- Email: [email protected]
Competition and transitions of the ground state properties of the impurity Anderson model with multi-electron occupancy
Solid State Communications, Vol. 101, No. 11, pp. 791-796, 1997 Q 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/97 $17.00+.00
PII:
s0038-1098(%)00739-9
COMPETITION AND TRANSITIONS OF THE GROUND STATE PROPERTIES OF THE IMPURITY ANDERSON MODEL WITH MULTI-ELECTRON OCCUPANCY 0. Sakai,a S. Suzuki” and Y. Shim&u& “Department of Physics, Tohoku University, Sendai 980-77, Japan department of Applied Physics, Tohoku University, Sendai 980-77, Japan (Received
11 November
1996; accepted 26 November
1996 by T. Tsuzuki)
To study the Kondo effect of the multi-electron occupancy systems (KEMES), the impurity Anderson model with tetrahedral crystalline field and intraatomic exchange interaction is investigated as an example. The low energy properties and the magnetization are examined based on Wilson’s numerical renormalization group method. When the hybridization strength is increased by fixing the lowest atomic state as the non-Kramers doublet (f2)F5, the ground state shows successively the doublet-like, the crystalline-field-singlet-like and the f O-singlet-like properties. The first one is the non-Fermi liquid state of the two-channel Kondo type. The last two are Fermi-liquid states, separated by transition with anomalous lowering of the low energy scale. These results suggest more varieties of KEMES than those previously expected. 0 1997 Elsevier Science Ltd. All rights reserved Keywords: D. Kondo effects, D. heavy fermions, D. valence fluctuations.
The study on the Kondo effect of magnetic impurities in metal has a very long history [l]. It has been established that the system has the singlet ground state and the low energy properties are given as the local Fermi liquid [2]. Yamada and Yosida have developed the local Fermi liquid theory based on the perturbation expansion of the Coulomb interaction for the impurity Anderson model [3,4]. They pointed out that the impurity system will not show transition. Actually, the opposite extreme limits, the weak perturbation limit and the s-d limit, have singlet ground state. This fruitful approach has brought many important developments [l] and has been justified by analytic solution [5]. Nozieres and Blandin have pointed out that in a restricted case of the multi-electron occupancy system, the ground state can have non-Fermi liquid properties, which is classified to the two channel Kondo model (TCKM) type [6]. But this case has been considered to be a pathological one. However, Cox has noted the possibility that a model for the electronic state of ions with f 2con@uration, such as df in the crystalline field, can be mapped on the TCKM [7,8]. Recently, Koga and Shiba have studied the problem extensively and shown that there are appreciable parameter regions where the
low energy properties are actually mapped on the TCKM even when realistic model for the crystalline field splitting is used [9, lo]. These approaches stand on the Schrieffer-Wolff (SW) transformation restricting the effective manifold to f 2 or to f 3 configurations. When the atomic electron-electron interactions are very weak in the Anderson model, the ground state should be the Fermi liquid state. Therefore, there arises the question that the non-Fermi liquid state does really appear for the Anderson model when the main atomic contiguration is the multi-electron occupancy state. If it does, how does the transition between the Fermi liquid and the non-Fermi liquid states occur when the atomic interaction becomes gradually strong. The aim of the present report is to check these points and show that nonFermi liquid state really appears. The transition between the two states occurs as the simple level crossing-like one in the case studied here. Moreover, the transition between singlet states, the crystalline-field-singlet (CFS)-like and the f O-likestates, occurs with anomalous lowering of the characteristic energy scale. We apply Wilson’s numerical renormalization group (NRG) method [2, 111 to the impurity Anderson model with crystalline field energy splitting of the localized
791
792
GROUND
STATE PROPERTIES
OF THE IMPURITY
orbits. The atomic Coulomb and the exchange interactions are assumed to be the j-j coupling type. In this report we vary the hybridization strength by fixing the atomic interaction parameters to avoid confusions arising from the modification of the atomic energy level structures. We show that the low energy excitation properties change suddenly at some critical values of the hybridization. This unexpected result suggests unknown contents of the magnetic impurity problem with multi-electron occupancy and may relate to complex behaviors of the U ion systems[12, 131. As an illustrative example, we consider the model with the j = 512 orbits in the tetrahedral point symmetry. The j = 5/2 orbits split into three doublets, (Fe + I’f’+ I’?‘) [14]. The model Hamiltonian is given by the form
ANDERSON
MODEL Vol. 101, No. 11
states are given as
iflr$? 5) = I+-5/2)( _+ al) + JT3/2)( -+ br),
(5)
IfIr,, 5 ) =
(6)
]*1/2),
1f11’$2’, I-t->=
]“5/2)(
T b,) + ]13/2)(
If 21’5,+)=]?1/2,
k al),
(7)
?3/2)(--&+]?3/2,
-t 5/2)( - b2) + 12512, + 1/2)(c2),
(8)
)f 2r4)= 1- i/2, - 3/z&@-t 11/2,- 5/2)&) -k(3/Z
l/2)( - dz) + 1512, - l/2)( - ez),
(9)
bl = a, a2 = 0.3395, b2 = where al - Js/6, 0.1874, c2 = 0.9218, d2 = 0.4041 and e2 = 0.5802. The right-hand side terms are expressed by using the magnetic quantum number nr of j = 5/2 manifold and the arguments in ]m,m’) refer to the ordering of the creation operators, so Im’, m) = - 1111, m’). If we restrict the effective manifold to I 2 1)~ If 2J?5,+->and excited states to f ‘, the effective Hamiltonian after the SW transformation is given as Herr
=J’ ELI + I)( -k N-42~4~c1,2
+ 42~2
k
+ b&f_
5,2a - 92 f
J’CU + N-
+
42~2))
+
t.c.1
WzAb+_3,2cu2
k
where nf = Cp,npr. The operator f&(CkpJ is the annihilation operator of the localized f-electron (conduction electron with wave number k) with the 7th component of the F-irreducible representation. The quantity e(F) is the single electron energy level of the localized f orbit, U and I are the Coulomb and the exchange interaction constants, respectively. The hybridization matrix V is assumed to be not dependent on r and ek is the conduction electron energy. We assume that the band has width from -DtoDwithD=l. We have chosen e#‘) = - 0.900, e(rg) = - 0.750 and e(I’\2)) = - 0.500 for U = 0.6 and1 = 8. The lowest energy atomic state is the doublet state which has mainly the character ((f 2J = 4)FS(0))I’,@,) (f 2-configuration, total angular momentum .7 = 4, F+educible representation of O-group, F&reducible representation of D4-group). The energy of this state is - 1.446. The first excited state has energy - 1.442 and has mainly the character ((f 2J = 4)I’,(O))I’,(D,). The third and the fourth excited states are I’,( - 1.286) and F,( - 1.167) of Dq, respectively, which are originated from the ((f 2J = 4)r3(0)) states in the cubic field. Other cotigurations, (f ‘J = 0), (f ‘J = 9/2) and (f 4J = 4) have higher energies 0.0, - 0.1 and 3.1, respectively. The main
+
Ct
1,2a3/2)}
+
h.c.1,
(10)
where J’ = V2/(AE) and the mean excitation energy, AE, is about 0.7. The operators are defined as (~~512= &5/2C2
