Spontaneous Magnetization in Maxwell QED2+1

Spontaneous magnetization in the Maxwell QED2+1 at finite fermion density is studied. It is shown that at low fermion densities the one-loop free energy has its minimum at some nonvanishing value of the magnetic field. The magnetization is due to the asymmetry of the fermion spectrum of the massive QED2+1 in an external magnetic field. † E-mail address: [email protected] ⋆ Talk presented at the 2nd International Sakharov Conference on Physics, Moscow, May 20-24, 1996 In (2+1)-dimensional quantum electrodynamics (QED2+1) an addition of the ChernSimons term θ 4 εμναF A to the bare Lagrangian drastically modifies the theory [1]: the gauge field becomes massive and, due to the presence of the totally antisymmetric tensor εμνα, the electric and magnetic components of the modified Maxwell equation are mixed up. As a consequence, in a static uniform magnetic field B the electric charge j cs = θB associated with the Chern-Simons term is induced. On the other hand, in a uniform magnetic field charge connected to fermions, j f = eB 4π (2N + 1) is induced [2, 3] (N is the number of filled Landau levels ), thus the electric neutrality condition in Maxwell-Chern-Simons QED2+1is j f + j 0 cs = 0. Recently Hosotani has shown that in QED2+1, with the Chern-Simons term in the bare Lagrangian a (self-consistent) neutral configuration with a uniform magnetic field has the energy lesser than that of the naive vacuum [4]. Since the condition j f + j 0 cs = 0 also implies masslessness of the gauge field, the spontaneous magnetization in the neutral Maxwell-Chern-Simons QED2+1has Nambu-Goldstone origin [5-8]. In this talk we shall discuss the possibility of the spontaneous magnetization in the Maxwell QED2+1with a finite fermion density. We shall consider QED2+1with two-component massive fermions with the Lagrangian L = − 4 FμνF μν + ψ̄(ı∂/+ eA/−m)ψ , (1) γ-matrices are the Pauli matrices, γ = σ3, γ 1 = ıσ1, γ 2 = ıσ2. Let us remind that with a uniform external magnetic field B = ∂1A2 − ∂2A1 the fermion energy spectrum (Landau levels) in this theory is the following [9, 10]: p (0) 0 = −m sign(eB), p 0 = ± √ m2 + 2|eB|n, n = 1, 2, . . . (2) (note the asymmetry of the spectrum. The degeneracy of all levels is |eB| 2π ). Addition of the term (-μψψ) to the Lagrangian (1) provides the parameter controlling the fermion density (μ is the chemical potential, therefore all levels with energies up to μ are filled). One may calculate the fermion density (induced charge) in QED2+1 with B, μ 6= 0 using either the corresponding Green function [3] or the spectral properties of the theory [11, 12]. The latter is most straightforward way: here, in case of discrete and equally degenerated levels the density is