Entanglement entropy of physical states in hypercuboidally truncated spin foam quantum gravity

Spin foam models (SFM) are certain proposals for the construction of transition amplitudes for states defined on graphs. A prime example are the spin foam models for quantum gravity, which have been developed as expressions for the physical inner product of spin network states in loop quantum gravity (LQG), but also topological BF theory, or even (pure) lattice gauge theory, can be formulated in terms of spin foam models [1–8].

SFM therefore deliver proposals for physical states in LQG, in that they can be used to define a rigging map, i.e. a bona fide projector from kinematical to physical states satisfying the constraints [9, 10].¹

As these SFM are defined on discrete structures, the question of the continuum limit naturally arises. This limit is captured in a refinement of both the bulk lattices (2-complex), as well as the boundary graphs, and leads to a notion of cylindrical consistency, allowing to construct the full continuum Hilbert space as an inductive limit over graphs [11–18]. This programme is a form of background-independent renormalisation, in which the coarseness of lattices plays the role of the scale, since in quantum gravity usual parameters such as e.g. lattice lengths are part of the dynamical fields themselves, which encode the geometry of space-time. In this framework there are several choices of renormalisation scheme, and the precise choice of boundary states, in particular ones stable under coarse graining, is an active field of research [19–22].

One of the most widely used models for the transition of LQG spin network states is the EPRL-FK model, which is defined on 2-complexes dual to 4d triangulations, and its KKL-extension to general 2-complexes, allowing the use of arbitrary polytopes [23–25]. It relies on a specific implementation of the so-called simnplicity constraints on topological SO(4)-BF theory, building on a classical equivalence of GR with BF theory, in which the bivector field B is constrained to be simple. This model has received much attention since its inception, although the question of its renormalisation is still very much open.

In [26], a specific truncation of the EPRL-FK-KKL model was introduced in order to construct a toy model, which serves as a laboratory for renormalisation². Additionally to Newton's constant G_N and the cosmological constant Λ, the model depends on the Immirzi parameter γ (usually kept fixed) and a crucial parameter α introduced in the face amplitude. The parameter α can, in the hypercuboidal setting, be associated with a factor of ${\sqrt{\mathrm{d}\mathrm{e}\mathrm{t}\;g}}^{\beta }$ for some β related to α, and has also been considered e.g. in the dynamical triangulation path integral approach [28].

In the hypercuboidal setting, G_N does not flow since there is no curvature, so Λ is fixed and α becomes the only interesting coupling. The model restricts the fluctuating geometries to specific (hyper-)cuboidal geometries [29, 30]. Interestingly, it was found that the RG flow already of this simple model is non-trivial, and induces a flow of the face amplitude, which governs the powers of volume factors in the path integral measure [31, 32]. Using frustal geometries, where also G_N and Λ begin to flow due to the excitation of curvature degrees of freedom, it was found that the UV-attractive fixed point is non-Gaussian (NGFP), in the sense that it lies at specific non-zero values of Newton's coupling and the cosmological constant [33, 34].

In the hypercuboidal setting, the UV-attractive NGFP separates two regions of phase space with vastly different geometric behaviour. Specifically, there are different (geometrically equivalent) states which receive different weights in terms of regular/irregular subdivision of polytopes. It is at the fixed point where these geometries are all treated equally, indicating a restoration of diffeomorphism symmetry. That this symmetry is broken in the EPRL-FK model has been known for some time [35]. In particular, it is broken in Regge calculus (RC), which arises in a certain limit of the EPRL-FK model, and while the symmetry is restored in classical RC even for flat configurations, it is broken even for those in the quantum theory due to the form of the path integral measure [26]. It was conjectured for some time that symmetries broken due to discretisation get restored at the coarse graining fixed point), and the properties of the NFGP are an indication for this mechanism in the 4d quantum gravity theory [12, 36].

Still, many features of the NGFP are yet to be understood³. To alleviate this somewhat, in this article we consider the entanglement entropy (EE) of physical states, at and away from the fixed point. EE is a very general concept, which is of great interest for general many-body systems [37]. For a physical system with local degrees of freedom, is measures the entanglement of degrees of freedom inside of a spatial region A with the ones outside of A. Here in particular the scaling behaviour of the EE is of interest: while generic states in the Hilbert space of a theory lead to an EE scaling with the region volume, many ground states for interesting physical Hamiltonian operators result in EE scaling with only the surface [37, 38]. It is this property which is used to identify and construct such states, for instance by a multiscale-entanglement renormalisation ansatz (MERA), or further developments building on this concept [39, 40].

Also in LQG the concept of entropy has been considered in relation to black holes [41], and in terms of EE of bipartitie systems for quite some time [42–48], in particular in view of isolated horizons. The spin network functions, which are defined on graphs thought of as embedded in (or building up) 3-dimensional space, have a geometrical interpretation which is ideally suited to discuss degrees of freedom associated to specific regions in space. However, for a single spin network, the only entropy between nodes is between Gauss-gauge degrees of freedom. If counting only gauge-invariant degrees of freedom, then the EE vanishes, and the state essentially factorises over the nodes of the graph (see section 3).

However, for physical states this picture changes. A physical state arises as the image of a kinematical one under the rigging map, and it can be represented as a superposition of different spin networks. The precise superposition depends on the parameters of the model, i.e. on the coupling constants.

In this article, we will compute the entanglement entropy ${S}_{\text{EE}}^{\left(\alpha \right)}$ for physical states in the hypercuboidal truncation of the EPRL-FK-KKL model. We will work in the large spin region of state space, in which the expressions for the path integral amplitude will become numerically manageable⁴. We will then investigate ${S}_{\text{EE}}^{\left(\alpha \right)}$ near the fixed point, and discuss its behaviour in relation to the restoration of diffeomorphism symmetry.

The plan of the article is as follows: in section 2 we recap the EPRL-FK-KKl model, as well as the hypercuboidal truncation. In section 2.6 we discuss the construction of physical states, using a dynamical embedding as rigging map. In section 3 we review the concept of entanglement entropy, and derive expressions for ${S}_{\text{EE}}^{\left(\alpha \right)}$ for different physical states, and in particular some scaling behaviour, which will help us to numerically compute the α-dependence numerically in section 4. Finally, we will interpret and discuss our findings in section 5.

A spin network function on a graph Γ is defined on an oriented graph Γ, and is labelled by a collection of spins ${k}_{\ell }\in \frac{1}{2}\mathbb{N}$ on the links ℓ of Γ, as well as invariant tensors (figures 1 and 2)

$$\begin{equation}{\iota }_{n}\;\in \;{\text{Inv}}_{SU\left(2\right)}\left[\underset{\left[n,\ell \right]=1}{\otimes }{V}_{{k}_{\ell }}\otimes \underset{\left[n,\ell \right]=-1}{\otimes }{V}_{{k}_{\ell }}^{{\dagger}}\right]\end{equation} \tag{ 2.1 }$$

along nodes n in Γ, where [n, ℓ] = ±1, depending on whether a link ℓ is outgoing/incoming to the node n. A widely-used overcomplete basis of the intertwiner spaces (2.1) for fixed spins is given by the Livine–Speziale-coherent intertwiners, which depend on 3d normal vectors ${\to {n}}_{\ell }$ satisfying the closure constraint

$$\begin{equation}{G}_{n}\;=\;\sum _{\left[n,\ell \right]=1}{k}_{\ell }{\to {n}}_{\ell }\;-\;\sum _{\left[n,\ell \right]=-1}{k}_{\ell }{\to {n}}_{\ell }\;=\;0.\end{equation} \tag{ 2.2 }$$

