Double frustration and magnetoelectroelastic excitations in collinear multiferroic materials

We discuss a model scenario for multiferroic systems of type II (collinear spins) where the electric dipolar order competes with a frustrated magnetic order in determining the elastic distortions of the lattice ion positions. High magnetic frustration due to second-neighbors exchange and small spin easy-axis anisotropy lead to the appearance of the so-called quantum magnetic plateau states. Increasing the magnetic field above the plateau border produces composite excitations, where fractionalized spin tertions arise together with spontaneous dipolar flips (in the form of domain walls) and enhanced localized elastic distortions. This peculiar magnetoelectric effect may be described by magneto-electric-elastic quasiparticles that could be detected by x-ray and neutron diffraction techniques. Our results are supported by extensive DMRG computations on the spin sector and self-consistent equations for the lattice distortions.

Figure 1

Schematic picture for the pantograph mechanism coupling electric dipoles to the lattice. Black dots represent magnetic sites and blue spheres represent a charge distribution, giving rise to dipolar moments. Green double arrows represent these dipolar moments that might point up or down. Displacements of magnetic sites, indicated by blue arrows, produce lattice bond length distortions ${\delta }_i$ that modify the strength of local dipoles.

Figure 2

Qualitative picture of magnetoelastic distortions at the classical $\uparrow \uparrow \downarrow$ and the quantum $•-•\uparrow$ magnetic configurations discussed in Ref. [19] [(a) and (b), respectively], and a picture of the electroelastic distortions at the dipolar $\Uparrow \Uparrow \Downarrow$ configuration (c) found in Ref. [15]. Spins are represented by single arrows in tones of red, dipoles by double arrows in tones of green. Three color tones (light, medium, dark) are used to facilitate the recognition of variables every three sites, in relation to data given below. Ionic and dipole displacements are indicated by blue arrows (nondistorted positions indicated with gray faded symbols). The size change of dipoles in (c) is demonstrative of the pantograph mechanism effect. In (a), the bond length follows a long-short-short pattern (L-S-S in the figure), providing a magnetic energy gain by getting closer (farther) antiparallel (parallel) spins. In (b), the magnetic energy gain is obtained by tightening singlet bonds following a short-long-long pattern (S-L-L in the figure). In (c), a dipolar energy gain stems from getting closer (farther) antiparallel (parallel) dipoles, with the same S-L-L bond distortions as in (b). Electroelastic distortions in (c) are compatible with the quantum magnetoelastic distortions in (b) but not with the classical ones in (a).

Figure 3

Schematic phase diagram in the frustration ratio ($J_2/J_1$) and the easy-axis anisotropy ($\mathrm{\Delta }$) plane. The colored regions indicate the parameter regimes where $M=1/3$ plateaus are observed in magnetization curves. The robust magnetic order giving rise to the plateau is mostly a collinear $\uparrow \uparrow \downarrow$ classical structure (yellow region) but turns into a quantum $•-•\uparrow$ state (orange region) when high frustration and low anisotropy enhance the quantum fluctuations. The circles mark the four points detailed in this paper (unnumbered ones are described in Supplemental Material [30]).

Figure 4

(a), (d) Magnetization curves $M$ versus. $h_z=g{\mu }_B\left|\mathit{B}\right|$ ($g$ being the gyromagnetic ratio of the electron in the specific material and ${\mu }_B$ the Bohr magneton) showing the existence of the $M=1/3$ plateau in cases 1(a) and 2(b) discussed in detail in this paper. The widths $W_p$ of the main magnetization plateaus are computed for several finite chain lengths and extrapolated to infinite size as shown in the insets. The smooth lines suggest the magnetization curves in the thermodynamic limit. For case 1, the plateau is narrower than case 2, due to the enhancement of quantum fluctuations favored by isotropic interactions and high magnetic frustration.(b), (c) Local profile of site and bond observables in the $M=1/3$ state for $\mathrm{\Delta }=1, J_2/J_1=0.5$ (case 1), obtained with the self-consistent numerical computations; a section of six sites (indicated by vertical lines) is taken from the $L=174$ chain with periodic boundary conditions. In the upper plot, different light-medium-dark tones are used to emphasize the period three structure of the spin expectation values $\langle S_i^z\rangle$ (in units of $\hslash$, in red circles), the bond distortions ${\delta }_i$ (in units of the lattice constant $a$, in blue squares drawn between sites) and dipolar moments $p_i$ (divided by $2p_0$, in green thin-diamonds between sites). The lower plot shows the nearest-neighbor spin correlations $\langle {\mathit{S}}_i·{\mathit{S}}_{i+1}\rangle$ in red diamonds; a detail of longitudinal $\langle S_i^z{S}_{i+1}^z\rangle$ and transverse $\langle S_i^{+}{S}_{i+1}^{-}\rangle =\langle S_i^x{S}_{i+1}^x\rangle +\langle S_i^y{S}_{i+1}^y\rangle$ correlations is given in triangles. Data shows the formation of local quantum singlets alternating with partially decoupled spins up every three sites. The cartoon picture in (c) qualitatively collects these numerical results and shows the compatibility of the short-long-long (S-L-L) quantum magnetoelastic and electroelastic patterns in Fig. 2. (e), (f) The $M=1/3$ ground state for $\mathrm{\Delta }=4, J_2/J_1=0.8$ (case 2). Symbols and colors follow the conventions in (b). Spin data shows the classical $\uparrow \uparrow \downarrow$ plateau features, with $\langle S_i^z\rangle$ not reaching $\pm 0.5$ because of small quantum fluctuations. Although the spin structure is classical, the elastic distortions do not fit the classical magneto-elastic pattern in Fig. 2 but the long-null-short (L-0-S) sequence depicted qualitatively in (f). The interaction between dipoles and spins, mediated by distortions, frustrates the magnetoelastic order producing the breaking of inversion symmetry.

