# Universal fluctuating regime in triangular chromate antiferromagnets

K. Somesh  School of Physics, Indian Institute of Science Education and
Research Thiruvananthapuram-695551, India  Y. Furukawa  Ames Laboratory and
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA  G. Simutis  Laboratory for Muon Spin Spectroscopy, Paul Scherrer
Institut, 5232 Villigen PSI, Switzerland  F. Bert  Université Paris-Saclay,
CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France  M. Prinz-
Zwick  N. Büttgen  Experimental Physics V, Center for Electronic Correlations
and Magnetism, University of Augsburg, D-86159 Augsburg, Germany  A. Zorko
Jožef Stefan Institute, Jamova c. 39, SI-1000 Ljubljana, Slovenia  Faculty of
Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
A. A. Tsirlin<EMAIL_ADDRESS>Experimental Physics VI, Center for
Electronic Correlations and Magnetism, University of Augsburg, 86135 Augsburg,
Germany  P. Mendels<EMAIL_ADDRESS>Université
Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France
R. Nath<EMAIL_ADDRESS>School of Physics, Indian Institute of Science
Education and Research Thiruvananthapuram-695551, India

###### Abstract

We report x-ray diffraction, magnetic susceptibility, heat capacity, 1H
nuclear magnetic resonance (NMR), and muon spin relaxation ($\mu$SR)
measurements, as well as density-functional band-structure calculations for
the frustrated $S=3/2$ triangular lattice Heisenberg antiferromagnet (TLHAF)
$\alpha$-HCrO2 (trigonal, space group: $R\bar{3}m$). This compound undergoes a
clear magnetic transition at $T_{\rm N}\simeq 22.5$ K, as seen from the drop
in the muon paramagnetic fraction and concurrent anomalies in the magnetic
susceptibility and specific heat. Local probes (NMR and $\mu$SR) reveal a
broad regime with slow fluctuations down to $0.7\,T_{\rm N}$, this temperature
corresponding to the maximum in the $\mu$SR relaxation rate and in the NMR
wipe-out. From the comparison with NaCrO2 and $\alpha$-KCrO2, the fluctuating
regime and slow dynamics below $T_{\rm N}$ appear to be hallmarks of the TLHAF
with $ABC$ stacking that leads to a frustration of interlayer couplings
between the triangular planes. This interlayer frustration is a powerful lever
to generate spin states with persistent dynamics and may bear implications to
spin-liquid candidates with the triangular geometry.

##  I Introduction

Experimental realization of quantum spin liquid (QSL) remains one of the major
challenges in condensed-matter physics [1, 2]. An identification of this
exotic entangled state in real-world materials could open the way to
understanding and utilizing fractionalized excitations, which are particularly
interesting in the context of quantum computing [3]. Theoretical studies
established several promising settings for the QSL that were indeed realized
[4] in antiferromagnets with the kagome [5], triangular [6], and honeycomb [7]
structures, all of them based on the two-dimensional interaction geometries.
Relatively less attention has been given to the interlayer couplings that are
often neglected in theoretical models but inevitably present in real
materials. Moreover, these interlayer couplings are responsible for the
formation of three-dimensional long-range magnetic order, so they may be in
fact decisive for whether a given material becomes magnetically ordered or
develops a fluctuating spin-liquid-like state.

Recent work on triangular antiferromagnets [6] exposed the $A$Yb$X_{2}$
compound family ($A$ = Na, K and $X$ = O, S, Se) as the largest stock of the
QSL candidates known to date. In fact, all members of this family evade long-
range magnetic order [8, 9, 10, 11, 12, 13], while some of them additionally
show spectral signatures of fractionalized spinon excitations [14] or
excitations of a Dirac QSL [15, 16]. One potentially important but hitherto
neglected feature of these compounds is the $ABC$-type stacking of their
triangular layers that causes frustration of the interlayer couplings, as
shown in Fig. 1. Each magnetic site is coupled to three sites of the adjacent
layer. This arrangement prevents simple ferro- or antiferromagnetic order
along $c$.

Here, we explore the effect of this intra and inter-layer frustration using
the $A$CrO2 chromates. These compounds are convenient model systems that
contain semiclassical spins $\frac{3}{2}$ and, thanks to the half-filled
$t_{2g}$ shell of Cr3+, feature isotropic Heisenberg interactions within the
triangular planes. Such an interaction regime is known to induce $120^{\circ}$
order in the plane, which is indeed observed in LiCrO2 [17, 18, 19, 20], but
other members of the same family do not show a clear magnetic ordering. In
NaCrO2, a transition at $T_{\rm N}\simeq 41$ K is followed by a broad
fluctuating regime with slow dynamics extending down to $T^{*}\simeq 30$ K
[21] for which the occurrence of Berezinskii-Kosterlitz-Thouless (BKT) was
suggested [22]. On the other hand, $\alpha$-HCrO2 with the shortest interlayer
distance was recently claimed to evade magnetic order and lie in the vicinity
of a spin-liquid phase [23]. Using a combination of thermodynamic and local
probes applied to $\alpha$-HCrO2, we show that in fact all these chromates –
NaCrO2, $\alpha$-HCrO2, as well as $\alpha$-KCrO2 [24] – develop a very
similar phenomenology. They enter a broad universal fluctuating regime below
$T_{\rm N}$, followed by the formation of a static and likely incommensurate
spin state below $T^{*}$. Our findings reveal a new mechanism of creating
fluctuating spin-liquid-like states that reside on the triangular planes but
crucially rely on the interlayer frustration.

Figure 1:  (a) Crystal structure of $\alpha$-HCrO2 consisting of triangular
layers formed by CrO6 octahedra and H atoms at the interstitial sites. (b)
Triangular Cr layers showing possible exchange couplings $J_{1}$, $J_{2}$,
$J_{3}$, and $J_{\perp}$ between the Cr3+ ions. (c) Coupling of H with three
Cr3+ ions from each layer where spins are arranged in a 120∘ structure.

##  II Methods

A polycrystalline sample of $\alpha$-HCrO2 was prepared by an hydrothermal
method. It involves two steps. First, the intermediate oxide Cr8O21 was
prepared by heating CrO3 (Aldrich, $99.99$% pure) granules for 12 hours at 250
∘C in air. In the second step, the intermediate product was crushed into fine
powder, put into a 1 M LiOH solution in deionized water, heated at 200 ∘C in a
teflon-lined stainless steel autoclave for about one week, and subsequently
furnace-cooled. The phase purity of the product was confirmed by powder x-ray
diffraction (XRD) using a PANalytical x-ray diffractometer (Cu Kα radiation,
$\lambda_{av}=1.54182$ Å). The temperature-dependent powder XRD measurement
was performed over the temperature range 15 K$\leq T\leq 300$ K using the low-
temperature attachment (Oxford Phenix) to the diffractometer.

Magnetization ($M$) measurements were performed as a function of temperature
($T$) and applied field ($H$) using a superconducting quantum interference
device (MPMS-3, Quantum Design). Heat capacity [$C_{\rm p}(T)$] as a function
of $T$ and $H$ was measured on a small piece of sintered pellet using the
relaxation technique in the physical property measurement system (PPMS,
Quantum Design).

The NMR measurements were carried out using pulsed NMR techniques on 1H
(nuclear spin $I=1/2$ and gyromagnetic ratio $\gamma_{\rm N}/2\pi=42.575$
MHz/T) nuclei in the temperature range 1.6 K $\leq T\leq 250$ K and at
different radio frequencies. The spectra were obtained either by Fourier
transform (FT) of the NMR echo signal or by sweeping the magnetic field. The
NMR shift $K=(\nu-\nu_{\rm ref})/\nu_{\rm ref}$ was determined by measuring
the resonance frequency of the sample ($\nu$) with respect to nonmagnetic
reference H2O (resonance frequency $\nu_{\rm ref}$). The 1H spin-lattice
relaxation rate ($1/T_{1}$) was measured by the conventional inversion
recovery method. The spin-spin relaxation rate ($1/T_{2}$) was measured
through the decay of the echo integral with variable spacing between the
$\pi/2$ and $\pi$ pulses.

