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Effect of lattice distortion on non-Fermi liquid state of two channel Kondo model
Physica B 281&282 (2000) 404}405
E!ect of lattice distortion on non-Fermi liquid state of two channel Kondo model O. Sakai!,",*, Y. Shimizu#, S. Suzuki! !Department of Physics, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan "Department of Physics, Graduate School of Science, Tokyo Metropolitan University, Hachioji 192-0397, Japan #Department of Applied Physics, Tohoku University, Sendai 980-8579, Japan
Abstract We studied the speci"c heat and the magnetic susceptibility of a model containing two competing interactions: the two-channel Kondo model type one with the conduction electrons and the single channel one with the low-energy excitation system, such as the phonons. The speci"c heat shows non-monotonic temperature dependence having negative value region. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Two-channel Kondo model; Dynamical Jahn}Teller e!ect; Dilute uranium alloy
The two-channel Kondo model (TCKM) has the nonFermi liquid (NFL) ground state with residual entropy 1 ln 2 (see for example, Ref. [1]). In actual systems, the 2 local &spin' usually denotes the quadrupolar moment of the non-Kramers doublet. It will couple with phonons through the dynamical Jahn}Teller-type interaction, and form the &spin' singlet state of the single-channel Kondo model (SCKM) type [2]. We study the speci"c heat and the magnetic susceptibility for the system containing these two competing interactions: the TCKM type with conduction electrons and the SCKM type with phonons. In this report, we imitate the phonon freedom by band electrons whose band width is chosen di!erent from that of the real conduction electrons. The following Hamiltonian is examined by Wilson's numerical renormalization group (NRG) method [3]: 2 H" + + ok c` k ck #+ ek b` k bk la la k a a l/1 ka a #Js+ s #J@(q q !q q ). cl x bx y by l
(1)
* Correspondence address. Department of Physics, Graduate School of Science, Tokyo Metropolitan University, Hachioji 1920397, Japan. Tel.: #81-22-217-6439; fax: #81-426-77-2483. E-mail address: [email protected] (O. Sakai)
Here ck denotes the annihilation operator of the conducla tion electron with wave number vector k, the channel number l, spin a and energy ok . bk denotes the a band electron introduced to represent the phonon system. We call this band as b-band hereafter. The matrix s is the local spin freedom, s and s are spins of the cl b conduction electron and b-electron on the local spin site, respectively. The real spin is denoted by l [4]. When q "$1 is ascribed to the C` doublet of D symmetry z 2 5 4 [5], s has the correspondence to the distortion modes, b q JQ 2 2 and iq JQ . We put the coupling conby xy bx x ~y stants for q and q terms to be equal to simplify the bx by analysis. The density of states for the conduction electron is set to be #at in the interval DoD(D, and we choose D"1. The density of states for the b-band, o , is given as b o "g in 1'DeD'D@ and o "gm in D@'DeD, where g is b b the factor to normalize the integrated intensity of o to be b 1 (see for example, Ref. [6]). In the present report, we choose D@"10~10 and m&102. Therefore, b-band acts mainly in the low-energy region, and will simulate the acoustic phonon system. In Fig. 1, we show the speci"c heat for J"0.3. The dot}dashed line gives the result for J@"0 and m"1. It has a peak at about ¹&10~7, and the entropy is integrated as 0.52 at ¹"10~3. This value is large compared with the expected one (1 ln 2"0.35) of the TCKM-type 2 NFL state [7]. But the analysis of the energy #ow chart
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 0 5 2 - 2
O. Sakai et al. / Physica B 281&282 (2000) 404}405
Fig. 1. Temperature dependence of the speci"c heat. J"0.3. The solid line is the LFL case (D@"10~10, m"2]102), the dashed line is the NFL case (D@"10~10, m"0.8]102) and the dot}dashed line is the NFL case with J@"0 and m"1, see the texts. The discretization parameter K"16 is used, and about 400 states are retained in each iteration step of NRG.
