Out-Of-Time-Order Operators and the Butterfly Effect

SU-ITP-17/03

Out-of-time-order Operators and the Butterﬂy Eﬀect

Jordan S. Cotler, Dawei Ding, and Geoffrey R. Penington Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305 (Dated: April 28, 2017) Out-of-time-order (OTO) operators have recently become popular diagnostics of quantum chaos in many-body systems. The usual way they are introduced is via a quantization of classical Lyapunov growth, which measures the divergence of classical trajectories in phase space due to the butterfly effect. However, it is not obvious how exactly they capture the sensitivity of a quantum system to its initial conditions beyond the classical limit. In this paper, we analyze sensitivity to initial conditions in the quantum regime by recasting OTO operators for many-body systems using various formulations of quantum mechanics. Notably, we utilize the Wigner phase space formulation to derive an ~–expansion of the OTO operator for spatial degrees of freedom, and a large spin 1/s– expansion for spin degrees of freedom. We find in each case that the leading term is the Lyapunov growth for the classical limit of the system and argue that quantum corrections become dominant at around the scrambling time, which is also when we expect the OTO operator to saturate. We also express the OTO operator in terms of propagators and see from a different point of view how it is a quantum generalization of the divergence of classical trajectories.

PACS numbers: 03.65.-w, 05.45.Mt

INTRODUCTION sensitivity as a Poisson bracket:

∂x(t) Quantum chaos attempts to generalize well-established = {x(t), p0}Poisson , (2) classical diagnostics of chaos to quantum-mechanical sys- ∂x0 tems [1–4]. Although classical diagnostics are well- where p0 is the initial momentum. We can now proceed understood, finding satisfying quantum diagnostics is an by canonical quantization to obtain the quantity ongoing field of research [5–8]. One reason is that na¨ıve quantum generalizations of classical measures of chaos 1 [xb(t), pb] . often prove unsatisfactory. i~ For instance, we can diagnose classical chaos by looking at the sensitivity of dynamics to initial conditions. We are often interested in expectation values of this ob- That is, for chaotic systems, two nearby initial states servable with respect to some density matrix ρ, but can- will diverge quickly under time evolution. This phe- cellations could occur due to terms with differing sign. nomenon is commonly known as the butterfly effect. We Hence, it is natural to take the norm squared of this op- could try to generalize this notion to the quantum case in erator: the following way. Given a chaotic quantum system, for 1 1 1 [x(t), p] · − [x(t), p]† = − [x(t), p]2. (3) 0 b b b b 2 b b an initial state |Ψi and some perturbed state |Ψ i, one i~ i~ ~ might na¨ıvely expect that their inner product diminishes quickly with time. However, due to the unitarity of time The above is an example of an out-of-time-order (OTO) evolution, the inner product actually stays constant: operator, which first appeared in [10] and was studied in [11–17] in the context of black hole physics and large- 0 † 0 hΨ |U (t) U(t)|Ψi = hΨ |Ψi . N quantum field theories. It is so named since it con- arXiv:1704.02979v3 [quant-ph] 27 Apr 2017 Hence we cannot characterize quantum chaos by looking tains terms that are not time-ordered such as xb(t) pbxb(t) pb. at the evolution of the overlap between states [9]. Note that in recent literature, any operator of the form † There has been greater success in characterizing quan- [Wc(t), Vb] · [Wc(t), Vb] is referred to as an OTO operator tum chaos by considering the time evolution of observ- (see, for instance, [18, 19]). The corresponding many- ables instead of states. To do this, we first mathemat- body, higher dimensional version of Eqn. (3) is given by ically express the butterfly effect as the sensitivity of a system’s trajectory x(t) to its initial condition x0. Specif- 1 − [x (t), p ]2 , (4) ically, a system exhibits the butterfly effect if 2 bi bj ~

∂x(t) λt ∼ e , (1) where xi, pj are the position and momentum of the ith ∂x0 and jth coordinate, respectively. Eqn. (4) quantiﬁes the where λ > 0 is known as a Lyapunov exponent. To gen- sensitivity of the position of the ith coordinate to the eralize this notion to quantum systems, we re-write the initial position of the jth coordinate. 2

It is not obvious how exactly the OTO operator char- a classical phase space distribution with small quantum acterizes the sensitivity of a quantum system to its initial effects causing it to be negative in some places [23]. The conditions beyond the classical limit. Indeed, the OTO Wigner distribution satisfies the correspondence princi- is known to saturate at the scrambling or Ehrenfest time, ple [25] while ∂x(t) of a chaotic system should grow exponen- ∂x0 lim ρ(~x,~p,t) = ρclass(~x,~p,t), (6) tially forever. In this paper, we consider the OTO op- ~→0 erator in different formulations of quantum mechanics to make explicit how it quantifies sensitivity to initial where intuitively the quasiprobability distribution tends conditions – the butterfly effect. We find that using the to a classical probability distribution evolving under clas- Wigner phase space representation, an OTO operator can sical Hamiltonian dynamics as the negative regions ap- be written as a semiclassical expansion with the lead- proach zero size. Note that what is physically meant by 2 taking ~ → 0 is that ~ is small compared to the length ing classical term equal to ∂x(t) . We further argue ∂x0 and momentum scales considered. that the quantum corrections become dominant around The map given in Eqn. (5) for density matrices can be the scrambling time, providing a heuristic explanation for applied to general operators to obtain the corresponding the ending of the exponential growth at that time scale. phase space functions in the Wigner representation. It As an example, we consider the BFSS matrix model and induces a non-commutative product known as the Moyal correctly predict its scrambling time. Additionally, we star product given by rewrite OTO operators in terms of propagators in the Schrödingerformulation and see in another way how they A(~x,~p) ?B(~x,~p) capture sensitivity to initial conditions. Our analysis is i ≡ A(~x,~p) exp ~ ∂ ~ · ∂~ − ∂ ~ · ∂~ B(~x,~p). repeated for an OTO operator for spin degrees of free- 2 ~x ~p ~p ~x dom. (7)

