Extremely correlated Fermi liquids: Self-consistent solution of the second-order theory

We present detailed results from a recent microscopic theory of extremely correlated Fermi liquids, applied to the $t$-$J$ model in two dimensions, developed recently by Shastry [Phys. Rev. Lett. 107, 056403 (2011); Phys. Rev. B 87, 125124 (2013)]. The second-order theory in the parameter $\lambda$, related to the density, is argued to be quantitatively valid in the overdoped regime for $0\le n\lesssim 0.75$, with $n$ denoting the particle density. The calculation involves the self-consistent solution of equations for an auxiliary Fermi liquid Green's function and an adaptive spectral weight. We present numerical results at low as well as high $T$, at various low to intermediate densities in the normal phase, using a minimal set of band parameters relevant to the cuprate superconductors. We display the momentum space occupation function $m_k$, energy dispersion curves locating the peaks of spectral functions, the optical conductivity, relaxation rates for quasiparticles, and the electronic spectral functions on an absolute scale. The line shapes have an asymmetric shape and a broad background that is also seen in experiments, and our calculations validate approximate recent versions of the theory. The results also display the experimentally noted high-energy kink and provide an in-depth understanding of its origin and dependence on band parameters.

Figure 1

The momentum distribution function $m_k$ is plotted along three principle lines of the BZ. The left and right figures are at 130 and 605 K, respectively. In each case the FS is the same as in the noninteracting problem. The Luttinger-Ward crossing $\mathrm{Re}{\mathcal{G}}^{-1}(\vec{k},0)=0$ is indicated for each density by the vertical dashed lines. For each density and each temperature the Luttinger-Ward crossings correspond well with the condition $m_k=\frac{1}{2}$.

Figure 2

$T=130$ K. The three dispersions defined in Eq. (19) are plotted along principle directions for three different densities. The upper insets show the bandwidth of the dispersions as a function of the density. The bare bandwidth is $2\text{eV}$, but each of these dispersions shrinks compared to that scale. The bandwidth renormalization due to $\mathrm{Re}\overline{\Phi }$ in Eq. (19) is $k$ dependent, and so $E_k$ has a different shape than ${\epsilon }_k$. Note that $E_k\sim E_k^{*}$ near the FS. However, $E_k^{*}$ differs from $E_k$ near the $\Gamma$ point for each of the densities. The lower inset shows the evolution of the real part of the denominator of $\mathbf{g}(k,\omega )$ with $\omega$ to illustrate the origin of the difference between $E_k$ and $E_k^{*}$. In the inset $E_k$ is determined by the zero crossings of the curves. At low $k$ notice that a relatively flat feature develops with a shallow minimum near $\omega =-0.3\text{eV}$. The minimum corresponds to the peak $E_k^{*}$. For increasing $k$, the flat feature quickly disappears and the zero crossing moves quickly upward in frequency, producing the observed kink in $E_k$.

Figure 3

$L=60$, $(n,T)=(0.75,300)$ K. Density plot of $A(k,\omega )$ of the minimal model (top) and the refined model Ref. 28 (bottom). (Here and below, red denotes high intensity and blue denotes low intensity). $E_k$, $E_k^{*}$, and $E_{MDC}\left(k\right)$ spectra are white, green, and black, respectively. Near $k_F$ we see that the three spectra coincide. In the region near $k=(0,0)$, $E_{MDC}\left(k\right)$ is at a significantly higher energy scale than $E_k$ or $E_k^{*}$, signifying the high-energy kink (waterfall) effect. Also, the EDC peak loses weight in this regime. A new feature arises at near $k=(\pi ,\pi )$ resembling an inverted waterfall.

Figure 4

$L=60$, $(n,T)=(0.75,300)$ K. (Top) $t^{\prime }/t=0.4$ is used to model electron-doped high-$T_c$ superconductors. The kink feature is prominent here. (Bottom) Uses $t^{\prime }/t=-0.4$ to crudely model a holelike FS. In this case the kink near $(0,0)$ is lost, unlike in Fig. 3, correlating with a flat (bare) band dispersion.

