A late times approximation for the SYK spectral form factor
aa r X i v : . [ h e p - t h ] F e b A late times approximation for the SYK spectral formfactor
Matteo A. Cardella ∗ Dipartimento di Fisica, Universit`a degli Studi di Milano and INFN,via Celoria 16, 20133 Milan, Italy
February 24, 2021
Abstract
We find a late times approximation for the SYK spectral form factor from a large N steepest descent version of the path integral over two replica collective fields. Main ingredientsare a suitable uv regularization of the two replica kinetic operator, the property of its Fouriertransform and some spectral analysis of the four point function two replica ladder kernel. ∗ [email protected] Z ( β − iT ) Z ( β + iT ),an analytic continuation to time T of the square of the imaginary time thermal partition functionat inverse temperature β . SFF is related to the two point correlation function of pairs of energyeigenvalues of the quantum system. It gives information in the time variable T on correlationsamong pairs of eigenvalues at the energy scale related to T . For example, for a certain coarsegraining energy scale, a correlation dependence h ρ ( E ) ρ ( E ′ ) i ∝ ( E − E ′ ) − corresponds to a linearlygrowing in time T form factor, at the related times scale. On the other hand, a h ρ ( E ) ρ ( E ′ ) i → typical of certain en-sembles of Random Matrix Theory. In particular, [14] have shown the existence of a linear rampin T rising behavior that saturates at late times to a constant plateau. Such a behavior is ob-served in various quantum chaotic systems. In mathematics, there is the so called Montgomeryphenomenon, that involves the positions of the Riemann zeta function zeros on the critical line.A repulsion among too closed zeros of the zeta function on the critical line is observed, that isaccurately reproduced by a correlator for pairs of zeroes involving a Sinc function. It accounts forthe ramp regime and the plateau saturation shown by the Riemann zeta spectral form factor.In [16] the SYK spectral form factor is obtained as a path integral over two replica of collectivecontinuous bilocal fields with a complex coupling. Various properties observed numerically [14],were deduced [16] by a replica non-diagonal conformal complex saddle. Since SYK is not exactlyconformal even in the strongly coupled regime, the ramp part of the SYK SFF was reproduced [16]by employing an effective action in the two time reparametrization soft modes. This effective actionis given by a complex linear combination of two Schwarzians local actions. The two Schwarziansaction was introduced in the paper [16] a bit out of the blue. The authors were based on numerical see also [15] for earlier related work. Recent works related to the SYK SFF include [17],[18],[19]. T approximation for the SYK spectral form factor is obtained, by workingon a large N approximation of the two replica path integral [21]. The large T approximation isobtained here by spectral analysis of the four point function two replica ladder kernel. In particular,the explicit form of the ladder kernel invariant eigenfunctions is obtained. Then, an argument isprovided, based on the Fourier transform of a uv regularization of the two replica kinetic operator,that the ladder kernel invariant eigenfunction is the relevant quantity that governs the spectralform factor at large times. Finally, by extracting the uv behavior of this eigenfunction, the form ofthe regularized source, needed for computing the large T effective action is derived. The argumentpresented at the end of this paper provides an alternative route w.r.t. one developed