Sachdev-Ye-Kitaev Model Research Articles

Quantum chaos on edge

Recently, the physics of many-body quantum chaotic systems close to their ground states has come under intensified scrutiny. Such studies are motivated by the emergence of model systems exhibiting chaotic fluctuations throughout the entire spectrum [the Sachdev-Ye-Kitaev (SYK) model being a renowned representative] as well as by the physics of holographic principles, which likewise unfold close to ground states. Interpreting the edge of the spectrum as a quantum critical point, here we combine a wide range of analytical and numerical methods to the identification and comprehensive description of two different universality classes: the near edge physics of “sparse” and the near edge of “dense” chaotic systems. The distinction lies in the ratio between the number of a system's random parameters and its Hilbert space dimension, which is exponentially small or algebraically small in the sparse and dense case, respectively. Notable representatives of the two classes are generic chaotic many-body models (sparse) and invariant random matrix ensembles or chaotic gravitational systems (dense). While the two families share identical spectral correlations at energy scales comparable to the level spacing, the density of states and its fluctuations near the edge are different. Considering the SYK model as a representative of the sparse class, we apply a combination of field theory and exact diagonalization to a detailed discussion of its edge spectrum. Conversely, Jackiw-Teitelboim gravity is our reference model for the dense class, where an analysis of the gravitational path integral and random matrix theory reveal universal differences to the sparse class, whose implications for the construction of holographic principles we discuss. Published by the American Physical Society 2024

Sparsity-Independent Lyapunov Exponent in the Sachdev-Ye-Kitaev Model.

The saturation of a recently proposed universal bound on the Lyapunov exponent has been conjectured to signal the existence of a gravity dual. This saturation occurs in the low-temperature limit of the dense Sachdev-Ye-Kitaev (SYK) model, N Majorana fermions with q body (q>2) infinite-range interactions. We calculate certain out-of-time-order correlators (OTOCs) for N≤64 fermions for a highly sparse SYK model and find no significant dependence of the Lyapunov exponent on sparsity up to near the percolation limit where the Hamiltonian breaks up into blocks. This provides strong support to the saturation of the Lyapunov exponent in the low-temperature limit of the sparse SYK. A key ingredient to reaching N=64 is the development of a novel quantum spin model simulation library that implements highly optimized matrix-free Krylov subspace methods on graphical processing units. This leads to a significantly lower simulation time as well as vastly reduced memory usage over previous approaches, while using modest computational resources. Strong sparsity-driven statistical fluctuations require both the use of a much larger number of disorder realizations with respect to the dense limit and a careful finite size scaling analysis. The saturation of the bound in the sparse SYK points to the existence of a gravity analog that would enlarge substantially the number of field theories with this feature.

Non-commutative probability insights into the double-scaling limit SYK model with constant perturbations: moments, cumulants and q-independence

Extending the results of Wu (2022 J. Phys. A 55 415207), we study the double-scaling limit Sachdev–Ye–Kitaev model with an additional diagonal matrix with a fixed number c of nonzero constant entries θ. This constant diagonal term can be rewritten in terms of Majorana fermion products. Its specific formula depends on the value of c. We find exact expressions for the moments of this model. More importantly, by proposing a moment-cumulant relation, we reinterpret the effect of introducing a constant term in the context of non-commutative probability theory. This gives rise to a q~ dependent mixture of independences within the moment formula. The parameter q~ , derived from the q-Ornstein–Uhlenbeck (q-OU) process, controls this transformation. It interpolates between classical independence ( q~=1 ) and Boolean independence ( q~=0 ). The underlying combinatorial structures of this model provide the non-commutative probability connections. Additionally, we explore the potential relation between these connections and their gravitational path integral counterparts.

