- Email: [email protected]
Kondo necklace model in approximants of Fibonacci chains
Journal of Magnetism and Magnetic Materials 441 (2017) 85–87
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Review articles
Kondo necklace model in approximants of Fibonacci chains Daniel Reyes a,⇑, H. Tarazona b, G. Cuba-Supanta b, C.V. Landauro b, R. Espinoza b, J. Quispe-Marcatoma b a b
Instituto Militar de Engenharia, Praça General Tibúrcio, 80, 22290-270 Praia Vermelha, Rio de Janeiro, Brazil Faculty of Physical Sciences, National University of San Marcos, P.O.-Box 14-0149, Lima 14, Peru
a r t i c l e
i n f o
Article history: Received 30 January 2017 Accepted 11 May 2017 Available online 19 May 2017
a b s t r a c t The low energy behavior of the one dimensional Kondo necklace model with structural aperiodicity is studied using a representation for the localized and conduction electron spins, in terms of local Kondo singlet and triplet operators at zero temperature. A decoupling scheme on the double time Green’s functions is used to find the dispersion relation for the excitations of the system. We determine the dependence between the structural aperiodicity modulation and the spin gap in a Fibonacci approximant chain at zero temperature and in the paramagnetic side of the phase diagram. Ó 2017 Elsevier B.V. All rights reserved.
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . Bond operator mean-field approximation Analysis of results . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction In the last years many studies on the low energy properties of quantum spin systems with modulated spatial structure have been attracting wide interest due to the huge sensitivity to structural details in such systems [1–5]. Even, in real quasicrystalline systems [6] with structural [1,2] and chemical [8,7] aperiodicity, the physical properties strongly differ from their related crystalline phases [9,10]. Among these systems, it has been recently observed a quantum criticality heavy fermion like behavior on the Yb15Al34Au51 icosahedral quasicrystal compound and its approximant crystal Yb14Al35Au51 [11–14]. Besides several authors have claimed that this behavior is due to strong correlation of the critical Yb-valence fluctuation and small Brillouin zone reflecting the large unit cell [15,16]. The heavy fermion behavior is mainly due to the competition of two effects: the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between the magnetic ions, which favors long-range magnetic ⇑ Corresponding author. E-mail address: [email protected] (D. Reyes). http://dx.doi.org/10.1016/j.jmmm.2017.05.030 0304-8853/Ó 2017 Elsevier B.V. All rights reserved.
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85 86 86 87 87 87
order, and the Kondo effect which tends to screen the moments and produces a nonmagnetic ground state. These effects are contained in the Kondo necklace model [17] which neglects charge fluctuations and considers only spin fluctuations. This model was previously used in one dimension, at zero and at finite temperature. These works showed that the pure Kondo necklace model (KNM) in a chain lattice at zero [18] and finite temperature [19], does not present a long-range magnetic order. On the other hand, at higher dimensions, it is found that the model presents a quantum critical point (QCP) where the nonmagnetic gapped phase goes to zero and at the same time appears a magnetic gapless phase [20–22]. This model was also employed for studying these systems under a magnetic field [23,24], considering thermal and magnetic entanglement [25], dimensional crossover [26,27], anisotropy [28–30] and for studying the role of an aperiodic exchange modulation in heavy fermions materials [8]. Based on a previous work [8] we introduce in this paper a structural aperiodicity, employing the Fibonacci sequence, in the one dimensional KNM for studying its effects on the quantum critical behavior spin gap. The Fibonacci chain is an example of a one dimensional quasicrystal. It can be constructed from two difference segments, one long and one short, which are denoted by L
86
D. Reyes et al. / Journal of Magnetism and Magnetic Materials 441 (2017) 85–87
and S, respectively. The constructing rule is to replace S with L, and L with LS such that we obtain the sequence LSLLSLSL . . .. The KNM replaces the hopping term of the conduction electrons by an XY interaction among the conduction electron spins [17] and is given by,
