# Double frustration and magneto-electro-elastic excitations in collinear
multiferroic materials

D. C. Cabra  IFLySiB-CONICET-UNLP and Departamento de Física, Universidad
Nacional de La Plata, 1900 La Plata, Argentina  A. O. Dobry  IFIR-CONICET-UNR
and Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Universidad
Nacional de Rosario, 2000 Rosario, Argentina  C. J. Gazza  IFIR-CONICET-UNR
and Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Universidad
Nacional de Rosario, 2000 Rosario, Argentina  G. L. Rossini  IFLP-CONICET-UNLP
and Departamento de Física, Universidad Nacional de La Plata, 1900 La Plata,
Argentina

###### Abstract

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 magneto-electric effect
may be described by magneto-electric-elastic (MEE) 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.

##  I Introduction

Multiferroic materials, in which magnetic and dipolar order coexist and
interact, are currently under intense investigation. Elastic distortions of
ionic positions, in the form of structural transitions, are deeply connected
with the onset of magnetic and/or ferroelectric order. Some of these materials
do not show signs of spin-orbit correlations, instead they exhibit collinear
spin order and electric polarization below a common transition temperature;
they are called improper type-II collinear multiferroics [2007-Cheong, ;
2008-vdBrink-Khomskii, ; 2009-Khomskii, ]. Several materials with such
transitions are described in [key-3, ; key-4, ; key-5, ; key-6, ; key-7, ;
key-8, ; key-9, ; key-10, ; key-11, ; key-12, ; key-13, ].

We have recently proposed a microscopic model in [2021-pantograph-II, ] (cited
as Ref. I in the following) describing the interaction between spins and
electric dipoles in a quasi one-dimensional multiferroic system via elastic
distortions. A key ingredient of this model, shown in Figure 1, is the
variation of dipolar moments according to the distance between neighboring
spin sites; as dipoles are shortened when spin separation is increased, the
model reminds a pantograph mechanism. Thus the dipole-dipole interaction
produces a back reaction of the dipolar degrees of freedom on the elastic
distortions, and consequently in the magnetic sector. A salient feature is the
observation of an ordered dipolar phase with period three, which shows up in
the presence of an appropriate homogeneous external electric field (see Figure
3 in Ref. I) because of the long range character of dipolar interactions (c.f.
[2019-pantograph-I, ] where only nearest neighbors dipolar interactions are
discussed). Some properties of this dipolar phase, denoted as
$\Uparrow\Uparrow\Downarrow$ in the following [footnote-1, ], motivate the
present work.

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.

We investigate here the commensurability interplay of the $P=1/3$ period three
dipolar order $\Uparrow\Uparrow\Downarrow$ (see Figure 2-C below) with the
period three magnetic configurations observed in many frustrated magnetic
materials with $M=1/3$ magnetization plateaus ($M$ expressed as a fraction of
saturation). In most studies the $M=1/3$ plateau state is found to form a
collinear $\uparrow\uparrow\downarrow$ (see Fig 2-B) classical pattern
[2003-Okunishi-a, ], but a quantum $\bullet\\!\\!-\\!\\!\bullet\uparrow$ order
(where $\bullet\\!\\!-\\!\\!\bullet$ stands for a spin singlet, see Figure
2-A) has been also predicted for spin $S=1/2$ modulated isotropic Heisenberg
chains [2005-Hida-Affleck, ]. The robustness of magnetic plateau states, given
by an energy gap in the magnetization spectrum, makes them good candidates for
technological applications. Relatedly, the characteristics of the elementary
$\Delta S^{z}=1$ excitations at the high field border of the plateaus deserve
detailed analysis.

We find that the dipolar order introduced by the long range dipolar
interactions indeed competes with the magnetic order set by the magnetic
frustration at the M=1/3 plateau, opening a non trivial scenario which we dub
_double frustration_. Our analysis predicts unusual effects due to this
scenario. For low anisotropy and high magnetic frustration, favoring quantum
fluctuations, the double frustration stabilizes a quantum structure at the
$M=1/3$ plateau state. In contrast, either for higher easy-axis anisotropy or
for lower magnetic frustration, or both, the double frustration competition
leads to the spontaneous parity symmetry breaking of the classical collinear
$M=1/3$ plateau state. We must stress that, in a more general situation with
charge order along the chain, parity breaking would imply the appearance of
longitudinal electric polarization [2008-vdBrink-Khomskii, ]. These results
are clear signals of the role of dipolar interactions and may guide the search
for materials realizing strong magneto-electric effects.

After analyzing the plateau states, we discuss the excitations caused by the
increase of the magnetic field. We show that the $\Delta S^{z}=1$ magnon on
top of the $M=1/3$ state fractionalizes into three $S^{z}=1/3$ spatially
separated solitons, where elastic distortions adapt to the magnetic order.
This change in the distortion pattern induces, in the dipolar sector, a
spontaneous unit polarization change which in turn fractionalizes into three
sharp domain walls. This emergent magneto-electric effect, that is the
polarization change induced by a magnetic field mediated by elastic
distortions, is one of the main results in the present paper.

Both the nature of the plateau state structure and the appearance of
intertwined magnetic and electric fractional excitations, mediated by the
lattice, are experimentally accessible by neutron scattering for the spin-
channel and by X-ray scattering for the lattice distortions.

The paper is organized as follows: in Section II we describe the microscopic
degrees of freedom, the Hamiltonian and the parameters that define our model
scenario. We also discuss the computational method. In Section III we describe
the double frustration effect, that is a competence of the elastic order
convenient to magnetic frustration with the elastic order driven by dipolar
interactions, stressing the differences between classical and quantum regimes.
We devote Section IV to describe the effects of increasing the magnetic field,
which leads to fractional magnetic and dipolar excitations as well as
localized elastic domain walls with clear different behavior with respect to
the classical or quantum plateau state structure. Conclusions are presented in
Section V.

##  II The model scenario

We briefly summarize here the model proposed in Ref. I. It describes spin
$S=1/2$ magnetic atoms $\boldsymbol{S}_{i}$ in a linear chain with elastic
degrees of freedom $\delta_{i}$ describing the variation of the bond length
between neighboring magnetic atoms $\boldsymbol{S}_{i}$ and
$\boldsymbol{S}_{i+1}$, from a regular lattice constant length $a$ to
distorted lengths $a+\delta_{i}$. Electric dipolar moments $p_{i}$, normal to
the chain direction, are located amid magnetic atoms $\boldsymbol{S}_{i}$ and
$\boldsymbol{S}_{i+1}$; when the magnetic lattice is distorted the distance
between dipoles $p_{i}$ and $p_{i+1}$ also changes from $a$ to $a+\eta_{i}$,
with $\eta_{i}=\left(\delta_{i}+\delta_{i+1}\right)/2$. A fixed chain length
condition is assumed, imposing a null constraint on the sum of local
distortions.

