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arXiv:2112.13489 · Physical review. B./Physical review. B

Lindbladian dynamics of the Sachdev-Ye-Kitaev model

Anish Kulkarni, Tokiro Numasawa, Shinsei Ryu

Published Aug 22, 2022• 50 citations
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Abstract

We study the Lindbladian dynamics of the Sachdev-Ye-Kitaev (SYK) model, where the SYK model is coupled to Markovian reservoirs with jump operators that are either linear or quadratic in the Majorana fermion operators. Here, the linear jump operators are nonrandom, while the quadratic jump operators are sampled from a Gaussian distribution. In the limit of large $N$, where $N$ is the number of Majorana fermion operators, and also in the limit of large $N$ and $M$, where $M$ is the number of jump operators, the SYK Lindbladians are analytically tractable, and we obtain their stationary Green's functions, from which we can read off the decay rate. For finite $N$, we also study the distribution of the eigenvalues of the SYK Lindbladians.

DOI
10.1103/physrevb.106.075138

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We study the Lindbladian dynamics of the Sachdev-Ye-Kitaev (SYK) model, where the SYK model is coupled to Markovian reservoirs with jump operators that are either linear or quadratic in the Majorana fermion operators. Here, the linear jump operators are nonrandom, while the quadratic jump operators are sampled from a Gaussian distribution. In the limit of large NNN, where NNN is the number of Majorana fermion operators, and also in the limit of large NNN and MMM, where MMM is the number of jump operators, the SYK Lindbladians are analytically tractable, and we obtain their stationary Green's functions, from which we can read off the decay rate. For finite NNN, we also study the distribution of the eigenvalues of the SYK Lindbladians.
A bstract We investigate the growth of operator size in the Lindbladian Sachdev-Ye-Kitaev model with q -body interaction terms and linear jump terms at finite dissipation strength. We compute the operator size as well as its distribution numerically at finite q and analytically at large q . With dissipative (productive) jump terms, the size converges to a value smaller (larger) than half the number of Majorana fermions. At weak dissipation, the evolution of operator size displays a quadratic-exponential-plateau behavior. The plateau value is determined by the ratios between the coupling of the interaction and the linear jump term in the large q limit. The operator size distribution remains localized in the finite size region even at late times, contrasting with the unitary case. Moreover, we also derived the time-independent orthogonal basis for operator expansion which exhibits the operator size concentration at finite dissipation. Finally, we observe that the uncertainty relation for operator size growth is saturated at large q , leading to classical dynamics of the operator size growth with dissipation.
We study the open quantum dynamics of the Sachdev-Ye-Kitaev (SYK) model described by the Lindblad master equation, where the SYK model is coupled to Markovian reservoirs with jump operators that are either linear or quadratic in the Majorana fermion operators. Of particular interest for us is the time evolution of the dissipative form factor, which quantifies the average overlap between the initial and time-evolved density matrices as an open quantum generalization of the Loschmidt echo. We find that the dissipative form factor exhibits dynamical quantum phase transitions. We analytically demonstrate a discontinuous dynamical phase transition in the limit of a large number of fermion flavors, which is formally akin to the thermal phase transition in the two-coupled SYK model between the black-hole and wormhole phases. We also find continuous dynamical phase transitions that do not have counterparts in the two-coupled SYK model. While the phase transitions are sharp in the limit of a large number of fermion flavors, their qualitative signatures are present even for a finite number of fermion flavors, as we show numerically.
A bstract We use Krylov complexity to study operator growth in the q -body dissipative Sachdev-Ye-Kitaev (SYK) model, where the dissipation is modeled by linear and random p -body Lindblad operators. In the large q limit, we analytically establish the linear growth of two sets of coefficients for any generic jump operators. We numerically verify this by implementing the bi-Lanczos algorithm, which transforms the Lindbladian into a pure tridiagonal form. We find that the Krylov complexity saturates inversely with the dissipation strength, while the dissipative timescale grows logarithmically. This is akin to the behavior of other 𝔮-complexity measures, namely out-of-time-order correlator (OTOC) and operator size, which we also demonstrate. We connect these observations to continuous quantum measurement processes. We further investigate the pole structure of a generic auto-correlation and the high-frequency behavior of the spectral function in the presence of dissipation, thereby revealing a general principle for operator growth in dissipative quantum chaotic systems.
We present an exact solution in the large-NNN limit of the Lévy Sachdev-Ye-Kitaev (LSYK) model introduced in Ref. [1], wherein the couplings are drawn from a Lévy Stable distribution parameterized by a tail exponent μ∈[0,2]μ\in [0, 2]μ∈[0,2]. Starting from the Hamiltonian and its associated partition function, we highlight the key differences from the standard Gaussian SYK model and derive the large-NNN Schwinger-Dyson equations via a bosonic oscillator representation of the action. These equations are solved both numerically and analytically in the large-qqq and infrared limits. We subsequently analyze the chaotic properties of the model by computing the Krylov exponent from the large-qqq Green's function and extracting the Lyapunov exponent from the 444-point function. The parameter μμμ continuously interpolates between a free theory at μ=0μ= 0μ=0 and the conventional, maximally chaotic Gaussian SYK model at μ=2μ= 2μ=2, with non-maximal chaos persisting throughout the intermediate regime 0<μ<20 < μ< 20<μ<2. Thermodynamic quantities, including the entropy, free energy, average energy, and specific heat capacity, are computed and compared with their Gaussian SYK counterparts. The interpretations of the thermodynamics are discussed with respect to the holographic dual and non-Fermi liquid theory. Finally, we discuss an alternative representation of the LSYK model based on a distinct decomposition of the Lévy Stable distribution, which establishes a non-trivial connection to Gaussian SYK, and provide supporting analytical and numerical results in the appendices.
We study stability of the Sachdev-Ye-Kitaev (SYK_{4}) model with a large but finite number of fermions N with respect to a perturbation, quadratic in fermionic operators. We develop analytic perturbation theory in the amplitude of the SYK_{2} perturbation and demonstrate stability of the SYK_{4} infrared asymptotic behavior characterized by a Green function G(τ)∝1/τ^{3/2}, with respect to weak perturbation. This result is supported by exact numerical diagonalization. Our results open the way to build a theory of non-Fermi-liquid states of strongly interacting fermions.
We propose the Sachdev-Ye-Kitaev Lindbladian as a paradigmatic solvable model of dissipative many-body quantum chaos.It describes N strongly coupled Majorana fermions with random all-to-all interactions, with unitary evolution given by a quartic Hamiltonian and the coupling to the environment described by M quadratic jump operators, rendering the full Lindbladian quartic in the Majorana operators.Analytical progress is possible by developing a dynamical mean-field theory for the Liouvillian time evolution on the Keldysh contour.By disorder-averaging the interactions, we derive an (exact) effective action for two collective fields (Green's function and self-energy).In the large-N, large-M limit, we obtain the saddle-point equations satisfied by the collective fields, which determine the typical timescales of the dissipative evolution, particularly the spectral gap that rules the relaxation of the system to its steady state.We solve the saddle-point equations numerically and find that, for strong or intermediate dissipation, the system relaxes exponentially, with a spectral gap that can be computed analytically, while for weak dissipation, there are oscillatory corrections to the exponential relaxation.In this letter, we illustrate the feasibility of analytical calculations in strongly correlated dissipative quantum matter.