Robust Correlation-Induced Localization Under Time-Reversal Symmetry Breaking

This paper demonstrates that in a one-dimensional disordered system with long-range correlated hopping, time-reversal symmetry breaking induces a robust transition from algebraic localization to delocalization, fundamentally altering wavepacket dynamics from subdiffusive to diffusive behavior.

The Gist

Imagine you are trying to walk through a crowded, chaotic city where the streets are full of random obstacles. In the world of quantum physics, this is called Anderson Localization. Usually, if the city is messy enough, a traveler (a quantum particle) gets stuck in one spot, unable to move forward. They are "localized."

However, this paper explores a very specific, weird version of that city.

The Setup: A City with "Long-Range Teleportation"

In this model, the city isn't just messy; it has a special rule. You can't just walk to the next block. Instead, you have a chance to "teleport" to blocks far away, but the chance of teleporting drops the farther you go (like a signal fading over distance).

Crucially, these teleportation links are correlated. It's not random chaos; the path from your house to the park is mathematically linked to the path from your house to the library. In the past, scientists found that this specific kind of "organized chaos" actually helps you get stuck even more than random chaos would. It's like having a map that accidentally guides you into a dead end.

The Twist: Breaking the "Mirror Rule"

Now, imagine the city has a rule called Time-Reversal Symmetry (TRS). Think of this as a "Mirror Rule." If you walk a path forward, the universe guarantees that walking that exact same path backward is equally likely. It's like a perfectly symmetrical dance.

The researchers asked: What happens if we break this Mirror Rule?

They introduced a "twist" (represented by a parameter called $\theta$) to the teleportation links. Now, walking forward feels different than walking backward. It's like the city has a slight wind blowing in one direction, or the streets are slightly tilted.

The Big Discovery: The "Goldilocks" Zone of Stuckness

The team found a surprising result:

1. The "Stuck" Phase is Tough: Even when they broke the Mirror Rule (tilted the streets), the particles stayed stuck (localized) for a while. The "organized chaos" that traps them is so strong that a little bit of tilt doesn't free them.
2. The Breaking Point: However, there is a limit. If the tilt ($\theta$) gets too steep (specifically, if it exceeds a value related to how far the teleportation reaches), the "stuck" phase collapses. Suddenly, the particles are free to roam the whole city again.
3. The Critical Threshold: They calculated exactly how much tilt it takes to break the trap. It depends on how far the teleportation reaches. If the teleportation reaches very far, the trap is weaker and breaks easily. If it reaches only a short distance, the trap is stronger.

The Dynamic Difference: The "Core" vs. The "Tail"

Here is where it gets really interesting. The researchers looked at two different ways of measuring the particle's movement:

• The Static View (The Map): If you look at where the particle usually sits, it looks stuck in both the "tilted" and "flat" versions of the city. It's still in a corner.
• The Dynamic View (The Movie): If you watch the particle move over time, the story changes completely.
  • Without the tilt (Mirror Rule intact): The particle moves very slowly, like a snail dragging its feet. This is called sub-diffusion. It's stuck, but it's trying to wiggle out slowly.
  • With even a tiny tilt (Mirror Rule broken): The particle suddenly starts moving at a normal, steady walking pace. This is diffusion.

The Analogy: Imagine a person trapped in a maze.

• No Tilt: They are pacing back and forth in a small circle, bumping into walls, but never making progress. They are "sub-diffusive."
• Tiny Tilt: The moment you tilt the floor slightly, they stop pacing and start walking straight down a corridor. They are now "diffusive."

The paper shows that the "core" of the particle's location stays stuck (the person is still in the maze), but the "tails" (the parts of the person reaching out) start flowing freely once the symmetry is broken.

Why Does This Matter?

This isn't just about math puzzles. This kind of physics helps us understand:

• Light in Disordered Materials: How light travels through fog or complex materials where atoms interact over long distances.
• Quantum Computers: Understanding how to keep quantum information "stuck" (localized) so it doesn't leak away, or conversely, how to let it flow when we want it to.
• New States of Matter: It reveals a new type of transition where a system goes from "stuck" to "flowing" just by changing the directionality of its connections, not by changing the amount of mess.

