Gauge invariance and effective actions inD=3at finite temperature

Abstract

For background gauge field configurations reducible to the form Aµ = (Ã3, A( x)) whereÃ3 is a constant, we provide an elementary derivation of the recently obtained result for the exact induced Chern-Simons (CS) effective action in QED3 at finite temperature. The method allows us to extend the result in several useful ways: to obtain the analogous result for the 'mixed' CS term in the Dorey-Mavromatos model of parity-conserving planar superconductivity, thereby justifying their argument for flux quantization in the model; to the induced CS term for a τ -dependent flux; and to the term of second order in A( x) (and all orders inÃ3) in the effective action.PACS numbers: 11.10.Wx 11.15 11.30.Er I. INTRODUCTION.Recently, there has been significant progress [1][2][3][4] in resolving a puzzle concerning the gauge invariance of induced Chern-Simons (CS) terms at finite temperature, T . In the non-Abelian case, for example, general arguments imply that the coefficient of the CS term must be quantized at zero temperature [5], and also at non-zero temperature [6,7], if the action is to be invariant under topologically non-trivial ('large') gauge transformations. On the other hand, simple perturbative calculations [8][9][10][11][12][13][14][15][16][17], give the result that the effect of moving to T = 0 is simply to multiply the zerotemperature CS term by a smoothly varying function of T (typically, tanh( β|M| 2 ), where β = 1 kT and M is the mass of the fermion(s) in the theory). Plainly, the perturbative result contradicts the quantization requirement [18][19][20]. A similar difficulty arises in the Euclidean case for T = 0 due to the S 1 topology of the compactified Euclidean time.Apart from its theoretical interest, the resolution of this puzzle is important in some physical applications. To give one specific example, consider the Dorey-Mavromatos (DM) model [21] of two-dimensional superconductivity without parity violation. This model employs two U (1) gauge fields, one the electromagnetic field A µ , the other a 'statistical' gauge field a µ , which is also massless.There are N f ≥ 2 flavours of four-component fermions, the mass term is parity conserving, and A µ and a µ have opposite parity. At zero temperature a 'mixed Chern-Simons (MCS) term' is generated by a fermion loop with one external A and one external a leg, the leading contribution to the action, in powers of derivatives, being * CONICET

Large gauge Ward identity

“…This mechanism extends [7,8,9,10,11,12] to 2+1 dimensional Abelian theories for a restricted class of static backgrounds A µ = (A 0 (t), A( x)). However, this is not the full answer, since for these backgrounds (and for their trivial non-abelian generalizations) the "large" gauge transformations in fact have zero winding number -the shift in the Chern-Simons actions comes from a total derivative term, not from the winding number piece.…”

The relation between Pauli and γ representation in QED3 and induced Chern-Simons term

“…On the other hand, Chern-Simons theory has induced many significative studies in QED 3 [15][16][17][18]. Besides introducing Chern-Simons term into the normal QED 3 Lagrangian, it was also found that in QED 3 with two-component fermions a Chern-Simons like term can also be induced from radiative corrections [19][20][21][22][23][24][25] (Appendix A). However, this result does not hold for QED 3 with four-component fermions due to the property of γ matrix algebra.…”

Finite temperature effective action in monopole background

“…Recent finite temperature studies [1][2][3][4][5][6][7][8] of 2+1 dimensional Chern-Simons systems [9] have revealed several interesting new features of induced parity violating terms at finite temperature. It is well known [10,11] that at T=0, for fermions in the presence of a static magnetic flux Φ = d 2 x eB 2π there is an induced charge…”

“…where β = 1 T is the inverse temperature. The corresponding parity-odd part of the (Euclidean) finite temperature induced effective action is [1][2][3][4][5][6][7][8] S P −odd = −iΦ arctan tanh( βm 2 ) tan( 1 2 β 0 eA 0 dτ )…”

“…The perturbative expansion in the second line of (4) shows that at nonzero temperature there is an infinite series of parity violating terms, of which the Chern-Simons term, with temperature dependent charge (3), is only the first. The entire series is required at finite T to show that S P −odd is invariant under "large" gauge transformations A 0 → A 0 + 2π eβ N , N ∈ Z [1][2][3][4][5][6][7][8]. The higher terms are non-extensive (i.e.…”