Induced order and collective excitations in three-singlet quantum magnets

The quantum magnetism in a three-singlet model (TSM) with singlet crystalline electric field (CEF) states interacting on a lattice is investigated, motivated by its appearance in compounds with 4f^2 and 5f^2 electronic structure. Contrary to conventional (semi-classical) magnetism there are no preformed moments above the ordering temperature Tm. They appear spontaneously as induced or excitonic moments due to singlet-singlet mixing at Tm. In most cases the transition is of second order, however for large matrix elements between the excited states it turns into a first order transition at a critical point. Furthermore we derive the excitonic mode spectrum and its quantum critical soft mode behaviour which leads to the criticality condition for induced order as expressed in terms of the control parameters of the TSM and discuss the distinctions to the previously known two-singlet case. We also derive the temperature dependence of order parameters for second and first order transitions and the exciton spectrum in the induced magnetic phase.


I. INTRODUCTION
In ordinary (semi-classical) magnets the individual magnetic moments at every lattice site exist already above the ordering temperature T m 1 .This holds even in strongly frustrated local-moment systems which may have a vanishing ordering temperature when fine-tuned to a spin-liquid regime where quantum fluctuations destroy the moment of the ground state but nevertheless the Curie-Weiss signature of local moments remains for elevated temperatures [2][3][4] .There are, however, true quantum magnets which do not have freely rotating magnetic moments above T m in the semi-classical sense as witnessed by an absence of the Curie-Weiss type susceptibility for some region above T m .In these compounds with partly filled 4f or 5f electron shells the degenerate ground state with integer (non-Kramers) total angular momentum J, created by spin-orbit coupling splits due to the local crystalline electric field (CEF) into a series of multiplets 5 .They belong to irreducible representatations Γ i which may comprise singlets, doublets or triplets depending on the symmetry of the CEF and the concrete CEF potential.For tetragonal or lower symmetry it is possible that the ground state and lowest excited states are all singlets without magnetic moment meaning Γ i |J|Γ i = 0. Nevertheless magnetic order occurs below the transition temperature T m .This order cannot be interpreted in the usual semiclassical way as an alignment of preexisting moments which then have collective semi-classical spin wave excitations as Goldstone modes.In the latter case quantum effects enter only through the possible reduction of the saturation moment due to zero point fluctuations leading to spin wave contribution to the ground state energy.
For the CEF systems with split singlet low lying states the local moments instead appear only simultaneously with the magnetic order as a true quantum effect due to the mixing of singlet states caused by inter-site exchange interactions.This 'induced' or 'excitonic' magnetic order has been observed primarily in various Pr (4f 2 ) and U (5f 2 ) compounds with two f-electrons which lead to CEF schemes with singlet ground state and possibly also low energy excited singlets.However it can also be found in f-electron compounds with higher even f-occupation, like e.g.Tb (4f 8 ).In the cubic (O h ) symmetry cases with singlet ground state the excited states must be degenerate as in fcc Pr 6,7 , PrSb 8 , Pr 3 Tl 9 and TbSb 10 (singlettriplet).Examples with hexagonal (D 6h ) structure are metallic Pr (singlet-doublet) 5,[11][12][13] and UPd 2 Al 3 14 (singlet-singlet).Tetragonal (D 4h ) cases are Pr 2 CuO 4 15 (singlet-doublet) and URu 2 Si 2 [16][17][18] (three singlets).The lower the symmetry the more likely one can have multiple low-energy singlets.The most promising class in this respect has orthorhombic symmetry (D 2h , D 2 ) which has only singlets left as in PrCu 2 19,20 , PrNi 21,22 , Tb 3 Ga 5 O 12 23 and Pr 5 Ge 4 24 .In the U-compounds, however, the situation may be more complicated due to only partial localisation of 5f-electrons 25,26 .Since there is no degeneracy in the local 4f or 5f basis states there can also be no continuous symmetry for the exchange Hamiltonian.Therefore the collective excitations in the ordered phase may not be interpreted as spin waves resulting from coupled local spin precessions but rather as dispersive singlet-singlet (or singlet-doublet and singlet-triplet) excitation modes due to intersite exchange, commonly termed 'magnetic excitons'.These are already present above the ordering temperature.The ordering is characterised by a softening of one of these modes at T m and a subsequent stiffening again further below.This type of excitonic magnetism has been considered analytically primarily within the two-singlet model 5,14,27,28 .A fully numerical treatment for a multilevel CEF-system is also possible 29 .However, for a deeper understanding of induced excitonic moment ordering and their finite temperture properties analytical investigations are desirable.In particular the influence of physical parameters like splittings, nondiagonal matrix elements and exchange which define dimensionless control parameters on the transition temperature, saturation moment and mode softening are rendered understandable only when explicit analytical expressions can be derived.This becomes quite involved beyond the two-singlet model.The latter is, however, an over-simplificiation as very often more levels, in particular another singlet state are present, as ,e.g,. in PrNi, PrCu 2 and URu 2 Si 2 .
Therefore in this work we give a detailed analytical treatment of induced moment behaviour in the physically impor-tant three-singlet model (TSM) relevant for non-Kramers felectron systems in lower than cubic symmetry, in particular we investigate the case of orthorhombic symmetry.We will focus on the mode spectrum, transition temperature and saturation moment and how they are influenced by the larger set of control parameters of this extended model.We show that under suitable conditons temperature variation induces hybridization of exciton modes in addition to the changes of intensity pattern.Furthermore we derive an algebraic equation that completely determines the transition temperature for the effective two control parameters and arbitrary splitting ratio of the TSM.In the symmetric TSM explicit closed expressions for T m are presented.Furthermore we give a comparative treatment of the exciton mode dispersions within random phase approximation (RPA) response function formalism and Bogoliubov quasiparticle picture and show that they give largely equivalent results, also for the phase boundary between disordered and excitonic phase.Finally within the RPA formalism we will investigate the change of mode dispersions and intensity in the induced moment phase.This work is mainly theoretically motivated with the aim to analyze and understand the three-singlet model and its significance for excitonic magnetism in detail.

