Quantum Epidemiology: Operator Growth, Thermal Effects, and SYK
QQuantum Epidemiology: Operator Growth, ThermalEffects, and SYK
Xiao-Liang Qi a and Alexandre Streicher aba Stanford Institute for Theoretical Physics, Stanford University,Stanford, CA 94305, USA b Department of Physics, University of California,Santa Barbara, CA 93106, USA [email protected] | [email protected]
Abstract
In many-body chaotic systems, the size of an operator generically grows in Heisen-berg evolution, which can be measured by certain out-of-time-ordered four-point func-tions. However, these only provide a coarse probe of the full underlying operator growthstructure. In this article we develop a methodology to derive the full growth structureof fermionic systems, that also naturally introduces the effect of finite temperature.We then apply our methodology to the SYK model, which features all-to-all q -bodyinteractions. We derive the full operator growth structure in the large q limit at alltemperatures. We see that its temperature dependence has a remarkably simple formconsistent with the slowing down of scrambling as temperature is decreased. Further-more, our finite-temperature scrambling results can be modeled by a modified epidemicmodel, where the thermal state serves as a vaccinated population, thereby slowing theoverall rate of infection. a r X i v : . [ h e p - t h ] N ov ontents q Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 The Large- q solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Size renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 In chaotic quantum many-body systems, operators grow in size as time evolves. For example,in spatially local systems one expects that the extent of an operator O ( t ) grows as ddt Volume [ O ( t )] ∝ Surface Area [ O ( t )] (1.1)since the new terms generated by taking [ H, O ( t )] will live on the boundary of the domainof O ( t ) [1–4]. Consequently, the extent grows linearly with an effective "speed of light" (cid:39) vt .Up to exponential error all operators outside the effective light-cone will commute with O ( t ) .This effective speed of light is known as the Lieb-Robinson velocity [1]. This highlights thefact that space can be a derived concept in quantum mechanics, as without the Hamiltonianthere may not be a sense in which one piece of the Hilbert space factorization is closer toanother. 1ow we would like to contrast this behavior with that exhibited by q -local systems, wherethe Hamiltonian couples all the degrees of freedom together in q -body interactions. Conse-quently, there is no notion of spatial locality, and we accordingly refer to such interactions ascoupling together “internal” degrees of freedom. Yet, there remains structure in the evolutionof operators in these systems, as we often find that the sizes of operators grow exponentially ddt Size [ O ( t )] ∝ Size [ O ( t )] (1.2)where by size we mean the number of simple operators multiplied together in a typical pieceof O ( t ) . The intuition behind this growth is that the percentage of the q -body interactionsutilized in [ H, O ( t )] is proportional to the size of O ( t ) , and almost all the resultant operatorsobtained from [ H, O ( t )] are bigger than O ( t ) [5–11]. Systems with both spatial locality and a large number of internal degrees of freedom–suchas (chaotic) field theories in the large- N limit – display both linear spatial growth and expo-nential internal size growth [12–16]. The growth of an evolving simple operator W ( t ) can beprobed using another simple operator V , using the (anti-)commutator squared (cid:104)| [ W ( t ) , V ] | (cid:105) or their corresponding out-of-time-order correlator (OTOC) (cid:10) W † ( t ) V † W ( t ) V (cid:11) [12, 17–25].In order to develop the coarse-grained profile of operator growth, one must compute manyOTOCs. The “chaos bound” [26] obeyed by OTOCs suggests that after an initial dissipationtime, they de-correlate no faster than exponentially, with a rate λ L no larger than πT where T = 1 /β is the temperature. This implies that presence of the thermal state ρ ∝ exp ( − βH ) slows down the effective growth rate of operators as temperature is decreased.The Heisenberg evolution of operator O ( t ) is independent of temperature, so the entireeffect of temperature must be contained in the matrix elements of O . Therefore, the naturalfinite temperature generalization of operator size has remained an open question (one recentproposal is given in [27]).In this article, we address this issue by characterizing not only the average size of anoperator but its entire size distribution. We then can define the effective size distribution ofan operator at finite temperature by how it changes the size of the square root of thermaldensity operator ρ / (we explain why this is a natural choice in section (3)).This definition leads to some nontrivial general results, independent of the details ofthe specific physical system. In particular, we observe that in generic fermion systems, theeffective size of a single fermion operator is “thermally renormalized” to a value δ β = G ( β/ smaller than , where G ( τ ) is the thermal two-point function. The size of the thermaloperator ρ / itself is N (1 − δ β ) , determined by the same renormalization factor δ β . To gaina more explicit understanding, we will work in the context of the SYK model [28, 29], a q -local Hamiltonian built out of N flavors of Majorana fermions, which saturates the chaosbound at low temperatures [20, 29, 30]The remainder of the article is organized as follows. We begin in section (2) by buildingup the notion of operator “size”. First, we show that one may expand any operator O ( t ) inan orthonormal operator basis of the unique products of Majorana flavors. In the doubledtheory, the operator basis maps to an orthonormal basis of states in the doubled Hilbert Eq. (1.2) actually can be considered as the same formula as in Eq. (1.1) applied to a completely connectedgraph, so that the area of a region is proportional to its volume [9]. n in the doubled theory counting the average number offlavors in an operator basis state. We are then able to demonstrate that four-point functionsmeasure the average “size” of an operator. Therefore, the de-correlation of a thermal OTOCis exactly equivalent to the growing average size of the operator ψ ( t ) ρ / − N N (cid:88) j =1 Tr (cid:0) ρ / ψ ( t ) ψ j ψ ( t ) ρ / ψ j (cid:1) = 1 − N n (cid:2) ψ ( t ) ρ / (cid:3) (1.3)The average size of the operator ψ ( t ) ρ / starts at n (cid:2) ψ ρ / (cid:3) ≈ n (cid:2) ρ / (cid:3) = N (1 − δ β ) andthen grows sigmoidally in time, eventually saturating (scrambling) at a value of N/ .Up to this point we have been discussing the average size of the operator ψ ( t ) ρ / . Infact, the entire size distribution of this operator has physical significance. Hence, in section(3) we construct generating functions for operator size distributions by inserting a weightingfactor exp ( − µn ) . First, we study the size distribution of the thermal operator ρ / by settingup a generating function Z µ (cid:2) ρ / (cid:3) , which is similar to a grand canonical partition function.Next, we show that the fractional distance to scrambling for the operator ρ / is alwaysgiven by δ β ≡ − n/n ∗ = G ( β/ < . Then, we set up the generating function for thesize distribution of ψ ( t ) ρ / , which we find naturally splits into a product of Z µ (cid:2) ρ / (cid:3) anda modified two-point function G µ ( t ) . We show that the µ -expansion of G µ ( t ) determinesthe growth distribution induced by multiplying ρ / by ψ ( t ) , and that G µ ( t ) is simply thetwo-point function for the original theory with a µ -dependent twisted boundary condition.We conclude the section by