The Moyal star product is induced by the Wigner rep- SPATIAL DEGREES OF FREEDOM resentation in the sense that W ? W = W . From Ab Bb Ab Bb Eqn. (7), we see that the Moyal star product naturally Phase Space Representation gives an ~–expansion. It is also convenient to deﬁne the phase space representation of the commutator called the In this section we use the Wigner phase space formu- Moyal bracket: lation [20–22] to study OTO operators for spatial de- 1 grees of freedom, that is, those of the form given in Eqn. {{A, B}} ≡ (A?B − B?A) (8) (4). This formulation is particularly useful for studying i~ 2 the quantum-classical correspondence and we will use it = {A, B}Poisson + O(~ ). (9) to obtain a semiclassical expansion of the OTO. For a nice overview of this formulation of quantum mechanics, In the limit ~ → 0, the Moyal bracket becomes the Pois- see [23]. son bracket, thus manifestly satisfying the correspon- Recall that in classical mechanics on phase space dence principle. we have a time-dependent probability distribution We will now rewrite OTO operators as phase space ρclass(~x,~p,t) over the phase space variables which sat- functions in the Wigner formalism: isﬁes Hamilton’s equations of motion. Phase space can 1 − [x (t), p ]2 7→ {{X (~x,~p,t),P (~x,~p, 0)}}?2 . (10) be generalized to the quantum mechanical setting via the 2 bi bj i j ~ Weyl correspondence [24]: given a density matrix ρb(t) of N particles in d spatial dimensions, we can compute the Note that although the phase space functions for position associated Wigner distribution on a 2 d N–dimensional and momentum have the initial conditions Xi(~x,~p,t = phase space with coordinates (~x,~p): 0) = xi and Pj(~x,~p,t = 0) = pj, they do not evolve by Hamilton’s classical equations of motion: instead, they ρ(~x,~p,t) ≡ W (~x,~p) ρb(t) are given by the solutions to the classical equations of 1 motion with corrections [23]. For instance, −2i~p·~y/~ ~ ≡ dN d~y h~x + ~y| ρ(t) |~x − ~yi e . (π ) ˆ dN b ~ R ∞ (5) cl X 2k (k) Xi(t) = xi (t) + ~ xi (t) (11) The Wigner function is a quasiprobability distribution, k=1 meaning that it integrates to unity but can take on neg- is the solution to the Moyal equation of motion ative values. In fact it is only negative on ~-scale cells of ˙ 2 phase space, and is otherwise positive – it is much like Xi(t) = {{H,Xi(t)}} = {H,Xi(t)} + O(~ ) . (12) 3

These ~ corrections depend on the dynamics under con- system. That is, we will provide an ~-expansion for the sideration and are not easily characterized for general OTO but will suppress the ~ dependence of Xi(t). systems. As a result, we will leave the corrections pack- aged implicitly into Xi(t), since we would like to derive Using d N-dimensional multi-index notation, we obtain an expression independent of the speciﬁc dynamics of the the expansion:

∞ 2` X 1 i X 2` X ~m {{X (~x,~p,t),P (~x,~p, 0)}}?2 = ~ (−1)|~n| ∂~n∂ ~m−~nY (t) ∂ ~m−~n∂~nY (t) (13) i j (2`)! 2 ~m ~n x p ij x p ij `=0 | ~m|=2` ~n≤ ~m

∂Xi(t) where |~v| ≡ v1 + ··· + vd N and Yij(t) ≡ . butterfly effect: ∂xj To better understand Eqn. (13), let us consider the 2 ∂xcl(t) simplified setting of a single particle in one dimension. ∼ e2λt. (15) In this case, the leading terms of Eqn. (13) reduce to ∂x