II. THE THREE SINGLET MODEL
We keep the specifications of the three-singlet model (TSM) illustrated in the inset of Fig. 1 as general as possible, as far as splittings and magnetic matrix elements of J z are concerned.However, having orthorhombic CEF system in mind the latter are assumed to be of uniaxial character due to Γ i |J x,y |Γ j = 0 (Sec.II A, Eq. ( 4)).The three singlets are denoted by |i (i = 1 − 3) with increasing level energies E i = 0, ∆, ∆ 0 , or shifted energies Êi = E i − ∆ = −∆, 0, ∆ which are more convenient for finite-temperature properties; here we defined ∆ = ∆ 0 − ∆.The CEF Hamiltonian in can be written in terms of standard basis operators L ij = |i j| as The total angular momentum component J z in this representation is given by J z = ij i|J z |j L ij .Without restriction this leaves us with three possible independent matrix elements α = 0|J z |1 , β = 0|J z |2 and α = 1|J z |2 (Sec.II A).The latter plays only a role at finite temperature T when excited states are populated with population numbers p i = Z −1 exp(−E i /T ) where Z = j exp(−E j /T ) is the threesinglet partition function.The N f = 3 CEF wave functions may each be gauged by an arbitrary phase factor exp(iφ n ) (n = 1..N f ).Furthermore there are 1 2 N f (N f − 1) = 3 excitation matrix elements between those states.Therefore in the TSM all matrix elements α, β, α may be chosen as real without loss of generality.We will pay particular attention to the special case of the (fully) symmetric three singlet model which is defined by ∆ = ∆; α = α in Fig. 1.The magnetic properties of the model are characterized by the three possible a † < l a t e x i t s h a 1 _ b a s e 6 4 = " 6 W q n h 7 S q 6 H A y G P w d 7 z K j I w V c 1 r c = " > A A A C 0 n i c h V F L S 8 N A E J 7 G V 1 t f V Y 9 e g k X w V F I p 6 L H 4 w o t Q o S 9 p q 2 z S b Q z N i 8 x N u 7 R H B 5 j q E Z X p k i q o w 0 I l L / R K b 1 p V G 2 u P 2 t M 3 V U s l O T v 0 a 2 n P X 5 z 5 k m 0 = < / l a t e x i t > b † < l a t e x i t s h a 1 _ b a s e 6 4 = " x q 9 h / 6 E P 2 X K 1 8 w 0 S a V q O j H d g j S c = " > A A A C 0 n i c h V F L S 8 N A E J 7 G V 1 t f V Y 9 e g k X w V F I p 6 L H 4 w o t Q o S 9 p q + y m 2 x i a x b D y 9 V E P x m z + H e q s U z 8 q m q V i 6 b p U K J + m A 8 / S H u 3 T I a Z 6 T G W 6 p C r q k N N 8 o V d 6 0 2 6 0 s f a o P X 1 T t U y a s 0 u / l v b 8 B c u 5 k a 0 = < / l a t e x i t > 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " g q S V 5 4 p 6 K r j H q y l L F M q Q a p o Q U / A J q + P e j B m 8 + 9 Q 5 5 3 G c c k s l 8 q 3 5 W L l L B 1 4 l v b p g I 4 w 1 R O q 0 B X V U I e D 6 l 7 o l d 6 0 G 0 1 q E 2 3 6 T d U y a c 4 e / V r a 8 x e / X 4 + z < / l a t e x i t > Êi < l a t e x i t s h a 1 _ b a s e 6 4 = " O Z e J S b v a y 5 S Y H L C 3 G a e J v E j 2 dimensionless control parameters which characterize the intersite-coupling strenghts of the three transitions with I e (q) denoting the Fourier transform of the inter-site exchange in Eqs.(9,10) at the wave vector of incipient induced order where it is at maximum value.It may be q = 0 ferromagnetic (FM) , general incommensurate or q = Q = (π, π, π) antiferromagnetic (AF).In the following we focus on the latter case.As we shall see now in the paramagnetic phase one of the three matrix elements or control parameters must vanish due to the requirements of time reversal symmetry.This leaves us with the three possible cases of TSM's depicted in Fig. 2.

A. The orthorhombic three singlet model
To realize the general TSM in a concrete CEF for given J is not so straightforward as it may seem.This is connected with the angular momentum structure of CEF eigenstates and their behaviour under time reversal.As already indicated in the introduction cubic symmetry does not allow the TSM.In tetragonal D 4h symmetry the TSM is only realized in one specific form (Fig. 2(c)) (see discussion in Appendix A).Therefore we relax to orthorhombic symmetry D 2h where all the CEF states have to be singlet Γ i , (i=1-4) representations and the TSM is naturally possible.As mentioned before there are several physical realisations in the orthorhombic symmetry class.The decomposition e.g. for J = 4 leads to nine r U 2 0 h 6 1 p 2 + q l k l z t u j X 0 p 6 / A M P a k a g = < / l a t e x i t > (1) 1 w 9 g P 7 H v B d b 7 9 4 a x U I 4 q H M H q U E w J x S v g I d 2 B M S / T j p n T W u Z n R l 2 F 1 K U j 0 Y 2 J + j y B R H 0 a P z q n i P j A + i I i 0 5 l g 9 q C h i / M Q L + D A l l F B 9 M p T B V l 0 3 I F l w n K h 4 s S K D H o + b P T 6 q A d j V v 8 O d d a p F P K q l t e u t W z x O B 5 4 k n Z p j 3 K Y 6 i E V 6 Z J K q M O A 8 g u 9 0 p t 0 K z 1 I j 9 L T N 1 V K x D k 7 9 G t J z 1 8 r c 5 O 9 < / l a t e x i t > (2) 4 < l a t e x i t s h a 1 _ b a s e 6 4 = " w 9 g P 7 H v B d b 7 9 4 a x U I 4 q H M H q U E w J x S v g I d 2 B M S / T j p n T W u Z n R l 2 F 1 K U j 0 Y 2 J + j y B R H 0 a P z q n i P j A + i I i 0 5 l g 9 q C h i / M Q L + D A l l F B 9 M p T B V l 0 3 I F l w n K h 4 s S K D H o + b P T 6 q A d j V v 8 O d d a p F P K q l t e u t W z x O B 5 4 k n Z p j 3 K Y 6 i E V 6 Z J K q M O A 8 g u 9 0 p t 0 K z 1 I j 9 L T N 1 V K x D k 7 9 G t J z 1 8 r c 5 O 9 < / l a t e x i t > (2) 4 < l a t e x i t s h a 1 _ b a s e 6 4 = " w 9 g P 7 H v B d b 7 9 4 a x U I 4 q H M H q U E w J x S v g I d 2 B M S / T j p n T W u Z n R l 2 F 1 K U j 0 Y 2 J + j y B R H 0 a P z q n i P j A + i I i 0 5 l g 9 q C h i / M Q L + D A l l F B 9 M p T B V l 0 3 I F l w n K h 4 s S K D H o + b P T 6 q A d j V v 8 O d d a p F P K q l t e u t W z x O B 5 4 k n Z p j 3 K Y 6 i E V 6 Z J K q M O A 8 g u 9 0 p t 0 K z 1 I j 9 L T N 1 V K x D k 7 9 G t J z 1 8 r c 5 O 9 < / l a t e x i t > (a) < l a t e x i t s h a 1 _ b a s e 6 4 = " D m l 2 s B a T / r < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 4 6 n M 1 i l u J H P S + d 9 e L X x s I u P R + 2 j g q S V 5 4 q 6 K r j L q y l L F c q f q p o Q S + C T V 4 f 9 W D M 5 t + h z j q N w 7 J Z K V d u K s X q S T r w L O 3 S H p U w 1 S O q 0 i X V U I e N 6 l 7 o l d 6 0 a 0 1 q Y 2 3 y T d U y a c 4 O / V r a 8 x f m o 4 9 X < / l a t e x i t > (c) < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 7 P z )) or (Γ4, Γ (1,2 1 )) representations.In the paramagnetic phase only two elements (thick lines) can be nonzero due to Θinvariance.Transitions with dashed lines vanish because they connect states with equal Θ; they are however induced and nonzero in the magnetic state.Similar diagrams hold for the representation pair (Γ2, Γ3) (cf.Eq. ( 4)).There always have to be two (inequivalent) representations of the same type in the TSM for the Jz structure presented.
(3Γ 1 ⊕ 2Γ 2 ⊕ 2Γ 3 ⊕ 2Γ 4 ) singlets.They may be grouped according to their behaviour under time reversal symmetry operation Θ 30 .For a CEF state |ψ written as linear combination The orthorhombic singlets for J = 4 may be expressed as 23 linear combinations of |M ± = |M ± | − M with only real coefficients according to This means that |Γ 1,2 K = |Γ 1,2 are even Θ = 1) and |Γ 3,4 K = −|Γ 3,4 are odd (Θ = −1) under time reversal.Because J z is also odd it has matrix elements only among singlets with opposite Θ.From those only two are different from zero: At the same time one observes that Γ 1 |J x,y |Γ 4 = Γ 2 |J x,y |Γ 3 = 0 so that we can restrict to J z in the model for inter-site interactions (Sec.III B).If the singlet representations in the TSM would all be different then only one matrix element of J z could be non-zero.However, in the J = 4 D 4h decomposition given above each singlet representation occurs at least twice.Then two matrix elements of the TSM containing two singlets with equal symmetry can be non-zero.Because these have necessarily equal Θ the third J z matrix element is always zero as long as time reversal symmetry holds.In the induced magnetic phase when Θ is broken it will also be non-zero as shown in Appendix B (Eq. (B7); this is es-sential to obtain the proper temperature dependence of order parameter and soft mode energy.
In the paramagnetic phase we are then left with the three posible cases of dipolar matrix element sets (α, β, α) as illustrated in Fig. 2. Since the orthorhombic CEF is characterized by nine arbitrary CEF parameters one may reasonably expect that every sequence in Fig. 2 and similar ones with (Γ 2 , Γ 3 ) singlets can in principle be realized.We note that in the higher D 4h symmetry only the model type of Fig. 2(c) seems possible (AppendixA).Rather than discussing each possible case presented in Fig. 2 individually it is more economic to treat the general TSM (inset of Fig. 1) keeping in mind that always one in the set of matrix elements (α, β, α) must vanish to reproduce any of the possible cases in Fig. 2 allowed by Θ.

