#  Induced quantum magnetism in CEF singlet ground state models:
Thermodynamics and excitations

Peter Thalmeier  Max Planck Institute for the Chemical Physics of Solids,
01187 Dresden, Germany  Alireza Akbari  Institut für Theoretische Physik III,
Ruhr-Universität Bochum, D-44801 Bochum, Germany  Asia Pacific Center for
Theoretical Physics (APCTP), Pohang, Gyeongbuk, 790-784, Korea

###### Abstract

We present a comparative investigation of singlet ground state induced
magnetism for singlet, doublet and triplet excited CEF states of non-Kramers
f-electrons relevant primarily for Pr- and U- based compounds. This type of
magnetic order is of intrinsic quantum nature because it requires the
superposition of singlet ground state with excited states due to non-diagonal
matrix elements of the effective intersite exchange to generate local moments.
In contrast to conventional magnets the local moments and their ordering
appear simultaneously at the transition temperature. It is finite only if the
control parameter proportional to the ratio of exchange strength to level
splitting exceeds a critical value marking the quantum critical point of the
models. We determine the dependence of transition temperature, saturation
moment, renormalised level splitting, specific heat jumps and low-temperature
susceptibility as function of control parameter. Furthermore the temperature
dependence of these quantities is calculated for control parameters above and
below the quantum critical point and the distinction to conventional magnetism
is discussed. In addition we investigate the dynamical properties of the three
models, deriving the magnetic exciton dispersion and their critical behaviour.
In particular the conditions for true and arrested soft-mode behaviour at the
ordering wave vector are identified.

##  I Introduction

The majority of magnets with local moments can be understood within the
context of semiclassical approach sufficiently below the ordering temperature
$T_{m}$ [1, 2]. The formation of a collective moment and molecular field is
due to the exchange coupling of stable local moments that exist already above
$T_{m}$ and order spontaneously at this temperature, breaking time reversal
and possibly spatial symmetries. The ordered moment is treated as a classical
variable and the quantum effects are included by considering the effect of a
small number of bosonic excitations, i.e. magnons that reduce the ordered
moment below the classical value by a certain amount. This reduction is
moderate, e.g. in the three dimensional case, unless lower dimension and/or
the effect of frustration leads to possibly divergent corrections heralding
the breakdown of magnetic order and the appearance of a nonmagnetic ’spin
liquid’ ground state. Except for 1D magnetic chains such state can exist only
in tiny parts of the exchange parameter space [3] and mostly the semiclassical
picture is valid, even when the ordered moment reduction by quantum
fluctuations and transition temperature suppression may be quite large [4].

There is, however a class of magnetic materials with non-Kramers 4f- or 5f
ions (with integer total angular momentum J) where the quasi-classical picture
of ordering is per se invalid. This is the case when the crystalline electric
field (CEF) splits the $(2J+1)$\- fold degenerate J-multiplet into a series of
CEF multiplets such that the ground state $|0\rangle$ is a singlet which has
no magnetic moment, i.e. $\langle 0|{\bf J}|0\rangle=0$. In this situation the
appearance of magnetic order is a true quantum effect, it can only happen if
at least one of the J components has non-diagonal matrix elements with nearby
excited multiplets. Then a sufficiently strong intersite exchange may lead to
the creation of an ordered moment by spontaneous formation of a new ground
state which is a superposition of singlet and excited multiplet state. In this
way the local moments at any site and their collective order appear
simultaneously and are selfconsistently tied to each other. The condition for
the strength of the intersite coupling is determined by a dimensionless
control parameter composed of coupling strength, level splitting and the
nondiagonal matrix elements (Eq. 6). The critical value of the control
parameter, equal to one, defines the quantum critical point (QCP) separating
the paramagnetic (less than one) from the induced magnetic moment (larger than
one) regime.

The thermodynamic signatures, as expressed by behaviour of critical
temperature, saturation moment, specific heat anomalies and susceptibility are
quite distinct from the quasi-classical magnets. Furthermore the dynamical
characteristics show pronounced differences. In the latter coherent
propagating magnons; i.e. quasi-classical precessing moments naturally can
only appear in the ordered phase; requesting the presence of a molecular
field. On the other hand in the induced moment magnets collective ’magnetic
exciton’ modes are present already in the paramagnetic phase due to the
possibility of dispersive inelastic CEF excitations between singlet ground
state and excited multiplet. The dispersion of magnetic excitons is strongly
temperature dependent controlled by the thermal population difference of CEF
levels. Under suitable conditions it may turn into a soft mode at the
incipient ordering wave vector as a precursor to spontaneous induced order.
However, it is frequently arrested at a finite energy at the ordering
temperature. Below the ordering temperature it reemerges as a strongly
renormalised magnonic mode (possibly split into several branches) where the
induced order parameter together with the molecular-field renormalised CEF
level splitting determine the dispersion.

This type of induced moment quantum magnetism, different from the common
semiclassical variety has been known for some time but its possibility and
distinction to quasi-classical magnetism is not commonly appreciated. It
appears primarly in compounds with J=4 4f or 5f ions such as PrSb [5],
Pr3Tl[6, 7], PrCu2[8], PrNi[9], Pr- metal under pressure [10, 11, 12] but also
Tb (J=6) compound TbSb [13]. Futhermore 5f candidates for induced order are
UGa2 [14], UPd2Al3 [15, 16, 17] and URu2Si2[18] (and references cited
therein), also Fe-subsituted [19]. The latter example (when considered within
the localized 5f scenario) and also the (Kramers ion) compound YbRu2Ge2 show
that the induced order mechanism not only works for magnetism but also for
multipolar [20, 21] and quadrupolar [22, 23] order, respectively. Finally
another aspect of the singlet ground state induced magnetism has been
discovered: For CEF singlet ground state f-electrons on suitable 2D lattices
like honeycomb or kagome the magnetic excitons may develop a nontrivial
topological character with non-vanishing Chern number which would entail the
existence of magnetic excitonic edge modes in the paramagnetic state [24] This
possibility has previously only been considered for ferro- or antiferro-
ordered 2D lattices [25, 26].

Although the above mentioned compounds are known to have a singlet CEF ground
state the type of the first excited state whether singlet, doublet or triplet
(in cubic site symmetry only) is not always known with certainty, in
particular in the U-compounds. However the quantitative and even qualitative
aspects of induced moment magnetic order will be different in these three
cases. Although they have been considered before in the references above there
is no systematic comparison concerning their thermodynamic (specific heat,
susceptibility etc.) or dynamic properties like distinct magnetic exciton
dispersion and differences in temperature dependence and soft mode behaviour.
In this work we undertake this effort to improve understanding of these
physical properties of singlet ground state quantum magnets and give a better
foundation for possible judgement of experimentally observed properties. We
will do this for three generally applicable models: The singlet-singlet,
doublet or triplet models (SSM, SDM, STM respectively) which may be commonly
realised in uniaxial (the two former) or cubic symmetry (the latter), see also
Appendix A.

Our approach is analytical based on molecular field (MFA) and random phase
(RPA) approximations as far as it can be carried out. We emphasise as a
unifying concept the central role of the control parameter of the three models
defining the QCP and separating paramagnetic and magnetic phases. All physical
properties will be expressed in terms of these control parameters. It has the
further advantage that one can continuously approach the common magnetic order
with (quasi-) degenerate ground state by tuning the control parameter to large
values far away from the QCP. In particular we shall focus on the temperature
behaviour of susceptibility and specific heat in the ordered phase and on the
concomitant control parameter dependence of ordered moment, ordering
temperature and specific heat anomalies which show a clear distinction between
the induced order quantum regime and the (asymptotic) quasi-classical regime.
Furthermore we calculate the magnetic excitation spectrum of the three models
across the disordered and induced order regimes and demonstrate their striking
differences. In particular we give an explicit demonstration how the simple
soft mode picture is modified in the STM model leading to an arrested soft
mode at the true transition temperature which is also of more general
significance. This investigation is not focused on a specific material but
rather on the comparative analysis of generally important singlet ground state
CEF models. To keep the results from the three models distinct and avoid
confusion we dedicate separate sections to them. This entails some redundancy
but supports clarity of presented results.

##  II Models for induced quantum magnetism

We consider three types of singlet- ground state level systems. The first two
are a singlet-singlet model (SSM) and a singlet-doublet model (SDM) frequently
appropriate for uniaxial symmetry and a singlet-triplet model (STM) only
possible for cubic symmetry. They correspond to simplified low energy CEF
schemes consisting of just two levels whose splitting is considerably smaller
than the excitation energies to the higher lying CEF states. Such systems are
encountered for even-numbered electron f- electron shells with integer total
angular momentum like Pr, U (J=4) , Tm, Tb (J=6) and Ho (J=8) with examples
given above. For the theoretical treatment of magnetic order and excitations
it is a prerequisite to know the matrix structure of the angular momentum
operators in the reduced SSM, SDM and STM level schemes. For uniaxial
symmetries there are two possibilities, corresponding to Ising-type where only
one operator, by convention $J_{z}$, or xy-type where two operators
($J_{x},J_{y}$) have nonzero matrix elements between the ground and excited
CEF states of the reduced level scheme. Which one is realised depends on the
Stevens parameters $B_{n}^{m}$ of the CEF potential and consequently the type
of irreducible representations of low lying CEF states. Here we consider the
Ising-type SSM, the xy-type SDM and the cubic STM cases. The CEF singlet
ground state is denoted by $|0\rangle$. In the SSM the excited singlet is
denoted by $|1\rangle$ and in the SDM the doublet states by $|1\sigma\rangle$
, $\sigma=\pm$, for STM see below. In all cases the CEF splitting energy is
$\Delta$ (Fig. 1). The angular momentum operators for the xy-type models
within these low energy CEF subspaces have the general structure (in
$|0\rangle,|1\rangle$ sequence):

Ising-type SSM: $\displaystyle
J_{z}\\!=\\!\frac{m_{s}}{2}\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)=m_{s}S_{x};$ (1)

where ${\bf J}$\- operators refer to the free $|JM\rangle$ states and ${\bf
S}$ are the pseudo-spin operators in the reduced subspace of CEF states.
Likewise in the singlet-doublet xy- type model the general form of J operators
is given by (in $|0\rangle,|1+\rangle,|1-\rangle$ sequence):

xy-type SDM: $\displaystyle
J_{x}=\frac{m_{d}}{\sqrt{2}}\left(\begin{array}[]{ccc}0&1&1\\\ 1&0&0\\\
1&1&0\end{array}\right)=m_{d}S_{x};$ (2) $\displaystyle
J_{y}=\frac{m_{d}}{\sqrt{2}}\left(\begin{array}[]{ccc}0&i&-i\\\ -i&0&0\\\
i&0&0\end{array}\right)=m_{d}S_{y},$

Here we defined $m_{s}$, $m_{d}$ in such a way that $S_{x},S_{y}$ correspond
to the canonical (pseudo-) spin matrices for $S=\frac{1}{2},1$, respectively
(with reordered sequence of states for the latter). The numerical values of
$m_{s},m_{d}$ are to be obtained from the diagonalisation of the full CEF
Steven’s Hamiltonian in a concrete case.
For the cubic STM we take as a model the most important
$\Gamma_{1}-\Gamma_{4}$ level scheme of $J=4$ whose states are fully
determined by symmetry and therefore do not depend on the CEF potential
parameters. Since the cubic axes are equivalent we restrict to the $J_{z}$
matrix where indices $n=0,1-3$ correspond to the singlet $\Gamma_{1}$ ground
state $|\psi_{0}\rangle$ and triplet $\Gamma_{4}$ excited states
$|\psi_{1-3}\rangle$, respectively. It is given by

cubic- type STM: $\displaystyle
J_{z}=\frac{1}{2}\left(\begin{array}[]{cccc}0&0&m_{t}&0\\\
0&m^{\prime}_{t}&0&0\\\ m_{t}&0&0&0\\\ 0&0&0&-m^{\prime}_{t}\\\
\end{array}\right)=m_{t}(\frac{1}{2}\sigma_{x})\oplus
m^{\prime}_{t}(\frac{1}{2}\tau_{z}),$ (3)

where $\sigma_{x}$ and $\tau_{z}$ are Pauli matrices in the $(0,2)$ and
$(1,3)$ subspaces, respectively. For $J=4$ we have [27]
$m_{t}=\frac{4}{3}\sqrt{15}=5.16$ for the nondiagonal element and
$m^{\prime}_{t}=1$ for the diagonal one. The ratio $m^{\prime}_{t}:m_{t}=0.19$
controls to which extent the induced magnetic order is influenced by the
excited magnetic $\Gamma_{4}$ triplet. This influence occurs only in the STM
and has also important consequences for the softening behaviour of the
magnetic exciton spectrum as a precursor to the induced order (Sec. IV.3).
This ratio is fixed by symmetry in the case of $J=4$ because
$\Gamma_{1},\Gamma_{4}$ representations occur only once. For higher $J=6,8$
they occur multiple times and therefore the ratio $m^{\prime}_{t}:m_{t}$
depends on the CEF potential parameters which may then be be considered as an
additional variable parameter.

The effective intersite exchange interactions (mediated, e.g. by conduction
electrons) together with the CEF potential is described by the Hamiltonian

$H\\!=\\!\sum_{\Gamma,i}\epsilon_{\Gamma}|\Gamma,i\rangle\langle\Gamma,i|-\\!\sum_{\langle
ij\rangle}J_{ij}{\bf J}_{i}\cdot{\bf J}_{j},$ (4)

where $\Gamma=0,1$ or $0,(1+,1-)$ or $0,(1,2,3)$ labels the CEF states of the
three models, respectively and $i,j$ denote the lattice sites. We restrict to
only nearest neighbour $\langle n.n.\rangle$ exchange interaction
$J_{ij}=I_{0}$ for in-plane bonds and $J_{ij}=\kappa I_{0}$ for out-of plane
bonds (along c-axis), thus $\kappa$ controls the real-space anisotropy of the
model and may be tuned continuously. We disregard possible spin-space
ansiotropies of the exchange term to limit the number of parameters. The two
CEF energy levels are given by $\epsilon_{0}=0$ and $\epsilon_{1}$ ,
$\epsilon_{(1+,1-)}$ or $\epsilon_{(1-3)}=\Delta$, respectively. Frequently it
will be convenient to use the more symmetrical shifted energy levels
$\epsilon_{\Gamma}-\Delta/2$ (without introducing a new symbol). For
simplicity we consider the localised f electrons and CEF states on a simple
tetragonal lattice. The corresponding (normalised) Fourier transformed n.n.
exchange function is then given by (wave vectors in units of $\pi/a$ or
$\pi/c$)

$\hat{I}({\bf q})=\frac{{\rm sign}[I_{0}]}{(2+\kappa)}(\cos q_{x}+\cos
q_{y}+\kappa\cos q_{z}),$ (5)

where $\hat{I}({\bf q})=I({\bf q})/I_{e}$ and the effective exchange coupling
strength is given $I_{e}=\frac{z}{3}|I_{0}|(2+\kappa)$ with $z=6$ denoting the
n.n. coordination. According to the sign of the exchange part in Eq. (4) we
use the convention that $I_{0}>0$ corresponds to FM coupling and $I_{0}<0$ to
AF coupling.

