the recorded events don’t carry any names to identify the event.

Figure 17: Read and write steps

For the channel communication in Figure 0.A.3, no more happens-before
information is communicated. Especially the forward channel carries only the
communicated value $v$ (cf. rule R-Send and R-Rec). To realize the boundedness
of the channels, the semantics still maintains the two parts of a channel: the
forward channel for communication, and the backward channel for “flow-
control.” The backward channel $c_{b}$ does not carry any information, just
the number of entries representing still empty slots in the forward channel.
We use the unit value $()$ for that, and initially, the backward channel is
filled with a number of $()$’s corresponding to the capacity of the channel
(see rule R-Make).

For channel communication, the semantics distinguished between synchronous
communication, i.e., a “rendezvous” over a channel of capacity 0, and
asynchronous communication, with a channel of non-zero, but finite capacity.
For the asynchronous case, both sending and receiving

Figure 18: Channel communication

##  Appendix 0.B Rules for the select statement

Rules dealing with the select statement semantics are given on Figure 0.B. The
R-Sel-Send and R-Sel-Rec rules apply to asynchronous channels and are
analogous to R-Send and R-Rec. The R-Sel-Sync rules apply to open synchronous
channels (i.e. the forward and backward queues are empty). The R-Sel-Rec$\bot$
is analogous to R-Rec$\bot$. Finally, the default rule (R-Sel-Def) applies
when no other select rule applies.

Figure 19: Operational semantics: Select statement

## Index

  * $(\\!\\!|E_{\mathit{hb}},\,z{:=}v|\\!\\!)$ §3.3
  * $\mathop{\leftarrow c}$ (receive) §3.1
  * $\bullet$ (empty configuration) §3.3
  * A-normal form §3.1
  * backward channel §3.2.3
  * bounded channel §2, §2, §3.2.3
  * $c\leftarrow v$ (send) §3.1
  * channel
    * backward §3.2.3
    * forward §3.2.3
  * channel name §3.1
  * choice
    * mixed §3.1
  * close a channel §3.1
  * data race §1, Definition 1, Definition 2
    * manifest Definition 1, Definition 2
      * trace Definition 2
  * DRF-SC §1
  * error control footnote 6
  * flow control §3.2.3
  * forward channel §3.2.3
  * Go §3.1
  * $\mathrel{\mathtt{let}}$ §3.1
  * $\mathrel{\mathtt{load}}z$ §3.1
  * local variable §3.1
  * $m(\\!\\!|z{:=}v|\\!\\!)$ ((recorded) write event) §3.3
  * $\mathrel{\mathtt{make}}(\mathrel{\mathtt{chan}}T,v)$ §3.1
  * memory §3.3
  * MESI §4.1
  * mixed choice §3.1
  * panic §3.1
  * properly manifest Definition 1, Definition 2
  * PSO footnote 1
  * $R$ (run-time configuration) §3.3
  * $r$ (local variable) §3.1
  * $R$ (run-time configuration) §3.3
  * run-time configuration §3.3
  * sequential composition §3.1
  * $\mathrel{\mathtt{stop}}$ §3.1
  * structural congruence §3.3
  * value §3.1
  * $z:=v$ §3.1


# Competing magnetic phases in the frustrated spin-1/2 chain compound
$\beta$-TeVO4 probed by NMR

M. Pregelj<EMAIL_ADDRESS>Jožef Stefan Institute, Jamova cesta 39, 1000
Ljubljana, Slovenia  A. Zorko  Jožef Stefan Institute, Jamova cesta 39, 1000
Ljubljana, Slovenia  Faculty of Mathematics and Physics, University of
Ljubljana, Jadranska u. 19, 1000 Ljubljana, Slovenia  D. Arčon  Jožef Stefan
Institute, Jamova cesta 39, 1000 Ljubljana, Slovenia  Faculty of Mathematics
and Physics, University of Ljubljana, Jadranska u. 19, 1000 Ljubljana,
Slovenia  M. Klanjšek  Jožef Stefan Institute, Jamova cesta 39, 1000
Ljubljana, Slovenia  N. Janša  Jožef Stefan Institute, Jamova cesta 39, 1000
Ljubljana, Slovenia  P. Jeglič  Jožef Stefan Institute, Jamova cesta 39, 1000
Ljubljana, Slovenia  O. Zaharko  Laboratory for Neutron Scattering and
Imaging, PSI, CH-5232 Villigen, Switzerland  S. Krämer  Laboratoire National
des Champs Magnétiques Intenses, LNCMI-CNRS (UPR3228), EMFL, Université
Grenoble Alpes, UPS and INSA Toulouse, Boîte Postale 166, 38042 Grenoble Cedex
9, France  M. Horvatić  Laboratoire National des Champs Magnétiques Intenses,
LNCMI-CNRS (UPR3228), EMFL, Université
Grenoble Alpes, UPS and INSA Toulouse, Boîte Postale 166, 38042 Grenoble Cedex
9, France  A. Prokofiev  Institute of Solid State Physics, Vienna University
of Technology, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria

###### Abstract

In frustrated spin-1/2 chains the competition between the nearest- and next-
nearest-neighbor exchange interactions leads to a rich phase diagram that
becomes even richer in the presence of perturbations in their material
realizations. These effects are still largely unexplored, so that new insight
into static and dynamic magnetism, in particular by sensitive local probes, is
highly desired. Here we present a comprehensive 17O nuclear magnetic resonance
study of $\beta$-TeVO4, where the anisotropy of the main exchange interactions
and additional weak interchain exchange interactions complement the
theoretical phase diagram. Our results confirm the dynamical nature of the
intriguing spin-stripe phase that has been reported in previous studies. In
addition, we find that the magnetic order in the high-field phase, which
develops just below the magnetization saturation, is consistent with an
unusual type of spin-density-wave (SDW) order with different alignments of the
magnetic moments on the neighboring chains. This is reminiscent of the
ordering in the SDW phase, realized in the absence of the magnetic field, and
is thus most likely stabilized by magnetic anisotropy.