Figure 5

$n=0.75$. The LDOS of the physical $\mathcal{G}$ (auxiliary $\mathbf{g}$) is in black (dotted blue), and the bare DOS is the dashed red curve. The renormalized band displays narrowing, and a long tail at $\omega <0$. The LDOS develops a second spectral peak for $\omega >0$ from a strongly $k$-dependent feature in the self-energy.

Figure 6

$n=0.75$. The spectral function ${\rho }_{\mathcal{G}}$ [$=A(k,\omega )$] at several $k$ points along the $\langle 11\rangle$ direction and $T$. We used $L_x=36$; the insets show all positive $k_x$'s and the main figures display a third of the allowed $k_x$'s. The inset in each case zooms out to reveal the heights. The linewidth near $k_F$ is seen to be strongly effected by rising $T$; the incoherent parts have very little $T$ dependence. The tails exhibit a secondary broad peak near $\omega =-0.4\text{eV}$, giving rise to the high-energy kink (waterfall).

Figure 7

$n=0.75$. The QP relaxation rate at the FS along $\langle 11\rangle$ obtained from ${\rho }_{\Sigma }(k,E_k^{*})$, and the rate obtained from the optical conductivity as in Eq. (22). The $T^2$ behavior of an FL is visible at low temperature, crossing over at a modest temperature ($\sim$150) K, partly due to the shrinking bandwidth, as seen directly in Fig. 2. The inset shows the dc resistivity obtained from the inverse of Eq. (21). It similarly displays a $T^2$ behavior crossing over to a linear behavior, as well as a lack of saturation that persists to higher $T$ than shown.

Figure 8

$n=0.75$, $T=60,90,130,190,280,410,605$ K. The optical conductivity is calculated on an absolute scale and illustrates how increasing $T$ rapidly fills up the regime $200\le \omega \le 1000$ cm${}^{-1}$. The rise of conductivity at very low $\omega$ is also inferred from the dc resistivity displayed in Fig. 7. The phase of the complex $\sigma$ falls off rapidly beyond 4000 cm${}^{-1}$.

Figure 9

An explicit comparison of optical conductivity with measurements of Puchkov et al. from Ref. 12 with the author's kind permission. The data pertains to an overdoped thallium-based cuprate with $T_c=23$ K, with a density $n\approx 0.75$. We note the similarity of magnitude and variation with $\omega$ and $T$. It is worth noting (to be reported elsewhere) that the vertical scale can be brought into better agreement with an adjusted hopping, as with the Fermi velocity.

Figure 10

$(n,T)=(0.75,130)$ K. The spectral functions for the two self-energies $\Phi$ and $\Psi$, i.e., ${\rho }_{\overline{\Phi }}$ (top) and ${\rho }_{\Psi }$ (bottom), at several $k$ points along the $\langle 11\rangle$ direction. Both are roughly quadratic and symmetric at low frequency but have a strongly $k$-dependent curvature. In the plot of ${\rho }_{\overline{\Phi }}$, the minimum width $\eta$ chops off the bottom of the low-frequency minimum.

Figure 11

$(n,T)=(0.75,130)$ K. (Top) The spectral function ${\rho }_{\Sigma }$ of the Dyson-Mori self-energy $\Sigma$ from Eq. (24), at several $k$ points along the $\langle 11\rangle$ direction. As with ${\rho }_{\overline{\Phi }}$, ${\rho }_{\Sigma }$ has inherited a strong $k$ dependence. (Bottom) $k$ dependence of the fit parameters from ${\rho }_{\Sigma }=a+b{\omega }^2(1+c\omega )$ at low frequencies $\left|\omega \right|\le 75$meV. Observe the softening of the quadratic coefficient with increased $k$. The cubic term ${\rho }_{\Sigma }\propto {\omega }^3$ produces particle-hole asymmetry, as argued in Ref. 6, and grows in magnitude with increasing $k$ beyond $k_F$.

Figure 12

$n=0.75$. The decay rate of QP near $k_F$ along the nodal line from ${\Gamma }_k={\rho }_{\Sigma }(k,E_k^{*})$. The strong and $T$-dependent asymmetry makes quasiparticles longer lived at $k>k_F$. With increased $T$ the minimum of $\Gamma$ moves to $k>k_F$.