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arXiv:2310.11951 · Physical review. E

Exact zeros of fidelity in finite-size systems as a signature for probing quantum phase transitions

Yumeng Zeng, Bozhen Zhou, Shu Chen

Published Jun 13, 2024• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.109.064130

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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 F(\ensuremathγ,\ensuremathγ\ifmmode~\else˜\fi)\mathcal{F}(\ensuremath{\gamma},\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\gamma}})F(\ensuremathγ,\ensuremathγ\ifmmode~\else˜\fi​) always approaches zero in the thermodynamical limit, due to the Anderson orthogonality catastrophe, no matter whether the parameters of two ground states (\ensuremathγ\ensuremath{\gamma}\ensuremathγ and \ensuremathγ\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\gamma}}\ensuremathγ\ifmmode~\else˜\fi​) 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 \ensuremathγ\ensuremath{\gamma}\ensuremathγ and \ensuremathγ\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\gamma}}\ensuremathγ\ifmmode~\else˜\fi​ belong to different phases. On the other hand, no exact zero of fidelity can be observed if \ensuremathγ\ensuremath{\gamma}\ensuremathγ and \ensuremathγ\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\gamma}}\ensuremathγ\ifmmode~\else˜\fi​ 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.
The dynamical quantum phase transition is characterized by the emergence of nonanalytic behaviors in the rate function, corresponding to the occurrence of exact zero points of the Loschmidt echo in the thermodynamical limit. In general, exact zeros of the Loschmidt echo are not accessible in a finite-size quantum system except for some fine-tuned quench parameters. In this work, we study the realization of the dynamical singularity of the rate function for finite-size systems under the twist boundary condition, which can be introduced by applying a magnetic flux. By tuning the magnetic flux, we illustrate that exact zeros of the Loschmidt echo can be always achieved when the postquench parameter is across the underlying equilibrium phase transition point, and thus the rate function of a finite-size system is divergent at a series of critical times. We demonstrate our theoretical scheme by calculating the Su-Schrieffer-Heeger model and the Creutz model in detail and exhibit its applicability to more general cases. Our result unveils that the emergence of dynamical singularity in the rate function can be viewed as a signature for detecting dynamical quantum phase transition in finite-size systems. We also unveil that the critical times in our theoretical scheme are independent on the systems size, and thus it provides a convenient way to determine the critical times by tuning the magnetic flux to achieve the dynamical singularity of the rate function.
We study the quantum fidelity (ground-state overlap) near quantum phase transitions of the Ising universality class in one-dimensional (1D) systems of finite-size LLL. Prominent examples occur in magnetic systems (e.g., spin-Peierls, the anisotropic XYXYXY model) and in 1D topological insulators of any topologically nontrivial Altland-Zirnbauer-Kitaev universality class. The rescaled fidelity susceptibility is a function of the only dimensionless parameter LMLMLM, where 2M2M2M is the gap in the fermionic spectrum. We present analytic expressions for the fidelity susceptibility for periodic and open boundaries conditions with zero, one, or two edge states. The latter are shown to have a crucial impact and alter the susceptibility both quantitatively and qualitatively. We support our analytical solutions with numerical data.
Fidelity has been widely used to detect various types of quantum phase transitions (QPTs). However, challenges remain in locating the transition points with precision in several models with unconventional phases. We propose the fidelity map approach to detect QPTs with higher accuracy and sensitivity than the conventional fidelity measures. Our scheme extends the fidelity concept from a single-dimensional to a multidimensional quantity and uses a metaheuristic algorithm to search for the critical points that globally maximize the fidelity within each phase. We verify the scheme in several interacting models that possess unconventional phases and QPTs, namely, the spin-1 Kitaev-Heisenberg model, the spin-12\frac{1}{2}21​ XXZ model, and the interacting Su-Schrieffer-Heeger model. The fidelity map can capture a wide range of phase transitions accurately even in small systems, thus providing a convenient tool to study QPTs in unseen models without prior knowledge of the symmetry of the system.
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.
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.
Lee-Yang theory is central to the analysis of thermal phase transitions. However, the underlying mechanism of the theory and the nature of Lee-Yang zeros in quantum many-body systems remains elusive. Here, we develop a unified framework for understanding quantum phase transitions from fidelity zeros induced by symmetry breaking. These zeros, arising from transitions between symmetry sectors, obey the Lee-Yang theorem and give rise to fidelity edges near critical points. Quantum criticality is further characterized through the finite-size scaling of fidelity zeros. As concrete examples, we analytically and numerically investigate fidelity zeros in one- and two-dimensional ferromagnetic Ising models under a complex magnetic field. Our results provide new insights into the mechanism of Lee-Yang theory and open avenues for exploring unexplored landscapes of phase transitions in quantum many-body systems.