Instead of the individual model parameters contained in the Hamiltonian of Eq.
(4) it will turn out that in each of the three models one has only a single
dimensionless control parameter given by

$\displaystyle\xi_{s,t}=\frac{m_{s,t}^{2}I_{e}}{2\Delta};\quad\xi_{d}=\frac{2m_{d}^{2}I_{e}}{\Delta},$
(6)

characterising the relative strength of effective intersite exchange $I_{e}$
and local CEF splitting $\Delta$. The slightly different definition of
$\xi_{d}$ is due to two facts: In the SDM two excited states are connected to
the ground states and the prefactor of the matrix is chosen differently to
comply with pseudospin conventions for $S=1$. As it will turn out the above
definition of the control parameters leads to identical quantum critical point
$\xi^{c}_{s,d,t}=1$ for all three models. We will commonly suppress the
$s,d,t$ indices when we talk about generic properties of all three models. We
will strive to express all relevant calculated quantities in terms of $\xi$
that controls the quantum phase transition from paramagnetic to induced moment
phase. The conventional quasi-classical magnetism in 3D is approached when
$\xi\gg 1$ and the CEF splitting becomes very small as compared to the
exchange energy scale so that the first term in Eq. (4) corresponds
effectively to a quasi-degenerate multiplet.

The molecular field (MF) treatment of Eq. (4) delivers the basic quantities of
induced order parameter, transition temperature and level splitting and state
superposition in the induced moment phase. Assuming that we either have
possible ferromagnetic (FM) or antiferromagnetic (AFM) induced order (the only
two possibilities for n.n. exchange) the effective molecular fields are given
by $h_{e}^{\lambda}=I_{e}\langle J_{\alpha}\rangle_{\lambda}$ ($\alpha=z$ for
Ising- SSM and STM) and we choose the moment direction for the xy- SDM along
$\alpha=x$. Furthermore $\lambda=A,B$ is the sublattice index such that
$h_{e}^{\lambda}=h_{e}$ for FM and $h_{e}^{A}=-h_{e}^{B}=h_{e}$ for AFM cases.
Then, e.g. for the SDM the molecular field Hamiltonian for each sublattice is
given by

$H^{\lambda}_{\rm
MF}=\sum_{i}[\sum_{\Gamma}\epsilon_{\Gamma}|\Gamma,i\rangle\langle\Gamma,i|-h_{e}^{\lambda}J_{x}(i)].$
(7)

After determination of the MF eigenvectors, energy eigenvalues and their
thermal occupations the induced moments $\langle J_{\alpha}\rangle$ may be
computed selfconsistently in the following.

##  III Thermodynamic properties: order parameter, specific heat and
susceptibility

The basic mechanism of induced order in singlet ground state systems is in
contrast to the quasi-classical case with (Kramers) degenerate ground state as
outlined in the Introduction. In the three singlet models there are no
preformed ground state moments, therefore magnetic order is only possible due
to a nondiagonal quantum mixture of singlet ground state with the excited
multiplet states due to inter-site exchange. Then on-site moments and their
collective order appear simultaneously from the nonmagnetic state. This is a
distinctly quantum mechanical origin of magnetic order caused by the
superposition of nonmagnetic ground and excited states forming spontaneously a
new magnetic ground state below $T_{m}$. It is natural for this mechanism to
work that the intersite exchange driving the moment formation from the split
states must be sufficiently large to overcome the CEF splitting stabilising
the nonmagnetic state as controlled by the dimensionless parameters defined in
Eq, (6). The technical treatment of the SSM, SDM and STM models will be rather
similar. But for the sake of clarity of results we treat them in the separate
following (sub-) sections.

Figure 1: Behaviour of singlet ground and excited multiplet level energies
$\epsilon_{\Gamma}-\frac{\Delta}{2}$ for the three models under the induced
magnetic order (shifted by $\frac{\Delta}{2}$). Note that for SDM we use
$\epsilon_{1}=\epsilon_{-}$ and $\epsilon_{3}=\epsilon_{+}$ in the related
parts of the text. Here $\xi_{s,d,t}=1.5$ corresponding to
$T_{m}=0.62,0.51,0.47$ consecutively. Nondiagonal matrix elements $m_{s,d,t}$
of moment operator necessary for induced order are indicated. In the STM a
diagonal triplet matrix element $\pm m^{\prime}_{t}=\pm 1$ occurs which leads
to an enhancement of the transition temperature (Fig. 2(a)) and an arrested
mode softening (Fig. 9). Energy and temperature unit is $\Delta\equiv 1$ here
and in all following graphs.

###  III.1 The Ising-type singlet-singlet model

In the SSM the MF Hamiltonian of Eq. (7) has the sublattice-independent
eigenvalues

$\epsilon_{\pm}=\pm\frac{1}{2}\Delta_{T}=\pm\frac{\Delta}{2}[1+\gamma^{2}\langle
J_{z}\rangle^{2}]^{\frac{1}{2}}$ (8)

leading to a renormalised (dimensionless) singlet-singlet splitting in the
ordered state according to

$\displaystyle\hat{\Delta}_{T}$
$\displaystyle=\frac{1}{\Delta}(\epsilon_{+}-\epsilon_{-})=\frac{\Delta_{T}}{\Delta}=[1+\gamma^{2}\langle
J_{z}\rangle^{2}]^{\frac{1}{2}};$ (9) $\displaystyle\gamma$
$\displaystyle=m_{s}I_{e}/\Delta=(2/m_{s})\xi_{s},$

which depends on temperature $T$ through the order parameter $\langle
J_{z}\rangle$ as shown in Fig. 1(a). The latter also leads to a coherent
mixing of ground and excited states as given by the rotation in singlet space:

$\displaystyle|\epsilon_{+}\rangle$
$\displaystyle=\cos\theta_{\lambda}|1\rangle-\sin\theta_{\lambda}|0\rangle,$
(10) $\displaystyle|\epsilon_{-}\rangle$
$\displaystyle=\sin\theta_{\lambda}|1\rangle+\cos\theta_{\lambda}|0\rangle,$

where $\tan 2\theta_{\lambda}=\gamma\langle J_{z}\rangle_{\lambda}$ with
$\theta_{\lambda}=\theta$; $\langle J_{z}\rangle_{\lambda}=\langle
J_{z}\rangle$ for FM and $\theta_{A,B}=\pm\theta$; $\langle
J_{z}\rangle_{A,B}=\pm\langle J_{z}\rangle$ for AFM. The mixing of singlets is
the essential mechanism to create a selfconsistent ground state induced moment
out of the singlet states due to the nondiagonal dipolar matrix element
$m_{s}$. This means that the mixing angle $\theta_{\lambda}$ and $\langle
J_{z}\rangle_{\lambda}$ can only be simultaneously nonzero.

####  III.1.1 Order parameter and transition temperature for SSM

From Eq. (10) the selfconsistent equation for the order parameter induced by
nondiagonl matrix element $m_{s}$ may be derived. Its temperature dependence
originates from the thermal populations $p_{\pm}=Z^{-1}\exp(\mp\Delta/2T)$ of
the two singlet states with energies $\epsilon_{\pm}$ where
$Z=2\cosh(\Delta_{T}/2T)$ is the partition function. For the evaluation of the
order parameter $\langle J_{x}\rangle$ and later susceptibility
$\hat{\chi}_{\alpha\alpha}$ components we need the form of $J_{\alpha}$ in Eq.
(1) in the eigenvector basis of Eq. (10). as given by

$J_{z}=\frac{m_{s}}{2}\left(\begin{array}[]{cc}\sin 2\theta&\cos 2\theta\\\
\cos 2\theta&-\sin 2\theta\end{array}\right).$ (11)

The matrix elements for $J_{z}$ are determined by $\cos
2\theta=[1+\gamma^{2}\langle J_{z}\rangle^{2}]^{-\frac{1}{2}}$ and $\sin
2\theta=\gamma\langle J_{x}\rangle[1+\gamma^{2}\langle
J_{z}\rangle^{2}]^{-\frac{1}{2}}$. For $\langle J_{z}\rangle=m_{s}\langle
S_{x}\rangle$ we then can write

$\displaystyle\langle J_{z}\rangle_{T}$
$\displaystyle=\frac{m_{s}}{2}\frac{1}{\xi_{s}}[\xi_{s}^{2}f_{s}^{2}(T)-1]^{\frac{1}{2}}=\frac{m_{s}}{2}\frac{1}{\xi_{s}}[\hat{\Delta}_{T}^{2}-1]^{\frac{1}{2}},$
(12) $\displaystyle f_{s}(T)$
$\displaystyle=\tanh[\bigl{(}\frac{\Delta_{0}}{2T}\bigr{)}f_{s}(T)]=p_{-}-p_{+}.$

Here $\Delta_{0}=\xi_{s}\Delta=m_{s}^{2}I_{e}/2$ is the zero temperature
splitting and $\hat{\Delta}_{T}\equiv\Delta_{T}/\Delta=\xi_{s}f_{s}(T)$ the
normalized T-dependent splitting. It is determined by the difference of
thermal level occupations $f_{s}(T)$ for $T\leq T_{m}$ as obtained from the
solution of the second equation for a given $\xi_{s}$. In the paramagnetic
regime $(T>T_{m})$ we simply have $f_{s}(T)\equiv
f_{s}^{0}(T)=\tanh\frac{\Delta}{2T}$. This means $f_{s}(0)=1$ and the
transition temperature $T_{m}$ is reached for
$f_{s}(T_{m})=\frac{1}{\xi_{s}}$, leading to

$\displaystyle
T_{m}=\frac{\Delta}{2\tanh^{-1}(\frac{1}{\xi_{s}})}=\left\\{\begin{array}[]{r
l}\frac{\Delta}{|\ln\frac{\delta}{2}|}&\xi_{s}=1+\delta\;\;\;(\delta\ll
1)\\\\[8.5359pt] \frac{1}{2}\xi_{s}\Delta&\xi_{s}\gg 1\end{array}\right.,$
(13)

where the asymptotic limits are given to the right, the upper one
corresponding to closeness to the QCP, the lower one approaching the
conventional degenerate ground state magnetism. This means for a finite
$T_{m}$ for induced quantum magnetic order one must have a control parameter
$\xi_{s}>1$, i.e. according to Eq. (6) an intersite exchange $I_{e}$
sufficiently large compared to the singlet-singlet CEF splitting. Therefore
$\xi^{c}_{s}=1$ marks the quantum critical point (QCP) between paramagnetism
$(\xi_{s}<\xi^{c}_{s})$ and induced magnetic order $(\xi_{s}>\xi^{c}_{s})$.
The normalized moment and the saturation moment $\langle J_{z}\rangle_{0}$ are
given by

$\displaystyle\langle\hat{J}_{z}\rangle_{T}$ $\displaystyle=\frac{\langle
J_{z}\rangle_{T}}{\langle
J_{z}\rangle_{0}}=\frac{[\xi_{s}^{2}f_{s}^{2}(T)-1]^{\frac{1}{2}}}{[\xi_{s}^{2}-1]^{\frac{1}{2}}},$
(14) $\displaystyle\langle J_{z}\rangle_{0}$ $\displaystyle=m_{s}\langle
S_{x}\rangle_{0}=\frac{m_{s}}{2}\frac{1}{\xi_{s}}[\xi_{s}^{2}-1]^{\frac{1}{2}}$
$\displaystyle=\left\\{\begin{array}[]{r
l}m_{s}\sqrt{\frac{\delta}{2}}&\xi_{s}=1+\delta\;\;\;(\delta\ll
1)\\\\[8.5359pt] \frac{m_{s}}{2}&\xi_{s}\gg 1\end{array}\right..$

It is also instructive to consider the ratio of (normalised) transition
temperature to (pseudospin) moment given by

$\displaystyle\frac{T_{m}/\Delta}{\langle S_{x}\rangle_{0}}$
$\displaystyle=\frac{\xi_{s}}{[\xi_{s}^{2}-1]^{\frac{1}{2}}\tanh^{-1}\bigl{(}\frac{1}{\xi_{s}}\bigr{)}}$
(15) $\displaystyle=\left\\{\begin{array}[]{r
l}\frac{1}{\sqrt{\frac{\delta}{2}}|\ln\frac{\delta}{2}|}&\xi_{s}=1+\delta\;\;\;(\delta\ll
1)\\\\[8.5359pt] \xi_{s}&\xi_{s}\gg 1\end{array}\right..$

Close to the QCP (upper limit) this ratio decreases steeply to zero, as
opposed to the conventional magnetism limit $\xi_{s}\gg 1$ where $\langle
S_{x}\rangle_{0}\approx\frac{1}{2}$ corresponding to proportionality of
transition temperature $T_{m}\sim I_{e}$ to the exchange strength in this
limit.

####  III.1.2 Internal energy and specific heat for SSM

In the paramagnetic phase the internal energy of the SSM is simply
$U(T)=-\frac{\Delta}{2}\tanh\bigl{(}\frac{\Delta}{2T}\bigr{)}$ leading to a
Schottky specific heat (second line in Eq. (18)). In the ordered phase we have
to include the direct contribution of the order parameter. Then the
temperature dependent internal energy (per site) in MF approximation is given
by

$U(T)=\sum_{\sigma}p_{\sigma}\epsilon_{\sigma}(T)+\frac{1}{2}I_{e}\langle
J_{x}\rangle_{T}^{2},$ (16)

where the first part is the contains the MF energy levels of Eq. (8) and their
thermal occupations. This may be evaluated by using the expressions derived
before as

$U(T)=\left\\{\begin{array}[]{l
l}-\frac{\Delta}{4}(\xi_{s}f_{s}^{2}(T)+\frac{1}{\xi_{s}})&T\leq
T_{m}\\\\[8.5359pt] -\frac{\Delta}{2}f_{s}^{0}(T)&T>T_{m}\end{array}\right..$
(17)

Then $U(0)=-\frac{\Delta}{4}(\xi_{s}+\frac{1}{\xi_{s}})$ at zero temperature.
For $\xi_{s}\rightarrow 1^{+}$ this approaches the paramagnetic ground state
energy $\epsilon_{0}=-\frac{\Delta}{2}$. The specific heat $C_{V}(T)=(\partial
U(T)/\partial T)_{V}$ may now be calculated using Eq. (12) leading to

$\displaystyle\left\\{\begin{array}[]{l
l}C_{V}(T)=\bigl{(}\frac{\Delta_{T}}{2T}\bigr{)}^{2}\bigl{(}\cosh^{2}\frac{\Delta_{T}}{2T}-\frac{\Delta_{0}}{2T}\bigr{)}^{-1}&T\leq
T_{m}\\\\[8.5359pt]
C_{V}^{0}(T)=\bigl{(}\frac{\Delta}{2T}\bigr{)}^{2}\cosh^{-2}\frac{\Delta}{2T}&T>T_{m}\end{array}\right.$
(18)

for magnetic and paramagnetic phases, respectively. where $C_{V}^{0}(T)$ is
the background of the Schottky anomaly for a two-level system (N=1 in Appendix
B). Since the specific heat for $T\leq T_{m}$ contains the effects of the
order parameter slope $\partial\hat{\Delta}(T)/\partial T$ which is
discontinuous at $T_{m}$ (zero above and finite below) there is a jump $\delta
C_{V}=C_{V}^{-}-C_{V}^{+}$ in the specific heat starting from the paramagnetic
value $C_{V}^{+}=C^{0}_{V}(T_{m}^{+})$ just above $T_{m}$ to the value
$C_{V}^{-}=C_{V}(T_{m}^{-})$ just below. From the above equations we get

