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The spin glass–Kondo competition in heavy fermion systems: study of the quantum critical point
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 46–47
The spin glass–Kondo competition in heavy fermion systems: study of the quantum critical point A. Theumanna,*, B. Coqblinb a
! Instituto de Fisica, Universidade Federal do Rio Grande do Sul, Av. Bento Goncalves 9500, 91501-970, Porto Alegre RS, Brazil b # Laboratoire de Physique des Solides, Universit!e Paris-Sud, Batiment 510, 91405 Orsay, France
Abstract We study here the Kondo-spin glass competition in a theoretical model with an intra-site Kondo-like exchange interaction and an inter-site interaction with random couplings between localized spins and with an additional transverse field. We obtain two second-order transition lines from the spin-glass state to the paramagnetic one and then to the Kondo state, where both end in two distinct quantum critical points. r 2003 Elsevier B.V. All rights reserved. PACS: 05.50.+q; 6460.Cn Keywords: Spin glass–Kondo competition; Quantum critical point
It is well known that there is a strong competition between the Kondo effect and the RKKY interaction between different magnetic atoms in heavy fermion systems. The Doniach diagram [1] gives a good description of this competition and the Neel temperature is passing through a maximum and tends to zero at the ‘‘quantum critical point’’ (QCP), with a second-order transition at zero temperature. Moreover, the disorder can yield a spin glass (SG) phase in addition to the Kondo behavior, which is mainly a non-Fermi liquid (NFL) one, at low temperatures around the QCP in disordered Cerium or Uranium alloys and compounds. This is the case of the magnetic phase diagrams of CeNi1x Cux [2] or Ce2 Au1x Cox Si3 alloys [3]. The three phases antiferromagnetic (AF), SG and NFL have been obtained at low temperature for different concentrations in UCu5x Pdx [4] or U1x Lax Pd2 Al3 [5]. Thus, a SGKondo transition has been observed experimentally with increasing concentration around the QCP. The Kondo-SG competition has been studied in the framework of an Hamiltonian with an intra-site Kondo-
type exchange interaction treated within the mean-field approximation and an inter-site quantum Ising exchange interaction with random couplings among localized spins as in the Sherrington–Kirkpatrick spinglass model [6]. This previous work has explained the SG–Kondo transition, but however a first-order transition with no QCP has been obtained there [6], as well as in the following study of the competition between SG, ferromagnetic and Kondo phases [7]. In the present work, we add a transverse field G in the x direction, which represents a simple quantum mechanism of spin flipping, in order to have a better description of the spinglass state. The Hamiltonian of the model is H ¼ Hk þ HSG ; X f X ek nks þ e0 nis Hk ¼ k;s
þ JK
X
i;s
½Sfiþ s ci
þ Sfi sþ ci ;
ð2Þ
i
HSG ¼
X i;j
*Corresponding author. Tel.: +55-51-3316-6454; fax: +5551-3316-7286. E-mail addresses: [email protected] (A. Theumann), [email protected] (B. Coqblin).
ð1Þ
Jij Sfiz Sfjz 2G
X
Sfix :
ð3Þ
i
The first term of Eq. (2) describes the conduction band, the second term the f -band, the third term the intra-site Kondo interaction (with a positive JK value); then, the first term of Eq. (3) describes the inter-site
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.11.039
ARTICLE IN PRESS A. Theumann, B. Coqblin / Journal of Magnetism and Magnetic Materials 272–276 (2004) 46–47 6.5 6.0 5.5 5.0 4.5
PARAMAGNETIC
Tc/J
4.0
T2
3.5
Tk
3.0 2.5
KONDO
2.0 1.5
T1
1.0
SG
0.5 0.0 0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
Jk/J
Fig. 1. Phase diagram in the plane T=J vs. JK =J; showing the two transitions from SG to paramagnetic and from paramagnetic to Kondo state, as explained in the text. The dash–dot line represents the ‘‘pure’’ Kondo temperature TK and the dotted line represents the residual magnetic moment on the line T ¼ T2 :
interaction between f -magnetic moments and the second term the additional transverse field. The coupling Jij is an independent random variable with a gaussian distribution which has a zero mean value and a variance 8J 2 : Then, we study the problem within the functional integral formalism and the replica symmetric theory in the static approximation, as previously performed [6]. We will publish detailed calculations elsewhere. The saddle-point equations for the SG- and Kondoorder parameters are obtained by minimizing the free energy with respect to these order parameters. Taking here a parametrization G ¼ aJK2 [8], we obtain Fig. 1 which shows the phase diagram in the plane T=J vs.
