A Real-Space Formulation of the Zak Phase via Weyl m-Functions

This paper establishes a novel real-space formula for the Zak phase of one-dimensional periodic Jacobi operators using Weyl $m_+$-functions, which bypasses Floquet-Bloch theory to explicitly highlight boundary term dependencies and recover classical quantization under inversion symmetry.

The Gist

Imagine you are walking along a long, repeating path made of stepping stones. Every few steps, the pattern of the stones repeats itself. In the world of quantum physics, this path is a "material," and the stones are atoms. Electrons move along this path like waves.

For a long time, scientists have used a specific map (called Floquet-Bloch theory) to understand how these electron waves behave. This map requires you to look at the entire infinite path at once and calculate a "phase" (a kind of angle or twist in the wave) as you go around the whole loop. This is called the Zak phase. It's a crucial number that tells us if the material is "topologically" special—meaning it might have special, protected states at its edges (like a road that forces traffic to the side).

The Problem:
Usually, to calculate this Zak phase, you need to know the "global" picture of the infinite path. It's like trying to calculate the total twist of a ribbon by looking at the whole ribbon at once. It's mathematically heavy and relies on the idea of an infinite, repeating world.

The New Discovery:
The authors of this paper, Habib Ammari and Clemens Thalhammer, found a clever shortcut. They discovered a way to calculate this same "twist" (the Zak phase) by looking only at the very edge of the material, without needing to see the whole infinite path or use the old "global map."

The Analogy: The "Edge Impedance" Meter
Think of the material as a long hallway.

• The Old Way: To know how the hallway twists, you had to walk the entire length, measure the angle at every step, and add them all up.
• The New Way: The authors found a special "meter" (called the Weyl m-function) that you can plug in right at the doorway (the edge).

This meter measures something called "surface impedance." In our analogy, imagine the hallway has a specific "resistance" to how waves bounce off the door. The authors proved that if you measure this resistance at the door and track how it changes as you tune the energy of the wave, you can calculate the total twist of the entire hallway.

How it Works (The Magic Trick):

1. Real Space vs. Momentum: The old method worked in "momentum space" (a mathematical world of frequencies). The new method works in "real space" (the actual physical location of the atoms).
2. The Formula: They derived a formula where the Zak phase is just an integral (a sum) of how this edge-meter behaves as you move through the energy levels.
3. The Surprise: This formula shows that the Zak phase isn't just a property of the "bulk" (the middle) of the material. It actually depends on boundary terms—how you cut the material or where you place the edge. It's like saying the total twist of a rope depends on how you hold the ends, not just how the rope is knotted in the middle.

What They Tested:
To prove their new formula works, they tested it on two famous models:

• The SSH Model: A classic model of a chain of atoms with alternating strong and weak links. Their new formula gave the exact same answer as the old, complicated method.
• The Rice-Mele Model: A more complex version with uneven energy levels. They used computers to show their new formula matched the standard results perfectly.

The "Mirror" Discovery:
The paper also looked at materials that are perfectly symmetrical (like a mirror image of themselves). In these cases, the Zak phase is usually "quantized," meaning it can only be 0 or $\pi$ (like a light switch being either off or on).
Using their new edge-based formula, they showed why this happens. Because of the mirror symmetry, the "edge meter" behaves in a very specific, predictable way that forces the total twist to snap to these specific values. They did this using only the geometry of the edge, without needing the complex global maps.

In Summary:
This paper provides a new, simpler way to calculate a fundamental property of 1D quantum materials. Instead of needing to analyze the entire infinite system, you can now calculate the "twist" of the system just by looking at the behavior of waves at the edge. It connects the abstract math of "spectral theory" (how waves resonate) directly to the physical reality of boundaries, offering a fresh perspective on how topological materials work.

Technical Summary

Technical Summary: A Real-Space Formulation of the Zak Phase via Weyl m-Functions

Problem Statement
The Zak phase is a fundamental concept in the study of topological properties of one-dimensional quantum systems, serving as a geometric phase accumulated by Bloch wave functions across the Brillouin zone. It provides a critical link between bulk material properties and edge phenomena, known as the bulk-boundary correspondence. Traditionally, the calculation of the Zak phase relies on Floquet-Bloch theory and momentum-space formulations involving the Berry connection of eigenfunctions. While the Zak phase is generally defined within $[0, 2\pi)$, it becomes quantized in systems possessing specific symmetries, such as inversion or chiral symmetry.

A gap exists in the literature regarding a formulation of the Zak phase that operates entirely within real space, independent of momentum-space representations. Furthermore, the dependence of the Zak phase on boundary conditions and the choice of the fundamental cell is often implicit in standard formulations. This paper addresses the need for a real-space representation that utilizes the spectral theory of discrete operators, specifically Jacobi operators, to express the Zak phase without relying on Floquet-Bloch theory.

