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arXiv:0802.0577 · Physical Review A

Chirality quantum phase transition in the Dirac oscillator

A. Bermúdez, Miguel Angel Martin-Delgado, Alfredo Luis

Published Jun 11, 2008• 112 citations
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Abstract

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.

DOI
10.1103/physreva.77.063815

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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.
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.
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.
We present quantum Monte Carlo simulations for the chiral Heisenberg Gross-Neveu-Yukawa quantum phase transition of relativistic fermions with N=4 Dirac spinor components subject to a repulsive, local four fermion interaction in (2+1)D. Here we employ a two-dimensional lattice Hamiltonian with a single, spin-degenerate Dirac cone, which exactly reproduces a linear energy-momentum relation for all finite size lattice momenta in the absence of interactions. This allows us to significantly reduce finite size corrections compared to the widely studied honeycomb and π-flux lattices. A Hubbard term dynamically generates a mass beyond a critical coupling of U_{c}=6.76(1) as the system acquires antiferromagnetic order and SU(2) spin rotational symmetry is spontaneously broken. At the quantum phase transition, we extract a self-consistent set of critical exponents ν=0.98(1), η_{ϕ}=0.53(1), η_{ψ}=0.18(1), and β=0.75(1). We provide evidence for the continuous degradation of the quasiparticle weight of the fermionic excitations as the critical point is approached from the semimetallic phase. Finally, we study the effective "speed of light" of the low-energy relativistic description, which depends on the interaction U, but is expected to be regular across the quantum phase transition. We illustrate that the strongly coupled bosonic and fermionic excitations share a common velocity at the critical point.
In this set of lectures, I shall review some recent work on quantum critical phenomena which involve the changes in phase of the ground state of a quantum antiferromagnetic system when external parameters are varied through some critical values. I will also draw analogies between antiferromagnetic phase transitions and chiral phase transitions in lower (particularly 2 + l)-dimensional gauge theories and also some 2+1-dimensional four-fermi theories where chiral symmetry breaking patterns depend on the coupling constants and also the matter and gauge field content of the theory. I will begin by reviewing some arguments that, in the infrared limit, the dynamics of Goldstone bosons in the confining and chiral symmetry breaking phase of 2 + 1 dimensional QCD is identical to the effective spin wave dynamics in 2 dimensional generalized quantum antiferromagnets. I also discuss the possibility that certain quantum critical behaviors in generalized antiferromagnets can be analyzed using the principle of universality and are in the universality class of a particular type of relativistic quantum field theory with gauge and chiral symmetry. I show how the epsilon expansion can be used and argue that, for a large range of symmetry groups and other parameters, the phase transitions are expected to be fluctuation induced first order transitions. For the small range of parameters where they are second order, the epsilon expansion and universality give information about the critical behavior.
The fluctuations of massless Dirac fermion can not only turn a first-order bosonic phase transition (in the Landau sense) to a quantum critical point, but also work reversely to enhance the first-order transition itself, depending on the implementation of finite-size effects in the coupling corrections. Here, we report a case study of the latter by employing quantum Monte Carlo simulation upon a lattice model in which the bosonic part featuring the Landau-Devonshire first-order phase transition and Yukawa coupled to the Dirac fermions. We find that the parameter range for the first-order phase transition becomes larger as the Yukawa coupling increases, and the microscopic mechanism of this phenomena is revealed, at a quantitative level, as the interplay between the critical fluctuations and the finite-size effects. Moreover, the scaling behavior at the separation point between the first-order and the continuous phase transitions is found to belong to the chiral tricritical Gross-Neveu universality. Our results demonstrate that the interplay of massless Dirac fermions, critical fluctuations, and the finite-size effects could trigger a plethora of interesting phenomena, and therefore great care is called for when making generalizations.
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.