††preprint: APS

##  I Introduction

Frustrated spin-1/2 chains, i.e., chains with nearest- and next-nearest-
neighbor Heisenberg exchange interactions, $J_{1}$ and $J_{2}$, respectively,
have been extensively studied [1, 2, 3, 4] due to their rich phase diagram [5,
6, 7]. Recently, the focus has been on compounds with ferromagnetic $J_{1}$
[8, 9, 10, 4], where a spin-nematic state, exhibiting long-range order of
magnetic quadrupoles born out of magnon pairing [11, 12, 13], has been
predicted [5, 6, 7]. Moreover, besides this enigmatic state, ferromagnetic
frustrated spin-1/2 chains show a plethora of other intriguing phases [5, 6].
To begin with, the zero-field magnetic ground state is characterized by long-
range order of vector-chiral (VC) correlations. With increasing the magnetic
field, a spin-density-wave (SDW) phase develops, which is characterized by
incommensurate amplitude modulation of the ordered magnetic moments. Just
below the magnetization saturation, the emergence of the spin-nematic state is
anticipated, but its realization is in real systems typically hindered by
perturbations, e.g., small additional intra- or inter-chain interactions or
magnetic anisotropy [4, 14, 15]. On the other hand, perturbations may lead to
novel phases with intricate magnetic properties that further enrich the phase
diagram of the frustrated spin-1/2 chain [16] and are yet to be explored
experimentally.

An intriguing compound containing zigzag chains of magnetic V4+ ($S$ = 1/2)
ions is $\beta$-TeVO4 [17]. This system exhibits a magnetic phase diagram
[Fig. 1(a)] that matches very well the theoretical phase diagram of
$J_{1}$-$J_{2}$ spin-1/2 chain with ferromagnetic $J_{1}$ $\sim$ $-38$ K and
antiferromagnetic $J_{2}$ $\sim$ $-J_{1}$ [18]. Yet, the exchange anisotropy
and weak interchain interactions lead to some unexpected properties. The
system orders below $T_{N1}$ = 4.6 K, where the SDW phase develops, whereas
the VC magnetic ground state is established below $T_{N3}$ = 2.3 K. Between
$T_{N2}$ = 3.3 K and $T_{N3}$, however, an extraordinary spin-stripe phase
emerges, where the two SDW order parameters that constitute the VC phase have
different magnetic ordering vectors [18]. This phase is characterized by
unusual spin dynamics in the MHz frequency range detected by muon spin
relaxation [19], which still lacks an independent experimental confirmation.
Applying the magnetic field at lowest temperatures induces the same spin-
stripe phase between $B_{C1}$ $\approx$ 2.5 T and $B_{C2}$ $\approx$ 5 T,
which is followed by the SDW phase at higher fields (Fig. 1). In the SDW phase
the magnetic ordering vector monotonically decreases with increasing magnetic
field [15], which complies with theoretical predictions [6]. However, this
dependence is broken at $B_{C3}$ $\approx$ 18 T, where, instead of the
anticipated spin-nematic phase, a high-field (HF) phase with yet unknown
incommensurate dipolar magnetic order has been found [15, 20].

Here we explore the magnetic phase diagram of $\beta$-TeVO4 by 17O nuclear
magnetic resonance (NMR). Previous low-magnetic-field NMR study [21] that has
been focused on 125Te nuclei was not able to probe the magnetism below
$\sim$10 K. This was due to direct involvement of the Te ions in the main
superexchange paths, leading to a very strong hyperfine interaction and
consequent extremely rapid relaxation. In contrast to Te, one of four
crystallographically inequivalent O sites experiences sufficiently small
hyperfine fields, allowing us to successfully probe the NMR response also in
the long-range-ordered magnetic phases. In fact, by extending the measurement
range down to 0.5 K and up to 21.8 T, we are able to confirm the dynamical
nature of the spin-stripe phase, reveal details about the high-field magnetic
structure, and observe potential magnetic-field-induced closing of the magnon
gap in the SDW phase. Our study, hereby, represents an important complementary
insight into the magnetic properties of $\beta$-TeVO4 and illuminates the
influence of anisotropic interactions on magnetic phases in the
$J_{1}$-$J_{2}$ spin-1/2 chain compounds from a local perspective.

Figure 1: (a) Magnetic phase diagram of $\beta$-TeVO4 in magnetic field
applied along the $a$ crystallographic axis. Inset: a part of the V4+ ($S$ =
1/2) zigzag chain, consisting of corner sharing VO5 pyramids that include four
crystalographically different O sites. $J_{1}$ is bridged by O atoms, whereas
$J_{2}$ is bridged via O-Te-O paths. (b) 17O NMR spectra for the highest-
frequency O site measured at different positions in the phase diagram [marked
by corresponding stars of the same color in (a)]. The horizontal square
brackets denote the U singularities corresponding to the same satellite line
split by the incommensurate SDW order. Only one side of the bracket can be
resolved in the HF phase.