noting that on average, the size increase induced by a singlefermion is given by the fractional scrambling distance δ β , which leads us to propose that δ β should be interpreted as a thermally renormalized unit of size.Everything in sections (2) and (3) applies to general fermionic systems. In sections (4)and (5), we apply this methodology to the large- q SYK model. Solving the large- N saddlepoint equation in the large q limit with our µ -dependent twist, we obtain the full operatorgrowth structure. After a dynamical renormalization of coupling constant J (which aftera short amount of time essentially amounts to replacing the coupling with a smaller β -dependent constant) and a renormalization of size unit from to δ β , we observe that thefull growth structure has the same functional form as the infinite temperature case. Thedynamical renormalization of the coupling is the signature of the slowdown of the effectivegrowth rate as temperature is decreased.We conclude the section by discussing how to understand this finite temperature slow-down of scrambling in an epidemic model, where the thermal factor effectively vaccinates alarge subset of the population, thereby slowing down the overall infection rate. We end thepaper by discussing implications and future directions in section (6). As an operator O ( t ) evolves in time, it becomes supported along operators of increasingsize. This can be inferred from the Heisenberg equation of motion ˙ O ( t ) = i [ H, O ( t )] . Now,in order to properly discuss how much one operator is supported along another, we needan operator inner product. When the Hilbert space is finite-dimensional it is natural to use3igure 1: Illustration of the purification procedure that maps operators to states in a dou-bled Hilbert space. (a) A maximally entangled state | (cid:105) (Eq. (2.2) which can be viewed asmany EPR pairs between the two systems. (b) The mapping between operator O and thecorresponding state |O(cid:105) obtained by applying O to the left system (see Eq. (2.3)).the Frobenius inner product: (cid:104)O A |O B (cid:105) ≡ Tr( O † A O B ) . We may then expand operators in anorthonormal operator basis, which amounts to inserting a complete set of operators { Γ I }O ( t ) = (cid:88) I Γ I Tr (cid:16) Γ † I O ( t ) (cid:17) ≡ (cid:88) I c I ( t ) Γ I (2.1)Note that at this point we have set up a Hilbert space of operators. If the original Hilbertspace H has dimension L , the operator Hilbert space is H ⊗ H , with dimension L . Since H ⊗ H is isomorphic to H ⊗ H , one can always maps each operator to a quantumstate in a “doubled” system with Hilbert space dimension L . More explicitly, this mappingis defined by considering two copies of the original physical system, named as L and R , andintroducing a maximally entangled state | (cid:105) (see Fig. 1(a)). For any maximally entangledstate, there is a basis choice of the form {| n (cid:105) L ⊗ | m (cid:105) R } such that | (cid:105) = (cid:88) m,n δ mn | n (cid:105) L ⊗ | m (cid:105) R = (cid:88) n | n (cid:105) L ⊗ | n (cid:105) R (2.2)For later convenience we have chosen the norm of the state to be (cid:104) | (cid:105) = L .Then the operator-to-state mapping is defined by O → |O(cid:105) ≡ O L ⊗ I R | (cid:105) (2.3)where O L is the operator O acting on the Hilbert space of the left system, as is illustrated inFig. 1(b). It is easy to verify that the inner product LR (cid:104)O A |O B (cid:105) LR = Tr( O † A O B ) is determinedby the Frobenius inner product of the corresponding operators. Our orthonormal basis ofoperators { Γ I } will thereby serve as an orthonormal basis of states | Γ I (cid:105) . Thus, the problemof understanding how O ( t ) is distributed across a particular choice of basis operators isequivalent to understanding how the two-sided state |O ( t ) (cid:105) is distributed across a particularchoice of two-sided basis states. Since the choice of maximally entangled state | (cid:105) is notunique, the operator-to-state mapping has an ambiguity of U ( L ) acting on the R system.Since the same transformation is performed to |O(cid:105) and the basis vector | Γ I (cid:105) , all our discussionwill be independent from this freedom of basis choice.4 .1.1 Orthonormal Basis of Operators One of the simplest algebras with an interesting finite dimensional representation is thealgebra of N flavors of Majorana fermions, where N is even: { ψ i , ψ j } = 2 δ ij (2.4)Note that this implies that ψ j = 1 , which will be convenient for our purposes, unlike themore common convention where { ψ i , ψ j } = δ ij and thus ψ j = 1 / . Such operators aretraceless, Hermitian, and unitary. Furthermore, the algebra is invariant under taking anysingle ψ i → − ψ i , and so the product of any subset of the N fermions is also traceless. Thus,it is easy to construct an orthogonal operator basis by taking unique ordered products ofMajorana fermions Γ I ≡ Γ i i ...i k = i k ( k − ψ i ...ψ i k ≤ i < i < ... < i k ≤ N (2.5)where the pre-factor has been inserted so that the resultant Γ I matrices are Hermitian. Allnontrivial Γ I (with k > ) are traceless. Since the product Γ I Γ J is also a string of fermions,which is only trivial when I = J (when the two strings are identical and Majorana fermionspairwise cancel), we have Tr (cid:16) Γ † I Γ J (cid:17) = Tr (Γ I Γ J ) = δ IJ Tr (1) (2.6)Furthermore, the basis operators Γ I have simple algebraic relations, since they either com-mute or anti-commute according to the relation Γ I Γ J = ( − | I || J | + | I ∩ J | Γ J Γ I (2.7)where | I | is the number of elements in the multi-index I . The purification isomorphism is quite simple to realize, as has been discussed by [32, 33]. Weconsider two copies of the original system, which contains N Majorana fermions labeled by ψ Lj and ψ Rj , j = 1 , , ..., N . We then define a maximally entangled state | (cid:105) , (cid:0) ψ Lj + iψ Rj (cid:1) | (cid:105) = 0 , ∀ j (2.8)We may think of this state as a vacuum (all spins down, all bits set to 0) with regards to aset of entangled complex fermions operators c j | (cid:105) = 0 c j ≡ ψ Lj + iψ Rj { c j , c k } = (cid:110) c † j , c † k (cid:111) = 0 (cid:110) c j , c † k (cid:111) = δ jk (2.9)where c † j = ( c j ) † . Since state | (cid:105) is the ground state of a quadratic Hamiltonian H = (cid:80) j c † j c j ,it is straightforward to compute the entanglement entropy [34] and verify that the stateis maximally entangled between L and R . As we discussed earlier, the choice of | (cid:105) is not5igure 2: (a) The mapping of a Majorana string Γ I in Eq. (2.5) to a state in the doubledsystem. Each fermion operator ψ Li creates a fermion (red dot) while the fermions that areabsent in Γ I stays in the vacuum state with fermion number (black dot). (b) Illustrationof the relation between average size of operator O and OTOC.unique, but this choice is convenient for our purpose. The basis operators Γ I are mapped tostates in the doubled system of N Majorana fermions: | Γ I (cid:105) ≡ Γ LI | (cid:105) = i k ( k − ψ Li ...