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With these, the Livine–Speziale coherent intertwiner is given by

$$\begin{equation}{\iota }_{\ell }\;=\;{\int }_{SU\left(2\right)}dg\;g{\unicode{9657}}\left[\underset{\left[n,\ell \right]=1}{\otimes }\vert {k}_{\ell },\;{\to {n}}_{\ell }\rangle \otimes \underset{\left[n,\ell \right]=-1}{\otimes }\langle {k}_{\ell },\;{\to {n}}_{\ell }\vert \right]\end{equation} \tag{ 2.3 }$$

where $\vert k,\to {n}\rangle ={g}_{\to {n}}\vert k,\;k\rangle$ is the Perelomov coherent state defined by the action of ${g}_{\to {n}}\in SU\left(2\right)$, the SU(2)-rotation which rotates ${\to {e}}_{z}$ into $\to {n}$, on the highest weight vector of the representation k.⁵

The spin foam state sum is defined on a 2-complex Δ, consisting of 2d faces, 1d edges, and 0d vertices. The 2-complex functions as a cobordism between two graphs (see figure 3), which arise on its boundary, as those edges and vertices which touch only one face and edge, respectively.

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Consider an oriented 2-complex Δ with a (not necessarily connected) boundary graph Γ. Then a state is an assignment of spins k_f to 2d faces f of Δ, and of intertwiners ι_e to edges e, from the tensor product of spins k_f on faces f meeting at e. The vertices and edges on the boundary form the nodes n and links ℓ of Γ, and touch exactly one face f_ℓ and edge e_n in the bulk respectively. Therefore, via ${k}_{\ell }\equiv {k}_{{\ell }_{f}}$ and ${\iota }_{n}\equiv {\iota }_{{n}_{e}}$ the state {k_f, ι_e} induces a spin network ${\psi }_{{\Gamma},\left\{{k}_{\ell }\right\},\left\{{\iota }_{n}\right\}}$ on the boundary. The spin foam amplitude assigned to the state {k_f, ι_e} is given by

$$\begin{equation}{\mathcal{Z}}_{{\Delta}}\left[{\psi }_{{\Gamma},\left\{{k}_{\ell }\right\},\left\{{\iota }_{n}\right\}}\right]\;=\;\sum _{{j}_{f},{\iota }_{e}}\prod _{f\subset {F}_{\text{bulk}}}{\mathcal{A}}_{f}\prod _{e\in {E}_{\text{bulk}}}{\mathcal{A}}_{e}\prod _{v\in {V}_{\text{bulk}}}{\mathcal{A}}_{v}\;\prod _{{f}_{\ell }\in {F}_{\text{bdy}}}{\mathcal{B}}_{{f}_{\ell }}\prod _{{e}_{n}\in {E}_{\text{bdy}}}{\mathcal{B}}_{{e}_{n}},\end{equation} \tag{ 2.4 }$$

where ${\mathcal{A}}_{f}$, ${\mathcal{A}}_{e}$ and ${\mathcal{A}}_{v}$ are, respectively, the face-, edge-, and vertex amplitude of the model, while the ${\mathcal{B}}_{{f}_{\ell }}$ and ${\mathcal{B}}_{{e}_{n}}$ are the boundary amplitudes, which are usually chosen in such a way that ${\mathcal{Z}}_{{\Delta}}$ behaves naturally under glueing [7].

The summation in (2.4) ranges only over spins and intertwiners in the bulk, while those on the boundary are being kept fixed, as they are determined by the boundary state.

In this article we work with the EPRL-FK-KKL model, which amounts to a specific choice for the amplitudes, described in detail in [26]. The model depends on the Barbero–Immirzi parameter γ, which in our case we allow to take values in γ ∈ (0, 1). To be precise, the sum in (2.4) is restricted to range over those k_f such that⁶

$$\begin{equation}{j}_{f}^{{\pm}}\;{:=}\;\frac{\vert 1{\pm}\gamma \vert }{2}{k}_{f}\;\in \;\frac{1}{2}\mathbb{N}.\end{equation} \tag{ 2.5 }$$

In what follows we are only interested in the amplitudes for a specific subset of states, which comprise the hypercuboidal truncation of the model.

2.1. Hypercuboidal truncation

The model truncated on hypercuboids is essentially a restriction to a specific set of allowed spins and intertwiners on a 2-complex Δ dual to the 2-skeleton of a 4-dimensional hypercubic lattice. The intertwiners in question are so-called quantum quboids, which in the large spin limit have the geometric interpretation of 3d cuboids with fixed areas (see figure 4). Each quantum cuboid is completely determined by three spins, and is given by

$$\begin{equation}{\iota }_{{k}_{1},{k}_{2},{k}_{3}}\;{:=}\;{\int }_{SU\left(2\right)}dg{\unicode{9657}}\left[\underset{i=1}{\overset{3}{\otimes }}\vert {k}_{i},{\to {e}}_{i}\rangle \otimes \langle {k}_{i},\;{\to {e}}_{i}\vert \right]\end{equation} \tag{ 2.6 }$$

where ${\to {e}}_{i}$ are the unit vectors pointing in the ith direction in ${\mathbb{R}}^{3}$. The quantum cuboids are only defined when faces on opposite sides have equal spin, which restricts the sum in (2.4) to a highly symmetric set, which has been described in detail in [26]. Still, there are quasi-local propagating degrees of freedom, which have, however, no interpretation as curvature. At each vertex there are 24 faces meeting, but due to the high amount of symmetry, quadruples of them have equal spin. As a consequence, a vertex amplitude ${\mathcal{A}}_{v}$ depends only on six spins k₁, ..., k₆ (see figure 5).

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A finite 2-complex of this form has eight boundaries (two in each major axis direction in ${\mathbb{R}}^{4}$), and a face f touching only one of those gets assigned the boundary amplitude

$$\begin{equation}{\mathcal{B}}_{f}\;=\;{\left({\mathcal{A}}_{f}\right)}^{\frac{1}{2}},\end{equation} \tag{ 2.7 }$$

while a face f touching two (i.e. on a 'corner' of the lattice) gets assigned

$$\begin{equation}{\mathcal{B}}_{f}\;=\;{\left({\mathcal{A}}_{f}\right)}^{\frac{1}{4}},\end{equation} \tag{ 2.8 }$$

since this is then only a quarter of a full rectangular face. The edges never end in corners, so we define for all boundary edges e that

$$\begin{equation}{\mathcal{B}}_{e}\;=\;{\left({\mathcal{A}}_{e}\right)}^{\frac{1}{2}},\end{equation} \tag{ 2.9 }$$

which is just the inverse of the norm of the boosted quantum cuboid intertwiner at that edge. Due to the high amount of symmetry, one can split up and rearrange all the amplitudes, associating them to vertices, writing

$$\begin{equation}\mathcal{Z}\;=\;\sum _{{k}_{f}}\prod _{v}{\hat{\mathcal{A}}}_{v}\end{equation} \tag{ 2.10 }$$

where the dressed vertex amplitude is given by

$$\begin{equation}{\hat{\mathcal{A}}}_{v}\;{:=}\;{\mathcal{A}}_{v}\prod _{e\supset v}{\left({\mathcal{A}}_{e}\right)}^{\frac{1}{2}}\prod _{f\supset v}{\left({\mathcal{A}}_{f}\right)}^{\frac{1}{4}}.\end{equation} \tag{ 2.11 }$$

This way, the boundary amplitudes are correctly taken care of.

