Conformal Quasicrystals and Holography

Latham Boyle ,1 Madeline Dickens,2 and Felix Flicker2,31Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada2Department of Physics, University of California, Berkeley, California 94720, USA

3Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Department of Physics,Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, United Kingdom

(Received 18 May 2018; revised manuscript received 22 August 2019; published 14 January 2020)

Recent studies of holographic tensor network models defined on regular tessellations of hyperbolicspace have not yet addressed the underlying discrete geometry of the boundary. We show that the boundarydegrees of freedom naturally live on a novel structure, a “conformal quasicrystal,” that provides a discretemodel of conformal geometry. We introduce and construct a class of one-dimensional conformalquasicrystals and discuss a higher-dimensional example (related to the Penrose tiling). Our constructionpermits discretizations of conformal field theories that preserve an infinite discrete subgroup of the globalconformal group at the cost of lattice periodicity.

DOI: 10.1103/PhysRevX.10.011009 Subject Areas: Condensed Matter Physics,Interdisciplinary Physics,String Theory

I. INTRODUCTION

A central topic in theoretical physics over the past twodecades has been holography: the idea that a quantumtheory in a bulk space may be precisely dual to anotherliving on the boundary of that space. The most concrete andwidely studied realization of this idea has been the AdS/CFT correspondence [1–3], in which a gravitational theoryliving in a (dþ 1)-dimensional negatively curved bulkspacetime is dual to a nongravitational theory living on itsd-dimensional boundary. Over the past decade, investiga-tions of this duality have yielded mounting evidence thatspacetime and its geometry can in some sense be regardedas emergent phenomena, reflecting the entanglement pat-tern of some other underlying quantum degrees of freedom(d.o.f.) [4].Recently, physicists have been interested in the possibility

that discrete models of holography would permit them tounderstand this idea in greater detail, by bringing in the toolsof condensed matter physics and quantum informationtheory—much as lattice gauge theory led to conceptualand practical progress in understanding gauge theories in thecontinuum [5]. Recent years have seen a surge of interest indiscrete models of holography based on tensor networks(TNs). Such holographic TNs relate the quantum d.o.f.living on a discretized (dþ 1)-dimensional hyperbolic

space to corresponding d.o.f. on the discretized d-dimen-sional boundary of that space. They bridge condensedmatterphysics, quantum gravity, and quantum information.In condensed matter physics, holographic TNs provide

a computationally efficient description of the highlyentangled ground states of quantum many-body systems(particularly scale-invariant systems or systems near theircritical points). The entanglement pattern is described bytensors living on an emergent discrete hyperbolic geometryin one dimension higher [6–11]. On the quantum gravityside, the TNs are a kind of UV regulator of the physics inthe bulk, and a proposed way to represent the fact thatspacetime and its geometry may be regarded as emergentphenomena, reflecting the entanglement pattern of someother underlying quantum d.o.f. [12–24]. Meanwhile,quantum information theory provides a unifying languagefor these studies in terms of entanglement, quantumcircuits, and quantum error correction [25].These investigations have gradually clarified our under-

standing of the discrete geometry in the bulk. There hasbeen a common expectation, based on an analogy withAdS/CFT [1–3], that TNs living on discretizations of ahyperbolic space define a lattice state of a critical system onthe boundary and vice versa. Initially, this led Swingle tointerpret Vidal’s Multiscale Entanglement RenormalizationAnsatz (MERA) as a discretized version of a time slice ofthe AdS geometry [7,12]. However, MERA has a preferredcausal direction, while any discretization of a spacelikemanifold should not. To fix this issue, it was proposed thatMERA was related first to a different object known as thekinematic space of AdS [18,26] and, more recently, to alightlike geometry [27].

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Meanwhile, new TN models (e.g., the holographicquantum error-correcting codes of Ref. [16], the hyper-invariant networks of Ref. [21], or the matchgate networksof Ref. [24]) were introduced to more adequately captureAdS. The key feature of these new TNs is that they live onregular tilings of hyperbolic space [28]. The symmetries ofsuch a tessellation form an infinite discrete subgroup of thecontinuous symmetry group of the AdS time slice, much asthe symmetries of an ordinary 3D lattice or crystal form aninfinite discrete subgroup of the continuous symmetrygroup of 3D Euclidean space. Physically, such TNsrepresent a discretization of the continuous bulk space,much as a crystal represents a discretization of a continuummaterial. In this paper, we restrict our attention to TNs ofthis type.Despite progress in describing the discrete bulk geom-