ψ Li k | (cid:105) = i k ( k − c † i ...c † i k | (cid:105) = c † i k ...c † i | (cid:105) (2.10)Therefore each basis operator Γ I is mapped to a particular fermion configuration in the dou-bled system, with fermions i , i , ..., i k , as is illustrated in Fig. 2(a). Essentially, the identityoperator maps to the vacuum and nontrivial operators are mapped to excitations in thedoubled theory. At this point, we can discuss the number operator n j ≡ c † j c j , which returns when appliedto basis states containing the flavor j and zero otherwise n j ≡ c † j c j = 12 (cid:0) iψ Lj ψ Rj (cid:1) (cid:104) Γ I | n j | Γ J (cid:105) = δ j ∈ J (cid:104) Γ I | Γ J (cid:105) (2.11)Thus, we see that for a generic operator O , the expectation value of n j returns the percentageof basis operators in O containing flavor j . Furthermore, we note that this expectation valueis closely related to a one-sided four-point function (see Fig. 2), since (cid:104)O| (2 n j − |O(cid:105) = (cid:104)O| iψ Lj ψ Rj |O(cid:105) = (cid:104) | (cid:0) O L (cid:1) † ψ Lj iψ Rj O L | (cid:105) = − (cid:104) | (cid:0) O L (cid:1) † ψ Lj O L iψ Rj | (cid:105) = (cid:104) | (cid:0) O L (cid:1) † ψ Lj O L ψ Lj | (cid:105) = Tr L (cid:16)(cid:0) O L (cid:1) † ψ Lj O L ψ Lj (cid:17) ⇒ (cid:104)O| (2 n j − |O(cid:105) = Tr (cid:0) O † ψ j O ψ j (cid:1) (2.12)Here we have assumed O to be fermionic. In the first two steps, we simply plugged inthe definitions of n j and |O(cid:105) . In the third step, we anti-commuted iψ Rj through O L , as rightfermionic operators anti-commute with left fermionic operators. Then, we used the definitionof | (cid:105) (2.8) to replace − iψ Rj | (cid:105) with ψ Lj | (cid:105) . Afterwards, we had an expectation value of onlyleft operators for a maximally entangled state, so we traced out the right Hilbert spaceentirely, leaving us with an infinite temperature four-point function of the left-only system.6he relationship between operator quantities and one-sided correlators is simpler in termsof the anti-commutator squared, since we have
14 Tr (cid:16) {O , ψ j } † {O , ψ j } (cid:17) = 12 Tr (cid:0) O † O (cid:1) + 12 Tr (cid:0) O † ψ j O ψ j (cid:1) = 12 (cid:104)O|O(cid:105) + 12 (cid:104)O| (2 n j − |O(cid:105)⇒
14 Tr (cid:16) {O , ψ j } † {O , ψ j } (cid:17) = (cid:104)O| n j |O(cid:105) ≡ n j [ O ] (2.13)where we used (2.12) to replace Tr (cid:0) O † ψ j O ψ j (cid:1) with (2 n j − . One should note that if O isbosonic, the right-hand side of (2.12) will acquire a minus sign and the anti-commutators in(2.13) will be replaced with commutators. We denote the average value of n j in operator O as n j [ O ] .We can also define a total number operator (a.k.a. size operator) that returns the numberof flavors or size of a basis state n ≡ N (cid:88) j =1 n j = N (cid:88) j =1 c † j c j (cid:104) Γ I | n | Γ J (cid:105) = | I | (cid:104) Γ I | Γ J (cid:105) (2.14)with | I | the number of Majorana fermion operators in the string Γ I . Consequently, (cid:104)O| n |O(cid:105) is the average number of flavors in the operator O , or the average size of the operator O .By flavor averaging Eq. (2.13), we see that the flavor-averaged anti-commutator squaredmeasures the average size of the operator O N N (cid:88) j =1 Tr (cid:16) {O , ψ j } † {O , ψ j } (cid:17) = (cid:104)O| n |O(cid:105) N ≡ n [ O ] N (2.15)where the anti-commutators are replaced with commutators if O is bosonic.Alternatively, we may flavor average Eq. (2.12) in order to relate the flavor-averagedfour-point function to the average size ( − |O| N N (cid:88) j =1 Tr (cid:0) O † ψ j O ψ j (cid:1) = (cid:104)O| (cid:18) − nN (cid:19) |O(cid:105) ≡ (cid:104)O| δ |O(cid:105) ≡ δ [ O ] (2.16)where ( − |O| is 1 if O is bosonic and − if O is fermionic.Noting that the number of unique products of k Majoranas Eq. (2.5) goes as (cid:0) Nk (cid:1) , wesee that the most common size is that of N/ . Indeed, a generic operator should be equallysupported across all unique products, leading to a distribution of the form P k = (cid:0) Nk (cid:1) / N .Thus, a totally scrambled operator has a size n ∗ = N/ , so we see that the flavor-averagedfour-point function measures the average fractional distance − n/N an operator’s size isfrom this scrambled value. Therefore we define the fractional scrambling distance operator δ ≡ − nn ∗ = 1 − nN (2.17)7 .3 Operator Size Generating Function By defining the number operator n , we can now go beyond the average operator size probedby four-point functions. Rather than just the average, we study all moments systematicallyby introducing a generating function [10] Z µ [ O ] = (cid:104)O| e − µn |O(cid:105) (2.18)By taking derivatives of the generating function we can obtain all moments of n : (cid:104)O| n k |O(cid:105) = ( − k k ! ∂ k Z ∂µ k (cid:12)(cid:12)(cid:12)(cid:12) µ =0 (2.19)A more useful expansion is a Taylor expansion in e − µ : Z µ [ O ] = N (cid:88) n =0 e − µn P n [ O ] (2.20)in which the coefficients P n [ O ] is the percentage of terms in O having size n . The main goal of the current work is to understand the role of temperature in operatorgrowth. After all, the dynamics of the operator ψ ( t ) under Heisenberg evolution has noknowledge about temperature.One natural way to introduce temperature is to consider the operator ρ / where ρ = Z − β exp ( − βH ) is thermal state at inverse temperature β . The purification of ρ / is thethermofield double (TFD) state | T F D (cid:105) (the other factor of ρ / from the full ρ is used tomake (cid:104) T F D | ) | T F D (cid:105) = Z − / β e − β ( H L + H R ) | (cid:105) = Z − / β e − β H L | (cid:105) = (cid:12)(cid:12) ρ / (cid:11) (3.1)where the Hamiltonians H L , H R are required to satisfy the condition ( H L − H R ) | (cid:105) = 0 . Thisstate is a natural choice for studying thermodynamic properties, because for each operator O , we can consider the corresponding operator O ρ / , and its average size will be directlymeasured by the finite temperature four-point function: δ (cid:2) O ρ / (cid:3) = 1 − n (cid:2) O ρ / (cid:3) N/ − |O| N N (cid:88) j =1 Tr (cid:0) ρ / O † ψ j O ρ / ψ j (cid:1) (3.2)Now, we are interested in uncovering the size distribution of a time evolved “thermal”operator O ( t ) ρ / . To do so, we will define a generating function for its size moments Z µ (cid:2) O ( t ) ρ / (cid:3) . We shall find that such a generating function naturally factorizes into aproduct of the generating function for the Gibbs state Z µ (cid:2) ρ / (cid:3) and a “connected” piece G µ [ O ( t )] . By using this to extract the size distribution of ρ / from the size distribution of O ( t ) ρ / , we find a natural definition for a “thermal” size of O ( t ) .8 .1 Thermal State We begin by studying ρ / . Taking O = ρ / in Eq. (2.16), we find the following relationbetween the thermal two-point function and the size operator G (cid:18) β (cid:19) = 1 N N (cid:88) j =1 Z − β Tr (cid:18) e − βH ψ j (cid:18) β (cid:19) ψ j (cid:19) = 1 − n (cid:2) ρ / (cid:3) N/ δ (cid:2) ρ / (cid:3) (3.3)This relation tells us that the most de-correlated value of the Euclidean two-point function- G ( β/ - is equal to the fractional distance the operator ρ / is from being scrambled δ β ≡ δ (cid:2) ρ / (cid:3) = G (cid:18) β (cid:19) (3.4)which implies that the average size of ρ / is given by n (cid:2) ρ / (cid:3) = N (cid:18) − G (cid:18) β (cid:19)(cid:19) (3.5)In the high temperature limit β → , one expects G ( β/ (cid:39) G (0) = 1 , since the fermionssquare to one (2.4), which is consistent with the fact that ρ / approaches identity and thesize shrinks to zero. On the contrary, in the low temperature limit β → ∞ , if G ( β/ → ,the size of ρ / approaches the scrambled (typical) value N/ . This result is very general sincethe imaginary time two-point function G ( τ ) decays in most physical systems. For example,in all systems with a unique ground state and an excitation gap, G ( τ ) decays exponentiallyat low temperature limit, so that G ( β/ → when β → ∞ . In a conformal field theory, G ( τ ) decays in power law in the zero temperature limit, which also leads to the same length n β →∞ = N/ .To learn more than just the average size, we construct the generating function Z µ (cid:2) ρ / (cid:3) = (cid:10) ρ / (cid:12)(cid:12) e − µn (cid:12)(cid:12) ρ / (cid:11) = (cid:104) T F D | e − µn | T F D (cid:105) (3.6)Therefore learning about the operator distribution of ρ / is equivalent to learning about thefermion number distribution in the thermofield double state. If we take O = ψ ( t ) in Eq. (3.2), we see that the average size of ψ ( t ) ρ / is entirelyequivalent to an out-of-time-order correlator (a.k.a. OTOC) [17–20]: − N N (cid:88) j =1 Tr (cid:0) ρ / ψ ( t ) ψ j ψ ( t ) ρ / ψ j (cid:1) = 1 − N n (cid:2) ψ ( t ) ρ / (cid:3) ≡ δ (cid:2) ψ ( t ) ρ / (cid:3) (3.7)Therefore, the statement that the OTOC de-correlates exponentially is equivalent to thestatement that the average size of the operator ψ ( t ) ρ / grows exponentially. If the OTOCvanishes in long time, that implies that the size of ψ ( t ) ρ / reaches the scrambled value n ∗ = N/ . 