# Elementary excitation in the spin-stripe phase in quantum chains

Matej Pregelj<EMAIL_ADDRESS>Jožef Stefan Institute, Jamova 39, 1000
Ljubljana, Slovenia  Andrej Zorko  Jožef Stefan Institute, Jamova 39, 1000
Ljubljana, Slovenia  Matjaž Gomilšek  Jožef Stefan Institute, Jamova 39, 1000
Ljubljana, Slovenia  Centre for Materials Physics, Durham University, South
Road, Durham, DH1 3LE, UK  Martin Klanjšek  Jožef Stefan Institute, Jamova 39,
1000 Ljubljana, Slovenia  Oksana Zaharko  Laboratory for Neutron Scattering,
PSI, CH-5232 Villigen, Switzerland  Jonathan S. White  Laboratory for Neutron
Scattering, PSI, CH-5232 Villigen, Switzerland  Hubertus Luetkens  Laboratory
for Muon Spin Spectroscopy, PSI, CH-5232 Villigen, Switzerland  Fiona Coomer
ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 OQX,
United Kingdom  Tomislav Ivek  Institute of Physics, Bijenička c. 46, HR-10000
Zagreb, Croatia  David Rivas Góngora  Institute of Physics, Bijenička c. 46,
HR-10000 Zagreb, Croatia  Helmuth Berger  Ecole polytechnique fédérale de
Lausanne, CH-1015 Lausanne, Switzerland  Denis Arčon  Jožef Stefan Institute,
Jamova 39, 1000 Ljubljana, Slovenia  Faculty of Mathematics and Physics,
University of Ljubljana, Jadranska c. 19, 1000 Ljubljana, Slovenia

###### Abstract

Elementary excitations in condensed matter capture the complex many-body
dynamics of interacting basic entities in a simple quasiparticle picture. In
magnetic systems the most established quasiparticles are magnons, collective
excitations that reside in ordered spin structures, and spinons, their
fractional counterparts that emerge in disordered, yet correlated spin states.
Here we report on the discovery of elementary excitation inherent to spin-
stripe order that represents a bound state of two phason quasiparticles,
resulting in a wiggling-like motion of the magnetic moments. We observe these
excitations, which we dub “wigglons”, in the frustrated zigzag spin-1/2 chain
compound $\beta$-TeVO4, where they give rise to unusual low-frequency spin
dynamics in the spin-stripe phase. This provides insights into the stripe
physics of strongly-correlated electron systems.

††preprint: APS

##  I Introduction

The concept of elementary excitations provides an elegant description of
dynamical processes in condensed matter Nakajima  _et al._ (1980). Its use is
widespread and represents the theoretical foundation for our understanding of
vibrational motions of atoms in crystals as phonons Nakajima  _et al._ (1980),
the excitations of the valence electrons in metals as plasmons Pines (2016),
the bound states of an electron and an electron hole in semiconductors as
excitons Frenkel (1931), etc. In magnetic systems, this approach inspired the
spinon picture of fractional excitations in spin liquids Balents (2010), the
phason description of the modulation-phase oscillations in amplitude modulated
structures Blanco  _et al._ (2013), and the magnon picture of collective spin
excitations in ordered states Bloch (1930). The latter led to further
intriguing discoveries, including longitudinal Higgs modes in two-dimensional
antiferromagnets Jain  _et al._ (2017) and magnon bound states in
ferromagnetic spin-1/2 chains Torrance and Tinkham (1969). Yet, for systems
where several order parameters interact, the elementary excitations remain
mysterious. A prominent example are the elusive excitations that cause the
melting of charge-stripe order in high-temperature superconductors Parker  _et
al._ (2010); Fernandes  _et al._ (2014); Fradkin  _et al._ (2015); Wang  _et
al._ (2016); Klauss (2012); Kivelson  _et al._ (2003); Vojta (2009) and
promote enigmatic charge fluctuating-stripe (nematic) states Kivelson  _et
al._ (2003); Vojta (2009); Anissimova  _et al._ (2014).

