Temperature dependence of the conductivity sum rule in the normal state due to inelastic scattering

We examine the temperature dependence of the optical sum rule in the normal state due to interactions. To be concrete, we adopt a weak-coupling approach that uses an electron-boson exchange model to describe inelastic scattering of the electrons with a boson, in the Migdal approximation. While a number of recent works attribute the temperature dependence in the normal state to that which arises in a Sommerfeld expansion, we show that in a wide parameter regime this contribution can be quite small. Instead, most of the temperature dependence arises from the zeroth-order term in the “expansion,” through the temperature dependence of the spectral function and the interaction parameters contained therein. For low boson frequencies, this circumstance causes a linear $T$ dependence in the sum rule. We develop some analytical expressions and understanding of the temperature dependence.

Figure 1

(Color online) The negative of the kinetic energy vs temperature for a variety of coupling strengths. Note that the magnitude of the temperature variation increases considerably with coupling strength, and, as explained in the text, this cannot be attributed to the impact of interactions on the Sommerfeld term.

Figure 2

(Color online) The negative of the kinetic energy vs temperature for a variety of boson frequencies. Note that with decreasing phonon frequency, not only is the magnitude of the temperature variation increased, but it clearly crosses over from quadratic to linear in temperature.

Figure 3

(Color online) The mass enhancement parameter vs temperature for (a) ${\omega }_E=2\mathrm{meV}$ and (b) ${\omega }_E=10\mathrm{meV}$. Note the (very) slight deterioration of the approximate result as the boson frequency increases. Also note that for low boson frequency, the mass enhancement parameter becomes negative for high temperatures (but “high” being defined relative to the boson, not the electronic energy scale). Counterintuitively, the result for infinite bandwidth (dashed green curve) becomes poorer as the boson energy scale decreases.

Figure 4

(Color online) The negative of the kinetic energy vs temperature showing the breakdown for the various contributions. The solid (red) curves are for small boson frequency, ${\omega }_E=2\mathrm{meV}$, with $\lambda =1$. The two thin lines show how the zero-temperature value is modified by the Sommerfeld term by a very small amount; the full numerical result is given by the thick red curve. A similar contribution is found for the noninteracting case (actually, there it is a bit larger) by the dotted (blue) curves. However, here the relative effect is much larger because only the Sommerfeld term is responsible for the temperature variation. Finally, for higher boson frequency, the dashed (green) curves illustrate the amount of temperature variation due to the Sommerfeld term compared with the rest.

Figure 5

(Color online) Various approximations of the kinetic energy (without the Sommerfeld expansion terms) compared with the numerical result (solid red curve). The green (dashed) curve is given by Eq. (22), where only the imaginary part of the self-energy is included. It is not so far off the numerical result. However, the (relatively small) degree to which a discrepancy exists is misleading, as an obvious improvement (introduction of a cutoff, see Fig. 6) results in the blue (dotted) curve. This latter “improved” result is in far worse agreement with the numerics.

Figure 6

(Color online) Imaginary part of the self-energy for various temperatures. At high temperatures, the use of a constant to describe the imaginary part of the self-energy works very well; there is structure at the cutoffs barely visible on the scale shown here, but they are not very important. Despite this accurate analytical representation of the imaginary part of the self-energy, this approximation does not work well for the kinetic energy, as explained in Fig. 5.

Figure 7

(Color online) Real part of the self-energy for various temperatures. Note that even at the lowest temperature shown (green, dashed curve), the overall slope at the origin looks positive on this scale; in fact, there is some structure there, and the slope is in fact negative, but this plays almost no role in our results. This is because the most important contributions to the kinetic energy come from near the bottom of the band. Note that beyond the band edges, the approximation given by Eq. (28) is indistinguishable from the numerical result on this energy scale (a closer look would reveal some discrepancies—the numerical result has two logarithmic divergences at each band edge at low temperature, not one).

Figure 8

(Color online) Comparison of $K_{0c}\left(T\right)$ approximation (solid green circles) with numerical results (solid red curve). Agreement is excellent. Also shown are the results for $K_{0c}^2\left(T\right)$ alone, i.e., without accounting for the $\delta$-function part. Clearly both real and imaginary parts of the self-energy are necessary for accurate results.

Figure 9

(Color online) Comparison of analytical model $\left[K_{0c}\left(T\right)\right]$ (symbols) with numerical results (curves) for several boson frequencies. Note the expanded vertical scale. Clearly, the model deteriorates as the boson frequency increases (also note that the result is also no longer linear). Nonetheless, the discrepancy for ${\omega }_E=25\mathrm{meV}$ is still less than 2%.