Scrambling in Yang-Mills
SScrambling in Yang-Mills
Robert de Mello Koch a,b, , Eunice Gandote b, and Augustine Larweh Mahu b,c, a School of Physics and Telecommunication Engineering , South China Normal University, Guangzhou 510006, China b National Institute for Theoretical Physics,School of Physics and Mandelstam Institute for Theoretical Physics,University of the Witwatersrand, Wits, 2050,South Africa c Department of Mathematics,,University of Ghana, P. O. Box LG 62, Legon, Accra, Ghana.
ABSTRACT
Acting on operators with a bare dimension ∆ ∼ N the dilatation operator of U ( N ) N = 4 super Yang-Mills theory defines a 2-local Hamiltonian acting on a graph. Degrees offreedom are associated with the vertices of the graph while edges correspond to terms in theHamiltonian. The graph has p ∼ N vertices. Using this Hamiltonian, we study scramblingand equilibration in the large N Yang-Mills theory. We characterize the typical graph andthus the typical Hamiltonian. For the typical graph, the dynamics leads to scrambling in atime consistent with the fast scrambling conjecture. Further, the system exhibits a notionof equilibration with a relaxation time, at weak coupling, given by t ∼ pλ with λ the ’t Hooftcoupling. [email protected] [email protected] [email protected] a r X i v : . [ h e p - t h ] S e p ontents Black holes in general relativity exhibit incredibly fast relaxation time scales. Since theAdS/CFT correspondence claims an equivalence between conformal field theories in d di-mensions and theories of quantum gravity on negatively curved spacetimes[1, 2, 3], themechanism behind these extremely rapid thermalization rates should be coded into the dy-namics of large N and strongly coupled conformal field theories. Motivated by this issuewe study scrambling and equilibration in N = 4 super Yang-Mills theory, with gauge group U ( N ). There are at least two features of our study that must be improved before we canmake contact with the physics of black holes. First, operators in the conformal field theorycorresponding to a black hole necessarily have a very large dimension ∆ ∼ N . The genericoperator is constructed using the complete collection of fields in the theory. Although ouroperators have a dimension of order N , they are special in that they are constructed usingthree complex adjoint scalars and two complex adjoint fermions. Second, the link to classicalgravity emerges in the strong coupling limit of the field theory. Our analysis is limited toweak coupling. However, we will see that our simplified system is already interesting.Recall that the AdS/CFT correspondence identifies the dimensions of operators in theconformal field theory with the energies of energy eigenstates in the dual gravitational the-ory. This has been pursued in exquisite detail in the planar limit of N = 4 super Yang-Mills1heory[4], where the identification of the dilatation operator D with a Hamiltonian is par-ticularly fruitful because D is the Hamiltonian of an integrable spin chain. The energy of aspin chain state equals the dimension of the corresponding operator. The dynamics of theworldsheet string theory is also integrable [5] and there is an exact match between stringtheory energies and operator dimensions [6]. Although integrability allows us to go beyondweak coupling, the planar limit is not the correct arena for the questions we consider. In-deed, integrable systems do not thermalize in the conventional way: they do not thermalizeto a Gibbs ensemble. Integrable systems thermalize into a “generalised Gibbs ensemble”due to the existence of many extensive conserved charges. This is well understood for inte-grable systems relaxing after a quantum quench[7]. Further, completely integrable modelscan never exhibit chaos, but the holographic dual to a black hole is expected to exhibitchaotic dynamics [8].An interesting extension beyond the planar limit considers operators whose bare dimen-sion grows parametrically with N as we take N → ∞ . The mixing problem of these heavyoperators has new complications absent in the planar limit: single trace operators can anddo mix so multi trace structures must be included in the problem and they all mix in anon-trivial way. A second complication is that the sheer number of non-planar diagrams isso big that it overcomes the usual higher genus suppression and we must sum more thanjust the planar diagrams [9, 10, 11]. The final complication arises because as the numberof fields in the multi trace operator grows beyond N there are trace relations which expressthe equality of naively distinct multi trace structures . Starting with [13] methods based ongroup representation theory were employed to address all three of these issues in a singlecomplex matrix model. A linear basis for multi-matrix invariants, the restricted Schur poly-nomials, which we use in this work, is constructed in [14, 15] (see also [16]). Although wewill not use them in our study, note that closely related bases were introduced and studiedin [17, 18, 19, 20]. The restricted Schur polynomials are labeled by a collection of Youngdiagrams, one for each species of field appearing in the operator, plus one more denoted R forthe complete collection of fields. They diagonalize the free field theory two point function,explicitly take all finite N trace relations into account and mix only weakly at one loop.Summing the complete set of ribbon graphs contributing to a free field theory correlator isreduced to rather straight forward manipulations in group theory: the computation of pro-jection operators and matrices representing permutations, as well as commutators, productsand traces of them.Our focus is on operators constructed using O ( N ) fields. The majority of the fieldsappearing in the operators we study are a single complex adjoint scalar (say φ ). There are For example, invariants of a single matrix are written in terms of the eigenvalues of the matrix. Given N independent invariants, the eigenvalues and hence all invariants are determined. As a consequence, thereare relations between invariants expressed as a collection of terms that sum to zero. Each term is of a fixeddegree in the matrix and different terms have different trace structures. An example of a relation of thistype is provided by the Cayley-Hamilton Theorem and by the Mandelstam relations [12]. φ and φ ) as well as fermion ( ψ and ψ ) fields, alltransforming in the adjoint of U ( N ). We will use n i to denote the number of φ i fields and m i to denote the number of ψ i fields. In the limit where the row lengths of the Young diagram R labeling the restricted Schur polynomials are all different, with the difference (cid:29) φ fields and one more, denoted R , for the completecollection of fields) and a graph for the remaining fields [22, 23]. The mixing problem can bediagonalized on the Young diagram labels, leaving a Hamiltonian describing dynamics on agraph[24, 25]. Vertices of the graph correspond to rows (for a short and wide diagram) orcolumns (for a tall and thin diagram) of the Young diagram R and hence they correspond todual giant and giant graviton branes. As a consequence of the displaced corners conditionthe branes are separated in spacetime. Edges stretching between vertices correspond to openstrings that stretch between branes. In a suitable adiabatic limit, reviewed in Appendix A,these modes are frozen, i.e. they do not evolve in time. Each brane can be excited, whichis represented as a closed loop made out of a single edge attached to a given vertex. Inthe adiabatic limit these excitations of a particular brane are the only dynamical degrees offreedom. These degrees of freedom live at the vertices of the graph and they are able to hopto any other site as long as there is an edge in the graph that connects the two sites [24, 25].Thus, the spin chain of the planar mixing problem is replaced by dynamics on a graph,when the mixing problem of heavy operators is considered. It is noteworthy that dynamicson a graph naturally emerges in this way. Indeed, models describing the dynamics on graphswere used to examine the fast scrambling conjecture[26], first in [27], which was followedby a number of interesting articles[28, 29, 30, 31, 32, 33] . The logic of [27] is elegant andworth summarizing. Consider a state of some subsystem S , and denote the complementarysubsystem to S by S c . By saying that information is scrambled we mean it is hiddenin complicated correlations between subsystems S and S c . Using this observation one canargue that scrambling subsystem S is the same as signaling to S c . Thus, bounds on signalingare immediately bounds on scrambling. With this insight, [27] appeals to classic methodsof Lieb-Robinson[34] which bound signaling by proving bounds on commutators [ O A ( t ) , O B ]where O A and O B are observables localized on disjoint subsystems A and B of a latticespin system. In this way [27] bound the signaling time for Hamiltonians with dense