47
JK =J for D=J ¼ 12 and aJ ¼ 0:01348: We see, in Fig. 1 for those values of the parameters, that the second-order critical line T1 ðJK Þ that separates the paramagnetic phase from the SG state decreases continuously with increasing JK and ends at a QCP corresponding to a JK c value called JK1 ; while the second-order critical line T2 ðJK Þ that separates the paramagnetic phase from the c Kondo state ends at a second QCP JK2 : We have here c c JK2 > JK1 and in order to obtain such a behavior, we need D=J > 11 and ao0:16=D: For other values of the parameters, we may obtain that the two critical lines intersect and that it is not really a QCP. The physical case shown in Fig. 1, where the two QCP are very close to each other and almost equal, corresponds well to experimental phase diagrams obtained in some Cerium or Uranium disordered alloys. The present work can account for the occurrence of second-order transitions and QCP between SG and paramagnetic and then to the Kondo state, which represents an improvement with respect to the previous work [6] where a first-order SG–Kondo transition was obtained. A full paper with detailed calculations will be published.
References [1] [2] [3] [4] [5] [6] [7] [8]
S. Doniach, Physica B 91 (1977) 231. J. Garcia Soldevilla, et al., Phys. Rev. B 61 (2000) 6821. S. Majumdar, et al., Solid State Comm. 121 (2002) 665. R. Vollmer, et al., Phys. Rev. B 61 (2000) 1218. V.S. Zapf, et al., Phys. Rev. B 65 (2001) 024437. A. Theumann, et al., Phys. Rev. B 63 (2001) 054409. S.G. Magalhaes, et al., Eur. Phys. B 30 (2002) 419. J.R. Iglesias, et al., Phys. Rev. B 56 (1997) 11820.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 46–47
The spin glass–Kondo competition in heavy fermion systems: study of the quantum critical point A. Theumanna,*, B. Coqblinb a
! Instituto de Fisica, Universidade Federal do Rio Grande do Sul, Av. Bento Goncalves 9500, 91501-970, Porto Alegre RS, Brazil b # Laboratoire de Physique des Solides, Universit!e Paris-Sud, Batiment 510, 91405 Orsay, France
Abstract We study here the Kondo-spin glass competition in a theoretical model with an intra-site Kondo-like exchange interaction and an inter-site interaction with random couplings between localized spins and with an additional transverse field. We obtain two second-order transition lines from the spin-glass state to the paramagnetic one and then to the Kondo state, where both end in two distinct quantum critical points. r 2003 Elsevier B.V. All rights reserved. PACS: 05.50.+q; 6460.Cn Keywords: Spin glass–Kondo competition; Quantum critical point
It is well known that there is a strong competition between the Kondo effect and the RKKY interaction between different magnetic atoms in heavy fermion systems. The Doniach diagram [1] gives a good description of this competition and the Neel temperature is passing through a maximum and tends to zero at the ‘‘quantum critical point’’ (QCP), with a second-order transition at zero temperature. Moreover, the disorder can yield a spin glass (SG) phase in addition to the Kondo behavior, which is mainly a non-Fermi liquid (NFL) one, at low temperatures around the QCP in disordered Cerium or Uranium alloys and compounds. This is the case of the magnetic phase diagrams of CeNi1x Cux [2] or Ce2 Au1x Cox Si3 alloys [3]. The three phases antiferromagnetic (AF), SG and NFL have been obtained at low temperature for different concentrations in UCu5x Pdx [4] or U1x Lax Pd2 Al3 [5]. Thus, a SGKondo transition has been observed experimentally with increasing concentration around the QCP. The Kondo-SG competition has been studied in the framework of an Hamiltonian with an intra-site Kondo-
type exchange interaction treated within the mean-field approximation and an inter-site quantum Ising exchange interaction with random couplings among localized spins as in the Sherrington–Kirkpatrick spinglass model [6]. This previous work has explained the SG–Kondo transition, but however a first-order transition with no QCP has been obtained there [6], as well as in the following study of the competition between SG, ferromagnetic and Kondo phases [7]. In the present work, we add a transverse field G in the x direction, which represents a simple quantum mechanism of spin flipping, in order to have a better description of the spinglass state. The Hamiltonian of the model is H ¼ Hk þ HSG ; X f X ek nks þ e0 nis Hk ¼ k;s
þ JK
X
i;s
½Sfiþ s ci
þ Sfi sþ ci ;
ð2Þ
i
HSG ¼
X i;j
*Corresponding author. Tel.: +55-51-3316-6454; fax: +5551-3316-7286. E-mail addresses: [email protected] (A. Theumann), [email protected] (B. Coqblin).