etry, the question of which discrete geometric spaces aresuitable for the boundary has received little attention. In adiscrete model of the AdS/CFT correspondence, we expectto be able to construct the discrete boundary geometryentirely from the data of the bulk tessellation, in analogywith the continuum case, where it is well known that thedata of any asymptotically AdS spacetime define a con-formal manifold on its boundary [2]. While this expectationseems natural, we are unaware of a corresponding discreteconstruction in the literature. In condensed matter physics,the discrete boundary geometry is typically assumed to be aperiodic lattice; but, as we observe here, there is no naturalway in which an ordinary regular tessellation of hyperbolicspace defines a periodic lattice on its boundary, and nonatural way for the discrete symmetries of the bulk to act onsuch a boundary lattice [29,31]. Thus, we expect that, inimplementing a discrete version of AdS/CFT, it will benatural to replace the periodic boundary lattice with adifferent discrete object. Such a replacement is reasonable,from a Wilsonian viewpoint, as the choice of underlyinglattice becomes irrelevant at a critical point [32].In this paper, we argue that the d.o.f. on the boundary of

a regular tessellation of hyperbolic space naturally live on aremarkable structure—a “conformal quasicrystal” (CQC)—built entirely from the data of the bulk tessellation. Thisprovides a new clue about the type of boundary theory thatshould appear in a discrete version of holography. Far frombeing a simple periodic lattice, each CQC locally resemblesa self-similar quasicrystal [33–37] (like the Penrose tiling[38], shown in Fig. 1) and possesses discrete symmetriesworthy of a discretization of a conformal manifold. In thispaper, we focus on the d ¼ 1 case but with a view towardshigher dimensions (which we discuss briefly near the end).We propose to use our framework to construct discretiza-tions of CFTs that preserve an infinite discrete subgroup ofthe global conformal group at the cost of exact discretetranslation invariance (which is replaced by quasitransla-tional invariance). We end with suggestions for futureresearch and a discussion of how our results will hopefully

provide an important clue in the ongoing effort to formulatea discrete version of holography, and also lead to animproved analytical and numerical understanding of thestructure of condensed matter systems at their criticalpoints, and other conformally invariant systems.

II. QUASICRYSTALS IN d = 1

We begin by briefly introducing self-similar quasicrys-tals (SSQCs) in d ¼ 1 (see Refs. [33–37] for furtherdetails). For general d, a SSQC may be thought of as aspecial quasiperiodically ordered set of discrete points inEuclidean space (like the vertices in an infinite 2D Penrosetiling, shown in Fig. 1). The same SSQC can be constructedin two very different ways: either (i) by a cut-and-projectmethod based on taking an irrationally sloped slice througha periodic lattice in a higher-dimensional Euclidean spaceand projecting nearby points onto the slice, or (ii) byrecursive iteration of an appropriate substitution (or “infla-tion”) rule. Restricting to d ¼ 1 in the following sections,we exploit the second (inflation) perspective to develop anovel construction whereby 1D SSQCs emerge on theboundary of a lattice in 2D hyperbolic space.Let Π be a (possibly infinite) string of two letters, α and

β, and let sαðα; βÞ and sβðα; βÞ be two finite strings of α’sand β’s. We can act on Π with the inflation rule

τ∶ ðα; βÞ ↦ (sαðα; βÞ; sβðα; βÞ); ð1Þ

which replaces each α or β in Π by the string sαðα; βÞ orsβðα; βÞ, respectively, to obtain a new stringΠ0. The inversemap is the corresponding deflation rule:

FIG. 1. The Penrose tiling [35]. The two different tiles (red andblue) cover the entire Euclidean plane in an aperiodic mannerwithout gaps. Centers of approximate fivefold symmetry can beseen. The tiling can be created through a cut-and-project methodfrom a five-dimensional hypercubic lattice or a four-dimensionalA4 lattice, or by applying “inflation rules” to a finite-size startingconfiguration [33–35].

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τ−1∶ (sαðα; βÞ; sβðα; βÞ) ↦ ðα; βÞ: ð2Þ

Note that, while the inflation rule has a well-defined actionon any string Π of α’s and β’s, the corresponding deflationmap only has a well-defined action on a string Π that maybe uniquely partitioned into substrings of the form sαðα; βÞand sβðα; βÞ.Any inflation rule τ induces a matrixMτ that encodes the

growth of Nα and Nβ (the number of α’s and β’s,respectively, in the string). For example, under τ∶ðα; βÞ ↦ðαβα; αβαβαÞ, which corresponds to inflation rule 3b inTable 1 of Ref. [36], we have

�N0α

N0β

�¼ Mτ

�Nα

Nβ

�; Mτ ¼

�2 3

1 2

�: ð3Þ

If λ is the largest eigenvalue of Mτ (λ ¼ 3þ ffiffiffi2

p), we can

represent Π geometrically as a sequence of 1D line seg-ments or “tiles” of length Lα and Lβ, where ðLα; LβÞ is thecorresponding left eigenvector of Mτ. If Nα

k and Nβk denote

the number of α’s and β’s after k successive applications ofτ to some finite initial string, then in the limit k → ∞,ðNα

k; NβkÞ is the corresponding right eigenvector of Mτ and

it grows like ðNαkþ1; N

βkþ1Þ ¼ λðNα

k; NβkÞ.

