Topological solitons and bulk polarization switch in collinear type-II multiferroics

We introduce a microscopic model for collinear multiferroics capable of reproducing, as a consequence of magnetic frustration and easy-axis anisotropy, the so-called “uudd” (or antiphase) magnetic ordering observed in several type-II multiferroic materials. The crucial role of lattice distortions in the multiferroic character of these materials enters into the model via an indirect magnetoelectric coupling, mediated by elastic degrees of freedom through a pantograph mechanism. Long-range dipolar interactions set electric dipoles in the antiferroelectric order. We investigate this model by means of extensive density matrix renormalization group computations and complementary analytical methods. We show that a lattice dimerization induces a spontaneous ${\mathbb{Z}}_2$ ferrielectric bulk polarization, with a sharp switch off produced by a magnetic field above a critical value. The topological character of the magnetic excitations makes this mechanism robust.

Figure 1

Schematic ground state diagram for the spin $S=\frac{1}{2}$ anisotropic frustrated antiferromagnetic chain. Zig-zag miniatures of the spin chain are used to visually emphasize the prevalence of first-neighbor or second-neighbor antiferromagnetic correlations or singlet correlations indicated by ellipses. A diffuse line separates the quantum behavior at low anisotropy from the classical behavior at high anisotropy. Most of the materials we are interested in are located in the frustrated anisotropic region (lower right corner). As representative points, we numerically explore in detail a highly frustrated regime given by $J_2/J_1=0.8$, in the isotropic case $\gamma =1$ (red star) and a high easy-axis anisotropy $\gamma =\frac{1}{8}$ (blue star).

Figure 2

Cartoon of degrees of freedom and dipole-elastic coupling mechanism. Shaded symbols show the undistorted (regular) lattice; full colored symbols represent a general distortion configuration. Magnetic ions are described as a chain of sites (in black), while dipoles (double arrows) associated with charged ions (in blue) are located at chain bonds. The asymmetry of charged ions with respect to the chain axis indicates the presence of dipolar moments normal to the magnetic chain, pointing toward the furthermost vertex. Displacements of magnetic ions $u_i$ modify bond lengths by a distortion ${\delta }_i=u_{i+1}-u_i$. Also, the distance between adjacent dipoles is distorted by ${\eta }_i=({\delta }_i+{\delta }_{i+1})/2$. A dipole strength is enlarged (shortened) when the bond is shortened (enlarged) while its orientation is given by an Ising variable, as described in Eq. (5).

Figure 3

Dipole-elastic phase diagram, computed for $\beta =0.2$. Double line arrows describe the dipole ordered pattern in each region. Elastic distortions follow the dipole pattern periodicity, except in the zero field line $\epsilon =0$ and the saturation region $\Uparrow \Uparrow \Uparrow \Uparrow$ where magnetic ions are equally spaced.

Figure 4

Dipolar pattern in the dimerized electroelastic phase $\Uparrow \Downarrow \Uparrow \Downarrow$ for a finite electric field pointing upwards (symbols as in Fig. 2). Dipoles pointing along the field are larger than dipoles in the opposite direction. The alternation of bond length distortions is the mechanism for bulk polarization. The system acquires a linear electrical polarization (paraelectric behavior). Distortions are magnified for visual effect.

Figure 5

Magnetization curves obtained by density matrix renormalization group (DMRG) self-consistent solution of Eq. (11) for the isotropic case ($\gamma =1$) and a high easy-axis anisotropy ($\gamma =1/8$), setting $J_1=0.5$, $J_2=0.4$, $J_e=0.2$, $\alpha =\beta =0.2$, and zero electric field. A plateau at $M=0$ is observed in both cases, though the spin structure found for the isotropic case (quantum dimerized plateau) is very different from the one found in the anisotropic case (classical $\uparrow \uparrow \downarrow \downarrow$ plateau), see discussion below. A prominent plateau at magnetization fraction $M=\frac{1}{3}$ is also observed in both cases. The insets show the finite sized scaling of the width of the main plateaus.

