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arXiv:2003.05401 · Journal of Physics A Mathematical and Theoretical

Correlated disorder in the SYK 2 model

Pak Hang Chris Lau, Chen-Te Ma, Jeff Murugan, Masaki Tezuka

Published Jan 21, 2021• 16 citations
arXivHTMLPDF

Abstract

Abstract We study the SYK 2 model of N Majorana fermions with random quadratic interactions through a detailed spectral analysis and by coupling the model to two- and four-point sources. In particular, we define the generalized spectral form factor (SFF) and level spacing distribution function by generalizing from the partition function to the generating function. For N = 2, we obtain an exact solution of the generalized SFF. It exhibits qualitatively similar behavior to the higher N case with a source term. The exact solution helps understand the behavior of the generalized SFF. We calculate the generalized level spacing distribution function and the mean value of the adjacent gap ratio defined by the generating function. For the SYK 2 model with a four-point source term, we find a Gaussian unitary ensemble behavior in the near-integrable region of the theory, which indicates a transition to chaos. This behavior is confirmed by the connected part of the generalized SFF with an unfolded spectrum. The departure from this Gaussian random matrix behavior as the relative strength of the source term is increased is consistent with the observation that the four-point source term alone, without the SYK 2 couplings turned on, exhibits an integrable spectrum from the SFF and level spacing distribution function in the large N limit.

