# Proximity-induced sequence of field transitions in Kitaev candidate
BaCo2(AsO4)2

Pavel A. Maksimov  Bogolyubov Laboratory of Theoretical Physics, Joint
Institute for Nuclear Research, Dubna, Moscow region 141980, Russia  M.N.
Miheev Institute of Metal Physics of Ural Branch of Russian Academy of
Sciences, S. Kovalevskaya St. 18, 620990 Ekaterinburg, Russia
<EMAIL_ADDRESS>
(September 3, 2024)

###### Abstract

We study field-induced phase transitions of the minimal exchange model
proposed earlier for BaCo2(AsO4)2, a candidate for Kitaev honeycomb model,
using numerical minimization of classical spin clusters. We show that
experimentally observed sequence of step-like transitions in magnetic field is
realized in the phase diagram of the minimal model. Surprisingly, intermediate
up-up-down plateau phase is stabilized only in the proximity of a double-
zigzag$-$zigzag phase boundary. We systematically map out the region of
stability of experimentally observed cascade of transitions and argue that
BaCo2(AsO4)2 exchange parameters are close to a region of strong phase
competition, which can explain suppressed saturation field.

###### pacs:

75.10.Jm, 75.25.+z, 75.30.Kz

Introduction.- Kitaev $S=1/2$ honeycomb exchange model [1] is exactly solvable
and has a quantum spin liquid (QSL) ground state, a correlated state without
long-range magnetic order [2, 3]. The model involves exchanges on three types
of bonds of the honeycomb lattice with compass-like Ising interactions of
three different spin axes [4], and was shown to host exotic Majorana and vison
excitations. The search for materials that can host QSL is invigorated by the
promise of application of its excitations in topologically protected quantum
computation [5, 6, 1, 7].

Honeycomb $d^{5}$ Mott insulators with strong spin-orbit coupling and edge-
sharing ligand octahedra were proposed by G. Jackeli and G. Khaliullin [8] to
host Kitaev honeycomb model. Subsequent experiments on $\alpha$-RuCl3 [9, 10]
and Na2IrO3 [11, 12], indeed, indicated the presence of Kitaev-type
anisotropic exchanges, which was also supported by ab initio calculations [13,
14, 15, 16, 17, 18]. However, the existence of additional interactions allowed
by symmetry of the lattice [19, 20], as well as sizeable third-neighbor
exchanges, yields antiferromagnetic zigzag magnetic order in these materials.
Nonetheless, the search for Kitaev interactions encouraged ongoing synthesis
and analysis of new materials based on honeycomb [21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31] and triangular lattices [32, 33, 34].

Figure 1: A sketch of summarized results presented below in this paper. (a)
Field phase diagram of the minimal model (1) for $J_{3}=0.34|J_{1}|$ as a scan
along $J_{z\pm}^{(3)}$, obtained with numerical minimization of classical
spins. Honeycomb lattice of BaCo2(AsO4)2 with the direction of magnetic field
used in Hamiltonian (1) are shown in the inset. (b) Magnetization along
$x$-axis, calculated for $J_{z\pm}^{(3)}=0.2|J_{1}|$ with double-zigzag, up-
up-down and polarized phases shown. Measurements from Ref. [35] are also
reproduced.

BaCo2(AsO4)2 (BCAO) was recently proposed as a platform for Kitaev honeycomb
model [35] in a new class of compounds with $d^{7}$ Co2+ ions, which were
introduced as an alternative path towards Kitaev interactions [36, 37, 38,
39]. In Co-based materials spin-orbit coupling yields lowest effective doublet
in the $L=1$, $S=3/2$ state, while additional hopping paths due to active
$e_{g}$ orbitals were suggested to increase Kitaev exchange [40]. This
mechanism was supported by experimental evidence in various honeycomb
materials [41, 42, 43, 44, 45, 46], as well as triangular lattice Kitaev
compounds [47].