Two strings Π and Π0 are locally isomorphic (or locallyindistinguishable) if every finite substring contained in oneis also contained in the other, so it would be impossible todistinguish them by inspecting any finite segment.We say that Π is a τ quasicrystal if it is equipped with an

inflation rule τ acting on the two interval types α and β [39]such that (i) the matrix Mτ has determinant �1, withits largest eigenvalue λ given by an irrational Pisot-Vijayaraghavan (PV) number, and (ii) the k-fold deflationmap τ−k is well defined on Π for all positive integers k.Condition (i) ensures that Π is quasiperiodic and crystalline(in the sense that the Fourier representation of its densityprofile exhibits delta function diffraction peaks like acrystal), and has τ−1 as the unique deflation rule thatinverts τ [33,40]. Condition (ii) implies that a particular τquasicrystal is locally isomorphic to every other τ quasi-crystal and, in particular, is locally isomorphic to its owndescendants under inflation. In this sense, every τ quasi-crystal is self-similar under inflation by τ. If a τ quasicrystalΠ is mapped to itself by (k successive applications of) τ,we say it is (k-fold) self-same, an even stronger scale-invariance property than self-similarity.

III. UNIT-CELL ASSIGNMENT ON THEHYPERBOLIC DISK

A regular fp; qg tessellation is constructed from regularp-sided polygons (with p geodesic edges), where q suchpolygons meet at each vertex. Such a tessellation exists forany p; q > 2: When ð1=pÞ þ ð1=qÞ is greater than, equal

to, or less than 1=2, respectively, it is a tessellation of thesphere, Euclidean plane, or hyperbolic plane [28,41].Consider a regular fp; qg tessellation of the hyperbolic

plane, ð1=pÞ þ ð1=qÞ < 12. In this paper, we focus on the

case where p and q are both finite. Let S (the “seed”) be afinite, simply connected patch of tiles, and let it also beconvex, which, here, means that, if θc is the interior angle ofS at a vertex c, then θc < 2πð1 − 1=qÞ for each boundaryvertex (when q > 3) or θa þ θb < 4πð1 − 1=qÞ for allnearest-neighbor pairs ha; bi of boundary vertices (whenq ¼ 3). On the boundary of S, we can assign different unit-cell types between which the d.o.f. live. The assignmentprocedure, illustrated in Fig. 2(a), is the following.Consider the union UðSÞ of all tiles that share a vertexwith the boundary of S (excluding the tiles in S itself). Inthe interior of UðSÞ, we can draw a curve that crosses everytile of UðSÞ exactly once. This curve partitions every tileinto two pieces: one contained in the interior of the curveand one in the exterior. If the interior part of the tile has nvertices, then we call it a type-n unit cell. Once every cellon the boundary of S has been labeled in this manner, wecan write down a corresponding stringQðSÞ of cell types asin Fig. 2(c). These strings are inside square brackets toemphasize that they are cyclically ordered. When S isconvex, only two cell types appear on its boundary (type 1and type 2 when q > 3, and type 2 and type 3 when q ¼ 3),but since the seed S in Fig. 2(a) is not convex, a type-4 cellalso appears.In a concrete holographic TN model, a tensor is placed

on each vertex of S, and contractions are performedbetween nearest-neighbor tensors along the tessellationedges. Extra tensors can be added to the edges, as in the

FIG. 2. Illustration of unit-cell and letter assignment on theboundary of a simply connected tile set S, in the f7; 3g tiling.(a) Unit-cell assignment. (b) Letter assignment. (c) Cyclicallyordered strings of unit cells and letters. (d) Correspondencebetween unit cells and letters in the f7; 3g case.

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hyperinvariant networks of Evenbly [21], or bulk ancillad.o.f. to the vertex tensors, as in the holographic quantumerror correcting codes of Ref. [16]. The boundary d.o.f. aredefined on legs placed on the edges of the tessellationemanating from S, which are cut by the partitioning curvein Fig. 2(a). Thus, the distance in the network between twoadjacent d.o.f. on the boundary of S depends on the typeof unit cell between them. In this sense, the cell typesdefined here behave analogously to the interval types Lα

and Lβ between d.o.f. placed on the vertices of a d ¼ 1

quasicrystal.There is an alternative way to label the boundary of S

using only two symbols, which treats the q ¼ 3 and q > 3cases more uniformly and continues to apply even when Sis nonconvex. Decompose every tile into the fundamentaldomains of the tiling symmetry group, which are righttriangles of angles π=2, π=q, and π=p. This decompositionis shown in Fig. 2(b). Now, consider the set of such righttriangles in the interior of S that share a vertex with theboundary of S. Within this set, we identify and label twotypes of isosceles triangles—A and B. Each A shares onevertex and no edges with the boundary of S, and each Bshares two vertices and one edge [see Fig. 2(b)]. Thisprocedure, called letter assignment, produces a cyclicallyordered string ΛðSÞ, distinct from QðSÞ, containing onlyA’s and B’s [see Fig. 2(c)].There is a way to extract QðSÞ from ΛðSÞ and vice versa

by mapping each type-n unit cell to a string of A’s and B’s.This correspondence is

1 ↔ A−1

2 ↔ Aq−3=2BAq−3=2

3 ↔ Aq−3=2BAq−2BAq−3=2

..

