedded
in any finite dimension [29, 30, 31]. This is clearly in contrast to the
quasiperiodic AB graph, which is constructed from a tiling of the plane.
Classical lattice statistical mechanics problems with DSI fixed points are
rather unusual; the only other example we are aware of is from a recent work
on DSI percolation transition [32].

Although one aspect of our RG transformation — the exact reproduction of the
dimer constraint at successive RG scales — was motivated by graph theoretic
considerations in a recent paper by us and others [33], our construction of an
explicit RG transformation and the determination of the fixed-point
Hamiltonian and its correlation structure represent a substantial advance,
confirming and substantially expanding on the ideas proposed in that previous
work. The fixed-point structure we identify and explore represents a striking
departure from the known universality classes studied in classical statistical
mechanics. Parallel and complementary work using machine-learning methods to
identify the effective degrees of freedom discover them to be hardcore dimers
without using any prior information about the model [34]. While this feature
is built into our RG transformation by hand, this enables us to explicitly
calculate effective Hamiltonians, characterise fixed points, and study
critical phenomena associated with our model.

The rest of this paper is organized as follows. In section II, we briefly
introduce the dimer model on the Ammann-Beenker tiling and review how the
effective dimer constraint emerges at all scales. In Section III.1, we
explicitly construct a Renormalisation Group transformation to coarse-grain
the dimer problem on the AB graph. We implement this RG transformation with
the aid of extensive Monte Carlo simulations, and show that the distribution
of effective dimers thus obtained reaches a fixed point under the RG
transformation. In Sec. IV, we calculate the effective Hamiltonian describing
the coarse-grained dimers. We show that the couplings flow to a fixed point
under our RG transformation, establishing a fixed point with DSI. This enables
us to explicitly write down the fixed point Hamiltonian controlling the dimer
model on the AB tiling. In Sec. V, we investigate the properties of the fixed
point Hamiltonian. We first explore the consequences of the fixed point theory
on long-range connected correlations of dimers. We show that dimer correlation
functions are consistent with power-laws with log-periodic modulations, which
signals a critical point with DSI. Finally we investigate the loop ensemble
defined by the overlap graphs of two decoupled dimer models; we show
conclusive numerical evidence of the these loops being critical, albeit with
discrete, instead of continuous, scale-invariance. We calculate the critical
exponents associated with this loop-ensemble. In Sec. VI, we summarise our
progress and outline some future directions directly motivated by our results.

##  II Dimers on the AB tiling

We begin with a brief introduction to the properties of AB tilings. We also
review the results of Ref. [33] on the dimer problem on the AB tiling, to
motivate the RG perspective adopted in the balance of the paper.

###  II.1 AB graphs and discrete scale symmetry

Figure 2: Inflation rules $\sigma$ which generate the AB tilings. Given each
square or rhomb tile, $\sigma$ first decomposes each tile into a smaller set
of tiles as shown, followed by rescaling so that the new tiles are of the same
size as before. One can also consider the inverse process $\sigma^{-1}$,
called deflation which involves composing the smaller tiles back to form the
larger tiles, followed by similar rescaling.

Figure 3: _Left_ : The AB∗ graph is obtained from the AB graph by removing all
the 8-vertices (marked by solid blue circles). The vertices in the AB∗ graph
can be partitioned into mutually exclusive sets: loops of 16 sites called the
stars, and quasi-1D objects which we call ladders. The stars can be matched
easily. The ladders can be further decomposed into two kinds of segments (they
are shaded with pink and green). By matching each of these segments
independently, the whole AB∗ graph can be matched up. _Right_ : Once the AB∗
graph is matched up, it remains to match up the 8-vertices by pairing them up
with alternating paths. The 8-vertices lie on the positions of another,
larger, AB graph with edge lengths larger by a factor of $\delta_{s}^{2}$; the
edges of this graph are shown by thick grey lines. It can be shown that pairs
of monomers corresponding to edges of the larger AB graph can always be
matched up by augmenting alternating paths between them; thereby reducing the
problem to that of matching up vertices of the larger AB tiling. This can be
iterated repeatedly to match up the whole AB graph, hinting at possible
effective matching problems of AB graphs at successive lengthscales.

The AB tiling is a quasiperiodic tesselation of the plane by square and
rhombus plaquettes. The sides and corners of these plaquettes define the edges
and vertices of a quasiperiodic graph, that will be the focus of our study.
(Henceforth we use graph and tiling interchangeably; the meaning will be clear
from context.) Although there are many different prescriptions to generate
quasiperiodic tilings, we focus on the approach where DSI is most clearly
evident: “inflation”, a procedure that generates a larger quasiperiodic tiling
from a given “seed” tiling. Consider an initial such seed, denoted
$\mathcal{G}^{0}$. The “inflation map” $\sigma$ provides a rule for decorating
each tile of $\mathcal{G}^{0}$ with new edges and vertices (and deleting some
edges), followed by a spatial rescaling by a factor $\delta$, such that the
area of each distinct tile type is preserved, generating a larger tiling
$\mathcal{G}^{1}=\sigma(\mathcal{G}^{0})$. Continuing this process iteratively
generates an AB tiling of the infinite 2D plane as the number of iterations,
$N\to\infty$.

In the $N\to\infty$ limit, a single inflation transformation $\sigma$ maps a
tiling to a $\pi/2$ rotated version of itself scaled by a factor $\delta$.
Applying the transformation twice, i.e $\sigma^{2}$ maps a tiling to an
identically oriented one that is bigger by a factor $\delta^{2}$. Evidently,
the existence of this inflation procedure demonstrates DSI of the
quasiperiodic tiling.

Since the RG ethos is to eliminate degrees of freedom, it is more natural to
consider the inverse “deflation” maps $\sigma^{-1}$ and $\sigma^{-2}$, which
shrink the tiling. While both $\sigma^{-1}$ and $\sigma^{-2}$ generate DSI, we
will focus on the latter, as it leads to a more natural RG rule due to the
behaviour of vertices under this transformation, which we now describe.

To understand the special properties of $\sigma^{-2}$, first observe that AB
tilings have 7 “types” of vertices, each associated with a specific
configuration of tiles surrounding it: $3$, $4$, $5_{A}$, $5_{B}$, $6$, $7$
and $8$, the names reflecting the coordination numbers of the vertices. All
vertices of a specific type develop a specific neighbourhood of tiles under
inflation ($5_{A}$ and $5_{B}$ imply the development of different
neighbourhoods of tiles). Crucially, the action of $\sigma$ changes the type
of existing vertices while also adding new vertices and removing some edges.
Under the action of $\sigma^{2}$, vertices of all coordination numbers on the
original tiling $\mathcal{G}_{0}$ map to $8$-fold coordinate vertices
(henceforth, $8$-vertices) of the double-inflated tiling
$\sigma^{2}(\mathcal{G}_{0})$. Inverting the map, an immediate corollary is
that only $8$-vertices survive two deflations. In other words, given an AB
tiling $\mathcal{G}$, the 8-vertices lie on the vertices of another AB tiling
$\sigma^{-2}(\mathcal{G})$, whose lengths are larger by a factor of
$\delta_{s}^{2}$. We can therefore define a “decimation” transformation where
removing all vertices but the 8-vertices gives us the same tiling rescaled by
$\delta_{s}^{2}$. This is the decimation we will use to define our RG
transformation in the next section to generate effective Hamiltonians on a
sequence of decimated graphs with lengthscales given by integer powers of
$\delta_{s}^{2}$. This RG transformation is naturally defined with the double
deflation $\sigma^{-2}$, rather than the single deflation $\sigma^{-1}$ for
which no comparable simplification of the vertex mapping is possible.

Since 8-vertices will play an important role in our discussion, it is helpful
to introduce some additional nomenclature relating to them. We mentioned that
only $8$ vertices survive two deflations; we can generalize this idea to
define an order-$n$ 8-vertex, or an $8_{n}$ vertex, as an $8$ vertex which
survives exactly $n+2$ deflations, remaining an $8$-vertex for $n$ of these
deflations. $8_{n}$-vertices map to $8_{0}$ vertices under $n$ deflations,
while $8_{0}$-vertices map to $8_{n}$ vertices under $n$ inflations. Like all
vertices in a quasiperiodic tiling, 8-vertices are associated with a “local
empire”, shown in Fig. 1, comprising the set of tiles which appear in the
neighbourhood of _each_ 8-vertex, and are simply connected to them. We see
that the local empire of the 8-vertex exhibits $D_{8}$-symmetry associated
with eightfold rotations. Since an $8_{n}$ vertex is obtained by inflating an
$8_{0}$-vertex $n$ times, and the inflations preserve the $D_{8}$ symmetry of
the local empire, it is clear that an $8_{n}$ vertex has a $D_{8}$-symmetric
local empire whose radius scales as $\delta_{s}^{n}$.

###  II.2 Structure of perfect dimer covers on AB graphs

We now summarize some key properties of perfect dimer coverings or perfect
matchings on AB graphs. Perfect matchings are configurations where every
vertex participates in exactly one dimer. The existence of such covers on the
AB tiling is nontrivial: other quasiperiodic graphs are known to host a finite
density of monomers, i.e. vertices that participate in no dimers, even in
maximum dimer covers [35] (those with the largest possible number of dimers).
On periodic graphs that admit perfect dimer coverings, one can first construct
one on a suitably chosen finite patch and then extend it to the whole graph by
periodic repetition, but this construction is obstructed by quasiperiodicity.
One must instead adopt different strategies; since the construction of perfect
dimer coverings on AB tilings is central to the rest of out discussion, we now
briefly review the arguments involved.

Let us begin with an infinite AB graph $\mathcal{G}$ a for which a perfect
matching is to be constructed. To proceed, we first consider an auxiliary
graph, $\mathcal{G}^{*}$, which is obtained from $\mathcal{G}$ by removing all
the 8-vertices from it. (We will denote by AB∗ graphs constructed by removing
8-vertices from AB graphs.) A perfect dimer cover on $\mathcal{G}^{*}$ and
other AB∗ graphs can be constructed by first decomposing the vertices of the
graph into non-overlapping quasi-1D objects, termed stars and ladders,
displayed in Fig. 3. The stars are rings of 16 vertices which can be easily
perfectly matched. The ladders are aperiodically repeating sequences of two
different kinds of segments (displayed as plaquettes shaded with green and
pink), and a perfect dimer cover on ladders can be constructed by
independently assigning perfect dimer covers to these segments. Applying these
procedures on each star and ladder gives us a perfect matching on
$\mathcal{G}^{*}$. It is possible to prove that in any perfect dimer cover on
an AB∗ graph, all dimers lie entirely within the ladders and the stars; there
is never a dimer on the edges that link vertices belonging to different stars
or ladders. This leads to a decoupling of the dimer partition function on AB∗
graphs into the partition functions on individual ladders and stars, enabling
transfer matrix computations of its statistical mechanical properties [33].

Figure 4: $8_{n}$-regions in AB graph for $n=0,1,2,3$: for each region,
exactly one dimer connects vertices within the region to vertices outside.
This imposes an effective dimer constraint on the region as a whole, making
the $8_{n}$-regions effective vertices.

Lifting a perfect matching from the AB∗ graph to the full AB graph results in
an imperfect matching with a monomer (i.e., a vertex with no dimers touching
it) on each 8-vertex of AB, since these were absent by construction on AB∗. To
construct a perfect matching on the AB graph $\mathcal{G}$, we have to
eliminate these monomers. To do so, we must find “alternating paths” between
the two monomers and “augment” them. Here, an alternating path between two
vertices is a sequence of edges terminating at the two vertices such that
every alternate edge hosts a dimer; augmenting an alternating path between
monomers involves flipping the occupancy state of dimers on each edge on the
path, thereby annihilating the two monomers while increasing the number of
dimers by one. As explained above, the 8-vertices of the AB graph
$\mathcal{G}$ lie on the vertices of the twice deflated graph
$\sigma^{-2}(\mathcal{G})$. It can be shown that starting from the perfect
matching in the auxiliary $\mathcal{G}^{*}$ graph, one can always annihilate
monomers at any pair of 8-vertices which corresponds to an edge of the twice-
deflated graph $\sigma^{-2}(\mathcal{G})$(Fig. 3). We are left with the task
of annihilating these monomers pairwise, which is equivalent to the problem of
constructing a perfect matching of vertices (a dimer cover) on the graph
$\sigma^{-2}(\mathcal{G})$. Thus, the problem of constructing perfect
matchings in $\mathcal{G}$ has been reduced to the problem of constructing
perfect matchings in $\sigma^{-2}(\mathcal{G})$. This can be treated exactly
as $\mathcal{G}$ was treated, i.e., by first matching the AB∗ graph
corresponding to $\sigma^{-2}(\mathcal{G})$ leaving monomers on the
$8$-vertices of $\sigma^{-2}(\mathcal{G})$. Annihilating these monomers are
now equivalent to the matching problem in $\sigma^{-4}(\mathcal{G})$.
Iterating this process gives us a perfect matching in the infinite tiling
$\mathcal{G}$. For finite patches of tiling, we can iterate this procedure
until the graph $\sigma^{-2n}(\mathcal{G})$ has an $O(1)$ number of vertices,
and then match them up (possibly leaving behind a $O(1)$ number of monomers
depending on the boundaries of the starting graph).

To understand this construction of perfect matchings, it is instructive to
recall the notion of $8_{n}$ vertices, i.e. $8$-vertices which survive exactly
$n+2$ deflations, as defined in the preceding subsection. The iterative
procedure outlined above matches $8$-vertices order by order: the first
iteration matches up everything except 8-vertices, the second iteration
matches all the remaining vertices except the $8$-vertices of
$\sigma^{-2}(\mathcal{G})$, i.e., the $8_{0}$\- and $8_{1}$-vertices. The
$n$-th iteration matches up $8_{n-2}$ and $8_{n-1}$-vertices with
$8_{k}$-vertices for $k\geq n$ remaining unmatched, to be matched in
subsequent iterations.

A consequence of this construction is that we can associate $D_{8}$-symmetric
local empires of an $8_{n}$-vertex with “effective vertices”. An effective
vertex is a simply-connected region surrounding an $8$-vertex, such that there
is exactly one dimer between vertices in the region and the rest of the graph.
It is this one-dimer constraint on the region as a whole that leads us to
consider it an effective vertex: if one coarse-grains this region to a single
vertex, then the dimer constraint is reproduced exactly on the coarse-grained
vertex. The largest such region in the local empire of an $8_{n}$ vertex has
radius $\sim\delta_{s}^{2n}$, and we refer to it as an $8_{n}$-region. The
structure of these $8_{n}$ regions are such that an $8_{n}$-region nests
$8_{k}$ regions for all $k<n$. The effective vertices ($8_{n}$-regions) for
the first few $n$ are displayed in Fig. 4.

For proofs of the above statements, which involve the use of ideas from
bipartite matching theory, the reader is referred to Ref. 33. The matching
theory approach automatically singles out the stars and ladders of AB∗, and
also allows for a clear proof of the effective vertex property based on the
associated graph decompositions.