The muon spin rotation/relaxation ($\mu$SR) experiments were carried out at
the $\pi$M3 beam line using the GPS spectrometer at the Paul Scherrer
Institute (PSI), Switzerland [25]. The zero-field (ZF) and the weak
transverse-field (wTF) $\mu$SR measurements were performed at temperatures
ranging from $\sim 1.5$ K to 50 K. The 100% spin-polarized $\mu^{+}$ (spin-1/2
and gyromagnetic ratio $\gamma_{\mu}=135.5$ MHz/T) with the momentum of $\sim
28.6$ MeV/c were implanted into the sample. Because of their positive charge,
muons come at rest in well-defined sites where their electrostatic energy is
minimized, i.e. in oxides typically 1 Å away from an oxygen O2- site. They
usually interact with surrounding moments through a dipolar coupling. In a
paramagnetic state, the electronic moments fluctuate fast ($\sim 10^{-12}$ s)
on the $\mu$SR time scale and the static field sensed by the muons has only a
nuclear origin with a typical value of a few Gauss, while dynamical electronic
fields lead to motional narrowing. On the contrary, when electronic moments
slow down, the evolution of the ZF polarization with time witnesses both their
dynamics and freezing. At the base temperature ($T\ll T_{\rm N}$), very high
counting statistics ($\sim 115$ Mevents) were taken in order to track the
fast-decaying polarization measured in zero-field, where the contribution from
static electronic moments dominates.

Density-functional (DFT) band-structure calculations were performed in the
FPLO code [26] using the Perdew-Burke-Ernzerhof flavor of the exchange-
correlation potential [27]. Correlation effects in the Cr $3d$ shell were
taken into account of the mean-field DFT+$U$ level using the on-site Coulomb
repulsion $U_{d}=2$ eV, Hund’s coupling $J_{d}=1$ eV, and atomic limit for the
double-counting correction [28, 29]. Experimental structural parameters for
LiCrO2 [30], NaCrO2 [31], and $\alpha$-KCrO2 [31] have been used. In the case
of $\alpha$-HCrO2, the position of hydrogen remains somewhat uncertain. We
thus considered two limiting cases: i) hydrogen placed in the middle between
the CrO2 layers, with the $R\bar{3}m$ symmetry preserved; ii) hydrogen
displaced toward one of the layers. In the latter case, the symmetry is
reduced to $R3m$, and the individual CrO6 octahedra are slightly distorted,
because the Cr–O distances can be shorter or longer depending on how strongly
the oxygen atom binds to the hydrogen. Room-temperature atomic positions from
Ref. [32] have been used, as they feature the largest distortion and allow one
to test its effect on magnetism. Exchange parameters $J_{ij}$ and single-ion
anisotropy $D$ enter the spin Hamiltonian,

$\mathcal{H}=\sum_{\langle
ij\rangle}J_{ij}\mathbf{S}_{i}\mathbf{S}_{j}-\sum_{i}D(S_{i}^{z})^{2},$ (1)

where the summation is over atomic pairs $\langle ij\rangle$, and
$S=\frac{3}{2}$. The parameters of this Hamiltonian were obtained by a mapping
procedure [33] from total energies of magnetically ordered states.

##  III Results

###  III.1 X-ray Diffraction

Figure 2:  Powder XRD data measured at (a) $T=300$ and (b) $T=15$ K. The red
solid line represents the Rietveld fit of the data. The Bragg positions are
indicated by pink vertical bars and the solid blue line at the bottom denotes
the difference between the experimental and calculated intensities. (c) The
variation of the lattice parameters ($a$, $c$, and $V_{\rm cell}$) as a
function of temperature. The solid line denotes the fit of the $V_{\rm
cell}(T)$ by Eq. (2).

In order to confirm the phase purity and to check the presence of any
structural distortions, powder XRD data were collected at various
temperatures. The Rietveld refinements of the XRD patterns were executed using
the `FullProf` software package [34] with the initial parameters taken from
Ref. [32]. Figure 2 presents the powder XRD patterns at $300$ and $15$ K along
with the Rietveld fits. At room temperature, all the peaks could be indexed
based on the space group $R\bar{3}m$ (No. 166) suggesting phase purity of the
sample. The obtained lattice parameters at room temperature are
$a=b=2.9836(1)$ Å, $c=13.4232(1)$ Å, and unit cell volume $V_{\rm cell}\simeq
103.48$ Å3, which are comparable with the previous report [32]. No extra
features or peaks were observed in the XRD scans implying the absence of any
structural transitions down to 15 K. On the other hand, around 30 K both
lattice parameters ($a$ and $c$) clearly deviate from their anticipated low-
temperature behavior [see Fig. 2(c)]. Such weak kinks, humps, and changes of
slope are commonly observed in magnetic compounds in the vicinity of their
magnetic transitions, where lattice symmetry does not change but a
magnetoelastic coupling leads to anomalies in thermal expansion [35, 36].

The temperature variation of the lattice constants ($a$ and $c$) and unit cell
volume ($V_{\rm cell}$) as shown in Fig. 2(c) are found to decrease
monotonically upon cooling down to $T\approx 45$ K. $V_{\rm cell}(T)$ above 45
K was fitted by the equation [37]

$V(T)=\gamma U(T)/K_{0}+V_{0},$ (2)

where $V_{0}$ is the cell volume in the zero temperature limit, $K_{0}$ is the
bulk modulus, and $\gamma$ is the Gr$\ddot{\rm u}$neisen parameter. $U(T)$ is
the internal energy, which can be derived in terms of the Debye approximation
as

$U(T)=9nk_{\rm B}T\left(\frac{T}{\theta_{\rm
D}}\right)^{3}\int_{0}^{\theta_{\rm D}/T}\dfrac{x^{3}}{e^{x}-1}dx.$ (3)

Here, $n$ is the number of atoms in the unit cell and $k_{\rm B}$ is the
Boltzmann constant. Using this approximation [see Fig. 2(c)], the Debye
temperature ($\theta_{\rm D}$) and other parameters were estimated to be
$\theta_{\rm D}\simeq 195$ K, $\gamma/K_{0}\simeq 2.95\times 10^{-5}$ Pa-1,
and $V_{0}\simeq 102.784$ Å3.

###  III.2 Magnetization

Figure 3:  Temperature dependence of magnetic susceptibility [$\chi(T)$]
measured in an applied field of $0.5$ T and in the temperature range 1.8
K$\leq T\leq 350$ K. The solid and dashed lines denote the fits by Eq. (4) and
Padé approximation, respectively. Arrows signal the limit of validity of HTSE
or Padé approximation. Inset: Zero-field-cooled (ZFC) and field-cooled (FC)
magnetic susceptibilities as a function of temperature measured in an field of
$100$ Oe.

Previous magnetic susceptibility $\chi(T)$ and 1H NMR measurements were
restricted to temperatures above the possible magnetic ordering [38, 39]. An
antiferromagnetic transition at $T_{\rm N}\sim 25$ K has been reported from
the heat capacity measurements [40].

The temperature-dependent magnetic susceptibility $\chi(T)$ ($\equiv M/H$)
measured in an applied field of $0.5$ T is shown in Fig. 3. As expected in the
paramagnetic regime, $\chi(T)$ increases with decreasing temperature in a
Curie-Weiss manner but becomes flatter toward lower temperatures when spin-
spin correlations set in. A hump around 27 K coincides with the lattice
anomaly discussed above. At even lower temperatures, $\chi(T)$ shows an upturn
around 20 K and a splitting at 12 K between field-cooled (FC) and zero-field-
cooled (ZFC) curves measured in a weak applied field. While the latter
signature may be indicative of spin freezing, our frequency-dependent AC
susceptibility measurement (see, Supplementary Materials) rules out the
possibility of a conventional spin-glass transition.

Traditionally, $\chi(T)$ is fitted by the sum of a temperature independent
term ($\chi_{0}$) and of a Curie-Weiss law, $\chi_{0}+\frac{C}{T-\theta_{\rm
CW}}$ in order to obtain the Curie constant $C$ and the characteristic CW
temperature $\theta_{\rm CW}$. $C$ yields the effective moment while
$\theta_{\rm CW}$ represents the energy scale of the total exchange
interactions. $\theta_{\rm CW}$ is given by: $|\theta_{\rm
CW}|=\frac{JzS(S+1)}{3k_{\rm B}}$, where $z=6$ is the number of nearest-
neighbours of Cr3+ ions and $J~{}(=J_{1})$ is the intra-layer exchange
coupling [41]. This requires the $T$ range of the measurements to fall into
the high-temperature regime, $T\gg|\theta_{\rm CW}|$ which is not the case
here. Indeed, our $\chi(T)$ data are limited up to 350 K and $\theta_{\rm CW}$
for HCrO2 is in the range $220-270$ K [38].