twobody interactions to no faster than O (log n ) with n the number of degrees of freedom. Theresulting bound refers to the maximum degree D V of any vertex of the interaction graph. These studies use an “interaction graph”. Degrees of freedom live at the vertices of the interactiongraph. The interaction graph has an edge between two vertices if and only if the Hamiltonian includes aninteraction term for these degrees of freedom. There is a simple relation between the graph that emergesfrom Yang-Mills theory, called a Gauss graph in [23] and the interaction graph: dropping the closed loopsfrom the Gauss graph one obtains the interaction graph. We stick to this terminology in this article. Dense means the number of interacting pairs of degrees of freedom scales like n . V also appears in the assumption that each term in the Hamiltonian is bounded by c/D V with c some constant that does not scale with the size of the system. The Lieb-Robinsonbound then says that a suitably normalized commutator is bounded by ∼ D V e ct . Usingthis bound, its now possible to show that for times ∼ log D V the reduced density matrix oneach site i is approximately a pure state. D V is the maximum vertex degree, so we expect D V ∼ n . Since scrambling requires entanglement, this bounds the scrambling time to be atleast ∼ log n .In this paper we study the dynamics of the Hamiltonian defined by the mixing problem forheavy operators, described by dynamics on a graph. The relevant Hamiltonian is describedin Section 2. Our Hamiltonian describes the physics of bound states of giant gravitons andtheir excitations. The number of giants in the boundstate is large enough to backreact andproduce a new spacetime geometry[35]. By choosing the right boundstate of giant gravitonsexcited in a particular way, we would produce operators dual to black holes. A black holestate would have a number of general features that one could look for . First, the massof the black hole in AdS translates, upon using the standard AdS/CFT dictionary, into ascaling dimension for operators that grows as ∆ ∼ N . To explain the entropy of the blackhole, the number of operators should be ∼ e bN with b some constant that does not dependon N . We verify these expectations in Section 3. Each operator is labeled by a differentgraph and hence by a different Hamiltonian. By numerically generating the complete set ofgraphs for finite values of N (where numerical analysis is still possible), we give evidencethat there is a “typical” graph and that almost every graph, at large N , looks like thetypical graph. This typical graph defines a typical Hamiltonian and it is the dynamics ofthis typical Hamiltonian that we consider. In Section 4 we study scrambling, establishinga Lieb-Robinson bound which ensures that the system does not scramble faster the boundimplied by the fast scrambling conjecture. We also explore entanglement generation forthe typical dynamics. The leads to a puzzle: the recurrence time is much smaller than weexpect. In Section 5 we show for a conveniently chosen initial non-equilibrium state, thatthe system evolves to thermal equilibrium and we estimate the thermalization time scale.The puzzle of the recurrence time is also resolved: we argue that as far as the dynamics isconcerned, the typical Hamiltonian is rather special and does not give a reliable descriptionof the physics. Small fluctuations in the typical Hamiltonian are important and must beincluded. In Section 6 we discuss our results and outline some future directions. As reviewed in Appendix A the dynamics we consider is of a system of bosons, hopping ona lattice. The lattice is defined by a directed graph G = ( V, E ), where V is a set of verticesand E a set of directed edges. In what follows we always use p = | V | to denote the total For a very readable and informative discussion we recommend [36, 37, 38]. i ∈ V we have a bosonicFock space F i . The full Fock space is a tensor product F ≡ F ⊗ F ⊗ · · · ⊗ F p . Associatedto the i th Fock space is a pair of oscillators b i , b † i and a vacuum state | (cid:105) i . The oscillators b i , b † i act as the identity on all F j with j (cid:54) = i , and in the usual way on F i . The algebra of thebosonic operators is [ b i , b † j ] = δ ij (2.1)The i th Fock space vacuum obeys b i | (cid:105) i = 0 and the vacuum of the full Fock space F isgiven by | (cid:105) = | (cid:105) ⊗ | (cid:105) ⊗ · · · ⊗ | (cid:105) p (2.2)To write the Hamiltonian describing the dynamics of these bosons, it is useful to introducethe p × p matrix N ij . The matrix elements N ij count how many edges stretch between vertices i and j , regardless of orientation. As far as the Hamiltonian is concerned, we can ignore theorientation of edges which corresponds to treating G as an undirected graph. Thus, N ij is asymmetric matrix with zeros on the diagonal, that completely determines the graph G . Asan example, consider the following graph N ij = (2.3)In terms of this matrix, the Hamiltonian we study is given by H = g Y M (4 π ) p (cid:88) i,j =1 ,i (cid:54) = j ( √ r i − √ r j ) N ij + 2 g Y M (4 π ) p (cid:88) i =1 r i l i k i b † i b i − g Y M (4 π ) p (cid:88) i,j =1 ,i (cid:54) = j (cid:114) r i r j l i l j N ji b † j b i (2.4)where r i = N + l i k i = p (cid:88) l =1 ,l (cid:54) = i ( N ) il (2.5)The parameters of the model are N , l i , g Y M , p and the matrix N ij . N sets the rank of thegauge group of the Yang-Mills theory and g Y M is the coupling constant. We study the large N limit, at weak ’t Hooft coupling. The parameters l i , i = 1 , , · · · , p are positive integers oforder ∼ N . In the CFT they set the row lengths of Young diagram R labeling our operator.They are ordered so that l i > l j if j > i . The displaced corners approximation requires5hat | l i − l j | (cid:29) i (cid:54) = j . In the holographic dual l i is the angular momentum of thecorresponding dual giant graviton. We are interested in the limit in which p goes to infinity.If we take p = (cid:15)N with (cid:15) (cid:28)
1, we can consider operators with differences in lengths ofadjacent rows of R of order ∼ (cid:15) − (cid:29) N ij . We focus on graphs with number of edges | E | ≈ p = (cid:15) N . In this case, thebare dimension of our operator is ∆ ∼ (cid:15)N and the number of fields scale as n ∼ (cid:15)N and n ∼ n ∼ m ∼ m ∼ (cid:15) N as we take N → ∞ .The spectrum of the Hamiltonian has an interesting structure. The first term in theHamiltonian is an order ∼ λ = g Y M N . This term isa constant, determined by the number of edges and the specific vertices the edges stretchbetween. The second term is a constant, equal to the total number of bosons hopping inthe graph. Since the Hamiltonian preserves particle number we can restrict the dynamics toa subspace with fixed total number of particles. We work on the subspace with N b bosonshopping on the graph. These first two terms give the largest contribution to the energyeigenvalues. The remaining terms give a much smaller correction to the first two terms.These small corrections resolve the degeneracies of the multiparticle Fock space. In Section4.1 we estimate the size of the terms in the Hamiltonian. The first two terms are of size ∼ λ ,and that the remaining terms are of size ∼ (cid:15) λ . The dimension of the multi particle Fockspace grows very rapidly: for N b bosons hopping on a graph with p vertices the dimensionof the relevant subspace of Fock space is given by dim p,N b = ( p + N b − N b !( p − E , which correctsthe bare dimension, itself of order E = (cid:15)N . The pattern for the possible E values presentin the above spectrum, is a set of levels separated by gaps of order ∼ λ , with each level acollection of an enormous numbers of nearly degenerate states, with splitting ∼ (cid:15) λ . Usinga measuring apparatus that can resolve energy differences ∼ N , but not the much smallerscales ∼ λ or ∼ (cid:15) λ we would only resolve a coarse grained version of the physics. Aftercoarse graining its not possible to distinguish between these almost degenerate states, so wenaturally obtain macrostates with a large entropy. This is a promising start to explain theblack hole entropy. One check of this idea is to count the total number of operators thatcan be defined. Since there is an operator associated to every graph (see Appendix A) thenumber of graphs should be large enough ( ∼ e bN ) for this idea to work.An important technical comment is in order: the studies of the fast scrambling conjecturegiven in [27, 28, 30, 31] which were an important motivation for this study, make use of theassumption that the Hamiltonian (and other operators) have a finite norm. The Hamilto-nian defined in (2.4), is unbounded. Thus, it seems that the methods of finite dimensionalquantum mechanics can not be used and a careful treatment of the system with the methodsof functional analysis [39] is necessary. This conclusion is too hasty and too pessimistic.6he Hamiltonian in (2.4) conserves particle number. Thus, if we restrict to initial stateswith finite particle number, the whole evolution happens in a finite dimensional subspace ofthe Fock space. In this case the Hamiltonian H and all relevant observables can be repre-sented by bounded operators on this subspace so that we are back in the framework of finitedimensional quantum mechanics[40].Our Hamiltonian is derived by evaluating the action of the dilatation operator on aspecific class of heavy operators in N = 4 super Yang-Mills. It is interesting to note that aclosely related model was suggested and studied in [41] as a toy model of black hole dynamics.See also [42, 43] for related work. The graphs arising from the operator mixing problem of Yang-Mills theory were called Gaussgraphs in [23]. Gauss graphs are graphs with directed edges and any number of vertices.In addition, at every vertex in the graph, the number of edges terminating on the vertex isequal to the number of edges departing from the vertex. We call this the Gauss constraint.By removing edges that have both endpoints at a single vertex (so these edges form a closedloop) we obtain the interaction graph. Edges of the interaction graph are always stretchedbetween distinct vertices. A directed graph obeying the Gauss constraint is called a balanceddirected graph [44] in the mathematics literature. In this section we describe an algorithmthat can be used to generate the complete set of interaction graphs, given that each graphhas p vertices and E edges. The number of interaction graphs grows extremely rapidly sothat is makes sense to talk about the “typical graph”. We characterize properties of thetypical graph, using numerical results. For each interaction graph there is a Hamiltonian.By characterizing the typical graph we are characterizing the typical Hamiltonian. We canthen study the scrambling time and relaxation rates of this typical Hamiltonian. The key difficulty in generating interaction graphs entails respecting the Gauss constraint.Consider some interaction graph G . Our first observation is that any closed oriented path,made from edges belonging to G , respects the Gauss constraint. Deleting the edges thatmake up this path produces a new graph G (cid:48) , which itself also obeys the Gauss constraint,i.e. G (cid:48) is also an interaction graph. We can now repeat the procedure: construct any closedpath, made from edges belonging to G (cid:48) . Delete this new path to find a new interaction graph G (cid:48)(cid:48) . This procedure can be repeated until all edges in G have been deleted, and so G hasbeen decomposed into a collection of closed paths. To generate the interaction graph G wefollow the reverse process in which we “grow” G by dressing a bare set of vertices with closedoriented paths. 7ts easy to understand why this decomposition is always possible: choose any given edgein the graph and consider the vertex that this edge ends on. The Gauss constraint guaranteesthat there is always an edge leaving this vertex, that can be joined with the edge we haveto produce the second edge in the path. We can keep growing the path in this way. Thegrowing process terminates when the last edge we consider can be joined with the first edgein the path, producing a closed path. The point is that the Gauss constraint implies thatany edges left after a closed path is deleted, belong to a closed path and hence as long asthere are edges left, we can keep making closed paths.Figure 1: The interaction graph shown can be decomposed into two paths, each of length 3as shown on the left, or into two paths, one of length 2 and one of length 4 as shown on theright.The decomposition of an interaction graph into closed paths is not unique. Indeed,consider the example shown in Figure 1. The interaction graph shown, with a total of 6edges, can be decomposed into two paths of length 3, or into one path of length 4 and onepath of length 2. N E N G N C F N G N C F N G N C F Table 1: A table showing how many interaction graphs N G with p vertices can be constructedusing N E edges. N C of the graphs are connected. The fraction F = N C N G tells us the probabilitythat a graph selected at random is connected. The first three columns have p = 4, the middlethree columns have p = 5 and the last three columns have p = 6.8he algorithm we use to generate interaction graphs is as follows:1. Partition the total number of edges E in the graph into a sum of path lengths in allpossible ways. The Gauss constraint forces paths to have a length of at least 2. Forexample, a graph with E = 4 edges can be realized as two paths of length 2 or onepath of length 4. We assume that the interaction graph has a total of p vertices andthat these vertices are labeled as 1 , , ..., p .2. Each path can be labeled with an ordered sequence of integers, which records the orderin which the different vertices are traversed as one travels on the path. Each path visitsany given vertex at most once. Thus, the integers appearing in a given path label aredistinct. In addition, since the path is closed, cyclic shuffling of the integers in the pathdoes not lead to a new path. This makes it clear that the paths of length L can belabeled by permutations that are a single cycle of length L . We now need to sum overcombinations of all possible paths consistent with the partition constructed in step 1.3. The resulting list of interaction graphs will have some duplicates, since the decompo-sition of a given interaction graph into a collection of paths is not unique. The finalstep in the algorithm simple deletes the duplicate graphs.For examples of the number of graphs obtained when using this algorithm, see Table 1. It isnoteworthy that the number of interaction graphs grows very rapidly. For example, there areroughly 25 million interaction graphs with 15 edges and 6 vertices. Such enormous numbersjustify a statistical approach to the problem.Before leaving this subsection, we will explain how to count the number of interactiongraphs, using methods from information theory used to count Markov types[45]. This count-ing will enable us to understand the number of interaction graphs as N → ∞ . Since thegraph is a label for the operator, this will allow us to count the number of orthogonal op-erators we have and thereby to verify that the growth is enough to explain the entropy ofa black hole. Introduce the matrix E ij , i, j = 1 , ..., p . The off diagonal matrix elements E ij denote the number of edges running from vertex i to vertex j . Clearly E ij is not in generala symmetric matrix . The diagonal matrix elements vanish E ii = 0. Our task is to countthe number of matrices obeying the equations p (cid:88) i =1 E ij = p (cid:88) i =1 E ji (3.1)and p (cid:88) i =1 p (cid:88) j =1 E ij = N E (3.2) By orthogonal operators, we mean operators which diagonalize the two point function. Thus they wouldbe orthogonal in the Zamolodchikov norm of the conformal field theory. The relation between E ij and the matrix N ij appearing in (2.4) is N ij = E ij + E ji . p verticesand N E edges, denoted N p,N E .Let E be the set of all integer matrices obeying (3.1) and let E N E be the subset of matricesbelonging to E that obeys (3.2). Given a pair of p × p matrices E and F , we define F ∗ E ≡ p (cid:89) i,j =1 i (cid:54) = j F E ij ij (3.3)We would like to evaluate the generating function Z Gauss ( F ) = (cid:88) E ∈E F ∗ E = (cid:88) n E ≥ (cid:88) E ∈E nE F ∗ E (3.4)Evaluating this generating function at F ij = z and using the obvious fact (cid:88) E ∈E nE F ∗ E = (cid:88) E ∈E nE z (cid:80) pi =1 (cid:80) pj =1 E ij = (cid:88) E ∈E nE z n E = N p,N E z n E (3.5)we find Z Gauss ( F ij = z ) = (cid:88) n E ≥ N p,N E z N E (3.6)We will now give a useful integral representation for Z Gauss ( F ) that uses nothing morethan the residue theorem. First, introduce the diagonal matrix D X = x · · · x · · · · · · x p −