In order to describe the several materials mentioned in the Introduction, the
model includes easy-axis anisotropic antiferromagnetic interactions $J_{1}$
and $J_{2}$ between nearest (NN) and next-to-nearest (NNN) neighbors
respectively, which produce the magnetic frustration. Electric dipoles
interact via long range Coulomb interactions. Both the magnetic and dipolar
sectors are coupled to the lattice in the most natural way. The spin sector
does it by a standard, spin-Peierls type, distance dependence of the NN
exchange coupling linearly expanded as
$J_{1}(i,i+1)=J_{1}\left(1-\alpha\delta_{i}\right)$. On the other hand, the
dipolar sector couples to elastic distortions through the $1/r^{3}$ distance
dependence of long range dipole-dipole interactions and by the geometric
mechanism mentioned in the Introduction, affecting the charge distribution in-
between magnetic atoms as they are displaced. This so called pantograph effect
is modeled by a linear expansion of the dipole moments
$p_{i}=p_{0}\left(1-\beta\delta_{i}\right)\sigma_{i}$ where $p_{0}$ is the
undistorted dipole magnitude, $\beta$ measures the pantograph electro-elastic
coupling and $\sigma_{i}=\pm 1$ is the Ising variable describing dipole
orientations. The elastic distortions have an energy cost given by a stiffness
constant $K$, providing an indirect magneto-electric coupling. More details on
the model can be found in Ref. I.

The complete Hamiltonian reads

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{i}\left[J_{1}\left(1-\alpha\delta_{i}\right)\left(\mathbf{S}_{i}\cdot\mathbf{S}_{i+1}\right)_{\Delta}+J_{2}\left(\mathbf{S}_{i}\cdot\mathbf{S}_{i+2}\right)_{\Delta}\right]$
(1) $\displaystyle+$
$\displaystyle\frac{K}{2}\sum_{i}\delta_{i}^{2}+\lambda_{dip}\sum_{i<j}\frac{1}{r_{ij}^{3}}p_{i}p_{j}$

where $\left(\mathbf{S}_{i}\cdot\mathbf{S}_{j}\right)_{\Delta}$ stands for the
anisotropic spin-spin product, $\lambda_{dip}>0$ is the Coulomb coupling
constant and $r_{ij}$ is the distance between dipoles $p_{i}$ and $p_{j}$,
which depends on distortions. With the previous notation, expanding the
dipolar $1/r^{3}$ dependence up to linear order in distortions and truncating
up to second neighbours (see Ref. I), the explicit model Hamiltonian is given
by

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{J_{1}}{\Delta}\left(1-\alpha\delta_{i}\right)\left(S_{i}^{x}S_{i+1}^{x}+S_{i}^{y}S_{i+1}^{y}+\Delta
S_{i}^{z}S_{i+1}^{z}\right)+\sum_{i}\frac{J_{2}}{\Delta}\left(S_{i}^{x}S_{i+2}^{x}+S_{i}^{y}S_{i+2}^{y}+\Delta
S_{i}^{z}S_{i+2}^{z}\right)$ (2) $\displaystyle+$
$\displaystyle\frac{K}{2}\sum_{i}\delta_{i}^{2}$ $\displaystyle+$
$\displaystyle
J_{e}\sum_{i}\left(\sigma_{i}\sigma_{i+1}+\frac{1}{8}\sigma_{i}\sigma_{i+2}\right)$
$\displaystyle-$ $\displaystyle
J_{e}\sum_{i}\delta_{i}\left[\left(\beta+\frac{3}{2a}\right)\left(\sigma_{i-1}\sigma_{i}+\sigma_{i}\sigma_{i+1}\right)+\frac{1}{8}\left(\beta+\frac{3}{4a}\right)\left(\sigma_{i-2}\sigma_{i}+\sigma_{i}\sigma_{i+2}\right)+\frac{3}{16a}\sigma_{i-1}\sigma_{i+1}\right]$

where $\Delta\geq 1$ is the easy-axis anisotropy and
$J_{e}=\lambda_{dip}p_{0}^{2}/a^{3}$ is the effective dipolar interaction
coupling, ($J_{1}$, $J_{2}$ and $J_{e}$ in energy units). Magnetic couplings
$J_{1}$ and $J_{2}$ are divided by $\Delta$ in order to reach the Ising regime
in the highly anisotropic $\Delta\to\infty$ limit.

Following Ref. I we are interested on a parameter region where the magnetic
and dipolar couplings are of the same order of magnitude, so both the spin and
dipole configurations are relevant to determine the ground state of the
system. Also the magneto-elastic coupling $\alpha$ and the electro-elastic
coupling $\beta$ are similar, in order to provide an efficient elastically
mediated magneto-electric interaction. We then avoid the multiplicity of
parameters in the Hamiltonian (II) by taking $Ka^{2}$ as the energy unit and
fixing $J_{1}$, $J_{e}$, $\alpha$ and $\beta$ at convenient values detailed
below. Only $J_{2}$ and $\Delta$ will be varied to explore the incidence of
magnetic frustration and easy-axis anisotropy (measured as $\gamma=1/\Delta$
in Ref. I) in the ground state properties of the system. Different values of
$J_{1}$, $J_{e}$, and $\beta$ can be studied similarly.

External electric and magnetic fields $\boldsymbol{E}$ and $\boldsymbol{B}$
will be adjusted to drive the system to the peculiar double frustration
scenario we discuss in the present paper. This is the region where the
electric field polarizes the otherwise antiferroelectric dipolar sector
(driven by $J_{e}$) up to $1/3$ of saturation, provoking the period three
$\Uparrow\Uparrow\Downarrow$ dipolar pattern (see Figure 3 in Ref. I) and the
magnetic field sets the spin degrees of freedom in the $M=1/3$ plateau region
(see Figure 6 in Ref. I). For a magneto-elastic, spin-Peierls, chain (not
coupled to electric dipoles), this plateau is known to appear together with an
energetically favorable period three elastic distortion [2006-Vekua-etal, ;
2007-Gazza-etal, ; 2007-Rosales-et-me, ]. On the other hand, for the electro-
elastic chain obtained from the Hamiltonian (II) when the spin sector is
decoupled ($\alpha=0$), the $\Uparrow\Uparrow\Downarrow$ dipolar pattern also
comes along with period three elastic distortions (as discussed in Ref. I)
bringing closer (farther) antiparallel (parallel) dipoles. This situation
might be reversed, for instance in the presence of itinerant electrons, since
they may induce RRKY-like interactions between dipoles leading to
ferroelectric effective couplings (see e.g. [2019-Jin, ]).

