Scaling theory of magnetic ordering in the Kondo lattices with anisotropic exchange interactions

30 Pages·0.23 MB·English

by  V. Yu. Irkhin

Scaling theory of magnetic ordering in the Kondo lattices with
anisotropic exchange interactions
9
9 V.Yu. Irkhin∗ and M.I. Katsnelson
9
1 Institute of Metal Physics, 620219 Ekaterinburg, Russia

n
a
J

2
2 Abstract

]
l
e
- Thelowest-order scaling consideration of themagnetic state formation in the
r
t
s
. Kondolatticesisperformedwithinthes f modelwithinclusionofanisotropy
t −
a
m for both the f f coupling and s f exchange interaction. The Kondo
− −
-
d
renormalizations of the effective transverse and longitudinal s f coupling
n −
o
c parameters, spin-wave frequency, gap in the magnon spectrum and ordered
[

moment are calculated in the case of both ferro- and antiferromagnets. The
2
v
0 anisotropy-driven change of the scaling behavior (e.g., critical value of g for
9
3
entering the strong-coupling region and the corresponding critical exponents)
5
0
8 is investigated numerically for N = 2 and analytically in the large-N limit.
9
/ Thedependenceoftheeffective Kondotemperatureonthebares f coupling
t
a −
m
parameter g weakens in the presence of anisotropy. The relative anisotropy
-
d
n parameters for both the s f and f f coupling are demonstrated to de-
− −
o
c crease during the renormalization process. The role of next-nearest exchange
:
v
i interactions for this effect in the antiferromagnet is discussed.
X
r
a
Typeset using REVTEX
1
I. INTRODUCTION
Anomalous rare earth and actinide compounds, including so-called Kondo lattices and
heavy-fermion systems, are studied extensively by both experimentalists and theorists [2–6].
It is now a common point of view that the most interesting peculiarities of electronic and
magnetic propeties of these systems are due to the interplay of the on-site Kondo effect
and intersite magnetic interactions. Whereas the one-impurity Kondo problem, being itself
very difficult and rich, is now studied in detail [7], the Kondo-lattice problem is still a
subject of many investigations [4–6,8]. Usually this problem is studied within the standard
s f exchange or Anderson models. On the other hand, strong effects of crystal field and
−
anisotropic interactions are expected in anomalous 4f and 5f-systems (see, e.g., [9]). These
effects can lead to anisotropic terms in the Hamiltonian. It is well known that the change
of symmetry of the s f exchange interaction modifies qualitatively the infrared behavior
−
in the one-impurity case [10,7,11]. Thus one could expect that similar effects should take
place in the lattice case. Therefore a question arises whether anisotropic contributions are
important also in the problem of competition of the Kondo effect and magnetism. It should
be noted that this question is relevant not only for magnetic systems, but also for models
with pseudospin degrees of freedom (e.g., for strongly anharmonic crystals demonstrating
band Jahn-Teller effect [12]).
TheoreticalinvestigationsoftheKondo-latticeproblemuseasarulemethodsappropriate
forcalculatinglow-temperaturepropertiesinthestrong-coupingregime(1/N-expansion[13],
slave-boson technique). However, thesemethods arenot convenient forthedescription ofthe
transition to the weak-coupling regime (in particular, even derivation of the standard Kondo
logaruithmsishereanon-trivialproblem[14]). Inourpreviouspaper[8]wehaveproposedan
alternative approach which starts from the weak-coupling regime and is based on summing
up leading divergent terms by the renormalization group method. We have investigated the
formation of magnetic state in the periodic s f exchange and Coqblin-Schrieffer models
−
with the f-subsystem being described by the isotropic Heisenberg Hamiltonian. The aim of
2
the present paper is the investigation of formation of the magnetic Kondo-lattice state for
various magnetic phases with account of the anisotropy in both the localized-spin subsystem
and s f exchange interaction.
