Emergent Einstein Equation in p-adic CFT Tensor Networks
EEmergent Einstein Equation in p -adic CFT Tensor Networks Lin Chen,
1, 2, 3
Xirong Liu,
1, 2, 3 and Ling-Yan Hung
1, 2, 3, 4, ∗ State Key Laboratory of Surface Physics, Fudan University, 200433 Shanghai, China Shanghai Qi Zhi Institute, 41st Floor, AI Tower,No. 701 Yunjin Road, Xuhui District, Shanghai, 200232, China Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China Institute for Nanoelectronic devices and Quantum computing, Fudan University, 200433 Shanghai, China (Dated: February 25, 2021)We take the tensor network describing explicit p -adic CFT partition functions proposed in [1], andconsidered boundary conditions of the network describing a deformed Bruhat-Tits (BT) tree geom-etry. We demonstrate that this geometry satisfies an emergent graph Einstein equation in a unique way that is consistent with the bulk effective matter action encoding the same correlation functionas the tensor network, at least in the perturbative limit order by order away from the pure BT tree.Moreover, the (perturbative) definition of the graph curvature in the Mathematics literature [2–4]naturally emerges from the consistency requirements of the emergent Einstein equation. This couldprovide new insights into the understanding of gravitational dynamics potentially encoded in moregeneral tensor networks.
PACS numbers: 11.15.-q, 71.10.-w, 05.30.Pr, 71.10.Hf, 02.10.Kn, 02.20.Uw
The AdS/CFT [5] provided deep insights of (quantum)gravity. One extremely important breakthrough inspiredby the Ryu-Takayanagi entanglement formula [6] is therealization that (semi-classical) geometries as solutions ofthe gravity theory are basically geometrization of the en-tanglement structure of wave-functions of the dual CFT.(see for example [7] for a review and references therein.)Tensor networks (TN) widely used to construct many-body wave-functions are also geometrization of patternsof entanglement of many-body wave-functions. This ledto proposals that TNs similar to the MERA capturesthe microscopic mechanism behind the AdS/CFT corre-spondence [8]. Toy models have been constructed [9, 10]that recreate many aspects of the AdS/CFT correspon-dence, most notably the RT formula and the error cor-recting property [11]. TN also provides deep insight inthe surface-state correspondence [12], kinematic space[13], the complexity of the wave-functions and their evo-lution [14–18] and the island formula that is probably thekey to the black hole information paradox [19, 20].Nonetheless, reconstruction of gravitational dynamicsusing TN remains a tremendous challenge. There is someprogress by considering relative entropies[21, 22] and alsocomplexity optimisation[16–18], although it is not com-pletely clear how bulk matter or generic time dependencecan be included (some progress is recently made in [23].).In this paper, we present the emergence of a graph
Einstein equation based on the proposed TN [1] of thep-adic AdS/CFT [24, 25]. Our strategy is to parametrisethe form of curvatures and stress tensor using an ansatzthat is based only on locality and symmetry. We willshow that if an Einstein equation exists at all the self-consistency constraints are stringent enough to return a unique one from the TN. This is perhaps the first such ∗ [email protected] quantitative demonstration involving both matter andtime [26], and where both the CFT and the TN can beexplicitly defined, taking advantage of the simplicity of p -adic CFT. Lightning review of p -adic CFTs A one dimensional p -adic CFT lives in the p -adic num-ber field Q p for any