ð1Þ
Jij Sfiz Sfjz 2G
X
Sfix :
ð3Þ
i
The first term of Eq. (2) describes the conduction band, the second term the f -band, the third term the intra-site Kondo interaction (with a positive JK value); then, the first term of Eq. (3) describes the inter-site
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.11.039
ARTICLE IN PRESS A. Theumann, B. Coqblin / Journal of Magnetism and Magnetic Materials 272–276 (2004) 46–47 6.5 6.0 5.5 5.0 4.5
PARAMAGNETIC
Tc/J
4.0
T2
3.5
Tk
3.0 2.5
KONDO
2.0 1.5
T1
1.0
SG
0.5 0.0 0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
Jk/J
Fig. 1. Phase diagram in the plane T=J vs. JK =J; showing the two transitions from SG to paramagnetic and from paramagnetic to Kondo state, as explained in the text. The dash–dot line represents the ‘‘pure’’ Kondo temperature TK and the dotted line represents the residual magnetic moment on the line T ¼ T2 :
interaction between f -magnetic moments and the second term the additional transverse field. The coupling Jij is an independent random variable with a gaussian distribution which has a zero mean value and a variance 8J 2 : Then, we study the problem within the functional integral formalism and the replica symmetric theory in the static approximation, as previously performed [6]. We will publish detailed calculations elsewhere. The saddle-point equations for the SG- and Kondoorder parameters are obtained by minimizing the free energy with respect to these order parameters. Taking here a parametrization G ¼ aJK2 [8], we obtain Fig. 1 which shows the phase diagram in the plane T=J vs.
47
JK =J for D=J ¼ 12 and aJ ¼ 0:01348: We see, in Fig. 1 for those values of the parameters, that the second-order critical line T1 ðJK Þ that separates the paramagnetic phase from the SG state decreases continuously with increasing JK and ends at a QCP corresponding to a JK c value called JK1 ; while the second-order critical line T2 ðJK Þ that separates the paramagnetic phase from the c Kondo state ends at a second QCP JK2 : We have here c c JK2 > JK1 and in order to obtain such a behavior, we need D=J > 11 and ao0:16=D: For other values of the parameters, we may obtain that the two critical lines intersect and that it is not really a QCP. The physical case shown in Fig. 1, where the two QCP are very close to each other and almost equal, corresponds well to experimental phase diagrams obtained in some Cerium or Uranium disordered alloys. The present work can account for the occurrence of second-order transitions and QCP between SG and paramagnetic and then to the Kondo state, which represents an improvement with respect to the previous work [6] where a first-order SG–Kondo transition was obtained. A full paper with detailed calculations will be published.
References [1] [2] [3] [4] [5] [6] [7] [8]
S. Doniach, Physica B 91 (1977) 231. J. Garcia Soldevilla, et al., Phys. Rev. B 61 (2000) 6821. S. Majumdar, et al., Solid State Comm. 121 (2002) 665. R. Vollmer, et al., Phys. Rev. B 61 (2000) 1218. V.S. Zapf, et al., Phys. Rev. B 65 (2001) 024437. A. Theumann, et al., Phys. Rev. B 63 (2001) 054409. S.G. Magalhaes, et al., Eur. Phys. B 30 (2002) 419. J.R. Iglesias, et al., Phys. Rev. B 56 (1997) 11820.