Electron-lattice coupling and partial nesting as the origin of Fermi arcs in manganites

A tight-binding model for $e_{\text{g}}$ electrons coupled to Jahn-Teller lattice distortions is studied via Monte Carlo simulations. By focusing on the periodicity of the cooperative Jahn-Teller distortions, and the one-particle spectral function, our results clarify the physical origin of the Fermi-arcs phase observed in layered manganites. In a range of parameters where no broken symmetry phase exists, the nearly nested Fermi surface favors certain correlations between Jahn-Teller distortions. The spectral weight near the Brillouin zone edge is suppressed, leading to the pseudogap in the density of states. We discuss the stability of this phase as a function of temperature and coupling strength for different hole dopings.

Figure 1

(Color online) Energy bands for the 2D two-orbital model in the spin ferromagnetic state, with no electron-lattice coupling [Eq. (2)]. The different horizontal lines correspond to the chemical potentials for $x=0.4$ (pink, slashed), $x=0.5$ (red, continuous), and $x=0.6$ (blue, dashed), the three different hole dopings studied in this work.

Figure 2

(Color online) Fermi surface for the same dopings as in Fig. 1: $x=0.6$ (blue, dark gray), $x=0.5$ (red, medium gray), and $x=0.4$ (pink, light gray). The better nesting (i.e., smaller angle between tangents of different pieces of the Fermi surface) for $x=0.4$, as compared to $x=0.6$, implies that a smaller electron-lattice coupling is needed to induce the pseudogap phase. For $x=0.5$ the nested portions of the Fermi surface are separated by $k_{\text{nest}}\approx 2\pi /4a$, which is commensurate with the systems size that can be studied with the Monte Carlo technique.

Figure 3

(Color online) Bands (top two panels) and Fermi surface (bottom) projected over the two $e_{\text{g}}$ orbitals, as indicated. The higher overlap of neighboring $x^2\text{−}{y}^2$ orbitals within the plane leads to the larger bandwidth of the corresponding projection. Notice how the symmetry of the orbitals prevents mixing in the (1,1) direction $\left(M\text{−}\Gamma \right)$. Consequently, in that direction, all spectral weight of the Fermi surfaces in Fig. 2 comes from the $x^2\text{−}{y}^2$ orbital, except for the small electron pocket for $x=0.4$ (light gray in Fig. 2), which is almost completely of $3z^2\text{−}{r}^2$ character. The spectral weight near the Brillouin zone face, on the contrary, comes from both orbitals.

Figure 4

(Color online) Total density of states for $x=0.5$ and different electron-lattice couplings ($\lambda$’s) from Monte Carlo simulations at $T=0.01$. The Fermi energy $\left(E_{\text{F}}\right)$ for the different parameters is taken as zero energy. As $\lambda$ increases, the density of states at the Fermi energy decreases, and, for a large enough $\lambda$, a gap opens. Experiments (Refs. 1, 3, 13) and the present work indicate that layered manganites are in the pseudogap regime with a strong reduction in the density of states at the Fermi energy. A small imaginary part has been given to each eigenvalue arising from the Monte Carlo simulations in order to obtain a smooth curve.

Figure 5

Cartoon characterizing the different electronic states of the system, as a function of $\lambda$ at low $T$. As the values at which different electronic structures are found depend on doping and temperature, no precise boundary for them is shown. The values of $\lambda$ indicated correspond to simulations with $x=0.5$, and $T=0.01$.

Figure 6

(Color online) Monte Carlo average of thermally excited distortions, as a function of their wave vector. For each $\left(k_x,k_y\right)$ point, the size of the dot is proportional to the thermal average of $Q_2^2\left(k_x,k_y\right)$ (see text). The main graph shows the thermal average of the $Q_2$ mode, for $\lambda =1.39$ and $T=0.01$. The inset at the top-right shows the Fermi surface of the undistorted system, where the wave vectors labeled in the main panel are shown connecting some points of the Fermi surface. For the thermal-averaged Fermi surface with these parameters, see Fig. 7. The finite size of the simulation lattice limits the allowed wave vectors of the distortions, and the number of them is the same as the number of sites (144 unless otherwise noted).

Figure 7

(Color online) Spectral function arising from the Monte Carlo simulations. The same parameters as in Fig. 6 are used ($\lambda =1.39$, $T=0.01$, and $x=0.5$). (a) Energy cut at the Fermi energy, and (b) $A\left(\mathbf{k},E\right)$ for $\mathbf{k}$ along the high symmetry directions. Already for this coupling the distortions shown in Fig. 6 result in a reduction in the spectral weight near the Brillouin zone face (antinodal direction) that can also be observed in both projections.

Figure 8

(Color online) Results of the Monte Carlo simulation for $\lambda =1.5$, $T=0.01$, and $x=0.5$, using a $12\times 12$ lattice. The spectral function cut at the Fermi energy is shown in (a), and its value along the high symmetry directions in (b). For these parameters, a gap is already formed along the antinodal direction $\left(X\text{−}M\right)$, while in the nodal $\left(M\text{−}\Gamma \right)$ direction there is a coherent band. This turns into a Fermi-arc-like Fermi surface, with hardly any spectral weight left near the Brillouin zone face, and a well-defined coherent peak near the zone diagonal (pseudogap regime). (c) shows the $Q_1$ and $Q_2$ (see text) distortions in $\mathbf{k}$ space. A larger electron-lattice coupling as compared to Fig. 6 make distortions with more wave vectors noticeable under this scale (same as Fig. 6). However, the distortions with the appropriate nesting wave vectors are more favored by $\lambda$, and they produce the arclike Fermi surface shown in (a).