Early neutron-scattering experiments on BaCo2(AsO4)2 indicated a spiral ground
state with $\mathbf{Q}=(0.25,0)$ ordering vector [48], which is stabilized in
the classical Heisenberg model by the competition of first- and third-neighbor
interactions, $J_{1}$ and $J_{3}$. However, a more refined polarized neutron-
diffraction data [49] pointed to a previously undetected double-zigzag ground
state, which has the same periodicity but a collinear
$\uparrow\uparrow\downarrow\downarrow$ structure. It was shown later that this
enigmatic state can be either stabilized by quantum fluctuations in the narrow
region of the isotropic $J_{1}$-$J_{3}$ model [50], or by extension of the
Heisenberg Hamiltonian by third-neighbor bond-dependent Kitaev-type coupling
[51].

Generally, candidate materials for Kitaev interactions can require a large
exchange parameter phase space to describe their magnetic properties, and its
experimental extraction of the couplings may become highly challenging, see as
an example the story of $\alpha$-RuCl3 [52]. Field-induced behavior,
especially if it involves multiple phase transitions, can provide extremely
useful information for the investigation of magnetic Hamiltonian [53]. In this
paper, we address peculiar experimentally observed feature of BaCo2(AsO4)2 \-
a field-induced sequence of phase transitions, which includes an up-up-down
(UUD) $\uparrow\uparrow\downarrow$ $1/3$-plateau state [48, 35]. This sequence
of transitions can be also seen in the magnetic spectrum of BaCo2(AsO4)2 in
THz measurements [54, 55]. Moreover, the saturation field appears to be
strongly suppressed relative to order of exchange interactions:
$g\mu_{B}H_{s}/|J_{1}|\simeq 0.03$. Using numerical minimization of classical
spin clusters up to 144 sites, we study magnetic field phase transitions of a
model proposed in Ref.[51] for BaCo2(AsO4)2, and show that $1/3$-plateau
appears in the cascade of field-induced phases only in the vicinity of zero-
field $\mathbf{Q}=(1/3,0)$ state, which is intermediate between double-zigzag
and zigzag states. Moreover, we map out a region, where parameters of the
model can fit experimentally observed ranges of field-induced phases.

Figure 2: (a) Honeycomb lattice with three types of nearest and third-neighbor
bonds indicated with color. Crystallographic and cubic axes, $\\{x,y,z\\}$ and
{x,y,z}, correspond to two representations of the extended Kitaev-Heisenberg
model, see text. Direction of magnetic field is also shown. (b) Brillouin zone
(BZ) of the honeycomb lattice with high-symmetry points marked. Colorbar
illustrates the value of the ordering vector $\mathbf{Q}$ in the phase digram
in (c). (c) Phase diagram of the model (1) in zero field, reproduced from
Ref.[51]. Representative scans used in the Fig. 3 are shown with dashed
arrows. (d) Sketches of primary phases in the phase diagram in (c).

The model and field phase diagram.- There are multiple attempts at
establishing exchanges in BaCo2(AsO4)2 both from neutron-scattering data [56]
and ab initio calculations [57, 51]. We would like to argue that the model
proposed in Ref.[51] sufficiently describes both a unique zero-field ground
state and the peculiar sequence of field-induced phase transitions.

The exchange Hamiltonian used in this paper, which was coined as a “minimal”
model for BaCo2(AsO4)2 [51], is given by

$\displaystyle\hat{\cal H}_{\text{min}}=-$ $\displaystyle
g\mu_{B}H\sum_{i}S^{x}_{i}$ (1) $\displaystyle+$ $\displaystyle\sum_{\langle
ij\rangle_{1}}J_{1}\Big{(}S^{x}_{i}S^{x}_{j}+S^{y}_{i}S^{y}_{j}+\Delta_{1}S^{z}_{i}S^{z}_{j}\Big{)}$
$\displaystyle+$ $\displaystyle\sum_{\langle
ij\rangle_{3}}J_{3}\Big{(}S^{x}_{i}S^{x}_{j}+S^{y}_{i}S^{y}_{j}+\Delta_{3}S^{z}_{i}S^{z}_{j}\Big{)}$
$\displaystyle-$ $\displaystyle
J_{z\pm}^{(3)}\Big{(}\Big{(}S^{x}_{i}S^{z}_{j}+S^{z}_{i}S^{x}_{j}\Big{)}c_{\alpha}+\Big{(}S^{y}_{i}S^{z}_{j}+S^{z}_{i}S^{y}_{j}\Big{)}s_{\alpha}\Big{)},$