given prime number p . i.e. coordi-nates x ∈ Q p . p -adic numbers Q p are field extensions ofthe rational numbers alternative to the reals R . This canbe readily generalized to an n -dimensional p -adic CFT byconsidering field extension of Q p to Q p n [24, 25]. To avoidclutter we will take n = 1 although all the expressionscan be generalized for generic n , basically by replacing p → p n . A p -adic number x can be uniquely expressedas an infinite series x = p v X i =0 a i p i ; v, a i ∈ Z ; 0 ≤ a i ≤ p − a = 0 . (1)The p -adic norm is defined as | x | p = p − v , (2)which satisfies various axioms of norms [27]. Conformalsymmetry is defined as the transformation x → x = ax + bcx + d , a, b, c, d ∈ Q p . (3)It furnishes the matrix group PGL(2 , Q p ), the direct ana-logue of SL(2 , R ) in 1d conformal transformation in realspace-time. There are two pieces of algebraic data re-quired to specify completely a p -adic CFT [28].• First, the spectrum of primary operators O a withconformal dimensions ∆ a , which transform underconformal symmetry as O a ( x ) → ˜ O a ( x ) = (cid:12)(cid:12)(cid:12)(cid:12) ad − bc ( cx + d ) (cid:12)(cid:12)(cid:12)(cid:12) − ∆ a p O a ( x ) (4). a r X i v : . [ h e p - t h ] F e b • Second, OPE coefficients C abc defined as [29] O a ( x ) O b ( x ) = X c C abc | x − x | ∆ c − ∆ a − ∆ b p O c ( x ) . (5)The OPE coefficients define an associative operatorfusion i.e. P c C abc C cde = P c C bdc C cae . There is aunique identity operator I so that C ab = C a b = δ ab , and that there is a unique dual of a whichwe denote as a ∗ satisfying C ab = δ ba ∗ . To avoidclutter we work with theories where a ∗ = a and C ab = δ ab . TN on the Bruhat-Tits tree and the p -adicAdS/CFT The partition function of a generic p -adic CFT can beconstructed explicitly in the form of a TN covering theBruhat-Tits (BT) tree [1], the latter of which is a discrete p + 1 valent tree graph. The BT tree is the analogueof the AdS space whose isometry is the correspondingconformal symmetry group. The TN covers the tree, suchthat at each vertex of the tree sits a p + 1 index tensor T a ··· a p +1 . The tensor is given by a fusion tree of p + 1operators, expressed in terms of the OPE coefficient ofthe p -adic CFT. T a ··· a p +1 = X b ··· b p − C a a b C b a b · · · C b p − a p a p +1 . (6)In the special case where p = 2, T abc = C abc . Twotensors at two vertices connected by an edge are con-tracted with the edge index weighted by p − ∆ a , where∆ a is the conformal dimension of the corresponding pri-mary labeled a . The asymptotic boundary of the treeis Q p , analogous to of the real line being the asymp-totic boundary of the AdS space. The partition func-tion is defined by setting appropriate boundary condi-tions at the cut-off surface which is then taken to ap-proach the asymptotic boundary. Specifically, the dan-gling legs at the cut-off surface Λ is projected along thevector | V f i ≡ P a δ a | a i = | i . To compute correlationfunctions, one inserts operator O a ( x b ) by projecting theboundary leg at x b along | a i instead. The TN reproducesthe correct correlation function in the form of Witten-likediagrams [1] . The boundary insertions of O a ( x b ) sourcethese p − ∆ a weighted paths in the TN, that coincides with“bulk-boundary propagators” G a ( x b , v i ) on the tree, untilthey meet at some vertices v i in the bulk. These bulk-boundary propagators are solutions of the graph Klein-Gordon equation ( (cid:3) v + m a ) G a ( u, v ) = δ u,v , G a ( u, v ) = ζ p (2∆ a ) p − ∆ a d ( u,v ) p ∆ a , (7) m a = − ζ p (∆ a − n ) ζ p ( − ∆) , ζ p ( s ) ≡ − p − s (8) (cid:3) u φ ( u ) ≡ X h uv i ,v ∼ u ( φ ( u ) − φ ( v )) , u ∼ v ≡ nearest neighbour. (9) Here h uv i denotes an edge ending at vertices u, v , and d ( u, v ) is the distance between u, v . Here, every link has FIG. 1. The tensor network representation of a p = 2-adic CFT. The diagram depicts three bulk operator insertion.Bulk insertions pushed to the asymptotic boundary asymp-totes to boundary insertions. The boundary condition V Λ i are chosen to be the fixed point tensor V af = δ a . Each vertextensor is C abc and each edge of the tensor with index a isweighted by p − ∆ a . unit length d e = 1. Operators labeled a, b . . . traversingdifferent paths can meet at a vertex if they can fuse to theidentity operator. The emergence of Witten-diagramssuggests that the TN is recreating the p -adic AdS/CFTcorrespondence proposed in [24, 25], connecting a p -adicCFT and a dual bulk theory containing some bulk fields φ a in 1-1 correspondence with the primary operators liv-ing on the BT tree. In fact, we can define bulk field φ a ( x )insertion by fusing an extra a leg to the bulk vertex x .The bulk correlation functions h φ a ( x ) φ b ( y ) · · · i are thusdefined as evaluation of the tensor network with the extralegs inserted at the appropriate vertices. See fig 1.One can readily show that these results are consistentwith an emergent bulk matter field theory living on theBT tree, with action in the large mass limit [30] S m = X h xy i "X a ( φ a ( x ) − φ a ( y )) + m a p + 1 ( φ a ( x ) + φ a ( y ) ) + X a,b,c ˜ C abc p + 1) [ φ a ( x ) φ b ( x ) φ c ( x ) + φ a ( y ) φ b ( y ) φ c ( y )] + · · · , ˜ C abc ≡ C abc r p ∆ a +∆ b +∆ c ζ p (2∆ a ) ζ p (2∆ b ) ζ p (2∆ c ) , (10) where · · · corresponds to higher point interaction terms.For p = 2 the action truncates at cubic level. We deliber-ately present the action as a sum over edges anticipatingcoupling the theory to a dynamical metric shortly. Distances and Curvatures in a Tensor Network
The tensor network constructed appears to be describ-ing a pure BT space (analogue of pure AdS) when theboundary edges are projected to the identity vector apartfrom locations where operators are inserted. As in theusual story of AdS/CFT, a most natural way to deformthe background geometry is to change the boundary con-ditions which would drive an RG flow in the CFT. Specif-ically, each boundary link i at the cutoff surface Λ is pro-jected to a generic vector | V Λ i i [1]. Where there is trans-lation invariance, we can take | V Λ i i = | V Λ i for all bound-ary legs i . Explicitly the vector | V Λ i is parametrized as | V Λ i = X a V a Λ | a i . (11)When the vectors | V Λ i are contracted with the p danglinglegs of a tensor T a ··· a p +1 at the cut-off surface, it wouldgenerate a new vector | V Λ − i| V Λ − i ≡ X a V a Λ − | a i = X a ··· a p +1 V a Λ p − ∆ a · · · V a p Λ p − ∆ ap T a ··· a p a p +1 | a p +1 i (12)which is fed into the next layer of tensors recursively.The flow of these vectors suggests that the geometrydescribed by the tensor network is deformed from thepure BT background. We note that the original bound-ary condition describing the un-deformed CFT partitionfunction V af = δ a (13)is indeed a fixed point vector under this flow, which re-covers the original pure BT space supposedly dual to theundeformed p -adic CFT partition function. In this case,all the edges contribute equally to the partition functionand we can assign unit length to every edge i.e. d e = 1.When we depart from pure BT background, one needsto assign a general length d e to each edge on the graph.We