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arXiv:2603.18592

Distribution of fidelity zeros in two-band topological models

Siyan Lin, Zhen-Yu Zheng, Shu Chen

quant-phSubmitted Mar 19, 2026• Announced Mar 20, 2026• 0 citations
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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.

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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=±2u = \pm2u=±2 are identified in the same way, whereas the critical point at u=0u = 0u=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.
The short time evolution of a topological system after a sudden quench is considered, calculating the Loschmidt rate and studying in detail the distribution of Fisher zeros. Singular behavior in the Loschmidt rate has been predicted and found in several systems and, in particular, in one dimension (1D) one finds sharp features in the time evolution of the Loschmidt rate. These are associated with critical times that may be seen as the imaginary part of the Fisher zeros, under the condition that their real part vanishes. In the complex plane one finds lines of zeros that cross the imaginary axis at certain points. In a 2D system, in general, the Fisher zeros occupy areas instead of lines, as shown before, with the consequences that a continuous set of critical times results along the imaginary axis and that one does not find a well defined singular behavior in the Loschmidt rate. Here we show that several 2D topological systems also display very narrow regions (effectively lines) of zeros, in addition to finite extent areas of zeros. As a consequence, well defined sharp critical times are found, under appropriate conditions. An example is a 2Dp2\text{D}\phantom{\rule{4pt}{0ex}}p2Dp-wave superconductor in the presence of a magnetic field, but other systems share the same property, such as a Haldane insulator, a system with an anomalous Hall effect, and topological insulators. Details on the momentum distribution of the real and imaginary parts of the Fisher zeros are discussed, clarifying the existence of the lines of zeros and taking as an example the ppp-wave superconductor in magnetic field. The time dependence of the Loschmidt rate phase and of the Pancharatnam geometrical phase is considered, for different times after the quench and as a function of momentum. Pairs of vortices and antivortices in the phase associated with the geometric phase and the Loschmidt rate phase, that have been proposed for some 2D systems, are also shown to appear in the regime where we find lines of zeros.
This dissertation puts forward the concept of fidelity as a way to understand and characterize quantum phase transitions involving topological phases. It lies at the interphase between Condensed Matter and Quantum Phase Transitions. ❧ We find that the fidelity metric, which characterizes the degree of distinghuishability between two quantum states, can serve as a useful tool in the understanding of transitions involving topological phases. ❧ The methodology does not require a complete knowledge of the physical mechanism involved in the occurrence of the critical point, such as the identification of a local order parameter and a symmetry breaking mechanism, the fundamental pillars of critical phenomena. Topological phases are unique in that they cannot be classified under these methodologies, due to their lack of local order parameters. It is one of the purposes of this thesis to understand how well we can characterize the phase boundaries between these phases using fidelity concepts. ❧ In the first part, we introduce two quantum models that exhibit a phase transition between an ordered and a topological phase, as quantified by the topological entanglement entropy. In the thermodynamic limit, the different components of the fidelity metric exhibit a divergence, signalling the occurrence of the critical point. ❧ In the second part, we introduce models that have phase boundaries between two distinct topological phases, and we show that the fidelity is successful at identifying the occurrence of a critical region. Moreover, a finite size scaling analysis is carried out, from which we determine the value of critical exponents.
Fisher zero lines of the Loschmidt amplitude in non-Hermitian quenches can exhibit anomalous structures in the complex time plane, such as divergence, separation, and the restriction of all branches or a specific branch to the real or imaginary axis. Here, we use the biorthogonal framework to reveal the origin of these structures and determine the corresponding critical momentum conditions by analyzing the values of mk=h^0⋅h^1m_k=\hat{\mathbf{h}}_0\cdot\hat{\mathbf{h}}_1mk​=h^0​⋅h^1​, the dot product between the normalized prequench and postquench complex Bloch vectors. When Fisher zeros continuously cover the real-time axis, the density of Fisher zeros is further introduced to identify the critical times within these critical regions, at which the rate function becomes nonanalytic. We verify our results in a generalized non-Hermitian SSH model by quenching the Peierls phase and the non-Hermitian gain-loss strength, and also use the Jordan form to analyze the dynamics at postquench exceptional points, where the conventional Fisher zero formula is not applicable.