where $c_{\alpha}\equiv\cos\varphi_{\alpha}$ and
$s_{\alpha}\equiv\sin\varphi_{\alpha}$ with the bond-dependent phases
$\varphi_{\alpha}\\!=\\!\\{0,2\pi/3,-2\pi/3\\}$ for the three types of first
and third-neighbor bonds ${\bm{\delta}}_{\alpha}$,
$\bm{\delta}^{(3)}_{\alpha}$, shown in Fig. 2(a). Note that this model is
written relative to crystallographic $\\{x,y,z\\}$ axes shown in Fig. 2(a),
which are defined by the plane of the honeycomb lattice. In Eq. (1) we also
include magnetic field parallel to $x$. The bond-independent terms in this
paper are assumed to be of easy-plane $XY$ type: $\Delta_{1}=\Delta_{3}=0$.
The bond-dependent term $J_{z\pm}^{(3)}$ is related to exchanges of the
extended Kitaev-Heisenberg model

$\displaystyle\hat{\cal H}_{\text{KH}}=\sum_{\langle
ij\rangle_{n}}\mathbf{S}_{i}^{\intercal}\hat{J}^{(n)}_{\alpha}\mathbf{S}_{j},\leavevmode\nobreak\
\hat{J}^{(n)}_{\text{X}}=\begin{pmatrix}\text{J}_{n}+K_{n}&\Gamma^{\prime}_{n}&\Gamma^{\prime}_{n}\\\
\Gamma^{\prime}_{n}&\text{J}_{n}&\Gamma_{n}\\\
\Gamma^{\prime}_{n}&\Gamma_{n}&\text{J}_{n}\end{pmatrix},$ (2)

which is written relative to the cubic axes {x,y,z} defined by ion-ligand
bonds, see Fig. 2(a). There are three types of bonds $\alpha=\text{X,Y,Z}$,
shown in Fig. 2(a), for both $n=1$ nearest-neighbor and $n=3$ third-neighbor
interactions. Interaction on X bond is given in Eq. (2), the exchange matrix
on Y and Z bonds is obtained through a cyclic permutation. Note that the
models in crystallographic and cubic axes are equivalent and related to each
other through a linear transformation [58, 59, 51].

Figure 3: (a) Intensity plots of magnetization and magnetic susceptibility for
two representative values of $J_{3}$, shown as scans along $J_{z\pm}^{(3)}$,
with phases indicated. (b) Sketches of the phases above with their ordering
vectors shown in the BZ.

Phase diagram of the $J_{1}$-$J_{3}$-$J_{z\pm}^{(3)}$ model (1) for $J_{1}<0$
and $H=0$ is shown in Fig. 2(c) and was calculated in Ref. [51] in classical
and quantum $S=1/2$ limits. In the classical limit, for $J_{z\pm}^{(3)}=0$
there is a transition from a ferromagnetic (FM) to zigzag (ZZ) state via
intermediate spiral states, whose ordering vectors changes gradually from
$\Gamma$ to $M$ [60]. The magnitude of the ordering vector of the phases in
Fig. 2(c) is indicated with color, and the corresponding colorbar is shown in
Fig. 2(b). When $J_{z\pm}^{(3)}$ is increased, double-zigzag state becomes
stabilized in a wide region of phase diagram both in the classical limit and
for $S=1/2$ [51]. Note that there are intermediate phases with various
periodicity between FM and double-zigzag, such as $\mathbf{Q}=(1/6,0)$
“triple-zigzag”, as well as $\mathbf{Q}=(1/3,0)$ UUD state between double-
zigzag and ZZ. The structure of these states and their ordering vectors in the
Brillouin zone are shown in Fig. 3(b).

In order to explore field phase transitions and how these intermediate phases
affect the field behavior, we perform two calculations for representative
values of third-neighbor interaction, $J_{3}=0.2|J_{1}|$ and
$J_{3}=0.45|J_{1}|$ (shown with dashed arrows in Fig. 2(c)), where we plot
phase diagram in the $J_{z\pm}^{(3)}-H$ axes. We perform numerical
minimization of classical energy of Hamiltonian (1) on various clusters up to
$12\times 12$ sites using Mathematica software to identify the phases, with a
subsequent minimization on smaller clusters to obtain phase boundaries. The
result is shown in Fig. 3(a) as intensity plot of magnetization $M_{x}$, and
magnetic susceptibility $\chi=dM_{x}/dH$.

