SMDTN_Ncomm_v22.pdf

Previous address: ]Laboratory for Solid State Physics, ETH Zürich, Switzerland

# Pressure-tuned quantum criticality in the large-$D$ antiferromagnet DTN

Kirill Yu. Povarov<EMAIL_ADDRESS>[  Dresden High Magnetic Field
Laboratory (HLD-EMFL) and Würzburg-Dresden Cluster of Excellence ct.qmat,
Helmholtz-Zentrum Dresden-Rossendorf (HZDR), 01328 Dresden, Germany  David E.
Graf  National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA
Andreas Hauspurg  Dresden High Magnetic Field Laboratory (HLD-EMFL) and
Würzburg-Dresden Cluster of Excellence ct.qmat, Helmholtz-Zentrum Dresden-
Rossendorf (HZDR), 01328 Dresden, Germany  Institut für Festkörper- und
Materialphysik, Technische Universität Dresden, 01062 Dresden, Germany  Sergei
Zherlitsyn  Dresden High Magnetic Field Laboratory (HLD-EMFL) and Würzburg-
Dresden Cluster of Excellence ct.qmat, Helmholtz-Zentrum Dresden-Rossendorf
(HZDR), 01328 Dresden, Germany  Joachim Wosnitza  Dresden High Magnetic Field
Laboratory (HLD-EMFL) and Würzburg-Dresden Cluster of Excellence ct.qmat,
Helmholtz-Zentrum Dresden-Rossendorf (HZDR), 01328 Dresden, Germany  Institut
für Festkörper- und Materialphysik, Technische Universität Dresden, 01062
Dresden, Germany  Takahiro Sakurai  Research Facility Center for Science and
Technology, Kobe University, Kobe, 657-8501, Japan  Hitoshi Ohta  Molecular
Photoscience Research Center, Kobe University, Kobe 657-8501, Japan  Graduate
School of Science, Kobe University, Kobe 657-8501, Japan  Shojiro Kimura
Institute for Materials Research, Tohoku University, Sendai 980-8578, Japan
Hiroyuki Nojiri  Institute for Materials Research, Tohoku University, Sendai
980-8578, Japan  V. Ovidiu Garlea  Neutron Scattering Division, Oak Ridge
National Laboratory, Oak Ridge, TN 37831, USA  Andrey Zheludev  Laboratory for
Solid State Physics, ETH Zürich, Switzerland  Armando Paduan-Filho  Instituto
de Fisica, Universidade de São Paulo, 05315-970 São Paulo, Brazil  Michael
Nicklas  Max Planck Institute for Chemical Physics of Solids, 01187 Dresden,
Germany  Sergei A. Zvyagin<EMAIL_ADDRESS>Dresden High Magnetic Field
Laboratory (HLD-EMFL) and Würzburg-Dresden Cluster of Excellence ct.qmat,
Helmholtz-Zentrum Dresden-Rossendorf (HZDR), 01328 Dresden, Germany

###### Abstract

Strongly correlated spin systems can be driven to quantum critical points via
various routes. In particular, gapped quantum antiferromagnets can undergo
phase transitions into a magnetically ordered state with applied pressure or
magnetic field, acting as tuning parameters. These transitions are
characterized by $z=1$ or $z=2$ dynamical critical exponents, determined by
the linear and quadratic low-energy dispersion of spin excitations,
respectively. Employing high-frequency susceptibility and ultrasound
techniques, we demonstrate that the tetragonal easy-plane quantum
antiferromagnet NiCl${}_{2}\cdot$4SC(NH2)2 (aka DTN) undergoes a spin-gap
closure transition at about $4.2$ kbar, resulting in a pressure-induced
magnetic ordering. The studies are complemented by high-pressure-electron
spin-resonance measurements confirming the proposed scenario. Powder neutron
diffraction measurements revealed that no lattice distortion occurs at this
pressure and the high spin symmetry is preserved, establishing DTN as a
perfect platform to investigate $z=1$ quantum critical phenomena. The
experimental observations are supported by DMRG calculations, allowing us to
quantitatively describe the pressure-driven evolution of critical fields and
spin-Hamiltonian parameters in DTN.