$\displaystyle C_{V}^{+}$
$\displaystyle=\frac{1}{\xi_{s}^{2}}(\xi_{s}^{2}-1)\bigl{(}\tanh^{-1}\frac{1}{\xi_{s}}\bigr{)}^{2},$
(19) $\displaystyle C_{V}^{-}$
$\displaystyle=\frac{\frac{1}{\xi_{s}^{2}}(\xi_{s}^{2}-1)\bigl{(}\tanh^{-1}\frac{1}{\xi_{s}}\bigr{)}^{2}}{1-\frac{1}{\xi_{s}}(\xi_{s}^{2}-1)\tanh^{-1}\frac{1}{\xi_{s}}}=\frac{C_{V}^{+}}{1-\lambda_{s}C_{V}^{+}},$
$\displaystyle\lambda_{s}$
$\displaystyle=\frac{\xi_{s}}{\tanh^{-1}\frac{1}{\xi_{s}}},$
$\displaystyle\delta C_{V}$
$\displaystyle=C_{V}^{+}\frac{\lambda_{s}C_{V}^{+}}{1-\lambda_{s}C_{V}^{+}}$
$\displaystyle\simeq\left\\{\begin{array}[]{l
l}\frac{1}{2}\delta^{2}|\ln\frac{\delta}{2}|^{3}&\xi_{s}=1+\delta\;\;\;(\delta\ll
1)\\\\[8.5359pt] \frac{3}{2};&\xi_{s}\gg 1\end{array}\right.,$

where we used
$\cosh^{2}\bigl{(}\frac{\Delta}{2T_{m}}\bigr{)}=\xi_{s}^{2}/(\xi_{s}^{2}-1)$
and $\bigl{(}\frac{\Delta}{2T_{m}}\bigr{)}=\tanh^{-1}\frac{1}{\xi_{s}}$ from
Eq. (13). The relative jump size compared to the paramagnetic value is then

$\displaystyle\frac{\delta
C_{V}}{C_{V}^{+}}=\frac{\lambda_{s}C_{V}^{+}}{1-\lambda_{s}C_{V}^{+}}\simeq\left\\{\begin{array}[]{l
l}\delta|\ln\frac{\delta}{2}|&\xi_{s}=1+\delta\;\;\;(\delta\ll
1)\\\\[8.5359pt] \frac{3}{2}\xi_{s}^{2}&\xi_{s}\gg 1\end{array}\right..$ (20)

On approaching the QCP from above $(\xi_{s}\rightarrow 1^{+})$ both
paramagnetic (Schottky) value $C_{V}^{+}$ and jump value $\delta C_{V}$ vanish
and their ratio also vanishes $\delta C_{V}/C_{V}^{+}\rightarrow 0$. For large
$\xi_{s}$, approaching conventional magnetism the ratio increases
$\sim\frac{3}{2}\xi_{s}^{2}$ because the Schottky value $C_{V}^{+}\simeq
1/\xi_{s}^{2}$ vanishes due to $\Delta/T_{m}\rightarrow 0$. This means the
absolute jump value approaches $\delta C_{V}\rightarrow\frac{3}{2}$ in this
limit which agrees with the value known from conventional magnets with twofold
degenerate ground state [2]. The $\xi$-dependence of the specific heat and its
jump for the three models is illustrated in Figs. 3,4 and discussed in Sec. V.

Figure 2: (a) Critical temperature for induced magnetic order as function of
control parameter $\xi$. The QCP value $\xi_{c}=1$ separates paramagnetic (PM,
$\xi<1$) from induced moment (IM, $\xi>1$) regime. For STM both exact $T_{m}$
(Eq. (46)) and approximate $T_{m}^{0}$ (Eq. (47)) are shown. (b) Temperature
dependence of population functions $f_{s,d,t}$ and corresponding normalised
order parameter $\langle S_{x}\rangle_{s,d}$ and $\langle S_{z}\rangle_{t}$
for SSM , SDM and STM cases scaled by $T_{m}=0.62,0.51,0.47$, respectively.

####  III.1.3 The static homogeneous SSM susceptibility

It is particularly instructive to consider the temperature dependence of the
susceptibility across the induced order phase transition both in the
paramagnetic and induced order (FM or AFM) phases with field applied parallel
to the moment direction $\langle J_{z}\rangle$. The general expression for the
single-ion susceptibility
$\hat{\chi}_{\alpha\alpha}^{0}=\chi_{\alpha\alpha}^{0}/(g_{J}\mu_{B})^{2}$
$(\alpha=x,y,z)$ for split singlets only is given by [10]

$\displaystyle\begin{aligned} \hat{\chi}^{0}_{\alpha\alpha}=&\sum_{n\neq
m}|\langle
m|J_{\alpha}|n\rangle|^{2}\frac{p_{n}-p_{m}}{\epsilon_{m}-\epsilon_{n}}\\\
&+\frac{1}{T}[\sum_{n}|\langle n|J_{\alpha}|n\rangle|^{2}p_{n}-\langle
J_{\alpha}\rangle^{2}]\\\
=&\;\hat{\chi}^{0VV}_{\alpha\alpha}+\hat{\chi}^{0C}_{\alpha\alpha},\end{aligned}$
(21)

where the first part is the vanVleck term (VV) and the second a pseudo-Curie
(C) term active at temperatures comparable to the CEF splitting due to
(possibly molecular field induced) diagonal matrix elements. It vanishes,
however exponentially for low temperatures.. The collective (RPA)
susceptibilities are then obtained via

$\displaystyle\hat{\chi}_{\alpha\alpha}(T)=\frac{\hat{\chi}^{0}_{\alpha\alpha}(T)}{1\mp
I_{e}\hat{\chi}^{0}_{\alpha\alpha}(T)}$ (22)

with $I_{e}=\frac{z}{3}|I_{0}|(2+\kappa)$. Here the upper (lower) sign
correspond to FM $(I_{0}>0)$ or AFM $(I_{0}<0)$ cases, respectively. For the
evaluation of susceptibilities in the ordered phase we need the forms of
$J_{z}$ in Eq. (1) in the eigenvector basis according to Eqs. (10,11). Then,
using the Eqs. (21,22) we obtain in the paramagnetic state:

$T>T_{m}:$
$\displaystyle\hat{\chi}^{VV0}_{zz}(T)=\frac{1}{I_{e}}\xi_{s}f_{s}^{0}(T);$
(23) $\displaystyle\hat{\chi}^{C0}_{zz}(T)=0;$
$\displaystyle\hat{\chi}_{zz}(T)=\frac{1}{I_{e}}\frac{\xi_{s}f_{s}^{0}(T)}{1\mp
I_{e}f_{s}^{0}(T)};$
$\displaystyle\hat{\chi}_{zz}(T_{m}^{+})=\left\\{\begin{array}[]{r
l}\rightarrow\infty&(-):{\rm FM}\\\\[8.5359pt] \frac{1}{2I_{e}}&(+):{\rm
AFM}\end{array}\right.;\;\;\;$ $\displaystyle\hat{\chi}_{zz}(T\gg
T_{m})\approx\frac{m_{s}^{2}}{4}\frac{1}{T}.$

In the induced moment phase we obtain for the $zz$ component, using the
abbreviation $f_{s}=f_{s}(T)$ and recalling that $\Delta_{0}=\xi_{s}\Delta$
and $(\Delta_{0}/2T_{m})=\xi_{s}\tanh^{-1}\frac{1}{\xi_{s}}$:

$T\leq T_{m}:$
$\displaystyle\hat{\chi}^{VV0}_{zz}(T)=\frac{1}{I_{e}}\frac{1}{\xi^{2}_{s}f^{2}_{s}};\;\;\;$
(24)
$\displaystyle\hat{\chi}^{C0}_{zz}(T)=\frac{1}{I_{e}}\bigl{(}\frac{\Delta_{0}}{2T}\bigr{)}\frac{(\xi^{2}_{s}f^{2}_{s}-1)(1-f^{2}_{s})}{\xi^{2}_{s}f^{2}_{s}};$
$\displaystyle\hat{\chi}_{zz}(T)=$ $\displaystyle\hskip
2.84544pt\frac{1}{I_{e}}\frac{1+\bigl{(}\frac{\Delta_{0}}{2T}\bigr{)}(\xi^{2}_{s}f^{2}_{s}-1)(1-f^{2}_{s})}{\xi^{2}_{s}f^{2}_{s}\mp[1+\bigl{(}\frac{\Delta_{0}}{2T}\bigr{)}(\xi^{2}_{s}f^{2}_{s}-1)(1-f^{2}_{s})]};$
$\displaystyle\hat{\chi}_{zz}(T_{m}^{-})=\left\\{\begin{array}[]{r
l}\rightarrow\infty&(-):{\rm FM}\\\\[8.5359pt] \frac{1}{2I_{e}}&(+):{\rm
AFM}\end{array}\right.;$
$\displaystyle\hat{\chi}_{zz}(0)=\left\\{\begin{array}[]{r
l}\frac{1}{I_{e}}\frac{1}{\xi_{s}^{2}-1}&(-):{\rm FM}\\\\[8.5359pt]
\frac{1}{I_{e}}\frac{1}{\xi_{s}^{2}+1}&(+):{\rm AFM}\end{array}\right..$

The complicated algebraic structure in this case is mostly due to the second
pseudo-Curie term. As $f_{s}(0)=1$ and $f_{s}(T_{m})=1/\xi_{s}$ we notice that
it vanishes both for $T=0$ and $T=T_{m}$ but is nonzero in between. The total
susceptibility $\hat{\chi}_{zz}(0)$ is also non-vanishing for T=0 because the
induced saturation moment $\langle J_{z}\rangle_{0}$ (Eq. (14)) is not fully
polarised for moderate $\xi_{s}$. This is a most characteristic difference to
conventional magnets with degenerate ground state where the fully developed
saturation moment $\langle J_{z}\rangle_{0}=m_{s}S\;(S=\frac{1}{2})$ can no
longer be polarised and therefore $\hat{\chi}_{zz}(0)=0$. The induced moment
case approaches that conventional limit for $\xi_{s}\gg 1$. These results are
presented in Fig. 5 and discussed further in Sec. V.

###  III.2 The xy-type singlet-doublet model

The SDM model consisting of a singlet ground state $|0\rangle$ with
$\epsilon_{0}=0$ and a doublet $|1\pm\rangle$ at $\epsilon_{1\pm}=\Delta$ is
defined by the Hamiltonian of Eq. (4). When convenient we also use shifted
values $\epsilon_{\Gamma}-\frac{\Delta}{2}$ (without introducing new symbols).
The MF treatment according to Eq. (7) leads to the split three-singlet
eigenvalues given by

$\epsilon_{1,3}=\frac{\Delta}{2}\Big{[}1\mp[1+2\bigl{(}\frac{2\gamma\langle
J_{x}\rangle}{\Delta}\bigr{)}^{2}\Big{]}^{\frac{1}{2}};\;\;\;\epsilon_{2}=\Delta,$
(25)

now with $\gamma=\frac{m_{d}}{\sqrt{2}}I_{e}$. Although the singlet ground
state mixes with both of the excited states only the $|\epsilon_{1,3}\rangle$
show repulsion whereas the energy of $|\epsilon_{2}\rangle$ remains unaffected
because the corresponding eigenstate is the antisymmetric combination
$\frac{1}{\sqrt{2}}(|1+\rangle-|1-\rangle)$ as shown below. Altogether the
orthonormal eigenstates in row order of increasing energies $\epsilon_{1,2,3}$
are given by the columns of the unitary $(U^{\dagger}U=1)$ matrix

$\displaystyle U=\left(\begin{array}[]{ccc}\sin\theta_{1}&0&\sin\theta_{3}\\\
-\frac{1}{\sqrt{2}}\cos\theta_{1}&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\cos\theta_{3}\\\
-\frac{1}{\sqrt{2}}\cos\theta_{1}&-\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\cos\theta_{3}\end{array}\right),$
(26)

where the matrix elements expressed by the mixing angles $\theta_{i}$ are
given by (i=1,3)

$\displaystyle\cos\theta_{i}=$
$\displaystyle\Big{[}1+\frac{1}{2}\Bigl{(}\frac{2\gamma\langle
S_{x}\rangle}{\epsilon_{i}}\Bigr{)}^{2}\Big{]}^{-\frac{1}{2}},$ (27)
$\displaystyle\sin\theta_{i}=$
$\displaystyle\frac{1}{\sqrt{2}}\Big{(}\frac{2\gamma\langle
S_{x}\rangle}{\epsilon_{i}}\Big{)}\Big{[}1+\frac{1}{2}\Big{(}\frac{2\gamma\langle
S_{x}\rangle}{\epsilon_{i}}\Big{)}^{2}\Big{]}^{-\frac{1}{2}}.$

From the structure of $U$ we can see that $|\epsilon_{2}\rangle$ is just the
antisymmetric combination of the original degenerate excited doublet
components with unchanged $\epsilon_{2}=\Delta$ whereas
$|\epsilon_{1,3}\rangle$ have components of all states mixed together and
their energies $\epsilon_{1,3}$ repel symmetrically leading to three singlets
in the ordered state (Fig. 1(b)).

Figure 3: Specific heat in units of $k_{B}$/site. Note in this figure $T$ is
normalised to the CEF splitting $\Delta$. (a) Temperature dependences of
internal energy $U$ and specific heat $C_{V}$ for the SSM. On approaching the
QCP for $\xi\rightarrow 1^{+}$ the jump $\delta C_{V}$ due to induced order
shifts downwards with $T_{m}$ and becomes progressively smaller. It is
superposed on the background Schottky peak due to the CEF splitting. (b)
Complementary specific heat for the three models with same control parameter
$\xi=1.5$ in each case. While the jump increases the transition temperature
decreases with $T_{m}=0.62,0.51,0.47$ consecutively. The $\xi-$ dependence of
the jump $\delta C_{V}$ is shown in Fig. 4. Figure 4: Dependence of absolute
specific heat jump $\delta C_{V}=C_{V}^{-}-C_{V}^{+}$ at $T_{m}$ on the
control parameter of the three models evolving in accordance with the size of
the normalised induced saturation order parameter $\langle
S_{\alpha}\rangle_{0}$. Here $\alpha=x$ for SSM and SDM and $\alpha=z$ for STM
are identical. Close to the QCP with $\xi=1+\delta$ $(\delta\ll 1)$ the moment
vanishes $\sim(\delta/2)^{\frac{1}{2}}$ with singular slope while the specific
heat jump vanishes more gradually according to $\delta
C_{V}\sim\frac{1}{2}\delta^{2}|\ln\frac{\delta}{2}|^{3}$.

