Scrambling the spectral form factor: Unitarity constraints and exact results

The exact spectral form factor in the Gaussian unitary ensemble for any N agrees with recent numerical studies.

“We find the spectral form factor in a number of illustrative models, notably we obtain the exact answer in the Gaussian unitary ensemble for any N with excellent agreement with recent numerical studies.”

For the thermofield double state, the exponential lower bound holds for S(β,t) ≥ 1/2 and t ≥ 0 up to the half lifetime t_h when S(β,t_h)=1/2.

“and holds for S ( β, t ) ≥ 1 / 2 and t ≥ 0 up to the half lifetime t h when S ( β, t h ) = 1 / 2.”

The spectral form factor is closely related to the survival probability of the thermofield double state.

“In this paper we exploit the fact that the spectral form factor is closely related to the survival probability of the thermofield double state (TDS).”

The purity of ρ_L determines the long-time average of the survival probability.

“It is the purity of ρ L that determines the long-time average of the survival probability,”

The survival probability of a thermofield double state under unitary time-evolution is related to the analytic continuation of the partition function.

“To this end, we consider the survival probability of a thermofield double state under unitary time-evolution which is related to the analytic continuation of the partition function.”

• Also stated in: § II. PARTITION FUNCTION AND SURVIVAL PROBABILITY OF A THERMOFIELD DOUBLE STATE · p. 4 · boosted

  “This survival probability is evidently related to the analytic continuation of the partition function, and thus the spectral form factor introduced above (2).”

For a TDS the mean energy and its fluctuations are given by the corresponding quantities in the canonical state at inverse temperature β.

“For a TDS the mean energy and its fluctuations are given by”

In a thermal state, the purity can be written in terms of the partition function as (24).

“In a thermal state, the purity can be written in terms of the partition function as”

In systems with a continuous spectrum bounded from below, the long-time asymptotic behavior of S(β,t) is governed by |M|^2.

“In systems with a continuous spectrum bounded from below, the long-time asymptotic behavior of S ( β, t ) is governed by | M | 2 .”

For the integral to be finite, it has to be the case that (27) with C,γ>0 and q<1.

“For this to be true it has to be the case that”

If ρ(E) ∼ E^k near E=E0, the survival probability at long times scales as (29).

“Indeed, if ρ ( E ) ∼ E k near E = E 0 , the survival probability at long times scales as”

The paper provides an exponential lower bound to the survival probability with a rate governed by the inverse of the energy fluctuations of the initial state.

“We provide an exponential lower bound to the survival probability with a rate governed by the inverse of the energy fluctuations of the initial state.”

The short-time asymptotic behavior of the survival probability reads (17).

“Exploiting the power-series expansion of the time-evolution operator, the short-time asymptotic behavior of the survival probability reads”

The fidelity decay in the RMT can be analytically evaluated for any N using the method of orthogonal polynomials.

“In fact we can go much further and analytically evaluate the fidelity decay in the RMT for any N using the method of orthogonal polynomials.”

Including the one-loop fluctuation determinant around the saddle gives a prefactor leading to algebraic decay at late times.

“Considering in addition the one-loop fluctuation determinant around the saddle [11, 56] gives the prefactor”

• Also stated in: § 1. Fidelity decay in AdS d +1 /CFT d · p. 22 · plain

  “leading to algebraic decay (at late times).”

• cites [11] E. Dyer and G. Gur-Ari, arXiv:1611.04592 (2016). sourceNot checked yet

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• cites [56] S. Giombi, A. Maloney and X. Yin, J. High Energy Phys. 0808 (2008) 007. sourceNot checked yet

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The survival probability in the CFT1 realized by the xp model is precisely given by (34).

“The survival probability in the CFT 1 realized by the xp model then reads as”

• Also stated in: § A. Quantum harmonic oscillator and the xp model in AdS 2 · p. 12 · boosted

  “i.e., it is precisely given by (34).”

The only effect of the coupling constant λ in the CSM spectrum is to renormalize the zero-point energy.