The question arises whether the elastic distortions compete or collaborate in
lowering the ground state energy of the magneto-electro-elastic multiferroic
system. We show below that they do compete, with profound consequences both in
the magnetic plateau configuration and the magnetic excitations at the high
field border of the plateau.

Our results are based on extensive numerical computations following an
iterative self-consistent method [1997-Feiguin, ] where the spin sector is
solved exactly by Density Matrix Renormalization Group (DMRG) techniques
[1992-White, ]. At each iteration, for a given configuration of dipoles
$\\{\sigma_{i}\\}$ and a quantum state for the spins
$\\{\boldsymbol{S}_{i}\\}$, the lattice distortions $\delta_{i}$ are obtained
by minimizing the elastic energy under a fixed chain length condition.
Unconstrained distortions $\delta_{i}^{\text{free}}$ are computed through the
local zero gradient conditions

$\displaystyle K\delta_{i}^{\text{free}}$ $\displaystyle=$
$\displaystyle\alpha
J_{1}/\Delta\langle\left(S_{i}^{x}S_{i+1}^{x}+S_{i}^{y}S_{i+1}^{y}+\Delta
S_{i}^{z}S_{i+1}^{z}\right)\rangle$ (4) $\displaystyle+$ $\displaystyle
J_{e}\left(\beta+\frac{3}{2a}\right)\left(\sigma_{i-1}\sigma_{i}+\sigma_{i}\sigma_{i+1}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{8}J_{e}\left(\beta+\frac{3}{4a}\right)\left(\sigma_{i-2}\sigma_{i}+\sigma_{i}\sigma_{i+2}\right)$
$\displaystyle+$ $\displaystyle
J_{e}\frac{3}{16a}\sigma_{i-1}\sigma_{i+1}-\beta\varepsilon\sigma_{i},$

where $\epsilon=2p_{0}E$ is the normalized electric field, while the
constraint is imposed as

$\delta_{i}=\delta_{i}^{\text{free}}-\overline{\delta^{\text{free}}}.$ (5)

where the bar stands for average value along the chain. Interestingly, the
self-consistent conditions in Eqs. (4, 5) also allow for a qualitative
analysis of the influence of spin-spin and dipole-dipole correlations on the
elastic distortions.

##  III Double frustration effect

###  III.1 Qualitative description

The elastic distortions associated with the $M=1/3$ magnetic plateau
configuration, and those associated with the $\Uparrow\Uparrow\Downarrow$
dipolar pattern, can be qualitatively described considering the nearest
neighbor (NN) interactions in Eq. (4,5). We then provide the numerical
evidence for the outcoming picture in the following subsection.

$M=1/3$ magnetic plateaus come in two flavors, dubbed classical and quantum
[2005-Hida-Affleck, ]. In the so called classical plateau spin components
parallel to the magnetic field have non vanishing $\langle S_{i}^{z}\rangle$
expectation value in an ordered pattern with two positive, one negative terms
that we represent by $\uparrow\uparrow\downarrow$. These expectation values
are reduced by quantum fluctuations in the isotropic $\Delta=1$ case, but
approach $\pm 0.5$ in the highly easy-axis anisotropic case $\Delta\gg 1$.
Spin-spin correlations
$\langle\boldsymbol{S}_{i}\cdot\boldsymbol{S}_{i+1}\rangle$ are positive
between ferromagnetic (parallel) neighbors $\uparrow\uparrow$ and negative
between antiferromagnetic (antiparallel) neighbors $\uparrow\downarrow$ and
$\downarrow\uparrow$, approaching the Ising correlations $\pm 0.25$ for
$\Delta\gg 1$. From Eq. (4), the correlation
$\langle\boldsymbol{S}_{i}\cdot\boldsymbol{S}_{i+1}\rangle$ affects the bond
distortion $\delta_{i}$; the $\uparrow\uparrow\downarrow$ spin configuration
favors distorted long bonds between ferromagnetic neighbors and short bonds
between antiferromagnetic neighbors, that is a "long-short-short" (L-S-S)
distortion pattern (see Figures 1 and 2-A). Notice that the antiferromagnetic
coupling $J_{1}\left(1-\alpha\delta_{i}\right)$ gets stronger for "satisfied"
antiferromagnetically aligned neighbors and weaker for "frustrated"
ferromagnetically aligned neighbors.

In contrast, in the so called quantum plateau two neighboring spins (out of
three) tend to form singlets while the third one points up, in a configuration
that we represent by $\bullet\\!\\!-\\!\\!\bullet\uparrow$ (see Figure 2-B).
In an ideal case the spins forming a quantum singlet would have $\langle
S_{i}^{z}\rangle=0$ and the third one $\langle S_{i}^{z}\rangle=0.5$, with
singlet correlation
$\langle\boldsymbol{S}_{i}\cdot\boldsymbol{S}_{i+1}\rangle=-0.75$ and
vanishing correlation between the spin up and its neighbors; the real
situation may be characterized as a quantum plateau when the spin expectation
and spin-spin correlation values show a tendency to such pattern. Again from
Eq. (4) one can see that a very negative singlet-like correlation strongly
favors a short bond at the expense of long bonds (according to Eq. (5)) where
spin correlations are close to zero, giving rise to a "short-long-long"
(S-L-L) distortion pattern. Notice that the singlets are more likely to appear
in the isotropic case $\Delta=1$, while the easy-axis anisotropy $\Delta>1$
diminishes transverse correlations and favors the classical configuration.

