Krylov Complexity in Conformal Field Theory

Krylov complexity in conformal ﬁeld theory

Anatoly Dymarsky1, 2 and Michael Smolkin3 1Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506 2Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Moscow, Russia, 143026 3The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (Dated: April 21, 2021) Krylov complexity, or K-complexity for short, has recently emerged as a new probe of chaos in quantum systems. It is a measure of operator growth in Krylov space, which conjecturally bounds the operator growth measured by the out of time ordered correlator (OTOC). We study Krylov complexity in conformal field theories by considering arbitrary 2d CFTs, free field, and holographic models. We find that the bound on OTOC provided by Krylov complexity reduces to bound on chaos of Maldacena, Shenker, and Stanford. In all considered examples including free and rational CFTs Krylov complexity grows exponentially, in stark violation of the expectation that exponential growth signifies chaos.

Quantum chaos and complexity play increasingly im- in terms of thermal 2pt function, as discussed below. portant role in understanding dynamical aspects of quan- Hence, the bound on OTOC in terms of KO(t) is the tum field theory and quantum gravity. The notion of bound on thermal 4pt function in terms of thermal 2pt quantum chaos is difficult to define and there are dif- function. In this sense it is similar to the proposals of ferent complementary approaches. The conventional ap- [8] and also [9], which derives the Maldacena-Shenker- proach in the context of quantum many-body systems is Stanford (MSS) bound on chaos [2] rooted in spectral statistics, Eigenstate Thermalization Hypothesis (ETH), and absence of integrability [1]. In λ ≤ 2π/β (2) the context of field theory and large N models another from the ETH. From the effective field theory point of well-studied signature of chaos is the behavior of the out view the 4pt function is independent from the 2pt one, of time ordered correlator (OTOC) [2]. These approaches hence such a bound could only be very general and apply focus on different aspects of quantum dynamics and usu- universally. One may not expect that a general theory ally apply to different systems. It is an outstanding prob- would saturate the bound, casting doubt on the proposal lem to develop a uniform approach to chaos which would that the exponent λK that controls the growth of Krylov connect and unite them. Dynamics of quantum oper- complexity is indicative of the Lyapunov exponent λ. In- ators in Krylov space has been recently proposed as a deed, we will see that in case of CFT models the conjec- potential bridge connecting dynamics of OTOC with the tural bound (1) holds but reduces to MSS bound (2) such conventional signatures of many-body chaos [3]. that λK would remain finite even when λ would approach Krylov space is defined as the linear span of nested zero or may not be well defined. commutators [H..., [H, O]], where H is the system’s There is another aspect of Krylov complexity which Hamiltonian and O is an operator in question. Accord- makes it an important topic of study in the context of ingly, time evolution O(t) can be described as dynamics quantum field theory and holographic correspondence. in Krylov space. Krylov complexity KO(t) defined be- Krylov complexity is one of the family of q-complexities arXiv:2104.09514v1 [hep-th] 19 Apr 2021 low in (7) is a measure of operator size growth in Krylov introduced in [3]. At the level of definition it is not re- space. For the chaotic systems it is expected to grow ex- lated to circuit complexity, but a number of recent works λK t ponentially [3], KO(t) ∝ e , the point we further elu- [10–12] found qualitative agreement between the behav- cidate below. For systems with finite-dimensional local ior of KO(t) with the behavior of circuit and holographic Hilbert space, e.g. SYK model [4–6], it has been shown complexities [13]. We further comment on possible simi- that at infinite temperature λK bounds Lypanunov ex- larity in the case of CFTs in the conclusions. ponent governing exponential growth of OTOC To conclude the introductory part, we remark that Krylov complexity, and dynamics in Krylov space in λ ≤ λK . (1) general, is fully specified by the properties of thermal This inequality conjecturally applies at finite tempera- 2pt function. Our results therefore should be seen in ture β > 0. From one side connection of Krylov com- a broader context of studying thermal 2pt function in plexity to OTOC is not that surprising given that the holographic settings with the goal of elucidating quan- latter measures spatial operator growth [7]. From an- tum gravity in the bulk [14–21]. other side, dynamics in Krylov space is fully determined To remind the reader, we briefly introduce main no- 2