−
In Sect.2 we discuss the theoretical model and calculate the logarithmic Kondo correc-
tionstothespin-wave spectrumofanisotropicmetallicferro-andantiferromagnets. InSect.3
wederive thelowest-order scaling equations fortheeffective transverse andlongitudinals f
−
exchange parameters and renormalized magnon frequencies. In Sect.4 we present a simple
analytical solution with magnon spectrum renormalizations being neglected, which is possi-
ble in the large-N limit of the Coqblin-Schrieffer model. In Sect.5 we discuss results of the
numerical solution of the full scaling equations for N = 2 in the presence of the anisotropy
in localized-spin system only and investigate new features which occur in comparison with
the isotropic case. In Sect.6, influence of the anisotropic s f coupling is considered.
−
II. THEORETICAL MODELS AND KONDO CORRECTIONS TO THE
SPECTRUM OF SPIN EXCITATIONS
To treat the Kondo effect in a lattice we use the s d(f) exchange model
−
H = t c† c +H +H = H +H (1)
k kσ kσ f sf 0 sf
kσ
X
where t is the band energy. We consider the pure spin s d(f)exchange Hamiltonian with
k
−
H = J S S +η J Sz Sz K (Sz)2, (2)
f q −q q q −q q − i
q q i
X X X
Hsf = − [IkSkz−k′(c†k↑ck′↑ −c†k↓ck′↓)+I⊥(Sk+−k′c†k↓ck′↑ +Sk−−k′c†k↑ck′↑)] (3)
kXk′αβ
whereS andS arespinoperatorsandtheirFouriertransforms,η > 0andK > 0arethepa-
i q
rameters ofthetwo-site andsingle-site easy-axis magneticanisotropy inthef-subsystem, re-
spectively. Notethatour consideration canbeformallygeneralized ontheCoqblin-Schrieffer
model with arbitrary N (cf. Ref. [8]) or to a more general form of the s f coupling pa-
−
rameter matrix [11]. For simplicity, we neglect k-dependence of the s f parameter which
−
3
occurs in the degenerate s f model due to the angle dependence of the coupling (see Ref.
−
[8]).Of course, in fact the f f exchange has usually the Ruderman-Kittel-Kasuya-Yosida
−
(RKKY) origin and is determined by the same s f coupling. Thus, generally speak-
−
ing, the anisotropy of the s f coupling and f-subsystem are not independent. However,
−
crystal-field effects are known to be more important in formation of magnetic anisotropy in
rare-earth metals than anisotropic exchange interactions [17] (in this case, the anisotropic
s f coupling is obtained by expansion in the parameter k r , r being the f-shell radius,
F f f
−
and contains, unlike (3), terms of the type (kS)(k′S)). On the other hand, the situation,
where anisotropy occurs in the s f coupling only, may be also considered: this corresponds
−
to the strong “direct” f f exchange (e.g., superexchange) interaction which is character-
−
istic for some f-compounds. In the Coqblin-Schrieffer model, which is more appropriate for
cerium compounds, crystal field results in occurrence of anisotropic s f coupling [15,16]
−
and new excitation branches [9,8]. For simplicity, we restrict ourselves to treatment of a
single magnon mode in the simplest s f model (1).
