Motion-driven quantum dissipation in an open electronic system with nonlocal interaction

This paper investigates motion-driven quantum dissipation in two infinite parallel metallic plates modeled by 1+2 dimensional Dirac fields with nonlocal interactions, demonstrating that their relative motion induces anisotropic vacuum excitations and a threshold-dependent dissipative force analogous to the Schwinger effect.

The Gist

Imagine two giant, invisible sheets of metal floating parallel to each other, very close together. In this paper, the authors are studying what happens when one of these sheets slides smoothly over the other, like a piece of paper sliding across a table.

Usually, we think of friction as just rough surfaces grinding together. But in the quantum world (the world of tiny particles like electrons), things are stranger. The authors wanted to know: If these metal sheets slide past each other, does the motion itself create new particles out of "nothing," and does this create a drag force?

Here is a breakdown of their findings using simple analogies:

1. The Setup: A Quantum Dance Floor

Think of the electrons inside these metal sheets as dancers on a floor.

• The Left Sheet (L-plate): This is the stationary dance floor. The dancers are standing still.
• The Right Sheet (R-plate): This is a moving dance floor. It is sliding past the first one.
• The Connection: Even though the sheets don't touch physically, the dancers on one sheet can "feel" the dancers on the other sheet through a special, invisible connection (a "nonlocal potential"). It's like if the dancers on the moving floor could whisper to the dancers on the stationary floor, telling them to start moving.

2. The "Magic" of Motion (Creating Particles)

In the quantum world, a "vacuum" isn't truly empty; it's like a calm ocean with tiny, invisible waves.

• When the sheets are still ($v=0$): The ocean is calm. The dancers are quiet. Nothing happens. The energy distribution is perfectly round and uniform (isotropic).
• When the sheets slide ($v > 0$): The motion acts like a wind blowing across the ocean. This wind is strong enough to turn those tiny, invisible waves into real, visible waves.
  • The Result: The sliding motion "excites" the electrons, creating new particles out of the vacuum.
  • The Shape: When the sheets slide, the pattern of these new particles stretches out in the direction of the slide, just like a rubber band being pulled. It is no longer round; it is stretched (anisotropic).

3. The Threshold: The "Speed Bump"

One of the most interesting findings is that nothing happens until the sliding speed reaches a specific limit.

• The Analogy: Imagine trying to push a heavy box. If you push gently, it doesn't move. You have to push harder than a certain amount to get it to budge.
• The Finding: The authors found a "speed bump" (a threshold). If the sliding speed is too slow (specifically, slower than twice the speed of the electrons inside the metal), nothing happens. No new particles are created, and there is no extra drag.
• The Breakthrough: Once the speed crosses this threshold, the "wind" becomes strong enough to create particles. The faster they slide past this point, the more particles are created.

4. The Drag Force (Quantum Friction)

Because creating these new particles requires energy, the moving sheet has to "pay" for it.

• The Energy Transfer: The external force pushing the moving sheet pumps energy into the system. This energy is used to create the new particles.
• The Friction: This energy loss feels like a drag or friction force. The moving sheet feels a resistance pulling it backward.
• The Relationship: The authors found that once the speed is high enough to cross the threshold, this drag force increases linearly with speed. It's like a car driving on a road where the air resistance gets stronger the faster you go, but only after you hit a certain speed.

Summary

The paper describes a scenario where motion creates matter. By sliding two metal plates past each other, the authors showed that:

1. Motion creates particles: The sliding motion turns the empty space between the plates into a place where new electrons appear.
2. There is a speed limit: This only happens if the plates move fast enough (faster than a specific threshold).
3. It costs energy: Creating these particles creates a friction-like force that resists the motion.

In short, the paper proves that in the quantum world, simply sliding two surfaces past each other can generate energy and particles, creating a unique type of "quantum friction" that only kicks in after a certain speed is reached.

Technical Summary

Technical Summary: Motion-driven Quantum Dissipation in an Open Electronic System with Nonlocal Interaction

Problem Statement
This study investigates the quantum excitations and dissipation mechanisms arising from the relative motion of two infinite parallel metallic plates. While open quantum systems (OQS) typically involve coupling to external environments, this work focuses on a specific class of OQS where dissipation arises internally through the relative motion of two coupled subsystems. The authors model the electrons in these plates as 1+2 dimensional massive Dirac fermions, a framework relevant to topological insulators (e.g., Bi2Se3, Sb2Te3). Unlike previous semiclassical models of electronic friction or dynamical Casimir effects that rely on vacuum electromagnetic fluctuations, this paper posits a direct nonlocal interaction between the plates. The core problem is to quantify how this internal relative motion, driven by an external force, induces on-shell particle production and dissipative forces within the quantum vacuum of the system.

