Group Field Theory and Tensor Networks: holographic entanglement entropy in full quantum gravity LOOPS 17

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1 Group Field Theory and Tensor Networks: holographic entanglement entropy in full quantum gravity LOOPS 17 Warsaw Goffredo Chirco AEI GC, D.Oriti, M.Zhang arxiv: v2

2 A new common framework in Quantum Gravity major advances in our understanding of quantum gravity come from insights and techniques from quantum information theory and quantum statistical mechanics Holographic principle Equivalence principle Bekenstein 81,t Hooft 93, Susskind 94 Einstein => Penrose 71, Rovelli & Smolin 95 => gauge field theory/gravity duality => AdS d+1 d 1 String theory: AdS/CFT duality H d => relational texture of spacetime ST dynamical record of frames transformations SpinNetworks 1 LQG, SF, GFTs: quantum space(time) geometry described by discrete, pregeometric degrees of freedom, of combinatorial and algebraic nature /2 2 3/ /2 3/2 1/2 1/2 3/ /2 1/2 3/2 2 in both scenarios quantum entanglement becomes a tool to characterise the quantum texture of spacetime in terms of the structure of correlations of some microscopic states as well as the emergence of a continuum geometric description for spacetime geometry

3 LQG: space(time) from entangled states of quantum geometry gravity as a lattice gauge theory on a superposition of SU(2)/SL(2,C) spinnetwork graphs LQG structural level: a c b = 2j+1 1 X p 2j +1 c=1 => space geometry from pregeometry, ent & coarse graining => (study of continuum limit) Girelli Livine 05, Livine Terno diffeos compatible definition of entanglement: localisation and boundary charges holographic dualities? hu 1,j,a,cihU 2,j,c,bi maximally mixed state Dittrich, Bahr, Steinhaus, MartinBenito... Charles Livine 2016, GC Mele, Vitale, Oriti Donnelly 2012 Freidel Donnelly 16 Freidel Perez Pranzetti 16 Delcamp Dittrich Riello, Geiller 1617 Area law for entanglement entropy as a signature of good semiclassical behaviour: Bianchi Myers 2012 Bianchi Guglielmon Hackll Yokomizo 16 GC Rovelli Haggard Riello Ruggiero 1415, Hamma Hung Marciano Zhang 15 GC Anzà 16, Han et al. 16 & BH entropy: Rovelli, Perez, de Lorenzo, Smerlak, Husain, Bodendorfer, Oriti, Pranzetti Sindoni \infty

4 Bulk reconstruction in AdS/CFT similar structural behaviour in AdS/CFT: gravitational theories are equivalent to nongravitational theories defined as quantum manybody systems or quantum field theories: a dual nongravitational theory lives on the boundary of its original gravitational spacetime holographic entanglement entropy S A = Area( A) 4G N ΣA ɤA Σ ΣB => a series of attempts of spacetime bulk geometry reconstruction from the structure of correlations of the boundary state Md+2 Σ=ΣA ΣB RyuTakayanagi 2015] [Hubeny, Rangamani] eternal AdS BH = classically connected spacetime QIT toy models for the bulk/boundary correspondence: holographic quantum errorcorrecting codes [Pastawski, Yoshida, Harlow Preskill] [Van Raamsdonk 2009] [Cao Carroll Michalakis 2016]

5 Surface/State correspondence one of main direct interests for our community: again key role played by networks: CMT Tensor Network techniques to express quantum wave functions in terms of network diagrams => help understanding holography via geometrization of algebraically complicated quantum states A e.g. MERA states: particular example of TN states designed to support the entanglement of a CFT with geometrical interpretation (hyperbolic space) A generalisation conjecture: AdS/CFT correspondence to be interpreted as a Multiscale Entanglement Renormalisation Algorithm MiyajiTakayanagi 2015] [Vidal 06, Swingle 09] surface/state correspondence: in any spacetime described by Einstein gravity, each codim2 convex surface corresponds to a quantum state in the dual theory. This largely extends the holographic principle as it can be applied to gravitational spacetime without any boundaries Freidel, Donnelly, Pranzetti, Dittrich,