∂X(t)2 2 ∂X(t) However, expectation values of the OTO operator are − ~ det Hess + O( 4) (14) ∂x 4 ∂x ~ known to initially grow exponentially but then saturate to a finite value [26]. This suggests that the OTO oper- where Hess( · ) is the Hessian on the two-dimensional ator stops growing and saturates because the expansion phase space (i.e., the matrix of second derivatives with is no longer dominated by the leading classical term. respect to the initial conditions X(0) = x and P (0) = p). We now make a heuristic argument that the classical In Appendix A, we detail how the O(~2) term in the term stops dominating at the time known as the scram- many-body case can be expressed in terms of a higher- bling time, when the exponential growth ends. The ar- dimensional analog of the Hessian. As expected, the first gument is similar to the reasoning used in [3] to explain term in Eqn. (14) contains the square of Eqn. (2) since the evolution of the Wigner representation of the density X(t) = xcl(t) + O(~2), and so we manifestly reproduce matrix becoming dominated by non-classical terms after classical chaotic behavior as ~ → 0. The O(~2) correc- exactly the same timescale, but we consider the evolu- tions in Eqn. (13) are interesting because they correspond tion of operators rather than states. The two arguments to higher derivatives with respect to the initial condi- are related by the usual duality in quantum mechanics tions of X(t). In fact, O(~2k) terms contain products of between the Schrödingerand Heisenberg pictures of time (2k + 1)th derivatives with respect to initial conditions. evolution. Both points of view describe the same physical behavior and so both must contain the same phenomena. The two leading quantum corrections are of order ~2 Saturation of Chaos and are

(1) cl 2 cl Here we argue that the semiclassical expansion of the 2 ∂x (t) ∂x (t) ~ ∂x (t) 2~ − det Hess (16) OTO operator can explain the operator’s early time ∂x ∂x 4 ∂x growth and heuristically justify the timescale at which 2 (1) this growth saturates – namely the scrambling time. (In where ~ x (t) is the leading quantum correction to the the original literature on semiclassical expansions, this Moyal trajectory X(t) as per Eqn. (11). is known as the Ehrenfest time, but we shall use scram- We ﬁrst consider the correction in the right half of bling time throughout this paper.) We will focus mostly Eqn. (16), which comes from higher derivatives of the on the expansion for a single particle in one dimension classical trajectory. We shall refer to this as the “Hessian for simplicity, but the generalization to more dimensions correction.” Consider the quantity and particles is straightforward. Note that although by s v deﬁnition a single classical particle in one dimension is f 2 u f f A(f) = = u always integrable, it can still be exponentially sensitive det Hess(f) t ∂2f ∂2f ∂z2 ∂z2 to initial conditions and thus is still instructive. 1 2 2 ∂xcl(t) The leading term ∂x in the expansion of the where z1, z2 are local coordinates on phase space that OTO operator in the Wigner representation simply de- diagonalize the Hessian. Up to order one factors, A(f) scribes the sensitivity of the position at time t to the characterizes the maximum possible area of an ellipse initial condition in the classical limit. For a chaotic sys- (with z1, z2 forming the principal or minor axes) within tem, it should grow exponentially for all time due to the which the second-order terms in the Taylor series of f 4

λt with respect to z1, z2 are small compared to f itself. The e (at least as long as it is evolving approximately clas- Hessian correction is small compared to the leading clas- sically). As a result we instead expect sical term when 2 x˙ (1)(t) 2 cl ~ ~ 2λt ∂x (t) cl ∼ 2 2 e , (22) A . (17) x˙ (t) LxLp ∂x ~ and so, just as for the Hessian correction, we should ex- Hence, the Hessian correction can be ignored when there pect that the trajectory correction will become significant is an ellipse larger than the Planckian area around the at the scrambling time. Furthermore, although we only ∂xcl(t) initial conditions (x, p) where is well-approximated 2 ∂x considered the leading corrections at order ~ , a simi- by a Taylor expansion truncated to first order. lar analysis holds for all other orders. As a result, the Liouville’s theorem states that classical time evolu- semiclassical expansion will entirely break down at the tion on phase space is area-preserving. However for scrambling time. chaotic systems, an ellipse will get mapped to increas- The classical term in the OTO operator expansion is ingly warped and complicated regions of phase space at given at the scrambling time by long times. This mapping should become highly nonlinear for all ellipses of area A when ∂x (t )2 (L L )2 cl scr 2λtscr x p ∼ e ∼ 2 . (23) L L ∂x ~ eλt x p (18) A Note that we cannot directly predict the saturation of the OTO at the scrambling using the Wigner expansion. where Lx and Lp are the characteristic length and momentum scales at the energy determined by the initial Instead, we can only see the end of the semiclassical exponential growth of the OTO. Going beyond the scram- conditions (x, p). LxLp is therefore the effective accessi- cl bling time and seeing the saturation will require different ble area of phase space. This suggests that A ∂x (t) ∂x . techniques and approximations. ~ and the expansion of the OTO operator is no longer It is interesting that both corrections naturally become dominated by its leading term when large at the same timescale. Essentially, the behavior is classical until the scrambling time when the phase space λt LxLp e & . (19) description becomes dominantly quantum. Not only does ~ the semiclassical expansion of the OTO break down, but This timescale is exactly the scrambling time. even the Moyal trajectory X(t) ceases to have any direct We now consider the correction in the left half of Eqn. physical meaning. (16), which we shall refer to as the “trajectory correc- With multiple dimensions or particles, we have to sum tion.” It was argued in [3] when considering the evolu- over Hessian corrections to the OTO operator for all tion of states in the Wigner picture that smooth func- pairs of conjugate dimensions, as discussed in Appendix tions f on phase space should evolve approximately clas- A. Performing a linear canonical transformation so that sically for chaotic systems, that is, f(t) ∼ f cl(t), until the Lyapunov growth is diagonal, we see that the quan- the scrambling time tum correction corresponding to the pair with the largest Lyapunov exponent λ will be the first to become sig- max −1 LxLp nificant. Similarly, the largest corrections to the Moyal tscr ∼ λ log (20) ~ trajectory will come from derivatives associated with the same conjugate dimensions (call them x, p). The scram- where quantum corrections begin to dominate. We shall bling time is therefore briefly review this argument for the specific case of X(t).

