On The Finite Temperature Chern-Simons Coefficient

We compute the exact finite temperature effective action in a 0+1dimensional field theory containing a topological Chern-Simons term, which has many features in common with 2+1-dimensional Chern-Simons theories. This exact result explains the origin and meaning of puzzling temperature dependent coefficients found in various naive perturbative computations in the higher dimensional models. There are many examples in physics in which the classical Lagrange density contains a term that is not strictly invariant under a certain transformation (for example, a ‘large’ gauge transformation), but the classical action changes by a constant that takes discrete values associated with the ‘winding number’ of the transformation. For such a system the ∗[email protected] †[email protected] ‡[email protected] 1 quantum theory is formally invariant provided the amplitude exp(i(action)) is invariant; thus, invariance of the quantum theory can be maintained provided the coefficient of the noninvariant term in the Lagrange density is chosen to take appropriate discrete values. This argument is familiar in the theory of the Dirac magnetic monopole, and in Chern-Simons theories [1]. It is important to ask what happens to this discretization condition when quantum interactions are taken into account. For example, the quantum effective action may contain induced terms of the same noninvariant form, but with a new coefficient. This subject of induced topological terms is relatively well understood in various examples of zero temperature quantum field theory [2–4]. However, there is currently a great deal of confusion in the corresponding theories at nonzero temperature. Typically [5–9], a naive perturbative computation that mimics the zero temperature computation leads to an induced topological term equal to the zero temperature induced topological term, but multiplied by an extra factor of tanh(β|m|/2). Here β = 1/T is the inverse temperature, and m is a relevant mass scale. Clearly, this coefficient cannot take only discrete values for all T , even though formal arguments suggest that it should. This dilemma has recently been emphasized [6,9] for the particular case of 2 + 1-dimensional fermion and/or Chern-Simons theories, for which quantum effects may lead to induced Chern-Simons terms. (Related features also appear in monopole and Aharonov-Bohm systems [10–12]). There is one opinion that anyonic superfluidity should break down at any finite temperature due to this anomaly [13]. There is an opposite opinion that there is no such temperature dependent anomaly due to some ‘nonperturbative’ physics. However, we feel that the discussion thus far has missed an essential point. To illustrate this, we consider a simple analogue of the Chern-Simons system, which has the advantage that it may be solved exactly and yet it still retains the essential topological complexities of the problem. Consider a 0 + 1-dimensional field theory of Nf fermions ψj , j = 1 . . .Nf , minimally coupled to a U(1) gauge field A. It is not possible to write a Maxwell-like kinetic term for