For $J_{3}=0.2$ and small values of $J_{z\pm}^{(3)}$ there is a direct
transition from a FM to polarized state. For $0.4<J_{z\pm}^{(3)}/|J_{1}|<0.6$
ground state is the canted triple-zigzag state, which further gradually cants
towards saturation for $H>0$. At large $J_{z\pm}^{(3)}$ zero-field double-
zigzag with three $C_{3}$-symmetric domains, selects only two domains of
canted double-zigzag state, closest to the field direction. This canted
double-zigzag state is illustrated in Fig. 3(b). This situation is similar to
the selection of zigzag domains in magnetic field in $\alpha$-RuCl3 [61],
where three domains are degenerate in zero field but for $H>0$ only some are
selected due to the presence of Kitaev terms [62]. Note that there is a region
with intermediate triple-zigzag state at high fields before double-zigzag
reaches saturation.

For $J_{3}=0.45$ we observe even more intricate field behavior. First, for low
values of $J_{z\pm}^{(3)}$, when the zero-field state is zigzag, as magnetic
field is increased there is only canting of spins in the ZZ state towards the
direction of the field without phase transition, which is illustrated by the
absence of singularities in intensity plot of magnetic susceptibility $\chi$.
As $J_{z\pm}^{(3)}$ increases, $\mathbf{Q}=(1/3,0)$ UUD phase becomes ground
state, and it is presented as $1/3$-plateau as magnetic field is increased,
until there is a transition to a canted “V” state at larger fields. Note that
this state is analogous to the “V” state of triangular lattice antifferomagnet
in magnetic field [63, 64, 65]. For large enough $J_{z\pm}^{(3)}$, deep in the
double-zigzag state, magnetic field only induces canting of spins towards the
$x$ direction until saturation.

The most interesting situation occurs in the vicinity of transition between
UUD and double-zigzag state around $J_{z\pm}^{(3)}=0.9|J_{1}|$, where UUD
state extends its region of stability as magnetic field increases, which
results in the transition from double-zigzag state to UUD state, similar to
what is observed in experiments on BCAO [48, 35, 54, 55]. Notably, this
transition occurs only near the zero-field phase boundary between double-
zigzag and UUD state. Therefore, the observation of this transition allows us
to map out the region of parameters applicable to BCAO. However, classical
analysis shows that at high field there is an intermediate “V” state before
full polarization, which is not observed experimentally. We would like to
argue that quantum fluctuations may be responsible for modifications to the
field phase diagram.

Quantum effects and parameters fit.- Quantum fluctuations in the case of
$S=1/2$ magnetic moments, such as the lowest doublet of BCAO, are known to
strongly affect magnetic phase diagram compared to classical calculations –
$1/3$ plateau of triangular lattice antiferromagnet being a prime example
[65]. Since quantum calculations can be demanding, it was shown that their
effect can be simulated by including biquadratic coupling to the classical
model, which can be derived as first order of quantum fluctuations [66, 67,
68, 69] or used ad hoc for various frustrated magnets [65, 70, 71, 72].

We also explore how quantum fluctuations, approximated by biquadratic coupling

$\displaystyle-B\sum_{\langle
ij\rangle_{1}}\left(\mathbf{S}_{i}\mathbf{S}_{j}\right)^{2},$ (3)

affect the field phase diagram of model (1). The results for $B=0.04|J_{1}|$
are shown in Fig. 4 for two representative values of $J_{3}$. This value is
the lowest to suppress “V” state, and is of the same order as earlier
estimates [67, 68, 69]. One can see that collinear UUD state is stabilized in
a wider region of parameters, while the “V” state is not present in the field
phase diagram in Fig. 4 compared to Fig. 3, leading to an experimentally
observed double-zigzag$\rightarrow$UUD$\rightarrow$saturation sequence in a
wide region of the phase diagram.