2.2. The asymptotic expression of the amplitudes

In the following, we will give large spin asymptotic formulas for the hypercuboidal amplitudes. These have been computed and discussed in [26, 32].

In the large spin limit, the asymptotic expression of the amplitude is, as is typical for the Euclidean signature EPRL-amplitude, given by terms involving the Regge action of the polytope geometry given by the boundary state, and the Hessian determinant D of the quantum action. [52]. For hypercuboids, the Regge action vanishes, due to which the asymptotic amplitude becomes independent of Newton's constant, and can be written, up to a k_i-independent factor, as

$$\begin{equation}{\mathcal{A}}_{v}\left({k}_{1},\;\dots ,\;{k}_{6}\right)\;=\;{\left(\frac{1}{D}+\frac{1}{{D}^{{\ast}}}\right)}^{2},\end{equation} \tag{ 2.12 }$$

where D is the determinant of the 21 × 21 Hessian matrix. It depends on the six spins k_i via

$$\begin{align*}{D}^{2}\hfill & =2\left({k}_{1}^{2}\left({k}_{2}+{k}_{4}\right)+{k}_{2}{k}_{4}\left({k}_{2}+{k}_{4}\right)\right.\hfill \\ \hfill & \quad \left.+{k}_{1}\left({k}_{2}^{2}+\left(1+i\right){k}_{2}{k}_{4}+{k}_{4}^{2}\right)\right)\left({k}_{1}^{2}\left({k}_{3}+{k}_{5}\right)\right.\hfill \\ \hfill & \quad \left.+{k}_{3}{k}_{5}\left({k}_{3}+{k}_{5}\right)+{k}_{1}\left({k}_{3}^{2}+\left(1+i\right){k}_{3}{k}_{5}+{k}_{5}^{2}\right)\right)\left({k}_{3}{k}_{4}{k}_{5}\right.\hfill \\ \hfill & \quad \left.+{k}_{2}\left({k}_{4}{k}_{5}+{k}_{3}\left({k}_{4}+{k}_{5}\right)\right)\right)\left({k}_{2}^{2}\left({k}_{3}+{k}_{6}\right)+{k}_{3}{k}_{6}\left({k}_{3}+{k}_{6}\right)\right.\hfill \\ \hfill & \quad \left.+{k}_{2}\left({k}_{3}^{2}+\left(1+i\right){k}_{3}{k}_{6}+{k}_{6}^{2}\right)\right)\left({k}_{4}^{2}\left({k}_{5}+{k}_{6}\right)\right.\hfill \\ \hfill & \quad \left.+{k}_{5}{k}_{6}\left({k}_{5}+{k}_{6}\right)+{k}_{4}\left({k}_{5}^{2}+\left(1+i\right){k}_{5}{k}_{6}+{k}_{6}^{2}\right)\right)\left({k}_{3}{k}_{4}{k}_{6}\right.\hfill \\ \hfill & \quad \left.+{k}_{1}\left({k}_{4}{k}_{6}+{k}_{3}\left({k}_{4}+{k}_{6}\right)\right)\right)\left({k}_{2}{k}_{5}{k}_{6}+{k}_{1}\left({k}_{5}{k}_{6}+{k}_{2}\left({k}_{5}+{k}_{6}\right)\right)\right),\hfill \end{align*}$$

where the branch cut to define D is put on the negative real axis. The omitted prefactor in (2.12) contains all of the dependence of γ, and can be ignored in what follows. The edge amplitude is given by the inverse norm squared of the SU(2) × SU(2) intertwiner⁷ which can, up to an irrelevant factor, be written in the asymptotic limit as

$$\begin{equation}{\mathcal{A}}_{e}\left({k}_{1},{k}_{2}{k}_{3}\right)\;=\;\left({k}_{2}+{k}_{3}\right)\left({k}_{3}+{k}_{1}\right)\left({k}_{1}+{k}_{2}\right).\end{equation} \tag{ 2.13 }$$

The asymptotic face amplitude is given by

$$\begin{equation}{\mathcal{A}}_{f}\left(k\right)\;=\;{k}^{2\alpha },\end{equation} \tag{ 2.14 }$$

where α is a free parameter in the model, which has been introduced in [26]. This parameter will play a crucial role in the following investigations.

It should be noted that, in the large spin expressions, all explicit dependencies of the Barbero–Immirzi parameter γ go into irrelevant prefactors, as all of the individual amplitude functions are homogeneous functions of the j^± in (2.5) of various degrees.

2.3. The state sum with the asymptotic amplitude

2.4. Geometricity of the vertex amplitude

The set of spins k_f distributed among the faces of the lattice, which comply to the hypercuboidal symmetry, contains many elements with non-metric interpretation, in the sense that they do not allow for a reconstruction of the 4d metric from the spins. The presence of these configurations results from the insufficient implementation of the volume simplicity constraint on non-simplicial vertex amplitudes. These non-metric configurations also appear on more general vertices, and are generally characterised by the fact that, unlike twisted geometries [53–55], they are not suppressed in the large-spin regime. They feature face-non-matching, but 2d angle-matching, i.e. there is a conformal mismatch between touching faces, preventing glueing in 4d [26, 27, 56].

The conditions on the spins to remove these non-metric configurations can, for the hypercuboid, be formulated in terms of the Hopf link volume constraint in [56], which result, for each vertex v in the conditions

$$\begin{equation}{k}_{1}{k}_{6}\;=\;{k}_{2}{k}_{5}\;=\;{k}_{3}{k}_{4}.\end{equation} \tag{ 2.15 }$$

For large values of α the non-metric configurations appear to be dynamically suppressed [26], but in general one can demand their absence from the start. In what follows, we will consider both cases of present and absent non-metric degrees of freedom in the path integral.

2.5. Kinematical and dynamical embedding maps

A central part of background-independent renormalization is the relation of degrees of freedom on different graphs. This is connected to the 'rescaling' of degrees of freedom in the traditional context, and to the 'block spin transformations' in the lattice theory context.

In SFM this is encoded in the embedding map, which maps the Hilbert space of a coarse graph Γ to a refined graph Γ'.

$$\begin{equation}{\phi }_{{{\Gamma}}^{\prime }{\Gamma}}\;:\;{\mathcal{H}}_{{\Gamma}}\;\rightarrow \;{\mathcal{H}}_{{{\Gamma}}^{\prime }},\end{equation} \tag{ 2.16 }$$

where Γ can be either the boundary of Δ, or the boundary of a single vertex v in Δ, in order to renormalise single amplitudes (figure 6).

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In [31, 32, 34], a kinematical embedding map was used which commutes with the electric fluxes of the EPRL-FK model. For a quantum cuboid state on the coarse graph Γ with spins K_I on squares I, and an embedding into quantum cuboid states with fine spins k_i, the map can be written as

$$\begin{equation}{\phi }_{{{\Gamma}}^{\prime }{\Gamma}}{\psi }_{\to {K}}\;=\;\frac{1}{{N}_{\to {K}}}\sum _{{k}_{i}}\left(\prod _{\text{coarse}\;\text{squares}\;I}\delta \left({K}_{I}-\sum _{i\subset I}{k}_{i}\right)\right)\;{\psi }_{\to {k}},\end{equation} \tag{ 2.17 }$$

where ${N}_{\to {K}}$ is a normalisation constant, since the embedding map is an isometry. This kinematical embedding map is an intermediate step in constructing the continuum theory.

