Matteo A. Cardella
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 ingredients are a suitable uv regularization of the two replica kinetic operator, the property of its Fourier transform and some spectral analysis of the four point function two replica ladder kernel.
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The moments of the spectral form factor in SYK
本文识别SYK模型中描述谱形状因子各矩的鞍点，揭示低阶矩遵循随机矩阵统计而高阶矩因谱边缘涨落产生偏差，并通过稀疏化模型和$q=2$情形加以验证。
We investigate the spectral form factor of the sparse Sachdev-Ye-Kitaev model. We use numerical methods to establish that at intermediate times the connected part of the spectral form factor is the dominant one. These connected contributions arise from fluctuations around the disconnected geometry, not from a new saddle point. A similar effect was previously conjectured in SYK but required a value of $N$ out of reach of current numerical simulations.
In this letter, we study the return amplitude, which is the overlap between the initial state and the time evolved state, in the Sachdev-Ye-Kitaev (SYK) model. Initial states are taken to be product states in a spin basis. We numerically study the return amplitude by exactly diagonalizing the Hamiltonian. We also derive the analytic expression for the return amplitude in random matrix theory. The SYK results agree with the random matrix expectation. We also study the time evolution under the different Hamiltonian that describes the traversable wormholes in projected black holes in the context …
In this note we study the spectral form factor in the SYK model in large $q$ limit at infinite temperature. We construct analytic solutions for the saddle point equations that describe the slope and the ramp regions of the spectral form factor time dependence. These saddle points are obtained by taking different approaches to the large $q$ limit: the slope region is described by a replica-diagonal solution and the ramp region is described by a replica-nondiagonal solution. We find that the onset of the ramp behavior happens at the Thouless time of order $q \log q$. We also evaluate the one-loo…
Many-body spectral transitions through the lens of the variable-range SYK2 model
研究变程SYK2模型中距离依赖相互作用对多体谱统计的影响，发现单粒子临界性如何传递到多体系统，并揭示新的谱区域和二次平台的出现。
Hybrid Brownian SYK-Hubbard Model: from Spectral Function to Quantum Chaos
本文提出混合布朗运动SYK-Hubbard模型，研究强相互作用系统中复杂关联的出现及量子混沌特性。
Probing quantum chaos through singular-value correlations in sparse non-Hermitian SYK model
本研究针对稀疏非厄米SYK模型，通过奇异值分解分析其奇异值谱，发现奇异值统计与厄米高斯系综一致，且稀疏度增加会引发混沌行为偏离，还定义了可给出稀疏临界阈值的奇异复杂度，关联全息对偶破缺问题。
Size Operator and Spectral Clustering in the Two Coupled SYK Model
该研究在双耦合SYK模型中，通过分析有限N下按算符尺寸划分的谱簇，揭示算符尺寸是连接有限N谱与大N涌现引力物理的桥梁，还解释了虫洞复兴动力学及虫洞-黑洞转变的微观机制。
The late time ramp from chord diagrams in the double-scaled SYK model
本研究在双标度SYK模型中通过弦图解析计算谱形状因子的晚期斜坡，重现随机矩阵理论局域本征值关联并拓展到有限q情形，验证了渐近行为，还给出弦的喇叭配分函数与粘合因子。
Exploring the Spectral Edge in SYK Models
研究SYK模型中低温度下淬火熵的幂律行为，通过数值模拟发现其能级间距统计与RMT集合一致，并探讨了超对称虫洞中物质填充的影响。
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 2- and 4-point sources. In particular, we define the generalized spectral form factor 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 spectral form factor. It exhibits qualitatively similar behavior to the higher $N$ case with a source term. The exact solution helps understand the behavior of the generalized spectral form…
Spectral form factor of quadratic $R$-para-particle SYK model with Random Matrix Coupling
本文研究了二次R-para-particle SYK模型在不同随机矩阵耦合下的谱形式因子，揭示了GUE与CUE在特定时间范围内的对应关系，并探讨了辛族和正交族的行为。
The Sachdev-Ye-Kitaev (SYK) model attracts attention in the context of information scrambling, which represents delocalization of quantum information and is quantified by the out-of-time-ordered correlators (OTOC). The SYK model contains $N$ fermions with disordered and four-body interactions. Here, we introduce a variant of the SYK model, which we refer to as the Wishart SYK model. We investigate the Wishart SYK model for complex fermions and that for hard-core bosons. We show that the ground state of the Wishart SYK model is massively degenerate and the residual entropy is extensive, and tha…
Sachdev-Ye-Kitaev (SYK) model, which describes $N$ randomly interacting Majorana fermions in 0+1 dimension, is found to be an solvable UV-complete toy model for holographic duality in nearly AdS$_2$ dilaton gravity. Ref. [1] proposed a modified model by coupling two identical SYK models, which at low-energy limit is dual to a global AdS$_2$ geometry. This geometry is an "eternal wormhole" because the two boundaries are causally connected. Increasing the temperature drives a Hawking-Page like transition from the eternal wormhole geometry to two disconnected black holes with coupled matter field…