In turn, the NN dipolar correlations are related to lattice distortions
through the second line of Eq. (4): bond distortion $\delta_{i}$ is influenced
by the correlations of the dipole $\sigma_{i}$ located at the bond $i$ with NN
dipoles at both sides. The $\Uparrow\Uparrow\Downarrow$ configuration then
favors short bonds where the dipole $\Downarrow$ is located, at the expense of
generating long bonds where the dipoles point $\Uparrow$ to fulfill the
constraint in Eq. (5), preferring to induce a S-L-L distortion pattern (see
Figure 2-C). Recalling that dipoles remain always midway between adjacent
magnetic atoms, in terms of dipole positions these magnetic lattice
distortions make antiparallel dipoles get closer, and parallel dipoles get
further away.

Figure 2: Qualitative picture of magneto-elastic distortions at the classical
$\uparrow\uparrow\downarrow$ and the quantum
$\bullet\\!\\!-\\!\\!\bullet\uparrow$ magnetic configurations discussed in
[2005-Hida-Affleck, ] (A and B respectively), and a picture of the electro-
elastic distortions at the dipolar $\Uparrow\Uparrow\Downarrow$ configuration
(C) found in Ref. I. 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 (non distorted positions indicated with gray faded symbols).
The size change of dipoles in C is demonstrative of the pantograph mechanism
effect. In panel 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 panel B the magnetic energy gain is obtained
by tightening singlet bonds following a "short-long-long" pattern (S-L-L in
the figure). In panel C a dipolar energy gain stems from getting closer
(farther) antiparallel (parallel) dipoles, with the same S-L-L bond
distortions as in panel B. Electro-elastic distortions in C are compatible
with the quantum magneto-elastic distortions in B, but not with the classical
ones in A.

From this qualitative discussion, the electro-elastic dipolar configuration
$\Uparrow\Uparrow\Downarrow$ found in Ref. I is compatible with the quantum
magnetic plateau configuration but competes with the classical plateau
configuration, which is usually the one observed in homogeneous $J_{1}-J_{2}$
magnetically frustrated spin chains in a wide variety of regimes (isotropic
with [2006-Vekua-etal, ; 2007-Gazza-etal, ] and without [2003-Okunishi-a, ]
elastic coupling, anisotropic [2003-Okunishi-b, ]). Then, the coupling to
dipolar degrees of freedom through lattice distortions introduces a second
frustration mechanism. Our numerical analysis below provides clear surprising
effects due to this double frustration scenario. In a regime of low anisotropy
and high magnetic frustration, favoring quantum fluctuations, this second
frustration is responsible for the stabilization of a quantum $M=1/3$ plateau
state. In contrast, for higher easy-axis anisotropy and/or lower magnetic
frustration, the second frustration competition leads to a spontaneous parity
symmetry breaking in the classical $M=1/3$ plateau state.

###  III.2 Numerical DMRG analysis

We have performed an extensive numerical computation of the ground state of
the model in Eq. (II), in the presence of magnetic and electric fields driving
the system to magnetization $M=1/3$ and polarization $P=1/3$. In order to
evaluate the role of magnetic frustration and easy-axis anisotropy we explored
the $J_{2}/J_{1}-\Delta$ plane, fixing the remaining parameters at
$J_{1}=0.5Ka^{2}$, $J_{e}=0.2Ka^{2}$ and $\alpha=\beta=0.2$ with $K=a=1$;
correspondingly the electric field is taken as $\epsilon=0.16$ (see Figure 3
in Ref. I).

The ground state is found through an iterative numerical analysis based on
DMRG to solve the magnetic sector in the adiabatic Eq. (4), along the lines
stated in [1997-Feiguin, ] and implemented in a similar context in
[2019-pantograph-I, ; 2021-pantograph-II, ]. At each point chosen in the
$J_{2}/J_{1}-\Delta$ plane the ground state of the system is found as follows:
starting from the $\delta_{i}$ and $\sigma_{i}$ configuration that solves the
electro-elastic part of the Hamiltonian, the quantum ground state of the spin
system is obtained by the DMRG algorithm. Therefore, we re-obtain the set of
$\delta_{i}$ from Eqs. (4,5) and prove different $\sigma_{i}$ to minimize the
total energy, until convergence. We use periodic boundary conditions, and we
have kept the truncation error less than $O(10^{-12})$, during up to more than
100 sweeps in the worst cases. This assures that errors of the DMRG
computation are smaller than symbol sizes in each figure. The DMRG
computations were implemented using the ITensor software library [ITensor, ].

We have covered a wide region of the $J_{2}/J_{1}-\Delta$ plane. From this
exploration, we found distinct regimes that we describe below. We paid
attention to the isotropic case $\Delta=1$, mainly for theoretical reasons,
and to high values of $\Delta$ where one expects a classical behavior which
may be in closer relation to real materials. Regarding the frustration ratio
$J_{2}/J_{1}$, we distinguish moderate and highly frustrated values (see
Figure 6 in Ref. I). Representative selected points are:

  * •

$\Delta=1$, $J_{2}/J_{1}=0.5$. Due to the isotropic Heisenberg interaction and
the high magnetic frustration ($J_{2}/J_{1}=0.5$ is the maximally frustrated
point in the case of Ising interactions) quantum fluctuations are enhanced at
this point.

  * •

$\Delta=4$, $J_{2}/J_{1}=0.8$. Easy-axis anisotropy and low magnetic
frustration inhibit quantum fluctuations, favoring classical behavior.

  * •

$\Delta=1$, $J_{2}/J_{1}=0.8$. Selected as a point with isotropic Heisenberg
interaction and low frustration.

  * •

$\Delta=4$, $J_{2}/J_{1}=0.5$. Selected as a high magnetic frustration point,
with weaker transverse spin interactions softening quantum fluctuations.

We found important qualitative differences between the first case (Case 1 in
the following). and the other three. For this reason we provide details on
that and the second one (Case 2 in the following) and defer the others for
Supplemental Material [SM, ]. Our complete analysis leads to the schematic
phase diagram shown in Figure 3.

Figure 3: Schematic phase diagram in the frustration ratio ($J_{2}/J_{1}$) and
the easy-axis anisotropy ($\Delta$) 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 $\bullet\\!\\!-\\!\\!\bullet\uparrow$ state (orange region)
when high frustration and low anisotropy enhance the quantum fluctuations. The
circles mark the four points detailed in this work (unnumbered ones in
Supplemental Material [SM, ]).

We have checked at each of these points that the magnetization curves indeed
show plateaus at $M=1/3$, with different widths. For Cases 1 and 2 these are
shown in Figure 4.