III. RESPONSE FUNCTION FORMALISM, MAGNETIC EXCITON BANDS AND INDUCED TRANSTION A. Local dynamic susceptibility of the TSM
The most direct way to understand the magnetic ordering in the TSM is provided by the response function formalism, the resulting magnetic exciton bands and their soft-mode behaviour.The dynamic response function for the isolated TSM is in general given by defining the occupation differences of levels by p ij = p i − p j this is evaluated explicitly as (∆ 0 = ∆ + ∆) where the p ij are given by with The occupation differences fulfil the relation p 12 = p 02 − p 01 .For T ∆, ∆ 0 when p 01 , p 02 ≤ 1 this means p 12 p 01 , p 01 .For the two-singlet model (i, j = 0, 1) one simply has p ij = tanh β 2 ∆ ij .In the TSM the expressions f ij in the denominators of Eq.( 7) are a correction taking into account the presence of the third level in the partition function.

B. Collective magnetic exciton modes
The relevant part of the inter-site exchange interaction of three-singlet states is given by (l, l denote lattice sites R l , R l ): with the Fourier component J q z = N − 1 2 l exp(iqR l )J l z .The transverse J q x , J q y do not contribute to the collective mode dispersion because of their vanishing matrix elements in the orthorhombic TSM's of Fig. 2. The Fourier transform of the exchange interaction may be expressed (assuming only next neighbor coupling I 0 ) as in the simple orthorhombic lattice of dimension D = 3 and coordination z = 2D.The momentum units are 1 a , 1 b and 1 c parallel to the respective orthogonal axes.For the AF case with I 0 < 0 on which we focus we also introduce the effective AF exchange I e ≡ I e (Q) = −zI 0 > 0 where Q = (π, π, π) denotes the AF wave vector.Then within RPA approximation 5 the collective dynamic susecptiblity (zzcomponent only) of coupled three singlet levels is obtained as Its poles as defined by 1 − I e (q)χ 0 (iω n ) = 0 give the dispersive collective magnetic exciton modes of the TSM which are determined by a cubic equation in ω 2 .We first derive its general solution and then a more intuitive restricted one for the low temperature case.
If we had only one of each contribution in Eq. ( 6) we would obtain isolated exciton modes given by where we used the effective T-dependent transition strengths defined by These uncoupled modes are hybridized into new eigenmodes when more matrix elements are present.We derive these expressions already in sight of the magnetic case of Sec.VI B where all the α 2 T , β 2 T , α 2 T as modified by the molcular field are nonzero.The hybridized modes may be expressed in terms of the following auxiliary quantities:  (20,16)).The flat band (dashed green) corresponding to 1 ↔ 2 excitations has vanishing intensity (Fig. 4).The hybridization gap is determined by the cross-coupling ∼ αβ (Eq.( 22)).Dash-dotted blue/red lines show the dispersion for the main ω1,2 modes in Bogoliubov approach (Eq.( 36)).Inset: crossing wave number qc.Here r = 0.3, ξα = 0.2, ξ β = 0.71 (ξs = 0.91 < 1 subcritical), ξα = 0., (scheme as Fig. 2(a)).
With the definition of The dispersions of of the coupled modes (i = 1, 2, 3) are given by where φ 1 = 0, φ 2 = 2π 3 , φ 3 = 4π 3 .These expressions give the RPA mode dispersions for any splittings and matrix elements of the TSM and also for abitrary temperature.In terms of these modes the collective RPA susceptibility may be written as This leads to a spectral function which determines the structure function in inelastic neutron scattering (INS): with the momentum and temperature dependent intensities R λ (λ = 1 − 3) of exciton modes given by T ∆ 0 (ω λ (q) 2 − ∆ 2 )(ω λ (q) 2 − ∆2 ) At low temperatures we can find an approximate and more intuitive solution for the dispersions: For T ∆, ∆ 0 when p 12 p 01 , p 02 we can neglect the second term in Eq. ( 6), i.e. the influence of transitions starting from the thermally excited states on the dynamics.Then the mode dispersions are obtained in concise form as with ∆ av = (∆∆ 0 ) 1 2 .The two dispersive modes stemming from the ground-to excited state transitions may anti-cross if their dispersion is sufficiently strong, i.e. if I 0 inter-site exchange is sufficiently large and matrix element α or β large and sufficiently different.This happens when the decoupled dispersions fulfil ω e (q c ) = ω e (q c ) ≡ ω c .For q = (q, q, q) along ΓR in the orthorhombic BZ where dispersion is maximal one obtains if the modulus of the argument is smaller than one.At the anti-crossing point q c of the two exciton modes (Figs.3,4) the splitting otained from Eq.( 20) is then given by The anti-crossing happens because both inelastic transitions start from the same ground state and the splitting is therefore ∼ |α T β T |.The dispersion as well as the splitting decrease with increasing T due to the reduction of effective transition strength α T ∼ p 01 and β T ∼ p 02 (Fig. 1).The intensities determining the spectral functions now take on the simplified form ) where λ = 2, 1 for λ = 1, 2, respectively.A discussion of exciton mode dispersions and intensities is given at the end of Sec.IV.
information on the Bloch functions of these modes.For that purpose a direct (approximate) diagonalization of the Hamiltonian using pseudo-unitary Bogoliubov and subsequent unitary transformations may be performed that also contain the eigenvectors of exciton modes.Therefore we also apply this alternative approach to the problem.In this context the local CEF excitation standard basis operators |i j| in the TSM are mapped to bosons (altough the former have more complicated commutation relations).This can be justified as long as the temperature fulfills T ∆, ∆ 0 and only the two excitations from the ground state have to be considered 27,31 corresponding to the TSM of Fig. 2(a).Defining a † i = |1 0| and b † i = |2 0| and using the Fourier the Hamiltonian H = H CEF + H ex may be written in its bosonic form by using the definition where (k suppressed on right side.):Here we defined This Hamiltionian may be approximately diagonalised by pseudounitary Bogoliubov transformations in each particlehole subspace of a, b -type operators and a subsequent unitary rotation in the space of isolated A,B normal modes.The former are given by which preserve the bosonic commutation relations for the A k , B k .The above transformation diagonalizes each diagonal 2 × 2 block in Eq. ( 24) when the conditions are fulfilled.This leads to the transformed Hamiltionian (in A,B particle space only) in terms of A,B uncoupled normal mode coordinates given by Here ω A k and ω B k are the uncoupled normal mode frequencies ; which are indeed equivalent to the uncoupled exciton modes ω e , ω e , respectively of the RPA response function approach in Eq. ( 12) for the low temperature limit.Furthermore they satisfy the relations The coupling term in Eq.( 28) obtained through the transformation described by Eq. ( 26) is given by It may be evaluated, using Eq.( 27) as x e H B g A 7 w t / F q p q i H d 8 I Z y W g L W R z c E J E q 7 e N e S E Y T 3 k l W D j 2 C / M S 9 l 5 j 9 b 4 a J Z E 4 q H E O a Y M x J x i v g M f X h M S / S T T 2 n t c y P T L q K q U f H s h u B + g K J J H 1 a P z x n s I T A B t K i 0 r n 0 t M F h y v c I E / A g D V S Q T H n K o M q O u 5 B M S i 5 Z v J S R g S + E T K a P e r B m / e 9 S Z 5 X a Y U k v l 8 r X 5 W L l J F 1 4 l n Z p j w 6 w 1 Here full lines correspond to ξa = 0 , dashed/dashed-dotted lines to ξa = −0.2,0.2 (black) and ξa = −0.3,0.3 (blue and red, respectively).Inset: PM/AF phase boundary Tm(ξα, ξ β ) = 0 for symmetric case r = 1 from RPA (black) and Bogoliubov (green) theories.Now a further unitary transformation in A, B particle space can be employed according to These are the normal mode exciton coordinates that diagonalise the Hamiltonian in Eq. ( 24) (up to residual two-exciton interactions) provided the condition is fulfilled, leading to where the exciton mode frequencies are finally given by which essentially corresponds to the RPA result of Eq. ( 20) for zero temperature.Obviously the direct diagonalization route to obtain the exciton modes is more elaborative than the response function formalism.On the other hand it also provides the Bloch functions χ † 1,2k |0 whose creation operators are, according to Eqs. (26,33) explicitly given by b t G a y g q l 4 i W K J v V C 2 v j 1 W Q / H b P w e 6 q R T 3 y 4 a p W K p W i q U 9 5 K B p 7 G O D W x x q j s o 4 x g V 1 m H j F s 9 4 w a t W 1 e 6 0 B + 3 x i 6 q l k p w 1 / F j a 0 y e F w p E 4 < / l a t e x i t >

