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A dynamic theory of the nonmagnetic phase of singlet magnetsF.P. OnufrievaL . Mechniko vState University of Odessa(Submitted 1June 1987)Zh. Eksp. Teor. Fiz. 94,232-246 (February 1988)A nonlinear microscopic theory of the low-temperature nonmagnetic phase of singlet magnets isconstructed, which makes it possible to calculate the spectrum of collective oscillations, the spincorrelation functions, the asymptotic behavior of the thermodynamic functions, the magnitude ofthe critical field and the criterion for ferromagnetism.

1. INTRODUCTIONIt is well known that many magnetic compounds con-

sisting of ordered arrays of magnetic ions which interact viaan exchange interaction of fixed sign and intensity neverthe-less remain nonmagnetic at all temperatures down to T= 0.The reason that magnetic order is absent in these cases is theexistence of a single-ion anisotropy (S IA) which is strongcompared to the exchange force.

As was shown in Ref. 1, these structures are spin-or-dered despite the absence of magnetic order; furthermore,this ordering has tensor characteristics. For this reason, theproperties of such structures differ from those of pure (dis-ordered) paramagnets (PMs); n particular, their propertiesare found to be close to those of antiferromagnets (AFMs)(specifically, uniaxial AFMs in a field parallel to the anisot-ropy axis). In order to describe such tensor structures, ageneralized Maleev-Dyson transformation was proposed inRef. 1 and a spin-wave theory was developed; however, inview of its crudeness, this theory can only be regarded as afirst step in the investigation of such systems.

The goal of this article is to construct a first-principlesmicroscopic theory of these PMs at low temperatures, whichcorrectly takes into account the quasiparticle interactions.The model we will investigate is a spin structure with iso-tropic exchange and an "easy plane" (EP) SIA in an exter-nal field perpendicular to the EP. This structure is describedby the Hamiltonian

A .J(1)

(The sign ofD is chosen so that the ground state of an isolat-ed ion is a singlet). For the case that the ratio {=D /Wo(Jo 2, 0 exceeds some critical value tC,in the molecu-lar-field approximation lC,1 when S = 1) , this modelgives rise to a nonmagnetic quadrupole-ordered (Q O)ground state in the region of fields 0

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3. LOW-TEMPERATURE MODIFICATION OF THEDIAGRAMMATICTECHNIQUEFOR HUBBARD OPERATORSA. Rules for the diagram technique

It is well-known that in the presence of tensor interac-tions, in particular SIA, Wick's theorem for spin opera-tors-which lies at the base of the Vaks-Larkin-Pikin(VLP) technique-does not hold; however, an analogoustheorem holds for Hubbard operators under the conditionthat the Hamiltonian of the system can be written in localcoordinates. The diagrammatic technique of Refs. 3 ,4 isbased on this modified Wick's theorem.In the low-temperature limit, the diagrammatic tech-nique for Hubbard operators can be simplified and is close tothe diagrammatic technique for Bose system^.^ Just as in thelow-temperature variant of the technique for spin opera-t o r ~ , ~his simplification is possible because the contributionfrom diagrams containing disconnected elements in blocks isfound to be exponentially small at low temperatures, and weneed take into account only diagrams containing Green'sfunction lines which are continuously linked with each oth-er. The difference lies in the fact that these Green's functionsare defined by Hubbard operators and not by spin operators.In what follows, the following matrix Green's function willbe important:

with components

Gb.(l, T; ' , T ' ) = < T X , ~ ' ( ~ ) X ~ ~ ~a') >,G,,(l,; l', T') =(TXlO' T)X,.'O(T')>. ( 5

In zero-order approximation its Fourier transform equals

where

There are also different diagrams of vertex-block type whichin the present case are determined by pairing of the Hubbardoperators.On the whole, the topology of diagrams and generalrules of the diagrammatic technique are found to be close tothose for the low-temperature modification of the VLP tech-

nique. These rules are as follows:1. Each diagram is made up of lines corresponding tothe Green's function ( 7) linked continuously with each oth-er and included in a block.2. A factor b, or b, is associated with the block, de-pending on the subscript of the Green's functions standing atthe right end of the link.3. Within the block all the lines are "thick," i.e., eachbare Green's function, as determined by Eqs. (7) , s replacedby a spin-wave function. The equations for the Green's func-tion in the spin-wave approximation take the form

where the spin-wave frequencies and functions u, ,u, equal

they are ^obtained from the relation between the Green'sfunctiosG(k,w , and Larkin's diagrams for the irreduciblepart of Z (k , w, :

