Quantum magnetism of iron-based ladders: Blocks, spirals, and spin flux

Motivated by increasing experimental evidence of exotic magnetism in low-dimensional iron-based materials, we present a comprehensive theoretical analysis of magnetic states of the multiorbital Hubbard ladder in the orbital-selective Mott phase (OSMP). The model we used is relevant for iron-based compounds of the ${A\mathrm{Fe}}_2{X}_3$ family (where $A=\mathrm{Cs}$, Rb, Ba, K are alkali metals and $X=\mathrm{S}$, Se are chalcogenides). To reduce computational effort, and obtain almost exact numerical results in the ladder geometry, we utilize a low-energy description of the Hubbard model in the OSMP—the generalized Kondo-Heisenberg Hamiltonian. Our main result is the doping vs interaction magnetic phase diagram. We reproduce the experimental findings on the ${A\mathrm{Fe}}_2{X}_3$ materials, especially the exotic block magnetism of ${\mathrm{BaFe}}_2{\mathrm{Se}}_3$ (antiferromagnetically coupled $2\times 2$ ferromagnetic islands of the $\uparrow \uparrow \downarrow \downarrow$ form). As in recent studies of the chain geometry, we also unveil block magnetism beyond the $2\times 2$ pattern (with block sizes varying as a function of the electron doping) and also an interaction-induced frustrated block-spiral state (a spiral order of rigidly rotating ferromagnetic islands). Moreover, we predict new phases beyond the one-dimensional system: a robust regime of phase separation close to half filling, incommensurate antiferromagnetism for weak interaction, and a quantum spin-flux phase of staggered plaquette spin currents at intermediate doping. Finally, exploiting the bonding/antibonding band occupations, we provide an intuitive physical picture giving insight into the structure of the phase diagram.

Figure 1

Schematic representation of (a) the ladder geometry, (b) density of states in the orbital-selective Mott phase, (c) the generalized Kondo-Heisenberg model, (d)–(i) the unveiled exotic magnetic orders.

Figure 2

Single-particle spectral function $A_{\gamma }(\mathbf{q},\omega )$ of the two-orbital Hubbard model (1) in the vicinity of the Fermi level ${\epsilon }_{\text{F}}$. The itinerant ($\gamma =0$) orbital is presented as blue-green color, whereas the localized ($\gamma =1$) as dark purple. Both $q_{\perp }=0,\pi$ components of each orbital are displayed. The frequency resolution was chosen to be $\mathrm{\Delta }\omega =0.02$ eV with the broadening $\eta =2\mathrm{\Delta }\omega$. The results were obtained for a ladder of $L=72$ sites, filling $n_{\text{H}}=2.75$, and interaction $U/W=1$.

Figure 3

Properties of the band structure. (a) Noninteracting ($U=0$) band structure of (3) for $t_0^{\perp }=t_0^{\parallel }$. The dispersion concerns only the itinerant $\gamma =0$ orbital, as the $\gamma =1$ orbital is completely localized. The dashed line marks the Fermi level ${\epsilon }_{\text{F}}$ at half filling, $n_{\text{K}}=1$. (b) $n_{\text{K}}\text{−}{t}_0^{\perp }$ phase diagram of the noninteracting ladder. The dashed line marks the point where the Fermi level touches the tip of the antibonding ($q_{\perp }=\pi$) band, while the dot marks the phase boundary between the one- and two-band regimes for $t_0^{\perp }=t_0^{\parallel }$. (c) DMRG results for the antibonding band filling $n_{\text{K}}^a$ vs the interaction $U$ and the total filling $n_{\text{K}}$ at fixed $t_0^{\perp }=t_0^{\parallel }$. The plot is composed of $37\times 40$ data points obtained for the generalized Kondo-Heisenberg ladder of $L=72$ sites. The sketches show the one-band ($n_{\text{K}}^a\simeq 1$) and two-band (both bands fractionally occupied) regimes. The dashed line is a contour at $n_{\text{K}}^a\simeq 1$.