2.6. The physical inner product

The spin foam state sum (2.4) is used as a proposal for the physical inner product of quantum gravity. As it is defined, it provides a linear map on the boundary Hilbert space ${\mathcal{H}}_{{\Gamma}}$. If the boundary graph consists of two separate components ${\Gamma}={{\Gamma}}_{\text{in}}\;\cup \;{\overline{{\Gamma}}}_{\text{out}}$, then this can be rewritten as a transition map

$$\begin{equation}{\mathcal{Z}}_{{\Delta}}\;:\;{\mathcal{H}}_{{{\Gamma}}_{\text{in}}}\;\rightarrow \;{\mathcal{H}}_{{{\Gamma}}_{\text{out}}}.\end{equation} \tag{ 2.18 }$$

Here by ${\overline{{\Gamma}}}_{\text{out}}$ we denote the graph Γ_out with reversed link orientations, which turns the Hilbert space to its dual. We therefore have

$$\begin{equation}{\langle {\psi }_{\text{in}}\vert {\psi }_{\text{out}}\rangle }_{\text{phys}}\;=\;{\mathcal{Z}}_{{\Delta}}\left[{\psi }_{\text{in}}\otimes {\psi }_{\text{out}}^{{\dagger}}\right].\end{equation} \tag{ 2.19 }$$

The definition (2.19) depends on the choice of Δ, which needs to be removed to make the physical inner product well-defined. The traditional idea is to sum over all possible Δ, which naively is not well-defined and comes with many problems [57], while a reorganisation in terms of a GFT might be a possibility (see e.g. [58]). Another way to remove the dependence on Δ is to make the model Δ-dependent to ensure mutual consistency of the physical inner products. This line of thinking led to the programme of background-independent renormalisation [15, 16, 59], and it is a way to at the same time encompass the continuum limit of the boundary Hilbert space, by refining boundary and bulk simultaneously.

The physical inner product functions as a bona fide projector from the kinematical⁸ to the physical Hilbert space via the rigging map $\eta \,:\mathcal{D}\to {\mathcal{D}}^{{\ast}}$ from a dense subspace $\mathcal{D}\subset {\mathcal{H}}_{\text{kin}}$ to its algebraic dual, given by

$$\begin{equation}\eta \left(\psi \right)\left[\phi \right]\;=\;{\langle \psi \vert \phi \rangle }_{\text{phys}}.\end{equation} \tag{ 2.20 }$$

The physical Hilbert space is then derived by dividing ${\mathcal{D}}^{{\ast}}$ by the kernel of (2.19) and completion [10]. As such, physical states arise as (equivalence class of) linear combination of kinematical states on different graphs. The coefficients are given by the physical inner product itself.

In the hypercuboidal model, due to the high amount of symmetry, there are no transitions between Hilbert spaces on different graphs, i.e. by construction a priori Γ_in = Γ_out. However, one can use the kinematical embedding map (2.17) to define a transition between differently refined graphs. As such, one can compute the physical state η(ψ_in) of a single quantum cuboid (i.e. a graph with one node and toroidally compactified links, describing a torus geometry) by first embedding it into a finer graph Γ' with more nodes, and then mapping it with the spin foam state sum. This will give the projection of η(ψ_in) to one specific graph Γ'

$$\begin{equation}\eta {\left({\psi }_{\text{in}}\right)}_{\left\vert {\mathcal{H}}_{{{\Gamma}}^{\prime }}\right.}\;=\;{\mathcal{Z}}_{{\Delta}}{\circ}{\iota }_{{{\Gamma}}^{\prime }{\Gamma}}\left[{\psi }_{\text{in}}\right]\end{equation} \tag{ 2.21 }$$

where ${\mathcal{Z}}_{{\Delta}}$ is interpreted as the map (2.18).

It is these (projections of) physical states on refined graphs Γ' which we are considering in what follows. We are in particular interested in the entanglement entropy of these states regarding a separation of the fine graph into two halves, each containing half the nodes of Γ'.

Entanglement entropy is a property of quantum states which has received increased interest in recent years. Measuring the entanglement of degrees of freedom within a region A with those outside of A, in particular its scaling property with increasing the size of A is important. While for generic states the entanglement entropy S_A grows with the volume of A, there is a specific class of states for which it only grows with its surface area. These arise e.g. as the ground states of physically interesting Hamiltonians. These also play a crucial role in the renormalisation procedure for discrete systems [40].

Assume that a Hilbert space $\mathcal{H}$ can be decomposed as $\mathcal{H}={\mathcal{H}}_{A}\otimes {\mathcal{H}}_{B}$, where ${\mathcal{H}}_{A}$ contains all degrees of freedom associated to a region A, and ${\mathcal{H}}_{B}$ those outside of A. The reduced density matrix of a state |ψ⟩ w.r.t. A is then given by

$$\begin{equation}{\hat{\rho }}_{A}\;{:=}\;{\text{tr}}_{B}\left(\vert \psi \rangle \langle \psi \vert \right),\end{equation} \tag{ 3.1 }$$

and the entanglement entropy S_A between A and B is

$$\begin{equation}{S}_{A}\;=\;-{\text{tr}}_{A}\left({\hat{\rho }}_{A}\;\mathrm{ln}\;{\hat{\rho }}_{A}\right).\end{equation} \tag{ 3.2 }$$

If $\mathcal{H}$ does not factorise according to the region A and its complement B, but rather takes on the form of a sum

$$\begin{equation}\mathcal{H}\;=\;\underset{i}{\oplus }\left({\mathcal{H}}_{A}^{\left(i\right)}\otimes {\mathcal{H}}_{B}^{\left(i\right)}\right),\end{equation} \tag{ 3.3 }$$

then

$$\begin{equation}\vert \psi \rangle \;=\;\sum _{i}{q}_{i}\vert {\psi }_{i}\rangle \end{equation} \tag{ 3.4 }$$

with normalised states |ψ_i⟩, each of which has an associated entanglement entropy

$$\begin{equation}{S}_{A}^{\left(i\right)}\;=\;{\text{tr}}_{A,i}\left({\hat{\rho }}_{{A}_{i}}\;\mathrm{ln}\;{\hat{\rho }}_{A,i}\right)\end{equation} \tag{ 3.5 }$$

with ${\hat{\rho }}_{A,i}={\text{tr}}_{B,i}\left(\vert {\psi }_{i}\rangle \langle {\psi }_{i}\vert \right)$. The total entanglement entropy associated to A can then be defined as [60]

$$\begin{equation}{S}_{A}\;=\;\sum _{i}{p}_{i}{S}_{A}^{\left(i\right)}\;-\;\sum _{i}{p}_{i}\;\mathrm{ln}\left({p}_{i}\right),\end{equation} \tag{ 3.6 }$$

with ${p}_{i}={\vert {q}_{i}\vert }^{2}$, i.e. the weighted sum of the individual entanglement entropies, plus the von Neumann entropy of the state decomposition according to (3.3). It can be shown that the expression is symmetric under exchange of A and B. See in particular [61–63].

