One- and two-band models for colossal magnetoresistive manganites studied using the truncated polynomial expansion method

Considerable progress has been recently made in the theoretical understanding of the colossal magnetoresistance (CMR) effect in manganites. The existence of inhomogeneous states has been shown to be directly related with this phenomenon, both in theoretical studies and experimental investigations. The analysis of simple models with two competing states and a resistor network approximation to calculate conductances has confirmed that CMR effects can be theoretically reproduced using nonuniform clustered states. However, a direct computational study of the CMR effect in realistic models has been difficult since large clusters are needed for this purpose. In this paper, the recently proposed truncated polynomial expansion method (TPEM) for spin-fermion systems is tested using the double-exchange one-orbital, with finite Hund coupling $J_H$, and two-orbital, with infinite $J_H$, models. Two dimensional lattices as large as $48\times 48$ are studied, far larger than those that can be handled with standard exact diagonalization (DIAG) techniques for the fermionic sector. The clean limit (i.e., without quenched disorder) is analyzed here in detail. Phase diagrams are obtained, showing first-order transitions separating ferromagnetic metallic from insulating states. A huge magnetoresistance is found at low temperatures by including small magnetic fields, in excellent agreement with experiments. However, at temperatures above the Curie transition the effect is much smaller confirming that the standard finite-temperature CMR phenomenon cannot be understood using homogeneous states. By comparing results between the two methods, TPEM and DIAG, on small lattices, and by analyzing the systematic behavior with increasing cluster sizes, it is concluded that the TPEM is accurate enough to handle realistic manganite models on large systems. Our results contribute to the next challenge in theoretical studies of manganites, namely a frontal computational attack of the colossal magnetoresistance phenomenon using double-exchange-like models, on large clusters, and including quenched disorder.

Figure 1

(Color online) Dependence of the TPEM algorithm results on the number of terms $M$ in the expansion. Shown are the spin structure factors at the momenta characteristic of (a) the FM state and (b) the flux state, normalized to 1 and 0.5, respectively, for the perfect states. Results are obtained on a $12\times 12$ lattice and they are compared with the numbers gathered using the exact diagonalization algorithm, all calculated at density $⟨n⟩=0.5$. (a), (c) correspond to $J_{\mathrm{AF}}=0.0$ and (b), (d) to $J_{\mathrm{AF}}=0.1$. Measurements were taken every $10$ steps of a MC run of $2000$ total iterations, after discarding $2000$ steps for thermalization. A random starting configuration is used for each $T$. In (a), at the most difficult temperature, $T=0.06$, where critical fluctuations are strong, the result shown was confirmed using several different starting configurations, including ordered ones. The average $S(0,0)$ obtained by this procedure was very similar among the several starts. The shown error bars at this temperature and $M=30$ mainly arise from the expected critical fluctuations. More details are provided in Sec. 3B3. In (b), a good convergence at $T=0.04$ is only achieved by using 40 moments with ${\epsilon }_{\mathrm{pr}}={10}^{-7}$ and ${\epsilon }_{\mathrm{tr}}={10}^{-5}$ and the result is shown with an orange star just below the exact result. In (c), the TPEM parameters used are $M=30$, ${\epsilon }_{\mathrm{pr}}={10}^{-5}$, and ${\epsilon }_{\mathrm{tr}}={10}^{-3}$, except at $T$=0.06, where the convergence is achieved by using 40 moments with ${\epsilon }_{\mathrm{pr}}={10}^{-7}$ and ${\epsilon }_{\mathrm{tr}}={10}^{-5}$. In (d), all the results shown were obtained with $M=30$, ${\epsilon }_{\mathrm{pr}}={10}^{-5}$, and ${\epsilon }_{\mathrm{tr}}={10}^{-3}$, and solid lines represent the DIAG results while the dashed lines are the TPEM results.

