Information Theoretic Signatures of Localization and Mobility Edges in Quasiperiodic Systems

This paper introduces an information-theoretic framework based on Tsallis entropy and its energy-dependent gradient susceptibility to effectively distinguish between uniform localization transitions and mobility edge phenomena in one-dimensional quasiperiodic lattice models by identifying sharp, system-size-independent spectral variations unique to the latter.

The Gist

Imagine you are standing in a vast, crowded concert hall. Your goal is to figure out how the crowd is behaving. Are they all dancing freely in the open space? Are they all huddled in tight, isolated groups? Or is it a mix: some people dancing wildly in the aisles while others are stuck in small, cramped corners?

In the world of quantum physics, this "crowd" is made of electrons (or waves) moving through a material. The "dancing" is their ability to move around (called delocalization), and the "huddling" is them getting stuck in one spot (called localization).

For a long time, physicists had a standard way to check this: the Inverse Participation Ratio (IPR). Think of the IPR as a camera that takes a photo of one single person in the crowd and asks, "Are you moving or stuck?" It's good at checking individuals, but it struggles to tell you about the whole room at once. If the room is a chaotic mix of movers and stayers, the IPR gets confused, and the answer depends heavily on how big the room is.

The New Idea: Listening to the "Entropy" of the Room

This paper introduces a new, smarter way to listen to the crowd using a concept called Tsallis Entropy.

Think of Entropy as a measure of "disorder" or "spread."

• High Entropy: The crowd is spread out evenly everywhere (like a gas filling a room).
• Low Entropy: The crowd is clumped together in a few spots.

The authors use a special "knob" on their measuring device, called $q$.

• If you turn the knob to $q > 1$, the device becomes very sensitive to the clumps. It shouts, "Look! There's a huge group stuck in the corner!"
• If you turn the knob to $q < 1$, the device becomes sensitive to the scattered individuals. It whispers, "Hey, look at those few people wandering far away."

By turning this knob, the researchers can tune their sensitivity to see different parts of the crowd's behavior.

The Big Discovery: The "Slope" of the Room

The real magic of this paper isn't just measuring the crowd; it's measuring how the crowd changes as you move from one end of the room to the other.

The authors invented a new tool called Entropy-Gradient Susceptibility. Imagine walking down the aisle of the concert hall and measuring the "spread" of the crowd at every step.

• Scenario A: The "All-or-Nothing" Hall (The AA Model)
  Imagine a room where, if the music gets loud enough, everyone suddenly stops dancing and sits down at the exact same time.

  • The Measurement: As you walk down the aisle, the "spread" of the crowd changes smoothly. It goes from "everyone dancing" to "everyone sitting" gradually.
  • The Result: Your measuring device shows a gentle, smooth hill. No sharp spikes. This tells you: "This is a uniform transition. Everyone is doing the same thing."

• Scenario B: The "Mixed" Hall (The Mobility Edge)
  Now imagine a different room. As the music gets louder, the people in the front of the hall sit down, but the people in the back keep dancing. There is a sharp line (a "Mobility Edge") separating the stuck group from the moving group.

  • The Measurement: As you walk down the aisle, you suddenly hit a wall where the crowd behavior changes drastically. One second it's "all dancing," the next it's "all sitting."
  • The Result: Your measuring device sees a massive, sharp spike (a peak) right at that boundary. It screams, "Something is changing here! We have a mix of two different worlds!"

Why This Matters

The authors tested this on three different types of "rooms" (mathematical models):

1. The AA Model: Everyone transitions together. The new tool showed a smooth curve, correctly identifying there is no "Mobility Edge."
2. The SSH Model & GAA Model: These have a mix of stuck and moving states. The new tool showed a sharp, distinct peak.

The Key Advantage:
The old tools (like the IPR) are like trying to guess the size of a room by counting people in a single corner; if the room gets bigger, your count changes wildly.
The new tool (Entropy-Gradient Susceptibility) is like looking at the shape of the crowd distribution. It gives a clear, sharp answer that doesn't change even if you make the room (the system) much larger. It is robust, reliable, and works for a wide range of settings (the $q$ knob).

The Takeaway

This paper gives physicists a new, sharper pair of glasses. Instead of just asking "Is this electron stuck?", they can now ask, "How does the behavior of electrons change as we look at different energy levels?"

If the answer is a smooth slide, it's a simple transition. If the answer is a sharp cliff (a peak in their new graph), they have found a Mobility Edge—a fascinating state where order and chaos coexist in the same material. This helps us understand complex materials better, potentially leading to new technologies in electronics and quantum computing.

Technical Summary

1. Problem Statement

The paper addresses the challenge of distinguishing between global localization transitions (where all eigenstates in a spectrum localize simultaneously) and mobility edge (ME) phenomena (where localized and extended eigenstates coexist at different energies within the same spectrum) in one-dimensional quasiperiodic systems.

• Limitations of Conventional Diagnostics: Standard measures like the Inverse Participation Ratio (IPR) or Lyapunov exponents characterize individual eigenstates but struggle to capture spectral heterogeneity (the coexistence of phases).
  • The mean IPR is highly sensitive to system size and rare states, requiring complex finite-size scaling.
  • These measures often fail to provide a direct, robust signal for the specific energy range where mobility edges exist without indirect analysis.
• Goal: To develop an information-theoretic framework that directly probes spectral heterogeneity and provides a size-stable diagnostic for identifying mobility edges.

2. Methodology

The authors propose a novel diagnostic tool based on Tsallis entropy and its spectral variation.

