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Replica symmetry breaking in a quantum spin glass-antiferromagnetic Kondo lattice
ARTICLE IN PRESS
Physica B 403 (2008) 1395–1397 www.elsevier.com/locate/physb
Replica symmetry breaking in a quantum spin glass-antiferromagnetic Kondo lattice S.G. Magalhaesa,, F.M. Zimmera, B. Coqblinb a
Departamento de Fı´sica, Universidade Federal de Santa Maria, 97105 900 Santa Maria, RS, Brazil Laboratoire de Physique des Solides, Universite´ Paris-Sud, CNRS, UMR 8502, 91405 Orsay, France
b
Abstract The competition between the Kondo effect and the glassy magnetic order has been studied in a theoretical model of a Kondo lattice with an intrasite Kondo interaction. The spin glass (SG) and the antiferromagnetic (AF) orderings are described by two Kondo sublattices with infinite-range Ising SG interactions among localized spins and the disordered interactions can occur with spins of same sublattices and between spins of distinct sublattices. A transverse field G is introduced in the effective model as a quantum mechanism to produce spin flipping. The problem is formulated in a Grassmann path integral formalism. The disorder is treated within the replica trick in one-step replica symmetry breaking (1S-RSB). The static ansatz is adopted to get a mean-field expression for the free energy and order parameters. Results show a transition from the AF order to an RSB region with a finite staggered magnetization (mixed phase) when temperature T decreases for low values of the Kondo interaction. The SG phase is not observed below the mixed phase for 1S-RSB solution, in contrast with previous replica symmetry (RS) results. The G field suppresses the Neel temperature leading it to a quantum critical point. r 2007 Elsevier B.V. All rights reserved. PACS: 05.50.þq Keywords: Kondo lattice; Spin glass; Antiferromagnetism
The competition between RKKY interaction and Kondo effect has a fundamental role in Ce and U compounds [1]. The presence of disorder can deeply affect such competition and, therefore, it can lead to a quite intriguing situation in their global phase diagram. For instance, the CeAu1x Cox Si3 alloys have a phase diagram which displays the sequence of spin glass (SG), antiferromagnetism (AF) and Kondo phases when the chemical disorder is increased by increasing x [2]. Moreover, the Neel temperature decreases until reaching a quantum critical point (QCP) at some particular value of the Co content, with no evidence of non-fermi liquid behavior. Earlier theoretical effort [3] has studied the competition among SG, AF and Kondo effect based on a framework previously introduced to study the presence of SG in disordered Kondo lattice [4]. The model used has been Corresponding author. Tel.: +55 55 3220 8862; fax: +55 55 32208032.
E-mail address: [email protected] (S.G. Magalhaes). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.10.147
basically composed of two kinds of interactions: an intralattice one between localized and itinerant spins and a gaussian random coupling (with average 2J 0 =N) only between Ising spin operators in distinct sublattices. A transverse field G is introduced as a quantum mechanism to produce spin flipping [3]. If it is assumed that J 0 and G behave as J 2K (where J K is the strength of the Kondo intralattice interaction), this approach could be used to reproduce some important aspects of the experimental phase diagram of the CeAu1x Cox Si3 such as the sequence of phases and the QCP for the Neel temperature when J K is increased. Nevertheless, this approach has two fundamental approximations. The first one is the static approximation (SA) to treat the correlation functions which is reasonable if the main goal of the problem is to describe the phase boundary behavior. The second one is the replica symmetry (RS) approximation used to treat the SG order parameters. The RS has an essential shortcoming since it
ARTICLE IN PRESS S.G. Magalhaes et al. / Physica B 403 (2008) 1395–1397
1396
produces unstable SG results. Therefore, the proper way to treat the SG part in the problem must overcome the RS approximation. The purpose of this paper is to implement one-step replica symmetry breaking (1S-RSB) using the socalled Parisi solution [5] to study the competition among SG, AF and Kondo effect. Furthermore, we also generalized the model of Ref. [3] to include frustrated interaction between localized spins of the same sublattice. We consider in this work a model given by two-Kondo sublattices [3] with random coupling between localized spins of same and distinct sublattices. The hopping of conduction electrons is allowed only inside the same sublattice [3]. Therefore, the Hamiltonian is given by " X XX X f f y H mN ¼ tij d^ d^j;p;s þ n^ 0;p i;p
