Conformal Quasicrystals and Holography

Latham Boyle1, Madeline Dickens2 and Felix Flicker2,3 1Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada, N2L 2Y5 2Department 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

Recent studies of holographic tensor network models defined on regular tessellations of hyperbolic space have not yet addressed the underlying discrete geometry of the boundary. We show that the boundary degrees of freedom naturally live on a novel structure, a conformal quasicrystal, that provides a discrete model of conformal geometry. We introduce and construct a class of one-dimensional conformal quasicrystals, and discuss a higher-dimensional example (related to the Penrose tiling). Our construction permits discretizations of conformal field theories that preserve an infinite discrete subgroup of the global conformal group at the cost of lattice periodicity.

I. INTRODUCTION dom [12–24]. Meanwhile, quantum information theory provides a unifying language for these studies in terms of entanglement, quantum circuits, and quantum error A central topic in theoretical physics over the past two correction [25]. decades has been holography: the idea that a quantum These investigations have gradually clarified our un- theory in a bulk space may be precisely dual to another derstanding of the discrete geometry in the bulk. There living on the boundary of that space. The most concrete has been a common expectation, based on an analogy and widely-studied realization of this idea has been the with AdS/CFT [1–3], that TNs living on discretizations AdS/CFT correspondence [1–3], in which a gravitational of a hyperbolic space define a lattice state of a critical theory living in a (d + 1)-dimensional negatively-curved system on the boundary and vice-versa. Initially, this bulk spacetime is dual to a non-gravitational theory liv- led Swingle to interpret Vidal’s Multiscale Entanglement ing on its d-dimensional boundary. Over the past decade, Renormalization Ansatz (MERA) as a discretized ver- investigations of this duality have yielded mounting evi- sion of a time-slice of the AdS geometry [7, 12]. How- dence that spacetime and its geometry can in some sense ever, MERA has a preferred causal direction, while any be regarded as emergent phenomena, reflecting the en- discretization of a spacelike manifold should not. To fix tanglement pattern of some other underlying quantum this issue, it was proposed that MERA was related, first degrees of freedom [4]. to a different object known as the kinematic space of AdS Recently, physicists have been interested in the possi- [18, 26] and, more recently, to a light-like geometry [27]. bility that discrete models of holography would permit Meanwhile, new TN models (e.g. the holographic them to understand this idea in greater detail, by bring- quantum error-correcting codes of [16], the hyper- ing in the tools of condensed matter physics and quan- invariant networks of [21], or the matchgate networks of tum information theory — much as lattice gauge theory [24]) were introduced to more adequately capture AdS. led to conceptual and practical progress in understand- The key feature of these new TNs is that they live on ing gauge theories in the continuum [5]. Recent years regular tilings of hyperbolic space [28]. The symmetries have seen a surge of interest in discrete models of holog- of such a tessellation form an infinite discrete subgroup raphy based on tensor networks (TNs). Such holographic of the continuous symmetry group of the AdS time-slice, TNs relate the quantum degrees of freedom (DOF) liv- much as the symmetries of an ordinary 3D lattice or crys- ing on a discretized (d + 1)-dimensional hyperbolic space tal form an infinite discrete subgroup of the continuous to corresponding DOF on the discretized d-dimensional symmetry group of 3D Euclidean space. Physically, such arXiv:1805.02665v2 [hep-th] 12 Dec 2019 boundary of that space. They bridge condensed matter TNs represent a discretization of the continuous bulk physics, quantum gravity, and quantum information. space, much as a crystal represents a discretization of In condensed matter physics, holographic TNs pro- a continuum material. In this paper, we restrict our at- vide a computationally efficient description of the highly- tention to TNs of this type. entangled ground-states of quantum many-body systems Despite progress in