Geometric Phases and Criticality in Spin-Chain Systems

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 correspond 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 ultra-cold atoms in optical lattices is presented.PACS numbers: 03.65. Vf, 05.30.Pr, 42.50.Vk Since the discovery by Berry [1], geometric phases in quantum mechanics have been the subject of a variety of theoretical and experimental investigations [2]. Possible applications range from optics and molecular physics to fundamental quantum mechanics and quantum computation [3]. In condensed matter physics a variety of phenomena have been understood as a manifestation of topological or geometric phases [4,5,6,7,8]. An interesting open question is whether the geometric phases can be used to investigate the physics and the behavior of condensed matter systems. Here we show how to exploit the geometric phase as an essential tool to reveal quantum critical phenomena in many-body quantum systems. Indeed, quantum phase transitions are accompanied by a qualitative change in the nature of classical correlations and their description is clearly one of the major interests in condensed matter physics [9,10]. Such drastic changes in the properties of ground states are often due to the presence of points of degeneracy and are reflected in the geometry of the Hilbert space of the system. The geometric phase, which is a measure of the curvature of the Hilbert space, is able to capture them, thereby revealing critical behavior. This provides the means to detect, not only theoretically, but also experimentally the presence of criticality without having to undergo a quantum phase transition.In this letter we analyze the XY spin chain model and the geometric phase that corresponds to the XX criticality. Since the XY model is exactly solvable and still presents a rich structure it offers a benchmark to test the properties of geometric phases in the proximity of a quantum phase transition. Indeed, we observe that, an excitation of the model obtains a non-trivial Berry phase if and only if it circulates a region of criticality. The generation of this phase can be traced down to the presence of a conical intersection of the energy levels located at the XX criticality. This geometric interpretation reveals a relation between the critical exponents of the model. The insights provided here shed light into the understanding of more general systems, where analytic solutions might not be available. A physical implementation is proposed with ultra-cold atoms superposed by optical lattices [11,12]. It utilizes Raman activated tunneling transitions as well as coher...

Quantum Critical Scaling of the Geometric Tensors

“…Another intriguing relation between QPTs and geometrical objects i.e. Berry-phases, was suggested in Refs [13] and [14]. There it was argued that loops in the parameter space, encircling a critical line give rise to a non zero GS Berry phase even for an arbitrary small loop size.…”

The effect of inertia on the Dirac electron, the spin Hall current and the momentum space Berry curvature

“…The Berry phase approach has been successfully adopted to explain various quantum mechanical/semiclassical aspects of different systems in condensed matter physics [22][23][24][25][26][27]. In particular, the studies on momentum space Berry phase associated to spin Hall effect [28][29][30][31]has become a topic of great recent interest .…”

Loschmidt echo and Berry phase of a quantum system coupled to anXYspin chain: Proximity to a quantum phase transition

“…The connection between geometric Berry phase (BP) and QPT for the case of spin-XY model has also been studied [10,11,12], through which a remarkable relation between the BP and criticality of spin chains is established. In addition, a characterization of QPT in terms of the overlap between two ground states obtained for two different values of external parameters has been presented [13].…”