Magnetic Catalysis of Chiral Symmetry Breaking in QED at Finite Temperature

The catalysis of chiral symmetry breaking by a magnetic field in the massless weakcoupling phase of QED is studied. The dynamical mass of a fermion (energy gap in the fermion spectrum) is shown to depend essentially nonanalytically on the renormalized coupling constant α in a strong magnetic field. The temperature of the symmetry restoration is calculated analytically as Tc ≈ mdyn, where mdyn is the dynamical mass of a fermion at zero temperature. To be published in Proceedings of APCTP-ICTP Joint International conference on ”Recent Developments in Non-perturbative Methods”, held on May 26-30, 1997 at Seoul, Korea. E-mail: [email protected] 1 In this talk I will discuss dynamical chiral symmetry breaking in a magnetic field and at finite temperature. The talk is based on the recent papers with V.Miransky and I.Shovkovy [1, 2, 3]. The dynamics of fermions in an external constant magnetic field in QED was considered by Schwinger long ago [4]. In that work, while the interaction with the external field was considered in all orders in the coupling constant, the quantum dynamics was treated perturbatively. There is no dynamical chiral symmetry breaking in QED in this approximation. Also, chiral symmetry breaking is not manifested in the weak coupling phase of QED in the absense of a magnetic field, even if it is treated nonperturbatively. We will show that a constant magnetic field B changes the situation drastically, namely, it leads to dynamical chiral symmetry breaking in QED for any arbitrary weak interaction. The essence of this magnetic catalysis is that electrons are effectively 1+1 dimensional when their energy is much less than the Landau gap √ |eB| what was pointed out recently in Refs.[5, 6]. The lowest Landau level (LLL) plays here the role similar to that of the Fermi surface in the BCS theory of superconductivity, leading to dimensional reduction in dynamics of fermion pairing. The dynamical mass of fermions (energy gap in the fermion spectrum) is: mdyn ≃ C √ eB exp [