note that there are two characteristic vectors at anedge e = h xy i bounded by vertices x and y . One is | V xy i introduced above corresponding to repeating theflow described in (12), contracting all the tensors fromthe cutoff surface all the way down to the vertex x . Theother vector | ˜ V xy i follows from analogously contractingall the tensors below the vertex y . This misleading notionof being “above” or “below” is illustrated in figure 2.The distinction of | V xy i and | ˜ V xy i comes from bound-ary conditions very far away, and so we expect the edgelength d h xy i to depend on them symmetrically. It is alsonatural to expect that | V xy i and | ˜ V xy i fully determine d h xy i . To have better analytic control, we consider per-turbing around the undeformed CFT by perturbing theboundary conditions | V Λ i around the fixed point tensor: V a Λ i = δ a + λv a Λ i , (14)where λ (cid:28)
1, but v a Λ i is completely arbitrary. Theboundary conditions are not assumed to respect anytranslation symmetries. In the perturbative limit, theflowed vector in the interior would admit the general ex-pansion V axy = δ a + ω axy , ω axy ≡ λ (1) axy + λ (2) axy + · · · , (15)˜ V axy = δ a + ˜ ω axy , ˜ ω axy ≡ ˜ λ (1) axy + ˜ λ (2) axy + · · · , (16) FIG. 2. Vector V axy and ˜ V axy following from the contraction oftensors above (colored red) and below (colored green) the edge h xy i respectively. Boundary conditions V Λ i differing from thefixed point vector drives an RG flow. The curvature of thepatch centred at x depends on the edge lengths d h xy i i sym-metrically. where λ ( n ) , ˜ λ ( n ) are of order λ n . The edge length d e = h xy i ( V ae , ˜ V ae ) would also admit an expansion aboutthe pure BT space as d e = 1 + j e , where j e = A a ( ω ae + ˜ ω ae ) + B ab ( ω ae ω be + ˜ ω ae ˜ ω be ) + C ab ω ae ˜ ω be + D abc ( ω ae ω be ω ce + ˜ ω ae ˜ ω be ˜ ω ce )+ E abc ( ω ae ω be ˜ ω ce + ˜ ω ae ˜ ω be ω ce ) + O ( ω ) , (17)for some constants A a , B ab , C ab · · · . Having assignededge lengths, one can compute the curvature of thisgraph. There are various proposals. e.g.[2–4]. Con-sider the curvature R x of a patch surrounding a vertex x . The graph curvature should be a symmetric functionof lengths d h xy i =1 , ··· p +1 i of the p + 1 edges connected to x .Generally therefore, in the small j e limit we expect R x = a + a X i j xy i + b X i j xy i + c X i = k j xy i j xy k + O ( j ) , (18)again for constants a , , b, c · · · . Emergent Action and Einstein equation
In the above, we have defined distances and curvatureswhich are, up to some undetermined coefficients, deter-mined by the TN. On the other hand, the TN encodesa bulk scalar field theory where the expectation valuesof the fields φ a ( x ) can be readily computed. We wouldtherefore like to inquire if the geometry and the expecta-tion values of φ a ( x ), both read off from the TN, can berelated by some graph Einstein equation.To look for such a relation, we need an emergent co-variant effective action that describes the bulk theory,and obtain the equations of motion following from it,before we could even check if such an equation is satis-fied by the TN. The matter part of the covariant actionshould reduce to (10) in the pure BT limit to be consis-tent with the correlation functions. We need to upgrade(10) by coupling it to the background geometry via theedge lengths d e . This is attempted in [4], although ithas made a choice of treating the mass term as a lo-cal term blind to d e . This is unlike in continuous fieldtheories where every term contains at least the volumeform and