Exact spectrum and anomalous relaxation in the open disorder-free Sachdev-Ye-Kitaev system

Abstract: 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.

• The paper establishes that the open, disorder-free SYK model exhibits finite Liouvillian gaps reflecting anomalous relaxation.
• It employs an analytic solution of the GKSL master equation, leveraging integrability and operator growth in a clean four-body Majorana Hamiltonian.
• The study reveals that noncommutative thermodynamic and dissipationless limits yield robust relaxation dynamics even in integrable systems.

Exact Spectrum and Anomalous Relaxation in the Open Disorder-Free SYK Model

The paper "Exact spectrum and anomalous relaxation in the open disorder-free Sachdev-Ye-Kitaev system" (2606.08079) investigates the interplay between integrability, dissipation, and operator relaxation in a non-random variant of the SYK model, employing analytic methods that yield a complete spectral resolution of the associated Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) quantum master equation. This study elucidates the profound emergence of anomalous relaxation—characterized by a nonvanishing Liouvillian gap in the thermodynamic, dissipationless limit—within an integrable, disorder-free context, extending the paradigm of quantum chaotic relaxation into new theoretical territory.

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Model and Analytic Solution Construction

The considered Hamiltonian is the clean (uniform coupling) SYK model with four-body Majorana terms:

$$H_4 = -\sum_{1 \leq i<j<k<l \leq N} \gamma_i \gamma_j \gamma_k \gamma_l$$

with Hermitian Majorana fermions $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$. This system is integrable for arbitrary $N$ and does not, in the closed limit, exhibit conventional quantum chaos. The dissipative dynamics are governed by a GKSL equation with jump operators $L_j = \sqrt{\Gamma} \gamma_j$:

$$\dot{\rho} = \mathcal{L}[\rho] = -[H_4, \rho] + \sum_{j=1}^N \left( L_j \rho L_j^\dagger - \frac{1}{2} \{L_j^\dagger L_j, \rho\} \right)$$

The analytic solution exploits the integrability and symmetry structure of the model. By mapping the problem to a basis of complex fermions diagonalizing the two-body Hamiltonian $H_2$, and leveraging commutation properties and superoperator structure, the authors explicitly derive the full Liouvillian spectrum and provide explicit forms for the projection superoperators. This yields spectral decompositions valid for arbitrary system size $N$, explicitly tracking the evolution of density operators.

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Liouvillian Spectrum Structure and Exceptional Points

A striking feature of the spectrum is the transition from complex to real eigenvalues as $\Gamma$ increases, signaling the appearance of non-Hermitian "exceptional points" in parameter space—a hallmark of many dissipative or open quantum systems.

Figure 1: Eigenvalues of the Liouvillian $\mathcal{L}$ for $N=16$ at various $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$0; increasing dissipation causes the spectrum to transition from complex (oscillatory) to real (purely relaxational) bands.

At small $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$1, the eigenvalues cluster along lines with constant real parts, inherited from the symmetric structure and integrability of $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$2. As dissipation is strengthened, the spectrum collapses onto the real axis, and the system's relaxation dynamics shift from underdamped to overdamped. The spectral gap—the real part of the eigenvalue with the smallest magnitude—ultimately dictates the asymptotic decay rate for typical two-point functions.

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Dissipative Form Factor and Dynamical Regimes

The dissipative form factor (DFF), given by $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$3, generalizes the spectral form factor (SFF) of closed models to open quantum dynamics. Calculations up to large $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$4 demonstrate regime structure in how the DFF approaches the SFF as $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$5, with different scaling behaviors in time manifesting for various $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$6 and $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$7.

Figure 2: DFF at $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$8 for multiple $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$9 values, showing (a) exponential decay at large $N$0, with DFF converging to the SFF at $N$1 as $N$2, (b) crossover among constant, $N$3, and $N$4 scaling regimes seen in the dissipationless case.