3.1. Entanglement entropy in LQG

In LQG, entanglement entropy has been considered for quite some time [42–48], as a spin network's degrees of freedom are localise on a graph Γ, which can be naturally split into regions. Initial computations have identified some entanglement between gauge degrees of freedom, while on the gauge-invariant level, the states carry no entanglement entropy, which can also be seen as follows: Consider the boundary Hilbert space

$$\begin{equation}{\mathcal{H}}_{{\Gamma}}\;=\;\underset{\left\{{j}_{\ell }\right\}}{\oplus }\left(\underset{n}{\otimes }{\mathcal{H}}_{n}^{\left\{{j}_{\ell }\right\}}\right),\end{equation} \tag{ 3.7 }$$

where ${\mathcal{H}}_{n}^{\left\{{j}_{\ell }\right\}}$ is the space of intertwiners for fixed spins on the node n. Consider a separation of nodes N(Γ) of Γ into A and B, then any state with fixed spins

$$\begin{equation}\psi \;=\;\underset{n}{\otimes }{\iota }_{n}\;=\;\underset{n\in A}{\otimes }{\iota }_{n}\otimes \underset{n\in B}{\otimes}{\iota }_{n}\end{equation} \tag{ 3.8 }$$

factorises over the nodes, and has therefore vanishing entanglement entropy (3.6).

For linear combinations, however, the situation changes. In particular, in the hypercuboidal truncation model presented in section 2, the projections of physical states (2.21) can be represented as finite linear combinations of spin networks. We will therefore write them as states within the kinematical Hilbert spaces. For a separation of nodes into regions A and B = N(Γ)\A, the entanglement entropy S_A will in general not vanish.

We consider two cases in what follows:

3.2. Case 1

First we consider the projection of one quantum cuboid state to two quantum cuboids: Γ_in consists of one node, with three loops as links, defining a toroidally compactified geometry depending on thee spins K₁, K₂, and K₃. The refined state Γ_out consists of two nodes connected by one link, resulting from one cuboid dissected into two (figures 7 and 8). There are five independent spins k₁, ..., k₅. The (projected) physical state is, after normalisation, given by

$$\begin{equation}\eta \left[{\iota }_{{K}_{1},{K}_{2},{K}_{3}}\right]\;=\;\frac{1}{{N}_{{K}_{1},{K}_{2},{K}_{3}}^{\left(\alpha \right)}}\sum _{{k}_{1},\dots ,{k}_{5}}{c}_{{k}_{1},\dots ,{k}_{5}}^{\left(\alpha \right)}{\iota }_{{k}_{1},{k}_{2},{k}_{3}}\otimes {\iota }_{{k}_{1},{k}_{4},{k}_{5}}\end{equation} \tag{ 3.9 }$$

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with

$$\begin{equation}{c}_{{k}_{1},\dots ,{k}_{5}}^{\left(\alpha \right)}\;=\;\sum _{{k}_{6},\dots ,{j}_{11}}{\hat{\mathcal{A}}}_{{k}_{1},{k}_{2},{k}_{3},{k}_{6},{k}_{7},{k}_{8}}^{\left(\alpha \right)}{\hat{\mathcal{A}}}_{{k}_{1},{k}_{4},{k}_{5},{k}_{9},{k}_{10},{k}_{11}}^{\left(\alpha \right)}\end{equation} \tag{ 3.10 }$$

and

$$\begin{equation}{\left({N}_{{K}_{1},{K}_{2},{K}_{3}}^{\left(\alpha \right)}\right)}^{2}\;=\;\sum _{{k}_{1},\dots ,{k}_{5}}{\left\vert {c}_{{k}_{1},\dots ,{k}_{5}}^{\left(\alpha \right)}\right\vert }^{2}.\end{equation} \tag{ 3.11 }$$

From the properties of the kinematical embedding map (2.17) we can see that the sums in (3.9) and (3.11) is restricted to

$$\begin{equation}{K}_{1}\;=\;{k}_{1},\quad {K}_{2}\;=\;{k}_{2}+{k}_{4},\quad {K}_{3}\;=\;{k}_{3}+{k}_{5}.\end{equation} \tag{ 3.12 }$$

Due to correct glueing of the 4-dim hypercuboids, the sum (3.10) has to range over

$$\begin{equation}{k}_{6}\;=\;{k}_{9},\quad {k}_{7}\;=\;{k}_{10}\end{equation} \tag{ 3.13 }$$

We consider two more simplifications:

• Volume-simplicity: We impose the Hopf-link-volume-simplicity constraint discussed in section 2 [56], leading to the additional conditions
  $$\begin{equation}\begin{array}{cc}\hfill {k}_{1}{k}_{8}\;& =\;{k}_{2}{k}_{7}\;=\;{k}_{3}{k}_{6},\hfill \\ \hfill {k}_{1}{k}_{11}\;& =\;{k}_{4}{k}_{10}\;=\;{k}_{5}{k}_{9}.\hfill \end{array}\end{equation} \tag{ 3.14 }$$
  Note that the Hopf-link constraints impose face-matching for each vertex separately, but this also leads to face-matching on the boundary Hilbert spaces, effectively restricting the Hilbert spaces to those which correspond to torsion-free geometries [26].
• Ischoric transition: We furthermore restrict the allowed transitions to those which fix the total 4d-volume V, which in this case is given by
  $$\begin{equation}V\;=\;\frac{{k}_{1}{k}_{8}+{k}_{2}{k}_{7}+{k}_{3}{k}_{6}}{3}+\frac{{k}_{1}{k}_{11}+{k}_{4}{k}_{10}+{k}_{5}{k}_{9}}{3}\end{equation} \tag{ 3.15 }$$
  Note that this leads to the constraints
  $$\begin{equation}{k}_{8}+{k}_{11}\;=\;{K}_{6},\quad {k}_{6}\;=\;{K}_{4},\quad {k}_{7}\;=\;{K}_{5}\end{equation} \tag{ 3.16 }$$
  where K₄, K₅, K₆ are the spins of the coarse hypercuboid, which satisfy
  $$\begin{equation}V\;=\;{K}_{1}{K}_{6}\;=\;{K}_{2}{K}_{5}\;=\;{K}_{3}{K}_{4}\end{equation} \tag{ 3.17 }$$
  due to the volume-simplicity constraint.