Figure 2

(Color online) $\epsilon$ dependence of the TPEM algorithm results $(M=30)$ for a $12\times 12$ lattice, compared with the exact (DIAG) results. Both are obtained at density $⟨n⟩=0.5$, using (a) $J_{\mathrm{AF}}=0.0$ and (b) $J_{\mathrm{AF}}=0.1$. Measurements were taken at every $10$ steps of a $2000$ MC steps total run, after $2000$ steps for thermalization. In (b), the convergence at $T=0.04$ is achieved by using 40 moments with ${\epsilon }_{\mathrm{pr}}={10}^{-7}$ and ${\epsilon }_{\mathrm{tr}}={10}^{-5}$, as shown with an orange star just below the exact result. The starting configurations and error bar convention is as in Fig. 1.

Figure 3

(Color online) Lattice size dependence of the TPEM algorithm results for the lattices and parameters shown, working at density $⟨n⟩=0.5$ and using (a) $J_{\mathrm{AF}}=0.0$ and (b) $J_{\mathrm{AF}}=0.1$. Measurements were taken at every $10$ steps of a $2000$ MC total steps run, after discarding $2000$ MC steps for thermalization. For the $20\times 20$ lattice and larger, the starting configuration used was a perfect FM state. In (b), the starting configuration used is a random one except at $T=0.04$ where the starting configuration is chosen to be a perfect flux state for faster convergence. In addition, for lattices except $40\times 40$, the convergence at $T=0.04$ is achieved by using 40 moments with ${\epsilon }_{\mathrm{pr}}={10}^{-7}$ and ${\epsilon }_{\mathrm{tr}}={10}^{-5}$, while for other temperatures $M=30$ with ${\epsilon }_{\mathrm{pr}}={10}^{-5}$ and ${\epsilon }_{\mathrm{tr}}={10}^{-3}$ were sufficient. For the lattice $40\times 40$, $M=40$ with ${\epsilon }_{\mathrm{pr}}={10}^{-6}$ and ${\epsilon }_{\mathrm{tr}}={10}^{-4}$ were used for all temperatures.

Figure 4

(Color online) Spin structure factor vs $T$ for $J_{\mathrm{AF}}=0.05$ using the lattice sizes and parameters shown in the figure.

Figure 5

(Color online) Analysis of MC error bars (standard deviation) with TPEM for parameters $M=30$, ${ϵ}_{\mathrm{pr}}={10}^{-5}$, and ${ϵ}_{\mathrm{tr}}={10}^{-3}$. (a) and (b) results using a FM starting configuration; (c) and (d) using a PM starting configuration. (a) $T=0.01$, $⟨\mathrm{mag}⟩=0.934$ with ${\sigma }_{\mathrm{mag}}=0.003$ and $⟨S(0,0)⟩=0.872$ with ${\sigma }_{S(0,0)}=0.005$. (b) $T=0.06$, $⟨\mathrm{mag}⟩=0.603$ with ${\sigma }_{\mathrm{mag}}=0.007$ and $⟨S(0,0)⟩=0.371$ with ${\sigma }_{S(0,0)}=0.008$. (c) $T=0.01$, $⟨\mathrm{mag}⟩=0.940$ with ${\sigma }_{\mathrm{mag}}=0.001$ and $⟨S(0,0)⟩=0.883$ with ${\sigma }_{S(0,0)}=0.002$. (d) $T=0.06$, $⟨\mathrm{mag}⟩=0.616$ with ${\sigma }_{\mathrm{mag}}=0.006$ and $⟨S(0,0)⟩=0.388$ with ${\sigma }_{S(0,0)}=0.007$. The approximate number of independent Monte Carlo steps, obtained with the data-blocking technique (Ref. 59), is 100 for case (d).

Figure 6

(Color online) Phase diagram at $⟨n⟩$=0.5, varying temperature and $J_{\mathrm{AF}}$. Results are shown for a $12\times 12$ lattice using both DIAG and TPEM techniques, and for larger lattices using TPEM, as indicated. The tilting of the first-order low-temperature FM-flux line emerges from a combination of finite-$T$ MC simulation results and the $T=0$ energy comparison between FM and flux phases. Its intuitive origin remains to be clarified in future work.