We therefore used high-temperature series expansion (HTSE), now available up
to 11th order [42] and Padé approximants [43] to fit our data,

$\chi=\chi_{0}+\chi_{\rm spin}(T)$ (4)

and

$\displaystyle\begin{split}\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\chi_{\rm
spin}(T)=\frac{N_{\rm A}g^{2}\mu_{B}^{2}}{k_{\rm
B}T}\times\Bigg{[}\frac{5}{4}-\frac{75}{8}x+\frac{225}{4}x^{2}$\mbox{}\hfil\lx@intercol\\\
&-\frac{18765}{64}x^{3}+\frac{712175}{512}x^{4}-\frac{6328661}{1024}x^{5}+\frac{643219519}{24576}x^{6}\\\
&-\frac{36677316665}{344064}x^{7}+\frac{1154751891527}{27525127}x^{8}\\\
&-\frac{1888217340683}{1179648}x^{9}+\frac{11768319087087113}{1981808640}x^{10}\Bigg{]},\end{split}$
(5)

with $x=\frac{J}{k_{\rm B}T}$. Figure 3 depicts the fitting of $\chi(T)$ data
in the validity $T$-range of Eq. (4), $k_{\rm B}T>8J$, corresponding to
$T>195$ K. Fixing $g=2$ from the ESR data of Ref. [22], we obtain $J/k_{\rm
B}=(24\pm 0.2)$ K and $\chi_{0}=-(5.35\pm 0.45)\times 10^{-5}$ cm3/mol. An
extension of the validity of the fit down to $T\sim 3.5J$ can be obtained
using Padé approximants which are now available. The [6,5] Padé extension of
our fit is shown as a dashed blue line in Fig. 3 and perfectly matches with
our data in its validity domain, $T\gtrsim 85$ K, with $J/k_{\rm B}=24$ K and
$\chi_{0}=-5.35\times 10^{-5}$ cm3/mol.

###  III.3 Heat Capacity

Figure 4:  Upper panel: Temperature-dependent heat capacity $C_{\rm p}(T)$ of
$\alpha$-HCrO2 measured in zero applied magnetic field. The solid line
represents the simulated phonon contribution $C_{\rm ph}(T)$ and the dashed
line represents the magnetic contribution $C_{\rm mag}(T)$. Inset: Plot of
$C_{\rm p}(T)/T$ vs $T^{2}$ below $T_{\rm N}$, highlighting the shoulder at
$T^{*}\simeq 12$ K. Lower panel: $C_{\rm mag}/T$ and $S_{\rm mag}/R\ln 4$ in
the left and right $y$-axes, respectively, are plotted as a function of
temperature. Inset: Logarithmic plot of $C_{\rm mag}$ vs $T$, with solid lines
showing the fits with a power-law, $C_{\rm mag}=aT^{\alpha}$, in different
temperature regimes.

The heat capacity ($C_{\rm p}$) data are shown in Fig. 4 (upper panel) as a
function of temperature measured in zero magnetic field. At high temperatures,
$C_{\rm p}(T)$ is entirely dominated by the contribution of phonon excitations
($C_{\rm ph}$) and the value of $C_{\rm p}$ at $300$ K is about $\sim 62$
J/mol K. This value is close to the expected Dulong-Petit lattice heat
capacity of $C_{\rm v}=3nR\simeq 74.8$ J/mol K, where $n$ is the number of
atoms per formula unit [44]. In $\alpha$-HCrO2, H is the lightest element
expected to have a very high Debye frequency compared to other elements, and
therefore we chose $n=3$ for the calculation. At around $24$ K, the heat
capacity shows a clear and very broad anomaly that confirms the magnetic
transition. With further decrease in temperature, $C_{\rm p}(T)$ decreases
gradually towards zero. At low temperatures, $C_{\rm p}(T)$ is dominated by
the magnetic contribution $C_{\rm mag}$.

In order to estimate the phonon part of the heat capacity, the $C_{\rm p}(T)$
data were fitted above 100 K by a sum of Debye contributions

$C_{\rm
p}(T)=9R\displaystyle\sum\limits_{n=1}^{3}c_{n}\left(\frac{T}{\theta_{Dn}}\right)^{3}\int_{0}^{\frac{\theta_{Dn}}{T}}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx.$
(6)

Here, $R$ is the molar gas constant, $\theta_{Dn}$ are the characteristic
Debye temperatures, and $c_{n}$ are the integer coefficients indicating the
contributions of different atoms to $C_{\rm p}(T)$. A similar approach has
been chosen previously to estimate the phonon contribution in different types
of frustrated magnets [45]. Figure. 4 (upper panel) presents the fit of
$C_{\rm p}(T)$ by Eq. (6) with $c_{1}=1$, $c_{2}=1$, and $c_{3}=2$. Here,
$c_{1}$, $c_{2}$, and $c_{3}$ represent the number of H, Cr, and O atoms,
respectively. The sum of $c_{n}$ is thus equal to $4$, the number of atoms per
formula unit. We have used three different Debye temperatures: $\theta_{D1}$
for H1+, $\theta_{D2}$ for Cr3+, and $\theta_{D3}$ for O2-. Finally, the
high-$T$ fit was extrapolated down to $2$ K and $C_{\rm{mag}}(T)$ was
estimated by subtracting $C_{\rm{ph}}(T)$ from $C_{\rm{p}}(T)$ [see Fig. 4
(upper panel)].

$C_{\rm{mag}}(T)/T$ is plotted as a function of temperature in the lower panel
of Fig. 4. It extends up to 100 K, even though the leading exchange coupling
$J$ is about 24 K only. For the validation of the fitting procedure, we
calculated the total magnetic entropy ($S_{\rm mag}$) by integrating $C_{\rm
mag}(T)/T$ between $2$ K and high-temperatures as $S_{\rm{mag}}(T)=\int_{\rm
2\,K}^{T}\frac{C_{\rm{mag}}(T^{\prime})}{T^{\prime}}dT^{\prime}$. The obtained
magnetic entropy at 150 K is $S_{\rm{mag}}\simeq$ $10.4$ J/mol K. This value
of $S_{\rm mag}$ corresponds to $\sim 90\%$ (see the lower panel of Fig. 4) of
the expected theoretical value $S_{\rm mag}=R\ln(2S+1)=11.5$ J/mol K for a
$S=3/2$ system. Moreover, the entropy at $T_{\rm N}$ is found to be only one-
third of the total entropy and the remaining entropy is distributed above
$T_{\rm N}$ due to short-range (in-plane) spin correlations. This is in
contrast with the conventional LRO where the entropy is recovered completely
just above $T_{\rm N}$, and confirms the strongly frustrated nature of
$\alpha$-HCrO2 [18, 46].

The broad maximum in $C_{\rm mag}$ around 24 K confirms the magnetic
transition, whereas a broad shoulder in $C_{\rm p}/T$ vs $T^{2}$ (inset of the
upper panel of Fig. 4) suggests another magnetic instability at $T^{*}\simeq
12$ K that coincides with the temperature at which the ZFC and FC
susceptibilities bifurcate. This feature is quite similar to that reported for
CuCrO2 and $\alpha$-KCrO2 previously [47, 24].

The logarithmic plot of $C_{\rm mag}$ vs $T$ below $T_{\rm N}$ (see, lower
inset of Fig. 4) reveals a nearly linear behavior, although a closer
inspection of the data suggests that different power-law exponents are
obtained in different temperature intervals. Using $C_{\rm mag}=aT^{\alpha}$,
we find $a\simeq 0.0052$ J-mol-1-K-4 and $\alpha\simeq 2.2$ for 12.5 K$\leq
T\leq 20$ K, $a\simeq 0.0026$ J-mol-1-K-4 and $\alpha\simeq 2.6$ for 4 K$\leq
T\leq 10$ K, and $a\simeq 0.0025$ J-mol-1-K-4 and $\alpha\simeq 2.8$ for 2
K$\leq T\leq 3$ K. Considering $\alpha=2$ and $\alpha=3$ expected for
antiferromagnets in 2D and 3D, respectively, we conclude that spin
correlations gradually evolve from a 2D behavior immediately below $T_{\rm N}$
toward a 3D behavior well below $T^{*}$.

###  III.4 1H NMR

Figure 5:  Upper panel: FT 1H NMR spectra at different temperatures for the
polycrystalline $\alpha$-HCrO2 sample measured at $H=1.33473$ T. The vertical
dashed line corresponds to the 1H resonance frequency of the non-magnetic
reference sample. Lower panel: 1H NMR shift $K$ as a function of $T$. Inset:
1H NMR shift vs $\chi$ with temperature as an implicit parameter. The solid
line represents the linear fit.