.

n ↔ Aq−3=2ðBAq−2Þn−2BAq−3=2;

where n ≤ p. Here, if fðA;BÞ is any finite substring, then(fðA;BÞ)x denotes its x-fold repetition whenever x ∈ Z≥0,with (fðA; BÞ)0 the empty substring. Fractional and neg-ative exponents behave in the usual way, e.g., AxA−y ¼Ax−y for all x; y ∈ Q.

IV. INFLATION RULESON THE HYPERBOLIC DISK

We can append the ring of tilesUðSÞ to the set S to obtaina new larger set S0 ¼ Sþ UðSÞ, as in Fig. 3. Accordingly,we add a layer of tensors to our TN to obtain a holographicTN model on S0 whose d.o.f. are defined between the unitcells on the boundary of S0. The vertices that were formerlyon the boundary (i.e., the ones that were not completelysurrounded by the tiles of S) are now in the interior (i.e., arecompletely surrounded by the tiles of S0). We dub thisprocedure “vertex completion.”

Vertex completion induces a mapping τΛðp; qÞ∶ΛðSÞ →ΛðS0Þ. Remarkably, τΛðp; qÞ is an inflation rule on A and Band is the same for every S. If we define the stringless

sL ≔ A1=2Bðp−2Þ=2; sR ≔ Bðp−2Þ=2A1=2; ð4Þ

then τΛðp; qÞ can be written

A ↦ ðsLsRÞ−1 ¼ A−1=2B2−pA−1=2; ð5aÞ

B ↦

�sLðsRsLÞq−3=2B−1ðsRsLÞq−3=2sR ðq oddÞðsRsLÞq−2=2B−1ðsRsLÞq−2=2 ðq evenÞ: ð5bÞ

FIG. 3. Demonstration of how vertex completion inducesinflation rules on boundary letters and unit cells, for the f7; 3gtiling. The induced letter inflation rule ðA; BÞ ↦ ðA−1B−5; B4AÞis the same for every tile set S, convex or not. When S is convex,only type-2 and type-3 unit cells appear, and there is an inducedunit-cell inflation rule ð2; 3Þ ↦ ð223; 23Þ. Note that vertexcompletion preserves convexity and that every tile set S becomesconvex after finitely many vertex completions.

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This version of the substitution rule is the canonical one, inthe sense that it respects the symmetry of the parent tile(A or B) under reflection about its midline, but anotherequivalent version (which is asymmetric) is simpler andmore convenient:

A ↦ A−1B2−p; B ↦ Bp−3AðBp−2AÞq−3: ð6Þ

Note that two substitution rules fα; βg ↦ fsα; sβg andfα; βg ↦ fs̄α; s̄βg are equivalent if s̄α ¼ usαu−1 and s̄β ¼usβu−1 for some finite string u ¼ uðα; βÞ. In particular, thetwo substitution rules (5) and (6) both correspond to thesubstitution matrix

MτΛðp;qÞ ¼� −1 q − 2

2 − p ðp − 2Þðq − 2Þ − 1

�: ð7Þ

The corresponding induced map τQðp; qÞ∶QðSÞ →QðS0Þ is also an inflation rule on two unit-cell typeswhenever S is convex. The rule τQðp; qÞ can be extractedby combining τΛðp; qÞ with the cell-to-letter mapping.When q > 3,

1 ↦ 1q−4=2ð2 1q−3Þp−32 1q−4=2; ð8aÞ

2 ↦ 1q−4=2ð2 1q−3Þp−42 1q−4=2; ð8bÞ

and when q ¼ 3,

2 ↦ 3122p−53

12; ð9aÞ

3 ↦ 3122p−63

12: ð9bÞ

The matrices MτQðp;qÞ and MτΛðp;qÞ are related by a changeof basis. They have unit determinant and eigenvalues

λ�ðp; qÞ ¼ γp;q � ðγ2p;q − 1Þ1=2; ð10Þ

with

γp;q ≔ðp − 2Þðq − 2Þ

2− 1: ð11Þ

These results are consistent with the previous findings ofRef. [42]. Note that the largest eigenvalue λþðp; qÞ isirrational and PV whenever ð1=pÞ þ ð1=qÞ < 1

2. When S is

nonconvex, τQðp; qÞ acts on the many cell types ofQðSÞ ina complicated way. However, every tile set S maps to aconvex set under finitely many vertex completions, andvertex completion preserves convexity once established.

