# Magnetism, non-Fermi-liquid behavior and deconfinement in Kondo lattices

V. Yu. Irkhin e-mail<EMAIL_ADDRESS>M. N. Mikheev Institute of
Metal Physics, 620077 Ekaterinburg, Russia

###### Abstract

A formulation of the effective hybridization model is developed which
describes Fermi-liquid Kondo and fractionalized Fermi-liquid (deconfinement
spin-liquid-like) states with account of possible magnetic ordering. A unified
consideration of charge and magnetic fluctuation is performed within
perturbation theory in hybridization and within gauge field treatment. The
corresponding contributions to specific heat and resistivity are compared.

##  1 Introduction

Anomalous $f$-systems, which are usually described as Kondo lattices, show
unusual behavior of thermodynamic and transport properties, e.g., heavy-
fermion behavior [1]: giant electron specific heat, $\gamma=C/T\sim 1/T_{K}$
with $T_{K}$ the Kondo temperature. Such systems demonstrate typically
magnetic instabilities and/or pronounced spin fluctuations [2, 3]. At the same
time, a number of such systems with frustrated magnetic structures demonstrate
spin-liquid features with $\gamma\sim 1/J$ where $J$ is the Heisenberg
exchange interaction, which is small in comparison with electron energies. One
more interesting phenomenon is a possibility of a non-Fermi-liquid behavior
where $\gamma$ diverges at low temperatures.

A scaling theory of the Kondo lattices developed in Ref. [2] demonstrates that
during the scaling procedure the process of magnetic moment compensation
terminates somewhere at the boundary of the strong coupling region, which can
results in the formation of a finite (although possibly small) saturation
moment. Thus a unified energy scale is established, both the effective spin-
fluctuation frequency (i.e., $J$) and $T_{K}$ being strongly renormalized.
Frustration parameters become renormalized too [4]. Therefore. the relative
role of magnetism and on-site Kondo effect is determined not only by bare
parameters, but also by their mutual renormalizations, and a description of
the ground state is a complicated problem which cannot be solved within
perturbation theory.

A ground state theory of magnetic Kondo lattices was developed in Refs. [5, 6,
7] using the mean-field approximation in the pseudofermion representation [8].
In the present paper we generalize the approach of Ref. [6] and formulate the
effective hybridization model describing the competition of magnetic and non-
magnetic Kondo and spin-liquid states. We also go beyond the mean-field theory
by taking into account fluctuations contribution, including gauge field ones.

##  2 The model of effective hybridization

We start from the $s-d(f)$ exchange (“Kondo-Heisenberg”) model. The
corresponding Hamiltonian reads

$H=\sum_{ij\sigma}t_{ij}c_{i\sigma}^{\dagger}c_{j\sigma}-I\sum_{i\alpha\beta}\mathbf{S}_{i}\mbox{\boldmath$\sigma$}_{\alpha\beta}c_{i\alpha}^{\dagger}c_{j\beta}+\sum_{ij}J_{ij}\mathbf{S}_{i}\mathbf{S}_{j}.$
(1)

Here $c_{i\sigma}^{\dagger}$, $c_{j\sigma}$ and $\mathbf{S}_{i}$ are operators
for conduction electrons and localized moments, $I$ is the $s-d(f)$ exchange
parameter.

To construct ground state phase diagrams, Hartree-Fock and mean-field
considerations were developed in Refs.[9, 10, 11]. The direct Zubarev’s
decoupling of the $s-d(f)$ exchange interaction was used in these works to
derive the hybridization term

$\displaystyle-I\sum_{\sigma\sigma^{\prime}}c_{i\sigma}^{\dagger}c_{i\sigma^{\prime}}\mbox{\boldmath$\sigma$}_{\sigma\sigma^{\prime}}\mathbf{S}_{i}\rightarrow
V(f_{i\sigma}^{\dagger}c_{i\sigma}+h.c.)$
$\displaystyle-\frac{1}{2}V^{\prime}\sum_{\sigma\sigma^{\prime}}(f_{i\sigma}^{\dagger}\mbox{\boldmath$\sigma$}_{\sigma\sigma^{\prime}}c_{i\sigma^{\prime}}+h.c.)-\frac{4}{3I}V^{2}+\frac{4}{I}V^{\prime
2}$ (2)

This procedure leads to incorrect value of exponent in the Kondo temperature
$T_{K}$ in the paramagnetic state, which is a crucial scale describing heavy-
fermion behavior: $T_{K}=D\exp(2/3I\rho)$ instead of $T_{K}=D\exp(1/2I\rho)$
($D$ is of order of bandwidth, $\rho$ is the density of states at the Fermi
level). This violation was demonstrated already in old papers [12] (see
discussion in Ref. [13]).