We use random matrix theory to explore late-time chaos in supersymmetric quantum mechanical systems. Motivated by the recent study of supersymmetric SYK models and their random matrix classification, we consider the Wishart-Laguerre unitary ensemble and compute the spectral form factors and frame potentials to quantify chaos and randomness. Compared to the Gaussian ensembles, we observe the absence of a dip regime in the form factor and a slower approach to Haar-random dynamics. We find agreement between our random matrix analysis and predictions from the supersymmetric SYK model, and discuss …
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 …
We present a replica path integral approach describing the quantum chaotic dynamics of the SYK model at large time scales. The theory leads to the identification of non-ergodic collective modes which relax and eventually give way to an ergodic long time regime (describable by random matrix theory). These modes, which play a role conceptually similar to the diffusion modes of dirty metals, carry quantum numbers which we identify as the generators of the Clifford algebra: each of the $2^N$ different products that can be formed from $N$ Majorana operators defines one effective mode. The competiti…
In finite entropy systems, real-time partition functions do not decay to zero at late time. Instead, assuming random matrix universality, suitable averages exhibit a growing "ramp" and "plateau" structure. Deriving this non-decaying behavior in a large $N$ collective field description is a challenge related to one version of the black hole information problem. We describe a candidate semiclassical explanation of the ramp for the SYK model and for black holes. In SYK, this is a two-replica nonperturbative saddle point for the large $N$ collective fields, with zero action and a compact zero mode…
A Study of the SYK$_{2}$ Model with Twisted Boundary Conditions
本文研究热回路上具有扭曲边界条件的 SYK$_2$ 模型，并通过 OTOC 与 SFF 分析其可积性和混沌特征。
We investigate the thermalization of Sachdev-Ye-Kitaev (SYK) models coupled via random interactions following quenches from the perspective of entanglement. Previous studies have shown that when a system of two SYK models coupled by random two-body terms is quenched from the thermofield double state with sufficiently low effective temperature, the Rényi entropies do not saturate to the expected thermal values in the large-$N$ limit. Using numerical large-$N$ methods, we first show that the Rényi entropies in a pair SYK models coupled by two-body terms can thermalize, if quenched from a state w…
Why Cooper pairs live in AdS2: a spectral analysis of the Yukawa-SYK model
本研究通过对Yukawa-SYK模型的谱分析，为库珀对在AdS2中的存在提供了微观依据，明确了超导不稳定性相关涨落的谱性质，阐明了双域库珀对场与AdS2中标量物质场的映射关系。
We show analytically that the spectral density of the $q$-body Sachdeev-Ye-Kitaev (SYK) model agrees with that of Q-Hermite polynomials with Q a non-trivial function of $q \ge 2$ and the number of Majorana fermions $N \gg 1$. Numerical results, obtained by exact diagonalization, are in excellent agreement with the analytical spectral density even for relatively small $N \sim 8$. For $N \gg 1$ and not close to the edge of the spectrum, we find the macroscopic spectral density simplifies to $ρ(E) \sim \exp[2\arcsin^2(E/E_0)/\log η]$, where $η$ is the suppression factor of the contribution of int…
The Sachdev-Ye-Kitaev (SYK) model describes a strange metal that shows peculiar non-Fermi liquid properties without quasiparticles. It exhibits a maximally chaotic behavior characterized by out-of-time-ordered correlators (OTOCs), and is expected to be a holographic dual to black holes. Recently, a striking similarity between the SYK model and the Hund-coupling induced spin-freezing crossover in multi-orbital Hubbard models has been pointed out. To further explore this connection, we study OTOCs for fermionic Hubbard models, which are prototypical models for strongly correlated electrons in so…
We give qualitative arguments for the mesoscopic nature of the Sachdev-Yee-Kitaev (SYK) model in the holographic regime with $q^2/N\ll 1$ with $N$ Majorana particles coupled by antisymmetric and random interactions of range $q$. Using a stochastic deformation of the SYK model, we show that its characteristic determinant obeys a viscid Burgers equation with a small spectral viscosity in the opposite regime with $q/N=1/2$, in leading order. The stochastic evolution of the SYK model can be mapped onto that of random matrix theory, with universal Airy oscillations at the edges. A spectral hydrodyn…
