Disorder averaging in random lattice models with periodic boundary conditions: Application to models with uncorrelated and correlated disorder

This paper develops disorder averaging techniques within the modern theory of polarization to calculate polarization cumulants and delocalization indicators for random lattice models with periodic boundary conditions, validating the method on fully localized Anderson and power-law correlated de Moura-Lyra models while highlighting significant pairwise degeneracies in the latter.

The Gist

The Big Picture: Finding the "Traffic Jam" in a Random City

Imagine you are trying to understand how electricity moves through a material. In a perfect crystal (like a diamond), the atoms are arranged in a neat, repeating grid, and electrons flow like cars on a smooth, empty highway. This is a conductor.

But in real materials, things are messy. There are impurities, missing atoms, or random bumps in the road. This is disorder. When the disorder gets bad enough, the electrons get stuck, unable to move past a certain point. This is localization (or an insulator).

The big question physicists have asked for decades is: Where is the line between the flowing traffic and the total traffic jam? And, if the road is messy in a specific, correlated way (like a pattern of potholes), does that change where the jam happens?

This paper introduces a new, high-tech "GPS" to find that line, even when the road is chaotic.

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The Problem: The "Periodic Boundary" Paradox

Usually, to study these materials, scientists simulate a giant, infinite road. But computers can't handle infinity, so they simulate a small loop (a circular track).

Here's the catch: In a perfect circle, you can't easily measure how "stuck" the electrons are using old-school methods. It's like trying to measure the length of a race track by running in circles; you never reach a finish line to see how far you got.

The authors use a modern theory called the Modern Theory of Polarization. Instead of asking "Where is the electron?", they ask, "What is the shape of the electron's wave?" They treat the electron's position not as a specific point, but as a geometric phase—think of it like the angle of a compass needle. If the needle spins wildly, the electron is free (metal). If the needle is stuck pointing one way, the electron is trapped (insulator).

The New Tool: Disorder Averaging

The paper's main innovation is a technique to handle randomness.

Imagine you are trying to predict the weather. You can't just look at one day; you need to look at 1,000 different days and average the results to see the pattern.

• The Old Way: Scientists would look at one specific random arrangement of atoms, calculate the result, and hope it represented the whole system.
• The New Way (This Paper): The authors developed a method to simulate thousands of different "random cities," calculate the "traffic jam" score for each, and average them out. This gives a much clearer picture of the true physics, ignoring the noise of any single bad luck scenario.

They also created a new "Delocalization Indicator." Think of this as a degeneracy detector.

• In a trapped system (insulator), the energy levels are like distinct, separate rungs on a ladder.
• In a free system (metal), the rungs sometimes merge or come incredibly close together.
• The authors found that by slightly twisting the rules of the road (changing boundary conditions), they could see if these rungs merged. If they did, the system was likely conducting electricity.

The Experiments: Two Test Cases

The team tested their new GPS on two different types of "roads":

1. The Anderson Model (The "Random Pothole" Road)

This is the classic model where every bump on the road is completely random and unrelated to the next.

• The Result: As expected, when the bumps get big enough, the electrons get stuck.
• The Surprise: When they looked at a crowd of electrons (many-body system) instead of just one, the "traffic jam" behaved differently. It was harder to localize a crowd than a single car. This confirms that electrons interact with each other, which changes how they get stuck.

2. The de Moura-Lyra Model (The "Patterned Pothole" Road)

This is a more complex road where the bumps aren't random; they follow a specific mathematical pattern (power-law correlation).

• The Controversy: For years, scientists argued about this model. Some said there was a "Mobility Edge"—a specific zone where some electrons could flow while others were stuck. Others said the whole road eventually gets stuck or free depending on the pattern.
• The Verdict: The authors' new method clarified the picture.
  • Global Transition: They confirmed that for a certain parameter (called $\alpha$), the whole system switches from stuck to free. There isn't a complex "half-stuck" zone for the whole system.
  • The "Pairing" Anomaly: However, they found something weird near the center of the energy band when the pattern is strong ($\alpha > 2$). The energy levels start to pair up and merge.
  • The Analogy: Imagine a dance floor. Usually, people dance alone. But in this specific zone, everyone suddenly finds a partner and dances in pairs. If you have an odd number of people, one person is left out, and the "dance" (conductivity) behaves differently than if you have an even number. This explains why previous studies were confused; the answer depends on whether you have an even or odd number of electrons!

