Dissipation- versus Chaos-Induced Relaxation in Non-Markovian Quantum Many-Body Systems

This paper investigates an open Sachdev-Ye-Kitaev model coupled to a pseudogapped fermionic bath using the Keldysh formalism, revealing a rich dynamical phase diagram where non-Markovian dissipation competes with internal chaos to produce distinct regimes of power-law and exponential relaxation, as well as an intermediate crossover phase.

The Gist

Imagine you have a very chaotic, crowded room full of people (the Quantum System) who are constantly bumping into each other, shouting, and changing the room's energy. This room is naturally chaotic; if you leave it alone, the chaos eventually settles down into a calm, average state. This is like a system "relaxing" or finding equilibrium.

Now, imagine this room is inside a larger building with an open window (the Environment or Bath). The air coming through the window can either help the room calm down quickly or mess with the process in strange ways.

This paper is about what happens when the air coming through that window isn't just "normal" air, but has a very specific, weird structure. The researchers wanted to see: Does the room calm down fast and smoothly, or does it get stuck in a weird, slow pattern?

Here is the breakdown using simple analogies:

1. The Two Forces at Play

The paper studies a tug-of-war between two forces:

• The Chaos Inside (Internal Dynamics): The people in the room are naturally chaotic. If they just interact with each other, they settle down quickly, like a shaken soda can fizzing out. This usually leads to exponential decay (a fast, smooth drop to zero).
• The Environment Outside (Dissipation): The window lets energy escape. Usually, we assume the wind outside is steady and forgets everything instantly (this is called a "Markovian" bath). In that case, the room cools down fast.

The Twist: In this study, the "wind" outside is pseudogapped. Imagine the window is covered with a special filter that blocks out low-frequency sounds (like a deep hum) but lets high-pitched sounds through. This filter has "memory." It doesn't just let energy out; it remembers what happened a moment ago and reacts differently. This is Non-Markovian behavior.

2. The Three Ways the Room Settles Down

The researchers found that depending on how "thick" the filter is (how much it blocks low frequencies), the room behaves in three distinct ways:

A. The "Slow Drift" (Bath-Driven Power-Law)

• The Scenario: The filter is very selective. It blocks almost all the low-energy "hum" from escaping.
• The Analogy: Imagine trying to drain a bathtub, but the drain is clogged with a sponge that only lets water out very slowly and unevenly. The water level doesn't drop in a straight line; it drops in a long, slow curve that takes forever to finish.
• The Result: The system relaxes algebraically (a power law). It's a slow, lingering decay. The environment is the boss here, forcing the system to move at a snail's pace.

B. The "Fast Reset" (Chaos-Driven Exponential)

• The Scenario: The filter is so thick it blocks everything below a certain frequency. The room is effectively isolated from the low-energy part of the outside world.
• The Analogy: The window is completely sealed. The only thing that matters is the chaos inside the room. The people inside bump into each other, and the energy dissipates quickly and smoothly, just like a normal chaotic system.
• The Result: The system relaxes exponentially. It's fast and predictable, driven entirely by the internal chaos.

C. The "Weird Middle Ground" (Pre-Relaxation)

• The Scenario: The filter is in the middle. It blocks some low frequencies but not all.
• The Analogy: Imagine the bathtub drain is partially clogged. At first, the water drains fast (because the initial rush is strong). But as the water level gets lower, the clog starts to matter more, and the draining slows down dramatically, switching from a fast drop to a slow drip.
• The Result: The system starts relaxing fast (exponential) but then gets stuck in a slow, long tail (algebraic). It's a crossover. The system tries to be chaotic, but the environment's memory eventually takes over.

3. Why Does This Matter?

In the real world, many quantum systems (like those in quantum computers or materials like graphene) aren't isolated. They are always interacting with their environment.

• Old Thinking: We usually assumed the environment was "boring" and just made things decay fast.
• New Discovery: The paper shows that if you engineer the environment (like designing a special filter or a specific material), you can control how a quantum system relaxes.
  • Want it to settle down fast? Make the environment "forgetful."
  • Want it to hold onto energy longer or decay slowly? Give the environment "memory" (a pseudogap).

The Big Picture

Think of the system as a dancer.

• Internal Chaos is the dancer's own rhythm.
• The Environment is the music and the floor.
• If the floor is slippery and the music is steady, the dancer stops quickly.
• But if the floor is sticky (has memory) and the music has a weird, deep bass that gets cut off, the dancer might start spinning fast, then suddenly get stuck in a slow, dragging motion.

This paper maps out exactly when the dancer spins fast, when they drag slowly, and when they do a mix of both, just by changing the "stickiness" of the floor. This helps scientists design better quantum devices by understanding how to control these "sticky" environments.

Technical Summary

Here is a detailed technical summary of the paper "Dissipation- versus Chaos-Induced Relaxation in Non-Markovian Quantum Many-Body Systems" by Almeida et al.

1. Problem Statement

The paper addresses the fundamental problem of how interacting quantum many-body systems relax to a steady state when coupled to an external environment.

• Context: In isolated systems, relaxation is driven by internal quantum chaos (Eigenstate Thermalization Hypothesis). In open systems, this competes with environmental dissipation.
• Gap in Knowledge: Most theoretical treatments assume Markovian baths (weak coupling, short memory), leading to exponential decay governed by a spectral gap. However, many realistic platforms (e.g., graphene, engineered reservoirs) exhibit non-Markovian dynamics due to structured environments with non-trivial density of states (DOS).
• Specific Focus: The authors investigate the relaxation dynamics of a strongly correlated system coupled to a pseudogapped fermionic bath, where the bath's DOS vanishes as a power law ($D(\omega) \sim |\omega|^\nu$) at low frequencies. The goal is to determine how this memory effect competes with internal chaos to alter relaxation mechanisms.