For our purposes, the central message to take from these considerations is
that the one-dimer constraint on $8_{n}$ regions effectively reproduces the
hard-core dimer constraint at successive scales of coarse-graining. This
remarkable feature strongly suggests that the dimer problem on an AB tiling
$\mathcal{G}$ can be recast as an effective dimer problems at each successive
RG scales. Specifically, the effective dimer problem at scale
$\delta_{s}^{2n}$ concerns $8_{2n}$ and $8_{2n-1}$ regions, which act as
“effective vertices” on the AB tiling $\sigma^{-2n}(\mathcal{G})$. In the
remainder of this paper, we make this intuition concrete by constructing a
explicit RG transformation and investigating the effective dimer models at
different scales.

##  III The RG transformation and the fixed point

We now present an explicit real-space coarse-graining transformation for the
dimer model on AB graphs, and show that the model has a fixed point under this
transformation.

###  III.1 The coarse-graining transformation

Consider a statistical mechanics model with microscopic degrees of freedom
$s_{i}$. A configuration $\mathcal{S}=\\{s_{i}\\}$ is associated with an
weight $W(\mathcal{S})$. The probability of the configuration $\mathcal{S}$ is
given by $P(\mathcal{S})=W(\mathcal{S})/Z$, where
$Z=\sum_{\mathcal{S}}W(\mathcal{S})$ is the partition function.

Figure 5: Left: A dimer configuration (blue dimers) on the AB graph
$\mathcal{G}$. Centre: A dimer configuration (brown dimers) on the AB∗ graph
$\mathcal{G}^{0\backslash 1}$ obtained by removing the $8$-vertices (light-
blue solid circles) from $\mathcal{G}$. Right: Symmetric difference between
brown and blue dimers — emptying edges where the blue and brown dimers
coincide gives us a configuration of non-overlapping loops and strings with
alternating blue and brown dimers. The strings connect the $8$-vertices, which
are the vertices of the coarse-grained graph $\mathcal{G}^{1}$; they imply an
‘effective dimer’ of the coarse-grained graph (shown with thick grey lines) in
our RG transformation.

Given a configuration $\mathcal{S}$, we introduce a RG transformation
$\Lambda$. This is a rule to generate an “effective” configuration
$\mathcal{S}^{{\mathrm{eff}}}$ in terms of effective degrees of freedom
$s^{\mathrm{eff}}_{i}$, defined in terms of the underlying “microscopic”
variables $s_{i}\in\mathcal{S}$. More precisely, the transformation $\Lambda$
is defined via a conditional probability
$P_{\Lambda}(\mathcal{S}^{\mathrm{eff}}|\mathcal{S})$, with
$\sum_{\mathcal{S}^{\mathrm{eff}}}P_{\Lambda}(\mathcal{S}^{\mathrm{eff}}|\mathcal{S})=1$
ensuring that the partition function is preserved [1]: we may then write

$\displaystyle Z=\sum_{\mathcal{S}}W(\mathcal{S})$
$\displaystyle=\sum_{\mathcal{S}}\left(\sum_{\mathcal{S}^{\mathrm{eff}}}P_{\Lambda}(\mathcal{S}^{\mathrm{eff}}|\mathcal{S})\right)W(\mathcal{S})$
$\displaystyle=\sum_{\mathcal{S}^{\mathrm{eff}}}\left(\sum_{\mathcal{S}}P_{\Lambda}(\mathcal{S}^{\mathrm{eff}}|\mathcal{S})W(\mathcal{S})\right)$
(1)
$\displaystyle\equiv\sum_{\mathcal{S}^{\mathrm{eff}}}W_{{\mathrm{eff}}}(\mathcal{S}^{\mathrm{eff}}).$
(2)

which defines the weight of an effective configuration
$\mathcal{S}^{\mathrm{eff}}$,

$W_{\mathrm{eff}}(\mathcal{S}^{\mathrm{eff}})=\sum_{\mathcal{S}}P_{\Lambda}(\mathcal{S}^{\mathrm{eff}}|\mathcal{S})W(\mathcal{S}).$
(3)

Intuitively, $P_{\Lambda}(\mathcal{S}^{\mathrm{eff}}|\mathcal{S})$ can be
viewed as a “projector” that is nonzero if and only if the microscopic
configuration $\mathcal{S}$ contributes to the effective configuration
$\mathcal{S}^{\text{eff}}$.

We now construct such an RG transformation $\Lambda$ which coarse-grains dimer
configurations on the AB tiling. The transformation is motivated by the
structure of perfect-dimer-covered configurations reviewed in Sec. II. Before
formally defining $\Lambda$, we first present an intuitive rule to construct
coarse-grained dimer configurations from microscopic dimer configurations,
motivating the rest of the discussion. We begin by reviewing the construction
of perfect-dimer covers. Recall that in constructing such a cover, we first
match all the non-8-vertices (which make up the corresponding auxiliary AB∗
graph), and then match pairs of 8-vertices by augmenting alternating paths
between them. These 8-vertices lie on the vertices of the coarse-grained graph
$\sigma^{-2}(\mathcal{G})$. The second step of this process, of constructing
and augmenting alternating paths between 8-vertices, resembles that of
constructing a dimer cover of the coarse-grained $\sigma^{-2}(\mathcal{G})$
(which consists of only the $8$-vertices of $\mathcal{G}$). This suggests that
the effective dimers on the coarse-grained graph $\sigma^{-2}(\mathcal{G})$
should somehow correspond to alternating paths between $8$-vertices on
$\mathcal{G}$. However, this implies that all the several alternating paths
between two fixed $8$-vertices in $\mathcal{G}$ will map to the same effective
dimer on $\sigma^{-2}(\mathcal{G})$. It therefore follows that the weights of
the latter should be set entropically by the whole ensemble of the former.
Constructing this entropic weight represents the central challenge in
specifying the RG transformation $\Lambda$.

Starting with an AB graph $\mathcal{G}$ with dimer configuration
$\mathcal{S}$, our goal is therefore to write down an effective dimer
configuration $\mathcal{S}^{{\mathrm{eff}}}$ on the coarse-grained graph
$\sigma^{-2}(\mathcal{G})$ by “identifying” alternating paths between
$8$-vertices on $\mathcal{G}$ which would have been augmented to obtain the
microscopic configuration $\mathcal{S}$.

To accomplish the goal of extracting the alternating paths between 8-vertices
on $\mathcal{G}$ for a given $\mathcal{S}$, we overlap the dimers in
$\mathcal{S}$ with the dimers in a perfect matching $\mathcal{S}^{*}$ of the
AB∗ graph (i.e., the graph obtained from the AB graph $\mathcal{G}$ by
removing all 8-vertices). In such overlap configurations, all non-8-vertices
have two dimers (one each from $\mathcal{S}$ and $\mathcal{S}^{*}$), while
8-vertices have a single dimer (from $\mathcal{S}$): in other words, pairs of
$8$-vertices lie on the ends of chains of alternating dimers from
$\mathcal{S}$ and $\mathcal{S}^{*}$, defining an effective dimer between them.
The effective dimer configurations constructed by this procedure will
evidently depend on the auxiliary dimer configuration $\mathcal{S}^{*}$; to
remove this dependence we will average over $\mathcal{S}^{*}$ when defining
the RG transformation. It is also sometimes desirable to perform “larger” RG
transformations, i.e., directly write down effective dimer configurations on
the graph $\sigma^{-2n}(\mathcal{G})$; effective dimers here would correspond
to alternating paths between $8_{k}$-vertices with $k\geq 2n$, obtained by
overlapping dimer-covers on an AB graph with dimer-covers on an auxiliary
graph with all $8_{k}$ vertices for $k\geq 2n$ removed. Since such overlaps
lead to exactly one alternating path connecting each 8-vertex to another, and
the effective dimers so obtained, by construction, obey the hardcore
constraint.

To place this intuitive discussion on firmer footing, let us first establish
some notation. We will use $\mathcal{G}_{n}$ as a shorthand for the graph
$\sigma^{-2n}(\mathcal{G})$, with edge lengths $\delta^{2n}$, that hosts
effective dimers obtained after $n$ steps of RG. Vertices of $\mathcal{G}_{n}$
correspond to $8_{k}$-vertices of $\mathcal{G}_{0}$ with $k\geq 2n$. We use
the binary variable $s^{n}_{i}$ to denote dimer occupancy in the edge $e_{i}$
of the graph $\mathcal{G}_{n}$, and the set $\mathcal{S}^{n}=\\{s^{n}_{i}\\}$
denotes a dimer configuration on the graph $\mathcal{G}_{n}$. We also use the
notation $\mathcal{G}_{m\backslash n}$, for $n>m$, to denote the graph
obtained by removing all vertices from $\mathcal{G}_{m}$ which also belong to
$\mathcal{G}_{n}$, i.e., removing the $8_{k}$-vertices for $k\geq 2n$. In this
language, the AB∗ graph introduced in Sec. II obtained by removing
$8$-vertices from the AB graph is therefore the graph
$\mathcal{G}_{0\backslash 1}$.

Adopting this notation, consider a dimer configuration
$\mathcal{S}=\\{s^{0}_{i}\\}$ constituting a perfect matching in an AB graph
$\mathcal{G}$, and another dimer configuration $\mathcal{S}^{0\backslash
1}=\\{s^{0\backslash 1}_{i}\\}$ constituting a perfect matching in the
corresponding AB∗ graph $\mathcal{G}_{0\backslash 1}$. We consider the
“transposition” (borrowing the terminology from the literature on valence bond
wave functions [36, 37]) of these sets of dimers $\mathcal{T}$, given by

$\mathcal{T}=\\{t_{i}|t_{i}\equiv(s^{0}_{i}+s^{0\backslash
1}_{i})~{}(\text{mod}~{}2)\\},$ (4)

where $t_{i}$ is 1 iff either $s^{0}_{i}$ or $s^{0\backslash 1}_{i}$, but not
both, are 1. This corresponds to superposing the dimers from both
$\mathcal{S}$ and $\mathcal{S}^{0\backslash 1}$ on the same graph and then
emptying edges with two dimers. As outlined earlier, the set $\mathcal{T}$
decomposes into different kinds of non-overlapping components: there are
closed loops of alternating dimers from the sets $\mathcal{S}$ and
$\mathcal{S}^{1\backslash 0}$; further, there are also open chains of such
alternating dimers which connect the $8$-vertices (lying on the coarse-grained
graph). In our proposed RG transformation such an open chain between two
$8$-vertices imply an effective dimer between them. Fig. 5, shows how
transposition between dimer configurations on $\mathcal{G}_{0}$ and
$\mathcal{G}_{0/1}$ can be used to construct an effective dimer configuration.

The effective dimer configuration
$\mathcal{S}^{\text{eff}}\equiv\mathcal{S}^{1}$ will depend not only on the
microscopic dimer configuration $\mathcal{S}^{0}$ but also the auxiliary
configuration $\mathcal{S}^{0\backslash 1}$ on the corresponding AB∗ graph;
this dependence is taken into account by defining the coarse-graining rule
$\Lambda$ as an average over all such auxiliary configurations
$\mathcal{S}^{0\backslash 1}$. Formally, we write

$\displaystyle
Z=\sum_{\mathcal{S}^{0}}W(\mathcal{S}^{0})=\sum_{\mathcal{S}^{1},\mathcal{S}^{0}}P_{\Lambda}(\mathcal{S}^{1}|\mathcal{S}^{0})W(\mathcal{S}^{0}),\text{with}$
$\displaystyle
P_{\Lambda}(\mathcal{S}^{1}|\mathcal{S}^{0})=\sum_{\mathcal{S}^{0\backslash
1}}\vartheta_{\mathcal{S}^{1}}(\mathcal{S}^{0},\mathcal{S}^{0\backslash
1})P(\mathcal{S}^{0\backslash 1}).$ (5)

The function
$\vartheta_{\mathcal{S}^{1}}(\mathcal{S}^{0},\mathcal{S}^{0\backslash 1})$
equals 1 if the transposition graph of $\mathcal{S}^{0}$ and
$\mathcal{S}^{0\backslash 1}$ lead to the effective dimer configuration
$\mathcal{S}^{1}$, 0 otherwise. $P(\mathcal{S}^{0\backslash 1})$ is the
uniform probability distribution over perfect dimer covers of the auxiliary
AB∗ graph $\mathcal{G}_{0\backslash 1}$. This leads to effective dimer
configurations on the graph $\mathcal{G}_{1}$. The effective dimer partition
function can be seen as being defined by Eq. (5) to be an average over a
“double ensemble” of perfect matchings: one over the AB graph
$\mathcal{G}_{0}$, and the other over the auxiliary AB∗ graph
$\mathcal{G}_{0\backslash 1}$.

The generalisation to “larger” RG transformations is now immediate. We simply
obtain a transposition between perfect matchings on the graphs
$\mathcal{G}_{0}$ and $\mathcal{G}_{0\backslash n}$, to obtain effective dimer
configurations on the graph $\mathcal{G}_{n}$ with edge lengths
$\delta_{s}^{2n}$. The average over auxiliary configurations
$\mathcal{S}^{0\backslash n}$ defines the $n$-step RG transformation

$\displaystyle
P_{\Lambda}(\mathcal{S}^{n}|\mathcal{S}^{0})=\sum_{\mathcal{S}^{0\backslash
n}}\vartheta_{\mathcal{S}^{n}}(\mathcal{S}^{0},\mathcal{S}^{0\backslash
n})P(\mathcal{S}^{0\backslash n}),$ (6)

where $P(\mathcal{S}^{0\backslash n})$ denotes the uniform probability
distribution over perfect dimer covers on the auxiliary graph
$\mathcal{G}_{0\backslash n}$.