The leading quantum correction to the Moyal equation of −1 LxLp 2 tscr = λmax log . (24) motion occurs at order ~ and contains two extra deriva- ~ tives with respect to each of x and p. By dimensional analysis, we might na¨ıvely assume that this means This is consistent with results in the literature [16]. We note that although for one particle in one dimension LxLp 2 x˙ (1)(t) 2 was the accessible area of phase space, this is no longer ~ ∼ ~ . (21) cl 2 2 true for higher dimensions. This is because the only part x˙ (t) LxLp of phase space that determines the scrambling time is the However, in each term of the leading correction to two-dimensional subspace involving the largest Lyapunov Moyal’s equation, there are two derivatives acting on exponent. All the additional dimensions are effectively X(t) and two others acting on the Hamiltonian. The two irrelevant for the breakdown of semiclassical behavior. derivatives acting on X(t) expose its exponential sensi- There is recent interest in studying chaos for systems tivity to initial conditions by producing two factors of with a large number of matrix degrees of freedom. One 5 such system is the BFSS matrix model [27], which is a backward in time. The chaos kernel can be imagined as chaotic quantum system of great importance in quantum the amplitude of a spacetime arc, as shown in Fig. 1. gravity. Chaos in the classical BFSS model was studied in [28]. It was shown that it is necessary to take the y ’t Hooft limit for the largest Lyapunov exponent to con- t verge to a fixed value at large N, the size of the matrices. In this limit, the ratio of the product of the accessible length and momentum scales to the Planck scale grows linearly with N. This is easiest to see if you take ~ the ’t Hooft limit by making the Planck scale N rather than ~, while keeping the coupling fixed. If you take the more traditional, but equivalent, approach of absorb- 1 ing the factor of N into the coupling, the Planck scale will be fixed but you instead get growth in the accessible size of phase space. Since the ratio LxLp is a di- x z ~ mensionless, physical quantity, it is the same from both points of view. If we apply our arguments to this spe- FIG. 1: An arc in (coordinate space)×(time). C(~x,~y, ~z, t) cific example, we find correctly that the scrambling time is the corresponding probability amplitude. −1 tscr ∼ λmax log(N/~). It is somewhat surprising at first glance why the scram- Let µ (~x,~z, t) be the first moment of the chaos kernel bling time for the BFSS model and other fast-scrambling, i with respect to the ith intermediate position coordinate: large N quantum systems should be proportional to log (N/~) when the entropy grows as N 2 log (1/~). Why should the scaling of the scrambling time be so different from that of the entropy when we increase the num- µ (~x,~z, t) ≡ d~yy C(~x,~y, ~z, t) . (27) i ˆ i ber of quantum states by adding more matrix degrees of freedom? Our approach makes the intuition behind Physically, µ is the first moment of the ith intermedi- this much clearer: even though the full volume of phase i ate position coordinate averaged over all spacetime arcs space grows exponentially with N, the area of the two- starting at ~x and ending at ~z. We can then write the dimensional subspace involving the largest Lyapunov ex- many-body OTO operator as ponent only grows linearly with N in units of the Planck scale in the ’t Hooft limit. As we discussed above, it is 1 the area of this subspace that determines the scrambling − [x (t), p ]2 = d~x d~x d~x |~x ih~x | 2 bi bj 1 2 3 1 3 time, rather than the entire volume of phase space. ~ ˆ

∂ ∂ × µ (~x , ~x , t) µ (~x , ~x , t) , (28) + i 2 1 + i 3 2 ∂x21,j − ∂x32,j − Propagator Representation x21,j x32,j

Now we will further study the structure of OTO oper- ± 1 where xab,j ≡ 2 (xa,j ± xb,j) and xa,j is the jth coor- ators by expressing them in terms of propagators in the dinate of ~xa. Each partial derivative can be physically standard Schrödingerformalism. We will find that a nat- interpreted as the sensitivity of µi to changes in the av- ural kernel-like quantity arises which we will refer to as erage position of the two ends of the arc while keeping the chaos kernel. The OTO operator can be interpreted the difference in the positions of the ends constant. We roughly as the sensitivity of the first moment of the chaos can imagine this change by a rigid 1D-motion of a rod kernel to initial conditions. connecting the two positions. Note that the expression We first fix some notation for the propagator: is manifestly Hermitian since

K(~x,~y, t) ≡ h~y|U|~xi, (25) ∂ ∂ ∂ where |~xi, |~yi are the position eigenstates of N particles in µi(~x2, ~x1, t) = + µi(~x2, ~x1, t), + ∂x ∂x ∂x21,j − 2,j 1,j d spatial dimensions and U is the time evolution operator x21,j with time parameter t. Then, we define the chaos kernel (29) ∗ as and µi (~x2, ~x1, t) = µi(~x1, ~x2, t). Eqn. (28) captures the sensitivity of the quantum state C(~x,~y, ~z, t) ≡ K(~x,~y, t)K∗(~z, ~y, t) . (26) at time t to its initial conditions, just as Eqn. (2) does for This is the probability amplitude of ~x → ~y, i.e., moving a classical system. If we take ~x1 = ~x2 for simplicity, the forward in time, multiplied by that of ~y ← ~z, i.e., moving relationship between Eqn. (28) and the butterfly effect 6 becomes especially clear: Spins via Phase Space Representation  