in [21] forobtaining the correct form of the source that enters in the computation of the late times SFFeffective action.In [21], we obtained the following large N approximation for the SYK spectral form factor Z ( β + iT ) Z ( β − iT ) ∼ e N J h s α | ( δ αγ − ˜ K cαγ ) − ˜ K cγβ | s β i + N J / α h s α || G cα | i , (1)which is valid in the strongly coupled nearly conformal regime. Eq. (1) was derived [21] bysteepest descent method on the path integral for the spectral form factor, along contours that gothrough the replica non diagonal conformal saddle. The following replica indexing [21] is used, α = 1 , . . . , α = 1 = LL , α = 2 = RR , α = 3 = LR , α = 4 = RL . The bilocal field s α ( t, t ′ )is a regularized version of the two replica kinetic operator δ ij δ ( t − t ′ ) ∂ t , i, j = L, R . The uvregularization replaces the singular Dirac delta kernel δ ( t − t ′ ) by a smearing function U ( ξ ( t, t ′ )).It implements the impossibility to resolve times intervals shorter then the SYK time scale 1 /J [22]. Moreover, the derivative operator ∂ t is substituted by a sum of contributions with differentscalings [21] that are analyzed in terms of the spectrum of the four point function two replicasymmetrized ladder kernel at the conformal point˜ K cαβ = ˜ K cij,kl ( t , t ; t , t ) ≡ J J / ij J / kl | G cij ( t ) | G cik ( t ) G cjl ( t ) | G ckl ( t ) | , i, j = L, R. (2)In the following, we will extract a late times T approximation for (1). To this aim,Let us expand the source s α ( t, t ′ ) = X h c h ψ hα ( t, t ′ ) U ( ξ α ( t, t ′ )) . (3)in terms of eigenfunctions of ˜ K cαβ ˜ K cαβ ( t , t ; t, t ′ ) ψ hβ ( t, t ′ ) = ˜ k SF F ( h ) ψ hα ( t , t ) . (4)In (4) h labels the eigenvalues of the SL diag (2) Casimir, the diagonal subgroup of SL L (2) × S R (2)that survives the spontaneous breaking of the twofold time diffeomorphisms by the conformal3eplica non-diagonal saddle [16]. Details and properties of the smearing function U ( ξ α ( t, t ′ )) aregiven in [21]. For our purposes, we find convenient to introduce the Fourier transform ˜ U ( η ) U ( ξ ) = Z dη π e − iξη ˜ U ( η ) . (5)Since the smearing function U ( ξ ( t, t ′ )) has support over ∆ ξ ∼ O ( T ), its Fourier transform˜ U ( η ) is non vanishing over an interval of length ∆ η ∼ O (cid:0) T (cid:1) , centered in η = 0. Therefore, ˜ U ( η )becomes very narrow in the large T limit. We use (3) and (5) in the first term at the exponent ofthe spectral form factor (1) h s α | (cid:16) δ αγ − ˜ K cαγ (cid:17) − ˜ K cγβ | s β i = X h,h ′ c h c h ′ Z dtdt ′ ( ψ h ′ α ( t, t ′ )) ∗ U ( ξ α ( t, t ′ )) Z dη π ˜ k SF F ( h + iη )1 − ˜ k SF F ( h + iη ) ψ h + iηα ( t, t ′ ) ˜ U ( η ) , (6)where [21] e − iη α ξ α = 1( ζ α ( | t − t ′ | )) iη α . (7)In the large T limit, one expands ˜ k SF F ( h + iη ) up to the first order in eta, ˜ k SF F ( h + iη ) =˜ k SF F ( h ) + iη ˜ k ′ SF F ( h ). Then, by evaluating the η integral in (6) by residues theorem, one finds h s α | (cid:16) δ αγ − ˜ K cαγ (cid:17) − ˜ K cγβ | s β i = X h,h ′ c h c h ′ Z dtdt ′ ( ψ h ′ α ( t, t ′ )) ∗ U ( ξ α ( t, t ′ )) ψ h + iη h α ( t, t ′ ) ˜ U ( η h ) , (8)where η h = i ˜ k SF F ( h ) − k ′ SF F ( h ) . (9)Since ˜ U ( η ) has support around η = 0 over an interval of length O (cid:0) T (cid:1) , the only non vanishing˜ U ( η h ) in (8) are those for | η h | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ k SF F ( h ) − k ′ SF F ( h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∆ η O (cid:18) T (cid:19) (10)For large enough time T , the only non vanishing term in (8) reduces to the η h = 0 one,which corresponds by eq. (9) to ˜ k SF F ( h ∗ ) = 1. Therefore, we have the following large times T approximation h s α | (cid:16) δ αγ − ˜ K cαγ (cid:17) − ˜ K cγβ | s β i ∼ Z T dt Z T dt ′ ( ψ h ∗ α ( t, t ′ )) ∗ U ( ξ α ( t, t ′ )) ψ h ∗ α ( t, t ′ ) , T → ∞ . (11)Therefore, we have shown that in the large T limit, the only rescaled smeared source which iseffective at the exponent for the spectral form factor (1) has the following form4 α ( t, t ′ ) = ψ h ∗ α ( t, t ′ ) U ( ξ α ( t, t ′ )) , (12)where ψ h ∗ α ( t, t ′ ) is the invariant eigenfunction of the four points ladder symmetrized kernel (2)˜ K cαβ ( t , t ; t, t ′ ) ψ h ∗ β ( t, t ′ ) = ψ h ∗ α ( t , t ) . (13)Let us recall, (see [21] for more details), that the collective fields action in the spectral formfactor path integral splits as follows Z ( β + iT ) Z ( β − iT ) = Z D G α D Σ α e NI c ( G α , Σ α )+ R G α σ α , (14)where I c is the time reparametrization invariant two replica critical action, without the con-formal symmetry breaking replica diagonal kinetic term δ ij δ ( t − t ′ ) ∂ t .On the other hand, I s = X α =1 , Z dtdt ′ G α ( t, t ′ ) σ α ( t, t ′ ) (15)is the conformal symmetric breaking source term, where σ α ( t, t ′ ) is a regularized version of thereplica diagonal kinetic kernel δ ij δ ( t − t ′ ) ∂ t , which is related to the previously introduced s α ( t, t ′ )by σ α ( t, t ′ ) = | G cα ( t, t ′ ) | s α ( t, t ′ ) , α = 1 , . (16)The effective action in the time reparametrization soft modes f L ( t ), f R ( t ) is computed from thenon conformal source action (15). In particular, it depends only on the uv | t − t ′ | → σ α ( t, t ′ ) . The reason for that is that the failure of time reparametrization is a uv effect,as for large times intervals, in the deep ir, the conformal saddle is a very accurate approximationfor the actual Green function. As we have shown above, in the large T regime, the spectral formfactor is dominated by a source (12) proportional to the invariant eigenfunction ψ h ∗ α ( t, t ′ ) (13) ofthe symmetrized four point ladder kernel ˜ K cαβ . In the following we compute explicitly the diagonalcomponents α = 1 , ψ h ∗ α ( t, t ′ ) and then extract their | t − t ′ | → T limit. By the same method presented here,one can also compute the off diagonals components α = 3 , ψ h ∗ α ( t, t ′ ), we will not do it here,since only the diagonal components of the source matter σ α ( t, t ′ ) in the source action (15).Let us consider the conformal Schwinger-Dyson equations for the spectral form factor [23],[21].An infinitesimal variation of the bilocal fields by a time reparametrization is still a solution of theconformal saddle point equations, due to conformal symmetry. After some manipulation, one getsto the following condition (1 − K cαβ ) δ ǫ G cβ = 0 , (17)where K cαβ is the following non symmetrized version of the four points ladder kernel K cij,kl ( t , t ; t , t ) = 3 J J kl G cik ( t ) G cjl ( t )( G ckl ( t )) (18)5nd δ ǫ G cα is the first order variation of G cα ( t, t ′ ) by a linear diffeomorphism t → t + ǫ ( t ). Eq.(17) provides also a way to compute the invariant eigenfunction of the symmetrized kernel (2),since one finds that ψ h ∗ α ( t, t ′ ) = J / α | G cα | δ ǫ G cα (19)by (17) is the invariant eigenfunction of ˜ K cαβ .We are now in the position of being able to compute the diagonal components α = 1 , ψ h ∗ α ( t, t ′ ) by using eq. (19). A linear time reparametrization t → t + ǫ ( t ) on G α ( t, t ′ ), α = 1 , ǫ ( t ) δ ǫ G α ( t, t ′ ) = 14 ( ǫ ′ ( t ) + ǫ ′ ( t ′ )) G α ( t, t ′ ) + ( ǫ ( t ) − ǫ ( t ′ )) ∂ t G α ( t, t ′ ) (20)We use the diagonal components of the spectral form factor conformal