00 0 · · · x p (3.7)which we will use below. Next, introduce the generating function Z ( F ) = (cid:88) E F ∗ E (3.8)where the sum above is over all matrices E ij with zeros on the diagonal and non-negativeintegers off the diagonal. There are two reasons for why it is useful to introduce this newgenerating function. First, it is a simple task to evaluate the sum and obtain an explicitanswer Z ( F ) = p (cid:89) i,j =1 i (cid:54) = j (1 − F ij ) − (3.9)10econd, it is possible to express Z Gauss ( F ) as a contour integral over Z ( F ). To see this, notethat the term that is independent of x i , i = 1 , · · · , p in Z ( D − X F D X ) = (cid:88) E p (cid:89) i,j =1 i (cid:54) = j F E ij ij p (cid:89) k =1 x (cid:80) pl =1 ,l (cid:54) = k E kl − (cid:80) pl =1 ,l (cid:54) = k E lk k = p (cid:89) i,j =1 i (cid:54) = j (cid:18) − z x i x j (cid:19) − (3.10)is obviously Z Gauss ( F ). Thus we have Z Gauss ( F ij = z ) = (cid:18) πi (cid:19) p (cid:73) dx x · · · (cid:73) dx p x p p (cid:89) i,j =1 i (cid:54) = j (cid:18) − z x i x j (cid:19) − (3.11)As an example, when p = 4 we find Z Gauss ( F ij = z ) = z − z + 3 z + 2 z − z + 2 z + 3 z − z + 1(1 − z ) ( z + 1) ( z + 1) ( z + z + 1) = 1 + 6 z + 8 z + 27 z + 48 z + 112 z + 192 z + 378 z + 624 z + 1092 z +1728 z + 2802 z + 4248 z + 6516 z + 9528 z + O ( z ) (3.12)which nicely confirms our numerical results in Table 1. For p = 5 we have Z Gauss ( F ij = z ) = n ( z )(1 − z ) (1 + z ) (1 + z ) (1 + z + z ) (1 + z + z + z + z ) (3.13)where n ( z ) = z − z + 7 z + 3 z + 2 z + 17 z + 35 z + 29 z + 45 z + 50 z + 72 z +50 z + 45 z + 29 z + 35 z + 17 z + 2 z + 3 z + 7 z − z + 1 (3.14)Expanding (3.13) we again confirm the results in Table 1.Starting from (3.11) we can now explore the growth of the number of interaction graphsas we take p → ∞ . Setting x i = e iθ i we have Z Gauss ( F ij = z ) = (cid:18) π (cid:19) p (cid:90) π − π dθ · · · (cid:90) π − π dθ p p (cid:89) i,j =1 i (cid:54) = j (cid:0) − ze i ( θ i − θ j ) (cid:1) − (3.15)The integrand is invariant under the simultaneous shift θ i → θ i − a , i = 1 , , · · · , p . Usingthis symmetry to carry out the integral over θ we obtain Z Gauss ( F ij = z ) = (cid:18) π (cid:19) p − (cid:90) π − π dθ · · · (cid:90) π − π dθ p p (cid:89) i =2 (cid:0) − ze iθ i (cid:1) − (cid:0) − ze − iθ i (cid:1) − p (cid:89) i,j =2 i (cid:54) = j (cid:0) − ze i ( θ i − θ j ) (cid:1) − (3.16)In terms of the function L ( z, θ , · · · , θ p ) = p (cid:88) i =2 log (cid:2)(cid:0) − ze iθ i (cid:1) (cid:0) − ze − iθ i (cid:1)(cid:3) + p (cid:88) i,j =2 i (cid:54) = j log (cid:2) − ze i ( θ i − θ j ) (cid:3) (3.17)we can write the number of interaction graphs as N p,N E = 1 i (2 π ) p (cid:73) dzz N E (cid:90) π − π dθ · · · (cid:90) π − π dθ p e − L ( z,θ , ··· ,θ p ) (3.18)To determine the asymptotic behavior of this integral we will use a saddle point evaluationas usual. Using the equivalent form L ( z, θ , · · · , θ p ) = p (cid:88) i =2 log (cid:2) − z cos θ i + z (cid:3) + 12 p (cid:88) i,j =2 i (cid:54) = j log (cid:2) − z cos( θ i − θ j ) + z (cid:3) (3.19)it is simple to verify that L ( z, θ , · · · , θ p ) assumes its minimum value at θ = · · · = θ p = 0.An equally simple computation shows that, at this minimum, L ( z, θ , · · · , θ p )+( N E +1) log z is minimized at z = N E + 1 p − p + N E + 1 (3.20)Setting p = (cid:15)N and N E = (cid:15) N and working to leading order in the saddle point approxi-mation, we find at large N that N p,N E ∼ e (cid:15) N log(2) (3.21)Assuming that the interaction graphs do indeed label microstates of a black hole, this is thecorrect growth to reproduce the expected black hole entropy. Given this algorithm we can now easily generate collections of graphs, and then use theseto numerically characterize the properties of interaction graphs. We would like to employthe notion of typicality . Something is typical if it happens in the vast majority of cases: thetypical lottery ticket loses, after 1000 coin flips we typically find the ratio of the number ofheads to the number of tails is close to 1 and so on. We would like to characterize the typicalinteraction graph.Our goal now is to make the above intuitive notions mathematically precise. For usefulbackground see [46]. What does it mean for an interaction graph to be typical? Consider12n element x of a set S , x ∈ S . Typicality is a relational property of x , which x possesseswith respect to S . Typicality refers to an attribute P and a (probability) measure for thisattribute µ P . For our discussion, S is the set of all interaction graphs, with a given number ofvertices p and edges E , denoted S p,E . As discussed in the previous section, we can consider p ∼ (cid:15)N . Thus, at large N we know that p is enormous and the number of interactiongraphs explodes. We will also assume that we are in the “dense graph” regime specified byallowing the total number of edges to scale as E ∼ p ∼ (cid:15) N . Thus, we are interested incharacterizing the typical graph in the set S p,p of interaction graphs.We will define the measure µ P simply by counting. This assumes that every graph isequally likely. In this case the probability µ P that a given graph has property P is simplygiven by counting the number of graphs with property P and then dividing by the totalnumber of graphs. When µ P tends towards 1, P becomes a property of a typical graph. Inwhat follows we are interested in determining some of the properties of a typical graph in S p,p .One interesting attribute P is whether or not the graph is connected. For a Hamiltoniandefined using a disconnected graph, the bosons hopping on the graph are confined to a givenconnected component. A state that is not initially entangled can never build up entanglementbetween Hilbert spaces defined on vertices of different disconnected components of the graph.Its only on a connected graph that an initial state that is not entangled can evolve intoa maximally entangled state, entangling all of the Hilbert spaces defined at the differentvertices. The trend shown in Table 1 is exactly what one expects: for a fixed numberof vertices, as the number of edges increases the probability that the graph is connected(denoted by F in Table 1) increases. Our numerical results imply that just as E approaches p , this probability of being connected approaches 1 . With this numerical evidence, weassume in what follows that the typical graph in S p,p is connected.As discussed in the introduction, when deriving the Lieb-Robinson bound for dynamicson a graph an important parameter which enters the bound is the maximum degree D V of any vertex in the graph . Thus, a second interesting attribute P for the questions weconsider is the maximum degree D V . In Figure 2 we have given histograms for the differentvalues of D V on the sets S ,E . In this case, the largest value D V can attain is 4, when agiven vertex connects to all of the remaining vertices. The first histogram has E = 6 edges.There are significant fractions of graphs with all possible allowed values D V = 1 , , ,
4. As E increases a definite pattern emerges: D V = 4 becomes the most probable value for E ≥ E = 16 edges. It is clear that by the time we reach E = p = 25 edges, the overwhelming majority of graphs will have D V = 4 = p −
1. Basedon this numerical evidence, we assume in what follows that the typical graph in S p,p has D V = p −
1. Since we work at large p we simplify this to D V = p . For p = 4 and E = 20 we find that a graph is connected with probability 0.99. Our graphs can have multiple edges between a given pair of vertices. D V counts how many other vertices V is connected to and not the number of edges with an endpoint on V D V , for Gaussgraphs with 5 vertices and E = 6 , , , ,
14 and 16 edges.It is interesting to ask how this typical value of D V is reached. It maybe that most graphshave a single vertex with a large value for D V and the remaining vertices have much smallervalues for their degree. In this case, since there are p vertices, we will find that the averagevertex degree stays close to 1. The opposite extreme is that the degree of all vertices isincreasing roughly equally, so that most graphs in S p,p have an average vertex degree whichis close to the maximum value of p −
1. Numerically we find that the average vertex degreeis an increasing function with the number of edges E (see Figure 4) and that when E ∼ p we find an average value close to the maximum allowed value. Of course we can not probelarge values of p (already p = 6 requires very long run times), but this conclusion makessense: nothing has introduced an asymmetry between the p vertices, so we would expect thedegree of each vertex to be roughly equal. Thus, from now on we assume that most verticesin a typical graph in S p,p have the maximum degree.The conclusions we have reached in this section regarding the typical interaction graphhave a number of interesting implications. Recall that each vertex in the graph is a giantgraviton brane and each edge is an open string excitation of the brane. By characterizingthe typical graph we are learning about the typical excited state of this p giant gravitonsystem. The typical state of a system of p giant graviton branes, excited by stretching p open strings between the branes, has roughly the same number of open strings endpointsglued to each brane. In terms of the Gauss graph operators, the differences between the rowlengths of R and those of r are roughly constant, equal to p . This is good news: if one simplypiled all the excitations into a small number of rows of R one might imagine a situation inwhich Young diagram R satisfies the distant corners approximation, but the approximationbreaks down for r . This might invalidate the derivation given in [25] which assumes that thecorners of both R and r are distant. Fortunately this does not happen for the typical graph This would correspond to piling many edges onto one vertex of the interaction graph. p = 4 verticesversus the number of edges E . For E = 16 the