Adiabatic Ramp Dynamics Across the ETH--MBL Transition in Disordered XXZ Spin Chain

Using exact diagonalization and time-dependent numerical methods, this study demonstrates that in a disordered XXZ spin chain, adiabatically ramped interactions preserve localized dynamical behavior at slow rates while faster driving rates induce significant excitation generation and entropy growth, thereby highlighting the strong dependence of nonequilibrium dynamics on ramp speed across the ETH-MBL transition.

The Gist

The Big Picture: A Quantum Game of "Freeze" vs. "Mix"

Imagine you have a box full of tiny, spinning magnets (these are the "spins" in the paper). In a normal, orderly world, if you shake this box, the magnets would eventually mix up completely, reaching a state of "thermal equilibrium" where everything is jumbled and random. This is how most things work in nature; they forget their starting position and settle into a messy average. Physicists call this the ETH (Eigenstate Thermalization Hypothesis) regime.

However, if you make the box very "rough" or "bumpy" (adding disorder), something strange happens. The magnets get stuck in their spots. They can't move past each other, and they remember exactly where they started, even after a long time. This is called MBL (Many-Body Localization). It's like the magnets are frozen in place, refusing to mix.

The Experiment:
The researchers wanted to see what happens when you slowly change the rules of the game while the magnets are spinning. Specifically, they slowly turned up the "interaction" between the magnets (making them push or pull on each other harder) over time. They asked: If we change the rules slowly enough, will the magnets stay in their frozen state, or will they eventually break free and start mixing?

The Three Ways to Change the Rules (The "Ramps")

To test this, the scientists didn't just change the rules at a constant speed. They tried three different "driving protocols" (ways of speeding up the change), like three different ways of pressing the gas pedal in a car:

1. Linear Ramp: Pressing the gas pedal steadily and evenly, like a car accelerating at a constant rate.
2. Quadratic Ramp: Starting slow and then pressing the gas harder and harder as time goes on (like a car that gets faster the longer you drive).
3. Exponential Ramp: Starting very gently and slowly, then suddenly speeding up very quickly at the end (like a rocket launch).

What They Measured: The "Messiness" Meter

To see if the magnets were mixing or staying frozen, the researchers measured two things:

1. Diagonal Entropy (The "Confusion" Score): This measures how many different possible states the system is "confused" about. If the system stays perfectly frozen in its original state, the confusion is zero. If it starts mixing and exploring new states, the confusion goes up.
2. Entanglement Entropy (The "Connection" Score): This measures how much the magnets are "talking" to each other across the chain. In a frozen state, they barely talk to their neighbors. In a mixed state, they are all deeply connected.

The Results: Frozen vs. Flowing

The study looked at two types of environments:

• The "Smooth" World (ETH): Low disorder.
• The "Rough" World (MBL): High disorder.

1. In the Smooth World (ETH):
When they changed the rules, the magnets mixed up easily. As they drove the change faster (pressed the gas harder), the "Confusion" and "Connection" scores went up significantly. The system lost its memory of the start and became a hot, messy soup. The faster they drove, the more "excited" the system got.

2. In the Rough World (MBL):
Even when they changed the rules, the magnets stayed stuck. The "Confusion" and "Connection" scores stayed very low, almost flat. No matter how fast they drove the change, the system refused to mix. It kept its memory of the starting position. This proves that the "frozen" state is very robust and hard to break, even when you try to shake it up.

3. The Effect of the "Gas Pedal" Style:
While the outcome (frozen vs. mixed) was the same regardless of how they drove, the amount of mess created differed slightly:

• The Linear drive (steady press) created the most mess.
• The Quadratic drive (slow start, fast end) was a bit more contained.
• The Exponential drive (gentle start, sudden end) was the smoothest, creating the least amount of sudden "shock" to the system.

The Bottom Line

The paper concludes that disorder is a powerful shield. Even if you try to force a quantum system to change its state by slowly turning up the interactions, if the system is in the "Many-Body Localized" (frozen) phase, it will resist. It won't thermalize. It will keep its secrets.

The researchers found that while the speed of the change matters (driving faster creates more heat/mess), the shape of the change (linear vs. exponential) only changes the details, not the fundamental result. Whether you drive a car gently or aggressively, if the road is icy enough (high disorder), the car will still slide and stay in place.

In short: The study confirms that in a disordered quantum world, you can't easily force a system to "forget" its past, even if you try to nudge it very carefully. The "frozen" state is incredibly stubborn.