+
CT3/2a2)lA
ad
a%312
=
&3/2C2
-
C75/2a2YA
with A = &a; + cz). The wave number k is dropped in these expressions. The notation t-c. means the term given as the time reversal counter part and h.c. the term with Hermite conjugate. In this Hamiltonian, the channels of the conduction electrons specified by (m = - 512, - l/2, 3/2) and (5/2, l/2, - 3/2) are classified into different flavor groups. They do not mix with each other and have matrix elements showing one-to-one correspondence. This type of Hamiltonian can give the TCKM anomalies [9]. If we consider the cubic case and take the effective manifold of the triplet, Fs(0), by adding the If 21?4) state as IO), the matrix elements of the local ion state are expressed as the operator in the spin S = 1 manifold. The interaction with the conduction electrons are given as the cubic invariant products of these operators with those of the conduction electrons [15]. The matrix elements among the j 2 1) states are given by the quadrupole operators, e = (3s: - 2)& and $221 = Sz - S$. The quadrupole operators of the types I’$ and G21
Vol. 101, No. 11 GROUND STATE PROPERTIES OF THE IMPURITY ANDERSON MODEL has the following properties in the j = 5/2 manifold: matrix elements between the different flavor channels are null; (mlYlm’>= (- m(YI -m’) [9]. Thus when the IO) state is expelled by the D4 field, the Hamiltonian is reduced to the form (10). The logarithmic discretization process and conversion to the linear chain type Hamiltonian are done following [ 111. From this we lead a sequence of effective Hamiltonian which gives low-lying many-body states at each renormalization step. From the analysis of the flow chart of the renormalized energy levels (FCEL), we can get insights of low energy properties of the original nondiscretized Hamiltonian (1). In Fig. 1 we show the FCEL in several cases of the hybridization strength for the fixed atomic parameters. The states are classified by their charge and the modulus 4 of the total magnetic quantum number. In this calculation we have chosen very large discretization parameter A, A = 8, to avoid confusions caused by the numerical error due to the smallness of the number of states keeping in the iteration. We have retained about 300 states. The calculations with smaller A, A = 4, show small but gradual change of the energy levels even in very large L region. In the weak hybridization case, P(= ?rV2/2D)< I’,i(-0.0375 X x), the ground state in the odd L series is the doublet (P5) even in the large L region [16]. In the TCKM, the FCEL of even L series and odd L series have common level structures indicating the intermediate coupling fixed point [17]. This is contrasted to the Fermi liquid case. The present system has six channels. As noted by Koga and Shiba, the extra two channels decouple from the other fours which show the features of the intermediate coupling [9]. The level structures in large L limit can be expressed as those of the coexistence of the non-interacting channel and the intermediate coupling channels, which are mutually independent. For the hybridization, PC1< I’ < PC2(-0.039040 X rr), the singlet state having P4 symmetry becomes the lowest energy level for L 2 1, though the Ps state is the lowest in the atomic Hf term. The step at which the alternation of the symmetry starts does not shift from L = 1 to large L side even when I’ is varied. This change occurs between states with same charge, Q = - 1. The level structures in large L region can be expressed as the coexistence of the non-interacting electron system and the ion with CFS. In some sense, we should interpret that this CFS is stabilized by the crystal field effect induced from the hybridization [18]. However, the excitation gap will be reduced and the scattering of electron will be enhanced by the Kondo effect [19]. We note that the FCEL for I’ near PC1cases almost coincides either Fig. 1A or Fig. 1B even when P is varied. The change of the ground state properties seems
793
to occur suddenly without anomalies, such as the lowering of the characteristic energy scale. In Fig. lC, we show the FCEL for P > Pc2. The energy of the levels with Q =-3 decreases gradually. These have charge - 2 compared with that of the lowest energy states (with Q = - 1) in small L region. In the large L region, the singlet state which has character similar to that of the f O-state (the two holes bound state), becomes the lowest energy state. The same possibility has been already pointed out previously for the problem of Tm3+-like ion case [20, 211. This state will show low energy properties similar to those of the singlet state of the usual Kondo effect. We note that at F just above Pcz, the transition region to the large L limit shift to very large L side as seen from Fig. 2. In the interval, L = 9-19, the energy level structures seem to be characterized by the competition of the crystalline field splitting and the Fermion excitation energy. But this type of fixed point is unstable and turns into the f O-like states. For I’ just below PcZ, the FCEL shows the similar behavior in the intermediate L region, but turns into the CFS-like state in the large L limit. The transition at PcZoccurs with anomalous lowering of the low energy scale characterized as II’- Palo(, where cr is grossly estimated as l-2 at present. The similar behavior has been already noted for the transitions between the J = O-like state and the Kondo singletlike state of the Sm2+-like system [19, 201, and for the transition between the spin-pair-singlet-like state and the Kondo singlet-like states for the two impurity problem [22, 231. But in the latter case the inclusion of the parity splitting of orbits suppressed the anomalous transition [23]. So, at first glance, the energy splitting of the single electron orbit seems to be relevant for the suppression. But, however, in the present model the splitting of the orbital energy is included. Therefore, at present we can not say clearly what quantity controls the nature of the transition. We calculate the thermal average of the magnetization of the f-electrons, M = (mf> 5 (C “,= _ j mn&, by adding the Zeeman term, H’ = - mfHz, to the Hamiltonian. In Fig. 3, we show the temperature dependence of the magnetization in four typical regions of the hybridization strength. This calculation is done by using parameter A = 4, to get fine mesh of the data points. In this case Per and PC2 are estimated as 0.0433 X ?r and 0.044995 X u, respectively. These magnetization curves will be proportional to the susceptibility when the magnetic field is very weak. In Fig. 3A of the doublet ground state case, the magnetization increases proportional to -1nT in the region T >>Hz. This is the characteristic behavior of the non-Fermi liquid region [24]. In Fig. 3B of the CFS case, the magnetization increases initially with decreasing temperature, and saturates after showing maximum at about
794
(A)
GROUND STATE PROPERTIES OF THE: IMPURITY ANDERSON MODEL Vol. 101, No. 11 Q ,2M
Q .2M
----(21, o.o.-1)(22. 0.0. 1) -(lo, l.O.-2)(20, 1.0, 2) ._ 17,-2.0,-3 18,-2.0, 3 I 15,-2.0,-l It 18.-2.0, 1I “(1%-l.O.-4)(14,-l.O,-4) :~li,-l.o.-e)(l2.-i.o, 2) \y o.-1.0. O)(lO.-1.0, 0). ‘( 7,-3.0,~2)( 8.-3.0, 2) c,--{ 6, o.o,-3)( 6, 0.0, a) -‘_( 3.-2.0, S)( 4,-2.0,-s) ----( f,-1.0, 2)( 2,-l.O.-2)
(16, (18. (14, (12, (IO.
