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arXiv:2601.01893 · Physica A Statistical Mechanics and its Applications

Longitudinal-field fidelity susceptibility analysis of the J 1 - J 2 transverse-field Ising model around J 2 / J 1 ≈ 0 . 5

Yoshihiro Nishiyama

Published Jan 2, 2026• 0 citations
arXivHTMLPDF

Abstract

The square-lattice J 1 - J 2 transverse-field (TF) Ising model was investigated with the exact diagonalization (ED) method. In order to analyze the TF-driven phase transition, we applied the longitudinal-field fidelity susceptibility χ F ( h ) , which is readily evaluated via the ED scheme. Here, the longitudinal field couples with the absolute value of the magnetic moment | M | rather than the raw M so that the remedied fidelity susceptibility exhibits a peak around the critical point; note that the spontaneous magnetization does not appear for the finite-size systems. As a preliminary survey, the modified fidelity susceptibility χ F ( h ) is applied to the analysis of criticality for J 2 = 0 , where a number of preceding results are available. Thereby, properly scaling the distance from the multi-criticality, η = 0 . 5 − J 2 , the χ F ( h ) data were cast into the crossover-scaling formula, and the multi-critical exponent for χ F ( h ) is estimated. The result is cross-checked by the numerically evaluated β -function behavior.

DOI
10.1016/j.physa.2025.131245

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The square-lattice J 1 - J 2 transverse-field (TF) Ising model was investigated with the exact diagonalization (ED) method. In order to analyze the TF-driven phase transition, we applied the longitudinal-field fidelity susceptibility χ F ( h ) , which is readily evaluated via the ED scheme. Here, the longitudinal field couples with the absolute value of the magnetic moment | M | rather than the raw M so that the remedied fidelity susceptibility exhibits a peak around the critical point; note that the spontaneous magnetization does not appear for the finite-size systems. As a preliminary survey, the modified fidelity susceptibility χ F ( h ) is applied to the analysis of criticality for J 2 = 0 , where a number of preceding results are available. Thereby, properly scaling the distance from the multi-criticality, η = 0 . 5 − J 2 , the χ F ( h ) data were cast into the crossover-scaling formula, and the multi-critical exponent for χ F ( h ) is estimated. The result is cross-checked by the numerically evaluated β -function behavior.
The inner product between the ground-state eigenvectors with proximate interaction parameters, namely, the fidelity, plays a significant role in the quantum dynamics. In this paper, the critical behaviors of the transverse- and longitudinal-field fidelity susceptibilities for the d=2 quantum (transverse-field) Ising model are investigated by means of the numerical-diagonalization method; the former susceptibility has been investigated rather extensively. The critical exponents for these fidelity susceptibilities are estimated as α(F)((t))=0.752(24) and α(F)((h))=1.81(13), respectively. These indices are independent, and suffice for obtaining conventional critical indices such as ν=0.624(12) and γ=1.19(13).
The critical properties of the one-dimensional spin-1/2 transverse Ising model in the presence of a longitudinal magnetic field were studied by the quantum fidelity method. We used exact diagonalization to obtain the ground-state energies and corresponding eigenvectors for lattice sizes up to 24 spins. The maximum of the fidelity susceptibility was used to locate the various phase boundaries present in the system. The type of dominant spin ordering for each phase was identified by examining the corresponding ground-state eigenvector. For a given antiferromagnetic nearest-neighbor interaction J_{2}, we calculated the fidelity susceptibility as a function of the transverse field B_{x} and the longitudinal field B_{z}. The phase diagram in the (B_{x},B_{z})-plane shows three phases. These findings are in contrast with the published literature that claims that the system has only two phases. For B_{x}<1, we observed an antiferromagnetic phase for small values of B_{z} and a paramagnetic phase for large values of B_{z}. For B_{x}>1 and low B_{z}, we found a disordered phase that undergoes a second-order phase transition to a paramagnetic phase for large values of B_{z}.
The behavior of the ground-state fidelity susceptibility in the vicinity of a quantum critical point is investigated. We derive scaling relations describing its singular behavior in the quantum critical regime. Unlike in previous studies, these relations are solely expressed in terms of conventional critical exponents. We also describe in detail a quantum Monte Carlo scheme that allows for the evaluation of the fidelity susceptibility for a large class of many-body systems and apply it in the study of the quantum phase transition for the transverse-field Ising model on the square lattice. Finite-size analysis applied to the so-obtained numerical results confirms the validity of our scaling relations. Furthermore, we analyze the properties of a closely related quantity, the ground-state energy's second derivative, which can be numerically evaluated in a particularly efficient way. The usefulness of both quantities as alternative indicators of quantum criticality is examined.
We analyze the fidelity per lattice site for two different ground states of the one-dimensional quantum Ising model in a transverse field near the critical point. It is found that, in the thermodynamic limit, the fidelity per lattice site is singular, and the derivative of its logarithmic function with respect to the transverse field strength is logarithmically divergent at the critical point. The scaling behavior is confirmed numerically by performing a finite-size scaling analysis for systems of different sizes, consistent with the conformal invariance at the critical point. This allows us to extract the correlation length critical exponent, which turns out to be universal in the sense that the correlation length critical exponent does not depend on either the anisotropic parameter or the transverse field strength.
We study critical behaviors of the reduced fidelity susceptibility for two neighboring sites in the one-dimensional transverse field Ising model. It is found that the divergent behaviors of the susceptibility take the form of square of logarithm, in contrast with the global ground-state fidelity susceptibility which is power divergence. In order to perform a scaling analysis, we take the square root of the susceptibility and determine the scaling exponent analytically and the result is further confirmed by numerical calculations.
Abstract The transverse-field XY spin chain with competing antiferromagnetic long-range interactions, J r ∝ 1 / r α (r: distance between spins), and the exponent α was investigated numerically. The main concern is to clarify the character of the transverse-field-driven phase transition for the small-α regime around the XX-symmetric point, η = 0 (η: XY-anisotropy parameter). To cope with the negative-sign problem, we employed the exact-diagonalization method, which enables us to evaluate the fidelity susceptibility χ F . Because the fidelity susceptibility does not assume any order parameter after a phase transition, it detects the multi-criticality around the XX-symmetric point in a systematic manner. As a preliminary survey, with η = 0.5 and α = 2 fixed, we made the scaling analysis of χ F . The scaling behavior of χ F shows that the transverse-field-driven phase transition belongs to the 2D-Ising universality class. Thereby, with the properly crossover-scaled η, the χ F data is cast into the crossover scaling formula for the small-α regime. The multi-critical exponents fed into the scaling formula are argued through referring to related studies.