Magnetic insulators with their short-range interactions and well-controlled
effective Hamiltonians offer an ideal playground for studying quantum critical
phenomena induced by various stimuli [1, 2, 3]. Nowadays, great attention is
attracted by antiferromagnetic (AF) ordering, which can be induced, e.g., by
magnetic field in gapped quantum magnets [4, 5]. On the other hand, especially
tantalizing is the possibility of magnetic ordering, accompanied by
spontaneous spin-symmetry break-down without the aid of magnetic field, while
keeping the spectrum degenerate in absence of Zeeman splitting [6]. Such
transitions are known to appear in certain quantum antiferromagnets under
applied pressure. As Fig. 1a illustrates, both the magnetic field and pressure
can be regarded as a tuning parameter. Contrary to the field-induced
transitions, the pressure-induced case belongs to a another class of
universality [7, 8, 9]. The key distinction is the dynamic critical exponent
$z$ that links the characteristic energy of spin excitations to the momentum
relative to the critical one:

$\omega\propto|q-q_{c}|^{z}.$ (1)

Field-induced phase transitions are characterized by $z=2$ and a quadratic
low-energy excitation spectrum, while the pressure-induced case has $z=1$ and
a linear spectrum [6, 8]. The realizations of these two scenarios are
illustrated in Fig. 1b. The $z=1$ regime is very remarkable. It should have
mean-field critical exponents [8] and the universal scaling of dynamic
fluctuations at the same time [10]. This quantum critical point (QCP) is also
a natural habitat for a well-defined order-parameter amplitude mode [11, 12].

Figure 1: Quantum phase transitions in a spin-1 AF with large planar
anisotropy. a The generic phase diagram ($\eta$ is the tuning parameter; QCP
is the quantum critical point). The critical value of the tuning parameter is
labeled $\eta_{c}$. b Schematic view of magnetic excitation spectra for the
field- and pressure-induced quantum phase transitions ($q$ is the momentum;
$\hbar\omega$ is the excitation energy; $z$ is the dynamic critical exponent,
see Eq. 1).

There is a number of gapped quantum antiferromagnets, demonstrating field-
induced AF ordering with $z=2$ [5]. If the spin Hamiltonian of a system has
axial symmetry with respect to the applied field, the field-induced AF
ordering can be formally described as the Bose-Einstein condensation of
magnons by mapping the spin-1 system into a gas of semi-hard-core bosons [4,
13]. In order to perform accurate comparison with the theory, high-symmetry
spin systems are highly demanded. The compound NiCl${}_{2}\cdot$4SC(NH2)2
(dichloro-tetrakis thiourea-nickel(II), known as DTN) is one of them, having
tetragonal crystal structure (space group $I$4) and easily accessible critical
fields.

A good realization of the true $z=1$ QCP is actually hard to find. Most of the
materials known to show such a transition (TlCuCl3 [11, 12], KCuCl3 [14, 15],
(C4H12N2)Cu2Cl6 [16], and (C9H18N2)CuBr4 [17]) suffer from unwanted
anisotropies. For TlCuCl3 and (C9H18N2)CuBr4 biaxial anisotropy was
experimentally detected [18, 19]. In KCuCl3 the presence of biaxial anisotropy
follows from symmetry considerations, and is indirectly evidenced by a
peculiar orientation of sublattice magnetization in the ordered phase [14].
Such anisotropies eventually lead to the criticality of the Ising universality
class [20], different from the target QCP. This does not seem to be the case
for CsFeCl3 [21, 22], but there one has to deal with geometric exchange
frustration.