∗ tions of Krylov space. More details can be found in where τ > 0 is the location of first singularity of C0(τ) [3, 22]. Starting from an operator O one introduces iter- along the imaginary axis, if any. It was anticipated long ative relation ago that the high frequency behavior of f 2(ω) for a local operator in many-body system can be used as a signa- 2 On+1 = [H, On] − bn−1On−1, (3) ture of chaos. In particular exponential behavior (11) where positive real Lanczos coefficients b are uniquely was proposed as a signature of chaos in classical systems n in [23]. An equivalent formulation in terms of the singu- fixed by the requirement that On are mutually orthogonal with respect to scalar product larity of C0(τ) was proposed as a signature of chaos for quantum many-body systems in [24] based on the rigor- −βH/2 −βH/2 Tr(e One Om) ∝ δnm. (4) ous bounds constraining the magnitude of C0(τ) in the complex plane. A further step had been taken in [3] who Lanczos coefficients depend on the choice of the system proposed the universal operator growth hypothesis: in Hamiltonian H, the operator O0 = O, and inverse tem- generic, i.e. chaotic quantum many-body systems Lanc- perature β. Time evolution of the operator can be rep- 2 zos coefficients bn associated with a local O exhibit max- resented in terms of Krylov space, imal growth rate compatible with locality, ∞ π iHt −iHt X b ≈ n + o(n), n 1. (12) O(t) ≡ e Oe = ϕ(t)nOn, (5) n 2τ ∗ n=0 This is stronger than the exponential behavior (11), i.e. it where normalized “wave-function” ϕ (t) satisfies dis- n implies the latter, and reduces to it upon an additional cretized “Schrödinger” equation 2 assumption that the behavior of bn as a function of n is dϕn sufficiently smooth for n → ∞. Modulo similar assump- −i = bnϕn+1 + bn−1ϕn−1, (6) 2 dt tion of smoothness of bn ref. [3] proved that in this case Krylov complexity grows exponentially as with the initial condition ϕn(0) = δn,0. It describes λK t hopping of a quantum-mechanical “particle” on a one- KO(t) ∝ e , (13) dimensional chain. Krylov complexity is defined as ∗ the averaged value of an “operator”n ˆ measured in the where λK = π/τ . “state” ϕ, where for convenience index n is shifted by 1, In field theory C0(τ) necessarily has singularity at ∞ τ = β/2, implying exponential decay of the power spec- X ∗ K (t) ≡ (O|nˆ|O) = 1 + n|ϕ (t)|2. (7) trum (11) with τ = β/2. Assuming sufficient smooth- O n 2 n=0 ness of bn, one immediately arrives at (12) [25, 26] (also see [3, 24]), and exponential growth of Krylov complexity One can similarly define K-entropy [10] with λK = 2π/β. Hence the conjectural bound on OTOC ∞ (1) reduces to the MSS bound (2). This logic applies to X 2 2 SO(t) ≡ − |ϕn| ln |ϕn| . (8) any quantum field theory, including free, integrable or ra- n=0 tional CFT models. Similarly, one can conclude that for field theories universal operator growth hypothesis (12) Lanczos coefficients, and hence KO(t), are encoded in trivially holds, but the exponential behavior of Krylov thermal Wightman 2pt function complexity can not be regarded as an indication of chaos. We stress, these conclusions are premature as one needs C0(τ) = hO(−i(τ + β/2))O(0)iβ to justify the smoothness assumption by e.g. evaluating −( β −τ)H −( β +τ)H ∝ Tr(e 2 Oe 2 O). (9) 2 bn explicitly. Without this assumption asymptotic behavior of b2 is not determined by the high frequency tail Precise relation evaluating b2 in terms of C and its n n 0 of f 2(ω), or the singularity of C (τ), as is shown explicitly derivatives is discussed in Supplemental Material. We 0 by a counterexample in [24]. We justify the smoothness only note here that b2 do not change under multiplica- n assumption by considering several different CFT models tion of C by an overall constant. 0 and evaluating Lanczos coefficients. In full generality for a physical system with local in- i) In case of 2d CFTs thermal 2pt function of primary teractions C (τ) is analytic in the vicinity of τ = 0. This 0 operators O is fixed by conformal invariance implies that power spectrum Z 1 2 iωt C0 = , (14) f (ω) = dt e C0(it) (10) cos(πτ/β)2∆ decays at large ω at least exponentially, where ∆ is the dimension of O. This C0 has been thor- oughly analyzed in [3] in the context of SYK model. In 2 −τ ∗ω 2 2 f (ω) ∼ e , ω → ∞, (11) particular they found bn = (n + 1)(n + 2∆)(π/β) and 3

