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arXiv:2102.09083 · Journal of Statistical Mechanics Theory and Experiment

Two-dimensional easy-plane SU(3) magnet with the transverse field: anisotropy-driven multicriticality

Yoshihiro Nishiyama

Published Mar 1, 2021• 1 citation
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Abstract

Abstract The two-dimensional easy-plane SU(3) magnet subjected to the transverse field was investigated with the exact-diagonalization method. So far, as to the XY model (namely, the easy-plane SU(2) magnet), the transverse-field-driven order–disorder phase boundary has been investigated with the exact-diagonalization method, and it was claimed that the end-point singularity (multicriticality) at the XX -symmetric point does not accord with large- N -theory’s prediction. Aiming to reconcile the discrepancy, we extend the internal symmetry to the easy-plane SU(3) with the anisotropy parameter η , which interpolates the isotropic ( η = 0) and fully anisotropic ( η = 1) cases smoothly. As a preliminary survey, setting η = 1, we analyze the order–disorder phase transition through resorting to the fidelity susceptibility χ F , which exhibits a pronounced signature for the criticality. Thereby, with η scaled carefully, the χ F data are cast into the crossover-scaling formula so as to determine the crossover exponent ϕ , which seems to reflect the extension of the internal symmetry group to SU(3).

DOI
10.1088/1742-5468/abe412

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Abstract The two-dimensional easy-plane SU(3) magnet subjected to the transverse field was investigated with the exact-diagonalization method. So far, as to the XY model (namely, the easy-plane SU(2) magnet), the transverse-field-driven order–disorder phase boundary has been investigated with the exact-diagonalization method, and it was claimed that the end-point singularity (multicriticality) at the XX -symmetric point does not accord with large- N -theory’s prediction. Aiming to reconcile the discrepancy, we extend the internal symmetry to the easy-plane SU(3) with the anisotropy parameter η , which interpolates the isotropic ( η = 0) and fully anisotropic ( η = 1) cases smoothly. As a preliminary survey, setting η = 1, we analyze the order–disorder phase transition through resorting to the fidelity susceptibility χ F , which exhibits a pronounced signature for the criticality. Thereby, with η scaled carefully, the χ F data are cast into the crossover-scaling formula so as to determine the crossover exponent ϕ , which seems to reflect the extension of the internal symmetry group to SU(3).
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.
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 one-dimensional spin-1/2 XXZ model in a transverse magnetic field is studied. It is shown that the field induces a gap in the spectrum of the model with the easy-plane anisotropy. Using conformal invariance, the field dependence of the gap is found at small fields. The ground state phase diagram is obtained. It contains four phases with the long-range order of different types and a disordered phase. These phases are separated by critical lines, where the gap and the long-range order vanish. Using scaling estimates, the mean-field approach, and numerical calculations in the vicinity of all critical lines, we find the critical exponents of the gap and the long-range order. It is shown that the transition line between the ordered and disordered phases belongs to the universality class of the transverse Ising model.
The susceptibility of a quantum X-Y model in a transverse field Gamma is studied in the vicinity of the multicritical point ( Gamma c ,T=0). For dimensions d between two and four we obtain a Hartree type approximation an equation for the corresponding crossover function which is solvable analytically in three dimensions. The critical line Gamma c (T)- Gamma c (0) approximately T psi is asymptotically described by the shift exponent psi =d/2 in accord with self-consistent spin wave theory. For the variable of the crossover scaling function T/( Gamma - Gamma c (T)) phi one obtains the crossover exponent phi =(4-d)/2 which is characteristic for the Gaussian multicritical point. A dimensional-analysis argument suggests that this is valid beyond the approximation.
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 infinite-disorder fixed point of the random transverse-field Ising model is expected to control the critical behavior of a large class of random quantum and stochastic systems having an order parameter with discrete symmetry. Here we study the model on the square lattice with a very efficient numerical implementation of the strong disorder renormalization group method, which makes us possible to treat finite samples of linear size up to L=2048L=2048L=2048. We have calculated sample dependent pseudocritical points and studied their distribution, which is found to be characterized by the same shift and width exponent: $\ensuremath{\nu}=1.24(2)$. For different types of disorder the infinite-disorder fixed point is shown to be characterized by the same set of critical exponents, for which we have obtained improved estimates: x=0.982(15)x=0.982(15)x=0.982(15) and $\ensuremath{\psi}=0.48(2)$. We have also studied the scaling behavior of the magnetization in the vicinity of the critical point as well as dynamical scaling in the ordered and disordered Griffiths phases.