One might expect that the large RG transformation described above can also by
implemented by composing two smaller RG transformations, i.e., by first
obtaining effective configurations at an intermediate scale $\mathcal{S}^{m}$
for some $0<m<n$, and then using that to obtain $\mathcal{S}^{n}$. We compose
the two transformations $P_{\Lambda}(\mathcal{S}^{m}|\mathcal{S}^{0})$ and
$P_{\Lambda}(\mathcal{S}^{n}|\mathcal{S}^{m})$ as follows:

$\displaystyle P_{\Lambda}(\mathcal{S}^{m}|\mathcal{S}^{0})$
$\displaystyle=\sum_{\mathcal{S}^{0\backslash
m}}\vartheta_{\mathcal{S}^{m}}(\mathcal{S}^{0},\mathcal{S}^{0\backslash
m})P(\mathcal{S}^{0\backslash m}),$ $\displaystyle
P_{\Lambda}(\mathcal{S}^{n}|\mathcal{S}^{m})$
$\displaystyle=\sum_{\mathcal{S}^{m\backslash
n}}\vartheta_{\mathcal{S}^{n}}(\mathcal{S}^{m},\mathcal{S}^{m\backslash
n})P(\mathcal{S}^{m\backslash n}),\text{ and}$
$\displaystyle\overline{P_{\Lambda}}(\mathcal{S}^{n}|\mathcal{S}^{0})$
$\displaystyle=\sum_{\mathcal{S}^{m}}P_{\Lambda}(\mathcal{S}^{n}|\mathcal{S}^{m})P_{\Lambda}(\mathcal{S}^{m}|\mathcal{S}^{0}).$
(7)

Note that the distribution $P(\mathcal{S}^{m\backslash n})$ over effective
dimer configurations on the graph $\mathcal{G}^{m\backslash n}$
$P(\mathcal{S}^{m\backslash n})$ is not uniform, but can in principle be
obtained by coarse-graining $P(\mathcal{S}^{0\backslash n})$, the distribution
over perfect dimer covers on $\mathcal{G}^{0\backslash n}$, as follows:

$\displaystyle P_{\Lambda}(\mathcal{S}^{m\backslash
n}|\mathcal{S}^{0\backslash n})$ $\displaystyle=\sum_{\mathcal{S}^{0\backslash
m}}\vartheta_{\mathcal{S}^{m\backslash n}}(\mathcal{S}^{0\backslash
n},\mathcal{S}^{0\backslash m})P(\mathcal{S}^{0\backslash m}).$ (8)

It is tempting to assume that the composed transformation,
$\overline{P_{\Lambda}}(\mathcal{S}^{n}|\mathcal{S}^{0})$ of Eq. (7) is equal
to the transformation $P_{\Lambda}(\mathcal{S}^{n}|\mathcal{S}^{0})$ from Eq.
(6): such composed transformations naturally lend themselves to RG iterations,
where successive RG transformations lead to effective descriptions at larger
and larger scales. However, analysis of our numerical implementation of the RG
reveals that this is not in fact the case — and indeed, this scale-composition
law is rarely true of similar real-space ‘block’ RG schemes. Nevertheless, the
direct and composed transformations are equal “in the RG sense”: the effective
Hamiltonians they generate only differ by irrelevant operators
333Specifically, we have numerically verified that all sizeable dimer
correlations are equal to within statistical error for the two different
transformations. The differences are in the occupancies of NNN dimers, which
remain very low in both cases and do not affect correlations further
“downstream” in the RG and are in this sense irrelevant variables.. Therefore
to streamline our discussion we will often ignore the distinction between
$P_{\Lambda}(\mathcal{S}^{n}|\mathcal{S}^{0})$ and
$\overline{P_{\Lambda}}(\mathcal{S}^{n}|\mathcal{S}^{0})$.

###  III.2 Numerical evidence of the fixed point

Figure 6: Left: An $8_{1}$-region, with edges in orange, and NNN-edges in
light blue. While the microscopic model does not have any dimers on NNN edges,
our RG transformation generates dimers on the NNN edges. Right: The labelling
scheme to uniquely identify all (including NNN) symmetry inequivalent edges of
the $8_{1}$ region. We use these labels to report dimer densities in Tab. 1 as
we track them as the Hamiltonian flows to a fixed point.

We now study the effective dimer configurations generated by the
transformations presented in Eqs. (5) and (6) from the preceding section. Our
main workhorse for this section (and the rest of the paper) is Monte Carlo
Renormalisation Group (MCRG) [23]. The basic idea involves sampling
microscopic configurations $\mathcal{S}$ using standard Monte Carlo
techniques, and then coarse-graining each configuration $\mathcal{S}$ to an
effective configuration $\mathcal{S}^{{\mathrm{eff}}}$, thereby generating
samples of effective configurations $\mathcal{S}^{\mathrm{eff}}$. This allows
the calculation of observables in terms of effective degrees of freedom. In
usual critical phenomena, correlations between microscopic and coarse-grained
degrees of freedom also allow the estimation of critical exponents. To
implement MCRG for our dimer model, we use the standard directed loop
algorithm for classical dimer models [38, 39]. We generate samples of
effective configurations $\mathcal{S}^{n}$ on the graph $\mathcal{G}^{n}$ by
coarse graining the configurations $\mathcal{S}^{0}$ on graph
$\mathcal{G}^{0}$ using the rule $\Lambda$ described in Sec. III.1. To do so,
we first generate independent samples from two different ensembles: the
ensemble of perfect matchings on the graph $\mathcal{G}^{0}$, and the ensemble
of perfect matchings on the graph $\mathcal{G}^{0\backslash n}$(obtained by
removing from $\mathcal{G}^{0}$ all vertices which belong to
$\mathcal{G}^{n}$). Given these two dimer configurations, we transpose them to
obtain effective configurations $\mathcal{S}^{n}$ on the graph
$\mathcal{G}^{n}$, as described in Eq. (6) and illustrated in Fig. 3.

Due to the lack of translational invariance and the intrinsic impossibility of
imposing periodic boundary conditions, investigations on quasiperiodic graphs
such as AB must work with specific samples and boundary conditions. For some
purposes, it is convenient to work with periodic approximants to these graphs.
Here, instead, given the focus of our investigations of the dimer problem, we
wish to work with samples which preserve the discrete scale symmetry of the
graph and its $D_{8}$-symmetries to the maximum extent possible given the
finite-size testriction. We choose the $8_{n}$ regions, or effective vertices,
introduced in Sec. II (cf. Fig. 4) as the finite graphs for our numerical
investigations. This choice offers two distinct advantages apart from the
explicit $D_{8}$-symmetry. Firstly, matching problems in any large patch of an
AB graph have descriptions in terms of effective matching problems between
$8_{n}$ regions. $8_{n}$ regions are “effective vertices”, i.e., they
collectively have exactly one dimer between them and the vertices in the rest
of the graph; intuitively, the matching problem on any large graph coarse-
grains to an effective matching problem of an $O(1)$ number of these $8_{n}$
regions. Secondly, each $8_{n}$ region has exactly one monomer in its maximum
matching. This ensures that the choice of boundary conditions do not introduce
an unnecessarily large number of monomers whose effects, while expected to be
irrelevant in the thermodynamic limit, might significantly modify results on
finite-size samples.

$8_{n}$ regions are inflations of each other; coarse-graining a dimer
configuration on an $8_{2n+1}$ region using the $n$-step RG transformation
$\Lambda^{n}$ (Eq. (6)) leads to an effective dimer configuration on an
$8_{1}$-region. We compare the effective dimer observables on the
$8_{1}$-regions obtained from coarse-graining $8_{2n+1}$ regions with the
coarse-graining rule $\Lambda^{n}$ for different $n$, and see that they reach
a fixed point with increasing $n$. We consider $n=(0,1,2,3)$; the
corresponding $8_{2n+1}$-regions are graphs with 48, 2497, 87425 and 2993281
vertices respectively.

Figure 7:  Our RG transformation generates effective dimers between vertices
of opposite sublattice; the most important new contributions are from next-
next-neighbour (NNN) vertices, i.e., ones separated by three edges. Some
representative NNN vertex pairs which can host an effective dimer are marked
with blue lines for an $8_{1}$-region. Note that our RG transformation does
not generate effective dimers between vertices of the same sublattice;
therefore, next-neighbour (NN) vertex pairs (marked with green lines)
separated by two edges never host an effective dimer.

In Fig. 6, we display the $8_{1}$-region on which we compare effective dimer
models obtained by coarse-graining microscopic dimer models on different
$8_{2n+1}$-regions. We also assign unique labels to all the symmetry-
inequivalent edges to set up the notation. Note, that the RG transformation
“generates” dimers between vertices that are separated by three edges;
vertices separated by two edge-distances belong to the same sublattice, and
dimers between such vertices can never be generated by our RG transformation.
We will call these dimers between vertices separated by three edges next-next-
neighbour(NNN)-dimers (see Fig. 7).

We first look at the effective dimer densities. The dimer density on an edge
$e_{i}$ is defined as the fraction of dimer configurations that host a dimer
on the edge $e_{i}$. We see that for all edges, the dimer densities reach a
fixed point density within $n=3$ RG steps. These dimer densities, and their
convergence to a fixed point, are shown in Tab. 1. We also note that the
densities of generated NNN dimers remain quite low ($\sim 10^{-3}$) as the
fixed point is approached.

While this provides strong evidence that the effective dimer densities are at
a fixed point, we expect that the fixed point also forces the same fate for
the entire dimer probability distribution, as probed by different correlation
functions. To explore this, we also compute connected correlation functions of
effective dimers; the connected dimer correlation between two dimers on edges
$e_{i}$ and $e_{j}$ is given by

$C^{n}(e_{i},e_{j})=\langle s^{n}_{i}s^{n}_{j}\rangle-\langle
s^{n}_{i}\rangle\langle s^{n}_{j}\rangle.$ (9)

We find that dimer correlations also flow to a fixed point within a few RG
iterations; for brevity, we tabulate the numerical values of dimer
correlations showing their convergence to the fixed point in Appendix A.

We remark that the choice of $8_{1}$-region, to which we coarse-grain dimer
configurations on $8_{2n+1}$-regions in this analysis, is driven primarily by
the numerical advantages offered by its relatively smaller size and is not an
essential property of the RG. We have verified that starting with dimer models
on $8_{2n+2}$-regions and coarse-graining them to effective dimer models on
$8_{2}$-regions also leads to a fixed point of effective dimer densities with
increasing $n$.

Edge | Effective-dimer density after $n$ RG steps
---|---
|  n=0 |  n=1 |  n=2 |  n=3
0 |  0.1244 |  0.1243 |  0.1242 |  0.1243
1 |  0.2074 |  0.1942 |  0.1921 |  0.1922
2 |  0.0521 |  0.0458 |  0.0456 |  0.0455
3 |  0.4811 |  0.5188 |  0.5237 |  0.5235(1)
4 |  0.4378 |  0.4374 |  0.4375 |  0.4375(1)
5 |  0.4798 |  0.4915 |  0.4916 |  0.4917(1)
6 |  0.2809 |  0.2897 |  0.2918 |  0.2916(1)
7 |  0.2393 |  0.2125 |  0.2115 |  0.2116(1)
8 |  0.0000 |  0.0007 |  0.0005 |  0.0005
9 |  0.0000 |  0.0005 |  0.0004 |  0.0004
10 |  0.0000 |  0.0063 |  0.0051 |  0.0051
Table 1: Starting with the graph $\mathcal{G}$ being an $8_{1}$ region of Fig.
6, we consider a series of inflated graphs $\mathcal{G}_{n}=\sigma^{2n}$ for
$n=0,1,2,3$. We use our RG transformation $n$ times for the graph
$\mathcal{G}_{n}$ to coarse grain it back to the $8_{1}$-region
$\mathcal{G}_{0}$ and compute effective dimer densities there. We use the
edge-labelling for symmetry-inequivalent edges introduced in Fig. 6 to report
the dimer densities, and we see that they reach a fixed point with increasing
$n$, implying that the coarse-grained dimer problem on the $8_{1}$-region has
approached a fixed point. In Tab. 3, reported in Appendix A, we report dimer
correlations and see that they also approach a fixed point with increasing
$n$.

##  IV The effective Hamiltonian

Having provided strong evidence suggesting that the distribution of effective
dimers converges to a fixed point under the RG transformation defined by Eq.
(6), we now proceed to calculate the Hamiltonians which describe the
distribution of effective dimers. An expression for a Hamiltonian governing
effective dimers provides a direct characterisation of the fixed point.
Additionally, this yields another advantage: an expression for such effective
Hamiltonians would allow us to sample effective dimer distributions on a graph
directly, opening up the possibility to use the RG to effectively probe larger
system sizes.

###  IV.1 Strategy to calculate effective Hamiltonian

The microscopic dimer model is purely entropic and ascribes no energy cost to
any allowed configuration, i.e. each dimer configuration has exactly the same
weight. On the other hand, the definition of our RG transformation (Eqs. (5)
and (6)) already points to the reason why effective dimer configurations
generated by it might acquire weights relative to each other. In our
transformation, effective dimers correspond to open paths between $8$-vertices
obtained by transposing two different dimer covers: a perfect dimer cover on
the AB tiling, and one on the auxiliary AB∗-tiling (which does not have
8-vertices). The relative weights of effective dimer configurations are then
set by the total number of such open paths for each effective dimer
configuration, counted in the double ensemble defined by perfect matchings on
both the AB and AB∗ graphs. Evidently, there is no a priori reason for these
weights to be identical given the lack of translational invariance. However,
calculating such weights in full generality is a formidable challenge.

The difficulty can be alleviated by the assumption, to be justified later, of
“weight factoring”: namely, that the weight function of an effective dimer
configuration is well approximated by the product of weights of dimers at each
edge, or equivalently, that the interactions between effective dimers are
negligible. For effective dimer configurations
$\mathcal{S}^{n}=\\{s^{n}_{i}\\}$, if $W_{n}(e_{k})$ denotes the weight of a
dimer on edge $e_{k}$, this amounts to assuming that the partition function of
the weighted dimer model can be written as

$\displaystyle
Z\sim\sum_{\mathcal{S}^{n}}\prod_{k}(W_{n}(e_{k}))^{s^{n}_{k}}.$ (10)

With this assumption, the problem of calculating $W_{n}(e_{k})$, the weight of
occupation of a specific edge by an effective dimer, can be posed in a
tractable manner, as follows.

Consider, in full generality, an effective configuration $\\{s^{n}_{i}\\}$ on
the graph $\mathcal{G}_{n}$ . We first calculate the effective dimer density
$\langle s^{n}_{k}\rangle$ on the edge $e_{k}$, which, under our assumption of
decoupled dimer weights, reduces to

$\displaystyle\langle
s^{n}_{k}\rangle=W_{n}(e_{k})\sum_{\mathcal{S}^{n}/e_{k}}\prod_{m\neq
k}(W_{n}(e_{m}))^{s^{n}_{m}}.$ (11)

The sum over $\mathcal{S}^{n}/e_{k}$ is a sum over all dimer configurations
with the condition that there is always a dimer on $e_{k}$. The weight
$W_{n}(e_{k})$ has the expression:

$\displaystyle W_{n}(e_{k})=\frac{\langle
s^{n}_{k}\rangle}{\sum_{\mathcal{S}^{{\mathrm{eff}}}/e_{k}}\prod_{m\neq
k}(W_{n}(e_{m}))^{s^{n}_{m}}}.$ (12)

The numerator of this expression is directly accessible to Monte Carlo
simulations; we calculate the expectation value of an effective dimer by
coarse-graining microscopic dimers as described in Sec. III.1. From Eq. (11),
it is clear that the denominator is the dimer density $s^{\mathrm{eff}}_{k}$
in a different dimer model with effective dimer weights
$W_{\mathrm{eff}}(e_{m})$ on all edges $e_{m}$ for $m\neq k$, and weight 1 for
an effective dimer on $e_{k}$. This observation allows us to devise a Monte-
Carlo algorithm which calculates the weights $W_{n}(e_{k})$ in Eq. 12 for all
edges $e_{k}$ in one go, starting from dimer configurations in the microscopic
graph $\mathcal{G}_{m}$ for $m<n$ . This algorithm to calculate the weights
$W_{n}(e_{k})$ overcomes a key technical challenge, allowing us to calculate
and investigate effective Hamiltonians directly. In the interest of
maintaining the linearity of presentation, we defer a detailed presentation of
the algorithm and its subtleties to Appendix B, and move directly to a
discussion of the effective Hamiltonian.