∂ The Wigner phase space formulation can be extended  µi(~x2, ~x1, t)  ∂x+ to spins [29–31]. Although our final result for the spin 21,j x− 21,j ~x2=~x1=~x OTO operator is straightforward, we will need to make many definitions to apply the spin phase space formalism. ∂ Let Ab be an operator which acts on the Hilbert space = d~yyi C(~x,~y, ~x,t) 2s+1 of a spin-s particle. The spin Weyl correspondence ∂xj ˆ C associates to each operator A a phase space function W : b Ab ∂ 2 = d~yyi Prob(~x → ~y, t) S → C defined by ∂xj ˆ ∂ W (θ, ϕ) ≡ tr A wW (θ, ϕ) (35) = hxi(t)i , (30) Ab b b ∂xj where where Prob(~x → ~y, t) = |h~y|U|~xi|2 is the probability den- √ 2s ` sity of transitioning from ~x to ~y in time t and hxi(t)i is 2 π X X ∗ (s) average ith coordinate at time t with the initial condition wW (θ, ϕ) ≡ √ Y`m(θ, ϕ)Tb , (36) b 2s + 1 `m `=0 m=−` xi(0) = xi. The propagator representation of OTO operators pro- In the equation above, Y are the spherical harmonics vides a complementary perspective to the Wigner phase `m and T (s) are the irreducible tensor operators [32] space representation. While the propagator represen- b`m tation does not have a manifest classical limit, it does r s (s) 2` + 1 X 0 show how the OTO operator can be expressed by objects Tb ≡ Csm |s, m0ihs, m| , (37) `m 2s + 1 sm,`m (namely, derivatives of the first moment of the chaos ker- m,m0=−s nel) with intuitive classical analogs. sm0 where Csm,`m are Clebsch-Gordan coefficients:

SPIN DEGREES OF FREEDOM CJm ≡ (hj m | ⊗ hj m |) |Jmi. (38) j1m1,j2m2 1 1 2 2

Next we repeat our above analyses for spin degrees of It will be useful to decompose Ab into irreducible tensor freedom. For motivation, we review a classical spin sys- operators as tem. In particular, we consider a top with angular mo- 2s ` mentum J~ driven by a Hamiltonian expressible in terms X X (s) Ab = A`mTb . (39) of the components of J~. Then, |J~|2 is conserved so that `m `=0 m=−` our dynamics are eﬀectively on the sphere S2 with coordinates (θ, ϕ). For |J~| = ps(s + 1), the Poisson bracket We denote by deg(Ab) the largest ` such that A`m 6= 0 for is given by some m. This is called the degree of non-linearity of Ab. 1 ∂f ∂g ∂f ∂g The spin Weyl correspondence induces the spin analog {f, g}Poisson = − . of the Moyal star product. It is given by ps(s + 1) sin θ ∂ϕ ∂θ ∂θ ∂ϕ

(31) jmax √ X (−1)j Now we consider the sensitivity of the z-component of the W ? W ≡ 2s + 1 Ab Bb j!(2s + j + 1)! angular momentum with respect to the initial azimuthal j=0 angle: h i × F −1(L2) S+(j)F (L2)W S−(j)F (L2)W e e Ab e Bb ∂J z(t) ∂ = ps(s + 1) cos θ(t) . (32) (40) ∂ϕ0 ∂ϕ0 2 In terms of Poisson brackets this is where jmax ≡ min{deg(Ab), deg(Bb)}, L is the Casimir operator on the sphere z z {J (t),J (0)}Poisson . (33) ∂2 ∂ 1 ∂2 L2 ≡ − + cot θ + (41) Hence, a spin OTO operator has the form ∂θ2 ∂θ sin2 θ ∂ϕ2 z z 2 2 −[Jb (t), Jb (0)] . (34) L Y`,m = `(` + 1)Y`,m, (42)

While our analysis extends straightforwardly to other Fe is a function such that types of commutators of spin operators, we restrict our 2 p attention on the OTO operator in Eqn. (34). Fe(L )Y`,m ≡ (2s + ` + 1)!(2s − `)! Y`,m , (43) 7 and the S± are diﬀerential operators deﬁned by the spin OTO operator, we obtain

?2 ( j−1 Q ∂ i ∂ ?2 ∂WJˆz (t) ±(j) m=0 m cot θ − ∂θ ∓ sin θ ∂ϕ j > 0 W (t),W (0) = , (45) S ≡ . Jbz Jbz ∂ϕ 1 j = 0 (44) where the bracket {{·, ·}} is deﬁned as in the spatial case, We can now use the formalism to obtain a semiclassical and the equality follows from generalizing Theorem 5 expansion. Applying the spin Weyl correspondence to in [29]. We can now use Eqn. (40) to obtain

deg(J z (t)) √ b j ∂W (t) ∂W (t) ?2 X (−1) −1 2 +(j) 2 Jbz −(j) 2 Jbz W z (t),W z (0) = 2s + 1 Fe (L ) S Fe(L ) S Fe(L ) . Jb Jb j!(2s + j + 1)! ∂ϕ ∂ϕ j=0 (46)

