# Magnetic field-induced phases and spin Hamiltonian in Cs2CoBr4

L. Facheris  Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich,
Switzerland  S. D. Nabi  Laboratory for Solid State Physics, ETH Zürich, 8093
Zürich, Switzerland  K. Yu. Povarov  Laboratory for Solid State Physics, ETH
Zürich, 8093 Zürich, Switzerland  Present address: Dresden High Magnetic Field
Laboratory (HLD-EMFL) and Würzburg-Dresden Cluster of Excellence ct.qmat,
Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  Z. Yan
Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland  A.
Glezer Moshe  U. Nagel  T. Rõõm  National Institute of Chemical Physics and
Biophysics, Akadeemia tee 23, 12618 Tallinn, Estonia  A. Podlesnyak  Neutron
Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, USA  E. Ressouche  K. Beauvois  Université Grenoble Alpes, CEA, IRIG,
MEM, MDN, 38000 Grenoble, France  J. R. Stewart  P. Manuel  D. Khalyavin  F.
Orlandi  ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, Didcot,
OX11 0QX, United Kingdom  A. Zheludev<EMAIL_ADDRESS>http://www.neutron.ethz.ch/ Laboratory for Solid State Physics, ETH Zürich,
8093 Zürich, Switzerland

###### Abstract

Magnetic structures and spin excitations are studied across the phase diagram
of the geometrically frustrated $S=3/2$ quantum antiferromagnet Cs2CoBr4 in
magnetic fields applied along the magnetic easy axis, using neutron
diffraction, inelastic neutron scattering and THz absorption spectroscopy. The
data are analyzed, where appropriate, using extended $SU(4)$ linear spin wave
theory. A minimal magnetic Hamiltonian is proposed based on measurements in
the high field polarized state. It deviates considerably from the previously
considered models. Additional dilatometry experiments highlight the importance
of magnetoelastic coupling in this system.

##  I Introduction

The title compound is a member of a well-known family of frustrated triangular
antiferromagnets that includes such thoroughly-studied systems as Cs2CuCl4 [1,
2, 3, 4], Cs2CuBr4 [5, 6] and Cs2CoCl4 [7, 8, 9]. Nevertheless, recent
magnetic, thermodynamic and neutron scattering studies of Cs2CoBr4 have
demonstrated its uniqueness [10, 11, 12]. The excitation spectrum is a complex
hierarchy of bound states of fractional excitations, reminiscent of Zeeman
ladders in Ising spin chains [13, 14, 15, 16]. Indeed, the magnetic lattice in
Cs2CoBr4 is a distorted triangular one, with interactions along one particular
crystallographic direction somewhat stronger than the transverse coupling.
However, the situation in Cs2CoBr4 is far more complex than in the one-
dimensional (1D) Ising model. The material features dominant easy-plane,
rather than Ising anisotropy. The observed bound states are not confined to
single “chains”, and also propagate in two dimensions [12]. In applied
magnetic fields Cs2CoBr4 demonstrates a complex phase diagram with as many as
five distinct ordered phases [10, 11]. Apart from the Néel “stripe” ground
state, those include an incommensurate spin density wave (SDW) phase and a
commensurate fractional magnetization plateau.

Despite these rather spectacular experimental findings, the physics of
Cs2CoBr4 remains poorly understood. To date, the spin Hamiltonian could only
be guessed from bulk measurements and a crude spin wave theory (SWT) analysis
of neutron spectra collected in zero applied field [10, 11]. The obvious
limitation of this approach is that this spectrum is Zeeman-ladder like and
actually has little resemblance of anything that semiclassical SWT predicts.
Another limitation is the complete frustration of inter-chain interactions in
the stripe phase. This suppresses the transverse dispersion of magnetic
excitations and makes it insensitive to any details of inter-chain coupling.
The third is that the quantum nature of spins involved and reduced
dimensionality most likely result in a substantial re-normalization of
exchange constants: even if the spectrum were adequately described by SWT, the
values of the exchange constants obtained in any fit to the data would be
significantly different from the actual constants in the Hamiltonian. The
final limitation of the previous analysis is the reliance on a
pseudospin-$1/2$ projection of the actual $S=3/2$ Hamiltonian appropriate for
describing the Co2+ ions in this material [8, 11]. The approximation would be
excellent if the single-ion anisotropy were orders of magnitude stronger than
exchange interactions and Zeeman energies of applied fields, but that is not
quite the case for Cs2CoBr4. All in all, one has to admit that the spin
Hamiltonian remains unknown.

The second gap in our understanding is of a technical nature. Due to the
geometry of split-coil magnets used in neutron experiments, it is rather
difficult to simultaneously apply a large magnetic field along the chain axis
and measure at large momentum transfers along that same direction.
Unfortunately, precisely this type of experiment is needed to accurately
determine the magnetic structures and understand the spin dynamics in applied
fields. As a result, the nature of the two high-field phases has not been
clarified to date.

The present work aims to address these issues. We report comprehensive
magnetic diffraction and spectroscopy experiments on Cs2CoBr4 in applied
magnetic fields. First, we determine the magnetic structures of the two
remaining high-field ordered phases, thereby completing the exploration of the
$H-T$ phase diagram. In this context we also report dilatometric measurements
that reveal the importance of magneto-elastic effects across the phase
boundaries. Next, we apply generalized $SU(4)$ SWT to analyze the neutron and
THz spectra collected in the high-field paramagnetic state. Since SWT is known
to work well in that regime, it provides reliable estimates of the actual
exchange constants. The results lead us to a substantial revision of the
“minimal model” for Cs2CoBr4. Finally, we use the exchange parameters measured
in high field to make sense of the spectra collected in several low-field
phases, including the plateau state.

##  II Structural considerations and minimal model

Before describing the experiments performed in this work we summarize the
basic facts about the crystal structure of Cs2CoBr4 and consider the single-
ion magnetic anisotropy and possible exchange interactions.