The concepts of operator size and computational complexity play important roles in the study of quantum chaos and holographic duality because they help characterize the structure of time-evolving Heisenberg operators. It is particularly important to understand how these microscopically defined measures of complexity are related to notions of complexity defined in terms of a dual holographic geometry, such as complexity-volume (CV) duality. Here we study partially entangled thermal states in the Sachdev-Ye-Kitaev (SYK) model and their dual description in terms of operators inserted in the inter…
Free energy and spectral edge of the SYK model
研究具有随机 4 体相互作用的 N 个马约拉纳费米子的 Sachdev--Ye--Kitaev 模型，通过引入新有限浴插值计算每个固定正温度下 SYK 自由能极限，确定施温格 - 戴森压力零温度斜率并转移到谱边缘，得出模型最大特征值相关结论。
It has been recently proposed by Maldacena and Qi that an eternal traversable wormhole in a two dimensional Anti de Sitter space (${\rm AdS}_2$) is the gravity dual of the low temperature limit of two Sachdev-Ye-Kitaev (SYK) models coupled by a relevant interaction (which we will refer to as spin operator). In this paper, we study spectral and eigenstate properties of this coupled SYK model. We have found that level statistics in the tail of the spectrum, and for a sufficiently weak coupling, shows substantial deviations from random matrix theory which suggests that traversable wormholes are n…
Entanglement spreading and emergent locality in Brownian SYK chains
本研究利用量子纠错工具，在一维布朗SYK链模型中探究量子信息扩散与清晰光锥的涌现，发现强耦合下信息关联呈陡峭跃迁，其底层由FKPP方程的畴壁解描述，验证了全息理论中涌现定域性的相关机制。
A class of tensor models were recently outlined as potentially calculable examples of holography: their perturbative large-$N$ behavior is similar to the Sachdev-Ye-Kitaev (SYK) model, but they are fully quantum mechanical (in the sense that there is no quenched disorder averaging). These facts make them intriguing tentative models for quantum black holes. In this note, we explicitly diagonalize the simplest non-trivial Gurau-Witten tensor model and study its spectral and late-time properties. We find parallels to (a single sample of) SYK where some of these features were recently attributed t…
We study the level statistics of a generalized Sachdev-Ye-Kitaev (SYK) model with two-body and one-body random interactions of finite range by exact diagonalization. Tuning the range of the one-body term, while keeping the two-body interaction sufficiently long-ranged, does not alter substantially the spectral correlations, which are still given by the random matrix prediction typical of a quantum chaotic system. However a transition to an insulating state, characterized by Poisson statistics, is observed by reducing the range of the two-body interaction. Close to the many-body metal-insulator…
The Late-time Ramp of the Double-Scaled SYK Model from the large-$n$ tail of Cactus Diagram
本研究从仙人掌图大n尾部推导双标SYK模型有限温度谱形式因子的后期斜坡，重现随机矩阵理论预测的线性斜坡，为斜坡提供了微观起源。
Parisi's hypercube, Fock-space fluxes, and the microscopics of near-AdS$_2$/near-CFT$_1$ duality
本文证明Parisi超立方体模型与DS-SYK模型在可观测量关联函数上等价，可作为近AdS$_2$/近CFT$_1$全息的微观模型，并提出Fock空间图上大量无序均匀通量是近CFT$_1$微观机制的更广泛特征。
Probing Evaporating Black Holes with Modular Flow in SYK
研究弱耦合到另一个SYK模型的费米子相关函数的模流效应，揭示模参数临界值后相关函数从一边边界转移到另一边的奇异结构，通过欧几里得时间的复制技巧和微扰分析，重构双侧AdS₂黑洞时空的对偶模流，识别量子极值表面和岛。
We study the Wishart-Sachdev-Ye-Kitaev (WSYK) model consisting of two $\hat{q}$-body Sachdev-Ye-Kitaev (SYK) models with general complex couplings, one the Hermitian conjugate of the other, living in off-diagonal blocks of a larger WSYK Hamiltonian. The spectrum is positive with a hard edge at zero energy. We employ diagrammatic and combinatorial techniques to compute analytically the low-order moments of the Hamiltonian. In the limit of large number $N$ of Majoranas, we have found striking similarities with the moments of the weight function of the Al-Salam-Chihara $Q$-Laguerre polynomials. F…
In this paper, we develop a general effective theory for two copies of the Sachdev-Ye-Kitaev (SYK) model with a time-dependent bilinear coupling. For a quantum quench problem with an initial state of the thermofield double state, we show how the evolution of the system is described by a complex reparametrization field with a classical Hamiltonian. We study correlation functions in this system and compare the large-$q$ theory with the bulk low energy effective theory. In particular, we study the special case of a ``rescued black hole'', which describes how a time-evolved thermofield double stat…
We show that in Jackiw-Teitelboim (JT) gravity, late-time two-point functions can get a leading non-decaying contribution from a spacetime with the topology of a Mobius strip (a disk with one crosscap). There is an interesting interplay between this contribution and the standard "plateau". The two can add together or cancel, depending on topological weighting factors. We match this behavior to Random Matrix Theory (RMT) and the N mod 8 periodicity of Sachdev-Kitaev-Ye (SYK) results.