Note that similar to the spatial case, W (t) can be ex- where Hess 2 ≡ ∇∇ is the Hessian tensor on the sphere. Jbz S ∂ 1 ∂ panded in 1/s, where the leading term is the solution In the (θ,b ϕb) basis, ∇ = ∂θ θb + sin θ ∂ϕ ϕb . Hence we in- to the classical equations of motion. However, again the triguingly find that Eqn. (49) has precisely the same form corrections depend on the specific dynamics, and so we as Eqn. (14), even up to constant factors. Furthermore, absorb them into W (t). Also note that the number of Jbz using the above scrambling time analysis and replacing terms in Eqn. (46) is directly related to non-linearity of ~ with 1/(2s + 1), we expect the spin OTO operator to z the operator Jb (t). This implies higher derivative cor- saturate at times tscr ∼ log(2s + 1) for large s. rections are induced by chaotic dynamics that increase deg(Jbz(t)), which is initially unity at t = 0. (Non- linearity can be phrased in terms of an expansion in a Spins via Propagator Representation basis of operators. For a discussion about the time dependence of such expansions, see Appendix B.) This is analogous to the spatial case where chaotic dynamics make We now consider a more general system with N spin-s Xi(t) nonlinear in ~x and ~p, and thereby generate higher particles and use propagators to express the spin OTO z z z derivative corrections in the OTO operator in Eqn. (13). operators −[Jbi (t), Jbj ], where Jbi is the z-component of Eqn. (46) has a particularly enlightening form in the the spin of the ith particle. We will find that the spin semiclassical limit where s 1. In this limit, we can OTO operator can be interpreted as the sensitivity of a approximate the star product using the small expansion quantity similar to µi defined in Eqn. (27). 1 To define a propagator for spins, we use spin coherent parameter ε ≡ 2s+1 [30]: states which are defined as [29] W ? W Ab Bb s 1 2 2π dψ h ε i X 2s s+m θ s−m θ −imϕ = W exp S ~ S~ − S ~ S~ W + O(ε3), |s, ni ≡ cos sin e |s, mi, Ab − + + − Bb s + m 2 2 ˆ0 2π 2 m=−s (47) (50) where s is the spin quantum number, n ∈ S2 and θ, ϕ are where (θ, ϕ, ψ) are the Euler angles, and its polar and azimuthal angles, respectively. The |s, mi are eigenvectors of J z with eigenvalues m. Note that the spin coherent states form a complete basis, thereby ∂ ∂ 1 ∂ S ≡ ie∓iψ ± cot θ + i ∓ . (48) allowing us to insert resolutions of the identity: ± ∂ψ ∂θ sin θ ∂ϕ 2s + 1 In the limit of large s, we can use Eqn. (47) to expand I = dn |s, nihs, n| . (51) 4π ˆS2 Eqn. (46) to second order in ε: We denote the spin propagator by ?2 W (t),W (0) Jbz Jbz s ∂W (t)2 ∂W (t) K (~n, m~ , t) ≡ hs, m~ |U|s, ~ni, (52) Jbz 2 Jbz 3 = − ε det Hess 2 + O(ε ) , ∂ϕ S ∂ϕ NN (49) where |s, ~ni ≡ a=1 |s, nii. Then, we define the spin 8

chaos kernel for the ith particle as other than ϕmi :

s ~ µi (~n, k, t) Cs(~n, m~ , ϕ , ~k, t) N i mi 2s + 1 Y s ~ ~ ~ 2π = dm` sin θmi dθmi Ci (n, m, ϕmi , k, t) s ∂ s ∗ ~ 4π ˆ = dϕmi K (~n, m~ , t) (K ) (k, m~ , t), (53) `6=i ˆ0 ∂ϕmi 2s + 1N ∂ = dm~ Ks(~n, m~ , t) (Ks)∗(~k, m~ , t). 4π ˆ ∂ϕmi s (54) where ϕmi means Ci is not a function of ϕmi . Geometri- s cally, Ci is the complex area enclosed by the loop traced s s ∗ ~ 2 µs(~n, ~k, t) can be interpreted as the average sensitivity of out by (K (~n, m~ , t), (K ) (k, m~ , t)) ∈ C as ϕmi varies i from 0 to 2π. Eqn. (53) is analogous to the chaos kernel the amplitude to go back in time to the angles ~k given defined in Eqn. (26) and we can likewise define a quantity that the initial angles were ~n. analogous to the first moment of µi defined in Eqn. (27) Putting everything together, the spin OTO operator is s by integrating Ci over all the degrees of freedom of m~ given by