##  II Experimental

The same single-crystal sample (1.00$\times$3.10$\times$7.55 mm3) was used as
in our previous NMR study [20]. The crystal was grown by chemical vapor
transport reaction using TeO2 and VO2 as starting materials and TeCl4 as a
transport agent. The enrichment of 8% 17O in the final product was achieved by
a preceding partial oxidation of V2O3 to VO2 by 17O2. The 17O NMR was measured
using a custom-built spectrometer at Jožef Stefan Institute, Slovenia, as well
as a high-field spectrometer at the Laboratoire National des Champs
Magnétiques Intenses (LNCMI), Grenoble, France, altogether covering the
magnetic field, $B$, range between 2.3 and 24 T. The shape of the coil was
adapted to fit the crystal (flat elongated plate), while a goniometer was used
to adjust the orientation of the sample with respect to the magnetic field
direction, which was limited to the crystallographic $ab$ plane. The
transverse magnetization, $M_{\perp}$, decay controlled by the spin-spin
relaxation $1/T_{2}$ was measured with a standard Hahn-echo pulse sequence and
fitted by a single-exponential function $\exp(-2\tau/T_{2})$, where $\tau$ is
the time between the $\pi/2$ and $\pi$ pulses. The decay of the longitudinal
magnetization, $M_{z}$, driven by the spin-lattice relaxation $1/T_{1}$ was
fitted to the expression for the outer-satellite line of a spin $I$ = 5/2
nucleus [22]:

$\begin{split}M_{z}(t)=&\,M_{0}-M_{1}[1/35\exp(-t/T_{1})\\\
&+3/14\exp(-3t/T_{1})+2/5\exp(-6t/T_{1})\\\
&+2/7\exp(-10t/T_{1})+1/14\exp(-15t/T_{1})],\end{split}$ (1)

where $t$ is the time after the initial inversion pulse, $M_{0}$ = $M_{z}(t$
$\to$ $\infty)$ the equilibrium magnetization and $M_{1}$ = $M_{0}-M_{z}(t$ =
0) the amplitude of the magnetization inversion.

##  III Results

Figure 2: (a) The temperature dependence of the 17O NMR spectrum measured at
9.4 T. The horizontal square brackets denote the U singularities corresponding
to the highest-frequency satellite. Five-pointed forks denote individual
quintets obtained by fit (see text for details). (b) Shift of the lowest- and
highest-frequency satellite lines (symbols), marked by arrows in (a), in
comparison to magnetic susceptibility data from Ref. [18]. The dotted line
represents the $(T-T_{N1})^{\beta}$ model with $\beta$ = 0.37(3). (c)
Clogston-Jaccarino plot for the lowest- and highest-frequency satellite lines
above $T_{N1}$. Solid lines represent linear fits.

To determine the coupling between the 17O nuclei and the magnetic moments of
the magnetic ions as well as to obtain an overview of the magnetic behavior in
the paramagnetic and SDW states, we first measured temperature dependence of
the 17O NMR spectrum of $\beta$-TeVO4 in the magnetic field $B$ = 9.4 T
applied along the crystallographic $a$ axis, i.e., perpendicular to the VO5
chains. At room temperature, the spectrum comprises twenty sharp peaks [Fig.
2(a)] corresponding to 17O ($I$ = 5/2) quintets from four crystallographically
inequivalent O sites split by quadrupole interaction. We note that some of the
spectral lines overlap. On cooling three quintets shift to lower frequencies
while the fourth quintet shifts to higher frequencies, i.e., in agreement with
our high-filed study [20]. This response becomes obvious when plotting the
temperature dependence of the positions of two representative spectral lines
[Fig. 2(b)] marked by arrows in Fig. 2(a). The two shifts are significantly
different, as the line moving to lower frequencies shifts $\sim$7-times more
than the one shifting to higher frequencies, indicating substantially weaker
hyperfine coupling for the latter. Both shifts exhibit a pronounced maximum at
$\sim$12.7 K, perfectly matching the temperature dependence of the magnetic
susceptibility $\chi$ [Fig. 2(b)]. In fact, plotting the magnetic hyperfine,
i.e., Knight, shift $K$ = $(\nu-\nu_{L})/\nu_{L}$, where $\nu_{L}$ =
${}^{17}\gamma B$ is the Larmor frequency, ${}^{17}\gamma$ = 5.772 MHz/T is
the 17O gyromagnetic ratio and $B$ is the applied magnetic field, as a
function of $\chi$, i.e., the so called Clogston-Jaccarino plot [Fig. 2(c)],
we find a clear linear dependence, where the slope is directly proportional to
the strength of the hyperfine interactions [23]. The derived hyperfine
couplings for $B||a$ are $A_{\text{low}}$ = $-$1.12(2) T/$\mu_{B}$ and
$A_{\text{high}}$ = 0.163(2) T/$\mu_{B}$ for the low and the high frequency
site, respectively. The bigger value is comparable with the values derived for
the 125Te NMR amounting up to 5.8 T/$\mu_{B}$ [21], leading to a similar loss
of the NMR signal below 10 K. The other coupling value is $\sim$7 times
smaller, which explains why this site could be used for the low-temperature
NMR study.

