Dynamical Effects of Chern-simons Term

We study how the Chern-Simons term effects the dynamically generated fermion mass in (2+1)D Quantum Electrodynamics in the framework of large N expansion. We find that when the Chern-Simons term is present half of the fermions get mass M + m and half get M − m. The parity-preserving mass m is generated only when N < Ñc. Both the critical number, Ñc, of fermion flavor and the magnitude, m, reduce when the effect of the Chern-Simons term dominates. To be published in the proceedings of the Mount Sorak Winter School, Korea Supported in part by the Korean Science and Engineering Foundation through the SRC program of SNU-CTP. 1 Motivation One of the most drastic effects of the Chern-Simons term, LCS = κǫμνλA ∂A, in field theories, is the generation of fractional spin. The Chern-Simons term induces fractional spin to a particle coupled to the Chern-Simons gauge field, Aμ [1]: the spin is given as s = e 4πκ . (1) Due to this effect, the Chern-Simons theory is not only interesting, field theoretically, but also has some applications in the condensed matter systems like the fractional or integer quantum Hall system [2]. The Chern-Simons term is topological in a sense that it does not involve a metric and thus does not contribute to energy-momentum tensor of the system. But, it modifies the equations of motion and breaks parity and Time-reversal symmetry, which must have dynamical significances. One nice example for this dynamical effect is the existence of a stable vortex solution found recently [3] in a system described by a Lagrangian density, L = 1 2 |Dμφ| 2 − V (|φ|) + LCS, (2) where Dμ = ∂μ − ieAμ and V (|φ|) is a φ 6 potential. In (2+1)-dimensions, the kinetic energy of the static soliton is scale-invariant, while the potential energy is not. Therefore, without a gauge field or other balancing force, the static soliton is unstable against collapsing to the center of the soliton [4]. Namely, it is energetically favorable for the soliton to collapse to φ = v, where v is the minimum of the potential. One would think that adding the Chern-Simons term does not do any good in stabilizing the soliton, since it does not contribute to the energy of the soliton. But, this is not true, since not only the energy of the soliton has now a term depending on the gauge field, 1 2 A2μ |φ| , but also the Chern-Simons term modifies the equation of motion, and thus a stable vortex solution is possible. 1 In this talk, I would like to present another effect of the Chern-Simons term [5], namely the dynamical effect of the parity-noninvariance of the Chern-Simons term. 2 Parity and Mass in (2 + 1)-dimensions Consider a (2 + 1)D QED, described by a Lagrangian density L = ψ̄i 6 Dψ − 1 4 F 2 μν + Lmass (3) where ψ is a two-component spinor and the gamma matrices are chosen as γ = σ, γ = iσ, γ = iσ. The mass terms for the fermion and the photon are Lmass = −mψ̄ψ + κǫμνλA ∂A (4) In (2+1)-dimensions, parity is defined to be a coordinate transformation, P : x = (x, y, t) 7→ x = (−x, y, t), under which the fields transform as following: A(x) 7→ A 0,2 (x) = A(x) A(x) 7→ A 1 (x) = −A(x) (5) ψ(x) 7→ ψ(x) = eσψ(x) (6) One can therefore easily see that the both mass terms are odd under parity (and also under time-reversal). If either of the mass terms is absent at tree level, it will be generated radiatively, since the parity, which forbids the mass term, is broken by the other mass term explicitly. For example, when the (topological) mass term for the gauge field is absent, the fermion mass term will generate it radiatively with a coefficient κ = e 2 8π m |m| [6]. Similarly, when the fermion mass term is absent, the Chern-Simons term will generate it at one-loop; one needs a counter-term to remove the divergence in the fermion mass, δm = − 6 π e κ |M |, where M is the Pauli-Villas regulator [7]. 2 When the number of the fermion flavors is even, the system has another obvious discrete symmetry, Z2, which interchanges half of the fermions with another half; for i = 1, · · · , N 2 , Z2 mixes the fermions fields as ψi(x) 7→ ψN 2 +i(x) ψN 2 +i 7→ ψi(x) (7) If we define a new parity, P4 ≡ PZ2, combining the old one with Z2, then the fermions can have “parity(P4)-invariant” mass, miψ̄iψi, with mi = { m, if 1 ≤ i ≤ N 2 −m, if N 2 + 1 ≤ i ≤ N (8) With this form of fermion mass, the Chern-Simons term will not be generated radiatively. We call this “parity(P4)-even mass”. On the other hand, this P4-invariant fermion mass can be generated dynamically due to a non-perturbative effect, though Lmass is not present in the Lagrangian; namely P is spontaneoulsy broken, while P4 is not. Appelquist et. al. [8] showed, using 1/N -expansion, that, if 1/N > 1/Nc with Nc = 32/π , the fermions condensate and thus the parity-even fermion mass is generated dynamically and the mass is given as meven = αe − 2 16 / 1 N (9) When the Chern-Simons term is present, this parity-even mass will be affected. As described below, due to the Chern-Simons term, 1/Nc increases (one needs a stronger interation to form a fermion condensate) and the magnitude of the parityeven fermion mass decreases.