####  III.2.1 Order parameter and transition temperature for SDM

For deriving the selfconsistency equation for induced moment $\langle
J_{x}\rangle$ and later the susceptibilities $\hat{\chi}_{\alpha\alpha}$ we
need again the $J_{\alpha}$ matrices in the eigenvector basis given by $U$,
i.e. we have to transform $J_{\alpha}\rightarrow U^{\dagger}J_{\alpha}U$.
After some algebra this results in

SSD: $\displaystyle
J_{x}=\frac{m_{d}}{\sqrt{2}}\left(\begin{array}[]{ccc}m_{0}&0&m_{1}\\\
0&0&0\\\ m_{1}&0&-m_{0}\end{array}\right);$ (28) $\displaystyle
J_{y}=\frac{m_{d}}{\sqrt{2}}\left(\begin{array}[]{ccc}0&im_{2}&0\\\
-im_{2}&0&-im^{\prime}_{2}\\\ 0&im^{\prime}_{2}&0\end{array}\right),$

where the transformed matrix elements are combinations of the sines and
cosines of mixing angles $\theta_{1,3}$ in U. They may be transformed to the
simple expressions

$\displaystyle J_{x}:\;\;\;m_{0}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{\hat{\Delta}_{T}}(1-\hat{\Delta}_{T}^{2})^{\frac{1}{2}};\;\;\;m_{1}=-\frac{\sqrt{2}}{\hat{\Delta}_{T}},$
(29) $\displaystyle J_{y}:\;\;\;m_{2}$ $\displaystyle=$
$\displaystyle(1+\frac{1}{\hat{\Delta}_{T}});\;\;\;m^{\prime}_{2}=(1-\frac{1}{\hat{\Delta}_{T}}),$

where $\hat{\Delta}_{T}=\xi_{d}f_{d}(T)$ is defined below. They fulfil the sum
rules
$\frac{1}{2}(m_{0}^{2}+m_{1}^{2})=\frac{1}{2}(m_{2}^{2}+m^{{}^{\prime}2}_{2})=1$.
Note that $J_{x}$ matrix elements are nonzero where those of $J_{y}$ vanish
and vice versa. Using Eq. (28) leads to an order parameter given by

$\langle
J_{x}\rangle_{T}=\frac{m_{d}}{\sqrt{2}}\frac{1}{\xi_{d}}[\xi_{s}^{2}f_{d}^{2}(T)-1]^{\frac{1}{2}}=\frac{m_{d}}{\sqrt{2}}\frac{1}{\xi_{d}}[\hat{\Delta}_{T}^{2}-1]^{\frac{1}{2}}.$
(30)

Similar to Eq. (12) here $\Delta_{0}=\xi_{d}\Delta$ is the renormalised zero
temperature splitting and $\hat{\Delta}_{T}=\xi_{d}f_{d}(T)$. The
selfconsistent equation for the temperature dependence of the occupation
difference $p_{1}-p_{3}=f_{d}(T)$ for $T\leq T_{m}$ is now more complicated
due to the thermal population of the $|\epsilon_{2}\rangle$ state at the
unrenormalised energy $\Delta$ resulting from the lower state of the excited
doublet split by $\langle J_{x}\rangle$ (Fig. 1(b)). We obtain

$\displaystyle f_{d}(T)=$
$\displaystyle\frac{\tanh[\bigl{(}\frac{\Delta_{0}}{2T}\bigr{)}f_{d}(T)]}{1+\tilde{f}_{d}(T)}$
(31) $\displaystyle=$
$\displaystyle\frac{\sinh[\bigl{(}\frac{\Delta_{0}}{2T}\bigr{)}f_{d}(T)\bigr{]}}{\frac{1}{2}\exp(-\frac{\Delta}{2T})+\cosh[\bigl{(}\frac{\Delta_{0}}{2T}\bigr{)}f_{d}(T)]}$

with
$\tilde{f}_{d}(T)=\frac{1}{2}\exp(-\frac{\Delta}{2T})/\cosh[\bigl{(}\frac{\Delta_{0}}{2T}\bigr{)}f_{d}(T)]$.
If we would (wrongly) ignore the lower doublet state (setting
$\tilde{f}_{d}=0$) the expression would become formally identical to that of
the SSM case in Eq. (12). Thus $\tilde{f}_{d}$ describes the correction due to
the doublet nature of the excited CEF level. In the paramagnetic regime we
simply have to replace $\Delta_{0}f_{d}\rightarrow\Delta$ and get
$f_{d}^{0}(T)=2\tanh(\frac{\Delta}{2T})/[3-\tanh(\frac{\Delta}{2T})]$. As for
the SSM the critical temperature for the induced moment $\langle J_{x}\rangle$
to appear is given by $f_{d}(T_{m})=\frac{1}{\xi_{d}}$ with the solution

$T_{m}=\frac{\Delta}{2\tanh^{-1}\bigl{(}\frac{3}{1+2\xi_{d}}\bigr{)}}$ (32)

that depends in a different manner on the control parameter $\xi_{d}$ as
compared to the singlet model (Eq. (13)). This is again caused by the finite
thermal population of the lower doublet level at $T_{m}$. Nevertheless the QCP
where $T_{m}$ vanishes, is identical to the SSM, given by $\xi^{c}_{d}=1$
(determined by the request that $3/(1+2\xi_{d})<1$). Furthermore the
normalised T-dependent and saturation moments for the SDM are similarly given
by

$\displaystyle\langle\hat{J}_{x}\rangle_{T}$ $\displaystyle=\frac{\langle
J_{x}\rangle_{T}}{\langle
J_{x}\rangle_{0}}=\frac{[\xi_{d}^{2}f_{d}^{2}(T)-1]^{\frac{1}{2}}}{[\xi_{d}^{2}-1]^{\frac{1}{2}}},$
(33) $\displaystyle\langle J_{x}\rangle_{0}$ $\displaystyle=m_{d}\langle
S_{x}\rangle_{0}=\frac{m_{d}}{\sqrt{2}}\frac{1}{\xi_{d}}[\xi_{d}^{2}-1]^{\frac{1}{2}}$
$\displaystyle=\left\\{\begin{array}[]{r
l}\frac{m_{d}}{\sqrt{2}}\sqrt{\frac{\delta}{2}}&\xi_{d}=1+\delta\;\;\;(\delta\ll
1)\\\\[8.5359pt] \frac{m_{d}}{\sqrt{2}}&\xi_{s}\gg 1\end{array}\right..$

####  III.2.2 Internal enery and specific heat for SDM

The internal energy of the paramagnetic SDM model is
$U(T)=-\frac{\Delta}{2}(3\tanh\bigl{(}\frac{\Delta}{2T}\bigr{)}-1)/(3-\tanh\bigl{(}\frac{\Delta}{2T}\bigr{)}$,
which results in a correspondingly modified SDM Schottky-type specific heat
(see N=2 case in Appendix B)

$C_{V}(T)=2\frac{4\bigl{(}\frac{\Delta}{2T}\bigr{)}^{2}}{[3\cosh\bigl{(}\frac{\Delta}{2T}\bigr{)}-\sinh\bigl{(}\frac{\Delta}{2T}\bigr{)}]^{2}}.$
(34)

For the magnetically ordered case it is determined by an expression
corresponding to Eq. (16) for the SSM, now summing over three MF levels and
occupations. Using the shifted eigenvalues
$\epsilon_{\Gamma}-\frac{\Delta}{2}$ this leads again to

$U(T)=-\frac{\Delta}{4}\Bigl{[}\bigl{(}\xi_{d}f_{d}^{2}(T)+\frac{1}{\xi_{d}}\bigr{)}-2\tilde{f}_{d}(T)\bigl{(}1+\tilde{f}_{d}(T)\bigr{)}^{-1}\Bigr{]}.$
(35)

The first part is like SSM case and the second one corrects for the additional
doublet level. For zero temperature we again have
$U(0)=-\frac{\Delta}{4}(\xi_{d}+\frac{1}{\xi_{d}})$. Due to the population
effect of the additional level analytical derivation and discussion of
$C_{V}(T)=(\partial U(T)/\partial T)_{V}$ is no longer reasonably feasible,
considering the complicated MF equation Eq. (31) for $f_{d}(T)$. Therefore in
the ordered phase, once this function has been determined the specific heat is
obtained by numerical differentiation of $U(T)$.

####  III.2.3 The static homogeneous SDM susceptibility

The basic expressions for single-ion and collective susceptibilities in Eqs.
(21,22), now with level energies and matrix elements corresponding to the SDM
(Eqs. (25,2)) are used to calculate these quantities. For the paramagnetic
state the expressions for the susceptibility components are completely
equivalent to Eq. (23) with the replacement
$(m_{s},\xi_{s},f^{0}_{s})\rightarrow(m_{d},\xi_{d},f^{0}_{d})$. For the
magnetic phase we obtain different expressions

$(xx):T\leq T_{m}:$
$\displaystyle\hat{\chi}^{VV0}_{xx}(T)=\frac{1}{I_{e}}\frac{1}{\xi^{2}_{d}f^{2}_{d}};$
(36)
$\displaystyle\hat{\chi}^{C0}_{xx}(T)=\frac{1}{I_{e}}\bigl{(}\frac{\Delta_{0}}{2T}\bigr{)}\frac{(\xi^{2}_{d}f^{2}_{d}-1)((1+\tilde{f}_{d})^{-1}-f^{2}_{d})}{\xi^{2}_{d}f^{2}_{d}};$
$\displaystyle\hat{\chi}_{xx}(T)=$ $\displaystyle\hskip
14.22636pt\frac{1}{I_{e}}\frac{1+\bigl{(}\frac{\Delta_{0}}{2T}\bigr{)}(\xi^{2}_{d}f^{2}_{d}-1)((1+\tilde{f}_{d})^{-1}-f^{2}_{d})}{\xi^{2}_{d}f^{2}_{d}\mp[1+\bigl{(}\frac{\Delta_{0}}{2T}\bigr{)}(\xi^{2}_{d}f^{2}_{d}-1)((1+\tilde{f}_{d})^{-1}-f^{2}_{d})]};$
$\displaystyle\hat{\chi}_{xx}(T_{m}^{-})=\left\\{\begin{array}[]{r
l}\rightarrow\infty&(-):{\rm FM}\\\\[8.5359pt] \frac{1}{2I_{e}}&(+):{\rm
AFM}\end{array}\right.;$
$\displaystyle\hat{\chi}_{xx}(0)=\left\\{\begin{array}[]{r
l}\frac{1}{I_{e}}\frac{1}{\xi_{d}^{2}-1}&(-):{\rm FM}\\\\[8.5359pt]
\frac{1}{I_{e}}\frac{1}{\xi_{d}^{2}+1}&(+):{\rm AFM}\end{array}\right..$

Obviously the values for $T=0,T_{m}$ are equivalent to the SSM case but the
T-dependence in between is modified by the presence of the
$(1+\tilde{f}_{d})^{-1}$ terms due to the lower doublet state.

For the transverse (yy) component the Curie terms are absent since $J_{y}$ is
unchanged in the eigenvector basis and therefore has no diagonal matrix
elements. And then we get the simple result

$\displaystyle\hat{\chi}_{yy}(T)=\left\\{\begin{array}[]{r
l}\rightarrow\infty&(-):{\rm FM}\\\\[8.5359pt] \frac{1}{2I_{e}}&(+):{\rm
AFM}\end{array}\right..$ (37)

In the FM case the (yy) component diverges for all $T<T_{m}$ because there is
no in-plane exchange anisotropy and the ordered moment direction can be
rotated from $x$ to any other direction by arbitrary small field. This is
connected with the existence of a Goldstone mode for the whole ordered region
(Sec. IV.1). In the AFM phase the two sublattice moments tilt in addition to
their rotation ($\perp$ to the field) leading to the finite susceptibility
which is constant throughout the ordered phase as in conventional magnets. The
transverse $(yy)$ susceptibility is then simply a constant
$\hat{\chi}_{yy}=1/2I_{e}$ for the AF case. The comparison of susceptibilities
for all models is presented in Fig. 5 for various positions of the control
parameters with respect to the QCP. For a further discussion see Sec. V.

###  III.3 The cubic singlet-triplet model

In the STM model the three cubic axis are equivalent and for convenience we
choose $\langle J_{z}\rangle$ direction for the induced magnetic moment. The
corresponding molecular field Hamiltonian with symmetric shifted CEF
$\Gamma_{1},\Gamma_{4}$ levels at $(-\frac{\Delta}{2},\frac{\Delta}{2})$ may
be written as

$H_{\rm
MF}^{\lambda}=\sum_{i}[\frac{\Delta}{2}(P_{4}(i)-P_{1}(i))-h_{e}^{\lambda}J_{z}(i)].$
(38)

Here $P_{1,4}$ are projectors to $\Gamma_{1}$ singlet and $\Gamma_{4}$ triplet
subspaces spanned by $|\psi_{0}\rangle$ and $|\psi_{1,2,3}\rangle$ states,
respectively and $h_{e}^{\lambda}=I_{e}\langle J_{z}\rangle$ is the molecular
field. Using Eq. (3) the eigenvalues and -vectors of $H_{\rm MF}$ may be
obtained easily because only $|\psi_{0,2}\rangle$ are mixed by a nondiagonal
matrix element (Eq. (3)). We obtain the four level scheme totally split by the
molecular field:

$\epsilon_{\pm}=\pm\frac{\Delta}{2}(1+\gamma^{2}\langle
J_{z}\rangle^{2})^{\frac{1}{2}};\;\;\;\epsilon_{1,3}=\frac{1}{2}(\Delta\mp\delta_{T}),$
(39)

where $\gamma=m_{t}I_{e}/\Delta=\frac{2}{m_{t}}\xi_{t}$. The symmetric
normalised splittings of $\Gamma_{1}-\Gamma_{4}$ $|\psi_{0,2}\rangle$ and
$\Gamma_{4}$ $|\psi_{1,3}\rangle$ states and their temperature dependence are
then given by $\hat{\Delta}_{T}=(1+\gamma^{2}\langle
J_{z}\rangle^{2})^{\frac{1}{2}}$ and
$\hat{\delta}_{T}=\delta_{T}/\Delta=I_{e}\langle J_{z}\rangle/\Delta$,
respectively (Fig. 1(c)). Since only the $|\psi_{0,2}\rangle$ mix we have
again

$\displaystyle|\epsilon_{+}\rangle$
$\displaystyle=\cos\theta_{\lambda}|\psi_{2}\rangle-\sin\theta_{\lambda}|\psi_{0}\rangle,$
(40) $\displaystyle|\epsilon_{-}\rangle$
$\displaystyle=\sin\theta_{\lambda}|\psi_{2}\rangle+\cos\theta_{\lambda}|\psi_{0}\rangle,$

with $\tan 2\theta_{\lambda}=\gamma\langle J_{z}\rangle_{\lambda}$. The wave
functions of split $\epsilon_{1,3}$ triplet levels are unchanged. In the
ordered phase the moment operator is then given by

$\displaystyle\begin{aligned}
J_{z}=\frac{1}{2}\left(\begin{array}[]{cccc}-m_{t}\sin 2\theta&0&m_{t}\cos
2\theta&0\\\ 0&1&0&0\\\ m_{t}\cos 2\theta&0&m_{t}\sin 2\theta&0\\\ 0&0&0&-1\\\
\end{array}\right).\end{aligned}$ (41)

The order parameter is determined by the thermal trace of the diagonal
elements and all states will contribute to $\langle J_{z}\rangle$, but only
the elements $\sim\sin\theta$ are induced by the order itself, whereas the
constant diagonal elements are regular but thermally activated contributions.