On further cooling, the spectral lines shifting to lower frequencies loose
intensity and finally disappear below $\sim$7 K [Fig. 2(a)], which is due to
extremely fast spin-spin ($T_{2}$) relaxation driven by stronger hyperfine
couplings of these lines compared to the lines that shift to higher
frequencies. Consequently, at 4.5 K, i.e., just above the magnetic transition,
the NMR spectrum reduces to only five lines, namely, a single quintet
corresponding to the O site with the weakest hyperfine coupling. This
particular O site is most likely the apical O1 site that is isolated from the
main exchange pathways [inset in Fig. 1(a)]. Below $T_{N1}$, each of the five
lines broadens into a U-shaped spectrum [Fig. 2(a)], characteristic of
incommensurate amplitude-modulated magnetic order in the SDW phase with the
width of the U splitting determined by the amplitude of the projection of the
magnetic-moments modulation on the external magnetic field. The position of
the outer satellite line below $T_{N1}$ [Fig. 2(b)], therefore, reflects the
amplitude of the ordered magnetic moments in the SDW phase, which scales with
temperature as $(T-T_{N1})^{\beta}$, yielding $\beta$ = 0.37(3) [see Figs.
2(b) and 3(c)]. The latter value is in agreement with the critical exponents
for the order parameter in three-dimensional (3D) spin lattices, ranging from
0.33 to 0.36 for the Ising and Heisenberg case, respectively, [24, 25].
Indeed, the 3D nature of critical correlations is in line with the existence
of sizable interchain, i.e., 3D, interactions identified in previous studies
[26, 27].

Figure 3: (a) The temperature dependence of the 17O NMR spectrum measured at
2.35 T at low temperatures. Five-pointed forks denote individual quintets
obtained by fit (see text for details). (b) The peak frequencies demonstrating
the splitting of the five spectral lines due to the establishment of the
incommensurate magnetic order. Solid lines are guides to the eye. Colored
dashed lines mark the positions of the spectral lines in the paramagnetic
state at 5 K marked by arrows in (a). (c) The temperature dependence of the
normalized NMR splittings (solid symbols) compared to the magnetic order
parameter $\sigma$ derived from neutron diffraction (empty symbols) [19]. The
solid line is the fit of the $(T-T_{N1})^{\beta}$ model, with $\beta$ =
0.37(3).

To explore all the low-field ordered phases in the same applied field, we
performed additional 17O NMR measurements at 2.35 T. The temperature
dependence of the spectrum measured below 5 K is shown in Fig. 3(a). The
relative positions of the five spectral lines of the remaining quintet are
significantly different from those collected at 9.4 T. This is a consequence
of the 17O Larmor frequency at 2.35 T being strongly affected by the
quadrupolar interaction [28], determined by the 17O quadrupolar moment and the
local electric-field-gradient (EFG) tensor. Nevertheless, below $T_{N1}$,
similarly as for measurements at 9.4 T, also at 2.35 T each of the five lines
splits into two that monotonically shift away from each other on cooling [Fig.
3(b)], as expected for the SDW, i.e., amplitude-modulated, phase [29]. In
fact, the temperature dependence of the U-splitting [Fig. 3(c)] follows the
same $(T-T_{N1})^{\beta}$ dependence as the outer satellite at 9.4 T [Fig.
2(b)], while both dependencies exactly match the evolution of the magnetic
order parameter obtained from neutron diffraction at zero field [Fig. 3(c)]
[19].

A detailed inspection of the U-split lines in the SDW phase [Fig. 1(b)]
provides further insight into quadrupolar effects. Namely, while the U
splittings of different lines in the quintets are rather similar at 9.4 T,
they differ significantly at 2.35 T, yet again indicating that quadrupolar
effects at 2.35 T are strong and that first-order perturbation theory no
longer applies. It is thus convenient to consider the edge singularities of
the U-shaped lines as two quintets corresponding to two magnetically-
inequivalent O1 sites; one that on cooling shifts to higher and one that
shifts to lower frequencies [Fig. 2(a) and 3(a)]. The observed behavior can
then be described by exact diagonalization of the quadrupolar Hamiltonian
$H=\hbar\gamma{\bf I}\cdot{\bf
B_{\text{eff}}}+h\nu_{Q}[3I_{z}^{2}-I(I+1)+\eta(I_{+}^{2}+I_{-}^{2})/2]/6$,
where $h$ is the Planck constant, $\hbar$ = $h/2\pi$, I =
$(I_{x},\,I_{y},\,I_{z})$ is the nuclear spin vector, $I_{\pm}$ = $I_{x}$
$\pm$ $iI_{y}$, B${}_{\text{eff}}$ is the effective magnetic field, $\nu_{Q}$
is the quadrupolar frequency, and $\eta$ is the EFG asymmetry parameter [30].
Adjusting $\nu_{Q}$, $\eta$ as well as the orientation and the size of
B${}_{\text{eff}}$, we can exactly reproduce the two quintets measured in the
paramagnetic phase, i.e., at 5 K, in the field of 2.35 and 9.4 T as well as
the four (two in 2.35 T and two in 9.4 T) quintets in the SDW phase [five-
pointed forks in Fig. 2(a) and 3(a)]. The derived parameters are $\nu_{Q}$ =
1.05(5) MHz, $\eta$ = 0.62(5), and the orientations of the two principal EFG
axes pointing along (0.7, 0, $-$0.7) and (0.5, 0.7, 0.5) in the $a^{*}bc$
crystalographic system, which is in line with the fact that the O1 local
environment possesses no symmetry restrictions on the EFG tensor. We note that
in our calculations a misalignment of $\sim$3∘ between different experiments
has been considered. Besides, we find that the local fields for the U-edge
singularities in the SDW phase deviate almost symmetrically from the
paramagnetic value by $\sim$ $\pm$0.05 T in size and $\sim$ $\pm 2^{\circ}$ in
orientation. Considering the hyperfine coupling constant $A_{\text{high}}$
this suggests that the amplitude of the modulated magnetic moments is
$\sim$0.3 $\mu_{B}$, i.e., in agreement with the neutron diffraction results
[31]. We stress that the obtained parameters are not exact and should be
refined by more involved angular-dependent measurements, yet, they still
provide a reasonable estimate of the EFG strength at the O1 site. Finally, we
point out that in contrast to the results at 9.4 T, the U-shape of the
spectral lines at 2.35 T is much less pronounced, as edge singularities are
far more dominant, indicating that middle parts of the spectrum associated
with the reduced static magnetic moments are suppressed. This can be explained
by enhanced $T_{2}$ relaxation, driven phason excitations of the SDW order
[32, 33], that have for small wave vectors (for the NMR case) most pronounced
effect on the reduced magnetic moments, i.e., on the middle of the U spectrum.

