Specific heat of a one-dimensional interacting Fermi system: Role of anomalies

We revisit the issue of the temperature dependence of the specific heat $C\left(T\right)$ for interacting fermions in one dimension. The charge component $C_c\left(T\right)$ scales linearly with $T$, but the spin component $C_s\left(T\right)$ displays a more complex behavior with $T$ as it depends on the backscattering amplitude, $g_1$, which scales down under renormalization group transformation and eventually behaves as $g_1\left(T\right)\sim 1∕\log T$. We show, however, by direct perturbative calculations that $C_s\left(T\right)$ is strictly linear in $T$ to order $g_1^2$ as it contains the renormalized backscattering amplitude not on the scale of $T$, but at the cutoff scale set by the momentum dependence of the interaction around $2k_F$. The running amplitude $g_1\left(T\right)$ appears only at third order and gives rise to an extra $T∕{\log }^{3}T$ term in $C_s\left(T\right)$. This agrees with the results obtained by a variety of bosonization techniques. We also show how to obtain the same expansion in $g_1$ within the sine-Gordon model.

Figure 1

A model interaction potential with two cutoffs: ${\overline{\Lambda }}_{\mathrm{b}}$ near $q=0$ and ${\Lambda }_{\mathrm{b}}$ near $q=2k_F$.

Figure 2

One-loop diagrams for the interaction vertices. In the RG regime (external momenta are much smaller than ${\Lambda }_{\mathrm{b}}$), the renormalizations of $g_{1\perp }$ and $g_{1\parallel }$ are given by diagrams (a), (b), and (d), while the renormalizations of $g_{2\perp }$ and $g_{2\parallel }$ are given by diagrams (c) and (d). All diagrams give rise to $\log T$ terms. For external momenta of order ${\Lambda }_{\mathrm{b}}$, only diagram (a) gives rise to the logarithmic term $\log ({\Lambda }_{\mathrm{f}}∕{\Lambda }_{\mathrm{b}})$, as the two fermions in the particle-hole bubble can have momenta in the whole range between ${\Lambda }_{\mathrm{b}}$ and ${\Lambda }_{\mathrm{f}}$. For all other diagrams, the interaction constraints the internal momenta to be of the same order as the external momentum, and there is no momentum space for the logarithm.

Figure 3

First-order diagrams for the thermodynamic potential. Here and in the rest of the figures, the dashed line represents the interaction. Diagram (1b) does not contribute to the temperature dependence of the thermodynamic potential.

Figure 4

Second-order diagrams for the thermodynamic potential.

Figure 5

Third-order diagrams that potentially give logarithmic contributions. In diagram (3a), momenta $k$ and $p$ are counted from $\pm k_F$, respectively.

Figure 6

Fourth-order diagram with four bubbles.

Figure 7

Integration contour for Eq. (A5).