9he size distribution of ψ ( t ) ρ / can be uncovered through the generating function Z µ (cid:2) ψ ( t ) ρ / (cid:3) = (cid:104) ψ ( t ) ρ / | e − µn | ψ ( t ) ρ / (cid:105) = (cid:104) T F D | ψ L ( t ) e − µn ψ L ( t ) | T F D (cid:105) (3.8)The operator e − µn can be viewed as an Euclidean time evolution with time µ and Hamiltonian n , so that the generating function (3.8) is related to the two-point function in a system withtime-dependent Euclidean evolution: G µ ( τ a , τ b ) = (cid:104) | T (cid:2) e − β ( H L + H R ) / e − µn ( β/ ψ L ( τ a ) ψ L ( τ b ) (cid:3) | (cid:105)(cid:104) | T [ e − β ( H L + H R ) / e − µn ( β/ ] | (cid:105) (3.9)where T is the Euclidean time ordering symbol and ψ L ( τ a,b ) are imaginary time evolvedfermion operators. Note that the denominator in (3.9) is, up to a factor of thermal partitionfunction Z β that cancels with the numerator, exactly the size generating function Z µ (cid:2) ρ / (cid:3) in Eq. (3.6). Therefore, the size generating function Z µ (cid:2) ψ ( t ) ρ / (cid:3) naturally factorizes intothe product of Z µ (cid:2) ψ ( t ) ρ / (cid:3) = G µ (cid:18) β + it, β − it (cid:19) Z µ (cid:2) ρ / (cid:3) (3.10)This equation clarifies that the two-point function G µ ( β/ + + it, β/ − + it ) measuresthe size change ψ ( t ) induces upon ρ / through multiplication. We can see this directly byapplying the expansion in Eq. (2.20) to both sides of this equation, in order to obtain thefollowing convolution formula for the size distribution of ψ ( t ) ρ / P n (cid:2) ψ ( t ) ρ / (cid:3) = (cid:0) K β [ ψ ( t )] ∗ P (cid:2) ρ / (cid:3)(cid:1) n = n (cid:88) m =0 K βm [ ψ ( t )] P n − m (cid:2) ρ / (cid:3) (3.11)with K βm [ ψ ( t )] defined by the expansion of G µ ( β/ + + it, β/ − + it ) in powers of e − µ : G µ (cid:18) β + it, β − it (cid:19) = N (cid:88) m =0 e − mµ K βm [ ψ ( t )] (3.12)In this sense, K βm can be viewed as the “growth distribution” caused by applying ψ ( t ) tothe thermal state ρ / .Note that the discussion above can be generalized to arbitrary operators. For an arbitraryoperator O , as long as we normalize it such that (cid:104) T F D |O † L O L | T F D (cid:105) ≡ (cid:10) O † O (cid:11) β = 1 , theexpansion of the two-point function G µ [ O ] ≡ Z µ (cid:2) O ρ / (cid:3) Z µ [ ρ / ] (3.13)measures the effective size distribution of O when applied to the thermal state. We are interested in studying the generating function Z µ [ O ] for O = ρ / and O = ψ ( t ) ρ / .As we discussed earlier, inserting the operator exp ( − µn ) corresponds to changing the imag-inary time evolution. The computation can be simplified by noticing that exp ( − µn ) is a10 s i ze d i s t r i bu ti on 𝐾 𝜓 & 𝑡𝑃 𝜌 &/+ 𝑃 𝜓 & (𝑡)𝜌 &/+ Figure 3: Schematic illustration of the size distribution P n (cid:2) ψ ( t ) ρ / (cid:3) for the operator ψ ( t ) ρ / (black curve) which naturally decomposes into a convolution of a growth dis-tribution K βn [ ψ ( t )] (blue dashed curve) with the size distribution P n (cid:2) ρ / (cid:3) of the operator ρ / (red dashed curve). This is due to the factorization relation of their respective generatingfunctions (3.10).Gaussian operator, such that its action by conjugation to fermion operators ψ L,Ri leads to asimple linear superposition: e µn (cid:18) ψ L iψ R (cid:19) e − µn = (cid:18) cosh ( µ ) − sinh ( µ ) − sinh ( µ ) cosh ( µ ) (cid:19) (cid:18) ψ L iψ R (cid:19) (3.14)As a result, inserting the operator-weighting term exp ( − µn ) is equivalent to twisting theboundary condition of the fermion fields at τ = β/ .It is convenient to “de-purify” the system and return to the single copy of fermion fields,but with a twisted boundary condition. The single field is defined by continuously stitchingthe left and right fields together: ψ i ( τ ) = (cid:40) ψ Li ( τ ) 0 ≤ τ ≤ β/ iψ Ri ( β − τ ) β/ ≤ τ < β (3.15)with the requirement of course that ψ ( τ + β ) = − ψ ( τ ) . This stitching transforms the pu-rified action for the two fields into the original action for this single field; however, thetwist condition must accompany the fields. In conclusion, the two-sided path integral in thepresence of the factor exp ( − µn ( β/ equals the original path integral where the fields aretwisted according to lim τ → β/ + (cid:18) ψ ( τ ) ψ ( β − τ ) (cid:19) = (cid:18) cosh ( µ ) − sinh ( µ ) − sinh ( µ ) cosh ( µ ) (cid:19) lim τ → β/ − (cid:18) ψ ( τ ) ψ ( β − τ ) (cid:19) (3.16)Therefore, we conclude that calculating the two point function G µ is equivalent to calculatingthe original two-point function, but with the following twisted boundary conditions lim τ / → β/ + G µ ( τ , τ )lim τ / → β/ − G µ ( τ , τ ) = (cid:18) cosh ( µ ) − sinh ( µ ) − sinh ( µ ) cosh ( µ ) (cid:19) lim τ / → β/ − G µ ( τ , τ )lim τ / → β/ + G µ ( τ , τ ) (3.17)11igure 4: (a) The twisted boundary condition on the imaginary time circle. When τ crosses β/ from below, ψ ( τ ) becomes a superposition of ψ ( β/ (cid:15) ) and ψ (3 β/ − (cid:15) ) (see Eq.(3.16). (b) The various symmetry and boundary conditions on the twisted two point functionin the ( τ , τ ) plane. First, G µ is odd under reflections across the red dotted line and evenunder reflections across the blue dashed lines. Thus, it is sufficient to solve the saddle-pointequations in the fundamental domain < τ − τ < β/ and β/ < τ + τ < β . The blacklines are the locations of the twisting boundary conditions (3.17), which reduce in the large q limit (4.7) and divide our fundamental domain into two regions. Region I is where neitherof the two fermions have crossed a twist, while in Region II the fermions are on oppositesides of the twist.We note that while these conditions break time translation invariance, they preserve aset of discrete symmetries. Specifically, if the original Hamiltonian is time-reversal invariant,then G µ ( τ , τ ) has reflection symmetry across the lines τ ± τ = nβ/ for all integers n ∈ Z G µ ( τ , τ ) = G µ (cid:18) nβ − τ , nβ − τ (cid:19) = ( − n +1 G µ (cid:18) τ + nβ , τ − nβ (cid:19) (3.18)Thus, we need only to solve for G µ ( τ , τ ) in the fundamental domain < τ − τ < β/ and β/ < τ + τ < β , as shown in Fig. 4(b) by the union of regions I and II.12 .4 Thermally Renormalized Unit of Size As an interesting application of our formalism, let us note how ψ ( t ) affects ρ / by taking t = 0 and consider the change of average size by a single fermion operator ψ . ∆ n β [ ψ ] ≡ n (cid:2) ψ ρ / (cid:3) − n (cid:2) ρ / (cid:3) = 12 N (cid:88) i =1 (cid:0) (cid:104) T F D | ψ L iψ Li ψ Ri ψ L | T F D (cid:105) − (cid:104)
T F D | iψ Li ψ Ri | T F D (cid:105) (cid:1) = (cid:104) T F D | iψ R ψ L | T F D (cid:105) = G (cid:18) β (cid:19) (3.19)At infinite temperature β → , G ( β/