At the transition into the spin-stripe phase, i.e., at $T_{N2}$, the spectrum
dramatically changes. The signal is almost completely lost, reflecting even
stronger $T_{2}$ relaxation, which corroborates the dynamical nature of the
spin-stripe phase on the MHz time scale [19]. We note that the spectra in Fig.
3(a) were measured with the shortest possible $\tau$ = 20 $\mu$s allowed by
our spectrometer. On further cooling, some spectral features reappear,
exhibiting an extended array of singularities at the lowest accessible
temperature of 1.7 K. However, this temperature point lies at the boarder
between the spin-stripe and the VC phase, so the resulting spectrum does not
fully represent the magnetic order in the VC phase. Hence, based on the
obtained spectrum we cannot learn much about the VC phase, except for the fact
that the $T_{2}$ relaxation in this phase is slowing down, supporting the
static long-range order in this phase.

Figure 4: A representative low-temperature (a) $T_{2}$ and (b) $T_{1}$
measurements (symbols), i.e., transverse- and longitudinal-magnetization
($M_{\perp}$ and $M_{z}$) decays, respectively, for the highest-frequency
satellite of the only-remaining low-temperature O site measured at 9.4 T and
the corresponding fits (lines; see text). (c) The temperature dependence of
the spin-lattice and the spin-spin relaxation rates, $1/T_{1}$ and $1/T_{2}$,
respectively, measured for the highest-frequency satellite. The solid line
shows a fit of the $1/T_{1}$ $\propto$ $T^{5}$ model. Inset: the temperature
dependence of $T_{1}/T_{2}$.

Next, we explore the spin dynamics through the temperature dependencies of the
spin-lattice and the spin-spin relaxation rates, $1/T_{1}$ and $1/T_{2}$,
respectively. Both relaxation rates were measured for the outer satellite line
[marked by the blue arrow in Fig. 2(a)], i.e., the line corresponding to the
$m$ = $-$5/2 to $m$ = $-$3/2 transition [Fig. 4(c)]. Above the magnetic
transition, the nuclear magnetization curves [Fig. 4(a) and (b)] yield well-
defined and almost temperature independent $1/T_{1}$ and $1/T_{2}$, as
expected for a paramagnet with strong antiferromagnetic correlations [34]. In
fact, the temperature independence of the $T_{1}/T_{2}$ $\approx$ 7 [inset in
Fig. 4(c)] ratio suggests that the $T_{2}$ relaxation is driven by
fluctuations of nearby electronic spins, i.e., by the Redfield process [35,
36], yielding

$\begin{split}T^{-1}_{2z}=&\frac{1}{2}(T^{-1}_{1x}+T^{-1}_{1y}-T^{-1}_{1z})|_{\omega=0}\\\
&+[I(I+1)-m(m-1)-1/2]T^{-1}_{1z}(\omega).\end{split}$ (2)

Taking $m$ = 5/2, as the relaxation rates were measured on the outer
satellite, the second term in Eq. (2) yields a prefactor 4.5, which suggests
that the $\omega=0$ contribution [the first term in Eq. (2)] is rather
anisotropic, implying $T^{-1}_{1x}+T^{-1}_{1y}$ $\approx$ $6T^{-1}_{1z}$. This
can be due to the anisotropy of the spin fluctuations as well as that of the
hyperfine couplings, since both contributions are allowed by a low symmetry of
the O1 site. In addition, the measurements at 2.35 T [Fig. 4(c)] show that
relaxation times at 10 K are the same as those obtained at 9.4 T. This
indicates that in the paramagnetic state relaxation processes are nearly
frequency independent, which is indeed expected in the extreme-narrowing limit
ensured by strong antiferromagnetic correlations.

At $T_{N1}$, the temperature-independent trend is abruptly broken, as on
further cooling $1/T_{1}$ decreases proportional to $T^{p}$ with $p$ = 5.0(2)
for 9.4 T [solid line in Fig. 4(c)]. The $T^{5}$ dependence of $1/T_{1}$ has
been observed in several quasi one-dimensional spin systems [37, 38] and is,
in fact, predicted for a three-magnon process for $T\gg\Delta$, where $\Delta$
is the excitation gap of magnons [39]. The fact that neutron scattering
experiment in the absence of the magnetic field revealed a magnon gap of
$\sim$7 K (0.6 meV) (see supplementary information to Ref. 19) suggests that
this gap is almost completely closed by the applied magnetic field of 9.4 T
($\sim$6.3 K), pushing the system into the limit $T\gg\Delta$ in agreement
with our NMR experiment. On the other hand, $1/T_{2}$ below $T_{N1}$ gradually
increases, which indicates slowing down of the spin-spin relaxation due to the
establishment of long-range magnetic ordering. We note that the data collected
at 2.35 T exhibit less pronounced temperature dependencies than at 9.4 T and
are most likely affected by the proximity of the $T_{N2}$ magnetic transition
[Fig. 1(a)], occurring less then 1.5 K below $T_{N1}$. Moreover, at 2.35 T we
were unable to consistently measure relaxation times below $T_{N2}$, as there
is no distinct spectral feature [Fig. 3(a)] that would allow for reliable
measurements at the same NMR transition. At last, we point out the absence of
critical fluctuations in the paramagnetic state just above the $T_{N1}$
transition. Considering a low symmetry of the O1 site [inset in Fig.
1(a)][17], a symmetry-based filtration of critical fluctuations is highly
unlikely thus implying that the critical region is very narrow, i.e., in line
with a three-dimensional nature of the $T_{N1}$ magnetic transition.

