We consider the Brownian SYK model of $N$ interacting Majorana fermions, with random couplings that are taken to vary independently at each time. We study the out-of-time-ordered correlators (OTOCs) of arbitrary observables and the Rényi-$2$ tripartite information of the unitary evolution operator, which were proposed as diagnostic tools for quantum chaos and scrambling, respectively. We show that their averaged dynamics can be studied as a quench problem at imaginary times in a model of $N$ qudits, where the Hamiltonian displays site-permutational symmetry. By exploiting a description in terms of bosonic collective modes, we show that for the quantities of interest the dynamics takes place in a subspace of the effective Hilbert space whose dimension grows either linearly or quadratically with $N$, allowing us to perform numerically exact calculations up to $N = 10^6$. We analyze in detail the interesting features of the OTOCs, including their dependence on the chosen observables, and of the tripartite information. We observe explicitly the emergence of a scrambling time $t^\ast\sim \ln N$ controlling the onset of both chaotic and scrambling behavior, after which we characterize the exponential decay of the quantities of interest to the corresponding Haar scrambled values.
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…
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…
In this work, we introduce a symmetry-based approach to study the scrambling and operator dynamics of Brownian SYK models at large finite $N$ and in the infinite $N$ limit. We compute the out-of-time-ordered correlator (OTOC) in the Majorana model without charge conservation and the complex model with charge conservation, and demonstrate that in both models taking the random average of the couplings gives rise to emergent symmetry structures. The random averaging exactly maps the operator dynamics of the Majorana model and the complex model to the imaginary time dynamics of an SU(2) spin and a…
We study the contour dependence of the out-of-time-ordered correlation function (OTOC) both in weakly coupled field theory and in the Sachdev-Ye-Kitaev (SYK) model. We show that its value, including its Lyapunov spectrum, depends sensitively on the shape of the complex time contour in generic weakly coupled field theories. For gapless theories with no thermal mass, such as SYK, the Lyapunov spectrum turns out to be an exception; their Lyapunov spectra do not exhibit contour dependence, though the full OTOCs do. Our result puts into question which of the Lyapunov exponents computed from the exp…
We study a class of SYK-type models in large N limit from the gravity dual side in terms of Schwarzian action analytically. The quantum correction to two point correlation function due to the Schwarzian action produces transfer of degree of freedom from the quasiparticle peak to Hubbard band in density of states (DOS), a signature strong correlation. In Schwinger-Keldysh (SK) formalism, we calculate higher point thermal out-of-time order correlation (OTOC) functions, which indicate quantum chaos by having Lyapunov exponent. Higher order local spin-spin correlations are also calculated, which c…
We study the operator growth in open quantum systems with dephasing dissipation terms, extending the Krylov complexity formalism of Phys. Rev. X 9, 041017. Our results are based on the study of the dissipative $q$-body Sachdev-Ye-Kitaev (SYK$_q$) model, governed by the Markovian dynamics. We introduce a notion of ''operator size concentration'' which allows a diagrammatic and combinatorial proof of the asymptotic linear behavior of the two sets of Lanczos coefficients ($a_n$ and $b_n$) in the large $q$ limit. Our results corroborate with the semi-analytics in finite $q$ in the large $N$ limit,…
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 …
The commutator between operators at different space and time has been a diagnostic for locality of unitary evolution. Most existing results are either for specific tractable (random) Hamiltonians(Out-of-Time-Order-Correlators calculations), or for worse case Hamiltonians (Lieb-Robinson-like bounds or OTOC bounds). In this work, we study commutators in typical Hamiltonians. Draw a sample from any zero-mean bounded independent random Hamiltonian ensemble, time-independent or Brownian, we formulate concentration bounds in the spectral norm and for the OTOC with arbitrary non-random state. Our bou…
Topological Control of Quantum Chaos Diagnostics: OTOCs, Spectral Statistics, and Information Scrambling in Ising Model
School of Quantum Physics and Matter Institute for Research in Fundamental Sciences (IPM)（量子物理与物质学院，基础科学研究院（IPM））；Ershad Damavand, Institute of Higher Education (EDI)（厄尔沙德达马万德高等教育研究所（EDI））；Department of Mathematics and Computer Science, Amirkabir University of Technology（阿米尔卡比尔理工大学数学与计算机科学系）；Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University（阿尔扎哈拉大学数学科学学院数学系）；Department of Physics, CT.C, Islamic Azad University（伊斯兰阿扎德大学CT.C物理学系）；Institute of Biosocial and Quantum Science and Technologies, CT.C, Islamic Azad University（伊斯兰阿扎德大学CT.C生物社会与量子科学与技术研究所）