In a Nutshell

The paper proves that a specific type of "organized mess" can trap particles very effectively, even if you try to break the symmetry of the universe. But, if you break that symmetry just enough, the trap snaps open, and the particles start flowing like water, even though they still look like they are stuck if you take a snapshot. It's a delicate balance between being trapped and being free, controlled by the "tilt" of the world.

Technical Summary

1. Problem Statement

The paper investigates Anderson localization in one-dimensional disordered systems characterized by long-range correlated hopping. While standard Anderson localization in 1D implies that all states are localized for any amount of uncorrelated disorder, specific exceptions exist (e.g., Aubry-André model, Power-Law Banded Random Matrix models) where delocalization can occur.

The authors focus on a model proposed by Logan, Wolynes, Burin, and Maksimov, where hopping amplitudes decay as a power law ($1/r^a$) but are fully correlated (translation-invariant). Previous studies showed that these correlations can induce algebraic localization even when standard perturbative arguments (Levitov's criterion) predict delocalization (specifically for $a < 1$).

The central question addressed is: How robust is this correlation-induced localization against the breaking of Time-Reversal Symmetry (TRS)? Since Anderson localization relies on coherent backscattering and quantum interference between time-reversed paths, breaking TRS (e.g., via complex hopping phases) is expected to suppress interference and potentially destabilize the localized phase.

2. Model and Methodology

The Model:
The system is defined by a 1D tight-binding Hamiltonian with periodic boundary conditions:
$$H = \sum_n \epsilon_n |n\rangle\langle n| + \sum_{n \neq m} j_{n-m} |n\rangle\langle m|$$

• Disorder: Onsite potentials $\epsilon_n$ are independent and identically distributed (i.i.d.) Gaussian random variables with variance $W$.
• Hopping: The hopping terms $j_{n-m}$ are complex and fully correlated:
  $$j_{n-m} = \frac{J_0 e^{i\theta \text{sign}(n-m)}}{|n-m|^a}$$
  • $a > 0$: Controls the power-law decay of hopping.
  • $\theta \in [-\pi/2, \pi/2)$: A tunable phase parameter.
  • TRS Breaking: When $\theta = 0$, the hopping is real and symmetric (TRS preserved). When $\theta \neq 0$, the hopping amplitudes to the right and left differ in phase, breaking TRS and inducing chiral propagation.

Methodological Approaches:

1. Analytical Extension of Matrix Inversion Trick (MxIT): The authors extend the MxIT method (originally developed for $\theta=0$ by Nosov et al.) to the TRS-broken case. This technique maps the problem to an effective model with a bounded spectrum, allowing perturbative expansion to converge even in regimes where the original spectrum diverges.
2. Spectral Analysis: They analyze the momentum-space spectrum $\tilde{E}_p$ derived via discrete Fourier transform. The behavior of this spectrum (specifically whether it is unbounded in one or both directions) determines the validity of the localization mechanism.
3. Numerical Simulations:
   • Static: Calculation of eigenstate decay profiles ($|\psi(n)|^2$) to determine localization exponents.
   • Dynamic: Simulation of wavepacket spreading using the generalized $q$-th central moment $\sigma_q(t)$, specifically focusing on the mean-squared displacement $\sigma_2(t)$, to distinguish between ballistic, diffusive, and subdiffusive transport.