βb ˜ n case when asymptotically hn → 0, mass eventually ap- 35 proaches zero for large x, describing propagation of a quantum “particle” with the speed of light x(t) ∼ t with 30 KO

100 20 50

15 10 10 5

n 0 2 4 6 8 1 0.5 FIG. 1. Lanczos coeﬃcients bn for free massless scalar φ in πt/β d = 4 (∆ = 1, blue), d = 5 (∆ = 3/2, orange), d = 6 (∆ = 2, 0.0 0.5 1.0 1.5 2.0 2.5 green) dimensions, and for the composite operator φ2 in d = 5 dimensions (∆ = 3, red); dashed lines of the appropriate color FIG. 2. Krylov complexity KO shown in logarithmic scale for show asymptotic behavior of bn as given by (21). free scalar in d = 4 (blue), d = 5 (orange), d = 6 (green) dimensions and for Generalized Free Field with ∆ = 10 (brown). Blue curve is known analytically, ln(1 + 2 sinh2(πt/β)). All four curves exhibit an apparent linear growth of ln KO ∝ 2 2 KO(t) = 1 + 2∆ sinh (πt/β). In other words bn depen- 2πt/β at late times. dence on n is smooth and Krylov complexity grows exponentially with λ = 2π/β. K respect to an auxiliary spatial continuous coordinate x ii) In case of free massless scalar in d dimensions, which is related to n via [28] as well as Generalized Free Field of conformal dimension ∆ [19], thermal 2pt function is given by, n ∝ (e(2π/β)x − 1). (17) From this follows that for late times Krylov complexity C0 = cd (ζ(2∆, 1/2 + τ/β) + ζ(2∆, 1/2 − τ/β)) . (15) will grow exponentially

2π/β(t−t0) Coefficient cd ensures canonical normalization in case of KO(t) ≈ e (18) free massless scalar and is not important in what follows. where t is the characteristic time “quantum particle” de- In the latter case ∆ = d/2 − 1. 0 scribed by ϕn(t) will spend near the edge of the Krylov For (15) with general ∆ explicit expression for Lanczos space n ∼ O(1). From the analytic expression for KO in coefficients is not known. In the special case of d = 4, case of 2d CFTs we conclude that t0 is growing negative C0 reduces to (14) with ∆ = 1, and the rest applies. For for large ∆, t0 ∼ − ln ∆. The only scenario to avoid ex- d = 6, ∆ = 2, and Lanczos coefficients can be evalu- ponential growth of KO with t is for ϕn(t) to be localized ated using connection to integrable Toda hierarchy [22], near the edge n ∼ O(1), which would presumably require yielding (see Supplemental Material) erratic behavior of bn for small n. Numerical simulation of K for massless scalar in g g O b2 = (π/β)2 (n + 2)(n + 3) n−1 n+1 , (16) d = 6 shown in Fig. 2 confirms exponential behavior (18) n g2 n with t0 of order one. Thus, despite “staggering” Krylov n+1 gn = Hn+2 + (−1) Φ(−1, 1, 3 + n) + ln(2). complexity for free massless scalar in d = 6 behaves qual- itatively similar to d = 4 case.

