Dynamical Generation of Fermion Mass and Magnetic Field in Three-Dimensional QED with Chern-Simons Term

Abstract

We study dynamical symmetry breaking in three-dimensional QED with a Chern-Simons (CS) term, considering the screening effect of N flavor fermions. We find a new phase of the vacuum, in which both the fermion mass and a magnetic field are dynamically generated, when the coefficient of the CS term k equals Ne 2 ͞4p. The resultant vacuum becomes the finite-density state half filled by fermions. For k Ne 2 ͞2p, we find the fermion remains massless and only the magnetic field is induced. For k 0, spontaneous magnetization does not occur and should be regarded as an external field. [S0031-9007(98)06439-4] Field theoretical models in (2 1 1)-dimensional spacetime have attracted much attention as effective theories at long distance in planar condensed matter physics. Especially, quantum electrodynamics in 2 1 1 dimensions (QED 3 ) has been intensively studied in connection with the effective theories of high-T c superconductivity [1], as well as the probe for (3 1 1)-dimensional quantum chromodynamics. In 2 1 1 dimensions, there can be a topological gauge action known as Chern-Simons (CS) term. This term connects a magnetic field B with an electric charge density e͗c y c͘ for fermion field c. From this peculiar property, the CS term is used in the field theoretical understanding of the fractional quantum Hall effect [2].As a natural extension of QED 3 , a theory in which gauge field action includes both the CS term and the Maxwell term was proposed by Ref. [3]. In this theory (CS-QED 3 ), the coefficient k for the CS term gives the photon a gauge invariant mass which explicitly violates the parity symmetry. Some years ago, Hosotani [4] showed that spontaneous magnetization occurs in CS-QED 3 , through breaking the most secret symmetry-Lorentz invariance. In this theory, the Gauss law kB 2e͗c y c͘ follows from the equation of motion. Thus the magnetized vacuum corresponds to state with finite fermion density ͗c y c͘ fi 0. In Ref.[5] a chemical potential term m c y c was introduced as an explicit breaking term for Lorentz symmetry, and the condensation ͗c y c͘ at m ! 0 limit was studied. It was found that the above vacuum is stable if and only if the fermion bare mass is zero.In this paper, we examine the possibility that Lorentz symmetry is broken in a theory in which fermion mass and magnetization are both spontaneously generated. We regard the fermion bare mass term 2mcc as an explicit breaking term for flavor U͑2N͒ symmetry [6] as well as a chemical potential term m c y c for Lorentz symmetry. Solving the Schwinger-Dyson (SD) equation, we clarify whether or not two condensations, ͗c y c͘ and ͗cc͘, are dynamically realized at the symmetric limit m ! 0, m ! 0. They signal fermion mass generation and magnetization, respectively. For k Ne 2 ͞4p, we find a new phase of vacuum, in which both massive fermions and a magnetic field are dynamically generated so that both symmetries are simultaneously broken. The vacuum stability is also examined by calculating the Cornwall-Jackiw-Tomboulis (CJT) potential [7] and 2-loop effe...

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