Figure 9

(Color online) Monte Carlo simulation results for $\lambda =1.58$, $T=0.01$, and $x=0.5$, using a $12\times 12$ lattice: (a) $A\left(\mathbf{k},E_{\text{F}}\right)$; (b) $Q_1$ and $Q_2$ distortions in $\mathbf{k}$ space. The system is close to opening a complete gap: spectral weight at the Fermi energy survives only near the (0,0) to $\left(\pi ,\pi \right)$ diagonal, or nodal direction. As $\lambda$ increases, distortions with more wave vectors appear on the system (see Fig. 8). Notice that if $Q_2$ induces orbital correlations with period $4a$ (wave vector $\pi /2a$), charge correlations with period $2a$ can be expected that favor $Q_1$ modes also with period $2a$ (wave vector $\pi /a$).

Figure 10

(Color online) Dispersion relation (a) and quasiparticle (QP) weight (b) along the nodal direction, for different $\lambda$’s. The dashed lines indicate the position of the Fermi energy for the different couplings. To overcome momentum quantization, a standard fitting procedure of momentum distribution curves has been followed (see text). This induces a small oscillatory error in the determination of the QP weight. The $\lambda =0$ (dotted lines) curves, determined with the same procedure has been included to illustrate the error and for comparison purposes. Numerical accuracy is not enough to determine the effective mass for large $\lambda$’s. Curvature rapidly changes around $k_{\text{F}}$, where an $s$ shape dispersion relation is found, similar to the one reported in experiments (Refs. 1, 5). As spectral weight and curvature change along the Fermi surface we do not expect this curvature to directly determine transport properties. QP weight at $k_{\text{F}}$ is lost as $\lambda$ increases and the system develops a pseudogap, but this takes place at larger $\lambda$’s in the nodal direction, as compared to the antinodal direction [see Figs. 7a, 7b].

Figure 11

Schematic representation of the evolution of the different electronic states as a function of temperature. The transformation from a normal metal with a smooth density of states at the Fermi energy to the pseudogap (PG) state, with a minimum of the density of states at the Fermi energy, is continuous (see Fig. 4). Our results indicate that temperature favors the pseudogap, as explained in the text. For $x=0.5$, the phase with a gap is an ordered phase and the continuous line separating the PG and the ordered phase corresponds to a first order transition.

Figure 12

Evolution of the $Q_2$ Jahn-Teller distortions in $\mathbf{k}$ space as a function of temperature for an $8\times 8$ (left column) and $12\times 12$ (right column) lattices, for $\lambda =1.4$, at half-doping. In general, distortions are overestimated at low temperatures for small lattice sizes (specially at temperatures lower than the ones shown), see legend. Finite size effects are small, and the squarelike feature responsible for the Fermi arcs consistently appears for these sizes and $T$ range.

Figure 13

(Color online) Spectral function at the Fermi energy for some of the temperatures in Fig. 12 ($\lambda =1.4$, $x=0.5$). At low temperatures the system is undistorted for this $\lambda$. The nearly perfect nesting of the antinodal portions of the Fermi surface makes the distortions with the proper nesting vector much more likely to be thermally excited, producing the Fermi arcs. The color scale is the same for all temperatures. At higher temperatures, the entropy contribution to the free energy dominates, and the spectral weight along the nodal and antinodal directions becomes similar.

Figure 14

(Color online) Evolution of the system as a function of $\lambda$ for $T=0.001$, $x=0.4$, using a $12\times 12$ lattice. (a) shows Fermi surface changes. Both the detailed way in which the Fermi surface changes, and the temperature at which the changes take place depend on doping. For this temperature and the lower two values of $\lambda$, the system is essentially undistorted at half-doping. Longer nested portions of the Fermi surface (see Fig. 2) for $x=0.4$ result in smaller coherent quasiparticle peaks at lower temperatures. As $\lambda$ grows, the Fermi surface shrinks to small Fermi arcs, making different dopings much more alike (compare $\lambda =1.5$, and $\lambda =1.6$ with Figs. 8, 9, 15). $Q_2$ Jahn-Teller distortions in $\mathbf{k}$ space are shown in (b). The behavior is similar to half-doping, but for this doping the nesting wave vectors cannot be well approximated by a wave vector commensurate with the lattice. For low values of $\lambda$, the distortions with the nearest commensurate wave vector components, $2/3\pi /a$, are favored, and as $\lambda$ increases, distortions with the next nearest one, $\pi /2a$ are also present in the system.

Figure 15

(Color online) Monte Carlo simulations results as a function of $\lambda$ for $T=0.01$, $x=0.6$, using a $12\times 12$ lattice: The spectral function at the Fermi level is shown in (a), and lattice distortions in (b). For this doping, larger $\lambda$’s are needed to achieve the reduction in the spectral weight along the antinodal direction, and the $\lambda =1.35$ spectral function appears essentially undistorted. Notice how $\lambda =1.6$ is large enough such that distortions cannot be understood within the perturbation theory arguments presented in the text. Like the cuprates, in the strong coupling regime, the Fermi surface is very similar for dopings as different as 10%–20% (see Figs. 9, 14)