Figure 8:  Left: an $8_{2}$-region. Right: The effective Hamiltonian of an
$8_{2}$-region, given in terms of dimer weights of its symmetry-inequivalent
edges. These are obtained using the methods of Sec. IV.1, by coarse-graining
the microscopic dimer problem (with all dimer weights 1) on an $8_{4}$-region.

###  IV.2 Structure of the effective Hamiltonian

We now use the algorithm described above to calculate the weights of dimers
$W_{n}(e_{k})$ after $n$ rescalings. We begin with the microscopic dimer
problem on an $8_{4}$ region and use the algorithm of Appendix B to calculate
weights $W_{1}(e_{k})$ of dimers on the coarse-grained $8_{2}$ region. We
display the $8_{2}$ region, and the calculated weights $W_{1}(e_{k})$ for a
set of its symmetry inequivalent edges in Fig. 8. These weights specify the
effective Hamiltonian $\mathcal{H}_{1}$ after one RG step on the $8_{2}$
graph, and hence the partition function is given by Eq. (10). To illuminate
the structure of this effective Hamiltonian, we note that the weight of an
edge depends very strongly on the “type” of the edge. The type $T(e_{k})$ of
an edge $e_{k}$ is determined by the the vertex types to which the edge
connects: recall that there are $7$ types of vertices, introduced in Sec. II:
$3$, $4$, $5_{A}$, $5_{B}$, $6$, $7$ and $8$. For example, if an edge $e_{k}$
connects an $8$-vertex to a $3$-vertex, the edge type $T(e_{k})=(3,8)$. To
check this quantitatively we calculate $\overline{W_{1}(t)}$, the average
weight of all edges with type $T(e_{k})=t$. In Fig. 9, we plot the difference
of edge weight and the type average $W_{1}(e_{k})-\overline{W_{1}(T(e_{k}))}$.
We see that the differences are small, and typically are $\lesssim 2\%$ of the
edge weights themselves.

To see intuitively why this strong edge-type dependence emerges, recall that
(as described in Sec. IV.1) the weights of effective dimers in
$\mathcal{G}_{n+1}$ measure the number of open alternating paths between
8-vertices of $\mathcal{G}_{n}$ obtained in the double ensemble of two
matchings: one on $\mathcal{G}_{n}$ and another on $\mathcal{G}_{n\backslash
n+1}$. It is reasonable to expect that these weights will depend on the
structure of $\mathcal{G}_{n}$ in the neighbourhood of the pair of 8-vertices
in question. Since the local neighbourhood of each of these 8-vertices are
determined by their vertex-type in $\mathcal{G}_{n+1}$(Sec. II), it follows
that the weight of an effective dimer on an edge $e_{k}$ will depend strongly
on the type of the two vertices connected by $e_{k}$.

Figure 9: We calculate the effective dimer weights $W_{1}(k)$ obtained by
coarse-graining the microscopic dimer problem once on an $8_{4}$-region to an
effective problem on an $8_{2}$ region. Here, we show the difference of
$W_{1}(k)$ from the edge-type average $W_{1}(T(e_{k})$, sorted by the edge-
type $T(e_{k})$ on the $x$-axis. The figure shows that the differences are
small, indicating that the dimer weight obtained under RG for an edge is set
primarily by the edge-type. We ascribe the small differences to the presence
of dimer-interactions.

This motivates us to propose an approximation $\overline{\mathcal{H}_{n}}$ to
the effective Hamiltonian $\mathcal{H}_{n}$ after $n$ RG steps:
$\overline{\mathcal{H}_{n}}$ involves using the edge-type average weight
$\overline{W_{n}(t)}$ for all edges $e_{k}$ with an edge-type $T(e_{k})=t$,
i.e. it asserts that the edge weights are functions solely of the edge type.
The small deviations of the calculated weights $W_{1}(k)$ from the edge-type
average $\overline{W_{1}}(T(e_{k}))$ are then presumably the result of weak
interactions between effective dimers, arising from the constraint that the
open alternating paths in the double ensemble that correspond to effective
dimers cannot intersect. To wit, if two effective edges share a vertex, the
only interactions between them arise out of the hardcore constraint, but two
parallel edges that share a plaquette, might be associated with interactions
arising out of the non-intersection constraints of the underlying alternating
paths. We can incorporate this effect with
$\overline{\overline{\mathcal{H}}}$, a better approximation to the effective
Hamiltonian. In $\overline{\overline{\mathcal{H}_{n}}}$, we allow the weight
of an effective dimer on an edge $T(e_{k})$ to depend not only on the edge-
type $T(e_{k})$, but also the types of the two edges parallel to $e_{k}$.
Thus, $\overline{\overline{\mathcal{H}_{n}}}$ is parameterised by the weights
$\overline{\overline{W_{n}}}(t,t^{||}_{1},t^{||}_{2})$.

Note that both types of effective Hamiltonians lead to identical interactions
for different edges that have similar local environments, and as such attempt
to capture the notion of ‘translational invariance’ as closely as possible in
the quasiperiodic environment. These approximations leads to significant
simplifications. Specifying $\overline{\mathcal{H}_{n}}$ requires just 10
parameters for graphs of all sizes, viz. a dimer weight for each of the 10
edge types, which is a considerable reduction from specifiying independent
weights for each symmetry-inequivalent edge. Similarly,
$\overline{\overline{\mathcal{H}_{n}}}$ requires a larger number of parameters
but still $O(1)$ as opposed to $O(N)$. Specifying the effective Hamiltonian
with $O(1)$ parameters is crucial to perform the RG iteratively, and allows us
to track the effective Hamiltonian as the RG proceeds.

To do this, we have coarse-grained the microscopic dimer problem on an
$8_{4}$-region to an $8_{2}$ region to calculate the dimer weights
$W_{1}(e_{k})$ after one step of RG for all edges of the $8_{2}$-region. Now,
using the approximation $\overline{\overline{\mathcal{H}_{1}}}$, i.e., the
average weights $\overline{\overline{W_{1}}}$, we can sample the effective
Hamiltonian $\overline{\overline{\mathcal{H}_{1}}}$ directly on an $8_{4}$
graph. Coarse-graining this to an $8_{2}$ graph again, we calculate the
weights $W_{2}(e_{k})$ after two steps of RG. We can now use $W_{2}(e_{k})$ to
construct the approximate effective Hamiltonian
$\overline{\overline{\mathcal{H}_{2}}}$ given in terms of weights
$\overline{\overline{W_{2}(t}})$. Coarse-graining
$\overline{\overline{\mathcal{H}_{2}}}$ on an $8_{4}$-graph now give us the
weights $W_{3}(e_{k})$ after 3 RG steps. Iterating this procedure allows us to
track the RG flow of the effective Hamiltonian $\mathcal{H}_{n}$, in terms of
the dimer weights $\overline{\overline{W_{n}}}$, with increasing RG step $n$.
In this process, we find that the weights of dimers maintain a strong
dependence on the edge-type for all RG steps, i.e., for all $n$, the weights
$W_{n}(e_{k})$ are narrowly distributed around the edge-type average
$\overline{W_{n}}(T(e_{k}))$ (as displayed in Fig. 9 for the first RG step).
To track the effective Hamiltonian $\mathcal{H}_{n}$, we display the weights
$\overline{W_{n}}(t)$ for all edge-types as a function of the RG step $n$ in
Tab. 2. It is evident that our effective Hamiltonians flow to a fixed point
Hamiltonian. Thus, we have not only provided strong evidence for a fixed-point
in the dimer problem on the AB graph, but also calculated a simple and
explicit expression for the fixed-point Hamiltonian given in terms of weights
of dimers.

We close this section by mentioning, that in Appendix C, we have shown that
the effective Hamiltonian approximations used here (both
$\overline{\mathcal{H}}$ and $\overline{\overline{\mathcal{H}}}$) in terms of
the effective dimer weights provide an accurate description of the
distribution of effective dimers. To do this, we have first calculated
effective dimer observables obtained directly from a dimer model with the
calculated effective dimer weights, and compared them with effective dimer
observables obtained by coarse-graining the microscopic Hamiltonian using MCRG
simulations. Note that this automatically implies that the initial hypothesis
that the effective Hamiltonian involves the effective dimers simply acquiring
weights in the partition function (with dimer-dimer interactions being
negligible) is also accurate. We have also demonstrated in Appendix C that the
errors introduced by making the approximations $\overline{\mathcal{H}}$ vs
$\overline{\overline{\mathcal{H}}}$ are irrelevant in an RG sense, i.e.,
differences between them disappear with further RG transformations. Therefore
it suffices to consider the simplified effective Hamiltonian
$\overline{\mathcal{H}}$ when analyzing universal properties.

Edge-type $t$ | Effective dimer weight $\overline{W_{n}(t)}$ after $n$ RG steps
---|---
| $n=0$ | $n=1$ | $n=2$ | $n=3$ | $n=4$
$(3,4)$ | 1.000 | 0.426(1) | 0.486(1) | 0.485(1) | 0.485(1)
$(3,6)$ | 1.000 | 1.002 | 1.003 | 1.001 | 1.002
$(3,7)$ | 1.000 | 1.003 | 0.999 | 1.001 | 1.001
$(3,8)$ | 1.000 | 1.000 | 1.000 | 1.000 | 1.000(1)
$(3,5_{B})$ | 1.000 | 0.547(2) | 0.590(2) | 0.589(2) | 0.588(1)
$(4,5_{A})$ | 1.000 | 1.505(1) | 1.461(1) | 1.456(1) | 1.459(1)
$(4,6)$ | 1.000 | 1.499 | 1.456(1) | 1.455(1) | 1.455(1)
$(4,7)$ | 1.000 | 1.501 | 1.450 | 1.458 | 1.456
$(4,5_{B})$ | 1.000 | 1.315(1) | 1.313(2) | 1.312(1) | 1.312(3)
$(5_{A},5_{B})$ | 1.000 | 2.157(1) | 2.021(1) | 2.022(7) | 2.020(3)
Table 2:  Starting with dimer weights $\overline{W_{0}(t)}$ for the
microscopic Hamiltonian, at each step we coarse-grain the approximate
effective Hamiltonian $\overline{\overline{\mathcal{H}}}$ on an
$8_{4}$-region, given by the dimer weights $\overline{W_{n}(t)}$, to an
$8_{2}$-region, calculating the new effective dimer weights by the methods
introduced in Sec. IV.1. We track effective Hamiltonians
$\overline{\mathcal{H}}$ under RG in terms of the weights
$\overline{W_{n}(t)}$; showing that they flow to fixed point values
$\overline{W_{n}(t)}^{*}$ under our RG transformation.

###  IV.3 Relevance of dimer and monomer fugacities at the fixed point

The fact that a Hamiltonian parameterised in terms of the weights
$\overline{W_{n}}(t)$ of effective dimers on each edge-type flows to the fixed
point under a numerical procedure already indicates that the fixed point is
stable both to small perturbations in the edge-type weights
$\overline{W_{n}}(t)$, i.e. that these perturbations are irrelevant in the RG
sense. This is confirmed by a more quantitative calculation, presented in
Appendix D. We expect other perturbations (like weights for next-nearest-
neighbour dimers which break the bipartiteness of the graph) to be relevant,
but calculating scaling dimensions of such operators is beyond the scope of
our methods.

We can, however, perform a simple analysis of the effect of a small monomer
fugacity at the fixed point. Consider perturbing the effective Hamiltonian
near the fixed point by a chemical potential $\mu$, which adds a cost for
deviating from the maximum matching condition by introducing monomers. The
effective Hamiltonian with monomer fugacity $\mathcal{H}_{n}^{\prime}$ at the
$n$-th step of RG is given by

$\displaystyle\mathcal{H}_{n}^{\prime}=\mathcal{H}_{n}+N^{n}_{m}\mu^{n},$ (13)

where $N^{n}_{m}$ is the number of monomers, and $\mathcal{H}_{n}$ is the
effective Hamiltonian for dimers studied in Sec. IV. The key point is that by
the definition of our coarse-graining transformation, the number of monomers
cannot change under RG; effective dimer configurations obtained by the
transposition procedure described in Sec. III.1 preserve the number of
monomers, i.e.,

$\langle N^{n+1}_{m}\rangle=\langle N^{n}_{m}\rangle.$ (14)

Since the linear dimension of the system decreases by a factor of
$\delta_{s}^{2}$ in one step of our RG transformation, near the fixed point we
expect

$\langle\mathcal{H}_{n}^{\prime}\rangle=\delta_{s}^{2}\langle\mathcal{H}_{n+1}^{\prime}\rangle.$
(15)

From Eq. (13), the number of monomers can be expressed as

$\langle
N^{n}_{m}\rangle=\frac{d\langle\mathcal{H}_{n}^{\prime}\rangle}{d\mu^{n}}.$
(16)

Combining Eqs. (14), (15) and (16), it is easy to see that

$\mu^{n+1}=\delta_{s}^{2}\mu^{n}.$ (17)

We have therefore shown that small perturbations of monomers are _relevant_ at
the fixed point. Since the rescaling associated with the RG transformation is
exactly $\delta_{s}^{2}$, it is clear that the RG eigenvalue is given by
$y_{\mu}=1$. It might be worth comparing with the familiar classical dimer
models on the square and honeycomb lattices: throughout the critical phase,
perturbations in monomer fugacity is relevant, driving the system to a short-
ranged fixed point with infinite monomer fugacity; while the monomer fugacity
is marginal exactly at the Kosterlitz-Thouless transition to a disordered
phase [39].

##  V Criticality

So far, we have established that the dimer model on the AB tiling flows to a
fixed point under an RG with discrete “block-spin” type transformations. We
now investigate the nature of the fixed point. A natural question is whether
the fixed point Hamiltonian exhibits critical scaling behaviour. For systems
with continuous scale invariance, this manifests itself in the familiar
feature that various observables show power-law scaling. Discrete scale
invariance modifies this picture: as we briefly review [40], such power-law
scaling forms acquire log-periodic modulations when the relevant fixed points
are only invariant under a discrete, rather than continuous, set of scale
transformations.

For a fixed point under an RG transformation $x\rightarrow bx$, one expects,
for the singular part of an observable $O(x)$,

$O(x)=b^{-\alpha}O(bx).$ (18)

If the above equation were to hold for all $b$, we have the familiar power-law
solution $O(x)\sim x^{-\alpha}$. However, if Eq. (18) were only required to
hold for a discrete set of scales $b=\lambda^{n}$ for integer $n$, then
substituting $O(x)=x^{-\omega}$ gives us $\omega=\alpha+i2\pi
k/\log(\lambda)$, for integer $k$. Therefore, the algebraic form $O(x)\sim
x^{-\alpha}$ acquires a modulation which is log-periodic in the discrete scale
$\lambda$,

$O(x)\sim x^{-\alpha}\mathcal{P}_{O}(\log x/\log\lambda),$ (19)

where $\mathcal{P}_{O}$ is a periodic function with a period of $1$. This log-
periodic structure is a distinctive signature of DSI. Therefore, we now
attempt to identify such power-law forms with log-periodic modulations for
various observables computed from our fixed point Hamiltonian.