We study the generalization of the Sachdev-Ye-Kitaev (SYK) model to a $1+1$ dimensional chiral SYK model of $N$ flavors of right-moving chiral Majorana fermions with all-to-all random 4-fermion interactions. The interactions in this model are exactly marginal, leading to an exact scaling symmetry. We show the Schwinger-Dyson equation of this model in the large $N$ limit is exactly solvable. In addition, we show this model is integrable for small $N\le6$ by bosonization. Surprisingly, the two point function in the large $N$ limit has exactly the same form as that for $N=4$, although the four po…
We study a simplified version of the Sachdev-Ye-Kitaev (SYK) model with real interactions by exact diagonalization. Instead of satisfying a continuous Gaussian distribution, the interaction strengths are assumed to be chosen from discrete values with a finite separation. A quantum phase transition from a chaotic state to an integrable state is observed by increasing the discrete separation. Below the critical value, the discrete model can well reproduce various physical quantities of the original SYK model, including the volume law of the ground-state entanglement, level distribution, thermody…
We systematically investigate scrambling (or delocalizing) processes of quantum information encoded in quantum many-body systems by using numerical exact diagonalization. As a measure of scrambling, we adopt the tripartite mutual information (TMI) that becomes negative when quantum information is delocalized. We clarify that scrambling is an independent property of integrability of Hamiltonians; TMI can be negative or positive for both integrable and non-integrable systems. This implies that scrambling is a separate concept from conventional quantum chaos characterized by non-integrability. Fu…
Many-body states that are invariant under particle relabelling, the permutation symmetric states, occur naturally when the system dynamics is described by symmetric processes or collective spin operators. We derive expressions for the reduced density matrix for arbitrary subsystem decomposition for these states and study properties of permutation symmetric states and their subsystems when the joint system is picked randomly and uniformly. Thus defining a new random matrix ensemble, we find the average linear entropy and von Neumann entropy which implies that random permutation symmetric states…
The delocalization or scrambling of quantum information has emerged as a central ingredient in the understanding of thermalization in isolated quantum many-body systems. Recently, significant progress has been made analytically by modeling non-integrable systems as stochastic systems, lacking a Hamiltonian picture, while honest Hamiltonian dynamics are frequently limited to small system sizes due to computational constraints. In this paper, we address this by investigating the role of conservation laws (including energy conservation) in the thermalization process from an information-theoretic …
Recent studies of out-of-time ordered thermal correlation functions (OTOC) in holographic systems and in solvable models such as the Sachdev-Ye-Kitaev (SYK) model have yielded new insights into manifestations of many-body chaos. So far the chaotic behavior has been obtained through explicit calculations in specific models. In this paper we propose a unified description of the exponential growth and ballistic butterfly spreading of OTOCs across different systems using a newly formulated "quantum hydrodynamics," which is valid at finite $\hbar$ and to all orders in derivatives. The scrambling of…
While classical chaos has been successfully characterized with consistent theories and intuitive techniques, such as with the use of Lyapunov exponents, quantum chaos is still poorly understood, as well as its relation with multi-partite entanglement and information scrambling. We consider a benchmark system, the kicked top model, which displays chaotic behaviour in the classical version, and proceed to characterize the quantum case with a thorough diagnosis of the growth of chaos and entanglement in time. As a novel tool for the characterization of quantum chaos, we introduce for this scope t…
Sachdev-Ye-Kitaev physics from the Hubbard model: A Floquet engineering approach
Universidad Complutense de Madrid（马德里康普顿斯大学）；Collège de France（法兰西公学院）；CNRS（法国国家科学研究中心）；PSL Research University（巴黎文理研究大学）；Sorbonne Université（索邦大学）；International Solvay Institutes（国际索尔维研究所）
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…
We present a systematic construction of probes into the dynamics of isospectral ensembles of Hamiltonians by the notion of Isospectral twirling, expanding the scopes and methods of ref.[1]. The relevant ensembles of Hamiltonians are those defined by salient spectral probability distributions. The Gaussian Unitary Ensembles (GUE) describes a class of quantum chaotic Hamiltonians, while spectra corresponding to the Poisson and Gaussian Diagonal Ensemble (GDE) describe non chaotic, integrable dynamics. We compute the Isospectral twirling of several classes of important quantities in the analysis …
