Strange metal from Gutzwiller correlations in infinite dimensions

Recent progress in extremely correlated Fermi liquid theory (ECFL) and the dynamical mean field theory (DMFT) enables us to accurately compute in the $d\to \infty$ limit the resistivity of the $t-J$ model after setting $J\to 0$. This is also the $U=\infty$ Hubbard model. Since $J$ is set to zero, our study isolates the dynamical effects of the single occupation constraint enforced by the projection operator originally introduced by Gutzwiller. We study three densities $n=.75,.8,.85$ that correspond to a range between the overdoped and optimally doped Mott insulating state. We delineate four distinct regimes separated by three crossovers, which are characterized by different behaviors of the resistivity $\rho$. We find at the lowest temperature $T$ a Gutzwiller correlated Fermi liquid regime with $\rho \propto T^2$ extending up to an effective Fermi temperature that is dramatically suppressed from the noninteracting value by the proximity to half filling, $n\sim 1$. This is followed by a Gutzwiller correlated strange metal regime with $\rho \propto (T-T_0)$, i.e., a linear resistivity extrapolating back to $\rho =0$ at a positive $T_0$. At a higher temperature scale this crosses over into the bad metal regime with $\rho \propto (T+T_1)$, i.e., a linear resistivity extrapolating back to a finite resistivity at $T=0$ and passing through the Ioffe-Regel-Mott value where the mean free path is a few lattice constants. This regime finally gives way to the high $T$ metal regime, where we find $\rho \propto T$, i.e., a linear resistivity extrapolating back to zero at $T=0$. The present work emphasizes the first two, i.e., the two lowest temperature regimes, where the availability of an analytical ECFL theory is of help in identifying the changes in related variables entering the resistivity formula that accompanies the onset of linear resistivity, and the numerically exact DMFT helps to validate the results. We also examine thermodynamical variables such as the magnetic susceptibility, compressibility, heat capacity, and entropy and correlate changes in these with the change in resistivity. This exercise casts valuable light on the nature of charge and spin correlations in the Gutzwiller correlated strange metal regime, which has features in common with the physically relevant strange metal phase seen in strongly correlated matter.

Figure 1

A schematic view of the different regimes of temperature dependent resistivities found in the calculations of Refs. [5, 6, 7, 8]. The various temperature scales are schematic. At the lowest $T$ we have a Gutzwiller-correlated-Fermi liquid regime (GCFL) with $\rho \propto T^2$. This quadratic variation terminates at a characteristic Fermi temperature $T_{\mathrm{FL}}\left(\delta \right)$, which is found to be surprisingly small relative to $T_{\mathrm{BR}}=\delta D$, the Brinkman-Rice temperature scale ($2D$ is the bandwidth). Upon warming we reach the Gutzwiller-correlated-strange metal (GCSM) regime, which is the main focus of this work. This gives way at higher $T$ to the so-called bad-metal regime with a resistivity that increases linearly beyond the Ioffe-Regel-Mott value ${\rho }_0$ characteristic of disordered metals. The temperature scale of this regime is $T_{\mathrm{BR}}$ discussed above. Finally at the highest $T$ we reach the high $T$ regime with $\rho \propto T$ that can be extrapolated back to pass through the origin. We thus find a total of four regimes separated by three crossovers. It should be noted that in both theories considered here, the approximate range of the temperatures scales are $T_{\mathrm{FL}}\sim 0.004\text{–}0.01D$, and the crossover to the bad-metal regime occurs at $T\sim 0.04\text{–}0.06D$ for the densities considered ($n=0.75$ to $n=0.85$).

Figure 2

Comparison of the resistivity computed using the ECFL (symbols) and the DMFT (dashed). ${\sigma }_0=1/{\rho }_0$ is the Ioffe-Regel-Mott conductivity. As $n$ gets closer to unity, the ECFL scheme employed systematically underestimates $Z$ relative to the exact DMFT values (see Fig. 6 of Ref. [7]). This lowers the effective Fermi temperature $T_{\mathrm{FL}}$ and simultaneously enhances the magnitude of $\rho$ for $T>T_{\mathrm{FL}}$, a feature that is prominently visible above. It should be possible to improve the quantitative agreement between the two theories in the future [28].

Figure 3

ECFL calculation of the resistivity and related objects. Panel (a): The resistivity as a function of the temperature using the exact formula, Eq. (1), compared with the approximation, Eq. (7), for $n=0.75,0.8,0.85$ (bottom to top). Equation (7) is an excellent approximation at all densities for all temperatures. Panel (b): Parameters resulting from a low-frequency expansion of the imaginary part of the self-energy in the vicinity of the Fermi surface, plotted as a function of temperature, for $n=0.75,0.8,0.85$ (bottom to top). $B_0$ is the self-energy on the Fermi surface, while $B_2$ is the quadratic-frequency term. The ratio $\frac{B_2{\pi }^2{T}^2}{B_0}\to 1$ as $T\to 0$ and is approximately constant as a function of temperature. Panel (c): $\varphi \left[A\right(0\left)\right]=\varphi [\mu -\Re e\mathrm{\Sigma }(0\left)\right]$, plotted as a function of the temperature, for $n=0.75,0.8,0.85$ (bottom to top). $\varphi \left[A\right(0\left)\right]$ is practically independent of temperature and has very weak density dependence.