V. CONFORMAL QUASICRYSTALS ON THEBOUNDARY OF THE DISK

Consider any finite, simply connected tile set S of thefp; qg tessellation. By repeatedly carrying out layers ofvertex completions, we generate a sequence of tile setsfSngn∈Z≥0

, with initial element S0 ¼ S, and fQðSnÞgn∈Z≥0.

Concurrently, we can consider a family of holographic TNmodels defined on each Sn whose d.o.f. live on theboundary of Sn. In the limit n → ∞, the tile set coversthe entire hyperbolic disk, and we can ask about the unit-cell assignment Q∞ðSÞ ≔ limn→∞QðSnÞ living at the diskboundary. We can interpret Q∞ðSÞ as a type of emergentquasicrystal harboring the d.o.f. To see this case, recall thatfor all n above some n� > 0, QðSnÞ contains only two celltypes, and thus τQðp; qÞ acts on these two cell types as aninflation rule. Taking QðSn�Þ as an initial string anditerating τQðp; qÞ infinitely many times generates theinfinite, cyclically ordered string Q∞ðSÞ, which is self-same under τQðp; qÞ. Thus, Q∞ðSÞ is locally isomorphic toa τQðp; qÞ quasicrystal, which is the promised quasicrys-talline interpretation. We call any Q∞ðSÞ obtained in thismanner a fp; qg conformal quasicrystal (CQC).

VI. DISCRETE CONFORMAL GEOMETRY

Our construction naturally endows CQCs with a discreteanalog of conformal geometry. In the continuum, con-formal geometry refers to those properties of a (pseudo-)Riemannian manifold that are invariant under position-dependent rescalings (or Weyl transformations) of themetric tensor gμνðxÞ ↦ Ω2ðxÞgμνðxÞ. Thus, length scalesare not well defined in conformal geometry. Conformalproperties are preserved under conformal maps, diffeo-morphisms fixing the metric up to a Weyl transformation.These include all the diffeomorphisms in d ¼ 1 and just theangle-preserving diffeomorphisms in d ≥ 2. In physics,conformal geometries underlie conformal field theories(CFTs), which are quantum field theories that are sym-metric under such conformal maps [43,44]. Here, we showthat one can relate any two CQCs with the same fp; qg viaa discrete analog of a Weyl transformation. Hence, CQCsinherit discrete conformal geometry, defined to be theproperties fixed under discrete Weyl transformations.Let MðSÞ ≠ S denote a finite, simply connected tile set

obtained from S in any manner. Here, Q∞ðSÞ andQ∞(MðSÞ) are locally isomorphic, but they differ in theirglobal structures. To relate Q∞ðSÞ and Q∞(MðSÞ), a mapΩ such that Ω(Q∞ðSÞ) ¼ Q∞(MðSÞ) is needed. We saythat any such Ω is a “discrete Weyl transformation.” Here,Ω acts upon the global structure of the CQC by a position-dependent set of inflations and deflations. The TN model iscorrespondingly transformed by adjoining or removingtensors from the edges in a position-dependent way.The intuition behind our definition comes from

an analogy with Weyl transformations in continuum

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AdS/CFT. There, a Weyl transformation corresponds to achange of choice of how to approach the boundary [3].More precisely, the line element for (dþ 1)-dimensionalhyperbolic space can be written in global coordinates as

ds2 ¼ dρ2 þ sin2ðρÞdΩ2d

cos2ðρÞ ; ð12Þ

where ρ ∈ ½0; π=2Þ is the radial coordinate and dΩ2d is the

line element on the unit d sphere Sd. Now, consider a familyof d-dimensional hypersurfaces of spherical topology,parametrized by ρðx; ϵÞ ¼ ðπ=2Þ − ϵfðxÞ, where x arethe coordinates on Sd and fðxÞ is an arbitrary positivefunction on Sd. To approach the boundary, we take ϵ → 0þin an x-independent way. As ϵ → 0þ, a constant-ϵ hyper-surface has an induced line element:

ds2ϵðxÞ ∼2

ϵ2dΩ2

dðxÞf2ðxÞ : ð13Þ

Thus, we see that changing the hypersurface profilefðxÞ ↦ Ω−1ðxÞfðxÞ induces the Weyl transformation

ds2ϵðxÞ ↦ Ω2ðxÞds2ϵðxÞ: ð14Þ

In our construction, the choice of initial tile set S isanalogous to the choice of hypersurface profile fðxÞ, whileiterating vertex completions infinitely many times corre-sponds to taking ϵ → 0þ. Thus, it is natural to interpret S ↦MðSÞ as inducing a Weyl-like transformation Q∞ðSÞ ↦Ω(Q∞ðSÞ) ¼ Q∞(MðSÞ).