Figure 4:  Magnetization curves showing the existence of the $M=1/3$ plateau
in Case 1 (upper plot) and Case 2 (lower plot) discussed in detail in this
work. The widths 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 the Case 1 the plateau is narrower than the Case 2, due to the enhancement
of quantum fluctuations favored by isotropic interactions and high magnetic
frustration.

Case 1 ($\Delta=1$, $J_{2}/J_{1}=0.5$): we have found that the _double
frustration_ caused by the dipolar degrees of freedom is able to radically
change the otherwise classical magnetic plateau structure to a quantum one.
This is a manifestation of a _strong magneto-electric effect._ The local
profile of the relevant variables has been computed in a chain of $N_{s}=174$
sites, with periodic boundary conditions and a magnetic field $h_{z}=0.6$
setting $M=1/3$ (meaning
$S_{total}^{z}=\left(N_{s}\cdot\frac{1}{2}\right)/3$).

The local results show a repeated structure every three sites, as expected; a
detail of a portion of the chain is shown in the upper panel of Figure 5. We
have used three color tones (light-medium-dark) to identify the corresponding
period three sub-lattices. We have also drawn vertical lines in these plots to
indicate the magnetic sites, drawing site variables ($\langle
S_{i}^{z}\rangle$) markers upon these lines and bond variables ($\delta_{i}$,
$p_{i}$ and spin-spin correlations) markers between them.

One can see in the sequence of $\langle S_{i}^{z}\rangle$ (red circles in the
upper plot, with light-medium-dark tones every three sites) a repetition of
one spin up ($\langle S^{z}\rangle\approx 0.4$, in medium red) followed by two
spins with almost vanishing expectation value ($\langle S^{z}\rangle\approx
0$, in dark and light red). The nearest neighbors spin correlations (red
diamonds in the lower plot, with corresponding light-medium-dark tones every
three bonds) take a very negative value (below $-0.6$) every three bonds,
indicating the tendency to form two sites local quantum singlets just between
sites with almost vanishing $\langle S^{z}\rangle$, with low antiferromagnetic
correlations between them and sites with spin up. The longitudinal and
transverse correlations are shown with up-triangles and horizontal-triangles
for more detail: typical singlet correlations (in dark red) get equal
contributions from each spin component $\langle
S_{i}^{x}S_{i+1}^{x}\rangle=\langle S_{i}^{y}S_{i+1}^{y}\rangle=\langle
S_{i}^{z}S_{i+1}^{z}\rangle\approx-0.2$, while the other bonds show almost
uncorrelated $z$-components $\langle S_{i}^{z}S_{i+1}^{z}\rangle\approx 0$.
The elastic distortions (blue squares in the upper plot, also with light-
medium-dark tones) are negative in the dark bonds and positive in the rest
forming a "short-long-long" (S-L-L) bond distortion pattern. The magnetic ions
in the spin quantum singlets get closer, augmenting the spin exchange
$J_{1}\left(1-\alpha\delta_{i}\right)$ for better magnetic energy gain at the
expense of elastic energy cost. These together are clear signals of the
$\bullet\\!\\!-\\!\\!\bullet\uparrow$ quantum plateau structure (see Figure
2-B, where the same tones of red are used for spin sites). The
$\Uparrow\Uparrow\Downarrow$ dipole amplitudes (green diamonds in the upper
plot) pin the dipoles pointing down in the short bonds; this makes
antiparallel dipoles get closer and parallel dipoles get farther, in a pattern
that minimizes the electro-elastic energy (see Figure 2-C, where the same
tones of green are used for dipoles). The same elastic distortions thus
contribute to the gain of both electric and magnetic energy. Notice that this
ground state breaks the translation invariance of the Hamiltonian in Eq. (II)
but maintains the inversion symmetry with respect to spin up sites or dipole
down bonds (in contrast to Case 2 discussed below). In consequence the ground
state is three-fold degenerate.

One should recall that a different (classical) magnetic order has been
previously observed at the $M=1/3$ plateau of the isotropic, frustrated
$J_{1}-J_{2}$ antiferromagnetic-elastic spin $S=1/2$ chain [2007-Gazza-etal, ]
in the absence of local dipoles. We can say that the distortions associated to
the dipolar order dominate and destroy the otherwise collinear
$\uparrow\uparrow\downarrow$ classical magnetic plateau order of the
isotropic, frustrated $J_{1}-J_{2}$ antiferromagnetic-elastic spin chain.
Instead they give rise to a $\bullet\\!\\!-\\!\\!\bullet\uparrow$ quantum
magnetic plateau order, elastically compatible with the dipolar order, where
the formation of spin quantum singlets lowers the magnetic energy. This is one
of the main results in this work.

The ground state obtained in the Case 1 may be visually summarized in the
cartoon description provided at the bottom of Figure 5.

Figure 5:  The $M=1/3$ ground state for $\Delta=1$, $J_{2}/J_{1}=0.5$ (Case 1)
obtained with the self-consistent numerical computations. Different light-
medium-dark tones are used to emphasize the period three structure. The upper
plot shows the local profile of the site observables (spin expectation values
$\langle S_{i}^{z}\rangle$ in red circles, vertical lines indicating sites)
and bond observables (distortions $\delta_{i}$ in blue squares and dipolar
moments $p_{i}$ (divided by $2p_{0}$) in green thin-diamonds, drawn between
sites). The lower plot shows the nearest neighbors spin correlations
$\langle\boldsymbol{S}_{i}\cdot\boldsymbol{S}_{i+1}\rangle$ 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 at the bottom qualitatively collects these numerical results and shows
the compatibility of the "short-long-long" (S-L-L) quantum magneto-elastic and
electro-elastic patterns in Figure 2.

Case 2 ($\Delta=4$, $J_{2}/J_{1}=0.8$): in the anisotropic, less frustrated
case we have observed qualitatively different magnetic and electric orders,
again with a period three structure. Numerical results are shown in Figure 6
together with the corresponding cartoon picture.

The spins clearly adopt the $\uparrow\uparrow\downarrow$ classical plateau
structure. This is seen in the sequence of $\langle S_{i}^{z}\rangle$ with two
positive, one negative values close to $0.5$ (red circles in the upper plot)
and mainly in the almost vanishing transverse correlations $\langle
S_{i}^{+}S_{i+1}^{-}\rangle$ (horizontal-triangles in the lower plot);
longitudinal correlations close to $0.25\,(-0.25)$ (up-triangles in the lower
plot) correspond to collinear parallel (antiparallel) spins. In this state the
magnetic sector could be well described by classical Ising spins, neglecting
the quantum fluctuations. However, we find in the next section that the
magnetic excitations above this plateau state show a clear quantum behaviour.