o.o,-2)(17. 0.0. q-,_l*O 0.0, 1)(16, 0.0,-l) --,‘= 0.0, l)(lS, 0.0,-l).--;; o.o,-S)(ll, 0.0, Sq’, 0.0, l)( 0, o.o,-l)~&qjj~ ( B.-1.0. 0) p1/‘,/ ( 7,-2.0, I)( 8,-2.0,-l)+ ( S,-2.0,~B)( 4.-2.0, 3)” ( 8,-1.0,-4)-,_c ( 2.-1.0,-2)( l,-1.0, 2) --8.2 Shell No L(odd) @I (18, (18, (14, (12. (IO.
Q ,2Y
0.0,~3)(17, 0.0. a)-, Ia0 0.0, 1)(15, O.O,-1)*-k: 0.0, 1)(13, 0.0.-l)+‘_;; o.o.-3)(11, 0.0, a)-3,’ 0.0, l)( 0, 0.0,-l) g,r*g ( a,-1.0, O)M&‘, ( 7.-2.0. l)( 6.-2.0.-l) ‘/= ( S,-2.0,-3)( 4,-2.0, 3)’ ( 3,-i.O.-2)( 2,-1.0, 2).. ( l,-l.O,-4) -&I
Q .ZM
----
‘( ‘:( ‘( (
8,-2.0, I)( 7.-2.0.-l) 4,-2.0, S)( 5,-2.0,-s) 2, 0.0. l)( 3, 0.0,-l) l.-l.O,-4)
Shell No L(odd) Q .2x (lo,-1.0, O)*..l.O (18, 0.0, 3)(17, 0.0.-s)-, ‘. (16. o.o,-1)(15, 0.0, 1) -2:: (14. o.o.-1)(13, 0.0. 1)---s; (12, 0.0, S)(ll, o.o.-S)-&’ (10, 0.0. 1)( 9, 0.0,-l)** P.9 ( 8,-1-O, O)-+, ( 7,-2.0.-l)( 6,-2.0, I)*‘/’ ( 5,-2.0. 3)( 4,-2.0,~a)+’
Q ,a ,~(17,-2.0, 3)(1S.-2.0.-a)
GI
( 2.-l.O.-2)
t ;:-::8:-tjY3.* ShetlNo L(odd)
Fig. 1. Flow chart of the renormalized energy levels (FCEL) for odd L. The energy levels are normalized by CL- ‘I”/2 in each L-step and L = 0 corresponds to the first shell orbit state. The discretization &&I =D(l -tfP)A-parameter A = 8 is used and about 300 states are retained in iteration step. The quantity Q [ = total electron number - half of the total orbital number ((2j + l)(L + 2)/2)] is the charge of the state and M is the modulus 4 of the total magnetic quantum number. In A (l’fu = 0.036) of the doublet ground state case, the excitation energy of the I’4 singlet (Q = -1, M = -2) from the I’S doublet ground state (Q = -1, M = + 1) increases with L. In B (r/x = 0.038) of the crystalline-field-singlet (CFS)-like case, the excitation energy of the I’S state increases with L. In C (T/n = 0.040) of the f O-likecase, energy of states with Q = -3 decreases with L
Vol. 101, No. 11 GROUND
STATE PROPERTIES
OF THE IMPURITY
ANDERSON
MODEL
795
Q ,a (13,
0.0,
3)(17,
0.0,~3)*,
(13,
0.0.
1)(15,
0.0,-l).-:,‘=
_l.O
(14,
0.0,
1)(13.
0.0,-l).---;
(12.
0.0,
3)(11,
0.0,-3)-L-y
(10.
o.o,-l)(
9,
0.0,
( a,-1.0, ( 7.-2.0.
l)(
l)-$,@.$ O)‘$“,
6,-2.0,-l)‘,,’
( S.-2.0.-3)(
4.-2.0,
3)’
( 3.-l-0.-2)(
2,-1.0,
2)x__
( l.-l.O,-4)
--9.a Shell No L(odd)
Fig. 2. FCEL of thef O-like singlet case in very near the transition point (I’&( = 0.0390405)), I’/?r = 0.039041. The intermediate L region appears in the interval, 9-19. When I’ approaches to .lYca,the large L end of the intermediate L region shifts to larger L side. that of Fig. 3C and the normalized value of it, M/Hz, decreases gradually with Hz for Hz > 10m6. In Fig. 4, we show the magnetization at T = 0 as a function of P. When l’ increases, it shows sharp decreases from about 10’ to lo4 at Pcl. In P > PC1 region, the magnetization has magnitude lo4 as seen from the inset. It increases gradually in CFS region with increasing I’ and shows steep increase near PCs. We have
T = 10M3, the effective crystalline field splitting energy. In Fig. 3C of the region near Pcz,, the magnetization at the lowest temperature is larger than that of Fig. 3B, although the hybridization is increased. The calculated magnetization data shows large scattering both in T and in Hz dependence near T = 10 - 5. Calculations keeping many states seem to be necessary in this region. In Fig. 3D of the f O-like region, the magnetization is larger than 3.0
1.0
2.0
n s
0.5
m xi0
’
0aaao0boDaaM000ooe
0.0
a% 0,
0
. 0.
n
10°F’-2
T/D
T/D 1.0
x106
*
$
0.5
..you 10-n
10-C.