The form of the interaction matrix is

[The potentials V "", V "', V '", V bb are defined by Eq.( 3 ) ] and the zero-order approximation for Z (k , w , ) :

30 )k, on ) & (O ' (k, on).In this case, we introduce through the relations

G,, Gab K,, K,,( Gba Gbb)=(Kb., Kbb (! ) (11)the Green's function K,, defined without the factors b,, bbwhich are characteristic of the block as a whole. The Green'sfunction (8) is represented by boldface continuous lineswith subscripts p, v at their ends ( p , ~a,b) .

4. Wavy lines with subscripts denote the interactionsV," ( p v = a,b) or V" as defined by the Hamiltonian (3).5. The types of vertex blocks are detemined by all possi-ble pairings in the SU(3) algebra. There are three types ofvertices: vertices with one entering or leaving line, vertices inwhich two lines intersect and finally vertices in which threelines intersect. Since the internal vertices, i.e., vertices of thelatter two types, are always linked by interaction lines in thelow-temperature technique, by indicating the subscripts ofthe latter we simultaneously and unequivocally indicate thetypes of operators corresponding to a given internal vertex.The type of operator corresponding to an external vertex isdetermined by the Green's function subscript.6. A factor of 2 goes with vertices in which threeGreen's functions of the same type ( p = v = v') converge,

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while for all the others this factor is 1 .7. For low T the relations b , =:- , b b-- 1 are approxi-

mately valid.B.Green's function and p olarization operator in the Bornapproximation

Taking into account the quasiparticle inte~actionseadsto a renormalization of the Green's function K ( k , w , ), de-fined as usual by the Dyson equation

hwhereK is the matrix of the unperturbed spin-wave Green'sfunction ( 8 ) , and f i is the matrix polarization operatorwhose components n,, are described as usual by graphswhich cannot be divided along a single spin-wave line, andwhich have entering vertices of typep and exiting vertices oftype v. The solution to the Dyson equation is

The explicit form of the graphs for components of thepolarization operator I I , , , I I , , in Born approximation isshown in Fig. 1 . To it correspond the analytic expressions

The equations for I I , , and n,, are easily obtained from theequations for no, nd I I , , respectively by replacing the sub-script a by the subscript b and conversely. The correlationfunctions entering into ( 1 3) are defined by the relations

nP= K ( o n w , p= i m z Kb b p, on)imnr,r-0-

0"r-0' "'"

pp= i m z K,(p. on)iun' = imZ K b a ( p , n) im"7-0-

0" r-0' "n

and equal

C.Spectrum of collective excitations, critical field andcriterion for ferromag netism in Born approximationThe spectrum of collective excitations renormalized by

the Born _approximation s determined by the zeroes of thefunction R - ,which is related to the polarization operatorby Eq. ( 1 2 ) ,or in explicit form by the equation

F I G . 1. Grap hical representation of compon ents of the polarization oper-ator in Born approximation.

Substituting the values of the components of the polarizationoperator, we obtain explicit expressions for the two branchesof the spectrum in the Born approximation, which we writeout in the limit of small quasimomentum:

{1f -6("HC1 T )1. ( 1 7 )The constants entering into ( 17) equal

.6(' )HClT )=--I { (3 -W. ) (u . '+u- . l ) +6ypupup)2 5 N

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The remaining symbols are: y, =J k / J , , H + h /D. Themeaning of the notations in Eq. ( 1 7 ) becomes clear if wenote that the condition for softening of the frequency w , fork = 0 determines the line of critical fields:

i.e., a line of second-order orientational phase transitions(OPTS) from the nonmagnetic phase to the angular ferro-magnetic ( FM , ) phase (see Fig. 2 ) .