(EFC) [3] explicitly shows that the low-energy "xed point is the NFL one. The present numerical result of the speci"c heat is not so reliable in quantitative sense. But we expect correct qualitative behaviors will be extracted. The broken line is the result for J@"0.3 and m"0.8]102. The entropy is 0.46 at ¹"10~3, and EFC indicates the NFL state. We note the speci"c heat has a negative region around ¹&10~11. The present speci"c heat is calculated as the di!erence from the system without the local spin, therefore it can be negative. In the temperature region ¹&10~10&D@, the coupling with b-band takes place, and the speci"c heat increases as if the residual entropy is released to go to the singlet ground state. But the true ground state is NFL in this case. The speci"c heat will have negative region to compensate this increase, because the integrated entropy should be that of the TCKM. The solid line is the speci"c heat for m"2]102. The EFC indicates the local Fermi liquid (LFL) ground state. The integrated entropy is 0.50 at ¹"10~3, which is very small compared with the value (ln 2"0.69) expected for the LFL state. The present calculation seems to stress the negative hallow to be too deep. But we may anticipate that the residual entropy of NFL states is released by a non-monotonous way accompanied by the negative speci"c heat region. This is contrasted to the cases that the entropy is released by static perturbations such as the lowering of the crystalline symmetry [8]. In Fig. 2, we show the magnetic susceptibility under the Zeeman "eld H@"!q H . In the NFL case, it z z increases logarithmic way with decreasing ¹ in the region, ¹)10~7, with small spikes near the negative
405
Fig. 2. The magnetic susceptibility for the LFL and NFL cases. The susceptibility is calculated as Sq T/H with H "3]10~14. z z z
hallows of the speci"c heat. In the LFL case, it increases in the temperature range ¹'10~11, the low ¹ side of the deep hallow. It is known that a dilute uranium system, U Th Ru Si , shows the TCKM like increase of the x 1~x 2 2 magnetic susceptibility and the c coe$cient, but does not show sign of the residual entropy [9]. This behaviors should be re-analyzed by considering the non-monotonic temperature dependence of the speci"c heat. Acknowledgements This work was partly supported by Grant-in-Aids No. 11125201 from the Ministry of Education, Science, Sports and Culture of Japan. References [1] D.L. Cox, A. Zawadowski, Adv. Phys. 47 (1998) 599. [2] F. Guinea, V. Hakim, A. Muramatsu, Phys. Rev. B 32 (1985) 4410. [3] H.B. Krishnamurthy, J.W. Wilkins, K.G. Wilson, Phys. Rev. B 21 (1980) 1003. [4] M. Koga, H. Shiba, J. Phys. Soc. Japan 64 (1995) 4345. [5] O. Sakai, S. Suzuki, Y. Shimizu, Solid State Commun. 101 (1997) 791. [6] O. Sakai, Y. Shimizu, T. Kasuya, Prog. Theor. Phys. (Suppl. 108) (1992) 73. [7] Y. Shimizu, O. Sakai, S. Suzuki, Physica B 259}261 (1999) 366. [8] Y. Shimizu, A. Hewson, O. Sakai, J. Phys. Soc. Japan 68 (1999) 2994. [9] H. Amitsuka, K. Kuwahara, M. Yokoyama, K. Tenya, T. Sakakibara, M. Mihalik, A.A. Menovsky, these Proceedings (SCES '99).
E!ect of lattice distortion on non-Fermi liquid state of two channel Kondo model O. Sakai!,",*, Y. Shimizu#, S. Suzuki! !Department of Physics, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan "Department of Physics, Graduate School of Science, Tokyo Metropolitan University, Hachioji 192-0397, Japan #Department of Applied Physics, Tohoku University, Sendai 980-8579, Japan
Abstract We studied the speci"c heat and the magnetic susceptibility of a model containing two competing interactions: the two-channel Kondo model type one with the conduction electrons and the single channel one with the low-energy excitation system, such as the phonons. The speci"c heat shows non-monotonic temperature dependence having negative value region. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Two-channel Kondo model; Dynamical Jahn}Teller e!ect; Dilute uranium alloy
The two-channel Kondo model (TCKM) has the nonFermi liquid (NFL) ground state with residual entropy 1 ln 2 (see for example, Ref. [1]). In actual systems, the 2 local &spin' usually denotes the quadrupolar moment of the non-Kramers doublet. It will couple with phonons through the dynamical Jahn}Teller-type interaction, and form the &spin' singlet state of the single-channel Kondo model (SCKM) type [2]. We study the speci"c heat and the magnetic susceptibility for the system containing these two competing interactions: the TCKM type with conduction electrons and the SCKM type with phonons. In this report, we imitate the phonon freedom by band electrons whose band width is chosen di!erent from that of the real conduction electrons. The following Hamiltonian is examined by Wilson's numerical renormalization group (NRG) method [3]: 2 H" + + ok c` k ck #+ ek b` k bk la la k a a l/1 ka a #Js+ s #J@(q q !q q ). cl x bx y by l
(1)
* Correspondence address. Department of Physics, Graduate School of Science, Tokyo Metropolitan University, Hachioji 1920397, Japan. Tel.: #81-22-217-6439; fax: #81-426-77-2483. E-mail address: [email protected] (O. Sakai)
