Temperature dependence of the spin susceptibility of a clean Fermi gas with repulsion

Spin susceptibility of a clean Fermi gas with repulsion in any dimension is considered using a supersymmetric low energy theory of interacting spin excitations and renormalization scheme recently proposed by Aleiner and Efetov. We generalize this method to include the coupling to the magnetic field. As a result, we obtain for the correction $\delta \chi$ to the Pauli susceptibility a nonanalytic temperature dependence of the form $T^{d-1}{\gamma }_b^2\left(T\right)$ in dimensions $d=2$, 3, where ${\gamma }_b\left(T\right)$ is an effective $d$-dependent logarithmically renormalized backscattering amplitude. In one dimension, $\delta \chi$ is proportional to ${\gamma }_b\left(T\right)$, and we reproduce a well known result obtained long ago by a direct calculation.

Figure 1

Basic building blocks of the perturbation theory.

Figure 2

A closed loop of fermionic Green’s function contrasted with the spin mode representation.

Figure 3

A particular diagram for the susceptibility formulated in the fermionic and the spin mode representation.

Figure 4

Relevant diagrams for the renormalization of the interaction amplitudes as found in Ref. 1.

Figure 5

Logarithmic correction to ${\mathcal{S}}_0$ in $d=1$.

Figure 6

Diagrams for the corrections to ${\mathcal{S}}_{b2}$. Logarithmic corrections from $\delta {\mathcal{S}}_{b2}^{\left(1\right)}$ and $\delta {\mathcal{S}}_{b2}^{\left(2\right)}$ cancel each other in any dimension. $\delta {\mathcal{S}}_{b2}^{\left(3\right)}$ gives a logarithmic correction in $d=1$, but for the susceptibility it gives corrections beyond our accuracy.

Figure 7

Logarithmic correction to ${\mathcal{S}}_{b0}$ in $d=1$.

Figure 8

Logarithmic corrections to $\delta {\mathcal{S}}_{b1}$ in $d=1$.

Figure 9

These two diagrams with renormalized external vertices $\left(\delta {\chi }_2\right)$ and interaction amplitude $\left(\delta {\chi }_3\right)$ determine the correction to the susceptibility in $d=1$ together with the diagram of Fig. 7.

Figure 10

These are the diagrams responsible for the nonanalytic temperature dependence of the spin susceptibility.