Quantum localization in incommensurate tight-binding chains

This paper investigates quantum localization in two coupled tight-binding chains with incommensurate periods, revealing a mobility edge and demonstrating that while weak magnetic fields enhance localization, strong fields tend to delocalize most states.

The Gist

Imagine you are at a music festival with two long, circular stages placed one above the other. On the top stage (Chain A), there are 2,584 musicians. On the bottom stage (Chain B), there are 1,597 musicians.

Here is the catch: The stages are the same physical size, but the musicians are spaced out differently. Because the numbers of musicians are different, the spacing between them never quite lines up. If you look down from the top, a musician on the top stage will rarely be directly above a musician on the bottom stage. This "mismatch" is what physicists call incommensurability.

This paper explores what happens when these two mismatched stages are connected by invisible bridges (hopping amplitudes) that allow the musicians to pass notes to each other.

The Main Discovery: The "Mobility Edge"

In a normal, perfectly ordered crystal (like a grid of identical musicians), sound waves (or electrons) can travel freely everywhere. In a completely messy, random system, sound gets stuck in one spot (localization).

But this system is special. It's not random, but it's not perfectly ordered either. The researchers found a surprising phenomenon called a Mobility Edge.

Think of the musicians' energy levels as a ladder:

• The Bottom Rungs (Low Energy): The musicians here are like a well-rehearsed choir. They can all sing together, and the sound travels freely across the entire stadium. They are delocalized.
• The Top Rungs (High Energy): Suddenly, the sound gets stuck. The musicians here are like soloists trapped in their own little corners, unable to pass their notes to neighbors. They are localized.

The "Mobility Edge" is the invisible line on the ladder where the sound suddenly stops traveling and gets trapped. The paper shows that this transition is abrupt, like a light switch flipping from "on" to "off," rather than a slow fade.

The Role of the Magnetic Field

The researchers also tested what happens if you turn on a giant magnet around the stadium. The effect depends entirely on the direction of the magnet:

1. Magnet Parallel to the Stages: It's like a gentle breeze blowing along the tracks. It doesn't really change anything. The musicians stay in their same patterns.
2. Magnet Perpendicular (Pointing Down): This is the interesting one.
   • Weak Magnet: It acts like a strict conductor. It actually makes the "trapped" musicians even more stuck. It forces the sound to localize even more tightly.
   • Strong Magnet: If you crank the magnet up too high, it breaks the rules entirely. The "trapped" musicians suddenly break free, and the sound starts traveling everywhere again. The strong magnet delocalizes the system.

Why Does This Happen?

The "trapping" isn't caused by broken equipment or random noise (disorder). It happens purely because of the geometry. The fact that the two rings have different numbers of musicians creates a complex, repeating pattern that naturally confuses the waves, causing them to get stuck at high energies.

Real-World Connection

While this sounds like a thought experiment, the authors suggest this could be tested in real life using polaritons. Imagine a special type of light-matter particle trapped in a tiny, etched semiconductor chip. Scientists can arrange these particles in a grid and control how they jump between spots. By tweaking the chip's design to create the "mismatched" spacing and applying magnetic fields, they could watch this "Mobility Edge" phenomenon happen in real-time.

Summary

• The Setup: Two mismatched rings of atoms.
• The Surprise: High-energy states get stuck (localized) while low-energy states flow freely.
• The Switch: There is a sharp boundary (Mobility Edge) between flowing and stuck states.
• The Magnet: A weak magnetic field makes things more stuck; a strong one frees them up.

It's a beautiful example of how the shape and arrangement of a system can create order out of chaos, trapping energy in specific zones without any random messiness.

Technical Summary

1. Problem Statement

The paper investigates quantum localization phenomena in a specific class of quasi-periodic systems: two coupled tight-binding chains with incommensurate lattice periods.

• Context: While the Anderson model (random disorder) and the Aubry-André (AA) model (quasi-periodic on-site potential) are well-studied, the AA model typically exhibits a global localization transition where all states become localized above a critical potential strength.
• Gap: There is a need to understand localization in systems where incommensurability arises geometrically rather than through an external potential, and to determine if such systems exhibit mobility edges (energy thresholds separating extended and localized states) without an intermediate critical phase.
• Specific Question: How does the inter-chain hopping in a geometrically incommensurate system affect the localization of electronic states, and how do external magnetic fields influence this behavior?

2. Methodology

A. The Model

The authors construct a minimal model consisting of two coupled 1D chains (Chain A and Chain B) arranged as concentric loops (or linear chains with periodic boundary conditions).

• Geometry: The chains have the same physical length but different numbers of atoms ($N_A$ and $N_B$). The ratio of lattice constants $\rho = N_A/N_B$ is set to approximate an irrational number, specifically the Golden Ratio ($\phi \approx 1.618$), using high-order Fibonacci numbers ($N_A=2584, N_B=1597$).
• Hamiltonian: The system is described by a tight-binding Hamiltonian with three components:
  1. Intra-chain hopping ($\hat{H}_A, \hat{H}_B$).
  2. Inter-chain hopping ($\hat{\tilde{H}}$).
• Hopping Mechanism: Unlike standard models with constant nearest-neighbor hopping, this model assumes exponential distance-dependent hopping for all pairs of atoms (both intra- and inter-chain). The hopping amplitude decays as $t \exp(-r/\lambda)$, where $r$ is the Euclidean distance between sites.
  • This creates a "geometrical disorder" in the inter-chain hopping terms because the relative positions of atoms in the two chains vary quasi-randomly due to the incommensurate periods.
• Magnetic Field: The authors incorporate external magnetic fields using the Peierls substitution. They analyze two orientations:
  1. Parallel field ($B_\parallel$): Along the $z$-axis (perpendicular to the chain plane).
  2. Perpendicular field ($B_\perp$): Radial field (approximating a field perpendicular to the chain plane in a cylindrical geometry).