Figure 5: 17O NMR spectra measured in high-magnetic fields in the
incommensurate HF phase and in the saturated state.

Finally, we performed 17O NMR also in high magnetic fields at 0.5 K. We
measured NMR spectra in the HF phase, i.e., at 20.5 T, as well as above the
magnetization saturation, at 21.8 T (Fig. 5). This complements the data from
our previous study [20], where spectra were measured at fixed frequency by
sweeping the magnetic field. Clearly, the $T_{2}$ relaxation at these fields
is significantly reduced, as in the saturated phase all four quintets are
observed again. Three quintets are found below the Larmor frequency,
$\nu_{L}$, while one quintet is found above $\nu_{L}$, in agreement with the
response observed in the paramagnetic phase at 9.4 T. Moreover, the large
separation of the quintets at 21.8 T allows us to distinguish the difference
in the EFG strengths at different oxygen sites, which are responsible for the
splittings of the lines within the individual quintets. In particular, for
$B||a$, these splittings amount to $\sim$0.25, 0.13, 0.35 and 0.19 MHz for the
O sites sorted from the lowest to the highest frequency, respectively. We
note, however, that these values reflect the EFG strengths along the applied
magnetic field, i.e., along the $a$ axis, and can change significantly for
different field orientation.

In the HF phase, however, the contributions from the four O sites overlap. In
particular, the low-frequency part of the spectrum is a complex mixture of
several overlapping contributions that cannot be disentangled. On the other
hand, the highest frequency U-shaped feature [Fig. 1(b)] is most likely
associated with the high-frequency O site (assigned to O1 site). In fact, it
appears that this part of the spectrum is composed of two sets of overlapping
U-split quintets [Fig. 1(b)], which might develop due to symmetry reduction of
the magnetic order in the HF phase. Namely, the two neighboring chains
probably exhibit different alignments of the SDW-modulation eigenaxes, which
would lead to different projections of the magnetic moments on the external
magnetic field. Similar ordering was found by neutron diffraction in the SDW
phase in the absence of the magnetic field [31], where magnetic moments lie
along the $a$ axis on one chain and along the $c$ axis on the other. The
possible similarity between the orders in these two phases is also supported
by the fact that both phases exhibit similar magnetic ordering vectors [15],
highlighting the importance of the magnetic anisotropy in this system.

##  IV Summary and conclusions

By employing 17O nuclear magnetic resonance (NMR), we have investigated
magnetic properties of $\beta$-TeVO4 for $B||a$. We have determined the
hyperfine coupling constants for the two out of four crystalographically
different O sites, i.e., the two that exhibit the smallest and the largest
frequency shifts, which differ by a factor of $\sim$7\. The shift of one 17O
quintet is considerably smaller than those of the other three, implying that
it probably corresponds to the isolated O1 site, positioned on the top of the
VO5 pyramid. For this site, we have also estimated the local EFG tensor. In
addition, we have determined the critical exponent for the magnetic order
parameter, $\beta$ = 0.37(3), which is in agreement with the theoretical
predictions for 3D spin models, and thus corroborates sizable interchain
interactions [27]. Furthermore, in the spin-stripe phase we have confirmed the
presence of strong dynamics in the MHz range that is driven by the unusual
low-energy bound-phason excitations emerging due to a weak forth-order
exchange coupling term [19]. On the other hand, in the SDW phase, the
temperature dependence of the $1/T_{1}$ relaxation at 9.4 T exhibits the
$T^{5}$ dependence, implying that the magnon gap in this phase is almost
completely closed by the applied magnetic field. Finally, the U shape of the
high-frequency part of the spectrum in the HF phase suggests that the
neighboring chains might exhibit different alignments of the magnetic moments,
similarly as found in the zero-field VC magnetic ground state. Still, detailed
understanding of the HF phase calls for future density-functional-theory
calculations of the EFG tensors at all O sites, which would allow for more
involved modeling of the HF NMR spectrum. For such calculations the derived
hyperfine-coupling constants and EFG parameters should serve as an important
point of reference, allowing an independent verification of the calculation
precision.

###### Acknowledgements.

This work has been funded by the Slovenian Research Agency (projects J1-9145,
J1-2461, N1-0148 and J2-2513, and program No. P1-0125) and the Swiss National
Science Foundation (project SCOPES IZ73Z0_152734/1). We acknowledge the
support of the LNCMI-CNRS, member of the European Magnetic Field Laboratory
(EMFL).