The Sachdev-Ye-Kitaev (SYK) model is a concrete solvable model to study non-Fermi liquid properties, holographic duality and maximally chaotic behavior. In this work, we consider a generalization of the SYK model that contains two SYK models with different number of Majorana modes coupled by quadratic terms. This model is also solvable, and the solution shows a zero-temperature quantum phase transition between two non-Fermi liquid chaotic phases. This phase transition is driven by tuning the ratio of two mode numbers, and a Fermi liquid non-chaotic phase sits at the critical point with equal m…
We introduce randomness into a class of integrable models and study the spectral form factor as a diagnostic to distinguish between randomness and chaos. Spectral form factors exhibit a characteristic dip-ramp-plateau behavior in the $N>2$ SYK$_2$ model at high temperatures that is absent in the $N=2$ SYK$_2$ model. Our results suggest that this dip-ramp-plateau behavior implies the existence of random eigenvectors in a quantum many-body system. To further support this observation, we examine the Gaussian random transverse Ising model and obtain consistent results without suffering from small …
Dynamics of operator size distribution in q-local quantum Brownian SYK and spin models
本研究推导了q-局域布朗SYK和自旋模型中算子尺寸分布的精确主方程，给出大N解析解，揭示分布对初始条件的依赖及晚期平均尺寸的普适衰减行为。
We construct three models to describe the scenario where two eternal black holes are separated by a flat space, and can eventually be entangled by exchanging radiations. In the doubly holographic setup, we compute the entanglement entropy and the mutual information among the subsystems and obtain the dynamic phase structure of the entanglement. The formation of entanglement between the two black holes is delayed by the space where the radiations must travel through. Finally, if the two black holes exchange sufficient Hawking modes, the final state is characterized by a connected entanglement w…
Disorder-free Sachdev-Ye-Kitaev models: Integrability and a precursor of chaos
本文提出两种无无序SYK模型，证明其可积性，并发现其OTOC早期指数增长，呈现混沌前兆，但无随机矩阵行为。
Out-of-time-order correlators (OTOCs) are a standard measure of quantum chaos. Of the four operators involved, one pair may be regarded as a source and the other as a probe. A usual approach, applicable to large-$N$ systems such as the SYK model, is to replace the actual source with some mean-field perturbation and solve for the probe correlation function on the double Keldysh contour. We show how to obtain the OTOC by combining two such solutions for perturbations propagating forward and backward in time. These dynamical perturbations, or scrambling modes, are considered on the thermofield do…
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 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 the scale fluctuations for which we give a simple estimate. Our numerical results for $N=32$ show that only the lowest eight polynomials are needed to eliminate the nonuniversal part …
Two types of quantum chaos: testing the limits of the Bohigas-Giannoni-Schmit conjecture
本文区分了本征基混沌与谱混沌两类量子混沌，通过分析泊松哈密顿量系综并利用 Metropolis 算法修改 SYK 模型耦合，发现存在具有本征基混沌而无谱混沌的 $k$-局域哈密顿量，从而指出 ETH 与 RMU 的联系需以 $k$-局域性为前提，挑战了 BGS 猜想。
We argue that in a large class of disordered quantum many-body systems, the late time dynamics of time-dependent correlation functions is captured by random matrix theory, specifically the energy eigenvalue statistics of the corresponding ensemble of disordered Hamiltonians. We find that late time correlation functions approximately factorize into a time-dependent piece, which only depends on spectral statistics of the Hamiltonian ensemble, and a time-independent piece, which only depends on the data of the constituent operators of the correlation function. We call this phenomenon "spectral de…
Exact spectrum and anomalous relaxation in the open disorder-free Sachdev-Ye-Kitaev system
