Schwinger-Keldysh field theory for operator Rényi entropy and entanglement growth in non-interacting systems with sub-ballistic transports

1. Problem Statement

The paper addresses the challenge of understanding the relationship between quantum information propagation (operator growth and entanglement generation) and transport phenomena (diffusion, ballistic transport, localization) in quantum many-body systems.

• Context: In generic interacting thermalizing systems with diffusive transport, state entanglement entropy ($S_A$) typically grows ballistically ($\sim t$), while higher-order R´enyi entropies ($S^{(n>1)}_A$) grow diffusively ($\sim \sqrt{t}$). This behavior is linked to the decay of Schmidt values (eigenvalues of the reduced density matrix), where the maximum eigenvalue decays diffusively while typical ones decay ballistically.
• Gap: The behavior of these quantities in non-interacting systems exhibiting sub-ballistic transport (e.g., anomalous diffusion, sub-diffusion) or localization (e.g., in quasiperiodic or disordered systems) is less understood analytically. Furthermore, standard measures like Out-of-Time-Order Correlators (OTOCs) or state entanglement entropies often lack explicit spatial resolution or are state-dependent, making it difficult to isolate pure transport characteristics from initial state properties.
• Goal: The authors aim to develop a unified theoretical framework to compute subsystem operator R´enyi entropy and state entanglement entropy in non-interacting systems, specifically to diagnose transport regimes (ballistic, diffusive, sub-diffusive, localized) and understand the dynamics of operator growth.

2. Methodology

A. Definition of Subsystem Operator R´enyi Entropy

The authors define a state-independent measure of operator growth: the subsystem operator R´enyi entropy $S^{(2)}_{OA}(t)$.

• Construction: For a local operator $O$ (e.g., particle number at site $l$), they evolve it in time: $O(t) = e^{iHt}Oe^{-iHt}$.
• Partial Trace: They define a reduced operator $\rho_{OA}(t) = \text{Tr}_B[O(t)] / Z_O$, where $B$ is the complement of subsystem $A$.
• Entropy: The entropy is defined as $e^{-S^{(2)}_{OA}(t)} = \text{Tr}_A[\rho_{OA}^2(t)]$.
• Significance: Unlike state entanglement, this quantity depends on the operator's spatial location and encodes both spectral and spatial information, making it a direct probe of transport.

B. Schwinger-Keldysh (SK) Field Theory Formalism

To compute these entropies efficiently for large systems, the authors construct a unified Schwinger-Keldysh coherent-state path integral framework.

• Replica Trick: They utilize an identity involving fermionic displacement operators to express the trace of squared reduced operators (or density matrices) as a path integral over two replicas ($\alpha=1, 2$).
• Action Construction:
  • For Operator Entropy: The path integral involves an infinite-temperature boundary condition (trace over the full Fock space). The action includes a "kick" term representing the insertion of the local operator.
  • For State Entropy: The path integral involves a vacuum (or specific particle-number) boundary condition corresponding to the initial pure product state (e.g., N´eel state).
• Gaussian Integration: Since the systems are non-interacting, the action is quadratic in Grassmann fields. The authors integrate out the dynamical fields to obtain exact expressions for the entropies in terms of single-particle Green's functions.
  • $S^{(2)}_{OA}(t)$ is expressed via infinite-temperature Green's functions ($G_{ij}$).
  • $S^{(n)}_A(t)$ is expressed via vacuum Green's functions ($G_{0,ij}$).
• Resulting Formulas: The entropies are derived as traces of logarithms of correlation matrices constructed from these Green's functions (e.g., $S^{(2)}_{OA}(t) = -\text{tr}\ln[(I-C_O)^2 + C_O^2]$).

