# Tensor network models of AdS/qCFT

Alexander Jahn  1,2 Zoltán Zimborás  3,4,5 and Jens Eisert1,6 1Dahlem Center
for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
2Institute for Quantum Information and Matter, California Institute of
Technology, Pasadena, CA 91125, USA
3Institute for Particle and Nuclear Physics, Wigner Research Centre for
Physics, 1121 Budapest, Hungary
4BME-MTA Lendület Quantum Information Theory Research Group, 1111 Budapest,
Hungary
5Faculty of Informatics, Eötvös Loránd University, 1117 Budapest, Hungary
6Department of Mathematics and Computer Science, Freie Universität Berlin,
14195 Berlin, Germany

###### Abstract

The study of critical quantum many-body systems through conformal field theory
(CFT) is one of the pillars of modern quantum physics. Certain CFTs are also
understood to be dual to higher-dimensional theories of gravity via the anti-
de Sitter/conformal field theory (AdS/CFT) correspondence. To reproduce
various features of AdS/CFT, a large number of discrete models based on tensor
networks have been proposed. Some recent models, most notably including toy
models of holographic quantum error correction, are constructed on regular
time-slice discretizations of AdS. In this work, we show that the symmetries
of these models are well suited for approximating CFT states, as their
geometry enforces a discrete subgroup of conformal symmetries. Based on these
symmetries, we introduce the notion of a quasiperiodic conformal field theory
(qCFT), a critical theory less restrictive than a full CFT and with
characteristic multi-scale quasiperiodicity. We discuss holographic code
states and their renormalization group flow as specific implementations of a
qCFT with fractional central charges and argue that their behavior generalizes
to a large class of existing and future models. Beyond approximating CFT
properties, we show that these can be best understood as belonging to a
paradigm of discrete holography.

##  1 Introduction

Quantum field theories constitute a central cornerstone of modern theoretical
physics, describing a large part of physical phenomena from condensed matter
to high-energy settings. From the start of the field, symmetries have played a
central role in identifying and describing physical theories, perhaps most
prominently in the case of _conformal field theories_ (CFTs) [1, 2, 3] which
possess an invariance under local scale transformations. These theories,
describing the physics of second-order phase transitions as well as being an
integral part of the _anti-de Sitter/conformal field theory correspondence_
(AdS/CFT) [4, 5], have properties that are highly restricted by their
symmetries. Unlike the well-studied $1{+}1$-dimensional case, analytical
techniques of continuum CFTs in higher dimensions are scarce and few exact
results exist. To circumvent these issues, two popular approaches exist:
First, AdS/CFT allows for many results of strongly interacting CFTs to be
mapped to weakly interacting gravitational theories, usually under the
assumption of additional symmetries. Second, one may study the properties of a
_discretized_ CFT model numerically. For this latter approach, tensor network
models have shown particular promise: The _multi-scale entanglement
renormalization ansatz_ (MERA) [6] has been shown to accurately reproduce a
large range of CFT properties in $1{+}1$ dimensions [7, 8] and is extensible
to higher dimensions [9]. The MERA has also been proposed as a discrete
realization of AdS/CFT [10, 11] albeit possessing a geometry that does not
directly match with that of its higher-dimensional gravitational theory [12,
13, 14]. In a parallel development, tensor network approaches have
incorporated _holographic quantum error-correcting codes_ , reproducing some
properties of critical theories, yet lacking a clear connection to CFTs (for a
recent review, see Ref. [15]).

In this work, we show that tensor networks on regular hyperbolic tilings
provide a natural CFT ansatz by extending ideas from continuum AdS/CFT to the
discrete setting. Crucially, their boundary states obey a discrete subset of
the continuum CFT symmetries _directly from the tensor network geometry_ ,
reproducing many of the properties of continuum CFT ground states. Rather than
breaking continuum translation invariance at a fixed lattice scale, this
approach tolerates soft lattice effects appearing at all scales in a fractal
manner, following a renormalization group (RG) prescription that allows fine-
and coarse-graining to arbitrary scales. Building on previous work that
identified the boundary symmetries of these tilings with those of a
_quasicrystal_ that suggested a notion of discrete conformal geometry [16], we
show how this picture can be modified and extended to describe tensor network
states whose symmetries are captured by a _multi-scale quasicrystal ansatz_
(MQA) with contributions from all tiling layers, each corresponding to a
length scale in the boundary state. Further characterizing the resulting
symmetries, we argue that these states realize instances of what we call a
_quasiperiodic conformal field theory_ (qCFT) with inherently discrete
structure that exhibits discrete analogues of scale and translation invariance
found in a more constrained continuum CFT. These boundary transformations
follow from invariance transformations of the hyperbolic disk that preserve
regular hyperbolic tilings, characterized by Fuchsian groups. As this
discretization produces specific relationships between bulk and boundary
symmetries, such tensor network models should be understood as realizations of
an _AdS/qCFT correspondence_ between discrete theories rather than merely as
an approximation of continuum AdS/CFT. Using examples from previous literature
in which tensor network models with these symmetries have already been
constructed, we illustrate the range of realizable qCFTs models.

Figure 1: AdS embeddings and Majorana dimers. a-b, embeddings of the MERA (a)
and regular $\\{5,4\\}$ tiling (b) into AdS3 space-time with coordinates
$(t,\rho,\phi)$, with the AdS boundary located at the surface of the cylinder.
The MERA corresponds to a light-cone embedding [14], while regular
discretizations such as the $\\{5,4\\}$ tiling can be identified with a time-
slice. Both tilings can be divided into radial _inflation layers_ , which are
drawn in bold. c-d, the $\\{5,4\\}$ tiling with _Majorana dimers_
corresponding to boundary states of the hyperbolic pentagon code [17]. States
at the zeroth and first inflation step are shown in the Poincaré disk (top)
and half-plane projection (bottom).