Why Does This Matter?

1. Better Materials: Understanding how disorder affects conductivity helps us design better batteries, supercapacitors, and electronic devices.
2. Solving Old Mysteries: They settled a long-standing debate about the de Moura-Lyra model, showing that the "mobility edge" isn't a simple line but a complex region where electrons pair up.
3. A New Toolkit: They gave the physics community a new set of mathematical tools (the "geometric cumulants" and "degeneracy indicators") to study messy, disordered systems without needing to simulate infinite sizes.

The Takeaway

The authors built a sophisticated statistical microscope. By averaging out the chaos of random materials and looking at the "shape" of electron waves, they proved that while disorder usually stops electricity, the way electrons interact with each other and pair up can create surprising new behaviors. They didn't just find the traffic jam; they figured out exactly why the cars are stuck and how the pattern of the road changes the rules of the game.

Technical Summary

1. Problem Statement

The study of disordered systems, particularly regarding the transition between localized (insulating) and delocalized (metallic) phases, faces significant challenges when using Periodic Boundary Conditions (PBC).

• The Core Issue: In standard quantum mechanics, polarization is often treated as an operator expectation value. However, under PBC, the position operator is ill-defined due to the periodicity of the lattice. Consequently, standard measures of localization (like the variance of the position operator) cannot be directly calculated.
• The Gap: While the Modern Theory of Polarization (MPT) successfully treats polarization in crystalline systems as a geometric phase (Berry phase), its application to disordered systems via disorder averaging has been limited. Existing methods often struggle to calculate higher-order statistical moments (like variance and kurtosis) and the geometric Binder cumulant in a way that is robust against disorder fluctuations and finite-size effects.
• Specific Models: The paper addresses two specific models:
  1. The standard Anderson model (uncorrelated disorder), which is fully localized in 1D.
  2. The de Moura-Lyra Model (dMLM), which features power-law correlated disorder. This model is historically controversial, with conflicting literature regarding the existence of a "mobility edge" (an energy separating localized and extended states) for correlation parameters $\alpha > 2$.

2. Methodology

The authors develop a rigorous framework for disorder averaging within the context of MPT, utilizing PBC.

• Geometric Phase Formalism:
  • Polarization is defined via the polarization amplitude $Z_q = \langle \Psi | \exp(i \frac{2\pi q}{L} \hat{X}) | \Psi \rangle$, where $\hat{X}$ is the total position operator.
  • $Z_q$ acts as a discrete characteristic function for the probability distribution of the total position.
• Statistical Moments and Cumulants:
  • Since the argument of $Z_q$ is discrete, continuous derivatives cannot be used. The authors employ finite difference approximations to define approximate moments ($\tilde{M}_n$) and cumulants.
  • Centering: To isolate fluctuations due to disorder rather than system displacement, the authors take the absolute value $|Z_q|$, effectively centering the distribution.
  • Key Quantities Derived:
    • Variance ($\tilde{M}_2$): Serves as a gauge for localization.
    • Size Scaling Exponent ($\gamma$): Defined by fitting $\tilde{M}_2 \propto L^\gamma$.
    • Geometric Binder Cumulant ($U_4$): Derived from the fourth moment and variance ($U_4 = 1 - \frac{1}{3}\frac{\tilde{M}_4}{\tilde{M}_2^2}$). This is a critical indicator of critical points and phase transitions.
• Disorder Averaging Strategy:
  • The authors generate thousands of independent disorder realizations.
  • They calculate the geometric quantities for each realization, then average them over energy intervals (for single-particle states) or particle densities (for many-body systems).
• Many-Body Extension:
  • For interacting (non-interacting fermion) systems, the ground state is constructed as a Slater determinant of the $N$ lowest energy single-particle states.
  • The polarization amplitude is calculated using the determinant of a matrix of single-particle overlaps.
• Degeneracy Indicator:
  • A novel indicator is introduced based on Peierls phases. By applying a phase shift $\Phi = \pi/L$ (switching between periodic and anti-periodic boundary conditions), the authors check for ground state degeneracy.
  • In the metallic (delocalized) phase, the ground state can become degenerate depending on the parity of particle number and system size, causing $|Z_1|$ to shift. In the insulating phase, $|Z_1|$ remains stable. This difference serves as a sensitive probe for delocalization.