Figure 1: Magnetic structures and magnetic phase diagram. a Magnetic structure
models corresponding to the spin-density-wave (SDW), spin-stripe and vectro
chiral (VC) phases in $\beta$-TeVO4. Note the difference in modulation periods
of the two ordered components in the spin-stripe phase, $d$ and $d^{\prime}$,
associated with the magnetic wave vectors k and k+$\Delta$k, respectively. b
The corresponding excitations extend over a wide frequency range, from low-
energy dielectric dynamics up to high-energy spinon excitations, i.e., from
$\sim$105 Hz up to $\sim$1012 Hz. The derived frequency of the magnetic-order
dynamics $\nu_{\text{stripe}}$ in the spin-stripe phase corresponds well to
the product of intensities of the two neutron reflections, associated with the
fourth-order coupling term $F_{4}$, and to the phenomenological model (solid
line). The error bars represent an uncertainty of 1 s.d.

Here we study elementary excitations of the spin-stripe phase in the
frustrated spin-1/2 chain compound $\beta$-TeVO4 Savina  _et al._ (2011);
Gnezdilov  _et al._ (2012); Pregelj  _et al._ (2015); Savina  _et al._ (2015);
Weickert  _et al._ (2016); Pregelj  _et al._ (2016, 2018), which contains
localized V4+ ($S$ = 1/2) magnetic moments Pregelj  _et al._ (2015, 2016).
This intriguing order involves two superimposed orthogonal incommensurate
amplitude-modulated magnetic components with slightly different modulation
periods (Fig. 1a), corresponding to two magnetic order parameters, that result
in a nanometer-scale spin-stripe modulation Pregelj  _et al._ (2015). We show
that in the low-frequency (megahertz) range this, otherwise long-range-ordered
state, is in fact dynamical due to the presence of a low-energy excitation
mode that results from the binding of two phasons from the two orthogonal
magnetic components. This type of elementary excitation, which we dub a
“wigglon”, is inherent to the spin-stripe order (Fig. 1) and provides insights
into the dynamics of stripe phases that may be found when there are two or
more order parameters coupled together.

Figure 2: $\mu$SR results. a-d The $\mu$SR polarization data. Lines are fits
to the model Eq. (1). The arrows indicate the first minimum due to
oscillations caused by static local magnetic fields at 4.4 K and 1.6 K. e The
derived mean value of the static local magnetic fields at the muon stopping
site. f The derived relaxation rate associated with the local magnetic-field
fluctuations. The line is a fit to a simple linear model
$\lambda\,\propto\,a+bT$. The apparent gap in the model at 1 K is just a gap
in the horizontal scale. g The derived stretching exponent, reflecting the
distribution of the local magnetic-field fluctuations. The error bars
represent an uncertainty of 1 s.d.

The peculiar spin-stripe order in $\beta$-TeVO4 evolves from a spin-density-
wave (SDW) phase, which develops below $T_{N1}$ = 4.65 K and is characterized
by a single collinear incommensurate amplitude-modulated magnetic component
(Fig. 1a). On cooling, a second superimposed incommensurate amplitude-
modulated component with a different modulation period and orthogonal
polarization emerges (Fig. 1a) at $T_{N2}$ = 3.28 K and the spin-stripe order
is formed. Finally, at $T_{N3}$ = 2.28 K the modulation periods of the two
incommensurate amplitude-modulated components become equal and a vector-chiral
(VC) phase (Fig. 1a) is established. For frustrated spin-1/2 chains with
ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor
exchange interactions, which in $\beta$-TeVO4 amount to $J_{1}$ $\approx$
$-$38 K and $J_{2}$ $\approx$ $-$0.8 $J_{1}$, respectively Pregelj  _et al._
(2015), the SDW and VC phases are predicted theoretically Sudan  _et al._
(2009); Hikihara  _et al._ (2008), while the intermediate spin-stripe phase is
not. The formation of the latter has been associated with exchange
anisotropies and interchain interactions Pregelj  _et al._ (2015, 2016), but
still awaits a comprehensive explanation.