4N 4 ! z z 2 2s + 1 Y ∂ s s ∂ s − [Jb (t), Jb ] = − dn` |~n1ih~n4| µ (~n2, ~n1, t) f (~n2, ~n3) µ (~n4, ~n3, t) , i j 4π ˆ ∂ϕ+ i ∂ϕ+ i `=1 21,j − 43,j − ϕ21,j ϕ43,j (55)

where ϕ± ≡ 1 (ϕ ± ϕ ) and f s(~n, m~ ) ≡ tower of higher derivatives with respect to the initial con- ab,j 2 (na)j (nb)j 1 ditions of a time-evolved observable. The Wigner phase hs, ~n|s, m~ i. For s = 2 , for instance, we have space formalism is central to this analysis, and provides a clear way of detailing semiclassical expansions in either ~ N 1 i Y θnl θml − ϕ −ϕ or 1/(2s+1). Our analysis has provided an estimate and f 2 (~n, m~ ) = sin sin e 2 ( nl ml ) 2 2 geometric interpretation of the scrambling time, which l=1 agrees with known results and has a clearer interpreta- i θnl θml ϕ −ϕ + cos cos e 2 ( nl ml ) . (56) tion in some respects. Additionally, we have recast OTO 2 2 operators in terms of propagators, and have found that they reflect the sensitivity of the propagation of the sys- We observe tem along a spacetime arc when its initial and final points s ∗ s are varied. All of these analyses can be extended to quan- (µi ) (~n2, ~n1, t) = −µi (~n1, ~n2, t) (57) tum field theories [33]. by integrating by parts and noticing that the boundary Perhaps what is most striking is that the OTO oper- term vanishes by periodicity. Furthermore, ators for both spatial and spin degrees of freedom have (f s)∗(~n, m~ ) = f s(m~ , ~n), (58) such similar form in their phase space representations. For example, the leading terms in both expansions so Eqn. (56) is manifestly Hermitian. have the same form, as do the leading higher derivative We see that Eqn. (55) has a similar form as Eqn. corrections which are Hessians. This universality of form (28) and admits a similar interpretation. We find again is not obvious in other treatments of the OTO operator that the OTO operator has essentially the same structure in which semiclassical expansions are not manifest. The across spatial and spin degrees of freedom. unifying ingredient is phase space, which is a natural setting for both classical and quantum chaos [6, 34–39]. This suggests it may be interesting to explore other DISCUSSION recent developments in quantum chaos using the phase space formalism. We have shown how the OTO operator reflects sensitivity to initial conditions beyond the classical limit. In Acknowledgements. JC is supported by the particular, for both spatial and spin degrees of freedom, Fannie and John Hertz Foundation and the Stanford the quantum corrections to the OTO operator form a Graduate Fellowship program. DD is supported by 9 the Stanford Graduate Fellowship program. We would [26] K. Hashimoto, K. Murata, and R. Yoshii, arXiv preprint like to thank Xingshan Cui, Jonathan Dowling, Nicole (2017), 1703.09435. Younger-Halpern, Patrick Hayden, Edward Mazenc, [27] T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, Stephen Shenker, Joseph Várilly, Cosmas Zachos and Physical Review D 55, 5112 (1997). [28] G. Gur-Ari, M. Hanada, and S. H. Shenker, arXiv Wojciech Zurek for valuable discussions and feedback. preprint (2015), 1512.00019. [29] J. C. Várillyand J. M. Gracia-Bond´ıa,Annals of Physics 190, 107 (1989). [30] A. Klimov and P. Espinoza, Journal of Physics A: Math- ematical and General 35, 8435 (2002). [1] R. L. Devaney, A first course in chaotic dynamical sys- [31] B. Koczor, R. Zeier, and S. J. Glaser, arXiv preprint tems (Westview Press, 1992). (2016), 1612.06777. [2] B. Hasselblatt and A. Katok, A first course in dynam- [32] D. A. Varshalovich, A. N. Moskalev, and V. K. Kher- ics: with a panorama of recent developments (Cambridge sonskii, Quantum theory of angular momentum (World University Press, 2003). Scientific, 1988). [3] W. H. Zurek and J. P. Paz, Physical Review Letters 72, [33] T. Curtright and C. Zachos, Journal of Physics A: Math- 2508 (1994). ematical and General 32, 771 (1999). [4] F. M. Cucchietti, D. A. Dalvit, J. P. Paz, and W. H. [34] H. Korsch and M. Berry, Physica D: Nonlinear Phenom- Zurek, Physical Review Letters 91, 210403 (2003), quant- ena 3, 627 (1981). ph/0306142. [35] J. S. Hutchinson and R. E. Wyatt, Chemical Physics Let- [5] M. C. Gutzwiller, Chaos in classical and quantum me- ters 72, 378 (1980). chanics, Vol. 1 (Springer Science & Business Media, [36] B. D. Greenbaum, S. Habib, K. Shizume, and B. Sun- 2013). daram, Chaos: An Interdisciplinary Journal of Nonlinear [6] F. Haake, Quantum signatures of chaos, Vol. 54 (Springer Science (2005), quant-ph/0401174. Science & Business Media, 2013). [37] A. Bracken and J. Wood, Physical Review A 73, 012104 [7] H.-J. Stöckmann, Quantum Chaos: An Introduction (2006), quant-ph/0511227. (Cambridge University Press, 2007). [38] R. N. Maia, F. Nicacio, R. O. Vallejos, and F. Toscano, [8] S. Wimberger, Nonlinear Dynamics and Quantum Physical Review Letters 100, 184102 (2008), 0707.2423. Chaos: An Introduction (Springer, 2014). [39] S. Chaudhury, A. Smith, B. Anderson, S. Ghose, and [9] Other notions of distance between states in the Hilbert P. S. Jessen, Nature 461, 768 (2009). space such as the relative complexity [40] have the po- [40] A. R. Brown and L. Susskind, arXiv preprint (2017), tential to characterize chaos, but are hard to work with 1701.01107. directly. [10] A. Larkin and Y. N. Ovchinnikov, Sov Phys JETP 28, 1200 (1969). [11] A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, Jour- nal of High Energy Physics 2013, 1 (2013), 1207.3123. [12] S. H. Shenker and D. Stanford, Journal of High Energy Physics 3, 1 (2014), 1306.0622. [13] S. H. Shenker and D. Stanford, Journal of High Energy Physics 2014, 46 (2014), 1312.3296. [14] S. H. Shenker and D. Stanford, Journal of High Energy Physics 2015, 1 (2015), 1412.6087. [15] A. 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Appendix A: OTO Operator to Second Order

