Microscopic model for magnetoelectric coupling through lattice distortions

We propose a microscopic magnetoelectric model in which the coupling between spins and electric dipoles is mediated by lattice distortions. The magnetic sector is described by a spin $S=1/2$ Heisenberg model coupled directly to the lattice via a standard spin-Peierls term and indirectly to the electric dipole variables via the distortion of the surrounding electronic clouds. Electric dipoles are described by Ising variables for simplicity. We show that the effective magnetoelectric coupling which arises due to the interconnecting lattice deformations is quite efficient in one-dimensional arrays. More precisely, we show using bosonization and extensive density matrix renormalization group numerical simulations that increasing the magnetic field above the spin-Peierls gap, a massive polarization switch-off occurs due to the proliferation of soliton pairs. We also analyze the effect of an external electric field when the magnetic system is in a gapped (plateau) phase and show that the magnetization can be electrically switched between clearly distinct values. More general quasi-one-dimensional models and two-dimensional systems are also discussed.

Figure 1

Description of the pantograph mechanism and electroelastic phase diagram. (a) Typical distortion of basic structural units containing magnetic and electric variables. (b) Minimal model linking ion displacements (blue arrows) with spin and dipolar variables (black and red arrows), when a shear strain relates a distance reduction between magnetic ions with a dipolar strength enlargement. (c) The two possible dimerized chain configurations (related by one site translation, ${\mathbb{Z}}_2$), producing net polarizations in the presence of an antiferroelectric dipolar phase. (d) Dipolar phases and electroelastic distortions under an external electric field ($\alpha =0, \beta =0.2$). Following the dotted line at a given $J_e$ one finds jumps in the polarization at $E_{c1}$ and $E_{c2}$.

Figure 2

Numerical results for $E=0$ (setting $J_e=0.5, \alpha =1$, and $\beta =0.2$). Left panel: Magnetization curve (infinite size extrapolation in solid red) and net polarization (in solid blue), both relative to saturation. A finite magnetic field is necessary to overpass the spin gap, dropping off the spontaneous polarization. Right panels: local magnetization and distortion profiles for $S^z=0$ and the first two magnetized excitations $S^z=1,2$ indicating that equidistant soliton pairs (analytically fitted by dashed lines) proliferate as the magnetic field is increased.

Figure 3

Magnetization curves for $E\ne 0$ (setting $J_m=1, J_e=0.5$, and $\beta =0.2$); the insets show the infinite length extrapolation of the main plateau widths, in order to assess their presence in the thermodynamic limit. A plateau at $M=0$ is always present; when $E$ drives the dipolar system into a quadrumerized phase a second plateau opens at $M=1/2$. $h^{\pm }$, the lower and upper boundaries of the $M=1/2$ plateau, are marked for later discussion (see Fig. 6).

Figure 4

Distortion and magnetization profiles at the $M=0$ plateau for $E\ne 0$ ($J_m=1, J_e=0.5, \alpha =1$, and $\beta =0.2$, in a $N=60$ sites chain with periodic boundary conditions). The differences in local dipolar profiles are clearly seen for $E$ below or above $E_{c1}$ [panels (a) and (c)]. The soliton-antisoliton pair for the first magnetic excitation is glued together by the electric field [panels (b) and (d)]; blue and green curves in panel (b) describe the repulsive soliton-antisoliton pair for $E=0$, added for comparison.

Figure 5

Quantum state signatures of the $M=1/2$ plateau (parameters as in Fig. 4, $E=0.45$; we show again results for 60 sites, zoomed to visualize clearly the ordered direct product of singlets and spin-up sites). Local magnetization and nearest-neighbor correlations are compatible with a factorized quantum state of alternating up-up and singlet states. This is depicted graphically by black, red, and blue bonds indicating, respectively, singlet, weak antiferromagnetic, and ferromagnetic correlations. The lowest-energy spin excitation is a localized triplet; the green line in panel (b) indicates the local increase in $S^z$.

Figure 6

Magnetoelectric response to the electric field in the quadrumerized scenario (schematic). Under appropriate applied magnetic fields $h^{\pm }$, as the electric field produces a polarization jump [see Fig. 1 at $J_e=0.5]$ the magnetization switches from $M^{\pm }$ to $M=1/2$, the value at the plateau (see Fig. 3).