Together with (3.14), this leads in total to a sum over one single boundary spin and no bulk spin, i.e.

$$\begin{equation}\eta \left[{\iota }_{{K}_{1},{K}_{2},{K}_{3}}\right]\;=\;\frac{1}{{N}^{\left(\alpha \right)}}\sum _{k=0}^{{K}_{2}}{c}_{k}^{\left(\alpha \right)}\;{\iota }_{{K}_{1},k,k{K}_{3}/{K}_{2}}\otimes {\iota }_{{K}_{1},\left({K}_{2}-k\right),\left({K}_{2}-k\right){K}_{3}/{K}_{2}}\end{equation} \tag{ 3.18 }$$

with

$$\begin{equation}{c}_{k}^{\left(\alpha \right)}\;=\;{\hat{\mathcal{A}}}_{{K}_{1},k,k\frac{{K}_{3}}{{K}_{2}},\frac{V}{{K}_{3}},\frac{V}{{K}_{2}},k\frac{V}{{K}_{1}{K}_{2}}}^{\left(\alpha \right)}{\hat{\mathcal{A}}}_{{K}_{1},{K}_{2}-k,\left({K}_{2}-k\right)\frac{{K}_{3}}{{K}_{2}},\frac{V}{{K}_{3}},\frac{V}{{K}_{2}},\left({K}_{2}-k\right)\frac{V}{{K}_{1}{K}_{2}}}^{\left(\alpha \right)}\end{equation} \tag{ 3.19 }$$

and

$$\begin{equation}{\left({N}^{\left(\alpha \right)}\right)}^{2}\;=\;\sum _{k}{\left\vert {c}_{k}^{\left(\alpha \right)}\right\vert }^{2}.\end{equation} \tag{ 3.20 }$$

From the form (3.18) one can see that the physical state lies in the direct product Hilbert space for fixed k₁ = K₁:

$$\begin{equation}\eta \left[{\iota }_{{K}_{1},{K}_{2},{K}_{3}}\right]\;\in \;{\mathcal{H}}_{A}\otimes {\mathcal{H}}_{B}\end{equation} \tag{ 3.21 }$$

with

$$\begin{equation}{\mathcal{H}}_{A}\;=\;\underset{{k}_{2},{k}_{3}}{\otimes }\text{span}\left({\iota }_{{k}_{1},{k}_{2},{k}_{3}}\right),\quad {\mathcal{H}}_{B}\;=\;\underset{{k}_{4},{k}_{5}}{\otimes }\text{span}\left({\iota }_{{k}_{1},{k}_{4},{k}_{5}}\right),\end{equation} \tag{ 3.22 }$$

With this, using (3.18) and tracing subsequently over degrees of freedom in A and B, one straightforwardly arrives at

$$\begin{equation}{S}_{A}^{\left(\alpha \right)}\;=\;\mathrm{ln}{\left({N}^{\left(\alpha \right)}\right)}^{2}\;-\;\frac{1}{{\left({N}^{\left(\alpha \right)}\right)}^{2}}\sum _{k}\;{\left\vert {c}_{k}^{\left(\alpha \right)}\right\vert }^{2}\mathrm{ln}\left({\left\vert {c}_{k}^{\left(\alpha \right)}\right\vert }^{2}\right).\end{equation} \tag{ 3.23 }$$

3.3. Case 2

In the second case we consider the subdivision of one quantum cuboid into four. Two of them, respectively, form the regions A and B (see figure 9). The final state therefore is of the form

$$\begin{equation}\eta \left[{\psi }_{{K}_{1},{K}_{2},{K}_{3}}\right]\;=\;\sum _{{k}_{i}}\;{c}_{{k}_{i}}^{\left(\alpha \right)}{\iota }_{{k}_{1},{k}_{2},{k}_{5}}\otimes {\iota }_{{k}_{3},{k}_{2},{k}_{6}}\otimes {\iota }_{{k}_{1},{k}_{4},{k}_{8}}\otimes {\iota }_{{k}_{3},{k}_{4},{k}_{7}}\end{equation} \tag{ 3.24 }$$

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Due to the embedding map (2.17) the coefficients will be zero unless

$$\begin{equation}\begin{array}{c}\hfill {k}_{1}+{k}_{3}\;=\;{K}_{1},\\ \hfill {k}_{2}+{k}_{4}\;=\;{K}_{2},\\ \hfill {k}_{5}+{k}_{6}+{k}_{7}+{k}_{8}\;=\;{K}_{3},\end{array}\end{equation} \tag{ 3.25 }$$

while geometricity will enforce

$$\begin{equation}\begin{array}{c}\hfill {k}_{5}\;=\;\frac{{k}_{1}{k}_{2}}{{K}_{1}{K}_{2}}{K}_{3},\quad {k}_{6}\;=\;\frac{{k}_{3}{k}_{2}}{{K}_{1}{K}_{2}}{K}_{3},\\ \hfill {k}_{7}\;=\;\frac{{k}_{3}{k}_{4}}{{K}_{1}{K}_{2}}{K}_{3},\quad {k}_{8}\;=\;\frac{{k}_{1}{k}_{4}}{{K}_{1}{K}_{2}}{K}_{3}.\end{array}\end{equation} \tag{ 3.26 }$$

From this one can see that k₁ and k₂ are the only independent variables in the coefficients of the physical state (3.24).

The bulk consists of four hypercuboids, i.e. in the 2-complex there are 4 vertices. Similarly to case 1, the isochoric constraint of fixing the total 4-volume V = V₁ + V₂ + V₃ + V₄, which results in

$$\begin{align*}\hfill {c}_{{k}_{i}}^{\left(\alpha \right)}\;\equiv \;{c}_{{k}_{1},{k}_{2}}^{\left(\alpha \right)}\;& =\;{\hat{\mathcal{A}}}_{{k}_{1},{k}_{2},{k}_{5},{K}_{4},\frac{{k}_{1}}{{K}_{1}}{K}_{5},\frac{{k}_{2}}{{K}_{2}}{K}_{6}}{\hat{\mathcal{A}}}_{{k}_{3},{k}_{2},{k}_{6},{K}_{4},\frac{{K}_{1}-{k}_{1}}{{K}_{1}}{K}_{5},\frac{{k}_{2}}{{K}_{2}}{K}_{6}}\hfill \\ \hfill & \quad {\times}{\hat{\mathcal{A}}}_{{k}_{1},{k}_{4},{k}_{8},{K}_{4},\frac{{k}_{1}}{{K}_{1}}{K}_{5},\frac{{K}_{2}-{k}_{2}}{{K}_{2}}{K}_{6}}{\hat{\mathcal{A}}}_{{k}_{3},{k}_{4},{k}_{7},{K}_{4},\frac{{K}_{1}-{k}_{1}}{{K}_{1}}{K}_{5},\frac{{K}_{2}-{k}_{2}}{{K}_{2}}{K}_{6}}\hfill \end{align*}$$

with

$$\begin{equation}{K}_{4}\;=\;\frac{V}{{K}_{3}},\quad {K}_{5}\;=\;\frac{V}{{K}_{2}},\quad {K}_{6}\;=\;\frac{V}{{K}_{1}}.\end{equation} \tag{ 3.27 }$$

With (3.25) and (3.26) these can be written entirely in terms of k₁ and k₂. In the graph Γ_out there are four nodes, two of which we regard as being in region A, while the other two are in region B (see figure 9). Therefore the physical state is in

$$\begin{equation}\eta \left[{\psi }_{{K}_{1},{K}_{2},{K}_{3}}\right]\;\in \;\sum _{{k}_{1}}{\mathcal{H}}_{{k}_{1}}^{\left(A\right)}\otimes {\mathcal{H}}_{{k}_{1}}^{\left(B\right)}\end{equation} \tag{ 3.28 }$$

with

$$\begin{equation}{\mathcal{H}}_{{k}_{1}}^{\left(A\right)}\;=\;\underset{{k}_{2},{k}_{5},{k}_{6}}{\oplus }\text{span}\left({\iota }_{{k}_{1},{k}_{2},{k}_{5}}\otimes {\iota }_{{K}_{1}-{k}_{1},{k}_{2},{k}_{6}}\right)\end{equation} \tag{ 3.29 }$$

$$\begin{equation}{\mathcal{H}}_{{k}_{1}}^{\left(B\right)}\;=\;\underset{{k}_{4},{k}_{7},{k}_{8}}{\oplus }\text{span}\left({\iota }_{{k}_{1},{k}_{4},{k}_{8}}\otimes {\iota }_{{K}_{1}-{k}_{1},{k}_{4},{k}_{7}}\right)\end{equation} \tag{ 3.30 }$$