∂Xi(t) Recall that Yij(t) ≡ . To second order, we can write Eqn. (13) as ∂xj

?2 {{Xi(~x,~p,t),P (~x,~p, 0)}}   2 d N 2 2 2 2! d N 2 2 2 2 2 ~ X ∂ Yij(t) ∂ Yij(t) ∂ Yij(t) X ∂ Yij(t) ∂ Yij(t) ∂ Yij(t) ∂ Yij(t)  ≈ (Yij(t)) −  2 2 − + 2 −  4  ∂p ∂x ∂xk∂pk ∂pk∂pk0 ∂xk∂xk0 ∂xk∂pk0 ∂pk∂xk0  k=1 k k k,k0=1 k6=k0   2 d N 2 ~ X = (Yij(t)) −  det2 Hess(Yij(t)) (A1) 4 2k,2k0 k,k0=1 where det2 denotes the principal minor of order 2 with (i, j) being the indices of the bottom right entry. We can i,j visualize the O(~2) term by partitioning the 2 d N × 2 d N Hessian matrix into 2 × 2 submatrices in the natural way:

  H1,1 ··· H1,d N  H ··· H   2,1 2,d N  Hess (Yij(t)) =  . . .  . (A2)  . .. .   . .  Hd N,1 ··· Hd N,d N

Then, the leading higher derivative correction is proportional to the sum of the determinants of all the 2×2 submatrices. Note that Eqn. (A1) manifestly simpliﬁes to Eqn. (14) in the case of a single particle in one dimension.

Appendix B: Comments on Operator Growth

Consider a lattice spin system of N spin-1/2 particles (each on their own site) evolving with some unitary time evolution. In this system, two operators localized to diﬀerent sites will commute. For example, the angular momentum z z z z operators Jbi and Jbj which act on sites i and j (for i 6= j) commute: [Jbi , Jbj ] = 0. However, working in the Heisenberg z z picture, we see that in contrast [Jbi (t), Jbj (0)] does not necessarily equal zero, and in fact will equal a sum of operators that will spread across all sites as a function of time until it saturates at some distribution over all operators. The z reason is because Jbi (t) has an expansion of the form

N N N z 1 X X α α X X αβ α β X X αβγ α β γ Jbi (t) = a(t) + bi (t) Jbi + cij (t) Jbi Jbj + dijk (t) Jbi Jbj Jbk + ··· i=1 α∈{0,x,y,z} i,j=1 α,β∈{0,x,y,z} i,j,k=1 α,β,γ∈{0,x,y,z} (B1)

0 1 z where Jbi ≡ b, and so Jbi (t) spreads across the space of operators as it evolves in time, eventually overlapping more z z z z and more with Jbj (0) = Jbj . So [Jbi (t), Jbj (0)] and related commutators provide a natural measure of quantum chaos, since their time dependence captures how local degrees of freedom are spreading in the system. z z Often, we want to compute quantities like tr(ρ [Jbi (t), Jbj (0)]) for some state ρ, such as a Gibbs state. However, for z z many states ρ we expect that there will be lots of cancellation in tr(ρ [Jbi (t), Jbj (0)]), since in light of Eqn. (B1), all of the terms not proportional to the identity can have expectation values which are either positive or negative, and their time-dependent coeﬃcients can also be positive or negative. As remarked in the introduction, the cancellation z z 2 obscures the operator growth under time evolution, and to remedy this, we instead consider tr(ρ [Jbi (t), Jbj (0)] ) which is manifestly positive. In the spin example, the salient feature which characterizes quantum chaos is the growth of operators under time 1 2 evolution. However in the case of the − 2 [x(t), p(0)] operator, writing ~ b b

X m n n m xb(t) = cmn(t)(xb pb + pb xb ) (B2) m,n 11 where the cmn(t) are real, we ﬁnd that

1 X − tr(ρ [x(t), p]2) ≈ m2 c (t)2 tr(ρ (xm−1pn + pnxm−1)2) (B3) 2 b b mn b b b b ~ m,n where “≈” signiﬁes that we have dropped terms which are not manifestly positive that may cancel out. From Eqn. 2 (B3), we see that the OTO operator captures how much the xb(t) operator is spreading. There is an m weighting m−1 each term that contains a xb ; this increases the rate of growth of the expectation of the OTO operator as xb(t) m spreads to operators containing xb ’s for progressively larger values of m.