10-Z
Fig. 3. Temperature dependence of the normalized magnetization for several values of the magnetic field, Hz. In this figureA = 4 is used and P&r and I’& are estimated as 0.0433 and 0.044995, respectively. In A (P/n = 0.042) of the doublet-like case, -In T increase is seen in the region T >>Hz. B (l% = 0.0434) is the CFS-like case, C (P/n = 0.0452) is thef O-like case very near the transition point PC2 and D (I% = 0.047) is the f O-like case. The data denoted by circles, diamonds and triangles in A and B are calculated for Hz = 10e9, 10m8 and 10e7, respectively, and in C and D for Hz = lo-*, low7 and 10M6.
GROUND
796
STATE PROPERTIES
OF THE IMPURITY
;I0
8
I
'
1.0
s
1
“106
p
A
4. 5.
:
L g
2. 3.
0
I
0.5
E
0
0
1.0
0
0.0 4.7.
i
0.0
4.6
4.4
~~
r/s
4.8 x10-*
6.
la 4.2
1. See for example, Hewson, A.C., The Kona!o Prob-
%
I
2.0
A
MODEL Vol. 101, No. 11
REFERENCES
a
3.0
ANDERSON
4.4
4.6
4.8
r/n Fig. 4. Normalized magnetization at T = 0 plotted as a function of F/x. The data denoted by circles, diyond$ triangles and gquares are calculated for HZ = lo- , lo- , lo- and lo- , respectively. The inset shows the magnetization in the singlet state region by using smaller scale. expected anomalous lowering of the low energy scale near the transition point, but the magnetization does not show sign of the divergence of the susceptibility. This situation is similar to the transition of the two impurity s-d model noted by Jones et al., for which the divergence occurs in the specific heat and the staggered susceptibility, but does not in the uniform susceptibility. From the calculations shown in this report, we can conclude that the Anderson Hamiltonian shows various types of ground state properties and transitions between them when the ratio of the atomic interactions and the hybridization strength is varied. In some cases the transition is accompanied by the anomalous lowering of the characteristic energy scale and in some cases it occurs as the simple level crossing-like behavior. At present, we can not pick out the relevant quantities which control the type of transitions. We may expect various types of transitions, such as the direct transition between the f O-like and the non-Fermi liquid state [9] and others [25, 261. The contents of the Kondo effect for ions with multi-electron occupancy seem to involve more varieties than those expected previously. Acknowledgements-We would like to thank to Professor H. Shiba for stimulation and valuable comments. This work is supported partly by Grants-in Aids No. 06244104 and No. 07640498 from the Ministry of Education, Science and Culture. The numerical computation was performed at the Computer Center of Tohoku University, the Computer Center of Institute for Molecular Science (Okazaki National Research Institute) and the Computer Center of Institute for Solid State Physics (University of Tokyo). We dedicate this report to the late retired Professor E. Hirahara.
7. 8. 9. 10. 11. 12.
13. 14. 15.
16.
17. 18. 19. 20. 21. 22. 23. 24. 25.
26.
lem to Heavy Fermions. Cambridge University Press, 1993. Wilson, K.G., Rev. Mod. Phys., 47, 1975, 773. Yosida, K. and Yamada, K., Progr. Theor. Phys., 53,1975,1286. Yamada, K., Progr. Theor. Phys., 54, 1975, 316. Okiji, A. and Kawakami, N., in Theory of Heavy Fermions and Valence Fluctuations (Edited by T. Kasuya and T. Saso), p. 46. Springer-Verlag (1985). Nozieres, P. and Blandin, A., J. Phys. (Paris), 41, 1980, 193. Cox, D.L., Phys. Rev. Lett., 59, 1987, 1240. Cox, D.L., Physica, B186-188, 1993,312. Koga, M. and Shiba, H., J. Phys. Sot. Jpn., 64, 1995,4345. Koga, M. and Shiba, II., J. Phys. Sot. Jpn., 65, 1996, 3007. Krishna-Murthy, H.R., Wilkins, J.W. and Wilson, K.G. Phys. Rev., B21,1980, 1003,1044. Sakai, O., Suzuki, S., Shimizu, Y., Kusunose, K. and Miyake, K., Solid State Commun., 99, 1996, 461 (and references therein). See for example, Physica B206&207, 1995, for references. We follow the notations in Butler, P.H., Point Group Symmetry Applications. Plenum Press, 1981. In this cubic case, we found that the ground state always has the properties classified later to the f ‘like singlet state. In general, the symmetry of the lowest energy level of FCEL does not directly indicate the symmetry properties of the ground state of the original Hamiltonian. However as checked later in the calculations of the magnetization, the symmetry of the lowest level in the odd L step seems to reflect the ground state properties in the present case. Cragg, D.M., Lloyd, P. and Noziere, P., J. Phys., C13,1980, 803. Takahashi, H. and Kasuya, T., J. Phys., C18,1985, 2697,2709. Sakai, O., Shimizu, Y. and Kaneko, N., Physica, B186-H&1993,322. Sakai, O., Shim& Y. and Kasuya, T., Progr. Theor. Phys. Suppl., 108,1992,73. Shim& Y., Sakai, 0. and Kasuya, T., Physica, B163,1990,401. Jones, B.A., Varma, C.M. and Wilkins, J.W., Phys. Rev. Lett., 61, 1988, 125. Sakai, 0. and Shimizu, Y., J. Phys. Sot. Jpn., 61, 1992,2333,2348. See for example, Affleck, I. and Ludwig, A.W.W., Phys. Rev., B48, 1993,7297. Kuramoto, Y., in Transport and Thermal Properties of f-Electron Systems (Edited by 6. Oomi, H. Fujii andT.Fujita),p.237.PlenumPress,NewYork, 1993. Shim& Y., Sakai, 0. and Kuramoto, Y., Physica, B206&207,1995,135.
