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Distribution of fidelity zeros in two-band topological models

Siyan Lin, Zhen-Yu Zheng, Shu Chen

Abstract

We investigate the distribution of fidelity zeros in two-band topological models by extending the phase transition driving parameter into the complex plane. Within the biorthogonal formulation, we unveil that fidelity zeros are related to momentum modes for which the real part of the energy gap vanishes. Guided by this relation, we analyze the Kitaev chain, the Haldane model, and the Qi-Wu-Zhang (QWZ) model. In finite-size systems the zeros form discrete lines parallel to the imaginary axis, while in the thermodynamic limit they accumulate into extended regions in the complex parameter plane. For the Kitaev and Haldane models, the accessible interval of the real part of the complexified parameter is bounded by the critical points of the corresponding topological transitions. For the QWZ model, the transitions at $u = \pm2$ are identified in the same way, whereas the critical point at $u = 0$ is signaled by fidelity zeros crossing the real axis. These results extend the fidelity-zero framework to topological quantum phase transitions and clarify how critical information is encoded in complexified parameter space.

Distribution of fidelity zeros in two-band topological models

Abstract

We investigate the distribution of fidelity zeros in two-band topological models by extending the phase transition driving parameter into the complex plane. Within the biorthogonal formulation, we unveil that fidelity zeros are related to momentum modes for which the real part of the energy gap vanishes. Guided by this relation, we analyze the Kitaev chain, the Haldane model, and the Qi-Wu-Zhang (QWZ) model. In finite-size systems the zeros form discrete lines parallel to the imaginary axis, while in the thermodynamic limit they accumulate into extended regions in the complex parameter plane. For the Kitaev and Haldane models, the accessible interval of the real part of the complexified parameter is bounded by the critical points of the corresponding topological transitions. For the QWZ model, the transitions at are identified in the same way, whereas the critical point at is signaled by fidelity zeros crossing the real axis. These results extend the fidelity-zero framework to topological quantum phase transitions and clarify how critical information is encoded in complexified parameter space.
Paper Structure (8 sections, 27 equations, 5 figures)

This paper contains 8 sections, 27 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Fidelity zeros and their connection to the energy spectrum
  3. Distribution of fidelity zeros
  4. Kitaev chain
  5. Haldane model
  6. QWZ model
  7. Conclusion
  8. Verification of analytic expressions for fidelity zeros

Figures (5)

  • Figure 1: (a) Fidelity $\mathcal{F}(\mu+\delta\mu/2,\mu-\delta\mu/2)$ and (b) the real part of the eigenenergies $\operatorname{Re}(E_{k,\pm})$ as a function of $\mu_R$, with fixed $\Delta=0.6,~\mu_I=0.5,~\delta\mu=0.001(1+\mathrm{i})$ and $L=16$. The positions where $\operatorname{Re}(E_{k,\pm})$ vanishes coincide precisely with the fidelity zeros, as indicated by the red dashed lines.
  • Figure 2: (a) The minimum real part of the energy gap $E_{\text{min}}(\mu)$ and (b) the minimum single-mode fidelity $\mathcal{F}_{\text{min}}(\mu+\delta\mu/2,\mu-\delta\mu/2)$ as a function of $\mu$ in the complex plane for the Kitaev chain. We fix $\Delta=0.6$ for both panels. In panel (b) we fix $\delta\mu=0.01(1+\mathrm{i})$ and $L=16$.
  • Figure 3: (a) The minimum real part of the energy gap $E_{\text{min}}(M)$ and (b) the minimum single-mode fidelity $\mathcal{F}_{\text{min}}(M+\delta M/2,M-\delta M/2)$ as a function of $M$ in the complex plane for the Haldane model. In panel (b) we fix $\delta M=0.01(1+\mathrm{i})$ and $L=16$.
  • Figure 4: (a) The minimum real part of the energy gap $E_{\text{min}}(\mu)$ and (b) the minimum single-mode fidelity $\mathcal{F}_{\text{min}}(u+\delta u/2,u-\delta u/2)$ as a function of $u$ in the complex plane for the QWZ model. In panel (b) we fix $\delta u=0.01(1+\mathrm{i})$ and $L=16$.
  • Figure 5: Comparison between fidelity zeros computed from the analytic expressions (green dots) and those obtained numerically (dark blue lines), for (a) the Haldane model and (b) the QWZ model with $L=8$.