##  II Results

We explore the spin dynamics in these phases by employing the local-probe
muon-spin-relaxation ($\mu$SR) technique, which is extremely sensitive to
internal magnetic fields and can distinguish between fluctuating and static
magnetism in a broad frequency range (from $\sim$100 kHz to $\sim$100 GHz)
Yaouanc and Dalmas de Reotier (2011). We used a powder sample obtained by
grinding single crystals (see Methods) to ensure that on average 1/3 of the
muon polarization was parallel to the local magnetic field. In the case of
static local fields, $B_{\text{stat}}$, the corresponding 1/3 of the total
muon polarization is constant, resulting in the so-called “1/3-tail” at late
times in the $\mu$SR signal Yaouanc and Dalmas de Reotier (2011). The
remaining muon polarization precesses with the angular frequency
$\gamma_{\mu}B_{\text{stat}}$ ($\gamma_{\mu}$ = 2$\pi\times$135.5 MHz/T),
leading to oscillations in the time dependence of the $\mu$SR signal around
the “1/3-tail” Yaouanc and Dalmas de Reotier (2011); sup . The only way for
the muon polarization to relax below 1/3 at late times is thus provided by
dynamical local fields.

At $T$ = 4.7 K $>$ $T_{N1}$, the measured $\mu$SR polarization decays
monotonically (Fig. 2a), as expected in the paramagnetic state where
fluctuations of the local magnetic fields are fast compared to the muon
lifetime Yaouanc and Dalmas de Reotier (2011). The muon relaxation curve
changes dramatically at $T_{N1}$ (Fig. 2b), where the polarization at early
times suddenly drops, reflecting the establishment of static internal fields
in the SDW phase. The corresponding oscillations are severely damped, i.e.,
only the first oscillation at $t$ $<$ 1 $\mu$s can be clearly resolved (Fig.
2b), which indicates a wide distribution of $B_{\text{stat}}$, a hallmark of
the incommensurate amplitude-modulated magnetic order. Clearly, in
$\beta$-TeVO4, this static damping is sufficiently strong that the $\mu$SR
signal beyond $\sim$1 $\mu$s can be attributed solely to the “1/3-tail”. The
latter notably decays (Fig. 2b), which proves that the local magnetic field is
still fluctuating, as expected for incommensurate amplitude-modulated magnetic
structures Pregelj  _et al._ (2012). Remarkably, below $T_{N2}$, in the spin-
stripe phase, the “1/3-tail” is dramatically suppressed and the oscillation is
lost (Fig. 2c), revealing a significant enhancement of local-field
fluctuations. This indicates that the system enters an intriguing state that
is completely dynamical on the $\mu$SR timescale. Finally, below $T_{N3}$ the
slowly-relaxing “1/3-tail” and the oscillation reappear (Fig. 2d),
corroborating the establishment of a quasi-static VC state with almost fully
developed magnetic moments.

To quantitatively account for the $\mu$SR signal we model the $\mu$SR
polarization over the whole temperature range as a product of the two factors

$\displaystyle\begin{split}P(t)=\left[\frac{1}{3}+\frac{2}{3}\cos(\gamma_{\mu}B_{\text{stat}}t)e^{-(\gamma_{\mu}\Delta
t)^{2}/2}\right]e^{-(\lambda t)^{\alpha}}.\end{split}$ (1)

The exponential in the first factor in (1) accounts for the muon relaxation
due to a Gaussian distribution of static magnetic fields with a mean value
$B_{\text{stat}}$ and a width $\Delta$. Since oscillations of the $\mu$SR
polarization are almost completely damped already after the first visible
minimum, the parameters $B_{\text{stat}}$ and $\Delta$ must be comparable.
Indeed, the best agreement with experiment was achieved for
$\Delta/B_{\text{stat}}$ = 1.25(1) (Fig. 2a-d), which was kept fixed for all
temperatures. The second factor in (1) is the stretched-exponential function
that describes the decay of the “1/3-tail” due to additional local magnetic-
field fluctuations. Here, $\lambda$ is the mean relaxation rate while $\alpha$
is the stretching exponent accounting for a distribution of relaxation rates
Johnston (2006).