saddle [16],[21] G cα ( t, t ′ ) ∝ sgn ( t − t ′ ) (cid:12)(cid:12)(cid:12) ˜ β aux π sinh (cid:16) π ˜ β aux ( t − t ′ ) (cid:17)(cid:12)(cid:12)(cid:12) / , α = 1 , , (21)and its derivative ∂ t G cα ( t, t ′ ) ∝ − sgn ( t − t ′ ) (cid:12)(cid:12)(cid:12) ˜ β aux π sinh (cid:16) π ˜ β aux ( t − t ′ ) (cid:17)(cid:12)(cid:12)(cid:12) / cosh (cid:18) π ˜ β aux ( t − t ′ ) (cid:19) α = 1 , . (22)We express the linear time reparmetrization as a Fourier integral ǫ ( t ) = Z dω π e − iωt ˜ ǫ ( ω ) . (23)A variation of the Green function by a phase ǫ ( t ) = e − iωt , by eq. (20) gives δ ǫ G cα ∝ − i ω e − iωt + e − iωt ′ ) G cα ( t, t ′ ) + ( e − iωt − e − iωt ′ ) ∂ t G cα ( t, t ′ ) . (24)After a little manipulation, we get to the following fixed ω form for the diagonal componentsof the invariant eigenfunction of the symmetrized ladder kernel ψ h ∗ ,ωα ( t, t ′ ) = J / α | G cα ( t, t ′ ) | δ ǫ G cα = ie − iω t + t ′ sgn ( t − t ′ ) (cid:12)(cid:12)(cid:12) ˜ β aux π sinh (cid:16) π ˜ β aux ( t − t ′ ) (cid:17)(cid:12)(cid:12)(cid:12) π ˜ β aux sin (cid:0) ω t − t ′ (cid:1) tanh (cid:16) π ˜ β aux ( t − t ′ ) (cid:17) − ω (cid:18) ω t − t ′ (cid:19) ,α = 1 , . (25)On the other hand, by linearity (23), the more general invariant eigenfunction of ˜ K cαβ has thefollowing diagonal components ψ h ∗ α ( t, t ′ ) = Z dω π ˜ ǫ ( ω ) ψ h ∗ ,ωα ( t, t ′ ) , α = 1 , , (26)6ith ˜ ǫ ( ω ) being the Fourier transform (23) of ǫ ( t ). Therfore, we have gotten a continuousfamily of eigenfunctions { ψ h ∗ ,ωα ( t, t ′ ) } ω ∈ I ω of the ladder kernel with eigenvalue one, labeled by ω ∈ I ω = ( − ∆ ω , ∆ ω ), with ∆ ω = O (cid:0) T (cid:1) , and (26) is a generic linear superposition.We want to extract the uv behavior | t − t ′ | → T effective action for the spectral form factor through the action term(15). Let us keep in mind that the source (12) is uv regularized by the smeared function U ( ξ ( t, t ′ ))that vanishes for | t − t ′ | < /J . On the other hand, ˜ ǫ ( ω ) has support over an interval centered in ω = 0 of length | I ω | = O (cid:0) T (cid:1) . In the limit T → ∞ one can take the following approximation ψ h ∗ α ( t, t ′ ) = Z dω π ˜ ǫ ( ω ) ψ h ∗ ,ωα ( t, t ′ ) , ∼ | I ω | ω ∗ ˜ ǫ ( ω ∗ ) ˆ ψ h ∗ α ( t, t ′ ) = − iǫ ′ ( t ǫ ) ˆ ψ h ∗ α ( t, t ′ ) , (27)where ω ∗ ∈ I ω , t ǫ >
0, andˆ ψ h ∗ α ( t, t ′ ) = i sgn ( t − t ′ ) (cid:12)(cid:12)(cid:12) ˜ β aux π sinh (cid:16) π ˜ β aux ( t − t ′ ) (cid:17)(cid:12)(cid:12)(cid:12) π ˜ β aux (cid:16) π ˜ β aux ( t − t ′ ) (cid:17) − . (28)Eq. (28) is a valid approximation as far as ˜ β aux << T . On the other hand, ǫ ′ ( t ǫ ) is regarded tobe a renormalized quantity. Its meaning follows from eq. (23) by the integral averaged theoremthat was used in (27).By taking the uv | t − t ′ | → T → ∞ limit s α ( t, t ′ ) ∝ sgn ( t − t ′ ) (cid:12)(cid:12)(cid:12) ˜ β aux π sinh (cid:16) π ˜ β aux ( t − t ′ ) (cid:17)(cid:12)(cid:12)(cid:12) U ( ξ ( t, t ′ )) α = 1 , . (29)This is the quantity that enters in the computation of the effective action for the spectral formfactor in the T → ∞ limit through eq. (16) σ α ( t, t ′ ) = | G cα ( t, t ′ ) | s α ( t, t ′ ) in the conformal symmetrybreaking source term (15). Moreover, this same quantity governs the large T approximation forthe spectral form factor path integral (1). In [21] the uv form of the smeared source displayed ineq. (29), was obtained by a different argument. and used there for computing the two Schwarziansaction in the time reparametrization soft modes f L ( t ), f R ( t ). References [1] L. Dyson, M. Kleban and L. Susskind,
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