Here Hn is the harmonic number and Φ is Lerch tran- Next we numerically plot Lanczos coefficients for free 2 scalar in d = 5 with ∆ = 3/2, see Fig. 1. Similarly scendent. In this case bn demonstrate “staggering” or 2 to d = 6, b ’s exhibit staggering, which does not affect “dimerization” – the sequences of bn for even and odd n n can be combined into two families, each approximately asymptotic exponential behavior of KO, see Fig. 2. n˜ described by smooth functions bn = hn +(−1) hn, where To analyze general case (15) with ∆ 1 we can ap- ∗ hn ≈ (π/2τ )n + o(n) for n 1. This is shown in Fig. 1. proximate C0 with an exponential precision by Such a behavior was analyzed in [27, 28], where it was 1 1 ˜ C ∝ + . (19) shown that for smooth functions hn, hn in the large n 0 (β + 2τ)2∆ (β − 2τ)2∆ region “Schrödingerequation” (6) reduces to continuous Dirac equation with the space-dependent mass. In the By employing 1/∆ expansion we find for small n 4

16∆2 + 8(1 + 3n)∆ + 19n2/2 + 7n + O(n3)/∆ + ... for n even, β2b2 = (20) n 16(1 + n)∆ + 2(n + 1)(5n + 1) + O(n3)/∆ + ... for n odd.