In the present study, utilizing high-pressure tunneling-diode oscillator (TDO)
susceptibility, ultrasound-propagation measurements and high-field electron
spin resonance (ESR) techniques, we demonstrate a pressure-induced phase
transition in DTN, that we ascribe to the long-sought $z=1$ criticality. This
transition resides at an easily accessible pressure of about $4.2$ kbar.
Neutron-diffraction measurements confirm the absence of a structural
transition and reveal an undistorted tetragonal symmetry near this QCP. At
higher pressure, we actually find an irreversible distortion of the lattice
occurring. We describe the experimentally measured phase boundaries employing
a quasi-1D numerical approximation, circumventing a renormalization of the
spin-Hamiltonian parameters by quantum fluctuations.

Figure 2: Pressure dependence of lattice parameters of DTN, as measured by
neutron diffraction. a Pressure dependence of the lattice parameter $a$. b
Pressure dependence of the lattice parameter $c$. Open and closed symbols
correspond to 300 and $1.8$ K data, respectively. Dotted lines denote the
irreversible structural phase transition at $P_{\mathrm{irr}}\sim 6$ kbar. Red
lines are fit results (see text for details).

Results

Magnetism in DTN: brief introduction. Magnetism in DTN originates from the
spin-1 Ni2+ ions forming a tetragonal lattice (we refer the reader to the
Supplemental Material S1 for detailed crystallographic information). The
magnetic properties are defined by the competition between the strong single-
ion planar anisotropy $D$, and the antiferromagnetic exchanges interactions
$J_{c}$ and $J_{a}\equiv J_{b}$ along the corresponding $c$ and $a,b$
directions. The effective Hamiltonian is:

$\hat{\mathcal{H}}=\sum\limits_{\mathbf{r}}D(\hat{S}^{z}_{\textbf{r}})^{2}+J_{c}\hat{\mathbf{S}}_{\mathbf{r}}\hat{\mathbf{S}}_{\mathbf{r+c}}+J_{a}\left(\hat{\mathbf{S}}_{\mathbf{r}}\hat{\mathbf{S}}_{\mathbf{r+a}}+\hat{\mathbf{S}}_{\mathbf{r}}\hat{\mathbf{S}}_{\mathbf{r+b}}\right).$
(2)

Here, $\mathbf{r}$ runs along the nickel positions in the tetragonal
sublattice; $\mathbf{a,b}$, and $\mathbf{c}$ are the primitive lattice
translation vectors towards the neighboring spins. Importantly, the symmetry
prohibits any in-plane second-order anisotropic terms. The planar anisotropy
protects the spin-singlet state $\left|S^{z}=0\right\rangle$ on every magnetic
ion, thus preventing Neél order in zero field, otherwise favored by the
exchange interactions. The set of constants describing DTN at ambient pressure
was initially obtained from zero-field neutron spectroscopy and thermodynamic
studies [23, 24], and later refined as $D/k_{\mathrm{B}}=8.9$ K,
$J_{c}/k_{\mathrm{B}}=1.82$ K, and $J_{a}/k_{\mathrm{B}}=0.34$ K [25] based on
the spectroscopic properties at high magnetic fields (where the effect of
quantum renormalization is not present). The zero-field gap
$\Delta/k_{\mathrm{B}}\simeq 3.5$ K [26] is relatively small compared to the
excitation-doublet bandwidth of $\sim 8$ K. If the magnetic field is applied
along the $c$ ($z$) direction, the degeneracy of the doublet is removed due to
Zeeman splitting (as shown in Fig. 1b), triggering eventually the onset of
low-temperature magnetic order at $\mu_{0}H_{c1}\simeq 2.1$ T [27, 23]. At
stronger magnetic field, $\mu_{0}H_{c2}\simeq 12.2$ T, DTN undergoes the
transition into the fully spin-polarized state. We would like to stress, that
the zero-field gap in DTN is dominantly determined by the single-ion
anisotropy (in contrast to Haldane spin-chain materials), making DTN a rare
example of a large-$D$ spin-1 system.

