Classical and Quantum Gravity
Class. Quantum Grav. 37 (2020) 095011 (36pp) https://doi.org/10.1088/1361-6382/ab7bb9
Group field theory and holographic tensor
networks: dynamical corrections
to the Ryu–Takayanagi formula
Goffredo Chirco1,2, Alex Goeßmann3, Daniele Oriti4and
Mingyi Zhang1
1Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Am
Mühlenberg 1, 14476 Golm, Germany
2Romanian Institute of Science and Technology, Strada Virgil Fulicea 17, 400000,
Cluj-Napoca, Romania
3Institut für Mathematik, Technische Universität Berlin, Straße des 17.Juni 135,
10623 Berlin, Germany
4Arnold-Sommerfeld-Center for Theoretical Physics,
Ludwig-Maximilians-Universität, Theresienstrasse 37, 80333 München, Germany
E-mail: goffredo.chirco@aei.mpg.de,goessmann@tu-berlin.de,
Received 1 October 2019, revised 20 February 2020
Accepted for publication 2 March 2020
Published 9 April 2020
Abstract
We introduce a generalised class of (symmetric) random tensor network states
in the framework of group field theory. In this setting, we compute the Rényi
entropy for a generic bipartite state via a mapping to the partition function of
a topological 3D BF theory, realised as a simple interacting group field the-
ory. The expectation value of the entanglement entropy is calculated by an
expansion into stranded Feynman graphs and is shown to be captured by a
Ryu–Takayanagi formula. For the simple case of a 3D BF theory, we can
prove the linear corrections, given by a polynomial perturbation of the Gaussian
measure, to be negligible for a broad class of networks.
Keywords: quantum gravity, holographic entanglement entropy, random
tensor networks, group field theories, tensor models
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Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
1. Introduction
Tensor networks algorithms from condensed matter theory [1–5] have recently experienced
a massive impact in quantum gravity as new powerful tools for investigating the nature of
spacetime at the Planck scale and its holographic properties. In the AdS/CFT framework,
the Ryu–Takayanagi formula, together with the geometry/entanglementcorrespondence[6–9]
have led to a new constructive approach to holographic duality, today further captured by the
AdS/MERA conjecture [10], suggesting an interpretation of the geometry of the auxiliary ten-
sor network decomposition of the quantum many-body boundary state as a representation of
the dual bulk geometry [11,12]. The use of tensor networks in this sense has produced a new
constructive approach [13], where the key entanglement features of some holographic theory
can be captured by classes of tensor network states.
In non-perturbative approaches to quantum gravity, including loop quantum gravity (LQG)
and spin foam models [14–17] and their generalization in terms of group field theories (GFT)
[18–20], pre-geometric quantum degrees of freedom are encoded in random combinatorial
spin-network structures, labeled by irreducible representation of SU(2) and endowed with a
gauge symmetry at each node. Such spin-network states can be understood as peculiar sym-
metric tensor networks [21,22], and tensor network techniques have found a number of quan-
tum gravity applications [23–26]. A discrete spacetime and geometry is naturally associated
with such structures, at a semi-classical level, and their quantum dynamics is related to (non-
commutative) discrete gravity path integrals [27–30]. The outstanding issue is then to show
the emergence of continuum spacetime geometry and GR dynamics from the full quantum
dynamics of the same pre-geometric degrees of freedom, which in fact describe a quantum
spacetime as a peculiar sort of quantum many-body system [31–33]. In this sense, tensor
network techniques have been largely exploited in relation to the problem of spin foam renor-
malization in the context of loop quantum gravity [23–26], as well as quantitative tools to
analyze the entanglement structure of spin-networks and to look for classes of spin-network
states with correlation and entanglement properties compatible with well behaved geometries
in the semiclassical interpretation.
More recently, tensor network representation schemes have been exploited to extract infor-
mation on the non-local entanglement structure of the spin-network states and to understand
the effects of the local gauge structure on the universal scaling properties of the holographic
entanglement, in the background independent context [34]. Along this line, a precise dictionary
between random tensor networks and group field theory (GFT) states was defined by some of
the authors in [35], and used as a basis for a first derivation of the Ryu–Takayanagi formula
[6] in a non-perturbativequantum gravity context. This dictionary also implied, under different
restrictions on the GFT states, a correspondence between LQG spin-network states and tensor
networks, and a correspondence between random tensors models [36] and tensor networks.
To summarize the mentioned dictionary, GFT states define (generalized) gauge-symmetric
tensor networks provided with a field theoretic formulation and a quantum dynamics. The
field-theoretic nature of the GFT tensors provides a natural random interpretation, albeit cor-
responding to a probability measure which is in general different from that of the standard
random tensor network models. Also, the main features of the GFT networks—lattice topol-
ogy, tensor order, bond dimension—are not fixed, but dynamically induced by the specific GFT
model considered. In this sense, GFT defines a generalization of the usual tensor networks. It
follows that the correlation functions of the GFT-defined tensor network will strongly depend
on the choice of the model. As showed in [35], the analogy between standard random tensor
network models and GFT tensor network is particularly strong for the simplest case of a non-
interacting GFT theory, wherein the propagator of the theory induces maximally entangled
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Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
bonds among single GFT tensor-fields and the partition function corresponds to a Gaussian
distribution. The usual tensor network correlation functions are obtained whenever the usual
tensor network setting is reproduced within the more general GFT one, i.e. e.g. fixing bond
dimensions to be equal for all tensors and not dynamical, restricting to Gaussian measures,
etc.
The generalized tensor field-theoretic formalism of GFTs, as mentioned, has been success-
fully used to recover the statistical behavior of a special class of random tensor network (RTN)
states, in the large dimensional regime [13]. Idealized versions of RTNs, so-called pluri-perfect
tensors, recently attracted a great deal of attention in the holographic context, as they can simul-
taneously satisfy the Ryu–Takayanagi (RT) formula for a subset of boundary states [37], they
can be used to define bidirectional holographic codes [38], error correction properties of bulk
local operators [39], and to investigate sub-AdS locality. In this sense, a fully developed dic-
tionary hold strong promise for a new interplay between non-perturbative quantum gravity,
AdS/CFT and quantum information theory.
Conversely, the provided correspondence between group field theory (GFT) many-body
states and large dimensional random tensor network, has allowed to reproduce standard tech-
niques of random state averaging in the non-perturbativequantum gravity context, by means of
a mapping to the evaluation of the partition function of a simple group field theory, understood
as a peculiar statistical quantum many-body system. In the preliminary study provided in [35],
this analysis was limited to the case of a non-interacting GFT model. For a given GFT model,
interaction kernels give the vertices of the spin foam, which can be naively seen as nodes of
the bulk tensor network corresponding with the Feynman diagrams of the theory. In this sense,
it is natural to wonder whether the presence of ‘bulk’ interaction may leave an imprint on the
expression of the holographic area law, in relation to deformation of the minimal surface.
In this paper, some of the results presented in [35] are re-derived and presented from a
different perspective, hoping that this will facilitate their comprehension from a deeper and
different angle. We then perform the calculation of the entanglement entropy for the simple
interacting group field theory, corresponding to Boulatov’s model for topological 3D gravity
[40,41]. From the statistical-mechanics point of view, the interacting model realizes a gen-
eral non-Gaussian probability distribution over random tensor networks, which can be further
exploited to characterize deviations from the perfect tensor behavior. Section 2introduces the
statistical treatment of the group field theory field, i.e. the basic dynamical field of a given
GFT model, focusing on the relation, in terms of entanglement, between tensor fields and GFT
fields. In section 3the notion of entanglement entropy for a bipartite network state is quanti-
fied by the measure of Rényi entropy and it is shown how, via replica trick, this quantity can
be computed in terms of two auxiliary variables, Z(N)
0and Z(N)
A, respectively defining the Nth-
power of the whole and reduced tensor network density matrix partition functions, graphically
described in terms of stranded contraction patterns.
Building on the group field-theoretic description, we then consider a statistical evaluation of
the entanglemententropy,which, in the assumption of largedimension of the tensorial leg space
H, reduces by the concentration of measure phenomenon to the computation of a ratio of par-
tition functions expectation values. The quantities Z(N)
0and Z(N)
Acan be expanded in Feynman
amplitudes corresponding to stranded diagrams. Their asymptotic behavior for large dim[H]
is, however, captured only by the diagrams with maximal divergence degree. In section 4we
define strategies to identify such maximal divergent diagrams, where we derive bounds of the
divergencedegree by the topologyof the Feynmangraphs. We study the maximum face number
for different types of networks in a class of diagrams that we call locally averaged diagrams.
The maximally divergent diagrams in this class are analyzed by first restricting to the free sector
of the theory. In this setting, for the case of a unique minimal surface separating two boundary
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Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
regions the maxima are unique. Maximal divergent patterns for the case of multiple minimal
surfaces are further discussed in appendix. We therefore consider the effect of a specific class
of interactions in section 5, by looking at a minimal perturbation of the free case given by a
single interaction process. In this setting, we determine the maximal face numbers for the case
of interaction kernels involving nodes incident to minimal surfaces. We find that the divergence
degree of the diagrams induced by single local interaction processes is lower than in the free
theory case, and we can find maximal divergent pattern only for a set of situations, where the
network graph can be coarse-grained to a nontrivial tree structure.
2. Random group fields and tensor networks
Group field theories (GFTs) are quantum field theories defined on product spaces of groups,
defined by combinatorially non-local kernels [18,19,42]. GFTs provide a higher-rank gen-
eralization of matrix models with quantum states given by regular d-valent, graphs5labeled
by group or Lie algebra elements, which can be equivalently represented as (d−1)-cellular
complexes. The quantum dynamics is defined by a vacuum partition function, whose pertur-
bative expansion gives a sum of Feynman diagrams dual to d-cellular complexes of arbitrary
topology. The Feynman amplitudes for these discrete histories can be written either as spin
foam models or as simplicial gravity path integrals [17,43].
In the forthcoming derivation of the RT formula, we will consider GFTs defined in terms of a
(complex) bosonic field φ(g1,...,gd)onG×d/G, specifying to the case d=3andG=SU(2).
As a first step, we shall introduce group fields as generalizations of tensors, with the aim of
strengthening the concept of entanglement as a unifying construction principle for both. We
then introduce a generalization of random tensors in terms of GFT fields and derive group field
network states as random variables dependent on field configurations.
2.1. From tensors to group fields
A rank-dtensor6is an array of Ndcomplex numbers7, which is modeled by a field on the dth
Cartesian product of an index set with cardinality N. Each such index set can be enriched to
represent the cyclic group ZN={|1,...,|N}, with the group relation induced by:
|k◦|l:=|(k+l)modN∀|k,|l∈ZN(2.1)
Following this intuition, we redefine a rank-dtensor as a field on a product group and generalize
it afterward to the case of more general group fields.
Definition 1. A rank-dtensor |Twith index cardinality Nis a complex field on dcopies of
the cyclic group ZN:
|T:Z×d
N→C
Let Hd,Nbe the space of tensors with fixed rank dand index cardinality N. Neglecting the
structure of the cyclic group, Hd,Nis reduced to CNd. The linear structure, the scalar product
and the completeness of CNdestablish Hd,Nto be a Hilbert space. A basis of Hd,Nis chosen by
5Including disconnected graphs.
6Not to be confused with the rank of a matrix, which is the number of its singular values.
