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arXiv:cond-mat/0502272 · Physical Review Letters

Geometric Phases and Criticality in Spin-Chain Systems

Angelo C. M. Carollo, Jiannis K. Pachos

Published Oct 5, 2005• 224 citations
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Abstract

A relation between geometric phases and criticality of spin chains is established. As a result, we show how geometric phases can be exploited as a tool to detect regions of criticality without having to undergo a quantum phase transition. We analytically evaluate the geometric phase that corresponds to the ground and excited states of the anisotropic XY model in the presence of a transverse magnetic field when the direction of the anisotropy is adiabatically rotated. It is demonstrated that the resulting phase is resilient against the main sources of errors. A physical realization with ultracold atoms in optical lattices is presented.

DOI
10.1103/physrevlett.95.157203

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A relation between geometric phases and criticality of spin chains is established. As a result, we show how geometric phases can be exploited as a tool to detect regions of criticality without having to undergo a quantum phase transition. We analytically evaluate the geometric phase that corresponds to the ground and excited states of the anisotropic XY model in the presence of a transverse magnetic field when the direction of the anisotropy is adiabatically rotated. It is demonstrated that the resulting phase is resilient against the main sources of errors. A physical realization with ultracold atoms in optical lattices is presented.
A general formalism of the relation between geometric phases produced by circularly evolving interacting spin systems and their criticality behaviour is presented. This opens up the way for the use of geometric phases as a tool to probe regions of criticality without having to undergo a quantum phase transition. As a concrete example, a spin-1/2 chain with XY interactions is considered and the corresponding geometric phases are analysed. Finally, a generalization of these results to the case of an arbitrary spin system is presented.
A general formalism of the relation between geometric phases produced by circularly evolving interacting spin systems and their criticality behavior is presented. This opens up the way for the use of geometric phases as a tool to study regions of criticality without having to undergo a quantum phase transition. As a concrete example a spin-1/2 chain with XY interactions is presented and the corresponding geometric phases are analyzed. The generalization of these results to the case of an arbitrary spin system provides an explanation for the existence of such a relation.
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.
We study the connection between Berry phases and quantum phase transitions of generic quantum many-body systems. Consider sequences of Berry phases associated to sequences of loops in the parameter space whose limit is a point. If the sequence of Berry phases does not converge to zero, then the limit point is a quantum critical point. Quantum critical points are associated to failures of adiabaticity. We discuss the remarkable example of the anisotropic XY spin chain in a transverse magnetic field and detect the XX region of criticality.
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.