The crystal structure of $\alpha$-HCrO2 [Fig. 1(c)] features a single
crystallographic H site located in between two Cr-triangles. The 1H nucleus is
then coupled to three Cr3+ ions from each of the two adjacent layers. Thus,
one can probe static and dynamic properties of the Cr3+ spins by performing 1H
NMR. We have measured 1H NMR spectra by doing a Fourier Transform of the echo
signal at different temperatures, keeping the field persistent. Figure 5
(upper panel) presents the FT 1H NMR spectra at different temperatures. We
indeed observed a single spectral line, as expected for a $I=1/2$ nucleus. The
line position was found to increase weakly with decreasing temperature. The
lower panel of Fig. 5 presents the temperature variation of the NMR shift
($K$) for $T>30$ K. The slope of the linear fit using $K(T)=K_{0}+\frac{A_{\rm
hf}}{N_{\rm A}}\chi_{\rm spin}(T)$ (where $K_{0}$ is the temperature-
independent chemical shift) of the $K$ vs $\chi$ plot yields the hyperfine
coupling constant $A_{\rm hf}\simeq 1024$ Oe/$\mu_{\rm B}$ between the 1H
nucleus and the Cr3+ electronic spins (see, inset of Fig. 5). This value of
$A_{\rm hf}$ is almost six times smaller than 7Li in LiCrO2 [18, 21] but much
larger than the expected dipolar coupling for $\alpha$-HCrO2 111The dipolar
coupling constant at the H site is calculated using lattice sum with an
assumption of $1\mu_{\rm B}$ magnetic moment on each Cr3+ ion to be $\sim
0.150$ kOe/$\mu_{\rm B}$ per Cr3+ ion.. The latter also indicates a
substantial overlap of the hydrogen $1s$ orbital with the $3d$ orbitals of
Cr3+ ion via $2p$ orbitals of O. This further explains why the inter-layer
exchange coupling through H is significant, as estimated in Sec. F.

Figure 6:  Field-sweep 1H NMR spectra at different temperatures around $T_{\rm
N}$ measured on the polycrystalline $\alpha$-HCrO2 sample at $91.1$ MHz.
Inset: 31Cr NMR spectrum at $T=1.67$ K measured in zero field. Figure 7:
Integrated NMR intensity as a function of temperature.

Figure 6 shows the field-sweep NMR spectra around and below $T_{\rm N}$. As
the temperature approaches $T_{\rm N}$ from above, the NMR signal intensity
decreases. Below $T_{\rm N}$, the signal is reduced drastically, whereas the
line width does not show any significant increase. The loss of the NMR signal
continues well below $T_{\rm N}$ but never becomes complete, and below $\sim
10$ K the intensity quickly recovers. Below 10 K, the NMR line broadens and
develops a triangular shape which is independent of the applied field. This
broadening is indicative of local magnetic fields appearing at the 1H site,
but it is remarkable that such a broadening appears only below 10 K and not
immediately below $T_{\rm N}$, as in conventional 3D antiferromagnets. This is
a clear indication that the NMR signal is wiped out over a broad temperature
window, below $T_{\rm N}$ and a pure static state appears only below 10 K. The
wipe out effect is very well evident in the temperature dependent integrated
NMR intensity plot in Fig. 7. A rectangular line shape on the polycrystalline
sample is expected in an ordered collinear antiferromagnet [49, 50]. The
triangular line shape observed at low temperatures in $\alpha$-HCrO2 is
reminiscent of an incommensurate magnetic order [51, 52] that may appear in
triangular antiferromagnets upon a distortion of the commensurate
$120^{\circ}$ state. Further, as the intrinsic signal is lost below $T_{\rm
N}$ a very narrow central line becomes prominent at the zero shift position,
on top of the broad spectrum. This narrow line persists down to the lowest
measured temperature and can be attributed to the effect of defects and/or a
small amount of non-magnetic impurities.

We have also measured the 53Cr NMR ($I=3/2$ and $\gamma_{\rm N}/2\pi=2.40094$
MHz/T) spectrum by sweeping the frequency in zero field at $T=1.67$ K (see,
inset of Fig. 6). The observation of 53Cr zero field NMR signal is a direct
evidence of a static magnetic ordering of Cr3+ moments in $\alpha$-HCrO2 well
below $T^{*}$. From the peak position, the internal field at the Cr site is
estimated to be $|H_{\rm int}|\sim 26.4$ T. Such a large value of $H_{\rm
int}$ is comparable to the reported value $\sim 28.64$ T for YCrO3 and $\sim
27.0$ T for CuCrO2 [53, *Smolnikov674].

###  III.5 $\mu$SR

In order to probe the dynamics in the fluctuating regime in more detail, we
performed complementary $\mu$SR experiments. Owing to its much shorter time
window (10 ns - 15 $\mu$s), $\mu$SR is better suited than NMR in tracking the
persisting dynamics in slowly fluctuating magnets. In addition, $\mu$SR allows
us to probe _all_ sites, whereas only a weak fraction of the sites were
detected in NMR between 20 and 3 K due to the wipeout effect.

Figure 8:  Left: $T$-variation of the muon relaxation rate; the dashed line is
a guide to the eyes. Right: weak transverse field $\mu$SR asymmetry (see
text). The dashed line is a fit to a phenomenological broadened "step"
function $(A_{\rm tot}-A_{\rm b})/[1+e^{((T_{\rm N}-T)/\Delta T_{\rm
N})}]+A_{\rm b}$.

$\mu$SR experiments under a weak applied field perpendicular to the initial
muon spin polarization (wTF), were performed in order to track the spin
freezing and the static magnetic volume fraction versus temperature. Indeed,
only muons stopping close to paramagnetic (unfrozen) sites precess around the
applied field direction and produce long-lived oscillations of the
polarization reflected in the measured asymmetry. On the contrary, muons close
to magnetically frozen parts experience much larger fields with a large
distribution and yield a strongly damped contribution in the 0.1 $\mu$s range
to the polarization as is evident in zero-field experiments presented next.

The wTF asymmetry was fitted using a standard function for $t>0.5~{}\mu$s

$A_{\rm wTF}(t)=A_{\rm osc}\cos(\gamma_{\mu}B_{\rm
ext}t+\phi)e^{-(\sigma^{2}t^{2}/2)}+A_{\rm tail}e^{-\lambda t},$ (7)

where $A_{\rm osc}$ is the wTF oscillating asymmetry and $A_{\rm tail}$ is
introduced to account for the non-oscillating long-time relaxing frozen part
(1/3rd tail). The fitted values for $A_{\rm osc}$ are shown in Fig. 8 (right).
An abrupt loss of asymmetry is observed at a temperature of 22.5 K - taken as
the most precise definition of $T_{\rm N}$, within a $\Delta T_{\rm N}\sim 3$
K range around $T_{\rm N}$, indicating a uniform freezing in the sample. At
lower temperatures, a constant $A_{b}\sim 0.01$ asymmetry is found which
corresponds to $\sim 4\%$ muons sitting in a non magnetic part of the sample,
possibly defects or an impurity phase. Note that this is in-line with the
spectral weight of the persisting narrow peak found in the NMR experiment.

Figure 9:  $\mu$SR asymmetry in zero external field at various temperatures.
The solid lines are fits described in the text.

Typical zero-field asymmetry curves are presented in Fig. 9. Above $T_{\rm
N}$, in the fast fluctuation regime, muons mainly sense a weak $\sim$ 4.6 G
static nuclear magnetic field mainly originating from H nuclei. The decrease
of the $\mu^{+}$ polarization has the expected Kubo-Toyabe shape, Gaussian at
early times and modulated by a slowly exponential relaxing envelope associated
with electronic spin fluctuations. When decreasing the temperature below
$T_{\rm N}$, similar to the NaCrO2 case [21], the freezing of the electronic
moments induces a very fast decrease of the asymmetry on a $0.1\mu$s range
while the long-time 1/3rd tail monitors the relaxation induced by the spin
dynamics.

We first focus on the data obtained deep in the frozen regime. Figure 10 shows
the very early time behavior of the asymmetry at 1.5 K. The oscillating
behavior is made clear by the two visible wiggles and rules out a spinglass-
like random freezing. The damping is very large, associated with a large
distribution of the field at the muon site as depicted by the Fourier
transform presented in the inset. Finally, one also observes the beginning of
the 1/3rd tail ($t>0.06\ \mu$s).

Figure 10:  Early-time $\mu$SR asymmetry in zero external field at 1.5 K and
fit with a zero-order Bessel function $A[\frac{2}{3}J_{0}(t)e^{-\lambda
t}+1/3]$. Inset: Corresponding Fourier transform after subtraction of the
1/3rd tail.