7These numbers can be understood as coefficients in a basis decomposition, where for each rank a basis of cardinality
Nis chosen.
4
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
Figure 1. Superposition of leg space product states |i1×···×|idwith the weights by
Ti1...idresults in the tensor state |T. The nonentangled product states are represented by
nonconnected legs, in contrary to the (a) entangled node states |Tand (b) more general
group fields |φas one connected component.
|i1,...,id,definedas:
|i1,...,id(j1×···× jd)=δi1,j1·...·δid,jd∀(j1,...,jd)∈Z×d
N(2.2)
With respect to this basis, we decompose a tensor |Tinto its components Ti1...id,which
introduces an isomorphism to CNd:
|T=:
i1,...,id∈ZN
Ti1...id|i1,...,id(2.3)
By decomposition of the basis elements in (2.2), we factorize the Hilbert space Hd,Nin dspaces
H(l)
1,N, which we refer to as leg spaces :
Hd,N=
l=1,..,d
H(l)
1,N,|i1,...,id=|i1×···×|id(2.4)
Each leg space H(l)
1,Nhas an induced Hilbert structure and a basis {|il}, whose product forms
the basis elements (2.2) of the tensor space Hd,N. A generic tensor |Tis given by an arbitrary
superposition of the chosen basis elements and generally does not decompose as a product
state in the leg spaces. In this case, we say that |Tis entangled with respect to the given
Hilbert space factorization. Identifying Hd,Nwith the space CNd, this amounts to an impos-
sible decomposition of an array Ti1...idinto a product of one-dimensional arrays lT(l)
il.To
capture this behavior in a graphical notation, we model the states |i1×···×|id∈H
d,Nas a
collection of legs in figure 1and the state |Tas a node with associated dordered open legs l
representing the respective Hilbert spaces H(l)
1,N. The rank dof the tensor, which is the number
of its indices, is thus equal to the number of legs in this graphical notation. The dimensionality
Nof each leg space, which is the cardinality of the basis {|il}, is called the bond dimension
of each leg.
Moving one step further, one can consider the extension of the index set from a discrete
cyclic group to a general locally compact group G. Although this results in dealing with infinite
dimensional leg spaces H, we can use the left Haar measure μof G8to lift the decompositions
(2.3) to more general integrals.
8Since Gis locally compact, there exists a left-invariant Haar measure [44].
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Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
Definition 2. Let Gbe a locally compact group with the normed left Haar measure μof its
elements. A rank-dgroup field φis a μ×d-integrable field over dcopies of the group G:
φ:G×d→C,(g1×···×gd)→φ(g1,...,gd)∈C(2.5)
such that φ2:=G×d
φ(g1,...,gd)φ(g1,...,gd)dμ×d(g1,...,gd)<∞(2.6)
If we take Gas the cyclic group ZN, the notion of group fields reduces to tensors, where
integrability with respect to the discrete Dirac measure μis satisfied for all fields considered
in definition 1. Analogously to the space Hd,Nof tensors, the space of integrable group fields
carries a Hilbert structure induced by the space L2(G×d,μ×d)ofl2-normed fields. The demand
of integrability for φensures the finiteness of the inner product on L2(G×d,μ×d) by the Cauchy-
Schwarz inequality:
φ|ψ:=G×d
φ(g1,...,gd)ψ(g1,...,gd)dμ×d(g1,...,gd)⩽φ·ψ| <∞(2.7)
A group field φis therefore understood as a state |φin the infinite dimensional Hilbert
space L2(G×d,μ×d). Dirac’s delta symbols g1,...,gd|in the associated dual space, which we
used to define a basis decomposition (2.2) in case of tensors, do not correspond to elements in
L2(G×d,μ×d). For the following we understand them as distributions acting on the group field
space L2(G×d,μ×d):
g1,...,gd|∈(L2(G×d,μ×d)∗,g1,...,gd|φ=φ(g1,...,gd) (2.8)
Also the Hilbert space L2(G×d,μ×d) admits by construction a factorization9into leg Hilbert
spaces L2(G,μ)(l), which correspond to a product decomposition of the distributions (2.8)into
distributions gl|affecting single variables:
L2(G×d,μ×d)=
l=1,...,d
L2(G,μ)(l),g1,...,gd|=g1|×···×gd|(2.9)
With the established factorization of the group field Hilbert space, we are now able to generalize
the notion of entanglement with the use of Dirac distributions instead of basis decompositions:
Definition 3. A group field theory state |φ∈L2(G×d,μ×d) is called unentangled10 with
respect to the factorization L2(G×d,μ×d)=l=1,...,dL2(G,μ)(l), if there is a collection of group
fields {|φl∈L2(G,μ)}d
l=1such that |φ=|φ1×···×|φd, i.e. it holds:
g1,...,gd|φ=
d
l=1
gl|φl∀g1×···×gd∈G×d(2.10)
If there is no such collection, the field |φis called entangled.
Entanglement with respect to the decomposition of group field spaces into a collection of
field spaces on smaller product groups is thus an analogous concept to the case of tensors on
discrete groups, with the difference just lying in a more general perspective by distributions
9In the sense of the topological tensor product, which will be denoted by ⊗.
10 Also known as an elementary tensor.
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Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
compared to basis decompositions. For the sake of simplicity we will treat the Dirac distribu-
tions g1,...,gd|in the following as elements |g1,...,gdof the space L2(G×d,μ×d), thus their
evaluations (2.8) are denoted by a scalar product.
2.2. Free tensor models
The central idea of a tensor field theory is to implement random distributions on field spaces,
which amounts in the case of the group field space L2(G×d,μ×d) to a probability measure
with density dν(φ)[20]. 11Measurable observations of a quantum field theory correspond to
random variables, so-called observables, O(φ) with a probability character induced by the field
measure ν.
Well controllable field theories are Gaussian probability measures, which correspond to free
field theories. For the sake of simplicity, let us introduce Gaussian probability measures on the
space of tensors Hd,N, since we can exploit basis decompostions in this case. Associated with
each Gaussian probability measure is a covariance C, which is an endomorphism in the space
of field configurations. In the case of the finite dimensional configuration space Hd,N,Cis
described by a matrix C(i,j):
C∈L(Hd,N,Hd,N), Ti1...id→
j1,...,jd
C(i1,...,id,j1,...,jd)Tj1... jd(2.11)
Definition 4. Let μC(T) be a probability measure of fields Ton X=Z×d
N.μC(T) is called
free tensor model with covariance C, if the only nonvanishing expectation of observables are
given by linear combinations of so-called 2p-point Green functions:
p
k=1
Ti(k)
1...i(k)
d
Tj(k)
1... j(k)
d=
π∈SN
p
k=1
C(i(k)
1,...,i(k)
d,j(π(k))
1,...,j(π(k))
d) (2.12)
The 2p-point Green functions have a direct physical interpretation in terms of particle prop-
agation. The pfields Ti(k)
1...i(k)
d
located at their arguments (i(k)
1,...,i(k)
d)∈Z×d
Nrepresent the
incoming particles, which propagate to the outgoing particles Tj(k)
1...j(k)
d
located at ( j(k)
1,...,j(k)
d).
Each permutation π∈SNencodes the propagation of the kth incoming particle to the π(k)th
outgoing particle, where the sum over all propagation possibilities is taken. A typical choice for
the covariance, determining the structure of particle propagation, is given by the identification
of each field argument:
C(i1,...,id,j1,...,jd):=
d
l=1
δ(il,jl) (2.13)
Let us now assume the existence of a to Cinverse covariance, that is a covariance Ksuch that:
N
h1,...,hd=1
C(i1,...,id,h1,...,hd)K(h1,...,hd,j1,...,jd)=
d
l=1
δ(il,jl) (2.14)
11 The definition of dν(φ) is usually given by a field measure at each element of Xand gets therefore difficult if the
cardinality of the set Xis not finite [20]. The reason for this lies in the structure of the space F(X), the space of
complex functions on X. If and only if |X|<∞, the dimension of F(X) is finite and we can construct measures using
the Lebesgue measure. For infinite cardinality of X, this is no longer possible, and the use of other methods like limit
processes of measures is required [45]).
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Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
This is for instance the case for the choice (2.13). Using the inverse covariance K, one can
express the Gaussian tensor model by the Lebesgue measure dTi1...idof the field value at each
element of Z×d
N[20].
Theorem 1 (Wick).GivenabyKinvertible covariance C, the associated Gaussian tensor
model is given by the probability density:
dμC(T)=det(C)⎡
⎣
N
{hl}
dT({hl})dT({hl})
2π⎤
⎦e−{il},{jl}T({il})K({il},{jl})T({jl})
(2.15)
The exponentialof the weight, which transforms the Lebesguemeasures to the tensor model,
is called the tensor model action S[T]. In the free Gaussian theory, the associated action S0[T]
is called free:
S0[T]:=
{il},{jl}
T({il})K({il},{jl})T({jl}) (2.16)
2.3. Free and perturbed group field theory
Allowing for groups with infinite cardinality, beyond the restriction to the cyclic group consid-
ered above, results in a significantly richer theory. The covariance Cis generalized by its action
on the Dirac distributions g1,...,gd|, which replaced in (2.8) the finite basis decomposition:
C:g1,...,gd|→G×d
d
i=1
dgiC(g1,...,gd,g1,...,gd)g1,...,gd|(2.17)
C(g1,...,gd|)[φ]=G×d
d
i=1
dgiC(g1,...,gd,g1,...,gd)φ(g1,...,gd) (2.18)
A Gaussian group field theory with covariance Cis defined in analogy to definition 4by replac-
ing the cyclic group arguments {il,jl∈ZN}by general group arguments {gl,hl∈G}and the
tensors Tby general group fields φ. The covariance Cis invertible by the covariance K,if
C◦K=IdL2(G×d,μ×d). In this case, one can associate a free action to the covariance C,which,
in analogy to the tensor model case, is given by:
S0[φ]=G×2dd
l=1
dμ(gl)dμ(hl)φ({gl})K({gl}{hl})φ({hl}) (2.19)
Assuming the existence of a Gaussian measure dμC(φ) for the group field configurations
φassociated with the covariance C, we define the Lebesgue measure [Dφ] in analogy to the
tensorial case by
dμC(φ)=:[Dφ]e−S0[φ](2.20)
Pure Gaussian probability measures describe free field propagation, formalized by 2p-point
functions in definition 4. In order to include field interactions, a perturbation of the free
Gaussian measure is needed, which can be implemented by a control parameter λ:
S0[φ]→S0[φ]+λSint[φ](2.21)
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Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
The perturbed exponential weight of the probability distribution can be expanded in a series
of manipulated free Gaussian probabilities, which can be interpreted as different orders of
interactions:
[Dφ]e−S0[φ]−λSint[φ]=[Dφ]e−S0[φ]·e−λSint[φ]
=dμC(φ)1−λSint[φ]+λ2
2(Sint[φ])2+O(λ3)(2.22)
The result is a d-dimensional combinatorially non-local quantum field theory living on a prod-
uct group manifold [18]. By decomposition of observables O(φ)into2p-point functions and
use of an analog version of Wicks theorem (theorem 1) expectation values Oare expanded
into series of terms with interpretation in terms of Feynman graphs. Due to the defining combi-
natorial structure by the action terms S0and Sint, the Feynman diagrams of the theory are dual
to cellular complexes, and the perturbative expansion of the quantum dynamics defines a sum
over random lattices of arbitrary topology. A similar lattice interpretation can be given to the
quantum states of the theory. For group field theory models, where appropriate group theoretic
data are used and specific properties are imposed on the states and quantum amplitudes, the
same lattice structures can be understood in terms of simplicial geometries [18,35,42,46].