The results of our fits of the $\mu$SR data to Eq. (1) are summarized in Fig.
2. $B_{\text{stat}}$ (Fig. 2e) grows from zero at $T_{N1}$ to 9(1) mT at
$T_{N2}$, which is a value of dipolar fields typical encountered by muons in
spin-1/2 systems Yaouanc and Dalmas de Reotier (2011). In the spin-stripe
phase, $B_{\text{stat}}$ slightly decreases, while below $T_{N3}$ it starts
growing again and reaches a 15(1) mT plateau at the lowest temperatures. On
the contrary, the relaxation rate $\lambda$ does not change significantly
throughout the SDW phase, but it escalates by more than an order of magnitude
below $T_{N2}$, i.e., in the spin-stripe phase. Below $T_{N3}$, however, it
reduces and resumes following the same linear temperature dependence (solid
lines in Fig. 2f) as in the SDW phase. This is a characteristic of the
persistent spin dynamics De Reotier  _et al._ (2006) of the disordered part of
the magnetic moments in amplitude-modulated magnetic structures Pregelj  _et
al._ (2012). As these fluctuations are fast compared to the muon precession,
i.e., they do not suppress the minimum in the $\mu$SR signal described by the
first factor in Eq. (1), one can assume that $\lambda$ =
2$\gamma_{\mu}^{2}B_{\text{dyn}}^{2}$/$\nu_{\text{dyn}}$ Yaouanc and Dalmas de
Reotier (2011), where $B_{\text{dyn}}$ is the size of the fluctuating field
and $\nu_{\text{dyn}}$ is the corresponding frequency. Considering that in
amplitude-modulated magnetic structures $B_{\text{dyn}}$ is comparable to
$B_{\text{stat}}$, we can estimate that $\nu_{\text{dyn}}$ ranges between 0.1
and 1 GHz (Fig. 1b). Finally, the stretching exponent $\alpha$ in the SDW and
VC phase (Fig. 2g) amounts to 0.43(5) and 0.25(2), respectively, as expected
for broad fluctuating-field distributions in the incommensurate amplitude-
modulated magnetic structures Pregelj  _et al._ (2012).

While the $\mu$SR response in the SDW and VC phases is within expectations,
the spin-stripe phase shows a surprising enhancement of $\lambda$ (Fig. 2f)
and $\alpha$ (Fig. 2g) that reflects the severe decay of the “1/3-tail” in
this phase (Fig. 2c). This clearly demonstrates the appearance of an
additional relaxation channel that is related to the spin-stripe order only.
Moreover, the increase of $\lambda$ is accompanied with the loss of the
oscillation in the $\mu$SR signal, which indicates that the corresponding
fluctuations are associated with the ordered part of the magnetic moments. To
account for these experimental findings we introduce the dynamics of the
magnetic order into our minimal model of Eq. (1) via the strong collision
approach Yaouanc and Dalmas de Reotier (2011). Namely, we assume that in the
spin-stripe phase the static fields derived for the SDW phase fluctuate with a
single correlation time 1/$\nu_{\text{stripe}}$, where $\nu_{\text{stripe}}$
is the fluctuating frequency, and numerically calculate the resulting muon
polarization function in a self-consistent manner. Indeed, the resulting muon
polarization function $P_{\text{stripe}}(t)$ sup , with all other parameters
fixed to the values derived for the SDW phase, explains the response of the
$\mu$SR signal throughout the spin-stripe phase sup . The derived temperature
dependence of $\nu_{\text{stripe}}$ exhibits a continuous increase from 0.5(5)
MHz at $T_{N2}$ to 7.3(5) MHz at $T_{N3}$ (Fig. 1b).

Figure 3: Neutron diffraction results. a The temperature dependence of the
strongest magnetic reflection and the accompanying satellite reflections. The
temperature dependences of b the order parameters $\sigma_{\bf k}$ and
$\sigma_{{\bf k+\Delta k}}$ as well as c $\Delta q$ derived from the neutron
diffraction experiment (symbols) and calculated using a phenomenological model
(solid lines; see text). The error bars represent an uncertainty of 1 s.d.