High-pressure neutron diffraction. As the pressure increases, the parameters
of Hamiltonian Eq. (2) are expected to change. However, it is necessary to
check first, whether the lattice remains undistorted. To this end, we have
performed a series of structural neutron diffraction studies [28]. As shown in
Fig. 2, the DTN lattice is smoothly compressible up to about $6$ kbar.
Interestingly, the compression goes in a rather uniaxial manner, with
shrinking of the sample mostly along the $c$ axis (${\partial c}/{\partial
P}=-0.038\pm 0.003$ Å/kbar at $1.8$ K). This implies the compression of the
main Ni-Cl-Cl-Ni superexchange pathway and deformation of the Ni2+ local
environment. Such changes must affect $D$ and $J_{c}$ in the first place. The
lattice parameter $a$ remains nearly constant with ${\partial a}/{\partial
P}=(-4.1\pm 1.5)\cdot 10^{-3}$ Å/kbar. At $P_{\mathrm{irr}}\simeq 6$ kbar, an
irreversible structural transition occurs, evidenced by discontinuities in the
lattice parameters. More detailed discussion is given in the Supplemental
Material (S2 and S3).

High-pressure TDO measurements. The TDO susceptibility technique is well
estalished as a versatile tool for detecting field-induced phase transitions
in solids under applied pressure [29, 30, 31, 32].

The results of our measurements are shown in Fig. 3. At low pressures two
anomalies, corresponding to the boundaries of the long-range AF ordered phase
at $H_{c1}$ and $H_{c2}$, respectively, are well discernible. To extract the
critical fields in a reliable way we use a set of empirical functions (red
lines, a detailed description is given in the Supplemental Material). With
applied pressure, we observe a decrease and increase of the first and second
critical field, respectively. Above $4.2$ kbar, the low-field transition is
not visible anymore. This is a strong evidence of the spin-gap closure. The
critical field $H_{c2}$ increases linearly up to $P_{\mathrm{irr}}$, where a
discontinuity appears. The high-field part of the curve at $6$ kbar (about
$P_{\mathrm{irr}}$) features a double-dip structure. This reflects the
coexistence of two structurally different phases at the first-order
crystallographic transition. In the structurally distorted phase above
$P_{\mathrm{irr}}$, we observe only one anomaly.

Figure 3: Pressure dependence of the TDO frequency changes $\Delta f/f$ in
response to the magnetic ﬁeld (the data are offset for clarity). Results of
empirical fits are shown in red (see Supplemental Material for details).

High-pressure ultrasound measurements. In order to reveal the nature of the
pressure-induced gapless phase, we performed high-pressure ultrasound
measurements. The magnetic ordering in DTN at zero pressure was thoroughly
investigated with ultrasound by Chiatti  _et al._ [33, 34], firmly
establishing the connection between the long-range-order onset and the sound
velocity anomalies in DTN. These anomalies reflects the spin-susceptibility
divergence in the vicinity of the critical temperature [35], as a result of
the pronounced magnetoelastic coupling in DTN.

We studied the longitudinal sound mode along the $c$ direction (mode $c_{33}$)
at high pressures. The field-induced changes of its velocity are shown in Fig.
4a. Similar to our TDO data (Fig. 3), there are two anomalies at low
pressures. As revealed previously [33, 34], they evidence the transitions from
disordered gapped to AF ordered gapless state at $H_{c1}$, and the subsequent
transition to the fully spin-polarized state at $H_{c2}$.

At higher pressures (above $3.8$ kbar), we only observe the high-field phase
transition, while the gapless AF ordered phase becomes extended to zero
magnetic field. An additional evidence for the pressure-induced magnetic
ordering in zero field is our observation of the sound-velocity anomaly below
$1$ K (red arrows in Fig. 4b). As expected, the ordering temperature grows
with increasing pressure.

Figure 4: Pressure dependences of ultrasonic properties of DTN. a Relative
change of the sound velocity $\Delta v/v$ of the longitudinal acoustic mode as
function of magnetic field (zero-pressure data are taken from Ref. [33] and
were measured at $0.3$ K). The data are offset for clarity. b Relative change
of the sound velocity $\Delta v/v$ of the longitudinal acoustic mode as
function of temperature. The data are offset for clarity. The onset of
magnetic order is marked by red arrows.