Bosonic near-CFT1 models from Fock-space fluxes

We construct a family of near-CFT1 models with a conserved U(1) charge, whose basic degrees of freedom are canonical bosons. The Sachdev-Ye-Kitaev (SYK) model — the first microscopic model that realizes the near-CFT1 dynamics — is based on random p-local interactions among fermions. However, a bosonic near-CFT1 model has remained elusive in the p-local approach because such constructions generally suffer from unwanted orderings at low temperatures. Our construction is based on a recent insight that near-CFT1 dynamics can quite generally arise if we place a large amount of random fluxes in a many-body Fock space and p-locality is not essential. All such models are essentially solved by chord diagrams regardless of the nature of the underlying degrees of freedom. We further argue that such bosonic models do not suffer from energetic instablities or unwanted low-temperature orderings. For comparison we also consider a second class of charge-conserving models which are based on qubits. The thermodynamic scalings of these models are very similar to those of the double-scaled complex SYK model but are free of certain singularities the latter suffers from. We also show the level statistics of both models are described by random matrix theory universality down to very low energies.

NoRA: A Tensor Network Ansatz for Volume-Law Entangled Equilibrium States of Highly Connected Hamiltonians

Motivated by the ground state structure of quantum models with all-to-all interactions such as mean-field quantum spin glass models and the Sachdev-Ye-Kitaev (SYK) model, we propose a tensor network architecture which can accomodate volume law entanglement and a large ground state degeneracy. We call this architecture the non-local renormalization ansatz (NoRA) because it can be viewed as a generalization of MERA, DMERA, and branching MERA networks with the constraints of spatial locality removed. We argue that the architecture is potentially expressive enough to capture the entanglement and complexity of the ground space of the SYK model, thus making it a suitable variational ansatz, but we leave a detailed study of SYK to future work. We further explore the architecture in the special case in which the tensors are random Clifford gates. Here the architecture can be viewed as the encoding map of a random stabilizer code. We introduce a family of codes inspired by the SYK model which can be chosen to have constant rate and linear distance at the cost of some high weight stabilizers. We also comment on potential similarities between this code family and the approximate code formed from the SYK ground space.

Strange metal to insulator transitions in the lowest Landau level

We study the microscopic model of electrons in the partially-filled lowest Landau level interacting via the Coulomb potential by the diagrammatic theory within the GW approximation. In a wide range of filling fractions and temperatures, we find a homogeneous non-Fermi liquid (nFL) state similar to that found in the Sachdev-Ye-Kitaev (SYK) model, with logarithmic corrections to the anomalous dimension. In addition, the phase diagram is qualitatively similar to that of SYK: a first-order transition terminating at a critical end-point separates the nFL phase from a band insulator that corresponds to the fully filled Landau level. This critical point, as well as that of the SYK model—whose critical exponents we determine more precisely—are shown to both belong to the Van der Waals universality class. The possibility of a charge density wave (CDW) instability is also investigated, and we find the homogeneous nFL state to extend down to the ground state for fillings 0.2≲ν≲0.8, while a CDW appears outside this range of fillings at sufficiently low temperatures. Our results suggest that the SYK-like nFL state should be a generic feature of the partially filled lowest Landau level at intermediate temperatures. Published by the American Physical Society 2024

A symmetry algebra in double-scaled SYK

The double-scaled limit of the Sachdev-Ye-Kitaev (SYK) model takes the number of fermions and their interaction number to infinity in a coordinated way. In this limit, two entangled copies of the SYK model have a bulk description of sorts known as the “chord Hilbert space”. We analyze a symmetry algebra acting on this Hilbert space, generated by the two Hamiltonians together with a two-sided operator known as the chord number. This algebra is a deformation of the JT gravitational algebra, and it contains a subalgebra that is a deformation of the \mathfrak{sl}_2𝔰𝔩2 near-horizon symmetries. The subalgebra has finite-dimensional unitary representations corresponding to matter moving around in a discrete Einstein-Rosen bridge. In a semiclassical limit the discreteness disappears and the subalgebra simplifies to \mathfrak{sl}_2𝔰𝔩2, but with a non-standard action on the boundary time coordinate. One can make the action of \mathfrak{sl}_2𝔰𝔩2 algebra more standard at the cost of extending the boundary circle to include some “fake” portions. Such fake portions also accommodate certain subtle states that survive the semi-classical limit, despite oscillating on the scale of discreteness. We discuss applications of this algebra, including sub-maximal chaos, the traversable wormhole protocol, and a two-sided OPE.