VII. SYMMETRIES OF CONFORMALQUASICRYSTALS IN d = 1

Now, let us emphasize two different types of symmetrypossessed by any CQCQ∞ðSÞ of type fp; qg. First, it has akind of exact scale symmetry: invariance under τQðp; qÞ.Second, it is invariant, up to a discrete Weyl transformation,under an infinite discrete group called the triangle group,Δð2; p; qÞ. This group is the symmetry group of the fp; qgtiling; it is generated by reflections across the fundamentalright-triangle domains of the tiling [45]. Each such reflec-tion induces a conformal map of the boundary into itself.Under an element of Δð2; p; qÞ, S maps to a differentsimply connected S0, which produces Q∞ðS0Þ. Since S0 ¼MðSÞ for some M, it follows that Q∞ðSÞ and Q∞ðS0Þ arerelated by a discrete Weyl transformation, as required.Thus, we regard Δð2; p; qÞ as a discretization (i) of theisometry group of the hyperbolic disk that acts on the bulktessellation and (ii) of the conformal group of the circularboundary that acts on the CQCs.

VIII. DISCUSSION

Before discussing our results, let us briefly summarizeour findings.The past decade has seen exciting developments in our

understanding of quantum gravity: In particular, argumentsbased on holography and gauge-gravity duality have led toa tantalizing but still-fragmentary picture in which space-time emerges from the pattern of entanglement in anunderlying quantum system. In particular, there has beenmuch recent interest in attempts to make this picture moreconcrete, in the context of AdS/CFT, by developing adiscrete formulation of holography, based on replacing thecontinuous hyperbolic “bulk” space by a discrete regulartessellation of hyperbolic space (or some other tessellationthat respects a large discrete subgroup of the originalspace’s symmetries). These developments are part of afamily of ideas that are sometimes summarized by theslogan “it from qubit”.Previous works have focused on discretizing the bulk

geometry, but they have not thought through the corre-sponding implications for the boundary geometry. In thispaper, we have shown that when one discretizes the bulkgeometry in a natural way (e.g., on a regular tessellation),one also induces a remarkable discretization of the lower-dimensional “boundary” geometry into a fascinating newkind of discrete geometric structure, which we call a“conformal quasicrystal.” The main goals of this paperwere to point out the existence of these new structures, todefine and explain their basic properties, and to emphasizethat they appear to be the natural discrete spaces living onthe boundary side of the holographic duality.We think these observations provide important clues for

the ongoing efforts to formulate a discrete version ofholography (perhaps in terms of tensor networks) [46]:In a correct discrete formulation of AdS/CFT, we expect theboundary theory to live on a conformal quasicrystal (ratherthan on an ordinary lattice, as imagined in previous works).Similarly, since the boundary theory in AdS/CFT is scaleinvariant, it suggests that the natural way to discretize andnumerically simulate scale-invariant systems (such asconformal field theories, or condensed matter systems neartheir critical points) is to discretize them on a conformalquasicrystal, rather than on a periodic lattice. Thus, wehope that this work opens the door to more efficientsimulation of the dynamics and quantum states of suchsystems (although much work remains in order to translatethis hope into practice).Let us now mention various directions for future work.

We expect our definitions and results to extend readily tohigher dimensions. Consider, for example, the self-dualf3; 5; 3g regular honeycomb in three-dimensional hyper-bolic space. This honeycomb is constructed by gluingtogether icosahedra such that the vertex figure at everycorner is a dodecahedron [28,41]. Now, imagine that wetake the initial seed S to be one such icosahedron and then

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carry out vertex completion, assigning boundary unit cells atevery step. At each step, we obtain a 2D layer of sphericaltopology with 12 points of exact fivefold orientationalsymmetry (or, more precisely,D5 symmetry), lying radiallyabove the 12 vertices of the initial icosahedron. Hence, afteriterating infinitelymany times, we obtain a layer of sphericaltopology at the boundary, tesselated by an infinite number oftiles. This layer still has 12 points ofD5 symmetry, and in thevicinity of any such point, it appears to be an infinite tiling ofthe 2D Euclidean plane. Since points of D5 symmetry areforbidden in an ordinary (periodic) 2D crystal, we expect theboundary to once again be quasicrystalline. Indeed, the 2Dquasicrystalline tilings with fivefold orientational order areall closely related to the Penrose tiling [33,35,37,38,49].(Note that, although the usual Penrose tiling has, atmost, onepoint of exactD5 symmetry, the Penrose-like CQC living atthe boundary of f3; 5; 3g can have 12 such points, asexplained above. This case is the higher-dimensional ana-logue of a phenomenon that already occurs in the 1D CQCsconstructed above: The CQC generated from a regular p-gon initial seed will have 2p points of perfect reflectionsymmetry, in contrast to the self-similar 1D quasicrystal thatit locally resembles, which has, at most, one such point.)The conjecture that f3; 5; 3g hosts a CQC generalization

