Realizing infrared power-law liquids in the cuprates from unparticle interactions

Recent photoemission experiments [1] reveal that the excitations along the nodal region in the strange metal phase of the cuprates, rather than corresponding to poles in the single-particle Green function, exhibit power-law scaling as a function of frequency and temperature. Because such power-law scaling is indicative of a scale-invariant sector, as a first step, we perturbatively evaluate the electron self-energy due to interactions with scale-invariant unparticles. We focus on a $G_{0}W$-type diagram with an interaction $W$ mediated by a bosonic scalar unparticle. We find that in the high-temperature limit, the imaginary part of the self-energy $\mathrm{Im}\mathrm{\Sigma }$ is linear in temperature. In the low-temperature limit, $\mathrm{Im}\mathrm{\Sigma }$ exhibits the same power law in both temperature and frequency, with an exponent that depends on the scaling dimension of an unparticle operator. Such behavior is qualitatively consistent with the experimental observations. We then expand the unparticle propagator into coherent and incoherent contributions and study how the incoherent part violates the density of states (DOS) and density-density correlation function sum rules (f-sum rule). Such violations, in principle, can be observed experimentally. Our work indicates that the physical mechanism for the origin of the power-law scaling is the incoherent background, which is generated from the Mott-scale physics.

Figure 1

The spectral function of unparticles when $0<\alpha <1$.

Figure 2

The Feynman diagram for the electron's self-energy. The solid line represents an electron propagator. The dashed line represents an unparticle propagator.

Figure 3

Log-log plots of the imaginary part of the self-energy as a function of temperature.

Figure 4

Plots of $\mathrm{Im}\mathrm{\Sigma }$ temperature power-law exponent $n$ vs $\alpha$ at low temperatures. Squares and circles correspond to the exponents obtained by fitting the low-$T$ parts of $\mathrm{Im}\mathrm{\Sigma }$ to power law in $d=3$ and $d=2$, respectively. Black lines are the plots of $n=d-2+2\alpha$.

Figure 5

Plots of the imaginary part of the self-energy as a function of frequency in the case $\alpha =0.7$ and $d=3$.

Figure 6

Plot of the $\mathrm{Im}\mathrm{\Sigma }$ frequency power-law exponent $n$ vs $\alpha$ at low frequency and temperature ($T=0.01m$). The spatial dimension in this case is $d=3$. Squares (circles) correspond to exponents obtained by fitting the positive (negative) $\omega$ part of $\mathrm{Im}\mathrm{\Sigma }$ to a power law. The black line is the plot of $n=d-2+2\alpha$.

Figure 7

(Left) Plot of the imaginary part of the fermionic density-density correlation function (Lindhard type) as a function of the transferred frequency $\omega$ for various values of $\alpha$. The plots are shown for $\left|\mathbf{q}\right|=0.5|{\mathbf{k}}_f|$. (Right) Violation of the sum rule integral $S$ as a function of the anomalous dimension $\alpha$. The dashed line shows the $\alpha =0$ value ($n{q}^2/m$), and the enclosed area between the red points and the dashed line shows the deviation from this value. The energy cutoff is fixed at $\frac{{\omega }_c}{2{\epsilon }_f}=2$.

Figure 8

Dependence of the sum rule integral $S$ on the energy cutoff $\frac{{\omega }_c}{2{\epsilon }_f}$. The value of the anomalous dimension parameter is fixed at $\alpha =0.09$.

Figure 9

The contour $C$ of the integral over $z$ in Eq. (A1). A solid dot represents a first-order pole. A dotted line represents a branch cut.