##  2 Results

###  2.1 Conformal symmetries and the Poincaré disk

A central insight underlying AdS/CFT is that the $SO(d,2)$ symmetries of pure
AdS space-time in $d{+}1$ dimensions are also the symmetries of a
$d$-dimensional CFT, whose space-time can be identified with the asymptotic
spatial boundary of the AdS bulk. Breaking these continuous bulk symmetries by
a discretization necessarily also breaks the boundary symmetries. Throughout
this paper, we consider $2{+}1$-dimensional AdS3 space-time, which can be
projected onto cylinder coordinates
$(t,\rho,\phi)\in(\mathbb{R},\,[0,1),\,[0,2\pi))$ with the metric

$\mathrm{d}s^{2}=\frac{-(1+\rho^{2})^{2}\,\mathrm{d}t^{2}+4\alpha^{2}\mathrm{d}\rho^{2}+4\alpha^{2}\rho^{2}\mathrm{d}\phi^{2}}{(1-\rho^{2})^{2}}\
,$ (1)

where the constant $\alpha$ is the _AdS radius_ determining the global
negative curvature of the geometry. The AdS boundary is located at $\rho\to
1$. Instead of discretizing the entire space-time, we focus on space-like
surfaces whose boundary is a slice of the AdS boundary at constant time $t$.
To describe the boundary quantum state on such a discretization, we consider a
_tensor network_ embedded into the discretized surface: Each simplex of the
discretization is associated with a tensor whose indices are contracted with
indices of neighboring simplices111Note that this picture is the geometric
dual of the Penrose graphical notation, where vertices represent tensors and
connected edges contractions.. The uncontracted indices are then located on
the edges of the tiling’s boundary, each index representing a site of a
quantum state. As shown in Fig. 1a-b, the bulk surface into which such a
tensor network is embedded is not automatically a time-slice of AdS space-
time. For example, the MERA can be identified with a path integral on a light-
cone geometry [14]. The geometry we consider here, however, is the AdS time-
slice as shown in Fig. 1b, the same setup underlying the Ryu-Takayanagi
formula [18] whose properties such tensor network constructions readily
reproduce. As with the MERA tensor network, these discretized time-slice
geometries can be iteratively tiled, corresponding to a tensor network that is
contracted in consecutive layers. Shown in Fig. 1c-d are the first two such
contraction steps for an instance of Majorana dimer states, which can be
contracted diagrammatically. We will use the dimer example in more detail
below to demonstrate how the tiling symmetries are realized in tensor network
states.

An AdS time-slice in the continuum, equivalent to the _Poincaré disk_ , is
invariant under $PSL(2,\mathbb{R})$ transformations. Describing a coordinate
point by the complex number $z=\rho\,e^{\,\mathrm{i}\phi}$, these
transformations take the form

$z\mapsto z^{\prime}=e^{\,\mathrm{i}\theta}\frac{w+z}{1+w^{\star}z}\,$ (2)

effectively shifting the origin of the Poincaré disk to the point $w$ with
$|w|<1$ and rotating by an angle $\theta$. We refer to these as _Möbius
transformations_ , as $PSL(2,\mathbb{R})$ is a subgroup of the Möbius group
$PGL(2,\mathbb{C})$. On the Poincaré disk boundary $\rho\to 1$, these
transformations are equivalent to _translations_ and non-uniform _local scale
transformations_. Furthermore, as length scales diverge at $\rho\to 1$,
changing the cutoff scale $\rho_{0}$ corresponds to a uniform _global scale
transformation_. These three transformations are exactly the spatial part of
conformal transformations in $1{+}1$ dimensions, and are shown in Fig. 2 in
their action on a regular boundary lattice. These local scale transformations
act similarly to the spatial part of boost transformations in the AdS
embedding spacetime, which also map between boundary regions of different size
[19]. Discretizing the bulk geometry restricts the symmetries on the boundary
to a discrete subgroup of the previously continuous symmetries. As in the
continuous case, we can characterize these boundary symmetries by the group of
invariance transformations in the bulk, which are characterized by a number of
symmetry groups. Specifically, we consider regular $\\{n,k\\}$ tilings (using
the _Schläfli notation_) that are composed of regular $n$-gons, $k$ of which
are adjacent at each corner. Symmetry groups of physical models on these
tilings have previously been identified and discussed in the context of tensor
networks [20, 21, 22] as well as physical bulk models and their band theory
[23, 24]. In any regular tiling – hyperbolic or not – the reflection along
edges of the tiling leaves the tiling invariant, and general mappings of the
tiling onto itself can be composed from such reflections, defining the
symmetries of a _Coxeter group_. As any polygon tiling can be further
subdivided into triangles, the symmetries of regular tilings extend to those
of a hyperbolic _triangle group_ , i.e., the group of reflections along edges
in a triangular tiling. Note that in the case of flat and hyperbolic tilings,
these are infinite symmetry groups assuming the tiling to be infinitely
extended; cutting it off after a finite number of tiles inherently breaks
these symmetries. For physical symmetry transformations that preserve
orientation, we restrict ourselves to a subgroup of index two of the triangle
group consisting of all transformations composed of an even number of
reflections, called _von Dyck groups_. In the hyperbolic case, these
transformations are now discrete subgroups of the continuous
$PSL(2,\mathbb{R})$ transformations, as Fig. 3c-d demonstrates. This property
defines a _Fuchsian group_. Let us consider these symmetries in explicit
examples. As we see in Fig. 3a-b, a Möbius transformation (2), such as a bulk
translation of the tiling center to a point $w$ (3a) or a rotation by an angle
$\theta$ (3b), generally does not map a regular hyperbolic tiling of infinite
size onto itself. The only tiling-preserving transformations which also
preserve orientation consist of bulk translations to a point $w$ that is the
center of an $n$-gon tile (equivalently, a lattice vector in the dual
$\\{k,n\\}$ tiling), followed by a suitable rotation with an angle $\theta$
that re-aligns the tiling up to a $\mathbb{Z}_{n}$ ambiguity. Such a
transformation is equivalent to an even number of reflections along a geodesic
edge of the triangularized tiling, demonstrated in Fig. 3c-d, which is exactly
the construction of Fuchsian groups via hyperbolic von Dyck groups mentioned
above. In order to understand these bulk tiling transformations as
discretizations of conformal transformations on the boundary, we need to study
their effect for a finite tiling cutoff. We will now discuss the construction
of such a cutoff in general and in a specific model before returning to these
specific transformations.