The numerical value of the correction S ( f ) o the barecritical field H,, at T=0 depends on a single parameter f ,which decreases monotonically as f grows on the interval(&, , 03 ) . The bare value ccr s f :: = 1 . Since S( ) = 0 ,and S ( 1 ), which we have calculated for a simple cubic (SC)lattice, equals

6 ( I ) =0.261, ( 2 1 )we obtain the upper and lower estimates O < S ( f )~ 0 . 2 6 1 .The value (2 1 ) allows us to determine more accuratelythe value of tC , tself in Born approximation. In fact, accord-ing to (18) and the condition a,, 0 )=0 for f = c , , wehave

l c r = f 7 ' ( f c r ). ( 2 2 )

FIG. 2. Phase diagram at T=0. 1-second-order O P T line h =i,,0 )between th_enonmagnet ic QO phase a nd the FM, phase, as described byEq. ( 18 ) (h,, (0 )r H ,, ( 0 ) , 2-the second-order O P T line h = h, , ( 0 )between the angular a_nd collinear (FMII ferromagnetic phases, de-scribed by the equation h C 2 / J ,= 2{ , see Ref. 1 ( h , ( 0 )= h , ( 0 ) . A s themulticritical point with coordinates ( 0 ,lc, .

Using estimate (2 1 ) at f =6 bz', we obtain the Born approx-imation value

a value which differs considerably from the bare f c , . Wemust then ask how accurate this value is; a discussion of thisquestion will be presented in Sec. 4 .

Let us note that the value of fc , is an important charac-teristic of the system. First of all, it constitutes a more pre-cise criterion for ferromagnetism, i.e., it is the value off =D /2J0 above which ferromagnetism is absent all the waydown to T =0 (in the absence of an external field). Second-ly, this quantity determines the critical point of the phasediagram (point A of Fig. 2 ) , .e., the point of nonanalyticityof the spectrum at k = 0 , T =0 .

In point of fact, it follows from ( 17) that a t the pointwhereg,, ( 0 ) = 0 the dispersion law of the soft mode is lin-ear in k for small k , whereas at the other points on the phaseline, say, point 1 where a,, 0 ) 0 , it is quadratic in k ask - 0 . (The nonanalyticity of the spectrum leads, e.g., to thefact that the values of such quasiparticle characteristics asthe sound velocity, scattering amplitude etc. depend at thispoint on the order in which the limits k - 0 , Af = c r - - 0are taken; this question will be discussed in detail in Sec. 4 . )What is important for us now is that the different qualitativecharacter of the dispersion law for H ,, -H g a k g H , , andH , , < a k < + 1 determines the different character of thethermal corrections to the spectrum and the critical field inthe two limiting cases: for 8%H,, ,where they equal

and for H ,, - H < 6 < H ,, ,where Eqs. ( 17) reduce to

and SA - ( T ) AA (T ) equal

((( p ) is the Riemann zeta function, a = ( 1 - k ) / k 2 ) . Forall the remaining cases the thermal corrections are exponen-tially small in T.

It also follows from ( 2 4 ) , 2 6 ) , nd ( 2 0 ) hat far fromthe point A the thermal corrections to the critical field are- 3 " , whereas near this point we have

H c , ( T ) - T . ( 2 7 )D.Born corrections to the free energy and thermodynamicfunctions

The basic spin wave corrections to the free energy aregiven by closed-loop graphs, with which is associated the

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standard expression for the free energy of noninteractingquasiparticlesP = ~n I - xp (- PWY,)I.

k)LThe correction due to interaction in the first Born approxi-mation is determined by the diagrams from Fig. 3. The cor-responding analytic expression is conveniently representedusing the components of the polarization operator ( 13 :AFw =- l z n, (k) k+nb b k) kB N k

=A F (0 ) +AF ( T ) ( 2 8 )For the temperature-independent corrections we obtain theequation

and for the temperature-dependent corrections the equation

( 3 0 )where

while the expression for rzBk,p) differs from ( 3 1 by thereplacement up+ , , up+u p .An explicit calculation of the temperature correctionAF( T ) gives for H,, %0,

FIG. 3. Graph for the correction to the free energy due to interac tions ofthe collective excitations (Born app roximation).

while for H,, ( 8 ,

In the same way, for the anharmonic corrections to the mag-netization and spin-induced specific heat-which are relat-ed to the free energy by the relations M , = b'F/b'h,C , = - Tb' 2F /a T2 -~ e obtain for H,, 9 3

and for H cl4

In the limit H,, - H < 8 < H c , , we obtain from ( 3 4 )