The physical state is therefore not in a Hilbert space which is a tensor product over the regions A and B, but rather a direct sum of those products. We therefore use the generalised expression (3.6) for the entanglement entropy. We write

$$\begin{equation}\eta \left[{\psi }_{{K}_{1},{K}_{2},{K}_{3}}\right]\;=\;\sum _{{k}_{1}}{q}_{{k}_{1}}{\psi }_{{k}_{1}}\end{equation} \tag{ 3.31 }$$

with real coefficients

$$\begin{equation}{q}_{{k}_{1}}\;=\;\frac{{N}_{{k}_{1}}}{N}\end{equation} \tag{ 3.32 }$$

and normalised states

$$\begin{equation}{\psi }_{{k}_{1}}\;=\;\frac{1}{{N}_{{k}_{1}}}\sum _{{k}_{2}}{c}_{{k}_{1},{k}_{2}}^{\left(\alpha \right)}{\iota }_{{k}_{1},{k}_{2},{k}_{5}}\otimes {\iota }_{{k}_{3},{k}_{2},{k}_{6}}\otimes {\iota }_{{k}_{1},{k}_{4},{k}_{8}}\otimes {\iota }_{{k}_{3},{k}_{4},{k}_{7}},\end{equation} \tag{ 3.33 }$$

where k₃, ..., k₈ are determined by k₁, k₂ via (3.25) and (3.26), and where

$$\begin{equation}\begin{array}{ccc}\hfill {N}_{{k}_{1}}^{2}\;& =\hfill & \hfill \sum _{{k}_{2}}{\left\vert {c}_{{k}_{1},{k}_{2}}^{\left(\alpha \right)}\right\vert }^{2}\\ \hfill {N}^{2}\;& =\hfill & \hfill \sum _{{k}_{1},{k}_{2}}{\left\vert {c}_{{k}_{1},{k}_{2}}^{\left(\alpha \right)}\right\vert }^{2}\end{array}\end{equation} \tag{ 3.34 }$$

For a fixed k₁, the entanglement entropy ${S}_{A}^{\left({k}_{1}\right)}$ can be computed similarly to (3.23), and yields

$$\begin{equation}{S}_{A}^{\left({k}_{1}\right)}\;=\;\mathrm{ln}\left({N}_{{k}_{1}}^{2}\right)-\frac{1}{{N}_{{k}_{1}}^{2}}\sum _{{k}_{2}}{\left\vert {c}_{{k}_{1},{k}_{2}}^{\left(\alpha \right)}\right\vert }^{2}\mathrm{ln}{\left\vert {c}_{{k}_{1},{k}_{2}}^{\left(\alpha \right)}\right\vert }^{2}\end{equation} \tag{ 3.35 }$$

The total entanglement entropy, with (3.6), can be computed to be

$$\begin{equation}\begin{array}{cc}\hfill {S}_{A}\;=& \sum _{{k}_{1}}\frac{{N}_{{k}_{1}}^{2}}{{N}^{2}}\left[\mathrm{ln}\;{N}_{{k}_{1}}^{2}-\frac{1}{{N}_{{k}_{1}}^{2}}\sum _{{k}_{2}}{\left\vert {c}_{{k}_{1},{k}_{2}}^{\left(\alpha \right)}\right\vert }^{2}\mathrm{ln}{\left\vert {c}_{{k}_{1},{k}_{2}}^{\left(\alpha \right)}\right\vert }^{2}\right]-\sum _{{k}_{1}}\frac{{N}_{{k}_{1}}^{2}}{{N}^{2}}\mathrm{ln}\frac{{N}_{{k}_{1}}^{2}}{{N}^{2}}\hfill \\ \hfill =& \mathrm{ln}\;{N}^{2}-\frac{1}{{N}^{2}}\sum _{{k}_{1},{k}_{2}}{\left\vert {c}_{{k}_{1},{k}_{2}}^{\left(\alpha \right)}\right\vert }^{2}\mathrm{ln}{\left\vert {c}_{{k}_{1},{k}_{2}}^{\left(\alpha \right)}\right\vert }^{2}.\hfill \end{array}\end{equation} \tag{ 3.36 }$$

In what follows we will present numerical results on the entanglement entropy for cases 1 and 2 from the last section. We work entirely in the large-spin-asymptotic regime, and assume that we can neglect the boundary contributions from small spins, that the spins are so large that the summations can be turned into integrals, even when taking the EPRL-FK quantisation condition (2.5) into account⁹, and one can use the asymptotic expressions for the amplitudes (2.12–2.14). Since the asymptotic formulas are homogenous under simultaneous scaling of the spins

$$\begin{equation}{\hat{A}}^{\left(\alpha \right)}\left(\lambda {k}_{1},\;\dots ,\;\lambda {k}_{6}\right)\;=\;{\lambda }^{\beta }{\hat{A}}^{\left(\alpha \right)}\left({k}_{1},\;\dots ,\;{k}_{6}\right)\end{equation} \tag{ 4.1 }$$

with β = 12α − 9 [26], the entanglement entropy S_A has a specific scaling behaviour with respect to K₁, K₂, K₃.

First we treat case 1: we have

$$\begin{equation}{S}_{A}^{\left(\alpha \right),\lambda }\;=\;\mathrm{ln}\;{\,N}^{2}\;-\;\frac{1}{{N}^{2}}{\int }_{0}^{{K}_{1}}\text{d}k\;{\left\vert {c}_{k}^{\left(\alpha \right)}\right\vert }^{2}\;\mathrm{ln}{\left\vert {c}_{k}^{\left(\alpha \right)}\right\vert }^{2}\end{equation} \tag{ 4.2 }$$

with

$$\begin{equation}{N}^{2}\;=\;{\int }_{0}^{{K}_{1}}\text{d}k\;{\left\vert {c}_{k}^{\left(\alpha \right)}\right\vert }^{2}.\end{equation} \tag{ 4.3 }$$

Denoting by ${S}_{A}^{\left(\alpha \right),\lambda }$ the entanglement entropy of the physical state $\eta \left[{\iota }_{\lambda {K}_{1},\lambda {K}_{2},\lambda {K}_{3}}\right]$, we get

$$\begin{equation}{S}_{A}^{\left(\alpha \right),\lambda }\;=\;{S}_{A}^{\left(\alpha \right)}\;+\;\mathrm{ln}\;\lambda \,\left(\text{case}\;\text{1}\right)\end{equation} \tag{ 4.4 }$$

Similarly, the scaling of the entanglement entropy in case 2 can be computed as

$$\begin{equation}{S}_{A}^{\left(\alpha \right),\lambda }\;=\;{S}_{A}^{\left(\alpha \right)}\;+\;\mathrm{ln}\;{\lambda }^{2}\,\left(\text{case}\;\text{2}\right).\end{equation} \tag{ 4.5 }$$

This scaling¹⁰ of S_A is very useful when it comes to numerical investigations.

Using the scaling behaviour (4.4) and (4.5), we can numerically evaluate the integrals (3.23), (3.36) for different values of initial spins K₁, K₂, K₃. We use numerical integration techniques from the GNU scientific library (GSL), which can be straightforwardly implemented in C++. In figures 10–12, we present the dependence on the coupling constant α for various fixed K_i, for either case.