Figure 2: Invariance transformations of a CFT ground state. The
transformations are shown in their effect on a regular lattice with periodic
boundary conditions and include translation of the lattice sites as well as
global (uniform) and local (non-uniform) rescaling of the lattice. Here we
consider local scaling transformations that preserve the total number of
lattice sites.

Figure 3: Symmetries of regular hyperbolic tilings. a-b, regular tilings of
the Poincaré disk breaking the continuous $PSL(2,\mathbb{R})$ symmetry under
transformations (2) that include a, bulk translations by $w$ not corresponding
to a lattice shift and b, bulk rotations that are not a multiple of $2\pi/k$.
c-d, triangle group transformations of triangulated tiling. An even number of
reflections $(1)\mapsto(2)\mapsto(3)$ along triangle geodesics (c) is
orientation-preserving (denoted by circular arrows on the central tile and its
reflections) and equivalent to a bulk translation+rotation (d).

###  2.2 Global scaling and inflation rules

A size cutoff for regular hyperbolic tilings is usually defined via an
iterative tiling construction that starts with a central tile. Equivalently, a
tensor network on such a geometry can be iteratively contracted, producing a
boundary state of increasingly large but finite Hilbert space dimension after
each iteration step. This fine-graining procedure can be described more
precisely by specifying the _inflation rule_ under which the bulk tiling is
filled layer-by-layer, or _inflated_ [16]. One choice for such an inflation
rule is _vertex inflation_ (also called _vertex completion_), where each
inflation step adds a closed layer of tiles adjacent to the vertices of the
previous tiling. This rule is a particularly natural one, as its behavior is
preserved under a mapping from an $\\{n,k\\}$ tiling to its dual $\\{k,n\\}$
one. A vertex inflation step also underlies the construction of
_hyperinvariant tensor networks_ [25] that realize certain properties of the
MERA in specific tensor networks on regular hyperbolic tilings. Such an
inflation method acts as a discrete global scale transformation on the
boundary states of a tensor network: Each inflation step increases the number
of boundary sites by a scale factor $\lambda$ that approaches a constant
independent of the initial starting region. While restricted to
$\lambda\in\mathbb{N}_{>1}$ for the MERA, this factor can take a large range
of fractional values for an $\\{n,k\\}$ tiling. Its asymptotical value after
many vertex inflation steps is given analytically by [26]

$\lambda=\frac{2+f(n,k)+\sqrt{(4+f(n,k))f(n,k)}}{2}\ ,$ (3)

where $f(n,k)=nk-2(n+k)$ is positive for any hyperbolic $\\{n,k\\}$ tiling.
The geometrical structure of the boundary layers is not asymptotically smooth
but _quasiperiodic_ , with self-similarity on all scales [16]. As a
consequence, the scaling factor $\lambda$ not only describes the growth of the
total system size but also that of any sufficiently fine-grained boundary
region. Specific vertex inflation rules can be written in terms of replacement
rules on letter sequences, with each letter denoting specific boundary
features [16]. For the $\\{5,4\\}$ tiling, we follow the notation of Ref. [26]
and denote with $a$ and $b$ the types of vertices that appear on tiling
boundaries throughout vertex inflation, with each vertex either connected to a
single boundary tile (type $a$) or sitting between two of them (type $b$). A
vertex inflation step then corresponds to applying the rule

$\displaystyle a$ $\displaystyle\mapsto abaab\ ,$ $\displaystyle b$
$\displaystyle\mapsto ab\ ,$ (4)

where the strings correspond to letter sequences. Fig. 4a shows the vertex
inflation layers of the $\\{5,4\\}$ tiling as colored bands, with the boundary
vertices on each layer denoted as circles and dots depending on their type.
For this tiling, the starting configuration at the $0$th inflation step
corresponds to the sequence $aaaaa$, resulting in a five-fold $\mathbb{Z}_{5}$
symmetry of the layers around the central pentagon. Rearranging a fifth of the
letter sequence at each layer (i.e., inflation steps for a single letter $a$)
into linear blocks leads to the visualization in Fig. 4b. We refer to this
approach of stacking the quasiperiodic symmetries on each inflation layer as
the _multi-scale quasicrystal ansatz_ , which serves as a simplified model of
tensor networks on regular hyperbolic lattices grown under vertex inflation.
As shown in Ref. [16], the letter sequence at the $n$th inflation step fully
describes the boundary geometry at that step, but boundary states produced by
tensor networks on such a geometry should be expected to have more complicated
symmetries, with contributions from all layers relevant for the final state.
As tensor networks on hyperbolic tilings typically produce _critical_ states
with divergent correlation lengths (see Ref. [27] for the Gaussian fermionic
case), the symmetries on all, and not merely short length scales, contribute
to the final result. This is sketched in the bottom row of Fig. 4b, where we
visualize the final state symmetries as a local multi-scale “product” of
contributions from each layer of the MQA. While the exact form of this product
depends on the tensor content, the multi-scale quasiperiodic structure is
fixed by the geometry.