are thus always sensitive to the metric. Ratherthan making such choices, we write down a more generalansatz that mimics the continuous covariant action moreclosely S covm = S cov + S cov + · · · S cov = X h xy i d k h xy i ( φ ax − φ ay ) + X h xy i d h xy i p + 1 m a (( φ ax ) + ( φ ay ) ) S cov = X h xy i (cid:18) h ( d h xy i ) H abc ( φ ax φ bx φ cx + φ ay φ by φ cy )+ r ( d h xy i ) R abc ( φ ax φ bx φ cy + φ ay φ by φ cx ) (cid:19) , (19)where k is a constant to be determined. For generality weconsidered more general cubic interactions other than the ex-actly local term with coupling H abc , and allowed also for near-est neighbour interaction with coupling R abc . The functions r ( d e ) and h ( d e ) should be regular in the pure BT background.Therefore in the perturbative limit they can be expanded as h ( d xy ) = h + h j xy + h j xy + . . . , (20) r ( d xy ) = r + r j xy + r j xy + . . . . (21) We introduce the graph Einstein Hilbert action making use of the graph curvature introduced in (18). S EH = X x R x ( d xy , d xy , . . . , d xy p +1 ) + X h xy i d xy Λ , (22)where we introduced also the cosmological constant termanalogous to R d d x √ g Λ. The total test effective actionis thus S tot = S EH + S covm . Now we are ready to varythese actions wrt each edge length d h xy i to obtain thegraph Einstein equation. Varying S EH we obtain the graph Einstein tensor G : G xy ≡ δS EH δd xy = Λ + 2 a + 4 bj xy + c ( X i ( y i = y ) j xy i + X i ( x i = x ) j x i y ) + O ( j ) . (23) Varying S covm we obtain T xy ≡ δS covm δd xy = k φ ax − φ ay ) + m a (( φ ax ) + ( φ ay ) )2( p + 1)+ (cid:18) h H abc ( φ ax φ bx φ cx + φ ay φ by φ cy ) + r R abc ( φ ax φ bx φ cy + φ ay φ by φ cx ) (cid:19) + · · · . (24) We would like to substitute the expectation values of φ a ( x ) into T . We provide detailed expressions of φ a ( x )in the supplementary material. Clearly φ a ( x ) ∼ λ , andthe omitted terms in (24) is thus of order λ and beyond.The graph Einstein equation is thus given by G xy + T xy = 0 . (25) We substitute the geometrical data and expectation valueof the stress tensor determined by the TN into (25). Re-quiring that (25) is satisfied order by order in λ turnsinto constraints of the undetermined parameters we haveintroduced. These constraints turn out to be very pow-erful because G xy and T xy are non-trivial functions of λ ( n ) axy i , ˜ λ ( n ) axy i and λ ( n ) ayx i , ˜ λ ( n ) ayx i . These edges h xy i i , h yx i i are marked in figure 2 for a given pair of connected ver-tices x, y . These variables λ ( n ) axy i , ˜ λ ( n ) ayx i are virtually in-dependent because they are distinct functions of genericboundary conditions arbitrarily far away. Therefore, wecan isolate the coefficient of each of these independentmonomials of λ ( n ) axy i , ˜ λ ( n ) ayx i , and require that it vanishesseparately. Up to some overall normalization, it fixes a unique form of the graph curvature, edge lengths and S covm such that (25) can in fact be satisfied. Solving con-straints up to order λ , we have2 bc = − p, A a = 0 , Λ + 2 a = 0 . (26)Substituting into G xy gives G xy = − c (cid:3) j xy which recov-ers the perturbative graph curvature defined in the Math-ematics literature [2–4] if we take c= -1. Moreover, thematter effective action is constrained to be m a = − p − p − ∆ a + p ∆ a , (27) k = 1 . (28)Equation (27) independently recovers the relation in (8). B ab and C ab introduced in (17) are in turn determined by b, c and ∆ a . Constraints at order λ again lead to uniqueexpressions for the matter couplings H abc , R abc and edgelength