2) = 1 , which restores the trivial result that ψ increasesthe size of the density operator (which is proportional to identity operator, with size ) by . At finite temperature, interestingly, the size change induced by a single fermion operatoris smaller than , and is given by the same imaginary time two-point function as the onethat determines the fractional scrambling distance δ β = 1 − n [ ρ / ] N/ in Eq. (3.4). In general,the size increase induced by ψ i is ∆ n β [ ψ i ] = G ii ( β/ , which may depend on i . The averagesize increase is exactly δ β . N (cid:88) i ∆ n [ ψ i ] = 1 N (cid:88) i G ii (cid:18) β (cid:19) = G (cid:18) β (cid:19) = δ β (3.20)Physically, the average size change due to applying a single fermion is generically δ β < at finite β , because in the presence of a nontrivial ρ / there is a chance that multiplying by ψ decreases the size, as is illustrated in Fig. 5, although the chance of increasing the size isalways bigger. The closer the length of ρ / is to the scrambling value n ∗ = N/ , the smalleris the size increase ∆ n β (cid:2) ψ ρ / (cid:3) . For a fully scrambled operator with n = N/ , δ = 0 ,multiplying a fermion ψ has equal chance of increasing or decreasing the size, so that theaverage size stays the same.It should be emphasized that the discussion above is not restricted to the thermal densityoperator. For any operator O , we can define the size change ∆ n O [ ψ i ] ≡ n [ ψ i O ] − n [ O ] (3.21)and obtain the following identity: N (cid:88) i ∆ n O [ ψ i ] = δ [ O ] ≡ − n [ O ] N (3.22)The only thing special for the thermal density operator is the relation of ∆ n to imaginarytime two-point function in a single-copy system.Furthermore, instead of ψ i we can consider a string Γ I ≡ Γ i i ...i k = i k ( k − ψ i ...ψ i k ≤ i < i < ... < i k ≤ N introduced in Eq. (2.5), and consider how the size of Γ I O is different In term of the probabilities K βm [ ψ ] in Eq. (3.12), we have ∆ n [ ψ i ] = (cid:80) Nm =0 mK βm [ ψ ] . ψ ( t ) is multiplied to ρ / , thereis a chance that some fermion flavors collide and the size increase is smaller than the size of ψ ( t ) itself.from O . We have ∆ n O [Γ I ] ≡ n [Γ I O ] − n [ O ] = k (cid:88) s =1 (cid:104)O| iψ Ri s ψ Li s |O(cid:105) (3.23)If we average over all Majorana strings Γ I with the same size k , we obtain C kN (cid:88) I ∆ n O [Γ I ] = kδ [ O ] (3.24)In the last equation, C kN = N ! k !( N − k )! is the number of strings with length k . This equationshows that the average size change induced by multiplying a string with length k is k times δ [ O ] , further confirms that each fermion in the string contributes additively.This observation suggests that at finite temperature (or more generally, for any densityoperator ρ ), the fractional scrambling distance δ , rather than , should be considered asthe fundamental unit of size, which is carried by each fermion operator. Indeed, as we willdiscuss in next section, our calculation in the SYK model in the large q limit suggestsuniversal behavior occurs when size is measured in this renormalized unit. In this section, we will study the operator size growth in the SYK model [28, 29]. Thismodel features q -local interactions with independently random couplings, where each of thecouplings is normal distributed H = i q/ (cid:88) ≤ i ... ≤ i q ≤ N J i ...i q ψ i ...ψ i q (cid:68) J i ...i q (cid:69) = J (cid:0) N − q − (cid:1) = J q (cid:0) N − q − (cid:1) { ψ i , ψ j } = 2 δ ij (4.1)14 a) (b) Figure 6: Two O ( J ) examples of the planar graphs of that survive the large- N limit. Onlygraphs of form (a), where melons are inserted into melons, survive the large q limit. Noticethat there are O ( q ) graphs of form (a), while there are only O ( q ) graphs of form (b). Thisis because there are O ( q ) locations at any depth of a given graph to insert another melon;however, there are typically only O (1) locations to thread another melon. Accordingly, wetake our coupling J to equal J / √ q . Consequently, the O ( q ) combinatorial enhancementgained for each new melon insertion is canceled by the J = J / (2 q ) factor accompanyingsaid melon. This q -scaling of the coupling isolates the infinite subset of the planar graphswhere the graphs are two copies of a tree that are then glued together (a.k.a. “doubletree”graphs) such as (a). All non-doubletree graphs such as (b) are suppressed in q since theyreceive factors of J / (2 q ) for each melon, but do not receive the necessary number of q combinatorial enhancements.At large N , the two-point function satisfies the saddle-point equations [ G ] − = [ G ] − − [Σ] Σ ( τ , τ ) = J q ( G ( τ , τ )) q − (4.2)where bracketed terms are Matsubara frequency matrices. One should note that since thefermions square to one, [ G ] − = − iω/ rather than − iω . q Approximation
In the language of Feynman diagrams, the Schwinger-Dyson equation (4.2) corresponds toonly keeping the leading “melon diagrams” as is shown in Fig. 6. All other diagrams are sub-leading in large N . In the large q limit, there are two types of diagrams. Those with melonsinserted into melons (such as Fig. 6(a)) receive a combinatorial q enhancement, as there aremany rungs upon which one may insert (hence the need for a q − factor in the self-energyto keep everything finite). In contrast, diagrams where melons are simply threaded together(such as Fig. 6(b)) do not receive this enhancement [35]. Thus, at large q only the formerdominate, which corresponds to the following truncation of the Schwinger-Dyson expansion: [ G ] = [ G ] + [ G ] [Σ] [ G ] Σ ( τ , τ ) = J q ( G ( τ , τ )) q − (4.3)15ombing the equations together and Fourier transforming, one obtains ∂ τ ∂ τ ( G − G ) = − J q G q − (4.4)The role played by G in this equation is to require that G → G as τ goes to integermultiples of β . Therefore, if we take G = G e σ/q with σ → at the τ boundaries, we obtainLiouville’s equation [30, 36] ∂ τ ∂ τ σ = − J e σ + O (1 /q ) (4.5)where the field σ is expected to be periodic in both of its arguments, as well as have kinkswhen τ approaches integer multiples of β .Now in order to find G µ , we will need to solve the above equations with the twistedboundary conditions (3.17). Furthermore, our twisted two-point function G µ also satisfiesthe reflection conditions (3.18). Thus, we need only solve for G µ in the fundamental domain < τ − τ < β/ and β/ < τ + τ < β , as shown in Fig. 4(b) by the union of regions Iand II. q solution In the large q limit, each commutator with the Hamiltonian increases the size of operator by ∼ q , so that it is natural to measure the operator size in unit of q . In the generating function,this corresponds to defining ˆ µ ≡ qµ (for reasons to be explained in the next subsection wewill actually use the slightly smaller variable ˆ µ = qδ β µ (4.16)), with ˆ µ kept finite in the large q limit. The derivative of the generating function over ˆ µ measures the size n in unit of q . Ifwe consider the large q limit with ˆ µ being kept finite, and use the large- q ansatz for twistedtwo-point function G µ ( τ , τ ) = G ( τ , τ ) e σ µ ( τ ,τ ) /q , (4.6)the boundary condition (3.17) reduces to lim τ / → β/ + G µ ( τ , τ )lim τ / → β/ − G µ ( τ , τ ) (cid:39) (cid:32) − ˆ µq − ˆ µq (cid:33) lim τ / → β/ − G µ ( τ , τ )lim τ / → β/ + G µ ( τ , τ ) (cid:39) e − ˆ µ/q lim τ / → β/ − G µ ( τ , τ )lim τ / → β/ + G µ ( τ , τ ) (4.7)Thus, to the leading order of q , the two equations for β/ and β/ decouple.As a reminder, these equations encode the effect of moving a Majorana fermion on theEuclidean circle (i.e. the two TFD half-circles) from one side of the twist operator (3.9) tothe other. The factor of exp ( − ˆ µ/q ) = exp ( − µ ) denotes the fact that such an action changesthe size of the doubled state by exactly one whole fermion. However, we shall find that forstates of large size such as the thermofield double (i.e. Gibbs state) each Majorana fermionactually increases the total average operator size by a smaller fraction δ β rather than ,16here the size change δ β is smaller when the state’s size is larger. As such, we shall find thatit will be appropriate to use the variable ˆ µ = qµδ β instead of ˆ µ = qµ when taking the large- q limit.The twisted two-point