Thus, staggering grows with ∆, but n dependence of bn the same conformal dimension ∆, i.e. in dimension d + 1. ¯ for odd and even n remain smooth. The same applies for bn for the composite operators ψψ 2 For large n pole structure of C0 suggests, see Supple- and φ . Corresponding comparison is delegated to Sup- mental Material, plemental Material. iv) In case of holographic CFT thermal two-point β b ≈ π(n + ∆ + 1/2). (21) n function can be calculated by solving wave equation in These approximations accurately describe bn for small the bulk [16, 17]. We perform this numerically in Sup- and large n correspondingly, as is shown in the left plemental Material to find that bn smoothly depend on panel of Fig. 3. Numerical simulation of KO(t) for n. This is shown in the right panel of Fig. 3 where we ∆ = 10 shown in Fig. 2 confirms exponential behav- superimposed bn for the holographic model with Lanczos ior with λK = 2π/β and t0 of order − ln ∆. In other coefficients for the Generalized Free Field of the same words staggering, exhibited by bn in case o free scalar effective dimension, determined by the singularity of C0 field, which grows with ∆, is not affecting dynamics at near τ → β/2. Smooth behavior perfectly matches the late times – KO grows exponentially with the exponent expectation that for holographic theories exhibiting max- λK = 2π/β, although dynamics at early times becomes imal chaos, λ = 2π/β, growth of Krylov complexity also more complicated. must be governed by the same exponent. Finally, we discuss composite operators Om for some Besides Krylov complexity we numerically plot growth integer m. By Wick theorem Wightman function simply of Krylov entropy (8) for several different models, shown m becomes C0 → C0 with an unimportant overall coeffi- in Fig. 4. In all cases it exhibits linear behavior for late t, cient. In the case of 2d CFT or free massless scalar in confirming scrambling of O in Krylov space. We conclude d = 4 we again obtain C0 of the form (14). In other that only early time dynamics is sensitive to pecularities cases Lanczos coefficients should be calculated numeri- of the model, while at late times dynamics in Krylov 2 cally. We plot bn for O = φ in free massless scalar space exhibits remarkable universality. theory in d = 5 in Fig. 1. Conclusions. In this paper we studied Lanczos coeffi- iii) In case of free fermions in d dimensions, cients and operator growth in Krylov space for local op- 1 τ 1 τ erators in various CFT models. For some models bn were C (τ) = r ζ2∆, − + ζ2∆, + ψ d 4 2β 4 2β calculated analytically, while for others we had to resort to numerical analysis. We also found asymptotic behav- 3 τ 3 τ − ζ 2∆, − − ζ 2∆, + , (22) ior of bn for large n (21). One of the main goals was to 4 2β 4 2β study if Krylov complexity is sensitive to the underlying where dimension of free fermion is ∆ = (d − 1)/2. We chaos. A general argument presented in the introduc- notice that Lanczos coefficients for free fermion in di- tion dictates that so far asymptotic behavior of bn as a mension d are very close to those for the free boson of function of n is sufficiently smooth, Lanczos coefficients exhibit universal operator growth hypothesis (12) and Krylov complexity grows exponentially (18). The only βb βb n n possible caveat is the possibility that for large n different 300 subsequences of bn would have different asymptotic, for a 50 example bn for even and odd n would grow as n with 200 different aeven 6= aodd. Another hypothetical possibility, 100 25 which will not affect (12) but may affect (18), is that erratic behavior of bn for small n will cause approximate n n 25 50 75 0 5 10 or complete localization of the operator “wave-function” ϕn, leading to large or infinite t0. We did not see any be- FIG. 3. Left panel. Lanczos coefficients bn for Generalized havior of this sort in any model we considered, including Free Field (15) with ∆ = 10 (blue) vs approximation for small arbitrary 2d CFTs, free bosons and fermions, composite n (20) (orange) and asymptotic behavior for large n (21) (red operators, generalized free field of arbitrary dimension, line). Right panel. Lanczos coefficients bn for Generalized Free Field (15) of dimension ∆ = 8.5 (blue) and for holo- and a holographic model in d = 4. On the contrary we R 3 graphic operator O = d x O of effective dimension ∆ = 8.5, observed linear growth of bn at large n in full agreement while O has dimension ∆ = 10 (orange). The same effective with (21) and exponential growth of Krylov complexity dimension means both sequences have the same asymptotic with λK = 2π/β. In other words for considered mod- behavior bn ≈ π(n + 9). els universal operator growth hypothesis of [3] trivially 5 holds, and the conjectural bound (1) on of OTOC at SUPPLEMENTAL MATERIAL finite temperature in terms of growth of Krylov complexity reduces to MSS bound [2]. At the same time Lanczos coefficients from the thermal 2pt function exponential growth of KO is not a signature of chaos as it grows with the same exponent λK = 2π/β for max- Recursion method is closely related to integrable Toda imally chaotic holographic CFTs as well as for rational chain [22]. In particular thermal 2pt function C0(τ) 2d CFTs and free field theories, for which Lypanunov should be understood as the tau-function of Toda hierar- exponent may not be even properly defined [29–31]. It chy, τ0 ≡ C0. Other tau-functions τn are related to C0 as would be interesting to extend our analysis for massive follows. One introduces (n + 1) × (n + 1), n ≥ 0, Hankel an interacting models, especially those exhibiting non- matrix of derivatives maximal chaos [32–35]. Nevertheless since a continuous (n) (i+j) deformation can not change asymptotic behavior of bn Mij = C0 (τ), (23) we provisionally conclude our results will remain valid in (k) the case of general interacting quantum field theory. where C0 (τ) stands for k-th derivative of C0. Then It is also instructive to compare behavior of Krylov (n) complexity with various notions of circuit and holo- τn(τ) = det M . (24) graphic complexities. While the latter are defined for Tau functions automatically satisfy Hirota bilinear rela- states and the former is a measure of operator growth, tion an analysis of [10–12] in the context of SYK-type mod- 2 els revealed some qualitative similarities. In this spirit τnτ¨n − τ˙n = τn+1τn−1, τ−1 ≡ 1. (25) we notice that KO exhibits essentially the same behavior for both free and holographic theories, similarly to com- At this point we can introduce qn via τn = P q0(τ) plexity action proposal, as discussed in [36, 37]. There exp( 0≤k≤n qn), such that C0(τ) = e . Functions are also bulk complexity proposals specific for conformal qn satisfy Toda chain equations of motion. Lanczos co- theories [38, 39], which exhibit robust universality due efficients are τ τ to extended symmetry. To complete the comparison, it 2 qn+1−qn n+1 n−1 bn = e = 2 . (26) would be important to go beyond thermodynamic limit τn by placing CFT on a compact background, e.g. d−1 × . S R 2 In this case one may hope to study qualitative behav- Defined this way bn are functions of τ. To evaluate Lanc- zos coefficients from (3) we need to take τ = 0 in (26). ior of KO beyond scrambling time t ∼ ln S, when the 2 exponential complexity growth should become linear. This prescription is equivalent to evaluation of bn from 2 We thank Pawe l Caputa, Mark Mezei and Alexander the moments of f (ω) (10) described in [3]. Zhiboedov for discussions. This work is supported by the There is an explicit family of solutions with the asymp- 2 2 BSF grant 2016186. totic behavior bn ∝ n [22] G(n + 2)G(n + 1 + 2∆) (π/β)n(n+1) τ (τ) = , n G(2∆)Γ(2∆)n+1 cos(πτ/β)(n+2∆)(n+1)