6 Frame a common field in Quantum Gravity H d => d 1 <=> 1 1/2 3/ /2 3 1/ /2 3/2 1 1/2 1 1/2 3/2 2 AdS d+1 1 1/2 LQGwise: surface/state correspondence continuum limit through renormalisation of general random cellular complexes dual to discrete space w/ possible QFT duals on boundaries think of TND path integral definition of boundary states through sums over spinfoam open graph states sketch a concrete realisation of this scenario by means of the generalised GFT formalism: => define a GFT Tensor Network analogue with nice properties => look for the holographic behaviour of entanglement entropy

7 Group Field Theories Group Field Theories (GFTs) are combinatorially nonlocal quantum field theories defined on a group manifold (g 1,...,g d ):G d! C d ( )/Z random function (field) probability measure provide a 2n quantisation scheme for LQG: e.g. d=3 g 1 g 2 with gauge invariance at the vertex v (g i )= (g i ) g 3 no embedding in a continuum manifold and no cylindrical consistency imposed on our quantum geometry wavefunctionals Fock construction through decomposition of spin network states in terms of elementary building blocks corresponding to tensor maps associated to nodes of the spin network graphs (quantum many body system) Ii = {a,b,c} i v i a,b,c j, ai j, bi j, ci i p = 1 2j +1 2j+1 p=1 e p i e p

8 GFTs random lattices as simplicial geometries F 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 (a priori) arbitrary topology Z = Z D D e S[, ] = X i ( i ) N i( ) A Aut( ) for GFT 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 S d [ ]= with Z dg i dg 0 i (g i ) K(g i g 0 1 i ) (g 0 i)+ Z K(g i,gi)= 0 dh Y (g i g 0 1 i h), G i Z Y Y V(g ij g 0 1 ji )= dh i (h i g ij ) G i i<j Z d+1 Y i6=j=1 dg ij V(g ij g 0 1 ji ) (g 1j ) (g d+1j ) 4body (e.g. Boulatov model => PonzanoRegge)

9 emphasise: GFT fields as tensors the singleparticle quantum state Z i = dg i (g i ) g i i2h d for g i i2h ' L 2 [G] G d behaves as generalised rankd tensor states, i.d. multi dimensional arrays of cnumbers g 1 g 2 v (~g) <=> T 1, 2..., d T ( ~ ) or T (v) g 1 g 2 g 3 g 3 where generally T : X! C with X = { ~ ~ =( 1,..., d)} (exploit the combinatorially tensorial nature) a Vparticle states can be decomposed into products of elements of singleparticle space Z i = V Y dg j i (g j i ) g1 i... g V i 2 H d V ' L 2 [G d V /G V ] j=1 can be seen as a Tensor Network state with (g j i )= (g1 1,g 1 2,...,g 1 d,g V 1,...g V d )

10 Tensor Networks In the tensor network methods, a quantum state is described in terms of a set of tensors. Consider a lattice L made of N sites, where each site is described by a complex vector space V of finite dimension d. i a pure state i2v N of the lattice can be expanded as [Vidal] i = dx ( ) i1,i 2,...,i N i 1,i 2,...,i N i i 1,i 2,...,i N =1 i s i V where denotes a basis of for site s in L i T (v) a TN decomposition for consists of a set of tensors and a network pattern or graph characterised by a set of vertices and a set of directed edges ( ) i 1 i 2 i 3 = e.g. N=3 T 1 T 4 T 2 T 3 auxiliary bond indices ( ) i1,i 2,...,i N =ttr! O T (v) the tensor trace contracts all bond indices, leaving only the physical indices v

11 Multiparticle state as a tensor network state we can understand the wavefunction on an open graph of V vertices or their dual polyhedra as a tensor network encoding the entanglement structure of the multiparticle state V (g j i ) <=> ( ) g 1,g 2,...,g V =ttr O v=1 (g i ) v! construct a representation with auxiliary group fields <=> g 1... g V g 1... glued by links convolution functions Mij M`i = M ij g i i g j i2h 2 a Vparticle states can be then decomposed as i Ò 2 hm` VO vi v