Figure 4: Global scale transformations. a, vertex inflation of the hyperbolic
$\\{5,4\\}$ tiling with $a$\- and $b$-type boundary vertices following the
rule (4), with an overall $\mathbb{Z}_{5}$ symmetry. b, the multi-scale
quasicrystal ansatz (MQA), a visualization of layered quasiperiodic
symmetries. Its layers consist of sequences of vertex types on one fifth of
the total system (one $a$ letter on first layer), the “product” of whose
quasiperiodic distribution (bottom row) determines the symmetries of the
boundary state. c, corresponding Majorana dimer state from a single initial
edge. Note the asymptotic pairing of dimers as discussed in Ref. [17].

###  2.3 Scale transformation with dimers

To demonstrate scale transformations on boundary states of a specific tensor
network, we consider tensors described by _Majorana dimer states_ : These are
Gaussian fermionic states, composed of paired Majorana modes, that are
characterized by efficiently contractible tensors and can also be interpreted
as ground states of Gaussian _stabilizer Hamiltonians_ in the context of
quantum error correction. Its most relevant example for our purposes is the
$[[5,1,3]]$ code that encodes one logical qubit in five physical spins or
fermions and can correct one Pauli-type error [28, 29]. The logical code
states $\bar{0}$ and $\bar{1}$ are encoded as Majorana dimer states on the
physical fermions. These share the same correlation pattern and can be
visualized as

$\displaystyle\left.\left|{\bar{0}}\right.\right\rangle_{5}\;$
$\displaystyle=\;\begin{gathered}\includegraphics[height=40.47212pt]{scode_n5d_0.pdf}\end{gathered}\
,$ $\displaystyle\left.\left|{\bar{1}}\right.\right\rangle_{5}\;$
$\displaystyle=\;\begin{gathered}\includegraphics[height=40.47212pt]{scode_n5d_1.pdf}\end{gathered}\
.$ (7)

A pentagon in this visualization represents a state on five qubits, or
equivalently, a five-leg tensor encoding this state. Each arrow from Majorana
mode $j$ to $k$ in this diagram corresponds to a term
$\,\mathrm{i}\operatorname{\gamma}_{j}\operatorname{\gamma}_{k}$ in the
stabilizer Hamiltonian while also representing a maximally entangled pair of
Majorana modes. We can use these states as the building blocks of a class of
holographic models known as _HaPPY codes_ [30] in which these code states are
placed on a regular hyperbolic tiling. When doing so, the contraction of these
states (or rather, the tensors encoding them) into a full tensor network is
equivalent to fusing endpoints of Majorana dimers between individual diagrams,
turning the contraction process into a geometric problem of tracing connected
dimers throughout the network. As a result, large tensor contractions can be
performed analytically. In particular, entanglement entropies and correlation
functions can be computed for various hyperbolic geometries, leading to
results consistent with those of conformal systems [17]. Given a tiling
cutoff, tensor contraction of the $[[5,1,3]]$ code states in Majorana dimer
form as shown in Fig. 1c-d thus requires connecting all dimers on tiles up to
the cutoff. Moving the cutoff closer to the Poincaré disk boundary leads to a
boundary dimer state supported on more modes, successively fine-graining the
resulting correlation structure [17, 26]. In the vertex inflation setup, this
is shown in Fig. 4c for four inflation steps. The dimer patterns follow the
same sequence of $a,b$ letters as the boundary geometry, with an $a$ letter
corresponding to a local crossing of a pairs of dimers, while dimers around a
$b$-type vertex form non-crossing singlets. The dimer representation of the
boundary state supports the qualitative picture of the MQA from Fig. 4b: Each
inflation step adds new (dimer) degrees of freedom at small scales while
preserving the long-range correlation structure, resulting in a boundary state
with symmetry contributions from all length scales. The inflation steps are
renormalization group (RG) steps not only in terms of correlations, as we have
just seen, but can be also be seen as fine-graining entanglement: Along with
the scale factor $\lambda_{\\{5,4\\}}=2+\sqrt{3}$, one first computes the
asymptotic scaling of the length of any discrete bulk geodesic $|\gamma_{A}|$
through each inflation step. Consider a tensor network of bond dimension
$\chi$. For a minimal cut $\gamma_{A}$ through this network with the same
endpoints as a boundary region $A$, its entanglement entropy [31] is upper-
bounded as

$S_{A}\leq\frac{|\gamma_{A}|}{s}\log\chi\ ,$ (8)

where $s$ is the geodesic length of each edge, which is constant for regular
tilings. One can bound the scaling of $S_{A}$, and by extension compute the
maximal central charge $c^{\text{max}}$ [32] of the boundary state, by
calculating the asymptotic growth of $|\gamma_{A}|$ for a given $\\{n,k\\}$
tiling [26]. Furthermore, it is possible to relate $c^{\text{max}}$ to the
Gaussian curvature $K=-1/\alpha^{2}$ of the embedding Poincaré disk. The AdS
radius $\alpha$ is then given by