We have also checked other values of $B$ up to $0.12|J_{1}|$, and confirmed
very weak changes of the results with the value of biquadratic coupling. This
is due to the fact that zero-field and field-induced phases in the studied
region of the phase diagram are nearly collinear. Therefore, biquadratic
coupling is a constant, and does not affect the transitions after quantum
effects are taken into account. Thus, in this paper we only present results
for $B=0.04|J_{1}|$.

Figure 4: Intensity plots of magnetic susceptibility for two representative
values of third-neighbor interaction $J_{3}\gtrsim 0.3|J_{1}|$ with phases
indicated for the model (1) with biquadratic coupling (3) included,
$B=0.04|J_{1}|$.

Moreover, we can map out the region of the $J_{3}-J_{z\pm}^{(3)}$ phase
diagram, where the values of critical magnetic fields fit experimental
measurements, $H_{c1}$ and $H_{c2}$. Here, $H_{c1}$ is the critical of field
of double-zigzag$\rightarrow$UUD transition, and $H_{c2}$ is the saturation
field. According to magnetization measurements [48, 35], $H_{c1}=0.26T$,
$H_{c2}=0.52T$. We break the parameter fit procedure into two steps. First, we
plot $H_{c1}/H_{c2}$ as an intensity plot in the $J_{3}-J_{z\pm}^{(3)}$ phase
diagram in Fig. 5(a), where one can see that such a sequence of field-induced
transitions occurs only in the proximity of the UUD state. When
$H_{c1}/H_{c2}$ reaches 1 in the phase diagram in Fig. 5(a), it implies the
absence of intermediate UUD state and a direct transition from canted double-
zigzag to polarized state. The range of parameters where
$0.4<H_{c1}/H_{c2}<0.6$ is shown as the red shaded region in Fig. 5(b) fitting
the experimentally observed ratio 0.5.

Next, we use $g=5.2$ [35] to map the region of phase diagram with saturation
field $H_{c2}=0.52T$. Various experimental estimates of nearest-neighbor
coupling range from $J_{1}=-3.3\text{ meV}$ [73] to $J_{1}=-7.6\text{ meV}$
[56] (see also Refs.[57, 51] for ab initio calculations with similar results),
which yields allowed region of values of $g\mu_{B}H_{c2}/|J_{1}|$ as
0.02–0.04. Parameters of Hamiltonian (1) which produce this range of critical
fields are shown as the white shaded region in Fig. 5(b). Combined with the
requirement for the ratio of critical fields $H_{c1}/H_{c2}$, we obtain the
region of exchange parameters for BaCo2(AsO4)2 that fit field-induced
transitions observed in experiment, shown as the intersection of two shaded
regions. We have also performed these calculations for the XXZ anisotropy
$\Delta_{1}=\Delta_{3}=0.4$, which is motivated by spin-wave theory fit to the
field-behavior of the $\mathbf{k}=0$ mode of BaCo2(AsO4)2 in the polarized
phase [56]. The region of optimal parameters for $\Delta=0.4$ is very similar
to the results in Fig. 5(b) due to the co-planar nature of the states in the
phase diagram, which makes classical energies of the zero-field and field-
induced states independent of choice of $\Delta$.

In order to illustrate the goodness of the fit, we plot magnetization for a
representative set of parameters $J_{z\pm}^{(3)}=0.2|J_{1}|$,
$J_{3}=0.34|J_{1}|$ in Fig. 1(b) together with experimental values from Ref.
[35] (we use $J_{1}=-7.6\text{ meV}$). There is a degree of discrepancy, which
can be attributed to the broadening of transitions due to the finite
temperature, while our calculations are performed at zero temperature.
However, one can still see that our calculations provide very good fit for the
experimental magnetization. Notably, the experimentally observed saturation
field is rather small, and we see from Fig. 5(b) that the reason for that is
the proximity of optimal parameter region to the region of multiple phases
competition. Note that compared to previous estimate of exchanges in BCAO
[51], this parameter set requires larger $\Gamma_{1}$ and
$\Gamma^{\prime}_{1}$ \- due to stronger easy-plane anisotropy, as well as
larger $K_{3}$, $\Gamma_{3}$, $\Gamma_{3}^{\prime}$ to describe the presence
of non-negligible $J_{z\pm}^{(3)}$.