B. Numerical Analysis

• Diagonalization: The authors perform exact diagonalization of the Hamiltonian matrix (size $4181 \times 4181$) to obtain energy eigenvalues ($E_n$) and eigenvectors ($|n\rangle$).
• Localization Measure: The Inverse Participation Ratio (IPR) is used to quantify localization:
  $$\text{IPR}(n) = \sum_{i} |\psi_n(i)|^4$$
  • $\text{IPR} \to 0$ (scaling as $1/N$) indicates extended states.
  • $\text{IPR} \to 1$ indicates localized states.
• Scaling Analysis: To distinguish between extended, localized, and critical states, the authors analyze the scaling of the IPR with system size $N$: $\text{IPR} \sim N^{-D}$.
  • $D=1$: Extended.
  • $D=0$: Localized.
  • $0 < D < 1$: Critical.
• Perturbation Theory: An analytical perturbative analysis (Appendix A) is conducted for weakly coupled chains to explain the origin of energy gaps and the locations of mobility edges.

3. Key Contributions

1. Geometric Incommensurability: The paper demonstrates that incommensurability can arise purely from the geometry of coupled lattices (different atom counts) affecting hopping amplitudes, rather than requiring an external on-site potential modulation (as in the AA model).
2. Existence of Mobility Edges: The study identifies a sharp mobility edge in the energy spectrum. Unlike the AA model where all states localize simultaneously, this system exhibits a coexistence of extended low-energy states and localized high-energy states.
3. Correlation with Spectral Gaps: The authors establish a direct link between the onset of localization and the presence of energy gaps in the spectrum. The mobility edge coincides with these gaps.
4. Magnetic Field Anisotropy: The paper reveals a strong anisotropy in the response to magnetic fields:
   • Parallel fields have negligible effect.
   • Perpendicular fields induce a complex transition: weak fields enhance localization, while strong fields cause delocalization.

4. Results

A. Energy Spectrum and Localization

• Spectral Gaps: As the inter-chain separation $d$ decreases, large energy gaps appear in the spectrum. These gaps are located at indices corresponding to Fibonacci numbers (e.g., $n = N_B$ and $n = N_A$).
• Mobility Edge:
  • Low Energy: States are extended ($D \approx 1$). The wave functions are delocalized across the chains.
  • High Energy: States are localized ($D \approx 0$). The wave functions exhibit sharp peaks and decay exponentially.
  • Transition: The transition from extended to localized is abrupt, occurring exactly at the spectral gaps. There is no intermediate critical phase (unlike the AA model where critical states exist at the transition point).
• Dependence on Parameters:
  • Increasing the inter-chain separation $d$ reduces localization (chains decouple).
  • Increasing the lattice constant $a$ (reducing intra-chain hopping relative to inter-chain) increases the percentage of localized states.
  • Localization is strongest when inter-chain hopping dominates intra-chain hopping.

B. Magnetic Field Effects

• Parallel Field ($B_\parallel$): No significant qualitative change in the IPR distribution or localization properties was observed.
• Perpendicular Field ($B_\perp$):
  • Weak Field: Enhances localization. Some previously extended states become localized, and IPR values for localized states increase.
  • Strong Field: Causes delocalization. As the field strength increases further, the majority of states (including those previously localized) become delocalized, eventually leading to a fully extended spectrum.

C. Perturbative Insight

The analytical appendix explains that the spectral gaps arise from "small denominators" in the perturbation expansion.

• Inter-chain coupling mixes states from the two chains.
• Degeneracies occur when $E_A(k_A) = E_B(k_B)$.
• Due to incommensurability, there are infinite solutions for momentum matching, but the matrix elements decay rapidly with the "Umklapp" order.
• The gaps appear at specific wave vectors determined by the ratio $\rho$, which correspond to the indices where the mobility edge is observed numerically.

5. Significance and Implications

• Theoretical Novelty: This work provides a new mechanism for generating mobility edges in 1D systems without introducing explicit disorder or on-site potentials. It challenges the universality of the AA model's critical phase by showing that geometric incommensurability can lead to a direct, abrupt transition between extended and localized phases.
• Experimental Relevance: The authors suggest that exciton-polariton systems in semiconductor micro-cavities are ideal platforms for testing these predictions. These systems naturally support tight-binding models with exponentially decaying hopping and can simulate effective gauge potentials (magnetic fields) via synthetic fields.
• Magnetic Control: The finding that a perpendicular magnetic field can tune a system from localized to delocalized (and vice versa) offers a potential control knob for quantum transport in quasi-periodic materials.
• Future Directions: The paper opens avenues for studying many-body localization (MBL) in such geometrically incommensurate systems and exploring the effects of orbital anisotropy (p-wave, d-wave) on localization.

In summary, the paper establishes that geometric incommensurability in coupled chains is a robust source of quantum localization, characterized by abrupt mobility edges and a unique, non-monotonic response to perpendicular magnetic fields.