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Communications in Theoretical Physics

Chirality Quantum Phase Transition in Noncommutative Dirac Oscillator

Shao‐Hua Wang, Yu-Long Hou, Qing Wang, Zheng‐Wen Long, Jian Jing

Published Dec 1, 2014• 0 citations
Publisher

Abstract

The charged Dirac oscillator on a noncommutative plane coupling to a uniform perpendicular magnetic held is studied in this paper. We map the noncommutative plane to a commutative one by means of Bopp shift and study this problem on the commutative plane. We find that this model can be mapped onto a quantum optics model which contains Anti—Jaynes—Cummings (AJC) or Jaynes—Cummings (JC) interactions when a dimensionless parameter ζ (which is the function of the intensity of the magnetic held) takes values in different regimes. Furthermore, this model behaves as experiencing a chirality quantum phase transition when the dimensionless parameter ζ approaches the critical point. Several evidences of the chirality quantum phase transition are presented. We also study the non-relativistic limit of this model and find that a similar chirality quantum phase transition takes place in its non-relativistic limit.

DOI
10.1088/0253-6102/62/6/10

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The charged Dirac oscillator on a noncommutative plane coupling to a uniform perpendicular magnetic held is studied in this paper. We map the noncommutative plane to a commutative one by means of Bopp shift and study this problem on the commutative plane. We find that this model can be mapped onto a quantum optics model which contains Anti—Jaynes—Cummings (AJC) or Jaynes—Cummings (JC) interactions when a dimensionless parameter ζ (which is the function of the intensity of the magnetic held) takes values in different regimes. Furthermore, this model behaves as experiencing a chirality quantum phase transition when the dimensionless parameter ζ approaches the critical point. Several evidences of the chirality quantum phase transition are presented. We also study the non-relativistic limit of this model and find that a similar chirality quantum phase transition takes place in its non-relativistic limit.
The charged Dirac oscillator on a noncommutative plane coupling to a uniform perpendicular magnetic field is studied in this paper. We map the noncommutative plane to a commutative one by means of Bopp shift and study this problem on the commutative plane. We find that this model can be mapped onto a quantum optics model which contains Anti–Jaynes–Cummings(AJC) or Jaynes–Cummings(JC) interactions when a dimensionless parameterζ(which is the function of the intensity of the magnetic field) takes values in different regimes. Furthermore, this model behaves as experiencing a chirality quantum phase transition when the dimensionless parameter ζ approaches the critical point. Several evidences of the chirality quantum phase transition are presented. We also study the non-relativistic limit of this model and find that a similar chirality quantum phase transition takes place in its non-relativistic limit.
A map from the charged Dirac oscillator on a noncommutative plane interacting with a homogeneous magnetic field onto a quantum optics model is proposed. We find that both anti-Jaynes-Cummings (AJC) and Jaynes-Cummings (JC) interactions are included in this mapped quantum optics model. A competition between AJC and JC interactions will happen when the intensity of the magnetic field changes. Contrary to previous studies, the competition is symmetrical since the phonons that take part in the competition are only associated with the excitation of the Dirac oscillator. Furthermore, this model will experience a quantum phase transition when the intensity of the magnetic field crosses the critical point. However, the critical point is different from the one in the model of the charged nonrelativistic harmonic oscillator interacting with a homogenous magnetic field on a noncommutative plane. It also deviates from its commutative counterpart because of spatial noncommutativity. Some characteristics of the quantum phase transition are studied. Then we investigate the nonrelativistic limit of this model. Our results show that its nonrelativistic counterpart inherits similar properties of the chirality quantum phase transition from the relativistic one.
We study a relativistic spin-1/2 fermion subjected to a Dirac oscillator coupling and a constant magnetic field. An interplay between opposed chirality interactions culminates in the appearance of a relativistic quantum phase transition, which can be fully characterized. We obtain analytical expressions for the energy gap, order parameter, and canonical quantum fluctuations across the critical point. Moreover, we also discuss the effect of this phase transition on the statistics of the chiral bosonic ensemble, where its super- or sub-Poissonian nature can be controlled by means of external parameters. Finally, we study the entanglement properties between the degrees of freedom in the relativistic ground state, where an interesting transition between a biseparable and a genuinely tripartite entangled state occurs.
In this paper we point out that the Jaynes-Cummings model without taking a renonance conditon gives a non-commutative version of the simple spin model (including the parameters xxx, yyy and zzz) treated by M. V. Berry. This model is different from usual non-commutative ones because the x-y coordinates are quantized, while the z coordinate is not. One of new and interesting points in our non-commutative model is that the strings corresponding to Dirac ones in the Berry model exist only in states containing the ground state (F×{∣0⟩}∪{∣0⟩}×F{\cal F}\times \{\ket{0}\} \cup \{\ket{0}\}\times {\cal F}F×{∣0⟩}∪{∣0⟩}×F), while for other excited states (F×F∖F×{∣0⟩}∪{∣0⟩}×F{\cal F}\times {\cal F} \setminus {\cal F}\times \{\ket{0}\} \cup \{\ket{0}\}\times {\cal F}F×F∖F×{∣0⟩}∪{∣0⟩}×F) they don't exist. It is probable that a non-commutative model makes singular objects (singular points or singular lines or etc) in the corresponding classical model mild or removes them partly.
This paper is devoted to a study of a kind of chirality appearing in a model in cavity or circuit quantum electrodynamics. The model especially describes an artificial atom fully coupled to a 1-mode photon in a cavity. This chirality is played in the fully coupled (FC) Hamiltonian by two modified Jaynes-Cummings (JC) Hamiltonians. The modified JC Hamiltonian as well as the standard one has the superradiant ground state energy for sufficiently large coupling strength. The superradiance is caused by a phase transition found by Preparata, called the Dicke-Preparata (DP) superradiance. Using the one common Hamiltonian formalism with the chirality, we theoretically prove the followings: (i) when the coupling strength makes the weak or standard strong coupling regime, the ground state energy of the FC Hamiltonian can be well approximated with that of the standard and modified JC Hamiltonians since the chirality does not work in the regime; (ii) once the coupling strength plunges into the ultra-strong coupling regime, the effect of the chirality is turned on and works in the ground state energy of the FC Hamiltonian. That is, there is a kind of a phase transition concerning the chirality associated with the DP superradiance between the strong coupling regime and the ultra-strong coupling one.
Appearance of adiabatic geometric phase shift in the context of noncommutative quantum mechanics is studied using an exactly solvable model of a two-dimensional simple harmonic oscillator in the Moyal plane, where momentum noncommutativity are also considered along with spatial noncommutativity. For that we introduce a modified form of Bopp's shift, that bridges the noncommutative phase-space operators with their effective commutative counterparts, having their dependence on the noncommutative parameters, and study the adiabatic evolution in the Heisenberg picture. An explicit expression for the geometric phase shift under adiabatic approximation is then found without using any perturbative technique. Lastly, this phase is found to be related to Hannay's angle of a classically analogous system, by studying the evolution of the coherent state of this system.