The Petz (lite) recovery map for the scrambling channel

Abstract We study properties of the Petz recovery map in chaotic systems, such as the Hayden–Preskill setup for evaporating black holes and the Sachdev–Ye–Kitaev (SYK) model. Since these systems exhibit the phenomenon called scrambling, we expect that the expression of the recovery channel $\mathcal {R}$ gets simplified, given by just the adjoint $\mathcal {N}^{\dagger }$ of the original channel $\mathcal {N}$ which defines the time evolution of the states in the code subspace embedded into the physical Hilbert space. We check this phenomenon in two examples. The first one is the Hayden–Preskill setup described by Haar random unitaries. We compute the relative entropy $S(\mathcal {R}\left[\mathcal {N}[\rho ]\right] ||\rho )$ and show that it vanishes when the decoupling is archived. We further show that the simplified recovery map is equivalent to the protocol proposed by Yoshida and Kitaev. The second example is the SYK model where the 2D code subspace is defined by an insertion of a fermionic operator, and the system is evolved by the SYK Hamiltonian. We check the recovery phenomenon by relating some matrix elements of an output density matrix $\langle{T}|\mathcal {R}[\mathcal {N}[\rho ]]|{T^{\prime }}\rangle$ to Rényi-two modular flowed correlators, and show that they coincide with the elements for the input density matrix with small error after twice the scrambling time.

Random coupling model of turbulence as a classical Sachdev-Ye-Kitaev model.

We point out that a classical analog of the Sachdev-Ye-Kitaev (SYK) model, a solvable model of quantum many-body chaos, was studied long ago in the turbulence literature. Motivated by the Navier-Stokes equationin the turbulent regime and the nonlinear Schrödinger equationdescribing plasma turbulence, in which there is mixing between many different modes, the random coupling model has a Gaussian-random coupling between any four of a large number N of modes. The model was solved in the 1960s, before the introduction of large-N path-integral techniques, using a method referred to as the direct interaction approximation. We use the path integral to derive the effective action for the model. The large-N saddle gives an integral equationfor the two-point function, which is very similar to the corresponding equationin the SYK model. The connection between the SYK model and the random coupling model may, on the one hand, provide new physical contexts in which to realize the SYK model and, on the other hand, suggest new models of turbulence and techniques for studying them.

Krylov complexity in large q and double-scaled SYK model

Considering the large q expansion of the Sachdev-Ye-Kitaev (SYK) model in the two-stage limit, we compute the Lanczos coefficients, Krylov complexity, and the higher Krylov cumulants in subleading order, along with the t/q effects. The Krylov complexity naturally describes the “size” of the distribution while the higher cumulants encode richer information. We further consider the double-scaled limit of SYKq at infinite temperature, where q ~ sqrt{N} . In such a limit, we find that the scrambling time shrinks to zero, and the Lanczos coefficients diverge. The growth of Krylov complexity appears to be “hyperfast”, which is previously conjectured to be associated with scrambling in de Sitter space.

Remarks on higher Schwarzians

The Schwarzian derivative has recently received renewed attention in connection with the study of the Sachdev–Ye–Kitaev model. In mathematics literature, various higher order generalizations of the Schwarzian derivative are known due to Aharonov, Bertilsson, and Schippers. Physical applications of the higher Schwarzian derivatives have not yet been discussed in any detail. In this work, we link Bertilsson's variant to the ℓ–conformal Galilei group, as well as discuss some of its interesting peculiarities. These include a recurrence relation, which allows one to construct the higher Schwarzians iteratively, a composition law, and symmetry transformations.

Keldysh wormholes and anomalous relaxation in the dissipative Sachdev-Ye-Kitaev model

We study the out-of-equilibrium dynamics of a Sachdev-Ye-Kitaev (SYK) model, $N$ fermions with a $q$-body interaction of infinite range, coupled to a Markovian environment. Close to the infinite-temperature steady state, the real-time Lindbladian dynamics of this system is identical to the near-zero-temperature dynamics in Euclidean time of a two-site non-Hermitian SYK with intersite coupling whose gravity dual has been recently related to wormhole configurations. We show that the saddle-point equations in the real-time formulation are identical to those in Euclidean time. Indeed, an explicit calculation of Green's functions at low temperature, numerical for $q = 4$ and analytical for $q = 2$ and large $q$, illustrates this equivalence. Only for very strong coupling does the decay rate approach the linear dependence on the coupling characteristic of a dissipation-driven approach to the steady state. For $q > 2$, we identify a potential gravity dual of the real-time dissipative SYK model: a double-trumpet configuration in a near-de Sitter space in two dimensions with matter. This configuration, which we term a Keldysh wormhole, is responsible for a finite decay rate even in the absence of coupling to the environment.