qn(τ) = 2n ln(π/β) − (2n + 2∆) ln(β cos(πτ/β)) + SO ln(n!Γ(n + 2∆)), (27)

an(τ) = (2n + 2∆)(π/β) tan(πτ/β), 5 2 2 (n + 2∆)(n + 1)(π/β) bn(τ) = 2 , 4 cos (πτ/β) where G(x) is the Barnes gamma function. This is the 3 solution which appears in the context of C0 for the 2d CFTs (14). 2

1 Free massless scalar in d dimensions

0 πt/β 0.0 0.5 1.0 1.5 2.0 For convenience we introduceτ ˜ = τ/β. Thermal 2pt function in coordinate space is given by an integral of FIG. 4. K-entropy SO shown in logarithmic scale for free Matsubara propagator, scalar in d = 4 (blue), d = 5 (orange), d = 6 (green) dimen- 2−d Z ∞ sions and for Generalized Free Field with ∆ = 10 (brown). All β d−3 cosh(yτ˜) Cφ(˜τ, d) = dy y . four curves exhibit an apparent linear growth at late times. d−1 d−1 y 2 sinh (4π) Γ 2 0 2 6

The integral over y can be evaluated yielding (15) with tion above that for integer d 2∆ = d − 2 and d2 Cφ(˜τ, d) ∝ 2 Cφ(˜τ, d − 2), (30) d−2 dτ˜ 2−d Γ 2 cd = β d . (28) and therefore for d = 6 4π 2 d2 1 C (τ) ∝ . (31) Numerically Lanczos coeﬃcients for (15) can be evalu- 0 dτ 2 cos2(πτ/β) ated from (24,26) using 2 To ﬁnd bn we look for the tau-functions of the form

2∆+2n −2n αn 2 (2n) 2(2 − 1)β (π/β) pn(cos (πτ/β)) Cφ (0) = cd Γ(2∆ + 2n)ζ(2∆ + 2n). τn = , n ≥ 1, (32) Γ2∆ cos(πτ/β)an (29) an = (n + a)(n + 1), αn = (n + α)(n + 1),

where pn(y) is a polynomial of degree n + 1 and p−1 = 1 Up to an overall coeﬃcient for d = 4 (15) reduces to such that τ−1 = 1. Then Hirota equations (25) become (14) with ∆ = 1. It follows from the integral representa- iterative equations for the polynomials

2 2 0 2 0 00 anpn − 4(1 − y)y (pn) + 2ypn((1 − 2y)pn + 2(1 − y)ypn) pn+1 = . (33) pn−1

To match with (31) we take a = 4, α = 2 (this value is ∝ (τ − τ ∗)−2∆ where τ ∗ = β/2, implying asymptotic in fact arbitrary) and p0 = 1 − 2y/3. Then behavior

n−1 n+1 2 −ωτ ∗ 2∆−1 Y X (−4)k Γ(2 + n)Γ(4 + n + k)yk f (ω) ∝ e ω (34) p (y) = an−k . n k (1 + k)2 Γ(4 + n)Γ(2 + 2k)Γ(2 + n − k) k=0 k=0 for large ω. Using saddle point approximation we can This can be evaluated for y = 1 which corresponds to estimate the moments ˜ τ = 0, ˜ 2k R ∞ 2 2k 2k dωf (ω)ω eτ ∗ H + ln(2) + (−1)n+1Φ(−1, 1, n + 3) M ≡ ∞ ≈ , (35) n+2 2k R ∞ 2 2∆−1 pn(1) = , dωf (ω) 2∆−1 (n + 2)(n + 3) ∞ eτ ∗

