Post-quench relaxation dynamics of Gross-Neveu lattice fermions

This paper numerically investigates the post-quench relaxation dynamics of a one-dimensional Gross-Neveu lattice fermion model, revealing that while closed systems exhibit revivals and thermalize in the thermodynamic limit according to the Eigenstate Thermalization Hypothesis, full equilibration of finite-momentum correlations requires system-reservoir coupling and is consistent with a Generalized Gibbs Ensemble description.

The Gist

Imagine a crowded dance floor where everyone is holding hands in a specific, synchronized pattern. This represents a quantum system (specifically, a chain of fermions) in a stable, ordered state. The "dance" is governed by the rules of physics, specifically the Gross-Neveu model, which describes how these particles interact.

In this paper, the authors ask a simple question: What happens if you suddenly change the music?

The Setup: The "Quench"

At time $t=0$, the researchers perform a "quantum quench." This is like suddenly switching the song from a slow waltz to a frantic techno beat. In physics terms, they instantly change the strength of the interaction between the particles.

Because the particles were dancing to the old song, they are now confused. They don't know how to move to the new beat. This creates a chaotic, non-equilibrium situation. The paper studies how the system tries to calm down and find a new rhythm (relaxation).

The Two Scenarios: The Isolated Room vs. The Open Door

The authors study two different versions of this dance floor to see how the chaos resolves.

1. The Closed System (The Soundproof Room)

Imagine the dance floor is in a perfectly soundproof room with no windows. No energy can get in or out.

• What happens: When the music changes, the dancers start flailing. At first, they seem to settle into a new, calmer rhythm. However, because the room is closed, the energy of their confusion has nowhere to go.
• The "Echo" Effect: After a while, the dancers accidentally bump into the walls and bounce back. This causes the whole group to suddenly snap back into their original, chaotic flailing pattern. In physics, this is called a revival.
• The Illusion of Calm: If you only look at the average movement of the crowd (the "order parameter"), it might look like they have settled down. But if you look closely at individual dancers (specific "correlation matrix elements"), you see they are still oscillating wildly, never truly stopping.
• The Lesson: In a closed system, the system never truly "forgets" its initial state. It gets stuck in a loop of oscillations, described by a special rulebook called the Generalized Gibbs Ensemble (GGE). It's like a pendulum that swings forever because there's no air resistance to stop it.

2. The Open System (The Open Door)

Now, imagine the dance floor has a door open to a quiet hallway (the "reservoir" or "bath").

• What happens: When the music changes and the dancers flail, they can now bump into people in the hallway or lose energy to the air outside.
• The Result: The "friction" from the outside world (represented by the coupling $\gamma$) drains the excess energy. The wild oscillations die out. The dancers eventually stop flailing and settle into a completely new, stable rhythm that matches the new song.
• The Lesson: Only when the system is connected to an environment can it truly "thermalize" (reach a state of true equilibrium). The door allows the system to forget its past and start fresh.

The Tools: The "Mean Field" and the "Lindblad Equation"

To figure this out, the authors used some heavy mathematical tools, but we can think of them simply:

• Self-Consistent Mean Field: Imagine trying to predict the dance by assuming everyone is doing the average move. It's a simplification, but it works surprisingly well for large crowds.
• Lindblad Master Equation: This is the mathematical rulebook for how the dance floor interacts with the hallway. It calculates exactly how much energy leaks out and how the dancers slow down.

Why Does This Matter?

This isn't just about dancing fermions. This research helps us understand:

1. Quantum Computers: If we want to build a quantum computer, we need to know how long a system stays stable before it "forgets" its state or gets corrupted by the environment.
2. New Materials: It helps us understand how materials (like superconductors) react when hit with a sudden laser pulse.
3. The Nature of Time: It shows that "relaxing" to a calm state isn't guaranteed. If you isolate a system perfectly, it might never truly settle down, even if it looks like it has.

The Bottom Line

The paper tells us that isolation is a trap. If you isolate a quantum system, it might look like it's calming down, but it's actually just stuck in a loop, remembering its past forever. To truly reach a new, stable state, the system needs to interact with the outside world.

In short: You can't just change the rules of a game and expect everyone to instantly adapt if they are trapped in a room with no exit. They need a way to let the old energy out to truly learn the new dance.

Technical Summary

1. Problem Statement

The paper investigates the non-equilibrium relaxation dynamics of a one-dimensional (1D) lattice version of the $N$-flavor Gross-Neveu (GN) model following a sudden quantum quench of the interaction parameter $g$. The study addresses two critical questions in quantum many-body physics:

• Closed Systems ($\gamma = 0$): How does an integrable system relax after a quench? Does it reach a true thermal equilibrium, or does it settle into a non-thermal steady state described by a Generalized Gibbs Ensemble (GGE)?
• Open Systems ($\gamma > 0$): How does coupling to a thermal environment (reservoir) affect the relaxation process, specifically regarding the equilibration of global order parameters versus local correlation functions?

The authors focus on the interplay between the Eigenstate Thermalization Hypothesis (ETH), quantum integrability, and dissipation induced by system-environment coupling.

