Low-energy physics of the $t\text{−}J$ model in $d=\infty$ using extremely correlated Fermi liquid theory: Cutoff second-order equations

We present the results for the low-energy properties of the infinite-dimensional $t\text{−}J$ model with $J=0$, using $O\left({\lambda }^2\right)$ equations of the extremely correlated Fermi liquid formalism. The parameter $\lambda \in [0,1]$ is analogous to the inverse spin parameter $1/\left(2S\right)$ in quantum magnets. The present analytical scheme allows us to approach the physically most interesting regime near the Mott insulating state $n\lesssim 1$. It overcomes the limitation to low densities $n\lesssim 0.7$ of earlier calculations, by employing a variant of the skeleton graph expansion, and a high-frequency cutoff that is essential for maintaining the known high-$T$ entropy. The resulting quasiparticle weight $Z$, the low $\omega ,T$ self-energy, and the resistivity are reported. These are quite close at all densities to the exact numerical results of the $U=\infty$ Hubbard model, obtained using the dynamical mean field theory. The present calculation offers the advantage of generalizing to finite $T$ rather easily, and allows the visualization of the loss of coherence of Fermi liquid quasiparticles by raising $T$. The present scheme is generalizable to finite dimensions and a nonvanishing $J$.

Figure 1

${\rho }_{dc}$ on absolute scale vs $T$ in Kelvin for particle density $n=0.85$. We have used the estimates $D=12000$ K and ${\rho }_0=258\mu \mathrm{\Omega }$ cm. The latter is obtained by using [57] ${\rho }_0\approx \frac{h{a}_{0}d}{e^2}$, where we estimate $a_{0}d\approx {10}^{-8}$ cm. The Fermi liquid behavior with quadratic resistivity in the blue dotted line breaks down above $T_{\text{FL}}\approx 30$ K, and is followed by a regime of linear resistivity.

Figure 2

A schematic depiction of the local spectral density of states ${\rho }_{GLocal}\left(\omega \right)$ [popularly called $A_{Local}\left(\omega \right)]$ for the large-$U$ Hubbard model, where the correlation split Hubbard bands are clearly separated. It shows three regions: (a) occupied electronic states, (b) unoccupied lower Hubbard band states, and (c) unoccupied upper Hubbard band states, with their respective weights as in Eq. (18). The $t\text{−}J$ model sends the region (c) off to infinity with weights given in Eq. (19). The area in region (b) is exactly $(1-n)$, and preserving this in an approximation is key to obtaining the correct low-energy scale.

Figure 3

Multiplication through the Tukey window ${\mathbf{W}}_T\left(\omega \right)$ [Eq. (34)] is used for providing a cutoff in our scheme (33). It is applied only to the auxiliary local Green's function ${\rho }_{\mathbf{g}L,m}\left(\omega \right)$, while the physical spectral functions ${\rho }_{\mathcal{G}L}\left(\omega \right)$ are unconstrained, apart from an overall window $\left|\omega \right|\le 5D$ used for numerical purposes. In this work, the upper cutoff used is ${\mathrm{\Omega }}_c^{(+)}=2D$, and the lower cutoff ${\mathrm{\Omega }}_c^{(-)}=1.5D$.

Figure 4

Left: The chemical potential at particle densities $n=0.4,0.5,0.6,0.7,0.75,0.775,0.8,0.825,0.85,0.875$ increasing from bottom to top. We confine ourselves to the limited regime $T\le 1.2$ since higher $T$ requires further adjustment of the cutoffs. Note that within this regime, the $\mu \left(T\right)$ curve turns around at a density around $n\sim 0.7$. For lower densities, $\mu$ decreases monotonically with increasing $T$, whereas at higher densities we have a shallow minimum followed by a regime of rising $\mu$. This change of behavior is expected from Eq. (31), and has important physical consequence of changing the sign of the Kelvin thermopower for correlated matter [55]. Right: The slope $d\mu /dT$ is calculated from the $\mu \left(T\right)$ curves at $T=1$, and contrasted with the exact values from Eq. (31). The points are taken from the same set of particle densities $n$ as the figure on left, increasing from left to right. Since there is yet some curvature in the figures at left when $T=1$, our procedure provides only a rough estimate. We note that these are in fair correspondence, especially at low hole density (see top right quadrant).