As discussed in some detail in Refs. [10, 11], the material is orthorhombic,
space group $Pnma$, with lattice parameters $a=10.137(1)$ Å $b=7.593(3)$ Å and
$c=13.281(1)$ Å at $\sim 100$ mK. There are four equivalent magnetic Co2+ ions
in the unit cell, each encased in a Br4 tetrahedron. The point symmetry of the
Co site allows for general single-ion anisotropy and gyromagnetic tensors with
the restriction that one principal axis of the corresponding ellipsoid must
coincide with the $\mathbf{b}$ direction. A previous analysis of high-
temperature susceptibility data suggested that the single-ion anisotropy term
in the spin Hamiltonian is of predominantly the XY-type, the hard axis
$\mathbf{z}$ being perpendicular to the crystallographic $\mathbf{b}$ axis
(denoted as $\mathbf{y}$), and forming an angle $\beta\approx 44^{\circ}$ with
the $\mathbf{a}$ direction. For the $S=3/2$ ion Co2+ the most general
symmetry-compatible single-ion Hamiltonian is

$\hat{\mathcal{H}}_{\text{single-
ion}}=-D{\hat{S}_{x}}^{2}-(D+d){\hat{S}_{y}}^{2},$ (1)

where $d$ is a small Ising-like contribution, always along the chains, i.e.,
the crystallographic $\mathbf{b}$-axis.

Figure 1: The five nearest-neighbor magnetic interactions in Cs2CoBr4. The
nodes of the lattice are Co2+ ions. The boundaries of a unit cell are shown in
black lines in the top-left corner.

The principal antiferromagnetic exchange constant referred to as $J$ in Refs.
[10, 11, 12] connects nearest neighbors along the $\mathbf{b}$-direction,
forming the above-mentioned chains. This interaction is the only one
corresponding to an obvious Co-Br-Br-Co supercharge pathway with a short (3.71
Å) Br-Br distance. Yet, at $7.593$ Å $=b$ it is only the 4-th shortest Co-Co
bond, and will thus be denoted below as $J_{4}$. Being a principal
crystallographic translation, this bond connects two CoBr4 tetrahedra with
identical orientations. The anisotropy axes of the corresponding interacting
ions coincide.

As in the much-studied Cs2CuCl4 [1], the distorted triangular lattice is
completed by a bond that is almost parallel to the $(b,c)$ plane and connects
magnetic sites in adjacent chains. We previously denoted it as $J^{\prime}$
[10, 11, 12]. It happens to be the 5th-shortest Co-Co distance of 7.65 Å and
will be henceforth denoted as $J_{5}$. The coordination tetrahedra of the Co2+
ions that it connects are related by a glide symmetry. The normal vectors to
the corresponding anisotropy planes form angles of $\pm\beta$ with the
$\mathbf{a}$ axis, which makes them almost orthogonal.

As will be shown below, these two coupling constants are not sufficient to
fully describe the dispersion of magnetic excitations in Cs2CoBr4. We
therefore also consider interactions across the three shortest Co-Co
distances, as shown in Fig. 1. The nearest-neighbor bond $J_{1}$ is only
$6.50$ Å and connects adjacent chains (and with them, the adjacent triangular
planes) along the $\mathbf{a}$ direction, in a zigzag manner. Together with
$J_{4}$ it forms zigzag ladders running along the $\mathbf{b}$ axis. Just like
$J_{5}$, this bond is frustrated by the AF coupling within each chain. The
sites that it connects are related by inversion, and therefore have identical
single-ion anisotropy and gyromagnetic tensors.

A very similar connection between triangular planes is provided by $J_{3}$
that spans over 6.86 Å. Together with $J_{1}$ and $J_{4}$, $J_{3}$ completes a
somewhat buckled triangular lattice in the $(a,b)$ plane. Unlike in the
previously discussed $(b,c)$-triangular lattice, the inter-chains bonds
$J_{3}$ and $J_{1}$ are not equivalent. It will be later argued that one of
them must be negligible compared to the other.

The last potential Co-Co exchange constant that we shall consider here is
$J_{2}$. It spans over 6.79 Å and also connects adjacent $(b,c)$-triangular
planes. Unlike all other inter-chain bonds discussed above, it is not
frustrated by antiferromagnetic in-chain $J_{4}$-interactions. However, like
$J_{5}$, it connects ions with almost orthogonal anisotropy planes.

As mentioned, there are good microscopic reasons for $J_{4}$ to be strong, but
the hierarchy of the remaining exchange constants is hard to guess based on
just the crystal structure. We note, however, that the next (6th) coupling
would have to span over a distance of more than 10 Å. Our “minimal model” for
Cs2CoBr4 will therefore be restricted to just $J_{1}$–$J_{5}$.

By symmetry considerations alone, all the above-mentioned interactions may
include anisotropic exchanges and also some off-diagonal Dzyaloshinskii-Moriya
terms. Taking all that into account would result in 45 independent exchange
parameters and clearly an impractical model. Instead we shall assume that all
interactions are of Heisenberg type, and that the single-ion term (1) is the
only source of magnetic anisotropy in Cs2CoBr4. While the assumption is
clearly arbitrary, below we show that this minimal model is able to adequately
describe many features of the spin excitation spectrum in a wide range of
applied fields.

##  III Experimental procedures

Magnetic propagation vectors in the high-field phases were determined with the
WISH time-of-flight diffractometer at Rutherford Appleton Laboratory (RAL, UK)
[17]. The sample environment consisted of a 3He-4He dilution refrigerator in
combination with a $10$ T vertical superconducting cryomagnet. A $221.4$ mg
single crystal was installed with the $(a,c)$ plane horizontal and the field
applied along the $\mathbf{b}$ axis. Thanks to the large detector bank, the
propagation vector of the “D” and “E” phases could be identified [18].

Following that, extensive nuclear scattering and magnetic diffraction data
sets were collected on the CEA-CRG D23 lifting-counter diffractometer at ILL
(France). The $36.6$ mg single crystal was mounted on the cold finger of a
3He-4He dilution refrigerator in a $6$ T superconducting cryomagnet. The field
was applied vertically, along the crystallographic $\mathbf{b}$ axis. Sample
mosaic was approximately $0.5^{\circ}$ FWHM. All the measurements were
performed at the base temperature of the dilution insert $<100$ mK. Neutrons
with $\lambda=1.28$ Å (copper monochromator) and $2.39$ Å (pyrolytic graphite
monochromator) were used for nuclear and magnetic studies, respectively.