## References

  * Castilla  _et al._ [1995] G. Castilla, S. Chakravarty, and V. J. Emery, Quantum Magnetism of CuGeO3, Phys. Rev. Lett. 75, 1823 (1995).
  * Seki  _et al._ [2008] S. Seki, Y. Yamasaki, M. Soda, M. Matsuura, K. Hirota, and Y. Tokura, Correlation between spin helicity and an electric polarization vector in quantum-spin chain magnet LiCu2O2, Phys. Rev. Lett. 100, 127201 (2008).
  * Nishimoto  _et al._ [2011] S. Nishimoto, S.-L. Drechsler, R. O. Kuzian, J. VanDenBrink, J. Richter, W. E. A. Lorenz, Y. Skourski, R. Klingeler, and B. Büchner, Saturation field of frustrated chain cuprates: broad regions of predominant interchain coupling, Phys. Rev. Lett. 107, 097201 (2011).
  * Cemal  _et al._ [2018] E. Cemal, M. Enderle, R. K. Kremer, B. Fåk, E. Ressouche, J. P. Goff, M. V. Gvozdikova, M. E. Zhitomirsky, and T. Ziman, Field-induced states and excitations in the quasicritical spin-1/2 chain linarite, Phys. Rev. Lett. 120, 067203 (2018).
  * Hikihara  _et al._ [2008] T. Hikihara, L. Kecke, T. Momoi, and A. Furusaki, Vector chiral and multipolar orders in the spin-1/2 frustrated ferromagnetic chain in magnetic field, Phys. Rev. B 78, 144404 (2008).
  * Sudan  _et al._ [2009] J. Sudan, A. Lüscher, and A. M. Läuchli, Emergent multipolar spin correlations in a fluctuating spiral: The frustrated ferromagnetic spin-1/2 Heisenberg chain in a magnetic field, Phys. Rev. B 80, 140402(R) (2009).
  * Hikihara  _et al._ [2010] T. Hikihara, T. Momoi, A. Furusaki, and H. Kawamura, Magnetic phase diagram of the spin-1 2 antiferromagnetic zigzag ladder, Phys. Rev. B 81, 224433 (2010).
  * Mourigal  _et al._ [2012] M. Mourigal, M. Enderle, B. Fåk, R. K. Kremer, J. M. Law, A. Schneidewind, A. Hiess, and A. Prokofiev, Evidence of a bond-nematic phase in LiCuVO4, Phys. Rev. Lett. 109, 027203 (2012).
  * Willenberg  _et al._ [2016] B. Willenberg, M. Schäpers, A. U. B. Wolter, S.-L. Drechsler, M. Reehuis, J.-U. Hoffmann, B. Büchner, A. J. Studer, K. C. Rule, B. Ouladdiaf, S. Süllow, and S. Nishimoto, Complex Field-Induced States in Linarite ${\mathrm{PbCuSO}}_{4}(\mathrm{OH}{)}_{2}$ with a Variety of High-Order Exotic Spin-Density Wave States, Phys. Rev. Lett. 116, 047202 (2016).
  * Orlova  _et al._ [2017] A. Orlova, E. L. Green, J. M. Law, D. I. Gorbunov, G. Chanda, S. Krämer, M. Horvatić, R. K. Kremer, J. Wosnitza, and G. L. J. A. Rikken, Nuclear magnetic resonance signature of the spin-nematic phase in LiCuVO4 at high magnetic fields, Phys. Rev. Lett. 118, 247201 (2017).
  * Chubukov [1991] A. V. Chubukov, Chiral, nematic, and dimer states in quantum spin chains, Phys. Rev. B 44, 4693 (1991).
  * Shannon  _et al._ [2006] N. Shannon, T. Momoi, and P. Sindzingre, Nematic order in square lattice frustrated ferromagnets, Phys. Rev. Lett. 96, 027213 (2006).
  * Zhitomirsky and Tsunetsugu [2010] M. E. Zhitomirsky and H. Tsunetsugu, Magnon pairing in quantum spin nematic, EPL 92, 37001 (2010).
  * Bush  _et al._ [2018] A. A. Bush, N. Büttgen, A. A. Gippius, M. Horvatić, M. Jeong, W. Kraetschmer, V. I. Marchenko, Y. A. Sakhratov, and L. E. Svistov, Exotic phases of frustrated antiferromagnet ${\mathrm{LiCu}}_{2}{\mathrm{O}}_{2}$, Phys. Rev. B 97, 054428 (2018).
  * Pregelj  _et al._ [2019a] M. Pregelj, A. Zorko, M. Klanjšek, O. Zaharko, J. S. White, O. Prokhnenko, M. Bartkowiak, H. Nojiri, H. Berger, and D. Arčon, Magnetic ground state of the frustrated spin-$\frac{1}{2}$ chain compound $\beta\text{--}\mathrm{TeVO}{}_{4}$ at high magnetic fields, Phys. Rev. B 100, 094433 (2019a).
  * Ueda and Onoda [2020] H. Ueda and S. Onoda, Roles of easy-plane and easy-axis XXZ anisotropy and bond alternation in a frustrated ferromagnetic spin-1/2 chain, Phys. Rev. B 101, 224439 (2020).
  * Meunier  _et al._ [1973] G. Meunier, J. Darriet, and J. Galy, L’oxyde double TeVO4 II. Structure cristalline de TeVO4-$\beta$-relations structurales, J. Sol. Stat. Chem. 6, 67 (1973).
  * Pregelj  _et al._ [2015] M. Pregelj, A. Zorko, O. Zaharko, H. Nojiri, H. Berger, L. Chapon, and D. Arčon, Spin-stripe phase in a frustrated zigzag spin-1/2 chain, Nat. Commun. 6, 7255 (2015).