i;p;s
p¼a;b
i;j s¼"#
i
X þ X z z ^ þ þ JK ðS^ i;p s^ V ip j p S^ i;p S^ j;p i;p þ S i;p s^i;p Þ þ þ
X
i
#
i;j
z z J ia j b S^ i;a S^ j;b
i;j
X x x þ 2G ðS^ i;a þ S^ i;b Þ,
ð1Þ
i
where i and j sum over N sites of each sublattice p (¼ a; b). The inter- and intra-sublattice exchange interactions J ia j b and V ip jp are random variables that follow Gaussian distributions with mean values 2J 0 =N and 0 and variances J 2 =32N and V 2 =16N, respectively. The spin operators are defined as in Ref. [3]. The partition function is treated in the fermionic path integral formalism in which the spin operators are represented by Grassmann fields. The free energy is obtained in the static approximation. The replica method and the Parisi’s scheme of 1S-RSB [5] are used to average over the random couplings V ip jp and J ia j b . The fundamental tool, which has allowed us to calculate the partition function, is to introduce a matrix formalism with a proper mixing of spinors of each sublattice [3]. The free energy is obtained as
hp ¼ h¯ p þ s1p vp þ s2p , with h¯ p ¼ J 0 M p0 ðpap0 Þ, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s1p ¼ 2½V 2 ðqp1 qp0 Þ þ J 2 ðqp0 1 qp0 0 Þ,
(4)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2½V 2 ðrp qp1 Þ þ J 2 ðrp0 qp0 1 Þxp qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2ðV 2 qp0 þ J 2 qp0 0 Þzp , ð5Þ pffiffiffiffiffiffi 2 and Dx ¼ dxex =2 = 2p (x ¼ zp ; vp or xp ). The saddle point equations for M p (sublattice magnetization), jlp j (Kondo order parameter), m (Parisi’s block size parameter), and qp0 and qp1 (related to the SG order parameter) are obtained directly from Eqs. (2)–(5). In the present fermionic formulation, rp is an additional saddle-point order parameter to be solved with previous ones. Numerical solutions for the saddle-point equations are found. In this context, the AF order (M A ¼ M B a0 and qp1 ¼ qp0 ), the mixed phase ðAF þ SGÞ and the Kondo state ðla ¼ lb a0Þ are obtained. The mixed phase is characterized by an RSB solution ðqp1 aqp0 Þ with finite staggered magnetization M s ¼ ðM A M B Þ=2. Phase diagrams T versus J K for J 0 ¼ 1:6 and two values of G (G ¼ 0 and 1) are shown in Fig. 1. In this work, the quantities T, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J K , J 0 and G are in units of V 2 þ J 2 . As we can see in Fig. 1, there is a transition from the AF order to an RSB region with M s 40 (mixed phase) when temperature T decreases for low values of the strength J K . The SG phase is not observed below the mixed phase for 1S-RSB solution, in contrast to the RS results [3]. The Kondo phase is obtained for high values of J K . The G field suppresses the Neel and the freezing temperatures. Therefore, when G is strong enough, it will lead the magnetic orders to QCPs. To conclude, we have presented here a mean field theory to study the competition between Kondo and RKKY interactions when disorder and frustration are present. In s2p ¼
2F ¼ J 0 M a M b þ J 2 =T½mðqa1 qb1 qa0 qb0 Þ
1
J0=1.6 and D=12. 2
T
þ ra rb qa1 qb1 X V2 ½mðq2p1 q2p0 Þ J K jlp j2 þ þ 2T p Z T 1 þ r2p q2p1 Dzp ln m Z 1 1 m Z 1 EðH p Þ Dvp Dxp e ,
PARA
ð2Þ 1
1
KONDO
AF
where Z
þbD
EðH p Þ ¼ bD
pffiffiffiffi x þ Hp dx ln cosh þ cosh D , bD 2
(3)
SG+AF 0 0
T is the temperature, b ¼ 1=T, 2D is the width of the conduction electron band with D ¼ ½ðx H p Þ2 =4þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðbJ K lp Þ2 , H p ¼ b G2 þ h2p , the internal field
2
4
6
8 Jk
10
12
14
Fig. 1. Phase diagram T versus J K for J 0 ¼ 1:6 when G ¼ 0:0 (full lines) and G ¼ 1:0 (dashed lines).
ARTICLE IN PRESS S.G. Magalhaes et al. / Physica B 403 (2008) 1395–1397
order to treat properly the frustration in the problem, we go beyond RS approximation using 1S-RSB [5]. The main result, as can be seen in Fig. 1, is that there is no more SG pure solution below the AF order, in contrast to the results of the RS treatment of Ref. [3] for the same values of J 0 . Thus, the presence of frustration appears only combined with spontaneous staggered magnetization in the mixed phase solution.
1397
References [1] B. Coqblin, et al., Philos. Mag. B 86 (2006) 2567. [2] S. Majundar, et al., Solid State Commun. 121 (2002) 665. [3] S.G. Magalhaes, F.M. Zimmer, B. Coqblin, J. Phys.: Condens. Matter 18 (2006) 3479. [4] A. Theumann, B. Coqblin, S.G. Magalhaes, A.A. Schmidt, Phys. Rev. B 63 (2001) 054409. [5] G. Parisi, J. Phys. A 13 (1980) 1101.