3. Key Contributions and Results

A. Static Phase Diagram (Localization-Delocalization Transition)

The authors derive a phase diagram in the $(\theta, a)$ parameter space, identifying three distinct regimes:

1. Regime I (Delocalized): $|\theta| \geq \pi a / 2$ and $a < 1$.
   • All eigenstates are extended (plane-wave-like).
   • Mechanism: The TRS breaking is strong enough that the momentum-space spectrum $\tilde{E}_p$ diverges in both directions (positive and negative momenta). This prevents the construction of the Matrix Inversion Trick (MxIT), leading to ergodic delocalization.
2. Regime II (Correlation-Induced Localized): $|\theta| < \pi a / 2$ and $a < 1$.
   • Eigenstates exhibit algebraic localization with a power-law decay $|\psi(n)|^2 \sim |n-n_0|^{-2(2-a)}$.
   • Robustness: The localization is robust against finite TRS breaking. The critical threshold is $|\theta_c| = \pi a / 2$.
   • Mechanism: For $|\theta| < \theta_c$, the spectrum diverges only in one direction. The MxIT remains valid, mapping the system to an effective model with hopping decaying as $1/r^{2-a}$, which satisfies Levitov's localization criterion.
3. Regime III (Standard Anderson Localized): $a > 1$.
   • Eigenstates decay as $|\psi(n)|^2 \sim |n-n_0|^{-2a}$.
   • Localization occurs for all $\theta$ because the spectrum is bounded for all $\theta$ when $a > 1$.

Key Finding: The correlation-induced localization is robust up to a critical TRS-breaking strength. The transition is driven by the topology of the energy spectrum (one-sided vs. two-sided unboundedness).

B. Dynamical Phase Diagram (Wavepacket Spreading)

The dynamics reveal a richer structure than the static phase diagram, particularly in the interplay between the core and the tails of the wavepacket.

• Time-Reversal Symmetric Case ($\theta = 0$):
  • The wavepacket exhibits subdiffusive growth in the intermediate time regime: $\sigma_2(t) \sim t^{2\beta}$ with $2\beta = \frac{1}{2-a} < 1$.
  • This arises from the interference of time-reversed paths and the presence of a small fraction of high-energy delocalized states.
• TRS-Broken Case ($\theta \neq 0$):
  • Even an infinitesimal breaking of TRS ($\theta \neq 0$) destroys the subdiffusive regime.
  • The dynamics become diffusive ($\sigma_2(t) \sim t$) in the intermediate time window, provided the system is in the localized phase (Regime II).
  • Physical Interpretation: While the core of the wavepacket remains localized (due to the algebraic decay of eigenstates), the tails diffuse. The breaking of TRS suppresses the constructive interference required for subdiffusion, allowing the tails to spread diffusively.

4. Significance and Implications

1. Robustness of Correlation-Induced Localization: The work establishes that localization driven by long-range correlations is not fragile to TRS breaking. It survives up to a well-defined critical phase $\theta_c = \pi a / 2$, challenging the intuition that any TRS breaking immediately leads to delocalization in 1D.
2. Spectral Topology as a Control Parameter: The paper identifies the "direction of unboundedness" of the energy spectrum as the fundamental determinant of the localization-delocalization transition. This provides a new analytical framework for understanding long-range disordered systems.
3. Static vs. Dynamic Decoupling: A crucial insight is the decoupling between static eigenstate properties and dynamical transport.
   • Static: The spatial decay exponent of eigenstates is insensitive to $\theta$ within the localized phase.
   • Dynamic: The transport behavior (subdiffusive vs. diffusive) is highly sensitive to $\theta$. This suggests that experimental probes of transport (e.g., wavepacket spreading) can detect TRS breaking even when the system appears statically localized.
4. Relevance to Physical Systems: The model mimics dipole-dipole interactions in disordered media (e.g., light localization in atomic clouds or exciton transport). The findings suggest that in such systems, breaking time-reversal symmetry (e.g., via magnetic fields or chiral interactions) can induce a transition from subdiffusive to diffusive transport without necessarily delocalizing the entire system immediately.

5. Conclusion

The paper provides a comprehensive analytical and numerical characterization of Anderson localization in 1D systems with long-range correlated hopping and broken TRS. It demonstrates that correlation-induced localization is robust against TRS breaking up to a critical threshold, governed by the spectral properties of the hopping matrix. Furthermore, it highlights a distinct dynamical signature where TRS breaking transforms subdiffusive transport into diffusive transport, offering a potential experimental probe for these correlation-induced phases.