Low-energy excitations and singular contributions in the thermodynamics of clean Fermi liquids

Using a recently suggested method of bosonization in an arbitrary dimension, we study the anomalous contribution of the low-energy spin and charge excitations to thermodynamic quantities of a two-dimensional (2D) Fermi liquid. The method is slightly modified for the present purpose such that the effective supersymmetric action no longer contains the high-energy degrees of freedom but still accounts for effects of the finite curvature of the Fermi surface. Calculating the anomalous contribution $\delta c\left(T\right)$ to the specific heat, we show that the leading logarithmic in temperature corrections to $\delta c\left(T\right)/T^2$ can be obtained in a scheme combining a summation of ladder diagrams and renormalization group equations. The final result is represented as the sum of two separate terms that can be interpreted as coming from singlet and triplet superconducting excitations. The latter may diverge in certain regions of the coupling constants, which should correspond to the formation of triplet Cooper pairs.

Figure 1

Decomposition of the interaction ${\mathcal{S}}_{\mathrm{int}}$, Eq. (2.6), into slow modes, $\left|\mathbf{q}\right|\lesssim q_0\ll p_F$. (a), (b), and (c) are the Hartree, Fock, and Cooper vertices, respectively. In our model, the Cooper vertex should be omitted to avoid double counting.

Figure 2

Diagrammatic building blocks of our low-energy field theory: (a) the propagator $g_{\mathbf{n}}\left(K\right)$, Eq. (2.54), and (b) the interaction vertices ${\mathcal{S}}_4$, ${\mathcal{S}}_3$, and ${\mathcal{S}}_2$ from Eqs. (2.55), (2.57), and (2.58).

Figure 3

Backscattering diagrams for the thermodynamic potential $\Omega$. Diagram (a) is the second-order diagram containing the leading backscattering contribution for $\mathbf{n}\sim -\tilde{\mathbf{n}}$, while diagram (b) represents an exemplary logarithmic renormalization to diagram (a), cf. Sec. 3c. Finally, diagram (c) gives for ${\mathbf{n}}_1\sim -{\mathbf{n}}_2\sim -{\mathbf{n}}_3\sim {\mathbf{n}}_4$ a backscattering contribution of higher order in the interaction and also includes a renormalizing building block ${\mathcal{S}}_4$. For weakly interacting fermions, diagram (c) can be neglected.

Figure 4

One-loop diagrams for the quartic action ${\mathcal{S}}_4$. Diagram (a) is logarithmic independently from the dimensionality of the system, whereas diagrams (b) and (c) are negligible in dimensions $d>1$ due to curvature effects. Diagrams (d) and (e) vanish after ${\overline{q}}_{\parallel }$ integration for any $d$ and, finally, diagram (f) vanishes due to supersymmetry. The contributions of diagrams (a${}^{\prime }$) and (a${}^{\prime \prime }$), each of which is logarithmic on its own, cancel each other.

Figure 5

Logarithmic one-loop corrections to the quadratic action ${\mathcal{S}}_2$, Eq. (2.58).

Figure 6

Diagrams (a) and (a${}^{\prime }$) represent the logarithmic one-loop corrections to the cubic action ${\mathcal{S}}_3$, Eq. (2.57).

Figure 7

Diagrammatical Bethe-Salpether equations for the ${\mathcal{S}}_4$ ladders. Dark gray blocks stand for the renormalized vertices, while light gray blocks stand for the bare quartic vertices. ${\mathcal{P}}^c$ and ${\mathcal{P}}^s$ are defined to project a general quartic vertex onto its vertex component of the form $\left[\Psi \Psi \right]\left[\Psi \Psi \right]$ or $\left[\Psi {\sigma }^k\Psi \right]\left[\Psi {\sigma }^k\Psi \right]$, respectively.

Figure 8

The nonanalytic correction to the specific heat of a two-dimensional Fermi liquid for a repulsive contact interaction as a function of ${\gamma }_{\pi }^{\mathrm{I}}L$. The logarithmic-linear plot in the background illustrates the behavior at large ${\gamma }_{\pi }^{\mathrm{I}}L$, i.e., in the asymptotic limit of very low temperatures $T$.

Figure 9

Diagrams for the third-order backscattering contribution to the thermodynamic potential.

Figure 10

Correspondence between the bosonic low-energy representation and conventional diagrammatics: (a) the bare backscattering diagram from Fig. 3a, (b) its representation in Hartree and Fock soft modes according to Table , and finally (c) the conventional fermionic diagrams.

Figure 11

The bosonic one-loop diagram Fig. 4a and its relation to conventional diagrams. The latter form particle-particle ladders, giving logarithms independently from the dimension $d$ of the system.

Figure 12

The bosonic one-loop diagram Fig. 4b and its related conventional diagrams. The latter are of particle-hole ladder and polarization bubble type, reflecting the absence of a logarithmic divergency in $d>1$ dimensions.