An introduction to the SYK model

Rosenhaus, Vladimir

doi:10.1088/1751-8121/ab2ce1

Cited by 162 publications

(155 citation statements)

References 182 publications

(332 reference statements)

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“…It drew considerable attention in recent years as an example of a strongly coupled system that is solvable at large N . An interesting feature of this model is that it displays a universal pattern of conformal symmetry breaking in the IR that is shared with near extremal black holes that have a long, nearly AdS 2 throat, and therefore it can be viewed as a toy model for these black holes [27,28], see [29,30] for reviews. The OTO correlator displays a maximal Lyapunov exponent λ L = 2πβ −1 to leading order in the inverse coupling.…”

Section: Syk Chainmentioning

confidence: 99%

Chaos in the butterfly cone

Mezei

Sárosi

2020

J. High Energ. Phys.

146

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A simple probe of chaos and operator growth in many-body quantum systems is the out of time ordered four point function. In a large class of local systems, the effects of chaos in this correlator build up exponentially fast inside the so called butterfly cone. It has been previously observed that the growth of these effects is organized along rays and can be characterized by a velocity dependent Lyapunov exponent, λ(v). We show that this exponent is bounded inside the butterfly cone as λ(v) ≤ 2πT (1 − |v|/v B ), where T is the temperature and v B is the butterfly speed. This result generalizes the chaos bound of Maldacena, Shenker and Stanford. We study λ(v) in some examples such as two dimensional SYK models and holographic gauge theories, and observe that in these systems the bound gets saturated at some critical velocity v * < v B . In this sense, boosting a system enhances chaos. We discuss the connection to conformal Regge theory, where λ(v) is related to the spin of the leading large N Regge trajectory, and controls the four point function in an interpolating regime between the Regge and the light cone limit. Finally, we comment on the generalization of the chaos bound to boosted and rotating ensembles and clarify some recent results on this in the literature.

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Section: Syk Chainmentioning

confidence: 99%

Chaos in the butterfly cone

Mezei

Sárosi

2020

J. High Energ. Phys.

146

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“…Recent years, the (0 + 1)-dimensional fermionic model with all-to-all random interactions referred to as Sachdev-Ye-Kitaev (SYK) model [64,65] has been studied very actively in the context of the nonequilibrium quantum many-body systems and its application to the AdS/CFT correspondence (see references in a recent review article [66]). The level statistics of the SYK model was numerically examined, and good agreements with the RMT have been observed.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Janossy densities for chiral random matrix ensembles and their applications to two-color QCD

Fuji

Kanamori

Nishigaki

2019

J. High Energ. Phys.

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We compute individual distributions of low-lying eigenvalues of massive chiral random matrix ensembles by the Nyström-type quadrature method for evaluating the Fredholm determinant and Pfaffian that represent the analytic continuation of the Janossy densities (conditional gap probabilities). A compact formula for individual eigenvalue distributions suited for precise numerical evaluation by the Nyström-type method is obtained in an explicit form, and the k th smallest eigenvalue distributions are numerically evaluated for chiral unitary and symplectic ensembles in the microscopic limit. As an application of our result, the low-lying Dirac spectra of the SU(2) lattice gauge theory with N F = 8 staggered flavors are fitted to the numerical prediction from the chiral symplectic ensemble, leading to a precise determination of the chiral condensate of a two-color QCD-like system in the future.

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“…The order parameter B takes form

B = i \sum_{n = 0}^{\infty} \frac{2}{ω_{n}^{2} + J^{2}} = \frac{i}{2 J} \tanh \frac{J}{2}

and diminishes as

\sim 1 / J

at large J . b)Sachdev–Ye–Kitaev (SYK) model. This is the model of “all with all” interactions between many zero‐dimensional Majorana fermions . Its Lagrangian reads as

L = \frac{1}{2} \sum_{i} μ_{i} \partial_{τ} μ_{i} - \frac{J_{i j k l}}{4!} \sum_{i j k l} μ_{i} μ_{j} μ_{k} μ_{l}

with

false⟨ J_{i j k l} J_{i j k l} false⟩ = 6 J^{2} / N^{3}

.…”

Section: Even‐ and Odd‐frequency Components Of The Correlation Functimentioning

confidence: 99%

“…Its Lagrangian reads as

L = \frac{1}{2} \sum_{i} μ_{i} \partial_{τ} μ_{i} - \frac{J_{i j k l}}{4!} \sum_{i j k l} μ_{i} μ_{j} μ_{k} μ_{l}

with

false⟨ J_{i j k l} J_{i j k l} false⟩ = 6 J^{2} / N^{3}

. The large‐N limit propagator is

F false(τ_{1}, τ_{2} false) = b \frac{sign (τ_{1} - τ_{2})}{{(J | τ_{1} - τ_{2} |)}^{1 / 2}}

where the term with

\partial_{τ}

in Equation was neglected and

b^{4} = 1 / 4 π

. The order parameter B reads as

B ≃ - \frac{2 b}{J^{1 / 2}}

As one can see from Equations and , the time‐independent correlator shows a different scaling with J for different types of interactions between Majorana fermions.…”

Section: Even‐ and Odd‐frequency Components Of The Correlation Functimentioning

confidence: 99%

Quantum Pairing Time Orders

2020

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The concept of the time‐independent correlators for the even‐ and odd‐frequency pairing states that can be defined for both bosonic and fermionic quasiparticles is proposed. These correlators explicitly capture the existence of two distinct classes of pairing states and provide a direct probe of the hidden Berezinskii order. This concept is illustrated in the cases of pairings for Majorana fermions and quasiparticles in Dirac semimetals. It is shown that the time‐independent correlator is able to effectively capture the energy scale relevant for pairing.

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An introduction to the SYK model

Cited by 162 publications

References 182 publications

Chaos in the butterfly cone

Chaos in the butterfly cone

Janossy densities for chiral random matrix ensembles and their applications to two-color QCD

Quantum Pairing Time Orders

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