⚛️ quantum physics

Distribution of fidelity zeros in two-band topological models

This paper investigates the distribution of fidelity zeros in two-band topological models by extending parameters into the complex plane, revealing that these zeros relate to momentum modes with vanishing real energy gaps and accumulate into regions that encode critical information about topological phase transitions in systems like the Kitaev chain, Haldane model, and QWZ model.

Original authors: Siyan Lin, Zhen-Yu Zheng, Shu Chen

Published 2026-03-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Siyan Lin, Zhen-Yu Zheng, Shu Chen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand how a complex machine works, like a car engine or a weather system. Usually, scientists study these systems by changing one dial at a time (like turning up the heat or changing the fuel) and watching how the machine reacts. If the machine suddenly changes its behavior—like an engine knocking or the weather turning into a storm—that's a phase transition.

In the world of quantum physics, these transitions happen at absolute zero temperature and are driven by "quantum fluctuations" rather than heat. Scientists have long used a tool called fidelity to detect these changes. Think of fidelity as a "similarity score." If you take the quantum state of a system at one setting and compare it to the state at a slightly different setting, the fidelity tells you how much they look alike. If the score drops to zero, it means the two states are completely different, signaling a major change in the system.

However, there's a catch. In the real world, you can only turn the dials to real numbers (like 1, 2, or 3.5). But what if you could turn the dial into a "ghost number"—a complex number that has a real part and an imaginary part? This is like turning a knob that exists in a parallel dimension.

This paper explores what happens when we do exactly that. The authors take three famous quantum models (the Kitaev chain, the Haldane model, and the QWZ model) and imagine their control knobs existing in this complex, "ghost" dimension. They are looking for Fidelity Zeros—points where the similarity score drops to zero in this imaginary world.

Here is the simple breakdown of their discovery:

1. The "Ghost Knob" and the Energy Gap

In quantum mechanics, particles have energy levels. Usually, there is a gap between the lowest energy level (the ground state) and the next one. Think of this gap as a floor you have to jump over to get to the next level.

The authors found a magical rule: Fidelity zeros only appear when this energy gap "closes" in a very specific way.

Imagine the energy gap is a bridge.
When you turn the "ghost knob" (the complex parameter), the bridge doesn't just disappear; it turns into a ghost bridge where the real height becomes zero.
When the real height of the bridge hits zero, the system gets confused about which state is the "ground state." It flips a switch.
This confusion causes the "similarity score" (fidelity) to crash to zero.

2. The Map of the Ghost World

The authors mapped out where these zeros appear in the complex plane (the world of real and imaginary numbers). They found a beautiful pattern:

In small systems: The zeros look like a row of discrete dots or short lines standing parallel to the "imaginary" axis. It's like a picket fence made of ghostly spikes.
In huge systems (Thermodynamic Limit): As the system gets bigger and bigger, those discrete lines merge into a solid, continuous region.

3. The "Fence" Around the Critical Point

The most exciting part is what this map tells us about the real world.

The "ghost zeros" only appear within a specific range of the real part of the knob.
The edges of this range correspond exactly to the critical points where the real-world phase transition happens.

The Analogy:
Imagine you are trying to find the exact moment a river freezes. You can't see the ice forming yet. But, if you could look into a "ghost dimension" parallel to the river, you would see a wall of fog (the fidelity zeros) appearing.

As you move your view along the river, the fog wall starts at a specific spot and ends at another.
The start and end points of that fog wall tell you exactly where the river freezes in the real world, even if you can't see the ice yet.

4. Testing the Theory

The authors tested this idea on three different quantum models:

The Kitaev Chain: A model for superconductors. They found the zeros appeared only when the chemical potential (the "knob") was between -1 and 1. The edges (-1 and 1) were exactly where the topological phase transition happened.
The Haldane Model: A model for the Quantum Hall Effect. Again, the zeros formed a region bounded by the critical points.
The QWZ Model: A more complex 2D model. Here, they found something special. Not only did the zeros mark the boundaries at $u = \pm 2$ , but they also crossed the "real axis" at $u = 0$ . This crossing signaled a critical point that was hidden in the other models. It's like the fog wall not only had edges but also had a "hole" right in the middle, signaling a different kind of change.

Why Does This Matter?

Previously, this "Fidelity Zero" method was mostly used for systems that break symmetry (like a magnet losing its magnetism). Topological systems (like the ones in this paper) are different; they don't break symmetry, they just change their "global shape" (like a coffee mug turning into a donut).

This paper proves that the "Ghost Knob" method works for these tricky topological systems too. It gives scientists a new, powerful tool to find the exact points where quantum matter changes its nature, by looking at where the "similarity score" vanishes in a complex, imaginary world.

In short: By looking at where quantum states become completely different in a "ghost" mathematical world, we can pinpoint exactly where the real-world quantum phase transitions happen. It's like using a shadow to find the object casting it.

1. Problem Statement

The study of Quantum Phase Transitions (QPTs) traditionally relies on order parameters (for symmetry-breaking transitions) or topological invariants (for topological transitions). While fidelity (the overlap between ground states) and fidelity susceptibility have proven effective for detecting critical points in symmetry-breaking systems, their application to topological quantum phase transitions (TQPTs) remains challenging.