Figure 7

(Color online) Examples of the criteria used in the calculation of the critical temperatures in Fig. 6. (a) and (c): Spin structure factors at the momenta of relevance vs temperatures, for many values of $J_{\mathrm{AF}}$, as indicated, using the DIAG technique on a $12\times 12$ cluster. (b) and (d): Same as in (a) and (c) except the technique used here is the TPEM. Shown are some of the results for $12\times 12$ and $20\times 20$, as indicated in the figure.

Figure 8

(Color online) (a) Density-of-states calculated for a perfect Flux state using both the DIAG and TPEM methods for a lattice of size $12\times 12$. In order to get the DOS accurately, even removing the Gibbs oscillations, one needs larger number of moments than what is usually required for other observables. (b) Monte Carlo results for the density-of-states obtained from simulations performed on a $40\times 40$ lattice. In this case the last configuration of the MC run has been used to calculate the density-of-states at $T=0.02$.

Figure 9

(a) Conductance and (b) resistance (1/conductance) vs temperature for a $12\times 12$ lattice, calculated with both DIAG and TPEM algorithms showing that the results agree. The couplings used are $J_{\mathrm{AF}}=0.0$ and $J_{\mathrm{AF}}=0.1$ as indicated, and the density is $⟨n⟩=0.5$. The convergence at $T=0.04$ is achieved by using 40 moments with ${\epsilon }_{\mathrm{pr}}={10}^{-7}$ and ${\epsilon }_{\mathrm{tr}}={10}^{-5}$, as discussed in previous figures captions.

Figure 10

(Color online) (a) Conductance and (b) resistance (1/conductance) vs temperature calculated with the TPEM algorithm at $J_{\mathrm{AF}}=0.0$ and $J_{\mathrm{AF}}=0.1$, $⟨n⟩=0.5$, and for the cluster sizes shown in the figure. The figure shows that the conductance does $not$ suffer from strong size effects. The convergence at $T=0.04$ for the $12\times 12$ lattice was achieved by using 40 moments with ${\epsilon }_{\mathrm{pr}}={10}^{-7}$ and ${\epsilon }_{\mathrm{tr}}={10}^{-5}$, as discussed elsewhere.

Figure 11

(Color online) (a) Conductance vs temperature for lattices of sizes $20\times 20$ and $32\times 32$, with and without magnetic fields. (b) Resistance vs temperature calculated by taking the inverse of the conductance in (a). Close to the first-order transition in the phase diagram, a small magnetic field can destabilize the insulating flux state into a metallic FM state. For each temperature, 1000 thermalization and 2000 measurements MC steps were used, with actual measurements taken at every 10 steps.

Figure 12

(Color online) Convergence of the structure factors, varying the parameters of the TPEM algorithm. (a) Magnetization $\mid M\mid =\sqrt{S(0,0)}$ vs $T∕t$ at $J_{\mathrm{AF}}=0.0$ using the DIAG method and TPEM with ${ϵ}_{\mathrm{pr}}={10}^{-5}$, ${ϵ}_{\mathrm{tr}}={10}^{-6}$, and the values of $M$ indicated. (b) Order parameter associated with the CE phase $\mid O_{\mathrm{CE}}\mid =\sqrt{S(\pi ,0)}$ vs $T∕t$ at $J_{\mathrm{AF}}=0.2$ using the DIAG method and TPEM with ${ϵ}_{\mathrm{pr}}={10}^{-5}$, ${ϵ}_{\mathrm{tr}}={10}^{-6}$, and the values of $M$ indicated. (c) Magnetization $\mid M\mid =\sqrt{S(0,0)}$ vs $T∕t$ at $J_{\mathrm{AF}}=0.0$ using the DIAG method and TPEM with $M=20$ and ${ϵ}_{\mathrm{tr}}={10}^{-6}$, varying ${ϵ}_{\mathrm{pr}}=ϵ$ as indicated. (d) Order parameter of the CE phase $\mid O_{\mathrm{CE}}\mid =\sqrt{S(\pi ,0)}$ vs $T∕t$ at $J_{\mathrm{AF}}=0.2$ using the DIAG method and TPEM with $M=20$, ${ϵ}_{\mathrm{tr}}={10}^{-6}$, varying ${ϵ}_{\mathrm{pr}}=ϵ$ as indicated. All calculations were done on a $12\times 12$ lattice, using $1000$ Monte Carlo steps for thermalization and $1000$ for measurements.