2.4. Closure constraint for group fields
Group field theories provide a generalizing structure for non-perturbative approaches to quan-
tum gravity [18,46]. Their interpretation as simplicial quantum geometries, in terms of spin-
networks [35], relies on the restriction to gauge invariant states, satisfying the so-called closure
constraint, given in generalization to arbitrary combinatorial dimension das:
φ0(hg1,...,hgd)!
=φ0(g1,...,gd)∀h∈G(2.23)
The closure constraint can be interpreted in two ways (equivalent, mathematically). It intro-
duces an SU(2) flat discrete connection at the level of the amplitudes, encoded in the elementary
line holonomies hl. This turns the spin foam amplitudes (GFT) into gauge theories on a lat-
tice with SU(2) as the local gauge group. Further, it imposes the closure of the sides that form
the triangle (similarly in higher dimensions), thus giving the correct geometric meaning to the
variables used.
For convenience in the forthcoming calculations, let us consider the imposition of such
constraint for random group fields by splitting a general group field φ∈L2(G×d,μ×d)intoa
field φ0, where the symmetry is realized by group averaging (2.23), and a gauge field χ:
φ0(g1,...,gd):=G
φ(hg1,...,hgd)dμ(h) (2.24)
χ(g1,...,gd):=φ(g1,...,gd)−φ0(g1,...,gd) (2.25)
The invariance of the Haar measure μunder shifts of the transformation variable hensures the
symmetry (2.23)forthefieldφ0:
φ0(ˆ
hg1,...,ˆ
hgd)=G
φ(hˆ
hg1,...,hˆ
hgd)dμ(h)=G
φ(hg1,...,hgd)dμ(hˆ
h−1) (2.26)
=φ0(g1,...,gd) (2.27)
We can then impose an equivalence relation ∼in L2(G×d,μ×d) and choose the symmetric
elements φ0as a representative of it:
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Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
φ(1) ∼φ(2) ⇐⇒ G
φ(1)(hg1,...,hgd)dμ(h)=G
φ(2)(hg1,...,hgd)dμ(h)=:φ0
⇐⇒ φ(1),φ(2) ∈[φ0] (2.28)
The field χit thereby redundant, in the sense that it labels the elements in the equivalence class
[φ0]:
φ∈[φ0]⇐⇒ ∃ χ∈L2(G×d,μ×d): φ=φ0+χ∧G
χ(hg1,...,hgd)dμ(h)=0
(2.29)
We can represent each equivalence class [φ0]bya(d−1)-dimensional group field. Therefore,
we define an equivalence relation in G×dto identify the relevant arguments of the representing
field:
(g1,...,gd)∼(ˆg1,...,ˆgd)⇐⇒ ∃ h∈G:(g1,...,gd)=(hˆg1,...,hˆgd)
(2.30)
(H1,...,Hd−1,e):=(g−1
dg1,...,g−1
dgd)∈[(g1,...,gd)] (2.31)
Since φ0is constant for all elements in a class [(g1,...,gd)] of arguments, it is already char-
acterized by its values for the representative (H1,...,Hd−1,e), which form an embedding of
G×(d−1) in G×d. We can thus think of φ0as an (d−1)-field by restricting it to this embedding.
Based on this observation, one can define d-dimensional group field theories depending just
on symmetric part φ0of the described group fields φ. We impose this dependence with the use
of propagation and interaction kernels respecting the symmetries [35]:
K({g(1)
i}{g(2)
i})=
G×2
dhdˆ
hK({hg(1)
i}{ˆ
hg(2)
i}) (2.32)
V({g(1)
i}...{g(d+1)
i})=
G×(d+1)
⎡
⎣
d+1
j=1
dhj⎤
⎦V({h1g(1
i}...{hd+1g(d+1)
i}) (2.33)
The action constructed by convolution of group fields weighted by a kernel Ksatisfying (2.32),
and analogously for kernel Vsatisfying (2.33), does just depend on the symmetric part φ0,since
for arbitrary fields χit holds:
S0[φ0+χ]=[dg(1)
idg(2)
idhdˆ
h]K({hg(1)
i}{ˆ
hg(2)
i})φ(g(1)
i)φ(g(2)
i)
=[d(hg(1)
i)d(ˆ
hg(2)
i)] K({hg(1)
i}{ˆ
hg(2)
i})G
dhφ(hg(1)
i)G
dˆ
hφ(ˆ
hg(2)
i)
=S0[φ0] (2.34)
If the propagation kernel Khas the demanded symmetry, it is not invertible. As a direct con-
sequence, there is no Gaussian probability measure on the space of in general d-dimensional
group fields, such that Karises as the inverse covariance. The reason for this lies in the gauge
freedom by the choice of the field χ, which does not affect the action. In order to get rid of
these freedom, we define the action Sjust on the space of symmetric fields φ0and strip the
10
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
gauge freedom χof. This corresponds to the reduction of the elements in an equivalence class
(2.28) of group fields to one representative, given by the effective (d−1)-dimensional group
field φ0. The action S0[φ0] thus defines an effective dimensional reduction.
2.5. Group field networks as observables
We shall now discuss the construction of generalized tensor networks, realized by entanglement
of individual groupfield states. By construction, such network states shall now be understoodas
random variables induced by the probabilistic character of their building blocks and interpreted
as observables of the associated group field theory.
Let us first recall the graphical visualisation of random fields φ∈L2(G×d,μ×d) as nodes
representing the entanglement of the state with respect to the decomposition of L2(G×d,μ×d)
into leg spaces L2(G,μ)=:H(see figure 1). Consider now a set Vof nodes, where each node
v∈Vis dressed with a random field φ(v)∈L2(G×d,μ×d)=:H⊗d. This corresponds to a state
Ψin the product of the node spaces:
Ψ=
v∈V
φ(v)∈H
⊗d·|V|=
v∈V
H⊗d
v(2.35)
Instead of looking at the decomposition of node Hilbert spaces H⊗dinto leg Hilbert
spaces, we lift our focus toward decomposition of many-particle Hilbert spaces H⊗d·|V|into
node Hilbert spaces. With respect to such decompositions, the state Ψis by construction
unentangled.
A simple method to construct network states, which are entangled with respect to the node
decomposition of the many-particle Hilbert space, is given by the projection of the unentangled
state in (2.35) from Hd·|V|to an entangled state in the product leg space. We can visualize
such a procedure as a gluing of the open tensor legs, which represents entanglement. Consider
for instance the unentangled product of two node states φ(1) ⊗φ(2) ∈H
⊗d⊗H
⊗d.Aspecific
gluing functional is chosen by an integrated delta distribution Macting on the ath leg of the
first and the bth leg of the second node:
φ(1) ⊗φ(2) →Mφ(1) ⊗φ(2)
:=G×2
[dga
1dgb
2]δ(ga
1gb
2)φ(1)(g1
1,...,gd
1)φ(2)(g1
2,...,gd
2) (2.36)
The resulting state can no longer be decomposed into a product of states in the open legs of
the first and the second node, and is thus entangled. In the established graphical representation
scheme of figure 1, this corresponds to a link among the ath and bth leg connected node states,
as sketched in figure 2(a).
It is useful to introduce a link state M[35], to redefine the gluing transformation (2.36)asa
scalar product in the leg spaces. In the basis given in (2.8), we have
M|=G×2
[dgadgb]δ(gagb)ga,gb|∈L2(G×2,μ×2) (2.37)
The gluing (2.36) then corresponds to a contraction of the link state M, such that
|Φ12:=M|φ(1) φ(2)(2.38)
11
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
Figure 2. (a) Entangling two node states φ(1) and φ(2) by action of the functional Mon
affected leg spaces results in an connected state. (b) The result of the repeated entangling
procedure along all edges of a graph Γ=(V,E∪∂Γ) is the sketched network state ΦΓ.
Notice that this is a stronger notion of gluing than the one used in GFT states, to define states
associated with closed graphs, however corresponds to the standard gluing prescription for
maximally entangled tensor networks states. More generally, one would consider a link con-
volution functional M(g†
ahlgb) in the link Hilbert space, the product space of two leg spaces.
With the adopted definition one effectively sets hl=efor all link l∈Γ. This assumption makes
our state |ΨΓlying in the flat vacuum of loop quantum gravity [47].
With the established entangling projections, we are now ready to introduce general network
states. Let there be an open graph Γ=(V,E∪∂Γ) consistent of a node set V,asetofEedges
incident with two vertices and a set ∂Γof open edges incident to single vertices. By itera-
tive projection with {M(e)}e∈Efor each edge comprising two legs of adjacent node states φ(v)
transforms (2.35) into the network state:
|ΦΓ=
e∈E
M(e)|
v∈V
|φ(v)∈L2(G×|∂Γ|,μ×|∂Γ|)=:H∂Γ(2.39)
We thus entangled the group field vertices iteratively by edgewise projections, where the
affected legs are determined by the graph Γ. The resulting network state |ΦΓis thus a col-
lection of entangled group fields, where the connectivity of Γcorresponds to the entanglement
structure of its node states (figure 2(b)).
3. Holographic entanglement entropy for GFT tensor networks
After introducing the pairwise entanglement projections to construct network states |ΦΓin the
previous section, we are now interested in the resulting global entanglement structure of such
networks, which we will quantify by the Rényi entanglement entropy. In a previous work [35],
building on the established dictionary between GFT states and (generalized) randomtensor net-
works, some of the authors have computed the Rényi entropy and derived the Ryu–Takayanagi
entropy formula by using a simple approximation to a complete definition of a random tensor
network evaluation seen as a GFT correlation function, along the lines given in [13]. Such a
derivation was limited to the case of a non-interacting GFT model, leaving open the question
about the effect of the interactions on the holographic scaling of the entropy. To answer this
question, here we perform the calculation of the entanglement entropy for a simple interact-
ing group field theory model, corresponding to Boulatov’s model for topological 3D gravity12
[40,41]. From the statistical-mechanics point of view, the interacting model realizes a general
non-Gaussian probability distribution over random tensor networks.
We first shortly review the main setting of the derivation of the Ryu–Takayanagi entropy
for the group field network state |ΦΓ. Thereby, we focus on the effects of the GFT interaction
12 It must be noted, however, that such model has limited gravitational features, since it corresponds to topological
gravity.
12
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
Figure 3. Sketched duality between a boundary theory to a theory in the bulk. The
Ryu–Takayanagi proposal states a proportionality between the entanglement entropy
S(ρA) on the marginal boundary state and the area Area(σmin) of the homotopic minimal
surface in the dual bulk. In the dashed network model, Area(σmin) corresponds to the
number of dual links.
terms combinatorics in the derivation of the leading contributions to the entanglement entropy.
We then provide a set of theorems aiming at a classification of the different combinatoric pat-
terns of interaction in the calculation of the Feymann diagrams divergences of the perturbative
expansion of the GFT partition function.
3.1. Replica trick and entanglement statistics
Let us consider the group field network state |ΦΓdefined on a d-valent graph Γ=(V,E∪∂Γ).