Recent extensions of fidelity analysis into the complex parameter plane (inspired by Lee-Yang theory) have successfully identified zeros in symmetry-breaking models (e.g., Ising model) via non-Hermitian symmetry breaking. However, it was an open question whether this fidelity-zero framework applies to topological systems, which lack local order parameters and spontaneous symmetry breaking. The core problem is to determine how critical information in topological transitions is encoded in a complexified parameter space and to establish a unified diagnostic tool for TQPTs using fidelity zeros.

2. Methodology

The authors employ a biorthogonal formulation within a non-Hermitian framework to analyze two-band topological models.

Complexification of Parameters: The driving parameter of the Hamiltonian ( $\gamma$ ) is extended from the real line to the complex plane ( $\gamma = \gamma_R + i\gamma_I$ ). This renders the Hamiltonian non-Hermitian.
Biorthogonal Basis: Since the Hamiltonian is non-Hermitian, the standard inner product is insufficient. The authors define the ground state using the biorthogonal basis constructed from left ( $|\Psi^L\rangle$ ) and right ( $|\Psi^R\rangle$ ) eigenstates. The ground state is identified as the state with the smaller real part of the energy.
Fidelity Definition: The many-body fidelity is defined as the product of overlaps over all momentum modes $k$ :
$F(\tilde{\gamma}, \gamma) = \prod_k \sqrt{|\langle \Psi^L_{k,g}(\tilde{\gamma}) | \Psi^R_{k,g}(\gamma) \rangle \langle \Psi^L_{k,g}(\gamma) | \Psi^R_{k,g}(\tilde{\gamma}) \rangle|}$
Minimum Single-Mode Fidelity ( $F_{min}$ ): To avoid the Anderson orthogonality catastrophe (where total fidelity vanishes in the thermodynamic limit), the authors focus on $F_{min}$ , the minimum overlap among all momentum modes. A zero in $F_{min}$ indicates a genuine zero in the total fidelity.
Theoretical Connection: The authors derive a fundamental link: a fidelity zero occurs if and only if the real part of the energy gap closes for a specific momentum mode, i.e., $\text{Re}[E_{k,+}(\gamma) - E_{k,-}(\gamma)] = 0$ .

3. Key Contributions

Establishment of a Spectral Condition: The paper proves that in the biorthogonal framework, fidelity zeros emerge precisely when the real part of the energy gap vanishes for specific momentum modes. This provides a direct spectral mechanism for the appearance of zeros, distinct from the symmetry-breaking mechanisms in previous works.
Extension to Topological Systems: The framework is successfully generalized to topological models (Kitaev chain, Haldane model, QWZ model), demonstrating that fidelity zeros are a universal diagnostic for TQPTs, not just symmetry-breaking transitions.
Finite-Size vs. Thermodynamic Limit: The authors characterize the distribution of zeros:
- Finite-size: Zeros form discrete lines parallel to the imaginary axis in the complex parameter plane.
- Thermodynamic Limit: These lines accumulate to form extended regions. The boundaries of these regions converge to the critical points of the topological phase transition.

4. Results

The authors analyzed three paradigmatic models:

A. Kitaev Chain (1D Topological Superconductor)

Model: Extended chemical potential $\mu = \mu_R + i\mu_I$ .
Critical Points: $\mu = \pm 1$ .
Findings: Fidelity zeros exist only when $|\mu_R| \leq 1$ . The zeros form lines parallel to the imaginary axis. The boundaries of the region where zeros exist ( $|\mu_R| = 1$ ) exactly match the topological phase transition points.
Implication: The presence or absence of fidelity zeros as a function of $\mu_R$ signals the topological phase boundary.

B. Haldane Model (Chern Insulator)

Model: Extended staggered potential $M = M_R + iM_I$ .
Critical Points: $M = \pm \frac{3\sqrt{3}}{4}t_2 \sin\theta$ .
Findings: Similar to the Kitaev chain, fidelity zeros appear only within the interval of $M_R$ corresponding to the topologically non-trivial phase. The boundaries of the zero-distribution region coincide with the critical points separating the trivial and non-trivial phases.

C. Qi-Wu-Zhang (QWZ) Model

Model: Extended potential $u = u_R + iu_I$ .
Critical Points: $u = 0, \pm 2$ .
Findings:
- The critical points at $u = \pm 2$ are identified by the boundaries of the region where fidelity zeros exist (similar to the other models).
- The critical point at $u = 0$ is unique: it is not marked by the appearance/disappearance of zeros, but by the crossing of fidelity zeros through the real axis. This mirrors the Lee-Yang paradigm where zeros approaching the real axis signal a critical point.

5. Significance

Unified Diagnostic Tool: The work demonstrates that the fidelity-zero framework is a robust, universal tool for detecting quantum criticality, applicable to both symmetry-breaking and topological transitions.
Mechanism Clarification: It clarifies that for topological systems, the mechanism driving fidelity zeros is the closing of the real part of the energy gap in the complexified spectrum, rather than non-Hermitian symmetry breaking.
Complex Parameter Space Encoding: The study reveals how critical information is encoded in the complex plane: the real part of the complex parameter determines the existence of zeros (defining the phase), while the imaginary part determines the location of the zeros within that phase.
Practical Application: The method offers a way to detect topological phase transitions by monitoring the distribution of zeros in the complex parameter space, which could be relevant for numerical studies and potential experimental probes involving complex parameters (e.g., via synthetic dimensions or driven systems).

In conclusion, the paper successfully bridges the gap between Lee-Yang type fidelity analysis and topological quantum matter, providing a rigorous theoretical foundation for using complexified fidelity zeros to map topological phase diagrams.