DOI
10.1088/1751-8121/abde77

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Abstract We study the SYK 2 model of N Majorana fermions with random quadratic interactions through a detailed spectral analysis and by coupling the model to two- and four-point sources. In particular, we define the generalized spectral form factor (SFF) and level spacing distribution function by generalizing from the partition function to the generating function. For N = 2, we obtain an exact solution of the generalized SFF. It exhibits qualitatively similar behavior to the higher N case with a source term. The exact solution helps understand the behavior of the generalized SFF. We calculate the generalized level spacing distribution function and the mean value of the adjacent gap ratio defined by the generating function. For the SYK 2 model with a four-point source term, we find a Gaussian unitary ensemble behavior in the near-integrable region of the theory, which indicates a transition to chaos. This behavior is confirmed by the connected part of the generalized SFF with an unfolded spectrum. The departure from this Gaussian random matrix behavior as the relative strength of the source term is increased is consistent with the observation that the four-point source term alone, without the SYK 2 couplings turned on, exhibits an integrable spectrum from the SFF and level spacing distribution function in the large N limit.
A bstract This paper investigates the spectral form factor (SFF) of the quadratic R -para-particle Sachdev-Ye-Kitaev ( R -PSYK 2 ) model with various random matrix ensemble couplings. We generalize previous work on Gaussian Unitary Ensemble (GUE) couplings to all three Gaussian ensembles (GUE, GOE, GSE) and three circular ensembles (CUE, COE, CSE). Through analytical and numerical methods, we establish precise correspondences between GUE and CUE results, demonstrating their SFFs satisfy KGUE(2t) {\mathcal{K}}_{\textrm{GUE}}(2t) KGUE​(2t) K GUE 2 t ≈ KCUE(t) {\mathcal{K}}_{\textrm{CUE}}(t) KCUE​(t) K CUE t in the time regime 1 ≪ t ≪ N . For the symplectic ensembles, we observe similar behavior with appropriate time rescaling, while we only provide the calculation method for the orthogonal ensembles.
Abstract We study the onset of RMT dynamics in the mass-deformed SYK model (i.e. an SYK model deformed by a quadratic random interaction) in terms of the strength of the quadratic deformation. We use as chaos probes both the connected unfolded Spectral Form Factor (SFF) as well as the Gaussian-filtered SFF, which has been recently introduced in the literature. We show that they detect the chaotic/integrable transition of the mass-deformed SYK model at different values of the mass deformation: the Gaussian-filtered SFF sees the transition for large values of the mass deformation; the connected unfolded SFF sees the transition at small values. The latter shows a closer agreement with the transition as seen by the OTOCs. We argue that the chaotic/integrable deformation affects the energy levels inhomogeneously: for small values of the mass deformation only the low-lying states are modified while for large values of the mass deformation also the states in the bulk of the spectrum move to the integrable behavior.
Abstract We present a detailed quantitative analysis of spectral correlations in the Sachdev-Ye-Kitaev (SYK) model. We find that the deviations from universal Random Matrix Theory (RMT) behavior are due to a small number of long-wavelength fluctuations (of the order of the number of Majorana fermions N) from one realization of the ensemble to the next one. These modes can be parameterized effectively in terms of Q-Hermite orthogonal polynomials, the main contribution being due to scale fluctuations for which we give a simple analytical estimate. Our numerical results for N = 32 show that only the lowest eight polynomials are needed to eliminate the nonuniversal part of the spectral fluctuations. The covariance matrix of the coefficients of this expansion can be obtained analytically from low-order double-trace moments. We evaluate the covariance matrix of the first six moments and find that it agrees with the numerics. We also analyze the spectral correlations in terms of a nonlinear σ-model, which is derived through a Fierz transformation, and evaluate the one and two-point spectral correlation functions to two-loop order. The wide correlator is given by the sum of the universal RMT result and corrections whose lowest-order term corresponds to scale fluctuations. However, the loop expansion of the σ-model results in an ill-behaved expansion of the resolvent, and it gives universal RMT fluctuations not only for q = 4 or higher even q-body interactions, but also for the q = 2 SYK model albeit with a much smaller Thouless energy while the correct result in this case should have been Poisson statistics. In our numerical studies we analyze the number variance and spectral form factor for N = 32 and q = 4. We show that the quadratic deviation of the number variance for large energies appears as a peak for small times in the spectral form factor. After eliminating the long-wavelength fluctuations, we find quantitative agreement with RMT up to an exponentially large number of level spacings for the number variance or exponentially short times in the case of the spectral form factor.
We introduce two disorder-free variants of the Sachdev-Ye-Kitaev (SYK) model, demonstrate their integrability, and study their static and dynamical properties. Unlike diagrammatic techniques, the integrability of these models allows us to obtain dynamical correlation functions even when the number of Majorana fermions is finite. From the solutions, we find that out-of-time-order correlators (OTOCs) in these models exhibit exponential growth at early times, resembling that of many-body quantum chaotic systems, such as those with disorder or external kick terms, despite their large N behavior differing from that of typical chaotic systems. Conversely, our analysis shows no evidence of random-matrix behavior in level spacing statistics or the spectral form factor. Our findings illustrate that the clean versions of the SYK models represent simple but nontrivial examples of disorder-free quantum many-body systems displaying chaoslike behavior of OTOCs.
Correlations between the energies of a system's spectrum are one of the defining features of quantum chaos. They can be probed using the spectral form factor (SFF). We investigate how each spectral distance contributes in building this two-point correlation function. Specifically, starting from the spectral distribution of kthk\mathrm{th}kth-neighbor-level spacing (knLS)(k\mathrm{nLS})(knLS), we provide analytical expressions for the kthk\mathrm{th}kth-neighbor spectral form factor (knSFF)(k\mathrm{nSFF})(knSFF). We do so for the three Gaussian random matrix ensembles and the ``Poisson'' ensemble of uncorrelated energy levels. We study the properties of the knSFFk\mathrm{nSFF}knSFF, namely its minimum value and the time at which this minimum is reached, as well as the energy spacing with the deepest knSFFk\mathrm{nSFF}knSFF. This allows us to quantify the contribution of each individual knLSk\mathrm{nLS}knLS to the SFF ramp, which is a characteristic feature of quantum chaos. In particular, we show how the onset of the ramp, characterized either by the dip or the Thouless time, shifts to shorter times as contributions from longer-range spectral distance are included. Interestingly, the even and odd neighbors contribute quite distinctively, the first being the most important to built the ramp. They respectively yield a resonance or antiresonance in the ramp. All of our analytical results are tested against numerical realizations of random matrices. We complete our analysis and show how the introduced tools help characterize the spectral properties of a physical many-body system by looking at the interacting XXZ Heisenberg model with local on-site disorder that allows transitioning between the chaotic and integrable regimes.
A bstract We elaborate further on the matter coupling of Dirac matter in the SYK framework, incorporating non-Gaussian coupling distributions and bulk fermion mass effects. Our study analyzes quartic matter couplings generated by a non-Gaussian distribution as an illustrative example. The introduction of bulk-fermion mass alters the boundary coupling between the Dirac and Majorana fermions. The averaged adjacent gap ratio is sensitive to the distribution of random couplings, which remains independent of the Hamiltonian’s symmetry. The generalization of the SYK model to non-Gaussian distributions and the inclusion of bulk fermion mass remain qualitatively similar to the Gaussian and massless cases. Key deviations are observed only in the time scales for the linear ramp in the spectral form factor and the saturation of entanglement entropy.