Figure 4

The exact resistivity [Eq. (1)] compared with the approximation Eq. (7), using the DMFT calculation for $n=0.75,0.8,0.85$ (bottom to top). Equation (7) is an excellent approximation at all densities for all temperatures. [See Fig. 3 for the corresponding figure in ECFL.]

Figure 5

Chemical potentials at $n=0.75,0.8,0.85$ for ECFL (symbols) and DMFT (dashed lines). The DMFT results are shifted by a density-dependent constant. After the shift, the chemical potentials almost coincide.

Figure 6

(a) Compressibility $\kappa =n^{-2}\partial n/\partial \mu$ of ECFL (symbols), DMFT (dashed lines), and free fermions (dotted lines). The DMFT results give a systematically higher value of compressibility than the ECFL theory. (b) $Z/\kappa$ for the lowest temperature $T=0.001D$ at the three densities considered for ECFL (blue) and DMFT (red). The ECFL result for the compressibility is proportional to the quasiparticle weight $Z$, unlike the DMFT result which displays some variation. The difference in compressibility between the two theories seems related to the density dependent shift in chemical potentials noted in Fig. 5.

Figure 7

The momentum distribution curves at three densities $n=.75,.8,.85$ (top to bottom at $\epsilon =-1$) at $T=.004$ D [panel (a)] and $T=.02$ D [panel (b)]. The ECFL curves are solid symbols and the DMFT curves are dashed lines.

Figure 8

The charge susceptibilities ${\chi }_c=dn/d\mu$, which are related to compressibility $\kappa$ as ${\chi }_c=n^2\kappa$. The numerically exact values versus bubble estimates [Eq. (12)] in panel (a) DMFT (full and dashed lines) and in panel (b) from ECFL (empty diamonds and solid circles).

Figure 9

Single particle decay rates, i.e., the spectral functions of self-energy $[{\rho }_{\mathrm{\Sigma }}\left(\omega \right)=-{\pi }^{-1}\Im m\mathrm{\Sigma }\left(\omega \right)]$ of ECFL (symbols) and DMFT (dashed lines) for a range of temperatures.

Figure 10

Local density of states ${\rho }_G^{\mathrm{local}}\left(\epsilon \right)$ of ECFL (symbols) and DMFT (dashed lines) at $T=0.002D$.

Figure 11

(a) Specific heat computed from the kinetic energy by differentiation as $C_V=\partial E_K/\partial T$ for ECFL (symbols), DMFT (dashed lines), and free fermions (dotted lines). For $n=0.8$ and $n=0.85$ the heat capacity shows a gentle maximum at a characteristic $T$. (b) The ratio $C_V/T$ versus $T$ of ECFL (symbols) and DMFT (dashed lines). Taking the ratio with $T$ wipes out the maximum seen in (a). (c) $\gamma \times Z$ at $T=0.001D$.

Figure 12

The entropy versus $T$ computed as ${\int }_0^{T}d{T}^{\prime }{C}_V\left(T^{\prime }\right)/T^{\prime }$ for ECFL (symbols), DMFT (dashed lines), and free fermions (dotted lines).

Figure 13

Resistivity (blue circle), specific heat (light blue square), and entropy (red triangle) as percentage of the ideal entropy at infinite temperature $S_{\mathrm{ideal}}$. The (Schottky) peak in the heat capacity is close to $T_{\mathrm{FL}}$, the onset point of the linear-$T$ resistivity, or the end of the crossover region.

Figure 14

Magnetic susceptibility (DMFT results). We note that the Stoner enhancement grows as $\delta \to 0$ and its $T$ dependence is Pauli like but with a somewhat enhanced $T$ dependence at higher $n$. The crossover to linear resistivity occurs (see Fig. 2) at fairly low $T\lesssim .005D$ at these densities but has no reflection on the variation of ${\chi }_{\mathrm{spin}}$. We may thus infer that spin disordering is not relevant to the linear resistivity seen here.

Figure 15

(a) Thermopower of ECFL (symbols) and DMFT (dashed lines). Both amplitudes and temperature derivatives are similar for $T\le .005$ but depart at higher T. (b) Electronic thermal resistivity ${\kappa }_e^{-1}$ of ECFL (symbols) and DMFT (dashed lines).

Figure 16

(a) Lorenz number of ECFL (symbols) and DMFT (dashed lines). The Lorenz number saturates to a constant ($\simeq 2.1$) which is typically expected for a Fermi liquid at low temperatures. (b) Figure of merit for ECFL (symbols) and DMFT (dashed lines). The low values of $ZT$ found here are typical of normal metals.