Figure 6

Schematic magnetic phases showing the appearance of plateaus from the competition between the frustrated exchange $J_2/J_1$ and the magnetic field $h$, as well as the spin-lattice coupling. We report separately the isotropic case (top panel, $\gamma =1)$ and a highly anisotropic case (bottom panel, $\gamma =1/8)$. $M$ is the magnetization relative to saturation; arrows indicate classical collinear order, and points in an ellipse indicate quantum singlet dimers. We do not intend to depict here the $J_2/J_1\to 0$ limit. The vertical lines correspond to magnetization curves at $J_2/J_1=0.2,0.5,\text{and}0.8$, where solid segments indicate plateau ranges ($J_1=0.5$, $J_e=0.2$, and $\alpha =\beta =0.2$).

Figure 7

Zoomed view of the $M=0$ plateau configuration in a chain of 84 sites with periodic boundary conditions ($J_1=0.5$, $J_2=0.4$, $J_e=0.2$, and $\alpha =\beta =0.2$, in the presence of an antiferroelectric dipolar background). Upper panels: local profile of $\langle S_i^z\rangle$ (blue circles) and distortions ${\delta }_i$ (orange squares), in the isotropic case (left panels, $\gamma =1$) and highly anisotropic case (right panels, $\gamma =1/8$). Distortions are scaled by a convenient factor for better visualization. Lower panels: local profile of spin correlations $\langle {\mathbf{S}}_i·{\mathbf{S}}_{i+1}\rangle$ in the isotropic and anisotropic cases. In the isotropic case, the vanishing of $\langle S_i^z\rangle$ and the enhanced alternated antiferromagnetic correlations are signals of a quantum dimer phase. In the anisotropic case, the consecutive $\langle S_i^z\rangle \approx \pm 0.5$ and the alternation of ferromagnetic and antiferromagnetic correlations $\langle {\mathbf{S}}_i·{\mathbf{S}}_{i+1}\rangle \approx \pm 0.25$ indicate a classical $\uparrow \uparrow \downarrow \downarrow$ phase. In both cases, distortions are negative (short bonds) when spin correlations are negative (antiferromagnetic).

Figure 8

Schematic picture for the quantum plateau state at $M=0$. The two-spin singlets represented by ellipses gain magnetic energy by shortening their distance, thus enlarging their exchange coupling. The influence of these distortions on the alternating dipole lengths (double arrows) produces a ferrielectric configuration with a finite bulk polarization.

Figure 9

Schematic picture for the $\uparrow \uparrow \downarrow \downarrow$ plateau state at $M=0$. The collinear spin configuration represented by black arrows gains magnetic energy by enlarging the exchange coupling of antiparallel nearest neighbors, shortening their distance. Same as in the quantum dimerized plateau, the influence of these distortions on the dipole strengths (double arrows) produces a ferrielectric configuration with a finite bulk polarization.

Figure 10

A magnetic soliton connects the two possible quantum dimer vacua. In this qualitative picture, red (cyan) circles represent odd (even) magnetic sites; squares represent the distortions of the bonds at the right of sites with the same color; dotted lines are a guide to follow odd and even site distortions; double arrows represent electric dipoles sitting amid magnetic sites in an antiferroelectric configuration. The sequence of spin singlets (ellipses, thick lines indicating enhanced NN exchange) is shifted by one lattice site across the soliton, defining a different dimerized domain. A spin $S=\frac{1}{2}$ (black arrow) indicates the fractional magnetization carried by the soliton. The sequence of short-long bonds is shifted accordingly. In the presence of the antiferroelectric dipolar background, the dimerization defines ferrielectric domains with opposite polarization.

Figure 11

Local observables in the $S_{\text{total}}^z=1$ excitation in a chain of 86 sites, in the isotropic case $\gamma =1$ (with periodic boundary conditions, $J_1=0.5$, $J_2=0.4$, $J_e=0.2$, and $\alpha =\beta =0.2$, in the presence of an antiferroelectric dipolar background). Notice that, using periodic boundary conditions, we have changed the chain length to 86 sites for commensurability of the density matrix renormalization group (DMRG) solution of the $S_{\text{total}}^z=1$ excitation with the lattice size. Upper panel: local profile of $\langle S_i^z\rangle$ (circles) and distortions ${\delta }_i$ (squares). Lower panel: local spin correlations $\langle {\mathbf{S}}_i·{\mathbf{S}}_{i+1}\rangle$ (triangles). The same colors red (cyan) in Fig. 10 are used here to visually distinguish odd (even) sites and bonds. The magnetic state develops a two-soliton profile for spin correlations, separating equal length domains. The dimerized distortions follow the same profile. The local magnetizations $\langle S_i^z\rangle$ do not vanish, being larger around the soliton regions, but show no clear order. Qualitative features agree with the cartoon in Fig. 10.