average vertex degree is 2.55. This indicatesthat most vertices that have been averaged over must assume the maximum value of p − S p,p has a mean vertex degreewhich is close to the maximum value, we find that the generic Hamiltonian defined by themixing problem for heavy operators, has all-to-all interactions. In this section we would like to explore how quickly entanglement is generated by the typicalgraph Hamiltonian. We restrict ourselves to the subspace of Fock space with a definite num-ber N b of bosons. Towards this end, in the next section we will formulate a Lieb-Robinsonbound for the typical graph Hamiltonian. The bound limits the growth of commutators[ O i (0) , O j ( t )] where O i ∈ F i so that we are bounding the growth of operators [50]. Thegrowth of operators is a reliable probe of scrambling[51, 27, 52, 53]. To obtain our bounds,we use arguments of [54], used to derive Lieb-Robinson bounds for general harmonic systemson general lattices. Information that has been scrambled is stored in the complicated corre-lations between many different subsystems. Consequently, scrambling is intimately relatedto the generation of entanglement. With this motivation, we consider in Section 4.2 a toymodel for our system, simple enough that we can compute the Von Neumann entropy asa function of time, using the reduced density matrix of a given Fock space F i and startingfrom an initially unentangled state. Although this explicitly shows the generation of entan-glement in the system, it also poses a puzzle: the recurrence time associated with the typicalHamiltonian is much smaller than expected. 15 .1 Lieb-Robinson bound for typical graph dynamics Trade the oscillator operators b i , b † i for a pair of Hermittian operators, α i and β i given by α i = b i + b † i √ β i = b i − b † i i √ O we have[ α i , O ] = 0 [ β i , O ] = 0 (4.2)then O is a multiple of the identity operator. Rewriting the Hamiltonian in terms of the α i , β i operators, we obtain the following result H = H + 12 p (cid:88) i,j =1 α i M ij α j + 12 p (cid:88) i,j =1 β i M ij β j (4.3)where H is an additive constant equal to H = 2 g Y M (4 π ) p (cid:88) i,j =1 ( √ r i − √ r j ) N ij − g Y M π p (cid:88) i =1 r i k i l i (4.4)and the matrix M ij is given by M ij = 2 g Y M (4 π ) r i k i l i δ ij − g Y M (4 π ) (cid:114) r i r j l i l j N ij (4.5)Recall that r i , l i and k i were introduced in Section 2. A simple computation shows that α i ( t ) = e iHt α i e − iHt = [cos( M t )] kj α j + [sin( M t )] kj β j (4.6) β i ( t ) = e iHt β i e − iHt = [cos( M t )] kj β j − [sin( M t )] kj α j (4.7)Using these results we immediately obtain the following commutators i [ α k ( t ) , β j ] = − cos( M t ) kj i [ α k ( t ) , α j ] = sin( M t ) kj i [ β k ( t ) , α j ] = cos( M t ) kj i [ β k ( t ) , β j ] = sin( M t ) kj (4.8)To proceed we would like to estimate the size of terms of the form ( M n ) kj . Recall that p = (cid:15)N . From our analysis of the typical graph, we know that all vertex degrees are close tothe maximal value of p , which implies that matrix elements N ij are typically non-zero andorder 1. Consequently, the size of the off diagonal elements of M ij are − g Y M (4 π ) (cid:114) r i r j l i l j N ij ∼ − g Y M (4 π ) (cid:114) r i r j l i l j = − (cid:15)λp (4 π ) (cid:114) r i r j l i l j ≡ (cid:15)λc ij p (4.9)16here c ij = − (cid:15)λ (4 π ) (cid:114) r i r j l i l j (4.10)is a small number, independent of N . For the diagonal elements of M ij , we use the fact thatfor the typical graph we have k i = the degree of the i th vertex = p independent of i andhence these matrix elements are of size2 g Y M (4 π ) r i k i l i = 2 (cid:15)λ (4 π ) r i l i ≡ (cid:15)λc i (4.11)which is independent of N . Using these results, we can bound the size of | ( M n ) ij | , for i (cid:54) = j .Choose the constant c > | c ij | for all i, j and larger than | c i | for all i . Wewill illustrate the computation with two examples and then state the general rule. For n = 1we are talking about an off diagonal element so that M ij = λ(cid:15)c ij p ⇒ | M ij | < cλ(cid:15)p (4.12)For n = 2 we have a product of two matrices. There is a single index summed. Thus, wehave p − | ( M ) ij | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p (cid:88) k =1 k (cid:54) = i,j c ik c kj (cid:15) λ p + c ij (cid:15) λ p ( c ii + c jj ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < p (cid:88) k =1 k (cid:54) = i,j | c ik | | c kj | (cid:15) λ p + | c ij | ( | c ii | + | c jj | ) (cid:15) λ p< ( p − c (cid:15) λ p + 2 c (cid:15) λ p = 3 c (cid:15) λ p + O (cid:18) p (cid:19) (4.13)We drop the p − term. Proceeding in this way its easy to see that | ( M n ) ij | < (2 n + 1) c n (cid:15) n λ n p (4.14)Consequently, for example, we can estimate | i [ α k ( t ) , β j ] | = | cos( M t ) kj | < ∞ (cid:88) n =0 t n (2 n )! | ( M n ) kj | < ∞ (cid:88) n =0 t n (cid:15) n λ n (2 n )! c n p (2 n + 1)= cosh( c(cid:15)λt ) + c(cid:15)λt sinh( c(cid:15)λt ) p (4.15)17he right hand side becomes of order 1 when e c(cid:15)λt ∼ p i.e. when t ∼ log p(cid:15)λc . At this time scale,the bounds for all of the commutators in (4.8) are order 1.Define the amount of time t sig as the smallest time needed to signal from site j to site k .This implies that for suitable operators O j acting on F j and O k acting on F k we have (cid:104) ψ (0) | [ O j (0) , O k ( t sig )] | ψ (0) (cid:105) > δ (4.16)with δ some O (1) number. The Lieb-Robinson bound then forces t sig > log p(cid:15)λc (4.17)This logarithmic scaling of signaling implies a logarithmic scaling of the scrambling time.Thus, our Hamiltonian does not scramble in a time less than ∼ log p consistent with the fastscrambling conjecture [26]. Recall that the matrix M ij for the typical interaction graph is given by M ij = v i v j (cid:18) δ ij − p (cid:19) v i = (cid:115) λ(cid:15) (4 π ) (cid:114) r i l i (4.18)where (cid:15), λ are both fixed and much smaller than 1 as we take N → ∞ . The ratios r i l i arelarger than 1 and fixed as we take N → ∞ . We will now make a simplifying assumption,that will yield a problem that is simple enough to solve. We assume that the v i ’s are sosimilar that we can simply set them to be equal to v . There are examples for which thisis indeed an accurate assumption, but this is besides the point. We make the assumptionbecause it leads to a simple model that nevertheless captures the scaling with p of matrixelements of the Hamiltonian and it also captures the all-to-all interactions property of thetypical Hamiltonian. In this case M has the following form M = v (1 + 1 p ) + v K = v (1 + 1 p ) + v − p − p − p . . . − p − p − p − p − p . . . − p − p − p − p − p . . . − p − p ... ... ... . . . ... ... − p − p − p . . . − p − p − p − p − p . . . − p − p (4.19)where is the p × p identity matrix. Starting from the initial state | ψ (0) (cid:105) = b † i | (cid:105) (4.20)18t is straight forward to find the probability that we have a particle on site i at time t |(cid:104) | b i | ψ ( t ) (cid:105)| = p + 2( p −
1) cos( pv t ) − p + 2 p ≡ p i ( t ) (4.21)The reduced density matrix obtained by tracing over all Fock spaces F j , j (cid:54) = i acts on thetwo dimensional subspace of F i with basis {| (cid:105) , b † i | (cid:105)} . The reduced density matrix is givenby ρ i ( t ) = (cid:20) − p i ( t ) 00 p i ( t ) (cid:21) (4.22)and the corresponding Von Neumann entropy is S i = − Tr ( ρ i log ρ i )= − p −
1) cos( v t ) + p − p + 2 p log (cid:18) p −
1) cos( v t ) + p − p + 2 p (cid:19) + 2( p − − cos( v t )) p log (cid:18) p − − cos( v t )) p (cid:19) (4.23)Figure 4: The above plot shows the Von Neumann entropy versus v t , for p = 100.The above plot shows that entanglement is generated and it exhibits the recurrence timeof T r = 2 π/v . At time T r the entanglement entropy returns to zero and the curve repeats.This recurrence time is much smaller than expected and it casts a doubt on the model.Typically the recurrence is doubly exponential in the size of the system [55, 56, 57]. Thissuggests a recurrence time of ∼ exp exp p . In the next section we will consider equilibrationfor dynamics described by the typical Hamiltonian, which will lead to an explanation forthis tiny recurrence time. 19 Equilibration during intervals
The classical idea of equilibration involves evolution towards thermal equilibrium. A closedfinite dimensional quantum system evolving unitarily has recurrent and time reversal invari-ant dynamics. It can never come to equilibrium in the classical sense, so we must relax theabove classical notion. At quantum level, equilibration will mean that a quantity, initialisedat a non-equilibrium value, evolves towards the equilibrium value and then stays close to itfor an extended time . This leads to two natural notions of equilibration (see [40]):1. Equilibration on average:
A time dependent observable equilibrates on average ifits value is for most times during the evolution close to some equilibrium value.2.