To a given partition A∪B=∂Γof the open legs we associate a factorization H∂Γ=HA⊗H
B
(figure 3). Given a network state |ΦΓ, and a density matrix ρ=|ΦΓΦΓ|,theNth Rényi
entanglement entropy of the reduced density matrix ρA:=TrB[ρ]isdefinedby
SN(A)=1
N−1ln TrA[ρN
A]
Tr∂Γ[ρ]N(3.1)
In the limit N→1, for positive real numbers N, the function SN(A) reduces to the entangle-
ment entropy S(A), which is the von-Neumann entropy of ρA. Due to the linear character in
the arguments, the approach by the Rényi formula in (3.1) simplifies the computation of the
entanglement (the so called replica trick) [35]. Exponentiation of equation (3.1) results in
e(1−N)SN(A)=TrA[ρN
A]
Tr∂Γ[ρ]N=:Z(N)
A
Z(N)
0
(3.2)
and the calculation of SN(A) reduces to the computation of the two partition functions
Z(N)
A:=TrA[ρN
A]=TrA(TrB∪E[
e∈E
ρ(e)
v∈V
ρ(v)])N(3.3)
Z(N)
0:=Tr∂Γ[ρ]N=TrA∪B∪E
e∈E
ρ(e)
v∈V
ρ(v)N
(3.4)
where in Z(N)
Aa trace over region Ais performed, after the N-times composition of the reduced
density ρA.
The effect of the trace TrB∪Ein Z(N)
Acan be represented by the action of a swap operator
Fpermuting the order of the leg space HAas the cyclic element in the permutation group
13
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
Figure 4. Stranded graph representation of the variables Z(2)
0and Z(2)
A. Each horizontal
bar corresponds to a node v∈Vwith field φ(v)and d=3 legs, or a node w∈∂A∪B
with field φwand one leg. Performing the traces (3.6)and(3.5) corresponds to contrac-
tion of the leg states, sketched by closure of the respective strands. While by definition
Z(2)
0corresponds to propagation operators Iin Aand B,Z(2)
Adiffers by the operator Fin A
(blue). Computation of the expectation of Z(2)
0and Z(2)
Aamounts to sums of stranded dia-
grams (dashed red box), appearing in the 2N|V|-point function, which replace the node
fields φ(v)and are contracted with the sketched scheme.
SN[35]. This is apparent by closing the open links in ∂Γby virtual single valent vertices w,
which are decorated with the density matrices ρ(w)=I(the identity) and ρ(w)=Facting on
H∂Γ. We thus have:
Z(N)
A=TrA∪B∪E
e∈E∪∂Γ
(ρ(e))⊗N
v∈V
(ρ(v))⊗N
w∈A
F
w∈B
I(3.5)
Z(N)
0=TrA∪B∪E
e∈E∪∂Γ
(ρ(e))⊗N
v∈V
(ρ(v))⊗N
w∈∂Γ
I(3.6)
The structure of these variables can be effectively captured by a stranded graph diagram
(figure 4), where each node vis denoted by a horizontal bar with dlegs and a node w∈∂Γby
a bar with one leg. The N-times tensor product of the node densities in Vand ∂Γis represented
by Nincoming copies, corresponding to the quanta in H⊗d, and by Noutgoing copies, corre-
sponding to the dual states in H⊗d. Performing the trace over all leg spaces after composition
with the link densities corresponds to the contraction of the connected legs on the incoming and
outgoingcopies and is represented by closed strands. While for the nodes w∈∂Γa contraction
scheme by the permutations Iand Fis determined by definition, Z(N)
0and Z(N)
Astill depend on
the node field configurations φ(v). We will in the following section discuss the computation of
expectation values, which corresponds to inserting sums over permutation patterns in figure 4.
3.2. Calculation setting for an interacting GFT model
The group field probability distribution dν(φ) has support on the space of integrable fields
L2(G×d,μ×d) with respect to the Haar measure μon G. Taking this space as product of leg
spaces H, the space of node tensors |φ(v)is given by H⊗d:=L2(G×d,μ×d). In case of finite
14
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
Figure 5. Stranded graph representation of the propagation kernel Kin (3.8)andthe
interaction kernel Vin (5.1)forthecaseofarankd=3 group field. Strands repre-
sent delta functions and ellipses gauge parameter hacting on the respective arguments.
The group element henforcing the symmetry of Kappears in all delta functions and its
graphically denoted by an ellipse crossing the strands.
groups G, the space L2(G×d,μ×d) has finite dimensions and the theory reduces to random
tensors, as we have discussed in section 2.
For the calculations in the following sections we need to specify the probability measure
dν(φ) of Haar-integrable group fields φby its weight. With respect to the uniform measure,
the weight is the exponential of the negative action S(φ), which consists in the perturbation of
a Gaussian weight S0with the term λSint, as described in (2.21). For any possible choice of the
kernels K, we further demand the symmetry condition
K(g1,...,gd,g1,...,gd)=K(h◦g1,...,h◦gd,h◦g1,...,h◦gd)∀h,˜
h∈G, (3.7)
This choice, corresponding to enforcing a kinematical closure constraint at the nodes, is the
minimal condition to put the networks dynamics in relation to topological 3d-gravity models
[14–17]. Because the symmetry (3.7) prevents the kernel Kfrom being invertible, we need to
restrict the probability measure (2.20) to the space of G-symmetric group fields, which were
discussed in section 2.4. However, we can safely treat the probability measures as defined
on the full space L2(G×d,μ×d) and ensure the symmetry constraint by projections, which are
implemented by averages (2.26) over the gauge parameters h.
Notice that while a simple kernel (2.13) is constructed by delta functions between the argu-
ments gand gof the same index, the symmetry (3.7) is enforced by group averaging due to the
invariance of the Haar measure μ:
Ksym(g1,...,gd,g1,...,gd)=dμ(h)
d
i=1
δ(h◦gi◦g−1
i) (3.8)
The kernel (3.8) corresponds graphically to a collection of strands, each representing a delta
function between incoming arguments gand outgoing arguments g(figure 5). The interaction
kernel Vis defined in an analogous way, a specific form for the rank-3 group field theory under
study is given in section 5.
Now, the field-theoretic description allows us to describe the entropy SN(A), as well as the
partition functions Z(N)
0and Z(N)
A, as random variables dependent on the field configuration φ(v),
taken to be identical for all vertices v. In particular, given the field-theoretic random character,
we expect the fluctuations of SN(A) around its average E[SN(A)] to be exponentially suppressed
in the limit of high bond dimension of the leg spaces, as a consequence of the phenomenon of
15
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
measure concentration, as suggested in [48].13 This has an important impact on the derivation
of the entanglement entropy, which effectively reduces to a computation of the expectation
value E[SN(A)].
Analogously, the variables Z(N)
0and Z(N)
Ashall concentrate around their averages, allowing
for a further approximation of E[SN(A)] in terms of the individual averages E[Z(N)
0]andE[Z(N)
A]
[13],
E[SN(A)] ≈1
N−1ln E[Z(N)
A]
E[Z(N)
0](3.9)
which are now defined in terms of the perturbed Gaussian measure for the interacting GFT,
E[Z(N)
A/0]=L2(G×d,μ×d)
dν(φ)Z(N)
A/0[φ]
=L2(G×d,μ×d)
[Dφ]Z(N)
A/0[φ]e
−[S0(φ)+λSint(φ)].(3.10)
By taking the perturbative expansion of E[Z(N)
A/0]inordersofλand applying Wicks theorem
for Gaussian random variables, the expectation (3.10) decomposes into a sum of contributions
with associated Feynman diagrams. Due to the linearity of the trace, as sketched in figure 4,
the integration over the group fields can be carried out independently before the contraction
with the links. The expectations E[Z(N)
A/0]define2N|V|-point functions of the group field theory,
which are then contracted by a pattern determined by the network geometry. The red box in
figure 4represents the sum of all Feynman diagrams of the 2N|V|-point functions, each of
which corresponds to a fully contracted diagram.
Each stranded Fenyman diagram Gconsists of edges lrepresenting field propagations,
which are bundles of parallel strands, and faces fdefined by closed strands with boundary ∂f,
given by an oriented and ordered collection of edges. A holonomy hlis associated with each
edge l, correspondingto the group element enforcing the symmetry constraint. Each face fcon-
tributes with a delta function of the group product of its edge holonomies and the amplitude of
adiagramGis given by:
A[G]=
l∈G
dhl
f∈G
δ⎛
⎝
l∈∂f
hl⎞
⎠(3.11)
From the infinite number of possible diagrams Grepresenting contracted propagation and
interaction processes, we want to identify the processes with dominant amplitudes in the limit
of high bond dimensions Dof the leg spaces L2(G,μ), which we generically treat as finite-
dimensional through of a sharp cut-off Λin the group representation, such that14.
13 An interestingly related application of quantum typicality and measure concentration concerning the area-scaling
behavior of entanglement in non-random spin-network states was considered in [49,50].
14 A stronger restriction consists in restricting to finite groups Gwith Delements. In this case, the Dirac distribution
δcan be understood as an element in L2(G,μ) and it holds δ(e)=Dfor the identity ein G. Nevertheless, the group
should remain non-abelian. More radically, one could generalise the derivation and regularize the divergences via ‘box’
normalization of δg∈L2[G,δμ] by using quantum groups. As shown in [51,52], the quantum deformation relates to
the cosmological constant Λin the semi-classical regime of the spinfoam formalism. Interestingly, the cosmological
constant Λin the link space dimension (3.12) would make our vacuum state a dS vacuum if Λ>0 and AdS vacuum
if Λ<0.
16
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
δ(e)=D(Λ).(3.12)
Therefore, the divergence degree Ωof Gis defined as the exponent of
A[G]=:δ(e)Ω[G].(3.13)
The dominant contributions to the averages (3.10) are the maxima of Ω[G]. Their number and
the maximal divergence degree determine the asymptotic behavior of the expectations such as
(3.10) and are thus of central interest in the derivation of the expected entanglement entropy
(3.9). To unravel the divergence degree Ωof a diagram, one needs to restrict the holonomy
parameters hlsuch that l∈∂fhl=e, which we refer to as fixing the face holonomy.Thisis
performed by changing the support of the integral (3.11) through action of a delta function,
which we call parameter evaluation. In the following sections we apply these procedures to
identify the maximal divergence degrees in the different settings of free and perturbed group
field theory.
4. Maximal divergent contributions: a general scheme
Diagrams Grepresenting contributions to the expectation values of Z(N)
A/0can have various
shapes, but only the maximal divergent contributions are relevant in the limit of high bond
dimensions D. Bounds of the divergence degree are given by the number of faces of each
diagram (3.11), which motivates us to study the maximum number of faces of diagrams con-
tributing to the expectation of Z(N)
0and Z(N)
A. In the free theory, we will find the maximal face
numbers in a class of diagrams that we call locally averaged diagrams and restrict our search
for the maximal divergence degree afterward to this class.
4.1. Local processes by independent node averaging
Let us first consider the case of a small number of interactions in a diagram G, corresponding
to a term in a small order of λin the perturbative expansion (2.22) of the expectation value
E[Z(N)
0/A]. In this case, the stranded structure sketched in figure 4is dominated by the contrac-
tion scheme determined by the network topology. In particular, the number of edges carrying
gauge parameters integrated in the amplitude (3.11) is proportional to the number of network
nodes. This allows for an estimation of the number of parameter evaluations needed to fix all
face holonomies, suggesting a correlation between the number of faces, which we will denote
by
Ω[G], and the divergence degree Ω[G]. We shall use this intuition to single out a class of
diagram, where the divergence Ωis expected to be maximal.