The $T=1.5$ K asymmetry evolution with time was fitted according to the model:

$A(t)=A_{0}\left[\frac{2}{3}P_{\rm
osc}(t)e^{-({\sigma^{\prime}}^{2}t^{2}/2)}+\frac{1}{3}\right]e^{-(t/T_{1})^{\alpha}}+A_{b},$
(8)

where $P_{\rm osc}(t)$ is an oscillating function corresponding to the
$\mu^{+}$ precession around the internal magnetic field ($B_{\mu}$) direction
at the average frequency $\nu=(\gamma_{\mu}/2\pi)B_{\mu}$, while the damping
of the oscillation is due to the width of the internal field distribution
$\sigma^{\prime}/\gamma_{\mu}$. The constant background, $A_{b}=0.01$, was
fixed from the remaining wTF asymmetry at $T\ll T_{\rm N}$ and $A_{0}$ was set
to $A_{0}=A_{\rm tot}-A_{b}$, with $A_{\rm tot}=0.255$, as determined from the
total asymmetry fitted above $T_{\rm N}$. We found a slightly better
$\chi^{2}$ when fitting the oscillations with a Bessel function rather than a
simple cosine. A $B_{\mu}\sim 2.75$ kG internal field is estimated from the
frequency of the oscillations. The fast early time damping reveals a large 1.4
kG HWHM distribution of $B_{\mu}$, either associated with an incommensurate
order (Bessel fit) or a sizeable disorder (cosine fit). The flatness of the
1/3rd tail on the timescale of Fig. 10, hence the weakness of the relaxation
rate $1/T_{1}$, indicates a purely static magnetic frozen phase in the low-$T$
limit.

We now focus on the relaxation effects evidenced on the 1/3rd tail at
$t>0.1-0.2~{}\mu$s as displayed in the main panel of Fig. 9. This was the
central focus of our $\mu$SR study, given the wipe-out observed in NMR. For
each temperature below 19.9 K, we could estimate the "unfrozen" contribution
so that the asymmetry of the corresponding 1/3rd tail was fixed from the wTF
data. The asymmetry evolution with time was fitted using the following
functions for $t>t_{1/3^{\rm rd}}$ taken such that $A(t)\leq A_{\rm tail}$ :

$A(t)=A_{\rm tail}\ e^{-{t/T_{1}}^{\alpha}}+A_{b}\quad(T<T_{N})$ (9)

and

$A(t)=A_{0}\ {\rm KT}(t)\ e^{-{t/T_{1}}^{\alpha}}+A_{b}\quad(T\gtrsim T_{N}),$
(10)

where KT($t$) is the Kubo-Toyabe function, Gausian like in the ${t\rightarrow
0}$ limit and $t_{1/3rd}$ is the time where the asymmetry falls to the value
expected for the 1/3rd tail, determined as $A_{\rm tail}+A_{b}$, see above.
The stretched exponent $\alpha$ was found to vary smoothly between 12 K and up
to the transition from its low-$T$ value $\sim 0.3$ to 1. This indicates a
broad distribution of relaxation times, in line with the distribution of
internal fields reported in Fig. 10. The error bars mainly come from the
estimate of $t>t_{1/3^{\rm rd}}$, the error bars on the background and on the
total asymmetry which slightly impact the relaxation rate. The fits were
straightforward between 1.5 and 17.3 K where the "unfrozen fraction" is zero
or marginal. For $T=19.9$ K, the unfrozen part has a contribution to the
relaxation which cannot be disentangled from the 1/3rd tail. $A_{\rm tail}$
was therefore replaced by $A_{\rm tail}+A_{\rm osc}$ in Eq. (11), leading to
an average value of the relaxation rate. The increase of the relaxation rate
is clearly visible, up to $T_{1}^{-1}$(15 K)$\sim$1 $\mu$s-1, followed by a
decrease at $T=17$ K (Fig. 8, left). The large $T$-range below $T_{\rm N}$
where slow fluctuations occur, clearly signals an unconventional dynamical
regime peaked around $0.7~{}T_{\rm N}$, at variance with the common phase
transitions where the relaxation is peaked at $T_{\rm N}$ 222Note that the
ratio of the NMR and $\mu$ SR relaxation rates $(1/T_{1})_{\rm
NMR}/(1/T_{1})_{\mu{\rm SR}}$ scales with the ratio of the coupling constants
squared. An order of magnitude can be calculated from the ratio of $\mu$SR
internal field found at 1.5 K and the product of the hyperfine constant by the
Cr3+ moment, assumed to be 3.87$\mu_{\rm B}$ or accordingly from the 5 kG HWHM
of the NMR spectrum at 1.5 K. At 15 K, this yields a rough estimate of
$(T_{1})_{\rm NMR}$, of the order of 9-25 $\mu$s, which explains well the
wipe-out..

###  III.6 Microscopic Analysis

Table 1:  Exchange couplings $J_{i}$ (in K) and single-ion anisotropy $D$ (in K) obtained from DFT+$U$ calculations for the spin Hamiltonian, Eq. (1). $J_{1}$, $J_{2}$, and $J_{3}$ are in-plane couplings between first, second, and third neighbors, respectively. $J_{\perp}$ is the frustrated interplane coupling. | $J_{1}$ | $J_{2}$ | $J_{3}$ | $J_{\perp}$ | $D$
---|---|---|---|---|---
LiCrO2 | 84 | $-0.3$ | 3.6 | 0.4 | 0.4
NaCrO2 | 48 | $-0.5$ | 2.7 | 0.1 | 0.6
$\alpha$-KCrO2 | 15 | 0.6 | 4.1 | 0.2 | 0.9
$\alpha$-HCrO2, $R\bar{3}m$ | 33 | $-0.2$ | 4.3 | 2.8 | 0.5
$\alpha$-HCrO2, $R3m$ | 28 | 1.0 | 4.3 | 2.0 | 0.5

DFT calculations are used to identify the trends in the microscopic magnetic
parameters across the $A$CrO2 series. Exchange couplings and single-ion
anisotropies computed for LiCrO2, NaCrO2, $\alpha$-KCrO2, and for the two
structural models of HCrO2 are summarized in Table 1. All compounds are
dominated by the nearest-neighbor in-plane coupling $J_{1}$. This coupling is
highly sensitive to the Cr–Cr distance $d$ and decreases from LiCrO2
($d=2.901$ Å) to NaCrO2 ($d=2.975$ Å) and eventually $\alpha$-KCrO2 ($d=3.044$
Å). The $J_{1}$ value in $\alpha$-HCrO2 lies in between those of the Na and K
compounds, despite its Cr–Cr distance of 2.968 Å, which is slightly shorter
than in the Na case. This additional reduction in the $J_{1}$ value may be
caused by the hydrogen atoms that are placed next to oxygen and change its
polarization [56]. The exact position of hydrogen along the O–H–O contact
($R\bar{3}m$ vs $R3m$ models) plays only a minor role, as evident from Table
1.

The computed values of $J_{1}$ are generally in good agreement with the
experimental estimates based on the magnetic susceptibility data (Table 2).
The coupling in LiCrO2 is also in accord with the spectroscopic measurements
of magnetic excitations [57] that suggested $J_{1}\simeq 70$ K (6.0 meV).

A weak easy-axis single-ion anisotropy ($D$) is present in all compounds and
does not change significantly across the series. Indeed, the calculated value
of $D$ for $\alpha$-KCrO2 matches well with the value estimated from the ESR
experiments (see, Supplementary Materials). On the other hand, the interlayer
coupling $J_{\perp}$ notably increases in $\alpha$-HCrO2, owing to the reduced
interlayer spacing (Table 2) and the covalent nature of the O–H bonds as
opposed to the purely ionic bonding between alkali metals (Li, Na, K) and
oxygen.

Table 2:  Comparison of different structural and magnetic parameters of $A$CrO2 ($A=$ H, Li, Na, and K) [58]. The exchange coupling $J~{}(=J_{1})$ for Li, Na, and K compounds is deduced using the HTSE given by Delmas et al [59] whereas for $\alpha$-HCrO2, we have used 11th order HTSE [Eq. (5)] suggested by Schmidt et al [42]. Compounds | Interplanar |  $J$ (K) |  $T_{\rm N}$ (K) | Refs.
---|---|---|---|---
| spacing (Å) |  |  |
$\alpha$-HCrO2 | 4.774 | 24 | 22.5 | this work
LiCrO2 | 4.807 | 78 | 62 | [22, 24]
NaCrO2 | 5.323 | 40 | 41 | [22, 21]
$\alpha$-KCrO2 | 5.963 | 24 | 23 | [60, 24]

##  IV Discussion

Figure 11:  Comparison of low temperature data of $\alpha$-HCrO2 with NaCrO2;
(a) magnetic heat capacity normalized to its maximum value vs $T/J$ (b)
evaluation of integrated NMR intensity with $T/J$, and (c) $\mu$SR relaxation
rate $1/T_{1,\mu}$ vs $T/J$ and the paramagnetic fraction vs $T/T_{\rm N}$ in
the left and right $y$-axes, respectively. Here the transition temperature is
taken to be $T_{\rm N}\simeq 22.5$ K and 41 K for $\alpha$-HCrO2 and NaCrO2,
respectively.

This discussion section is divided into two parts: the first part is devoted
to a discussion on the transition temperature with respect to frustration. The
second part is devoted to the extended dynamical regime below the transition
which is found to have a universal character among the $A$CrO2 ($A$=H, Na, and
K) series of chromates.

