Benchmark of a modified iterated perturbation theory approach on the fcc lattice at strong coupling

The dynamical mean-field theory approach to the Hubbard model requires a method to solve the problem of a quantum impurity in a bath of noninteracting electrons. Iterated perturbation theory (IPT) has proven its effectiveness as a solver in many cases of interest. Based on general principles and on comparisons with an essentially exact continuous-time quantum Monte Carlo (CTQMC) solver, here we show that the standard implementation of IPT fails away from half-filling when the interaction strength is much larger than the bandwidth. We propose a slight modification to the IPT algorithm that replaces one of the equations by the requirement that double occupancy calculated with IPT gives the correct value. We call this method IPT-$D$. We recover the Fermi liquid ground state away from half-filling. The Fermi liquid parameters, density of states, chemical potential, energy, and specific heat on the fcc lattice are calculated with both IPT-$D$ and CTQMC as benchmark examples. We also calculated the resistivity and the optical conductivity within IPT-$D$. Particle-hole asymmetry persists even at coupling twice the bandwidth. A generalization to the multiorbital case is suggested. Several algorithms that speed up the calculations are described in appendixes.

Figure 1

Noninteracting density of states for the fcc lattice with nearest-neighbor hopping only. The large particle-hole asymmetry caused by frustration is apparent.

Figure 2

Results obtained with CTQMC as impurity solver are plotted as a function of density and shown in blue with circles and line for $U=8t$ and in black with dots and line for $U=32t$. In all numerical results, energy units are such that $t=1$. Boltzmann's constant and the lattice spacing are also taken as unity. We obtain the zero-frequency limit from a poor man's approach: we take $\beta t=25,50$, and 75 and use the value of the function at the lowest Matsubara frequency in the three cases to perform the extrapolation. (a) Check for Luttinger's theorem: The effective chemical potential $\tilde{\mu }=\mu -{\Sigma }^{\prime }\left(0\right)$ is equal to the noninteracting chemical potential shown in red except at half-filling where there is a Mott gap for $U=32t$. (b) At $U=32t$ the imaginary part of the self-energy at zero frequency ${\Sigma }^{\prime \prime }\left(0\right)$ should be zero away from half-filling and infinite at half-filling. (c) The single-particle spectral weight $Z$ vanishes only at $n=1$, $U=32t$ where there is a Mott gap.

Figure 3

Fermi liquid parameters as a function of density. Zero-frequency results are obtained with the same extrapolation method as in Fig. 2. (a) Check of Luttinger's theorem. The effective chemical potential $\tilde{\mu }=\mu -{\Sigma }^{\prime }\left(0\right)$ should equal the noninteracting value, shown in red, when the theorem is satisfied. For $U=8t$, the brown asterisks ($*$) obtained with IPT $n=n_0$ satisfy the theorem. For $U=32t$, results for three different methods are shown: in khaki ($\square$) for IPT $n=n_0$, in cyan ($☆$) for IPT $D_{\mathrm{naive}}$, and in magenta ($◊$) for IPT ${\langle D\rangle }_{\mathrm{CTQMC}}$. (b) ${\Sigma }^{\prime \prime }\left(0\right)$ is plotted for $U=32t$ in magenta ($◊$) for IPT ${\langle D\rangle }_{\mathrm{CTQMC}}$ as above, and compared with the CTQMC results shown previously in Fig. 2 (black dots joined by a line). (c) Quasiparticle spectral weight $Z$ computed for different methods and displayed with the same symbols as in (a). We compare with the CTQMC results of Fig. 2, namely, blue symbols ($◯$) with line for $U=8t$ and black symbols ($·$) with line for $U=32t$. The results for IPT $n=n_0$ at $U=32t$ are unphysical since they predict an insulator away from half-filling.

Figure 4

CTQMC results at $U=32t$ for double occupancy $D-D_{\mathrm{naive}}$ plotted as a function of temperature. We define $D_{\mathrm{naive}}=0$ for fillings $n<1$ and $D_{\mathrm{naive}}=n-1$ for $n>1$. The various densities are represented by different symbols: $n=0.2$ [black ($◯$)], $0.4$ [blue ($\times$)], $0.6$ [red ($\square$)], $0.8$ [green ($◊$)], $1.0$ [yellow ($+$)], $1.2$ [cyan ($▿$)], $1.4$ [magenta (▵)], $1.6$ [brown ($⊲$)], and $1.8$ [khaki ($☆$)]. The largest deviations from the naive value, occurring close to $n=1$, are less than ${10}^{-2}$ in absolute value.

Figure 5

Double occupancy $D$ as a function of density obtained from CTQMC for $U=32t$ for three temperatures: $\beta =25/t$ [black ($\square$)], $\beta =10/t$ [blue ($◯$)], and $\beta =0.5/t$ [red ($·$)]. On this scale, the naive value of $D$ is very accurate. The inset is a zoom for densities $n\le 1$.