nd
< l a t e x i t s h a 1 _ b a s e 6 4 = " E B 2 v 4 3 T q a i T H d 0 q 6 A e 0 H 9 z 3 E u v 9 e c N Y K k c V j m h 1 K q a k 4 j n x E D d k / J d p x 8 x p L f 9 n R l 2 F 6 O J A d m O y P k 8 i U Z / G t 8 4 x I z 6 x v o x k c S K Z P W r o 8 j z k C z i 0 N V Y Q v f J U I S s 7 7 t B q 0 g q p 4 s S K G v V 8 2 u j 1 W Q / H X P g 9 1 F m n X s w X S v l S t Z Q r H 8 Y D T 2 I b O 9 j j V P d R x h k q r M P A L Z 7 x g l e l q t w p D 8 r j F 1 V J x D l b + L G U p 0 9 V t Z E k < / l a t e x i t > T ⇤ > 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " V O Z 2 F + U t r g q S E E + 8 s j f s 7 7 t B q 0 g q p 4 s S K G v V 8 2 u j 1 W Q / H X P g 9 1 F m n X s w X S v l S t Z Q r H 8 Y D T 2 I b O 9 j j V P d R x h k q r M P A L Z 7 x g l e l q t w p D 8 r j F 1 V J x D l b + L G U p 0 9 V t Z E k < / l a t e x i t > b t G a y g q l 4 i W K J v V C 2 v j 1 W Q / H b P w e 6 q R T 3 y 4 a p W K p W i q U 9 5 K B p 7 G O D W x x q j s o 4 x g V 1 m H j F s 9 4 w a t W 1 e 6 0 B + 3 x i 6 q l k p w 1 / F j a 0 y e F w p E 4 < / l a t e x i t > T ⇤ < l a t e x i t s h a 1 _ b a s e 6 4 = " (38) for the Bogoliubov transformation coefficients.Likewise we get for the coefficients of the subsequent unitary transformation.The comparison of the low excitonic modes at low temperature as obtained from response function and Bogoliubov approach is shown in Fig. for the (111) direction.Control parameters are chosen such that a crossing of uncoupled modes (dotted lines) of Eqs.(12,29) occurs at wave number q c .Their hybridisation leads to an anticrossing of the coupled modes (Eqs.(16,20,36)).The full line represents RPA result of Eqs.(16,20).The inset depicts the increase of the crossing wave number with temperature.Once it has reached the zone boundary the modes become gradually decoupled due the suppression of their dispersion.At the AF zone boundary vector Q (q/π = 1) the lower mode ω 1 (q) shows incipient softening.The dash-dotted line is obtained from the Bogoliubov result in Eq. ( 36) and is rather close to the full line.There .The transition for ξs = 1.02 and ξa = −0.2 is of second order (Fig. 5).The moment appears due to the mixing of excited |1 and |2 into the mf ground state |ψ0 (Eq.(B4)).Their coefficients v0, w0 are shown as broken blue lines (scheme as Fig. 2(a)).
are, however distinct differences close to the AF point: Because the effective hybridisation λk (Eq.( 32)) is enhanced by a feedback effect due to the mode softening, the latter happens more rapidly in the Bogoliubov approach.This will also lead to a difference in the phase boundary for the two techniques (inset of Fig. 5).The temperature dependence of spectral functions for TSM's of Fig. 2 (a),(b) obtained from RPA theory (Eq.( 19)) is presented in Fig. 4 (left and right columns, respectively) and shows distinctive features.Left: (i) With increasing temperature the anti-crossing region moves to larger wave vectors, concommitant with q c (T ) in Fig. 3 (ii) With decreasing temperature ω 1 (q) becomes an incipient soft mode.For slightly larger ξ β it would become unstable at lowest temperature.Correspondingly the dispersive width of ω 2 (q) increases for lower temperature.Right: (i) At larger temperature the mostly flat ω 3 (q) low-energy mode originating from transitions between thermally excited |1 and |2 states is still visible, its flatness is caused by the always small thermal population difference factor p 12 (Fig. 1).For this reason its spectral weight also decays exponentially at low temperature and therefore it has vanished from Fig. 4 (T /∆ 0 = 0.1).(ii) The ω 2 (q) mode now shows incipient soft mode behaviour due to slightly below-critical control parameters.

