Analysis of the singular band structure occurring in one-dimensional topological normal and superfluid fermionic systems: A pedagogical description

This paper provides a pedagogical analysis of the singular band structures in one-dimensional topological normal and superfluid fermionic systems, demonstrating how nonanalyticities in Bloch eigenfunctions lead to obstructions in symmetric Wannier functions and bulk-boundary correspondence through detailed examinations of both noninteracting and interacting cases.

The Gist

Imagine you are trying to organize a massive library of books (the electrons in a material). In a normal library, you can easily sort the books into neat, compact shelves where every book is close to its neighbors. This is what physicists call a "trivial" state: everything is orderly, localized, and easy to find.

But in some special materials, the "books" refuse to stay on the shelves. They want to stretch out across the entire library, or they seem to be glued to the walls instead of the shelves. This is the "topological" state.

This paper is a "user manual" for understanding why this happens in two specific types of one-dimensional systems (think of them as long, single-file lines of particles). The authors, Marcello and Giancarlo Calvanese Strinati, use two different stories to explain the same deep mathematical mystery.

Here is the breakdown in simple terms:

The Two Stories

The paper compares two very different physical systems to show they actually behave the same way:

1. The Superfluid Story (The Dancing Pair): Imagine a line of dancers (fermions) who suddenly decide to pair up and waltz together (forming a superfluid). They are holding hands and moving in a specific rhythm.
2. The Ladder Story (The Two-Track Train): Imagine a train track with two parallel rails. One rail has "spherical" cars (round), and the other has "peanut-shaped" cars (oval). The cars can hop between the rails.

The Core Problem: The "Twist" in the Map

In physics, we describe these particles using a map called the Brillouin Zone. Think of this map as a circle. As you walk around this circle, the "state" of the particle (its wave function) changes smoothly, like a color gradient.

• The Trivial Phase (The Flat Sheet): In a normal state, if you walk around the circle, the color changes smoothly and returns to the start without any issues. You can fold this map into a neat, compact package (a localized Wannier function). The particles stay close to their home spots.
• The Topological Phase (The Möbius Strip): In a special state, something weird happens. As you walk around the circle, the "color" or phase of the particle gets twisted. By the time you get back to the start, the particle is "upside down" relative to where it started.
  • The Analogy: Imagine a rubber band. If you twist it once and tape the ends together, you get a Möbius strip. You can't flatten a Möbius strip without tearing it. Similarly, you can't "fold" the particle's wave function into a neat, compact package. It is obstructed.

The "Quantum Critical Point" (The Tipping Point)

The paper studies what happens when you slowly change a dial (the chemical potential) on these systems.

• Before the dial: The particles are happy and compact (Trivial).
• Turning the dial: You hit a "Quantum Critical Point" (QCP). This is like a phase transition, similar to water turning to ice, but happening at absolute zero temperature.
• After the dial: The particles are now "stretched out" and messy (Topological).

At this exact tipping point, the energy gap between the particles closes, and the "map" of the system develops a sharp kink or discontinuity. This kink is the mathematical reason why the particles can no longer stay localized.

The Consequences: Why Should We Care?

The authors show that this "kink" in the math has real-world consequences:

1. The "Long Tail" Effect:

   • In the Trivial phase, the probability of finding a particle drops off exponentially as you move away from its home. It's like a flashlight beam that fades quickly.
   • In the Topological phase, the probability drops off much slower, like a power law (a gentle slope). The particle's "shadow" stretches far out into the distance. It's like a flashlight that never really turns off.

2. The "Interstitial" Shift (The Bulk-Boundary Correspondence):

   • Because the particles are so "twisted," they can't sit comfortably on the main lattice sites (the shelves).
   • To fix the math, you have to shift the center of the particle's location to the empty space between the atoms (the interstitial positions).
   • The Metaphor: It's like trying to park a car in a garage. In a normal garage, the car fits perfectly. In a topological garage, the car is too twisted to fit in the spot, so it ends up parked halfway between two spots, or sticking out the door. This "sticking out" is what creates the famous surface states (like the edge currents in topological insulators) that make these materials so useful for future electronics.