Figure 4

Schematic $n_{\text{K}}\text{−}U$ magnetic phase diagram of the generalized Kondo-Heisenberg ladder of $L=72$ sites. The vertical lines within the phase-separation regime mark special fillings $n_{\text{K}}=1.17,1.25$, where perfect block order is recovered (see the discussion in Sec. 3b). The phase diagram was inferred from extensive DMRG calculations performed at $37\times 40$ data points uniformly distributed over the range of the plot. The phase boundaries are necessarily approximate as they cannot be exactly determined from finite-size calculations.

Figure 5

Block-magnetic order. (a) Spin-spin correlations $\langle {\mathbf{S}}_{L_{\parallel }/2,1}·{\mathbf{S}}_{\mathbf{r}}\rangle$ as a function of distance with $\mathbf{r}=({\ell }_{\parallel },1)$ (intraleg) or $\mathbf{r}=({\ell }_{\parallel },2)$ (interleg). Top to bottom: $\pi /2$ block ($n_{\text{K}}=1.75$, $U/W=1$), $\pi /3$ block ($n_{\text{K}}=1.83$, $U/W=1.1$), mixed block ($n_{\text{K}}=1.81$, $U/W=1$). (b) Spin structure factor $S\left(\mathbf{q}\right)$ being the Fourier transform of the correlations shown in (a). (c) Bond correlations $\langle {\mathbf{S}}_{\mathbf{r}}·{\mathbf{S}}_{\mathbf{r}+\mathbf{1}}\rangle$ corresponding to (a) and (b). $\mathbf{1}$ connects the nearest-neighbor sites on the ladder. All results were obtained for a generalized Kondo-Heisenberg ladder of $L=72$ sites.

Figure 6

Block-spiral magnetic order. (a) Interaction $U$ evolution of the spin structure factor $S(q_{\parallel },0)$ for the $2\times 2$ block spiral ($n_{\text{K}}=1.75$). (b) Spin-spin correlations $\langle {\mathbf{S}}_{L_{\parallel }/2,1}·{\mathbf{S}}_{\mathbf{r}}\rangle$ as a function of distance corresponding to (a). (c) Chirality correlations $\langle {\mathit{\kappa }}_{L_{\parallel }/2,1}·{\mathit{\kappa }}_{\mathbf{r}}\rangle$ as a function of distance corresponding to (a). In (a) and (b), both the intraleg $[\mathbf{r}=({\ell }_{\parallel },1\left)\right]$ and interleg $[\mathbf{r}=({\ell }_{\parallel },2\left)\right]$ components are presented (as lines and symbols, respectively). All results were obtained for a generalized Kondo-Heisenberg ladder of $L=72$ sites.

Figure 7

Special features of the block-spiral phase. (a) Secondary Fourier mode (arrows) being a fingerprint of the block and block-spiral order. Shown are the representative cases of $\pi /3$ block ($n_{\text{K}}=1.83$, $U/W=1.1$) and $2\times 2$ block spiral ($n_{\text{K}}=1.75$, $U/W=2.2$). (b) Spectral function of the itinerant orbital $A_0(\mathbf{q},\omega )$ corresponding to the $2\times 2$ block spiral of (a). The frequency resolution was chosen to be $\mathrm{\Delta }\omega =0.02$ eV with the broadening $\eta =2\mathrm{\Delta }\omega$. All results were obtained for a generalized Kondo-Heisenberg ladder of $L=72$ sites.

Figure 8

Incommensurate antiferromagnet. (a) The filling $n_{\text{K}}$ evolution of the spin structure factor $S\left(\mathbf{q}\right)$ within the two-band regime at $U/W=1$. The main panel (inset) shows the antibonding $S(q_{\parallel },\pi )$ [bonding $S(q_{\parallel },0)$] component. (b) The filling $n_{\text{K}}$ evolution of the bond correlations $\langle {\mathbf{S}}_{\mathbf{r}}·{\mathbf{S}}_{\mathbf{r}+\mathbf{1}}\rangle$ corresponding to the structure factors shown in (a). $\mathbf{1}$ connects the nearest-neighbor sites on the ladder. Note the evident amplitude modulation of the AFM correlations. All results were obtained for a generalized Kondo-Heisenberg ladder of $L=72$ sites.