To further investigate the relation between the spin-stripe order witnessed
previously by neutron diffraction Pregelj  _et al._ (2015, 2016) and the
stripe dynamics observed by $\mu$SR, we performed additional neutron
diffraction measurements (see Methods). We measured the temperature dependence
of the strongest magnetic reflection and its satellites (Fig. 3a), the latter
being associated with the orthogonal $b$ magnetic-moment component (Fig. 1a)
Pregelj  _et al._ (2016) that emerges at slightly different wave vectors,
shifted by $\pm$$\Delta$k from the main magnetic wave vector k Pregelj  _et
al._ (2015). The intensity of an individual magnetic reflection, scales with
the square of the corresponding order parameter $\sigma_{\bf k}$ = $M_{\bf
k}/\mu_{B}$, where $M_{\bf k}$ denotes the sublattice magnetization component
associated with k and $\mu_{B}$ is the Bohr magneton Pregelj  _et al._ (2016).
This allows for the comparison of $\nu_{\text{stripe}}(T)$ with the
temperature evolution of $F_{4}$ $\sim$ $\sigma_{\bf-k}^{2}\sigma_{\bf
k+\Delta k}\sigma_{\bf k-\Delta k}$, i.e., the lowest-order term in the
magnetic free energy that couples all magnetic components with different
modulation periods (k, k+$\Delta$k, and k$-\Delta$k) Pregelj  _et al._ (2016).
We find a very good correspondence (Fig. 1b), which confirms a direct link
between the formation of spin stripes and the remarkable low-energy
excitations found in this state.

##  III Discussion

To put the observed excitations into context in terms of frustrated quantum-
spin chains, we plot a schematic temperature–frequency diagram of excitations
in $\beta$-TeVO4 in Fig. 1b. At the lowest energies, we find dielectric
dynamics, which peaks at $\sim$0.4 MHz and is most pronounced in the
multiferroic VC phase (see Supplementary information sup  for complementary
dielectric measurements). These are followed by strong $\nu_{\text{stripe}}$
fluctuations, which emerge at $T_{N2}$ and reach a maximum of 7.3(5) MHz close
to the $T_{N3}$ transition, after which they disappear. The persistent spin
dynamics, which can be significantly enhanced in frustrated spin-1/2 systems
due to quantum effects, exhibits even faster fluctuations at 0.1–1 GHz. The
collective magnon excitations, determined by the main exchange interactions
$J_{i}$ ($i$ = 1,2), develop in the VC phase above the gap, most likely
induced by spin–orbit coupling, which is also responsible for exchange
anisotropy, i.e., at frequencies of 0.1–1 THz Pregelj  _et al._ (2018); sup .
The diagram is completed by spinon excitations, which form a continuum
extending up to $\sim$$\pi J_{i}\sigma^{2}$ Lake  _et al._ (2005), in this
case up to $\sim$3 THz Pregelj  _et al._ (2018).

Figure 4: Wigglon excitation. a Schematic dispersion relations for individual
gapless phasons corresponding to $\sigma_{\bf k}$ and $\sigma_{{\bf
k}\,+\,{\bf\Delta k}}$ (dotted lines), as well as for the gaped two-phason
bound wigglon excitation (solid line). b Schematic representation of the
magnetic order in the spin-stripe phase, decomposed into two orthogonal
magnetic components (green and violet arrows) and summed together (magenta
arrows), for the two decoupled phason modes (left) and for the two-phason
bound wigglon excitation (right). Consecutive figures represent the time
evolution, while large arrows denote shifts of the central maximum (see
simulations in Supplementary videos 1 and 2). Note the uniform shifts for
decoupled phasons and nonuniform extensions/contractions induced by a wigglon.
c The position, amplitude and orientation of the marked maximum in the
magnetic structure (thick black and magneta arrows in b) as a function of time
for decoupled phason excitations (left) and a wigglon excitation (right).