Figure 5

The $T$ dependence of the chemical potential $\mathbf{\mu }$ and its three additive contributions from Eq. (23) at two densities. The physical chemical potential $\mathbf{\mu }$ (I, red), the auxiliary part: ${\mathbf{\mu }}^{\prime }$ (II, blue), the $u_0$ contribution: $(1/2+n/4)u_0$ (III, purple), and the small part from the integral $-\int f{\rho }_{\mathbf{g}L,1}$ (IV, magenta). The observed upturn in $\mathbf{\mu }$ at high $T$ for $n=0.8$, reflecting the physics of Mott holes near half-filling, is predominantly due to the upturn of the second chemical potential $u_0$. Its growth, in turn, causes the numerical issues requiring the implementation of a cutoff in this work.

Figure 6

The computed quasiparticle weight $Z$ (dots) versus the hole density $\delta =1-n$, compared with the exact numerical results from DMFT ([6] solid curve), which fits very well to the formula $Z\sim {\delta }^{1.39}$. We see that the present scheme accounts well the suppression of $Z$ near $\delta \sim 0$, even reproducing nonlinear vanishing near the Mott limit seen in [6]. This nonlinear feature goes beyond the predictions of both slave-boson mean field and Brinkman-Rice theory [56], and signifies an important correction to the mean field behavior.

Figure 7

Top left: $-\text{Im}\mathrm{\Sigma }\left(\omega \right)$ versus $\omega /Z$ at several densities $n=0.7,0.725,0.75,0.775,0.8,0.825,0.85,0.875,0.9$ from bottom to top. We see that the frequency dependence scales well with $Z$, with better behavior on the occupied side $\omega \le 0$. Top right: The symmetrized function $[-{\mathrm{\Sigma }}^{\prime \prime }\left(\omega \right)-{\mathrm{\Sigma }}^{\prime \prime }(-\omega )]/2$ exhibits the quadratic behavior at $\omega \sim 0$ expected from a Fermi liquid. Bottom left: The antisymmetric part is defined as $R=[{\mathrm{\Sigma }}^{\prime \prime }\left(\omega \right)-{\mathrm{\Sigma }}^{\prime \prime }(-\omega )]/[{\mathrm{\Sigma }}^{\prime \prime }\left(\omega \right)+{\mathrm{\Sigma }}^{\prime \prime }(-\omega )]$, so that if we assume Eq. (35) then $R=-\omega /\mathrm{\Delta }$. We show the computed $R$ multiplied by $\mathrm{\Delta }/Z$ at the above densities versus $\omega /Z$, with $n=0.9$ at the top and $n=0.7$ at the bottom for $\omega \le 0$. These collapse to a straight line with slope $-1$ in the range $\left|\omega \right|\le Z$, provided we allow for an additional mild density dependence of the ratio $\mathrm{\Delta }/Z$, as in Eq. (36). Bottom right: The energy scale ${\mathrm{\Omega }}_0$ [Eq. (35)] determining the magnitude of the $\text{Im}\mathrm{\Sigma }$ at $T=0$ is shown versus $Z^2$, and in the inset versus the hole density ${\delta }^2$. Here, ${\mathrm{\Omega }}_0$ is seen to scale better with $Z^2$ rather than with ${\delta }^2$.

Figure 8

The two figures on the left display the physical local spectral function ${\rho }_{\mathcal{G}L,0}=-\frac{1}{\pi }\text{Im}{\mathcal{G}}_{Loc,0}(\omega +i{0}^{+})$ from Eq. (25), and the two figures on the right show the Dyson self-energy $-\frac{1}{\pi }\text{Im}\mathrm{\Sigma }\left(\omega \right)$, plotted against the frequency $\omega /D$. The figures are at low $T$ for the six indicated values of the density, and display a region that is somewhat greater than the one, where it is expected to be reliable $\left|\omega \right|\lesssim ZD$. One sees a correlation between the quasiparticle weight $Z$ (Fig. 6) and the scale of variation of the decay rate. Densities $n>0.875$ have larger errors in $Z$ compared to the exact DMFT results (see Fig. 6), and therefore not shown. However, it is easy to picture them at low $\omega$, using the observation that scaling $\omega$ with $Z$ collapses ${\mathrm{\Sigma }}^{\prime \prime }$.