Due to their different structure at the boundary ∂Γ(figure 4), we distinguish between the
variables Z(N)
0and Z(N)
Aand proof optimality statements of different generality. A central aspect
is the notation of locality, which in our context is understood as propagation processes hap-
pening just between field copies associated with the same network node v. Such propagation
processes are indexed by the permutation group SNas introduced in (2.12), where the identity
Iand the cyclic permutation Fare of particular interest.
Theorem 2. Let us assume a network graph Γ=(V,E∪∂Γ), such that every node is path
connected to boundary links. The only diagram contributing to E[Z(N)
0]in the free sector, which
maximizes the face number
Ω, is given by local propagation with the identity Iat each node.
Proof. Let us treat the open links of the network as incident to single-valent boundary nodes
with fixed local propagation determined by the boundary conditions, thus in the case of Z(N)
0
by the symbol I. Each face of a diagram in the free theory includes at least one contraction
17
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
Figure 6. Sketch of the induction argument, which is exploited in the proof of theorem
2:ifinthekth network copy a local propagation Iis happening at a neighboring node u,
the same process must happen at vfor the blue strand to close directly.
of the incoming fields, since each outgoing field propagates to an incoming field. If thus each
face includes just one incoming contraction, as it is the case for the pattern {I}, the divergence
degree is maximal.
We prove the uniqueness of this maximal divergent case by induction through the network
starting from its boundary with fixed propagation I. Let us assume the fields of a neighbor v
to the node uare propagating locally with the symbol I, as sketched in figure 6.Forthekth
copy of the link ebetween vand uto be the single incoming contraction of the associated
face, the kth copy of the incoming and outgoing field of the node umust propagate into each
other, as sketched dashed in figure 6. Applying this argument to all Ncopies of the link, the
field copies of the node uhave to propagate locally with the symbol I, such that the link to v
contributes maximally to the face number. Since the network is path connected to the boundary,
the induction reaches all nodes.
In contrast to the homogeneousboundary situation of the variable Z(N)
0, the boundary regions
Aand Bare differently treated in the variable Z(N)
A. By assuming the boundary regions to be
connected by the bulk network, we will always find closed strand including more than one
contractionof incoming fields. To proof a similar result to theorem2, in this boundary situation,
we need to make further assumptions on the network graph.
Theorem 3. Let us assume a tree graph Γ=(V,E∪∂Γ)and a partition A ∪B=∂Γ,such
that we find a link e ∈E separating regions connected to A and B. Then all diagrams con-
tributing to E[Z(N)
A]in the free sector, which maximize the face number
Ω, are included in the
local averages.
Proof. After omission of the link ewe have two trees, the first connected just to the region
Aand the second to B. By the minimal path length kto the root node, taken to be the node
incident to e, we classify each node of both trees into a layer. Assuming a diagram maximizing
Ωwe will now proof in both trees the local propagation by induction from the deepest layer to
the layers with smaller indices. As in the proof of theorem 2, the induction starts with boundary
links, which are treated as additional network nodes with fixed propagation by Iin the tree to
Aand Fto B.
Let then vbe a node in the kth layer, which has two children u1and u2in the (k+1)th layer
with the same local propagation happening w.l.o.g. by I(figure 7(b)). Let us further assume,
that an incoming copy of the node field is involved in a nonlocal propagation, as sketched
in blue. We find a nonlocal propagating outgoing copy to the nonlocal propagating incoming
18
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
Figure 7. (a) Network with a tree structure, such that the subtrees to HAand HBare
connected by a single link. (b) Node vwith two neighbors u1and u2, which have fixed
local propagation by I. The blue and black dashed lines compare a nonlocal propagation
with a modification to a local propagation.
copy, such that the strands associated with the links from vto u1and u2connect both nonlocal
propagatingcopies of the node field v. This can be done simply by followingtwo strands, which
need to close in each situation. We now modify the process by a local propagation between the
identified pair of nonlocal propagating incoming and outgoing fields, sketched in figure 7(b) by
dark dashed lines. Due to the special choice of the nonlocal propagating pair at the node v,the
number of strands with the first two indices increases by two. This compensates the maximal
decrease of one in the number of closed strands associated with the third index. The modified
diagram has thus a higher number of faces, thus the assumption of a nonlocal propagation
contradicts the assumption of maximal face number
Ω.
Induction with decreasing layer number kreaches both subtrees of the network, thus at each
node a local process has to happen to maximize
Ω.
The statements concerning the unique maximum of the face number in the free theory moti-
vate the search for maximal divergent diagrams in the class of local processes. This class
corresponds to the statistics of independent node averaging, where we separately average the
density (ρ(v))⊗dat each network node v∈V:
E[Z(N)
0/A]≈TrE∪∂Γ
e∈E∪∂Γ
(ρ(e))⊗N
v∈V
E[(ρ(v))⊗N]
w∈B
I
w∈A
I/F(4.1)
This finite but by expectation dominant sector of possible diagrams corresponds to a differ-
ent interpretation of the variables (3.10) as composed of independent but identically distributed
random fields φ(v). The corresponding Feynman diagrams decompose into different subdia-
grams to each node vof the network, which are contracted by the network pattern sketched in
figure 4. At each node, we index the diagrams by an element πof the permutation group SN:
E[(ρ(v))⊗N]=
π∈SN
P(π)+O(λ) (4.2)
The sum over the permutation group SNcaptures all processes of the corresponding free theory
(λ=0), where we define for each permutation πan operator P(π) modeling the propagation
(3.8)ofthekth incoming node copy to the π(k)th outgoing. Perturbation of the free the-
ory would give rise to diagrams with interactions, which are controlled by the perturbation
parameter λ.
19
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
4.2. Maximal divergent diagrams for locally averaged networks
Insights into the structure of dominant patterns are gained in the sector of local node averages,
where we first restrict to the case λ=0, therefore to the sum term in (4.2). Each diagram Gcon-
tributing to one of the averages E[Z(N)
A/0] corresponds in this sector to a choice of a permutation
πv∈S
Nat each network node v∈V.
For such permutation pattern we can categorize each face of a diagram by the unique net-
work link e∈E∪∂Γthey include, which turns the number of faces
Ω[G] into a sum of all link
contributions. Each closed strand to a link e=(e1,e2) corresponds to a cycle in the permuta-
tion π−1
e1◦πe2∈S
N, where the difference of the cycle number χ(π−1
1◦π2) to the maximum N
defines a metric d(π1,π2) in the permutation group SN. The face number for a local permutation
pattern {πv}as a sum of the cycle numbers along each link is thus
Ω[{πv}]=
(e1,e2)∈E∪∂Γ
χ(π−1
e1◦πe2)=
(e1,e2)∈E∪∂Γ
(N−d(πe1,πe2)) (4.3)
Within our aim to identify permutation pattern maximizing the divergence degree Ω,we
first identify those maximizing the face number
Ω. While for Z(N)
0we have already found the
single maximum by {I}with theorem 2, we will now build on theorem 3to find the maximal
face number in the situation Z(N)
A.
Theorem 4. Let us assume a connected network graph Γ=(V,E∪∂Γ)and a partition
A∪B=∂Γ.Letσmin ⊂E be a link set with minimal cardinality, such that by omission of σmin
the graph Γreduces to two connected components, the first including the boundary A and the
second B (see figure 8). It then holds:
(a)A pattern maximizing the number
Ω[{πv}]of faces in the situation of Z(N)
Ais the associ-
ation of Iand Fto the components connected to A and B after omission of the links σmin
(figure 9(b)).
(b)If σmin is unique, the maximum of
Ω[{πv}]is also unique.
Proof. By a corollary of the maximal-flow-minimal-cut theorem [53,54], we find a number
of |σmin|disjoint paths Pkthrough the network starting with a link in Aand ending in B[13]
(figure 8). Taking just the links included in the paths into account we estimate with the triangle
inequality of the metric d:
Ω[{πv}]−N|E∪∂Γ|⩽−
|σmin|
k=1
(e1,e2)∈Pk
d(πe1,πe2) (4.4)
⩽−|σmin|(N−1) (4.5)
Associating the trivial permutation Iwith the nodes connected to Aafter omission of σmin and
Fto the other nodes results in a number N|E∪∂Γ|−|σmin|(N−1) of faces and is with (4.5)
maximal, which shows (a).
Let us assume a different pattern {πv}maximizing
Ω. If it contains just the symbols Iand
F, both regions would be separated by a different minimal set σmin,if(4.4)and(4.5) hold
straight. If other symbols are included, we set all further symbols to I, which does not change
Ω, if both inequalities hold straight. The resulting pattern thus also maximizes
Ω,wherethe
regions are separated by a minimal set σmin. Setting the symbols instead to Fwould result in
another minimal set σmin.Ifσmin is unique, the pattern (a) maximizing
Ωis also unique.
20
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
Figure 8. Disjoint paths through a network graph Γ, such that they start in a region A
and end in a region Bof open links. As a corollary of the maximal-flow-minimal-cut
theorem [53,54] the maximal number of such paths is |σmin|,whereσmin is a minimal
set of links separating Γaccording to the chosen partition of open links.
Figure 9. Permutation pattern with maximal face number
Ω(theorem 4) and divergence
degree (theorem 5). (a) In the boundary situation of Z(N)
0, the maxima are achieved by
association of the trivial permutation Iand (b) in the situation of Z(N)
Aby association of
Iand Fto regions separated by σmin.
The divergencedegree Ω[G] of the amplitude (3.11) is however smaller than the face number
Ωdue to the evaluation of the gauge integrals with parameters heassociated with each prop-
agation. However, we can use certain bounds oriented on the face number
Ωto find maximal
divergent patterns. We proceed by first introducing the network property of reducibility to a
tree via coarse-graining, which will then be exploited to prove the uniqueness of the divergence
degree maxima.
Definition 5. Let Γ=(V,E) be a graph with a disjoint partition mVm=Vof its nodes into
regions m. The corresponding coarse-grained graph Γ{Vm}(figure 10) consists of the regions
mas nodes and a number Em˜mof links between regions mand ˜mgiven by:
Em˜m=#{(e1,e2)∈E|e1∈Vm,e2∈V˜m}(4.6)
21
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
Figure 10. Coarse-graining of a permutation pattern by grouping neighbored nodes with
same permutation symbol to one region (a). A minimal graph Tconnects all nodes to
the boundary regions and reduces to T{Vm}in the coarse-grained graph Γ{Vm},sketched
in (b).
Γis called coarse-grainable to a tree, if there exists a disjoint partition mVm=Vof its
nodes such that for all mit holds Vm=Vand the corresponding coarse-grained graph Γ{Vm}is
at most minimal connected.
Network graphs Γ, however, have open links, which we will close by adding single-valent
nodes on their ends. The definition of coarse-graining is thus extended to open graphs by
partition of the virtual boundary nodes ∂Γtogether with the nodes V.
Theorem 5. With the same conditions on the network graph Γas in theorem 4it holds:
(a)A pattern maximizing the divergence degree Ω[{πv}]in the boundary situation of Z(N)
0is
the association of Ito all nodes (figure 9(a)).
(b)A pattern maximizing the divergence degree Ω[{πv}]in the boundary situation of Z(N)
Ais
the association of Iand Fto the separated regions connected to A and B after omission
of the links σmin (figure 9(b)).