function in large- q limit can thus be obtained by solving Liouville’sequation (4.5) with the µ -dependent boundary conditions (4.7). Here we will skip the tediousdetails and directly present the solution. When the times are such that the two fermions areon the same side of the twisted boundary, which corresponds to τ > β/ (region I in Fig.4(b), we find a seemingly time-translation invariant solution solution G µ ( τ , τ ) = (cid:18) sin γ µ sin ( α µ ( τ − τ ) + γ µ ) (cid:19) /q ≡ G µ ( τ − τ ) (4.8)However, when the times are such that the two fermions are on opposite sides of the twistedboundary at β/ and β/ , which for our domain amounts to the condition τ < β/ (regionII in Fig. 4(b), the time translation symmetry is explicitly broken G µ ( τ , τ ) = e − ˆ µ/q G µ ( τ − τ ) (cid:18) − ( − e − ˆ µ ) sin γ µ ( G µ ( τ − τ )) q/ sin (cid:0) α µ (cid:0) τ − β (cid:1)(cid:1) sin (cid:0) α µ (cid:0) τ − β (cid:1)(cid:1)(cid:19) /q (4.9)Here the parameters α µ and γ µ are functions of β J and µ , which are determined by theboundary condition G ( τ, τ ) = 1 as well as the reflection conditions (3.18) α µ β = β J sin γ µ , sin (cid:18) α µ β γ µ (cid:19) = e − ˆ µ sin (cid:18) α µ β (cid:19) (4.10)In the limit µ → , we recover the untwisted two-point function G µ =0 ( τ , τ ) = G µ =0 ( τ − τ ) = G ( τ − τ ) in the whole domain, and the equation for the parameters reduce to the ordinarycase [30]: α µ =0 ≡ α = J cos (cid:18) αβ (cid:19) , γ µ =0 ≡ γ = π − αβ (4.11)The asymptotic behavior at small values of β J and large values of β J respectively are givenby α = J (cid:18) − β J O (cid:0) β J (cid:1)(cid:19) α = πβ (cid:18) − β J + O (cid:18) β J (cid:19)(cid:19) (4.12) Before carrying further analysis to the SYK operator growth in next section, we need todiscuss an important modification to the two-point function solution due to higher order q effects. If we take τ → β/ (cid:15), τ → β/ − (cid:15) in Eq. (4.9), we obtain G µ (cid:18) β (cid:15), β − (cid:15) (cid:19) = e − ˆ µ/q (4.13)17his is the kernel that determines the size change induced by multiplying ψ to ρ / , whichhas been discussed in section (3.4). Taking the µ -derivative of G µ , we find ∆ n β [ ψ ] ≡ n (cid:2) ψ ρ / (cid:3) − n (cid:2) ρ / (cid:3) = − ∂ µ log G µ (cid:18) β (cid:15), β − (cid:15) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) µ =0 = 1 (4.14)However, we also know that the size change is directly determined by the two-point functiondue to Eq. (3.19): ∆ n β [ ψ ] = δ β = G (cid:18) β (cid:19) = (cid:18) α J (cid:19) /q (4.15)where α ≡ α µ =0 ( β J ) is the smallest positive root of Eq. (4.11).This discrepancy between the two calculations is because δ β → in the large q limit, andthe O ( q − ) difference is neglected in the approximation we made to the boundary condition.The easiest way to resolve this issue and makes a consistent large- q limit is by redefining ˆ µ = qµ to ˆ µ = qµδ β (4.16)in Eq. (4.7), which leads to the same substitution in Eqs. (4.8), (4.9), and (4.10). In thefollowing, we will always use this definition of ˆ µ .Physically, this substitution is a consequence of the size renormalization discussed insection (3.4). Each Majorana fermion increase the operator size by δ β rather than . Eachaction of the Hamiltonian increases the operator size by ∼ qδ β . Although in large q limit − δ β is order q − , it is important to keep track of this distance, since the same δ β alsomeasures the fractional scrambling distance of ρ / , as we discussed in section (3.1). The sizeof ρ / is n (cid:2) ρ / (cid:3) = N − δ β ) = N (cid:32) − (cid:18) α J (cid:19) /q (cid:33) (4.17)which decreases with increasing q , but is always large since we should always take the large N limit before taking large q . We are now equipped with everything we need to understand P n (cid:2) ψ ( t ) ρ / (cid:3) , the size distri-bution of ψ ( t ) ρ / . According to Eq. (3.10), the generating function Z µ (cid:2) ψ ( t ) ρ / (cid:3) for thisdistribution splits into a product of the thermal state’s generating function Z µ (cid:2) ρ / (cid:3) and G µ ( β/ + + it, β/ − + it ) . The latter is simply the twisted two-point function we discussedin the previous subsection with an analytic continuation. The generating function Z µ (cid:2) ρ / (cid:3) is the partition function of the system with the insertion exp ( − µn ( β/ divided by that of the original system (see Eq. (3.6)). This quantity can be18 �� � �� � �� � �� / �� / � β � [ ρ � / � ] ��� � = � � �� � �� � �� � �� / �� / � β � [ ρ � / � ] ��� � = �� Figure 7: Plots of the average size of ρ / ∝ exp ( − βH/ for different q , given by Eq. (5.2).determined by the twisted two-point function, since one has − ∂ µ ln Z µ = (cid:104) ρ / | ne − µn | ρ / (cid:105)(cid:104) ρ / | e − µn | ρ / (cid:105) = N (cid:18) − G µ (cid:18) β (cid:19)(cid:19) = N (cid:32) − sin /q γ µ sin /q (cid:0) α µ β + γ µ (cid:1) (cid:33) (5.1)In theory, we can integrate this equation to obtain Z µ (cid:2) ρ / (cid:3) . However, many importantproperties of the distribution can be inferred from just the first and second moment.The first moment is simply the average size n (cid:2) ρ / (cid:3) = N − δ β ) = N (cid:18) − G (cid:18) β (cid:19)(cid:19) = N (cid:32) − (cid:18) α J (cid:19) /q (cid:33) (5.2)The behavior of n (cid:2) ρ / (cid:3) for two different q values are plotted in Fig. 7 as a function of β J .Interestingly, we see that for larger q it takes larger values of β J to achieve the same averagesize. Thus, the exponentiation of increasingly heavy SYK Hamiltonians results in relatively lighter thermal states. This is unintuitive, so one might argue that it is simply due to the q -scaling nature of J , but that is only a constant shift in the log scale plot in Fig. 7, whichis nowhere large enough to account for the above discrepancy. The true origin of this effectis the power of /q in the two-point function. We conclude that this heavy-light relationshipis thus a non-trivial consequence of the large- N and large- q limit.For β J (cid:29) e q , one expects that the higher order corrections to Liouville’s equation (4.5)cannot be neglected, and so one should turn to the conformal approximation [30]. Theirexpression for the two-point function implies that the average size of ρ / when N (cid:29) β J (cid:29) is given by n (cid:2) ρ / (cid:3) ≈ N (cid:32) − c ( q ) (cid:18) πβ J (cid:19) /q (cid:33) c ( q ) = (cid:32) ( q −
2) tan πq π (cid:33) /q (5.3)The difference between the large q and low temperature n (cid:2) ρ / (cid:3) = N (cid:0) − G (cid:0) β (cid:1)(cid:1) is capturedby the factor c ( q ) . It monotonically increases from c (4) = (2 /π ) / ≈ . when q = 4 , and19symptotically approaches when q is large as − /q . We expect that q = 4 and large β J will be where our large q approximation will have the largest error. However, that this erroris at worst renews our confidence that the large q approximation captures importantanalytic features of the large- N SYK model.The second derivative of ln Z µ determines the width of the distribution: σ n (cid:2) ρ / (cid:3) = lim µ → ∂ µ ln Z µ (cid:2) ρ / (cid:3) = N ∂ µ G µ (cid:18) β (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) µ → ∝ N (5.4)Therefore the width of the distribution σ n ∝ √ N , such that the relative deviation from theaverage value σ n (cid:2) ρ / (cid:3) /n (cid:2) ρ / (cid:3) ∝ N − / is sharply peaked in the large N limit. This is aconsequence of large N factorization. As explicitly discussed in section (3.2), the generating function for the growth distribution K β (3.12) is determined by the twisted two-point function (4.9) G µ (cid:18) β + it, β − it (cid:19) = e − µδ β (cid:32) (cid:0) − e − qµδ β (cid:1) (cid:18) J α µ sinh α µ t (cid:19) (cid:33) − /q (5.5)where α