In comparing Eqs. ( 3 5 ) , ( 3 6 ) with the results of Ref. 1, wenote that the characteristics of the temperature dependenceof the Born and spin-wave corrections to the magnetization,susceptibility and specific heat are the same in the limitH,, (8, but for H,, - H < 8 4 H,, they differ qualitatively.It is interesting that in the latter case the Born correction tothe specific heat is found to vary as an integer power of T,which allows us to observe its contribution experimentallyagainst the background of the half-integral powers of thespin-wave results.We note that formally the Born correction SM , (T)goes to infinity at the OPT point H =H,, . However, thisresult must not be understood literally, because in the imme-diate vicinity of a second-unit OPT point our perturbation-theory calculation becomes inapplicable; it only points upthe strengthened role of anharmonicity in the present case. '

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As for the actual behavior of the system in the fluctuationregion, we can say the following: it is well-known that forarbitrarily small but finite T, n the immediate vicinity of theOPT point the quantum system behaves cl a~sica lly. ~orre-spondingly, in our case within this region we will have scal-ing behavior characteristic of the three-dimensional classi-cal XY model.

To conclude this section, we note that the quantitiesT;"(k,p), I?{@ (k ,p ), T;@(k,p) entering into (30) have themeaning of forward scattering amplitudes of a - andfi-quasi-particles. (Beginning at this point in the article, we willtransform the notation w +wg, w,+ -+wf.) In the long-wavelength limit they equal

(The upper sign refers to the casep = v, the lower to p #v. )We will say more about scattering amplitudes in the follow-ing section, where an alternate method will be used in theinvestigation-the quasiparticle formalism. In this sectionwe will also present an investigation of the ground statewhich is more accurate than the Born approximation, andcompare the results of the two different approaches used.4. QUASIPARTICLE FORMALISM

As usual, for low T the original spin Hamiltonian can bereduced to an effective Hamiltonian for interacting Boseparticles. In order to do this, we must use a transformationfrom spin to Bose operators, which in the case of systemswith tensor interactions is implemented by using the gener-alized Maleev-Dyson transformation presented in Ref. 1. Itconsists of a transformation to local coordinates and fromthem to Hubbard operators, and the introduction of Boseoperators into the latter according to the equations

Substituting (381 into ( 3 ) and passing to momentum space,we obtain

A. GroundstateIn order to treat the ground state more precisely, it is

convenient first of all to carry out the linear u- u transforma-tion (a , , b, +a,, fik ) so as to diagonalize the quadratic partof the Hamiltonian. In this procedure, the transformationcoefficients are determined by the condition that the nondia-gonal terms reduce to zero in the bare quadratic Hamilto-nian P 2 . e can proceed more consistently by introducinga self-consistent transformation which takes into accountthe fact that after reducing the anharmonic terms to normalform there appear corrections to the quadratic part whichthemselves depend on the. u-u transformation functions. Byimposing the condition that the full Hamiltonian, not thebare quadratic Hamiltonian, be diagonalized, we obtain in-tegral equations for the u- u transformation functions. More-over, because the original Hamiltonian contained a finitenumber of anharmonic terms, we can thus determine thesefunctions exactly.

What complicates things is the fact that the renormal-ized quadratic Hamiltonian is formally not self-adjoint. Inparticular, the coefficientsof the Hermitian-conjugate termsa,P , nd f i ?,a: are found to be different, which pre-vents us from selecting conditions on the functions u,, u,which reduce the coefficients of both terms to zero at thesame time.

The reason these coefficients are different is that trans-formations of Maleev-Dyson type from spin to Bose opera-tors are non-Hermitian. It is easy to confirm that in order toexactly diagonalize the quadratic part with a linear u-utransformation it is necessary also to choose non-Hermitiantransformations, i.e., to determine the relation between theoriginal operators a,, b, and the eigenoperators a,, Bk forthe collective excitations by the equations

without assuming that u , = U,, , = V, . In order to obtaincertain general relations, let us perform this transformation,assuming that the original Hamiltonian is of the form

while not yet specifying the form of 2 , B , ,and e, . Afterthis transformation, we obtain

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From the condition that the coefficients of the anomalousterms a,P -, nd B +,a: reduce to zero, we obtain theequations2iikvkUk+8kvkz+CkUk2=0, 2Ak~kvk+Bk~k~+CkVk~=0,