of the Penrose tiling on its boundary is supported by acalculation [50] of the ratio Nkþ1=Nk as k → ∞, where Nkis the number of boundary cells after k layers of vertexcompletion (this ratio is expected to be equal to the largesteigenvalue of the matrix Mτ associated with the inflationrule τ induced by the vertex completion). The limiting ratiois found to be Nkþ1=Nk → φ8, where φ is the golden ratio.Note that this result is irrational and PV (as expected for a2D quasicrystal) and is a power of φ (as expected for aquasicrystal with fivefold orientational order, in particular[33,35,37,38]). If we compare this result to the scaling ratiofor the standard Penrose tiling (Nkþ1=Nk → φ2), it suggestseither that a single layer of vertex completion in thef3; 5; 3g honeycomb corresponds to four iterations of theusual Penrose tiling inflation rule or that the Penrose tilingand f3; 5; 3g CQC are related in some way besides localisomorphism.We have seen that applying vertex completion to a

regular fp; qg tiling of hyperbolic space yields a τQðp; qÞquasicrystal. There are, however, other τ quasicrystals thatare not obtained in this way—like the example given abovein Eq. (3). We wonder whether every remaining inflationrule τ is also naturally induced by applying some appro-priate generalization of vertex completion to some appro-priate tiling of hyperbolic space. Tilings beyond the regularfp; qg type seem to be necessary.The remarkable conformal geometry of CQCs suggests

that their importance extends beyond the realm of holo-graphic TN models. We propose to use them as the under-lying spaces on which to discretize CFTs, in at least twoways. First, one can discretize a QFT partition function onto

a τ quasicrystal directly. After implementing a real-spacerenormalization-group procedure whereby d.o.f. are deci-mated according to τ−1, the renormalization-group fixedpoints can be identified with the discrete CFTs. Second, onecanwork, by analogy, with AdS/CFT. Consider the partitionfunction ZS;n½y� of a discretized field theory on the n-foldimage Sn under vertex completion of a simply connectedsubset S of the bulk tessellation, such that the fields takevalues y at the boundary of Sn. As n → ∞, we propose tointerpret ZS;∞½y� as the moment-generating function of adiscrete CFT on the boundary.From a practical standpoint, such CQC-based discreti-

zations may be important for studying scale-invariantphysical systems such as CFTs or condensed mattersystems at their critical points. These discretizations shouldpermit finite-size simulations of such systems, in ways thatpreserve an exact discrete subgroup of their scale sym-metry. This case is in contrast with ordinary lattice gaugetheory simulations or other periodic lattice models, whichinstead preserve an exact discrete subgroup of translationsymmetry, at the cost of breaking scale invariance. We canattempt to finitize a numerical calculation on a conformalquasicrystal by restricting ourselves to an annulus in thereciprocal space of the quasicrystal and imposing theboundary condition that observables on the inner and outerboundaries of the annulus are the same up to an appropriatescale factor. This method would provide the analog indiscrete CFTs of a periodic boundary condition in aperiodic lattice model, along the scale direction insteadof in real space. The resulting finite problem can then beattacked with a large arsenal of Monte Carlo and tensornetwork methods. Our hope is that this approach may leadto more accurate and efficient numerical simulation of suchsystems, perhaps allowing us to simulate much largersystems and leading to qualitative improvements in ourunderstanding.TN algorithms based on inflations and deflations may

also find use in the numerical description of quantumcritical states. We can imagine implementing a MERA-like coarse-graining scheme induced by the quasicry-stalline structure. Consider the simplest τ quasicrystals,the 1D Fibonacci quasicrystals generated by the inflationrule τ∶fα; βg ↦ fβ1=2β1=2; β1=2αβ1=2g. Given an initialFibonacci quasicrystal Π, we can use a deflationlike ruleto obtain a coarser quasicrystal cðΠÞ in two steps: (i) Sliceevery β interval into two halves, β1=2, resulting in a stringthat can be uniquely partitioned into substrings of the formβ1=2β1=2 and β1=2αβ1=2; (ii) glue these substrings together toform intervals α0 ¼ β1=2β1=2 and β0 ¼ β1=2αβ1=2 of respec-tive lengths Lα0 ¼ Lβ and Lβ0 ¼ Lα þ Lβ. Iterating thisprocedure yields a sequence fckðΠÞgk∈Z≥0

of successivelycoarser quasicrystals. We can now construct a MERA-likecircuit as follows. Embed each quasicrystal ckðΠÞ into thetwo-dimensional plane, such that ckþ1ðΠÞ is parallel to and