Equilibration during intervals:
A time dependent property equilibrates during a(time) interval if its value is close to some equilibrium value for all times in that interval.Results establishing equilibration during intervals imply bounds on the time it takes to equi-librate. These time scales are of central interest to us, so we use the second notion above.The conditions under which equilibration during intervals of quadratic bosonic Hamiltoni-ans can be guaranteed has been studied in [58, 59]. Since our system is a quadratic bosonicHamiltonian, these results are immediately applicable.We will study a subsystem, given for simplicity by a single site. The rest of the systembehaves like a heat bath allowing the subsystem to reach a state that maximizes its entropy.Our strategy is to start the entire lattice system in an initial non-equilibrium state and thento demonstrate that the state of a single site evolves to the equilibrium state, that is, thestate that maximizes the entropy.To carry out the computation, we need to know something about the state that maximizesthe entropy. The state (density operator ρ ) that maximizes the entropy H ( ρ ) = − Tr ( ρ log ρ ) (5.1)for given mean and second moments , is a Gaussian state [60]. Instead of discussing thedensity operator itself, it is useful to study the characteristic function χ ( β ). The charac-teristic function contains all the information necessary to reconstruct the density matrix sothat it is an alternative description of the system. The characteristic function is given bythe expectation value of the Weyl operators, defined by D ( β ) = e (cid:80) pi =1 β i b † i − β ∗ i b i (5.2) This requires a measure that quantifies how close the value of an obervable is to its “equilibrium value. Hamiltonians that are quadratic polynomials in the bosonic creation and annihilation operators. Define b ( α ) = (cid:80) li =1 ( α i b † i − α ∗ i b i ). We call (cid:104) b ( α ) (cid:105) the mean and (cid:104) b ( α ) b ( α (cid:48) ) (cid:105) the correlation matrix or thesecond moment.
20 density operator ρ is called Gaussian if its quantum characteristic function has the formTr ( ρ V ( z )) = e im · β − β ∗ · α · β (5.3)with m i , α ij constants independent of β i , β ∗ i . To see why Gaussian states maximize theentropy, recall that given any density matrix ρ , there is a Gaussian density matrix ˜ ρ withthe same mean and second moments[61]. Consider the quantity H ( ˜ ρ ) − H ( ρ ) = Tr ( ρ (log ρ − log ˜ ρ )) + Tr (( ρ − ˜ ρ ) log ˜ ρ ) (5.4)The first term on the right hand side is the relative entropy, which is non-negative[62]. Thesecond term on the right hand side vanishes because (i) log ˜ ρ is a quadratic polynomial in b i , b † i and (ii) ρ and ˜ ρ have the same first and second moments. This proves that H ( ˜ ρ ) − H ( ρ ) ≥ E is all sitesexcept for site i . The reduced density matrix ρ i = Tr E ( | φ (cid:105)(cid:104) φ | ) (5.6)is an operator acting in the Hilbert space associated to the i th site. Put m bosons on eachsite so that the initial state is | φ (cid:105) = | m (cid:105) ⊗ p | m (cid:105) ⊗ p = p (cid:89) i =1 ( b † i ) m √ m ! | (cid:105) i (5.7)Notice that this initial state is not entangled and is nothing like the maximally entangledequilibrium state. We will evaluate the characteristic function χ i ( α, t ) = Tr (cid:16) ρ i ( t ) e αb † i − α ∗ b i (cid:17) = (cid:104) φ | e αb † i ( t ) − α ∗ b i ( t ) | φ (cid:105) (5.8)From (4.6) and (4.7) we have b i ( t ) = ( e − itM ) ij b j ≡ U ij b j b † i ( t ) = ( e itM ) ji b † j = U ∗ ji b † j (5.9) M is a symmetric matrix so U is also symmetric. Using the initial state χ i ( t ) = (cid:104) φ | e (cid:80) pj =1 ( αU ij b j − α ∗ U ∗ ij b † j ) | φ (cid:105) = p (cid:89) j =1 j (cid:104) m | e αU ij b j − α ∗ U ∗ ij b † j | m (cid:105) j (5.10)we can evaluate each factor in this product. First, using the Baker-Campbell-Haussdorfformula [65] it is simple to verify that (no sum on j ) e αU ij b j − α ∗ U ∗ ij b † j = e αU ij b j e − α ∗ U ∗ ij b † j e − | α | | U ij | (5.11)21sing this identity we easily find j (cid:104) m | e αU ij b j − α ∗ U ∗ ij b † j | m (cid:105) j = j (cid:104) m | e αU ij b j e − α ∗ U ∗ ij b † j | m (cid:105) j e − | α | | U ij | = m (cid:88) k =0 m (cid:88) l =0 ( αU ij ) k ( − α ∗ U ∗ ij ) l k ! l ! m ! (cid:112) ( m − k )!( m − l )! j (cid:104) m − k | m − l (cid:105) j e − | α | | U ij | = m (cid:88) k =0 ( −| α | | U ij | ) k k ! k ! m !( m − k )! e − | α | | U ij | = L m ( | α | | U ij | ) e − | α | | U ij | (5.12)where L m ( · ) is a Laguerre polynomial. Thus, we find χ i ( t ) = p (cid:89) j =1 L m ( | α | | U ij | ) e − | α | | U ij | (5.13)We want to prove that, at late times, the characteristic function becomes a Gaussian, i.e.that at late times we have χ i ( t ) = e − cα (5.14)where c is a constant (independent of α ). Considerlog χ i ( t ) = − p (cid:88) j =1 | α | | U ij | + p (cid:88) j =1 log L m ( | α | | U ij | ) (5.15)Expand the log p (cid:88) j =1 log L m ( | α | | U ij | ) = p (cid:88) j =1 (cid:0) − L m ( | α | | U ij | ) (cid:1) + p (cid:88) j =1 ∞ (cid:88) k =2 (1 − L m ( | α | | U ij | )) k k (5.16)and use the expansion of the Laguerre polynomials L m ( x ) = m (cid:88) n =0 m !( m − n )! n ! n ! ( − x ) n = 1 − mx + O ( x ) (5.17)If we can argue that x = | α | | U ij | is small for late times t , then we can set p (cid:88) j =1 log L m ( | α | | U ij | ) = p (cid:88) j =1 m | α | | U ij | (5.18)and consequently log χ i ( t ) = − p (cid:88) j =1 | α | | U ij | − p (cid:88) j =1 m | α | | U ij | (5.19)22hich would establish the result. Determining how rapidly | α | | U ij | approaches zero willtell us how quickly the system equilibrates.Why should | α | | U ij | get small for large t ? The matrix M ij is real and symmetric, sothat it can be diagonalized. Denote the eigenvectors and eigenvalues of M ij , labeled by k = 1 , · · · p , as ( η k ) i and λ k respectively. In terms of these eigenvectors and eigenvalues wehave U ij = p (cid:88) k =1 ( η k ) i e − iλ k t ( η ∗ k ) j (5.20)so that | U ij | = p (cid:88) k =1 p (cid:88) l =1 ( η k ) i e − iλ k t ( η ∗ k ) j ( η ∗ l ) i e iλ l t ( η l ) j (5.21)There are two sums above which become infinite sums at large N . If the eigenvalues aredistinct, we are adding terms with different rapidly oscillating phases for large t , so that therewill be many cancellations and we expect the sum is small. To formulate a precise argumentwe need to know the λ k and ( η k ) i . We have not managed to solve for the eigenvectors andeigenvalues of M in general, but can do so for the toy model we introduced in Section 4.2.In this case M has the following form M = v (1 + 1 p ) + v K (5.22)where is the p × p identity matrix. Everything is an eigenvector of the identity matrix, sowe need only find the eigenvectors and eigenvalues of K . Notice that K has rank one. Recallthat the rank of K is the dimension of the vector space spanned by its columns. Since allof the columns of K are identical they span a one dimensional space. The rank is also equalto the number of non-zero eigenvalues, so K has only one non-zero eigenvalue equal to − | k = − (cid:105) has every component equal to 1 | k = − (cid:105) = 1 √ p (5.23)The remaining K eigenvectors span the subspace orthogonal to | k = − (cid:105) and have K eigenvalue equal to zero. Thus, | k = − (cid:105) is an eigenvector of M with eigenvalue equalto p while any vector orthogonal to | k = − (cid:105) is also an eigenvector of M with eigenvalueequal to 1 + p . In this case the eigenvalues are not distinct and terms in (5.21) will not ingeneral cancel. Thus we don’t expect | α | | U ij | to becomes small for large t and hence thesystem will not equilibrate. To get some insight into what is going on, consider an initialstate, with excitations localized on a pair of lattice sites i and j , given by | ψ (cid:105) = 1 √ (cid:16) b † i − b † j (cid:17) | (cid:105) (5.24)23e easily find H | ψ (cid:105) = (cid:18) H + v + v p (cid:19) | ψ (cid:105) + v √ p (cid:88) k =1 ( K ki − K kj ) b † k | (cid:105) (5.25)The second term in the last line above allows excitations to move from their original latticesite to a new lattice site. However the two contributions cancel so that | ψ (cid:105) is an eigenstateand the excitation does not disperse - it remains localized on sites i and j . The excitationcan move to every other site with exactly equal hopping strength, so that in the end theexcitations are blocked from moving anywhere and are instead localized. This is rathergeneric: all states in the Hilbert space orthogonal to the state | k = − (cid:105) = 1 √ p p (cid:88) i =1 b † i | (cid:105) (5.26)are eigenstates of the Hamiltonian and hence do not evolve in time . The intuitive picturebehind equilibration is as follows [63]: as time evolves, the system becomes correlated.From each site a wave front moving at the speed of sound for the lattice emerges, carryinginformation. The cumulative effect is an effective averaging process: information stored atone site becomes spread across the entire lattice. In our case, since the wave fronts areblocked from moving, we should not expect the system to equilibrate.The spectrum of the typical Hamiltonian gives us an explanation for why we found sucha small recurrence time. In the large p limit there is a single energy eigenvalue equal to v p = 0 + O ( p − ) and p − v − v p = v + O ( p − ). The energy ofthese degenerate states is the only energy in the problem and it clearly sets the recurrencetime we found. To get such a simple spectrum things must be fine tuned. Random deviationsfrom this typical Hamiltonian will lift the degeneracy leading to a spectrum that is morerealistic as we will soon see.So, the Hamiltonian associated to the typical interaction graph exhibits localization.However, even small fluctuations about this typical configuration should disrupt the local-ization. Lets again look at a simple example. Choose a matrix M ij which opens up a“conducting path” that passes through each vertex of the graph, as follows M = v (1 + 1 p ) + v K + v − p p . . . p p − p p . . . p − p . . . . . . − p p p . . . p − p = v (1 + 1 p ) + v K + v L At large p the Hamiltonian becomes a projector onto the space orthogonal to the | k = − (cid:105) so that theresult of applying the Hamiltonian to any state is an eigenstate. p matrixelements out of a total of p matrix elements, and we have only adjusted each element byan amount ∼ p − . The form of matrix L was chosen so that we can again solve for theeigenvectors and eigenvalues of M exactly. First note that [ L, K ] = 0 so that L and K can be simultaneously diagonalized. A simple computation shows that the eigenvectors andeigenvalues of L are given by( η k ) l = e ikl √ p λ k = 2 cos( k ) − p − pδ k, p v k = 2 nπp with n = 0 , , , · · · , p −
1. The eigenstate with n = 0 is the eigenstate of K with eigenvalue −
1. Using these eigenvalues and eigenvectors we have U ij = p − (cid:88) n =0 e i πnp ( i − j ) e − i nπp )+ p − pδk, − p v t = U ( i − j ) ∼ i i − j e − iv × πi i − j (cid:90) π dφe − i p cos( φ ) v t e i ( i − j ) φ = i i − j J i − j ( 2 v tp ) (5.27)where J l ( x ) is the Bessel function. This is the kind of result we want because we know that | J l ( x ) | < x − for all x ≥ v t eq p ∼ ⇒ t eq ∼ p v (5.28)the matrix elements U ij are becoming small enough to neglect and the density matrix isapproaching a Gaussian state. Thus, the system evolves to the state of maximum entropyand we come to equilibrium. Notice that this time is much much smaller than the enormousrecurrence time. The system now remains at equilibrium until we get close to the recurrencetime. Notice that t eq is significantly larger than the scrambling time. We have studied the one loop mixing problem for operators with a large enough bare di-mension that they could be dual to black holes or new spacetime geometries. This mixingproblem is significantly more complicated than the planar mixing problem. Despite this, aremarkably simple description emerges. The dilatation operator defines dynamics on a graphof the type that has recently been suggested as models for quantum dynamics of black holes[27, 28, 30, 31, 32, 33]. It is intriguing to see simple dynamics on graphs emerging naturallyfrom the mixing problem of very large dimension operators in Yang-Mills theory.Each operator has a number of labels, one of which is the interaction graph. Operatorsonly mix if they have the same interaction graph label. We have carried out a careful25ounting of the interaction graphs and find that the number of graphs matches the entropyof a black hole suggesting that we might think of these operators as dual to a black holemicrostate. By numerically generating lists of graphs we have characterized the “typicalinteraction graph” and the dynamics associated to it. We find a lattice model defined on p sites with p ∼ O ( N ) and with all-to-all interactions. Despite this non-locality, we haveproved that the scrambling time is bounded consistent with the fast scrambling conjecture.By considering a specific example, we have also given evidence that the system equilibratesin a time scale t ∼ pλ where p ∼ N .The idea that gravitational dynamics should emerge from the sector of heavy operators inthe Yang-Mills theory has been pursued in [66, 67, 68, 69, 70, 71]. Our study is a continuationof these ideas.There are a number of interesting directions that could now be pursued. Our analysishas all been limited to weak coupling. To make contact with black hole physics we need tomake progress in understanding the strong coupling limit of the theory, which is presently aformidable problem. However, one might look for BMN like [72] limits or for observables thatare protected by super symmetry, which has not yet been considered in the setting of heavyoperators. A more manageable problem is to generalize our analysis to generic operatorsconstructed using all of the fields in the field theory. By using only complex scalar fields φ i and not φ † i , we naturally construct operators that have dimension close to their R -charge.By including enough φ † i fields we would be able to construct operators with the quantumnumbers expected for near extremal or even Schwarzschild black holes. This generalizationshould be a straight forward technical exercise. The spectrum we have computed may findapplication in the arguments of [73] which explore how the thermodynamics of small blackholes is recovered from the dual conformal field theory. Our considerations of equilibrationmade use of two specific examples and a specific initial condition. Clearly a lot more is neededto properly understand the equilibration of our system and the associated time scales. Itwould also be interesting to explore situations in which we need to correct the distant cornersapproximation, which are required when the giant gravitons become coincident in space time.Thermal averages in the Yang-Mills theory involve averages over the complete ensemble ofgraphs. Corrections to the distant corners approximation would allow transitions betweendifferent graphs and the number of particles hopping on the graph would no longer beconserved.Finally, our goal was to gain some insights into the mechanism behind extremely rapidblack hole thermalization rates which must be present in the dynamics of large N Yang-Mills theories. Since our study has reduced to simple dynamics on graphs, perhaps the mostimportant lesson to be drawn is that the “toy models” considered in [27, 28, 30, 31, 32, 33]may in fact be better than one might have expected. The description in terms of a graphcertainly carries over to the case that more fields are included, but its validity at strongcoupling is yet to be established. 26 cknowledgements
This work is supported by the Science and Technology Program of Guangzhou (No.2019050001), by a Simons Foundation Grant Award ID 509116 and by the South AfricanResearch Chairs initiative of the Department of Science and Technology and the NationalResearch Foundation. We are grateful for useful discussions to Sanjaye Ramgoolam.
