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arXiv:2104.13845 · Physical review. B./Physical review. B

Unveiling quantum phase transitions by fidelity mapping

Ho-Kin Tang, Mohamad Ali Marashli, Wing Chi Yu

Published Aug 23, 2021• 19 citations
arXivHTMLPDF

Abstract

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-$\frac{1}{2}$ 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.

DOI
10.1103/physrevb.104.075142

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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.
The potential of fidelity in accurately detecting unconventional quantum phase transitions (QPTs) remains an open question. Specific attention is directed towards high-order phase transitions, including the infinite-order Berezinskii-Kosterlitz-Thouless (BKT) transition and the Gaussian transition. In this study, we extend the concept of fidelity susceptibility to fixed-point fidelity susceptibility in the thermodynamic limit and assess this measure using tensor-network algorithms. Our results demonstrate that the fixed-point fidelity susceptibility effectively identifies the BKT and Gaussian phase transitions, a capability lacking in the traditional finite-fidelity approach. Furthermore, we establish a correlation between the fixed-point fidelity method and the entanglement method, demonstrating their equivalence. This research not only confirms the efficacy of fidelity in detecting QPTs but also enhances the understanding of fidelity-related methodologies.
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.
Fidelity measures the similarity between two states and is widely adapted by the condensed matter community as a probe of quantum phase transitions in many-body systems. Despite its success in witnessing quantum critical points, information about the fine structure of a quantum phase one can get from this approach is still limited. Here, we proposed a scheme called fidelity spectrum. By studying the fidelity spectrum, one can obtain information about the characteristics of a phase. In particular, we investigated the spectra in the one-dimensional transverse-field Ising model and the two-dimensional Kitaev model on a honeycomb lattice. It was found that the spectra have qualitative differences in the critical and non-critical regions of the two models. From the distributions of them, the dominating k modes in a particular phase could also be captured.
The discovery of topological features of quantum states plays a central role in modern condensed matter physics and various artificial systems. Due to the absence of local order parameters, the detection of topological phase transitions remains a challenge. Machine learning may provide effective methods for identifying topological features. In this work, we show that manifold learning can successfully retrieve topological quantum phase transitions in momentum space. Our results show that the Chebyshev distance (CD) between two data points can successfully capture the main features of topological phase transitions, while the widely used Euclidean distance in general fails. The similarity matrix built upon the CD thus naturally exhibits uncorrelated cluster structures, corresponding to distinct sectors in the topological phase diagram. Then a diffusion map is applied to implement dimensionality reduction and to learn about topological quantum phase transitions in an unsupervised manner. Our demonstrations on the Su-Schrieffer-Heeger (SSH) model, the 2D Qi-Wu-Zhang (QWZ) model, and the quenched SSH model show the capability of generic unsupervised learning, when equipped with a suitable distance metric, in exploring topological quantum phase transitions.
The discovery of topological features of quantum states plays an important role in modern condensed matter physics and various artificial systems. Due to the absence of local order parameters, the detection of topological quantum phase transitions remains a challenge. Machine learning may provide effective methods for identifying topological features. In this work we show that the unsupervised manifold learning can successfully retrieve topological quantum phase transitions in momentum and real space. Our results show that the Chebyshev distance between two data points sharpens the characteristic features of topological quantum phase transitions in momentum space, while the widely used Euclidean distance is in general suboptimal. Then a diffusion map or isometric map can be applied to implement the dimensionality reduction, and to learn about topological quantum phase transitions in an unsupervised manner. We demonstrate this method on the prototypical Su-Schrieffer-Heeger (SSH) model, the Qi-Wu-Zhang (QWZ) model, and the quenched SSH model in momentum space, and further provide implications and demonstrations for learning in real space, where the topological invariants could be unknown or hard to compute. The interpretable good performance of our approach shows the capability of manifold learning, when equipped with a suitable distance metric, in exploring topological quantum phase transitions.
We investigate the topological-to-nontopological quantum phase transitions (QPTs) occurring in the Kitaev code under local perturbations in the form of a local magnetic field and spin-spin interactions of the Ising-type using fidelity susceptibility (FS) and entanglement as the probes. We assume the code to be embedded on the surface of a wide cylinder of height MMM and circumference DDD with $M\ensuremath{\ll}D$. We demonstrate a power-law divergence of FS across the QPT and determine the quantum critical points via a finite-size-scaling analysis. We verify these results by mapping the perturbed Kitaev code to the two-dimensional Ising model with nearest- and next-nearest-neighbor interactions and computing the single-site magnetization as the order parameter using the quantum Monte Carlo technique. We also demonstrate a finite-size odd-even dichotomy in the occurrence of the QPT in the Kitaev ladder with respect to the odd and even values of DDD, when the system is perturbed with only Ising interaction. Our results also indicate a higher robustness of the topological phase of the Kitaev code against local perturbations if the boundary is made open along one direction. We further consider a local entanglement witness operator designed specifically to capture a lower bound to the localizable entanglement on the vertical nontrivial loop of the code. We show that the first derivative of the expectation value of the witness operator exhibits a logarithmic divergence across the QPT and perform the finite-size-scaling analysis. We demonstrate similar behavior of the expectation value of the appropriately constructed witness operator also in the case of a locally perturbed color code with open boundaries.