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Supporting Information for Fast Scrambling on Sparse Graphs

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Supporting Information for Fast Scrambling on Sparse Graphs Supporting Information for Fast scrambling on sparse graphs A Bounding the Growth of Entanglement The supporting information contains the technical details of the results announced in the first section. Henceforth, we will assume that the Hamiltonian is k-local on a hypergraph with L vertices, and that the local Hilbert space dimension is 2M . Here we are only interested in models where M is strictly finite: e.g. M = 1;1 thus log N log L. ⇡ In this section, we sketch a proof that λ2 is finite under rather general circumstances: (i)thek-local Hamiltonian (2) is extensive, and (ii) the spectrum of the many-body Hamiltonian approaches a smooth distribution in the thermodynamic limit. The proof proceeds in two parts. Let ⇢v(t)bethereduced density matrix for vertex v at time t, and define S(1)(t)= tr [⇢ (t) log ⇢ (t)] ,S(2)(t)= log tr ⇢ (t)2 . (S1) v − v v v − v (2) M ⇥ ⇤ First, we show that if M log 2 Sv = δ, and 2 δ 1, − ⌧ δ 2(M 1)/2 M log 2 S(1) − δ3/2 +O δ2 . (S2) − v − 2 6 3 ⇣ ⌘ Secondly, we show that v(t) v0 (0) λ t |hO O i| > Ce− v⇤ , (S3) kOvkkOv0 k where and are local operators on vertex v, and λ is a v-dependent decay rate which is independent Ov Ov0 v⇤ of N for almost all vertices. With these two results in hand, we now consider the von Neumann entanglement entropy of a region 1 A consisting of L0 = pL of the vertices in the graph, with 0 <p6 2 .Since (1) 1 (2) ML0 log 2 S M log 2 S M log 2 S , (S4) − A > − v ⇡ 2 − v v A v A X2 h i X2 h i and (2) 2λ2vt M log 2 S e− , (S5) − v & if half of the vertices within A have decay rates λ2v 6 λ2, then such vertices must (on average) have 4a δ in order for ML0 log 2 Sv a. In the thermodynamic limit (L with M finite), we will 6 L0 6 − 0 λ2t !1 show that there exists a positive constant C N for which δ & Ce− ,whichproves(12). P / 1In SYK-like models, the local Hilbert spaces below should be interpreted as few fermion clusters on a given graph vertex, in contrast to our interpretation in (25), and in Appendix C. 1 A.1 Relating Renyi to von Neumann Entropy First we show (S2). If 0 6 pi 6 1 denote the eigenvalues of the reduced density matrix ⇢v,then (1) (2) 2 Sv = pi log pi,Sv = log pi . (S6) − − ! Xi Xi Also observe that pi = 1. Let 1+✏ p = i , (S7) P i n (2) and let M log 2 Sv = δ.If2δ<1, then by definition − 1 ✏2 =eδ 1 <δ+ δ2 < 2δ. (S8) 2M i − Xi Thus, for all i, ✏ p2M+1δ. (S9) | i| 6 Now, when 2M+1δ 1, ⌧ 1 1 1 S(1) = M log 2 ✏2 + ✏3 ✏4 + , (S10) v − 2M+1 i 6 2M i − 12 2M i ··· Xi · Xi · Xi and so 2 (1) 1 1 2 max( ✏i ) max( ✏i ) 1 2 M log 2 Sv ✏i 6 | | + | | + ✏i (S11) − − 2 2M 6 12 ··· 2M ⇣ ⌘ i ✓ ◆ i X X Taylor expanding the equality in (S8), and using (S9), we obtain (S2). A.2 Renyi Entropy and the Memory Function (2) Next, we derive an exact, albeit non-local in time, equation of motion for Sv (t). To do so, let us consider the vector space of all Hermitian operators which act on the Hilbert space of vertex v. The density matrix a a ⇢v) is in this vector space. A basis set for this vector space is I), T ) ,whereT denote the Hermitian | {| | v } v generators of SU(2M ), acting on vertex v alone. An (orthogonal) inner product for this basis is ( ) tr ( ) . (S12) O1|O2 ⌘ O1O2 Our goal is now to find a linear equation for ⇢ (t)). This can be done, as the Schr¨odinger equation | v is linear. However, as we are integrating out degrees of freedom, the resulting equation will be nonlocal. This is the essence of the memory function formalism [58, 59, 60]. Unlike in the usual of the memory function equations (in real time), here we will also need to keep track of additional terms related to the initial conditions of the entire many-body state. We will ultimately argue that these additional terms cannot generally modify (S3). The key point is as follows. First, let us temporarily expand the vector space to include the set of all Hermitian matrices. Our goal is to project the dynamics back on to the set of reduced density matrices on vertex v, which is an exponentially