“Surprisingly, the only effect of the coupling constant λ in the spectrum is to renormalize the zero-point energy”

The exact expression for the connected contribution can be mapped to the expectation value of two Wilson loops in the Gaussian matrix model.

“Let us remark that this exact expression can be mapped to the expectation value of two Wilson loops in the Gaussian matrix model, as shown in appendix (C 4).”

For N=1 the CSM survival probability recovers the behavior of a single-harmonic oscillator.

“For N = 1 one naturally recovers the behavior of a single-harmonic oscillator.”

The survival probability of the CSM is independent of the coupling constant λ.

“First, S ( β, t ) is independent of the coupling constant whose only effect in the spectrum is the renormalization of zeropoint-energy.”

The survival probability is dominated by an evolution during which the initial TDS is fully scrambled at an intermediate time τ in the sense that the time evolving state is completely orthogonal to the initial state.

“The survival probability is then dominated by an evolution during which the initial TDS is fully scrambled at an intermediate time τ in the sense that the time evolving state is completely orthogonal to the initial state,”

Adding the purity to the connected contribution leads to an approximately linear rise at late times until the plateau value of Z(2β) is approached.

“By adding the purity this leads to an approximately linear rise at late times until the plateau value of Z (2 β ) is approached.”

In contrast with the large N expression (54), the energy fluctuations ΔE^2 = 1/τ_Z^2 do not vanish in the low-temperature limit and saturate at ΔE^2 = 1/2 independent of N.

“In contrast with the large N expression (54), energy fluctuations ∆ E 2 = 1 /τ 2 Z do not vanish in the low temperature limit, and saturate at a constant value ∆ E 2 = 1 / 2 independent of N .”

The survival probability of the harmonic oscillator is an oscillatory function of time with a single nominal frequency ω.

“As a result, S ( β, t ) is an oscillatory function of time with a single nominal frequency ω .”

As the time of evolution goes by, a power-law behavior sets in and S(β,t) ∝ t^{-3}.

“As the time of evolution goes by, a power-law behavior sets in and S ( β, t ) ∝ t -3 .”

The CSM survival probability is a periodic function of time, with period 2πN/ω.

“As in the case of the single-harmonic oscillator, the survival probability is a periodic function of time, with period 2 πN/ω .”

The decay products Q|ψ(β,τ)> subsequently reconstruct the initial state at the later time t.

“yet the decay products ˆ Q | ψ ( β, τ ) 〉 subsequently reconstruct the initial state at the later time t .”

The spectral form factor in GUE can be identified with three terms (59).

“As a result, we identify three terms in the spectral form factor”

The paper presents a generalization of the bound that holds for arbitrary initial states, including mixed states described by an arbitrary density matrix.

“In this appendix, we present a generalization that holds for arbitrary initial states, including mixed states described by an arbitrary density matrix.”

The spectral form factor maps to a correlation function of two Wilson loops.

“establishing, as claimed, that the spectral form factor maps to a correlation function of two Wilson loops.”

The survival probability S(t) vanishes at infinity as a consequence of the Riemann-Lebesgue lemma.

“That S ( t ) vanishes at infinity as a consequence of the Rieman-Lebesgue lemma [26].”

• cites [26] V. A. Fock and S. N. Krylov, Zh. Eksp. Teor. Fiz. 17 , 93 (1947). sourceNot checked yet

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• AssumptionThe authors state this as an assumption: The Hamiltonian has a ground state energy E0 and is bounded from below.

For AdS3 (d=2), r+ = 2πl^2/β.

“The special case of AdS 3 we considered above has d = 2 and r + = 2 πℓ 2 β .”

For AdS3, the energy fluctuations ΔE are given by Eq. (80), in accordance with the value appearing in the holographic Zeno time (74).

“We start with the AdS 3 case, where we find”

• Also stated in: § 2. Bound on Fidelity Decay · p. 23 · plain

  “in accordance with the value for ∆ E appearing in the holographic Zeno time (74).”

For the general d-dimensional case, the energy fluctuations lead to a bound S(β,t) ≥ exp(-k'_d √c t) for a d-dimensional constant k'_d.