Inelastic neutron scattering measurements in applied fields were conducted on
the CNCS time-of-flight spectrometer at ORNL (USA) [19], using $3.32$ meV
incident energy neutrons. The $825$ mg single crystal sample was aligned with
the $(a,c)$ plane in the principal (horizontal) scattering plane of the
instrument. The sample environment consisted of a 3He-4He dilution
refrigerator and an $8$ T vertical field cryomagnet ($\mathbf{H}\|\mathbf{b}$
at all times). The spectrum was measured by rotating the sample over a
$190^{\circ}$ angular range around the $\mathbf{b}$ axis in steps of
$1^{\circ}$. Typical counting times were $\sim 5$ minutes at each sample
orientation. Energy resolution was $\approx 0.10$ meV at the elastic position.
All the measurements were performed at the base temperature of the dilution
insert $\sim 100$ mK. The data were reduced using MANTID [20] and HORACE [21]
software.

Below we also show some previously unpublished data from a zero-field
inelastic neutron experiment carried out at the LET spectrometer at RAL [12,
22]. The same $1.16$ g single crystal as used in Refs. [11, 12] was aligned to
have the $(b,c)$ plane horizontal. All the measurements were performed at the
base temperature of a 3He-4He dilution refrigerator. The sample was rotated
over a $180^{\circ}$ angular range around the $\mathbf{a}$ axis in steps of
$1^{\circ}$, typically counting $\sim 10$ minutes at each orientation. The
data were reduced with MANTID [20] and HORACE [21].

The THz absorption experiment was performed with a Martin-Puplett-type
interferometer and a 3He-4He dilution refrigerator using a 3He-cooled Si
bolometer at $300$ mK. The sample was a circular plate $1.1$ mm thick in the
$\mathbf{b}$ direction and $4$ mm in diameter. THz radiation propagating along
the crystal $\mathbf{b}$ axis was unpolarized and the apodized instrumental
resolution was $0.025$ meV. Due to the heating caused by the THz radiation,
the lowest stable attainable temperature was around $200$ mK.

Studies of magnetostriction were performed by measuring the capacitance
between two electrodes deposited as thin layers of conducting Silver paste on
the surface of a plate-shaped sample. Thin coaxial cables were attached to the
electrodes and carried the signal to an Andeen-Hagerling A2550A ultra-
precision capacitance bridge operating with a $1$ kHz AC probing field and an
excitation voltage of $15$ V. The measurements were performed with a 3He-4He
dilution refrigerator insert for a commercial Quantum Design Physical Property
Measurement System equipped with a $9$ T superconducting magnet. In the
experiment, the capacitance was tracked versus increasing field for a set of
fixed temperatures. The field was swept at $10$ Oe/s and the raw data were
averaged over $200$ Oe wide bins. In all the cases, the probing electric field
was $\mathbf{E}\parallel\mathbf{a}$, while the sample was installed with
$\mathbf{H}\parallel\mathbf{b}$. Only the real part of the complex capacitance
is discussed, as it showed a modulated response versus field, in contrast to
the imaginary part which was in all the cases flat.

##  IV Results and data analysis

We now report and briefly discuss the results of different experiments in
applied magnetic fields. In all cases, the latter was directed along the
crystallographic $\mathbf{b}$ axis.

###  IV.1 Low-temperature crystal structure

The crystal structure in zero field and a temperature of $\sim 0.1$ K was
previously determined in a neutron diffraction experiment at the D23 lifting-
counter diffractometer at ILL. Details of the measurements and structural
refinement are given in the Supplement of Ref. [11]. For the sake of
completeness, here we provide the resulting atomic coordinates, which were not
reported to date. These are listed in Table 1.

Table 1:  Crystal structure parameters of Cs2CoBr4 at $100$ mK determined through neutron diffraction, as described in the Supplement of Ref. [11]. Atom | $x$ | $y$ | $z$ | $B_{\text{iso}}$
---|---|---|---|---
Cs | 0.5240(6) | 0.75 | 0.3339(5) | 0.59(12)
Cs | 0.8604(8) | 0.75 | 0.6002(6) | 0.53(15)
Co | 0.2616(12) | 0.75 | 0.5801(10) | 0.73(27)
Br | 0.4970(6) | 0.75 | 0.6055(5) | 0.73(11)
Br | 0.1901(9) | 0.75 | 0.4050(4) | 0.74(11)
Br | 0.1694(4) | 0.4946(29) | 0.65279(16) | 0.88(5)

###  IV.2 Magnetic neutron diffraction

####  IV.2.1 Succession of magnetic propagation vectors

Figure 2:  A schematic representation of magnetic structures that occur in
Cs2CoBr4 in magnetic fields applied along the $\mathbf{b}$ axis. Only the Co-1
and Co-3 sites that form $J_{4}-J_{1}$ zigzag ladders are shown. For these,
the individual magnetic moments are in all cases confined to a plane which
coincides with the local anisotropy easy plane common to Co-1 and Co-3. The
ellipses represent additional weak in-plane anisotropy (parameter $d$ in Eq.
(1)), with an easy $\mathbf{b}$-axis.

As discussed in detail in Ref. [10], in Cs2CoBr4 applying a magnetic field
along the crystallographic $\mathbf{b}$ axis causes a cascade of transitions
between magnetic phases A-F at $H_{\text{AB}}=1.5$ T, $H_{\text{BC}}=2.7$ T,
$H_{\text{CD}}=4.0$ T, $H_{\text{DE}}=4.5$ T, and $H_{\text{sat}}=5.2$ T. The
corresponding phase boundaries in the $(H,T)$ plane are reproduced in symbols
in Fig. 10 below. All phase transitions except the one at $H_{\text{sat}}$ are
discontinuous. The magnetic structures of phases A, B and C have been
previously solved from neutron diffraction [11]: Phase A is the Néel or
“stripe” phase with propagation vector $(0,1/2,1/2)$. Phase B is an
incommensurate longitudinal spin density wave (SDW). The propagation vector is
$(0,\xi,0)$ and is strongly field-dependent, with $\xi$ proportional to the
$S=1/2$ pseudospin-magnetization. The pseudospin-plateau Phase C is a lock-in
version of Phase B, with propagation vector $(0,1/3,0)$. It can be seen as a
classical “up-up-down” (UUD) structure. All this is illustrated in Fig.
2(a-c), where only the Co-1 and Co-3 sites belonging to a single $J_{4}-J_{1}$
zigzag ladder are shown. Beyond $H_{\text{sat}}$ is the paramagnetic
“pseudospin-saturated” state.