V. SOFT MODE BEHAVIOUR AND CRITICAL CONDITION FOR MAGNETIC ORDER
When temperature is lowered the effective coupling parameters α 2 T = α 2 p 01 , β 2 T = β 2 p 02 for the TSM of Fig. 2(a) increase and with it the dispersive width of ω 1 , ω 2 modes.Eventually one of them may touch zero a the wave vector q FIG. 8. Same as Fig. 7, now for ξs = 0.85, 0.83 ( full and broken black/red lines ) and ξa = 2.5 when the transition is of first order (Fig. 6) with jump in Jz .Note that susceptibility above Tm does not diverge due to first order character.Ground state admixture coefficients v0, w0 (broken blue lines) also jump to finite values at Tm (scheme as Fig. 2(b)).
where I e (q) has its maximum, frequently (but not necessarily) at zone center q = 0 or boundary Q = (π, π, π).This mode softening signifies the onset of induced excitonic FM, AF quantum magnetism at T m , respectively.In distinction to common magnetic order the moments are not preformed already at larger temperature and order at T m , since there are only nonmagnetic singlet states available, but rather the creation and ordering of moments happens simultaneously at T m due to off-diagonal virtual transitions between the singlets.We first consider the soft mode condition within the RPA response function formalism.According to Eq. (17)  it is equivalent to the divergence of the static susceptibility χ(q, T m ) −1 → 0 which leads to the criticality condition This means that the static (iω n = 0) single-ion susceptibility given by Eq. ( 6) must reach a mininum value ≥ 1/I e (q) to achieve induced magnetic order at finite T m .We focus at the AF case (I 0 < 0) where this is first fulfilled for the AF wave vector q = Q.The procedure for FM (q = 0) or even incommensurate cases are analogous.
As a reference we recapitulate the well known expression for T m in the two-singlet model 5,7,14,28 (e.g.taking off the upper singlet-state |2 in Fig. 1).In this case (I e ≡ I e (Q) = −zI 0 > 0): where ξ α is now the only dimensionless control parameter of the model and at ξ c α = 1 a quantum phase transition from , transition temperature Tm/∆0 and their ratio as function of ξs.Matrix elements α, β, α as in Fig. 7 for weakly symmetric case with r = 1.Here the interaction constant Ie is varied leading to concommitant variation of ξs and ξa.The ratio shows steep decrease close to quantum critical point (scheme as Fig. 2(a)).paramagnetic ξ < ξ c α to magnetic ξ > ξ c α ground state appears.In the marginally critical case ξ α = 1 + δ (0 < δ 1) we can expand T m ∆/ ln 2 δ and thus the ordering temperature vanishes logarithmically when approaching the critical value ξ c β (δ → 0) from above.This is a characteristic behaviour of an induced excitonic quantum magnet.Now we consider the extended TSM cases of Fig. 2 with generally possible parameter sets.The critical equation for T m (Eq.( 41)) may be written with the use of control parameters of Eq. (2) as For convenience we now define the splitting ratio r = ∆/∆, meaning ∆ = ∆ 0 /(1 + r).Then r = 1 corresponds to the symmetric case with ∆ = ∆ (Fig. 1) and r = 1 to the general asymmetric case.Defining furthermore y = exp(∆/T ) the critical condition for induced order Eq. ( 40) can be written as then T m = ∆/ ln y m is the ordering temperature with y m given by the solution of the above algebraic equation.Note that even in the general asymmetric TSM described by Eq. ( 43) there are effectively two control parameters which are combinations of the three possible parameters in Eq. ( 2) according to ξ s = ξ α + ξ β and ξ a = ξ α − ξ α leading explicitly to the expressions We should remember that in the paramagnetic state one of the elements in the se (α, β, α) must be zero corresponding to the cases of Fig. 2. The solution of Eq. ( 43) for finite T m and general splitting ratio r is only possible numerically.However, for discrete values like r = 1 2 , 1, 2, 3 explicit solutions for T m can be obtained but except for r = 1 are not particularly instructive.We may also look at the limiting cases r → 0, ∞.The latter corresponds to the singlet-singlet model and recovers the solution in Eq. (41) while the former describes the singlet-doublet model 11 with splitting ∆.Its T m is also described by Eq. (41) but with the replacement ξ β → 2(α 2 + β 2 )I e /∆.Now we discuss two typical special cases of the TSM model where the solution for T m can be obtained in closed form from Eq. ( 43).These are considerably more complicated to derive than for the two level system but formally similar: i) weakly symmetric TSM r = 1 but α = α Then Eq. ( 43) reduces to a quadratic equation and from its two solutions y ± m the critical temperature may be obtained as Explicitly one obtains after some derivations: where ξ s , ξ a are given by Eq. ( 44) with r = 1.Instead of having directly the control parameter ξ s appearing in T m as in Eq.(41) it is replaced by a function η s (ξ s , ξ a ).A solution for finite T ± m exists only when η ± s (ξ s , ξ a ) ≥ 1.The physical solution is always T m ≡ T − m with η s = η − s .The second solution T + m does not exist for ξ a < 1 and for ξ a > 1 corresponds to the unphysical branch with T − m < T m (blue dashed line in Fig. 6).It is easy to show that η −1 s (1, ξ a ) = 1 for ξ a ≤ 1. Therefore the T m and the phase boundary position ξ s does not depend on ξ a in this case as is indeed demonstrated by Fig. 5 and inset of Fig. 6.
ii) fully symmetric TSM r = 1 and α = α This means that now ξ a = 0 and only one effective control parameter ξ s remains.T m is given by the same expression as above but with the simplification where now ξ s is the control parameter for the fully symmetric TSM that contains both matrix elements and the splitting ∆ = ∆ 0 /2.For a finite T m one must have η s > 1 and hence ξ s > 1.In the marginal critical case ξ s = 1 + δ the transition temperature shows similar logarithmic behaviour as before, but with T m ∆/ ln 1 δ .The systematic variation of T m (ξ s , ξ a ; r) is shown in Fig. 5 for ξ a < 1.The fully and weakly symmetric cases (r = 1) discussed in detail above correspond to the full and broken black lines in Fig. 5, respectively.In the asymmetric case (r = 1) the transition temperature T m changes considerably The softening of the critical mode ω2(q) at the zone boundary AF point Q = (π, π, π) is reversed below Tm into a hardening with decreasing temperature (cf.Fig. 11).Here we have above-critical values ξs = 1.02, ξa = −0.2(scheme as Fig. 2(a)).
with the splitting asymmetry r = ∆/∆, keeping the total splitting ∆ 0 constant.When r < 1 the central state |1 is shifted upwards leading to an increased effectiveness because the occupation difference p 01 increases, therefore T m increases.The reversed argument holds for r > 1. Furthermore when the asymmetric control parameter ξ a is larger or smaller than zero for a given r the value of T m moderately increases or decreases, respectively.
For ξ a > 1 when the coupling of thermally excited states becomes important a surprising new situation occurs (Fig. 6): Firstly the second order transition temperature T m now stays finite for ξ s < 1 and secondly at a certain critical point T m = T * it changes into a first order transition for T m < T * .This is of course no longer described by Eq. ( 43) and its special cases since it was obtained from the divergence of the susceptibility at T m .Below T * this is no longer true and T m has to be determined by solving directly the selfconsistency equations for the order parameter (Sec.VI).The resulting line of first order transitions is shown by red symbols and dashed line in the main Fig. 6 for ξ a = 2.5.For this value the 1 st order line stops at ξ s = 0.77.
Alternatively this variation can be combined in a contour plot of T m in the (ξ s , ξ a ) control parameter plane for fixed r, taken as the symmetric case r = 1 in the inset of Fig. 6.Firstly it shows that the sector of first order transitions bounded by the red symbols and broken line to the left and the T * > 0 line to the right widens when ξ a increases, i.e. the transitions between thermally excited states become more important.Secondly it shows explicitly the ξ s -independence of the second order PM/AF phase boundary defined by T m (ξ s , ξ a ; r) = 0 for ξ a < 1 as already noticed before.This property may be traced back directly to the fundamental equation for T m given by Eq. (43).In this respect it is instructive to compare the predictions of the soft mode conditons ω 2 (Q) = 0 at T m = 0 for RPA (Eq.( 20)) and Bogoliubov (Eq.( 36)) approaches for consistency (in the case α = 0 of Fig. 2(a)).They cannot be identical due to the slightly different expressions for the exciton mode dispersions.In the RPA case one simply obtains from the equivalent Eq. (43) in the limit y m → ∞: ξ s = ξ α + ξ β = 1, in accordance with previous discussion of symmetric models (inset of Fig. 6 for ξ a < 1).This means the effect of the two excitations is simply additive at the phase boundary.In comparison the Bogoliubov case leads to the more complicated relation For the special case ξ α = ξ β = ξ we obtain ξ = 0.5 in the RPA approach and ξ = 0.57 in the Bogoliubov approach.Furthermore in both cases the boundary points (ξ α , ξ β ) = (1, 0), (0, 1) are identical for both methods.The complete comparison of PM/AF phase boundaries T m (ξ α , ξ β ) = 0 is shown in the inset of Fig. 5 for both methods.It demonstrates a rather close agreement between the two technically rather different approaches.