PII:
s0038-1098(%)00739-9
COMPETITION AND TRANSITIONS OF THE GROUND STATE PROPERTIES OF THE IMPURITY ANDERSON MODEL WITH MULTI-ELECTRON OCCUPANCY 0. Sakai,a S. Suzuki” and Y. Shim&u& “Department of Physics, Tohoku University, Sendai 980-77, Japan department of Applied Physics, Tohoku University, Sendai 980-77, Japan (Received
11 November
1996; accepted 26 November
1996 by T. Tsuzuki)
To study the Kondo effect of the multi-electron occupancy systems (KEMES), the impurity Anderson model with tetrahedral crystalline field and intraatomic exchange interaction is investigated as an example. The low energy properties and the magnetization are examined based on Wilson’s numerical renormalization group method. When the hybridization strength is increased by fixing the lowest atomic state as the non-Kramers doublet (f2)F5, the ground state shows successively the doublet-like, the crystalline-field-singlet-like and the f O-singlet-like properties. The first one is the non-Fermi liquid state of the two-channel Kondo type. The last two are Fermi-liquid states, separated by transition with anomalous lowering of the low energy scale. These results suggest more varieties of KEMES than those previously expected. 0 1997 Elsevier Science Ltd. All rights reserved Keywords: D. Kondo effects, D. heavy fermions, D. valence fluctuations.
The study on the Kondo effect of magnetic impurities in metal has a very long history [l]. It has been established that the system has the singlet ground state and the low energy properties are given as the local Fermi liquid [2]. Yamada and Yosida have developed the local Fermi liquid theory based on the perturbation expansion of the Coulomb interaction for the impurity Anderson model [3,4]. They pointed out that the impurity system will not show transition. Actually, the opposite extreme limits, the weak perturbation limit and the s-d limit, have singlet ground state. This fruitful approach has brought many important developments [l] and has been justified by analytic solution [5]. Nozieres and Blandin have pointed out that in a restricted case of the multi-electron occupancy system, the ground state can have non-Fermi liquid properties, which is classified to the two channel Kondo model (TCKM) type [6]. But this case has been considered to be a pathological one. However, Cox has noted the possibility that a model for the electronic state of ions with f 2con@uration, such as df in the crystalline field, can be mapped on the TCKM [7,8]. Recently, Koga and Shiba have studied the problem extensively and shown that there are appreciable parameter regions where the
low energy properties are actually mapped on the TCKM even when realistic model for the crystalline field splitting is used [9, lo]. These approaches stand on the Schrieffer-Wolff (SW) transformation restricting the effective manifold to f 2 or to f 3 configurations. When the atomic electron-electron interactions are very weak in the Anderson model, the ground state should be the Fermi liquid state. Therefore, there arises the question that the non-Fermi liquid state does really appear for the Anderson model when the main atomic contiguration is the multi-electron occupancy state. If it does, how does the transition between the Fermi liquid and the non-Fermi liquid states occur when the atomic interaction becomes gradually strong. The aim of the present report is to check these points and show that nonFermi liquid state really appears. The transition between the two states occurs as the simple level crossing-like one in the case studied here. Moreover, the transition between singlet states, the crystalline-field-singlet (CFS)-like and the f O-likestates, occurs with anomalous lowering of the characteristic energy scale. We apply Wilson’s numerical renormalization group (NRG) method [2, 111 to the impurity Anderson model with crystalline field energy splitting of the localized
791
792
GROUND
STATE PROPERTIES
OF THE IMPURITY
orbits. The atomic Coulomb and the exchange interactions are assumed to be the j-j coupling type. In this report we vary the hybridization strength by fixing the atomic interaction parameters to avoid confusions arising from the modification of the atomic energy level structures. We show that the low energy excitation properties change suddenly at some critical values of the hybridization. This unexpected result suggests unknown contents of the magnetic impurity problem with multi-electron occupancy and may relate to complex behaviors of the U ion systems[12, 131. As an illustrative example, we consider the model with the j = 512 orbits in the tetrahedral point symmetry. The j = 5/2 orbits split into three doublets, (Fe + I’f’+ I’?‘) [14]. The model Hamiltonian is given by the form
ANDERSON
MODEL Vol. 101, No. 11
states are given as
iflr$? 5) = I+-5/2)( _+ al) + JT3/2)( -+ br),
(5)
IfIr,, 5 ) =
(6)
]*1/2),
1f11’$2’, I-t->=
]“5/2)(
T b,) + ]13/2)(
If 21’5,+)=]?1/2,
k al),
(7)
?3/2)(--&+]?3/2,
-t 5/2)( - b2) + 12512, + 1/2)(c2),
(8)
)f 2r4)= 1- i/2, - 3/z&@-t 11/2,- 5/2)&) -k(3/Z
l/2)( - dz) + 1512, - l/2)( - ez),
(9)
bl = a, a2 = 0.3395, b2 = where al - Js/6, 0.1874, c2 = 0.9218, d2 = 0.4041 and e2 = 0.5802. The right-hand side terms are expressed by using the magnetic quantum number nr of j = 5/2 manifold and the arguments in ]m,m’) refer to the ordering of the creation operators, so Im’, m) = - 1111, m’). If we restrict the effective manifold to I 2 1)~ If 2J?5,+->and excited states to f ‘, the effective Hamiltonian after the SW transformation is given as Herr
=J’ ELI + I)( -k N-42~4~c1,2
+ 42~2
k
+ b&f_
5,2a - 92 f
J’CU + N-
+
42~2))
+
t.c.1
WzAb+_3,2cu2
k
where nf = Cp,npr. The operator f&(CkpJ is the annihilation operator of the localized f-electron (conduction electron with wave number k) with the 7th component of the F-irreducible representation. The quantity e(F) is the single electron energy level of the localized f orbit, U and I are the Coulomb and the exchange interaction constants, respectively. The hybridization matrix V is assumed to be not dependent on r and ek is the conduction electron energy. We assume that the band has width from -DtoDwithD=l. We have chosen e#‘) = - 0.900, e(rg) = - 0.750 and e(I’\2)) = - 0.500 for U = 0.6 and1 = 8. The lowest energy atomic state is the doublet state which has mainly the character ((f 2J = 4)FS(0))I’,@,) (f 2-configuration, total angular momentum .7 = 4, F+educible representation of O-group, F&reducible representation of D4-group). The energy of this state is - 1.446. The first excited state has energy - 1.442 and has mainly the character ((f 2J = 4)I’,(O))I’,(D,). The third and the fourth excited states are I’,( - 1.286) and F,( - 1.167) of Dq, respectively, which are originated from the ((f 2J = 4)r3(0)) states in the cubic field. Other cotigurations, (f ‘J = 0), (f ‘J = 9/2) and (f 4J = 4) have higher energies 0.0, - 0.1 and 3.1, respectively. The main
+
Ct
1,2a3/2)}
+
h.c.1,
(10)
where J’ = V2/(AE) and the mean excitation energy, AE, is about 0.7. The operators are defined as (~~512= &5/2C2
+
CT3/2a2)lA
ad
a%312
=
&3/2C2
-
C75/2a2YA