####  III.3.1 Order parameter and transition temperature for STM

Using the diagonal elements of $J_{z}$ for the eigenstates and the split level
energies the MF equation for the order parameter $\langle J_{z}\rangle$ is
given by

$\displaystyle\langle J_{z}\rangle=\frac{\xi_{t}}{\hat{\Delta}_{T}}\langle
J_{z}\rangle P_{a}(T)+P_{b}(T)$ (42)

with the contributions resulting from the induced moment (a) and splitting of
the excited triplet (b), respectively. Here $P_{a},P_{b}$ the differences of
the thermal level populations $p_{i}(T)=Z^{-1}\exp(-\epsilon_{i}/T)$
$(i=\pm,1,3)$ with the partition function given by

$\displaystyle Z=$ $\displaystyle
2\bigl{[}\cosh(\frac{\epsilon}{2T})+\exp(-\frac{\Delta}{2T})\cosh(\frac{\delta_{s}}{2T})\bigr{]}$
(43) $\displaystyle\rightarrow
Z_{0}=2(2\cosh\frac{\Delta}{2T}-\sinh\frac{\Delta}{2T}).$

Then we obtain, with the expressions to the right of the arrows corresponding
to the paramagnetic state ($\hat{\delta}_{T}=0,\Delta_{T}=\Delta$):

$\displaystyle P_{a}(T)=$ $\displaystyle
p_{-}-p_{+}=\frac{\tanh\frac{\Delta_{T}}{2T}}{1+g(T)}\rightarrow\frac{\tanh\frac{\Delta}{2T}}{2-\tanh\frac{\Delta}{2T}},$
(44) $\displaystyle P_{b}(T)=$ $\displaystyle
p_{1}-p_{3}=\frac{g(T)\tanh\frac{\delta_{T}}{2T}}{1+g(T)}\rightarrow 0,$
$\displaystyle g(T)=$
$\displaystyle\exp(-\frac{\Delta}{2T})\frac{\cosh\frac{\delta_{T}}{2T}}{\cosh\frac{\Delta_{T}}{2T}}\rightarrow
1-\tanh(\frac{\Delta}{2T}).$

The latter encodes the influence of occupation and splitting of the two
$\epsilon_{1,3}$ triplet levels that have no matrix elements with the singlet
ground state but contribute to the moment formation for finite temperature
below $T_{m}$.

For the numerical determination of the induced moment we write, similar as for
SSM and SDM $\hat{\Delta}_{T}=\xi_{t}f_{t}(T)$. From Eq. (42) we obtain the MF
equation for $f_{t}(T)$ as

$\displaystyle f_{t}(T)=$
$\displaystyle\frac{\tanh[\bigl{(}\frac{\Delta_{0}}{2T}\bigr{)}f_{t}(T)]}{1+\tilde{f}_{t}(T)};$
(45) $\displaystyle\tilde{f}_{t}(T)=$ $\displaystyle
g(T)\bigl{[}1-\frac{2\xi_{t}}{m_{t}^{2}}\frac{1}{\hat{\delta}_{T}}\tanh(\frac{\Delta}{2T}\hat{\delta}_{T})\bigr{]}$

with $\hat{\delta}_{T}=(1/m_{t})(\hat{\Delta}_{T}^{2}-1)^{\frac{1}{2}}$. The
solution for $f_{t}(T)$ leads to $\hat{\Delta}_{T}$ and likewise to the
induced moment $\langle
J_{z}\rangle=\frac{m_{t}}{2}\frac{1}{\xi_{t}}(\hat{\Delta}_{T}^{2}-1)^{\frac{1}{2}}$
with saturation value $\langle
J_{z}\rangle_{0}=\frac{m_{t}}{2}\frac{1}{\xi_{t}}(\xi_{t}^{2}-1)^{\frac{1}{2}}$.
The latter is similar to SSM and SDM because at $T=0$ the $\Gamma_{4}$
contribution in the STM vanishes.

Figure 5:  (a) Temperature dependence of static homogeneous susceptibilities
(unit $1/I_{e}$) for different models and control parameters for the AF case
with $\langle J_{x}\rangle$ (SDM) or $\langle J_{z}\rangle$ (SSM,STM) moment
directions. In the xy-type SDM the transverse (yy) susceptibility is constant
below $T_{m}$ as in conventional degenerate ground state magnets. In contrast
the longitudinal susceptibilities at $T=0$ are generally finite, evolving from
the same as transverse value at the QCP to progressively lower values with
increasing $\xi$. (b) Control-parameter dependence of $T=0$ longitudinal
susceptibility as function of $\xi$, identical for the three models. It
vanishes asymptotically when approaching the quasi-degenerate case.

Letting $\langle J_{z}\rangle\rightarrow 0$ in Eq. (42) (note that then also
$P_{b}\rightarrow 0$) we arrive at an implicit equation for the transition
temperature of induced order given by the solution of

$T_{m}=\frac{\Delta}{2\tanh^{-1}(x_{m})};\;\;\;x_{m}=\frac{2-\bigl{(}\frac{I_{e}}{\Delta}\bigr{)}\tanh^{-1}(x_{m})}{1+\xi_{t}-\bigl{(}\frac{I_{e}}{\Delta}\bigr{)}\tanh^{-1}(x_{m})}.$
(46)

If we neglect the contribution of the thermally excited $\epsilon_{1,3}$
levels justified for
$\frac{I_{e}}{\Delta}=\frac{\gamma}{m_{t}}=\frac{2\xi_{t}}{m_{t}^{2}}=0.075\xi_{t}\ll
1$ then we arrive at the simple expression

$\displaystyle T_{m}^{0}=\frac{\Delta}{2\tanh^{-1}\frac{2}{1+\xi_{t}}},$ (47)

which is formally similar to Eqs. (13,32) of the SSM and SDM case and again
$\xi_{t}>\xi_{t}^{c}=1$ for $T^{0}_{m}>0$. We will show later that this zeroth
order approximation of transition temperature is identical to the temperature
where the soft longitudinal (polarisation parallel to $\langle J_{z}\rangle$)
exciton mode appears, However, the true transition temperature obtained from
the Eq. (46) is larger than $T_{m}^{0}$ due to the thermally excited Curie
contributions from $|\psi_{1,3}\rangle$ triplet states which stabilise the
moment $\langle J_{z}\rangle$ beyond $T_{m}^{0}$. This means that at the true
$T_{m}>T_{m}^{0}$ the complete softening of the exciton mode will not occur
but rather it will be arrested at a finite frequency (Sec. IV.3). This fact
for the STM was known for some time [28, 12, 6] but has not been considered in
detail sofar. We first calculate the approximate shift defined by
$T_{m}=T_{m}^{0}+\delta T_{m}$. Using the first iteration step of Eq. (46) and
expanding in the small parameter $(I_{e}/\Delta)=2\xi_{t}/m_{t}^{2}$ as given
above we obtain

$\delta T_{m}=\frac{1}{m_{t}^{2}}\frac{2\xi_{t}}{3+\xi_{t}}T_{m}^{0}.$ (48)

This is the upward shift of the transition temperature due to the effect of
diagonal matrix elements in Eq. (41). Their relative size is
$\frac{1}{2}:\frac{m_{t}}{2}=\frac{1}{m_{t}}$ which controls the size of the
correction $\delta T_{m}$. It is a very good approximation to the numerically
determined exact shift obtained from Eq. (46) shown in Fig. 2(a).

####  III.3.2 Internal energy and specific heat for STM

The internal energy of the STM in the paramagnetic phase is
$U(T)=-\frac{\Delta}{2}/(2\tanh\frac{\Delta}{2T}-1)/(2-\tanh\frac{\Delta}{2T})$
leading to a paramagnetic specific heat (corresponding to $N=3$ in Appendix
B):

$C_{V}(T)=\frac{3\bigl{(}\frac{\Delta}{2T}\bigr{)}^{2}}{[2\cosh\bigl{(}\frac{\Delta}{2T}\bigr{)}-\sinh\bigl{(}\frac{\Delta}{2T}\bigr{)}]^{2}}.$
(49)

In the ordered phase the internal energy can be calculated as previously using
a similar expression as Eq. (16) summing over all four states of the STM and
adding the MF constant term we obtain

$\displaystyle U(T)$
$\displaystyle=-\frac{\Delta}{2(1+g)}\Big{\\{}\hat{\Delta}_{T}\tanh\big{[}\bigl{(}\frac{\Delta}{2T}\bigr{)}\hat{\Delta}_{T}\big{]}$
(50)
$\displaystyle+g(T)\Big{[}\hat{\delta}_{T}\tanh\big{[}\bigl{(}\frac{\Delta}{2T}\bigr{)}\hat{\delta}_{T}\big{]}-1\Big{]}\Big{\\}}+\frac{\Delta}{4}\frac{1}{\xi_{t}}(\hat{\Delta}_{T}^{2}-1).$

For zero temperature this leads to the ground state energy in the universal
form $U(0)=-\frac{\Delta}{4}(\xi_{t}+\frac{1}{\xi_{t}})$ and the specific heat
$C_{V}(T)$ has to be obtained from $U(T)$ by numerical differentiation as in
the SDM case.

####  III.3.3 The static homogeneous longitudinal STM susceptibility

The static susceptibility may be calculated as before from Eq. (21). For the
induced moment phase $(T<T_{m})$ we obtain $(m^{\prime}_{t}=1)$:

$\displaystyle\hat{\chi}^{0}_{zz}(T)=[1+g(T)]^{-1}\Bigl{\\{}\frac{1}{I_{e}}\bigl{(}\frac{\xi_{t}}{\hat{\Delta}_{T}^{3}}\bigr{)}\tanh(\frac{\Delta}{2T}\hat{\Delta}_{T})$
(51)
$\displaystyle+\frac{1}{T}\Big{[}\bigl{(}\frac{m^{\prime}_{t}}{2}\bigr{)}^{2}g(T)+\bigl{(}\frac{m_{t}}{2}\bigr{)}^{2}(\hat{\Delta}^{2}_{T}-1)\bigl{(}\frac{1}{\hat{\Delta}_{T}^{2}}-\frac{1+g(T)}{\xi_{t}^{2}}\bigr{)}\Big{]}\Bigr{\\}}.$

Note that at zero temperature with $\hat{\Delta}_{0}=\xi_{t}$ and $g(0)=0$
only the first van Vleck term survives whereas the Curie contributions from
thermally excited split triplet states vanishes. In the paramagnetic regime
$(T>T_{m})$ with $\hat{\Delta}_{T}=1$ this reduces to the explicit expression

$\hat{\chi}^{0}_{zz}(T)=\bigl{(}\frac{\xi_{t}}{I_{e}}\bigr{)}\frac{\tanh(\frac{\Delta}{2T})}{2-\tanh(\frac{\Delta}{2T})}+\frac{1}{T}\bigl{(}\frac{m^{\prime}_{t}}{2}\bigr{)}^{2}\Bigl{[}\frac{1-\tanh(\frac{\Delta}{2T})}{2-\tanh(\frac{\Delta}{2T})}\Bigr{]}.$
(52)

Finally the homogeneous longitudinal MF-RPA susceptibility is again obtained
from $\hat{\chi}_{zz}(T)=\hat{\chi}^{0}_{zz}(T)/(1\mp\hat{\chi}^{0}_{zz}(T))$
with upper and lower signs corresponding to FM or AF exchange, respectively.
In the $T=0$ limit this recovers $\hat{\chi}_{zz}(0)=(1/I_{e})(\xi_{t}^{2}\mp
1)^{-1}$ which is identical to the longitudinal susceptibility in both SSM and
STM because in this limit there is no contribution from depopulated excited
states. In the large $\xi_{t}$ (quasi-degenerate) case that approaches the
conventional magnet the longitudinal susceptibility vanishes because the
saturation moment approaches the maximum value and no further polarisation by
external field is possible (Fig. 5), see also Sec. V.

##  IV Magnetic exciton dispersions and soft mode behaviour

In the two-level SSM, SDM and STM the dynamics is described by collective
modes termed ’magnetic excitons’. They correspond to propagating CEF
excitations which develop a dispersion due to effective intersite exchange.
Their dispersion is strongly temperature dependent controlled by the thermal
population difference of singlet ground state and excited multiplet. Within
RPA approach they may exhibit, as a precursor phenomenon, a complete softening
at the magnetic ordering wave vector when $T_{m}$ is approached from above.
Such magnetic exciton softening to varying degree has been found in quite a
number of singlet ground state CEF systems, in particular Pr compounds [6,
29]. It is frequently incomplete because the static susceptibility
contributions form higher lying CEF states lead to a magnetic transition
already before the softening of the lowest mode is achieved. Furthermore
dynamical effects beyond the RPA complicate this simple picture [10]. The
magnetic exciton formation has mostly been studied in the paramagnetic phase.
Here we give a complete theory for the three models also in the induced moment
regime. We focus on a representation that highlights the role of the control
parameters that measure the distance from the QCP and emphasise the connection
to thermodynamic properties.

The magnetic exciton modes may be calculated by the RPA dynamic response
function technique [30, 10, 17] or with the bosonic Bogoliubov transformation
approach [31, 12, 32, 24]. Here we prefer the former because it gives a more
reliable description of dispersions in the whole temperature range. The
starting point is the dynamical single-ion susceptibility tensor
($\alpha,\beta=x,y$) defined by

$\hat{\chi}_{0\alpha\beta}(i\omega_{n})=\sum_{n\neq m}\frac{\langle
m|J_{\alpha}|n\rangle\langle
n|J_{\beta}|m\rangle}{\epsilon_{m}-\epsilon_{n}-i\omega_{n}}(p_{n}-p_{m}),$
(53)

and the RPA cartesian susceptibility tensor is then obtained as

$\tensor{\hat{\chi}}({\bf q},i\omega_{n})=\bigl{[}1-I({\bf
q})\tensor{\hat{\chi}_{0}}(i\omega_{n})\bigr{]}^{-1}\tensor{\hat{\chi}_{0}}(i\omega_{n}),$
(54)

The poles of this response function determine the temperature-dependent
magnetic exciton modes. This formulation is valid both in the paramagnetic and
magnetic regimes provided the proper split level energies and matrix elements
for the three models are used. In the magnetic phase we will restrict to FM
case $(I_{e}>0)$ because for two - sublattice AFM order the dimension of the
susceptibility matrix is doubled to four, making it more involved for
analytical treatment. The AFM case has been considered before for the Ising-
type SSM [17]. For the models considered the susceptibility tensor will be
diagonal even in the ordered phase, so that the inversion in the above
equation poses no problem.

Figure 6:  (a) Magnetic exciton modes for FM case along simple tetragonal BZ
path R(001), A(111), $\Gamma$(000), M(110) for SSM at three different
temperatures. A softening at the FM $\Gamma$ -point occurs at $T_{m}$ with a
subsequent re-hardening below $T_{m}$. The normalised intensity $\hat{R}({\bf
q})$ is represented by the dash-dotted lines, showing the diverging intensity
(in units $(\Delta/I_{e})$) of the soft mode.

