Emergent dimensional reduction of the spin sector in a model for narrow-band manganites

The widely used double-exchange model for manganites is shown to support various “striped” phases at filling fractions $1/n$ ($n=3$, $4$, $5,\dots$), in the previously unexplored regime of narrow bandwidth and small Jahn-Teller coupling. Working in two dimensions, our main result is that these stripes can be individually spin flipped without a physically relevant change in the energy, i.e., we find a large ground-state manifold with nearly degenerate energies. The two-dimensional spin system thus displays an unexpected dynamically generated dimensional reduction into decoupled one-dimensional stripes, even though the electronic states remain two dimensional. Relations of our results with recent literature addressing compass models in quantum computing are discussed.

Figure 1

(a) Monte-Carlo snapshot for $J_{\mathrm{AF}}=0.19t_0$, $\lambda =0$, and $\beta t_0=1200$, in the regime of the SOS${}_{1/3}$ phase. It can be visualized as composed of domains of the $E$-AF phase illustrated in the cartoon (b). (a) NN domains (“stripes”) are separated by dashed lines, they have spins at right angles. (b) Dashed lines indicate the zigzag FM chains of the $E$-AF phase; in (a) shading (color) indicates a few of the short segments of the $E$-phase zigzag chains that survive in the SOS${}_{1/3}$ phase, later called “arrows.”

Figure 2

Zero temperature ($T=0$) energies (per site) of several magnetic phases, at doping $x=1/3$ and $\lambda =0$, vs $J_{\mathrm{AF}}$. Dots represent MCMC results at $T=0.002$ on a $6\times 6$ lattice that closely follows the variational results. S6 denotes a spiral phase with a period of 6 lattice spacings (Ref. 35), while a $12\times 12$ lattices shows one with period 12; this phase thus converges to FM order with increasing lattice sizes (Ref. 36).

Figure 3

Zero-$T$ phase diagram of model Eq. (1) at $x$ $=$ 1/3, varying $\lambda$ and $J_{\mathrm{AF}}$. Variational technique results are shown with shading (colors). The FM DE metallic phase dominates; large $J_{\mathrm{AF}}$ stabilizes AF phases. The “C${}_{1/3}$E${}_{2/3}$” phase is a variant (Refs. 6 and 11) of the CE phase at half filling (Ref. 1). The SOS${}_{1/3}$ phase at intermediate $J_{\mathrm{AF}}$ and small $\lambda$ is analogous to the $x$ $=$ 1/4 SOS${}_{1/4}$ phase (Ref. 11), and is highly degenerate. Full dots show where MCMC (for $6\times 6$ sites) confirmed the results; open dots are MCMC results that remained inconclusive due to metastabilities caused by phase competition. For details see Ref. 11.

Figure 4

Cartoon for the spin patterns for two quasidegenerate states of the SOS${}_{1/3}$ phase (a) and (b). Lines between domains with orthogonal spins illustrate the periodic pattern of Berry phases: thin lines denote positive sign and thick lines denote negative sign. The pattern in (b) is reached by flipping all spins in the top black stripe of (a), which entails changes in the signs of the Berry phases, but induces an energy difference of only ${10}^{-5}{t}_0$. The dotted lines connecting three ferromagnetically aligned spins indicate “arrows” of the $E$ phase, see Fig. 1. In the pattern in (c), “arrows” in adjacent domains point in opposite directions. The energy per site in this phase is ${10}^{-2}{t}_0$ higher than that of the SOS${}_{1/3}$ phase; flipping spins of one stripe induces energy differences of $\approx$5 $\times {10}^{-4}{t}_0$, i.e., at least 50 times larger than between (a) and (b).

Figure 5

Spin-structure factor $S(\mathbf{k})$ along the line $(\pi ,0)$ to $(0,\pi )$, calculated for the SOS${}_{1/3}$ phase by averaging over all $2^{16}$ degenerate realizations with stripes along one direction of a $24\sqrt{2}\times \sqrt{2}$ cluster ($\lambda$ $=$ 0). $S(\mathbf{k})\approx 0$ for all $\mathbf{k}$ except the line running from $(\pi ,0)$ to $(0,\pi )$. Results at $x$ $=$ 1/4, 1/5, and 1/6 are also shown.

Figure 6

(a) Zero-$T$ spin and orbital ordering of the SOS${}_{1/3}$ phase, the ground state for $0.17\lesssim J_{\mathrm{AF}}\lesssim 0.21$ ($\lambda =0$), see Fig. 3. For each site, the linear combination $|\alpha \rangle =\cos \alpha |3z^2-r^2\rangle +\sin \alpha |x^2-y^2\rangle$ is depicted that has the highest density, i.e., for the value of $\alpha$ maximizing $n_{i,\alpha }$. (b) The orbital with maximal density is shown for the modified SOS${}_{1/3}$ phase with the spin pattern given in Fig. 4c. While the depicted orbital is the one with the highest density, these cartoons do not fully describe the orbital states, as some density is also found in the orthogonal orbitals, see text.

Figure 7

(a) Density of states $N(\omega )$ ($\lambda =0$) for two patterns of the SOS${}_{1/3}$ phase, the phase shown in Fig. 4c, and the SOS${}_{1/2}$ phase (Ref. 12). The arrow indicates the dispersionless states of the SOS${}_{1/3}$ phases.

Figure 8

Spectral density $A(\mathbf{k},\omega )$ for two patterns of the SOS${}_{1/3}$ phase at $\lambda =0$. The $L_x\times L_y=24\times 24$ lattice is used, and the $E$-AF domains run in the $(1,1)$ direction, from $(0,0)$ to $(24,24)$. Periodic boundary conditions are employed, and peaks were broadened with a Lorentzian with a width $\delta =0.02t_0$.

Figure 9

Spectral density $A(\mathbf{k},\omega )$ for an artificial reference model where all hoppings are replaced by their absolute values; this phase has the same energy as the SOS${}_{1/3}$ phase. Parameters are as in Fig. 8.

Figure 10

Two orbital building blocks of the SOS${}_{1/3}$ phase. Black arrows give the SOS${}_{1/3}$ spin pattern and dashed lines indicate the stripes. Orbitals shown with shading/lines illustrate the unit cells corresponding to two arrows in adjacent stripes. Site A is a “center” site with occupancy in both the $x^2-y^2$ and the $3z^2-r^2$ orbitals, site B is a “wing” site, only the more highly occupied $3y^2-r^2$ orbital is shown. Site C is also a “wing” site, but here both orbitals are shown, the planar $y^2-z^2$ orbital is drawn with lines and the directional $3x^2-r^2$ with shading. An electron in the planar $y^2-z^2$ orbital can only hop along $+y$ to site B of the adjacent stripe: It cannot hop along the $x$-direction due to its orbital symmetry (Ref. 1), and it cannot hop along the $-y$ bond due to the AF spins.