describing the discrete bulk geom- (particularly scale invariant systems or systems near their etry, the question of which discrete geometric spaces are critical points). The entanglement pattern is described suitable for the boundary has received little attention. In by tensors living on an emergent discrete hyperbolic ge- a discrete model of the AdS/CFT correspondence, we ometry in one dimension higher [6–11]. On the quantum expect to be able to construct the discrete boundary ge- gravity side, the TNs are a kind of UV regulator of the ometry entirely from the data of the bulk tessellation, in physics in the bulk, and a proposed way to represent the analogy with the continuum case, where it is well-known fact that spacetime and its geometry may be regarded as that the data of any asymptotically AdS spacetime define emergent phenomena, reflecting the entanglement pat- a conformal manifold on its boundary [2]. While this ex- tern of some other underlying quantum degrees of free- pectation seems natural, we are unaware of a correspond- 2 ing discrete construction in the literature. In condensed matter physics, the discrete boundary geometry is typi- cally assumed to be a periodic lattice; but, as we observe here, there is no natural way in which an ordinary regular tessellation of hyperbolic space defines a periodic lattice on its boundary, and no natural way for the discrete symmetries of the bulk to act on such a boundary lattice [48] [49]. Thus we expect that, in implementing a discrete version of AdS/CFT, it will be natural to replace the periodic boundary lattice with a different discrete object. Such a replacement is reasonable, from a Wilsonian viewpoint, as the choice of underlying lattice becomes irrelevant at a critical point [30]. In this paper, we argue that the DOF on the boundary of a regular tessellation of hyperbolic space naturally live on a remarkable structure – a conformal quasicrystal FIG. 1: The Penrose tiling [33]. The two different tiles (red (CQC) – built entirely from the data of the bulk tessel- and blue) cover the entire Euclidean plane in an aperiodic lation. This is a new clue about the type of boundary manner without gaps. Centers of approximate five-fold sym- theory that should appear in a discrete version of holog- metry can be seen. The tiling can be created through a cut- raphy. Far from being a simple periodic lattice, each and-project method from a five-dimensional hypercubic lat- CQC locally resembles a self-similar quasicrystal [31–35] tice or a four-dimensional A4 lattice, or by applying ‘inflation (like the Penrose tiling [36], shown in Fig. 1) and pos- rules’ to a finite-size starting configuration [31–33]. sesses discrete symmetries worthy of a discretization of a conformal manifold. In this paper, we focus on the d = 1 case, but with a view towards higher dimensions and β; and let sα(α, β) and sβ(α, β) be two finite strings (which we discuss briefly near the end). We propose to of αs and βs. We can act on Π with the inflation rule use our framework to construct discretizations of CFTs τ :(α, β) 7→ (s (α, β), s (α, β)) (1) that preserve an infinite discrete subgroup of the global α β conformal group at the cost of exact discrete translation which replaces each α or β in Π by the string sα(α, β) invariance (which is replaced by quasi-translational in- 0 or sβ(α, β), respectively, to obtain a new string Π . The variance). We end with suggestions for future research, inverse map is the corresponding deflation rule: and a discussion of how our results will hopefully provide −1 an important clue in the ongoing effort to formulate a dis- τ :(sα(α, β), sβ(α, β)) 7→ (α, β). (2) crete version of holography, and also lead to an improved analytical and numerical understanding of the structure Note that, while the inflation rule has a well-defined ac- of condensed matter systems at their critical points, and tion on any string Π of αs and βs, the corresponding other conformally invariant systems. deflation map only has a well-defined action on a string Π that may be uniquely partitioned into substrings of the form sα(α, β) and sβ(α, β). II. QUASICRYSTALS IN d = 1. Any inflation rule τ induces a matrix Mτ that en- codes the growth of Nα and Nβ (the number of αs We begin by briefly introducing self-similar quasicrys- and βs, respectively, in the string). For example, un- tals (SSQCs) in d = 1 (see Refs. [31–35] for further