Calculation of dynamical and many-body observables for spin-fermion models using the polynomial expansion method

The calculation of two- and four-particle observables is addressed within the framework of the truncated polynomial expansion method (TPEM). The TPEM replaces the exact diagonalization of the one-electron sector in models for fermions coupled to classical fields such as those used in manganites and diluted magnetic semiconductors. The computational cost of the algorithm is $O\left(N\right)$—with $N$ the number of lattice sites—for the TPEM, which should be contrasted with the computational cost of the diagonalization technique that scales as $O\left(N^4\right)$. By means of the TPEM, the density of states, spectral function, and optical conductivity of a double-exchange model for manganites are calculated on large lattices and compared to previous results and experimental measurements. The ferromagnetic metal becomes an insulator by increasing the direct exchange coupling that competes with the double-exchange mechanism. This metal-insulator transition is investigated in three dimensions.

Figure 1

(Color online) (a) Kinetic energy of a single configuration of classical spins on a $8\times 8$ lattice for model Eq. (15) calculated with the DIAG and TPEM algorithms. The configuration was obtained after evolving the model with Monte Carlo with 600 steps for $J_H=4t$ and $\mu =-4t$, $T=0.02t$ and $J_{\mathit{AF}}=0$. ${ϵ}_{\mathit{pr}}={10}^{-5}$ and ${ϵ}_{\mathit{tr}}={10}^{-6}$. (b) Density of states for the model for the same configuration and parameters as in (a). Algorithms and different expansion cutoffs are indicated.

Figure 2

(Color online) DOS at three temperatures on a $20\times 20$ lattice calculated with the TPEM and the Monte Carlo algorithm, $M=200$, for model Eq. (15) at $J_H=8t$, $⟨n⟩=0.5$ for (a) $J_{\mathit{AF}}=0$ and (b) $J_{\mathit{AF}}=0.02$. The location of the chemical potential is indicated by the vertical dashed line.

Figure 3

(Color online) Spectral function for the double-exchange model calculated on a $20\times 20$ lattice with the TPEM at $J_H=8t$, $n=0.5$, $T=0.02t$ (FM system). The curves are parametrized in terms of $(k_x,k_y)$ from (0,0) to $(\pi ,0)$ to $(\pi ,\pi )$. The location of the Fermi energy is indicated by the vertical dashed line. There is a finite DOS at the Fermi energy. The dashed line is the spectrum calculated analytically for a perfect ferromagnet.

Figure 4

(Color online) For $J_{\mathit{AF}}=0.15$ and the same parameters as in Fig. 3, corresponding to an insulator, $A(k,\omega )$ shows a clear gap and a more complex form. The vertical dashed line indicates the position of the Fermi energy.

Figure 5

Optical conductivity on a $6^3$ lattice calculated with the TPEM and DIAG for the same configurations of classical fields at $J_H=8t$ and quarter filling. (a) $T=0.02t$ calculated with the DIAG, (b) $T=0.02t$ calculated with the TPEM, (c) $T=0.2t$ calculated with DIAG, and (d) $T=0.2t$ calculated with the TPEM. In (b) and (d), $M=200$.

Figure 6

(Color online) Maximum distance spin-spin correlation, $C\left(d_{\mathit{max}}\right)$, vs temperature for the ferromagnetic regime $(J_{\mathit{AF}}=0)$ on $6^3$ and $8^3$ lattices. $d_{\mathit{max}}=\sqrt{3}L∕2$, where $L=6$ or $L=8$.

Figure 7

(Color online) $\sigma \left(0\right)$ vs temperature at $J_H=8t$ and quarter filling on $6^3$ and $8^3$ lattices for (a) $J_{\mathit{AF}}=0$ and (b) $J_{\mathit{AF}}=0.1t$ showing metallic and insulating behavior with temperature respectively. The results were obtained with the TPEM and $M=200$.