Figure 5

(See online version to appreciate details.) Magneto-electro-elastic excitations (MEE) with $\mathrm{\Delta }{S}^z=1$ induced by the magnetic field on top of the $M=1/3$ plateau for $J_2/J_1=0.5, \mathrm{\Delta }=1$ (case 1). In the top panel, we show numerical results for the local observables (upper plot) and spin correlations (lower plot). Symbols and colors follow the conventions in Fig. 4. The magnetic sector shows regions which adopt the $•-•\uparrow$ quantum plateau order (two of them highlighted as $I$ and $II$), interpolated by soliton structures with a classical $\downarrow \uparrow \uparrow \downarrow$ center. The position of the $•-•\uparrow$ pattern is shifted to a different sublattice across each soliton. Dipolar domain walls $\Downarrow \Uparrow \Uparrow \Uparrow \Downarrow$ are formed at the core of the magnetic solitons, shifting the $\Uparrow \Uparrow \Downarrow$ order from one to another sublattice to accompany the magnetic configuration; these dipolar domain walls carry fractionalized spontaneous electric excitations. The lattice distortions also accompany the magnetic configuration but show singular localized excitations at each dipolar domain wall. Each region between $•-•\uparrow$ quantum plateau orders thus allocates a coupled magneto-electro-elastic excitation (the one between regions $I$ and $II$ is highlighted as MEE in the plots). A cartoon picture of these results is shown in Fig. 6.

Figure 6

Qualitative picture of the magneto-electro-elastic (MEE) excitation separating the quantum plateau regions $I$ and$\mathit{II}$ in Fig. 5. The excitation has a classical magnetic $\downarrow \uparrow \uparrow \downarrow$ core coinciding with the $\Downarrow \Uparrow \Uparrow \Uparrow \Downarrow$ dipolar domain wall and a highly enlarged bond. Local observables and color codes follow actual data in the upper plot in Fig. 5. The dotted line is a mirror plane showing the parity symmetry of the MEE excitation. The local order in quantum region $\mathit{II}$ is shifted by one site with respect to the order in quantum region $I$.

Figure 7

(See online version to appreciate details.) Decoupled electroelastic and magnetic and excitations with $\mathrm{\Delta }{S}^z=1$ induced by the magnetic field on top of the $M=1/3$ plateau for $J_2/J_1=0.8, \mathrm{\Delta }=4$ (case 2). The plot in the upper panel shows the local observables, that in the lower panel shows spin correlations. Magnetic solitons (one of them labeled as MS in the figure) having a singlet core interpolate between different classically $\uparrow \uparrow \downarrow$ ordered regions related by one-site translations (see, for instance, regions $I$ and $\mathit{II}$ in the figure). Dipolar excitations in the form of domain walls $\Downarrow \Uparrow \Uparrow \Uparrow \Downarrow$ (DW), accompanied by localized elastic excitations, are spatially decoupled from the magnetic solitons. They separate different mirror-related classically $\uparrow \uparrow \downarrow$ ordered regions (see, for instance, $\mathit{II}$ and III). All of the six degenerate magneto-electro-elastic classical plateau configurations show up in different regions of the system. A cartoon picture of these results is shown in Fig. 8.

Figure 8

Qualitative picture of the magnetoelastic and electroelastic excitations in Fig. 7. The cartoon in the upper panel describes the magnetoelastic solitonic excitation MS with a quantum singlet core interpolating the classical orders highlighted as $I$ and $\mathit{II}$ in Fig. 5; in passing the soliton, the L-0-S distortion pattern changes to S-0-L and the $\uparrow \uparrow \downarrow$ magnetic pattern is shifted by one site while the dipolar $\Uparrow \Uparrow \Downarrow$ pattern remains unaltered. In the lower panel, the electroelastic domain wall DWseparates the classical orders highlighted as $\mathit{II}$ and III; the S-0-L distortion pattern changes again to L-0-S and the dipolar $\Uparrow \Uparrow \Downarrow$ pattern is shifted by one site but the magnetic $\uparrow \uparrow \downarrow$ order remains the same at both sides of the domain wall. Dotted lines are added to make apparent the mirror symmetry of the excitation configurations.