12 Dictionary: GFT states (many body wavefunctions) as tensors networks Table A Group Fields Tensors quantum ~g i2h d ' L 2 [G d ] ii, i =1,...,d = D in H D one particle state 'i = '(~g) ~g i T n i = T 1 d ~ i2h n = H d D tensor state gluing functional hm g` = R dg1 dg 2 M(g 1 g`g 2 ) hg 1 hg 2 2 H 2 Mi = M 1 2 1i 2i 2 H` = H 2 D link state multiparticle state i2h V ' L 2 [G d V /G V ] N i tensor network state product state convolution g` N`2 hm g` N V n ' ni = R ) i N i N L ` hm` N N n T ni2 tensor network decomposition randomness 1 Z d (') field theory probability measure T U µ (UT 0 ) µ T 0 µ T 0 1 d 2 H T, U 2 U(dim(H T )) random tensor state

13 Dictionary: Spinnetworks as gauge invariant tensors networks Table B GFT network Spin Tensor Network Tensor Network node '(~g) '(g 1,g 2,g 3,g 4 ) ' j {m} / P {k} ˆ' j {m}{k} ij{k} T {µ} link M(g 1 g`g 2 ) M j mn M 1 2 sym '(h~g) ='(~g) Q v s Dj m s m 0 s (g)ii m 0 1 m0 v = i i m 1 m v Q v s U µ s µ 0 s T µ 0 1 µ0 v = T µ1 µ v state g` N ` hm g` N n ni ji i N`hM j` N n j n i n n i N i N L` hm` N N n T ni indices g i 2 G, g i i2h ' L 2 [G] m i 2 H j, SU(2) spinj irrep. µ i 2 Z n, nth cyclic group dim 1 dim H j =2j +1 dim Z n = n

14 Group Field Random Tensor Networks being tensor field theories, GFTS reproduce the structure of very interesting TN states: random tensor networks (RTS) i <ij>hm O NO ij T v i 1 maximally entangled link states v Mi = 1 p D 1 2 1i 2i 2 tensors T v are unit vectors chosen independently at random from their respective Hilbert spaces. the unique uniform unitarily invariant distribution is induced by the Haar measure on the unitary group by acting on an arbitrarily chosen generating vector: 0 v i T v i = U 0 v i (for arbitrary reference state define with U unitary ) 3 In the large bond dimension limit, RTS saturate the TN entropy bound, reproducing the holographic Ryu Takanayagi entropy formula A S(A) ' log(d) v A Hayden et al.arxiv: v1 F. Pastawski, B. Yoshida, D. Harlow and J. Preskill

15 A relevant example: Random Tensor Networks => holographic behaviour: random tensors networks provide explicit toy examples for the surface/state correspondence key features of the random character: the random average of an arbitrary function f ( T v ) of the the state T v > is equivalent to an integration over U according to the Haar probability measure averages enters linear traces operation, hence simplifying the derivation of the nontrivial entanglement properties of the states, induced as usual by partial tracing as the system dimension becomes large, random states have typical behaviour Hayden et al.arxiv: v1 => we expect to find a similar behaviour in our GFT setting along with Hayden s statistical approach

16 Group Field Random Tensor Networks Let s consider the boundary state associated to the open spin network graph N N i Ò 2NhM` VO ni 2 O H` n `2@N the boundary density operator is a linear function of independent pure states of each tensor # =tr` " O M`ihM` `2 v VO vih v to a subregion (A) of the boundary we associate a reduced state ˆ A =tr B [ ]/tr[ ] A