Figure 9

The temperature variation with the frequency $\omega /D$, of the spectral function ${\rho }_{\mathcal{G}L,0}$ on the left and the Dyson self-energy $-\frac{1}{\pi }\text{Im}\mathrm{\Sigma }$ on the right, at density $n=0.875$ (top), $n=0.85$ (middle), and at $n=0.6$ (bottom). With increasing $T$ we note the rapid broadening and shifting of ${\rho }_{\mathcal{G}L,0}$. Here, $-\frac{1}{\pi }\text{Im}\mathrm{\Sigma }$ displays a rapid destruction of the coherent Fermi liquid behavior observed at the lowest $T$, by the filling up of the minimum at $\omega =0$. Comparing the top two sets shows that at the lowest hole density, a small change in $T$ has a large effect, due to the low effective Fermi temperature. We also observe here, as well as in Fig. 8, that $-\frac{1}{\pi }\text{Im}\mathrm{\Sigma }$ has a strong asymmetric correction to the quadratic $\omega$ dependence of the standard Fermi liquid, as highlighted in the bottom left panel of Fig. 7. This is in accord with one of the basic analytical predictions of the ECFL theory, and also is found in the DMFT results.

Figure 10

(a), (b) $\frac{{\rho }_{dc}}{{\rho }_0}$ vs $\frac{T}{D}$ for $n=0.75,0.8,0.85$ from bottom to top. In (a), the blue dashed parabola tracks the FL regime $0<T<T_{\text{FL}}$ where $\frac{{\rho }_{dc}}{{\rho }_0}\propto {\left(\frac{T}{D}\right)}^2$. The magenta dashed line tracks the first linear regime $T_{\text{FL}}<T\lesssim 0.01D$. In (b), the blue dashed line tracks the second linear regime $T\gtrsim 0.07D$. (c) $\frac{{\rho }_{dc}}{{\rho }_0}$ vs $\frac{T}{DZ(T=0)}$ for $n=0.75,0.8,0.85$ (red, orange, green). The blue dashed parabola tracks the Fermi liquid regime, demonstrating that $T_{\text{FL}}=\left(cD\right)\times Z(T=0)$, with $c\approx 0.05$, and that $\frac{{\rho }_{dc}}{{\rho }_0}$ is a function of $\frac{T}{DZ(T=0)}$ for $T\lesssim 2T_{\text{FL}}$.

Figure 11

(a) $[-\text{Im}\mathrm{\Sigma }(0,T\left)\right]$ vs $\frac{T}{D}$ for $n=0.75,0.8,0.85$ from bottom to top. $[-\text{Im}\mathrm{\Sigma }(0,T\left)\right]$ is quadratic for $T\lesssim T_{\text{FL}}$ (tracked by the blue dashed parabola) and linear for $T_{\text{FL}}\lesssim T\lesssim 0.01D$ (tracked by the magenta dashed line). (b) $Z\left(T\right)$ vs $\frac{T}{D}$ for $n=0.75,0.8,0.85$ from top to bottom. $Z\left(T\right)$ is approximately constant for $T\lesssim T_{\text{FL}}$, and grows linearly for $T_{\text{FL}}\lesssim T\lesssim 0.01D$, with a slope on the order of the bandwidth (tracked by the magenta dashed line). The blue dashed curve is the fit to the functional form $Z\left(T\right)=\sqrt{\frac{1+aT+b{T}^2}{c+dT}}$ using a broader range of temperatures than the one shown here [Fig. 12]. This form works well for $T\gtrsim T_{\text{FL}}$. (c) $A(0,T)=\mathbf{\mu }\left(T\right)-\text{Re}\mathrm{\Sigma }(0,T)$ vs $\frac{T}{D}$ for $n=0.75,0.8,0.85$ from bottom to top. For $T_{\text{FL}}\lesssim T\lesssim 0.01D$, it is linear, as tracked by the dashed blue line.

Figure 12

Same plots as in Fig. 11 over a broader range of temperatures. (a) $[-\text{Im}\mathrm{\Sigma }(0,T\left)\right]$ continues to grow as $T$ is increased beyond $0.01D$, until it finally begins to saturate at higher temperatures. (b) The blue dashed curve is the fit to the functional form $Z\left(T\right)=\sqrt{\frac{1+aT+b{T}^2}{c+dT}}$, which works well for $T\gtrsim T_{\text{FL}}$. For $T\gtrsim 0.01D,Z\left(T\right)\propto \sqrt{T}$. (c) The blue dashed line tracks the second linear regime in $A\left(T\right)$ (with a slope slightly smaller than the first) for $T\gtrsim 0.01D$.