with A = &a; + cz). The wave number k is dropped in these expressions. The notation t-c. means the term given as the time reversal counter part and h.c. the term with Hermite conjugate. In this Hamiltonian, the channels of the conduction electrons specified by (m = - 512, - l/2, 3/2) and (5/2, l/2, - 3/2) are classified into different flavor groups. They do not mix with each other and have matrix elements showing one-to-one correspondence. This type of Hamiltonian can give the TCKM anomalies [9]. If we consider the cubic case and take the effective manifold of the triplet, Fs(0), by adding the If 21?4) state as IO), the matrix elements of the local ion state are expressed as the operator in the spin S = 1 manifold. The interaction with the conduction electrons are given as the cubic invariant products of these operators with those of the conduction electrons [15]. The matrix elements among the j 2 1) states are given by the quadrupole operators, e = (3s: - 2)& and $221 = Sz - S$. The quadrupole operators of the types I’$ and G21
Vol. 101, No. 11 GROUND STATE PROPERTIES OF THE IMPURITY ANDERSON MODEL has the following properties in the j = 5/2 manifold: matrix elements between the different flavor channels are null; (mlYlm’>= (- m(YI -m’) [9]. Thus when the IO) state is expelled by the D4 field, the Hamiltonian is reduced to the form (10). The logarithmic discretization process and conversion to the linear chain type Hamiltonian are done following [ 111. From this we lead a sequence of effective Hamiltonian which gives low-lying many-body states at each renormalization step. From the analysis of the flow chart of the renormalized energy levels (FCEL), we can get insights of low energy properties of the original nondiscretized Hamiltonian (1). In Fig. 1 we show the FCEL in several cases of the hybridization strength for the fixed atomic parameters. The states are classified by their charge and the modulus 4 of the total magnetic quantum number. In this calculation we have chosen very large discretization parameter A, A = 8, to avoid confusions caused by the numerical error due to the smallness of the number of states keeping in the iteration. We have retained about 300 states. The calculations with smaller A, A = 4, show small but gradual change of the energy levels even in very large L region. In the weak hybridization case, P(= ?rV2/2D)< I’,i(-0.0375 X x), the ground state in the odd L series is the doublet (P5) even in the large L region [16]. In the TCKM, the FCEL of even L series and odd L series have common level structures indicating the intermediate coupling fixed point [17]. This is contrasted to the Fermi liquid case. The present system has six channels. As noted by Koga and Shiba, the extra two channels decouple from the other fours which show the features of the intermediate coupling [9]. The level structures in large L limit can be expressed as those of the coexistence of the non-interacting channel and the intermediate coupling channels, which are mutually independent. For the hybridization, PC1< I’ < PC2(-0.039040 X rr), the singlet state having P4 symmetry becomes the lowest energy level for L 2 1, though the Ps state is the lowest in the atomic Hf term. The step at which the alternation of the symmetry starts does not shift from L = 1 to large L side even when I’ is varied. This change occurs between states with same charge, Q = - 1. The level structures in large L region can be expressed as the coexistence of the non-interacting electron system and the ion with CFS. In some sense, we should interpret that this CFS is stabilized by the crystal field effect induced from the hybridization [18]. However, the excitation gap will be reduced and the scattering of electron will be enhanced by the Kondo effect [19]. We note that the FCEL for I’ near PC1cases almost coincides either Fig. 1A or Fig. 1B even when P is varied. The change of the ground state properties seems
793
to occur suddenly without anomalies, such as the lowering of the characteristic energy scale. In Fig. lC, we show the FCEL for P > Pc2. The energy of the levels with Q =-3 decreases gradually. These have charge - 2 compared with that of the lowest energy states (with Q = - 1) in small L region. In the large L region, the singlet state which has character similar to that of the f O-state (the two holes bound state), becomes the lowest energy state. The same possibility has been already pointed out previously for the problem of Tm3+-like ion case [20, 211. This state will show low energy properties similar to those of the singlet state of the usual Kondo effect. We note that at F just above Pcz, the transition region to the large L limit shift to very large L side as seen from Fig. 2. In the interval, L = 9-19, the energy level structures seem to be characterized by the competition of the crystalline field splitting and the Fermion excitation energy. But this type of fixed point is unstable and turns into the f O-like states. For I’ just below PcZ, the FCEL shows the similar behavior in the intermediate L region, but turns into the CFS-like state in the large L limit. The transition at PcZoccurs with anomalous lowering of the low energy scale characterized as II’- Palo(, where cr is grossly estimated as l-2 at present. The similar behavior has been already noted for the transitions between the J = O-like state and the Kondo singletlike state of the Sm2+-like system [19, 201, and for the transition between the spin-pair-singlet-like state and the Kondo singlet-like states for the two impurity problem [22, 231. But in the latter case the inclusion of the parity splitting of orbits suppressed the anomalous transition [23]. So, at first glance, the energy splitting of the single electron orbit seems to be relevant for the suppression. But, however, in the present model the splitting of the orbital energy is included. Therefore, at present we can not say clearly what quantity controls the nature of the transition. We calculate the thermal average of the magnetization of the f-electrons, M = (mf> 5 (C “,= _ j mn&, by adding the Zeeman term, H’ = - mfHz, to the Hamiltonian. In Fig. 3, we show the temperature dependence of the magnetization in four typical regions of the hybridization strength. This calculation is done by using parameter A = 4, to get fine mesh of the data points. In this case Per and PC2 are estimated as 0.0433 X ?r and 0.044995 X u, respectively. These magnetization curves will be proportional to the susceptibility when the magnetic field is very weak. In Fig. 3A of the doublet ground state case, the magnetization increases proportional to -1nT in the region T >>Hz. This is the characteristic behavior of the non-Fermi liquid region [24]. In Fig. 3B of the CFS case, the magnetization increases initially with decreasing temperature, and saturates after showing maximum at about
794
(A)
GROUND STATE PROPERTIES OF THE: IMPURITY ANDERSON MODEL Vol. 101, No. 11 Q ,2M
Q .2M
----(21, o.o.-1)(22. 0.0. 1) -(lo, l.O.-2)(20, 1.0, 2) ._ 17,-2.0,-3 18,-2.0, 3 I 15,-2.0,-l It 18.-2.0, 1I “(1%-l.O.-4)(14,-l.O,-4) :~li,-l.o.-e)(l2.-i.o, 2) \y o.-1.0. O)(lO.-1.0, 0). ‘( 7,-3.0,~2)( 8.-3.0, 2) c,--{ 6, o.o,-3)( 6, 0.0, a) -‘_( 3.-2.0, S)( 4,-2.0,-s) ----( f,-1.0, 2)( 2,-l.O.-2)
(16, (18. (14, (12, (IO.