###  V.1 Dimer correlations

Figure 10:  We plot $\overline{C}(r)$, the average absolute value of dimer
correlations for all edge pairs at a separation $r$, calculated from MC
simulations of the fixed point Hamiltonian on the $8_{2}$-region, along with
$\overline{C}^{\mathrm{est}}$, an estimate of $\overline{C}(r)$ based on the
RG fixed point for an $8_{4}$-region, both on a double-log scale. A power-law
behaviour of $\overline{C}(r)$ modulated by an oscillatory function log-
periodic in $\delta_{s}^{2}$ is suggested by this. The inset shows
$\overline{C}^{\mathrm{est}}$ for an $8_{4}$-region scaled by $r^{\eta}$, for
$\eta=1.4$, suggesting a log-periodic function possibly affected by
significant boundary effects.

An obvious quantity to investigate is the connected dimer correlations
function $C_{ij}$, given by

$C_{ij}=\langle s(e_{i})s(e_{j})\rangle-\langle s(e_{i})\rangle\langle
s(e_{j})\rangle$ (20)

In earlier work (Ref. [33]), it was demonstrated that a subset of dimers have
intriguing long-range connected correlations resembling power laws, whereas
other dimers have correlations cut-off by lengthscales associated with the
effective vertices or $8_{n}$-regions. The correlation functions do not
approximate a smooth function, and further, do not show any translational or
continuous rotational symmetry— even at large lengthscales. To investigate the
nature of decay of these correlations quantitatively, we define
$\overline{C}(r)$ which is the average of absolute values of all connected
correlations between edge-pairs whose separations lie within the interval
$(r-\epsilon,r+\epsilon)$. The lack of symmetries noted above poses a
technical challenge to the numerical calculation of $\overline{C}(r)$: in
general, the calculation of $\overline{C}(r)$ involves the calculation of
$N_{e}^{2}$ dimer correlations for a system of $N_{e}$ edges— a formidable
challenge for Monte Carlo calculations in system sizes of interest (for
example the $8_{4}$ region, used in most numerical calculations reported so
far, has $N_{e}=50,000$ edges).

However, we can make use of the RG transformation near the fixed point to
alleviate this problem. Consider the free energy
$f^{n}(\\{K^{n}\\})=-\log(\sum_{\\{s^{n}_{i}\\}}\exp(-K^{n}_{i}s^{n}_{i}))$ as
a function of the couplings $K^{n}_{i}$ at the $n$-th RG iteration. We
consider different couplings for all edges $i$ to facilitate calculations of
dimer correlations later, even though the fixed point values of $K^{n}_{i}$
for many edges are the same, as explained in Sec. IV.2. The free energy
changes inhomogeneously under RG transformations,

$f^{s}(\\{K^{n}\\})=g(\\{K^{n}\\})+f(\\{K^{n+1}\\}),$ (21)

where $g(K)$ is the non-singular contribution to the free energy. If we are
interested only in the singular part of the free energy which controls
singular behaviour in all observables including correlation functions, we can
ignore the regular part $g(\\{K\\})$. The singular part of the connected dimer
correlations is now given by taking second derivatives with respect to the
couplings $K_{i}$ on both sides

$\displaystyle\langle s^{n}_{i}s^{n}_{j}\rangle^{s}_{c}$
$\displaystyle=\frac{\partial f(\\{K^{n+1}\\})}{\partial K^{n}_{i}\partial
K^{n}_{j}}$ $\displaystyle=\sum_{kl}\frac{\partial K^{n+1}_{k}}{\partial
K^{n}_{i}}\frac{\partial K^{n+1}_{l}}{\partial K^{n}_{j}}\langle
s^{n+1}_{k}s^{n+1}_{l}\rangle_{c}$ (22)

For a system with $N_{e}$ edges, (22) uses the RG transformation, encoded in
the derivatives $\frac{\partial K^{n+1}_{k}}{\partial K^{n}_{i}}$, to
calculate $N_{e}^{2}$ correlation functions on the LHS in terms of
$\frac{N_{e}}{\delta_{s}^{4}}^{2}$ correlation functions in the RHS. The
derivatives can be calculated using MCRG, by solving a chain-rule equation
[see also Eq. (34) in Appendix D],

$\langle s^{n+1}_{i}s^{n}_{j}\rangle_{c}=\sum_{k}\frac{\partial
K^{n+1}_{k}}{\partial K^{n}_{j}}\langle s^{n+1}_{i}s^{n+1}_{k}\rangle_{c}.$
(23)

We use this technique to make estimates of $\overline{C}(r)$, which we call
$\overline{C}^{\text{est}}(r)$, for the $8_{4}$ region, which affords access
to sizes $r\sim 100$. In Fig. 10, we display $\overline{C}^{\text{est}}(r)$
along with $\overline{C}(r)$ calculated directly from MC samples for the
modestly-sized $8_{2}$ region where such calculations are still feasible. The
data appears consistent with a power-law modulated by log-periodic function.
In the inset, we show $r^{\eta}\overline{C}^{\text{est}}(r)$ with $\eta\sim
1.4$, which shows an approximately periodic function on the logarithmic scale.
While this is suggestive of an underlying DSI critical point with
$\overline{C}(r)\sim r^{\eta}$, the second period is already dominated by
boundary effects. Since larger sizes remain inaccessible to the calculation of
$\overline{C}^{\text{est}}(r)$, systematic use of finite-size scaling theory
to conclusively establish criticality and determine the exponent $\eta$ is not
currently within the reach of our numerics. We instead consider an alternative
approach to probing the criticality and DSI at the fixed point, using the
structure of the structure of overlap loops of two decoupled dimer models.

###  V.2 Random geometry of overlap loops in the double dimer model

Figure 11: The average size $s(R)$ of a loop plotted against its radius $R$
reveals a power-law with (weak) periodic corrections on the log-log scale.
Plotting $s(R)R^{-d_{f}}$ against $R/L$, the data for different sizes collapse
to a finite-size scaling form (Eq. (25)) for a fractal dimension $d_{f}=1.38$,
revealing the weak log-periodic modulations to power-law behaviour.  Figure
12: The average number density $n(R)$ of a loop plotted against its radius $R$
reveals a power-law with periodic corrections on the log-log scale. Plotting
$n(R)R^{3}$ against $R/L$, the data for different sizes collapse to a finite-
size scaling form (Eq. (27)), revealing the log-periodic modulations to power-
law behaviour. A collapse with an exponent of $3$ (and no other free
parameters) confirms the expectations of the scaling theory presented in the
text. Figure 13: $P(s)$, the density of loops of size $s$ plotted against $s$
also reveals a power-law with periodic corrections on the log-log scale.
Plotting $P(s)s^{\tau}$ against $s^{-d_{f}}/L$, the data for different sizes
collapse to a finite-size scaling form (Eq. (29)) for $\tau=2.55$ and
$d_{f}=1.37$, from Fig. 11, revealing the log-periodic modulations to power-
law behaviour. The exponent $\tau$ and $d_{f}$ used in the scaling collapses
are consistent with the scaling relation $d_{f}(\tau-1)=2$, expected to hold
from the scaling theory presented in the main text.

To quantitatively probe criticality at the DSI fixed point, we consider the
seemingly-unrelated problem of a double dimer model. This is constructed by
superposing two independent dimer covers, which defines a configuration of
tightly-packed self-avoiding loops on the AB graph, with the additional
possibility that links can host loops of length 2, corresponding to the
overlapping of dimers at the same edge from both covers. Thus, the double
dimer model can be mapped to a loop gas. There is a long history of the
investigation of such loop gases, whose properties are intimately connected
with those of the underlying dimer model. For instance, for bipartite dimer
models with a height description (e.g. on periodic graphs), the long-
wavelength properties of loops in the double dimer model correspond to equal-
height contours of the fluctuating height field [41, 42], and have been
investigated in detail with a scaling theory [43, 44]. In such cases, the
ensemble of overlap loops is also rigorously known to be conformally invariant
[45].

Our primary motivation for studying the double dimer model and the associated
loop gas is the relative numerical accessibility of loop observables in Monte
Carlo simulations. Incontrovertible evidence of critical scaling of the loop
observables, defined in the configuration space of two decoupled dimer models,
provides a very strong suggestion of criticality in the single dimer model. To
test the critical scaling of this loop gas, we use the scaling theory of the
critical loop ensembles, largely following Kondev and Henley [43], but
additionally modifying their power-law scaling ansatzes with log-periodic
modulations appropriate for DSI. We present numerical evidence of DSI in the
loop gas by showing that Monte Carlo calculations of observables of the loop
gas are consistent with such a critical scaling theory of loops with log
periodic modulations.

Following the motivations outlined at the beginning of Sec. III.2, we choose
effective vertices, or $8_{n}$ regions, as our samples, specifically the
$8_{4}$, $8_{6}$ and $8_{8}$ regions. They have $15473$, $517825$ and
$17520593$ vertices respectively. In terms of units of edge separation, the
linear dimensions are $L\sim 100$, $L\sim 500$, and $L\sim 3000$ respectively.
In these simulations, we use the fixed-point values of dimer weights obtained
from the RG calculations described in Sec. III.1 and Sec. IV

For each loop, we define a loop size $s$ (not to be confused with dimer
occupancy variables defined earlier) and a loop radius $R$. The loop size is
the number of edges in a loop, while $R$ is operationally defined as the
radius of the smallest disk which completely covers the loop. At a critical
point with scale symmetry at discrete scales given by $\delta_{s}^{2n}$, from
Eq. (19) we expect

$s(R)=R^{d_{f}}\mathcal{P}_{d_{f}}(\log R/\log\delta_{s}^{2}),$ (24)

where $\mathcal{P}_{d_{f}}(\log R/\log\delta_{s}^{2})$ is a log-periodic
function. Our finite-size scaling hypothesis for $s$ is

$s(R,L)=R^{d_{f}}\mathcal{F}_{d_{f}}(RL^{-1}),$ (25)

where we expect the scaling function $\mathcal{F}_{d_{f}}(RL^{-1})$ to be
equal to $\mathcal{P}_{d_{f}}(\log R/\log\delta_{s}^{2})$ for $RL^{-1}\ll 1$,
and encode the effects of finite sample size and boundary termination for
$RL^{-1}\simeq 1$. We expect systems of different linear dimensions $L$ to
collapse to the same scaling function $\mathcal{F}_{d_{f}}$ only if these
systems are related by discrete scale transformations. Our samples, the
$8_{4}$, $8_{6}$, and $8_{8}$ regions, satisfy this condition. We display a
histogram of loop sizes $s$ plotted against their radius $R$ in Fig. 11. We
see that the histograms approximate a power-law with periodic modulations of
very small magnitude, establishing the fractal nature of our loops. We can
perform a finite-size scaling collapse (Eq. (25)) on plotting $sR^{-d_{f}}$
against $RL^{-1}$ with the exponent $d_{f}=1.384(1)$. The error bars reflect
the range of $d_{f}$ over which a good collapse can be obtained. The scaling
function clearly demonstrates the log-periodic modulations which are the
hallmark of DSI.

Next, we consider the quantity $n(R)$, the density of loops with radius in
$(R,R+dR)$. If we consider a patch of area $A$, the number of loops in
$(R,R+dR)$ in the patch is then $n(R)A\,dR$. Under a coarse-graining
transformation $R\rightarrow b^{-1}R$ with $b=\delta_{s}^{2}$, the same loops
lie in a patch of area $b^{-2}A$ with radii in $(b^{-1}R,b^{-1}(R+dR))$ . If
we have DSI, then the same function $n(R)$ describes the rescaled loop
ensemble, and so we have $n(R)A\,dR$ = $n(Rb^{-1})A\,dR\,b^{-3}$. Following
Eq. (19), we then have the scaling form 444Note that general loop ensembles
often have $n(R)\sim R^{-3+\alpha}$ with nonzero $\alpha$, e.g., contour loops
of fluctuating rough surfaces.

$n(R)=R^{-3}\mathcal{P}_{n}(\log R/\log\delta_{s}^{2}),$ (26)

and the corresponding finite-size scaling form

$n(R,L)=R^{-3}\mathcal{F}_{n}(RL^{-1}).$ (27)

As usual, we expect $\mathcal{F}_{n}$ to be equal to the periodic function
$\mathcal{P}_{n}$ when $RL^{-1}\ll 1$ and incorporate finite size effects when
$RL^{-1}\simeq 1$. We show, in Fig. 12, that plotting $n(R,L)R^{3}$ against
$RL^{-1}$ gives us an excellent scaling collapse, without any free parameters,
into a function periodic in $\log\delta_{s}^{2}$— thereby providing strong
evidence of critical scaling and DSI.

Finally, we look at the quantity $P(s)$, the area density of loops of size
between $s$ and $s+ds$. We use a similar argument about the total number of
loops in a patch of area $A$ remaining invariant under the rescaling
$R\rightarrow Rb^{-1}$ with $b=\delta_{s}^{2}$, which
implies$P(s)A\,ds=P(s/b^{d_{f}})A\,ds\,b^{-2-d_{f}}$, giving the scaling forms

$\displaystyle P(s)\sim s^{-\tau}\mathcal{P}_{P}(\log
s/\log\delta_{s}^{2d_{f}}),\text{ and}$ (28) $\displaystyle
P(s,L)=s^{-\tau}\mathcal{F}_{P}(sL^{-d_{f}}).$ (29)

The finite-size scaling function $\mathcal{F}_{P}$ is equal to the log-
periodic function $\mathcal{P}_{P}$ for $sL^{-d_{f}}\ll 1$, and the exponent
$\tau$ satisfies the scaling relation

$d_{f}(\tau-1)=2.$ (30)

We first plot $P(s)$ against $s$ in Fig. 13 to demonstrate a power-law with
log-periodic modulations, in the inset we show that plotting $P(s)s^{\tau}$
against the scaling variable $sL^{-d_{f}}$, with $d_{f}=1.384$ obtained above
and $\tau=1+2/d_{f}$ from Eq. (30) obtains a good scaling collapse. We note
that all scaling collapses (Figs. 11, 12 and 13) were obtained with a single
free parameter, the fractal dimension $d_{f}$ of the critical loops.