VI. THE INDUCED ORDER PHASE AND ITS EXCITATIONS
We now consider the phase with induced magnetic order in the TSM.To be specific we treat only the AF case corresponding to the soft mode with at Q = (π, π, π).The more direct treatment is based on the RPA approach with the inclusion of the mean field induced order.The alternative would be the exciton condensation picture for Bogoliubov quasiparticles.The latter is problematic to extrapolate to the disordered phase with temperatures considerably above T m due to the influence of thermally populated CEF singlets.This is no problem for the response function approach which will therefore be used here.As a necessary basis we need the mean field selfconsistency equation for the induced order parameter.The CEF molecular field Hamiltonian is given by with the exchange model of Eq. ( 10) the effective molecular field on the two AF sublattices A, B is h A,B e = I e J B,A z where I e = z|I 0 | (I 0 < 0 for AF exchange) and J A,B z = ± J z .The associated difference in free energy per site between induced moment state and paramagnetic state corresponding to Eqs. (48,1) is given by where the p i are the paramagnetic CEF level occupations (Sec.II) and p i the occupation of levels in the AF state, renormalized by the molecular field.Explicitly, The modified molecular field CEF energies E i and eigenstates |ψ i are derived and discussed in Appendix B.

A. Order parameter and saturation moment
Calculating the diagonal (elastic) matrix elements of J z within the mf eigenstates |ψ i the selfconsistency equation of the order parameter may be given as The primed quantities generally refer to the MF values in the ordered state with nonvanishing J z .The latter appears implicitly in Eq.( 50) through the mf energy levels E i and the coefficients (u i , v i , w i ), i = 0, 1, 2 of the wave functions |ψ i .The resulting temperature dependence of the AF order parameter J z below T m together with the paramagnetic inverse static RPA susceptibility χ −1 (0, T ) above T m is shown in Fig. 7 for a value of ξ a < 1 that results in a second order transition.The divergence of χ(0, T ) at T m triggers the appearance of the induced moment J z .The latter is due to the mixing of excited |1 , |2 CEF states into the mf ground state |ψ 0 (Eq.(B4)).The figure also displays the T-dependence of selfconsistent admixture coefficients v 0 , w 0 of excited states |1 , |2 into the molecular field ground state |ψ 0 according to Eq. (B4).In contrast the similar Fig. 8 presents the case of the first order transition (ξ a > 1) for two different T m .There the susceptibility χ(0, T ) no longer diverges at T m and the order parameter J z and admixture coefficients jump to a finite value.From tracing J z =0 for different ξ s the 1 st order transition line in the inset of Fig. 6 (red symbols) may be obtained.
The saturation moment at zero temperature is obtained from Eq. (50) as where on the r.h.s the index zero refers to the ground state |ψ 0 .For the TSM this equation cannot be solved explicitly for J z since the latter enters on the r.h.s in a complicated manner in the mixing coefficients and associated mf energies (see App. B).As a reference we give the expression for the two-singlet case (discarding the state |2 for the moment) where it can be derived 14 explicitly as Thus the saturation moment and its ratio to the transition temperature ( J z T =0 /α)/(T m /∆) = (2δ) 1 2 ln( 2 δ ) → 0 vanish when the induced magnet is close to the quantum critical point, i.e. δ → 0. This is in marked contrast to a conventional semiclassical (degenerate S = 1 2 ) magnet 1 where the corresponding ratio is constant, given by S z T =0 /(T m /I e ) = 1 in that case.This peculiar dependence of saturation moment and its ratio with the transition temperature on the control parameters is also apparent in the TSM (Eq.( 51)) as presented in Fig. 9.The saturation moment (now normalized to m 0 = (α 2 + β 2 ) 1 2 ) increases with square root-like behaviour above the critical parameter ξ c s = 1 approaching unity for ξ s 1.Because T m varies only logarithmically for ξ s → ξ c s the ratio of both quantities (blue line) first increases and the rapidly drops to zero.