“For the general d -dimensional case we find”

• Also stated in: § 2. Bound on Fidelity Decay · p. 23 · plain

  “leading to a bound of the form S ( β, t ) ≥ exp ( -k ′ d √ ct ) for a d dimensional constant k ′ d which follows upon taking the square root of (80).”

Whenever the spectrum of the Hamiltonian generating the time evolution is continuous, the vanishing survival probability of a thermofield double state at infinitely long time follows from the analytic properties of the density of states.

“Whenever the spectrum of the Hamiltonian generating the time evolution is continuous, the vanishing survival probability of a thermofield double state at infinitely long time of evolution follows form the analytic properties of the density of states.”

• AssumptionThe authors state this as an assumption: The Hamiltonian has a ground state energy E0 and is bounded from below.

The computation of the survival amplitude can be mapped exactly to the computation of a Wilson loop in the Gaussian matrix model.

“It is amusing to note that the computation of the survival amplitude can be mapped exactly to the computation of a Wilson loop in the Gaussian matrix model.”

The exact result for the spectral form factor for all times can be computed in the generalized unitary ensemble of random matrices using the method of orthogonal polynomials.

“In particular we have shown that the exact result for the spectral form factor for all times can be computed in the generalized unitary ensemble of random matrices using the method of orthogonal polynomials.”

The Paley-Wiener theorem implies that the integral (26) be finite.

“Taking the point of view of the inverse transform, in order for the spectral density to vanish for energies less than E 0 , the Paley-Wiener theorem implies that the integral”

The specific form of the long-time decay is set by the low-energy behavior of the spectral density.

“The specific form of the long-time decay is set by the low-energy behavior of the spectral density.”

Following Khalfin, the survival amplitude A(β,t) is the Fourier transform of a function with support on the half energy axis.

“Following Khalfin [23, 29], we note that A ( β, t ) is then the Fourier transform of a function with support on the half energy axis, ˜ ρ ( E ) ≡ ρ ( E )Θ( E -E 0 ).”

• cites [23] L. Fonda, G. C. Ghirardi, and A. Rimini, Rep. Prog. Phys. 41 , 587(1978). sourceNot checked yet

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• cites [29] L. A. Khalfin, Sov. Phys. JETP 6 , 1053 (1958) [Zh. Eks. Teor. Fiz. 33 , 1371 (1957)]. sourceNot checked yet

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The long-time asymptotics are characterized by a slower than exponential decay.

“In other words the long-time asymptotics are characterized by a slower than exponential decay.”

The origin of long-time deviations from exponential decay can be traced back to the possibility of state reconstruction under unitary quantum dynamics.

“Physically, the origin of long-time deviations from exponential decay can be traced back to the possibility of state reconstruction under unitary quantum dynamics.”

Under quite general conditions, a power-law decay holds.

“Under quite general conditions, a power-law decay holds, as has been demonstrated in single-particle [23], few-body [30] and many-body systems [25, 31-33].”

• cites [23] L. Fonda, G. C. Ghirardi, and A. Rimini, Rep. Prog. Phys. 41 , 587(1978). sourceNot checked yet

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• cites [30] T. Taniguchi and S. I. Sawada, Phys. Rev. E 83 , 026208 (2011). sourceNot checked yet

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• cites [25] E. J. Torres-Herrera and L. F. Santos, Phys. Rev. A 89 , 043620 (2014); Phys. Rev. A 90 , 033623 (2014); M. T´ avora, E. J. Torres-Herrera, L. F. Santos, Phys. Rev. A 95 , 013604 (2017). sourceNot checked yet

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• cites [31] A. del Campo, Phys. Rev. A 84 , 012113 (2011). http://dx.doi.org/10.1103/PhysRevA.84.012113 sourceNot checked yet

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• cites [32] A. del Campo, New J. Phys. 18 , 015014 (2016). http://dx.doi.org/10.1088/1367-2630/18/1/015014 sourceNot checked yet

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• cites [33] S. Bagrets, A. Altland, A. Kamenev, Nucl. Phys. B 911 , 191 (2016).Not checked yet

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Because of the close connection to the partition function, unitarity bounds on the dynamics of the system can be expressed in terms of equilibrium thermodynamics.