Finally, one might consider the loop correlation function $G(r)$ which is the
average probability that two sites separated by a distance $r$ is a part of
the same loop. For usual critical loop ensembles [43], one expects $G(r)\sim
r^{-2x}$, with the exponent $x$ satisfying the scaling relation $d_{f}=2-x$.
We expect similar forms to hold for our case, albeit with the usual log-
periodic modulations. Calculating loop-correlation functions are numerically
challenging, and we do not attempt to do so here. However our scaling analyses
already predict $G(r)\sim r^{-1.23}$ from the scaling relation relating $x$ to
the fractal dimension of the loops. Therefore, by studying the ensemble of
overlap loops, we have established that two decoupled copies of classical
dimer models are critical in an unusual way, exhibiting discrete instead of
continuous scale invariance. This strongly suggests that the underlying
classical dimer model also has similar critical properties.

##  VI Discussion

We have constructed a numerical real-space RG transformation for the classical
dimer model on the AB tiling. Implementing these transformations by large-
scale Monte Carlo simulations, we show that our model flows to a fixed point
under discrete “block-spin” type RG transformations. We have also introduced a
Monte-Carlo-based technique to calculate the effective Hamiltonians, using it
to track RG flows and explicitly construct the fixed point Hamiltonian in
terms of dimer weights on different types of edges. The ability to write down
and simulate the fixed point Hamiltonian directly has allowed us to make
further progress: most importantly, we show that dimer correlation functions
on accessible system sizes are consistent with the expectation of power-laws
modulated by log-periodic corrections — the hallmark of a critical point with
DSI. Finally, we studied the loop ensemble defined by the overlap graphs of
two-decoupled dimer models; loop observables in this auxiliary ensemble are
numerically more accessible, and this allows us to show conclusive evidence of
critical scaling with DSI, via the explicit computation of power laws with
log-periodic modulations which are in perfect agreement with a DSI-adapted
scaling theory of the critical loop ensemble.

Very recent work has discovered a DSI critical point (with similar lack of
emergent translational and rotational symmetries) in a quasiperiodic
percolation problem [32], in which the percolating bonds are chosen according
to a quasiperiodic pattern. DSI has also been observed before in critical
phenomena. Most such examples, however, involve quantum spin chains, where
older results [46, 47, 48, 49, 50, 51] focussed on quasiperiodic couplings
generated by certain binary substitution rules have been recently supplemented
by results on more general quasiperiodic modulations [52, 53, 54, 55]. DSI has
also been reported in classical statistical mechanics [31, 56], albeit on
heirarchical graphs. These graphs are closely connected to real-space
renormalisation transformations, often defined such that RG transformations of
the Migdal-Kadanoff type are exact on them by construction [28]. Hierarchical
graphs are unusual in terms of their geometry, and not straightforwardly
generalisable to other systems. While quasiperiodic systems such as those
considered in this paper have a built-in scale symmetry (and in this sense
resemble hierarchical graph) they are typically not amenable to exact real
space block-spin RG [26, 27]. Indeed, DSI is only manifest Monte Carlo
renormalisation procedure. To our knowledge, other statistical mechanical
models with uniform couplings on quasiperiodic graphs host conventional
critical phenomena with continuous scale invariance [57, 58, 59, 60, 61].
Further, the log-periodic modulations acquired by power-laws at critical
points of hierarchical graphs are typically very weak [31] — typically a
factor of $\sim 10^{-5}$ weaker than the background — whereas in the present
example DSI is clearly manifest at the critical point.

Another, possibly related feature of our model is that the theory at the fixed
point is described by hard-core dimers. This is in contrast to critical points
of constrained dimer and loop models which are described by continuum fields.
This has been confirmed by parallel work [62] using machine-learning-based
coarse-graining schemes [63, 62]. These schemes have “discovered” that the
effective degrees of freedom are hard-core dimers [34] based purely on
information theoretic considerations— i.e., without any prior knowledge about
the structure of perfect matchings.

Some future directions are naturally suggested by our work. The first concerns
loop models. The partition function of the double dimer model considered here,
expressed in terms of the overlap loops, is given by
$Z=\sum_{\mathrm{C}}1^{N_{d}}2^{N_{L}}$, where $N_{d}$ is the number of
doubled edges with two dimers (one from each copy of the dimer model), and
$N_{L}$ is the total number of other non-trivial loops. These are closely
connected to the $O(n)$ loop models [64, 65], that describe a configuration of
closely packed loops with $Z=\sum_{\mathrm{C}}n^{N_{L}}$; the configuration
space typically excludes doubled edges. The AB graph has been recently shown
to be perfectly packed by such loops [66]. The $O(2)$ loop model is expected
to have similar critical behaviour to the double dimer model considered here,
since the weights for all non-trivial loops are the same in both models. Since
we have shown that the double dimer model is critical, $O(n)$ loop models with
$n\leq 2$, i.e., ones with larger relative weight for longer loops, are
expected to be either critical or in the long-loop phase. It is clear that for
large $n$ where long loops are suppressed, loop models must be in a short-loop
phase. The question then is to understand the critical behaviour of $O(n)$
models as a function of $n$ to investigate a transition from a critical or
long-loop phase to a short-loop phase. (For $O(n)$ loop models on square and
honeycomb lattices, the model is critical for $n\in[0,2]$, and short-ranged
for $n>2$ [64].) For periodic bipartite lattices, other related models such as
the non-crossing [67] and bilayer [68, 42] dimer models and colouring problems
[41] also show a wealth of interesting critical behaviour that can be
understood in terms of height variables; the fate of such models on AB graphs
remains an important open question.

A second direction concerns the properties of nearest neighbour Resonating
Valence Bond (nnRVB) wavefunctions on these graphs. nnRVB wavefunctions often
exhibit a rich host of spin-liquid behaviours and algebraic correlations [69,
70, 71, 72]. RVB wavefunctions for $SU(g)$-spins can be characterised by an
overcomplete basis of dimer configurations, with each dimer representing an
$SU(g)$-singlet [73, 36, 74]. Observables of such RVB wave-functions are given
by estimators of overlap loops of dimer configurations, with distributions
given by the partition function $Z(g)=\sum_{\mathrm{C}}g^{N_{d}}(2g)^{N_{L}}$.
In particular, spin-correlation functions
$\langle\vec{S}_{j}\vec{S}_{j}\rangle$ are related to the loop-correlation
functions evaluated in $Z(g)$. On a square lattice, $Z(g)$ is short-ranged for
the $g=2$ case relevant to $SU(2)$-spins, resulting in short-range spin
correlations. However the valence-bond correlations $\langle
P_{ij}P_{kl}\rangle$, with the projector
$P_{ab}=\langle\vec{S}_{a}\cdot\vec{S}_{b}-1/4\rangle$, are still critical
[69, 70]. This can be understood from the observation that such valence bond
correlations, in the large-$g$ limit, are equal to the classical dimer
correlations [75], which are known to be critical for the square and honeycomb
lattices. The fateof such nnRVB wavefunctions for $SU(2)$-spins is still an
open question for the AB graph: even if spin-correlations are short-ranged
like the square and honeycomb lattice cases, nnRVB wavefunctions can still
host critical valence-bond correlations, as we have shown in Sec. V.1 that the
classical dimer-correlations appear critical. In this light, investigating
properties of nnRVB wave functions is likely to be a rewarding enterprise.

A final direction is to try to the drive the dimer model out of criticality. A
first step in this direction might incorporate aligning interactions [39], and
study how their presence affects the RG flow.

We close by noting that the AB graph can be deformed into closely-related
cousins which are likely to be numerically more accessible while sharing
similar properties. $D_{8}$-symmetric samples of the AB graph, such as those
considered in most numerical investigations of this paper, have been argued to
control the structure of maximum matchings in all samples in thermodynamic
limit. These samples are built of of eight equivalent sectors. Decreasing the
number of sectors leads to smaller graphs (though no longer tilings), while
preserving the scale-invariant structure of perfect matchings. Whether
classical dimer models on these deformed graphs leads to DSI fixed points
remains an open question. We expect such deformations to be useful in
numerically attacking questions on quantum-mechanical dimer models, as well as
in performing quantitative calculations of critical behaviour of dimer and
monomer correlations of the classical dimer model which remain inaccessible
even to our large-scale numerics on the $AB$-graph.

###### Acknowledgements.

We are grateful Tyler Helmuth and Kedar Damle for suggesting that we
investigate the double dimer model, and Abhishodh Prakash and Michele Fava for
crucial discussions throughout the period of the project. We thank Jerome
Lloyd, Steven Simon, and Felix Flicker for a previous collaboration studying
the AB tiling (Ref. [33]). SB thanks Doruk Gökmen, Sebastian Huber, Maciej
Koch-Janusz, Zohar Ringel, and Felix Flicker for collaboration on recent
related work (Ref. [34]). We acknowledge support from the European Research
Council under the European Union Horizon 2020 Research and Innovation
Programme via Grant Agreement No. 804213-TMCS.

##  Appendix A Fixed point behaviour of dimer correlations

Earlier, in Tab. 1, we showed how effective dimer densities of coarse-grained
dimer models on an $8_{1}$-region flows to a fixed point in the limit of
effective densities being obtained by coarse-graining large microscopic dimer
models. Specifically, we consider $8_{1+2n}$-regions and coarse-grain them to
$8_{1}$ regions, measuring the effective dimer densities and checking that
they flow to a fixed point with $n$. For a fixed point, we need not only the
dimer densities, but the whole probability measure of dimer distributions to
go to a fixed point with increasing $n$. Here, in Tab. 3, we show how
connected correlations of dimers also flow to a fixed point under RG
transformations. While we display the data for a particular choice of source
edge, ($e_{j}=3$ in the labelling convention of Fig. 6), the convergence to
the fixed point has no dependence on this choice.

Edge | Effective dimer correlations after $n$ RG steps
---|---
|  n=0 |  n=1 |  n=2 |  n=3
0 |  0.2021(1) |  0.2058 |  0.2066 |  0.2070(1)
1 |  -0.1351(1) |  -0.1424 |  -0.1434 |  -0.1437(1)
2 |  0.1347(1) |  0.1422 |  0.1432 |  0.1434(1)
3 |  -0.0792(1) |  -0.0839 |  -0.0851 |  -0.0853(1)
4 |  -0.0670(1) |  -0.0616 |  -0.0617 |  -0.0618(1)
5 |  -0.0555(1) |  -0.0564 |  -0.0566 |  -0.0567(1)
6 |  -0.0582(1) |  -0.0562 |  -0.0561 |  -0.0562(1)
7 |  -0.0582(1) |  -0.0563 |  -0.0560 |  -0.0561(1)
8 |  0.0490(2) |  0.0474 |  0.0475 |  0.0476(1)
9 |  0.0405(1) |  0.0434 |  0.0435 |  0.0435(1)
10 |  0.0177(1) |  0.0149 |  0.0148 |  0.0147(1)
11 |  0.0147(1) |  0.0137 |  0.0136 |  0.0136(1)
12 |  0.0151(1) |  0.0133 |  0.0131 |  0.0131(1)
13 |  0.0153(1) |  0.0133 |  0.0131 |  0.0129(1)
14 |  -0.0125(2) |  -0.0100 |  -0.0099 |  -0.0099(1)
15 |  0.0124(2) |  0.0098 |  0.0097 |  0.0098(1)
Table 3: Starting with the graph $\mathcal{G}$ being an $8_{1}$ region of Fig.
6, we consider a series of inflated graphs $\mathcal{G}_{n}=\sigma^{2n}$ for
$n=0,1,2,3$. We use our RG transformation $n$ times for the graph
$\mathcal{G}_{n}$ to coarse grain it back to the $8_{1}$-region
$\mathcal{G}_{0}$ and compute effective dimer correlations there. In labelling
for symmetry-inequivalent edges introduced in Fig. 6 to report the dimer
correlations $\langle s(e_{i})s(e_{j})\rangle-\langle s(e_{i})\rangle\langle
s(e_{j})\rangle$ for $e_{i}=3$, and sort $e_{j}$ in decreasing order of
magnitude of correlations. We see that they reach a fixed point with
increasing $n$, implying that the coarse-grained dimer problem on the
$8_{1}$-region has approached a fixed point. This complements Tab. 1 in the
main text, which tracks dimer densities as they flow to a fixed point.

##  Appendix B Alogrithm to calculate effective dimer weights

For concreteness, let us consider the effective configuration
$\\{s^{1}_{i}\\}$ on a graph $\mathcal{G}_{1}$, obtained by coarse-graining
(Eq. (5)) a microscopic configuration $\\{s^{0}_{i}\\}$ on the graph
$\mathcal{G}_{0}$ We will now describe a Monte Carlo procedure to estimate the
weights of effective dimers $W_{1}(e_{k})$. We start with the graph
$\mathcal{G}_{0}$, and add to it all the edges which belong to the coarse-
grained graph $\mathcal{G}_{1}$, with dimer weights on these new edges set to
1. The dimers on these new edges will be called “long-range dimers”. On the
configuration space of this auxiliary dimer model we impose the additional
constraint that there can be at most one long-range dimer. If we now coarse-
grain a configuration with a long-range dimer on the edge $e_{k}$ by
transposing with a dimer configuration on the corresponding AB∗ graph as
before, the transposition graph will have open alternating paths connecting
all the 8-vertices, except the pair corresponding to the edge $e_{k}$ which
already hosts the long-range dimer. Since the long-range dimer has a weight of
$1$, such configurations approximately sample the partition function described
by the denominator of Eq. (12).

Before we convert this into an operational algorithm for calculating dimer
weights, we must address another subtlety. In the context of the RG
transformations, the effective dimers correspond to open paths connecting
8-vertices obtained from transposition graphs of the double ensemble of
matchings in AB and the corresponding AB∗ graphs. In this context, the
partition function of the auxiliary dimer model with a single long-range dimer
fixed on $e_{k}$ is approximately equal to the partition function of the
effective dimer model obtained by fixing not only the effective dimer on the
edge $e_{k}$, but also the specific open path in the transposition graph which
resulted in the effective dimer; the entropy of the choice of open path is
captured by the weight $W_{1}(e_{k})$. The long-range dimer, then, is a proxy
for a single open path corresponding to an effective dimer on the edge
$e_{k}$. Crucially, the open paths obtained from transposition graphs obtained
from the double ensemble do not intersect each other (by construction); they
have an exclusion constraint. Our proxy long-range dimer on the edge $e_{k}$
of our auxiliary dimer problem, however, is completely permeable to open paths
between other $8$-vertices corresponding the other effective dimers. We
rectify this problem approximately: we maintain a precomputed list of
“shortest open paths” for each edge $e_{k}$, and calculate the partition
function with the long range dimer fixed at edge $e_{k}$ and the additional
constraint that the open paths corresponding to other effective dimers do not
intersect the shortest path corresponding to the edge $e_{k}$.

We distill the main ideas from the previous paragraphs into an algorithm to
calculate weights of effective dimers $W_{n+1}(e_{k})$ on the graph
$\mathcal{G}_{n+1}$, given the dimer weights $W_{n}(e_{k})$ on the graph
$\mathcal{G}_{n}$:

  1. 1.