B. Collective excitations in the AF phase
With J z determined we now may compute the renormalized excitation spectrum in RPA approach in the induced moment phase.For this purpose we need the renormalized local CEF energy differences E ij = E i − E j of molecular field states (Eq.(B2)) and the inelastic matrix elements between them which lead to renormalized matrix elements α , β , α which are now generally all non-vanishing because Θ is broken (Appendix B).In addition we define modified effective temperature dependent parameters α 2 T = α 2 p 01 , β 2 T = β 2 p 02 , α 2 T = α 2 p 12 analogous to Eq.( 13).With the replacements (∆, ∆ 0 , ∆) → (E 10 , E 20 , E 21 ) and (α 2 T , β 2 T , α2 T ) → (α 2 T , β 2 T , α 2 T ) the exciton mode frequencies in the induced AF ordered phase may be obtained from Eqs. (12,16) by substitution.
An example of the temperature dependence of the exciton dispersions ω 1,2 is presented in Fig. 10 using the parameter set of Fig. 7 (2 nd order case) for temperatures above, at and below T m .The flat mode ω 3 in Fig. 3 which has vanishing intensity is not shown here.The ω 1 dispersion displays the typical softmode behaviour when temperature is lowered down to T m (dashed, full lines).However, immediately below T m the dispersion shifts to finite frequency again (dash-dotted).
The corresponding continuous temperature dependence for the Q = (πππ) soft mode with the same parameter set and another subcritical one for comparison is shown in Fig. 11.In the latter (broken lines) the zone boundary ω 1 (Q) mode softens but then stays flat with lowering temperature while the upper mode ω 2 (Q) is practially constant, see also Figs. 3,4.When ξ s is above critical value (as in Fig. 10) ω 1 (Q) now actually hits zero, triggering the onset of AF order shown in Fig. 7. Once the molecular field h e becomes finite and increases the splittings between renormalized levels at E i the critical mode ω 1 (Q) is again stabilized to finite frequencies already seen in Fig. 10 for T < T m .On the other hand the upper mode for the parameters used shows very little temperature effect.It is certainly possible to fine-tune the parameters such that both modes of the TSM become critical or closely so, but this seems rather artificial and physically one normally has to deal with just one critical mode as is the case e.g. in the cubic singlet-triplet system Pr 3 Tl 9 .From the above disucussion it is clear that the softening of the critical mode in the case of a 1 st order transition is arrested at a finite energy value.

VII. SUMMARY AND CONCLUSION
In this work we have given a complete survey of a most general extended three-singlet model (TSM) of induced moment quantum magnetism.It consists of three nonmagnetic CEF singlets coupled by non-diagonal matrix elements of one of the angular momentum components constrained by time reversal Θ.Such low-lying TSM configurations occur frequently in rare earth or actinide compounds with 4f 2 or 5f 2 or other even occupation f-electron configurations.The model may be characterized by individual three (ξ α , ξ β , ξ α) but effectively two (ξ s , ξ a ) dimensionless control parameters, involving the CEF splittings, non-diagonal matrix elements and intersite exchange.
We used two approaches to calculate the elementary excitation spectrum as function of control parameters; the response function RPA formalism and the Bogoliubov quasiparticle approach.They agree on the basic properties of the magnetic exciton dispersions and their soft mode behaviour as function of (ξ s , ξ a ).While the latter approach is only practical at low temperature range but gives the Bloch states of exciton bands the RPA formalism covers all temperatures and in particular the mode softening as function of temperature and the criticality condition for the onset of induced magnetism as function of (ξ s , ξ a ).
As a new aspect of the TSM we showed that for suitable control parameters a temperature induced hybridization of modes takes place with an anticrossing of the two exciton dispersions resulting from excitations out of the ground state.On the other hand a possible thermally excited mode stays dispersionless and is only visible at elevated temperatures.
The criticality condition leads to an equation for the dependence of ordering temperature on control parameters which may be solved explicitly for T m in the weakly and partly sym-metric cases.For ξ a < 1 the condition for a finite induced ordering temperature is always given by ξ s > 1, independent of the values of ξ a and the splitting ratio r.Furthermore in this case the transition is always of second order as evidenced by the calculation of paramagnetic susceptibility and temperature dependent order parameter.
Another possibility not observed in the singlet-singlet case arises when we consider the phase diagram and magnetic ordering temperature T m (ξ s , ξ a ) for ξ a > 1 which means that the thermally excited nondiagonal processes are important.Then the second order transition at T m extends to control parameter ξ s < 1 and finally turns into a first order transition at the critical point T * , as is demonstrated by the behaviour of susceptibility and order parameter temperature dependence.
The latter is obtained from the mean-field selfconsistency equations.The resulting molecular field enters into the dynamics via renormallized local CEF energies and nondiagonal matrix elements.Their influence leads to a resurgent stiffening of the soft mode immediately below T m which mimics the order parameter.The stiffening is continuous when the transition at T m is of second order and has jumplike behaviour for the first order transitions.
These predicted features may play a role in real Pr-and Ubase singlet excitonic magnets and deserve further experimental investigations.This also refers to pressure experiments.The latter may change the CEF splittings and matrix elements and hence the control parameters and therefore may allow to tune between the different phases found in this three-singlet model investigation.
with α e = αh e , β e = βh e and αe = αh e .Here h e = I e J z is the molecular field and we abbreviate I e = I e (Q) = −zI 0 > 0 with Q = (π, π, π) the AF ordering vector.The eigenvalues E i (h e ) of the molecular field Hamiltionian are then again given by the solutions of the cubic secular equations (i = 0, 1, 2): where ϕ 0 = 2π 3 , ϕ 1 = 4π 3 , ϕ 2 = 0 and p =1 3 (3b − a 2 ); q = We formally keep the last term in c although it must vanish identically because one of the matrix elements has to be equal to zero due to time reversal symmetry (Sec.II A).The phases ϕ i are denoted such that for the paramagnetic case with h e = 0 we recover E i = Êi = −∆, 0, ∆ for i = 0, 1, 2 consecutively, corresponding to the sequence in Fig. 1.The associated molecular field orthornormal eigenvectors are These coefficients may be obtained for the general model by elimation from the eigenvalue equation H mf CEF |ψ i = E i |ψ i .
2 2 0 B Y P 4 t W b V / 1 h + l s 8 + G V N B S 3 S D Z u Z / e a b b 2 d 2 z N B 1 I m k Y 7 y l t Y X F p e S W d y a 6 u r W 9 s 5 r a 2 6 1E w E B a v W Y E b i K b J I u 4 6 P q 9 J R 7 q 8 G Q r O P N P l D b N / G s c b Q y 4 i J / C r c h T y j s d s 3 + k 5 F p O A b t h t u 8 t s m 4 u 7 X N 4 o G G r p s 0 4 x c f K U r E q Q + 6 A 2 d S k g i w b k E S e f J H y X G E X 4 W l Q k g 0 J g H Z o A E / A c F e f 0 Q F n k D s D i Y D C g f f x t n F o J 6 u M c a 0 Y q 2 8 I t L r Z A p k 7 7 2 B d K 0 Q Q 7 v p X D j 2 A / s c c K s / + 9 Y a K U 4 w p H s C Y U M 0 r x C r i k e z D m Z X o J c 1 r L / M y 4 K 0 k 9 O l b d O K g v V E j c p / W j c 4 a I A N Z X E Z 3 O F d O G h q n O Q 7 y A D 1 t D B f E r T x V 0 1 X E X l i n L l Y q f K D L o Cd j 4 9 V E P x l z 8 O 9 R Z p 3 5 Y K J Y K p e t S v n y S D D F 5 u 0 0 B Y P 4 t W b V / 1 h + l s 8 + G V N B S 3 S D Z u Z / e a b b 2 d 2 e O D Y Y W Q Y 7 y l t Y X F p e S W d y a 6 u r W 9 s 5 r a 2 6 6 E / k K a o m b 7 j y y Z n o X B s T 9 Q i O 3 J E M 5 C C u d w R D d 4 / j e O N o Z C h 7 X v V a B S I j s s s z + 7 Z J o s A 3 f D b d p d Z l p B 3 u b x R M N T S Z 5 1 i 4 u Q p W R U / 9 0 F t 6 p J P J g 3 I J s h a 1 _ b a s e 6 4 = " l B y y 4 5 9 e 8 m a U S z 4 2 G N 9 q b r D D q e 6 h h B N U W I e N W 7 z g F W 9 G 1 b g z H o 2 n b 6 o x l e R s 4 N c y n r 8 A J C O R C g = = < / l a t e x i t > 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " O 5 n t H G I R 4 t I E u y + C y 3 P p 7 3 0 m i n Y = " > A A A C 0 X i c h V F L S 8 N A E J 7 G V + u z 6 t F L s A i e S i I F P X g o W M W L U N E + o C 1 l k 2 5 j a F 4 k 2 0 I t g n j 1 5 l X / m P 4 W D 3 6 7 p o I W 6 Y b N z H 7 z z b c z O 1 b k u Y k w j P e M t r C 4 t L y S z a 2 u r W 9 s b u W 3 d + p J O I x t X r N D L 4 y b F k u 4 5 w a 8 J l z h 8 W