Figure 6

Density of states for $n=0.84$, $\beta t=25$, and $U=32t$ obtained with three methods: black (solid line) with CTQMC maxent, blue (dashed line) with IPT-${\langle D\rangle }_{\mathrm{CTQMC}}$, and red (dashed-dotted line) with IPT-$D_{\mathrm{naive}}$. (b) is a zoom of (a) around $\omega =0$. The value at zero frequency is improved when a more accurate value of $D$ is used in IPT.

Figure 7

Chemical potential as a function of temperature for different IPT approximations, compared with the reference CTQMC calculations as black dots with line obtained for $U=32t$ and different densities: (a) $n=0.80$, (b) $n=1.2$, and (c) $n=0.84$. The three different IPT approximations are given in khaki ($\square$) for IPT-$n_0$, in cyan ($☆$) for IPT-$D_{\mathrm{naive}}$, and in magenta ($◊$) for IPT-${\langle D\rangle }_{\mathrm{CTQMC}}$. The latter approximation in magenta ($◊$) is best, having a more or less doping and temperature independent offset $\delta \mu /t\sim 0.5$ when compared with the reference CTQMC in black.

Figure 8

Energy as a function of temperature obtained from IPT-$D$ (solid lines) and CTQMC (dashed lines) for $U=32t$. (a) Densities equal to, or below, half-filling $n=0.2$ [black ($◯$)], $0.4$ [blue ($\times$)], $0.6$ [red ($\square$)], $0.8$ [green ($◊$)], $0.84$ [cyan ($▿$)], $0.88$ [magenta (▵)], $0.92$ [brown ($⊲$)], and $1.0$ [khaki ($☆$)]. For densities above half-filling, displayed in (b), $n=1.08$ [black ($◯$)], $1.2$ [blue ($\times$)], $1.4$ [red ($\square$)], $1.6$ [green ($◊$)], $1.8$ [cyan ($▿$)], the quantity $U{D}_{\mathrm{naive}}$ is subtracted from the energy to allow the results to fit on the same scale. (c) is a zoom for $n=1.08$ and (d) a zoom for $n=1.4$.

Figure 9

Specific heat at constant filling as a function of temperature for $U=32t$. (a) Results from IPT-$D$ (solid line) for densities below half-filling: $n=0.2$ [black ($◯$)], $0.4$ [blue ($\times$)], $0.6$ [red ($\square$)], $0.8$ [green ($◊$)], $0.84$ [cyan ($▿$)], $0.88$ [magenta (▵)], $0.92$ [brown ($⊲$)], and $1.0$ [khaki ($☆$)]. (b) Specific heat from IPT-$D$ (solid line) for densities above half-filling $n=1.08$ [black ($◯$)], $1.2$ [blue ($\times$)], $1.4$ [red ($\square$)], $1.6$ [green ($◊$)], $1.8$ [cyan ($▿$)]. (c) Comparison between IPT-$D$ as solid lines and CTQMC as dashed lines for $n=0.6$ [red ($\square$)], $0.8$ [green ($◊$)], and $0.84$ [cyan ($▿$)]. The peak positions and shape coincide, even though the absolute values differ. In this case, since CTQMC values come from differentiation of Monte Carlo data, there is a rather large uncertainty, especially for peaks.

Figure 10

Transport function $X\left(\varepsilon \right)={\sum }_k{\left(\frac{\partial {\varepsilon }_k}{\partial k_x}\right)}^2\delta (\varepsilon -{\varepsilon }_k)$, as calculated in Appendix C.

Figure 11

Resistivity as a function of temperature for $U=32t$ as calculated by IPT-$D$ for different values of density: $n=0.2$ [black ($◯$)], $0.4$ [blue ($\times$)], $0.6$ [red ($\square$)], $0.8$ [green ($◊$)], $1.2$ [cyan ($▿$)], $1.4$ [magenta (▵)], $1.6$ [brown ($⊲$)], and $1.8$ [khaki ($☆$)]. The resistivities are largest close to half-filling where they exhibit low coherence temperatures.

Figure 12

Optical conductivity for $U=32t$ and $n=0.80$, as calculated from IPT-$D$ for three different temperatures using two analytical continuation approaches for each temperature. The maximum entropy results are represented with solid lines and the Padé analytical continuations with dashed lines. The broadest zero-frequency peak [red ($\times$)] is for the largest temperature $\beta =1/t$, and the narrowest one [black ($◯$)] for the lowest temperature $\beta =25/t$. The intermediate case $\beta =2.3/t$ is in blue ($\square$). The features in the optical conductivity can be identified with transitions between the Fermi level and peaks in the single-particle density of states.

Figure 13

The special points for a 3D Gaussian quadrature of fifth order over a cube of length $2h$.

Figure 14

Flow chart for the impurity solver loop in IPT. There is also an outer loop for $\Delta$ (see text).

Figure 15

Illustration of the Monte Carlo integration scheme used to obtain $N_0\left(\varepsilon \right)$ and $X\left(\varepsilon \right)$.