Measurements on the WISH instrument at RAL revealed the propagation vectors in
the two remaining phases: $(0,1/2,1/2)$ for Phase D and $(0,0.47,0)$ in Phase
E. Subsequent measurements on the D23 diffractometer have shown that the
propagation vector in Phase E is field-independent, unlike Phase B. In Fig. 3
we show the evolution of the magnetic order parameters in each phase. For
Phases A-C the data are from [11], and for Phases D and E from the analysis
described below.

Figure 3:  Magnetic order parameter (amplitude of the corresponding Fourier
component of magnetization) versus magnetic field applied along the
$\mathbf{b}$ axis, as deduced from the intensities of the indicated Bragg
reflections measured in Cs2CoBr4 at $T<100$ mK. The data for Phases A-C are
from Ref. [11].

####  IV.2.2 Spin-flop phase

The spin structure at 4.25 T in Phase D, where the propagation vector is
$(0,1/2,1/2)$, was determined from the analysis of 70 magnetic Bragg
intensities. These were collected in rocking curves with a 0.06∘ step and
counting $15$ s/point. The symmetry-based group theory analysis was performed
using the SARA$h$ software [23]. It restricts the possible structure to the
same two irreducible representations as in zero field [11]. Of these only
$\Gamma_{2}$ is consistent with the measured pattern of intensities, as is
also the case in zero field. Its basis vectors are tabulated in Table 2.

$\Gamma_{2}$ | $\Psi_{1}$ | $\Psi_{2}$ | $\Psi_{3}$ | $\Psi_{4}$ | $\Psi_{5}$ | $\Psi_{6}$
---|---|---|---|---|---|---
Co-1 | $(1,0,0)$ | $(0,1,0)$ | $(0,0,1)$ | $(1,0,0)$ | $(0,-1,0)$ | $(0,0,1)$
Co-2 | $(1,0,0)$ | $(0,-1,0)$ | $(0,0,-1)$ | $(1,0,0)$ | $(0,1,0)$ | $(0,0,-1)$
Co-3 | $(1,0,0)$ | $(0,1,0)$ | $(0,0,1)$ | $(-1,0,0)$ | $(0,1,0)$ | $(0,0,-1)$
Co-4 | $(1,0,0)$ | $(0,-1,0)$ | $(0,0,-1)$ | $(-1,0,0)$ | $(0,-1,0)$ | $(0,0,1)$
Table 2:  Components of the magnetic basis vectors (in $\mu_{B}$) of the
$\Gamma_{2}$ irreducible representation associated with the $(0,1/2,1/2)$
propagation vector at $4.25$ T (Phase D). The cell coordinates of the four Co
sites Co-1: $(0.262(1),0.75,0.580(1))$, Co-2: $(0.762(1),0.75,0.920(1))$,
Co-3: $(0.738(1),0.25,0.420(1))$ and Co-4: $(0.238(1),0.25,0.080(1))$,
following the convention in [11].

A refinement of this model was performed using the FullProf Suite [24]. A good
fit to the data ($R$-factor $=10.1\%$, Fig. 4) can be obtained with just 3
parameters $C_{1}=1.11(1)$, $C_{2}=0.01(3)$, and $C_{3}=1.14(1)$, representing
the mixing coefficients of the $\Psi_{1}$, $\Psi_{2}$, and $\Psi_{3}$ basis
vectors, respectively. We have considered that $\Gamma_{2}$-structures can
occur in several domains. Specifically, the generators of the little group
produce an equivalent domain where the spins on Co-3 and Co-4 are flipped. In
our analysis we included domain population as a fitting parameter. In the end,
the population of the second domain was found to be negligible, which is
likely due to a slight misalignment of the field away from the $\mathbf{b}$
axis.

At 4.25 T the refined staggered component of magnetic moment of Co-1 is
$\mathbf{m}=(\,1.11(1),0.01(3),1.14(1)\,)\mu_{B}$ in Cartesian coordinates.
The modulation is entirely transverse. Combined with a uniform magnetization
of 1.94 $\mu_{\mathrm{B}}$ that we expect to be induced by the applied field
[10], this is nothing else but a spin-flop version of the zero-field
structure, as illustrated in Fig. 2(d). The spins are confined to planes that
form an angle of 44∘ with the $\mathbf{a}$ axis. In other words, the spins lie
very close to the respective easy anisotropy planes, as one would expect. The
net ordered magnetic moment is $m=2.51(2)~{}\mu_{\mathrm{B}}$ on each site.

Figure 4:  Calculated versus observed integrated intensities (from the D23
experiment) for magnetic Bragg peaks at $4.25$ T for the spin-flop structure.

####  IV.2.3 Fan phase

The highest-field ordered Phase E with propagation vector $(0,0.47,0)$ was
solved from the analysis of $25$ magnetic reflections measured at $4.8$ T in a
similar manner as for Phase D. In this case, the form of the propagation
vector is the same as in the SDW and UUD states [11]. The magnetic structure
must belong to one of four 3-dimensional irreducible representations. Again,
only one is compatible with the measured distribution of intensities, namely
$\Gamma_{3}$, same as that realized in the SDW and UUD states (Table 3).

$\Gamma_{3}$ | $\Psi_{1}$ | $\Psi_{2}$ | $\Psi_{3}$
---|---|---|---
Co-1 | $(1,0,0)$ | $(0,1,0)$ | $(0,0,1)$
Co-2 | $(1,0,0)$ | $(0,1,0)$ | $(0,0,-1)$
Co-3 | $(e^{i\varphi},0,0)$ | $(0,-e^{i\varphi},0)$ | $(0,0,e^{i\varphi})$
Co-3 | $(e^{i\varphi},0,0)$ | $(0,-e^{i\varphi},0)$ | $(0,0,-e^{i\varphi})$
Table 3:  Components of the magnetic basis vectors (in $\mu_{B}$) of the
$\Gamma_{3}$ irreducible representation associated with the $(0,0.47,0)$
propagation vector at $4.8$ T (Phase E).