−
In the ferromagnetic (FM) state the spin-wave spectrum for the Hamiltonian (2) reads
ω = ω +ω (q), (4)
q 0 ex
ω (q) = 2S(J J ),ω = 2SηJ +(2S 1)K (5)
ex q 0 0 0
− − −
To find the Kondo logarithmic corrections to the spin-wave spectrum we calculate the
magnon Green’s functions in the model (1). For a ferromagnet we obtain to second or-
der in I (cf. the calculations in the isotropic case [18])
b b† = [ω ω 2 (J +J J J +ω /2S) b†b
hh q| qiiω − q − 0 q−p − p − q 0 h p pi
p
X
−1
n n ∂n
2S I2 k − k−q I2 k 2 (I2φ I2φ ) (6)
− Xk ⊥ω +tk −tk−q − k ∂tk ! − Xp k pqω − ⊥ p00 #
where we have taken into account kinematic requirements in the magnon anharmonicity
terms by introducing the factor of (2S 1)/2S at K (this replacement may be justified
−
by considering higher-order terms in the formal parameter 1/S), n = n(t ) is the Fermi
k k
function,
4
n (1 n )
k k+p−q
φ = − (7)
pqω
ω +t t ω
Xk k − k+p−q − p
The averages that enter (6) can be obtained from the spectral representation for the Green’s
function (6) to first order in 1/S and contain the singular contributions
δ b†b = SI2Φ (8)
h q qi ⊥ q
with
n (1 n )
ΦFM = k − k+q (9)
q (t t ω )2
Xk k − k+q − q
Expanding the denominators of (7) in the magnon frequencies we obtain the singular cor-
rection to the pole of the magnon Green’s function
δω = 2S(I2 +I2) (J J +J J +ω /2S)ΦFM
q − ⊥ k p − q−p q − 0 0 p
p
X
This result can be represented as
δω (q)/ω (q) = (I2 +I2)(1 α ) ΦFM (10)
ex ex − ⊥ k − q p
p
X
δω /ω = (I2 +I2) ΦFM (11)
0 0 − ⊥ k p
p
X
where 0 < α < 1. Passing into real space (see [8]) yields
q
2
α = J eikR [1 cosqR]/ J [1 cosqR], (12)
q R h itk=EF − R −
XR (cid:12) (cid:12) XR
(cid:12) (cid:12)
(cid:12) (cid:12)
In the approximation of nearest neighbors at the distance d, the quantity α does not depend
on q. For a spherical Fermi surface we have
2
α = α = eikR 2 = sinkFd (13)
q h itk=EF kFd !
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Hereafter we put α = const. Then we may use in further consideration of the scaling
equations a single renormalization parameter, rather than the whole function of q.
Now we consider a two-sublattice antiferromagnet (AFM) with the wavevector of mag-
netic structure Q,
5
Sz = ScosQR , Sy = Sx = 0
h ii i h ii h ii
(J = J < 0; 2Q is equal to a reciprocal lattice vector, so that cos2QR = 1, sin2QR =
Q min i i
0; only in this case we can retain the definitions of I and I in the local coordinate system).
⊥ k
Passing to the spin-deviation operators in the local coordinate system where
Sz = SˆzcosQR , Sy = SˆycosQR , Sx = Sˆx (14)
i i i i i i i i
we derive
1
H = const + [C b†b + D (b b +b†b† )]+... (15)
f q q q 2 q −q q q −q
q
X
C = S(J +J 2J (1+η))+(2S 1)K, D = S(J J ) (16)
q Q+q q Q q q Q+q
− − −
Diagonalizing (15) we obtain the spin-wave spectrum
ω2 = C2 D2 ω2 +ω2 (q) (17)
q q − q ≃ 0 ex
ω (q) = 2S(J J )1/2(J J )1/2, (18)
ex q Q Q+q Q
− −
ω2 = 2S(J J )[(2S 1)K 2SηJ ] (19)
0 0 − Q − − Q
where we have neglected a weak wavevector dependence of ω .
0
The Kondo correction to the spectrum (17) reads (cf. Refs. [8,19])
δω2 = 2 [I2ω2 +2S2I2(J +J 2J )(J +J J J ) (20)
q − ⊥ q ⊥ Q+q q − Q p Q+p − Q+q−p − q−p
p
X
+2(I2 I2)(C C D D )]ΦAFM
k − ⊥ q p−q − q p−q p
with
n (1 n )
ΦAFM = k − k+q (21)
q (t t )2 ω2
Xk k − k+q − q
For an antiferromagnet in the nearest-neighbor approximation (J = J ) we obtain
Q+q q
−
from (20)
δω2 (q)/ ω2 (q) = 2[I2 α(I2 I2)] ΦAFM (22)
ex ex − ⊥ − k − ⊥ p
p
X
δω2/ω2 = 2[I2 (1 α)s(I2 I2)] ΦAFM
0 0 − ⊥ − − k − ⊥ p
p
X
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where s = 4S2J2/ω2 1.
