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arXiv:0806.2476 · Journal of Physics A Mathematical and Theoretical

Finite-temperature scaling of magnetic susceptibility and the geometric phase in the XY spin chain

H T Quan

Published Sep 11, 2009• 12 citations
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Abstract

We study the magnetic susceptibility of 1D quantum XY model, and show that when the temperature approaches zero, the magnetic susceptibility exhibits the finite-temperature scaling behavior. This scaling behavior of the magnetic susceptibility in 1D quantum XY model, due to the quantum-classical mapping, can be easily experimentally tested. Furthermore, the universality in the critical properties of the magnetic susceptibility in quantum XY model is verified. Our study also reveals the close relation between the magnetic susceptibility and the geometric phase in some spin systems, where the quantum phase transitions are driven by an external magnetic field.

DOI
10.1088/1751-8113/42/39/395002

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We study the magnetic susceptibility of 1D quantum XY model, and show that when the temperature approaches zero, the magnetic susceptibility exhibits the finite-temperature scaling behavior. This scaling behavior of the magnetic susceptibility in 1D quantum XY model, due to the quantum-classical mapping, can be easily experimentally tested. Furthermore, the universality in the critical properties of the magnetic susceptibility in quantum XY model is verified. Our study also reveals the close relation between the magnetic susceptibility and the geometric phase in some spin systems, where the quantum phase transitions are driven by an external magnetic field.
We study the magnetic susceptibility of 1D quantum XY model, and show that when the temperature approaches zero, the magnetic susceptibility (MS) exhibits finite-temperature scaling behavior. This scaling behavior of MS in 1D quantum XY model, due to the quantum-classical map, can be regarded as a witness of macroscopic quantum entanglement, and can be experimentally tested. Furthermore, the universality in the critical properties of MS in quantum XY model is verified. Our study also reveals the close relation between MS and geometric phase in some quantum phase transition systems, which has been studied recently.
We show that the geometric phase of the ground state in the XY model obeys scaling behavior in the vicinity of a quantum phase transition. In particular we find that the geometric phase is nonanalytical and its derivative with respect to the field strength diverges at the critical magnetic field. Furthermore, the universality in the critical properties of the geometric phase in a family of models is verified. In addition, since the quantum phase transition occurs at a level crossing or avoided level crossing and these level structures can be captured by the Berry curvature, the established relation between the geometric phase and quantum phase transitions is not a specific property of the XY model, but a very general result of many-body systems.
Based on the obtained exact results we systematically study the quench dynamics of a one-dimensional spin-1/2 transverse field Ising model with zero- and finite-temperature initial states. We focus on the magnetization of the system after a sudden change of the external field and a coherent time-evolution process. With a zero-temperature initial state, the quench magnetic susceptibility as a function of the initial field strength exhibits strongly similar scaling behaviors to those of the static magnetic susceptibility, and the quench magnetic susceptibility as a function of the final field strength shows a discontinuity at the quantum critical point. This discontinuity remains robust and always occurs at the quantum critical point even for the case of finite-temperature initial systems, which indicates a great advantage of employing quench dynamics to study quantum phase transitions.
Jordan-Wigner transformation and Bogolyubov transformation are the main steps of the diagonalization of Hamiltonian and paly an important role in the statistical mechanics calculations for one-dimensional Heisenberg spin chain model. Many methods can be exploited as a tool to detect quantum phase transition, regions of criticality and scaling behavior in the vicinity of a quantum phase transition, such as geometric phase, fidelity susceptibility, order parameter, and entanglement entropy, which have direct relation with Bogolyubov transformation. We diagonalized the Hamiltonian in XY spin-chain systems with Dzyaloshinskii-Moriya interactions, the results shows that only the energy spectrum but not the coefficients of the Bogolyubov transformation depends on DM interaction. Therefore, the DM interaction may not influence the critical magnetic field of quantum phase transitions and not induce new critical regions in the XY spin model. Moreover, we further prove the ideas by the methods of geometric phases in this model.
We use finite-size methods to estimate the critical properties of s=1 xy chains with uniaxial single-ion anisotropy of strength D. Strong evidence is presented for a phase transition, belonging to the universality class of the 1D quantum 0(2) model, with the weak coupling phase gapless, an essential zero in the gap at the critical coupling D*, and critical exponents z(D*)≊1, η(D*)≊1/4. Like the quantum 0(2) system, one of the models turns out to be a 1D quantum analog of the classical and isotropic 2D xy model.
Abstract We investigate the quantum coherence of an XY spin chain with Dzyaloshinskii–Moriya interaction using a quantum renormalization group-based method, and show that the relative entropy of coherence increases with Dzyaloshinskii–Moriya interaction strength at zero temperature, as well as at finite temperature. We also study the quantum criticality of the system by making use of the relative entropy of coherence and violation of Bell inequality, and find that the first derivatives of the renormalized quantum coherence exhibit singularity near the critical point of the quantum phase transition. Finally, we explore the finite-size scaling behaviors of the derivatives of the quantum coherence at the critical point of the quantum phase transition, and obtain several universal finite-size scaling laws.