The phase shift $\varphi=84.6^{\circ}$ in Table 3 is not a free parameter, but
is determined by the propagation vector. No domains are possible. This leaves
three parameters needed to describe the structure, namely the mixing
coefficients $C_{1}$, $C_{2}$, and $C_{3}$ for $\Psi_{1}$, $\Psi_{2}$, and
$\Psi_{3}$ basis vectors from Table 3, respectively. The refinement converged
to a solution with $R$-factor $=8.0\%$ (Figure 5). The modulated component of
magnetization is $(\,1.18(3),0.42(16),1.00(4)\,)\mu_{B}$ for Co-1. Note that
the oscillating component lies almost exactly in the local anisotropy plane,
but is not strictly perpendicular to the applied field. Instead it forms an
angle of 75∘ with it. From magnetization data, we also know the uniform
$y$-axis component to be $2.46~{}\mu_{\mathrm{B}}$. The matching classical
structure (one with the same magnitude of magnetic moment on all sites) is a
tilted fan structure in which the ordered magnetic moment
$m=3.26(3)~{}\mu_{\mathrm{B}}$ undergoes a rotational oscillation around a
fixed direction $\mathbf{n}$, canted by an angle of 28∘ with respect to the
applied field. The angular amplitude of the oscillations is 35∘. In
neighboring chains within each $J_{4}-J_{1}$ ladder the canting direction is
opposite, as illustrated in Fig. 2(e). This canting may be caused by
interactions with adjacent ladders, where the spin oscillation planes are
almost orthogonal. An alternative explanation would be Dzyaloshinskii-Moriya
interactions.

Figure 5:  Calculated versus observed integrated intensities (from the D23
experiment) for magnetic Bragg peaks at $4.8$ T for the fan structure.

###  IV.3 Neutron spectroscopy

As mentioned in the introduction, the cryomagnet used in the experiment
severely limits the access to momentum transfers in the field direction. In
our cases, only reciprocal space vectors
$\mathbf{q}=h\mathbf{a}^{\ast}+k\mathbf{b}^{\ast}+l\mathbf{c}^{\ast}$ with
$-0.3<k<0.3$ can be reached. At the same time, unlike in previous studies, a
wide range of momentum transfers in the $(h,0,l)$ plane are readily
accessible.

####  IV.3.1 Paramagnetic (pseudo-polarized) phase

The data collected at the CNCS instrument in paramagnetic state at $T\sim 100$
mK and a magnetic field $\mu_{0}H=7$ T applied along the $\mathbf{b}$
direction are shown in Fig. 6(a,c,e,g,i,k). Here the material is in the
paramagnetic or “pseudospin-saturated” state [10]. The false color plots
represent measured scattered neutron intensity versus energy and momentum
transfers along different reciprocal-space directions. Integration along the
two transverse directions has been performed as follows: in the range $\pm
0.25$ r.l.u. for the $\mathbf{a}^{\ast}$ and/or $\mathbf{c}^{\ast}$ directions
and, where applicable, $\pm 0.1$ r.l.u. along the $\mathbf{b}^{\ast}$ axis.
These are raw data, without any background subtraction. The strong scattering
at low energies is due to incoherent scattering in the sample and spurious
scattering from the cryomagnet. The incident neutron energy is
$E_{\text{i}}=3.32$ meV.

Figure 6: (a,c,e,g,i,k) Inelastic neutron scattering intensity measured in
Cs2CoBr4 in a magnetic field $\mu_{0}H=7$ T applied along the crystallographic
$\mathbf{b}$ axis at $T\sim 100$ mK, in the paramagnetic pseudo-polarized
phase. The data are integrated along the transverse directions in reciprocal
space, as described in the text. (b,d,f,h,j,l) $SU(4)$ spin wave theory
calculation based on the minimal model parameters listed in Table 4.

One readily discerns two excitation sectors. At low energies we see
transitions within the lower-energy doublet of Co2+, which is dominated by
$S_{z}=\pm 1/2$ states due to the prevailing XY-type anisotropy. These will
henceforth be loosely referred to as “magnons”. At higher energies we see
transitions between the lower and higher doublets, the latter mainly composed
of $S_{z}=\pm 3/2$ Co2+ states. These we will denote as “crystal field” (CF)
excitations.

| distance,Å | anisotropy | value, meV | zero field | renormalized
---|---|---|---|---|---
$\bm{J_{1}}$ | 6.50 | $\|$ | 0.06(1) | $BJ_{1}$ | $B=0.55(2)$
$J_{2}$ | 6.79 | $\bot$ | -0.003(2) | $BJ_{2}$ |
$J_{3}$ | 6.86 | $\|$ | 0.00(1) | $BJ_{3}$ |
$\bm{J_{4}}$ | 7.59 | $\|$ | 0.135(5) | $AJ_{4}$ | $A=1.15(2)$
$J_{5}$ | 7.65 | $\bot$ | 0.020(5) | $BJ_{5}$ |
$\bm{D}$ | – | – | 0.70(5) | $D$ |
$\bm{d}$ | – | – | 0.20(5) | $Cd$ | $C=0.40(5)$
Table 4: Parameters of a minimal model Heisenberg Hamiltonian for Cs2CoBr4.
The 3rd column indicates whether the corresponding bond connects magnetic ions
with parallel or almost perpendicular anisotropy plane. The last column shows
the renormalization of exchange constants in zero applied field.

The first thing to note is that several branches show a sizable dispersion
along the $\mathbf{c}^{\ast}$ axis, whereas it was very narrow in zero field.
This is to be expected, since polarizing the spins in the chains by the
external field removes the frustration for the zigzag inter-chain coupling.
What could not be expected from our previous weakly-coupled triangular planes
model for Cs2CoBr4 is the sizable dispersion along the $\mathbf{a}^{\ast}$
direction. It immediately shows that inter-plane interactions are at least as
strong as the in-plane ones.

As demonstrated in Ref. [2], the value of inelastic data collected in the
saturated phase is that the true exchange constants can be determined by
analyzing it with SWT, which becomes exact. Strictly speaking, the latter is
only true at full saturation in the axially symmetric case. In our experiment
the field is applied inside the easy plane, so axial symmetry is lacking.
Related to this is that a field of $7$ T only corresponds to saturation in the
pseudospin model. The actual magnetization of Cs2CoBr4 continues to increase
[10]. Nevertheless, we can hope that SWT is reasonably accurate, and certainly
more reliable than in zero field.