Figure 13

(Color online) Lattice size dependence of the square root of the structure factors at the momenta characteristic of (a) a FM state, $\mathbf{k}=(0,0)$ and $J_{\mathrm{AF}}=0.0$, and (b) a CE phase, $\mathbf{k}=(\pi ,0)$ and $J_{\mathrm{AF}}=0.2$. Results were obtained with the TPEM with $M=20$, ${ϵ}_{\mathrm{pr}}={10}^{-5}$, ${ϵ}_{\mathrm{tr}}={10}^{-6}$. In addition, for (a) the DIAG method was also used on a $8\times 8$ lattice as indicated. In the simulation, $1000$ MC steps were used for thermalizations and $1000$ steps for measurements.

Figure 14

(Color online) (a) Total energy vs $J_{\mathrm{AF}}$ at low temperature $(T=0.01t)$ for different lattices as indicated. For the $12\times 12$ lattice the DIAG method was used, while for the others the TPEM was employed with $M=20$, ${ϵ}_{\mathrm{pr}}={10}^{-5}$, ${ϵ}_{\mathrm{tr}}={10}^{-6}$. In the simulation, $1000$ MC steps were used for thermalizations and $1000$ steps for measurements. (b) Phase diagram of Hamiltonian Eq. (8) varying temperature and $J_{\mathrm{AF}}$ $(\lambda =0)$. The three magnetically different regions: FM, PM, and CE are indicated. The phase diagram was calculated for different lattices as shown. For $12\times 12$ the DIAG method was used and for the others the TPEM with $M=20$, ${ϵ}_{\mathrm{pr}}={10}^{-5}$, ${ϵ}_{\mathrm{tr}}={10}^{-6}$. The critical temperatures were obtained from the calculation of structure factors, as shown in Fig. 13.

Figure 15

Phase diagram of Hamiltonian (8) varying temperature and $J_{\mathrm{AF}}$ for $\lambda =0.5$ and the harmonic parameters discussed in the text, i.e., with the inclusion of phonons. The critical temperatures were obtained from the calculation of structure factors on a $12\times 12$ lattice with the DIAG method. Note the similarity of this phase diagram with the result obtained at $\lambda =0.0$, suggesting that to simulate the competition between FM metallic and CE insulating regimes the presence of a robust electron-phonon coupling is not necessary.

Figure 16

(Color online) (a) DOS of a perfect CE phase (single spin configuration) obtained on a $12\times 12$ lattice calculated with the DIAG method (solid black line) and with the TPEM (dashed red line) using $M=100$. The chemical potential lies in the left gap (arrows) indicating that the system is an insulator. (b) DOS of the system with $J_{\mathrm{AF}}=0.2$ (CE-phase ground state) on a 20$\times$20 lattice calculated with the TPEM using $M=100$, as described in the text. The location of the chemical potential is indicated by the vertical dashed line.

Figure 17

(Color online) Upper panel: Conductance vs temperature for $J_{\mathrm{AF}}=0.0$ (ferromagnet) and $J_{\mathrm{AF}}=0.2$ (CE phase), different lattices sizes and algorithms, as indicated. Lower panel: Logarithm of the resistivity vs temperature for the same parameters as before.

Figure 18

(Color online) Effect of a small magnetic field on the resistivity of the CE phase, for couplings in the vicinity of the first order transition metal insulator. Shown is the resistivity vs $T$ at $J_{\mathrm{AF}}=0.175$, $\lambda =0$, on a $20\times 20$ lattice for $B=0.1t$ and without field $(B=0)$ for comparison. Note the large change in the resistivity at low temperature, compatible with a colossal magnetoresistance. The TPEM was used with $M=30$, ${ϵ}_{\mathrm{pr}}={10}^{-5}$, and ${ϵ}_{\mathrm{tr}}={10}^{-6}$. 1,000 Monte Carlo steps were done for thermalization and an additional 100 more for measurements.