For a purely 2D triangular Heisenberg antiferromagnet, one would expect a
transition only at $T=0$ into a three-sublattice structure with spins at 120∘.
In the presence of a weak interlayer exchange coupling, the transition is
still mainly driven by the growth of the 2D correlation length $\xi(T)$ when
cooling down: it diverges at $T=0$. Indeed, in a mean-field approach, the
transition is obtained by equating the thermal energy to the interaction of 2D
correlated patches through a small interlayer exchange coupling. This can be
approached by the following mean-field equation that leads to a finite
transition temperature,

$k_{\rm B}T_{\rm N}\sim J_{\perp}S^{2}[\xi(T_{\rm N})/a]^{2}.$ (11)

From Refs. [61, 62] we get

$\xi(T)\sim(T/J)^{-0.5}\exp(\alpha J/T).$ (12)

One can then expect a transition temperature modestly increasing with
$J_{\perp}$. From Tables I and II, one can notice that $J_{\perp}$ decreases
when the inter-layer distance increases from $\alpha$-HCrO2 to $\alpha$-KCrO2,
as expected, and $J_{\perp}$ is more than a factor 15-30 larger for
$\alpha$-HCrO2 as compared to NaCrO2 and $\alpha$-KCrO2. While the $J/T_{\rm
N}$ ratio is similar or even slightly larger for $\alpha$-HCrO2, the
transition temperature surprisingly does not scale at all with $J_{\perp}$. We
suggest that this failure of the mean field, well-established, approach for
low-dimensional systems can be assigned to a peculiar degree of interlayer
frustration in $\alpha$-HCrO2. Indeed, we first note that the geometry of the
inter-layer coupling [see Fig. 1(b)] leads to some frustration through the
tetrahedral units resulting from the $ABC$ type of stacking, common to all the
chromates mentioned above: one Cr3+ is coupled to six Cr3+ from the two
adjacent layers. Second, a higher degree of frustration induced by the
interlayer coupling, as found for $\alpha$-HCrO2, certainly prevents the
correlation length to grow according to Eq. (12) which leads to the apparent
paradox of "a larger $J_{\perp}$ yields to a lower $T_{\rm N}$". We note that
this frustration scenario is in line with the incommensurate structure
observed at 1.57 K through the NMR lineshape and also consistent with the
$\mu$SR time-evolution of the asymmetry at 1.5 K.

Now, coming to the intermediate $T$-range below $T_{\rm N}$, several of the
$A$CrO2 compounds show a very similar phenomenology. In Fig. 11, we compare
our results to those obtained in NaCrO2 [21], using $J$ as a scaling
parameter.
(i) Both $\alpha$-HCrO2 and NaCrO2 undergo a magnetic transition that
manifests itself in the specific heat and $\mu$SR, but does not lead to the
formation of a conventional 3D transition with a narrow critical regime
centered in the vicinity of $T_{\rm N}$. On the contrary, here the transition
at $T_{\rm N}$ is followed by a broad regime with slow fluctuations peaked
around 0.6-0.7$~{}T_{\rm N}$ [Fig. 11(c)]. Strikingly, this maximum largely
coincides with the bifurcation point of the FC/ZFC susceptibility reported at
12-15 K in Fig. 3 for $\alpha$-HCrO2. The extension of this regime can be
tracked through the $\mu$SR relaxation rate, which shows a striking similarity
in both compounds once the temperatures are normalized by $J$.
(ii) The heat capacity also displays a very broad maximum around $T_{\rm N}$
and a similar behavior below $T_{\rm N}$ for both compounds when the
temperature axis is scaled by their respective $J$.
(iii) For both compounds, the intensity of the NMR line progressively
decreases, goes through a minimum, and recovers below a temperature $T^{*}$
where slow fluctuations detected through $\mu$SR freeze out 333Note that the
difference in the extension of this regime can be due in the way
extrapolations of transverse relaxation $T_{2}$ plots have been performed
which is always fairly delicate when $T_{2}$ is short..

Similar plots including the data taken on $\alpha$-KCrO2 can be found in the
supplementary material section, they clearly demonstrate the universal
character of this scaling in $T/J$ for that series of chromates.

We are therefore led to the conclusion that a broad universal dynamical regime
emerges on a $T/J$ scale, typical of $ABC$-stacked triangular Heisenberg
antiferromagnets and characterized by a very progressive slowing down of spin
fluctuations.

Neutron scattering studies offered a possible explanation for this unusual
behavior as reported earlier for NaCrO2 [64, 65, 66]. Between 1.3 and 0.7
$T_{\rm N}$, the in-plane correlation length grows rapidly with a $\vec{q}$
vector typical of the $120^{\circ}$ pattern. On the contrary, the inter-layer
correlation length increases very progressively from 1 inter-layer spacing at
$T_{\rm N}$ to only 4 inter-layer spacings at 0.7 $T_{\rm N}$ where it levels
off. This $T$-range closely below $T_{\rm N}$ is dominated by 2D correlations.

The increase in $1/T_{1,\mu}$ up to its maximum could be such a hallmark of
the 2D regime, in fair agreement with the $T^{2}$ behavior of the specific
heat that we clearly observe in $\alpha$-HCrO2. At lower temperatures, a
release of the inter-layer frustration through a small incommensuration may
strengthen the 3D coupling. The consequence is a progressive slowing down of
the fluctuations as observed in $\mu$SR. This regime is typical of a 2D-3D
crossover. What remains unclear is certainly the universality observed in the
maximum of $1/T_{1,\mu}$ . It might be a coincidental result of two
counteracting mechanisms driven by $J_{\perp}$, namely, an increase of both
the coupling and the frustration between the layers.

Alternative scenarios, namely, the Berezinskii-Kosterlitz-Thouless (BKT)
scenario of vortex-antivortex binding or that of $Z_{2}$ vortex excitations,
have been proposed for NaCrO2 [21, 22]. Indeed the maximum in the $\mu$SR
relaxation rate at $T_{m}\simeq 0.7\;T_{\rm N}$ was found to coincide with the
divergence of the correlation length inferred from the broadening of the ESR
line according to the BKT model, $\xi(T)\sim\exp{\frac{b}{T-T_{m}}}$. The data
now available on $\alpha$-HCrO2 and $\alpha$-KCrO2 prove the failure of this
scaling in $T/T_{m}$ (see the Supplementary Materials for the tentative
scaling) and rules out such an interpretation. In addition, we note that a BKT
scenario requires a substantial XY anisotropy, which has never been observed
experimentally. This, therefore, calls for a different interpretation of the
ESR line broadening in terms of a growth of the correlation length when
entering the 3D regime.

A more general lesson from the $A$CrO2 chromates is that the fluctuating 2D
regime clearly sets in below $T_{\rm N}$, likely from the interlayer
frustration. This offers a new perspective on the persistent spin dynamics and
other spin-liquid phenomenology such as observed in the $A$Yb$X_{2}$ compounds
[15, 16] where exchange anisotropy also certainly plays a role. It also calls
for the improved theoretical understanding of how deep the frustration of weak
interlayer couplings may affect the ground state of a quasi-2D
antiferromagnet. Our experimental data for the pure Heisenberg triangular
chromates set an important benchmark for such studies.

In conclusion, besides the observation of a Néel transition, features
reminiscent of the pure 2D Heisenberg triangular lattice are manifested
through the dynamics as revealed here experimentally, between $T_{\rm N}$ and
$0.7\;T_{\rm N}$. The study of the $A$CrO2 chromates ($A$ = H, Na, and K)
points to an original and universal character of excitations. With their half
filled $t_{2g}$ orbitals, chromates are certainly the best representative of a
semi-classical $S=3/2$ Heisenberg model. The perfection of the equilateral
triangular lattice and the absence of disorder are strong assets to provide a
solid playground for the physics of frustration at play on the triangular
lattice in a context where more disordered compounds such as YbMgGaO4 or
$A$Yb$X_{2}$ QSL candidates are in the spotlight. Certainly, the
interpretation of the 2D-3D crossover resides in the details of each of these
compounds although the universal plot in $T/J$ might suggest that it is still
governed by the dynamics specific to the 2D frustrated character of the
triangular lattice.

##  V Summary

The static and dynamic properties of spin-$3/2$ TLHAF $\alpha$-HCrO2 are
studied in detail and compared with the iso-structural compounds (Na,K)CrO2.
$\chi(T)$ could be modeled using HTSE for spin-$3/2$ TLHAF which yields an
intra-layer coupling $J/k_{\rm B}\simeq 24$ K. The complementary band
structure calculations result in a similar value of $J$. They additionally
provide a sizable inter-layer coupling $J_{\perp}/k_{\rm B}\simeq 2.8$ K which
appears specific to $\alpha$-HCrO2. This brings in inter-layer frustration
which shifts down the transition temperature as compared to other members of
the chromates series. On the experimental side, a large hyperfine coupling
$A_{\rm hf}\simeq 1021$ Oe/$\mu_{\rm B}$ between the 1H nuclei and the Cr3+
spins corroborate the existence of such a sizable inter-layer coupling.