Therefore we prefer to use the formulation by Coleman and Andrei [8] which was
first applied to the magnetic Kondo lattices in [6]. Making the saddle-point
approximation for the path integral describing the spin-fermion interacting
system [8] one reduces the $s-f$ exchange interaction to the effective
hybridization:

$\displaystyle-I\sum_{\sigma\sigma^{\prime}}c_{i\sigma}^{\dagger}c_{i\sigma^{\prime}}(\mbox{\boldmath$\sigma$}_{\sigma\sigma^{\prime}}\mathbf{S}_{i}-\frac{1}{2}\delta_{\sigma\sigma^{\prime}})$
$\displaystyle\rightarrow
f_{i}^{\dagger}V_{i}c_{i}+c_{i}^{\dagger}V_{i}^{\dagger}f_{i}-\frac{1}{2I}{\rm
Sp}{}(V_{i}V_{i}^{\dagger}),$ (3)

where the vector notations are used

$f_{i}^{\dagger}=(f_{i\uparrow}^{\dagger},f_{i\downarrow}^{\dagger}),\qquad
c_{i}^{\dagger}=(c_{i\uparrow}^{\dagger},c_{i\downarrow}^{\dagger}),$

$V$ is the effective hybridization matrix which is determined from a minimum
of the free energy. Then we have from the minimization condition the correct
value of the Kondo temperature in the paramagnetic state,

$V=2I\sum_{\mathbf{k}}\langle
f_{\mathbf{k}\sigma}^{\dagger}c_{\mathbf{k}\sigma}\rangle$ (4)

and the dispersion of the $f$-states $V^{2}/t$ at the scale $V^{2}/D\sim
T_{K}$.

The mean-field Hamiltonian takes the form

$\displaystyle H$ $\displaystyle-$
$\displaystyle\mu\hat{n}=\sum_{\mathbf{k}\sigma}[(t_{\mathbf{k}}-\mu)c_{\mathbf{k}\sigma}^{\dagger}c_{\mathbf{k}\sigma}+Wf_{\mathbf{k}\sigma}^{\dagger}f_{\mathbf{k}\sigma}]$
(5) $\displaystyle+$
$\displaystyle\sum_{\mathbf{k}\sigma\sigma^{\prime}}\sum_{\mu=0...3}(V^{\mu}_{\sigma\sigma^{\prime}}\sigma^{\mu}_{\sigma\sigma^{\prime}}c_{\mathbf{k}\sigma}^{\dagger}f_{\mathbf{k}\sigma}+h.c.)$
$\displaystyle-$ $\displaystyle\frac{1}{2I}\sum_{i}{\rm
Sp}{}(V_{i}V_{i}^{\dagger})+H_{f}$

where $-W\sim T_{K}$ is the chemical potential for $f$-states.

Whereas in the usual picture of Kondo singlet formation only scalar boson is
present (see, e.g., Ref. [14]), the matrix triplet components create local
singlet and triplet states resulting from the Kondo coupling between localized
and itinerant spins. Thereby SU(2) invariance of the $s-f$ exchange
interaction is reflected. Although, as demonstrate numerical calculations [10,
11], as a rule the value of $V^{\prime}$ is probably small in comparison with
$V$, this can play important role in magnetization processes (“metamagnetic
behavior”), which occur in some heavy-fermion compounds, e.g., YbRh2Si2 [9].