Use weights $W_{n}$ to Monte Carlo-sample dimer configurations
$\\{s^{n}_{i}\\}$ on the graph $\mathcal{G}_{n}$. Independently sampling dimer
configurations on the corresponding AB∗ graph $\mathcal{G}_{n/n+1}$, use the
double ensemble to construct coarse-grained dimer configurations $s^{n+1}_{i}$
on the graph $\sigma^{-2(n+1)}(\mathcal{G})$, as described in Sec. III.1.
Calculate the dimer densities $\langle s^{n+1}_{k}\rangle$ on all edges.

  2. 2.

For each edge $e_{k}$ in $\mathcal{G}_{n+1}$, compute
$\mathcal{P}_{\mathrm{s}h}(e_{k})$, the shortest path between the 8-vertices
in $\mathcal{G}_{n}$ which correspond to the edge $e_{k}$.

  3. 3.

Starting from the graph $\mathcal{G}_{n}$, add edges present in
$\mathcal{G}_{n+1}$. Construct an auxiliary dimer model on this graph as
follows: All dimers on the edges in $\mathcal{G}_{n}$ have weights given by
$W_{n}$ as before, while the “long-range” dimers between 8-vertices have
weights of 1. Additionally, this new dimer model has the constraint that there
is at most 1 long-range dimer present in an allowed configuration.

  4. 4.

Monte Carlo-sample the auxiliary dimer model described above. Calculate the
fraction of configurations without long-range dimers $\bar{\rho}$. If a
configuration has a long-range dimer on the edge $e_{k}$, we sample a dimer
configuration on the AB∗ graph corresponding to $\mathcal{G}_{n/n+1}$. Now we
coarse-grain, by transposing these two dimer configurations, leading to open
paths linking up all 8-vertices in pairs, except the pair corresponding to the
edge $e_{k}$. If none of these open paths intersect the precomputed path
$\mathcal{P}_{\mathrm{s}h}(e_{k})$, we add to a buffer. The ratio of the
number of configurations in this buffer to the total number of configurations
gives us $\langle\rho_{k}\rangle$.

  5. 5.

The estimator for the weight $W_{n+1}(k)$ is given by

$W_{n+1}(k)=\frac{\langle
s^{n+1}_{k}\rangle}{\langle\rho_{k}\rangle/\bar{\rho}}$ (31)

##  Appendix C Details on effective Hamiltonian approximations

Figure 14: We display and compare the error accumulated by dimer-densities
calculated using the approxmations $\overline{\mathcal{H}}$ and
$\overline{\overline{\mathcal{H}}}$ to the effective Hamiltonian obtained by
coarse-graining the microscopic dimer problem on an $8_{4}$-region to an
effective problem on an $8_{2}$-region. Top (Bottom): The difference of
effective dimer-densities $\overline{\langle s_{1}(e_{k})\rangle}$
($\overline{\overline{\langle s_{1}(e_{k})\rangle}}$), calculated with the MC
simulations of the approximate effective Hamiltonian $\overline{\mathcal{H}}$
($\overline{\overline{\mathcal{H}}}$) on an $8_{2}$-region, to the exact
effective dimer-densities $\langle s_{1}(e_{k})\rangle_{\mathrm{ex}}$,
calculated directly from MCRG coarse-graining of an $8_{4}$-region.

In the main text, we hypothesized that the effective Hamiltonian involves
effective dimers getting weights $W_{n}(e_{k})$ on all edges $e_{k}$. An
effective Hamiltonian thus obtained, $\mathcal{H}_{n}$ has as one independent
parameters for each symmetry-inequivalent edge of the graph.

We proposed two approximations to the effective Hamiltonian that reduce the
number of free parameters to an $O(1)$ number. The first,
$\overline{\mathcal{H}_{n}}$, is based on the observation that the weight
$W_{n}(e_{k})$ depends strongly on the type of the edge, $T(e_{k})$, and
approximates $W_{n}(e_{k})$ by $\overline{W_{n}}(t)$, the average
$W_{n}(e_{k})$ over all edges $e_{k}$ with type $T(e_{k})=t$. The second,
$\overline{\overline{\mathcal{H}}}_{n}$, is based on incorporating the effect
that the alternating paths which correspond to effective dimers in the RG
transformation cannot intersect, and consequently might result in an
interaction between effective dimers on parallel edges. To account for this,
$\overline{\overline{\mathcal{H}}}$ allows the weight to depend not only on
the edge type $t$, but also $t^{||}_{1}$ and $t^{||}_{2}$, the types of the
two edges parallel to $e_{k}$. It replaces $W_{n}(e_{k})$ by
$\overline{\overline{W_{n}}}(t,t^{||}_{1},t^{||}_{2})$, the average weight
over all edges such that their edge-type is $t$ and they have parallel edges
with edge-types $t^{||}_{1}$ and $t^{||}_{2}$.

In this Appendix, our goal is to check that these are reasonable
approximations; i.e., to verify that the partition function of Eq. (10) with
the effective Hamiltonian calculated according to these approximations
correctly reproduces the effective dimer distributions obtained directly via
the RG transformation. To check these effective Hamiltonians, we first
calculate the effective dimer densities _exactly_ on an $8_{2}$-graph by
coarse-graining dimer distributions on an $8_{4}$ graph using MCRG; we denote
this by $\langle s_{1}(e_{k})\rangle^{\mathrm{ex}}$, where the subscript $1$
denotes the fact that the effective dimer densities are obtained after 1 RG
step. We compare this against the dimer densities calculated by MC sampling
from effective Hamiltonians $\overline{\mathcal{H}}$ and
$\overline{\overline{\mathcal{H}}}$. We denote the corresponding dimer
densities by $\overline{\langle s_{1}(e_{k})\rangle}$ and
$\overline{\overline{\langle s_{1}(e_{k})\rangle}}$ respectively. In Fig. 14,
we compare both these quantities to $\langle
s_{1}(e_{k})\rangle^{\mathrm{ex}}$. We find that while the dimer densities
obtained with $\overline{\mathcal{H}}$ accrue a maximum error of $\sim 0.05$,
the ones obtained with $\overline{\overline{\mathcal{H}}}$ have a maximum
error of $\sim 0.005$.

Figure 15: Top (Bottom): The difference of effective dimer-densities
$\overline{\langle s_{2}(e_{k})\rangle}$ ($\overline{\overline{\langle
s_{2}(e_{k})\rangle}}$), calculated with the MCRG coarse-graining of the
approximate effective Hamiltonian $\overline{\mathcal{H}}$
($\overline{\overline{\mathcal{H}}}$) on an $8_{4}$-region to an
$8_{2}$-region, to the exact effective dimer-densities $\langle
s_{2}(e_{k})\rangle_{\mathrm{ex}}$, calculated directly from MCRG coarse-
graining of the microscopic dimer problem on an $8_{6}$-region. The errrors,
accumulated are very small, suggesting that whatever errors are picked up in
our calculations of effective Hamiltonians $\overline{\mathcal{H}}$ and
$\overline{\overline{\mathcal{H}}}$ (showing up in errors of dimer-densities
in Fig. 14)) are irrelevant, being suppressed under further RG transformations
as shown here.

While both these approximations might be good enough depending on the purpose,
we argue that the errors incurred by approximating the effective Hamiltonians
with either $\overline{\mathcal{H}}$ or $\overline{\overline{\mathcal{H}}}$
are not _relevant_ in an RG sense, i.e., they do not increase under the RG
flow. To see this, we now exactly compute the effective dimer density $\langle
s_{2}(e_{k})\rangle^{\mathrm{ex}}$ on the $8_{2}$-graph, obtained by
implementing a “large” RG transformation (Eq. (6)) on dimer configurations on
the $8_{6}$-graph. We compare this with the densities $\overline{\langle
s_{2}(e_{k})\rangle}$ and $\overline{\overline{\langle s_{2}(e_{k})\rangle}}$,
obtained by coarse-graining the effective Hamiltonians
$\overline{\mathcal{H}}$ and $\overline{\overline{\mathcal{H}}}$ on an $8_{4}$
graph, the idea being that this extra RG transformation might help suppress
the irrelevant parts of any errors incurred in the approximations
$\overline{\mathcal{H}}$ and $\overline{\overline{\mathcal{H}}}$ to the
effective Hamiltonian in the first step. The comparisons, displayed in Fig.
15, are encouraging, in that errors in dimer densities are now upper bounded
by $\sim 0.0005$ for both $\overline{\mathcal{H}}$ and
$\overline{\overline{\mathcal{H}}}$.

Based on this evidence, we proceed with the assumption that the dimer weights
calculated using the algorithm presented at the end of Sec. IV.1, via the
partition function of Eq. (10), provide a close approximation of effective
Hamiltonians for the purposes of investigating the RG flow of the dimer model
on the AB tiling. To study the RG flow in terms of these effective
Hamiltonians, we start with the microscopic Hamiltonian on an $8_{4}$-graph
where all dimer weights $W_{0}(e_{k})=1$. Using the algorithm described in
Sec. IV.1 we calculate the weights of effective dimers $W_{1}$ on the coarse-
grained $8_{2}$-graph. Using the approximation
$\overline{\overline{\mathcal{H}}}$, which gives dimer weights of any edge of
any AB graph in terms of a finite number of parameters, we now sample this
effective Hamiltonian in an $8_{4}$-graph. This allows us to calculate the
weights $W_{2}(e_{k})$ on an $8_{2}$-graph— providing us with, via the
approximation $\overline{\overline{\mathcal{H}}}$, the effective Hamiltonian
on the second RG iteration. This process can be repeated to track the
effective Hamiltonian as a function of the RG iteration. While we have found
the choice of approximation between $\overline{\overline{\mathcal{H}}}$ and
$\overline{\mathcal{H}}$ to not affect much of what follows, we parameterize
the effective Hamiltonian at each stage in terms of the parameters
$\overline{W_{n}}(t)$; as before $\overline{W_{n}}(t)$ is calculated as the
mean of calculated weights $W_{n}(e_{k})$ for all edges $e_{k}$ whose edge-
type $T(e_{k})$ is equal to $t$. This parametrisation is reasonable in light
of the fact that the calculated weights $W_{n}(e_{k})$ for edges with same
edge-type $t$ are narrowly distributed around the mean $\overline{W_{n}}(t)$
(For $W_{1}$, this was displayed in Fig. 9). The effective Hamiltonian in
terms of these parameters as a function of the RG step $n$ is displayed in
Tab. 2. It is evident that our effective Hamiltonians flow to a fixed point
Hamiltonian, the fixed point is stable to whatever errors are picked up by the
numerical approximations in this calculation.

##  Appendix D Stability of the fixed point to perturbations in parameters of
the effective Hamiltonian

We use a standard MCRG trick to study at the stability of the fixed point to
simple perturbations in the dimer weights $\overline{W_{n}(t)}$ [76]. The idea
is to consider a set of couplings $\\{K^{n}_{i}\\}$ describing the effective
Hamiltonian at an RG-step $n$, which presumably flows to a set of fixed-point
values $K^{*}_{i}$. For us, the couplings are given by the logarithm of the
dimer weights, $K^{n}_{t}\sim\log\Big{(}\overline{W_{n}}(t)\Big{)}$. In this
Appendix, we approximate the partition function by the one obtained with an
effective Hamiltonian $\overline{\mathcal{H}}$ since, as shown in the previous
section, they are a good approximation and they describe RG flows faithfully.
Under this approximation, the partition function is given by

$\displaystyle Z$ $\displaystyle=\exp(\overline{\mathcal{H}}^{n}).$ (32)

$\overline{\mathcal{H}}^{n}$ is given in terms of couplings by

$\displaystyle\overline{\mathcal{H}}^{n}$
$\displaystyle=\sum_{e_{k}}K^{n}_{T(e_{k})}s^{n}(e_{k})$
$\displaystyle=\sum_{t}K^{n}_{t}S^{n}(t).$ (33)

As before, $T(e_{k})$ is the edge-type of edge $e_{k}$. In the last line, the
sum is over the ten different edge-types $t$, and we have defined $S^{n}(t)$
as the total number of dimers on edges with edge-type $t$. This is the
operator to which $K^{n}_{t}$ couples.

To investigate the stability of the fixed point, we calculate the stability
matrix $M^{n}_{pq}=\partial K^{n+1}_{p}/\partial K^{n}_{q}$. This can be done
using the ‘MCRG chain rule’:

$\displaystyle\frac{\partial\langle S^{n+1}_{p}\rangle}{\partial
K^{n}_{q}}=\sum_{r}\frac{\partial K^{n+1}_{r}}{\partial
K^{n}_{q}}\frac{\partial\langle S^{n+1}_{p}\rangle}{\partial K^{n+1}_{r}}$
(34)

The advantage of starting from Eq. (34) is that the derivatives of dimer
densities are directly accessible to MC simulations in terms of connected
correlation functions :

$\displaystyle\frac{\partial\langle S^{n+1}_{p}\rangle}{\partial K^{n}_{q}}$
$\displaystyle=\langle S^{n+1}_{p}S^{n}_{q}\rangle_{c}=\langle
S^{n+1}_{p}S^{n}_{q}\rangle-\langle S^{n+1}_{p}\rangle\langle
S^{n}_{q}\rangle$ (35) $\displaystyle\frac{\partial\langle
S^{n+1}_{p}\rangle}{\partial K^{n+1}_{q}}$ $\displaystyle=\langle
S^{n+1}_{p}S^{n+1}_{q}\rangle_{c}=\langle
S^{n+1}_{p}S^{n+1}_{q}\rangle-\langle S^{n+1}_{p}\rangle\langle
S^{n+1}_{q}\rangle$ (36)

Once these correlation functions are computed, the stability matrix
$M^{n}_{pq}=\partial K^{n+1}_{p}/\partial K^{n}_{q}$ can be computed from Eq.
(34) using techniques of linear algebra. The eigenvalues of the stability
matrix $\lambda_{k}$ computed close to a fixed point (large $n$) determine the
relevance of various couplings at the fixed point; $\lambda_{k}>1$ denote
relevant couplings which grow under RG, while $\lambda_{k}<1$ denote
irrelevant couplings which go to zero under RG flow. It is conventional to
express $\lambda_{k}=b^{y_{k}}$, where $b=\delta_{s}^{2}$ is the length
rescaling associated with the coarse-graining transformation. In this
language, $y_{k}>0$ and $y_{k}<0$ denote relevance and irrelevance
respectively.

We consider a dimer model on an $8_{4}$-region with weights
$\overline{W_{n}}(t)$ for $n=4$ where the system is already very near to the
fixed point (See Tab. 2). We coarse grain this to a dimer model on an
$8_{2}$-region and compute the stability matrix $M^{4}_{pq}$ as described
above. The eigenvalues computed are $-0.046(1)\pm i0.048(2),0.043\pm i0.032$,
$-0.005(4)$, $\pm i0.002$, $0.001$, $0.000$ and $0.000$. We find that the
eigenvalues are rather small in magnitude. Note that while imaginary
eigenvalues of RG transformations are unusual, they have been recorded in the
literature and often lead to non-trivial features like limit cycles in RG
flows. In our case, the small real part ensures that the fixed point is stable
to all small perturbations in dimer weights.