The Big Takeaway

The paper is a "pedagogical" (teaching) guide. It says: "Don't get lost in the scary, complex math of modern topology. Just look at these two simple examples. If you understand how the 'map' of the particle gets twisted and develops a kink, you understand the heart of topological materials."

They prove that whether you are looking at a superfluid of dancing pairs or a ladder of different-shaped cars, the geometry of the quantum world is the same: Twists in the map lead to long-range connections and particles that refuse to stay put.

This insight helps scientists predict where these "twisted" materials will appear and how to use them for things like quantum computers, where these "stretched out" states are incredibly stable and hard to destroy.

Technical Summary

1. Problem Statement

The paper addresses the fundamental connection between the topological properties of band structures in condensed matter systems and the spatial localization of Wannier functions. Specifically, it investigates how non-analytic points (discontinuities) in Bloch-function eigenvectors, arising from band crossings or touching points in the Brillouin Zone (BZ), lead to an "obstruction" in finding exponentially localized, symmetric Wannier functions.

While modern topological physics often relies on abstract concepts like Berry curvature and Chern numbers, the authors argue that the root cause of topological phenomena (such as the bulk-boundary correspondence) lies in the specific behavior of eigenvector discontinuities. The paper aims to provide a pedagogical, analytical, and numerical demonstration of these mechanisms in two distinct 1D systems:

1. A p-wave fermionic superfluid (interacting system).
2. A two-leg ladder model with non-interacting fermions (normal phase).

The goal is to explicitly trace how crossing a Quantum Critical Point (QCP) changes the eigenvector structure, thereby altering the spatial decay of Wannier functions from exponential (trivial phase) to power-law (topological phase).

2. Methodology

The authors employ a combination of analytical diagonalization and numerical integration to study two model systems at zero temperature.

A. Model 1: One-Dimensional p-wave Fermionic Superfluid

• Hamiltonian: A spin-polarized fermionic system on a 1D lattice with nearest-neighbor hopping ($t$) and a mean-field p-wave pairing term ($\Delta$).
• Diagonalization: The mean-field Hamiltonian is diagonalized using the Bogoliubov-de Gennes (BdG) formalism. The system is mapped to a $2 \times 2$ matrix $M(k)$ involving the dispersion $\xi(k)$ and the pairing gap $\Delta(k) \propto \sin(k)$.
• Symmetry: The system possesses particle-hole symmetry, which allows band touching in 1D (circumventing the von Neumann-Wigner theorem).
• Analysis: The authors analyze the eigenvectors $(u_k, v_k)$ across the BZ as the chemical potential $\mu$ is varied. They define Wannier-like lattice functions via Fourier transforms of the eigenvector components.

B. Model 2: Two-Leg Ladder (Normal Phase)

• Hamiltonian: Non-interacting spinless fermions on two coupled chains ("a" and "b").
• Orbital Configurations:
  • s-s orbitals: Both chains have s-like orbitals. This results in avoided crossings but no topological obstruction in the standard sense.
  • s-p orbitals: Chain "a" has s-like orbitals, and chain "b" has p-like orbitals. The inter-chain hopping acquires a momentum dependence ($\tau(k) \propto \sin(k)$), making the Hamiltonian mathematically isomorphic to the p-wave superfluid case.
• Mapping: The authors explicitly map the s-p ladder eigenvectors to the superfluid eigenvectors, demonstrating that the s-p system exhibits two Quantum Critical Points (QCPs) at $\mu_c = \pm 2t_c$, separating three distinct phases (Trivial-Topological-Trivial).

C. Numerical and Analytical Techniques

• Wannier Functions: Calculated as the Fourier transform of the Bloch eigenvectors.
• Asymptotic Analysis: The authors use methods from Fourier analysis (stationary phase and singularity analysis) to determine the large-$n$ behavior of the Wannier functions. They relate the decay rate (exponential vs. power-law) to the smoothness of the eigenvectors in the BZ.
  • Exponential decay: Occurs when eigenvectors are smooth (analytic) across the BZ.
  • Power-law decay: Occurs when eigenvectors have discontinuities (jumps in value or derivatives) at specific $k$ points (e.g., $k=0, \pm\pi$).
• Shifted Wannier Functions: They demonstrate that multiplying the integrand by a phase factor $e^{-ik/2}$ shifts the Wannier center to interstitial positions, restoring exponential decay even in the topological phase.

