Prediction of exotic magnetic states in the alkali-metal quasi-one-dimensional iron selenide compound ${\mathrm{Na}}_2{\mathrm{FeSe}}_2$

The magnetic and electronic phase diagram of a model for the quasi-one-dimensional alkali-metal iron selenide compound ${\mathrm{Na}}_2{\mathrm{FeSe}}_2$ is presented. The novelty of this material is that the valence of iron is ${\mathrm{Fe}}^{2+}$, contrary to most other iron-chain compounds with valence ${\mathrm{Fe}}^{3+}$. Using first-principles techniques, we developed a three-orbital tight-binding model that reproduces the ab initio band structure near the Fermi level. Including Hubbard and Hund couplings and studying the model via the density-matrix renormalization group and Lanczos methods, we constructed the ground-state phase diagram. A robust region where the block state $\uparrow \uparrow \downarrow \downarrow \uparrow \uparrow \downarrow \downarrow$ is stabilized was unveiled. The analog state in iron ladders, employing $2\times 2$ ferromagnetic blocks, is by now well established, but in chains a block magnetic order has not been observed yet in real materials. The phase diagram also contains a large region of canonical staggered spin order $\uparrow \downarrow \uparrow \downarrow \uparrow \downarrow \uparrow$ at very large Hubbard repulsion. At the block-to-staggered transition region, an exotic phase is stabilized with a mixture of both states: an inhomogeneous orbital-selective charge density wave with the exotic spin configuration $\uparrow \uparrow \downarrow \uparrow \downarrow \downarrow \uparrow \downarrow$. Our predictions for ${\mathrm{Na}}_2{\mathrm{FeSe}}_2$ may guide crystal growers and neutron-scattering experimentalists towards the realization of block states in one-dimensional iron selenide chain materials.

Figure 1

(a) Crystal structure of ${\mathrm{Na}}_2{\mathrm{FeSe}}_2$, the material that we predict should present exotic magnetic order. (b) Side view of a single Fe chain and the nearest-neighbor $t$ and next-nearest-neighbor $t^{\prime }$ hopping amplitudes used in our study. The dashed line indicates the primitive unit cell used in DFT calculations.

Figure 2

(a) Band structure and (b) projected density of states of the ${\mathrm{Na}}_2{\mathrm{FeSe}}_2$ single-chain compound obtained using DFT calculations. (c) Tight-binding (TB) band structure in the folded zone. (d) Tight-binding unfolded (TB-unfold) band structure used in the DMRG calculations. Parts of the original folded bands are marked with blue dashed lines. The zero in the vertical energy axis is the position of the Fermi level.

Figure 3

Schematic representation of the magnetic states observed in the phase diagram. (i) AF1: standard staggered antiferromagnetic phase $\uparrow \downarrow \uparrow \downarrow$; (ii) AF2: antiferromagnetically coupled ferromagnetic blocks resulting in $\uparrow \uparrow \downarrow \downarrow$ spin order. (iii) AF3: mixed ferro- and antiferromagnetic ordering $\uparrow \uparrow \downarrow \uparrow \downarrow \downarrow \uparrow \downarrow$ stable in a narrow region of couplings. At the bottom: schematic phase diagram of the ground state for fixed $J_H/U=0.25$.

Figure 4

Real-space spin correlation $S\left(r\right)=\langle {\mathbf{S}}_m·{\mathbf{S}}_l\rangle$, with $r=|m-l|$, for various values of the Hubbard interaction $U/W$, at fixed $J_H/U=0.25$ and using an $L=24$ cluster studied with DMRG. Results are shown for (a) the PM phase at $U/W=0.4$, (b) the block phase (AF2) at $U/W=4.0$, and (c) the staggered AF1 phase at $U/W=12.0$. The AF3 state will be discussed later in Fig. 6. The spin structure factor $S\left(q\right)$ is shown for three values of $L$ at (d) $U/W=4.0$ in the block AF2 phase and (e) $U/W=12.0$ in the AF1 phase.

Figure 5

(a) Spin structure factor $S\left(q_p\right)$ vs $U/W$ at $J_H/U=0.25$ for several values of the three dominant wave vectors shown in the legend. (b) Site-average electronic occupancy $n_{\gamma }$ for the three orbitals $\{\gamma =1,2,3\}$ vs $U/W$ using DMRG and a chain of $L=16$ sites. Inset: Site-average expectation value of the total spin squared vs $U/W$ at $J_H/U=0.25$.

Figure 6

(a) Electronic occupancy $\langle n_{\gamma ,i}\rangle$ for the three orbitals $\{\gamma =1,2,3\}$ vs site index $i$ at $U/W=9.1$, $L=24$, in the AF3 regime showing an orbital-selective charge density wave. (b) Spin correlation $S\left(r\right)=\langle {\mathbf{S}}_m·{\mathbf{S}}_l\rangle$ at $U/W=9.1$ using a $L=24$ cluster displaying the AF3 magnetic ordering $\uparrow \uparrow \downarrow \uparrow \downarrow \downarrow \uparrow \downarrow$. (c) The spin structure factor for three different values of $L=8,16,24$ and at $U/W=9.1$. Clear peaks at $q=3\pi /4$ are shown.

Figure 7

DOS of different orbitals corresponding to different phases at $J_H/U=0.25$ on a four-site three-orbital system using Lanczos diagonalization. (a) Corresponds to the PM phase at $U/W=0.4$, (b) is for the AF2 phase at $U/W=4.0$, while (c) is for the AF1 phase at $U/W=10.0$.

Figure 8

Local DOS of different orbitals corresponding to different phases at $J_H/U=0.25$ for system size $L=16$ using dynamical DMRG. (a) The AF2 phase at $U/W=4.0$ and (b) the AF3 phase at $U/W=9.1$. (c) Site-averaged local charge fluctuations and orbital-resolved charge fluctuations vs $U/W$ at $J_H/U=0.25$ and for $L=16$. The nonzero values indicate charge fluctuations are present in the entire AF2 phase, suggesting it is metallic.

Figure 9

Phase diagram of the three-orbital Hubbard model with the hopping amplitudes of ${\mathrm{Na}}_2{\mathrm{FeSe}}_2$, varying the Hund coupling and Hubbard interactions. Panel (a) depicts results based on DMRG, while panel (b) are results using the Lanczos method on an $L=4$ sites cluster and open boundary conditions. PM stands for paramagnetic phase, while AF2 for block phase with $\uparrow \uparrow \downarrow \downarrow$ order. The intermediate phase AF3 with $\uparrow \uparrow \downarrow \uparrow \downarrow \downarrow \uparrow \downarrow$ spin ordering appears using both methods in a narrow range of couplings. AF1 stands for the staggered antiferromagnetic phase $\uparrow \downarrow \uparrow \downarrow \uparrow$.

Figure 10

Spin structure factors $S\left(q\right)$ at $q=\pi /2$, $3\pi /4$, and $\pi$ vs $U/W$ for (a) $J_H/U=0.20$ and (b) $J_H/U=0.25$, using the Lanczos method for $L=4$ sites, and the three-orbital Hubbard model used here.