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arXiv:0806.3529 · Physical Review E

Geometric phases and quantum phase transitions in open systems

Alexander I. Nesterov, Sergei G. Ovchinnikov

Published Jul 18, 2008• 37 citations
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Abstract

The relationship is established between quantum phase transitions and complex geometric phases for open quantum systems governed by a non-Hermitian effective Hamiltonian with accidental crossing of the eigenvalues. In particular, the geometric phase associated with the ground state of the one-dimensional dissipative Ising model in a transverse magnetic field is evaluated, and it is demonstrated that the related quantum phase transition is of the first order.

DOI
10.1103/physreve.78.015202

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The relationship is established between quantum phase transitions and complex geometric phases for open quantum systems governed by a non-Hermitian effective Hamiltonian with accidental crossing of the eigenvalues. In particular, the geometric phase associated with the ground state of the one-dimensional dissipative Ising model in a transverse magnetic field is evaluated, and it is demonstrated that the related quantum phase transition is of the first order.
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Geometrical and topological properties of quantum ground state manifolds permit to characterize phases of matter and identify phase transitions. Here, we study the effect of a quantum dissipative environment on the geometrical properties of the ground state manifold of a single spin 1/2 in an external effective magnetic field. We show that the quantum phase transition at zero temperature in this model is associated with a universal metric singularity related to the divergence of the spin susceptibility. The absence of transition at finite temperature corresponds to a smooth variation of the associated metric with temperature, without singular points.
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.
By studying the topological invariance and Berry phase in non-Hermitian systems, we reveal the basic properties of the complex Berry phase and generalize the global Berry phases QQQ to identify the topological invariance for non-Hermitian systems. We find that QQQ can identify topological invariance in two kinds of non-Hermitian model, the two-level non-Hermitian Hamiltonian and the bipartite dissipative model. For the bipartite dissipative model, an abrupt change of the Berry phase in the parameter space reveals a quantum phase transition and is related to the exceptional points. These results give the basic relationships between the Berry phase and the quantum and topological phase transitions of non-Hermitian systems.
By using quantum renormalization group (QRG) approach, we first derive the effective Hamiltonian and QRG equations of the two-dimensional (2D) Ising models with two different time-dependent transverse magnetic fields analytically. Then we examine the nonanalytic and scaling behaviors of the linear-entropy-based uncertainty relation and quantum entanglement of the models near the critical point through numerical analysis. Moreover, we investigate the relation between the quantum critical point and the external magnetic field. Our results show that both the uncertainty relation and the quantum entanglement are feasible to detect the quantum phase transition (QPT), and the uncertainty relation may be a better indicator of QPT than quantum entanglement. Our findings could shed new light on the observable of the QPTs of the solid-state system with the uncertainty relation.