Pn −1 ˜ where Hn = k=1 k is the harmonic number and where k = k + ∆ − 1/2. Strictly speaking for validity P∞ k s Φ(z, s, α) = k=0 z /(k + α) is the Lerch transcendent. we need to require ∆ → ∞. In the case when ∆ is of From here we obtain (16). order one the expression above is valid only in the sense The same logic can be applied to d = 8, in which case of k-dependence in the limit of large k. Assuming smooth 2 a = 6, and p0 = 1 − y + 2y /15. Iterative relation (33) k-dependence of bk we will approximate it by gives β bk ≈ a k + b, k 1. (36) n−1 Y (n + 1)(n + 6) p (y) = an−k 1 − y + O(y2) , It is tempting to rewrite it as β b = a k˜, and identify b/a n k 6 k k=0 with ∆−1/2. To justify that we will use the formalism of but we were not able to ﬁnd closed form analytic expres- integral over Dyck paths which evaluates M2k in terms sion. of bn’s developed in [24]. At the level of quasiclassical approximation, which gives leading contribution in the limit of large k, the moments are given by

Pole structure of C0 controls asymptote of bn S M2k ≈ e , (37)

Under the assumption that the n-dependence of bn is where S is the on-shell value of action smooth, at least for large n 1, the asymptote of b n Z 1 is controlled by the poles of C0, or equivalently high fre- S[f(t)] = 2k dt (−p ln p − (1 − p) ln(1 − p) + b(2kf)) , 2 quency behavior of f (ω). For an operator of dimension 0 ∆ CFT correlator C0(τ) would have a pole singularity (38) 7

0 βb where p ≡ (1+f )/2, Lanczos coefficients bn are described n by the smooth function b(n) and f(t) satisfies boundary 35 conditions f(0) = f(1) = 0. For b(2kf) = 2kfa + b equation of motion reads 30 f00 2ka = (39) 25 (f0)2 − 1 2kaf + b with the solution 20 sin((π − 2r)t + r) b f(t) = − , (40) 15 (π − 2r) 2ka where r is defined from the equation 10 b sin(r) = . (41) n 2ka π − 2r 0 2 4 6 8

When k goes to infinity this gives the asymptote r ≈ FIG. 5. Lanczos coefficients for free scalar (15) in d = 5 bπ/2ka. Plugging the solution (40) back into action (38) dimensions (blue) and free fermion (22) in d = 4 dimensions yields (orange) – in both cases ∆ = 3/2 and bn are very close to each other and overlap in the plot. Also, Lanczos coefficients 4ka b 2 S = 2k ln + 2 ln cot(r/2). (42) for composite operator φ in d = 5 dimensions (green) and e(π − 2r) a composite operator ψψ¯ in d = 4 (red) – in both cases ∆ = 3 and bn are again very close to each other and overlap. Taking large k limit we finally arrive at

2b 2k a V(z) 4ka 2b 4ka M ≈ e a . (43) 2k eπ eb 800 Comparing with (35) k-dependence we ﬁrst obtain the result well-appreciated in the literature, 600 π a = , (44) 2τ ∗ and then also 400 b = ∆ − 1/2. (45) a 200 In this subsection we used the formalism of [24] which numerates bn starting from n = 1. In the main text z index n starts from zero, yielding a shift by one in (21). 0.0 0.5 1.0 1.5 2.0

FIG. 6. Eﬀective potential V (z) (52) for d = 4, ν = 8 (blue) Free massless fermion in d dimensions vs asymptotic expressions (54) (orange) and (55) (green).