As for Heisenberg term, we use the pseudofermion representation

$\mathbf{S}_{i}=\frac{1}{2}\sum_{\sigma\sigma^{\prime}}f_{i\sigma}^{\dagger}\mbox{\boldmath$\sigma$}_{\sigma\sigma^{\prime}}f_{i\sigma^{\prime}}$
(6)

with the subsidiary condition (constraint)

$f_{i\uparrow}^{\dagger}f_{i\uparrow}+f_{i\downarrow}^{\dagger}f_{i\downarrow}=1.$
(7)

In the spin-liquid state, $f_{i\sigma}^{\dagger}$ are essentially operators of
spinons – neutral fermions. After decoupling the Heisenberg term, we obtain
their dispersion on the scale $J$, so that the spinon Fermi surface is formed.
The constraint (7) can be enforced by a gauge field. Then in the case of U(1)
symmetry we can write down the Lagrangian

${\cal
L}_{f}=\sum_{i}{f}^{\dagger}_{i}(\partial_{\tau}+ia_{0})f_{i}-\sum_{ij}\chi_{ij}\left(e^{ia_{ij^{\prime}}}f^{\dagger}_{i\sigma}f_{j\sigma}+\mbox{h.c.}\right)$
(8)

where $\chi_{ij}=J\sum_{\sigma}\langle
f^{\dagger}_{i\sigma}f_{j\sigma}\rangle$, $a_{0}$ and $a_{ij}$ are time and
coordinate components of the gauge field. Thus the spinons acquire the
spectrum $\epsilon_{\bf k}=-\chi_{\bf k}$. The presence of hybridization
results in the formation of a unified “large” Fermi surface including both
conduction electrons and $f$-states.

The mean-field treatments [6, 14] seem to overestimate the saturated
ferromagnetic state, so that the problem of consistent mean-field description
is not solved. An additional effective exchange field was introduced in Ref.
[14] to avoid this difficulty. Besides that, the parameter values for phase
transitions on the mean-field phase diagrams obtained in Refs.[9, 10, 11]
(rather large $I$) were inappropriate for Kondo physics. Therefore we pass to
a qualitative analysis of fluctuation effects.

To take into account such effects, we can remember that $f$-states are in fact
described by many-electron Hubbard’s projection operators,
$f_{\sigma}\rightarrow X(0,\sigma)=|0\rangle\langle\sigma|$. Thus we can use
slave-particle representations, e.g., $X_{i}(0,\sigma)=f_{i\sigma}b_{i}$.
Fluctuations can destroy the condensate picture, so that $\langle b\rangle=0$
Then we obtain a spin-liquid-like state where spinon and electron Fermi
surfaces are distinct, i.e., a “small” Fermi surface. This state was called
fractionalized Fermi-liquid (FL∗) [14]. A topological “Kondo breakdown”
transition can occur between FL∗ and “usual” heavy Fermi liquid (FL) states.

To include transverse spin fluctuations, we can use the relation (7) and write
down

$f_{i\sigma}\rightarrow(f_{i\uparrow}^{\dagger}f_{i\uparrow}+f_{i\downarrow}^{\dagger}f_{i\downarrow})f_{i\sigma}=-S_{i}^{\sigma}f_{i-\sigma}$

with $S_{i}^{\sigma}=f_{i\sigma}^{\dagger}f_{i-\sigma^{\prime}}$ the spin
operators (Instead, we can proceed with equations of motion sequence to obtain
similar results.) This is analogous to using multiplication rules for
Hubbard’s operators
$X(0,\sigma)=X(0,-\sigma)X(-\sigma,\sigma)=X(0,-\sigma)S^{-\sigma}.$

Another way to treat all kinds of fluctuations is using the KotliarRuckenstein
boson representation [15] in rotationally invariant version [16], which was
adopted in Ref.[17] to magnetic state as

$\displaystyle f_{i\sigma}$ $\displaystyle\rightarrow$
$\displaystyle\frac{1}{\sqrt{2}}\sum_{\sigma^{\prime}\mu}f_{i\sigma^{\prime}}p_{i\mu}\sigma^{\mu}_{\sigma^{\prime}\sigma}$
(9) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\sum_{\sigma^{\prime}}f_{i\sigma^{\prime}}[\delta_{\sigma\sigma^{\prime}}p_{i0}+(\mathbf{p}_{i}\mbox{\boldmath$\sigma$}_{\sigma^{\prime}\sigma})].$

with $p_{i\mu}$ ($\mu=0...3$) the auxiliary Bose operators. This
representation satisfies exactly commutation relations for Hubbard’s
operators. One more possibility is using supersymmetric decoupling of the
Kondo interaction, which introduces both fermion and Schwinger boson operators
[18], so that

$\mathbf{S}_{i}=\frac{1}{{2}}\sum_{i\sigma\sigma^{\prime}}\mbox{\boldmath$\sigma$}_{\sigma\sigma^{\prime}}(f_{i\sigma}^{\dagger}f_{i\sigma^{\prime}}+b_{i\sigma}^{\dagger}b_{i{\sigma^{\prime}}}).$
(10)

##  3 Heavy-fermion and non-Fermi-liquid behavior, and magnetism

As discussed above, in the quantum critical regime, a topological
transformation from large Fermi surface (Kondo lattice state) to small Fermi
surface is possible, which can be accompanied by magnetic-order instability.
First we consider the non-magnetic state.