2. Methodology

The authors employ a combination of analytical approximations and numerical simulations:

• Model: A lattice Hamiltonian for $N$ flavors of spinless fermions with nearest-neighbor hopping and a displacement field $\Delta_j$ coupled via interaction strength $g$. In the continuum limit, this maps to the massless 1D GN model.
• Self-Consistent Mean Field (SCMF) Approximation: To handle the many-body problem, the authors use a time-dependent SCMF approach. This reduces the interacting fermion problem to a quadratic (quasi-free) Hamiltonian where the displacement field is determined self-consistently by the fermion expectation values. This allows for the treatment of large system sizes ($N \to \infty$ limit).
• Lindblad Master Equation (LME): To model dissipation, the system is coupled to a thermal reservoir with strength $\gamma$. The dynamics are governed by a time-dependent, non-linear LME.
  • The jump operators are derived microscopically from a tunneling model between the system and a metallic substrate.
  • Because the Hamiltonian becomes quadratic under SCMF and the jump operators are linear in fermion operators, the LME is mapped onto a closed set of linear differential equations for the time-dependent correlation matrix elements $\theta_{k,\alpha;(a,a')}(t)$.
• Simulation Protocol:
  • The system is initialized in an ordered phase ($m \neq 0$) at $t < 0$.
  • At $t=0$, the interaction $g$ is suddenly quenched to a new value $g_f$.
  • The authors numerically solve the coupled equations for the correlation matrix and the self-consistent order parameters $m(t)$ and $\delta J(t)$.
  • They compare results for closed systems ($\gamma=0$) and open systems ($\gamma > 0$) across various system sizes ($L$) and quench amplitudes.

3. Key Contributions

• Unified Framework: The paper successfully integrates the time-dependent SCMF approximation with the Lindblad master equation to study post-quench dynamics in a lattice GN model, a setup relevant for describing Peierls transitions in polymers and dynamical mass generation.
• Distinction between Global and Local Relaxation: The authors demonstrate a crucial decoupling between the relaxation of global observables (like the order parameter $m(t)$) and the relaxation of finite-momentum correlation matrix elements.
• Role of Integrability vs. Dissipation: The study clarifies how integrability (in the closed limit) prevents true thermalization of all observables, leading to persistent oscillations in correlation functions, whereas finite dissipation ($\gamma > 0$) drives the entire system to a unique stationary state.

4. Key Results

A. Closed System Dynamics ($\gamma = 0$)

• Order Parameter ($m(t)$): For finite system sizes $L$, $m(t)$ exhibits undamped oscillations and periodic revivals (recurrences) at times $t \propto L$. In the thermodynamic limit ($L \to \infty$), these revivals are suppressed, and $m(t)$ appears to relax to a stationary value consistent with the ETH.
• Correlation Matrix Elements: Despite the apparent relaxation of $m(t)$, the finite-momentum correlation matrix elements (e.g., $\theta_{k^*}$) exhibit persistent, unattenuated oscillations for arbitrarily long times.
• Interpretation: The system does not reach a true thermal equilibrium state. Instead, the asymptotic state is a Generalized Gibbs Ensemble (GGE) determined by the infinite set of conserved quantities inherent to the integrable (quadratic) mean-field Hamiltonian. The "relaxation" of $m(t)$ is an artifact of dephasing among many harmonics, masking the underlying non-equilibrium nature of the state.

B. Open System Dynamics ($\gamma > 0$)

• True Thermalization: When a finite coupling to the environment is introduced, the periodic revivals are suppressed. Both the order parameter $m(t)$ and all correlation matrix elements converge to stationary values as $t \to \infty$.
• Damping Mechanism: The relaxation is driven by the dissipative terms in the LME. The authors note that the relaxation rate in the self-consistent theory is time-dependent and slows down as the system approaches the final equilibrium state, differing from the constant decay rate predicted by non-self-consistent approximations.
• Phase Transitions: The study shows that quenching from a disordered to an ordered phase (or vice versa) can trigger the onset of symmetry breaking in real time, with the system settling into the equilibrium phase dictated by the post-quench parameters.

C. Analytical Insights

• The authors derive analytical solutions for the non-self-consistent limit, showing that the dynamics can be mapped to Bloch-like equations for pseudospin vectors.
• They confirm that in the thermodynamic limit, the decay of correlations follows an exponential law characterized by the Bessel function $K_0$, consistent with the ETH for local subsystems.

5. Significance

• Validation of ETH and GGE: The results provide a concrete numerical demonstration of how ETH applies to global order parameters in integrable systems while failing to describe the full dynamics of local correlations, which remain in a GGE state.
• Necessity of Dissipation: The paper highlights that for a quantum many-body system to reach a true thermal equilibrium where all observables become stationary, coupling to an external bath is essential. Relying solely on global observables can be misleading regarding the true equilibration status of a closed system.
• Methodological Advancement: The approach of solving the LME via correlation matrices under a time-dependent SCMF approximation offers a powerful tool for studying non-equilibrium dynamics in interacting lattice models, applicable to superconductors, topological insulators, and other correlated electron systems.
• Experimental Relevance: The findings are relevant for time-resolved spectroscopic experiments in correlated electronic systems and cold atom setups where quench protocols and environmental coupling can be precisely controlled.