## References

  * Cardy [1996] J. Cardy,  _Scaling and Renormalization in Statistical Physics_, Cambridge Lecture Notes in Physics (Cambridge University Press, 1996).
  * Barenblatt [2003] G. I. Barenblatt,  _Scaling_, Cambridge Texts in Applied Mathematics (Cambridge University Press, 2003).
  * Ma [2001] S.-K. Ma,  _“Modern Theory Of Critical Phenomena”_, 1st ed. (Routledge, 2001).
  * Goldenfeld [1992] N. Goldenfeld,  _“Lectures On Phase Transitions And The Renormalization Group (1st ed.)”_, 1st ed. (CRC Press, 1992).
  * Sachdev [2011] S. Sachdev,  _Quantum Phase Transitions_, 2nd ed. (Cambridge University Press, 2011).
  * Wilson and Kogut [1974] K. G. Wilson and J. Kogut, The renormalization group and the expansion, Physics Reports 12, 75 (1974).
  * Belavin  _et al._ [1984] A. Belavin, A. Polyakov, and A. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Physics B 241, 333 (1984).
  * Ginsparg [1990] P. Ginsparg, Applied conformal field theory, in  _Fields, Strings and Critical Phenomena, (Les Houches, Session XLIX, 1988)_, edited by E. Brézin and J. Zinn-Justin (North Holland, Amsterdam, 1990).
  * [9] J. Cardy.
  * Kasteleyn [1961] P. Kasteleyn, “The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice”, Physica 27, 1209 (1961).
  * Kenyon [2009] R. Kenyon, Lectures on dimers (2009), arXiv:0910.3129 [math.PR] .
  * Grunbaum and Shephard [1986] B. Grunbaum and G. C. Shephard,  _“Tilings and Patterns”_ (W. H. Freeman and Company, New York, 1986).
  * Baake and Grimm [2013] M. Baake and U. Grimm,  _“Aperiodic Order Volume 1: A Mathematical Invitation”_ (Cambridge University Press, 2013).
  * Lovász and Plummer [1986] L. Lovász and M. D. Plummer,  _“Matching Theory”_ (North-Holland (Amsterdam), 1986).
  * Blote and Hilborst [1982] H. W. J. Blote and H. J. Hilborst, Roughening transitions and the zero-temperature triangular ising antiferromagnet, Journal of Physics A: Mathematical and General 15, L631 (1982).
  * Nienhuis  _et al._ [1984] B. Nienhuis, H. J. Hilhorst, and H. W. J. Blote, Triangular SOS models and cubic-crystal shapes, Journal of Physics A: Mathematical and General 17, 3559 (1984).
  * Henley [1997] C. L. Henley, Relaxation time for a dimer covering with height representation, Journal of Statistical Physics 89, 483 (1997).
  * Moessner and Raman [2011] R. Moessner and K. S. Raman, Quantum dimer models, in  _Introduction to Frustrated Magnetism: Materials, Experiments, Theory_, edited by C. Lacroix, P. Mendels, and F. Mila (Springer Berlin Heidelberg, Berlin, Heidelberg, 2011) pp. 437–479.
  * Luck [1993] J. M. Luck, A classification of critical phenomena on quasi-crystals and other aperiodic structures, Europhysics Letters 24, 359 (1993).
  * Kadanoff [1966] L. P. Kadanoff, Scaling laws for ising models near ${T}_{c}$, Physics Physique Fizika 2, 263 (1966).
  * Migdal [1975] A. Migdal, A., Phase transitions in gauge and spin-lattice systems, Journal of Experimental and Theoretical Physics 42, 743 (1975).
  * Efrati  _et al._ [2014] E. Efrati, Z. Wang, A. Kolan, and L. P. Kadanoff, Real-space renormalization in statistical mechanics, Rev. Mod. Phys. 86, 647 (2014).
  * Swendsen [1979] R. H. Swendsen, Monte carlo renormalization group, Phys. Rev. Lett. 42, 859 (1979).
  * Wu and Car [2017] Y. Wu and R. Car, Variational approach to monte carlo renormalization group, Phys. Rev. Lett. 119, 220602 (2017).
  * Chung and Kao [2021] J.-H. Chung and Y.-J. Kao, Neural monte carlo renormalization group, Phys. Rev. Research 3, 023230 (2021).
  * Xiong  _et al._ [1999] G. Xiong, Z.-H. Zhang, and D.-C. Tian, Real-space renormalization group approach to the potts model on the two-dimensional penrose tiling, Physica A: Statistical Mechanics and its Applications 265, 547 (1999).
  * Xiong  _et al._ [1998] G. Xiong, Z.-H. Zhang, and D.-C. Tian, A real-space renormalization group approach to the ferromagnetic potts model on the two-dimensional octagonal quasi-periodic tiling, Journal of Physics: Condensed Matter 10, 10067 (1998).
  * Griffiths and Kaufman [1982] R. B. Griffiths and M. Kaufman, Spin systems on hierarchical lattices. introduction and thermodynamic limit, Phys. Rev. B 26, 5022 (1982).
  * Meurice  _et al._ [1995] Y. Meurice, G. Ordaz, and V. G. J. Rodgers, Evidence for complex subleading exponents from the high-temperature expansion of dyson’s hierarchical ising model, Phys. Rev. Lett. 75, 4555 (1995).
  * Derrida  _et al._ [1983] B. Derrida, L. De Seze, and C. Itzykson, Fractal structure of zeros in hierarchical models, Journal of Statistical Physics 33, 559 (1983).
  * Derrida  _et al._ [1984] B. Derrida, C. Itzykson, and J. M. Luck, Oscillatory critical amplitudes in hierarchical models, Communications in Mathematical Physics 94, 115 (1984).
  * Sommers  _et al._ [2022] G. M. Sommers, M. J. Gullans, and D. A. Huse, Self-dual quasiperiodic percolation (2022).
  * Lloyd  _et al._ [2022] J. Lloyd, S. Biswas, S. H. Simon, S. A. Parameswaran, and F. Flicker, Statistical mechanics of dimers on quasiperiodic ammann-beenker tilings, Phys. Rev. B 106, 094202 (2022).
  * Gökmen  _et al._ [2023] D. E. Gökmen, S. Biswas, S. D. Huber, Z. Ringel, F. Flicker, and M. Koch-Janusz, Machine learning assisted discovery of exotic criticality in a planar quasicrystal (2023).
  * Flicker  _et al._ [2020] F. Flicker, S. H. Simon, and S. A. Parameswaran, Classical dimers on penrose tilings, Phys. Rev. X 10, 011005 (2020).
  * Beach and Sandvik [2006] K. Beach and A. W. Sandvik, Some formal results for the valence bond basis, Nuclear Physics B 750, 142 (2006).
  * Sandvik and Evertz [2010] A. W. Sandvik and H. G. Evertz, Loop updates for variational and projector quantum monte carlo simulations in the valence-bond basis, Phys. Rev. B 82, 024407 (2010).
  * Sandvik and Moessner [2006] A. W. Sandvik and R. Moessner, Correlations and confinement in nonplanar two-dimensional dimer models, Phys. Rev. B 73, 144504 (2006).
  * Alet  _et al._ [2006] F. Alet, Y. Ikhlef, J. L. Jacobsen, G. Misguich, and V. Pasquier, Classical dimers with aligning interactions on the square lattice, Phys. Rev. E 74, 041124 (2006).
  * Sornette [1998] D. Sornette, Discrete-scale invariance and complex dimensions, Physics Reports 297, 239 (1998).
  * Kondev and Henley [1995a] J. Kondev and C. L. Henley, Four-coloring model on the square lattice: A critical ground state, Phys. Rev. B 52, 6628 (1995a).
  * Desai  _et al._ [2021] N. Desai, S. Pujari, and K. Damle, Bilayer coulomb phase of two-dimensional dimer models: Absence of power-law columnar order, Phys. Rev. E 103, 042136 (2021).
  * Kondev and Henley [1995b] J. Kondev and C. L. Henley, Geometrical exponents of contour loops on random gaussian surfaces, Phys. Rev. Lett. 74, 4580 (1995b).
  * Kondev  _et al._ [2000] J. Kondev, C. L. Henley, and D. G. Salinas, Nonlinear measures for characterizing rough surface morphologies, Phys. Rev. E 61, 104 (2000).
  * Kenyon [2014] R. Kenyon, Conformal invariance of loops in the double-dimer model, Communications in Mathematical Physics 326, 477 (2014).
  * Luck and Nieuwenhuizen [1986] J. M. Luck and T. M. Nieuwenhuizen, A soluble quasi-crystalline magnetic model: The xy quantum spin chain, Europhysics Letters 2, 257 (1986).
  * Hermisson  _et al._ [1997] J. Hermisson, U. Grimm, and M. Baake, Aperiodic ising quantum chains, Journal of Physics A: Mathematical and General 30, 7315 (1997).
  * Hermisson [2000] J. Hermisson, Aperiodic and correlated disorder in xy chains: exact results, Journal of Physics A: Mathematical and General 33, 57 (2000).
  * Hida [2004] K. Hida, New universality class in spin-one-half fibonacci heisenberg chains, Phys. Rev. Lett. 93, 037205 (2004).
  * Vieira [2005] A. P. Vieira, Aperiodic quantum xxz chains: Renormalization-group results, Phys. Rev. B 71, 134408 (2005).
  * Satija and Doria [1988] I. I. Satija and M. M. Doria, Quasiperiodic anisotropic xy model, Phys. Rev. B 38, 5174 (1988).
  * Crowley  _et al._ [2018] P. J. D. Crowley, A. Chandran, and C. R. Laumann, Quasiperiodic quantum ising transitions in 1d, Phys. Rev. Lett. 120, 175702 (2018).
  * Crowley  _et al._ [2022] P. J. D. Crowley, C. R. Laumann, and A. Chandran, Critical behaviour of the quasi-periodic quantum ising chain, Journal of Statistical Mechanics: Theory and Experiment 2022, 083102 (2022).
  * Agrawal  _et al._ [2020a] U. Agrawal, S. Gopalakrishnan, and R. Vasseur, Universality and quantum criticality in quasiperiodic spin chains, Nature Communications 11, 2225 (2020a).
  * Agrawal  _et al._ [2020b] U. Agrawal, S. Gopalakrishnan, and R. Vasseur, Quantum criticality in the 2d quasiperiodic potts model, Phys. Rev. Lett. 125, 265702 (2020b).
  * Andrade [2000] R. F. S. Andrade, Detailed characterization of log-periodic oscillations for an aperiodic ising model, Phys. Rev. E 61, 7196 (2000).
  * Sørensen  _et al._ [1991] E. S. Sørensen, M. V. Jarić, and M. Ronchetti, Ising model on penrose lattices: Boundary conditions, Phys. Rev. B 44, 9271 (1991).
  * Ledue  _et al._ [1995] D. Ledue, D. P. Landau, and J. Teillet, Static critical behavior of the ferromagnetic ising model on the quasiperiodic octagonal tiling, Phys. Rev. B 51, 12523 (1995).
  * Reid  _et al._ [1998] R. W. Reid, S. K. Bose, and B. Mitrovic, Monte carlo study of the xy-model on penrose lattices, Journal of Physics: Condensed Matter 10, 2303 (1998).
  * Boutry  _et al._ [1997] T. Boutry, D. Ledue, and D. P. Landau, Three-state potts model on the quasiperiodic octagonal tiling, in  _Computer Simulation Studies in Condensed-Matter Physics IX_ , edited by D. P. Landau, K. K. Mon, and H.-B. Schüttler (Springer Berlin Heidelberg, Berlin, Heidelberg, 1997) pp. 122–127.
  * Ledue  _et al._ [1997] D. Ledue, T. Boutry, D. P. Landau, and J. Teillet, Finite-size behavior of the three-state potts model on the quasiperiodic octagonal tiling, Phys. Rev. B 56, 10782 (1997).
  * Gökmen  _et al._ [2021] D. E. Gökmen, Z. Ringel, S. D. Huber, and M. Koch-Janusz, Statistical physics through the lens of real-space mutual information, Phys. Rev. Lett. 127, 240603 (2021).
  * Koch-Janusz and Ringel [2018] M. Koch-Janusz and Z. Ringel, Mutual information, neural networks and the renormalization group, Nature Physics 14, 578 (2018).
  * Blöte and Nienhuis [1994] H. W. J. Blöte and B. Nienhuis, Fully packed loop model on the honeycomb lattice, Phys. Rev. Lett. 72, 1372 (1994).
  * Batchelor  _et al._ [1996] M. T. Batchelor, H. W. J. Blöte, B. Nienhuis, and C. M. Yung, Critical behaviour of the fully packed loop model on the square lattice, Journal of Physics A: Mathematical and General 29, L399 (1996).
  * Singh  _et al._ [2023] S. Singh, J. Lloyd, and F. Flicker, Hamiltonian cycles on ammann-beenker tilings (2023).
  * Raghavan  _et al._ [1997] R. Raghavan, C. L. Henley, and S. L. Arouh, New two-color dimer models with critical ground states, Journal of Statistical Physics 86, 517 (1997).
  * Wilkins and Powell [2020] N. Wilkins and S. Powell, Interacting double dimer model on the square lattice, Phys. Rev. B 102, 174431 (2020).
  * Tang  _et al._ [2011] Y. Tang, A. W. Sandvik, and C. L. Henley, Properties of resonating-valence-bond spin liquids and critical dimer models, Phys. Rev. B 84, 174427 (2011).
  * Albuquerque and Alet [2010] A. F. Albuquerque and F. Alet, Critical correlations for short-range valence-bond wave functions on the square lattice, Phys. Rev. B 82, 180408 (2010).
  * Patil  _et al._ [2014] P. Patil, I. Dasgupta, and K. Damle, Resonating valence-bond physics on the honeycomb lattice, Phys. Rev. B 90, 245121 (2014).
  * Albuquerque  _et al._ [2012] A. F. Albuquerque, F. Alet, and R. Moessner, Coexistence of long-range and algebraic correlations for short-range valence-bond wave functions in three dimensions, Phys. Rev. Lett. 109, 147204 (2012).
  * Sutherland [1988] B. Sutherland, Systems with resonating-valence-bond ground states: Correlations and excitations, Phys. Rev. B 37, 3786 (1988).
  * Beach  _et al._ [2009] K. S. D. Beach, F. Alet, M. Mambrini, and S. Capponi, $\text{SU}(n)$ heisenberg model on the square lattice: A continuous-$n$ quantum monte carlo study, Phys. Rev. B 80, 184401 (2009).
  * Damle  _et al._ [2012] K. Damle, D. Dhar, and K. Ramola, Resonating valence bond wave functions and classical interacting dimer models, Phys. Rev. Lett. 108, 247216 (2012).
  * Landau and Binder [2021] D. Landau and K. Binder, Monte carlo renormalization group methods, in  _A Guide to Monte Carlo Simulations in Statistical Physics_ (Cambridge University Press, 2021) p. 416–429, 5th ed.