The novelty here is that the lattice distortions do not follow the pattern of
the magnetic correlations (compare with Figure 2-A). The first neighbor
dipolar terms in the self-consistent Eq. (4), following the
$\Uparrow\Uparrow\Downarrow$ configuration induced by the external electric
field (see Figure 2-C), are not compatible with such magnetic correlations and
force a competition in determining the lattice distortions. The resulting
distortion pattern does neither optimize the magnetic energy nor the dipolar
energy separately, but their sum with the elastic energy. It can be
qualitative described as "long-null-short" (L-0-S) distortion pattern
(following light-medium-dark blue squares in the upper plot of Figure 6). A
different, degenerate, ground state is obtained by inversion with respect to
any of the "short" bonds. Thus the ground state is six-fold degenerate.

Figure 6:  The $M=1/3$ ground state for $\Delta=4$, $J_{2}/J_{1}=0.8$ (Case
2). Symbols and colors follow the conventions in Figure 5. 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. However, the elastic distortions do not fit the classical
magneto-elastic pattern in Figure 2 but the "long-null-short" (L-0-S) sequence
depicted qualitatively in the bottom panel. The interaction between dipoles
and spins, mediated by distortions, frustrates the magneto-elastic order
producing the breaking of inversion symmetry.

The features of the ground state in Case 2 may be recognized in the cartoon in
the lower panel of Figure 6. The double frustration effect leads to a
compromising distortion pattern that breaks the inversion symmetry. This lack
of inversion symmetry in the distortion pattern of course modifies the
magnetic and dipolar couplings through the magneto-elastic coupling $\alpha$
and the pantograph electro-elastic coupling $\beta$. In consequence the spins
(dipoles) pointing up have slightly different values of $\langle S^{z}\rangle$
($p_{i}$), breaking the inversion symmetry observed in the quantum plateau
Case 1 and also in the $J_{1}-J_{2}$ magneto-elastic spin chain [2007-Gazza-
etal, ]. This spontaneous symmetry breaking is another important result,
consequence of the present double frustration effect. Were charge degrees of
freedom included, the induced charge order would result in a longitudinal
component of electrical polarization [2008-vdBrink-Khomskii, ].

The analysis of the other points indicated in Figure 3 show a classical
plateau behaviour similar to that observed in Case 2\. They correspond to
isotropic spin interactions and low frustration ($\Delta=1$,
$J_{2}/J_{1}=0.8$) and to a highly frustrated case with important easy-axis
anisotropy ($\Delta=4$, $J_{2}/J_{1}=0.5$). It appears that both isotropy and
high frustration are necessary to stabilize the quantum plateau. Further
numerical exploration indicates that there is a finite small region around
$\Delta=1$, $J_{2}/J_{1}=0.5$ where the $M=1/3$ plateau remains open and the
spins order in the quantum $\bullet\\!\\!-\\!\\!\bullet\uparrow$ structure, as
shown schematically in Figure 3.

As a summary of this Section, we have provided a qualitative description and
numerical evidence for a novel double frustration effect in a multiferroic
model scenario.

##  IV Composite excitations induced by a magnetic field

The $\Delta S^{z}=1$ spin excitations induced by a magnetic field on a plateau
state are the key to understand the high field plateau border. It is well
known that the excitation of $M=0$ plateau in one dimensional
antiferromagnetic spin chains is not a stable singlet-triplet excitation but
decays into two spinons [1980-Nakano-Fukuyama, ; 1981-Fadeev-Takhtajan, ;
1996-Kiryukhin, ; 1997-Feiguin, ; 2001-Dobry, ]. Each spinon carries spin
$S^{z}=1/2$ as a topological charge and may be described as a soliton
quasiparticle interpolating between different dimerized vacua; spatially, the
soliton profile can be seen as a smooth domain wall. We have discussed this
spin fractionalization phenomenom in the present multiferroic model in Ref. I.

The $\Delta S^{z}=1$ spin excitations on top of the $M=1/3$ magnetization
plateau in antiferromagnetic magneto-elastic spin chains is also known to
exhibit spin fractionalization [2007-Gazza-etal, ]. Remarkably, this goes
beyond the spinon description: the excitation decays into three $S^{z}=1/3$
noninteracting solitonic excitations (dubbed tertions). For a classical
$\uparrow\uparrow\downarrow$ plateau, it has been shown that the tertions have
a local singlet core causing the $\uparrow\uparrow\downarrow$ order on one
side to be shifted by one site with respect to the other side. In this way the
tertion interpolates between two different $\uparrow\uparrow\downarrow$
domains.

It has been observed in several systems that the high field plateau border is
characterized by a sudden finite magnetization jump when the magnetic field
takes a threshold value. The magnetization curves in Figure 4 suggest that
this might also occur in our model. If this is the case, the magnetic
excitation would fractionalize into a periodic lattice of self avoiding
solitons [1996-Kiryukhin, ; 1998-Lorenz, ]. Such a periodic magnetic structure
could be detected by unusual line shapes in neutron scattering data
[1999-Horvatic, ], as well as the associated lattice distortions could be
detected by X-ray measurements.

A natural question arises, whether these features are modified by the double
frustration effect in the present magneto-electro-elastic chain. We have
explored the numerical self-consistent solutions of Eqs. (4, 5) in periodic
chains with $N_{s}$ sites in the subspace of
$S_{total}^{z}=$$\left(N_{s}\cdot\frac{1}{2}\right)/3+1$ (that is one unit of
magnetization above $M=1/3$). We considered large chains in order to allow for
a most clear spatial separation of the three expected tertions 111For a
periodic pattern with magnetization
$S_{total}^{z}=$$\left(N_{s}\cdot\frac{1}{2}\right)/3+1$ to be commensurate
with the chain length, $N_{s}$ must be restricted to integers of the form
$9q+3$ [2007-Gazza-etal, ].. We report results on chains of $N_{s}=174$ sites,
where the plateau state has $S_{total}^{z}=29$ and the excited state has
$S_{total}^{z}=30$.