e a b b 2 d 2 7
M j j s T S M 9 4 y 2 t L y y u p b N 5 d c 3 N r e 2 C z u 7 j T g c C o f V n d A L R c u 2 Y u b x g N U l l x 5 r R Y J Z v u 2 x p j 0 4 T + L N E R M x D 4 N 7 O Y 5 Y x 7 f c g P e 5 Y 0 l A d 9 U u 7 x a K R s l Q S 5 9 3 z N FIG. 1. Inset: Designations of the general TSM with singlet states |i (i = 1 − 3) and respective energies Ei or Êi = Ei − ∆, here ∆ = ∆0 − ∆.Matrix elements of Jz are denoted by α, β, α (one of them must vanish in the nonmagnetic state) and boson excitation operators by a † , b † , c † .The special fully symmetric TSM is defined by r = ∆/∆ = 1 and α = α.Transition arrows correspond to the thermal occupation differences pij shown in the main figure.Here r = 0.5, the energy scale is ∆0 in all figures.

↵
< l a t e x i t s h a 1 _ b a s e 6 4 = " l B y y 4 5 9 e 8 m a U S z 4 2 G N 9 q b
where we defined s = sin φ k , c = cos φ k and sa,b = sinh θ a,b k , ca,b = cosh θ a,b k .They fulfil the standard bosonic commutation relations [χ nk , χ † n k ] = δ nn δ kk (n = 1, 2).We can give explicit expressions for the transformation coef-t e x i t s h a 1 _ b a s e 6 4 = " E 4 F Z R k f + 7 U 6 4 m D m 9 h / b I T D 9 7i G U = " > A A A C z 3 i c h V H L S s N A F D 2 N r 7 a + q i 7 d F I v g q i R S 0 G X x h R u h B f s A X y T p W G P T J C T T S i 0 V t +7 c 6 p / p t 7 j w Z E w F L e K E y b 1 z 7 r l n 7 p 1 r B a 4 T S V 1 / S 2 l T 0 z O z c + l M d n 5 h c W k 5 t 7 J a j / x e a I u a 7 b t + 2 L T M S L i O J 2 r S k a 5 o B q E w u 5 Y r G l Z n P 4 4 3 + i K M H N 8 7 l Y N A X H T N t u d c O 7 Y p C d W N y 2 E k R 1 e 5 g l 7 U 1 c p P O k b i F J C s i p 9 7 x z l a 8 G G j h y 4 E P E j 6 L k x E / M 5 g Q E d A 7 A J D Y i E 9 R 8 U F R s g y t 0 e W I M M k 2 u G / z d N Z g n o 8 x 5 q R y r Z 5 i 8 s d M j O P T e 4 j p W i R H d 8 q 6 E e 0 H 9 z 3 C m v / e c N Q K c c V D m g t K m a U 4 g l x i R s y / s v s J s x x L f 9 n x l 1 J X G N X d e O w v k A h c Z / 2 t 8 4 B I y G x j o r k c a i Y b W p Y 6 t z n C 3 i 0 N V Y Q v / J Y I a 8 6 5 Y p y 5 W K n y o y 6 E W w y e u j H o z Z / D 3 U W a d + U D L L p f J 1 u V g 5 S Q e e p W 3 a o T 1 M 9 Z A q d E l V 1 O F A 8 5 l e 6 F W r a k N t o j 1 + U b V M m r N F P 5 b 2 9 A m h M 5 A K < / l a t e x i t > PM AF ⇠ s < l a t e x i t s h a 1 _ b a s e 6 4 = " I j y R M f r e 1 5 d D t e x i t s h a 1 _ b a s e 6 4 = " E 4 F Z R k f + 7 U 6 4 m D m 9 h / b I T D 9 7 i G U = " > A A A C z 3 i c h V H L S s N A F D 2 N r 7 a + q i 7 d F I v g q i R S 0 G X x h R u h B f s A X y T p W G P T J C T T S i 0 V t + 7 c 6 p / p t 7 j w Z E w F L e K E y b 1 z 7 r l n 7 p 1 r B a 4 T S V 1 / S 2 l T 0 z O z c + l M d n 5 h c W k 5 t 7 J a j / x e a I u a 7 b t + 2 L T M S L i O J 2 r S k a 5 o B q E w u 5 Y r G l Z n P 4 4 3 + i K M H N 8 7 l Y N A X H T N t u d c O 7 Y p C d W N y 2 E k R 1 e 5 g l 7 U 1 c p P O k b i F J C s i p 9 7 x z l a 8 G G j h y 4 E P E j 6 L k x E / M 5 g Q E d A 7 A J D Y i E 9 R 8 U F R s g y t 0 e W I M M k 2 u G / z d N Z g n o 8 x 5 q R y r Z 5 i 8 s d M j O P T e 4 j p W i R H d 8 q 6 E e 0 H 9 z 3 C m v / e c N Q K c c V D m g t K m a U 4 g l x i R s y / s v s J s x x L f 9 n x l 1 J X G N X d e O w v k A h c Z / 2 t 8 4 B I y G x j o r k c a i Y b W p Y 6 t z n C 3 i 0 N V Y Q v / J Y I a 8 6

FIG. 6 .
FIG.6.Critical temperature as function of ξs for ξa = 2.5 > 1 for the weakly symmetric case (r = 1).Here full black line corresponds to 2 nd order transition obtained from physical solution of Eq. (43) as in Figs.5,7and red symbols and dashed line denote a first order transition (cf.Fig.8).Here T * denotes the critical point.The dashed blue line gives the lower unphysical solution of Eq. (43).Inset: phase diagram of induced AF order (weakly symmetric r = 1 TSM) in the control parameter plane (ξs, ξa).For ξa < 1 the Tm = 0 phase boundary does not depend on ξa (cf.Fig.5and Sec.V).For ξa > 1 T * maps the line of critical points between 1 st /2 nd order regime and the red symbols trace the first order transition line Tm = 0.The blue lower corner corresponds to the inverse logarithmic decrease of Tm/∆0 evident from Fig.5.