Interacting Fermions

Abstract

In this chapter we study interacting four-Fermi theories in two and three spacetime dimension. Their Lagrangian density contains—besides the ubiquitous Dirac term —a Lorentz invariant interaction term with four powers of the Fermi field.

Appendix: Covariant ζ-Function Regularization

Rather than applying a Poisson resummation formula to the sum over the Matsubara frequencies only, as we did in (17.83), we could resum the Gaussian sum over Matsubara frequencies and spatial momenta at once. With (16.143) this leads to an alternative representation for the ζ-function,

$$\displaystyle \begin{aligned} \zeta_{H}(s)=\frac{d_s}{2}\frac{\beta V_s}{(4\pi)^{d/2}} \frac{1}{\varGamma(s)} \int \mathrm{d} t\,t^{s-1-\frac{d}{2}} \mathrm{e}^{-t\sigma^2}\sum_{n} (-1)^n \mathrm{e}^{-\beta^2n^2/4t -n\beta\mu} \sum_{\boldsymbol{n}} \mathrm{e}^{-L^2\boldsymbol{n}^2/4t}\,.{} \end{aligned} $$

(17.218)

The term with vanishing n and n is identical to the zero-temperature, zero-density, and infinite-volume contribution ζ ^{(0, ∞)} in (17.86). The remaining UV-finite series, denoted by ζ ⁽²⁾, represents the sum ζ ^{(0, L)} + ζ ⁽¹⁾ of the ζ-functions in (17.86) and (17.90) and comprises the finite temperature, density, and volume corrections. It follows that the first contribution in the associated decomposition of the effective potential

$$\displaystyle \begin{aligned} U_{\mathrm{eff}}=U^{(0,\infty)}_{\mathrm{eff}}+U^{(2)}_{\mathrm{eff}}\, {} \end{aligned} $$

(17.219)

is the expressions in (17.87) and (17.89), but without the sums over Bessel functions. Again we use (5.30) to express the remaining alternating series in (17.218) as sum over modified Bessel functions. The derivative with respect to s at s = 0 finally yields

$$\displaystyle \begin{aligned} U^{(2)}_{\mathrm{eff}} =\frac{d_s\,\sigma^d}{(2\pi)^{d/2}} {\sum_{n,\boldsymbol{n}}}'(-1)^n \mathrm{e}^{-n\beta \mu}\, \frac{K_{d/2}\big(\beta\sigma\sqrt{n^2+\tau^2\boldsymbol{n}^2}\,\big)} {\big(\beta\sigma\sqrt{n^2+\tau^2\boldsymbol{n}^2}\,\big)^{d/2}},\quad \tau=\frac{L}{\beta}\,, {} \end{aligned} $$

(17.220)

where the prime at the summation symbol means omission of the term with vanishing n and n.

In the zero temperature and density limit, only terms with n = 0 contribute, and $U^{(2)}_{\mathrm {eff}}$ is equal to the sums over Bessel functions in (17.87) and (17.89). On the other hand, at finite temperature but in the thermodynamic limit L →∞, only terms with vanishing n contribute. Using an integral representation for the Bessel function, one can prove the useful relation given in Problem 17.6. This relation shows that for L∕β →∞, the sum (17.220) is identical to the integral (17.92),

$$\displaystyle \begin{aligned} \begin{array}{rcl} \lim_{L\to\infty} U^{(2)}_{\mathrm{eff}} & =&\displaystyle 2d_s\,\Big(\frac{\sigma}{2\pi\beta}\Big)^{d/2} \sum_{n\geq 1}(-1)^n \cosh(n\beta\mu) \frac{K_{d/2}(n\beta\sigma)}{n^{d/2}}{}\\ & =&\displaystyle \frac{d_s}{2\beta (2\pi)^{d-1}}\int\mathrm{d} \boldsymbol{p}\,\log\big(1+\mathrm{e}^{-\beta(\varepsilon_{\boldsymbol{p}}+\mu)}\big) +(\mu\to-\mu)= \lim_{L\to\infty}U^{(1)}_{\mathrm{eff}} \,. \end{array} \end{aligned} $$

(17.221)

More generally, for finite temperature and finite box size $U^{(2)}_{\mathrm {eff}}=U^{(0,L)}_{\mathrm {eff}} +U^{(1)}_{\mathrm {eff}}$.