der τ :(α, β) 7→ (αβα, αβαβα), which corresponds to details). For general d, an SSQC may be thought inflation rule 3b in Table 1 of [34], we have of as a special quasiperiodically-ordered set of discrete N 0 N 2 3 points in Euclidean space (like the vertices in an infi- α = M α ,M = . (3) N 0 τ N τ 1 2 nite 2D Penrose tiling, shown in Fig. 1). The same β β SSQC can be constructed in two very different ways: √ either (i) by a cut-and-project method based on tak- If λ is the largest eigenvalue of Mτ (λ = 3 + 2), we can ing an irrationally-sloped slice through a periodic lattice represent Π geometrically as a sequence of 1D line seg- in a higher-dimensional Euclidean space, and projecting ments or ‘tiles’ of length Lα and Lβ, where (Lα,Lβ) is α β nearby points onto the slice; or (ii) by recursive iteration the corresponding left eigenvector of Mτ . If Nk and Nk of an appropriate substitution (or ‘inflation’) rule. Re- denote the number of αs and βs after k successive appli- stricting to d = 1 in the following sections, we exploit cations of τ to some finite initial string, then in the limit α β the second (inflation) perspective to develop a novel con- k → ∞,(Nk ,Nk ) is the corresponding right eigenvector α β α β struction whereby 1D SSQCs emerge on the boundary of of Mτ and grows like (Nk+1,Nk+1) = λ(Nk ,Nk ). a lattice in 2D hyperbolic space. Two strings Π and Π0 are locally isomorphic (or locally Let Π be a (possibly infinite) string of two letters, α indistinguishable) if every finite substring contained in 3

finite, simply-connected patch of tiles; and let it also be convex, which here means that, if θc is the interior angle of S at a vertex c, then: θc < 2π(1−1/q) for each boundary vertex (when q > 3); or θa + θb < 4π(1 − 1/q) for all nearest-neighbor pairs ha, bi of boundary vertices (when q = 3). On the boundary of S, we can assign different unit cell types between which the DOF live. The assignment procedure, illustrated in Fig. 2a, is the following. Consider the union U(S) of all tiles that share a vertex with the boundary of S (excluding the tiles in S itself). In the interior of U(S), we can draw a curve that crosses every tile of U(S) exactly once. This curve partitions every tile into two pieces, one contained in the interior of the curve and one in the exterior. If the interior part of the tile has n vertices, then we call it a type-n unit cell. Once every cell on the boundary of S has been labeled in this manner, we can write down a corresponding string FIG. 2: Illustration of unit cell and letter assignment on the Q(S) of cell types as in Fig. 2c. These strings are inside boundary of a simply-connected tile set S, in the {7, 3} tiling. square brackets to emphasize that they are cyclically or- (a) Unit cell assignment. (b) Letter assignment. (c) Cycli- dered. When S is convex, only two cell types appear on cally ordered strings of unit cells and letters. (d) Correspon- its boundary (type-1 and type-2 when q > 3, and type-2 dence between unit cells and letters in the {7, 3} case. and type-3 when q = 3); but since the seed S in Fig. 2a is not convex, a type-4 cell also appears. one is also contained in the other so it would be impossi- In a concrete holographic TN model, a tensor is placed ble to distinguish them by inspecting any finite segment. on each vertex of S and contractions are performed We say that Π is a τ-quasicrystal if it is equipped with between nearest-neighbor tensors along the tessellation an inflation rule τ acting on the two interval types α edges. Extra tensors can be added to the edges, as in and β [50] such that: (i) the matrix Mτ has determinant the hyperinvariant networks of Evenbly [21], or bulk an- ±1, with its largest eigenvalue λ given by an irrational cilla DOF to the vertex tensors, as in the holographic Pisot-Vijayaraghavan (PV) number; and (ii) the k-fold quantum error correcting codes of Ref. [16]. The bound- deflation map τ −k is well-defined on Π for all positive ary DOF are defined on legs placed on the edges of the integers k. Condition (i) ensures that Π is quasiperiodic, tessellation emanating from S which are cut by the par- crystalline (in the sense that the Fourier representation titioning curve in Fig. 2a. Thus, the distance in the net- of its density profile exhibits delta function diffraction work between two adjacent DOF on the boundary of S peaks like a crystal), and has τ −1 as