In the present multiferroic model scenario we have confirmed that the $\Delta
S^{z}=1$ excitation induced by a magnetic field on top of the $M=1/3$ state
indeed fractionalizes into three $S^{z}=1/3$ spatially separated tertions. The
trial of different dipolar configurations has shown that the dipolar sector
suffers a spontaneous unit polarization change along the electric field
direction to minimize the energy cost of the distortions accompanying the
magnetic order. This polarization change induced by a magnetic field is an
emergent magneto-electric effect mediated by elastic distortions. We discuss
below the numerical data supporting these statements.

Case 1: We show in Figure 7 numerical results for the magnetically excited
state in the isotropic frustrated regime ($\Delta=1$, $J_{2}/J_{1}=0.5$).
Using the same color codes as in the plateau state (see Figure 5), in the
upper panel we show the local $\langle S_{i}^{z}\rangle$ in red circles, the
dipoles $p_{i}$ in green diamonds and the distortions $\delta_{i}$ in blue
squares; the light-medium-dark tones for sites 1, 2, 3 are repeated every
three sites to visually distinguish the associated sub-lattices. In the lower
panel we show the $\langle\boldsymbol{S}_{i}\cdot\boldsymbol{S}_{i+1}\rangle$
correlations. With the help of the color tones one can see a short wavelength
oscillation of each observable, with period three as in the plateau state, but
modulated by a long wavelength oscillation spanning three whole periods along
the chain. The regions around sites $\sim 26$ (marked as $I$ in the plots),
$\sim 26+N_{s}/3=\sim 84$ (marked as $II$ in the plots), and $\sim 26+2\times
N_{s}/3=\sim 142$ show locally the same features as the plateau state.
However, in the first one the spin up is located on the medium-red sub-lattice
(region $I$), in the second one it falls on the dark-red sub-lattice (region
$II$) and in the third region it corresponds to the light-red sub-lattice.
Thus, regarding the spin sector, each of these regions adopts one of the three
degenerate possible quantum plateau configurations, different because of a
relative shift of the spin up and the spin singlet positions by one site to
the right. In the sites between the plateau regions one can see a smooth sub-
lattice interpolation between observables; for instance, the $\langle
S_{i}^{z}\rangle$ in the medium-red sub-lattice evolves from a spin up in
region $I$ to a spin in a singlet in region $II$. These intermediate sites
then allocate the solitonic excitations interpolating between different vacua
(in the sense of degenerate plateau states related by translations); the
analysis of local $\langle S_{i}^{z}\rangle$ values shows that they carry a
fraction $S^{z}=1/3$ of the magnetic excitation. At the center of the soliton
the spins take a $\downarrow\uparrow\uparrow\downarrow$ configuration, that
may be called a classical core between quantum orders. One can of course
notice that the soliton regions occupy an important fraction of the chain
length. As the spatial width of the magnetic solitons is usually in inverse
relation with the plateau width (or the spin gap producing it [1997-Feiguin,
]), we expect that in a larger chain they will maintain their size and more
space will be left for better defined plateau regions.

Figure 7:  Magneto-electro-elastic excitations (_MEE_) with $\Delta S^{z}=1$
induced by the magnetic field on top of the $M=1/3$ plateau for
$J_{2}/J_{1}=0.5$, $\Delta=1$ (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 Figure 5. The magnetic
sector shows regions which adopt the $\bullet\\!\\!-\\!\\!\bullet\uparrow$
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 $\bullet\\!\\!-\\!\\!\bullet\uparrow$ pattern is
shifted to a different sub-lattice 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 sub-lattice 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 $\bullet\\!\\!-\\!\\!\bullet\uparrow$ 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 Figure 8.

The analysis of dipolar configurations necessary to minimize the system energy
has shown that the dipolar order $\Uparrow\Uparrow\Downarrow$ induced by the
electric field (fixed at the plateau value $\epsilon=0.16$) is altered by the
appearance of three dipolar domain walls with patterns
$\Downarrow\Uparrow\Uparrow\Uparrow\Downarrow$. Each of them can be seen as
the insertion of an extra $\Uparrow$ dipole, namely a dipolar excitation. As
the three domain walls accumulate one dipole flip with respect to the
homogeneous $\Uparrow\Uparrow\Downarrow$ order, we observe a spontaneous unit
dipolar excitation that appears to decay into three domain walls. A similar
behavior is reported in [2003-Okunishi-b, ] for magnetic excitations above the
$M=1/3$ plateau in the Ising limit. These dipolar excitations are localized at
the soliton cores, so that the dipolar order observed at the plateau state is
shifted by one site at each domain wall accompanying the shift of magnetic
plateau structures. Notice that magnetic tertions and dipolar domain walls
occur in the same positions, a fact that may be interpreted as a magneto-
electric coupling between magnetic and dipolar excitations.

The elastic sector evolves smoothly along the magnetic solitons, shifting the
S-L-L pattern by one site as the spins and dipoles do, with a noticeably
exception at the dipolar domain walls. A singularly long bond is formed there,
while the others abruptly interchange from "null" to "short". We interpret
this feature as a local elastic excitation, coupled to the magneto-electric
one. Thus a localized magneto-electro-elastic excitation (indicated as _MEE_
in Figure 5) shows up between quantum plateau regions.

A cartoon picture of these features is drawn in Figure 8. The spins, dipoles
and distortions are schematically indicated at the quantum plateau regions $I$
and $II$ (as labeled in Figure 7), separated by the _MEE_ excitation with a
classical magnetic $\downarrow\uparrow\uparrow\downarrow$ core coinciding with
the $\Downarrow\Uparrow\Uparrow\Uparrow\Downarrow$ dipolar domain wall.

Figure 8:  Qualitative picture of the magneto-electro-elastic (_MEE_)
excitation separating the quantum plateau regions _I_ and _II_ in Figure 7.
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 Figure 7. The dotted line is a
mirror plane showing the parity symmetry of the _MEE_ excitation. The local
order in the quantum region _II_ is shifted by one site with respect to the
order in the quantum region _I_.