A Gauss Graph Hamiltonian from Yang-Mills
In this section we review the results of [24, 25], where the Hamiltonian we study (2.4) wasderived. We consider the mixing problem for operators belonging to the su(2 |
3) sector of thetheory. Truncation to this subsector is consistent to all orders of perturbation theory[74].We choose this sector because it is the maximal closed subsector with finitely many fields.The fact there are finitely many fields simplifies the analysis and it is possible to obtainexplicit formulas for the action of the dilatation operator.A basis for these operators is given by the restricted Schur polynomials. The relevantrestricted Schur polynomials are labeled by 6 Young diagrams and some multiplicity labels.We study operators with ∆ ∼ N that are holographically dual to a system of giant gravitons.Operators with p long columns (rows) are dual to a system of p (dual) giant gravitons .These operators mix with each other, but not with operators labeled by Young diagramsof a different shape. We take n ∼ (cid:15)N with (cid:15) (cid:28)
1. There are bosonic φ , φ excitations,as well fermionic ( ψ and ψ ) excitations. Limit the number of excitations by requiring n ∼ n ∼ m ∼ m ∼ (cid:15) N . We use a collective label N A = ( n , n , m , m ) to refer to thenumber of excitations.We will now explain why our operators are labeled by Young diagrams. To construct allpossible gauge invariant operators, we can take a product of an arbitrary number of fields andthen contract all row indices with all column indices to obtain a gauge invariant operator.We can specify which row indices are to be computed with which column indices by giving apermutation. So we could label our operators with a permutation. Alternatively, by taking aFourier transform on the group, we can trade the permutation for the label of an irreduciblerepresentation, that is, for a Young diagram . This introduces the Young diagram R whichhas as many boxes as fields used to construct the operator, i.e. it has n + n + n + m + m boxes. For operators dual to giant gravitons[75], the Young diagram R has a small numberof long columns and for operators dual to dual giant gravitons[76, 77], the Young diagram R has a small number of long rows[9, 13, 78]. We will consider operators with a total of p Branes connected by an open string described using a spin chain have been considered in [79, 80, 81, 82,83]. This discussion is not quite the whole story. We should have one Young diagram for the row indices andone for the column indices. Projecting to the singlet then forces these two to agree. . Since we have five differenttypes of fields, this makes a total of 6 Young diagrams. Each box in the Young diagram R corresponds to a field, and we can specify how many fields of each species appear in a givenrow of R . This specifies the excitations of each dual giant graviton brane.The operators that are obtained by this construction have orthogonal two point functionsin the free field theory[14], provide a complete linear basis for local gauge invariant operators[15] and they mix only weakly when interactions are turned on [22]. The Hamiltonian westudy is derived by evaluating the action of the one loop dilatation operator in the su(2 | D = − g Y M (4 π ) (cid:32) (cid:88) i>j =1 Tr (cid:0) [ φ i , φ j ] (cid:2) ∂ φ i , ∂ φ j (cid:3)(cid:1) + (cid:88) i =1 2 (cid:88) a =1 Tr ([ φ i , ψ a ] [ ∂ φ i , ∂ ψ a ])+ Tr ( { ψ , ψ } { ∂ ψ , ∂ ψ } ) (cid:33) (A.1)on restricted Schur polynomials. It is useful to introduce the notation D ≡ − g Y M (4 π ) (cid:88) A>B =1 D AB (A.2)where D AB mixes fields of species A and B . A major simplification in this computationfollows by noting that at large N , corners on the right hand side of the Young diagram arewell separated. This is the displaced corners limit [21, 22]. The action of the symmetricgroup simplifies in this limit and there are new symmetries: swapping the row or columnindices of fields that belong to a given species and sit in the same row of R is a symmetry.To use these new symmetries we refine the number of fields of a species N A to produce a p dimensional vector (cid:126)N A , with each component recording how many fields are in a given row.For example, the number of φ fields n is refined to produce (cid:126)n , and the group swapping φ fields in a given row, the enhanced symmetry of the displaced corners limit, is H (cid:126)n = S ( n ) × S ( n ) × · · · × S ( n ) p (A.3) Recall that the permutation group swapping indices of all fields has appeared. R is a representationof this group. The representations for each species are a representation of the subgroup which swaps onlyindices of fields that are the same species. The representation of the subgroup can be embedded into R inmore than one way and this is why we need multiplicity labels. We divide on the left to account for the symmetry associated with the row indices and on the right toaccount for the symmetry associated with column indices. See (A.4).
28n this limit, the number of restricted Schur polynomials matches the order of the doublecoset, indicating that we can organize the local operators using the double coset [23]. Thefour double cosets relevant for labeling our operators are A ↔ σ A ∈ H (cid:126)N A \ S N A /H (cid:126)N A (A.4)These double cosets are the crucial ingredient needed to make the connection to physics ona graph. Indeed, the collection of graphs with n edges and p vertices, and with number ofedges terminating at each vertex recorded in (cid:126)n is described by a double coset [86]. By thisconnection each element of a double coset is described by a graph, so that we can label ouroperators by a graph. Diagonalizing D φ ,A ∈ { D φ φ , D φ φ , D φ ψ , D φ ψ } first, the resultingeigenoperators are the Gauss graph operators [22, 23], labeled by two Young diagrams (the R and r labels of the restricted Schur polynomial) and a graph (which takes the place offour Young diagrams). Vertices of graphs correspond to rows/columns of r , i.e. each vertexcorresponds to a giant graviton brane. Each A field type is a species of edge in the graphand there is an edge for each field. Edges are directed. We give the complete graph as agraph for each A , specified by four elements (cid:126)σ , one of each of the four double cosets in (A.4).These are the graphs that we call Gauss graphs.Intuitively its clear why the graph provides a useful description: it naturally accounts forthe symmetries of the displaced corners limit. Recall that each row of R corresponds to avertex and each edge in the graph corresponds to a field in the operator. The symmetry ofswapping row indices of fields in a given row is now the symmetry of swapping endpoints ofedges that end on the same vertex (an obvious symmetry of the graph) while the symmetryof swapping column indices of fields is the symmetry of swapping start points of edges thatstart on the same vertex.The elements of the double cosets in (A.4) correspond to the graphs we consider. Verticescan be dressed by closed edges with ends attached to the same vertex or by edges betweentwo distinct vertices. Fermi statistics forbids two or more parallel edges (edges with thesame orientation and endpoints) of the same fermion species [87]. We refined N A to producea vector (cid:126)N A . To describe the graph refine (cid:126)N A to produce a matrix ( N A ) i → j whose elementsdescribe the number of edges running from vertex i to vertex j . In terms of this matrix, theGauss Law constraint is (cid:80) k (cid:54) = i ( N A ) i → k = (cid:80) k (cid:54) = i ( N A ) k → i . The transformation from restrictedSchur basis to Gauss graph basis is derived in [88]. After the transformation, the dilatationoperator is most naturally written as a system of particles hopping on a lattice, with latticesites given by vertices of the Gauss graph [24]. Closed edges forming loops at a vertextranslate into particles at that site. The hopping strength is determined by the number ofedges of all other species stretched between the vertices. There are two distinct species ofbosons, for φ , φ , and two distinct species of fermions, for ψ , ψ .To simplify the discussion that follows, we will consider only the bosonic sector of thetheory. The bosons are described by oscillators (cid:2) a ij , ¯ a kl (cid:3) = δ il δ jk (cid:2) b ij , ¯ b kl (cid:3) = δ il δ jk (A.5)29ith all other commutators vanishing. The Fock space vacuum | (cid:105) obeys a ij | (cid:105) = 0 = b ij | (cid:105) for i, j = 1 , , · · · , p . our final result for the Hamiltonian of the lattice model, arising fromthe one loop dilation operator, is H = 2 g Y M (4 π ) (cid:88) A =1 p (cid:88) i>j =1 ( ˆ N A ) ij (cid:16)(cid:112) N + l R i − (cid:113) N + l R j (cid:17) + 2 g Y M (4 π ) (cid:88) A =1 4 (cid:88) B =1+ A p (cid:88) i,j =1 (cid:115) ( N + l R i )( N + l R j ) l R i l R j (cid:32) − ( ˆ N B ) ji (¯ a A ) jj ( a A ) ii − ( ˆ N A ) ji (¯ a B ) jj ( a B ) ii +2 δ ij (cid:16) (cid:88) l (cid:54) = i ( ˆ N A ) i → l + (¯ a A ) ii ( a A ) ii (cid:17)(cid:16) (cid:88) l (cid:54) = i ( ˆ N B ) i → l + (¯ a B ) ii ( a B ) ii (cid:17)(cid:33) (A.6)In the above formula, l R i is the length of the i th row of Young diagram R .Thus, in this non-planar limit the operator mixing problem translates into dynamics onan emergent lattice, described by a graph. We consider states with definite ( ˆ N A ) i → j , ( ˆ N A ) ij for i (cid:54) = j eigenvalues. These are constants of the motion. We can replace the operators( ˆ N A ) i → j , ( ˆ N A ) ij , for i (cid:54) = j by fixed non-negative integers ( N A ) i → j , ( N A ) ij for each state. 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