The form factor decays exponentially at early times for strong dissipation, consistent with $N$5. In contrast, the weak dissipation regime allows the DFF to closely track the SFF at various time scales, with step-wise convergence that depends on both $N$6 and the underlying operator growth in the system—a feature that is different than in disordered SYK models with dissipation.

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Anomalous Relaxation and Liouvillian Gap Scaling

A central result is the analytic demonstration that, when the thermodynamic limit is taken before the dissipationless limit, the Liouvillian gap—determining relaxation rate—remains finite. In contrast, if the order of limits is reversed, the gap closes, consistent with unitary, integrable dynamics. This noncommutativity is captured explicitly:

$N$7

This anomalous relaxation, usually associated with quantum chaotic models and non-Hermitian Ruelle-Pollicott resonances, is here shown analytically in a nonchaotic, disorder-free, integrable system—revealing that rapid operator growth, not spectral chaos per se, can underpin robust relaxation dynamics.

Figure 3: (a) Steady-state autocorrelation for $N$8 and $N$9 shows single-exponential decay; (b) Fitted decay rate $L_j = \sqrt{\Gamma} \gamma_j$0 as a function of $L_j = \sqrt{\Gamma} \gamma_j$1 for various $L_j = \sqrt{\Gamma} \gamma_j$2 is consistent with a nonzero lower bound in the thermodynamic limit as $L_j = \sqrt{\Gamma} \gamma_j$3.

Figure 4: Analytic computation of the asymptotic decay rate $L_j = \sqrt{\Gamma} \gamma_j$4 versus $L_j = \sqrt{\Gamma} \gamma_j$5 at $L_j = \sqrt{\Gamma} \gamma_j$6, illustrating the approach to the finite gap value.

Figure 5: Asymptotic decay rate $L_j = \sqrt{\Gamma} \gamma_j$7 versus $L_j = \sqrt{\Gamma} \gamma_j$8 for $L_j = \sqrt{\Gamma} \gamma_j$9, with the thermodynamic limit yielding $$\dot{\rho} = \mathcal{L}[\rho] = -[H_4, \rho] + \sum_{j=1}^N \left( L_j \rho L_j^\dagger - \frac{1}{2} \{L_j^\dagger L_j, \rho\} \right)$$0 at $$\dot{\rho} = \mathcal{L}[\rho] = -[H_4, \rho] + \sum_{j=1}^N \left( L_j \rho L_j^\dagger - \frac{1}{2} \{L_j^\dagger L_j, \rho\} \right)$$1.

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Implications and Outlook

This work provides the first full analytic resolution of dissipative dynamics in an integrable, disorder-free SYK model, establishing that the anomalous persistence of relaxation—long linked to quantum chaos—can emerge from the structure and operator growth dynamics of highly symmetric, nonrandom systems. The noncommutativity of limits and the explicit spectral resolution offer a template for understanding open dynamics far beyond random matrix or strongly chaotic models.

Practically, these results suggest new ways to engineer robust relaxation and information scrambling in synthetic quantum matter by tuning dissipation rather than randomness or chaos. The exceptional-point structure, and the closed-form analytic framework developed here, allow systematic exploration of the interplay between integrability, open system effects, and quantum many-body relaxation. Extending these methods to more general classes of nonrandom, interacting systems—including supersymmetric and higher $$\dot{\rho} = \mathcal{L}[\rho] = -[H_4, \rho] + \sum_{j=1}^N \left( L_j \rho L_j^\dagger - \frac{1}{2} \{L_j^\dagger L_j, \rho\} \right)$$2-body variants—in open environments, is a promising direction for future research.

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Conclusion

The analytic solution of the open, clean SYK model's Liouvillian spectrum demonstrates that anomalous relaxation is not exclusive to disordered or chaotic systems but can appear in integrable environments displaying rapid operator growth. These findings deepen the theoretical foundations of open quantum dynamics and illuminate the fundamental mechanisms behind dissipative relaxation in many-body settings. The techniques and structures uncovered in this study provide a foundation for an expanded understanding of nontrivial collective behavior in both theoretical and experimental open quantum systems.