###  IV.1 Magnetic excitons in the Ising- type SSM

Firstly we consider the SSM case, the calculation of the zz component of the
single ion dynamical susceptibility according to Eq. (53) leads to

$\displaystyle\hat{\chi}^{0}_{zz}(i\omega_{n})$
$\displaystyle=\bigl{(}\frac{m_{s}}{2}\bigr{)}^{2}\cos^{2}2\theta\tanh\bigl{(}\frac{\Delta_{T}}{2T}\bigr{)}\frac{2\Delta_{T}}{\Delta_{T}^{2}-i\omega_{n}^{2}},$
(55)

where $\Delta_{T}=\Delta\xi_{s}f_{s}(T)=\Delta\hat{\Delta}_{T}$ is the
renormalised CEF splitting and $\cos 2\theta=1/[\xi_{s}f_{s}(T)]$. As
described in Sec. II for simplicity we use the intersite exchange
$\hat{I}({\bf q})$ for the simple tetragonal lattice with lattice constants
$a,c$ set to unity and the corresponding real-space exchange anisotropy
defined by $\kappa$ (Eq. (5)). However the theory may be applied to any other
uniaxial Bravais lattice by using the corresponding exchange function
$\hat{I}({\bf q})$. With the above expression for
$\hat{\chi}^{0}_{zz}(i\omega_{n})$ the poles of the collective susceptibility
$\hat{\chi}_{zz}(i\omega_{n})$, i.e. the magnetic exciton dispersion in the
induced moment phase may be found as

$\displaystyle\omega({\bf q},T)\\!=$
$\displaystyle\Delta_{T}\Big{[}1-\frac{1}{\xi_{s}^{2}f_{s}(T)^{2}}\hat{I}({\bf
q})\Big{]}^{\frac{1}{2}}.$ (56)

For $T>T_{m}$ the singlet mixing angle $\theta$ vanishes and we obtain the
well known SSM paramagnetic exciton dispersion

$\displaystyle\omega({\bf q},T)$
$\displaystyle=\Delta\bigl{[}1-\xi_{s}f_{s}^{0}(T)\hat{I}({\bf
q})\bigr{]}^{\frac{1}{2}},$ (57)

where $\hat{I}({\bf q})=I({\bf q})/I(0)$ or $\hat{I}({\bf q})=I({\bf
q})/I({\bf q}_{m})$ is the exchange Fourier transform normalised to the
maximum value for FM $({\bf q}_{m}=0)$ or AFM $({\bf q}_{m}=(111))$ wave
vectors, respectively. We note that the SSM exciton mode at the FM or AFM wave
vector for $I_{0}>0$ or $I_{0}<0$ decreases continuously on approaching
$T_{m}$ as a precursor of the ordering and becomes soft at the transition
(Fig. 6) according to $\omega({\bf q},T_{m})=\Delta[1-\hat{I}({\bf
q})]^{\frac{1}{2}}$. On the other hand at zero temperature the mode dispersion
in Eq. (56) reduces to the simple form $(\Delta_{0}=\xi_{s}\Delta)$

$\omega({\bf q},0)=\Delta_{0}\bigl{[}1-\frac{1}{\xi_{s}^{2}}\hat{I}({\bf
q})]^{\frac{1}{2}}.$ (58)

Therefore at the ordering wave vector with $\hat{I}({\bf q}_{m})=1$ the
paramagnetic exciton mode becomes soft at $T_{m}$ and re-hardens below it
since $\xi_{s}>1$. These dispersions may be identified in inelastic neutron
scattering (INS) experiments. The intensity $R({\bf q},T)$ of a dispersive
exciton mode is not constant but varies with momentum and temperature (without
the Bose prefactor) according to

$\displaystyle R({\bf q},,\omega,T)$ $\displaystyle=\frac{1}{\pi}{\rm
Im}\hat{\chi}_{zz}({\bf q},i\omega_{n})=\hat{R}({\bf
q},T)\delta(\omega-\omega_{\bf q}),$ (59)

with the weight function given in the ordered phase by $(f_{s}=f_{s}(T))$:

$\displaystyle\hat{R}({\bf q},T<T_{m})=\frac{\Delta^{2}}{2I_{e}\omega({\bf
q},T)}=\frac{1}{2}\bigl{(}\frac{\Delta}{I_{e}}\bigr{)}[\xi_{s}f_{s}-\hat{I}({\bf
q})]^{-\frac{1}{2}}.$ (60)

and in the paramagnetic phase we obtain

$\displaystyle\hat{R}({\bf
q},T>T_{m})=\frac{1}{2}\bigl{(}\frac{\Delta}{I_{e}}\bigr{)}(\xi_{s}f_{s}^{0})[1-(\xi_{s}f_{s}^{0})\hat{I}({\bf
q})]^{-\frac{1}{2}}.$ (61)

At the transition temperature with $\xi_{s}f_{s}=\xi_{s}f_{s}^{0}=1$ then
$\hat{R}({\bf
q},T_{m})=\frac{1}{2}\bigl{(}\frac{\Delta}{I_{e}}\bigl{)}[1-\hat{I}({\bf
q})]^{-\frac{1}{2}}$ which diverges at the soft mode wave vector ($\Gamma$
point) The behaviour of $\hat{R}({\bf q},T)$ in the magnetic phase showing the
singularity is plotted in Fig. 6 with a broken line.

Figure 7: SDM spectral density (Eq. (77)) of magnetic exciton branches in the
induced moment phase with $\xi_{d}=2.5$ for different temperatures
$T<T_{m}=0.91$. To enhance visibility $\ln R({\bf q},\omega)$ is plotted and a
broadening of $\eta=0.005$ is used. Three modes appear due to the excited
doublet state. The high-energy ones are due to singlet-doublet excitations
while the lower one corresponds to thermally excited transitions between the
doublet components. It exhibits hybridisation and anti-crossing with one of
the high-energy modes and its intensity is appreciable only in that region
(see also Fig. 8(a)). The soft mode actually originates from the anti-crossing
high energy part. For $T\rightarrow T^{-}_{m}$ the hybridisation gap is closed
and $\omega^{-}_{y}({\bf q})$ vanishes for all wave vectors due to the
degeneracy of the excited doublet components (see also Fig. 8(b)). For
$T>T_{m}$ (not shown) only one fully gapped (degenerate
$\omega_{x},\omega_{y}$) branch remains.

###  IV.2 Magnetic excitons in the xy-type SDM

The calculation of exciton dispersion in the induced moment state of this
extended model is more involved. Firstly there is also a CEF excitation from
the thermally populated $\epsilon_{2}$ level that has to be taken into account
in principle, although it becomes insignificant at low temperatures.
Furthermore for $T<T_{m}$ two excitation branches from the singlet ground
state with different energies occur. On the other hand $J_{x}$, $J_{y}$
operators in the eigenvector basis of the induced moment phase (Eq. (2)) have
matrix elements for mutually exclusive transitions so that again no mixed
nondiagonal dynamic susceptibilities occur. The diagonal elementw
$\hat{\chi}^{0}_{\alpha\alpha}$ are given by

$\displaystyle\hat{\chi}^{0}_{xx}(i\omega_{n})$
$\displaystyle=\frac{m_{d}^{2}m_{1}^{2}\Delta_{T}f_{d}(T)}{\Delta_{T}^{2}-(i\omega_{n})^{2}},$
(62) $\displaystyle\hat{\chi}^{0}_{yy}(i\omega_{n})$
$\displaystyle=\frac{m_{d}^{2}m_{2}^{2}\Delta_{T}f^{12}_{d}(T)}{(\Delta^{21}_{T})^{2}-(i\omega_{n})^{2}}+\frac{m_{d}^{2}m_{2}^{{}^{\prime}2}\Delta^{32}_{T}f^{23}_{d}(T)}{(\Delta^{32}_{T})^{2}-(i\omega_{n})^{2}},$

where $\Delta_{T}=\epsilon_{3}-\epsilon_{1}=(\xi_{d}\Delta)f_{d}$ and the
matrix elements $m_{1},m_{2},m_{2}^{\prime}$ are given in Eq. (29). The
additional transition energies and their associated occupation differences for
the $yy$ case are defined by
$\Delta_{T}^{21}=\epsilon_{2}-\epsilon_{1}=\frac{1}{2}(\Delta_{T}+\Delta)$,
$\Delta_{T}^{32}=\epsilon_{3}-\epsilon_{2}=\frac{1}{2}(\Delta_{T}-\Delta)$ and
similar $f_{d}^{12}(T)=p_{1}-p_{2}$, $f_{d}^{23}(T)=p_{2}-p_{3}$. They fulfil
the constraints $\Delta_{T}^{21}+\Delta_{T}^{32}=\Delta_{T}$ and
$f_{d}^{12}+f_{d}^{23}=p_{1}-p_{3}=f_{d}(T)$. The diagonal RPA susceptibility
matrix elements are then simply given by $(\alpha=x,y)$

$\hat{\chi}_{\alpha\alpha}({\bf
q},i\omega_{n})=\frac{\hat{\chi}^{0}_{\alpha\alpha}({\bf
q},i\omega_{n})}{1-I({\bf q})\hat{\chi}^{0}_{\alpha\alpha}({\bf
q},i\omega_{n})}$ (63)

Their poles lead to three magnetic exciton branches, one for $xx$ and two for
$yy$ polarisation. We obtain

$\displaystyle xx:\;\;\;\omega_{x}({\bf q},T)=\left\\{\begin{array}[]{l
l}\Delta_{T}[1-\frac{1}{(\xi_{d}f_{d})^{2}}\hat{I}({\bf
q})]^{\frac{1}{2}};\;\;\;T<T_{m}\\\\[8.5359pt]
\Delta[1-\xi_{d}f_{d}\hat{I}({\bf
q})]^{\frac{1}{2}};\;\;\;T>T_{m}\end{array}\right..$ (64)

Figure 8:  (a) Hybridisation behaviour between transverse
$\omega_{y}^{\pm}({\bf q})$ modes as function of temperature in the induced
moment phase. Here $q_{cr}=|{\bf q}_{cr}|$ (in units of $\pi,(a,c=1)$) is the
wave vector of mode crossing along (111) with respect to $\Gamma$ point and
$\Delta_{h}$ the hybridisation gap that opens at the crossing point, see also
Fig. 7. The hybridisation gap is non-monotonic and achieves its maximum for an
intermediate temperature. It vanishes for low T due to thermal depopulation of
doublets and for $T_{m}$ where the modes themselves become soft and their gap
also has to close. Dashed lines are guides to the eye. (b) Soft mode $({\bf
q}=0)$ evolution with temperature from paramagnetic to ordered state
$(\xi_{d}=1.5)$. Two of the modes re-harden below $T_{m}=0.51$ and the
remaining one stays at zero energy.

This result corresponds formally to the Ising-type SSM (Eqs. (56,57)). because
the $J_{x}$ dipolar operator in Eq. (2) connects only to the $\epsilon_{3}$
level (which furthermore has no matrix element for $J_{y}$ so that no mixed
response function appears). Therefore it is formally equivalent to an Ising
type singlet singlet model (for $J_{x}$), the only instance where the doublet
nature of the excited state enters is through the population difference factor
$f_{d}(T)$, but not in the dynamics.

On the other hand for the $yy$ response function two more modes appear
originating from ground state to $\epsilon_{2}$ excited level and from this to
the upper level $\epsilon_{3}$. The latter is a thermally activated mode which
looses its intensity for low temperature. We obtain:

$\displaystyle yy:\;\;\;\omega^{y2}_{\pm}({\bf q},T)$
$\displaystyle=\frac{1}{2}B({\bf q},T)\pm[\frac{1}{4}B({\bf q},T)^{2}-C({\bf
q},T)]^{\frac{1}{2}};$ (65) $\displaystyle C({\bf q},T)$
$\displaystyle=\bigl{(}\frac{\Delta}{2}\bigr{)}^{4}(\hat{\Delta}_{T}^{2}-1)^{2}[1-\hat{I}({\bf
q})];$ $\displaystyle B({\bf q},T)$
$\displaystyle=\bigl{(}\frac{\Delta}{2}\bigr{)}^{2}\Bigl{\\{}2(1+\hat{\Delta}_{T}^{2})$
$\displaystyle-\bigl{(}\frac{\xi_{d}}{\hat{\Delta}_{T}}\bigr{)}\bigl{[}(\hat{\Delta}_{T}+1)^{2}f_{d}^{12}+(\hat{\Delta}_{T}-1)^{2}f_{d}^{23}\bigr{]}\hat{I}({\bf
q})\Bigr{\\}},$

where $\pm$ correspond to the high energy mode and thermally excited low
energy mode with maximum or vanishing intensity at zero temperature,
respectively.

We now discuss a few important special cases. At $T=T_{m}$ with
$\hat{\Delta}=1$ this reduces to $C({\bf q},T)=0$ and
$f_{d}^{23}=0,f^{12}_{d}=f_{D}=1/\xi_{d}$. Then we simply get

$\omega_{\pm}^{y}({\bf q},T=T_{m})=\left\\{\begin{array}[]{l
l}\Delta[1-\hat{I}({\bf q})]^{\frac{1}{2}}\equiv\omega_{x}({\bf
q},T_{m})\\\\[8.5359pt] 0\end{array}\right..$ (66)

Therefore in the paramagnetic case the $x$ and $y+$ modes are degenerate and
become soft at $T_{m}$ (Fig. 8(b)). The $y-$ mode corresponding to the
transitions between degenerate doublet states is a thermally activated zero
energy mode for all q (quasielastic mode in reality). At zero temperature for
saturated order $(\hat{\Delta}_{T}=\xi_{d},f_{d}=f_{d}^{12}=1,f_{d}^{23}=0)$
the doublet is split and both $y$-modes have nonzero energy given by

$\omega^{y}_{\pm}({\bf q},T=0)=\left\\{\begin{array}[]{l
l}\frac{\Delta}{2}(\xi_{d}-1)\\\\[8.5359pt]
\frac{\Delta}{2}(\xi_{d}+1)[1-\hat{I}({\bf
q})]^{\frac{1}{2}}\end{array}\right..$ (67)

This holds for $|{\bf q}|<|{\bf q}_{cr}|$, for larger $|{\bf q}|$ the $\pm$
mode labels are interchanged. Here $|{\bf
q}_{cr}|=\sqrt{3}\cos^{-1}(4\xi_{d}/(\xi_{d}+1)^{2})$ is the wave vector where
the two modes are crossing for $T=0$ along $(111)$ or $\Gamma A$ direction and
hybridizing for finite temperature (see Fig. 7(a)). At the ordering wave
vector ${\bf q}_{m}=0$ Eq. (67) reduces to $\omega^{y}_{-}(0,T=0)=0$ and
$\omega^{y}_{+}(0,T=0)=\frac{\Delta}{2}(\xi_{d}-1)$. The above equations
demonstrate that $\omega^{y}_{-}({\bf q}=0,T)$ vanishes at both $T=0,T_{m}$,
in fact it is a Goldstone soft mode within the whole ordered regime $T\leq
T_{m}$ since $\hat{I}({\bf q}_{m})=1$ in Eq. (65). This is due to the fact
that $(J_{x},J_{y})$ moments have continuous rotation symmetry in the SDM,
whereas in the Ising-type SSM without such continuous symmetry the soft mode
re-hardens immediately below $T_{m}$. The second transverse mode, however
shows stiffening for $T<T_{m}$ described by

$\displaystyle\omega_{+}({\bf q}=0,T)\\!=$ (68)
$\displaystyle\;\;\frac{\Delta}{2}\Bigl{\\{}2(1+\hat{\Delta}_{T}^{2})-\\!\frac{\xi_{d}}{\hat{\Delta}_{T}}\bigl{[}4\hat{\Delta}_{T}f_{d}^{12}(T)\\!+\\!(\hat{\Delta}_{T}-1)^{2}f_{d}(T)]\Bigl{\\}}^{\frac{1}{2}}.$

This frequency starts at zero for $T_{m}$ and below increases to the maximum
value $\frac{\Delta}{2}(\xi_{d}-1)$ at $T=0$ (Eq. (67)). The dispersion and
temperature behaviour of the exciton modes in the SDM model are presented in
Figs. 7 as spectral plots and will be further discussed in the following
section.