µ and γ µ depend on µ and β J through the constraints (4.10). This implies that the average size of the operator ψ ( t ) ρ / is given by n (cid:2) ψ ( t ) ρ / (cid:3) = n (cid:2) ρ / (cid:3) − ∂ µ ln G µ (cid:18) β + it, β − it (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) µ =0 ⇒ n (cid:2) ψ ( t ) ρ / (cid:3) = N − δ β ) + δ β (cid:32) (cid:18) J α sinh αt (cid:19) (cid:33) (5.6)where δ β = ( α/ J ) /q , and α ≡ α µ =0 ( β J ) is the smallest positive root of Eq. (4.11). Wesee that the difference in averages sizes of ψ ( t ) ρ / and ρ / is a simple when expressed inthe renormalized size unit δ β , which inspires us to define a notion of the “average growth” of ψ ( t ) as ∆˜ n β [ ψ ( t )] ≡ n (cid:2) ψ ( t ) ρ / (cid:3) − n (cid:2) ρ / (cid:3) δ β = 1 + 2 (cid:18) J α sinh αt (cid:19) (5.7)Now, scrambling occurs when the average size of ψ ( t ) ρ / given by Eq. (5.6) reaches n ∗ = N/ . This produces a slightly complicated expression for the scrambling time t ∗ ;however, it simplifies dramatically when phrased in terms of the average growth of ψ ( t ) .Manipulating the scrambling time equation n (cid:2) ψ ( t ∗ ) ρ / (cid:3) = N/ , we find that one mayequivalently state that scrambling occurs when the average growth of ψ ( t ) reaches n ∗ = N/ n β [ ψ ( t ∗ )] = 1 + 2 (cid:18) J α sinh αt ∗ (cid:19) = N (5.8)20 �������� � �� ����� - � �� - � �� - � �� � � � [ ψ � ( � )] ��� � = � ��� β(cid:3) = � (cid:1) t = (cid:1) t = (cid:1) t = (cid:1) t = (cid:1) t = Figure 8: After either dynamical renormalization (5.12) or time re-parametrization (5.13),the growth distribution K β [ ψ ( t )] takes the same form as the Heisenberg evolution of theoperator ψ ( t ) (i.e. the infinite temperature size distribution P [ ψ ( t )] ). This distribution isgiven by Eq. (5.14), and we plot it on a log-log scale. Note that it reaches out towards largeroperators exponentially quickly.This growth is consistent with the known result of large- q Lyapunov exponent [30]. λ L = 2 α (5.9) In the Lyapunov regime, we may expand the generating function of the growth distributionas G µ (cid:18) β + it, β − it (cid:19) = e − µδ β ∞ (cid:88) n =0 (cid:18) − /qn (cid:19) (cid:0) − e − qµδ β (cid:1) n (cid:18) J α sinh αt (cid:19) n (5.10)where δ β = ( α/ J ) /q , and α ≡ α µ =0 ( β J ) is the smallest positive root of Eq. (4.11). Groupingterms by powers of exp ( − µ ) and using the definition (3.12), we conclude that the growthdistribution is given by K βδ β (1+ qn ) [ ψ ( t )] = ( − n (cid:18) − /qn (cid:19) (cid:0) J α sinh ( αt ) (cid:1) n (cid:16) (cid:0) J α sinh ( αt ) (cid:1) (cid:17) n + q (5.11)where we note that ( − n (cid:0) − /qn (cid:1) is always positive for integer n . Thus, K β [ ψ ( t )] ≥ andso we have no negative probabilities in the size distribution of ψ ( t ) ρ / , since it is given by P (cid:2) ψ ( t ) ρ / (cid:3) = K β [ ψ ( t )] ∗ P (cid:2) ρ / (cid:3) as shown in section (3.2).Interestingly, we see that the growth distribution K β (5.11) has a functional form in-dependent of temperature, which we plot in Fig. (8). We can use either of two methods21 �������� � � � � � ������������ (cid:1) � (cid:1) ∼ ( � ) (cid:1) ��������� ��������������� β(cid:2) = β(cid:2) = β(cid:2) = β(cid:2) = β(cid:2) = β(cid:2) = Figure 9: For different values of β J , the effective coupling ˜ J ( t ) given by Eq. (5.12) slowsdown from J to α on a timescale of order α − . As always, α is the smallest root of Eq.(4.11).to expose this phenomenon. One option is to replace the coupling J with the dynamicallyrenormalized coupling ˜ J ( t ) (plotted in Fig. (9)): ˜ J ( t ) = arcsinh (cid:0) J α sinh ( αt ) (cid:1) t = α + log ( J /α ) t + O ( e − αt ) t (5.12)The other option is to re-parametrize time ˜ t = 1 J arcsinh (cid:18) J α sinh ( αt ) (cid:19) = α J t + log ( J /α ) J + O ( e − αt ) J (5.13)Both methods transform the finite temperature growth distribution into that of the Heisen-berg evolution of the operator ψ ( t ) (i.e. the infinite temperature size distribution P qn [ ψ ( t )] )[11]. For example, using ˜ t gives K βδ β (1+ qn ) (cid:2) ψ (cid:0) t (cid:0) ˜ t (cid:1)(cid:1)(cid:3) = ( − n (cid:18) − /qn (cid:19) tanh (cid:0) J ˜ t (cid:1) n cosh (cid:0) J ˜ t (cid:1) q = K β =01+ qn (cid:2) ψ (cid:0) ˜ t (cid:1)(cid:3) = P qn (cid:2) ψ (cid:0) ˜ t (cid:1)(cid:3) (5.14)This temperature-independence is fascinating since the Heisenberg evolution of ψ ( t ) wasobtained in [11] via fully-dressed Feynman graph calculations. In other words, the distribution P [ ψ ( t )] represents the simple tree graphs such as Fig. 6(a) constructed using the originalSYK Hamiltonian. However, we just showed how the growth distribution K β [ ψ ( t )] can beeasily transformed to P [ ψ ( t )] . Therefore, since P (cid:2) ψ ( t ) ρ / (cid:3) = K β [ ψ ( t )] ∗ P (cid:2) ρ / (cid:3) and P (cid:2) ρ / (cid:3) is well-peaked, we are led to the remarkable conclusion the growth dynamics oflarge- N , large- q SYK model is totally universal. In fact, if one waits an initial period α − to enter the Lyapunov regime, then one need simply use the effective size δ β and coupling ˜ J = α for the full growth structure of ψ ( t ) ρ / to match that of ψ ( t ) .22 .3 Finite Temperature Epidemic Model In this subsection we will discuss the physical interpretation of the SYK operator growthby relating it to an epidemic model. Intuition for operator scrambling behavior has beendeveloped by various authors [6, 11, 22, 37], resulting in an infection picture for operatorgrowth. An operator such as ψ ( t ) can be expanded in the strings of Majorana fermion Γ I .We consider the fermions already included in the string as “infected”. Heisenberg evolution of Γ I generates a term [Γ I , H ] which could contain a few more fermions. For example for SYKmodel with q -body interactions, in the large N limit most of the terms have one fermionreplaced by q − other fermions. In order for these q − fermions to be “infected”, they mustnot be already in Γ I . Therefore the infection rate depends on the infectable population.In the simplest infection model for a population of n ∗ individuals, the rate of infection isproportional to the number of unexposed people times the number of contagious people dn ( t ) dt = r (cid:18) − n ( t ) n ∗ (cid:19) n ( t ) (5.15)More generally, in various quantum circuit and Hamiltonian systems, both terms on theright-side of the equation may be raised to various powers or there may even be a sumof such terms, due to the potential multi-body nature of the interaction. For example, inSYK, upon a single commutation with the Hamiltonian, a size operator becomes a size q − operator, so we might expect various powers of q to appear in the above expression.Regardless, in either case sigmoidal behavior will be produced, which is consistent withgeneral expectations of four-point functions.Let us see just how well such a picture can apply to the SYK model. Taking the derivativeof Eq. (5.6) and using Eq. (5.8), we find that during the Lyapunov regime ( log N (cid:29) αt (cid:29) ) ddt (cid:0) n (cid:2) ψ ( t ) ρ / (cid:3)(cid:1) ≈ (2 J ) (cid:32) − n (cid:2) ρ / (cid:3) n ∗ (cid:33) q/ (cid:0) n (cid:2) ψ ( t ) ρ / (cid:3) − n (cid:2) ρ / (cid:3)(cid:1) (5.16)Comparing with the infection equation (5.15), we have the fundamental rate r = 2 J aswell one of the terms being raised to q/ due to the q -local nature of the interaction. How-ever, rather than (cid:0) − n (cid:2) ψ ( t ) ρ / (cid:3) /n ∗ (cid:1) q/ , which one may have expected by direct analogywith the infection equation, we have the static term (cid:0) − n (cid:2) ρ / (cid:3) /n ∗ (cid:1) q/ = δ q/ β . Duringthe Lyapunov regime, these two are the same to leading order in N . Lastly, it appearsthrough the final term that of the