Unconventional quantum phase transition in the finite-size Lipkin–Meshkov–Glick model

Gang Chen,J-Q Liang

doi:10.1088/1367-2630/8/12/297

Abstract

The Lipkin model of arbitrary particle number N is studied in terms of exact differential-operator representation of spin-operators from which we obtain the low-lying energy spectrum with the instanton method of quantum tunnelling. Our new observation is that the well-known quantum phase transition (QPT) can also occur in the finite-N model only if N is an odd number. We furthermore demonstrate a new type of QPT characterized by level-crossing which is induced by the geometric phase interference and is marvellously periodic with respect to the coupling parameter. Finally, the conventional QPT is understood intuitively from the tunnelling formulation in the thermodynamic limit.

Highlights

Lipkin model of arbitrary particle-number N is studied in terms of exact differential-operator representation of spin-operators from which we obtain the low-lying energy spectrum with the instanton method of quantum tunneling
The Lipkin-Meshkov-Glick (LMG) model of manybody two-level system[1, 2, 3], which was invented originally for the study of nuclear giant monopole[4], is exactly solvable[5, 6, 7] and has wide applications in the statistical mechanics of mutually interacting spins[8, 9] and the Bose-Einstein condensates[10]. It has been regarded as an important model to explore the novel relation between the quantum entanglement and quantum phase transition (QPT)[11, 12, 13, 14, 15]
As a matter of fact QPT originates from singularity of the energy spectrum[17] and any non-analyticity point in the ground-state energy of the infinite-lattice-system can be identified as a QPT

Summary

Introduction

Lipkin model of arbitrary particle-number N is studied in terms of exact differential-operator representation of spin-operators from which we obtain the low-lying energy spectrum with the instanton method of quantum tunneling. In spin language the deformed-phase corresponds to a long-range magnetic order and may be regarded as a ferromagnet, while the transverse-magnetic-field h lowering the potential-barrier induces quantum tunneling between two generate-states and eventually destroy the magnetic-order at the critical point h = hc. A similar QPT was observed with the ferromagnetic moment vanishing continuously at a quantum critical point of the transverse field in the low-lying magnetic excitations of the insulator LiHoFe4 [19].

Results

Conclusion