Figure 9: (a) STM exciton dispersion of longitudinal mode, At the $\Gamma-$
point first a softening (black arrow) is observed but is arrested at $T_{m}$
at finite energy; followed by a re-hardening below $T_{m}$ (red arrow). (b)
Arrested soft mode (${\bf q}=0$) temperature evolution. The extrapolated
paramagnetic mode touches zero at $T^{0}_{m}/T_{m}=0.975$ $(T_{m}=0.467)$
where $T_{m}^{0}$ is the approximate ordering temperature in (Eq. (47)) by
neglecting the excited triplet Curie terms.

###  IV.3 Magnetic excitons in the cubic STM

The dynamical longitudinal single ion susceptibility $\chi^{0}_{zz}({\bf
q},i\omega_{n})$ has only contributions from the transition between
$\epsilon_{\pm}$ states but no others, even in the magnetic phase as can be
seen from the $J_{z}$ matrix in Eq. (41). Therefore it is similar in structure
as in the SSM case with

$\chi^{0}_{zz}({\bf
q},i\omega_{n})=\frac{\xi_{t}}{I_{e}}\frac{1}{\hat{\Delta}_{T}}\frac{\Delta^{2}}{(\Delta_{T}^{2}-i\omega_{n}^{2})}P_{a}(T).$
(69)

Using this expression and inserting into Eq. (54) the resulting pole give the
longitudinal exciton mode dispersions $(\hat{\Delta}_{T}=\xi_{t}f_{t}(T))$:

$\displaystyle\omega({\bf q},T)=\left\\{\begin{array}[]{r
l}\Delta_{T}[1-\frac{\xi_{t}}{\hat{\Delta}^{3}_{T}}P_{a}(T)\hat{I}({\bf
q})]^{\frac{1}{2}};\;\;\;T<T_{m}\\\ \Delta[1-\xi_{t}P^{0}_{a}(T)\hat{I}({\bf
q})]^{\frac{1}{2}};\;\;\;T>T_{m}\end{array}\right..$ (70)

Here again $\hat{I}({\bf q})=I({\bf q})/I_{e}$ is the exchange normalised to
that at the ordering vector ${\bf q}_{m}$ for FM $({\bf q}_{m}=0)$ or AFM
$({\bf q}_{m}=(\pi,\pi,\pi)$ for which then $\hat{I}({\bf q}_{m})=1$. Using
the lower paramagnetic expression and $P_{a}^{0}(T)$ from Eq. (44) we see that
at the ordering vector the mode energy becomes soft with $\omega({\bf
q}_{m},T_{m}^{0})=0$ at the approximate transition temperature $T_{m}^{0}$ of
Eq. (47) which is lower than the real $T_{m}$ obtained from Eqs. (46). In
reverse this means that the mode energy will be arrested at a finite value at
$T_{m}>T_{m}^{0}$ and the softening is incomplete. Using the first iterative
approximation for $T_{m}$ from Eq. (46) this finite value is given by
approximately by

$\displaystyle\omega^{2}({\bf
q}_{m},T_{m})=\bigl{(}\frac{\Delta^{2}}{2m_{t}^{2}}\bigr{)}(\xi_{t}^{2}-1)\tanh^{-1}\bigl{(}\frac{2}{1+\xi_{t}}\bigr{)}$
(71) $\displaystyle\rightarrow\omega({\bf
q}_{m},T_{m})\simeq\bigl{(}\frac{\Delta}{m_{t}}\bigr{)}\delta^{\frac{1}{2}}|\ln\frac{\delta}{2}|^{\frac{1}{2}}$

where the second expression is the asymptotic form close to the QCP
$(\xi_{t}=1+\delta;\;\delta\ll 1)$. Therefore $\omega({\bf q}_{m},T_{m})$ is
directly proportional to the ratio $m^{\prime}_{t}:m_{t}=1/m_{t}$ of diagonal
(responsible for the shift of $T_{m}$) and non-diagonal (responsible for
induced order) matrix elements in the $J_{z}$ matrix of Eq. (3). In the $J=4$
STM used here it is $1/m_{t}=0.145\ll 1$ and therefore $\omega({\bf
q},T_{m})\ll\Delta$ for a reasonably sized $\xi_{t}$. But for a general
$\Gamma_{1}-\Gamma_{4}$ reduced level scheme $m^{\prime}_{t}/m_{t}$ depends on
the CEF potential parameters and may vary. Below $T_{m}$ the mode energy
increases sharply again due to the effect of the molecular field. The
dependence of the arrested soft mode frequency on this ratio of diagonal to
non-diagonal matrix elements and on the control parameter $\xi_{t}$ is shown
in the two panels of Fig. 10, respectively. We note that in the ordered phase
the components $J_{x},J_{y}$ transverse to the chosen moment direction
$\langle J_{z}\rangle$ have transitions between the excited triplet states.
Therefore, as in the SDM case the transverse exciton modes for the STM would
consist of several branches hybridising with each other, similar as in Fig. 7.

##  V Discussion of numerical results

In the following discussion we will focus on the most typical results for the
three models that show the characteristic distinction of induced moment
magnetism as compared to common quasi-classical magnetic order. Therefore we
will not present and discuss figures for all physical quantities for all three
models that can be obtained from the previous analysis.

As a prerequisite we depict the CEF level splittings in Fig. 1 for the three
singlet ground state models caused by the appearance of the T- dependent
induced order parameter $\langle J_{x}\rangle_{T}$ or $\langle
J_{z}\rangle_{T}$ below $T_{m}$. For SSM a symmetric repulsion of the two
singlets due to their mixing is observed such that the splitting increases to
$\Delta_{0}=\xi_{s}\Delta$ at $T=0$. For the SDM only the symmetric
combination of the excited doublet states mix with the ground state singlet
and show the similar repulsion to an increased $T=0$ splitting of
$\Delta_{0}=\xi_{d}\Delta$. The second antisymmetric doublet combination
remains isolated at the paramagnetic $\Delta$, nevertheless it influences the
thermodynamic properties by its thermal population. In the STM the repelling
singlet and triplet component behave similar as in STM again with a
$\Delta_{0}=\xi_{t}\Delta$. The other two states do not mix with the ground
state singlet but split due to their diagonal matrix element
$(m^{\prime}_{t}/2)$ with a splitting energy directly proportional to the
order parameter $\langle J_{z}\rangle$, therefore it starts with an infinite
slope at $T_{m}$ while that of the singlet ground state and upper triplet
component begin with a finite slope.

Figure 10: Arrested soft mode frequency as function of ratio of diagonal
$\Gamma_{4}$ $(m^{\prime}_{t}=1)$ to nondiagonal $\Gamma_{1}-\Gamma_{4}$
$(m_{t})$ matrix elements (a) for $\xi_{t}=1.5$ and as function of control
parameter (b) for J=4 ratio $m^{\prime}_{t}/m_{t}=0.19$. Diamonds: exact
numerical results from Eq. (70). Dashed lines: approximate result from Eq.
(71).

A central characteristic of induced moment magnetism is the fact that to
achieve a finite transition temperature the control parameter must exceed the
critical value $\xi_{c}=1$. This defines the QCP separating paramagnetism
$(\xi<\xi_{c})$ from induced quantum magnetism $(\xi>\xi_{c})$. The dependence
of $T_{m}$ on the control parameter is shown in Fig. 2(a) for the three models
as obtained from Eqs. (13,32,46). For STM we show the exact numerical solution
of $T_{m}$ from solving Eq. (46) (full line) as well as the zeroth order
approximation $T_{m}^{0}$ (broken line) given by Eq. (47). Their difference is
quite small because the ratio of elastic (triplet) to inelastic (singlet-
triplet) matrix elements is $m^{\prime}_{t}/m_{t}=0.19\ll 1$ and the
difference is well described by the first iterative improvement in Eq. (46).
Despite its smallness it plays an essential role in the arrested soft-mode
behaviour of the STM exciton modes as discussed below. For $\xi\rightarrow
1^{+}$ there is a logarithmic singularity of the slope of $T_{m}(\xi)$ and for
large $\xi$ the dependence on $\xi$ becomes linear (see Eq. (13)). The
logarithmic singularity has a practical consequence for identifying induced
moment magnetism. It shows that for achieving $T_{m}<0.2\Delta$ one must fine
tune $\xi$ very close to the QCP. Therefore if this inequality is observed it
is a priori unlikely that the mechanism of induced order can be invoked for a
given compound. Such interpretation holds also for nonmagnetic (multipolar)
induced order. In this case one should consider other mechanisms, e.g. based
on hybridisation and itineracy of f-electrons for the observed order.

For the same control parameter $\xi$ the $T_{m}$ of the three models are
quantitatively different. However, if we scale temperature with each
individual $T_{m}$ it turns out that the temperature dependence of their order
parameters is quite similar as shown in Fig. 2(b). The panel (b) depicts the
corresponding functions $f_{s,d,t}(T)$ which are the difference of thermal
populations between the states connected by the nondiagonal $m_{s,d,t}$ matrix
elements.These functions depend on $\xi_{s,d,t}$ and are central for the
calculation of most physical quantities. They are obtained from solving
numerically their selfconsistency equations. Below $T_{m}$ they may also be
interpreted as the T-dependent normalised energy splitting $\hat{\Delta}_{T}$
for the $m_{s,d,t}$\- connected levels below $T_{m}$.

A most characteristic feature of induced moment magnetism is the behaviour of
the specific heat $C_{V}(T)$ and in particular its jump at $T_{m}$. It is
presented in Fig. 3(a) for SSM for three different control parameters together
with the internal energy $U(T)$ and in complementary Fig. 3(b) for the three
models with the same $\xi$. The contribution from the order parameter
T-derivative that leads to the jump (Eq. (19)) is seen to be superimposed on
the Schottky type background specific head $C_{V}^{0}$ (Eqs. (18,34,49)). This
means on lowering temperature the entropy is released by the induced moment
transition as well as the single ion depopulation effect of the excited CEF
level. The jumps increase with the degeneracy of the excited CEF level while
the corresponding $T_{m}$ decreases (Fig. 3(b)).

On approaching the QCP from a above while $T_{m}$ shifts to lower value the
specific heat jump moves with it and becomes progressively smaller. This
dependence is also seen directly in Fig. 4 as continuous drop in $\delta
C_{V}$ close to the QCP with $\xi=1+\delta$ ($\delta\ll 1$). While the moment
behaves like $\langle S_{\alpha}\rangle\sim(\delta/2)^{\frac{1}{2}}$ (Eq.
(14)) with a singular slope at the QCP the specific heat jump $\delta
C_{V}\sim(\delta^{2}/2)|\ln(\delta/2)|^{3}$ (Eq. (19)) approaches zero more
gradually. The similar though quantitatively different behaviour for SDM and
STM from numerical calculations is also shown. The $\xi$\- dependent reduction
of $\delta C_{V}(T_{m})$ in accordance with the normalised ordered saturation
moment $\langle S_{\alpha}\rangle_{0}$ is distinct from quasi-classical
magnets where the saturation moment and specific heat jump are just constants
independent of interaction parameters. On the other hand for large $\xi$ one
moves to this quasi-classical regime with constant $\delta C_{V}=\frac{3}{2}$
(for effective $S=\frac{1}{2}$ for SSM). Finally we note that not only the
individual jump $\delta C_{V}(T_{m})$ and $C_{V}^{+}(T_{m})$ approach zero for
$\xi\rightarrow 1^{+}$ their ratio also tends to zero according to $\delta
C_{V}/C_{V}^{+}\sim\delta|\ln\frac{\delta}{2}|$.

A similar important feature of distinction to quasi-classical AF magnets is
seen in Fig. 5. In the SDM below $T_{m}$ the transverse (with respect to
moment direction) susceptibility stays constant while the longitudinal one
falls to zero at low temperature where the saturated maximum moment can no
longer be polarised. For the singlet ground state quantum magnet the
transverse behaviour is the same while the longitudinal one behaves
fundamentally different. When we start with $\xi$ close above the QCP
$\xi_{c}=1$ the latter is only slightly less than the transverse one because
the induced saturation moment $\langle S_{x}\rangle_{0}$ (Fig. 3(b)) is very
small and therefore may be easily polarised by the probe field leading to a
similar large longitudinal susceptibility. However, as seen in Fig. 3(b) when
the control parameter $\xi$ is increased the saturation moment increases
rapidly and hence the low temperature susceptibility drops to lower values as
demonstrated in Fig. 5(a). This behaviour is summarised in Fig. 5(b) which
shows the AF, paramagnetic and FM T=0 longitudinal susceptibilities as
function of $\xi$. Asymptotically, for very large $\xi$ they approach zero as
in the semiclassical magnets. We note that in the latter an intermediate value
for the $T=0$ susceptibility may occur for polycrystalline case where an
averaging over longitudinal and transverse susceptibility occurs.

We now turn to the dynamical properties of the three models as evidenced by
the magnetic exciton dispersions, their relation to the induced moments and
their critical behaviour. As mentioned before we treat only the FM case in
order to avoid the complications with the unit cell doubling in AF case. The
SSM case is presented in Fig. 6. It shows the single exciton branch for a
$\xi_{s}=1.5$ above the QCP for various temperatures. As the latter approaches
the $T_{m}$ for induced FM moment formation a complete softening $\omega({\bf
q}_{m},T)\rightarrow 0$ at the incipient ordering vector ${\bf q}_{m}=0$;
($\Gamma$ -point) is observed (red curve). Below $T_{m}$ the soft mode at
$\Gamma$ shows rapid re-hardening since there is no continuous rotation
symmetry that protects it as in the SDM case. We also include the intensity
$\hat{R}({\bf q})$ (dash-dotted lines) which show a pronounced peak at the
soft mode position and singular behaviour at $T_{m}$ but modest variation
elsewhere in the BZ.

The SDM model displays a more intricate behaviour of excitation spectrum
presented in Fig. 7 as spectral intensity plot for three different
temperatures in the induced moment phase. There are now three possible modes
appearing. Two of them correspond to dispersive excitations from the singlet
ground state to the split doublet states which are different for polarisation
parallel and perpendicular to the induced moment
($\omega_{x},\omega^{+}_{y}$). The remaining transverse low energy mode
$\omega^{-}_{y}$ originates from the thermally activated excitation between
the split doublet components. As one can see close to the $\Gamma$-point it
hybridises with the dispersive high energy transverse $\omega_{y}^{+}$ mode
and outside the hybridisation region is almost dispersionless and rapidly
looses intensity. Therefore the soft mode at $\Gamma$ corresponds to a
hybridised singlet-doublet and thermally activated doublet-doublet mode. It is
the Goldstone mode of the ordered phase since it is zero at the FM ordering
vector ($\Gamma$-point) for all temperatures. On approaching $T_{m}$ from
below the $\omega^{-}_{y}$ mode is pushed to zero energy due to the vanishing
$\Gamma_{4}$ splitting and essentially becomes a quasi-elastic excitation
around the $\Gamma$-point, For $T>T_{m}$ (not shown) only the fully gapped two
high energy $\omega_{x},\omega_{y}$ branches remain which are then degenerate
throughout the BZ.