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lies above ckðΠÞ for every k. Assign to every point inckðΠÞ a tensor, and contract each tensor only with itsnearest neighbors in ckþ1ðΠÞ and ck−1ðΠÞ. This methodresults in three- and four-legged tensors; if we choosethese to be isometries and unitary disentanglers, respec-tively, then we acquire the desired MERA-like circuit. Wecall this network, depicted in Fig. 4, the “quasi-MERA”due to its relationship with the quasicrystalline inflationrule τ.To see why the quasi-MERAmay be special, we contrast

the deflation-based decimation procedure of the τ quasi-crystal with the standard block decimation of the periodiccrystal. Unlike the periodic crystal, for every τ-quasicrystallayer, there is a local refinement (inflation) rule that isinverted by an unambiguous local coarse-graining (defla-tion) rule. To see why this cannot hold for an ordinarycrystal, consider the example of a 1D periodic lattice withintervals of length a. We can produce a more refined lattice(with intervals of length a=2) by the local rule of choppingeach interval in half, but the inverse coarse-grainingtransformation is not locally well defined: There is anambiguity about which pair of short intervals we need toglue together to make a long one, so N “joiners” at widelyseparated points on the lattice would make different,incompatible choices about which tiles to join, andOðNÞ defects would be produced. Hence, block decimatinga periodic lattice destroys information about how to locallyrecover the original lattice. Since the ordinary (binary)MERA architecture relies on this form of block decimation,its description of a quantum state is necessarily lossy, evenat criticality. By contrast, when coarse-graining theFibonacci quasicrystal as in Fig. 4, one can locally andunambiguously determine, from the tiles α and β in a givenlayer, where to place the larger intervals α0 and β0 in the nextlayer up. In this sense, the coarse-graining transformation

for a τ quasicrystal is lossless, in contrast to the blockdecimation of a periodic lattice. For this reason, we askwhether the quasi-MERA can give an exact description of aquantum critical state provided that the system is alreadydiscretized on the appropriate quasicrystal. This is a non-trivial statement that requires numerical checking andbenchmarking.Geometrically, the quasi-MERA construction of Fig. 4

generates a novel emergent “tiling” of the hyperbolic half-plane by hexagons, with three or four glued together at avertex. However, this network should be conceptuallyseparated from the TNs considered in the main paperbecause, unlike those TNs, this network can harbor unitarydisentanglers and has a preferred causal direction.Specifically, we do not believe that the quasi-MERA shouldbe interpreted as a discretization of a spacelike slice ofAdS2þ1. In view of Refs. [18] and [27], we ask whetherFig. 4 could be interpreted as a discretization of thekinematic space or some other causal geometry.We leave the investigation of these many exciting

possibilities for future studies.

ACKNOWLEDGMENTS

We thank Ehud Altman, Guifre Vidal, and MudassirMoosa for helpful discussions. L. B. acknowledges supportfrom a NSERC Discovery Grant. M. D. acknowledges theNational Science Foundation and the Perimeter Institute forTheoretical Physics for partial funding. F. F. acknowledgessupport from the English Speaking Union and NewCollege, Oxford. Research at the Perimeter Institute issupported by the Government of Canada through theDepartment of Innovation, Science and EconomicDevelopment, and by the Province of Ontario throughthe Ministry of Research and Innovation. Figures 2 and 3have been generated with the assistance of the KaleidoTile

software [51].

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FIG. 4. A “quasi-MERA” tensor network based on the 1DFibonacci quasicrystal. Vertices denote tensors, and edges denotecontractions between tensors. Three- and four-legged tensorsrepresent isometries and disentanglers, respectively. Dotted linesseparate layers related by the Fibonacci deflation rule.

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represent the natural discretizations of theories that arelocal in the bulk. Another fascinating possibility is to relaxthis locality restriction and consider fp;∞g tilings of thehyperbolic plane in which infinitely many tiles meet at eachvertex; in this case, tile edges are infinitely long, and the tilevertices live at infinity, on the boundary of the hyperbolicplane, where they can then induce a periodic structure atinfinity. Reference [22] contains a detailed exploration ofthis latter possibility and, in particular, in this context,presents an interesting toy model in which the discretesymmetries in the bulk may be related to a discretization ofthe conformal group of the boundary. The dual cases f∞; qgconsist of regular polygons with an infinite number of sides(apeirogons), with q meeting at each vertex. When q is aprime p, the boundary theory is then naturally described bythe p-adic numbers [30].

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[31] Yet another possibility is to replace the regular tiling inthe bulk by a random tensor network as in Ref. [17]. InAdS/CFT, a key role is played by the isometry group ofthe bulk AdS space, and a regular tiling has the advantagethat it preserves a large discrete subgroup of this sym-metry. However, for some purposes, a random tensornetwork may also have certain advantages, and it wouldbe interesting to investigate the mathematical structurethat it induces on the boundary. If the random tensornetwork in the bulk respects the properties of the regulartiling in a statistical sense, then the random structureinduced on the boundary should also respect the proper-ties of the corresponding conformal quasicrystal in astatistical sense, but we leave the investigation of thistopic to future work.

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