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One can see a clear behaviour of the entanglement entropy, which has a maximum around

$$\begin{equation}{\alpha }_{\mathrm{max}}\;\approx \;0.51\end{equation} \tag{ 4.6 }$$

where the precise value depends on the (ratios of the) K_i. The contour is qualitatively robust, however, and S_A decreases rapidly for smaller values of α. It also decreases for larger values, albeit more slowly.

This behaviour can be directly understood when taking a closer look at the coefficients ${c}_{k}^{\left(\alpha \right)}$ and ${c}_{{k}_{1},{k}_{2}}^{\left(\alpha \right)}$. In figure 13 we have depicted the graph of the coefficients for different α. One can clearly see that for small α, ${c}_{{k}_{1},{k}_{2}}^{\left(\alpha \right)}$ is sharply peaked around few values of k₁, k₂, indicating that the physical state is a superposition of only few states with definite spins, resulting in low entanglement entropy. For large values of α, the coefficients are also concentrated around specific values of spins k_i, albeit with a larger spread. There is an intermediate regime in which the coefficients are spread out over a much larger regions of spins, indicating that the physical state is a coherent superposition of a large number of different spins k₁, k₂, which leads to a large entanglement entropy.

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The geometric interpretation of this becomes clear when one considers the 3-volume. Since we are working in the isochoric framework, the total 4-volume V is fixed in the transition between the in- and out-state, and hence, in the hypercubic setting, also the spatial 3-volume is. Different values of k_i therefore correspond to different states in which the 3-volume

$$\begin{equation}{V}^{\left(3\right)}\;=\;\sqrt{{K}_{1}{K}_{2}{K}_{3}}\end{equation} \tag{ 4.7 }$$

is distributed differently between the regions A and B. In the case 1 one can directly see that

$$\begin{equation}{V}_{A}\;=\;\frac{k}{{K}_{1}}{V}^{\left(3\right)}\end{equation} \tag{ 4.8 }$$

$$\begin{equation}{V}_{B}\;=\;\frac{{K}_{1}-k}{{K}_{1}}{V}^{\left(3\right)}\end{equation} \tag{ 4.9 }$$

and for case 2:

$$\begin{equation}{V}_{A}\;=\;\sqrt{{k}_{1}{k}_{2}{k}_{5}}+\sqrt{{k}_{2}{k}_{3}{k}_{6}}\;=\;\frac{{k}_{2}}{{K}_{2}}{V}^{\left(3\right)}\end{equation} \tag{ 4.10 }$$

$$\begin{equation}{V}_{B}\;=\;\sqrt{{k}_{1}{k}_{4}{k}_{8}}+\sqrt{{k}_{3}{k}_{4}{k}_{7}}\;=\;\frac{{K}_{2}-{k}_{2}}{{K}_{2}}{V}^{\left(3\right)}\end{equation} \tag{ 4.11 }$$

From this and figure 13, one can see that the main contribution to the physical state for α < α_max comes from either of the four extremal cases

$$\begin{equation}\left({k}_{1},{k}_{2}\right)\;=\;\left(0,0\right),\;\left(0,{K}_{2}\right),\;\left({K}_{1},0\right),\;\left({K}_{1},{K}_{2}\right).\end{equation} \tag{ 4.12 }$$

Two of these cases correspond to V_A = 0, V_B = V⁽³⁾, and two to V_A = V⁽³⁾,V_B = 0.Conversely, the regime α > α_max leads to the main contribution coming from an area around

$$\begin{equation}\left({k}_{1},{k}_{2}\right)\;\approx \;\left(\frac{{K}_{1}}{2},\frac{{K}_{2}}{2}\right),\end{equation} \tag{ 4.13 }$$

which corresponds to

$$\begin{equation}{V}_{A}\;\approx \;{V}_{B}\;\approx \;\frac{{V}^{\left(3\right)}}{2},\end{equation} \tag{ 4.14 }$$

i.e. where the 3-volume is distributed equally between the regions A and B.

Between these two extremal cases there is an intermediate regime in which almost all possible distributions of 3-volume among A and B occur with roughly equal probability in the physical state. This is the state with maximal entanglement entropy.

These features appear to occur for various different initial spins K_i, and both cases 1 and 2. From investigations with larger lattices it can be expected that the peak of S_A at α = α_max becomes even more pronounced as the number of lattice sites increases. This is due to the fact that the coefficients ${c}_{{k}_{i}}^{\left(\alpha \right)}$ of the physical states, as product of more and more amplitudes $\hat{\mathcal{A}}$, become more and more sharply peaked.

In this article we have considered the entanglement entropy of physical states in the EPRL-FK spin foam model, where we have worked in the hypercuboidal truncation. In the large-spin regime this model depends on only one parameter α, and has a manageable set of degrees of freedom, which makes this truncation an interesting toy model for the full, untruncated, theory.

Rather than single spin network functions, we considered physical states as given by the spin foam transition, which in this setting arise as linear combination of spin networks. The boundary graph are dual to cubic 3d lattices, describing a Cauchy surface with the topology of a 3-torus. We have considered graphs with an even number of nodes, such that the 'universe' could be separated into two similar regions A and B. We then numerically computed the entanglement entropy S_A of these states with regards to the separation of space into A and B.

We were specifically interested in the dependence of S_A on the parameter α, for different physical states. We have found that S_A, generically has a maximum around α_max ≈ 0.51. It therefore lies in the 'critical regimes' of ∼0.5–0.65, where the model generically undergoes a qualitative change in behaviour. This regime has been found to have several interesting features, and in particular contains the fixed point of the background-independent RG flow [31, 33]. This is the point where the diffeomorphism symmetry gets restored, which is broken in the EPRL-FK model due to the discretisation [12, 26].

It is this restoration of diffeomorphism symmetry which we conjecture to be the reason for the maximising of the entanglement entropy in this region. In particular, the physical states are superpositions of spin network functions which—in the large spin regime we are considering—are all on the same orbit of the classical vertex translation symmetry group, which is the lattice version of the diffeomorphism group which arises e.g. in Regge calculus [12, 26, 64–66]. For α ≈ α_max, these diffeomorphically equivalent degrees of freedom of the kinematical (spin-network) states are becoming maximally entangled with one another, which here arises as another feature of the restoration of diffeomorphisms at the RG fixed point. It can in particular be regarded to be a consequence of the fact that the subdivision of space into two regions A and B is being performed not with regards to any physical property of the system, but with regards to the nodes of the graph, which functions as external structure here. Thus, the separation of space into A and B is not diffeomorphism-invariant, in line with the discussion in [26].

This findings could be used for future investigations, also in the full theory. Those points in parameter space with maximal entanglement entropy indicate interesting behaviour with regards to the diff symmetry, and therefore could be used to find e.g. fixed points of the RG flow. This would be far less effort than computing the RG flow in the full theory, which is still an unsolved problem.

In the future, it would be interesting to check whether the behaviour of the entanglement entropy is persistent when relaxing the hypercuboidal truncation of the model. In particular including curvature degrees of freedom is necessary in order to solidify the results of this analysis. We hope to come back to this point in another article.

This work was funded by the project BA 4966/1-2 of the German Research Foundation (DFG). The author acknowledges support by the DFG under Germanys Excellence Strategy EXC 2121 'Quantum Universe' 390833306.