Physica B 403 (2008) 1395–1397 www.elsevier.com/locate/physb
Replica symmetry breaking in a quantum spin glass-antiferromagnetic Kondo lattice S.G. Magalhaesa,, F.M. Zimmera, B. Coqblinb a
Departamento de Fı´sica, Universidade Federal de Santa Maria, 97105 900 Santa Maria, RS, Brazil Laboratoire de Physique des Solides, Universite´ Paris-Sud, CNRS, UMR 8502, 91405 Orsay, France
b
Abstract The competition between the Kondo effect and the glassy magnetic order has been studied in a theoretical model of a Kondo lattice with an intrasite Kondo interaction. The spin glass (SG) and the antiferromagnetic (AF) orderings are described by two Kondo sublattices with infinite-range Ising SG interactions among localized spins and the disordered interactions can occur with spins of same sublattices and between spins of distinct sublattices. A transverse field G is introduced in the effective model as a quantum mechanism to produce spin flipping. The problem is formulated in a Grassmann path integral formalism. The disorder is treated within the replica trick in one-step replica symmetry breaking (1S-RSB). The static ansatz is adopted to get a mean-field expression for the free energy and order parameters. Results show a transition from the AF order to an RSB region with a finite staggered magnetization (mixed phase) when temperature T decreases for low values of the Kondo interaction. The SG phase is not observed below the mixed phase for 1S-RSB solution, in contrast with previous replica symmetry (RS) results. The G field suppresses the Neel temperature leading it to a quantum critical point. r 2007 Elsevier B.V. All rights reserved. PACS: 05.50.þq Keywords: Kondo lattice; Spin glass; Antiferromagnetism
The competition between RKKY interaction and Kondo effect has a fundamental role in Ce and U compounds [1]. The presence of disorder can deeply affect such competition and, therefore, it can lead to a quite intriguing situation in their global phase diagram. For instance, the CeAu1x Cox Si3 alloys have a phase diagram which displays the sequence of spin glass (SG), antiferromagnetism (AF) and Kondo phases when the chemical disorder is increased by increasing x [2]. Moreover, the Neel temperature decreases until reaching a quantum critical point (QCP) at some particular value of the Co content, with no evidence of non-fermi liquid behavior. Earlier theoretical effort [3] has studied the competition among SG, AF and Kondo effect based on a framework previously introduced to study the presence of SG in disordered Kondo lattice [4]. The model used has been Corresponding author. Tel.: +55 55 3220 8862; fax: +55 55 32208032.
E-mail address: [email protected] (S.G. Magalhaes). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.10.147
basically composed of two kinds of interactions: an intralattice one between localized and itinerant spins and a gaussian random coupling (with average 2J 0 =N) only between Ising spin operators in distinct sublattices. A transverse field G is introduced as a quantum mechanism to produce spin flipping [3]. If it is assumed that J 0 and G behave as J 2K (where J K is the strength of the Kondo intralattice interaction), this approach could be used to reproduce some important aspects of the experimental phase diagram of the CeAu1x Cox Si3 such as the sequence of phases and the QCP for the Neel temperature when J K is increased. Nevertheless, this approach has two fundamental approximations. The first one is the static approximation (SA) to treat the correlation functions which is reasonable if the main goal of the problem is to describe the phase boundary behavior. The second one is the replica symmetry (RS) approximation used to treat the SG order parameters. The RS has an essential shortcoming since it
ARTICLE IN PRESS S.G. Magalhaes et al. / Physica B 403 (2008) 1395–1397
1396
produces unstable SG results. Therefore, the proper way to treat the SG part in the problem must overcome the RS approximation. The purpose of this paper is to implement one-step replica symmetry breaking (1S-RSB) using the socalled Parisi solution [5] to study the competition among SG, AF and Kondo effect. Furthermore, we also generalized the model of Ref. [3] to include frustrated interaction between localized spins of the same sublattice. We consider in this work a model given by two-Kondo sublattices [3] with random coupling between localized spins of same and distinct sublattices. The hopping of conduction electrons is allowed only inside the same sublattice [3]. Therefore, the Hamiltonian is given by " X XX X f f y H mN ¼ tij d^ d^j;p;s þ n^ 0;p i;p