Although, $\chi(T)$, $C_{\rm p}(T)$ and muon asymmetry measurements reveal the
onset of a magnetic transition at $T_{\rm N}$, which is also monitored through
a magnetoelastic coupling, a wide fluctuating crossover regime marked by a
broad peak in the muon relaxation rate $1/T_{1,\mu}$ centered at $T\simeq
0.7~{}T_{\rm N}$ and a minimum in the 1H NMR integrated intensity is singled
out. Apparently, this slow dynamical regime turns out to be a universal
character of the TLHAFs $A$CrO2 ($A=$ Na, K, and H) when the specific heat,
NMR intensity, $1/T_{1,\mu}$, and paramagnetic fraction are plotted against
the scaled (with respect to $J$ or $T_{\rm N}$) temperature. This supports a
scenario where a crossover from 2D to 3D correlations sets in around
$0.7~{}T_{\rm N}$ preceded by a typical 2D regime of the TLHAF.

###### Acknowledgements.

PM and RN thank J. Richter for discussions and for providing the HTSE and Padé
approximants used in the fit of our susceptibility data. KS and RN would like
to acknowledge SERB, India for financial support bearing sanction Grant No.
CRG/2019/000960. The work in Augsburg was supported by the German Research
Foundation (DFG) via the Project No. 107745057 (TRR80). Work at the Ames
Laboratory was supported by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences, Materials Sciences, and Engineering Division. The Ames
Laboratory is operated for the U.S. Department of Energy by Iowa State
University under Contract No. DEAC02-07CH11358. FB, PM, and GS acknowledge the
support of the French Agence Nationale de la Recherche under Grant No.
ANR-18-CE30-0022. The work of GS is funded by the Swiss National Science
Foundation Mobility grant P2EZP2-178604 and PALM LabEx grant
ANR-10-LABX-0039-PALM. Part of this work is based on experiments performed at
the Swiss Muon Source S$\mu$S, Paul Scherrer Institute, Villigen, Switzerland.
We would like to thank A. Amato and C. Wang for the technical assistance with
the GPS spectrometer. We thank C. Delmas (ICMCB, Bordeaux) for providing the
KCrO2 sample and N. Penin for mounting it. We thank A. Ozarowski for technical
assistance with ESR measurements. AZ acknowledges the financial support of the
Slovenian Research Agency through program No. P1-0125 and projects No. BI-
US/18-20-064, No. J1-2461, and No. N1-0148. The National High Magnetic Field
Laboratory where the ESR investigation was conducted is supported by National
Science Foundation through NSF/DMR-1644779 and the State of Florida.

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Supplementary Materials

"Universal fluctuating regime in triangular chromate antiferromagnets"

##  S-I AC Susceptibility

Figure S1:  Real part of the ac susceptibility ($\chi^{\prime}$) of
$\alpha$-HCrO2 vs $T$ measured in an ac field of $H_{\rm AC}=5$ Oe and at
three different frequencies.

The real part of the ac susceptibility ($\chi^{\prime}$) as a function of
temperature is shown in Fig. S1. A weak anomaly is observed at $T_{\rm N}$
which is frequency independent. No features are visible around 12 K where ZFC
and FC susceptibilities show bifurcation, ruling out the possibility of spin-
glass transition.

##  S-II NMR Relaxation Rates

Figure S2:  Upper panel: Temperature dependence of the 1H spin-lattice
relaxation rate ($1/T_{1}$) of $\alpha$-HCrO2, measured at two different
frequencies. Lower panel: Temperature variation of the 1H spin-spin relaxation
rate ($1/T_{2}$) for $\alpha$-HCrO2.

1H nuclear spin-lattice relaxation rate $1/T_{1}$ was measured at the central
position of the NMR spectral line. For a $I=1/2$ nucleus, the recovery of the
longitudinal magnetization is expected to follow a single-exponential
behavior. In $\alpha$-HCrO2, the recovery of the nuclear magnetization after a
saturation pulse was indeed fitted well by the stretch exponential function

$1-\frac{M(t)}{M_{0}}=Ae^{-(t/T_{1})^{\eta}},$ (S1)

where $M(t)$ is the nuclear magnetization at a time $t$ after the saturation
pulse, $M_{0}$ is the equilibrium magnetization, and $\eta$ is the exponent.
The $\eta$ parameter is close to 1 for $T>T_{\rm N}$ suggesting a single
relaxation time that confirms high homogeneity of the sample and its
magnetism. $\eta$ values below 1 were obtained at lower temperatures.

Temperature evolution of $1/T_{1}$ does not show any appreciable frequency
dependence (upper panel of Fig. S2). $1/T_{1}$ diverges around $T_{\rm N}$ and
develops a broad hump between $\sim 10$ and 30 K. At high temperatures
($T\gtrsim 30$ K), $1/T_{1}$ is almost temperature-independent, typical in the
high-temperature limit ($T\gg J/k_{\rm B}$) because spins are uncorrelated
[1].

The evolution of $1/T_{1}$ at lower temperature is remarkable. Conventional
antiferromagnets show a sharp peak at the magnetic ordering temperature
followed by a fast decay towards low temperatures caused by the suppression of
magnon excitations [2]. Such a decay is indeed observed in $\alpha$-HCrO2, but
only beyond a broad hump developing between 10 and 30 K. Similar features have
been seen in some other frustrated antiferromagnets, where they are ascribed
to the abundance of low-energy excitations [4]. However, those systems develop
static magnetic order below $T_{\rm N}$, which is not the case in
$\alpha$-HCrO2, as we show here. Moreover, because of much shorter time
window, $\mu$SR is more suitable than NMR for probing the slow dynamics,
especially in the fluctuating regime. In such a case, NMR can only detect a
fraction of the total nuclei with longer relaxation.

To measure the relaxation of transverse nuclear magnetization $M_{xy}(2t)$, a
spin-echo sequence was used with varying time ($t$) between the pulses. The
recovery curves were well fitted by an exponential function

$M_{xy}(2t)=M_{0}e^{-2t/T_{2}}.$ (S2)

The obtained spin-spin relaxation rate $1/T_{2}$ data are plotted as a
function temperature in the lower panel of Fig. S2. It also shows a very broad
peak (10-30 K) with the center of gravity at $T\simeq 17$ K, similar to the
$1/T_{1}$ data.

##  S-III Universal behavior of chromates $A$CrO2, $A$ = H, Na, and K

Figure S3:  Plot of low temperature data of $A$CrO2, $A$ = H, Na, and K; (a)
normalized magnetic heat capacity to its maximum value vs temperature for
$\alpha$-HCrO2 and NaCrO2, (b) evaluation of integrated NMR intensity with
temperature, and (c) temperature variation of $\mu$SR relaxation rate
$1/T_{1,\mu}$ and the paramagnetic fraction. For the latter and for the NMR
intensity, a 21% impurity phase has been subtracted. The temperature in the
$x$-axis is normalized by $T_{\rm N}\simeq 22.5$ K, 41 K, and 24.6 K for
$\alpha$-HCrO2, NaCrO2, and $\alpha$-KCrO2, respectively.

Similar $\mu$SR (PSI facility), 19K NMR experiments were performed for
$\alpha$-KCrO2 along those for $\alpha$-HCrO2. Given its high reactivity, the
sample which had been synthesized in the 70’s and kept in sealed ampoules
since then was transferred under He atmosphere into air tight sample holders
suited for each experiment and filled with He gas to ensure a good
thermalization. A 21% part of the sample was found to be degraded both from
19K NMR and $\mu$SR. The contribution from that part gives clear-cut signals
and can safely be separated from the relevant one in all three experiments.
Similar to Fig. 11, a comparison of our $\mu$SR and NMR results for the three
compounds is presented in Fig. S3, as plotted versus $T/T_{\rm N}$. Clearly,
the universal character of the $T/T_{\rm N}$ scaling is confirmed.

##  S-IV Failure of the BKT scenario

Figure S4:  Same plot as in Fig. 11. The temperature in the $x$-axis is
normalized by the presumed BKT transition temperature as reported in Ref. [3]
$T_{\rm m}/T_{\rm N}=0.37$ for $\alpha$-HCrO2 and 0.56 for NaCrO2,
respectively.