small subset of all possible basis vectors. Let the initial many-body state be 0). Since tr⇢ = 1 for all times, we know that 0) and I) have some overlap, so we may write the | | | initial state as 0) = 0 )+ 0). (S13) | | v | e 2 The first term is the initial condition for ⇢v, tensored with the identity on all other sites; the second term makes up the remainder of terms. Let be the Liouvillian: L )= [H, ]), (S14) L|O −| O with H the many-body Hamiltonian. The time evolution of the density matrix is given by e i t 0). − L | Denoting p as the projection operator onto I), T a) , and q =1 p, we thus wish compute {| | v } − i t ⇢ (t)) = pe− L 0). (S15) | v | More specifically, since S(2)(t) e− v =(⇢ (t) ⇢ (t)), (S16) v | v we wish to compute (2) dSv 2 d ⇢ (t)) = (⇢ (t) | v . (S17) dt −(⇢ (t) ⇢ (t)) v | dt v | v Using the identity t i t i qt i q(t s) i s pe− L = pe− L ip dse− L − pe− L , (S18) − L Z0 we obtain t d ⇢v(t)) iq qt iq qs | = ip qe− L 0) ip p ⇢ (t)) ds p qe− L q p ⇢ (t s)). (S19) dt − L | − L | v − L L | v − Z0 e This is a linear equation (albeit nonlocal in time) with a source, given by the first term on the right hand side. We may write its solution as the sum of a particular solution and a homogeneous solution: ⇢ (t)) = ⇢0(t)) + ⇢ (t)) (S20) | v | v | v The first term above is a homogeneous solution of (S19), obeying p ⇢0)= ⇢0); the second term is a e | v | v particular solution of (S19) obeying ⇢ (0)) = 0. For generic initial conditions, we expect the decay of | v ⇢v(t)) to I) can be no faster than the decay of the homogeneous term. So we expect that it suffices | ⇠| to bound the decay of ⇢0(t)). This is the first point at which we will sacrifice some rigor and simply | v e sketch out a proof of (S3). The fact that a generic linear equation decays no faster than the decay of the homogeneous solution is the formal version of the inequality we provided in (12). It is most natural to describe the dynamics of ⇢0(t)) via a Laplace transform. It is convenient to | v subtract out the identity component: 0 1 ⇢ (t)) ⇢ (t)) I). (S21) | v ⌘| v − 2M | Since I) is a null vector of , it decouples from the dynamics and we will omit this basis vector in what | L follows. The assumption that entropy saturates to maximal then implies that ⇢ (t)) 0 as t . v ! !1 Letting 1 zt ⇢ (z)) dt e− ⇢ (t)), (S22) | v ⌘ | v Z0 we find that z ⇢ (z)) ⇢ (0)) = (ip p + (z)) ⇢ (z)), (S23) | v −| v − L K | v 3 where the memory function 1 (z) p q (z +iq q)− q p. (S24) K ⌘ L L L The inverse Laplace transform then gives us the time-dependent reduced density matrix: dz zt 1 ⇢ (t)) = e (z + (z)+ip p)− ⇢ (0)) (S25) | v 2⇡i K L | v z0+iZ R The real number z0 is chosen somewhere where the contour can be closed: i.e. for z0 6 Re(z ), where ⇤ z is the complex number with smallest real part such that z + (z)+ip p is a singular matrix. For ⇤ K L the moment, let us assume that z is finite. We will justify this in the next subsection, together with ⇤ Re(z ) < 0. Standard theorems then give that (for generic initial conditions) as t ⇤ !1 zt dz zt 1 dz e z t ⇢v(t)) p e (z + (z)+ip p)− p ⇢v(0)) = C ⇢v(0)) = e ⇤ C ⇢v(0)) | ⇡ ⇤ 2⇡i K L ⇤| 2⇡i z z ⇤| ⇤| z0+iZ R z0+iZ R − ⇤ (S26) where p projects onto all vectors with this most singular ‘eigen’vector, and C is a suitable matrix. ⇤ ⇤ Subleading corrections to this equation decay with a faster exponential rate. We then conclude that (S3) 2 holds, with λv⇤ = 2Re(z ) and C =(⇢v(0) C ⇢v(0)). − ⇤ | ⇤ | A.3 Finiteness of the Memory Function It remains to justify why < Re(z ) < 0. First, let us suppose that E↵) denote the eigenfunctions of 1 ⇤ | q q in the image of q.Wewrite L p q = c )(E (S27) L ↵E↵ | ↵ ↵| ↵ X Let us further assume that the density of states of the many-body Hamiltonian is continuous, and is given by ⇢(E). We then find that the density of states of both and q q is given by L L ⇢ (E)= dE0 ⇢(E0)⇢(E0 + E). (S28) L Z Because q q and are identical matrices up to an exponentially small number of entries, corrections to L L their relative continuous spectra are e L, and such e↵ects are comparable, in the thermodynamic limit, / − to other “finite size” e↵ects that we neglect.
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