(c)The number of maximal divergent pattern is the same for both boundary situations Z(N)
0
and Z(N)
A.IfΓis not coarse-grainable to a tree, the maxima (a)and (b)are unique.
Proof. Let us find a minimal subgraph Tof the network graph Γ, which includes all nodes
Vand to each node a path to a boundary node. Thus Tis a forest with leaves in the boundary
and one can iteratively perform the integrals associated with its links in the amplitude (3.11).
Starting from the links in the boundary Aand Bwe find to each link t∈Ta different incident
node vt, which carries gauge parameters hti. By integration with respect to the gauge parameter
of this node, the delta functions associated with each face fcategorized by a link in Tvanishes.
We denote the stranded subdiagram of G, which is spanned by faces to the subgraph T,byT
and by G/Tthe diagram after omission of the faces to t∈Tas well as the evaluated gauge
parameters hti. It then holds:
22
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
l∈G
dhl
f∈G
δ⎛
⎝
l∈∂f
hl⎞
⎠=⎡
⎣
l∈G/T
dhl⎤
⎦
t∈T
i
dhti
f∈G/T
δ⎛
⎝
l∈∂f
hl⎞
⎠
f∈T
δ⎛
⎝
l∈∂f
hl⎞
⎠
(4.7)
=⎡
⎣
l∈G/T
dhl⎤
⎦
f∈G/T
δ⎛
⎝
l∈∂f
hl⎞
⎠(4.8)
After the integration procedure along the minimal subgraph Twe are left with a subdiagram
G/T, which has a reduced face number
Ω[G/T]. We use this face number as an upper bound
of the divergence degree Ω[G]:
Ω[G]⩽
Ω[G/T]=
Ω[G]−
t=(t1,t2)∈T
(N−d(πt1,πt2)) (4.9)
We rewrite (4.9) by grouping neighbored vertices with the same permutation symbol to regions
Vm. This coarse-graining procedure results in a graph Γ{Vm}(figure 10), where regions m
denote vertices and Em˜mnotes the number of links between two regions. The minimal sub-
graph Tcoarse-grains to T{Vm}by omitting links between nodes of the same region. With
d(ππv1,πv2)=0 between vertices in the same regions, we get:
Ω[G]⩽
Ω[G]−
t∈T
(N−d(πt1,πt2)) −N|V|+
(m,˜m)∈T{Vm}
d(πm,π˜m) (4.10)
=N[|E∪∂Γ|−|V|]−
(m,˜m)∈Γ{Vm}
Em˜md(πm,π˜m)+
(m,˜m)∈T{Vm}
d(πm,π˜m) (4.11)
While the maximal values of the second term in (4.11) have been identified in theorem 4,
we will optimize it here in combination with the third term in order to get a maximal upper
bound of
Ωfor a fixed network graph Γ. Since the second sum is always bigger than the third,
both terms taken together are smaller or equal to zero. In the boundary situation of Z(N)
0,this
maximal bound is saturated for the pattern {πv}={I}with a divergence degree Ω[{I}]=
N[|E∪∂Γ|−|V|]. Further maximal divergent processes are possible only if the second and
third term vanish together and the inequality (4.11) remains straight. But this would imply a
network graph which is coarse-grainable to a tree. The discussion of such graphs is the subject
of section 5.3.
The different situation of Z(N)
Aresults in a sharper bound for the face number
Ω. Under
the assumption of a unique minimal surface σmin, the second term is with theorem 4at most
|σmin|(1 −N), which is reached in a situation of vanishing T{Vm}.IfT{Vm}does not vanish, the
decrease of the second term needs to be compensated by the third term, where we observe a
direct correspondence to other maximal cases in the situation of Z(N)
0in section 5.3. In both
boundary situations, there is thus the same number of cdifferent maximal divergent pattern.
We note that unlike the discussed maxima of the face number
Ω, the maximum of the diver-
gence degree Ωis not unique and nontrivial examples are given in section 5.3.However,we
were able to show that the number cof maxima is the same for both boundary conditions if one
assumes a unique minimal surface σmin. The multitude of the maxima has thus no influence on
the entanglement entropy (3.9), which depends just on the quotient of the variables.
23
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
4.3. Ryu–Takayanagi formula in the free theory
In the limit of high bond dimensions D(Λ)=δ(e) of the leg space H, the most divergent contri-
butions determine the behavior of the fraction between the expectations of Z(N)
Aand Z(N)
0,which
have been determined in the previous sections. In the free theory the asymptotic behavior of
the Rényi entanglement entropy (3.1)isgivenby[35]:
E[SN(A)] ≈1
N−1ln E[Z(N)
A]
E[Z(N)
0]≈1
N−1ln c·δ(e)N[|E∪∂Γ|−|V|]−|σmin|(N−1)
c·δ(e)N[|E∪∂Γ|−|V|]
=|σmin|ln[δ(e)] +O1
δ(e)(4.12)
Since no dependence on the parameter Nindicating the order of the Rényi entropy is given in
the limit δ(e)1, we directly apply the replica trick to recover the von-Neumann entangle-
ment entropy S(A)=limN→1SN(A). Equation (4.12) is thus the entanglement entropy within
the approximation given by local averaging in the free theory.
The proportionality of the entropy to the cardinality of the minimal domain wall σmin has a
clear geometric interpretation, in the sense of discrete geometry, in the context of group field
theory. The graph Γis the dual of a 2D simplicial complex. Each node is dual to a triangle
and each link is dual to an edge of this complex, and the group field theory model endows the
simplicial complex with dynamical geometric data. The length of each edge j, in any given
eigenstate of the length operator, is a function15 of the irreducible representation jeassociated
to it, and to the dual link. If the quantum state is still defined on a fixed graph, but it is not
an eigenstate of the length operator, then one has to average over the possible assignments
of irreps je, with weights depending on the chosen state (i.e. on its decomposition into length
eigenstates). We have
Length(σmin)=
e∈σmin
e(je)=je|σmin|(4.13)
which can be interpreted as the length of a dual discrete minimal one-dimensional path.
Therefore we can write |σmin|=Length(σmin)/jeand in this sense our result constitutes
a Ryu–Takayanagi formula proposed in [6], if we consider the path integral averaging over
the open network Γas a simplified model of a bulk/boundary (spinfoam/network state) duality
[35]. There are two further points to notice. First, our chosen quantum state fixes the par-
allel transports associated with the dual links to the identity and it is therefore maximally
spread in the conjugate observables, which are in fact (including) the edge lengths associ-
ated to the same links. It is therefore a highly non-classical state to which it is not appropriate
to associate a semi-classical geometric interpretation. A more appropriate choice, to this end,
would be a coherent state peaking on both phase space variables on each link [55,56]or,
even better, a coherent state peaking on the collective variable Length(σmin) as a whole [57].
Second, a generic quantum state of the theory would also involve a superposition of combina-
torial structures, i.e. a superposition of states associated to different graphs. In particular, this
would be needed if the bulk is to admit a continuum geometric interpretation, going beyond
the geometry associated to a (fixed) simplicial complex, which amounts to a drastic truncation
of the allowed degrees of freedom of the fundamental (quantum gravity) model. In this case,
one would have to understand the quantity |σmin|itself as the result of an average over such
15 The exact form of the function depends on the quantization map chosen to define the quantum theory.
24
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
superposed graphs. Improving our derivation in both these directions would clearly be an
interesting, and potentially important, development.
5. Entropy corrections from group field interactions
We are now interested in the possible modifications of (4.12) induced by group field interaction
terms. These interaction processes correspond to further stranded diagrams which contribute
to the expectation value of Z(N)
A/0.
5.1. Interaction processes
In the free GFT calculations discussed above, the d-valence of the node tensors was arbitrary.
For the interacting case, we fix the valence to d=3 and we specify the interaction kernel to be
Vsym({g(1)
i}{g(2)
i}{g(1)
i}{g(2)
i})
=4
l=1
dhlδ(h1g(1)
1,h3g(1)
1)δ(h1g(1)
2,h4g(2)
2)δ(h1g(1)
3,h2g(2)
3)
×δ(h2g(2)
1,h4g(2)
1)δ(h2g(2)
2,h3g(1)
2)δ(h3g(1)
3,h4g(2)
3) (5.1)
Stranded Feynman graphs contributing to 2N-point functions are all possible combinations
of the building blocks sketched in figure 5. Allowing for just one interaction vertex results
already in a variety of possible combinations. If two fields of the interaction vertex propagate
into each other, that is combining them with a propagator, we have an effective propagator.
Since the divergence degree of the resulting diagram is the same as that of a propagator, the
process could be captured in a mass renormalization.If two pairs of interacting fields propagate
into each other, the process would disconnect and can be considered as a vacuum amplitude,
which also does not contribute to the divergence degree of a diagram.
In this sense, the only processes capable of extending the divergence degree consist in the
interaction of two incoming and two outgoing copies of network nodes. In the local averaging
sector of the network statistics discussed in this section, the 2N-point functions of interest are
the local averages E[(ρ(v))⊗N], thus all interacting fields are restricted to copies of the same
node v.
5.2. Maximal divergent diagrams in the linear perturbation order
In the following, we shall estimate the divergence degree Ωof diagrams with one interaction
process by determining the face number
Ω, with the same strategy already used in section 4.2
for the free case.
Theorem 6. A pattern with a single interaction happening between two incoming and two
outgoing fields of the same network node vleads at most to the same face number
Ωas in a
maximal case of the free theory. For the face number
Ωto reach the maximum, there must be a
pattern of the free theory maximizing
Ω, such that vis incident to a link in the domain wall σ.
Proof. Let v∈Vbe an arbitrary node in a network and its neighbors given by u1,u2,u3
connectedwith a link affecting the argumentof vwith the respective index (1, 2, 3).The by (5.1)
chosen interaction block Vconnects only strands of the same argument index i, which again
enables us to decompose the number of strands into separate link contributions, dependent on
the local processes {πi}at the neighboring nodes. Let us assume the fields participating in the
25
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
Figure 11. Stranded graph of a local pattern with an interaction happening at the net-
work node v. After relabeling of the network copies, the first two incoming copies of v
participate in the interaction and a further permutation ¯πensures the generality of the
process.
interaction process are of the first two copies, where a permutation operator to the symbol ¯π
acts on the outgoing fields (figure 11). It is sufficient to discuss just this case, since we can
renumber the copy index of the network, thus each ¯πrepresents N
2equivalent interaction
processes.
We assume now for a given ¯πa higher face number than for all reference free propagations
at vwith symbol π, including the choice π=¯π. The additional faces in the interacting case
must result from link contributions with field indices 2 and 3 since the link contribution with
index 1 is in both cases given by χ(¯π◦π−1
1).
For the link 3 to contribute not less in the interacting case than in the free case, ¯π◦π−1
3must
contain a cycle including positions 1 and 2, since only in that case the interaction of the index 3
does not decrease the number of cycles in ¯π◦π−1
3compared to the free case. Link 2 contributes
in the interacting case by one more than in the free case if and only if the additional action of
F2⊗IN−2increases the number of cycles, that is, if and only if in ¯π◦π−1
2the positions 1 and
2 are included in a cycle.