$\frac{s}{\alpha}=2\operatorname{arcosh}\left(\frac{\cos\frac{\pi}{n}}{\sin\frac{\pi}{k}}\right)=2\log\left(\frac{2k}{\pi}\cos\frac{\pi}{n}\right)+O(k^{-2})\
.$ (9)

Combining (8) and (9) leads to a discrete, tiling-dependent generalization of
the Brown-Henneaux formula $c=3\alpha/(2G)$ [33] relating AdS radius $\alpha$
and central charge $c$. In this identification the effective gravitational
constant scales as $G\leavevmode\nobreak\ \sim\leavevmode\nobreak\
1/\log\chi$; comparing with AdS/CFT, the limit of semiclassical bulk gravity
thus corresponds to a model with large bond dimension. Geometrically, this can
be represented as a flat subdivision of each tile into smaller tiles, each of
which carries only spin degrees of freedom on its edges. This discrete version
of the Brown-Henneaux formula is generally an upper bound $c<c^{\text{max}}$
that can only be saturated for tensor networks that are maximally entangling
across any minimal cut. However, given the specific tensors of a tensor
network on a regular hyperbolic geometry, we can analytically compute the
value of the effective central charge $c$ using the same approach. Consider
the HaPPY code in its Majorana dimer formulation in Figs. 1c-d and 4c: As each
dimer carries an entanglement entropy of $\frac{1}{2}\log 2$, an inflation
step is equivalent to an _entanglement renormalization_ step [34], adding
entanglement through the addition of dimers along the boundary. Relating the
growth of boundary regions to the growth of entanglement over cuts under an
inflation step allows us to analytically compute $c$. For example, for the
$\\{5,4\\}$ HaPPY model under vertex inflation one finds [26]

$\displaystyle c_{\\{5,4\\}}$ $\displaystyle=\frac{9\log
2}{\log\left(2+\sqrt{3}\right)}\approx 4.74\ .$ (10)

As this is a bond dimension $\chi=2$ model with only spin degrees of freedom
on the edges of each tile, we find a relatively small central charge. For a
more realistic model of (semiclassical) bulk gravity, a large bond dimension
would be required, e.g. by placing $N$ copies of the $[[5,1,3]]$ code tensor
on each tile, increasing $c$ by a factor of $N$. Note that the procedure
leading to (10) can be applied to any tensor network on a regular hyperbolic
tiling whose tensors have equal entanglement properties in the asymptotic
region near the Poincaré boundary, yielding a different central charge for
each model. A simpler version of this approach can also be used to compute the
change in qubit erasure probabilities with each inflation step: While the size
of boundary regions grows exponentially and the entanglement entropy linearly
with the number of inflation steps, the erasure probability of a logical qubit
deep in the bulk decays double exponentially if the single-qubit erasure
probability is below a threshold value [30]. Like the effective central
charge, this threshold is lattice-dependent.

###  2.4 Local scaling

Figure 5: Local scale transformations. a, bulk shift of $\\{5,4\\}$ tiling
center (thick circle) after two inflation steps, coarse-graining boundary
region $A$ (blue) to $B$ (yellow) on upper-left part of the tiling, as well as
fine-graining a corresponding region on the lower-right one. b, the same bulk
shift with an additional deflation step, coarse-graining boundary region $A$
to $C$ while leaving the lower-right part of the tiling unchanged. c, the
regions $A,B,C$ from a-b in the MQA of Fig. 4b, with each row representing an
inflation layer. Note that region $C$ corresponds to only two inflation
layers. d, the total HaPPY Majorana dimer state after two (first two rows) and
one inflation step (bottom row), with subsystems $A,B,C$ marked. e, the
subsystems $A,B,C$ of the HaPPY Majorana dimer state rescaled to the same size
for comparison. The same correlation structure appears on multiple length
scales in different subsystems of the state, with only small deviations.

The distinguishing feature of conformal transformations is a local rather than
merely global change in scale, i.e., a location-dependent fine- and coarse-
graining. These transformations can be represented by discrete lattices, such
as for the MERA tensor network [35]. Regular tilings also naturally
incorporate local scale transformations in the shape of non-uniform boundary
deformations that arise from the discretization of the bulk Möbius
transformations (2). As in the continuum, the corresponding bulk
transformation is a lattice-preserving shift $w$ of the center of the Poincaré
disk combined with a global rotation $\theta$ as shown in Fig. 3. We show in
Fig. 5a how such a bulk shift on a tiling after two inflation steps moves
parts of the tiling boundary while changing its resolution: While one region
of the boundary is fine-grained in a local application of the vertex inflation
rules, another region $A$ is coarse-grained to a region $B$ with fewer edges
through local vertex _deflation_ rules. The boundary geometry as a whole is
left invariant by this transformation, as each vertex in the hyperbolic tiling
can be arbitrarily defined as its “center”. In other words, the bulk shift
only changes the projection of the tiling onto the Poincaré disk. Note that
the effective local scale transformation on the boundary, following the same
inflation/deflation rules as the previously described global scale
transformation, thus exhibits the same scaling factor $\lambda$ for large
regions. Unlike arbitrary deformations of the tiling boundary, these
transformations thus preserve the entanglement scaling and effective central
charge $c$ computed above for global scale transformations. Combining the bulk
shift with a deflation step, as Fig. 5b shows, preserves the (previously
inflated) part of the boundary while deflating the rest of the boundary. Again
using the $\\{5,4\\}$ HaPPY code in its Majorana dimer form for illustration,
Fig. 5d shows how the local deflation step acts on the boundary state while
leaving it globally invariant. In Fig. 5e these subsystems are rescaled to the
same size to show that the local correlation structure is preserved (up to
coarse-graining). As there are as many such allowed transformations as there
are tiles at a given inflation step, these local scale transformations lead to
a fractal self-similarity of the boundary state as the number of inflation
steps is increased. We can also characterize these transformations using the
MQA picture introduced above, as is visualized in Fig. 5c. The two examples
from Fig. 5a-b correspond to a mapping from a large block $A$ of three
quasiperiodic layers to a smaller block $B$ contained within it (a) and to a
small block $C$ of only two layers (b). Close inspection shows that all of the
three blocks can be traced to similar initial letter sequences on different
layers: For example, the sequence $bab$ belonging to the red-shaded block $C$
in Fig. 5c is inflated to