利用无无序SYK模型的可积性，在Gorini-Kossakowski-Sudarshan-Lindblad形式下精确求解了Liouvillian谱，发现热力学极限下能隙在无耗散极限不消失，导致反常弛豫，并观察到谱特征如特征值实复转变。
The Sachdev-Ye-Kitaev model spectral form factor exhibits absence of information loss in the form of a ramp and a plateau, that are typical of random matrix theory. In a large $N$ collective fields description, the ramp was reproduced by Saad, Shenker and Stanford \cite{Saad:2018bqo}, by replica symmetry breaking saddles for a connected component of the analytically continued to real times thermal partition function two point function. We derive a two sides Schwarzians effective action for fluctuations around the ramp critical saddles, by adapting to the two replica system a method by Kitaev a…
We consider Random Matrix Theories with non-Gaussian potentials that have a rich phase structure in the large $N$ limit. We calculate the Spectral Form Factor (SFF) in such models and present them as interesting examples of dynamical models that display multi-criticality at short time-scales and universality at large time scales. The models with quartic and sextic potentials are explicitly worked out. The disconnected part of the Spectral Form Factor (SFF) shows a change in its decay behavior exactly at the critical points of each model. The dip-time of the SFF is estimated in each of these mo…
We argue that the late time behavior of horizon fluctuations in large anti-de Sitter (AdS) black holes is governed by the random matrix dynamics characteristic of quantum chaotic systems. Our main tool is the Sachdev-Ye-Kitaev (SYK) model, which we use as a simple model of a black hole. We use an analytically continued partition function $|Z(β+it)|^2$ as well as correlation functions as diagnostics. Using numerical techniques we establish random matrix behavior at late times. We determine the early time behavior exactly in a double scaling limit, giving us a plausible estimate for the crossove…
This is a collection of notes that are about spectral form factors of standard ensembles in the random matrix theory, written for the practical usage of current study of late time quantum chaos. More precisely, we consider Gaussian Unitary Ensemble (GUE), Gaussian Orthogonal Ensemble (GOE), Gaussian Symplectic Ensemble (GSE), Wishart-Laguerre Unitary Ensemble (LUE), Wishart-Laguerre Orthogonal Ensemble (LOE), and Wishart-Laguerre Symplectic Ensemble (LSE). These results and their physics applications cover a three-fold classification of late time quantum chaos in terms of spectral form factors…
We analyze a two-body nonhermitian two-site Sachdev-Ye-Kitaev model with the couplings of one site complex conjugated to the other site. This model, with no explicit coupling between the sites, shows an infinite number of phase transitions which is a consequence of the partition function factorizing into a product over Matsubara frequencies. We calculate the quenched free energy in two different ways, first in terms of the single-particle energies, and second by solving the Schwinger-Dyson equations. The first calculation can be done entirely in terms of a one-site model. The conjugate replica…
Quantum chaos and late-time equipartition of symmetry-resolved Krylov complexity
该研究针对有限维混沌量子多体系统，提出对称分辨Krylov复杂度的晚期维数加权均分规则，经多种模型数值验证，明确分辨精确对称性对Krylov复杂度诊断混沌算符增长的关键作用。
Fisher zeroes and the fluctuations of the spectral form factor of chaotic systems
针对量子混沌系统谱形式因子的涨落问题，本文通过Fisher零点揭示了尖峰的产生机制，并在引入能级排斥的修改型随机能级模型及SYK模型中验证了该机制。
Reappearance of Thermalization Dynamics in the Late-Time Spectral Form Factor
本文发现缺乏时间反演对称性的系统在晚期海森堡时间附近，其谱形式因子会短暂偏离随机矩阵理论结果，且该偏离完全由早期热化性质决定，并利用类黎曼-西格尔公式推导与验证了此现象。
A semiclassical Hilbert space for random matrix theory
该研究针对随机矩阵理论的热场双态，构建了以最大纠缠态为初始矢量的Krylov基半经典希尔伯特空间，发现其在低能连续极限下可约化为JT引力的Liouville哈密顿量，暗示该空间是对偶引力的体希尔伯特空间
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