Next, we try to identify the physical mechanism responsible for the unusual
spin-stripe excitations. Among numerous experimental and theoretical studies
of spin chains, considering different $J_{1}/J_{2}$ ratios in an external
magnetic field Sudan  _et al._ (2009); Hikihara  _et al._ (2008); Katsura  _et
al._ (2008) as well as in the presence of magnetic anisotropy Nersesyan  _et
al._ (1998); Furukawa  _et al._ (2010); Hüvonen  _et al._ (2009) and
interchain interactions Sato  _et al._ (2013); Nishimoto  _et al._ (2015);
Weichselbaum and White (2011), there appears to be no record of multi-k
magnetic structures that would resemble the spin-stripe order observed in
$\beta$-TeVO4, nor its associated excitations. Moreover, if persistent spin
dynamics or any other low-energy excitation inherent to the SDW (or VC) phase,
e.g., phasons Blanco  _et al._ (2013); Furukawa  _et al._ (2010), were also
primarily responsible for the spin relaxation in the spin-stripe phase, there
should be no significant difference between the SDW (or VC) and spin-stripe
dynamics. Contrary to this, spin dynamics in the spin-stripe phase is
completely different from the other ordered phases, as evidenced by the
drastically enhanced muon-spin relaxation rate (Fig. 2). Further comparison
with dynamical processes in spin systems that do develop multi-k magnetic
structures, e.g., skyrmion phases Han (2017), does not reveal any similarity
either. Namely, in contrast to our case, in such systems spin dynamics are
typically driven either by very slow domain fluctuations in the range between
1 Hz to 1 kHz Zang  _et al._ (2011), or stem from much faster collective
breathing, magnon or even electromagnon excitations in the GHz range Nagaosa
and Tokura (2013); Pimenov  _et al._ (2006); Mochizuki and Seki (2015).
Finally, dynamics in alternating patterns of spin and charge stripes in oxides
was found between 1 and 100 GHz Lancaster  _et al._ (2014). To summarize,
there exists no report of electron spin dynamics in the mid-frequency (MHz)
range, as observed in $\beta$-TeVO4. Such dynamics, therefore, seems to
originate from the peculiarities of the spin-stripe phase, which are also
responsible for the remarkable coincidence of the $\nu_{\text{stripe}}(T)$ and
$F_{4}(T)$ dependences (Fig. 1b).

To explain the sequence of the magnetic transitions as well as to clarify the
existence of the spin-stripe phase and its corresponding excitations we
undertake a phenomenological approach based on the classical Ginzburg–Landau
theory of phase transitions. We construct the following expression for the
magnetic free energy

$\displaystyle F$ $\displaystyle=$ $\displaystyle
A_{1}\left(T/T_{N1}-1\right)\sigma_{\bf
k}^{2}+A_{2}\left(T/T_{N2}-1\right)\sigma_{{\bf k+\Delta k}}^{2}+$ (2)
$\displaystyle B_{1}\sigma_{\bf k}^{4}+B_{2}\sigma_{{\bf k+\Delta
k}}^{4}+C\left(1-\Delta q\right)^{2}\sigma_{{\bf k+\Delta k}}^{2}+F_{4},$
$\displaystyle F_{4}$ $\displaystyle=$ $\displaystyle\left[D+Ef(\sigma_{\bf
k},\sigma_{{\bf k+\Delta k}})\Delta q^{2}\right]\sigma_{\bf-k}^{2}\sigma_{{\bf
k+\Delta k}}\sigma_{{\bf k-\Delta k}},$ (3)