$bab\mapsto ababaabab$ (11)

which belongs to the yellow-shaded block $B$, but also appears as a
subsequence under inflation of another sequence,

$aaa\mapsto aba\underline{ababaabab}aab\ .$ (12)

This sequence is indeed the starting point of the blue-shaded (initial) block
$A$ of both transformations considered in Fig. 5, showing how identical
sequences appear on different scales as a consequence of an invariance under
discretized Möbius transformations. The number of such transformations that
map the boundary system to part of itself is given by the number of tiles up
to the given inflation step (each defining a possible bulk shift) times the
number of inflation steps (possible rescalings). The former is propertional to
the number of boundary sites and the latter to the discretized length scales
encoded, as would be expected from a discretization of local scale
transformations. This self-similarity can also be explained by the MQA, where
we find that a given letter (say, $b$) on any inflation level produces the
same sequences “below” it and thus a similar correlation structure in the
corresponding boundary subsystem up to boundary effects. Of course, the block-
based MQA as shown in Figs. 4b and 5c is dependent on the exact assignment of
the inflation rules and does not take into account any “smearing” of
correlations that a tensor network with less localized correlations than
Majorana dimers would produce. However, the self-similarity argument is
indendendent of the letter assignment (e.g.
$a\,{\mapsto}\,abaab,b\,{\mapsto}\,ab$ instead of
$a\,{\mapsto}\,baaba,b\,{\mapsto}\,ba$) and self-similarity of subsystems is
still expected for subsystem sizes larger than the “smearing radius” by which
the tensors in the particular tensor network spread correlations. This self-
similarity, as in continuum CFTs, enforces a polynomial decay of correlation
functions, as short- and long-range correlations are equivalent up to a factor
corresponding to the scaling dimension of the fields considered. We elaborate
on the connection to continuum CFTs below.

###  2.5 Approximate translation invariance

Figure 6: Boundary translations. a, the $\\{5,4\\}$ tiling after three
inflation steps (gray-shaded area) with the tiling center (thick circle) on
the central tile. Colored areas correspond to a bulk shift of tiling center
and two deflation steps (similar to Fig. 5b) so that boundary regions $A,B,C$
of the smaller sets of tiles overlap with the boundary of the large (gray-
shaded) one. b, the regions $A,B,C$ from a in the MQA picture of Fig. 4b, with
each row representing an inflation layer. c, the total HaPPY Majorana dimer
state after three inflation steps, with the subsystems $A,B,C$ magnified. All
three show a similar correlation structure and quasiperiodic boundary
sequences, as they can be identified with parts of the same deflated geometry
related by bulk shifts in a.

Exact invariance under boundary translations, i.e., transformations (2) with
$w=0$, is manifestly broken by the discretization of an $\\{n,k\\}$ regular
hyperbolic tiling. The tiling can still be centered around either a vertex or
a tile center in the Poincaré disk to produce a global $\mathbb{Z}_{n}$ or
$\mathbb{Z}_{k}$ cyclic symmetry, but towards the boundary, these bulk
rotations corresponds to an asymptotically infinite translation. However, we
can use the same geometrical construction discussed above in the context of
local scale transformations to show that boundary states still exhibit at
least an approximate translation invariance under certain boundary
translations. We start from the observation in Fig. 5b that a shift of the
tiling center of a set of tiles after two inflation steps together with a
deflation step, i.e., a global scale transformation, is equivalent to instead
applying one or multiple deflation steps to only part of the boundary. Due to
the properties of hyperbolic geometry, the non-deflated, “joint” part of the
boundary (in the lower right of the figure) contains almost half of the edges
of the deflated (red-shaded) tiles. As the boundary features (sequence of
vertex types $a$ and $b$) obey a $\mathbb{Z}_{5}$ symmetry, i.e., are repeated
five times, this implies that the vertices of the joint boundary region follow
at least one fifth of the quasiperiodic sequence characterizing the boundary
geometry after one inflation step, i.e., $abaab$ or any cyclic permutation
thereof. This argument holds for any bulk shift, and so many different parts
of the tiling boundary after two inflation steps (blue-shaded tiles) can be
identified as a joint boundary region of a shifted and deflated tiling. It
follows that same sequence – $abaab$ – should appear in many different parts
of the boundary. Of course, this is wholly expected from the quasiperiodicity
of such sequences, as any inflated $a$ letter on any inflation level produces
it on the next. Note that this does not imply an exact invariance of
subsystems of a boundary state under translations preserving this boundary
sequence, as the finite size of the tiling implies that long-range
correlations (contributions from earlier inflation layers in the MQA) in these
subsystems can still differ. To show that short-range correlations are still
closely reproduced, we consider a larger tiling after three vertex inflation
steps in Fig. 6a, with three shifted-and-deflated sets of tiles contained
within it. This leads to three joint boundary regions labeled $A,B,C$ (not
related to the previous ones) between the large and smaller sets of tiles.
Again, we expect that these joint boundary regions are characterized by the
same sequence of vertex types. Indeed, we find that $A$ and $B$ follow the
sequence $ababaababa$, which appears around any inflated $a$ letter regardless
of the neighboring letters as

