We study the complex Sachdev-Ye-Kitaev (cSYK) numerically and investigate thermodynamic behavior of cSYK model across varying chemical potentials. We discover that the cSYK model remarkably mirrors the first-order phase transition seen in the van der Waals-Maxwell system, culminating at a non-mean-field critical point with distinctively different critical exponents. We analyze in detail the similarity between the van der Waals phase transition and the cSYK model, and further explore the mechanism by which the chemical potential drives the phase transition in the system. Exact diagonalization for the cSYK model reveals the significant impact of chemical potential on energy distribution, with observable energy gaps in the gapped phase. Quantum chaos indicators, including spectral form factors, suggest more stable energy states in the neutral case. Real-time dynamics, analyzed via analytical continuation of Schwinger-Dyson equations, show rapid decay in the gapless phase and prolonged oscillation lifetimes in the gapped regime. Spectral functions imply a shift from a continuous to a discrete energy level distribution, emphasizing the critical role of chemical potential in shaping spectral properties.
Abstract We study the complex Sachdev–Ye–Kitaev (cSYK) model numerically and investigate the thermodynamic behavior of the cSYK model across varying chemical potentials. We discover that the cSYK model remarkably mirrors the first-order phase transition seen in the van der Waals–Maxwell system, culminating at a non-mean-field critical point with distinctively different critical exponents. We analyze in detail the similarity between the van der Waals phase transition and the cSYK model, and further explore the mechanism by which the chemical potential drives the phase transition in the system. Exact diagonalization at finite N resolves the conserved U(1) charge sectors, showing that the chemical potential reshapes the density of states. Within each charge sector we find a strong level repulsion, while mixing sectors shows a Poisson distribution. The normalized spectral form factor displays a clear dip-ramp-plateau at low temperature for the neutral case, whereas for non-vanishing chemical potential the ramp is weakened by finite N reweighting of small edge sectors and only becomes visible at relatively high temperatures. Real-time dynamics, analyzed via analytical continuation of Schwinger–Dyson equations, show rapid decay in the gapless phase and prolonged oscillation lifetimes in the gapped regime. Spectral functions imply a shift from a continuous to a discrete energy level distribution, emphasizing the critical role of chemical potential in shaping spectral properties.
We analyze the relation between quantum chaotic behavior and phase transitions of the Yukawa--Sachdev-Ye-Kitaev model as a function of filling and temperature, which describes random Yukawa interactions between
N complex fermions and
M bosons in zero spatial dimensions for both the non-Fermi liquid and insulating states at finite temperature and chemical potential. We solve the ladder equations for the out-of-time-order correlator (OTOC) for both bosons and fermions. Despite the appearance of the chemical potential in the Hamiltonian, which explicitly introduces an additional energy scale, the OTOCs for the fermions and bosons in the non-Fermi liquid state turn out to be unaffected, and the Lyapunov exponents that diagnose chaos remain maximal. As the chemical potential increases, the system is known to experience a first-order transition from a critical phase to a gapped insulating phase. We postulate that the boundary of the region in parameter space where each phase is (meta)stable coincides with the curve on which the Lyapunov exponent is maximal. By calculating the exponent in the insulating phase and comparing to numerical results on the boundaries of stability, we show that this is plausible.
Abstract It has been known that the large-q complex Sachdev-Ye-Kitaev (SYK) model falls under the same universality class as that of van der Waals (mean-field) and saturates the Maldacena-Shenker-Stanford (MSS) bound, both features shared by various black holes. This makes the SYK model a useful tool in probing the fundamental nature of quantum chaos and holographic duality. This work establishes the robustness of this shared universality class and chaotic properties for SYK-like models by extending to a system of coupled large-q complex SYK models of different orders. We provide a detailed derivation of thermodynamic properties, specifically the critical exponents for an observed phase transition, as well as dynamical properties, in particular the Lyapunov exponent, via the out-of-time correlator calculations. Our analysis reveals that, despite the introduction of an additional scaling parameter through interaction strength ratios, the system undergoes a continuous phase transition at low temperatures, similar to that of the single SYK model. The critical exponents align with the Landau–Ginzburg (mean-field) universality class, shared with van der Waals gases and various AdS black holes. Furthermore, we demonstrate that the coupled SYK system remains maximally chaotic in the large-q limit at low temperatures, adhering to the MSS bound, a feature consistent with the single SYK model. These findings establish robustness and open avenues for broader inquiries into the universality and chaos in complex quantum systems. We provide a detailed outlook for future work by considering the ‘very’ low-temperature regime, where we discuss relations with the Hawking–Page phase transition observed in the holographic dual black holes. We present preliminary calculations and discuss the possible follow-ups that might be taken to make the connection robust.
We present an exact solution in the large-
N 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]. Starting from the Hamiltonian and its associated partition function, we highlight the key differences from the standard Gaussian SYK model and derive the large-
N Schwinger-Dyson equations via a bosonic oscillator representation of the action. These equations are solved both numerically and analytically in the large-
q and infrared limits. We subsequently analyze the chaotic properties of the model by computing the Krylov exponent from the large-
q Green's function and extracting the Lyapunov exponent from the
4-point function. The parameter
μ continuously interpolates between a free theory at
μ=0 and the conventional, maximally chaotic Gaussian SYK model at
μ=2, with non-maximal chaos persisting throughout the intermediate regime
0<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.
A bstract It is known that, for a range of chemical potentials, the infrared behavior of the complex Sachdev-Ye-Kitaev (cSYK) model is governed by a 1D Nearly Conformal Field Theory (NCFT 1 ), thereby realizing a continuous line of NCFTs. A finite chemical potential μ introduces an asymmetry parameter
E E into the cSYK fermion two-point function in the conformal limit. In this work, we compute the cSYK four-point function in the conformal limit for an arbitrary value of
E E at leading order in 1/ N . We show that the result is fully consistent with the NCFT 1 structure of the cSYK model and use it to extract the structure constants for correlation functions of two complex fermions with bilinear operators.
We study a disorder-free variant of the Sachdev-Ye-Kitaev (SYK) model with dissipation within the Gorini-Kossakowski-Sudarshan-Lindblad formalism. By utilizing the integrability of the clean SYK model, we derive an exact solution in a spectrum-resolved form, i.e., the eigenvalues and corresponding projection superoperators of the Liouvillian for arbitrary system size
N. We determine the scaling of the gap that governs the long-time decay of the two-point correlation functions. Importantly, the gap does not vanish in the dissipationless limit when the thermodynamic limit is taken first, despite the integrability of the model. This phenomenon, known as anomalous relaxation, suggests a possible connection with chaotic dynamics and quantum Ruelle-Pollicott resonances. We also find several spectral features, such as transitions in the Liouvillian spectrum from complex to real eigenvalues with increasing dissipation strength, as well as the convergence of the dissipative form factor to the spectral form factor in the dissipationless limit. These findings indicate that the present model offers a useful platform for exploring nontrivial open dynamics of many-body quantum systems.