the unique deflation depends on the type of unit cell between them. In this rule that inverts τ [31, 37]. Condition (ii) implies that sense, the cell types defined here behave analogously to a particular τ-quasicrystal is locally isomorphic to every the interval types Lα and Lβ between DOF placed on other τ-quasicrystal and, in particular, is locally isomor- the vertices of a d = 1 quasicrystal. phic to its own descendants under inflation. In this sense, every τ-quasicrystal is self-similar under inflation by τ. There is an alternative way to label the boundary of S If a τ-quasicrystal Π is mapped to itself by (k successive using only two symbols, that treats the q = 3 and q > 3 applications of) τ, we say it is (k-fold) self-same, an even cases more uniformly, and continues to apply even when stronger scale-invariance property than self-similarity. S is nonconvex. Decompose every tile into the fundamental domains of the tiling symmetry group, which are right triangles of angles π/2, π/q, and π/p. This decomposi- III. UNIT CELL ASSIGNMENT ON THE tion is shown in Fig. 2b. Now consider the set of such HYPERBOLIC DISK right triangles in the interior of S that share a vertex with the boundary of S. Within this set we identify and A regular {p, q} tessellation is constructed from regular label two types of isosceles triangles – A and B. Each A p-sided polygons (with p geodesic edges), where q such shares one vertex and no edges with the boundary of S, polygons meet at each vertex. Such a tessellation exists and each B shares two vertices and one edge (see Fig. 2b). for any p, q > 2: when (1/p)+(1/q) is greater than, equal This procedure, letter assignment, produces a cyclically- to, or less than 1/2, respectively, it is a tessellation of the ordered string Λ(S), distinct from Q(S), containing only sphere, Euclidean plane, or hyperbolic plane [28, 38]. As and Bs (see Fig. 2c). Consider a regular {p, q} tessellation of the hyperbolic 1 1 1 plane, p + q < 2 . We will focus in this paper on the case There is a way to extract Q(S) from Λ(S) and vice- where p and q are both finite. Let S (the ‘seed’) be a versa by mapping each type-n unit cell to a string of As 4 and Bs. This correspondence is

1 ↔ A−1 q−3 q−3 2 ↔ A 2 BA 2 q−3 q−2 q−3 3 ↔ A 2 BA BA 2 . . q−3 q−2 n−2 q−3 n ↔ A 2 (BA ) BA 2 , where n ≤ p. Here, if f(A, B) is any ﬁnite substring then (f(A, B))x denotes its x-fold repetition whenever 0 x ∈ Z≥0, with (f(A, B)) the empty substring. Frac- tional and negative exponents behave in the usual way, e.g. AxA−y = Ax−y for all x, y ∈ Q.

IV. INFLATION RULES ON THE HYPERBOLIC DISK

We can append the ring of tiles U(S) to the set S to obtain a new larger set S0 = S + U(S), as in Fig. 3. Accordingly, we add a layer of tensors to our TN to obtain a holographic TN model on S0 whose DOF are defined between the unit cells on the boundary of S0. The vertices that were formerly on the boundary (i.e. the ones that were not completely surrounded by the tiles of S) are now in the interior (i.e. are completely surrounded by the tiles of S0). We dub this procedure vertex completion. Vertex completion induces a mapping τΛ(p, q): 0 Λ(S) → Λ(S ). Remarkably, τΛ(p, q) is an inflation rule on A and B and is the same for every S. If we define the stringlets

1/2 (p−2)/2 (p−2)/2 1/2 sL := A B , sR := B A , (4) FIG. 3: Demonstration of how vertex completion induces then τΛ(p, q) can be written inflation rules on boundary letters and unit cells, for the {7, 3} tiling. The induced letter inflation rule (A, B) 7→ −1 −1/2 2−p −1/2 −1 −5 4 A 7→ (sLsR) = A B A , (5a) (A B ,B A) is the same for every tile set S, convex ( q−3 −1 q−3 or not. When S is convex, only type-2 and type-3 unit sL(sRsL) 2 B (sRsL) 2 sR (q odd), cells appear and there is an induced unit cell inflation rule B 7→ q−2 −1 q−2 (5b) (sRsL) 2 B (sRsL) 2 (q even). (2, 3) 7→ (223, 23). Note that vertex completion preserves convexity and that every tile set S becomes convex after finitely This version of the substitution rule is the canonical one, many vertex completions. in the sense that it respects the symmetry of the parent tile (A or B) under reflection about its midline, but another equivalent version (which is asymmetric) is simpler The corresponding induced map τQ(p, q): Q(S) → 0 and more convenient: Q(S ) is also an inflation rule on two unit cell types whenever S is convex. The rule τQ(p, q) can be extracted by −1 2−p p−3 p−2 q−3 A 7→ A B ,B 7→ B A(B A) . (6) combining τΛ(p, q) with the cell-to-letter mapping. When q > 3:

Note that two substitution rules {α, β} 7→ {sα, sβ} and q−4 q−4 −1 2 q−3 p−3 2 {α, β} 7→ {s¯α, s¯β} are equivalent ifs ¯α = usαu and 1 7→ 1 (2 1 ) 2 1 , (8a) s¯ = us u−1 for some ﬁnite string u = u(α, β). In par- q−4 q−3 p−4 q−4 β β 2 7→ 1 2 (2 1 ) 2 1 2 , (8b) ticular, the two substitution rules (5) and (6) both correspond to the substitution matrix and when q = 3:

1 p−5 1 −1 q − 2 2 7→ 3 2 2 3 2 , (9a) Mτ (p,q) = . (7) Λ 2 − p (p − 2)(q − 2) − 1 1 p−6 1 3 7→ 3 2 2 3 2 . (9b) 5

The matrices MτQ(p,q) and MτΛ(p,q) are related by change are symmetric under such conformal maps [40, 41]. Here, of basis. They have unit determinant, and eigenvalues we show that one can relate any two CQCs with the same {p, q} via a discrete analog of a Weyl transforma- 2 1/2 λ±(p, q) = γp,q ± (γp,q − 1) (10) tion. Hence CQCs inherit discrete conformal geometry, defined to be the properties fixed under discrete Weyl with transformations. (p − 2)(q − 2) Let M(S) 6= S denote a finite simply-connected tile set γ := − 1. (11) p,q 2 obtained from S in any manner. Q∞(S) and Q∞(M(S)) are locally isomorphic, but they differ in their global These results are consistent with the previous findings structures. To relate Q∞(S) and Q∞(M(S)), a map Ω of Ref. [39]. Note that the largest eigenvalue λ+(p, q) such that Ω(Q∞(S)) = Q∞(M(S)) is needed. We say 1 1 1 is irrational and PV whenever p + q < 2 . When S is that any such Ω is a discrete Weyl transformation.Ω nonconvex, τQ(p, q) acts on the many cell types of Q(S) acts upon the global structure of the CQC by a position- in a complicated way. However, every tile set S maps to dependent set of inflations and deflations. The TN model a convex set under finitely many vertex completions, and is correspondingly transformed by adjoining or removing vertex completion preserves convexity once established. tensors from the edges in a position dependent way. The intuition behind our definition comes from an analogy with Weyl transformations in continuum V. CONFORMAL QUASICRYSTALS ON THE AdS/CFT. There, a Weyl transformation corresponds to BOUNDARY OF THE DISK a change of choice of how to approach the boundary [3]. More precisely, the line element for (d + 1)-dimensional Consider any finite simply-connected tile set S of hyperbolic space can be written in global co-ordinates as the {p, q} tessellation. By repeatedly carrying out lay- 2 2 2 ers of vertex completions we generate a sequence of 2 dρ + sin (ρ)dΩd ds = 2 , (12) tile sets {Sn} , with initial element S0 = S, and cos (ρ) n∈Z≥0 {Q(Sn)} . Concurrently we can consider a family of n∈Z≥0 where ρ ∈ [0, π/2) is the radial co-ordinate and dΩ2 is holographic TN models defined on each S whose DOF d n the line element on the unit d-sphere S . Now consider a live on the boundary of S . In the limit n → ∞ the tile d n family of d-dimensional hypersurfaces of spherical topol- set covers the entire hyperbolic disk and we can ask about ogy, parameterized by ρ(x, ) = π −f(x) where x are the the unit cell assignment Q (S) := lim Q(S ) living 2 ∞ n→∞ n co-ordinates on S and f(x) is an arbitrary positive func- at the disk boundary. We can interpret Q (S) as a type d ∞ tion on S . To approach the boundary, we take → 0+ of emergent quasicrystal harboring the DOF. To see this, d in an x-independent way. As → 0+, a constant- hy- recall that for all n above some n∗ > 0, Q(S ) contains n persurface has induced line element: only two cell types and thus τQ(p, q) acts on these two cell types as an inflation rule. Taking Q(Sn∗ ) as an initial 2 dΩ2(x) string and iterating τ (p, q) infinitely many times gener- ds2(x) ∼ d , (13) Q 2 f 2(x) ates the infinite cyclically-ordered string Q∞(S) which is self-same under τQ(p, q). Thus Q∞(S) is locally isomor- so we see that changing the hypersurface profile f(x) 7→ phic to a τQ(p, q)-quasicrystal, which is the promised qua- Ω−1(x)f(x) induces the Weyl transformation sicrystalline interpretation. We call any Q∞(S) obtained in this manner a {p, q} conformal quasicrystal (CQC). 