$\displaystyle aaa$ $\displaystyle\to aba\underline{ababaababa}ab\ ,$ (13)
$\displaystyle baa$ $\displaystyle\to\underline{ababaababa}ab\ ,$ (14)
$\displaystyle aab$ $\displaystyle\to aba\underline{ababaabab(a)}\ ,$ (15)
$\displaystyle bab$ $\displaystyle\to\underline{ababaabab(a)}\ .$ (16)

This sequence appears once for every $a$ letter on the previous inflation
layer. The boundary region $C$, which contains more sites than $A$ or $B$,
even contains it twice, following the sequence

$\underline{ababa\overline{ababa}}\overline{ababa}\ .$ (17)

We thus find that bulk transformations reproduce just the quasiperiodic
symmetries following from the letter assignment of vertex types. Indeed, the
number of such boundary translations at a given length scale is equal to the
number of tiles on that inflation layer, each of which defines an allowed bulk
shift (followed by a suitable rescaling that matches up boundary regions as in
Fig. 6a). For the $\\{5,4\\}$ tiling the number of tiles on a given inflation
layer is equal to the number of $b$ letters on it, which implies that at a
large number of inflation steps, a constant fraction

$\frac{N_{b}}{N_{a}+N_{b}}\to 1-\frac{1}{\sqrt{3}}$ (18)

of all boundary lattice shifts corresponds to an approximate invariance
transformation. In the MQA setup for four inflation layers, shown in Fig. 6b,
we see how the different boundary regions $A,B,C$ correspond to whole blocks
of quasiperiodic layers. The blocks corresponding to $A$ and $B$ match on
three inflation layers, while $C$ only matches the others on two layers. As a
result, subsystems of these boundary states closely match, as we demonstrate
with the Majorana dimer setup in Fig. 6c, where we consider the dimer
subsystems corresponding to boundary regions $A,B,C$. This self-similarity of
dimer subsystems under boundary translations can also be confirmed
numerically: In Fig. 7, we take an arbitrary dimer subsystem of size $\ell$
and consider “dimer fidelity” with respect to subsystems under a boundary
translation $d$, that is, the percentage of matching dimers ($j\mapsto k$
mapped to $j+d\mapsto k+d$) between both states. As expected, approximately
matching subsystems are found for a large range of $d$. Approximate
translation invariance appears independent of $\ell$ as long as $\ell\gg 1$,
showing that the quasiperiodicity is a scale-independent effect.

Figure 7: Approximate translation invariance of subsystems of HaPPY code
dimers. a, two dimer subsystem of $\ell$ sites ($2\ell$ Majorana modes) with a
relative offset of $d$ sites for a total system of $L=1325$ sites (four
inflation steps), one fifth of which is shown. b,  Ratio of matching dimers
between both subsystems for different $\ell$ and $d$. Dimers approximately
match for a large number of values for $d$, with exact matching only occurring
at multiples of $L/5$ due to the pentagon tiling’s $Z_{5}$ symmetry.

###  2.6 Relationship to continuum CFT

The symmetries of quasiperiodic CFTs have an inherent lattice structure, as
they form a discrete subgroup of those of continuum CFTs. However, that does
not mean that tensor networks on regular hyperbolic tiling, via whose geometry
we defined qCFTs, are unable to produce states that resemble the more common
regular discretizations of (ground) states of a continuum CFT. On the
contrary, the enforcement of a subset of CFT symmetries is a stronger
condition than other tensor network models used to describe critical states,
such as the MERA, possess from geometry alone. While some choices for tensor
content and bond dimension thus result in qCFT models that clearly break CFT
properties such as full translation invariance, as seen in the Majorana dimer
models discussed above, others should still allow for a close approximation of
continuum CFTs. Even the dimer models can be identified with a class of CFT-
like critical systems: As we saw, the $\\{5,4\\}$ dimer model implements an RG
step that can be formulated in terms of Majorana dimer replacement rules (see
details in Ref. [26]). However, this is merely a fermionic generalization of
the RG step of _strongly disordered_ spin chains such as the _Fibonacci XXZ
chain_ , where such an RG step appears in the form of a replacement rule for
singlets [36, 37]. This results in the same $\propto 1/d$ correlation decay
with distance $d$ and $\propto\log\ell$ entanglement entropy scaling with
subsystem size $\ell$ expected from a continuum CFT, as shown in Fig. 8, with
a piecewise linear growth of entanglement characteristic for strong disorder.
Note that the average decay of correlation functions, given by a histogram of
the dimers/singlets over the boundary distance $d$ over which they are paired
up, appears “split” into two series in the dimer case; this is the result of
two types of dimer pairs that appear at each length scale (compare Fig. 6c). A
second, more realistic qCFT model to consider is that of generic tensor
networks on a regular $\\{3,7\\}$ geometry using _matchgate tensors_ as
studied in Ref. [17]. These tensors can encode any fermionic Gaussian state
and can be contracted with a computational effort that is only polynomial,
rather than exponential, in the number of contracted bonds. For the simplest
instance of this model at bond dimension $\chi=2$, correlation functions with
only soft deviations from translation invariance and site-averaged properties
(scaling dimensions and OPE coefficients) matching the continuum Ising CFT can
already be recovered [17]. Entanglement scaling and two-point correlation
decay is shown in Fig. 8 along the results for the dimer model. In particular,
the entanglement entropy smoothly follows the Calabrese-Cardy prediction [32],
rather than the piece-wise linear growth of the disordered Majorana dimer
states. This model illustrates that qCFT properties can closely approximate
continuum CFTs even at small bond dimension. To recover full translation
invariance, tensor networks at larger bond dimension become necessary. Such a
construction in terms of matchgate tensors will be explored in future work. As
a third example, consider the setup of _hyper-invariant_ tensor networks [25]
also based on vertex inflation on regular hyperbolic tilings, using tensors
with certain constraints to reproduce properties of MERA-type tensor networks.
In particular, these symmetries preserve the full tiling symmetries and thus
also classify as an instance of a qCFT. Indeed, it has recently been shown
that this ansatz also leads to boundary states that are consistent with
certain conformal properties [38], confirming that the class of qCFTs includes
at least close approximations of continuum CFTs.