Generalized $SU(4)$ SWT computations of the spectrum for our minimal model
were performed using the SUNNY software package [25]. The best fit to the data
is plotted in Fig. 6(b,d,f,h,j,l). To enable a direct comparison with the
experiment, the computation assumed a Gaussian energy resolution of
$\sigma=0.04$ meV standard deviation. The corresponding parameter values of
the minimal model are listed in Table 4. Given the rather arbitrarily chosen
Heisenberg model Hamiltonian, the limited experimental resolution, noise level
and computational issues, they are not the result of a rigorous least-squares
fit procedure. Rather, they are a proposed set of values chosen by trial and
error to optimize the overall agreement with experiment. The error bars are
estimated conservatively: changing any parameter by the corresponding
indicated amount results in an obviously worse reproduction of the
measurement.

This said, all parameters of the minimal model are well decoupled and can thus
be reliably determined: i) The spitting between the CF and magnon sectors is
primarily given by $D$. The orientation of the anisotropy plane (angle
$\beta$) has only a minor effect on the spectrum in the paramagnetic state,
and thus was fixed at the value quoted above. ii) The curvature of magnon
dispersion along $k$ is strongly influenced by $d$ and $J_{4}$. At the same
time, the overall energy shift of the spectrum is sensitive, in a mean field
manner, to the sum of all exchange constants, but hardly at all to $d$. iii)
$J_{5}$ is the only exchange pathway that provides connectivity along the
$\mathbf{c}$ axis. It therefore defines the bandwidth along $l$. iv) $J_{1}$
and/or $J_{3}$ are responsible for the splitting of that dispersion into two
bands, and also for the dispersion along the $h$ direction. The characteristic
oscillation of intensity between the two bands along $l$ is due to the
structure factor of the zigzag ladders formed by $J_{4}$ (legs) and either
$J_{1}$ or $J_{3}$ (rungs). If these exchange constants were comparable, a
contiguous triangular plane would be completed in the $(a,b)$ plane, and the
intensity oscillation would disappear. This implies that one of the two
exchange constants must be small. Since the two bonds are topologically
equivalent, it is not possible to say which one it is, based on the excitation
energies alone. As they are differently oriented, they can be told apart from
the intensity modulation they produce. Unfortunately, the projections of those
bonds onto the scattering plane of the experiment are quite close in length,
so the effect is very subtle. Assuming $|J_{1}|\gg|J_{3}|$ gives slightly
better fits, but this result can not be considered conclusive. Below we
postulate that $J_{1}$ is dominant, which is plausible given it corresponds to
nearest-neighbor Co-Co link. v) The final relevant parameter is the effective
gyromagnetic ratio $g=2.30(5)$. It was deduced independently from the field
dependence of the zone-center magnon energy measured with ESR (“p-mode” in
Ref. [26]) and in our THz absorption experiments described below.

All in all, the agreement between the SWT calculation and experiment is rather
good. In fact, the “magnon” part of the spectrum is reproduced almost
perfectly. For CF excitations the agreement is less satisfactory. In
particular, the data seem to show a weak flat band at about 1.8 meV energy
transfer that cannot be reproduced by SWT with any reasonable combination of
parameters.

####  IV.3.2 UUD phase

Inelastic neutron data collected at the CNCS instrument in the plateau Phase C
at $T\sim 100$ mK and a magnetic field of $3.2$ T applied along the
$\mathbf{b}$ direction, that is, near the middle of the UUD plateau phase, are
shown in Fig. 7(a,c,e,g). The integration is done as described in Sec. IV.3.1.
The scattering intensity in the accessible wave vector transfer range is much
smaller than in the saturated state shown in Fig. 7 (note the different color
range). Thus the data are a great deal noisier. Still, it is very apparent
that dispersion transverse to the chains practically disappears. Even the
chain-axis dispersion, as seen from the narrow experimental “keyhole” appears
almost flat.

Figure 7: (a,c,e,g) Inelastic neutron scattering intensity measured in
Cs2CoBr4 in a magnetic field $\mu_{0}H=3.2$ T applied along the
crystallographic $\mathbf{b}$ axis at $T\sim 100$ mK in the UUD-plateau phase.
The data are integrated along the transverse directions in reciprocal space,
as described in the text. The intensity units are consistent with those in
Fig. 6. (b,d,f,h) $SU(4)$ spin wave theory calculation based on the minimal
model parameters listed in Table 4.

An SWT analysis of the data is only possible if the actual magnetic structure
is the ground state of the classical model. For the plateau state at 3.2 T and
our the minimal model (Table 4), this happens to be the case. With our
parameters, the plateau remains stable to 4.1 T, at which point it is replaced
by the spin flop state. This is in remarkably good agreement with experiment,
where the corresponding phase transition occurs at $H_{\text{CD}}\sim 4.0~{}T$
[10]. SWT calculations in the plateau phase are therefore meaningful, at least
at the qualitative level. Next to each measured spectrum in Fig. 7 is the
corresponding SWT simulation based on the high-field parameter set.
Considering that this is not a fit, the agreement is not bad. The main
discrepancy is the intensity distribution between the two CF branches, the
lower mode being always the stronger one experimentally.

####  IV.3.3 Zero field

The zero field excitation spectrum (Phase A) has been previously measured in
Cs2CoBr4 on the LET spectrometer [11]. The horizontal positioning of the
$\mathbf{b}$ axis allowed the exploration of a wide range of momenta along the
chains, and provided data with a much better signal-to-noise ratio. In Fig.
8(a,c,e) we show the same data collected with an incident neutron energy of
$E_{i}=2.35$ meV. No background subtraction has been performed. At higher
energies the same plot includes new data measured simultaneously in the same
experiment with $E_{i}=3.71$ meV, by virtue of frame rate multiplication. This
representation makes the chain-axis dispersion apparent also for the CF modes.

Figure 8: (a,c,e) Inelastic neutron scattering intensity measured in Cs2CoBr4
in zero applied field at $T<100$ mK in the Néel (stripe) phase. High- (high-
contrast color map) and low-resolution (shaded) data are combined. The data
are integrated along the transverse directions in reciprocal space, as
described in Ref. [12]. (b,d,f) $SU(4)$ spin wave theory calculation based on
renormalized minimal model parameters listed in the last colum in Table 4.
Dashed lines in (a) are computed dispersion relations of all SWT modes.