Integration over the Matsubara propagator yields (22) with As is pointed out in the main text resulting Lanczos coef- ﬁcients are numerically very close to those for free scalar Γ d 1−d 2 of the same conformal dimension ∆. The same applies rd = β . (46) d 2 (4π) 2 for composite operators ψψ¯ and φ . We illustrate that by the plot in Fig. 5. It is valid only for −1/2 < τ/β < 1/2 and should be extended by antiperiodicity

Cψ(τ + β) = −Cψ(τ) (47) Holographic thermal 2pt function beyond that. To evaluate b2 numerically it is helpful to n 1,d know closed-form expression for the 2n-th derivative CFT on R at ﬁnite temperature β is holographi- cally described by a black brane background. To evalu- (2n) Γ(2∆ + 2n) 1 3 ate Wightman 2pt function C (τ) one needs to solve the C = rd ζ 2∆ + 2n, − ζ 2∆ + 2n, . 0 ψ 22n−1Γ(2∆) 4 4 wave-equation for a scalar ﬁeld in the bulk dual to an (48) operator O of dimension ∆. For simplicity we will con- 8

2 f β/πb n 35 2.5 30 2.0 25

1.5 20

15 1.0

10 0.5 5

ω n -30 -20 -10 10 20 30 5 10 15 20 25

2 FIG. 7. Power spectrum f given by (58) for d = 4, ν = 8 FIG. 9. “Holographic” Lanczos coefficients bn for 0 ≤ n ≤ 13 calculated using numerical integration from z1 to z2 (blue) evaluated using numerical integration from z1 to z2 (blue), ∗ ∗ and crude approximation by gluing ψ1 and ψ2 at z = z (red). bn evaluated using gluing ψ1 and ψ2 at z = z (orange) and Both expression are multiplied by the same overall coefficient asymptotic behavior given by (21), (β/π)bn ≈ n + ∆ − 1 (red 2 such that in the crude approximation case f (0) = 1. line). sider O at zero spatial momentum, i.e. our operator in and an effective potential question is 2 Z 2 (d − 1) 1 d−1 V (z) = f(r) ν − 1/4 + , (52) O(t) = √ d x O(t, x), (49) 4rd Volume ∆ = d/2 + ν. where overall normalization is chosen such that two-point function of O is finite. Then the power spectrum f 2(ω) The scalar in the bulk dual to O satisfies “Schrodinger” associated with equation −(β−τ)H/2 −(β+τ) C0(τ) = Tr(e Oe O) = d2ψ Z − + V ψ = ω2ψ, 0 ≤ z < ∞. (53) d−1 dz2 d x hO(i(τ + β/2, ~x)O(0,~0)iβ, (50) Near z → 0 the potential behaves as is given by the following procedure [16, 17]. One intro- ν2 − 1/4 duces tortoise coordinate in the bulk V (z) ≈ , (54) Z ∞ z2 dr 2 1 z = , f = r − d−2 , (51) r f r and at large z 2 π −4z V (z) ≈ 8 ν + 2 e 2 , (55) -2ln|a| 40 where we restricted to d = 4. The boundary behavior of ψ is therefore 30 ψ(z) ≈ zν+1/2, z → 0, (56) 20 ψ(z) ≈ a e−iωz +a ¯ eiωz, z → ∞, (57) 10 where a is a complex ω-dependent constant. The power spectrum of C0 is then given by ln(ω) 1 2 3 4 |a|−2 f 2(ω) ∝ , (58) -10 ω sinh(βω/2)

-20 where temperature is ﬁxed to be β = 4π/d. To obtain f 2(ω) numerically one needs to solve (53) FIG. 8. Plot of −2 ln |a| vs ln ω calculated using numerical with the boundary conditions (56,57). In practice the integration from z1 to z2 (blue) and crude approximation by asymptotic approximations (54,55) accurately describe ∗ gluing ψ1 and ψ2 at z = z (orange) vs linear ﬁt describing V (z) everywhere outside of a small region of z ∼ 1, as is large ω behavior (red dashed line). shown in Fig. 6 for d = 4 and ν = 8 which corresponds to 9

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