where $A_{i}$, $B_{i}$ ($i$ = 1,2), $C$, $D$, and $E$ are scaling constants.
The first four terms in (2) describe the evolution of two independent magnetic
order parameters $\sigma_{\bf k}$ and $\sigma_{{\bf k+\Delta k}}$ that emerge
at $T_{N1}$ and $T_{N2}$, respectively. The fifth term represents exchange
anisotropy that is responsible for a different magnetic wave vector for the
$\sigma_{{\bf k+\Delta k}}$ component, i.e., favoring $\Delta q$ = $\Delta
k/\Delta k_{0}$ $\neq$ 0, where $\Delta k_{0}$ $\equiv$ $\Delta k(T_{N2})$
represents the discrepancy between the native magnetic wave vectors for the
$\sigma_{{\bf k+\Delta k}}$ and $\sigma_{\bf k}$ components. The $F_{4}$ term
is associated with the coupling between the two order parameters and favors
fully developed magnetic moments, i.e., it acts against the discrepancy
between the two modulation periods ($\Delta q$ $\to$ 0). The function
$f(\sigma_{\bf k},\sigma_{{\bf k+\Delta k}})$ accounts for the size limitation
of the V4+ $S$ = 1/2 magnetic moments and thus smoothly changes from 0 to 1,
when $\sqrt{\sigma_{\bf k}^{2}+\sigma_{{\bf k+\Delta k}}^{2}}$ exceeds the
limiting value sup . Considering $\sigma_{{\bf k-\Delta k}}$ $\approx$
$\sqrt{0.2}\sigma_{{\bf k+\Delta k}}$ (Fig. 3 and Ref. Pregelj  _et al._ ,
2015), the minimization of (2) with respect to $\Delta q$, $\sigma_{\bf k}$
and $\sigma_{{\bf k+\Delta k}}$ sup  returns the corresponding temperature
dependences (Fig. 3b and 3c) that almost perfectly describe the observed
behavior. In particular, we find that in the vicinity of the paramagnetic
phase, where ordered magnetic moments are still small, a sizable exchange
anisotropy can impose different modulations for different magnetic-moment
components through the $C$ term. On cooling, however, the ordered magnetic
moments increase, causing the $F_{4}$ term to prevail and thus to stabilize
the VC phase with $\Delta q$ = 0 below $T_{N3}$. Finally, the derived
parameters allow us to calculate the temperature dependence of the $F_{4}$
term. Comparison of the derived temperature dependence $F_{4}(T)$ with
experimentally determined $\nu_{\text{stripe}}(T)$, i.e., assuming that
$h\nu_{\text{stripe}}$ = $c|F_{4}|$, where $h$ is the Planck constant, we
obtain a very good agreement for $c$ = 0.7 (Fig. 1b), corroborating the
connection between the $F_{4}$ term and the $\nu_{\text{stripe}}$ dynamics.

Having established the intimate relation between low-frequency excitations and
the spin-stripe phase, the open question that remains concerns the microscopic
nature of the spin-stripe excitation mode. In contrast to ordinary magnon and
spinon modes, these excitations arise from a fourth-order free-energy term
that couples magnetic components with different modulation periods (Fig. 1a).
Higher-order terms in the free energy impose an interaction between the basic
elementary excitations Torrance and Tinkham (1969); Kecke  _et al._ (2007);
Nawa  _et al._ (2017). The observed excitations are thus most likely bound
states of two elementary excitations of the incommensurate amplitude-modulated
magnetic components. The latter may either be two phasons, i.e., linearly
dispersing zero-frequency Goldstone modes that change the phase of the
modulation Blanco  _et al._ (2013); Overhauser (1971), two amplitudons, i.e.,
high-frequency modes that change the amplitude of the modulation Blanco  _et
al._ (2013), or a combination of the two. Given that $\nu_{\text{stripe}}$ is
very small compared to the exchange interactions, the spin-stripe excitation
is most likely a two-phason bound mode that has minimal energy at a certain
wave vector k${}_{\text{stripe}}$ (Fig.4a). Since k${}_{\text{stripe}}$, in
principle, differs from both k and k$\pm$$\Delta$k, the bound mode imposes
additional expansion and contraction of the modulation periods on top of the
phase changes induced by individual phasons (Fig. 4b). Consequently,
positions, sizes and orientations of maxima in the magnetic structure exhibit
completely different time dependences than for individual phason (Fig. 4c),
resulting in a wiggling-like motion of the magnetic moments (see simulation in
Ref. sup, ). Hence, we dubbed this type of spin-stripe excitations “wigglons”.
Moreover, the corresponding amplitude variation is reminiscent of the
longitudinal (amplitude) Higgs mode in the Ca2RuO4 antiferromagnet, which has
also been found to decay into a pair of Goldstone modes Jain  _et al._ (2017).
Finally, we point out that “wigglon” dynamics in the spin-stripe phase of
$\beta$-TeVO4 might share similarities with fluctuating-charge-stripe phases
Klauss (2012); Kivelson  _et al._ (2003), where the nematic response is
ascribed to fast stripe dynamics Anissimova  _et al._ (2014).