Electron spin resonance. The applied pressure in DTN should not only affect
the ground state but the spin dynamics as well. Electron spin resonance (ESR)
has recently proven to be a powerful tool to probe the excitation spectra in
strongly correlated spin systems under applied pressure [36, 29]. Selected
examples of ESR spectra in DTN are shown in Fig. 5a (adopting the naming
convention from Ref. [25], we label the observed modes B and C). Excitation
modes corresponding to $\Delta S^{z}=1$ transitions remain well visible up to
$6$ kbar. With the pressure increase both modes demonstrate a gradual shift
towards higher fields (as illustrated in Fig. 5b) until they vanish in the
structurally distorted phase above about $6$ kbar. The complete frequency-
field diagrams can be found in the Supplemental Material S6.

Figure 5: High-field ESR results. a Selected ESR spectra taken at a frequency
of $228$ GHz ($H\parallel c$, $T=4$ K) at different pressures [the sharp line
corresponds to DPPH (2,2-diphenyl-1-picrylhydrazyl), used as a marker]. b
Pressure dependencies of ESR fields for mode B and C, as revealed by
experiment.

Discussion

The results of our TDO and ultrasound experiments on DTN are summarized in the
phase diagram in Fig. 6. The data ensure that near $4$ kbar the spin gap does
close and antiferromagnetic order emerges. One can theoretically describe the
pressure-dependent spin Hamiltonian, consistent with these observations. The
saturation field $H_{c2}$ is known to follow the linear spin wave theory
(LSWT) description without any quantum renormalization [25]:

$g\mu_{\mathrm{B}}\mu_{0}H_{c2}=D+4J_{c}+8J_{a}.$ (3)

Given the overall smallness of $J_{a}$ and lack of a significant pressure-
induced length change in this direction, we can neglect possible variations in
this interaction. Thus, the phase-diagram evolution is due to the changes in
$D$ and $J_{c}$:

$g\mu_{\mathrm{B}}\mu_{0}\dfrac{\partial H_{c2}}{\partial P}=\dfrac{\partial
D}{\partial P}+4\dfrac{\partial J_{c}}{\partial P}.$ (4)

With the $g$-factor being $2.26$, the measured slope $\mu_{0}{\partial
H_{c2}}/{\partial P}=0.78\pm 0.03$ T/kbar (upper red line in Fig. 6 below the
instability pressure $P_{\mathrm{irr}}$) quantifies the linear pressure
dependence of $D$ and $J_{c}$.

Figure 6: Field-pressure phase diagram of DTN. Open and closed symbols
correspond to ultrasound and TDO data, respectively. Red lines are model fits
(see the text for details). The dotted line corresponds to the irreversible
structural phase transition at $P_{\mathrm{irr}}\sim 6$ kbar. Figure 7:
Results of model calculations. a Pressure dependence of the anisotropy
parameter $D/k_{\mathrm{B}}$. b Pressure dependence of the exchange coupling
parameter $J_{c}/k_{\mathrm{B}}$. c Pressure dependence of the transverse
exchange coupling parameter $J_{a}/k_{\mathrm{B}}$. The calculation results
are shown for the pressures, matching the experiments. d The Sakai–Takahashi
phase diagram for a quasi-1D spin-1 antiferromagnet [37]. The red open points
correspond to QMC calculations for the phase boundary [38]; the red line is a
prediction based on the ansatz as described in the text. The purple circles
correspond to estimates of the spin Hamiltonian parameters at given pressures.
The dotted cross marks the critical parameter values at $4.2$ kbar.

