Impact of a finite cut-off for the optical sum rule in the superconducting state

A single band optical sum rule derived by Kubo can reveal a novel kind of superconducting state. It relies, however, on a knowledge of the single band contribution from zero to infinite frequency. A number of experiments on the high temperature superconductors over the past five years have used this sum rule; their data has been interpreted in support of “kinetic energy-driven superconductivity.” However, because of the presence of unwanted interband optical spectral weight, they necessarily have to truncate their sum at a finite frequency. This work examines theoretical models where the impact of this truncation can be examined first in the normal state, and then in the superconducting state. The latter case is particularly important as previous considerations attributed the observed anomalous temperature dependence as an artifact of a noninfinite cut-off frequency. We find that this is, in fact, not the case, and that the sign of the corrections from the use of a noninfinite cut-off is such that the observed temperature dependence is even more anomalous when proper account is taken of the cutoff. This conclusion holds if a finite scattering rate is used, which is in contrast to the often considered but less realistic Mattis–Bardeen (dirty) limit. On the other hand, in these same models, we find that the strong observed temperature dependence in the normal state can be attributed to the effect of a noninfinite cut-off frequency.

Figure 1

(Color online) The real part of the optical conductivity (normalized to ${\sigma }_0\equiv \frac{{\omega }_p^2}{4\pi }\tau$) vs frequency at two different temperatures. In either case, we use the full Kubo formulation [labeled Kubo–see Eq. (1)] and the so-called Drude result [labeled Drude—see Eq. (7)], which is designed to be accurate at high frequency. In (a), we show the low frequency part, where discrepancies can be large, particularly at low temperatures. In (b), we show the high frequency tail, where the Drude result, based on the high frequency scattering rate [Eq. (6)], is very accurate. This high frequency accuracy means that the conductivity (and hence the spectral weight) beyond some frequency will be well represented by the simpler Drude expression. Here, $\nu \approx 2\mathrm{eV}$ suffices for high accuracy.

Figure 2

(Color online) As in Fig. 1, the real part of the optical conductivity (normalized to ${\sigma }_0\equiv \frac{{\omega }_p^2}{4\pi }\tau$) vs frequency at two different temperatures. Now, the Einstein boson frequency is $20\mathrm{meV}$, compared to that in Fig. 1, which was $60\mathrm{meV}$. The legend and curve designation are as in Fig. 1. Comparison with Fig. 1 illustrates that for lower boson frequency (i.e., overall lower effective high frequency scattering rate), the Drude approximation is more precise for lower frequencies.

Figure 3

(Color online) The optical spectral weight, integrated up to $1\mathrm{eV}$, vs temperature for a variety of model parameters. Curves are for results using the Kubo formula [Eq. (1)], while accompanying symbols are for the Drude approximation, given analytically in Eq. (7). The lowest three curves and symbols explore the dependence with boson frequency (fixed coupling strength) and show that, in the temperature range of interest, increasing the boson frequency results in less temperature dependence. Since increasing the boson frequency also increases the effective (infinite frequency) scattering rate [see Eq. (6)], the sum rule up to $1\mathrm{eV}$ is lower for higher boson frequency. Starting from the second lowest curve (indicated to have ${\omega }_E=40\mathrm{meV}$), the top three curves explore the dependence on coupling strength. Clearly, as coupling strength decreases, the temperature dependence diminishes until, with $\lambda =0$, and therefore, only elastic scattering remaining, there is no temperature dependence (top curve). That two of the curves merge at low temperature is not a coincidence; they have the same $\lambda {\omega }_E$ product, and therefore, have the same low temperature sum rule result [see Eq. (6) with an Einstein spectrum substituted for ${\alpha }^{2}F\left(\Omega \right)$].

Figure 4

(Color online) The Mattis–Bardeen result for the optical conductivity (left scale) and the optical spectral weight (right scale) as a function of frequency (note the bottom scale is for the conductivity and the top scale is for the spectral weight). In the Mattis–Bardeen limit, the normal state is a constant; this is indicated by the horizontal (green) line at unity. The full numerical result for the conductivity is labeled and shown by the solid (red) curve, including a delta-function contribution at the origin. The analytical result, given by Eq. (11), is also shown and is in remarkable agreement with the numerical result right down to $\nu =2{\Delta }_0$. The full numerical result for the optical spectral weight integrated up to a frequency $\nu$ is shown by the solid (pink) curve. The approximate result [Eq. (12)] is indiscernible from the numerical result. Note that as the frequency increases, the Mattis–Bardeen result always lies beneath the normal state.

Figure 5

(Color online) The real part of the optical conductivity (normalized to ${\sigma }_0\equiv \frac{{\omega }_p^2}{4\pi }\tau$) vs frequency for a variety of scattering rates. The solid red curves are in the normal state and are simple Drude results. The dotted blue curves are the results of the numerical integration in the superconducting state; they all have a gap at $\nu =2{\Delta }_0$. The high frequency expansions for the superconducting state [Eq. (14)] are shown by the dashed green curves. These are surprisingly very accurate right down to $\nu =2{\Delta }_0$. The Mattis–Bardeen result is also shown—it is quite obvious that the result in the superconducting state is always beneath that in the normal state. For the other scattering rates shown, however, the superconducting state crosses the normal state, a characteristic first noted by Chubukov et al. (Ref. 32) and emphasized in Fig. 6 below.

Figure 6

(Color online) The difference (superconducting − normal) in the real part of the optical conductivity (normalized to ${\sigma }_0\equiv \frac{{\omega }_p^2}{4\pi }\tau$) vs frequency for the cases shown in Fig. 5. The solid red curves are for the numerical result and the green curves are for the analytical high frequency result given by Eq. (14). The accuracy is remarkable (note the vertical scale). In particular, the frequency at which the difference crosses zero is given by the approximate expression fairly accurately. Note that the Mattis–Bardeen approximate result [Eq. (11)] is invisible under the numerical result.

Figure 7

(Color online) The normalized difference in the optical spectral weight for a variety of scattering rates, $1∕\tau$. Note that for any noninfinite scattering rate, this difference is negative for frequencies above approximately $1∕\tau$. A negative difference has the same sign as the conventional BCS result. The experimental results for various doping levels in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ (Ref. 6) are shown by the points, with a cut-off frequency ${\nu }_c=1.24\mathrm{eV}$ and an assumed gap of $14\mathrm{meV}$. Note that the underdoped and optimally doped results are positive.