A. Theoretical Framework

1. Tsallis Entropy ($S_q$): Instead of standard Shannon entropy, the authors use the non-extensive Tsallis entropy for a single-particle eigenstate $\psi$:
   $$S_q = k_B \frac{1 - \sum_i p_i^q}{q - 1}$$
   where $p_i = |\psi_i|^2$ is the probability distribution of the wavefunction amplitude, and $q$ is the entropic index.

   • Tunability: The parameter $q$ acts as a tunable probe.
     • $q > 1$: Enhances sensitivity to high-probability components (localized peaks).
     • $q < 1$: Enhances sensitivity to low-probability components (extended tails/rare states).
   • Normalization: A normalized entropy $\tilde{S}_q$ is defined to range between 0 (localized) and 1 (delocalized), allowing comparison across system sizes.

2. Entropy-Gradient Susceptibility ($\chi_q$):
   The core innovation is defining a susceptibility based on the energy dependence of the entropy:
   $$\chi_q = \left\langle \left| \frac{d\tilde{S}_q(E)}{dE} \right| \right\rangle_E$$
   This quantity measures the fluctuation of entropy across the energy spectrum.

   • Physical Intuition: In a system with a mobility edge, the spectrum contains a sharp transition from extended states (high entropy) to localized states (low entropy). The gradient of this transition creates a large value for $\chi_q$. In a global transition, the entropy changes smoothly across the spectrum, resulting in a low, broad susceptibility.

B. Models Studied

The framework is tested on three distinct 1D quasiperiodic models:

1. Aubry–André (AA) Model: Exhibits a global localization transition at $\lambda_c = 2t$. All states localize simultaneously; no mobility edge.
2. Quasiperiodic SSH Chain: A dimerized Su–Schrieffer–Heeger chain with onsite quasiperiodic potential. Known to host mobility edges.
3. Generalized Aubry–André (GAA) Model: An extension of the AA model with an exact analytical mobility edge condition ($\alpha E = 2t - \lambda$).

3. Key Results

A. Distinguishing Global Transitions vs. Mobility Edges

• AA Model (Global Transition):
  • The normalized entropy $\tilde{S}_q$ decreases monotonically as the potential strength $\lambda$ increases.
  • The entropy-gradient susceptibility $\chi_q$ shows only a broad, smooth crossover near the critical point. There is no sharp peak, reflecting the homogeneous nature of the transition.
• SSH and GAA Models (Mobility Edges):
  • In the regime where localized and extended states coexist, $\tilde{S}_q(E)$ exhibits a sharp step-like change across the energy spectrum.
  • Consequently, $\chi_q$ develops a pronounced, sharp peak at intermediate potential strengths.
  • This peak accurately tracks the energy-resolved localization and the boundaries of the mobility edge regime.

B. Robustness and System Size Independence

• Finite-Size Scaling: Unlike the IPR variance, which exhibits strong system-size dependence and sensitivity to rare states, the peak position of $\chi_q$ remains invariant with increasing system size ($L=600, 800, 1000$).
• Peak Sharpness: While the peak height increases and sharpens with larger $L$ (due to better resolution of the transition), the location of the peak remains stable, confirming it as a thermodynamic signature.

C. Dependence on the Entropic Index ($q$)

• The mobility-edge signature (the peak in $\chi_q$) persists across a broad range of $q$ values ($q \geq 1$).
• Role of $q$:
  • Increasing $q$ ($q > 1$) sharpens the contrast between localized and extended regions, leading to a narrower and more pronounced peak.
  • Decreasing $q$ ($q < 1$) broadens the peak due to the enhanced weighting of rare states, but the peak position remains largely stable for $q \geq 1$.
• This confirms that the observed signature is an intrinsic property of the spectral structure, not an artifact of a specific moment (like $q=2$, which relates algebraically to IPR).

D. Analytical Insights

The authors provide an analytical derivation showing that in the mobility-edge regime, the spectrum can be approximated as two populations: extended states ($S_E \approx 1$) and localized states ($S_L \approx 0$).

• The susceptibility scales as: $\chi_q \propto \frac{f(1-f)|S_E - S_L|}{\Delta E}$, where $f$ is the fraction of extended states and $\Delta E$ is the transition width.
• The susceptibility is maximized when $f \approx 0.5$ (equal coexistence of phases), explaining the peak. In global transitions, $f$ changes uniformly, preventing such a peak.

4. Significance and Contributions

1. New Diagnostic Tool: The paper introduces the entropy-gradient susceptibility, a derivative-based measure that directly quantifies spectral heterogeneity. It complements state-resolved measures like IPR by focusing on the collective structure of the spectrum.
2. Resolution of Mobility Edges: It provides a robust, size-stable method to identify mobility edges without the need for complex finite-size scaling or sensitivity to rare states, which plagues traditional methods.
3. Information-Theoretic Perspective: The work bridges statistical mechanics and condensed matter physics by demonstrating that fluctuations in information-theoretic quantities (Tsallis entropy) serve as a physical probe for quantum phase transitions.
4. Experimental Relevance: The authors suggest that this framework is experimentally accessible in platforms like ultracold atomic gases and photonic waveguide arrays, where wavefunction spatial distributions can be imaged to reconstruct spectral entropy variations.

Conclusion

The paper establishes that spectral heterogeneity in quasiperiodic systems manifests as a sharp peak in the energy-gradient of Tsallis entropy. This "entropy-gradient susceptibility" offers a superior, tunable, and robust diagnostic for distinguishing between global localization and mobility-edge physics, overcoming the limitations of conventional participation-ratio-based methods.