2 2 2 ds (x) 7→ Ω (x)ds (x). (14) In our construction, the choice of initial tile set S is anal- VI. DISCRETE CONFORMAL GEOMETRY ogous to the choice of hypersurface profile f(x), while iterating vertex completions infinitely many times corre- Our construction naturally endows CQCs with a dis- sponds to taking → 0+. Thus, it is natural to inter- crete analog of conformal geometry. In the contin- pret S 7→ M(S) as inducing a Weyl-like transformation uum, conformal geometry refers to those properties of a Q∞(S) 7→ Ω(Q∞(S)) = Q∞(M(S)). (pseudo-)Riemannian manifold that are invariant under position-dependent rescalings (or Weyl transformations) 2 of the metric tensor gµν (x) 7→ Ω (x)gµν (x). Thus, length VII. SYMMETRIES OF CONFORMAL scales are not well-defined in conformal geometry. Con- QUASICRYSTALS IN d = 1 formal properties are preserved under conformal maps, diffeomorphisms fixing the metric up to a Weyl trans- Now let us emphasize two different types of symme- formation. These are all the diffeomorphisms in d = 1 try possessed by any CQC Q∞(S) of type {p, q}. First, and just the angle-preserving diffeomorphisms in d ≥ 2. it has a kind of exact scale symmetry: invariance un- In physics, conformal geometries underlie conformal field der τQ(p, q). Second, it is invariant, up to a discrete theories (CFTs), which are quantum field theories that Weyl transformation, under an infinite discrete group 6 called the triangle group ∆(2, p, q). This is the sym- Thus, we hope that this work opens the door to more metry group of the {p, q} tiling; it is generated by re- efficient simulation of the dynamics and quantum states flections across the fundamental right-triangle domains of such systems (although much work remains in order of the tiling [42]. Each such reflection induces a confor- to translate this hope into practice). mal map of the boundary into itself. Under an element Let us now mention various directions for future work: 0 of ∆(2, p, q), S maps to a different simply-connected S We expect our definitions and results to extend read- 0 0 which produces Q∞(S ). Since S = M(S) for some M, ily to higher dimensions. Consider, for example, the self- 0 it follows that Q∞(S) and Q∞(S ) are related by a dis- dual {3, 5, 3} regular honeycomb in three-dimensional hy- crete Weyl transformation, as required. Thus we regard perbolic space. This is constructed by gluing together ∆(2, p, q) as a discretization (i) of the isometry group of icosahedra such that the vertex figure at every corner is the hyperbolic disk which acts on the bulk tessellation; a dodecahedron [28, 38]. Now imagine that we take the and (ii) of the conformal group of the circular boundary initial seed S to be one such icosahedron, and then carry which acts on the CQCs. out vertex completion, assigning boundary unit cells at every step. At each step we obtain a 2D layer of spherical topology with 12 points of exact five-fold orientational VIII. DISCUSSION symmetry (or, more precisely, D5 symmetry), lying ra- dially above the 12 vertices of the initial icosahedron. Before discussing our results, let us briefly summarize: Hence, after iterating infinitely many times, we obtain a The past decade has seen exciting developments in our layer of spherical topology at the boundary, tesselated by understanding of quantum gravity: in particular, argu- an infinite number of tiles. This layer still has 12 points ments based on holography and gauge/gravity duality of D5 symmetry, and in the vicinity of any such point it have led to a tantalizing but still-fragmentary picture in appears to be an infinite tiling of the 2D Euclidean plane. which spacetime emerges from the pattern of entangle- Since points of D5 symmetry are forbidden in an ordinary ment in an underlying quantum system. In particular, (periodic) 2D crystal, we expect the boundary to once there has been much recent interest in attempts to make again be quasicrystalline. Indeed, the 2D quasicrystalline this picture more concrete, in the context of AdS/CFT, tilings with 5-fold orientational order are all closely re- by developing a discrete formulation of holography, based lated to the Penrose tiling [31, 33, 35, 36, 45]. (Note that, on replacing the continuous hyperbolic “bulk” space by a although the usual Penrose tiling has at most one point discrete regular tesselation of hyperbolic space (or some of exact D5 symmetry, the Penrose-like CQC living at other tesselation which respects a large discrete subgroup the boundary of {3, 5, 3} can have 12 such points, as ex- of the original space’s symmetries). These developments plained above. This is the higher-dimensional analogue are part of a family of ideas which are sometimes