Methodology
The authors employ a field-theoretic approach using the path integral formalism and perturbation theory:

1. Microscopic Model: The system consists of a stationary L-plate and a moving R-plate. The electrons are described by 1+3 dimensional massive Dirac fields reduced to 1+2 dimensions on the plate surfaces. The free action includes a mass term ($m$) and Fermi velocity ($v_F$).
2. Interaction: The plates are coupled via a nonlocal potential $U(x,y)$, modeled as the Feynman propagator of a massless particle. This "Anderson mixed coupling" represents a direct-exchange interaction that propagates at a finite speed, distinct from local exchange models.
3. Galilean Boost: The relative motion is introduced by applying a Galilean boost to the R-plate along the $x_1$-axis with velocity $v$. This modifies the kinetic term of the R-plate's Dirac field in the total action.
4. Effective Action and Integration: To study the L-plate as an open system, the degrees of freedom of the R-plate are integrated out in the path integral. This yields an effective action for the L-electrons containing a non-local self-energy term dependent on the relative velocity.
5. Perturbative Calculation:
   • Vacuum Occupation: The vacuum occupation number $n_L(k)$ is calculated perturbatively to order $g^2$ (where $g$ is the coupling constant) by evaluating the exact propagator of the L-electrons.
   • Quantum Action: The Vacuum Persistence Amplitude (VPA) is expressed as $Z = e^{i\Gamma}$. The quantum action $\Gamma$ is expanded to order $g^2$. The imaginary part of $\Gamma$ ($\text{Im}\Gamma$) is extracted using the residue theorem on the complex energy plane, identifying poles corresponding to on-shell excitations.
   • Dissipative Force: The dissipative force is derived from the energy balance equation, equating the power pumped into the system by the external force to the dissipation power resulting from particle production.

Key Results

• Anisotropy of Vacuum Occupation: Numerical analysis of the vacuum occupation number $n_L(k)$ reveals that for zero relative velocity ($v=0$), the distribution is isotropic in momentum space. However, for non-zero velocity, the distribution becomes anisotropic, stretching along the direction of motion ($k_1$). This confirms that relative motion modifies the vacuum state, inducing excitations even when the system is close to the initial vacuum.
• Threshold Behavior: A critical finding is the existence of a velocity threshold for dissipation. Both the imaginary part of the quantum action ($\text{Im}\Gamma$) and the dissipative force ($F_{\text{diss}}$) vanish when the relative velocity $v$ is below a critical value $v_c = 2v_F$. Dissipation only occurs when $v > 2v_F$.
• Linear Friction Coefficient: Above the threshold, the dissipative force exhibits a linear dependence on the relative velocity $v$. This suggests the emergence of a linear friction coefficient at zero temperature, analogous to classical friction but arising from quantum particle production.
• Schwinger-like Mechanism: The process of relative motion inducing on-shell excitations is shown to be analogous to the Schwinger effect, where energy from an external source (the force maintaining motion) is converted into particle creation.

Significance and Claims
The paper claims to provide a rigorous quantum mechanical description of motion-driven dissipation in a system without an external thermal bath, relying solely on internal nonlocal coupling. The authors highlight that:

• The nonlocal interaction, modeled via a massless propagator, successfully transmits friction and induces dissipation without invoking electromagnetic vacuum fluctuations.
• The derived threshold ($v > 2v_F$) provides a clear kinematic condition for the onset of quantum friction in such Dirac systems, distinguishing it from regimes where only virtual fluctuations exist.
• The linear relationship between the dissipative force and velocity at zero temperature offers a theoretical basis for understanding quantum friction in topological materials.
• The methodology, specifically the use of nonlocal potentials to model inter-plate coupling, could be extended to other condensed matter systems with complex excitations, though the authors note that strongly correlated systems without well-defined quasi-particles would require non-perturbative approaches.

The study concludes that relative motion fundamentally alters the vacuum state of coupled electronic systems, leading to measurable dissipative forces and particle production governed by specific kinematic thresholds.