Figure 5: (a) Intensity plot of $H_{c1}/H_{c2}$, two critical fields of the
double-zigzag$\rightarrow$UUD$\rightarrow$saturated sequence of transitions.
When $H_{c1}/H_{c2}=1$ there is only a direct transition from canted double-
zigzag to polarized state. (b) Intensity plot of saturation field $H_{c2}$ in
the units of $g\mu_{B}H/|J_{1}|$. Blue shaded region indicates parameters with
$0.02<g\mu_{B}H_{c2}/|J_{1}|<0.04$. Red shaded region marks parameters with
$0.4<H_{c1}/H_{c2}<0.6$. The intersection of these regions gives optimal
parameter sets that fit experimentally observed values.

Moreover, in order to check that a rather large bond-dependent term
$J_{z\pm}^{(3)}$, which is necessary for explanation of the features studied
in this paper, does not introduce anomalies to the magnetic spectrum, we have
performed linear spin-wave calculations for $J_{z\pm}^{(3)}=0.4|J_{1}|$,
$J_{3}=0.34|J_{1}|$, $J_{1}=-7.6\text{ meV}$. The result is shown in Fig. 6
with energy in units of meV for a direct comparison with inelastic neutron
scattering data [74, 56]. While we cannot expect a complete agreement with
experimental data, we see that our calculation does reproduce multiple
features of the spectrum. For instance, there are low-energy modes with a
small gap around 1 meV (1.5 meV experimentally), as well as high-energy modes
around 11 and 13 meV, in agreement with a 12 meV mode in neutron scattering
data [74]. Moreover, there is a rather flat mode around 3.5 meV in Fig. 6,
which was also observed in neutron scattering data [56]. Note that even in the
presence of large $J_{z\pm}^{(3)}=3\text{ meV}$, the gap is rather small,
which means that the model proposed in this paper is physically viable.
Nonetheless, we should note that the spin-wave energies in Fig. 6 do not give
precise agreement with neutron scattering data and a careful fitting of
magnetic spectrum assumably requires all eight parameters of the extended
Kitaev-Heisenberg model, placing such fit beyond the scope of this work.

Figure 6: Spin-wave magnetic spectrum in the double-zigzag state for
$J_{1}=-7.6\text{ meV}$, $J_{z\pm}^{(3)}=0.4|J_{1}|$, $J_{3}=0.34|J_{1}|$.

Discussion.- Establishing parameters of a model with multiple allowed exchange
parameters, like extended Kitaev-Heisenberg model, can become cumbersome in a
lot of cases. Experiments in magnetic field, which can include several
transitions, may strongly assist in setting boundaries on exchanges. We
performed analysis of field-induced phases of the anisotropic exchange model
suggested previously for BaCo2(AsO4)2 using numerical minimization of
classical spins, and we observed a plethora of phase transitions in various
parts of the zero-field phase diagram. We showed that experimentally
established double-zigzag$\rightarrow$UUD$\rightarrow$saturated phase sequence
is stabilized by quantum fluctuations and only in the proximity of the double-
zigzag phase boundary. By scanning through the phase space, we fit critical
fields to the experimentally observed values, and narrow down the parameters
which provide the best agreement. As it turns out, the presence of third-
neighbor bond-dependent exchange $J_{z\pm}^{(3)}$ is necessary, and in our
analysis we are able to put limits on its values. We should note that previous
attempts at calculating exchanges for BaCo2(AsO4)2 are not in a precise
agreement with each other. However, they all point to an easy-plane $XXZ$
$J_{1}$-$J_{3}$ model, with the ratio of nearest- and third-neighbor exchanges
around 0.3, and the presence of small but non-negligible bond-dependent
corrections [73, 57, 51, 56]. Our parameter set supports this notion and
further promotes the idea that third-neighbor bond-dependent term is crucial
for the physics of BaCo2(AsO4)2. Moreover, recent measurements on a related
compound BaCo2(PO4)2 indicate a very similar field-induced behavior [75],
making our results applicable for a broad class of materials.

Acknowledgments. I would like to thank Sergey Streltsov and Sasha Chernyshev
for useful discussions. I acknowledge support of the Russian Science
Foundation via project 23-12-00159.

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