Figure 8: Physical qCFT properties. a, Average entanglement entropy scaling
for the HaPPY model (with basis state input) and matchgate tensor networks on
regular geometries. Effective central charges are $c_{\\{5,4\\}}\approx 4.74$
and $c_{\text{Ising}}=\frac{1}{2}$. b, Average correlation decay between
Majorana modes in both models and continuum $1/d$ decay (scale arbitrary). The
$\\{5,4\\}$ dimer result follows from the relative frequency of dimer pairs
over boundary distance $d$.

##  3 Discussion

In this work, we found that tensor networks on regular $\\{n,k\\}$ tilings
with equivalent tensors on all sites lead to boundary states that respect a
discrete subset of conformal symmetries, specifically invariance under local
and global scale transformations, as well as an approximate translation
invariance. Due to the associated quasiperiodic boundary symmetries, we call a
theory invariant under these transformations a _quasiperiodic conformal field
theory_ (qCFT) in analogy to the symmetry definition of a continuum CFT. As in
the continuum, states that are invariant under all discrete qCFT
transformations are associated with ground states, while excited states need
only respect the asymptotic symmetries at small scales. In the bulk
perspective, this corresponds to deforming either the regular geometry or
changing the tensor content around the center of the tiling while leaving the
near-boundary structure unchanged. Our approach expands upon the idea of
_conformal quasicrystals_ [16] which posits that tensor networks on regular
hyperbolic tilings describe boundary states on a quasiperiodic lattice, but
differs from it in several respects: Here we argued that the combined
symmetries of all tiling layers characterize the final boundary states, rather
than the boundary symmetries alone, as captured by our multi-scale
quasicrystal ansatz (MQA). Physical models with local couplings that are
quasiperiodically modulated between two or more values, such as the Fibonacci
XXZ model [37], more properly describe spin chains on quasicrystals but have
symmetries that are simpler than that of the qCFT framework described here.
Furthermore, we related discretized bulk symmetries to specific boundary
conformal transformations, whereas Ref. [16] considered deformations of the
initial tiling before inflation as defining such transformations. However,
such deformations only change the long-range (IR) symmetries of the boundary
state in the manner of a relevant CFT operator and thus cannot describe the
full ground state symmetries. In addition, we found that the choice of
inflation rule changes the effective central charge of the boundary state,
which leads to the restricted class of “allowed” discretized symmetry
transformation described here. We expect qCFTs to appear on the boundary of
all tensor network models on a regular bulk geometry [30, 39, 21, 25, 40],
potentially including p-adic AdS/CFT models [41, 42]. From their construction
on regular tilings, these models necessarily inherit the discretely broken
conformal symmetries we discussed, thus falling into the rubric of our
proposed AdS/qCFT correspondence. The HaPPY pentagon code, studied here in its
Majorana dimer form, shows that qCFTs are closely related to strongly
disordered critical models, which have been extensively studied in the
condensed matter literature [43, 44, 37, 45, 46, 47]. As the HaPPY code is a
model of quantum error correction, we find that AdS/qCFT includes exact
holographic codes, rather than the approximate codes found in AdS/MERA models
[48]. The similarity to strongly disordered models also suggests that boundary
dynamics can be described by an effective _local_ (though not nearest-
neighbor) Hamiltonian, which would allow for dynamical AdS/qCFT models.
Further work should also explore the role of qCFT excitations, where the
regular symmetries of the tensor network are (locally) broken. On a
mathematical side, a rigorous extension of the discrete breaking of
$PSL(2,\mathbb{R})$ symmetries through Fuchsian groups to the breaking of full
conformal symmetries by space-time lattices would help in the construction of
new tools to study time evolution in CFTs. Similar notions of discrete
conformal transformations should also appear when considering the geometry of
higher-dimensional regular hyperbolic tilings.

##  4 Acknowledgements

We thank Marek Gluza, Xiaoliang Qi, Sukhbinder Singh, Tadashi Takayanagi, and
Charlotte Verhoeven for helpful comments and discussions. This work has been
supported by the Simons Collaboration on It from Qubit, the Templeton
Foundation, the DFG (CRC 183, EI 519/15-1), and the FQXi. This work has also
received funding from the European Union’s Horizon 2020 research and
innovation programme under grant agreement No. 817482 (PASQuanS). This
research has been supported in part by the Perimeter Institute for Theoretical
Physics. Research at Perimeter Institute is supported by the Government of
Canada through the Department of Innovation, Science, and Economic
Development, and by the Province of Ontario through the Ministry of Research
and Innovation. We also acknowledge support from the National Research,
Development and Innovation Office (NKFIH) through the Quantum Information
National Laboratory of Hungary and Grants No. K124176, FK135220, K124351.

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