sum- of a phenomenon that already occurs in the 1D CQCs marized by the slogan “it from qubit.” constructed above: the CQC generated from a regular Previous works have focused on discretizing the bulk p-gon initial seed will have 2p points of perfect reflection geometry, but have not thought through the correspond- symmetry, in contrast to the self-similar 1D quasicrys- ing implications for the boundary geometry. In this pa- tal that it locally resembles, which has at most one such per, we have shown that when one discretizes the bulk point.) geometry in a natural way (e.g. on an regular tessela- The conjecture that {3, 5, 3} hosts a CQC generalization), one also induces a remarkable discretization of the tion of the Penrose tiling on its boundary is supported lower-dimensional “boundary” geometry into a fascinat- by a calculation [46] of the ratio Nk+1/Nk as k → ∞, ing new kind of discrete geometric structure which we call where Nk is the number of boundary cells after k layers a “conformal quasicrystal.” The main goal of this paper of vertex completion (this ratio is expected to be equal was to point out the existence of these new structures, to to the largest eigenvalue of the matrix Mτ associated to define and explain their basic properties, and to empha- the inflation rule τ induced by the vertex completion). 8 size that they appear to be the natural discrete spaces The limiting ratio is found to be Nk+1/Nk → ϕ , where living on the boundary side of the holographic duality. ϕ is the golden ratio. Note that this is irrational and We think this is an important clue for the ongoing ef- PV (as expected for a 2D quasicrystal) and is a power forts to formulate a discrete version of holography (per- of ϕ (as expected for a quasicrystal with 5-fold orienta- haps in terms of tensor networks) [51]: in a correct dis- tional order in particular [31, 33, 35, 36]). If we compare crete formulation of AdS/CFT, we expect the boundary this to the scaling ratio for the standard Penrose tiling 2 theory to live on a conformal quasicrystal (rather than (Nk+1/Nk → ϕ ), it suggests either that a single layer of on an ordinary lattice, as imagined in previous works). vertex completion in the {3, 5, 3} honeycomb corresponds Similarly, since the boundary theory in AdS/CFT is scale to four iterations of the usual Penrose tiling inflation rule, invariant, it suggests that the natural way to discretize or else that the Penrose tiling and {3, 5, 3} CQC are re- and numerically simulate scale invariant systems (such lated in some way besides local isomorphism. as conformal field theories, or condensed matter systems We have seen that applying vertex completion to a near their critical points) is to discretize them on a con- regular {p, q} tiling of hyperbolic space yields a τQ(p, q)- formal quasicrystal, rather than on a periodic lattice. quasicrystal. There are, however, other τ-quasicrystals 7 that are not obtained in this way – like the example given above in Eq. (3). We wonder whether every remaining inflation rule τ is also naturally induced by applying some appropriate generalization of vertex completion to some appropriate tiling of hyperbolic space. Tilings beyond the regular {p, q} type would seem to be necessary. The remarkable conformal geometry of CQCs suggests that their importance extends beyond the realm of holographic TN models. We propose to use them as the underlying spaces on which to discretize CFTs, in at least two ways. First, one can discretize a QFT partition function onto a τ-quasicrystal directly. After implementing a real space renormalization group procedure whereby DOF are decimated according to τ −1, the RG fixed points can be identified with the discrete CFTs. Second, one can work by analogy with AdS/CFT. Consider the partition FIG. 4: A ‘quasi-MERA’ tensor network based on the 1D function ZS,n[y] of a discretized field theory on the n-fold image S under vertex completion of a simply-connected Fibonacci quasicrystal. Vertices denote tensors and edges de- n note contractions between tensors. 3- and 4-legged tensors subset S of the bulk tessellation, such that the fields take represent isometries and disentanglers, respectively. Dotted values y at the boundary of Sn. As n → ∞, we propose lines separate layers related by the Fibonacci deflation rule. to interpret ZS,∞[y] as the moment-generating function of a discrete CFT on the boundary.

