Spectral Form Factor

Fig. 1: An illustration of the spectral form factor vs time plot for a SYK model with N=34 and β=5. (Source: C. Yan, following Cotler et al. [3])

A natural way to characterize the energy spectrum of a Hamiltonian H is to consider the thermal correlator of a Hermitian operator O

where Z is the partition function, β is inverse temperature, and t is time. But according to the Eigenstate Thermalization Hypothesis, the contribution to the correlator from the specific operator O varies smoothly as time becomes large, and we are more interested in the phases that cause oscillations. [1] To ease computation and to gain more physical intuition, with the analytical continuation of the partition function given by

we can define a spectral form factor, which is the two-point correlator without matrix elements of O, as [2]

Cotler et al. compared the g(t) graphs of the Sachdev-Ye-Kitaev (SYK) model and the Random Matrix Theory (RMT) and find a similarity when t is large. [3,4] Because there was an established correspondence between large AdS black hole and the SYK model, they concluded that large AdS black holes is a quantum chaotic system. In this report, we present some important features of the spectral form factor and some physical intuitions behind those.

Different Sectors of the Spectral Form Factor

Fig. 2: An illustration of the averaged SYK spectral form factor vs time plot. The average is over 90 samples with N=34 and β=5. Here td is the Thouless time and tp is the Heisenberg time. (Source: C. Yan, after Cotler et al. [3])

Shown in Fig. 1 is the spectral form factor plot of one sample in the SYK model. In both SYK and RMT, to make the spectral form factor into a smooth curve, we need some kind of average. As shown in Fig. 2, the smoothed-out g(t) graph consists of a slope, a dip, a ramp, and a plateau. Before the dip we call the early time behavior, and after the dip we call the late time behavior. Before the dip, the spectral form factor is dominated by disconnected pair correlations between density of states. After the dip, the spectral form factor probes connected pair correlations between density of states. As time increase, correlation between energy eigenvalues that are closer and closer to each other are being probed. The early time behavior of the SYK model and that of the RMT are different (although the qualitative behaviors of the slope region are the same, and the quantitative difference is indistinguishable by eye from the spectral form factor plots, and that is the reason why we have only included plots of SYK), but the later time behaviors are the same. We can understand that as a similarity between SYK and RMT as we zoom in to look at connected part of the correlation between energy eigenvalues that are close to each other.

Spectral Form Factor of SYK

The SYK model was shown to have some features of a gravity dual, and it has been recently a convenient simple condensed matter model to study features of gravity and black holes. [5] By studying the spectral form factor of SYK, we can make statements about the Hilbert space of black holes. The SYK Hamiltonian is given by

where ψ_i are Majorana fermions and J_abcd are totally antisymmetric random couplings satisfying a Gaussian distribution. In this context, we average the spectral form factor with respect to J. A single realization of the SYK Hamiltonian has a spectral form factor with large fluctuations during late time. Taking the ensemble average smoothes out these fluctuations. But even with the fluctuations, the general trend of the spectral form factor is still visible.

Spectral Form Factor of RMT

There is a well-established conjecture that RMT gives the characteristic behavior of eigenvalue spacing of quantum chaotic systems. [4] Thus, RMT spectral form factor is a good reference for those of other models to compare with. In RMT, a single realization of matrix M corresponds to a single realization of the Hamiltonian H in the SYK model, so we can write the partition function as instead

and average over a chosen matrix ensemble. There are three different random matrix ensembles, GOE, GUE, and GSE, meaning matrices in these ensembles satisfy different constraints. [6] For example, if we focus on GUE, we can average the spectral form factor over the GUE ensemble. The plateau in RMT spectral form factor comes from level repulsion between nearest neighbors in the energy eigenvalue spectrum; while the ramp comes from repulsion between eigenvalues that are further away from each other, specifically this ramp demonstrates spectral rigidity of RMT since to get the linear behavior the repulsion and the potential have to be balanced very precisely.

Comparison and Connection to Black Holes

A SYK Hamiltonian with N fermions in SYK corresponds to an L×L matrix in RMT, so we know that L∼ e^N. The ramp is explained as a perturbative effect in RMT. But since a perturbative expansion of RMT in terms of 1/L∼ e^-N is non-perturbative viewed as a perturbative expansion of SYK in terms of 1/N, the ramp becomes a non-perturbative effect in SYK. The plateau is non-perturbative even in RMT so it comes even more non-perturbative in SYK.

The plateau in the spectral form factor can demonstrate the discreteness of energy spectrum of the black hole microstates. [7] To see this, if the energy spectrum is continuous, the spectral form factor should decay indefinitely as time increases. On the other hand, if the energy spectrum is discrete as time becomes much larger than the smallest energy separation, e^i(E_m-E_n)t gives erratic fluctuations and sum to some fix small number in late times, which gives a plateau. The long time average of the spectral form factor is on the order of e^-aS. This is a non-perturbative effect.

©Cynthia Yan. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

References

[1] M. Srednicki, "Chaos and Quantum Thermalization," Phy. Rev. E 50, 2 (1994).

[2] E. Wigner, "Characteristic Vectors of Bordered Matrices With Infinite Dimensions," Ann. Math. 62, 548 (1955).

[3] J. S. Cotler et al., "Black Holes and Random Matrices," J. High Energy Phys. 2017, 118 (2017).

[4] M. L. Mehta, Random Matrices, 3rd Ed. (Academic press, 2004).

[5] A. Kitaev and S. J. Suh, "The Soft Mode in the Sachdev-Ye-Kitaev Model and Its Gravity Dual," J. High Energy Phys. 05, 183 (2018).

[6] F. J. Dyson, "Statistical theory of the Energy Levels of Complex Systems. I," J. Math. Phys. 3, 140 (1962).

[7] J. M. Maldecena, "Eternal Black Holes in Anti-de Sitter," J. High Energy Phys. 04, 021 (2003).