We consider now the choice π=(F2⊗IN−2)◦¯πfor the reference free propagation. Since
¯π◦π−1
2and ¯π◦π−1
3contain cycles connecting the first two positions, in π◦π−1
2and π◦π−1
2
the first two positions are in different cycles, thus it holds:
χ(π◦π−1
2)=χ(¯π◦π−1
2)+1, χ(π◦π−1
3)=χ(¯π◦π−1
3)+1, (5.2)
χ(π◦π−1
1)⩾χ(¯π◦π−1
1)−1 (5.3)
⇒
3
i=1
χ(π◦π−1
i)⩾
3
i=1
χ(¯π◦π−1
i)+1 (5.4)
The choice π=(F2⊗IN−2)◦¯πfor a free reference process results thus at least in the same
face number compared to ¯πin the interacting theory. There is thus no ¯π, such that the associated
interaction process leads to a higher face number
Ωcompared to all other free propagations.
Furthermore, there is no interaction pattern maximizing
Ωin case of π1=π2=π3,since
at most N−1 closed strands correspond to the index 3 in an interaction pattern, whereas a free
propagation with π=π1would result in Nclosed strands corresponding to each index. Hence,
if at van interaction process leads to a maximal face number
Ω, there must be a different
propagation process at two neighbors of v, thus vwould be incident to a domain wall σin a
reference pattern in the free theory.
26
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
Since for the boundary situation of Z(N)
0no domain walls appear in free permutation pattern
maximizing
Ω, local interaction can only lead to maximal face numbers in the situation Z(N)
A.
From the proof of theorem 6we identify 2 N
2possible interaction processes happening at one
of the 2|σmin|nodes incident to the minimal surface σmin, which maximize the face number
Ω. For the case of symmetric network states under study, these patterns will not result in an
increasing divergence degree compared to the free patterns, as we will show in the following.
Theorem 7. A pattern with a single interaction happening between two incoming and two
outgoing fields of the same network node vhas at most the same divergence degree Ωas in a
reference case of the free theory, where the interaction at vis replaced by a free propagation.
Proof. Let us again model the interaction process at vby ¯π, as sketched in figure 11.From
theorem 6we already know, that the face number
Ωof the diagram with an interaction does
not exceed the degree of a reference free propagation happening at v. In the free propagating
case there are two gauge parameters associated with the first two node copies, thus at most two
evaluations of delta functions can be induced by them.
Let us now assume, that the interaction process leads to a higher divergence degree than for
all replacements by free propagations, which implies that there can be at most one evaluation
of delta functions induced by the interaction block. Since all delta functions associated with
one network link can always be evaluated, the symbol ¯π−1◦π1would have a cycle contain-
ing the first two indices and the symbol ¯π−1◦π2would not. But then a replacement of ¯πby
(F2⊗IN−2)◦¯πwould increase the number of faces by two. If our assumption of maximal
divergence was correct, there have to be at least three evaluations in the modified interaction
process, such that the modified process does not exceed the divergence degree. But the mod-
ified interaction process cannot be maximal divergent with three evaluations since we find a
reference free propagation with at least the same face number and at most two evaluations.
Theorem 7enables us to follow the previous arguments in the optimization of the face
number
Ω, since the divergence degree of patterns in the free theory cannot be increased by
local modifications with interaction processes. We thus just consider the pattern in the free
theory, which maximizes the divergence degree Ω, and study the impact of the modification
by a local interaction process. Since only in this case the face number can stay constant, the
modification needs to take place at a node incident to a domain wall.
Theorem 8. Let Γ=(V,E∪∂Γ)be a connected network graph, which is not coarse-
grainable to a tree and which has a unique minimal set σmin ⊂E separating the boundary
regions A and B. Then, each diagram with a single local interaction process has a smaller
divergence degree compared to the unique maxima in the free theory.
Proof. In theorem 5we determined the unique pattern of the free theory maximizing the
divergence degree under the same assumptions as here. The dominant pattern to Z(N)
0does
not have a domain wall, thus with theorems 6and 7all patterns with a single local interaction
process have subleading divergence degree. Although the dominant pattern contributing to Z(N)
A
contains a domain wall σmin, an interaction happening at a node incident to σmin always results
in three evaluations induced by the interaction block and the diagram would be subleading.
Theorem 8therefore proves for a broad class of network graphs, that the local permutation
pattern of the free theory are maxima of the divergence degree, also if we include pattern
with—up to one—happening interaction process. For the simple case of a single O(λ)GFT
interaction term the free theory result
27
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
E[SN(A)] ≈|σmin|ln[δ(e)] +O1
δ(e)
is not modified. As we will discuss in the next section, there are, however, special network
architectures, which are coarse-grainable to a tree and allow for multiple maxima of Ω,even
if the minimal surface σmin is unique.
One main point to notice about the above result is the general link between the computed
entropy and the divergences of the quantum amplitudes of the group field theory model we
have been using. This is important from a conceptual standpoint, as we will discuss later on.
From a more technical perspective, it is also important because the perturbative divergences
of group field theory models are a well-explored subject [58,59], and in particular the diver-
gences of topological GFTs, as the Boulatov mode we have used, are well-understood [60–64].
They are associated to 3-cells of the 3D cellular complex identified by each GFT Feynman dia-
gram, or to the vertices of the dual 3D simplicial complex, and they have been fully classified.
Such generic divergences do not appear in our calculation, however, because Feynman dia-
grams at order λonly involve a single interaction vertex and no bubble. In fact, there is no real
3D dynamics at such linear order, even in the sense of topological gravity, thus any physical
interpretation of our present result should be attempted with caution. Nevertheless, generic
divergences will become crucial when going beyond this crude approximation, and the tech-
nical tools to do so are already available. Concerning the same perturbative approximation of
the underlying GFT model that we have relied on (and that the whole spin foam literature, for
example, relies on too), another important cautionary remark should be made. Working with
the perturbative approximation is tantamount to assuming that the quantity we are trying to
evaluate is analytic in the GFT coupling constant, and thus well approximated already at low
orders. From the point of view of spacetime structures, this means assuming that their role is
well approximated by simple cellular complex (made of few vertices, edges, faces etc), before
any coarse-graining of the same complexes, and of the associated quantum amplitudes.This
may not be the case. In fact, it is not expected to be the case in much GFT (and tensor models)
literature, where instead the goal is to extract emergent continuum gravitational (thus geomet-
ric) physics from the collective behavior of the underlying microscopic degrees of freedom
[65–69]. If the relation between the Rényi entropy of our states and the divergences of the
underlying GFT is generic, as we expect, the former is probably not analytic in the coupling
constant, and its value will be dictated by the most divergent contributions to the Feynman
expansion of the GFT model, which are obviously growing with the number of interaction ver-
tices involved, thus it will be ultimately dominated by higher powers of λ. This is one more
reason to go beyond the approximation adopted in the present work.
5.3. Networks coarse-grainable to a tree
We recall the upper bound (4.11) of the divergence degree in case of free local propagations,
where neighboring nodes with same propagation pattern were coarse-grained to regions m.It
has been shown in sections 4.2 and 5.2, that up to the linear order of the interaction a pattern
is maximal divergent if and only if this upper bound is maximal and straight. A maximum of
the upper bound is attained for the pattern also maximizing the face number
Ω, which consists
of a single region with Iin the boudary case of Z(N)
0and an additional region with Fin case of
Z(N)
A. Derivations from this pattern result in the same upper bound (4.11), only if the following
stays constant:
−
m˜m
Em˜md(πm,π˜m)+
m˜m∈Tc
d(πm,π˜m) (5.5)
28
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
Figure 12. Network with a tree structure after a coarse-graining procedure. The special
topology enables maximal divergent patterns with arbitrary symbol πm.
In case of multiple minimal surfaces σmin this bound stays constant for the pattern discussed in
appendix,whereTc, which is the minimal subgraph along which all associated gauge freedoms
are evaluated, increases.
If the minimal surface σmin is unique, (5.5) can just attain further maxima in both boundary
situations, if the decrease of the first term is compensated by the increase of the second. How-
ever, since Em˜m⩾1for(m,˜m)∈Tc, this directly implies a tree structure of the coarse-grained
graph. As sketched in figure 12, we thus find further maximal upper bounds (5.5)incaseofa
network graph, which is coarse-grainable to a tree. The number of maxima of the bound (5.5)
is furthermore equal in the boundary situation of Z(N)
0and Z(N)
A. Since in each case all delta
functions associated with the links between different regions will drop out in the amplitude
(3.11) by parameter evaluation, the choice of the permutation symbol at each region does not
modify Ω. The less amount of faces in this case is compensated by a remaining freedom of the
parameter hmin each region of the coarse-grained tree structure.
We can exploit the independence of the divergence degree Ωon the links connecting the
coarse-grained tree to construct maximal divergent pattern with a local interaction. Let us there-
fore assume the region with symbol π1consisting just of one network node u. Since in this case
all links incident to udo not contribute to the divergence degree, any local process happening
at u, thus also interaction processes, result in a maximal divergent diagram.
6. Discussion
6.1. Entanglement and geometry
Symmetric group field networks carrying discrete geometries in their spin-network decompo-
sition were taken as a state class of central interest. Along with the pre-geometric interpretation
of the fundamental quanta of a group field theory [70], we understand such network states as
a collection of abstract quanta of space connected by entanglement patterns partially reflect-
ing symmetry and topology of a quantum discrete geometries [71–73]. In this picture, the
quantum-many-body approach to quantum geometry has allowed to import new quantitative
tools to investigate the behavior of the quantum geometry states in non-perturbative quantum
gravity [34]. As an explicit example in this sense, the Ryu–Takayanagi formula, generally
intended as a proportionality between entanglement entropies of boundary states to the surface
of minimal areas in a dual bulk state [74], can be used as a guiding principle to select states
with an interesting semiclassical limit [75].
The kinematic states associates to quantum geometries in the background independent
approach to quantum gravity [16] consist of spin-network states with the structure of a tensor
network. Tensorial entanglement and building of discrete geometries by network states have
29
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
a natural correspondence, made precise in [35]. It is however not clear from first principles if
the geometric interpretation of holographic network models in terms of the Ryu–Takayanagi
formula is reflected by such discrete geometries. A connection between spin-networks and
holography is established in [34], where network states representing the holographic duality
are derived from a coarse-grainingof spin-network states. Spin-networks are thereby represent-
ing bulk degrees of freedom, where the coarse-grained tensor network represents the boundary
state. Also within this setup, the Ryu–Takayanagi proposal holds in the semiclassical regime
provided by a limit of increasing scale in the coarse-graining procedure [34].
A crucial ingredient for establishing the holographic properties of such network states in
quantum gravity states may be found in the random character of the tensor networks, as sug-
gested in [13]. The group-field theoretic derivation for the typical entanglement entropy for
a generalized open spin-network, as given in [35] points in this direction. The underlying
assumptions of a free theory and independent averaging at each network node capture just
a small corner of possible amplitudes, nevertheless we find further evidence for the dominance
of this corner in this work.
We were able to prove the validity of the independence assumption within a class of cases
in section 4.1. However, the central new aspect faced in this work consists in the computa-
tion of the correction terms to the entanglement entropy possibly induced by deviations from
the Gaussian random tensor description, introduced by considering an interacting group field
theory, hence a polynomially perturbed Gaussian measure for the tensor fields. Building on
theorem 7and two assumptions on the network graph, which is not coarse-grainable to a tree
and possesses a unique minimal surface, the main result of the work consists in the proof,
in theorem 8, that the linear order correction of the perturbation series produces no leading
amplitudes in the average of Z(N)
0and Z(N)
A, thus the Ryu–Takayanagi formula is not modified.