Here ck denotes the annihilation operator of the conducla tion electron with wave number vector k, the channel number l, spin a and energy ok . bk denotes the a band electron introduced to represent the phonon system. We call this band as b-band hereafter. The matrix s is the local spin freedom, s and s are spins of the cl b conduction electron and b-electron on the local spin site, respectively. The real spin is denoted by l [4]. When q "$1 is ascribed to the C` doublet of D symmetry z 2 5 4 [5], s has the correspondence to the distortion modes, b q JQ 2 2 and iq JQ . We put the coupling conby xy bx x ~y stants for q and q terms to be equal to simplify the bx by analysis. The density of states for the conduction electron is set to be #at in the interval DoD(D, and we choose D"1. The density of states for the b-band, o , is given as b o "g in 1'DeD'D@ and o "gm in D@'DeD, where g is b b the factor to normalize the integrated intensity of o to be b 1 (see for example, Ref. [6]). In the present report, we choose D@"10~10 and m&102. Therefore, b-band acts mainly in the low-energy region, and will simulate the acoustic phonon system. In Fig. 1, we show the speci"c heat for J"0.3. The dot}dashed line gives the result for J@"0 and m"1. It has a peak at about ¹&10~7, and the entropy is integrated as 0.52 at ¹"10~3. This value is large compared with the expected one (1 ln 2"0.35) of the TCKM-type 2 NFL state [7]. But the analysis of the energy #ow chart
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 0 5 2 - 2
O. Sakai et al. / Physica B 281&282 (2000) 404}405
Fig. 1. Temperature dependence of the speci"c heat. J"0.3. The solid line is the LFL case (D@"10~10, m"2]102), the dashed line is the NFL case (D@"10~10, m"0.8]102) and the dot}dashed line is the NFL case with J@"0 and m"1, see the texts. The discretization parameter K"16 is used, and about 400 states are retained in each iteration step of NRG.
(EFC) [3] explicitly shows that the low-energy "xed point is the NFL one. The present numerical result of the speci"c heat is not so reliable in quantitative sense. But we expect correct qualitative behaviors will be extracted. The broken line is the result for J@"0.3 and m"0.8]102. The entropy is 0.46 at ¹"10~3, and EFC indicates the NFL state. We note the speci"c heat has a negative region around ¹&10~11. The present speci"c heat is calculated as the di!erence from the system without the local spin, therefore it can be negative. In the temperature region ¹&10~10&D@, the coupling with b-band takes place, and the speci"c heat increases as if the residual entropy is released to go to the singlet ground state. But the true ground state is NFL in this case. The speci"c heat will have negative region to compensate this increase, because the integrated entropy should be that of the TCKM. The solid line is the speci"c heat for m"2]102. The EFC indicates the local Fermi liquid (LFL) ground state. The integrated entropy is 0.50 at ¹"10~3, which is very small compared with the value (ln 2"0.69) expected for the LFL state. The present calculation seems to stress the negative hallow to be too deep. But we may anticipate that the residual entropy of NFL states is released by a non-monotonous way accompanied by the negative speci"c heat region. This is contrasted to the cases that the entropy is released by static perturbations such as the lowering of the crystalline symmetry [8]. In Fig. 2, we show the magnetic susceptibility under the Zeeman "eld H@"!q H . In the NFL case, it z z increases logarithmic way with decreasing ¹ in the region, ¹)10~7, with small spikes near the negative
405
Fig. 2. The magnetic susceptibility for the LFL and NFL cases. The susceptibility is calculated as Sq T/H with H "3]10~14. z z z
hallows of the speci"c heat. In the LFL case, it increases in the temperature range ¹'10~11, the low ¹ side of the deep hallow. It is known that a dilute uranium system, U Th Ru Si , shows the TCKM like increase of the x 1~x 2 2 magnetic susceptibility and the c coe$cient, but does not show sign of the residual entropy [9]. This behaviors should be re-analyzed by considering the non-monotonic temperature dependence of the speci"c heat. Acknowledgements This work was partly supported by Grant-in-Aids No. 11125201 from the Ministry of Education, Science, Sports and Culture of Japan. References [1] D.L. Cox, A. Zawadowski, Adv. Phys. 47 (1998) 599. [2] F. Guinea, V. Hakim, A. Muramatsu, Phys. Rev. B 32 (1985) 4410. [3] H.B. Krishnamurthy, J.W. Wilkins, K.G. Wilson, Phys. Rev. B 21 (1980) 1003. [4] M. Koga, H. Shiba, J. Phys. Soc. Japan 64 (1995) 4345. [5] O. Sakai, S. Suzuki, Y. Shimizu, Solid State Commun. 101 (1997) 791. [6] O. Sakai, Y. Shimizu, T. Kasuya, Prog. Theor. Phys. (Suppl. 108) (1992) 73. [7] Y. Shimizu, O. Sakai, S. Suzuki, Physica B 259}261 (1999) 366. [8] Y. Shimizu, A. Hewson, O. Sakai, J. Phys. Soc. Japan 68 (1999) 2994. [9] H. Amitsuka, K. Kuwahara, M. Yokoyama, K. Tenya, T. Sakakibara, M. Mihalik, A.A. Menovsky, these Proceedings (SCES '99).