large population n (cid:2) ψ ( t ) ρ / (cid:3) , only the small population n (cid:2) ψ ( t ) ρ / (cid:3) − n (cid:2) ρ / (cid:3) possesses the ability to infect others. Notice that there remains thelarge population n (cid:2) ρ / (cid:3) who count as having been exposed, but do not infect others. It isthus natural to view this group as a vaccinated population.In other words, after waiting for the dynamical renormalization/time re-parametrizationto settle down, the physics of the four-point function is well-described by an infection model,with the caveat that only a small population n (cid:2) ψ ( t ) ρ / (cid:3) − n (cid:2) ρ / (cid:3) possess the ability toinfect. In this sense, the operator ρ / vaccinates a finite fraction of the N flavors. Nowregardless of whether any particular individual possess the ability to infect, it remains that alarge portion of the population has been exposed, and thus the probability for any contagious23igure 10: Illustration of different epidemics at infinite temperature (left panel) and finitetemperature (right panel). Green dots represents unexposed individuals, red dots representscontagious individuals, and blue dots represent vaccinated individuals. Finite temperaturefactors such as ρ / “use up” some of the available flavors for growth, resulting in collisionslike those depicted in Fig. (5). This effect ends up being well-modeled by an epidemic wherethese flavors or individuals count as having been exposed, but do not spread disease. As aresult of this large vaccinated population, it simply more rare for a contagious individualto encounter an unexposed individual, even at the start. Hence the rate of infection – theLyapunov exponent – slows down, as seen in the right figure.individual to encounter an unexposed individual is decreased. Consequently, the overall rateof infection slows down to λ L = 2 J (cid:32) − n (cid:2) ρ / (cid:3) N ∗ (cid:33) q/ = 2 J δ q/ β = 2 J (cid:18) G (cid:18) β (cid:19)(cid:19) q/ = 2 α (5.17)as illustrated in Fig. (10). The methodology developed in sections (2) and (3) is very powerful, as it applies to allfermionic systems. Specifically, determining the system’s full growth distribution amountsto calculating the twisted (3.17) two-point function G µ followed by inverse transforming in µ (3.12). The large- N saddle point technique and the large- q simplification enabled us toobtain a closed solution in SYK. Even if analytics are too difficult, this analysis can beeffectively implemented numerically for many classes of models.24hese techniques also allowed us to compute a four-point function, since Eq. (3.2) showsthat the average size of the operator ψ ( t ) ρ / (5.6) gives the value of a certain four-pointfunction. We can generalize and calculate arbitrary four-point functions by moving the twist(3.17) to other locations. This has the non-trivial consequence that the twisted two-pointfunction solves the ladder kernel [30, 31]. In practice solving the former can be substantiallyeasier than solving the latter. As an example, in [38] we use this “twisted” technique toderive an elegant expression for the large- q SYK four-point function at arbitrary couplingand temperature. Like the growth distribution methodology, this new method for calculatingfour-point functions works for all fermionic systems.The dynamical renormalization of the coupling (5.12) plays a central role in this work.It will be important to understand this in a deeper and more general context. The successof the modified infection model in capturing the thermal operator growth suggests that theprinciple underlying the finite temperature slowdown in SYK is competition for Majoranaflavors. The presence of various powers of the thermal state exp ( − βH ) “uses up” some finitefraction of the flavors. Consequently, when we apply a single fermion, there is a fractionalprobability for it to become absorbed and thus its size is renormalized (3.20) to a value basedupon the percentage δ β of “unused” flavors. Now, the renormalized coupling J ( t ) (5.12) slowsdown during time-evolution. We believe that this occurs due to the same principle, but havenot yet fully understood the mechanism. Our belief is motivated by the empirical observationthat the Lyapunov exponent is a power of the percentage δ β of “unused” flavors λ L = lim t (cid:29) t dissipation J ( t ) = 2 J ( δ β ) q/ ≡ J (cid:32) − n (cid:2) ρ / (cid:3) n ∗ (cid:33) q/ (6.1)This kind of sigmoidal operator growth is generic in many-body chaos. However, withoutMajorana fermions, the manner in which the thermal factors interfere with operator growthmust be more complicated, as there is not a bit-like notion of “using up” a flavor. Sigmoidalbehavior signals the existence of a competition for some finite resource. For SYK, this re-source was flavor; we only have N flavors with which to grow operators, so eventually wewill be led to flavor collisions as in Fig. (5). However, flavor competition is only one aspect ofcompetition for a more general resource. The question remains: what do operators competefor during evolution? Is there some sort of “operator entropy”? Perhaps when summed across“all operators” at finite temperature, there is always a fixed amount of total correlation withan initial simple operator due to unitarity. A better understanding of such a resource wouldgive an organization to operator dynamics at different energy scales.Our results have interesting implications for the holographic dual of the SYK model.This is simplest to understand when we explicitly express our results in terms of the doubledtheory. We defined an entangled orthonormal basis for the doubled theory using the eigen-states (2.10) of the size operator (2.14). Taking the state ψ L ( t ) | T F D (cid:105) , we related its sizewave-function squared (i.e. P n (cid:2) ψ ( t ) ρ / (cid:3) ≡ (cid:12)(cid:12) (cid:104) n | ψ L ( t ) | T F D (cid:105) (cid:12)(cid:12) ) to the size wave-function If we replace t → − t , then this is the precursor state ψ L ( − t ) | T F D (cid:105) , where the “boundary” operator ψ L acted upon the thermofield double state at time − t [3, 12, 39, 40]. P n (cid:2) ρ / (cid:3) ≡ |(cid:104) m | T F D (cid:105)| ) (cid:12)(cid:12) (cid:104) n | ψ L ( t ) | T F D (cid:105) (cid:12)(cid:12) = n (cid:88) m =0 K βn − m [ ψ ( t )] |(cid:104) m | T F D (cid:105)| (6.2)isolating the time-dependence into the growth distribution K β [ ψ ( t )] (3.11). Using this, wefound the “average growth” of ψ ( t ) (5.7) at low temperatures to be β J sinh ( πt/β ) /π ) ,which was shown in [41] to exactly match the classical momentum dynamics of a “boundary”particle falling into a near-extremal black hole. That is, the average growth of an SYKfermion exactly matches the average momentum of an infalling particle in a N AdS blackhole.It is a striking result of our analysis that the full size wavefunction squared of the SYKfermion precisely relates to the full momentum wavefunction squared of the infalling particle.The universal form (5.14) of the growth distribution K β [ ψ ( t )] precisely gives the squaredcoefficients of the AdS momentum bulk-to-boundary propagator. Exploring this connectionwill be an important focus of future work . Acknowledgements
We are grateful to Adam Brown, Tarun Grover, Yingfei Gu, Guy Gur-Ari, Matt Hastings,Andy Lucas, Daniel Ranard, Dan Roberts, Phil Saad, Steve Shenker, Eva Silverstein, DouglasStanford, Lenny Susskind, Brian Swingle, Aron Wall, and Ying Zhao for extremely useful dis-cussions. This work is supported by the National Science Foundation grant 1720504 (XLQ),and the Simons Foundation (XLQ and AS). This work is supported in part by the U. S.Department of Energy award de-sc0019380 (XLQ).
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