The arguments used are strongly based on the dominance of the stranded Feynman dia-
gram by the contraction structure (figure 4), which is fixed by the network topology and the
boundary conditions of the variables. Already at the level of the free theory, it would be inter-
esting to see what changes if the kinetic term is modified from a simple delta functions to a
more general propagator. For higher order perturbations, on the other hands, the diagrams will
be dominated by the bulk structure, given by interaction vertices, and we expect significant
changes in the leading order amplitudes. To calculate the impact of these sectors, one needs
to control the divergent degrees of such diagrams, similarly to what has been done in previous
renormalization studies [61].
One interesting general feature of our results is the relation we found between the Rényi
entropy and the (perturbative) divergences of the GFT model within which this is computed.
This implies a direct link between continuum physics and geometry, to the extent to which it
is captured by the Rényi entropy and the Ryu–Takayanagi formula, and the renormalization
group flow of the fundamental quantum gravity model, thus the collective, many-body physics
of its basic entities. Beyond the technical points we have already discussed, this relation is
evocative of (and clearly consistent with) the general perspective that sees continuum spacetime
and geometry as emergent from the collective behavior of the fundamental quantum gravity
degrees of freedom encoded in these models. This perspective, indeed, motivates a large part of
the literature and in particular the one concerned with GFT renormalization (both perturbative
and non-perturbative). Such overall coherence between the specific GFT realization of the
emergent spacetime (and geometry) perspective and the ideas inspiring the ‘geometry from
entanglement’scenario, within which the Ryu–Takayanagi result plays such an important role,
is very remarkable, albeit still tentative. We take it as a further motivation to proceed along the
research direction followed in the present work.
30
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
One more important aspect concerning the relation between the entanglement structure of
the GFT quantum states and the geometry of the corresponding emergent geometry is the
role of curvature in both the result and in the specific model used to obtain it. For the spe-
cific GFT model considered in the work, the computation of the entropy gets mapped to
the evaluation of a topological BF field theory, which does not generate curvature. A dif-
ferent choice of the model (group domain, tensor rank and kernels) can do so, and the role
of the generated curvature constraints on the entanglement structure of the network can be
studied. This very interesting aspect has been left for future work. We only note here that a
Ryu–Takayanagi-like formula, following our procedure, will most likely follow also in that
case, since it appears to be the result mostly of combinatorial aspects of the formalism, rather
than details of the probability measure; some of the precise details of the formula, however, like
proportionalitycoefficient and correction terms, will most likely be modified by the presence of
curvature.
Looking at the entropy calculation from the point of view of the GFT Feynman diagrams
and their associated dynamical amplitudes raises one further issue, that is whether one can also
identify a contribution to the final result that can be understood as due to bulk entanglement
entropy. For example, such contribution is found when considering 1-loop corrections for the
standard Ryu–Takyanagi formula in [76]. The question on the presence of a bulk entanglement
entropy is a very interesting one. In fact, the very original motivation in considering an interact-
ing GFT model was based on the idea that interaction kernels (at the various order in the field
theory perturbative expansion) could be considered as dynamically induced extra ‘bulk’ nodes,
which would necessarily increase the degree of entanglement of the state by increasing the
connectivity of the graph. In the present work we have been only partially able to explore this
perspective. On the one hand, the introduction of a non-vanishing interaction in the GFT model
leads to a departure from the main features of the standard random tensor network framework
(i.i.d distribution for the tensors, etc) and leads to corrections in the final entropy formula; these
corrections are coming indeed from the ‘bulk contributions’. On the other hand, the dynamical
‘bulk’ corrections we have computed are of a very limited and simple type: they do not amount
to full 1-loop corrections, from the point of view of the underlying GFT quantum dynamics,
nor they account for true bulk degrees of freedom. To account for them, one would need to
consider Feynman diagrams of order 2, at least, and such that at least one ‘face’ of the GFT
2-complex, corresponding to a Feynman diagram, is fully internal, i.e. not in touch with the
boundary and thus with its associated algebraic variables summed over in the evaluation of the
amplitude.
A closer connection between our GFT framework for the Ryu–Takanayagi entropy formula
and the AdS/CFT one in which the same formula has been explored so far would require to
explore all the previous points in greater detail. The AdS/CFT literature would also suggest
that strongly interacting GFT models should be found to be ‘more holographic’ than weakly
interacting ones. Indeed, we would expect this to be true even from the pure GFT perspective,
simply because stronger interactions can be expected to a higher degree of entanglement in
the quantum states satisfying the quantum dynamics, which should then translate into a more
pronounced holographic behavior. Having stated this general expectation, it is not easy to make
more conclusive statements or explicit calculations to confirm it, at present. One reason is
simply that less is understood about strongly interacting GFTs (and tensor models) that about
weakly interacting ones. This is true in general, and in particular about the relevant quantum
states and transition amplitudes, which may depart, in the strongly coupled regime, from the
weakly coupled ones we put in correspondence with standard tensor network structures in this
work. Another is that also the discrete geometric interpretation in terms of lattice geometries,
that we have associated to the same structures (spin networks and their GFT generalization,
31
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
spin foam amplitudes, etc), is solid and well understood in the regime that, from a GFT point of
view, corresponds to weak coupling. Also, in this regard, then, more work and deeper analyses
are needed.
6.2. Modification of the holographic entanglement scaling
Finally, an interesting remark concerns the role of symmetry in this result. Along with the result
obtained in [35], throughout the work we have treated the network states and their entangle-
ment variables on the base of a probabilistic distribution on group fields satisfying a symmetry
constraint. Although the symmetry constraint is important for a reformulation of group field
networks in terms of spin-network states [35], dropping it offers an interesting perspective on
their statistics. By considering analogous probability measures on more general group fields,
that do not satisfy the closure constraint discussed in section 2.4 and therefore defining an
action by a propagation and interaction kernel by dropping the gauge integrations in (3.8)and
(5.1) one finds:
Knon-sym(g1,...,gd,g1,...,gd)=
d
i=1
δ(gi◦g−1
i) (6.1)
Vnon-sym({g(1)
i}{g(2)
i}{g(1)
i}{g(2)
i})=δ(g(1)
1,g(1)
1)δ(g(1)
2,g(2)
2)δ(g(1)
3,g(2)
3)
×δ(g(2)
1,g(2)
1)δ(g(2)
2,g(1)
2)δ(g(1)
3,g(2)
3) (6.2)
Both kernels thus remain the combinatorial structure with the only difference lying in the
missing group averaging with respect to gauge parameters h, which would enforce symmetry
properties. The expectation values Enon-sym[Z(N)
A/0], now averaging on general group fields, are
analogously expanded into stranded graphs as in figure 4. Without integration of the gauge
parameters, which amounts to already trivial face holonomies, the amplitudes of stranded
graphs Greduces to:
A[G]=
f∈G
δ(e)=δ(e)
Ω[G](6.3)
The divergent degree of any diagram equals the face number
Ω[G] and we can directly apply
theorems 2–4and 6to determine the asymptotic behavior of Enon-sym[Z(N)
A/0]. Assuming a net-
work graph Γ=(V,E∪∂Γ) with a disjoint partition A∪B=∂Γsuch that the minimal surface
σmin is unique and the graph cannot be coarse-grained to a tree, in the sector of local averaging
one finds
Enon-sym[Z(N)
A]=1+2λN
2|σmin|+O(λ)21+O1
δ(e)δ(e)|V|−(N−1)|σmin|(6.4)
Enon-sym[Z(N)
0]=δ(e)|V|(6.5)
The expected Nth Rényi entanglement entropy (3.9) is thus estimated as:
Enon-sym[SN(ρA)] ≈ln[δ(e)]|σmin|−ln 1+2λN
2|σmin|
N−1(6.6)
≈|σmin|[ln[δ(e)] −λN](6.7)
In the non-symmetric case the linear order correction modifies the asymptotical scaling of the
Rényi entanglement entropy with the area of a minimal surface and establishes therefore a
32
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
new Ryu–Takayanagi proportionality [6]. The von-Neumann entropy S(A)[77] corresponds
to the limit N→1 the Rényi entropy, where the proportionality gets in facts corrected by the
coupling constant λof the perturbed group field theory. The active role of the GFT dynamics
in the rescaling of the area proportionality factor is intriguing. A more refined analysis of
the modified holographic entropy scaling is left for further investigation. It must be stressed,
however, that models without the symmetry under diagonal action of the gauge group in the
fields (or the corresponding symmetry implemented at the level of the dynamical kernels) do
not allow for the standard interpretation in terms of dynamical quantum geometry for each
given lattice generated in the Feynman expansion, and for each quantum state of the theory.
This urges caution and some different procedure in interpreting the resulting entanglement
entropy, computed via random tensor networks techniques, in geometric terms.
Acknowledgments
The authors are grateful to the support of the Albert Einstein Institute, where the main part of
this work was realized. A Goeßmann also acknowledges the recent support from the MATH
+research center and the Fritz Haber Institute. M Zhang acknowledges the support from the
A von Humboldt Foundation.
Appendix A. Networks with multiple maxima of the face number in the free
theory
The unique pattern maximizing the face number
Ωin the boundary conditions of Z(N)
0is with
theorem 2the association of Ito each node. For the different boundary conditions of Z(N)
A
theorem 4identifies the unique maximum of
Ωwith additional assumption of a unique minimal
surfaces σmin separating the regions Aand B. Droppingthis assumption can give rise to different
maxima, if the inequalities (4.4)and(4.5) hold straight. All links between nodes with different
permutationsymbols have thereforeto be included in |σmin|disjoint paths Pkbetween boundary
Aand Band the triangle equation needs to hold straight for the permutations along each path.
This directly implies the separation of the network into regions of constant permutations by
different minimal surfaces σmin.
An example of a sequence of permutation symbols (πi)N−1
i=0⊂S
N, for which the inequality
(4.5) holds straight, is the combination of the cyclic element FN−i∈S
N−ipermuting the first
N−icopies with the trivial element I∈S
ifor the last icopies:
πi:=FN−iIi∈S
N(A.1)
Since d(πi,πj)=j−iholds for this sequence, we have for a collection of indices 0 =i1⩽
i2⩽...⩽il=N−1:
l
k=1
d(πik,πik+1)=(N−1) =d(F,I)(A.2)
With this the triangle equations (4.5) would hold straight for a path Pkalong nodes with per-
mutation symbols πik. A pattern with maximal face number
Ω, sketched in figure A1, is thus
given by from Ato Bnumerated regions separated by different choices of σmin, where ele-
ments of the sequence {πik}determine the local propagations. The number of such patterns
depends besides the number of different minimal surfaces also on their arrangement, since this
33
Class. Quantum Grav. 37 (2020) 095011 G Chirco et al
Figure A1. Coarse-grained network with multiple minimal surfaces σmin. If the regions
of a permutation pattern are separated by minimal surfaces and the triangle equation
holds straight for the associated permutation processes, the face number is maximal.
determines the connectivity of the resulting regions and thus influence the triangle equations
to be stated along the disjoint paths. However, a multitude of cdifferent maximal divergent
diagrams contributing to the expectation of Z(N)
Aresults in an offset of ln[c]
N−1in the entanglement
entropy (4.12). Since the offset is independent from the leg space dimension D, a multitude of
different maximal pattern does not influence the asymptotic entanglement entropy in the limit
of high D.
ORCID iDs
Goffredo Chirco https://orcid.org/0000-0001-9538-4956
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