o.o,-2)(17. 0.0. q-,_l*O 0.0, 1)(16, 0.0,-l) --,‘= 0.0, l)(lS, 0.0,-l).--;; o.o,-S)(ll, 0.0, Sq’, 0.0, l)( 0, o.o,-l)~&qjj~ ( B.-1.0. 0) p1/‘,/ ( 7,-2.0, I)( 8,-2.0,-l)+ ( S,-2.0,~B)( 4.-2.0, 3)” ( 8,-1.0,-4)-,_c ( 2.-1.0,-2)( l,-1.0, 2) --8.2 Shell No L(odd) @I (18, (18, (14, (12. (IO.
Q ,2Y
0.0,~3)(17, 0.0. a)-, Ia0 0.0, 1)(15, O.O,-1)*-k: 0.0, 1)(13, 0.0.-l)+‘_;; o.o.-3)(11, 0.0, a)-3,’ 0.0, l)( 0, 0.0,-l) g,r*g ( a,-1.0, O)M&‘, ( 7.-2.0. l)( 6.-2.0.-l) ‘/= ( S,-2.0,-3)( 4,-2.0, 3)’ ( 3,-i.O.-2)( 2,-1.0, 2).. ( l,-l.O,-4) -&I
Q .ZM
----
‘( ‘:( ‘( (
8,-2.0, I)( 7.-2.0.-l) 4,-2.0, S)( 5,-2.0,-s) 2, 0.0. l)( 3, 0.0,-l) l.-l.O,-4)
Shell No L(odd) Q .2x (lo,-1.0, O)*..l.O (18, 0.0, 3)(17, 0.0.-s)-, ‘. (16. o.o,-1)(15, 0.0, 1) -2:: (14. o.o.-1)(13, 0.0. 1)---s; (12, 0.0, S)(ll, o.o.-S)-&’ (10, 0.0. 1)( 9, 0.0,-l)** P.9 ( 8,-1-O, O)-+, ( 7,-2.0.-l)( 6,-2.0, I)*‘/’ ( 5,-2.0. 3)( 4,-2.0,~a)+’
Q ,a ,~(17,-2.0, 3)(1S.-2.0.-a)
GI
( 2.-l.O.-2)
t ;:-::8:-tjY3.* ShetlNo L(odd)
Fig. 1. Flow chart of the renormalized energy levels (FCEL) for odd L. The energy levels are normalized by CL- ‘I”/2 in each L-step and L = 0 corresponds to the first shell orbit state. The discretization &&I =D(l -tfP)A-parameter A = 8 is used and about 300 states are retained in iteration step. The quantity Q [ = total electron number - half of the total orbital number ((2j + l)(L + 2)/2)] is the charge of the state and M is the modulus 4 of the total magnetic quantum number. In A (l’fu = 0.036) of the doublet ground state case, the excitation energy of the I’4 singlet (Q = -1, M = -2) from the I’S doublet ground state (Q = -1, M = + 1) increases with L. In B (r/x = 0.038) of the crystalline-field-singlet (CFS)-like case, the excitation energy of the I’S state increases with L. In C (T/n = 0.040) of the f O-likecase, energy of states with Q = -3 decreases with L
Vol. 101, No. 11 GROUND
STATE PROPERTIES
OF THE IMPURITY
ANDERSON
MODEL
795
Q ,a (13,
0.0,
3)(17,
0.0,~3)*,
(13,
0.0.
1)(15,
0.0,-l).-:,‘=
_l.O
(14,
0.0,
1)(13.
0.0,-l).---;
(12.
0.0,
3)(11,
0.0,-3)-L-y
(10.
o.o,-l)(
9,
0.0,
( a,-1.0, ( 7.-2.0.
l)(
l)-$,@.$ O)‘$“,
6,-2.0,-l)‘,,’
( S.-2.0.-3)(
4.-2.0,
3)’
( 3.-l-0.-2)(
2,-1.0,
2)x__
( l.-l.O,-4)
--9.a Shell No L(odd)
Fig. 2. FCEL of thef O-like singlet case in very near the transition point (I’&( = 0.0390405)), I’/?r = 0.039041. The intermediate L region appears in the interval, 9-19. When I’ approaches to .lYca,the large L end of the intermediate L region shifts to larger L side. that of Fig. 3C and the normalized value of it, M/Hz, decreases gradually with Hz for Hz > 10m6. In Fig. 4, we show the magnetization at T = 0 as a function of P. When l’ increases, it shows sharp decreases from about 10’ to lo4 at Pcl. In P > PC1 region, the magnetization has magnitude lo4 as seen from the inset. It increases gradually in CFS region with increasing I’ and shows steep increase near PCs. We have
T = 10M3, the effective crystalline field splitting energy. In Fig. 3C of the region near Pcz,, the magnetization at the lowest temperature is larger than that of Fig. 3B, although the hybridization is increased. The calculated magnetization data shows large scattering both in T and in Hz dependence near T = 10 - 5. Calculations keeping many states seem to be necessary in this region. In Fig. 3D of the f O-like region, the magnetization is larger than 3.0
1.0
2.0
n s
0.5
m xi0
’
0aaao0boDaaM000ooe
0.0
a% 0,
0
. 0.
n
10°F’-2
T/D
T/D 1.0
x106
*
$
0.5
..you 10-n
10-C.