As already mentioned, the lower-energy part of the spectrum looks nothing like
anything in SWT. This said, the lowest-energy bound state of the Zeeman ladder
is essentially a single spin flip, and can be at least qualitatively
associated with an SWT excitation. The CF modes can also be expected to be
SWT-like. Yet SWT calculations with parameters quoted in Table 4 are totally
incompatible with the measurements. Moreover, the Néel “stripe” phase with
propagation vector $(0,1/2,1/2)$ that is found in Cs2CoBr4 in zero applied
fields appears not to be the classical ground state. Instead, the strong
frustration in the zigzag ladders favors a helimagnetic spin arrangement, even
despite the slight easy-axis anisotropy. This is not necessarily a
contradiction. Due to the effective low dimensionality the SWT exchange
parameters are expected to be strongly renormalized, as they are, for
instance, in Cs2CuCl4 [2]. For weakly-coupled spin chains the upward
renormalization of the chain-axis exchange constant $J$ can be as large as
$\pi/2$ [27]. For inter-chain coupling $J_{\bot}$ it is downward, by a factor
of the order $\sqrt{J_{\bot}/J}$. Indeed, the inter-chain dispersion bandwidth
is $\sqrt{JJ_{\bot}}$ in SWT but of the order of $J_{\bot}$ in the real
quantum spin model [28]. With this in mind, we attempted to reproduce the main
features of the experimental data using an SWT calculation with an effective
set of parameters: $\tilde{J}_{4}=A\,J_{4}$, $\tilde{J}_{1-3,5}=B\,J_{1-3,5}$
and $\tilde{d}=C\,d$. A reasonable agreement (Fig. 8(b,d,f)) is obtained with
renormalization factors as listed in Table 4. The dispersion of all computed
modes is shown in dashed lines superimposed over the data in Fig. 8(a).

###  IV.4 THz absorption

Previously, THz absorption experiment provided an important independent
confirmation of the Zeeman-ladder type spectrum in Cs2CoBr4 [12]. We have
extended this experiment to cover a wide range of magnetic fields applied
along the $\mathbf{b}$ axis. The corresponding absorption coefficient $\alpha$
measured at $T=200$ mK is shown in the plots in Fig. 9(a,b) as a function of
field and photon energy. In the false color plot, white lines indicate
boundaries of field-induced magnetic phases, as determined in previous
thermodynamic measurements [10]. Our results are generally in agreement with
previous ESR experiments that investigated the lower-energy part of the
excitation spectrum (up to about 1 meV) at a higher temperature of 0.5 K [26].

Figure 9: (a) THz absorption spectra measured at $T=200$ mK for different
vales of magnetic field applied along the $\mathbf{b}$ axis. The various
magnetic phases of Cs2CoBr4 are labeled and color-coded. (b) Same data as a
false color plot. (c) $SU(4)$ spin wave theory calculation based on minimal
model parameters listed in Tab. 4 for the high-field plateau and “spin flop”
phases, and renormalized value for the low-field phase. Solid lines in (b) are
energies of all SWT modes with line opacity representing the corresponding
intensities.

In Fig. 9(b) the Zeeman ladder is the weak signal in the lower left part of
the plot. More prominent are sharp features in the low-energy part of the
spectrum in all other phases excluding the longitudinally modulated
incommensurate SDW. In contrast, the CF excitation sector appears quite broad
except in the Néel and paramagnetic/saturated phases. Another observation is
the large number of distinct modes in the paramagnetic phase. That is rather
surprising, since here we expect to be approaching the trivial regime of
single-ion transitions. At $T>T_{N}$, ESR also shows complicated behavior
beyond single-ion picture [26].

We also note some curious weakly field-dependent sharp modes that are seen at
about 1.95 meV and 2.7 meV in the high field phase, but are present at all
fields. These may be optical phonons. The “1.95 meV” mode shows a systematic
softening with increasing field in the magnetically ordered phases. The “2.7
meV” feature is less field-dependent, but splits in two components beyond
about 4 T and appears to eventually anti-cross with one of the magnetic
excitations. Substantial shifts of phonon frequencies in response to changing
magnetic correlations in quantum magnets is a common phenomenon, with the
“pantograph” mode in SrCu2(BO3)2 being a prominent example [29].

Quite revealing is a comparison of the measured spectra with SWT calculations
based on high-field parameters for the plateau, spin-flop and polarized phases
and renormalized values for the low-field stripe phase. No calculation could
be performed for either incommensurate phases. The results are shown in Fig.
9(c). The computed energies of all branches are also shown in red lines
overlaying the data in Fig. 9(b). Line opacity is linked to the calculated
absorption intensity of the corresponding mode. The first thing we note is
that several modes in the high field phase are very well reproduced. In
particular, the correctly predicted slope of the lowest-energy excitations
validate our choice of the $g$-factor value. At the same time, several
branches are entirely missing from the calculation. No choice of parameters in
the minimal model can account for these extra features in the data. In the UUD
and spin flop phases the agreement between the SWT calculation and experiment
is qualitative at best. In particular, the experimentally observed broadening
in the CF part of the spectrum is entirely unaccounted for. In the low-field
phase the behavior of the CF excitations is reproduced reasonably well,
whereas a total lack of agreement for the Zeeman-ladder part is to be
expected.

###  IV.5 Dilatometry

Figure 10:  False color plot of measured change in relative capacitance
$\Delta C/C$ in Cs2CoBr4 for $\mathbf{E}\parallel\mathbf{a}$ and
$\mathbf{H}\parallel\mathbf{b}$. Green solid symbols mark magnetic phase
boundaries deduced from specific heat data [10].

While the THz spectra contain some indications of magneto-elastic coupling in
Cs2CoBr4, dilatometric measurements make it very apparent. The measured change
in sample capacitance is plotted against field and temperature in Fig. 10. If
we assume that changes in the dielectric constant are negligible, the observed
capacitance change is to be attributed to sample expansion and contraction. In
the geometry of our experiment a positive $\Delta C$ is due to either a
contraction of the sample along the $\mathbf{a}$ direction, or to a dilation
along the $\mathbf{b}$ or $\mathbf{c}$ direction:

$\frac{\Delta C}{C}=\frac{\Delta b}{b}+\frac{\Delta c}{c}-\frac{\Delta a}{a}.$
(2)

The entire phase diagram is clearly visible. Unfortunately, it is not possible
to extract individual components of the magnetoelastic tensor from this single
measurement. Further experiments will be required to fully understand the
microscopic implications, but already the present data emphasize the magnitude
of the magneto-elastic effect.