Figure 12

$S_{\text{total}}^z=1$ excitation configuration in a chain of 86 sites with periodic boundary conditions ($J_1=0.5$, $J_2=0.4$, $J_e=0.2$, and $\alpha =\beta =0.2$, in the presence of an antiferroelectric dipolar background), in the anisotropic case $\gamma =\frac{1}{8}$. Upper panels: local profile of $\langle S_i^z\rangle$ (circles) and distortions ${\delta }_i$ (squares), in the isotropic case ($\gamma =1$) and highly anisotropic case ($\gamma =1/8$). Lower panels: local profile of spin correlations $\langle {\mathbf{S}}_i·{\mathbf{S}}_{i+1}\rangle$ (triangles). Four different colors, red, cyan, green, and blue, are used to visually help the location of data every four sites. The system develops two magnetic solitons, separating equal length domains. The $\uparrow \uparrow \downarrow \downarrow$ spin pattern, as well as the short-long distortion pattern, are shifted by one lattice site across each soliton.

Figure 13

Schematic description of the first soliton in Fig. 12, connecting two different $\uparrow \uparrow \downarrow \downarrow$ dimerized domains. The zig-zag chain design (following the solid lines) is solely intended to distinguish odd sites in the lower leg and even sites in the upper leg. Nearest-neighbor exchanges $J_1$ are represented by solid rung lines and next nearest-neighbor exchanges $J_2$ by dashed straight leg lines. Single arrows represent the $\langle S_i^z\rangle$ component of spins, using the same sublattice colors as in Fig. 12, and orange double arrows represent the electric dipoles between them. Notice that the $\uparrow \uparrow \downarrow \downarrow$ spin order along the chain can be seen as Néel configurations along each leg. The magnetic soliton reverses the spins in the lower leg (indicated as a twist in the dotted lines), leaving unchanged those in the upper leg. The lattice dimerization brings closer the antiparallel nearest-neighbor spins, enlarging their exchange couplings (thick solid rungs); thus, the magnetic twist produces different dimerization domains with ferrielectric polarizations in opposite directions.

Figure 14

Polarization curves (red solid lines, scale in the right axis in units of $p_0$) in an external magnetic field for the isotropic $\gamma =1$ and the anisotropic $\gamma =\frac{1}{8}$ models ($J_1=0.5$, $J_2=0.4$, $J_e=0.2$, and $\alpha =\beta =0.2$, in the presence of an antiferroelectric dipolar background). Magnetization curves in the low $M$ region (extrapolated as blue solid lines, scale in the left axis) are also plotted for comparison. In both cases, the system supports a finite spontaneous polarization at low magnetic fields, while $S_{\text{total}}^z=0$, but a sudden drop is observed once the system exits the $M=0$ magnetic plateau. The polarization curves follow from finite sized results and infinite sized extrapolation. Insets: finite sized scaling for the polarizations obtained for $S_{\text{total}}^z=0,1,\text{and}2$ shows almost no size dependence.

Figure 15

A magnetic field step, in any orientation and strong enough to magnetize the system, produces electric depolarization. In combination with a tiny poling field signal, it can be used to reverse the spontaneous polarization $\mathbf{P}$. This could be the basis for storing information in a dipolar memory bit.

Figure 16

A small electric field provokes the displacement of the solitons in the $S_z=1$ configuration, enlarging the domain with electric polarization along the field. The solid symbols correspond to an electric field $\epsilon =0.01$, with the rest of the parameters as in Figs. 11 and 12 (repeated here in shaded symbols for comparison). The same effect is observed both in the isotropic and the anisotropic case.