The analysis of the first critical field poses a greater challenge: since
$g\mu_{0}\mu_{B}H_{c1}=\Delta$, it requires evaluating the zero-field spin gap
for the given set of Hamiltonian (2) parameters. No exact theory is available
for that to the best of our knowledge. We overcome this challenge employing a
random-phase approximation (RPA) [39] based ansatz combined with density
matrix renormalization group (DMRG) calculations [40, 41]. The starting point
is the determination of the zero-field gap value $\Delta_{0}(D/J_{c})$ in the
$J_{a}=0$ limit of a single anisotropic chain, utilizing the latter technique.
As far as we know, only a particular range close to the quantum critical point
at $D/J_{c}\sim 1$ was thoroughly accessed with DMRG earlier [42]. The next
step of the model is to apply the RPA treatment to interacting chains in order
to estimate the critical value of the coupling $J_{a}^{\mathrm{crit}}$ that
closes the gap. The details of the calculations (based on large-$D$
approximation for the dynamic structure factor [43, 44,
*ZaliznyakLee_2005_NeutronSumRules] and the Kramers–Kronig relations) are
given in the Supplemental Material (S4).

The main result is the approximation for the order-disorder zero-field phase
boundary:

$J_{a}^{\mathrm{crit}}=\dfrac{\Delta_{0}^{2}}{4\mathcal{A}D}.$ (5)

We find that the numerical value $1/4\mathcal{A}\simeq 0.14$ provides an
excellent description of this phase boundary, previously obtained by extensive
quantum Monte–Carlo (QMC) simulations [37, 38]. This comparison can be seen in
Fig. 7d, where the calculation result using Eq. (5) is shown as the red line.
Then, it follows from our approach, that the value of the gap, renormalized by
$J_{a}$, is

$\Delta=\Delta_{0}\sqrt{1-\dfrac{J_{a}}{J_{a}^{\mathrm{crit}}}}.$ (6)

Hence, the first critical field can be expressed through the pressure-
dependent Hamiltonian parameters as

$g\mu_{0}\mu_{B}H_{c1}(P)=\Delta\left[\Delta_{0}\left(\frac{D(P)}{J_{c}(P)}\right),D(P)\right].$
(7)

The ambient pressure $H_{c1}$ evaluated by this procedure agrees with the
experimental value, ensuring the validity of the approach for the smaller gap
values as well.

The ability to predict both critical fields using Eqs. (3) and (7) finally
opens a route to a self-consistent treatment of the measured phase diagram
without any “effective” parameters. We numerically optimize the agreement
between (7) and the experimental $H_{c1}$ data by varying the choice of the
critical pressure $P_{c}$ at which the gap closes completely. This fixes a
unique combination of $\partial(D,J_{c})/\partial P$. The optimization results
in the critical pressure $P_{c}=4.2\pm 0.3$ kbar (see Supplemental Material S5
for more details). The magnetic phase diagram (Fig. 6) is fully captured with
$k_{\mathrm{B}}^{-1}\partial D/\partial P=0.16\pm 0.03$ K/kbar and
$k_{\mathrm{B}}^{-1}\partial J_{c}/\partial P=0.25\pm 0.01$ K/kbar. Figs. 7a-c
illustrate how the spin-Hamiltonian parameters of DTN are pressure-tuned up to
$P_{\mathrm{irr}}$. In the phase diagram of Fig. 7d, we show how the
corresponding ground state (illustrated by purple points) is changing from
gapped to long-range ordered at $P_{c}$ in accordance with the previous
discussion.

The rich high-pressure physics of DTN under pressure invites further
investigations, related to the interplay of quantum and thermal fluctuations
near $z=1$ quantum criticality [12, 46, *ScammellSushkov_PRB_2017_Paramagnons,
9, 48]. In particular, it would be important to access the order parameter
(e.g., with neutron diffraction or nuclear magnetic resonance) and the
extended field-temperature-pressure phase diagram [32]. This would open the
possibility to establish a direct connection between the minimalistic and
highly symmetric spin Hamiltonian of DTN and the effective field theory used
to describe the static and dynamic properties of critical quantum magnets [49,
50]. The undistorted symmetry also makes DTN a perfect candidate for
investigating of the effect of thermal and quantum fluctuations on the
amplitude mode [12] in the absence of Ising-type anisotropy. One can also
expect chemically substituted DTNX [51, *MannigPovarov_PRB_2018_DTNXGSWT] to
be a fruitful playground for the interplay of quantum $z=1$ criticality and
quenched disorder, similarly to (C4H12N2)Cu2(Cl1-xBrx)6 [53], (NH4)xK1-xCuCl3
[54], and Cs1-xRbxFeCl3 [55]. The higher symmetry and a simpler, well-
understood Hamiltonian make the case of DTNX much more appealing for such
studies.