Electron paramagnetic resonance (EPR) measurements by M. Hemmida et al. [22]
suggested the occurrence of Berezinskii-Kosterlitz-Thouless (BKT) transition
at temperatures below $T_{\rm N}$. The BKT transition is expected at $T_{\rm
m}\simeq 0.95T_{\rm N}$, $T_{\rm m}\simeq 0.56T_{\rm N}$, and $T_{\rm m}\simeq
0.37T_{\rm N}$ for LiCrO2, NaCrO2, and $\alpha$-HCrO2, respectively. It was
therefore suggested that the fluctuating regime persists below $T_{\rm N}$ and
down to $T_{\rm m}$. In Fig. S4, we plot a summary of our $\mu$SR results
similar to Fig. 11 and compare it to NaCrO2 using a $T/T_{\rm m}$ [3] rather
than a $T/T_{\rm N}$ scale. No commonality is found for this BKT scenario.

##  S-V Electron Spin Resonance on $\alpha$-KCrO2: a check of DFT+U
calculations

Our ESR investigation on $\alpha$-KCrO2 was conducted on a powder sample
sealed in a quartz tube on a custom-made homodyne-detection spectrometer
working in transmission mode at the NHMFL, Tallahassee, USA. The measurements
were performed in the Faraday configuration at the irradiation frequency of
108.8 GHz in the temperature range between 50 and 290 K. Field modulation with
modulation amplitude of 2 mT was used to enhance signal-to-noise ratio and
derivative ESR spectra were consequently recorded.

A typical ESR spectrum recorded at 290 K is shown in Fig. S5(a). It is of
Lorentzian shape, as typically observed in strongly exchange-coupled spins
systems [5]. The fit with an isotropic Lorentzian model is not far from the
experimental data ($\chi^{2}=4.65$), indicating that the $g$-factor anisotropy
is small. However, allowing for uniaxial $g$-factor anisotropy
$g(\theta)=(g_{c}^{2}\cos^{2}\theta+g_{ab}^{2}\sin^{2}\theta)^{1/2}$ and line-
width anisotropy $\Delta B(\theta)=(\Delta B_{c}^{2}\cos^{2}\theta+\Delta
B_{ab}^{2}\sin^{2}\theta)^{1/2}$ fits the experiment considerably better
($\chi^{2}=0.50$). Here $\theta$ denotes the angle between the high-symmetry
crystallographic $c$-axis and the applied magnetic field. Such uniaxial
anisotropy is due to trigonal distortion of CrO6 octahedra – it reflects a
three-fold rotational axis that is parallel to the crystallographic $c$-axis
and passes through the Cr site. The fit with the uniaxial model discloses a
small $g$-factor anisotropy [$g_{c}=1.995(3)$, $g_{ab}=1.989(3)$] and a large
line-width anisotropy [$\Delta B_{c}=151(1)$ mT, $\Delta B_{ab}=76(1)$ mT],
yielding $\Delta B_{c}/\Delta B_{ab}=2.0$. The corresponding peak-to-peak line
width $\Delta B_{pp}=53$ mT is in good agreement with previously reported
values of 49 and 52 mT, recorded at lower irradiation frequencies in $X$-band
and $Q$-band, respectively[6].

Figure S5: (a) The ESR spectrum of $\alpha$-KCrO2 recorded at 290 K and 108.8
GHz (black line). The red and green lines show powder simulations with
Lorentzian line shape assuming uniaxial anisotropy ($\chi^{2}=0.50$) and
isotropic symmetry ($\chi^{2}=4.65$), respectively. The inset shows a perfect
fit of the uniaxial model in a wider field window. (b) The temperature
dependence of the principal ESR line widths for the magnetic field along the
crystallographic $c$-axis and within the $ab$-plane. The solid lines are fits
of the two datasets with the model $\Delta B_{i}=\Delta
B_{i}^{\infty}+C_{i}/T$ above 100 K. The inset shows the temperature variation
of the ratio of the two principal line widths.

With decreasing the temperature below 290 K the ESR spectrum of $\alpha$-KCrO2
progressively broadens and becomes too broad for a reliable fit below $\sim
50$ K [Fig. S5(b)]. Interestingly, the ratio of the two principal line-widths
$\Delta B_{c}/\Delta B_{ab}=2.0$ found at 290 K remains unchanged down to
$\sim 100$ K, where it starts to decrease with decreasing temperature and then
notably increases below $\sim 70$ K [inset of Fig. S5(b)]. As the temperature
dependence of the ESR line width is a sign of developing spin correlations,
these thus seem to be isotropic above $\sim$100 K and become anisotropic below
this temperature. In the regime of isotropic spin correlations both line
widths follow the temperature dependence $\Delta B_{i}(T)=\Delta
B_{i}^{\infty}+C_{i}/T$, where the term $\Delta B_{i}^{\infty}$ denotes the
infinite-temperature limit of the line width and the term $C_{i}/T$
corresponds to the predicted lowest order correction in $1/T$ [5]. The
limiting line widths $\Delta B_{c}^{\infty}=65(1)$ mT and $\Delta
B_{ab}^{\infty}=31(1)$ mT that are found after fitting the line-width datasets
above 100 K are a direct measure of magnetic anisotropy present in
$\alpha$-KCrO2.

Within the well-established Kubo-Tomita theory [7], the ESR full-width-at-
half-maximum of an exchange-coupled spin system is determined by the second
moment $M_{2}$ and the fourth moment $M_{4}$ of the absorption line [8],

$\Delta B=\sqrt{2\pi}\frac{k_{B}}{g\mu_{B}}\sqrt{\frac{M_{2}^{3}}{M_{4}}},$
(S3)

with

$\displaystyle M_{2}$
$\displaystyle=\frac{\left\langle\left[\mathcal{H}^{\prime},S^{+}\right][S^{-},\mathcal{H}^{\prime}]\right\rangle}{\left\langle
S^{+}S^{-}\right\rangle},$ $\displaystyle M_{4}$
$\displaystyle=\frac{\left\langle\left[\mathcal{H}-\mathcal{H}_{Z},\left[\mathcal{H}^{\prime},S^{+}\right]\right][\mathcal{H}-\mathcal{H}_{Z},\left[\mathcal{H}^{\prime},S^{-}\right]]\right\rangle}{\left\langle
S^{+}S^{-}\right\rangle},$ (S4)

where $[A,B]$ denotes a commutator of operators $A$ and $B$,
$\left\langle...\right\rangle$ stands for canonical averaging and $S^{+}$,
$S^{-}$ are the ladder operators for spin $S=3/2$. The Hamiltonian of the
system $\mathcal{H}=\mathcal{H}_{ex}+\mathcal{H}_{Z}+\mathcal{H}^{\prime}$ is
divided into the exchange term $\mathcal{H}_{ex}$, the Zeeman term
$\mathcal{H}_{Z}$ and an anisotropy term $\mathcal{H}^{\prime}$. The two
moments can be calculated analytically at infinite temperature where spin
correlations between neighboring sites vanish.

In $\alpha$-KCrO2 the main magnetic anisotropy term is of the single-ion
anisotropy type

$\mathcal{H}^{\prime}=\sum_{i}D(S_{i}^{z})^{2},$ (S5)

due to the crystal-field symmetry being reduced from cubic at the Cr site. The
anisotropy axis is parallel to the anisotropy axis of the $g$ tensor and is
thus parallel to the crystallographic $c$ axis. We note that Dzyaloshinskii-
Moriya interaction, often being the dominant magnetic anisotropy term in
exchange coupled systems, is absent in $\alpha$-KCrO2 due to a center of
inversion in the middle of the Cr-Cr bond. Inserting magnetic anisotropy given
by Eq. (S5) into Eqs. (S4) yields the angular dependence of the ESR line width
from Eq. (S3),

$\Delta
B^{\infty}(\theta)=\sqrt{2\pi}\frac{k_{B}}{g\mu_{B}}\frac{6D^{2}\left(1+\cos^{2}\theta\right)}{5\sqrt{30}J_{1}}.$
(S6)

Here we neglect all exchange interactions beyond the dominant nearest-neighbor
exchange interaction $J_{1}$, because squares of exchange interactions appear
in the expression of $M_{4}$ [9].

The calculated ESR line-width anisotropy yields $\Delta B^{\infty}_{c}=2\Delta
B^{\infty}_{ab}$, which is in perfect agreement with the experiment above 100
K [inset in Fig. S5(b)]. The experimentally determined values $\Delta
B_{c}^{\infty}=65(1)$ mT, $\Delta B_{ab}^{\infty}=31(1)$ mT and the nearest-
neighbor exchange coupling $J_{1}=15$ K from our DFT+$U$ calculations for
$\alpha$-KCrO2 (Table I in the main text) then predict the single-ion
anisotropy $D=1.1(1)$ K, which agrees very well with the theoretical
prediction in Table I, thus verifying the validity of the theoretical
calculations.

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