Non-maximal chaos in some Sachdev-Ye-Kitaev-like models

We study the chaos exponent of some variants of the Sachdev-Ye-Kitaev (SYK) model, namely, the mathcal{N} = 1 supersymmetry (SUSY)-SYK model and its sibling, the (N|M)-SYK model which is not supersymmetric, for arbitrary interaction strength. We find that for large q the chaos exponent of these variants, as well as the SYK and the mathcal{N} = 2 SUSY-SYK model, all follow a single-parameter scaling law. By quantitative arguments we further make a conjecture, i.e. that the found scaling law might hold for general one-dimensional (1D) SYK-like models with large q. This points out a universal route from maximal chaos towards completely regular or integrable motion in the SYK model and its 1D variants.

Free energy subadditivity for symmetric random Hamiltonians

We consider a random Hamiltonian H:Σ→R defined on a compact space Σ that admits a transitive action by a compact group G. When the law of H is G-invariant, we show its expected free energy relative to the unique G-invariant probability measure on Σ, which obeys a subadditivity property in the law of H itself. The bound is often tight for weak disorder and relates free energies at different temperatures when H is a Gaussian process. Many examples are discussed, including branching random walks, several spin glasses, random constraint satisfaction problems, and the random field Ising model. We also provide a generalization to quantum Hamiltonians with applications to the quantum Sherrington–Kirkpatrick and Sachdev–Ye–Kitaev models.

Generalized black hole entropy in two dimensions

The Bekenstein–Hawking entropy of a black hole is proportional to its horizon area, hence in [Formula: see text] spacetime dimensions it is constant because the horizon degenerates into two points. This fact is consistent with Einstein’s gravity becoming topological in two dimensions. In [Formula: see text] gravity, which is non-trivial even in [Formula: see text], we find that the entropy is constant, as for Bekenstein–Hawking. As shown in Europhys. Lett. 139(6) (2022) 69001, arXiv: 2208.10146, two-dimensional [Formula: see text] gravity is equivalent to Jackiw–Teitelboim gravity, in turn, equivalent to the Sachdev–Ye–Kitaev model where the entropy becomes constant in the large [Formula: see text] limit. Several recently proposed entropies are functions of the Bekenstein–Hawking entropy and become constant in [Formula: see text], but in two-dimensional dilaton gravity entropies are not always constant. We study general dilaton gravity and obtain arbitrary static black hole solutions for which the non-constant entropies depend on the mass, horizon radius, or Hawking temperature, and constitute new proposals for a generalized entropy.

Absence of operator growth for average equal-time observables in charge-conserved sectors of the Sachdev-Ye-Kitaev model

Quantum scrambling plays an important role in understanding thermalization in closed quantum systems. By this effect, quantum information spreads throughout the system and becomes hidden in the form of non-local correlations. Alternatively, it can be described in terms of the increase in complexity and spatial support of operators in the Heisenberg picture, a phenomenon known as operator growth. In this work, we study the disordered fully-connected Sachdev-Ye-Kitaev (SYK) model, and we demonstrate that scrambling is absent for disorder-averaged expectation values of observables. In detail, we adopt a formalism typical of open quantum systems to show that, on average and within charge-conserved sectors, operators evolve in a relatively simple way which is governed by their operator size. This feature only affects single-time correlation functions, and in particular it does not hold for out-of-time-order correlators, which are well-known to show scrambling behavior. Making use of these findings, we develop a cumulant expansion approach to approximate the evolution of equal-time observables. We employ this scheme to obtain analytic results that apply to arbitrary system size, and we benchmark its effectiveness by exact numerics. Our findings shed light on the structure of the dynamics of observables in the SYK model, and provide an approximate numerical description that overcomes the limitation to small systems of standard methods.