The hybridisation of excitations from the ground state with thermally excited
transitions is a fundamental feature of the dynamics of singlet induced moment
systems. The evidence for the importance of the latter may also be seen in the
non-monotonic temperature dependence of the hybridisation gap
$\Delta_{h}(T)=\omega_{y}^{+}({\bf q}_{r})-\omega_{y}^{-}({\bf q}_{r})$
between the two modes with ${\bf q}_{r}(T)$ denoting the crossing wave vector
as plotted in Fig. 8(a): At low temperature $\Delta_{h}(T)$ vanishes because
of doublet depopulation, then achieves a maximum for and intermediate
$T/T_{m}\simeq 0.8$ and drops steeply to zero when the soft mode at $T_{m}$ is
approached and the doublet splitting tends to zero. The corresponding position
$|{\bf q}_{r}(T)|$ of the hybridisation gap first remains flat at the low-T
value given below Eq. (67) and then also drops to zero, i.e., approaches the
$\Gamma$-point. Furthermore the behaviour of the three ${\bf q}=0$ modes as
function of temperature is shown in (b). Above Tm the two modes of x,y
polarisation are degenerate and they become soft modes at $T_{m}$. While
$\omega_{y}^{-}(0)$ remains soft for all $T<T_{m}$ the other transverse and
longitudinal modes show a re-hardening below $T_{m}$.

Finally in the STM case we encounter another interesting and important feature
of the dynamics in singlet ground state magnetism. In this model we discuss
only the longitudinal modes (in the cubic case the moment may be oriented
along any axis chosen as z here). Its temperature dependent dispersion is
presented in Fig. 9(a). First it shows the usual softening at the ordering
vector ($\Gamma$-point) designated by the blue arrow. However, unlike in the
previous case it does not come down to zero energy but is arrested at a finite
energy at the transition temperature. Below $T_{m}$ it immediately re-hardens
(red arrow). The reason becomes clear when we consider Fig. 9(b) where the
$\Gamma$\- longitudinal mode is plotted in the paramagnetic and induced moment
regime. If we extrapolate the paramagnetic mode energy beyond $T_{m}$ it would
indeed become soft at the approximate transition temperature $T_{m}^{0}<T_{m}$
that does not contain the effect of the Curie type contributions from the
excited triplet with matrix element $m^{\prime}_{t}=1$. Because of their
presence the actual transition occurs at a higher temperature $T_{m}$ where
the mode has not yet become soft. And below $T_{m}$ it re-hardens immediately
due to the effect of the molecular field on the singlet-triplet splitting.
This behaviour is presented in more detail in Fig. 10. The left panel (a)
shows that the arrested $\omega(0,T_{m})$ increases rapidly linearly with the
ratio of elastic (intrinsic $\Gamma_{4}$) to inelastic
($\Gamma_{1}-\Gamma_{4}$) matrix elements whereas in (b) the square- root
increase close to the QCP according to the approximate expression in Eq. (71)
is observed.

This arrested softening effect is not constrained to only the Curie-type
contributions in the STM. If, for any of the singlet ground state models
discussed here the influence of even higher CEF multiplets are considered
there will always be such terms which increase the transition temperature to a
value $T_{m}$ before the softening of the mode connected with the nondiagonal
induced order from the inelastic transition is achieved. Therefore, Figs. 9,10
illustrate in the simplest possible model what will happen quite generally to
the temperature dependence of magnetic excitons: There will be some softening
observed but it is hardly ever complete, even on the RPA level due to the
influence of thermally excited Curie type contributions from higher level that
increase the transition temperature. To achieve a maximum softening effect at
$T_{m}$ one should have i) A control parameter $\xi$ close to the QCP such
that at $T_{m}\ll\Delta$ all thermally excited Curie contributions enhancing
$T_{m}$ are exponentially suppressed and ii) the diagonal matrix elements of
the first excited multiplet should be small. Furthermore beyond RPA one has to
consider scattering effects of magnetic excitons which in principle will lead
to a broadening of the approximate soft mode into a quasielastic peak before
the transition occurs [10].

##  VI Summary and conclusion

We have undertaken a comparative investigation of quantum critical and
thermodynamic properties as well as magnetic excitations in the most typical
singlet ground state quantum magnets. These models are realised approximately
in various non-Kramers f-electron compounds based containing frequently Pr or
U whose f-shells have total angular momentum J = 4 or other integer values
split into CEF multiplets with a nonmagnetic singlet ground state. In this
case the conventional quasi- classical magnetic order of Kramers degenerate
ground state compounds is not possible. The order can only appear via
spontaneous superposition with excited CEF states belonging to singlet,
doublet or triplet caused by intersite exchange and facilitated by a
nondiagonal matrix element of angular momentum operators between ground and
excited states. In this mechanism the moment and its ordering appear
simultaneously.

This type of quantum magnetism is governed by a control parameter
characterising the relative strength of intersite exchange to CEF splitting.
Below a critical value for this parameter the ground state is paramagnetic and
a spontaneous induced moment appears when the control parameter crosses the
quantum critical point. The behaviour of the induced moment and the transition
temperature near the QCP show a logarithmic singularity meaning a rather
sudden appearance of the moment and a finite ordering temperature.

There are two main distinctions in the thermodynamic properties of singlet
ground state magnets as compared to the quasi-classical ones with degenerate
ground state: Firstly, in the latter the specific heat discontinuity at the
ordering temperature is an interaction independent constant while in the
latter it strongly depends on the control parameter characterising the
interaction strength to splitting ratio. Therefore the size of the specific
heat jump tends to zero when approaching the QCP from above, concomitant with
the vanishing size of the ordered induced saturation moment. Secondly in the
ordered state of quasi-classical magnets the longitudinal susceptibility tends
to zero at low temperatures while in the singlet ground state quantum magnets
it has generally a nonzero value that depends on the control parameter and
reaches zero only for very large values corresponding to a quasi-degenerate
CEF level system. These two major distinctions should be helpful criteria in
distinguishing singlet ground state quantum magnetism from quasi-classical
magnetism.

While these thermodynamic properties are qualitatively similar, though
quantitatively different, for the three singlet ground state models
investigated, the dynamical properties as evidenced by the dispersion of
magnetic exciton modes are distinct. As opposed to magnons in the quasi-
classical case the magnetic excitons of singlet ground state magnets appear
already in the paramagnetic phase as dispersive collective CEF excitations but
change their quantitative appearance below the ordering temperature. In the
SSM with only one branch there is a soft mode at the incipient ordering vector
at the transition temperature which then re-hardens again due to the Ising-
character of the model. For the SDM several modes appear which become
nondegenerate below the ordering temperature, one of the transverse modes
turns into a Goldstone mode for all temperatures below $T_{m}$ while the other
two (longitudingal and transverse) modes show again a typical re-hardening
from the soft mode below the transition. Finally in the the STM an important
effect of the quasi-Curie type contributions to the static susceptibility
originating from the triplet states can be identified: Their influence on the
ordering temperature leads to an arrested, only partial exciton mode softening
at the true transition temperature with a further re-hardening of the mode
energy below it. Such partial softening is commonly observed in real singlet
ground state compounds because of the influence of higher lying multiplets not
contained in the two-multiplet simplified models analysed here.

##  Appendix A CEF singlet ground state models

Here we give possible realisations (among many others) of the three singlet
ground state models investigated. We focus on $J=4$ representations relevant
for Pr and U- compounds. The first two models are for uniaxial symmetry the
last one for cubic symmetry. The notations for CEF states on the left are used
in the main text while those to the right are in free ion
$|J,M\rangle=|M\rangle$ notation with the point group representation
indicated. Within these state spaces the magnetic dipolar operators have the
form given in Sec.II.

SSM: (72)
$\displaystyle|0\rangle=\cos\theta|0\rangle+\sin\theta\frac{1}{\sqrt{2}}(|+4\rangle+|-4\rangle)\;\;(\Gamma^{(1)}_{1})$
$\displaystyle|1\rangle=\frac{1}{\sqrt{2}}(|+4\rangle-|-4\rangle)\;\;(\Gamma_{2})\quad\quad\quad\quad\quad\quad\quad$

This Ising-type singlet-singlet model is relevant for Pr - compounds with
uniaxial symmetry (e.g. $D_{4h}$) and has in particular been confirmed for for
URu2Si2 [18, 19] from spectroscopic results. In this case the CEF mixing angle
$\theta$ is close to $\pi/2$. The Ising moment operator is $J_{z}=m_{s}S_{x}$
in pseudo spin presentation within this subspace (Eq. (1)) with
$m_{s}=8\sin\theta$. We finally remark that a singlet-singlet level scheme in
a uniaxial symmetry cannot support an xy-type SSM, and it is conceivable that
this is forbidden for any symmetry.

SDM: (73) $\displaystyle|0\rangle=|0\rangle\;\;(\Gamma_{1})$
$\displaystyle|1+\rangle=|+1\rangle\;\;(\Gamma^{+}_{6})$
$\displaystyle|1-\rangle=|-1\rangle\;\;(\Gamma^{-}_{6})\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

This xy-type singlet- doublet model was proposed for hexagonal (D6h) UGa2 from
spectroscopic results [14]. The moment operators $J_{\alpha}=m_{d}S_{\alpha}$
(Eq. (2)) $(\alpha=x,y)$ are then defined with the matrix element
$m_{d}=\sqrt{10}$.

STM: (74)
$\displaystyle|\psi_{0}\rangle=\frac{\sqrt{21}}{6}|0\rangle+\frac{\sqrt{30}}{12}(|+4\rangle+|-4\rangle)\;\;(\Gamma_{1})$
$\displaystyle|\psi_{1}\rangle=-(c|-3\rangle+d|+1\rangle)\;\;(\Gamma^{+}_{4})$
$\displaystyle|\psi_{2}\rangle=\frac{1}{\sqrt{2}}(|4\rangle-|-4\rangle)\;\;(\Gamma^{0}_{4})$
$\displaystyle|\psi_{3}\rangle=c|+3\rangle+d|-1\rangle\;\;(\Gamma^{-}_{4})$

where $c=\sqrt{\frac{1}{8}}$ and $d=\sqrt{\frac{7}{8}}$. This $J=4$ singlet-
triplet model is appropriate for cubic $(O_{h})$ Pr- and U- compounds [33,
27]. It leads to the $J_{z}$ moment operator defined in Eq. (3) with
$m_{t}=\frac{4}{3}\sqrt{15}$ and $m^{\prime}_{t}=1$ fixed by symmetry for
$J=4$. For larger integer $J$ the $\Gamma_{1,4}$ occur multiple times and then
$m_{t},m^{\prime}_{t}$ depend on CEF potential parameters.

##  Appendix B Generalized Schottky anomaly

In this Appendix we give the specific heat for a singlet ground state - N-fold
degenerate excited state multiplet two level system in the paramagnetic state.
In reality a maximum of $N=3$ is possible for CEF states in cubic point group
symmetry, except when further accidental degeneracy occurs. For the general
case we obtain

$C_{V}(T)=\frac{4N\bigl{(}\frac{\Delta}{2T}\bigr{)}^{2}}{\bigl{[}(N+1)\cosh\bigl{(}\frac{\Delta}{2T}\bigr{)}-(N-1)\sinh\bigl{(}\frac{\Delta}{2T}\bigr{)}\bigr{]}^{2}}.$
(75)

For the SSM case (N=1) one obtains the common Schottky anomaly given in the
second line of Eq. (18) and for the SDM (N=2) and STM (N=3) cases the explicit
expressions are given in Eqs. (34,49), respectively. The low and high
temperature limits of the above general expression are obtained as

$\displaystyle C_{V}(T)\simeq\left\\{\begin{array}[]{l
l}N\bigl{(}\frac{\Delta}{T}\bigr{)}^{2}e^{-\frac{\Delta}{T}};\;(T\ll\Delta)\\\\[8.5359pt]
\bigl{(}\frac{N}{N+1}\bigr{)}^{2}\bigl{(}\frac{\Delta}{T}\bigr{)}^{2};\;\;\;\;(T\gg\Delta)\end{array}\right..$
(76)

##  Appendix C Spectral function for the SDM

Here we give the magnetic exciton intensities that define the spectrum of the
dynamical magnetic susceptibility of the SDM. Because of the appearance of
three exciton branches it is more involved than for the single branch in the
SSM case (Eqs. (60,61)). For the trace over cartesian components $R({\bf
q},\omega,T)=(1/\pi)\sum_{\alpha}{\rm Im}\chi_{\alpha\alpha}({\bf q},\omega)$
we obtain the three mode contributions

$\displaystyle R({\bf q},\omega,T)=$ $\displaystyle\hat{R}_{xx}({\bf
q},T)\delta(\omega-\omega_{x})+\hat{R}^{+}_{yy}({\bf
q},T)\delta(\omega-\omega^{+}_{y})$ (77) $\displaystyle+\hat{R}^{-}_{yy}({\bf
q},T)\delta(\omega-\omega^{-}_{y})$

with the mode dispersions $\omega_{x}({\bf q},T)$ and $\omega^{\pm}_{y}({\bf
q},T)$ given in Eqs. (64, 65). The intensities of each branch are obtained as

$\displaystyle\hat{R}_{xx}({\bf q},T)=$
$\displaystyle\bigl{(}\frac{\Delta}{I_{e}}\bigr{)}\bigl{(}\frac{\Delta}{2\omega_{x{\bf
q}}}\bigr{)},$ (78) $\displaystyle\hat{R}^{+}_{yy}({\bf q},T)=$
$\displaystyle\bigl{(}\frac{\Delta}{I_{e}}\bigr{)}\Bigl{[}W_{21}^{+}\frac{\omega^{+2}_{y{\bf
q}}-(\Delta_{T}^{32})^{2}}{\omega^{+2}_{y{\bf q}}-(\omega^{-}_{y{\bf
q}})^{2}}+W_{32}^{+}\frac{\omega^{+2}_{y{\bf
q}}-(\Delta_{T}^{21})^{2}}{\omega^{+2}_{y{\bf q}}-(\omega^{-}_{y{\bf
q}})^{2}}\Bigr{]},$ $\displaystyle\hat{R}^{-}_{yy}({\bf q},T)=$
$\displaystyle\bigl{(}\frac{\Delta}{I_{e}}\bigr{)}\Bigl{[}W_{21}^{-}\frac{(\Delta_{T}^{32})^{2}-(\omega^{-}_{y{\bf
q}})^{2}}{\omega^{+2}_{y{\bf q}}-(\omega^{-}_{y{\bf
q}})^{2})^{2}}+W_{32}^{-}\frac{(\Delta_{T}^{21})^{2}-\omega^{-2}_{\bf
q}}{\omega^{+2}_{y{\bf q}}-(\omega^{-}_{y{\bf q}})^{2}}\Bigr{]},$

where $\Delta^{21}=\frac{1}{2}(\Delta_{T}+\Delta)$,
$\Delta^{32}=\frac{1}{2}(\Delta_{T}-\Delta)$ are excitation energies between
the molecular field states and $f^{12}_{d}=p_{1}-p_{2}$,
$f^{23}_{d}=p_{2}-p_{3}$ the corresponding occupation differences. Furthermore
we introduced weight coefficients defined by

$\displaystyle W_{21}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(1+\frac{1}{\hat{\Delta}_{T}})(\hat{\Delta}_{T}+1)\frac{\Delta_{0}f_{d}^{12}}{2\omega_{y{\bf
q}}^{\pm}},$ (79) $\displaystyle W_{32}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(1-\frac{1}{\hat{\Delta}_{T}})(\hat{\Delta}_{T}-1)\frac{\Delta_{0}f_{d}^{23}}{2\omega_{y{\bf
q}}^{\pm}}.$

An example of the spectral function $R({\bf q},\omega,T)$ for SDM is given in
Fig. 7 (using a finite broadening) and discussed in Sec.V.

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