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Geometric phases and criticality in spin systems

B Y Jiannis, K. Pachos, Angelo, C. M. Carollo

Published Jan 1, 2006• 17 citations
OpenAlex

Abstract

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.

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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.

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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.
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 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.
The geometric phase can act as a signature for critical regions of interacting spin chains in the limit where the corresponding circuit in parameter space is shrunk to a point and the number of spins is extended to infinity; for finite circuit radii or finite spin chain lengths, the geometric phase is always trivial (a multiple of 2 π ). In this work, two related signatures of criticality are proposed which obey finite-size scaling and which circumvent the need for assuming any unphysical limits. They are based on the notion of the Bargmann invariant, whose phase may be regarded as a discretized version of the Berry phase. As circuits are considered which are composed of a discrete finite set of vertices in parameter space, they are able to pass directly through a critical point, rather than having to circumnavigate it. The proposed mechanism is shown to provide a diagnostic tool for criticality in the case of a given non-solvable one-dimensional spin chain with nearest-neighbour interactions in the presence of an external magnetic field.
The geometric phase can act as a signature for critical regions of interacting spin chains in the limit where the corresponding circuit in parameter space is shrunk to a point and the number of spins is extended to infinity; for finite circuit radii or finite spin chain lengths, the geometric phase is always trivial (a multiple of 2pi). In this work, by contrast, two related signatures of criticality are proposed which obey finite-size scaling and which circumvent the need for assuming any unphysical limits. They are based on the notion of the Bargmann invariant whose phase may be regarded as a discretized version of Berry's phase. As circuits are considered which are composed of a discrete, finite set of vertices in parameter space, they are able to pass directly through a critical point, rather than having to circumnavigate it. The proposed mechanism is shown to provide a diagnostic tool for criticality in the case of a given non-solvable one-dimensional spin chain with nearest-neighbour interactions in the presence of an external magnetic field.
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 introduce a connection between entanglement induced by interaction and geometric phases acquired by a composite quantum spin system. We begin by analyzing the evaluation of cyclic (Aharonov-Anandan) and noncyclic (Mukunda-Simon) geometric phases for general spin chains evolving in the presence of time-independent magnetic fields. Then, by considering Heisenberg chains, we show that the interaction geometric phase, namely, the total geometric phase with subtraction of free spin contributions, is directly related to the global (Meyer-Wallach) entanglement exhibited by an initially separable state during its evolution in Hilbert space. This is analytically shown for N=2N=2N=2 spins and numerically illustrated for larger chains. This relationship promotes the interaction geometric phase to an indicator of global entanglement in the system, which may constitute a useful tool for quantum tasks based on entanglement as a resource to their performance.