expansion coefficients D abc , E abc up to an overallundetermined normalization. Detailed expressions arerelegated to the appendix. We are interested in the limit m a → ∞ = ⇒ ∆ a → ∞ . This is when the bulk effectiveaction S m exactly reproduces φ a ( x ) correlation functionsof the TN. In this limit, satisfyingly only the local termsurvives with an overall undetermined normalization,lim ∆ a →∞ ,d e → S covm = − h h X h xy i ˜ C abc p φ ax φ bx φ cx , (29)but otherwise in exact agreement with the effective action(10). Summary and Discussion
In this paper, we demonstrated that there is, up tosome overall normalization, a unique way of assigninglengths to the p -adic TN so that the geometry read offfrom the TN satisfies a graph Einstein equation, thatis consistent with the bulk effective action that repro-duces bulk correlation functions encoded by the TN, inthe perturbative limit away from pure BT geometry. Wehave made minimal assumptions other than locality inour ansatz for the action and also the dependence of edgelengths on the TN data. We note that in retrospect, theedge distance d h xy i = 1 + j xy defined in (17) can be writ-ten as d h xy i = 1 − h u x | u y i , (30)which, in the limit ∆ a → ∞| u x i = 1 p p + 1) X a ˜ φ ax | a i . (31)Up to the overall factor in front, this is simply the statecorresponding to one dangling leg inserted at a bulk ver-tex x in the TN. Indeed, d h xy i ends up being a Fisherinformation metric between these vertex states. This is arare quantitative demonstration of an emergent Einsteinequation from a TN that couples to matter, albeit in asimplified setting of p -adic CFTs. There are more pat-terns in the emergent graph Einstein equation that wewill report in a forth-coming accompanying paper. We believe this provides further evidence of geometry be-ing moulded by the correlation of matter, supporting theTN as the microscopic mechanism behind the AdS/CFTcorrespondence. We believe some ideas and methodol-ogy discussed here should admit generalization to TNdescribing more realistic CFTs. Acknowledgements.—
LYH acknowledges the support ofNSFC (Grant No. 11922502, 11875111) and the ShanghaiMunicipal Science and Technology Major Project (Shang-hai Grant No.2019SHZDZX01), and Perimeter Institute forhospitality as a part of the Emmy Noether Fellowship pro-gramme. Part of this work was instigated in KITP duringthe program qgravity20. LC acknowledges support of NSFC(Grant No. 12047515). We thank Bartek Czech, Ce ShenGabriel Wong, Qifeng Wu and Zhengcheng Gu for useful dis-cussions and comments. We thank Si-nong Liu and Jiaqi Loufor collaboration on related projects.[1] L.-Y. Hung, W. Li, and C. M. Melby-Thompson, JHEP , 170 (2019), arXiv:1902.01411 [hep-th].[2] Y. Lin and S.-T. Yau, Tohoku Math. J. (2) , 605(2011).[3] Y. Ollivier, Journal of Functional Analysis , 810(2009).[4] S. S. Gubser, M. Heydeman, C. Jepsen, M. Marcolli,S. Parikh, I. Saberi, B. Stoica, and B. Trundy, JHEP , 157 (2017), arXiv:1612.09580 [hep-th].[5] J. M. Maldacena, Int. J. Theor. Phys. , 1113 (1999),arXiv:hep-th/9711200.[6] S. Ryu and T. Takayanagi, Phys. Rev. Lett. , 181602(2006), arXiv:hep-th/0603001.[7] M. Van Raamsdonk, in Theoretical Advanced Study In-stitute in Elementary Particle Physics: New Frontiers inFields and Strings (2017) pp. 297–351, arXiv:1609.00026[hep-th].[8] B. Swingle, Phys. Rev. D , 065007 (2012),arXiv:0905.1317 [cond-mat.str-el].[9] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill,JHEP , 149 (2015), arXiv:1503.06237 [hep-th].[10] P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter,and Z. Yang, JHEP , 009 (2016), arXiv:1601.01694[hep-th].