i;p;s
p¼a;b
i;j s¼"#
i
X þ X z z ^ þ þ JK ðS^ i;p s^ V ip j p S^ i;p S^ j;p i;p þ S i;p s^i;p Þ þ þ
X
i
#
i;j
z z J ia j b S^ i;a S^ j;b
i;j
X x x þ 2G ðS^ i;a þ S^ i;b Þ,
ð1Þ
i
where i and j sum over N sites of each sublattice p (¼ a; b). The inter- and intra-sublattice exchange interactions J ia j b and V ip jp are random variables that follow Gaussian distributions with mean values 2J 0 =N and 0 and variances J 2 =32N and V 2 =16N, respectively. The spin operators are defined as in Ref. [3]. The partition function is treated in the fermionic path integral formalism in which the spin operators are represented by Grassmann fields. The free energy is obtained in the static approximation. The replica method and the Parisi’s scheme of 1S-RSB [5] are used to average over the random couplings V ip jp and J ia j b . The fundamental tool, which has allowed us to calculate the partition function, is to introduce a matrix formalism with a proper mixing of spinors of each sublattice [3]. The free energy is obtained as
hp ¼ h¯ p þ s1p vp þ s2p , with h¯ p ¼ J 0 M p0 ðpap0 Þ, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s1p ¼ 2½V 2 ðqp1 qp0 Þ þ J 2 ðqp0 1 qp0 0 Þ,
(4)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2½V 2 ðrp qp1 Þ þ J 2 ðrp0 qp0 1 Þxp qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2ðV 2 qp0 þ J 2 qp0 0 Þzp , ð5Þ pffiffiffiffiffiffi 2 and Dx ¼ dxex =2 = 2p (x ¼ zp ; vp or xp ). The saddle point equations for M p (sublattice magnetization), jlp j (Kondo order parameter), m (Parisi’s block size parameter), and qp0 and qp1 (related to the SG order parameter) are obtained directly from Eqs. (2)–(5). In the present fermionic formulation, rp is an additional saddle-point order parameter to be solved with previous ones. Numerical solutions for the saddle-point equations are found. In this context, the AF order (M A ¼ M B a0 and qp1 ¼ qp0 ), the mixed phase ðAF þ SGÞ and the Kondo state ðla ¼ lb a0Þ are obtained. The mixed phase is characterized by an RSB solution ðqp1 aqp0 Þ with finite staggered magnetization M s ¼ ðM A M B Þ=2. Phase diagrams T versus J K for J 0 ¼ 1:6 and two values of G (G ¼ 0 and 1) are shown in Fig. 1. In this work, the quantities T, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J K , J 0 and G are in units of V 2 þ J 2 . As we can see in Fig. 1, there is a transition from the AF order to an RSB region with M s 40 (mixed phase) when temperature T decreases for low values of the strength J K . The SG phase is not observed below the mixed phase for 1S-RSB solution, in contrast to the RS results [3]. The Kondo phase is obtained for high values of J K . The G field suppresses the Neel and the freezing temperatures. Therefore, when G is strong enough, it will lead the magnetic orders to QCPs. To conclude, we have presented here a mean field theory to study the competition between Kondo and RKKY interactions when disorder and frustration are present. In s2p ¼
2F ¼ J 0 M a M b þ J 2 =T½mðqa1 qb1 qa0 qb0 Þ
1
J0=1.6 and D=12. 2
T
þ ra rb qa1 qb1 X V2 ½mðq2p1 q2p0 Þ J K jlp j2 þ þ 2T p Z T 1 þ r2p q2p1 Dzp ln m Z 1 1 m Z 1 EðH p Þ Dvp Dxp e ,
PARA
ð2Þ 1
1
KONDO
AF
where Z
þbD
EðH p Þ ¼ bD
pffiffiffiffi x þ Hp dx ln cosh þ cosh D , bD 2
(3)
SG+AF 0 0
T is the temperature, b ¼ 1=T, 2D is the width of the conduction electron band with D ¼ ½ðx H p Þ2 =4þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðbJ K lp Þ2 , H p ¼ b G2 þ h2p , the internal field
2
4
6
8 Jk
10
12
14
Fig. 1. Phase diagram T versus J K for J 0 ¼ 1:6 when G ¼ 0:0 (full lines) and G ¼ 1:0 (dashed lines).
ARTICLE IN PRESS S.G. Magalhaes et al. / Physica B 403 (2008) 1395–1397
order to treat properly the frustration in the problem, we go beyond RS approximation using 1S-RSB [5]. The main result, as can be seen in Fig. 1, is that there is no more SG pure solution below the AF order, in contrast to the results of the RS treatment of Ref. [3] for the same values of J 0 . Thus, the presence of frustration appears only combined with spontaneous staggered magnetization in the mixed phase solution.
1397
References [1] B. Coqblin, et al., Philos. Mag. B 86 (2006) 2567. [2] S. Majundar, et al., Solid State Commun. 121 (2002) 665. [3] S.G. Magalhaes, F.M. Zimmer, B. Coqblin, J. Phys.: Condens. Matter 18 (2006) 3479. [4] A. Theumann, B. Coqblin, S.G. Magalhaes, A.A. Schmidt, Phys. Rev. B 63 (2001) 054409. [5] G. Parisi, J. Phys. A 13 (1980) 1101.