Our results reveal an intriguing spin-only manifestation of fluctuating-stripe
physics, which has, so far, been studied exclusively in the context of nematic
phases in high-temperature superconductors. We show that $\beta$-TeVO4
displays an intriguing spin-stripe order, which, due to a slow wiggling motion
of the magnetic moments, appears static on the neutron-scattering timescale
Pregelj  _et al._ (2015), i.e., at $\nu$ $>$ 10 GHz, while it is in fact
dynamical at MHz frequencies. The phenomenon is driven by sizable exchange
anisotropy, which prevails in a finite temperature range where it stabilizes
the dynamical spin-stripe phase that hosts an extraordinary type of
excitation, driven by the fourth-order coupling term in the magnetic free
energy. Our discovery draws attention to other frustrated spin-1/2 chain
compounds with complicated and unresolved magnetic phase diagrams Willenberg
_et al._ (2016); Bush  _et al._ (2018), where similar effects may be
anticipated to play a role. Finally, more details of the wigglon excitation,
such as their dependence on the applied magnetic field, should be explored by
complementary nuclear-magnetic-resonance measurements that are highly
sensitive to spin dynamics in the relevant MHz range even in a sizable applied
magnetic field.

##  IV Methods

Sample description. The single-crystal samples were grown from TeO2 and VO2
powders by chemical vapor transport reaction, using two-zone furnace and TeCl4
as a transport agent, as explained in Ref. Pregelj  _et al._ , 2015. Powder
samples were obtained by grinding single-crystal samples.

$\mu$SR experiments. The experiments were performed on the MuSR instrument at
the STFC ISIS facility, Rutherford Appleton Laboratory, United Kingdom, and on
the General Purpose Surface-Muon instrument (GPS) at the Paul Scherrer
Institute (PSI), Switzerland. The measurements at PSI were performed in zero-
field, while at ISIS a small longitudinal field of 4 mT was applied to
decouple the muon relaxation due to nuclear magnetism at long times. The dead
time at the GPS instrument is $\sim$0.01$\mu$s, whereas its is $\sim$0.1
$\mu$s at the MuSR instrument. For details on background subtraction see Ref.
sup, .

Neutron diffraction. Neutron diffraction measurements were performed on a
2$\times$3$\times$4 mm3 single-crystal on the triple-axis-spectrometer TASP at
PSI. To assure the maximal neutron flux the wavelength of 3.19 Å was chosen
for the experiment. An analyzer was used to reduce the background, while the
standard ILL orange cryostat was used for cooling.

##  V Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  VI Acknowledgments

###### Acknowledgements.

We thank T. Lancaster for fruitful discussions and valuable comments. We are
grateful for the provision of beam time at the Science and Technology
Facilities Council (STFC) ISIS Facility, Rutherford Appleton Laboratory, UK,
and S$\mu$S, Paul Scherrer Institut, Switzerland. This work has been funded by
the Slovenian Research Agency (project J1-9145 and program No. P1-0125), the
Swiss National Science Foundation (project SCOPES IZ73Z0_152734/1) and the
Croatian Science Foundation (project IP-2013-11-1011). This research project
has been supported by the European Commission under the 7th Framework
Programme through the ’Research Infrastructures’ action of the ’Capacities’
Programme, NMI3-II Grant number 283883, Contract No. 283883-NMI3-II. M.G. is
grateful to EPSRC (UK) for financial support (grant No. EP/N024028/1). We are
grateful to M. Enderle for local support at Institut Laue-Langevin, Grenoble,
France.

##  VII Competing interests

The authors declare no competing interests.

##  VIII Author contributions

M.P., A.Z., and D.A. designed and supervised the project. The samples were
synthesized by H.B. The $\mu$SR experiments were performed by A.Z., M.G., H.L.
and F.C and analyzed by M.P., M.G., and A.Z.. The neutron diffraction
experiments were performed by M.P., O.Z. and J.S.W. and analyzed by M.P. The
dielectric experiments were performed by T.I. and D.R.G. All authors
contributed to the interpretation of the data and to the writing of the
manuscript.

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