“In this case, because of its close connection to the partition function, we can express unitarity bounds on the dynamics of the system in terms of equilibrium thermodynamics.”

Unitary quantum dynamics implies that at late times the discrete nature of the spectrum forbids a continued decay and instead requires time averages of correlations to stabilize at ∼ e^{-S}.

“Unitary quantum dynamics, however, implies that at late times the discrete nature of the spectrum forbids a continued decay and instead requires time averages of correlations do stabilize at ∼ e -S , where S is the entropy of the thermal ensemble or simply the dimension of the associated Hilbert space.”

Universal features of the non-exponential behavior at short and long times of evolution follow from the analytic properties of the survival probability and its Fourier transform, both for systems with a continuous and for systems with a discrete energy spectrum.

“Further, we elucidate universal features of the non-exponential behavior at short and long times of evolution that follow from the analytic properties of the survival probability and its Fourier transform, both for systems with a continuous and for systems with a discrete energy spectrum.”

Under unitary evolution, the short-time decay of the survival probability is nonexponential.

“Under unitary evolution, the short-time decay of the survival probability is nonexponential [23].”

• cites [23] L. Fonda, G. C. Ghirardi, and A. Rimini, Rep. Prog. Phys. 41 , 587(1978). sourceNot checked yet

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For systems with a continuum spectrum, the survival probability of an arbitrary initial state vanishes identically at t→∞.

“For systems with a continuum spectrum, Fock and Krylov showed that the survival probability of an arbitrary initial state vanishes identically at t → ∞ [26].”

• cites [26] V. A. Fock and S. N. Krylov, Zh. Eksp. Teor. Fiz. 17 , 93 (1947). sourceNot checked yet

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The absence of a term linear in time in the short-time expansion is a signature of unitary dynamics.

“The absence of a term linear in time in this expansion is a signature of unitary dynamics.”

The Mandelstam-Tamm bound can be used to derive a lower bound on the survival probability by an exponential function.

“As was shown by Bhattacharyya [13], the Mandelstam-Tamm bound can be used to derive to lower bound the survival probability by an exponential function.”

• cites [13] K. Bhattacharyya, J. Phys. A: Math. Gen. 16 , 2993 (1983). http://dx.doi.org/10.1088/0305-4470/16/13/021 sourceNot checked yet

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In systems with a discrete spectrum, the quantum recurrence theorem states that there exist approximate revivals.

“By contrast in systems with a discrete spectrum the quantum recurrence theorem (QRT) [27, 28] states that there exist approximate revivals, in the sense that according to various notions of distance in Hilbert space the time-dependent state and the initial state become arbitrarily close for some time of evolution.”

• cites [27] P. Bocchieri and A. Loinger, Phys. Rev. 107 , 337 (1957).Not checked yet

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• cites [28] L. S. Schulman, Phys. Rev. A 18 , 2379 (1978). https://doi.org/10.1103/PhysRevA.18.2379Not checked yet

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The exponential bound is restricted to a short time of evolution comparable to that governing the short-time asymptotic.

“Nonetheless, this bound is restricted to a short time of evolution comparable to that governing the short-time asymptotic that we next discuss.”

The long-time behavior is associated with rare events in which the dynamics first leads to 'decay products' that subsequently evolve to reconstruct the initial state at time t.

“As a result, it is therefore associated with rare events in which the dynamics first leads to 'decay products', identified with the component ˆ Q | ψ ( β, τ 〉 , that subsequently evolve to reconstruct the initial state at time t .”

At early times the energy fluctuations of the initial unstable state set the speed of evolution in Hilbert space.

“Hence, at early times the energy fluctuations of the initial unstable state set the speed of evolution in Hilbert space.”

Poincaré recurrences are not expected until times of order t ∼ e^{e^S}.