Q 0 ≫
Introducing next-nearest-neighbor interactions and putting for simplicity I = I = I
k ⊥
we obtain
δω2 = 2[ω2 8S2α(2)(J(2) J )(J(2) J(2))] ΦAFM (23)
q − q − q q − Q q − Q p
p
X
where
1
J(1,2) = (J J )
q 2 q ∓ q+Q
correspond to the contribution of nearest and next-nearest neighbors, and α(2) is given
q
(12) with the sum over the next-nearest neighbors. Provided that next-nearest neighbor
exchange interaction is ferromagnetic (so that the AFM ordering is stable), J(2) J(2) > 0
q − Q
and the next-nearest neighbors result in decreasing the Kondo suppression of the magnon
frequency, as well as nearest neighbors in the FM case. Then, instead of (22), we can use
phenomenologically (e.g., in the long-wave limit) the expression
δω (q)/ ω (q) = (1 α′)I2 ΦAFM (24)
ex ex − − p
p
X
δω / ω = I2 ΦAFM
0 0 − p
p
X
with α′ α(2)J(2)/J(1). In the opposite case of AFM next-nearest neighbor exchange the
∝
situation is more complicated. In particular, the simple collinear antiferromagnetism can
become unstable, and formation of the spiral structure is possible. Note the difference
between the FM and AFM cases by a factor of 2, which is due to violation of time-reversal
symmetry in a ferromagnet (terms that are linear in ω give a contribution).
The quantities (9), (21) determine also the singular correction to the (sublattice) mag-
netization
1
δS¯/S = δ b†b = I2 ΦFM,AFM (25)
−S h q qi − ⊥ q
q q
X X
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III. SCALING EQUATIONS
Using the perturbation theory results we can write down the system of scaling equations
inthecase ofthe Kondolatticefor variousmagneticphases. Their derivation intheisotropic
case is described in detail in Ref. [8]. As well as in this paper, we apply the “poor man scal-
ing” approach [20]. In this method one considers the dependence of effective (renormalized)
model parameters on the cutoff parameter C ( D < C < 0, here and hereafter the energy is
−
calculated from the Fermi energy E = 0) which occurs at picking out the Kondo singular
F
terms.
The renormalization of I is obtained from renormalization of the magnetic splitting in
k
electron spectrum, and of I from renormalization of the second-order contribution to the
⊥
electron self-energy (see corresponding perturbation expressions for a ferromagnet in Ref.
[21]). The renormalized I chosen in such a way coincides with the three-leg vertex (with
⊥
two electron lines with opposite spins and one magnon line) which yields the most natural
definition in a magnetically ordered case and agrees with the one-impurity scaling consider-
ation [7] To find the equation for the effective coupling parameters Iα (C) (Iα( D) = Iα)
ef ef − ef
we have to calculate the contribution of intermediate electron states near the Fermi level
with C < t < C + δC in the sums that enter expressions for the self-energies (which
k+q
include, unlike Ref. [8], magnon frequencies with a gap). Then we obtain
ω ω
δIk (C) = 2ρI2η( ex, 0)δC/C (26)
ef ⊥ − C −C
ω ω
δI⊥(C) = 2ρI I η( ex, 0)δC/C
ef ⊥ k − C −C
where ρ is the density of states at the Fermi level, ω is a characteristic spin-fluctuation
ex
energy, ω is the gap in the spin-wave spectrum, η(x) is a scaling function which satisfies
0
the condition η(0) = 1 which guarantees the correct one-impurity limit. For both FM and
AFM phases we have
ω ω
ηFM,AFM(− Cex,−C0) = h(1−ωk2−k′/C2)−1itk=tk′=EF (27)
8
where the magnon frequencies are given by (4), (17). The corresponding analytical expres-
sions are presented in Appendix.