17 Group Field Random Tensor Networks we then look for the entanglement entropy of A/B: S EE = tr[ˆ A log ˆ A ]= lim N!1 S N (A) = 1 1 N log tr[ˆ N A ] ( Rényi via replika trick ) where e S N (A) =tr[ N A ]/(tr[ ]) N Z A /Z 0 KEY 1: calculating the Rényi entropy is hard, however we can use the random character of the field to calculate the expectation value of the Rényi: expand in the fluctuation S N (A) = log Z A + Z A Z 0 + Z 0 = log Z A Z 0 + 1X n ( 1) n 1 n Z n 0 Z n 0 Z n A Z n A! KEY 2: fluctuations are suppressed in the limit of large bond dimension random states in highdimensional bipartite systems: concentration of measure phenomenon applies, meaning that on a largeprobability set macroscopic parameters are close to their expectation values (bond/group dimension, => continuum limit)

18 Group Field Random Tensor Networks we then look for the entanglement entropy of A/B: S EE = tr[ˆ A log ˆ A ]= lim N!1 S N (A) = 1 1 N log tr[ˆ N A ] ( Rényi via replika trick ) where e S N (A) =tr[ N A ]/(tr[ ]) N Z A /Z 0 KEY 1: calculating the Rényi entropy is hard, however we can use the random character of the field to calculate the expectation value of the Rényi: expand in the fluctuation S N (A) = log Z A + Z A Z 0 + Z 0 = log Z A Z 0 + ' S N (A) KEY 2: fluctuations are suppressed in the limit of large bond dimension random states in highdimensional bipartite systems: concentration of measure phenomenon applies, meaning that on a largeprobability set macroscopic parameters are close to their expectation values (bond/group dimension, => continuum limit)

19 Random Group Field Tensor Networks since Z A = E(tr N A ) Z 0 E(tr ) N = Etr[ N P( A 0 ; N,d)] E(tr ) N + assuming factorised state permutation operator acting on the states in A = tr N ` Ǹ N n E( N n )P( 0 A ; N,d) tr N ` Ǹ N n E( N n ) ' S N (A) we can get SN (A) by computing the expectation values: " Z Y N E( N n )=E[( nih n ) N ]=E dg a dg 0 a a n(g a ) n (g 0 a) g a ihg 0 a!# in the standard field theory formalism we define the averaging via the path integral of some GFT model E f[, ] Z [D ][D ] f[, ] e S[, ] the average over the Nreplica of the wave functions (generalised tensors) associated to each network vertex can be interpreted as a GFT Npoint correlation function

20 Path integral averaging for the free theory we take the case S[, ]= Z dgdg 0 (g)k(g, g 0 ) (g 0 )+ S int [, ]+cc with K(g, g 0 )= (g g 0 ) <<1 and consider a perturbative expansion of the path integral in powers of " N # " Y Y N E (g a ) (g 0 a) E 0 a a Wick theorem C X P hn ( n ) 2S N (g a ) (g 0 a) with # P h ( ) = + O( ) NY a h a g a g (a) the free theory N points correlation function translates into a sum over all permutations among the group elements attached at each node tr[ 2 A]=tr[ P A ] B B A A A B A B

21 Path integral averaging for the free theory Z A and Z 0 correspond to summations of the combinatorial networks N A (h n, π n ) and N 0 (h n,π n ) Z N C AA V CV X Z " Y O dh n Tr O Ǹ n ` n X Z Y dh n N A (h n, n ) n 2S N n 2S N Z N 0 = C V X C V n 2S N X n n 2S N n Z " Y O dh n Tr O # Ǹ P hn ( n ) n ` n Z Y dh n N 0 (h n, n ), # P hn ( n )P( A; 0 N,d) at each node n we have a contribution P h n(π n ) links contribution the (Feynman )networks get divided into several regions, with same π n and h n. The links which connect different regions identify boundaries between each pair of different regions or domain walls. for different domain walls and different assignments of permutation groups to each region, we have different patterns for given network => Z A and Z 0 are sum of BF amplitudes with different 2complexes