10-Z
Fig. 3. Temperature dependence of the normalized magnetization for several values of the magnetic field, Hz. In this figureA = 4 is used and P&r and I’& are estimated as 0.0433 and 0.044995, respectively. In A (P/n = 0.042) of the doublet-like case, -In T increase is seen in the region T >>Hz. B (l% = 0.0434) is the CFS-like case, C (P/n = 0.0452) is thef O-like case very near the transition point PC2 and D (I% = 0.047) is the f O-like case. The data denoted by circles, diamonds and triangles in A and B are calculated for Hz = 10e9, 10m8 and 10e7, respectively, and in C and D for Hz = lo-*, low7 and 10M6.
GROUND
796
STATE PROPERTIES
OF THE IMPURITY
;I0
8
I
'
1.0
s
1
“106
p
A
4. 5.
:
L g
2. 3.
0
I
0.5
E
0
0
1.0
0
0.0 4.7.
i
0.0
4.6
4.4
~~
r/s
4.8 x10-*
6.
la 4.2
1. See for example, Hewson, A.C., The Kona!o Prob-
%
I
2.0
A
MODEL Vol. 101, No. 11
REFERENCES
a
3.0
ANDERSON
4.4
4.6
4.8
r/n Fig. 4. Normalized magnetization at T = 0 plotted as a function of F/x. The data denoted by circles, diyond$ triangles and gquares are calculated for HZ = lo- , lo- , lo- and lo- , respectively. The inset shows the magnetization in the singlet state region by using smaller scale. expected anomalous lowering of the low energy scale near the transition point, but the magnetization does not show sign of the divergence of the susceptibility. This situation is similar to the transition of the two impurity s-d model noted by Jones et al., for which the divergence occurs in the specific heat and the staggered susceptibility, but does not in the uniform susceptibility. From the calculations shown in this report, we can conclude that the Anderson Hamiltonian shows various types of ground state properties and transitions between them when the ratio of the atomic interactions and the hybridization strength is varied. In some cases the transition is accompanied by the anomalous lowering of the characteristic energy scale and in some cases it occurs as the simple level crossing-like behavior. At present, we can not pick out the relevant quantities which control the type of transitions. We may expect various types of transitions, such as the direct transition between the f O-like and the non-Fermi liquid state [9] and others [25, 261. The contents of the Kondo effect for ions with multi-electron occupancy seem to involve more varieties than those expected previously. Acknowledgements-We would like to thank to Professor H. Shiba for stimulation and valuable comments. This work is supported partly by Grants-in Aids No. 06244104 and No. 07640498 from the Ministry of Education, Science and Culture. The numerical computation was performed at the Computer Center of Tohoku University, the Computer Center of Institute for Molecular Science (Okazaki National Research Institute) and the Computer Center of Institute for Solid State Physics (University of Tokyo). We dedicate this report to the late retired Professor E. Hirahara.
7. 8. 9. 10. 11. 12.
13. 14. 15.
16.
17. 18. 19. 20. 21. 22. 23. 24. 25.
26.
lem to Heavy Fermions. Cambridge University Press, 1993. Wilson, K.G., Rev. Mod. Phys., 47, 1975, 773. Yosida, K. and Yamada, K., Progr. Theor. Phys., 53,1975,1286. Yamada, K., Progr. Theor. Phys., 54, 1975, 316. Okiji, A. and Kawakami, N., in Theory of Heavy Fermions and Valence Fluctuations (Edited by T. Kasuya and T. Saso), p. 46. Springer-Verlag (1985). Nozieres, P. and Blandin, A., J. Phys. (Paris), 41, 1980, 193. Cox, D.L., Phys. Rev. Lett., 59, 1987, 1240. Cox, D.L., Physica, B186-188, 1993,312. Koga, M. and Shiba, H., J. Phys. Sot. Jpn., 64, 1995,4345. Koga, M. and Shiba, II., J. Phys. Sot. Jpn., 65, 1996, 3007. Krishna-Murthy, H.R., Wilkins, J.W. and Wilson, K.G. Phys. Rev., B21,1980, 1003,1044. Sakai, O., Suzuki, S., Shimizu, Y., Kusunose, K. and Miyake, K., Solid State Commun., 99, 1996, 461 (and references therein). See for example, Physica B206&207, 1995, for references. We follow the notations in Butler, P.H., Point Group Symmetry Applications. Plenum Press, 1981. In this cubic case, we found that the ground state always has the properties classified later to the f ‘like singlet state. In general, the symmetry of the lowest energy level of FCEL does not directly indicate the symmetry properties of the ground state of the original Hamiltonian. However as checked later in the calculations of the magnetization, the symmetry of the lowest level in the odd L step seems to reflect the ground state properties in the present case. Cragg, D.M., Lloyd, P. and Noziere, P., J. Phys., C13,1980, 803. Takahashi, H. and Kasuya, T., J. Phys., C18,1985, 2697,2709. Sakai, O., Shimizu, Y. and Kaneko, N., Physica, B186-H&1993,322. Sakai, O., Shim& Y. and Kasuya, T., Progr. Theor. Phys. Suppl., 108,1992,73. Shim& Y., Sakai, 0. and Kasuya, T., Physica, B163,1990,401. Jones, B.A., Varma, C.M. and Wilkins, J.W., Phys. Rev. Lett., 61, 1988, 125. Sakai, 0. and Shimizu, Y., J. Phys. Sot. Jpn., 61, 1992,2333,2348. See for example, Affleck, I. and Ludwig, A.W.W., Phys. Rev., B48, 1993,7297. Kuramoto, Y., in Transport and Thermal Properties of f-Electron Systems (Edited by 6. Oomi, H. Fujii andT.Fujita),p.237.PlenumPress,NewYork, 1993. Shim& Y., Sakai, 0. and Kuramoto, Y., Physica, B206&207,1995,135.