Shared universality of charged black holes and the complex large- q Sachdev-Ye-Kitaev model

We investigate the charged $q/2$-body interacting Sachdev-Ye-Kitaev (SYK) model in the grand-canonical ensemble. By treating $q$ as a large parameter, we are able to analytically study its phase diagram. By varying the chemical potential or temperature, we find that the system undergoes a phase transition between low and high entropies, in the maximally chaotic regime. A similar transition in entropy is seen in charged AdS black holes transitioning between a large and small event horizon. Approaching zero temperature, we find a first-order chaotic-to-non-chaotic quantum phase transition, where the finite extensive entropy drops to zero. This again has a gravitational analogue -- the Hawking-Page (HP) transition between a large black hole and thermal radiation. An analytical study of the critical phenomena associated with the continuous phase transition provides us with mean field van der Waals critical and effective exponents. We find that all analogous power laws are shared with several charged AdS black hole phase transitions. Together, these findings indicate a connection between the charged $q\to\infty$ SYK model and black holes.

Holographic measurement and quantum teleportation in the SYK thermofield double

According to holography, entanglement is the building block of spacetime; therefore, drastic changes of entanglement will lead to interesting transitions in the dual spacetime. In this paper, we study the effect of projective measurements on the Sachdev-Ye-Kitaev (SYK) model’s thermofield double state, dual to an eternal black hole in Jackiw-Teitelboim (JT) gravity. We calculate the (Renyi-2) mutual information between the two copies of the SYK model upon projective measurement of a subset of fermions in one copy. We propose a dual JT gravity model that can account for the change of entanglement due to measurement, and observe an entanglement wedge phase transition in the von Neumann entropy. The entanglement wedge for the unmeasured side changes from the region outside the horizon to include the entire time reversal invariant slice of the two-sided geometry as the number of measured Majorana fermions increases. Therefore, after the transition, the bulk information stored in the measured subsystem is not entirely lost upon projection in one copy of the SYK model, but rather teleported to the other copy. We further propose a decoding protocol to elucidate the teleportation interpretation, and connect our analysis to the physics of traversable wormholes.

Binary-coupling sparse Sachdev-Ye-Kitaev model: An improved model of quantum chaos and holography

The sparse version of the Sachdev-Ye-Kitaev (SYK) model reproduces essential features of the original SYK model while reducing the number of disorder parameters. In this paper, we propose a further simplification of the model which we call the binary-coupling sparse SYK model. We set the nonzero couplings to be $\pm 1$, rather than being sampled from a continuous distribution such as Gaussian. Remarkably, this simplification turns out to be an improvement: the binary-coupling model exhibits strong correlations in the spectrum, which is the important feature of the original SYK model that leads to the quick onset of the random-matrix universality, more efficiently in terms of the number of nonzero terms. This model is better suited for analog or digital quantum simulations of quantum chaotic behavior and holographic metals due to its simplicity and scaling properties.

A proposal to extract and enhance four-Majorana interactions in hybrid nanowires

We simulate the smallest building block of the Sachdev-Ye-Kitaev (SYK) model, a system of four interacting Majorana modes. We propose a 1D Kitaev chain that has been split into three segments, i.e., two topological segments separated by a non-topological segment in the middle, hosting four Majorana Zero Modes at the ends of the topological segments. We add a non-local interaction term to this Hamiltonian which produces both bilinear (two-body) interactions and a quartic (four-body) interaction between the Majorana modes. We further tune the parameters in the Hamiltonian to reach the regime with a finite quartic interaction strength and close to zero bilinear interaction strength, as required by the SYK model. To achieve this, we map the Hamiltonian from Majorana basis to a complex fermion basis, and extract the interaction strengths using a method of characterization of low-lying energy levels and then finding the differences in energies between odd and even parity levels. We show that the interaction strengths can be tuned using two methods - (i) an approximate method of tuning overlapping Majorana wave functions (without non-local interactions) to a zero energy point followed by addition of a non-local interaction, and (ii) a direct parameter space optimization method using a genetic algorithm. We propose that this model could be further extended to more Majorana modes, and show a 6-Majorana model as an example. Since eigenspectral characterization of one-dimensional nanowire devices can be done via tunneling spectroscopy in quantum transport measurements, this study could be performed in experiment.