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Physical Review A

Scaling of the Berry phase close to the excited-state quantum phase transition in the Lipkin model

Zi-Gang Yuan, Ping Zhang, Shu‐Shen Li, Jian Jing, Ling-Bao Kong

Published Apr 25, 2012• 35 citations
Publisher

Abstract

We study the Berry phase of the ground state and excited states in the Lipkin model. Using the Schwinger representation of spin operators, we get the expression for the Berry phase of the ground state in the thermodynamic limit and numerically calculate that of the excited states in finite sizes. We find that the Berry phase of the ground state is nonanalytic, but its derivative (with respect to the control parameter $\ensuremath{\alpha}$, see text) is finite even in the thermodynamic limit. Whereas, for the excited states, the Berry phase tends to be nonanalytic, and its derivative approaches $\ensuremath{\infty}$ in the thermodynamic limit and in the vicinity of the critical point of the excited-state quantum phase transition.

DOI
10.1103/physreva.85.044102

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We study the Berry phase of the ground state and excited states in the Lipkin model. Using the Schwinger representation of spin operators, we get the expression for the Berry phase of the ground state in the thermodynamic limit and numerically calculate that of the excited states in finite sizes. We find that the Berry phase of the ground state is nonanalytic, but its derivative (with respect to the control parameter \ensuremathα\ensuremath{\alpha}\ensuremathα, see text) is finite even in the thermodynamic limit. Whereas, for the excited states, the Berry phase tends to be nonanalytic, and its derivative approaches \ensuremath∞\ensuremath{\infty}\ensuremath∞ in the thermodynamic limit and in the vicinity of the critical point of the excited-state quantum phase transition.
In our previous work [Phys. Rev. A 85 (2012) 044102], we studied the Berry phase of the ground state and exited states in the Lipkin model. In this work, using the Hellmann—Feynman theorem, we derive the relation between the energy gap and the Berry phase closed to the excited state quantum phase transition (ESQPT) in the Lipkin model. It is found that the energy gap is approximately linearly dependent on the Berry phase being closed to the ESQPT for large N. As a result, the critical behavior of the energy gap is similar to that of the Berry phase. In addition, we also perform a semiclassical qualitative analysis about the critical behavior of the energy gap.
We discuss the thermodynamic and finite-size scaling properties of the geometric phase in the adiabatic Dicke model, describing a quantum phase transition for an N qubit register coupled to a slow oscillator mode. We show that, in the thermodynamic limit, a non-zero Berry phase is obtained only if a path in parameter space is followed that encircles the critical point. Furthermore, we investigate the precursors of this critical behavior for a system with finite size and obtain the leading orders in the 1/ N expansion of the Berry phase and its critical exponent.
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 theoretically investigate the impact of the excited state quantum phase transition on the adiabatic dynamics for the Lipkin-Meshkov-Glick model. Using a time-dependent protocol, we continuously change a model parameter and then discuss the scaling properties of the system especially close to the excited state quantum phase transition where we find that these depend on the energy eigenstate. On top, we show that the mean-field dynamics with the time-dependent protocol gives the correct scaling and expectation values in the thermodynamic limit even for the excited states.