In the spin-liquid state, gauge field fluctuations play an important role. In
particular, there occurs a spinon contribution to specific heat. This
contribution retains in FL∗ state and in FL state near the topological
transition. After integrating out spinons one obtains the gauge field
propagator [14]

$\displaystyle D_{\alpha\beta}({\bf q},i\omega_{n})\equiv\langle
a_{\alpha}({\bf q},i\omega_{n})a_{\beta}(-{\bf q},-i\omega_{n})\rangle$
$\displaystyle=\frac{\delta_{\alpha\beta}-q_{\alpha}q_{\beta}/q^{2}}{\Gamma|\omega_{n}|/q+\chi_{f}q^{2}+\rho_{s}}\,.$
(11)

Here $\Gamma$ and $\chi_{f}$ are determined by the details of the spinon
dispersion, $\omega_{n}$ is an imaginary Matsubara frequency; $\rho_{s}\sim
b_{0}^{2}$ is the boson “superfluid” density which is zero in the FL∗ state
and provides a cutoff for singular gauge-fluctuation contribution. Physically,
this contribution is due to Landau damping connected with the spinon Fermi
surface. As demonstrate the calculation [14] with the use of (11), the
specific heat coefficient $\gamma=C/T$ diverges logarithmically at
$T\rightarrow 0$ in the FL∗ phase, which means the non-Fermi-liquid behavior
(in the 2D case, $C(T)\sim T^{2/3}$). At approaching the transition from the
FL side, we have $\gamma\sim\ln(1/b_{0})$. This contribution retains in the
insulating phase (in the absence of conduction electrons).

In the quantum critical regime, the fluctuations of order parameter field $b$
are also important. They yield additional contributions to thermodynamic
properties. Unlike gauge field (neutral spinon) ones, they contribute also to
electronic transport. The $b$-field fluctuation contribution to the
conduction-electron self-energy can be written in the form [19]

$\Sigma({\bf k},i\omega_{n})=T\sum_{\Omega_{n},\bf q}G_{f}({\bf
k+q},i\omega_{n}+i\Omega_{n})D_{b}({\bf q},i\Omega_{n})$ (12)

where $G_{f}({\bf k},i\omega_{n})=1/(i\omega_{n}-\epsilon_{\bf k})$ is the
spinon Green’s function. Due to mismatch of electron and spinon Fermi
surfaces, the decay of bosons into particle-hole pairs becomes possible above
an energy $E^{*}$, which can be small if the distance between the two Fermi
surfaces is small. Above this energy the theory belongs to the universality
class of the ferromagnet (formally, wave vector ${\bf Q}=0$) with the critical
exponent $z=3$ [20]. For $E>E^{*}$ and a spherical Fermi surface

$D_{b}(q,\Omega_{n})\approx(\rho
V^{2})^{-1}/[q^{2}/(4k_{F}^{2})+\pi|\Omega_{n}|/(2\alpha v_{F}q)]$ (13)

with $\alpha$ the ratio of the spinon to the conduction electron bandwidth.

Then at $T>E^{*}$ one obtains a $T\ln T$ contribution to specific heat.
Besides that, the fluctuation scattering results in the $T\ln T$ dependence of
resistivity $R(T)$ in this regime [19]. For $T<E^{*}$, the behavior of
thermodynamic and transport properties depends on that the Fermi surfaces
intersect or not [21]. In the first case we have $R(T)\sim T^{3/2}$ and
$R(T)\sim T\ln T$ in 3d and 2d cases, respectively, as in an antiferromagnet
near the quantum phase transition with $z=2$ (although in fact we are in the
charge channel). In the second case, $R(T)\sim T^{2}$.

In the FL states, this contribution is cut at the Kondo gap of order of
$T_{K}$, which is related to the boson condensate, $T_{K}\simeq\pi\rho
V^{2}\langle b\rangle^{2}$ in the vicinity of the critical point.