3. Key Contributions

1. Unified SK Framework: Development of a single field-theoretic formalism that simultaneously computes operator R´enyi entropy and state entanglement entropy for non-interacting systems, linking them directly to single-particle Green's functions.
2. State-Independent Transport Probe: Demonstration that the subsystem operator R´enyi entropy provides a robust, state-independent diagnostic for transport, capable of distinguishing between ballistic, diffusive, sub-diffusive, and localized regimes.
3. Schmidt Value Dynamics in Non-Interacting Systems: A crucial theoretical finding that in non-interacting systems, all Schmidt values (both maximum and typical) decay with the same temporal scaling (ballistic or sub-ballistic) dictated by the transport regime. This contrasts sharply with interacting diffusive systems, where a dichotomy exists (maximal eigenvalue decays diffusively, typical ones ballistically).
4. Analytical and Numerical Benchmarking: The method allows for the calculation of entropies in large systems ($L \sim 500+$) by avoiding full many-body diagonalization, relying instead on single-particle eigenstates.

4. Results

The authors applied their method to three models:

1. 1D Aubry-Andr´e (AA) Model: Quasiperiodic potential.
2. 2D Aubry-Andr´e Model: Generalization of the 1D case.
3. 2D Anderson Model: Random disorder.

A. Scaling of Saturation Times

They analyzed the time $t_{sat}$ required for entropies to saturate as a function of system size $L$. The scaling $t_{sat} \sim L^\alpha$ reveals the transport nature:

• Ballistic Phase ($V < V_c$): $\alpha \approx 1$. Entropy grows linearly with time ($\sim t$).
• Critical Point ($V = V_c$):
  • 1D AA: $\alpha \approx 2$, indicating diffusive transport ($\sim \sqrt{t}$ growth).
  • 2D AA: $\alpha \approx 2$, consistent with diffusive scaling for the system sizes studied ($L < \ell$, where $\ell$ is the mean free path).
• Localized Phase ($V > V_c$): No significant growth; entropy saturates to an $O(1)$ value immediately.

B. Temporal Growth of Entropies

• Operator R´enyi Entropy ($S^{(2)}_{OA}$):
  • Exhibits a characteristic logarithmic growth ($\sim \ln t$) before saturation in the metallic/critical phases.
  • Does not grow in the localized phase.
  • The onset time of growth scales with the distance of the operator from the entanglement cut.
• State Entanglement Entropy ($S_A$):
  • Shows super-ballistic growth ($\sim t^{1.5}$ or $t^2$) at very early times (microscopic scale) due to local equilibration of quasiparticle pairs.
  • Follows the transport regime at intermediate times: Ballistic ($\sim t$) in the metallic phase, Diffusive ($\sim \sqrt{t}$) at the critical point.
  • In 2D AA, anomalous super-ballistic growth was observed for small $V$ in the nearest-neighbor model, attributed to van-Hove singularities, which was corrected by adding next-nearest-neighbor hopping.

C. Schmidt Value Decay

Numerical analysis of the Schmidt values ($\Lambda_i$) for the reduced density matrix revealed:

• Ballistic: $\ln \Lambda_i \sim -t$ (Exponential decay).
• Diffusive (Critical 1D): $\ln \Lambda_i \sim -\sqrt{t}$.
• Super-diffusive (2D AA): $\ln \Lambda_i \sim -t^{0.6}$.
• Uniformity: Unlike interacting systems, there is no distinction between the decay of the maximum and typical Schmidt values; all follow the transport scaling.

5. Significance

• Bridge Between Transport and Information: The paper establishes a rigorous quantitative link between the growth of quantum information (entanglement/operator spread) and conventional transport coefficients (conductivity, diffusion constants) in non-interacting systems.
• Diagnostic Tool: The subsystem operator R´enyi entropy is proposed as a superior diagnostic for transport regimes compared to standard OTOCs or state entanglement, as it is state-independent and explicitly spatially resolved.
• Theoretical Insight: The finding that non-interacting systems lack the "dichotomy" of Schmidt value decay seen in interacting diffusive systems provides a fundamental distinction between integrable/non-interacting and chaotic/interacting dynamics.
• Future Applications: The unified SK framework is general and can be extended to interacting systems using approximations like Dynamical Mean Field Theory (DMFT) or Random Phase Approximation (RPA), offering a pathway to study operator growth in complex interacting matter.