To summarize, we have identified the material DTN as a unique platform for
studies of the exotic $z=1$ universality class in a three-dimensional magnetic
material. By means of TDO, ultrasound, and ESR measurements we have confirmed
the existence of pressure-induced criticality, separating gapped disordered
and gapless long-range ordered magnetic phases in the material. The nearly
ideal axial symmetry of the structure is retained at that point, which makes
DTN to stand out among the non-frustrated quantum magnets. In addition to
that, we have extracted the pressure dependence of the spin Hamiltonian
parameters from the data, and have achieved a quantitative theoretical
understanding of the transition mechanism.

Methods

Sample growth. Samples for the thermodynamic measurements and the electron
spin resonance spectroscopy were synthesized in the University of Saõ Paulo
from aqueous solution using a thermal-gradient method [56,
*Yankova_PhilMag_2012_ReviewXtals]. Samples used in the neutron diffraction
studies were synthesized at ETH Zürich. Fully deuterated chemicals (water and
thiourea) were used. The crystals were crushed into powder for these
experiments.

High-pressure neutron diffraction. Neutron-diffraction experiments with the
powder samples were performed at instrument HB2a at the High Flux Isotope
Reactor, Oak Ridge National Laboratory (Oak Ridge, Tennessee, USA). Up to a
few grams of deuterated DTN powder material were used. A Ge[113] or Ge[115]
vertically focussing wafer-stack monochromator was used to produce a neutron
beam with $2.41$ Å or $1.54$ Å wavelength, respectively. A 4He Orange cryostat
with aluminum He-gas pressure cell (capable of producing pressures up to $6$
kbar) and a CuBe clamp cell (capable of producing pressures up to $20$ kbar)
were used. The pressure in the gas cell was measured in situ using a
manometer. For the clamp cell a rock salt (halite) calibration curve was used
for the pressure estimate, resulting in an error bar of about $1$ kbar. After
initial data reduction, the FULLPROF package [58] was used for the intensity
profile analysis and structure determination. More details of these
experiments can be found in Ref. [28].

High-pressure TDO. High-pressure TDO measurements were conducted at the
National High Magnetic Field Laboratory, Florida State University
(Tallahassee, Florida, USA) in magnetic fields up to 18 T using a TDO
susceptometer [59]. Magnetic field was applied along the $c$ axis of the
crystal placed in a tiny copper-wire coil with $0.8$ mm diameter and $1$ mm
height. This assembly was immersed into Daphne 7575 oil and encapsulated in a
Teflon cup inside the bore of a piston-cylinder pressure cell made of a
chromium alloy (MP35N). The pressure created in the cell was calibrated at
room temperature and again at low temperature using the fluorescence of the
$R1$ peak of a small ruby chip as a pressure marker [60] with accuracy better
than $0.15$ kbar. The pressure cell was immersed directly into 3He, allowing
TDO measurements down to $350$ mK.

High-pressure ultrasound measurements. We performed ultrasound measurements
using the pulse-echo method with phase-sensitive detection technique [61, 35].
Overtone polished LiNbO3 transducers with 36∘ Y-cut (longitudinal sound
polarization) were used to generate and detect ultrasonic signals with a
frequency of about $35$ MHz. The transducers were bonded to natural $(001)$
crystal surfaces of a DTN sample with Thiokol 32, providing $k\parallel
u\parallel H\parallel c$ experiment geometry. The crystal dimensions were
$0.3\times 0.3\times 2.95$ mm3. We determined the low-$T$ sound velocity at
zero field and pressure to $v\simeq 2600$ m/s, in agreement with [33, 34]. A
commercially available piston-cylinder cell with CuBe outer sleeve, NiCrAl
inner sleeve, and tungsten carbide inner pistons (C&T Factory Co.,Ltd) was
adapted for ultrasound experiments following [62], with Daphne oil 7373 as a
pressure medium. The pressure cell (with a calibrated RuO2 temperature sensor
on the outer side) was thermally anchored to the 3He pot of the cryostat via a
copper rod. The pressure was estimated from the pressure-dependent
superconducting transition of tin [63,
*EilingSchilling_JPhysF_1981_SimpleSCpressure].