Case 2: In Figure 9 we show the qualitatively different results obtained for
the excitations of the classical plateau state in the anisotropic less
frustrated regime $J_{2}/J_{1}=0.8$, $\Delta=4$. Here the magnetic sector, in
the $S_{total}^{z}=$$\left(N_{s}\cdot\frac{1}{2}\right)/3+1$ excited subspace,
presents three regions with each $\uparrow\uparrow\downarrow$ classical
plateau order, periodically modulated along the chain. In each classical
plateau region the $\uparrow\uparrow\downarrow$ pattern lies in different sub-
lattices (for instance regions _I_ and _II_ in the Figure), separated by
solitons (one is highlighted as a magnetic soliton (_MS_) in the plot). Again
these solitons are tertions, carrying a fraction $S^{z}=1/3$ of the magnetic
excitation. They interpolate classical plateau regions, having a local quantum
spin singlet core; the same features have been observed in the excitations of
the $M=1/3$ classical plateau in magneto-elastic chains, in absence of dipolar
degrees of freedom [2007-Gazza-etal, ]. Besides, the solitons are narrower
than in Case 1, in accordance with a wider magnetic plateau (see Figure 4).

__

Figure 9:  Decoupled electro-elastic and magnetic and excitations with $\Delta
S^{z}=1$ induced by the magnetic field on top of the $M=1/3$ plateau for
$J_{2}/J_{1}=0.8$, $\Delta=4$ (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
_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 _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 Figure 10.

The dipolar sector again presents a unit spontaneous excitation (dipole flip)
fractionalized into three domain walls (one of them is highlighted as _DW_ in
the plot). But in this case the domain walls appear to decouple from the
magnetic solitons. Instead, they occur inside a $\uparrow\uparrow\downarrow$
plateau region signaling a parity change of the accompanying lattice
distortions (see for instance the regions highlighted as _II_ and _III_ in the
plot, with mirror symmetry with respect to the domain wall between them).
Singular elastic excitations (very long bonds) show up together with the
dipolar domain walls. One can thus observe electro-elastic excitations well
decoupled from magnetic $S^{z}=1/3$ excitations.

As the dipolar domain walls separate the two parity-related degenerate elastic
configurations compatible with the same magnetic order, all of the six
possible (degenerate) magneto-electro-elastic classical plateau configurations
are realized along the system length.

Figure 10:  Qualitative picture of the magneto-elastic and electro-elastic
excitations in Figure 9. The cartoon in the upper panel describes the magneto-
elastic solitonic excitation _MS_ with a quantum singlet core interpolating
the classical orders highlighted as _I_ and _II_ in Figure 7; 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 electro-elastic domain wall _DW_ separates the classical orders
highlighted as _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.

We have drawn a schematic description of these results in Figure 10: the
transition between the classical $\uparrow\uparrow\downarrow$ plateau regions
marked as _I_ and _II_ is given by a magnetic soliton _MS_ passing through a
spin singlet, while that between the classical plateau regions _II_ and _III_
is given by an electro-elastic excitation without disruption of the
$\uparrow\uparrow\downarrow$ magnetic order.

It is interesting to recall an argument based on the bosonized description of
the $M=1/3$ plateau in spin $S=1/2$ chains, discussed in [2007-Gazza-etal, ].
Along this argument the occurrence of the classical or quantum plateau are
related to the vacuum expectation value of a compactified bosonic field. This
explains why a soliton interpolating classical configurations must pass over a
quantum ordered region as found in that reference and also here in Case 2.
Conversely, the same argument suggests that a soliton interpolating quantum
configurations must pass over a classical order. As far as we know, our
findings in Case 1 are the first realization of this conjecture.

In comparing the location of dipolar domain walls
$\Downarrow\Uparrow\Uparrow\Uparrow\Downarrow$ with respect to the magnetic
order, one can extract as a thumb rule that they fit better in the elastic
distortions of a classical $\downarrow\uparrow\uparrow\downarrow$ magnetic
environment (see cartoons in Figures 8 and 10, second line). Thus, when
exciting a classical plateau state the dipolar domain walls are located in the
classical plateau regions, away from magnetic excitations. Instead, when
exciting a quantum plateau state the dipolar domain walls are located in the
classical core of the magnetic solitons forming a composite _MEE_
quasiparticle.

##  V Summary and conclusions

We have explored the interplay between frustrated magnetic and dipolar orders
in a one dimensional model for collinear type II multiferroic materials, where
electric and magnetic degrees of freedom are indirectly coupled by the lattice
distortions. More precisely, we have investigated the commensurability of the
$P=1/3$ period three dipolar order $\Uparrow\Uparrow\Downarrow$ with the
period three magnetic configurations observed in many frustrated magnetic
materials within $M=1/3$ magnetization plateaus.

Both from qualitative arguments and extensive DMRG computations we have found
that the dipolar order introduced by frustrating dipolar interactions competes
with the magnetic order set in turn by the magnetic frustration at the M=1/3
plateau. This opens a non trivial scenario which we dub _double frustration_.
Our analysis provides clear and surprising effects due to this double
frustration. In a regime of low anisotropy and high magnetic frustration,
favoring quantum fluctuations, the double frustration is responsible for the
stabilization of a _quantum_ $M=1/3$ plateau state. In contrast, in all other
cases (either introducing higher easy-axis anisotropy and/or reducing magnetic
frustration) the second frustration competition leads to the spontaneous
parity symmetry breaking in the order of the _classical_ $M=1/3$ plateau
state. From this parity breaking mechanism, and in the presence of charge
order along the chain, a longitudinal component of the polarization should
appear [2008-vdBrink-Khomskii, ]. Detection of different directions of the
polarization could be the clue to identify the underlying magneto-electric
effects operating in a given material.

We have also discussed the excitations caused by the increase of the magnetic
field. We have found that the $\Delta S^{z}=1$ magnon on top of the $M=1/3$
state fractionalizes into three $S^{z}=1/3$ spatially separated solitons,
encompassing elastic distortions adapted to the magnetic order. This change in
the distortion pattern induces, in the dipolar sector, a spontaneous unit
polarization change which in turn fractionalizes into three sharp domain
walls. Moreover, on top of the quantum plateau state these fractional
excitations form a composite magneto-electro-elastic quasiparticle. This
emergent magneto-electric effect, that is the polarization change induced by a
magnetic field mediated by elastic distortions, is one of the main results in
the present paper.

The nature of the plateau state structure and the appearance of intertwined
magnetic and electric fractional excitations, mediated by the lattice, are
experimentally accessible by neutron scattering for the spin-channel and by
X-ray scattering for the lattice distortions. The striking differences between
the present results and those for pure magneto-elastic chains are clear
signals of the role of dipolar interactions in multiferroic systems and may
guide the search for materials realizing strong magneto-electric effects.

## Acknowledgements

This paper was partially supported by CONICET (Grant No. PIP 2015-813 and No.
PIP 2015-364), Argentina.

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