From a practical standpoint, such CQC-based dis- 1/2 1/2 1/2 1/2 cretizations may be important for studying scale- substrings of the form β β and β αβ ; (ii) glue these substrings together to form intervals α0 = β1/2β1/2 invariant physical systems such as CFTs or condensed 0 1/2 1/2 matter systems at their critical points. These discretiza- and β = β αβ of respective lengths Lα0 = Lβ and tions should permit finite-size simulations of such sys- Lβ0 = Lα +Lβ. Iterating this procedure yields a sequence ck(Π) of successively coarser quasicrystals. We tems, in ways which preserve an exact discrete subgroup k∈Z≥0 of their scale symmetry. This is in contrast with ordi- can now construct a MERA-like circuit as follows. Em- nary lattice gauge theory simulations, or other periodic bed each quasicrystal ck(Π) into the two-dimensional lattice models, which instead preserve an exact discrete plane, such that ck+1(Π) is parallel to and lies above subgroup of translation symmetry, at the cost of breaking ck(Π) for every k. Assign to every point in ck(Π) a tensor, scale invariance. We can attempt to finitize a numerical and contract each tensor only with its nearest neighbors calculation on a conformal quasicrystal by restricting to in ck+1(Π) and ck−1(Π). This results in 3- and 4-legged an annulus in the reciprocal space of the quasicrystal and tensors; if we choose these to be isometries and unitary imposing the boundary condition that observables on the disentanglers, respectively, then we acquire the desired inner and outer boundaries of the annulus are the same MERA-like circuit. We call this network, depicted in up to an appropriate scale factor. This would provide the Fig. 4, the quasi-MERA due to its relationship with the analog in discrete CFTs of a periodic boundary condition quasicrystalline inflation rule τ. in a periodic lattice model, along the scale direction in- To see why the quasi-MERA may be special, we con- stead of in real space. The resulting finite problem can trast the deflation-based decimation procedure of the τ- then be attacked with a large arsenal of Monte Carlo and quasicrystal with the standard block decimation of the tensor network methods. Our hope is that this approach periodic crystal. Unlike the periodic crystal, for every may lead to more accurate and efficient numerical sim- τ-quasicrystal layer there is a local refinement (inflation) ulation of such systems, perhaps allowing us to simulate rule which is inverted by an unambiguous local coarse- much larger systems, and leading to qualitative improve- graining (deflation) rule. To see why this cannot hold for ments in our understanding. an ordinary crystal, consider the example of a 1D peri- TN algorithms based on inflations and deflations may odic lattice with intervals of length a. We can produce also find use in the numerical description of quantum a more refined lattice (with intervals of length a/2) by critical states. We can imagine implementing a MERA- the local rule of chopping each interval in half; but the like coarse-graining scheme induced by the quasicrys- inverse coarse-graining transformation is not locally well talline structure. Consider the simplest τ-quasicrystals, defined: there is an ambiguity about which pair of short the 1D Fibonacci quasicrystals generated by the inflation intervals to glue together to make a long one, so that N rule τ : {α, β} 7→ {β1/2β1/2, β1/2αβ1/2}. Given an ini- ‘joiners’ at widely-separated points on the lattice would tial Fibonacci quasicrystal Π, we can use a deflation-like make different, incompatible choices about which tiles to rule to obtain a coarser quasicrystal c(Π) in two steps: join, and O(N) defects would be produced. Hence, block (i) slice every β interval into two halves, β1/2, result- decimating a periodic lattice destroys information about ing in a string which can be uniquely partitioned into how to locally recover the original lattice. Since the or- 8 dinary (binary) MERA architecture relies on this form terpreted as a discretization of the kinematic space, or of block decimation, its description of a quantum state is some other causal geometry. necessarily lossy, even at criticality. By contrast, when We leave the investigation of these many exciting pos- coarse-graining the Fibonacci quasicrystal as in Fig. 4, sibilities for future studies. one can locally and unambiguously determine, from the tiles α and β in a given layer, where to place the larger intervals α0 and β0 in the next layer up. In this sense, the coarse-graining transformation for a τ-quasicrystal is Acknowledgments lossless, in contrast to the block decimation of a periodic lattice. For this reason, we ask whether the quasi-MERA can give an exact description of a quantum critical state We thank Ehud Altman, Guifre Vidal and Mudassir provided that the system is already discretized on the Moosa for helpful discussions. LB acknowledges sup- appropriate quasicrystal. This is a nontrivial statement port from an NSERC Discovery Grant. MD acknowl- that requires numerical checking and benchmarking. edges the NSF Graduate Research Fellowship Program Geometrically, the quasi-MERA construction of Fig. 4 and the Perimeter Institute for Theoretical Physics for generates a novel emergent ‘tiling’ of the hyperbolic half- partial funding. FF acknowledges support from a Linde- plane by hexagons, with 3 or 4 glued together at a vertex. mann Trust Fellowship of the English Speaking Union, However, this network should be conceptually separated and the Astor Junior Research Fellowship of New College, from the TNs considered in the main paper because, un- Oxford. Research at Perimeter Institute is supported by like those, this network can harbor unitary disentanglers the Government of Canada through the Department of and has a preferred causal direction. Specifically, we do Innovation, Science and Economic Development and by not believe that the quasi-MERA should be interpreted the Province of Ontario through the Ministry of Research as a discretization of a space-like slice of AdS2+1. In view and Innovation. Figs. 2 and 3 have been generated with of Refs. [18] and [27], we ask whether Fig. 4 could be in- the assistance of the KaleidoTile software [47].

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