Now we consider the situation of antiferromagnetic (AFM) ordering. As it was
concluded in Refs.[14, 22], the FL∗ state with the spinon Fermi surface should
be unstable with to a broken-symmetry AFM (spin-density wave, SDW) state. Th
exotic AFM order on the spinon Fermi surface is called AFM∗ or SDW∗ phase [14,
20]. The presence of an SDW∗ condensate does not cause a radical change in the
structure of the gauge fluctuation in comparison with the FL* state: the
spinons remain deconfined and coupled to a gapless U(1) gauge field. The gauge
field excitations coexist with the gapless Goldstone magnon mode and with a
Fermi surface of conduction electrons. Because of the broken translational
symmetry, there is no clear difference between small and large Fermi surfaces
now.

With increasing $s-d(f)$ coupling, the deconfined phase with small Fermi
surface can pass first into usual itinerant AFM state with a large Fermi
surface volume, and then into FL state. The spinon Fermi surface of the
FL∗-phase is expected to evolve smoothly into the FL region in some vicinity
of the topological transition, so that the SDW order can continue to FL in the
ground state, as discussed in Ref. [2]. Thus there is no sharp transition
between the FL and FL∗ regions, and there is instead expected to be a large
intermediate quantum-critical region [14].

To take into account transverse spin fluctuations, we use the rotationally
invariant representations discussed in Sect.2 and write down the corresponding
self-energy to second order in hybridization, we derive

$\displaystyle\Sigma_{\mathbf{k}}^{(2)}(E)\sim
V^{2}\sum_{\mathbf{q}}(u_{\mathbf{q}}-v_{\mathbf{q}})^{2}$ (14)
$\displaystyle\times\left(\frac{1-n_{\mathbf{k+q}}+N_{\mathbf{q}}}{E-\epsilon_{\mathbf{k+q}}-\omega_{\mathbf{q}}}+\frac{n_{\mathbf{k+q}}+N_{\mathbf{q}}}{E-\epsilon_{\mathbf{k+q}}+\omega_{\mathbf{q}}}\right)$
(15)

where $n_{\mathbf{k}}$ and $N_{\mathbf{q}}$ are Fermi and Bose functions,

$\omega_{\mathbf{q}}=2\overline{S}(J_{\mathbf{Q-q}}-J_{\mathbf{Q}})^{1/2}(J_{\mathbf{q}}-J_{\mathbf{Q}})^{1/2}$
(16)

is the magnon frequency. and

$u_{\mathbf{q}}^{2}=1+v_{\mathbf{q}}^{2}=\frac{1}{2}[1+\overline{S}(J_{\mathbf{q+Q}}+J_{\mathbf{q}}-2J_{\mathbf{Q}})/\omega_{\mathbf{q}}]$
(17)

are coefficients of the Bogoliubov transformation to the magnon operators in
the local coordinate system for the ordering with the vector $\mathbf{Q}$.
They satisfy $u_{\mathbf{q}}\mp v_{\mathbf{q}}\cong u_{\mathbf{q+Q}}\pm
v_{\mathbf{q+Q}}\propto\omega_{\mathbf{q}}{}^{\pm 1/2}$at $q\rightarrow 0.$
This contribution is similar to that of the usual perturbation theory in the
$s-d(f)$ model [23, 24], but works on the scale of the spinon bandwidth. It is
interesting that in the FL∗ state the quantity $V$ plays the role of the $s-f$
exchange parameter and mediates the RKKY-type interaction between $f$-states.
At the same time, the contribution (15) survives in the FL state where a
unified Fermi surface arises and a correlated $s-f$ band with narrow density-
of-states peaks is formed. However, in such a situation spectrum
$\epsilon_{\mathbf{k}}$ and the prefactor in (15) change.