Technical Summary

Technical Summary: Adiabatic Ramp Dynamics Across the ETH–MBL Transition in Disordered XXZ Spin Chain

Problem Statement
The paper addresses the non-equilibrium dynamics of isolated interacting quantum systems with disorder, specifically focusing on the crossover between the Eigenstate Thermalization Hypothesis (ETH) regime and the Many-Body Localization (MBL) phase. While ETH systems thermalize and lose memory of initial conditions, MBL systems retain initial state signatures due to the suppression of transport and dephasing. A critical open question involves understanding how these distinct dynamical regimes respond to time-dependent driving protocols. Specifically, the authors investigate how adiabatic control of interaction parameters in disordered systems affects excitation generation and entropy growth, noting that achieving a strict adiabatic limit in interacting, disordered many-body systems is complicated by interactions, avoided level crossings, and finite-size effects.

Methodology
The study employs exact diagonalization (ED) and time-dependent numerical evolution methods on a disordered spin-1/2 XXZ chain with open boundary conditions.

• Model: The Hamiltonian includes nearest-neighbor exchange ($J_{xy}=1$) and a tunable interaction term $J_{zz}(t)$, subject to onsite disorder fields $h_j$ drawn from a uniform distribution $[-h_0, h_0]$. Two disorder strengths are analyzed: $h_0 = 0.1$ (ETH regime) and $h_0 = 3.72$ (MBL regime).
• Driving Protocols: The interaction parameter $J_{zz}(t)$ is ramped from an initial value of 1 to a final value of 2 using three distinct protocols:
  1. Linear ramp: $J_{zz}(t) \propto (1/2 + vt)$
  2. Quadratic ramp: $J_{zz}(t) \propto (1/2 + vt^2)$
  3. Exponential ramp: $J_{zz}(t) \propto (1/2 e^{vt})$
     The ramp speed $v$ is varied to probe finite-rate driving effects.
• System Sizes and Averaging: Simulations are performed for system sizes $L = 10, 12,$ and $14$, restricted to the zero-magnetization sector. Disorder averaging is conducted over $\sim 10^3$ samples for $L=10, 12$ and $\sim 10^2$ samples for $L=14$ using the QuSpin package.
• Observables: The dynamics are diagnosed using:
  1. Diagonal Entropy Density ($s_d$): Measures the spread of the time-evolved state over the eigenbasis of the final Hamiltonian, quantifying non-adiabatic excitation generation.
  2. Entanglement Entropy Density ($s_{ent}$): Measures the growth of bipartite quantum correlations during the ramp.
  3. Adjacent-Gap Ratio ($r$): Used to characterize the static spectral properties and identify the ETH-MBL crossover.

Key Results

1. Spectral Characterization: The disorder-averaged adjacent-gap ratio $r$ transitions from the Gaussian Orthogonal Ensemble (GOE) value ($\approx 0.53$) at low disorder to the Poisson value ($\approx 0.39$) at high disorder. This crossover becomes sharper with increasing system size, confirming the identification of the ETH and MBL regimes.
2. Entropy Dynamics in the ETH Regime: In the ergodic phase, both diagonal and entanglement entropy densities increase significantly with ramp speed $v$. This indicates a strong departure from adiabatic evolution and efficient generation of many-body correlations. The magnitude of entropy growth increases with system size, consistent with the larger Hilbert space available for ergodic dynamics.
3. Entropy Dynamics in the MBL Regime: In the localized phase, both entropy measures remain small and largely insensitive to the ramp speed across the investigated range. This reflects the suppression of transport and the limited participation of many-body states, preserving the memory of the initial state. Finite-size variations are minor.
4. Protocol Dependence: While the qualitative distinction between ETH and MBL remains robust across all protocols, quantitative differences exist:
   • Linear ramps produce the largest changes in entropy measures.
   • Quadratic ramps result in more confined variations.
   • Exponential ramps exhibit the smoothest dependence on ramp speed with reduced fluctuations, suggesting a gentler accumulation of non-adiabatic excitations at early times.
5. Finite-Size Scaling: The scaling trends of entropy production are largely independent of system size ($L=10, 12, 14$) within the accessible range. This suggests that the observed dynamical signatures are not dominated by finite-size effects and are robust against variations in system size and the specific temporal shape of the drive.

Significance and Claims
The paper claims that diagonal entropy and entanglement entropy serve as complementary and robust probes for distinguishing the ETH-MBL crossover under finite-rate driving. The study demonstrates that while the specific driving protocol (linear, quadratic, or exponential) influences the magnitude and smoothness of the entropy response, the fundamental separation between ergodic and localized dynamics remains intact. The results highlight that localized dynamical behavior remains largely intact under sufficiently slow ramp evolution, whereas increasing the driving rate promotes stronger excitation generation in ergodic systems. The authors conclude that entropy production is a useful diagnostic tool for nonequilibrium dynamics in disordered interacting quantum systems, providing a clear distinction between thermalizing and localized phases regardless of the specific ramp geometry or system size within the studied limits.