  * Pregelj  _et al._ [2019b] M. Pregelj, A. Zorko, M. Gomilšek, M. Klanjšek, O. Zaharko, J. White, H. Luetkens, F. Coomer, T. Ivek, D. Góngora, H. Berger, and D. Arčon, Elementary excitation in the spin-stripe phase in quantum chains, npj Quantum Mater. 4, 22 (2019b).
  * Pregelj  _et al._ [2020] M. Pregelj, A. Zorko, D. Arčon, M. Klanjšek, O. Zaharko, S. Krämer, M. Horvatić, and A. Prokofiev, Thermal effects versus spin nematicity in a frustrated spin-1/2 chain, Phys. Rev. B 102, 081104(R) (2020).
  * Weickert  _et al._ [2016] F. Weickert, N. Harrison, B. L. Scott, M. Jaime, A. Leitmäe, I. Heinmaa, R. Stern, O. Janson, H. Berger, H. Rosner, _et al._ , Magnetic anisotropy in the frustrated spin-chain compound $\beta$-TeVO4, Phys. Rev. B 94, 064403 (2016).
  * Simmons  _et al._ [1962] W. W. Simmons, W. J. O’Sullivan, and W. A. Robinson, Nuclear spin-lattice relaxation in dilute paramagnetic sapphire, Phys. Rev. 127, 1168 (1962).
  * Clogston  _et al._ [1964] A. M. Clogston, V. Jaccarino, and Y. Yafet, Interpretation of knight shifts and susceptibilities of transition metals: Platinum, Phys. Rev. 134, A650 (1964).
  * Pelissetto and Vicari [2002] A. Pelissetto and E. Vicari, Critical phenomena and renormalization-group theory, Phys. Rep. 368, 549 (2002).
  * Chaikin and Lubensky [2000] P. M. Chaikin and T. C. Lubensky,  _Principles of condensed matter physics_ , Vol. 1 (Cambridge university press Cambridge, 2000).
  * Saúl and Radtke [2014] A. Saúl and G. Radtke, Density functional approach for the magnetism of $\beta$-TeVO4, Phys. Rev. B 89, 104414 (2014).
  * Pregelj  _et al._ [2018] M. Pregelj, O. Zaharko, U. Stuhr, A. Zorko, H. Berger, A. Prokofiev, and D. Arčon, Coexisting spinons and magnons in frustrated zigzag spin-1/2 chain compound $\beta$-TeVO4, Phys. Rev. B 98, 094405 (2018).
  * Bain and Khasawneh [2004] A. D. Bain and M. Khasawneh, From NQR to NMR: The complete range of quadrupole interactions, Concepts in Magnetic Resonance Part A 22A, 69 (2004).
  * Pregelj  _et al._ [2013] M. Pregelj, P. Jeglič, A. Zorko, O. Zaharko, T. Apih, A. Gradišek, M. Komelj, H. Berger, and D. Arčon, Evolution of magnetic and crystal structures in the multiferroic FeTe2O5Br, Phys. Rev. B 87, 144408 (2013).
  * Aimo  _et al._ [2009] F. Aimo, S. Krämer, M. Klanjšek, M. Horvatić, C. Berthier, and H. Kikuchi, Spin configuration in the 1/3 magnetization plateau of azurite determined by nmr, Phys. Rev. Lett. 102, 127205 (2009).
  * Pregelj  _et al._ [2016] M. Pregelj, O. Zaharko, M. Herak, M. Gomilšek, A. Zorko, L. C. Chapon, F. Bourdarot, H. Berger, and D. Arčon, Exchange anisotropy as mechanism for spin-stripe formation in frustrated spin chains, Phys. Rev. B 94, 081114(R) (2016).
  * Zumer and Blinc [1981] S. Zumer and R. Blinc, Nuclear spin-lattice relaxation in incommensurate systems, J. Phys. C 14, 465 (1981).
  * Blinc [1981] R. Blinc, Magnetic resonance and relaxation in structurally incommensurate systems, Phys. Rep. 79, 331 (1981).
  * Moriya [1956] T. Moriya, Nuclear magnetic relaxation in antiferromagnetics, Prog. Theor. Phys. 16, 23 (1956).
  * Horvatić and Berthier [2002] M. Horvatić and C. Berthier, NMR studies of low-dimensional quantum antiferromagnets, in  _High Magnetic Fields. Lecture Notes in Physics_, Vol. 595, edited by C. Berthier, L. P. Lévy, and G. Martinez (Springer Berlin Heidelberg., 2002) pp. 191–210.
  * Tokunaga  _et al._ [2015] Y. Tokunaga, D. Aoki, H. Mayaffre, S. Krämer, M.-H. Julien, C. Berthier, M. Horvatić, H. Sakai, S. Kambe, and S. Araki, Reentrant Superconductivity Driven by Quantum Tricritical Fluctuations in URhGe: Evidence from 59Co NMR in URh0.9Co0.1Ge, Phys. Revi. Lett. 114, 216401 (2015).
  * Mayaffre  _et al._ [2000] H. Mayaffre, M. Horvatić, C. Berthier, M.-H. Julien, P. Ségransan, L. Lévy, and O. Piovesana, NMR Evidence for a “Generalized Spin-Peierls Transition” in the High-Magnetic-Field Phase of the Spin Ladder Cu2(C5H${}_{1}2$N2)2Cl4, Phys. Rev. Lett. 85, 4795 (2000).
  * Jeong  _et al._ [2017] M. Jeong, H. Mayaffre, C. Berthier, D. Schmidiger, A. Zheludev, M. Horvatić, _et al._ , Magnetic-order crossover in coupled spin ladders, Phys. Rev. Lett. 118, 167206 (2017).
  * Beeman and Pincus [1968] D. Beeman and P. Pincus, Nuclear spin-lattice relaxation in magnetic insulators, Phys. Rev. 166, 359 (1968).