Low quasiparticle coherence temperature in the one-band Hubbard model: A slave-boson approach

We use the Kotliar-Ruckenstein slave-boson formalism to study the temperature dependence of paramagnetic phases of the one-band Hubbard model for a variety of band structures. We calculate the Fermi liquid quasiparticle spectral weight $Z$ and identify the temperature at which it decreases significantly to a crossover to a bad metal region. Near the Mott metal-insulator transition, this coherence temperature $T_{\mathrm{coh}}$ is much lower than the Fermi temperature of the uncorrelated Fermi gas, as is observed in a broad range of strongly correlated electron materials. After a proper rescaling of temperature and interaction, we find a universal behavior that is independent of the band structure of the system. We obtain the temperature-interaction phase diagram as function of doping, and we compare the temperature dependence of the double occupancy, entropy, and charge compressibility with previous results obtained with dynamical mean-field theory. We analyze the stability of the method by calculating the charge compressibility.

Figure 1

Temperature vs interaction phase diagram for the Hubbard model at half-filling for different band structures. We observe universal behavior, in that the results are independent of the details of the band structure when $T$ and $U$ are scaled by $U_c$ [defined in Eq. (18)]. $T_{\mathrm{coh}}$ (color points) is the coherence temperature, where the quasiparticle weight $Z=q$ goes to zero, calculated with the slave-boson method (cf. Fig. 3). At $T=0$ and for $U>U_c$, the system is a Mott insulator. For $U<U_c$ and $T<T_{\mathrm{coh}}$, the system is in a Fermi liquid phase. For $T>T_{\mathrm{coh}}$, the system is in the bad metal regime. Close to the metal-insulator transition, the coherence temperature is orders of magnitude smaller than the energy scale $U_c$, which is of the order of the energy of the uncorrelated system. There is very good agreement with the analytic approximation (dashed black line) found in Ref. [30] [cf. Eq. (20)]. Note also that $T_{\mathrm{coh}}\ll T_{\mathrm{F}}^{*}\equiv Z(T=0){\varepsilon }_{\mathrm{F}}^0=Z(T=0)\frac{W}{2}$ (dashed blue line), where ${\varepsilon }_{\mathrm{F}}^0$ is the uncorrelated Fermi energy and $W$ is the bandwidth. This shows that the large reduction in $T_{\mathrm{coh}}$ is not just due to the renormalization of the bandwidth. Our results are qualitatively consistent with DMFT calculations [26, 31].

Figure 2

Coherence temperature $T_{\mathrm{coh}}$ as a function of interaction strength $U$ for different fillings, using the semicircular DOS (see the text). Away from half-filling, the system is metallic, and there is no transition to a Mott insulator at $T=0$. Due to the particle-hole symmetry of the model, the $T_{\mathrm{coh}}$ curves for $n=0.9$, 0.8, and 0.7 are equivalent to those for $n=1.1$, 1.2, and 1.3, respectively. Away from $n=1$ we calculate the temperature $T_{\mathrm{coh}}$ as the minimum of $\frac{\partial Z}{\partial T}$ (cf. Fig. 4). Here, $T_{\mathrm{coh}}$ defines a crossover between Fermi liquid and bad metal regimes. $T$ and $U$ are rescaled by the correlation scale $U_n^{*}$ [see Eq. (13)]. Inset: $T=0$ phase diagram of the Hubbard model in the $U\text{−}n$ plane for the semicircular DOS case, using its bandwidth $W$ as the energy scale. At half-filling ($n=1$) and for $U>U_c$, the system is a Mott insulator, otherwise it is a Fermi liquid. The dashed line shows the behavior with $n$ of the zero-temperature correlation scale $U_n^{*}$ measured in units of $W$.

Figure 3

Fermi liquid quasiparticle weight $Z$, at half-filling, as a function of temperature, for several values of $U/U_c$ and band structures. Note the weak dependence on the shape of the DOS. An increment of the temperature $T$ or the interaction $U$ diminishes the quasiparticle weight $Z$. We identify the emergence of the bad metal with the collapse of $Z$ at the coherence temperature $T_{\mathrm{coh}}$. The dependence of $T_{\mathrm{coh}}$ on $U$ is shown in Fig. 1.