22 Path integral averaging for the free theory remark this simple form of the various functions entering the calculation of the entropy, with the emergence of BFlike amplitudes, is not generic: it follows from the choice of GFT kinetic term, from the approximation used in the calculation of expectation values (neglecting GFT interactions) and from the special type of GFT tensor network chosen we are interested in is the leading term of Z A and Z 0, while taking the dimension D of the bond Hilbert space much larger than 1. This leads us to seek the most divergent term of the amplitudes (bubble divergences) => the divergence degree of BF amplitudes discretised on a lattice has been the subject of a number of works, both in the spin foam an GFT literature Freidel, Gurau, Oriti 09, Bonzom and Smerlak 1012 the Nth Rényi entropy S N is associated to patterns with only one domain wall and h n =1 e (1 N)S N = Z N Z N 0 =[ (1)] (1 N)min(#`2@ AB ) 1+O( 1 (1)) + O( )

23 Path integral averaging for the free theory When N goes to 1, S N becomes the entanglement entropy S EE. The leading term of the entanglement entropy S EE is S EE =min(#`2@ab )ln (1) A which can be understood as the RyuTakayanagi formula in a GFT context, with the same interpretation for the area of the minimal surface that we have mentioned in the previous section, concerning the tensor network techniques. the interaction kernel will generally lead to nontrivial bulk corrections! A B A B tr[ ] 2 A B A B 4body Mingyi s talk later!

24 Results establish a precise dictionary between GFT states and (generalized) random tensor networks. Such a dictionary also implies, under different restrictions on the GFT states, a correspondence between LQG spin network states and tensor networks, and a correspondence between random tensors models and tensor networks. compute the Rényi entropy and derived the RT entropy formula: using directly GFT and spin network techniques, first using a simple approximation to a complete definition of a random tensor network evaluation seen as a GFT correlation function, but still using a truly generalized tensor network seen as a GFT state, and then considering directly a spin network state as a random tensor network. This elucidates further the correspondence and its potential the result shows that the same formalism allows to compute nonperturbative quantum gravity corrections to the RyuTakayanagi formula, by including the contributions from the GFT interaction term into the amplitude (as well as considering different choices for the GFT kinetic term). Mingyi s talk later!

25 Remarks/Speculations AdS/MERA/CFT may be extended, beyond AdS/CFT, to a more general space/tnr/qft correspondence: GFTs may play a role as auxiliary tensor field theories both fixing the entanglement structure of the boundary physical theory and providing a dual simplicial characterisation of the tensor network diagrams as discretised space dynamics induces entanglement: looks like standard multi scale renormalisation techniques may be associated to cmera within the field theory framework of GFTs: what would play the role of the MERAlike renormalisation scale (radial dimension)? the structural similarity had been noted before, and also exploited, in the context of renormalization of spin foam models treated as lattice gauge theories G. Vidal,(2008), S. Singh, R. N. C. Pfeifer, and G. Vidal, Tensor network decompositions in the presence of a global symmetry, G. Evenbly and G. Vidal, Tensor network states and geometry; M. Han and L.Y. Hung, Loop Quantum Gravity, Exact Holographic Mapping, and Holographic Entanglement Entropy, B. Dittrich, S. Mizera, and S. Steinhaus, Decorated tensor network renormalization for lattice gauge theories and spin foam models, C. Delcamp and B. Dittrich, Towards a phase diagram for spin foams, B. Dittrich, F. C. Eckert, and M. MartinBenito, Coarse graining methods for spin net and spin foam models, B. Dittrich, E. Schnetter, C. J. Seth, and S. Steinhaus, Coarse graining flow of spin foam intertwiners

26 Thank You!

27 Tensor Network decomposition TN representations of a quantum state structure network diagram where bond indices represent unphysical, auxiliary degrees of freedom that are introduced for the purpose of efficiently writing down a ground state variational parameters to be optimized, within a given structure of the tensor network, to find a best tensor network state which best describes the target quantum state e.g. Multiscale Entanglement Renormalization Ansatz (MERA): efficient TNR for variationally estimating the ground state of a critical quantum system (CFT). isometries implementing coarse graining A (dis)entanglers min A [# bonds( A )] log L network geometry resembles a discretization of spatial slices of an AdS spacetime and ``geodesics'' in the MERA reproduce the RyuTakayanagi formula for the entanglement entropy of a boundary region in terms of bulk properties => AdS/MERA A(L)