High-pressure ESR. High-pressure ESR studies of DTN were performed employing a
$25$ T cryogen-free superconducting magnet ESR setup (25T-CSM) at the High
Field Laboratory for Superconducting Materials, Institute for Materials
Research, Tohoku University (Sendai, Japan) [65, 66]. Gunn diodes were
utilized as microwave sources for frequencies up to $405$ GHz; the transmitted
radiation power was detected using a hot-electron InSb bolometer operated at
$4.2$ K. A DTN crystal was loaded into a Teflon cup filled with Daphne 7474
oil as a pressure medium. A two-section piston-cylinder pressure cell made
from NiCrAl (inner cylinder) and CuBe (outer sleeve) has been used. The key
feature of the pressure cell is the inner pistons, made of ZrO2 ceramics. The
applied pressure was calibrated against the superconducting transition
temperature of tin [63, *EilingSchilling_JPhysF_1981_SimpleSCpressure],
detected by AC susceptometer. The actual pressure during the experiment was
calculated using the relation between the load at room temperature and the
pressure obtained at around $3$ K; the pressure calibration accuracy was
better than $0.5$ kbar [67].

DMRG calculations. The DMRG calculations utilized the Julia version of the
ITensors package [68, 69]. A $249$-site anisotropic $S=1$ chain was simulated.
The calculations were performed on the hemera cluster (HZDR).

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

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Acknowledgements We thank A. Mannig, J. Möller, and G. Perren for their
involvement at the early stage of the ETH Zürich part of the project. This
work was supported by the Deutsche Forschungsgemeinschaft through ZV 6/2-2,
the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in
Quantum Matter - $ct.qmat$ (EXC 2147, project No. 390858490) and the SFB 1143
(project No. 247310070), as well as by HLD at HZDR, member of the European
Magnetic Field Laboratory (EMFL) [K.Yu.P., S.A.Zv., S.Zh., A.H., J.W.]. The
neutron-diffraction experiments at HB2a used resources at the High Flux
Isotope Reactor, a Department of Energy Office of Science User Facility
operated by the Oak Ridge National Laboratory. This work was partially
supported by the Swiss National Science Foundation, Division II [A.Z.]. A
portion of this work has been performed at the National High Magnetic Field
Laboratory, which is supported by the National Science Foundation Cooperative
Agreement DMR-1644779 and the State of Florida [D.E.G.]. ESR experiments were
performed at the High Field Laboratory for Superconducting Materials,
Institute for Materials Research, Tohoku University (proposal 19H0501 and
20H0501). Support of the ICC-IMR Visitor Program at Tohoku University is
acknowledged [S.A.Zv.]. This work was partially supported by the Brazilian
agencies CNPq (grant 304455-2021-0) and FAPESP (grant 2021-12470-8) [A.P.F.].
We also thank M. E. Zhitomirsky for fruitful discussions.

Author contributions S.A.Zv. conceived the project and performed the TDO
(together with D.E.G.) and ESR (together with T.S., S.K., and H.N.)
measurements; K.Yu.P. performed the analysis of this data and the numeric
simulations. V.O.G. has performed the neutron diffraction measurements and
analyzed the data. A.H. and S.Zh. have performed the ultrasound propagation
measurements. M.N. has contributed to the preliminary high-pressure
thermodynamic measurements. A.P.F. has synthesized the non-deuterated single
crystals used in the study. H.O. and H.N., A.Z., and J.W. administered at the
work at Kobe University and Tohoku University, ETH Zürich, and HZDR,
respectively. All the authors have contributed to the discussions and
manuscript writing.

Competing interests The authors declare no competing interests.

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