Figure 4

Same as Fig. 3 for $n=0.8$, with $U_0^{*}.8$ as the energy unit. Values of $U/U_n^{*}$ larger than 1 can be achieved because the system always remains metallic and does not go through a MIT when $U$ increases. Even though the double occupancy goes to zero for larger temperatures (cf. Fig. 10 in the Appendix), there are still holes that allow electronic movement, and the quasiparticle weight $Z$ remains finite, with a lower bound of $Z=\frac{1-n}{1-\frac{n}{2}}$. The inflection points in the curves for the semicircular DOS are used to calculate the coherence temperature $T_{\mathrm{coh}}$ in Fig. 2.

Figure 5

Entropy density at half-filling as a function of temperature for different interaction values $U/U_c$, using the semicircular DOS. The $ln\left(4\right)$ dotted line corresponds to the $d^2=0$ limit and to a regime with local charge and spin fluctuations, and the $ln\left(2\right)$ value corresponds to localized noninteracting spins. The increase of entropy with $U$, accompanied by the decrease of $d^2$ with $T$, is an analog of the Pomeranchuk effect in liquid ${}^3\mathrm{He}$ and is consistent with DMFT results in the same model [55]. For a comparison with DMFT results in the hypercubic lattice, see Fig. 11 in the Appendix.

Figure 6

Charge compressibility ${\chi }_c$ as a function of temperature for the semicircular DOS, using slave-boson (solid line) and DMFT (dashed line) calculations from Ref. [58]. The bandwidth $W$ is used as an energy scale in order to make the comparison easier. The decrease of charge compressibility with increasing interaction $U$ and temperature is consistent with DMFT results [58] and with finite-temperature Lanczos methods for the triangular lattice [59]. Also, we qualitatively reproduce the apparent kinklike feature found with DMFT at about $T\simeq 0.025W$ for $U=1.25W$ [58].

Figure 7

Double occupancy at $U=1.5U_n^{*}$ as a function of temperature, for several fillings. Inset: Dependence of $T=0$ double occupancy with the filling $n$. As we approach the Mott phase at $n=1$, effective interactions become stronger and $d^2$ decreases toward zero.

Figure 8

Coherence temperature $T_{\mathrm{coh}}$ (black circles) as a function of hole doping $\delta =1-n$ for $U=1.5U_n^{*}$ using the semicircular DOS. Near half-filling, $T_{\mathrm{coh}}$ is proportional to the doping level $T_{\mathrm{coh}}\simeq 0.18\delta U_n^{*}\simeq 0.3\delta W$. This behavior is similar to that obtained with DMFT [24]. The red curve shows the boundary of the region at which the system becomes unstable (shaded area) due to a negative charge compressibility ${\chi }_c$. This region with ${\chi }_c<0$ does not change significantly as we vary the value of $U$ or the band structure.

Figure 9

Double occupancy $d^2$ as a function of $T$ for the semicircular DOS, using slave-boson (solid line) and DMFT (dashed line) calculations using TRIQS. The decrease of $d^2$ with increasing $T$ indicates a higher degree of localization. The slave-boson method ceases to be valid for $T>T_{\mathrm{coh}}$, where the self-consistent equations (15, 16, 17) collapse to the trivial solution $d^2=0$. The obtained $T_{\mathrm{coh}}$ compares favorably with the minima in the DMFT results.

Figure 10

Same as Fig. 9 for $n=0.8$. Even though $d^2$ goes to zero at high $T$, the quasiparticle weight goes to a finite value because the system always remains metallic [cf. Eq. (11)].

Figure 11

Entropy density at half-filling as a function of temperature for different interaction values using the hypercubic lattice band structure for the slave-boson method (up) and DMFT from Ref. [54] (bottom).

Figure 12

Dependence of the chemical potential $\mu$ with the filling number $n$ for $T/U_n^{*}=0.005$ and 0.015, calculated using different band structures. Note the weak dependence of $\mu$ over the various band structures. Regions with negative derivative define the shaded area in Fig. 8.