Methodology
The authors focus on one-dimensional periodic Jacobi operators $J$ acting on $\ell^2(\mathbb{Z})$, defined by diagonal elements $b(n)$ and off-diagonal elements $a(n)$. The methodology centers on the spectral theory of these operators, specifically the use of Weyl m-functions.

1. Weyl m-Functions: The paper utilizes the Weyl m-functions, $m_+(\lambda, n_0)$ and $m_-(\lambda, n_0)$, which are defined via the resolvent of semi-infinite Jacobi operators ($J_+$ and $J_-$) or as the Borel transform of the associated spectral measures. These functions encode detailed spectral information, including the density of states and spectral gaps.
2. Analytic Continuation: The proof strategy involves extending the Berry connection, $A_n(k) = -i\langle u_n(k), \partial_k u_n(k) \rangle$, into the complex plane by considering complex Bloch momenta $k + i\varepsilon$. This allows the authors to express the eigenvectors and their derivatives in terms of the resolvent of the half-line operator $J_+$.
3. Spectral Integrals: By leveraging functional calculus, the authors relate the inner products involving the eigenvectors and their derivatives to specific spectral integrals ($I_0$ through $I_4$) involving the spectral measure of $J_+$.
4. Asymptotic Analysis: The derivation proceeds by taking the limit as the imaginary part of the energy approaches zero ($\varepsilon \to 0$). This involves analyzing the behavior of the spectral integrals near the real axis, distinguishing between the absolutely continuous spectrum (bands) and point spectrum.

Key Contributions and Results

• Real-Space Formula for the Zak Phase: The primary result is Theorem 3.1, which establishes a new formula for the Zak phase $\gamma_n$ of the $n$-th isolated band in terms of the Weyl m-function $m_+$. The formula is given by:
  $$\gamma_n = \int_{\lambda_{n,\min}}^{\lambda_{n,\max}} \frac{\Re(m'_+(\lambda + i0))}{\Im(m_+(\lambda + i0))} d\lambda - \pi$$
  where $\lambda_{n,\min}$ and $\lambda_{n,\max}$ are the lower and upper edges of the spectral band. This formulation is entirely in real space, relying only on the boundary values of the resolvent (via the m-function) and the spectral density, without explicit reference to Bloch momentum $k$.

• Dependence on Boundary Conditions: The authors highlight that this real-space representation explicitly demonstrates the dependence of the Zak phase on the choice of the fundamental cell (boundary conditions). The value of the Zak phase is shown to be sensitive to the specific definition of the Weyl m-function at the boundary.

• Recovery of Quantization: The paper demonstrates that for periodic Jacobi operators with inversion-symmetric fundamental cells, the Zak phase is quantized to integer multiples of $\pi$. Using the new formula (3.3) and the property that the modulus of the Weyl m-function is constant along the band for symmetric cells, the authors derive this quantization using only real-space symmetry arguments, avoiding momentum-space parity arguments typically used for Wannier functions.

• Validation via Examples:

  • Su–Schrieffer–Heeger (SSH) Model: The authors analytically compute the Weyl m-function for the SSH model and show that the new formula reproduces the standard Zak phase values ($0$ or $\pi$) distinguishing trivial and topological phases.
  • Rice-Mele Model: For the Rice-Mele model (which breaks inversion and chiral symmetry), the authors perform a numerical evaluation of the new formula. The results show excellent agreement with standard discrete implementations of the Zak phase, validating the computational utility of the approach.
  • Mirror Symmetric Cells: A numerical example of a symmetric trimer setup confirms that the new formula yields a value close to $\pi$, consistent with the theoretical prediction of quantization.

Significance and Claims
The paper claims that this work provides a novel computational tool and a deeper conceptual understanding of the Zak phase. By rephrasing the result in the language of dynamical systems and spectral theory (Weyl m-functions), the authors bridge the abstract mathematical framework of Jacobi operators with physically observable quantities like edge modes.

The significance lies in:

1. Independence from Floquet-Bloch Theory: The formulation does not require the construction of Bloch eigenfunctions or integration over the Brillouin zone, offering an alternative perspective rooted in real-space spectral properties.
2. Explicit Boundary Dependence: It clarifies that the Zak phase is not purely a bulk property but depends on boundary choices, a feature often implicit in traditional formulations.
3. Connection to Surface Impedance: The authors note a link between the Weyl m-functions and surface impedance, suggesting a physical interpretation of the real-space formula in terms of field-to-flux ratios.

The authors remain modest regarding the interpretation of the formula as a geometric or topological phase, noting that the precise topological interpretation of equation (3.3) remains an open question for future conceptual investigation. The work is presented as a rigorous mathematical derivation that aligns with and recovers classical results while offering a new real-space perspective.