3. Key Contributions

1. Pedagogical Clarification of Topological Obstruction: The paper demystifies the "obstruction to finding symmetric Wannier functions" by explicitly showing that it arises from the sign discontinuity of eigenvector components at the BZ boundaries when the system is in a topological phase.
2. Unified Treatment of Interacting and Non-Interacting Systems: By comparing the p-wave superfluid (interacting) and the s-p ladder (non-interacting), the authors show that the topological transition mechanism is identical in both cases, governed by the same mathematical structure of the Hamiltonian matrix.
3. Explicit Characterization of Spatial Decay:
   • Trivial Phase: Eigenvectors are smooth; Wannier functions decay exponentially ($e^{-\nu n}$).
   • Topological Phase: Eigenvectors have discontinuities (e.g., a jump of $\pi$ in phase at $k=\pi$); Wannier functions decay algebraically (power-law, e.g., $1/n$ or $1/n^2$).
4. Resolution of Contradictions in Literature: The authors address a discrepancy regarding the Cooper pair wave function in 1D p-wave superconductors. While previous works suggested a power-law decay in the topological phase, this paper shows that the pair wave function itself converges to a constant (or oscillates between constants) at large distances, while the deviation from this constant decays exponentially in the trivial phase and as a power law in the topological phase.
5. Bulk-Boundary Correspondence via Wannier Shifts: The paper provides a concrete mechanism for the bulk-boundary correspondence: the power-law decay of bulk Wannier functions in the topological phase implies that the "center of charge" cannot reside on the lattice sites. Shifting the Wannier functions to interstitial positions restores exponential localization, physically manifesting the topological edge states.

4. Key Results

• Phase Transitions:
  • Superfluid: A single QCP at $\mu = 2t$ separates a trivial phase ($\mu > 2t$) from a topological phase ($\mu < 2t$).
  • s-p Ladder: Two QCPs at $\mu_c = \pm 2t_c$ create a sequence: Trivial ($\mu_c > 2t_c$) $\to$ Topological ($-2t_c < \mu_c < 2t_c$) $\to$ Trivial ($\mu_c < -2t_c$).
• Eigenvector Behavior:
  • In the trivial phase, the components $|u_k|$ and $|v_k|$ do not cross, and the eigenvectors can be chosen to be smooth across the entire BZ.
  • In the topological phase, $|u_k|$ and $|v_k|$ cross at an intermediate $k$, forcing a discontinuity (a "kink" or jump) in the eigenvector components at the BZ boundaries ($k=0$ or $k=\pi$) to maintain continuity of the physical state.
• Wannier Function Decay:
  • Trivial: $|w_n| \sim e^{-\nu n} / n^{3/4}$.
  • Topological: $|w_n| \sim 1/n$ (for the component with the sign jump) and $|w_n| \sim 1/n^2$ (for the component with the derivative jump).
• Pair Wave Function ($g(n)$):
  • In the topological phase, the pair wave function approaches a uniform constant value $(\mu+1)/2$ at large $n$.
  • In the trivial phase, it oscillates between two values depending on whether $n$ is even or odd.
  • The convergence rate to these asymptotic values is exponential in the trivial phase and power-law in the topological phase.

5. Significance

• Historical Context: The paper highlights that the physical insights regarding singular points in band structures and their effect on Wannier tails were anticipated in 1978 by G. Strinati, predating the formal introduction of the Berry phase (1984). It re-establishes the importance of these "old" insights for modern topological physics.
• Conceptual Clarity: It bridges the gap between abstract topological invariants (Chern numbers, Berry phases) and real-space physical observables (localization length, decay rates). It proves that "topology" in 1D is fundamentally about the inability to define a smooth gauge for the Bloch functions over the entire BZ.
• Experimental Relevance: The analysis of the s-p ladder and the p-wave superfluid provides concrete, analytically tractable models for experimentalists working with cold atoms in optical lattices or nanowires, where these topological transitions and the associated changes in correlation functions can be directly measured.
• Methodological Value: The paper serves as a "how-to" guide for calculating and interpreting the spatial decay of Wannier functions as a diagnostic tool for topological phases, offering a complete analytical framework that avoids the need for complex numerical band-structure codes for simple 1D systems.