Figure 9

Phase separation. (a), (b) Bond correlations $\langle {\mathbf{S}}_{\mathbf{r}}·{\mathbf{S}}_{\mathbf{r}+\mathbf{1}}\rangle$ for two fillings (a) $n_{\text{K}}=1.11\in [1,1.17]$, (b) $n_{\text{K}}=1.19\in [1.17,1.25]$ lying within the intervals of unstable densities. $\mathbf{1}$ connects the nearest-neighbor sites on the ladder and the system size is $L=72$ sites. (c)–(e) Grand-canonical electron density $\langle \langle n_{\text{K}}\rangle \rangle$ vs $\mu$ for different values of the interaction $U/W=1,2,3$ and three system sizes $L=24,72,96$. The horizontal lines mark $\langle \langle n_{\text{K}}\rangle \rangle =1.17$ ($\pi /3$ block) and $\langle \langle n_{\text{K}}\rangle \rangle =1.25$ ($\pi /2$ block). All results were obtained for a generalized Kondo-Heisenberg ladder.

Figure 10

Spin structure factor and momentum distribution function in the spin-flux region. (a) Spin structure factor $S\left(\mathbf{q}\right)$ at $n_{\text{K}}=1.5$ and $U/W=1$ (inset), $U/W=2$ (main panel). (b) The same as in (a) but at $n_{\text{K}}=1.47$ (lines), $n_{\text{K}}=1.53$ (symbols), and $U/W=2$. The inset shows the bond correlations $\langle {\mathbf{S}}_{\mathbf{r}}·{\mathbf{S}}_{\mathbf{r}+\mathbf{1}}\rangle$ corresponding to $n_{\text{K}}=1.47$. Here, red (blue) color marks AFM (FM) bonds and $\mathbf{1}$ connects the nearest-neighbor sites on the ladder. (c) Momentum distribution function $n\left(\mathbf{q}\right)$ at $n_{\text{K}}=1.47,1.5,1.53$ and $U/W=2$. All results were obtained for a generalized Kondo-Heisenberg ladder of $L=72$ sites.

Figure 11

Spin flux. (a)–(d) Chirality correlations $\langle {\mathit{\kappa }}_{L_{\parallel }/2,1}·{\mathit{\kappa }}_{\mathbf{r}}\rangle$ at $n_{\text{K}}=1.5$, $U/W=2$ as a function of distance. Shown are all possible components: (a) intraleg, (b) interleg, (c) rung-rung, (d) leg-rung. The sketches show the proposed magnetic order and also highlight the bonds which are involved in the calculation. (e) Comparison of the distance-dependent rung-rung chirality $\langle {\mathit{\kappa }}_{L_{\parallel }/2,1}^{\perp }·{\mathit{\kappa }}_{\mathbf{r}}^{\perp }\rangle$ between the flux ($n_{\text{K}}=1.5$, $U/W=2$) and the $2\times 2$ block spiral ($n_{\text{K}}=1.75$, $U/W=2.2$). (f) Rung-rung chirality in the flux phase with $n_{\text{K}}=1.53$, $U/W=2$. The dark symbols correspond to $cos\left(2{\tilde{k}}_{\text{F}}^b{\ell }_{\parallel }\right)$ fit using the effective Fermi wave vector ${\tilde{k}}_{\text{F}}^b$ obtained from Fig. 10. (g) Interaction $U$ evolution of the rung-rung chirality for the spin flux with $n_{\text{K}}=1.5$. All results were obtained for a generalized Kondo-Heisenberg ladder of $L=72$ sites.