3. Key Contributions

1. Formalism Development: Successfully extended MPT to disordered systems by deriving explicit formulas for disorder-averaged variance, higher-order cumulants, and the geometric Binder cumulant using finite difference derivatives.
2. Validation: Validated the method against the 1D Anderson model, confirming that it correctly identifies the fully localized nature of the system (scaling exponent $\gamma \approx 0$ or $1$ depending on the regime, and $U_4$ behavior).
3. Resolution of the dMLM Controversy: Provided a definitive numerical analysis of the de Moura-Lyra model, clarifying the nature of the "mobility edge" and the global delocalization transition.
4. Many-Body Insights: Demonstrated that many-body localization properties differ from single-particle properties due to Fermi statistics (Pauli exclusion acting as short-range correlation), specifically showing different scaling behaviors for $\gamma$ in many-body systems compared to single-particle ones.

4. Key Results

A. Anderson Model (Uncorrelated Disorder)

• Single-Particle: The size scaling exponent $\gamma$ drops significantly below 1 (approaching 0) as disorder strength $W$ increases, confirming strong localization.
• Many-Body: For fermions, the localized phase exhibits $\gamma \approx 1$ (finite dielectric susceptibility), jumping to $\gamma \approx 2$ upon delocalization.
• Binder Cumulant: $U_4$ remains positive but does not reach the metallic limit of 0.5 in the localized regime, consistent with insulating behavior.

B. de Moura-Lyra Model (Correlated Disorder)

• Global Delocalization Transition: The results confirm a global delocalization transition at $\alpha \approx 1$. For $\alpha > 1$, all states are delocalized, contradicting the existence of a mobility edge in the traditional sense.
• The $\alpha > 2$ Anomaly:
  • In the region $\alpha > 2$ near the band center (half-filling), the authors discovered pairwise degeneracies in the energy spectrum.
  • Energy levels close gaps in pairs as $\alpha$ increases.
  • Many-Body Effect: When the system is filled with fermions, this leads to a stark difference between odd and even particle numbers.
    • Odd particles (half-filled bands): The delocalization indicators ($\gamma$ and $U_4$) reach their maximum values, indicating strong delocalization.
    • Even particles: The indicators are suppressed, showing less pronounced delocalization.
• Phase Diagram: The authors propose a refined phase diagram for dMLM:
  • $\alpha < 1$: Localized phase.
  • $\alpha > 1$: Delocalized phase.
  • Sub-phase for $\alpha > 2$: A distinct region near the band center characterized by persistent gap closures and parity-dependent delocalization sensitivity.

5. Significance

• Methodological Advancement: The paper provides a robust toolkit for studying disordered systems with PBC, which are computationally advantageous for mimicking thermodynamic limits but historically difficult to analyze for polarization properties.
• Clarification of Correlated Disorder: It resolves long-standing contradictions in the literature regarding the de Moura-Lyra model. It demonstrates that while a global delocalization transition exists at $\alpha=1$, the nature of the delocalized phase for $\alpha > 2$ is complex, involving spectral degeneracies that are only visible through many-body calculations and specific boundary condition probes.
• Connection to Many-Body Localization (MBL): By explicitly treating finite particle numbers and Slater determinants, the work bridges the gap between single-particle localization theories and many-body phenomena, highlighting how Fermi statistics interact with disorder to alter transport properties.
• Experimental Relevance: The findings are relevant for understanding materials with controlled correlated disorder (e.g., supercapacitors, metal-organic frameworks, and DNA strands), where the interplay between correlation length and system size dictates electronic transport.