In the case of weak antiferromagnetism we have an energy scale
$T^{*}=(\Delta/\alpha v_{F})J$ ($\Delta$ is the AFM splitting of the
spectrum), so that for $T^{*}<T<J$ the transitions between AFM subbands become
singular. In the general $d=3$ case we have Im$\Sigma(E)\propto E^{2}$. For
$d=2$ we derive (cf. Ref. [23])

$\mathrm{Im}\Sigma^{(2)}(E)\sim\lambda_{\mathbf{Q}}E\,\mathrm{sgn}E$ (18)

with

$\lambda_{\mathbf{q}}\sim
V^{2}(J_{0}-J_{\mathbf{Q}})\sum_{\mathbf{k}}\delta(\epsilon_{\mathbf{k}})\delta(\epsilon_{\mathbf{k+q}})$
(19)

The corresponding intersubband correction to specific heat is

$\delta C_{inter}(T)\sim\sum_{\mathbf{q\simeq
Q,}T\leq\omega_{\mathbf{q}}}\lambda_{\mathbf{q}}/\omega_{\mathbf{q}}^{2}.$
(20)

In the 2d (or “nesting” 3d) case the integral is logarithmically divergent at
$\mathbf{q\rightarrow Q}$ and the divergence is cut at $\max(T,T^{*}),$ si
that

$\delta
C_{inter}(T)\sim\lambda_{\mathbf{Q}}T\ln\frac{\bar{\omega}}{\max(T,T^{*})}.$
(21)

Thus for $T>T^{*}$ we obtain the $T\ln T$-dependence of specific heat. Since
the integral in (20) is determined by the magnon spectrum only, the result
(21) holds also in the case the frustrated (2d-like) magnon spectrum. It
should be noted that 2d-like spin fluctuations near the quantum-critical point
are observed in CePdAl [25], CeCu6-xAux [26] and YbRh2Si2 [27].

For general $d=3$ case we have the quadratic temperature dependence of spin-
wave resistivity, $R(T)\propto T^{2}$. For the 2d magnon spectrum or “nested”
3d situation one obtains

$R(T)\propto T\ln(1-\exp(-T^{*}/T))\simeq T\ln(T/T^{*})$ (22)

The role of magnetic fluctuations depends again on the relation between $J$
and $T_{K}$ (again, in frustrated and low-dimensional systems with strong
short-range the scale $J$ can be considerably larger than $T_{N}$). Note that
a gauge vector boson can be introduced in the vicinity of deconfined quantum
critical points, but it is characterized by large damping [28]. The behavior
of thermodynamic properties in quantum-disordered case, including quasi-linear
magnetic contributions to specific heat, can be found in Ref. [29].

Similar to Ref.[6], we can find singular corrections to sublattice
magnetization owing to spin-wave damping. The corrections due to hybridization
fluctuations in second order in $V$ turn out to be not only of order of
$J/T_{K}$, as in Ref. [6], but also simply of order of unity.

##  4 Conclusions

We have considered various contribution to thermodynamic and transport
properties in the vicinity of the quantum phase transitions in the Kondo
lattices. The gauge field contributions, being connected with the spinon Fermi
surface, result in specific heat and heat transport in the insulating phase.
This dependence can change for other classes of spin liquids, e.g.., for
pyrochlores with electron spectrum including Dirac points one has $C(T)\sim
T^{2}\ln T$ [30]. Experimental data indeed demonstrate considerable $T$-linear
specific heat in narrow-gap insulating and low-carrier $f$-systems which are
usually treated as Kondo lattices, e.g., in SmB6 and Ce3Bi4Pt3 [31], CeNiSn
[32], YbPtSb [33], Ce3Bi4Pd3 ($\gamma=360$ mJ/mol K2) [34] CeSbTe ($T_{N}=2.6$
K 41 mJ/mol K2, $T_{K}=$ 1.2 K) [35], Yb3Ir4Ge13 [36]. Moreover, in some such
systems, e.g., in CeNi2-δAs2-xPx [37], the non-Fermi-liquid behavior is
observed.

The Kondo breakdown transition has a topological nature and is connected with
a reconstruction of the Fermi surface. Such a transition is expected to occur
for the Kondo lattices (unlike the single-impurity Kondo model where only
crossovers take place). It is intimately related to magnetic transition,
although the detailed picture is complicated and is not finally understood. In
this connection, the YbRh2Si2 compound is widely discussed. At very low
temperatures this system demonstrates a non-trivial competition of Kondo
effect, magnetism and spin-liquid behavior in the quantum criticality. For
positive chemical pressure the Kondo breakdown is observed within the
magnetically ordered phase, whereas negative pressure results in their
separation, so that intermediate spin-liquid-type ground state occurs in an
extended range [38]. Related theoretical problems require further
investigations.

The research funding from the Ministry of Science and Higher Education of the
Russian Federation (the state assignment, theme “Quantum” No. 122021000038-7
is acknowledged.

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