[11] A. Almheiri, X. Dong, and D. Harlow, JHEP , 163(2015), arXiv:1411.7041 [hep-th].[12] M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi,and K. Watanabe, Phys. Rev. Lett. , 171602 (2015),arXiv:1506.01353 [hep-th].[13] B. Czech, L. Lamprou, S. McCandlish, and J. Sully,JHEP , 100 (2016), arXiv:1512.01548 [hep-th].[14] L. Susskind, Fortsch. Phys. , 24 (2016), [Addendum:Fortsch.Phys. 64, 44–48 (2016)], arXiv:1403.5695 [hep-th].[15] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle,and Y. Zhao, Phys. Rev. D , 086006 (2016),arXiv:1512.04993 [hep-th].[16] P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi, andK. Watanabe, Phys. Rev. Lett. , 071602 (2017), arXiv:1703.00456 [hep-th].[17] P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi, andK. Watanabe, JHEP , 097 (2017), arXiv:1706.07056[hep-th].[18] B. Czech, Phys. Rev. Lett. , 031601 (2018),arXiv:1706.00965 [hep-th].[19] G. Penington, JHEP , 002 (2020), arXiv:1905.08255[hep-th].[20] A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao,JHEP , 149 (2020), arXiv:1908.10996 [hep-th].[21] T. Faulkner, M. Guica, T. Hartman, R. C. My-ers, and M. Van Raamsdonk, JHEP , 051 (2014),arXiv:1312.7856 [hep-th].[22] T. Faulkner, JHEP , 033 (2015), arXiv:1412.5648 [hep-th].[23] J. Boruch, P. Caputa, and T. Takayanagi, (2020),arXiv:2011.08188 [hep-th].[24] S. S. Gubser, J. Knaute, S. Parikh, A. Samberg, andP. Witaszczyk, Commun. Math. Phys. , 1019 (2017),arXiv:1605.01061 [hep-th].[25] M. Heydeman, M. Marcolli, I. Saberi, and B. Stoica,Adv. Theor. Math. Phys. , 93 (2018), arXiv:1605.07639[hep-th].[26] Here time corresponds to an extra coordinate. p-adicCFT is basically a Euclidean theory with no distinctionof space and time.[27] A. Ostrowski, Acta Arith. , 271 (1916).[28] E. Melzer, Int. J. Mod. Phys. A , 4877 (1989).[29] Note that this expression is exact since p -adic CFT isknown to have no descendents.[30] The interaction vertex is essentially fixed at the meetingpoint of geodesics on the tree rather than summed over[1]. Therefore it agrees with the semi-classical limit of amassive field. Appendix A: Some details in the perturbativeexpansion
Here we would like to supply some extra details of thecomputations discussed in the main text. Further discus-sion of these results will appear in a forth-coming com-pany paper.The expression of the edge length expanded up to order λ is given by d e = 1 + B ab ( λ (1) ae λ (1) be + ˜ λ (1) ae ˜ λ (1) be ) + C ab λ (1) ae ˜ λ (1) be +2 B ab ( λ (1) ae λ (2) be + ˜ λ (1) ae ˜ λ (2) be ) + C ab ( λ (1) ae ˜ λ (2) be + λ (2) ae ˜ λ (1) be )+ D abc ( λ (1) ae λ (1) be λ (1) ce + ˜ λ (1) ae ˜ λ (1) be ˜ λ (1) ce )+ E abc ( λ (1) ae λ (1) be ˜ λ (1) ce + ˜ λ (1) ae ˜ λ (1) be λ (1) ce ) + O ( λ ) . (A1) The expectation value of the bulk field ˜ φ ax ≡ q p ∆ a ζ p (2∆ a ) φ ax at x and a neighbour y can be expressedin terms of the characteristic vectors V i , ˜ V i (see figure 3)which can be expanded in λ as follows ˜ φ ax = λ (1) a p − ∆ a + ˜ λ (1) a p − a + λ (2) a p − ∆ a + ˜ λ (2) a p − a + γ a + ˜ γ a p − ∆ a + λ (1) b ˜ λ (1) c p − ∆ b p − c C abc + O ( λ ) , (A2)˜ φ ay = ˜ λ (1) a p − ∆ a + λ (1) a p − a + ˜ λ (2) a p − ∆ a + λ (2) a p − a +˜ γ a + γ a p − ∆ a + ˜ λ (1) b λ (1) c p − ∆ b p − c C abc + O ( λ ) , (A3) where λ (1) a ≡ p X i =1 λ (1) ai , (A4)˜ λ (1) a ≡ p X i =1 ˜ λ (1) ai . (A5) λ (2) a ≡ X i λ (2) ai , (A6)˜ λ (2) a ≡ X i ˜ λ (2) ai , (A7) γ a ≡ X i = j,b,c λ (1) bi λ (1) cj C abc p − ∆ b p − ∆ c , (A8)˜ γ a ≡ X i = j,b,c ˜ λ (1) bi ˜ λ (1) cj C abc p − ∆ b p − ∆ c . (A9)We remind the readers