(46)and from the commutation relations the identity

Eqs. (46), (47) allow us to determine the explicit relationbetween the spectrum of collective excitations, as well as thefunctions u,, v,, U,, V, , with the coefficients i , , , , andEk

G:"=~~T/Z= (x.L~-B~c~)b ~ h , (48)

Let us turn now to the Hamiltonian (39 )-(42) . Thebare quadratic partZ2as the form (40) , whereAk=D-Jk, B~=c~ - - - J~ . (50)

The Hamiltonian 3, as the form (41 . Carrying out thetransformation (43) on this Hamiltonian, followed by a re-duction to normal ordering in the operators a,, fl, , andisolating the quadratic part and comparing the result withthe form (45) , we obtain corrections to the coefficients A,,B,, and C,:

Proceeding in the same way with the Hamiltonian Z 6 , eobtain

The total values of the coefficients in the quadratic Hamilto-nian equal

Thus, the renormalized excitation spectrum for T=0 is de-termined by Eq. ( 48 ) after substituting in (53 ). If in the

course of these transformations we extract terms which donot depend on the operators a, , 0, ,we obtain an expressionfor the ground-state energy:

The functions u, , v,, U,, V , also determine correctionsto the correlation functions for the spin and quadrupole op-erators at T= 0. They can be obtained by using the connec-tion between the spin and quadrupole operators and the op-erators a,, b , ,after passing in accordance with (43) to theoperators a , , p, and extracting the temperature-indepen-dent part. For example, the quantity AQ = Q - QhO)(whereQo= (3 S ' 2- 2) is the quadrupole order param-eter' ) characterizes the deviation of the ground state from astate with full quadrupole order (S" = 0, Qo= - 2 due tothe presence of zero-point energy of the oscillations; we ob-tain for this quantity

It follows from (49 ) that the functions u , , v,, U,, V, them-selves are determined by a system of three integral equations:

since all the parameters entering into the right side of (56)can be uniquely expressed in terms of the three independentcombinations standing on the left side.In view of the complicated form of the integral equa-tions, it is expedient to solve them via an iterative method.The zero-order approximation obtained by using the unper-turbed values A,, B, , and C, from (50) yields functionswhich coincide with those given by ( 9) . Using them, we canfind more precise values in first approximation to the coeffi-cients2,, 5 , and e, ; hese functions in turn determine thespectrum in first approximation via ( 48), and also by substi-tuting into the right side of (56) the transformation func-tions (43) in second approximation. The iterative process isrepeated until the desired accuracy is attained.It is obvious that the computed results will depend onthe parameter f . The maximum value of the anharmoniccorrection is attained for minimum f , .e., for f = fc , seeFig. 2) . Mathematically this follows from the equations wehave obtained; physically it corresponds to the increasinglyimportant role of anharmonicity as we approach the multi-critical point A. So as to estimate the upper bound of thecorrections, let us carry out an explicit calculation forf = c , for a SC lattice), thereby obtaining a self-consistentvalue for l c , tself.

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Carrying out this calculation, we verify that the iter-ation process attains an adequate degree of convergenceafter only three steps. The result:

can be considered exact to three decimal places and used forfurther calculati6ns based on the equations obtained in thispaper, e.g., for numerical estimates of the ground-state ener-gy, correlation functions etc.In particular, we obtain the following criterion for fer-romagnetism, along with an upper limit on the deviation ofthe ground state from the state of full quadrupole order

In conclusion, we note that if we use the zeroth-orderapproximation (9) in Eqs. (48)-(54) for the u-v transfor-mation functions and then limit ourselves only to terms de-rived fromZ4n these equations, we obtain exactly the re-sults of the previous section for the T= 0 spectrum and thecorrection AF(0) to the free energy.0.Thermal corrections and scattering amplitudes

Besides the terms quadratic ina,,0, nvestigated ear-lier, the Hamiltonian contains fourth- and sixth-order termswhich describe quasiparticle scattering processes, e.g.,p4has the form1$j4=- {Toma x ,k; p, q)a.+aPtaka.