2. Methodology

The study employs a controlled analytical and numerical approach using the Sachdev-Ye-Kitaev (SYK) model as the paradigmatic strongly correlated system.

• Model:
  • System: An open SYK model consisting of $N$ Majorana fermions with random all-to-all 4-body interactions (strength $J$).
  • Bath: A fermionic environment with a pseudogapped DOS, $D(\omega) \propto (1 - e^{-\omega^2/\Lambda^2})^{\nu/2}$, characterized by the pseudogap exponent $\nu$ and width $\Lambda$.
  • Coupling: Linear coupling between system and bath operators with strength $\mu$.
• Formalism:
  • Keldysh Path Integral: Used to handle non-equilibrium dynamics and integrate out the bath degrees of freedom.
  • Large-$N$ Limit: The disorder-averaged effective action is dominated by the saddle point, leading to a closed set of Schwinger-Dyson (SD) equations for the steady-state correlation functions.
• Key Quantities:
  • Retarded Green's Function ($G^R(t)$): Determines the time-domain relaxation behavior.
  • Spectral Function ($\rho_-(\omega)$): The imaginary part of the retarded Green's function in frequency space. The low-frequency behavior of $\rho_-(\omega)$ dictates the long-time decay of $G^R(t)$.
• Numerical Solution: The SD equations are solved self-consistently using an iterative algorithm on a frequency grid with exponential accumulation near $\omega=0$ to resolve critical low-frequency behaviors.

3. Key Contributions and Results

The authors uncover a rich dynamical phase diagram determined by the competition between the dissipation strength ($\mu$) and the bath structure exponent ($\nu$).

A. Analytical Insights (Strong Dissipation Limit)

In the limit of strong dissipation ($J \ll \Lambda \ll \mu$), the nonlinear interaction terms in the SD equations become negligible. The spectral function $\rho_-(\omega)$ exhibits distinct behaviors based on $\nu$:

• $\nu < 1$: $\rho_-(\omega)$ diverges as $|\omega|^{-\nu}$. This leads to algebraic (power-law) relaxation in time: $G^R(t) \sim t^{-(1+\nu)}$.
• **$1 < \nu < 2$:**$\rho_-(\omega)$is finite at$\omega=0$but possesses a non-analytic cusp ($\rho_-(0) - \rho_-(\omega) \sim |\omega|^\nu$). This results in a pre-relaxation regime where the system initially decays exponentially but crosses over to algebraic decay at very long times.
• $\nu > 2$: The non-analytic term is subdominant to the analytic $O(\omega^2)$ term. The system exhibits exponential relaxation, effectively behaving as if it were isolated (chaos-driven) because the bath is effectively gapped.

B. Dynamical Phase Diagram

Combining analytical limits with numerical solutions for finite $\mu$, the authors map out a phase diagram in the $(\mu, \nu)$ plane featuring three distinct regimes:

1. Bath-Driven Power-Law Regime ($\nu < \nu_c(\mu)$):
   • Characterized by a divergent spectral density $\rho_-(\omega) \to \infty$ as $\omega \to 0$.
   • Relaxation is purely algebraic with exponent $p = 1 + \nu$.
   • Dominated by the environment's memory effects.
2. Chaos-Driven Exponential Regime ($\nu > 2$):
   • The bath is effectively gapped; low-energy modes are suppressed.
   • Relaxation is exponential, driven by the internal chaotic dynamics of the SYK model.
   • The spectral gap and oscillation frequency saturate to values characteristic of the isolated system.
3. Intermediate Pre-Relaxation Regime ($\nu_c(\mu) < \nu < 2$):
   • Features a finite but non-analytic $\rho_-(0)$.
   • Dynamics show a crossover: an initial exponential decay (governed by an effective gap) followed by a late-time algebraic tail.
   • This regime arises from a diverging timescale $t^* \sim 1/\omega^*$, where $\omega^*$ is the scale where the non-analytic term overtakes the analytic term.

C. Robustness

The study confirms that these qualitative phases persist at finite bath temperatures ($\beta \neq 0$), although the crossover scales are quantitatively modified. The system equilibrates to the bath temperature, satisfying the fluctuation-dissipation relation.

4. Significance

• Mechanism of Relaxation: The paper demonstrates that the low-energy structure of an environment is not merely a technical detail but a decisive factor in determining whether a quantum system relaxes exponentially or algebraically.
• Beyond Markovian Approximation: It provides a rigorous framework for understanding non-Markovian relaxation in strongly correlated systems, moving beyond the standard weak-coupling/impurity paradigms.
• Experimental Relevance: The findings are directly applicable to platforms with structured reservoirs, such as graphene (where the DOS vanishes linearly) or engineered photonic/phononic crystals.
• Control of Quantum Dynamics: The results suggest that "environment engineering"—tailoring the spectral density of a bath—can be used as a practical tool to control relaxation timescales and induce specific dynamical phases (e.g., slowing down relaxation via pseudogaps) in quantum simulators.
• Pre-Relaxation Phenomena: The identification of a generic pre-relaxation phase where exponential decay crosses over to algebraic decay offers a new perspective on transient dynamics in open quantum systems.

In summary, the work establishes that non-Markovian environments can qualitatively reshape the relaxation landscape of quantum many-body systems, creating a competition between chaos-induced exponential decay and dissipation-induced algebraic relaxation.