The C-dependent renormalizations of the spin-wave frequencies and ground-state mo-
ment are obtained in the same way as in the isotropic case [8] from (10), (20) (22), (24),
(25) and expressed in terms of the same scaling functions. Introducing the dimensionless
coupling constants
gα (C) = 2ρIα (C), g = 2ρI
ef − ef α − α
k
(we will drop sometimes the index , but not , so that g (C) g (C)) and replacing
k ⊥ ef ≡ ef
g gα (C), ω ω (C), ω ω (C) in the right-hand parts of (26) and expressions for
α → ef ex → ex 0 → 0
δω (C),δω (C) and δS (C), we obtain the system of scaling equation
ex 0 ef
∂gk (C)/∂C = [g⊥(C)]2Λ (28)
ef − ef
∂g⊥(C)/∂C = gk (C)g⊥(C)Λ (29)
ef − ef ef
[gk (C)]2 +[g⊥(C)]2 /2 FM
∂lnω (C)/∂C = aΛ/2 { ef ef } (30)
ex
×[g⊥(C)]2 α [gk (C)]2 [g⊥(C)]2 AFM
ef − { ef − ef }
 [gk (C)]2 +[g⊥(C)]2 /2 FM
∂lnω (C)/∂C = bΛ/2 { ef ef } (31)
0
×[g⊥(C)]2 s(1 α) [gk (C)]2 [g⊥(C)] AFM
ef − − { ef − ef }
∂lnSef(C)/∂C = [ge⊥f(C)]2Λ/2 (32)
where
Λ = Λ(C,ω (C),ω (C)) = η( ω (C)/C, ω (C)/C)/C, (33)
ex 0 ef 0
− −
and
2(1 α) FM 2 FM
a =  − , b =  (34)
 1 α′ AFM 1 AFM
−
The integral of motion ofthe system (28), (29) reads
[gk (C)]2 [g⊥(C)]2 = µ2 = g2 g2 = const (35)
ef − ef k − ⊥
9
so that the equation (28) takes the form
∂g (C)/∂C = [g2 (C) µ2]Λ (36)
ef − ef −
IV. ANALYTICAL SOLUTION IN THE LARGE-N LIMIT
In the formal large-N limit in the Coqblin-Schrieffer model where 2 N in Eqs.(30)-
→
(32)) we can neglect renormalizations of magnon frequencies (note that the true large-N
limit in the FM case is somewhat different since symmetry of spin-up and spin-down state
contributions is violated for N > 2, see Ref. [8]). Note that the same approximation is valid
for N = 2 provided that g is well below the critical value g for entering the strong-coupling
c
region. Then, ontaking into account (A1),(A4), equation (36) can be integrated analytically
to obtain
1 C dC′ ω ω
0
[arctanh(µ/g (C)) arctanh(µ/g)] = G(C) = η( , ) (37)
µ ef − − C′ −C′ −C′
Z−D
GFM(C) = ln C/D ((1+w)/2)[(C/ω 1)ln 1 ω/C (C/ω +1)ln 1+ω/C ] (38)
| |− − | − |− | |
+(w/2)[(C/ω 1)ln 1 ω /C (C/ω +1)ln 1+ω /C ] 1
0 0 0 0
− | − |− | | −
1
GAFM(C) = ln C/D [(1+w2)(C2/ω2 1)ln 1 ω2/C2 (39)
d=3 | |− 2 − | − |
w2(C2/ω2 1)ln 1 ω2/C2 +1]
− 0 − | − 0 |
1
GAFM(C) = [θ(C2 ω2)+θ(ω2 C2)]ln( ( C2 ω2 + C2 ω2 )/D) (40)
d=2 − 0 − 2 | − 0| | − |
q q
+θ(C2 ω2)θ(ω2 C2)ln(ω /2D)
− 0 − ex
where
ω +ω FM
0 ex
ω =  (41)
 ω02 +ω2ex AFM
q
The scaling trajectories described by (37)-(40) are shown in Fig.1 for µ = 0,w = 0 and in
6
Fig.2 for w = 0,µ = 0. Note that these pictures describe adequately the case N = 2, since
6
g = 0.15 is considerably lower than the critical values.
10