“Poincar´ e recurrences are not expected until times of order t ∼ e e S [14, 15].”

• cites [14] J. L. F. Barbon and E. Rabinovici, J. High Energy Phys. 11 (2003) 047. sourceNot checked yet

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• cites [15] J. L. F. Barbon and E. Rabinovici, Fortsch. Phys. 52 , 642 (2004). sourceNot checked yet

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By combining the detailed decay with the very late RMT behavior, one may estimate the dip time in a CFT2 dual to the AdS3 black hole as exponentially large in the central charge c.

“By combining the detailed decay with the very late RMT behavior, one may estimate the 'dip' time in a CFT 2 dual to the AdS 3 black hole as [11]”

• Also stated in: § 1. Fidelity decay in AdS d +1 /CFT d · p. 22 · plain

  “which is exponentially large 6 in the central charge c .”

• cites [11] E. Dyer and G. Gur-Ari, arXiv:1611.04592 (2016). sourceNot checked yet

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The analytic RMT curves in Fig. 2 give a surprisingly good description of the holographic spectral form factor.

“In summary, then, the analytic RMT curves in Fig. 2 give a surprisingly good description of the holographic spectral form factor as well.”

The power-law t^{-3} behavior can be traced back to the compact support of Wigner's semicircle law in the light of Paley-Wiener theorem.

“This behavior can be traced back to the compact support of the Wigner's semicircle law in the light of Paley-Wiener theorem.”

The survival probability in this approximation is obtained by analytic continuation.

“The survival probability in this approximation is therefore obtained simply by analytic continuation”

For a system with a finite Hilbert space, complete decay would imply information has been lost, since discreteness of the spectrum forbids such complete decay.

“For a system with a finite Hilbert space this state of affairs would imply that information has been lost, since the discreteness of the spectrum forbids such complete decay, as explained in Section III C.”

The behavior of g(β,t) in random matrix theory is qualitatively very similar to the holographic case.

“As we shall see, the behavior of g ( β, t ) in random matrix theory is actually qualitatively very similar to the holographic case.”

By assuming that the latest-time behavior is dominated by random matrix theory, one infers on average a linear ramp up to an eventual plateau, as shown analytically in previous sections.

“Then, by assuming that the latest time behavior is dominated by random matrix theory, one infers - on average - a linear ramp up to an eventual plateau, as we have shown analytically in previous sections.”

• AssumptionThe authors state this as an assumption: The latest-time behavior of the CFT2 spectral form factor is dominated by random matrix theory.

Given that near the low-energy edge E=-λ, the density of states scales as ρ(E) ∼ (E+λ)^{1/2}, the power-law t^{-3} follows from Eq. (29).

“Actually, given that near the low-energy edge E = -λ , the density of states scales as ρ ( E ) ∼ ( E + λ ) 1 / 2 , the power-law t -3 follows from Eq. (29).”

The survival probability of the harmonic oscillator is shared by the xp model in AdS2.

“As it turns out, the survival probability of the harmonic oscillator is shared by the xp model in AdS 2 [39].”

• cites [39] J. Molina-Vilaplana, G. Sierra, Nucl. Phys. B877 (2013) 107-123. sourceNot checked yet

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The exact expression reveals three contributions, each responsible for one of the characteristic shapes in Figure 2.

“From the exact expression we can discern three contributions, each of which is responsible for one of the characteristic shapes seen in Figure 2.”

In holography the leading result for the partition function, valid at large central charge c, is given by an appropriately regularized version of the on-shell action of the Euclidean black hole or whichever other classical saddle dominates in the bulk.

“In holography the leading result for the partition function, valid at large central charge, which we denote generically by c , is given by an appropriately regularized version of the on-shell action of the Euclidean black hole or whichever other classical saddle dominates in the bulk, [55]”

• cites [55] S. W. Hawking and D. N. Page, Commun. Math. Phys. 87 577 (1983). http://dx.doi.org/10.1007/BF01208266 sourceNot checked yet

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The time after which the linear ramp joins the plateau is parametrically large in c compared to the dip time.