##  V Discussion

The most unexpected finding of this work is the hierarchy of exchange
constants. Triangular-lattice interactions that are known to dominate in
isostructural Cu-based systems [2, 5] play only a secondary role in Cs2CoBr4.
Instead, the basic building blocks in our material are zigzag spin ladders,
that can also be seen as zigzag chains with nearest-neighbor coupling $J_{1}$
and next nearest neighbor interactions $J_{4}$. This is a much simpler model
than an extended triangular plane, particularly since anisotropy planes for
all spin in each ladder are parallel. The dominant one-dimensional nature of
the system also clearly favors the formation of bound states in the presence
of an attractive interaction.

The above does not, however, invalidate the conclusions of our previous work:
i) At special wave vectors $\mathbf{qb}=\pi n$ the Zeeman ladder excitations
are confined to single chains (single ladder legs in the new terminology). ii)
At all other wave vectors the bound states do propagate in the triangular
plane. That must be due to the triangular lattice geometry. Indeed, of all
exchange constants considered here only $J_{5}$ provides connectivity along
the $\mathbf{c}$ direction and produces a corresponding dispersion. iii) Other
than at special wave vectors the bound states are not confined to individual
chains. In view of the findings in this study, it appears most likely that
these inter-chain correlations occur within individual zigzag ladders. In
other words, the transverse intensity modulation of the bound states reported
in Ref. [12] may be related to the structure factor of a single ladder.

Our work once again shows that measurements in the fully saturated phase are
indispensable for understanding magnetic interactions in any model compound
[2]. For Cs2CoBr4 they also highlight the importance of a complete $S=3/2$
model, rather than an effective $S=1/2$ projection. The latter corresponds to
the easy plane anisotropy being infinitely strong, while in fact it is
comparable in energy scale to the Zeeman energies explored in the present
study. For that reason, if one were to focus on effective $S=1/2$ exchange
parameters, those would have to be treated as field-dependent. Deducing true
Hamiltonian parameters for the Zeeman ladder phase from high field
measurements would then be entirely impossible.

The new minimal model helps make sense of the magnetic phase diagram. Even
though the spin wave language is not appropriate for describing the Néel-like
stripe phase, it is still explained quasi-classically in terms of kinks. The
“spin flop” and plateau phases are purely classical in nature appearing
naturally in SWT. They are a feature of frustrated zigzag ladders with
predominantly planar and additional weak easy-axis anisotropy. The main clue
to the nature of the incommensurate pre-saturation phase is the field-
independence of the propagation vector. This behavior has a simple classical
interpretation as well. In the classical planar zigzag ladder frustration is
resolved by forming a helimagnet. In fields applied in the anisotropy easy
plane this turns into an elliptical-umbrella structure or a planar spin-fan,
depending on anisotropy strength. Minimizing the classical exchange energy
with respect to the period of modulation for transverse spin components at a
fixed longitudinal magnetization gives a propagation vector that only depends
on the frustration ratio:

$\zeta=\frac{1}{2}-\frac{J_{1}}{4\pi J_{4}},$ (3)

in the limit of $4J_{4}\gg J_{1}$. Using the numbers from Table 4 gives
$\zeta=0.465$, almost exactly as seen experimentally. We conclude that the
only “truly quantum” phase in Cs2CoBr4 is the incommensurate SDW state.

It has to be re-iterated that our SWT analysis has its limitations. The
agreement of minimal model simulations with experiment is not perfect,
particularly in what concerns the CF sector. Part of it may be due to quantum
effects beyond the SWT paradigm and part to our simplified treatment of
magnetic anisotropy, where everything is reduced to a single-ion term. The
discrepancies in the high field regime are the most telling, since here the
SWT is expected to work best. It is well known that extended SWT sometimes
predicts excitations such as longitudinal modes that are not reproduced
experimentally (see, for example, Refs. [30, 31]) due to their decay into
transverse excitations [32]. In our case though, there seem to be more modes
visible in the measurements than predicted. It can not be excluded that some
essential component is still missing from our simple model for Cs2CoBr4. As
discussed above, one such component may be the very strong magneto-elastic
coupling that seems to affect not just bulk properties, but also the
excitations. A 0.1% change in lattice parameters may not appear very large,
but the magnetoelastic effect on exchange constants via bond angles close to
the Kanamori-Goodenough critical value may be considerable. Further studies in
that direction are needed.

##  VI Conclusion

Particularly for frustrated systems, the spin Hamiltonian of a material can
not be considered known until neutron spectroscopy has been performed in the
high field paramagnetic state. Cs2CoBr4 should be seen as a network of coupled
frustrated zigzag spin ladders, rather than a distorted triangular lattice
system. This opens the possibility to quantitatively study the exotic spin
excitations in this compound using matrix product states numerical methods,
which are known to be exceptionally effective in application to quasi-1D
systems. At the same time triangular-lattice inter-ladder interactions in
Cs2CoBr4 are significant and lead to new physics compared to that of a single
ladder. A new dimension to the problem is brought by strong magneto-elastic
interactions that will need to be explored more thoroughly in Cs2CoBr4.

###### Acknowledgements.

One of the authors (AZ) would like to thank Prof. Giamarchi (University of
Geneva) and Prof. Coldea (University of Oxford) for illuminating discussions.
This research was in part funded by a MINT grant “Tricoordinated frustrated
quantum magnets” of the Swiss National Science Foundation. It made use of
resources at the Spallation Neutron Source, a DOE Office of Science User
Facility operated by the Oak Ridge National Laboratory. We acknowledge support
by the Estonian Research Council Grants No. PRG736 and No. MOBJD1103, and by
European Regional Development Fund Project No. TK134. Experiments at the ISIS
Neutron and Muon Source were supported by beam time allocation from the
Science and Technology Facilities Council (DOI: [18, 22]). This research
benefited from neutron beam time at the instrument D23 at the Institute Laue
Langevin provided via the Collaboration Agreement between the French
Alternative Energies and Atomic Energy Commission (CEA) and the École
Polytechnique Fédérale de Lausanne (EPFL) in Switzerland and funded by the
Swiss State Secretariat for Education, Research, and Innovation (SERI).

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