The fidelity is widely used to detect quantum phase transitions, which is characterized by either a sharp change of fidelity or the divergence of fidelity susceptibility in the thermodynamical limit when the phase-driving parameter is across the transition point. In this work, we unveil that the occurrence of exact zeros of fidelity in finite-size systems can be applied to detect quantum phase transitions. In general, the fidelity $\mathcal{F}(\ensuremath{\gamma},\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\gamma}})$ always approaches zero in the thermodynamical limit, due to the Anderson orthogonality catastrophe, no matter whether the parameters of two ground states ($\ensuremath{\gamma}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\gamma}}$) are in the same phase or different phases, and this makes it difficult to distinguish whether an exact zero of fidelity exists by finite-size analysis. To overcome the influence of orthogonality catastrophe, we study finite-size systems with twist boundary conditions, which can be introduced by applying a magnetic flux, and demonstrate that exact zeros of fidelity can be always accessed by tuning the magnetic flux when $\ensuremath{\gamma}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\gamma}}$ belong to different phases. On the other hand, no exact zero of fidelity can be observed if $\ensuremath{\gamma}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\gamma}}$ are in the same phase. We demonstrate the applicability of our theoretical scheme by studying concrete examples, including the Su-Schrieffer-Heeger model, Creutz model, and Haldane model. Our work provides a practicable way to detect quantum phase transitions via the calculation of fidelity of finite-size systems.
We investigate the Lee--Yang zeros (LYZ) of inhomogeneous anisotropic XY spin chains with Dzyaloshinskii--Moriya (DM) interactions in the complex transverse-field plane, focusing on their fundamental connection to quantum phase transitions. We systematically study uniform chains, period-2 and period-3 modulated chains, and Fibonacci quasiperiodic chains of lengths 5 and 8. As the DM coupling strength DDD increases, the LYZ exhibit qualitatively distinct topological evolutions on the complex plane: the complex zeros of the uniform chain collapse toward the real axis; periodic chains feature either bifurcation of closed zero contours before all zeros become real or a single re-emergence of complex zeros; quasiperiodic chains exhibit repeated annihilation and revival of complex zeros. Analytical derivations demonstrate that this diverse behavior originates from DM-induced shifts of folded bands and the modulation of zero positions by anisotropic pairing at particle--hole band crossings. Our results establish a direct correspondence between LYZ topology and band deformation: isolated contact points of zeros with the real axis correspond to discrete quantum critical fields, while continuous real-zero intervals directly identify gapless chiral phases. Accordingly, beyond locating phase boundaries, LYZ can distinguish characteristic phases and serve as an intuitive probe for band folding and phase diagram restructuring.
In an attempt to theoretically investigate the quantum phase transition and criticality in topological models, we study a Kitaev chain with longer-range couplings (finite number of neighbors) as well as truly long-range couplings (infinite number of neighbors). We carry out an extensive topological characterization of the momentum space to explore the possibility of obtaining higher order winding numbers and analyze the nature of their stability in the model. The occurrences of phase transitions from even-to-even and odd-to-odd winding numbers are observed with decreasing longer-rangeness in the system. We derive topological quantum critical lines and study them to understand the behavior of criticality. A suppression of higher order winding numbers is observed with decreasing longer-rangeness in the model. We show that the mechanism behind such phenomena is due to the superposition and vanishing of the topological quantum critical lines associated with the higher winding number. Through the study of the Berry connection, we show the possible different behaviors of critical lines when they undergo superposition along with the corresponding critical exponents. We analyze the behavior of the long-range models through the momentum space characterization. We also provide the exact solution for the problem and discuss the experimental aspects of the work.