“Similarly the time after which the linear ramp joins the plateau is parametrically large in c compared to the dip time.”

The estimate for the dip time t_d follows from equating the power-law decay 1/t^3 from the disconnected part to the approximately linear rise ∼t from the connected part.

“The estimate for t d follows from equating the power-law decay 1 /t 3 coming from the disconnected part to the approximately linear rise ∼ t coming from the connected part.”

The exact eigenvalue density averaged over the GUE is given by (63).

“In fact, it is known [53] that the exact eigenvalue density averaged over the GUE is given by”

• cites [53] M. L. Mehta, Random Matrices. (Elsevier, San Diego, 2004) Third edition. sourceNot checked yet

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In holography the time evolution of the spectral form factor is characterized by three phases: Gaussian decay, power-law decay, and at very late times a linear ramp up to a plateau.

“As in RMT the time evolution of the spectral form factor in holography is characterized by three phases.”

• Also stated in: § V. SURVIVAL PROBABILITY IN ADS d +1 /CFT d · p. 21 · hedged

  “At very late times this decay is believed to give way to behavior typically associated with RMT: on average we see a linear ramp up to a plateau at the latest times which persists until the latest times.”

A better estimate of the decay of the spectral form factor in CFT2 is obtained by including the images under SL(2,Z) of the vacuum character, which results in a 1/t^3 power law.

“A better estimate of the decay of the spectral form factor in CFT 2 is obtained by including the images under SL (2 , Z ) of the vacuum character, which results in a 1 /t 3 power law [11].”

• cites [11] E. Dyer and G. Gur-Ari, arXiv:1611.04592 (2016). sourceNot checked yet

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For AdS3 the leading saddle point is the BTZ black hole.

“Let us illustrate this with the case of AdS 3 where the leading saddle point is given by the so-called BTZ black hole, and reads”

The suppression of the memory term, e.g. via nonunitary dynamics, leads to exponential behavior.

“The suppression of the memory term, e.g. via nonunitary dynamics, leads to exponential behavior, as seen from the Ersak equation (31).”

The specific heat of the holographic dual of the thermofield double state is given by the eternal black hole.

“Let us thus calculate the specific heat of the holographic dual of the thermofield double state, given, as we argued, by the eternal black hole [5].”

• cites [5] J. M. Maldacena, J. High Energy Phys. 04 (2003) 021. http://dx.doi.org/10.1088/1126-6708/2003/04/021 sourceNot checked yet

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In theories with a discrete spectrum, continued decay (perhaps for an intermediate time scale) should be interpreted as a sign of information loss.

“In theories with a discrete spectrum continued decay (perhaps for an intermediate time scale) should be interpreted as a sign of information loss, and so our results here help to sharpen this discussion, in particular in the context of holographic duality.”

In spacetime dimension d, the partition function is given by Eq. (78), with c the central charge and k_d a dimension-dependent constant.

“In spacetime dimension d one finds for the partition function”

• Also stated in: § 2. Bound on Fidelity Decay · p. 23 · plain

  “Again, c is the central charge, and k d is a dimension-dependent constant, which can be found for example in [57].”

• cites [57] R. Emparan, C. V. Johnson and R. C. Myers, Phys. Rev. D 60 , 104001 (1999). sourceNot checked yet

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The time evolution of g(β,t) is mapped to the decay amplitude of the fidelity of the TDS and the associated decay probability S(β,t).

“The time evolution of g ( β, t ) is thus mapped to the decay amplitude, as a function of time, of the fidelity of the TDS and the associated decay probability S ( β, t ).”

The results in the GUE should serve as an analytical toy model of the study of information loss and retrieval in black holes.

“Our results in the GUE should thus serve as an analytical toy model of the study of information loss and retrieval in black holes.”

The spectral form factor g(β,t) can be reformulated in terms of a decay amplitude of the thermofield double state.

“In this paper we have established that the spectral form factor g ( β, t ) can be reformulated in terms of a decay amplitude of a certain pure state, namely the thermofield double state.”