that V ai = δ a + ω ai , ω ai ≡ λ (1) ai + λ (2) ai + · · · , (A10)˜ V ai = δ a + ˜ ω ai , ˜ ω ai ≡ ˜ λ (1) ai + ˜ λ (2) ai + · · · , (A11)See figure 3 for the characteristic vector each V i corre-sponds to .The graph Einstein tensor δS EH /δd h xy i up to order λ FIG. 3. The edge xy is connected with { xy i } and { x i y } .When one restricts attention to this patch, { V ai } and { ˜ V ai } encodes all needed information. is given by G xy = X a,b p − a +∆ b ) ( λ (1) a λ (1) b + ˜ λ (1) a ˜ λ (1) b ) (cid:18) bB ab p ∆ a +∆ b + B ab c (cid:0) − p ∆ a +∆ b + p ∆ a +∆ b +1 + p (cid:1) + cC ab p a +∆ b (cid:19) + p − a +∆ b ) λ (1) a ˜ λ (1) b (cid:18) bC ab p ∆ a +∆ b + 2 B ab c ( p − (cid:0) p ∆ b + p ∆ a (cid:1) + cC ab (cid:0) p a + p b (cid:1)(cid:19) + X i c ( λ (1) ai λ (1) bi + ˜ λ (1) ai ˜ λ (1) bi )2 (cid:18) B ab (1 + p − ∆ a − ∆ b ) − C ab ( p − ∆ a + p − ∆ b ) (cid:19) + O ( λ ) . (A12)The stress tensor defined as δS covm /δd xy up to order λ is given by T xy = X a p − a ( λ (1) a λ (1) a + ˜ λ (1) a ˜ λ (1) a ) (cid:18) k ( p + 1) (cid:18) p ∆ a − (cid:19) + m a (cid:0) p a + 1 (cid:1)(cid:19) p + 1) (cid:0) p ∆ a − (cid:1)(cid:0) p ∆ a + (cid:1) + p − a (cid:18) m a p ∆ a − k ( p + 1) (cid:0) p ∆ a − (cid:1) (cid:19) ( p + 1) (cid:0) p ∆ a − (cid:1)(cid:0) p ∆ a + 1 (cid:1) λ (1) a ˜ λ (1) a + O ( λ ) . (A13)Let us give an example how the equations of motionleads to powerful constraints on the couplings. By requir-ing that G xy + T xy = 0, one can see that the monomial λ (1) ai λ (1) ai only appears in G but not in T . Therefore itscoefficient must be vanishing by itself. c (cid:18) B ab (1+ p − ∆ a − ∆ b ) − C ab ( p − ∆ a + p − ∆ b ) (cid:19) = 0 , (A14)We subsequently consider the coefficients of λ (1) a λ (1) a ,˜ λ (1) a ˜ λ (1) a and λ (1) a ˜ λ (1) a separately to obtain the con-straints (26, 27, 28) in the main text. In addition weget B aa = 12 c ( p + 1) (1 − p a ) . (A15)Then we expand G and T up to order λ to solve forconstraints at that order. We will not reproduce all theexpressions here, but present the solutions of the con-straints. R abc = ˜ C abc (cid:0) p ∆ a +∆ c − p ∆ b (cid:1) (cid:0) p ∆ b +∆ c − p ∆ a (cid:1) r ( p a −
1) ( p b −
1) ( p c − ,E abc = ˜ C abc c ( p + 1) ( p a −
1) ( p b −
1) ( p c − − p ∆ a +∆ b +2∆ c − p ∆ a − ∆ b − p ∆ b − ∆ a + 3 p ∆ a +∆ b − p a +∆ c − p b +∆ c − p − ∆ c + 3 p ∆ c ) . (A16)Having E abc , R abc , we can further get D abc = − ˜ C abc p − ∆ a − ∆ b − ∆ c c ( p + 1) ( p a −
1) ( p b −
1) ( p c − (cid:16) − p ∆ a +∆ b +∆ c − p a +∆ b +∆ c ) + p a +∆ b +∆ c + p ∆ a +3∆ b +∆ c + p ∆ a +∆ b +3∆ c + p a +∆ b ) + p a +∆ c ) + p b +∆ c ) (cid:17) , (A17) H abc = ˜ C abc p − ∆ a − ∆ b − ∆ c (cid:18) p ∆ a +∆ b +∆ c + p (cid:19) h ( p + 1) (cid:0) p a − (cid:1)(cid:0) p b − (cid:1)(cid:0) p c − (cid:1)(cid:16) − p ∆ a +∆ b +∆ c − p (cid:0) ∆ a +∆ b +∆ c (cid:1) + p a +∆ b +∆ c + p ∆ a +3∆ b +∆ c + p ∆ a +∆ b +3∆ c + p a +∆ b ) + p a +∆ c ) + p b +∆ c ) (cid:17) . (A18) A very simple relation between these coupling con-stants is found: D abc H abc = − h c ( p ∆ a +∆ b +∆ c + p ) . (A19)Finally, as already noted in the main text, on retro-spect d h uv i can be written as 1 − h u | v i for some emer-gent states | u i , | v i . Before taking the semi-classical limit∆ a → ∞ , the complete expressions for them are given by | u i ≡ X a ( ˜ φ au + ˜ φ bu ˜ φ cu ˜ R bca ) | ˜ a i , (A20) | v i ≡ X a ( ˜ φ av + ˜ φ bv ˜ φ cv ˜ R bca ) | ˜ a i , (A21)where h ˜ a | ˜ b i = − δ ab p ∆ a c ( p + 1) (1 − p a ) , (A22)˜ R abc ≡ r R abc s ζ p (2∆ a ) ζ p (2∆ b ) ζ p (2∆ c ) p ∆ a +∆ b − ∆ c ..