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 pro- vides 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 reg- ular tessellation of hyperbolic space defines a periodic lattice on its boundary, and no natural way for the dis- crete 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 ob- ject. 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 bound- ary 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: p 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 .