whereZi contains terms which do not preserve the numberof quasiparticles. Without writing out the full expressionsfor the scattering amplitudes at arbitrary anglesTga(x,k;p,q),rtp (x,k ;p,q) , Tgo(x,k;p,q), we note that inthe limit x =p, k =q, and also when using the zero-orderapproximation for the functions u, ,u, ,U , , V , , hese func-tions lead to the Born-approximation forward-scatteringamplitude (31 :Correspondingly, by investigating fourth-order scatteringprocesses in Born approximation we obtain thermal correc-tions to the spectrum, critical field, free energy, etc., whichprecisely coincide with the results obtained using the dia-gram technique. Taking into account 4-particle processes tohigher order leads only to changes in the numerical coeffi-cients in Eqs. (24)- (27), (32)-(36) without changing thecharacter of the temperature dependence (except for the im-mediate vicinity of the OPT point H =H,, ). Taking intoaccount 6-particle scattering processes leads to thermal cor-

'rections which are higher order in T.Let us dwell on the behavior of the scattering amplitudein some detail. We remark that, as follows from (37),rr(p,k;p,k) a l / k p at the point A (where H,, = 0) andrr(0,0;0,0) = constant at all the remaining points on theline H = H,, (see Fig. 2 ) . On the other hand, in tbe easy-plane phase ( the FM, phase in the designation of Fig. 2,where the continuous symmetry of the Hamiltonian ( 1 ) rel-ative to rotation around thez-axis is spontaneously broken),according to Adler's principles the scattering amplitudewith the involvement of Goldstone quasiparticles with fre-quency w, in the long-wavelength limit must have the form

Let us discuss how the above-mentioned behavior of thescattering amplitude at the boundary of the FM, phaseagrees with this requirement. We note first of all that if forthe FM, phase the Goldstone quasiparticles have the dis-persion law w, = ck, then at its boundary H = H,, the ve-locity of sound reduces to zero and w, =g k for small k.This follows from Eq. (58) of Ref. 1 for the FM, phase, andagrees with the behavior of the frequency of low-lying modeswhen we approach from the QO phase side [see Eq. ( 17)and is typical behavior for a second-order OPT. Correspond-ingly, from (61 we obtain the requirement I? (0,0;0,0)= constant, which is also satisfied by the amplitudes wehave found.The exception is at the multicritical point A , for whichthe behavior is characteristically unique. On the one hand, itfollows from ( 17) and the conditionK,0) = 0 that at thispoint the velocity of sound remains constant2' (w : = ck), sothat we must have r$"(0,0;0,0) =0 at the point h = 0,f = cr - .On the other hand, this is a point of nonanalyti-city in the spectrum at k =0, so that the result at zero quasi-momentum, generally speaking, will depend on the path ofapproach to this point. Direct calculations for the FM,phase carried out using the same diagram technique and tothe same order of approximation as in Sec. 3 for the QOphase (the corresponding computations will be presented ina separate paper), show that on the lineh = 0 the scatteringamplitude for Goldstone quasiparticles has the form (61)[and consequently r(0,0;0,0) = 01 for any fs(O,fC,); how-ever, as we approach the point A this form is valid over andever-decreasing interval of values of quasimomentum,namely those for which o,, w , Af =fc r - 6. At the pointh = 0 , f = lcrtself, Eq. (61) can retain its validity only fork = p = 0. However, this depends on the order in which thelimits are taken. If the limit is taken for finite values of Af,i.e., if we first set k = p =0, and then Af = 0, then the Adlerprinciple is fulfilled and r(0 ,0;0 ,0) = 0.When the limits aretaken in the opposite order we obtain a different answer, inwhich the form is r(p,k ;p,k) - /kp, which coincides exact-ly with the result (37 ) for the scattering amplitude of the QOphase if we set f = cr in it. Hence, for this value of f wesimultaneously satisfy both the conditions for joining theresults of the QO phase with the results for the FM, phaseand the Adler principle for the FM, hase.It should be noted that when we calculate the velocitiesof sound of both modes in the spectrum of the QO phase, i.e.,c = lim am:/dk , we find an analogous dependence of theresult on the order of taking the limits k-0,

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A6 =6-&, -0. Actually, it follows from ( 17) thatc- k /(H , + a k 2 , I 2 , and consequently c equals zero or aconstant depending on the order in which we let k-- 0, A6- 0(becauseH f , 0 as Af - 0 ) .In conclusion we point out that the theory we have con-structed applies equally to the cases of bulk FM or AFM. Inboth cases, for H H,, ,where ordering arises in theXYplane: ferromagne-tic for J,,>0 and antiferromagnetic for J ,