H¼t
X X ðrix rjx þ riy rjy Þ þ J i Si ri ; hi;ji
20 15
where we use the triplet operator Fourier transformation qffiffiffiffiffiffiP 1 ikrði;lÞ t a ði; lÞ ¼ NN ; N s is the number of localized moments k t a ðkÞe s per unit cell (approximant) and N is the number of unit cells. The singlet operators have been condensed, hsyi i ¼ hsi i ¼ s, following the standard bond operator mean-field approximation [20,8]. Also c ¼ x; y; Kk ¼ x0 þ 2Dk , kðkÞ ¼ F n cosðkLÞ þ F n1 cosðkSÞ; Dk ¼ ts2 kðkÞ=4, x0 ¼ 4J þ l , Ns ¼ F nþ1 ¼ F n þ F n1 where F n is the number of L distance’s and F n1 is the number of S distance’s between nearestneighbor sites, and L ¼ F n0 b; S ¼ F n0 1 b, have been defined in function of parameter n0 which generate Fibonacci numbers F n0 and pffiffiffi F n0 1 . In the limit n0 ! 1 L=S ¼ ð 5 þ 1Þ=2 (golden mean). The parameter b ¼ 1=ðF n F n0 þ F n1 F n01 Þ is chosen such that lattice parameter is kept to be unity and n is the approximant order. The wave-vectors k are taken in the first Brillouin zone and the lattice spacing was assumed to be unity. The mean-field Hamiltonian Eq. (2) can be diagonalized using the Green’s functions method [20,32] to obtain the thermal averages of the singlet and triplet correlation functions and the spectrum of excitations in the paramagnetic phase [8]. The energies of these modes are given by x0 ¼ 4J þ l , which is the z-branch dispersionless spectrum of the longitudinal spin triplet states, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and xk ¼ K2k ð2Dk Þ2 ¼ x0 1 þ ts2 kðkÞ=x0 that corresponds to the excitation spectrum of the transverse spin triplet states for both branches xx ¼ xy . Following the mean-field approximation, the parameters introduced can then be obtained by minimization of the ground state energy of Eq. (2), deriving the following saddle-point equations:
2ð2 s2 Þ ¼ I1 ðyÞ þ I2 ðyÞ 1 3 J l ¼ I2 ðyÞ I1 ðyÞ; 2y t 4 with
1 Ns p
Z p
dk pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 þ ykðkÞ 0 Z p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 dk 1 þ ykðkÞ; I2 ðyÞ ¼ Ns p 0
I1 ðyÞ ¼
ð3Þ
ð4Þ
Δ/J
3
4
5
(2,1)
(8,5)
Δ/J
15 10
0
L/S=F2 /F1 F3 /F2 F4 /F3 F6 /F5
(5,3)
5 0
ð2Þ
2
20
2. Bond operator mean-field approximation
k
1
0
where ri and Si are independent sets of spin-1/2 Pauli operators, representing the conduction electron spin and localized spin operators, respectively. The sum hi; ji denotes summation over nearestneighbor sites. The first term mimics electron propagations and the second term represents the magnetic interaction between conduction electrons and localized spins Si via the coupling Ji .
X 3 J þl Hmf ¼ NN s s2 J þ ls2 l þ t yk;z tyk;z 4 4 k i Xh þ Kk tyk;c tyk;c þ Dk t yk;c t yk;c þ tk;c tk;c
0
Zhang et. al.
5
ð1Þ
(3,2)
0.5 0.0
10
i
Considering the bond-operator representation [31] of the Hamiltonian in Eq. (1) the resulting effective Hamiltonian Hmf in one dimension with only quadratic operators can be written by,
1.0
1
2
3
4
5
t/J
0
1
2
3
4
5
t/J
Fig. 1. Spin-gap as a function of t=J for different approximants (2/1), (3/2), (5/3) and (8/5). Four ratios L=S are considered: F 2 =F 1 (cross lines), F 3 =F 2 (dashed lines), F 4 =F 3 (square lines), and F 6 =F 5 (dots lines). Also for L ¼ S we obtain the original spin gap (solid line) in the KNM at zero temperature [22].
where y ¼ ts2 =Jx0 . An equation for y can be obtained:
y¼2
t I1 ðyÞ 1 : J 2
ð5Þ
3. Analysis of results We will now obtain the numerical solutions to the zero temperature self-consistent Eqs. (3) using Eq. (5). In this paramagnetic case, the z-polarized branch of excitations has a dispersionless value xz ðkÞ ¼ x0 and the other two branches show a dispersion which has a minimum at the AF reciprocal vector k ¼ Q . The minimum value of the excitations defines the spin gap as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 D 1 l u t u kðpÞ; þ ¼ t1 þ 1 4 J J J þ lJ 4
ð6Þ
The spin gap D=J defines the energy scale for the Kondo singlet phase. In Fig. 1 for L ¼ S we obtain the original spin gap (solid line) in the KNM at zero temperature [22]. The analysis of the spin gap is important because the vanishing of this and the appearance of soft modes define the transition from the disordered Kondo spin liquid to the AF phase at a critical value ðJ=t ¼ ðJ=tÞc Þ. From Eq. (6) we can see that the influence of the structural aperiodicity on the model is controlled by the ratio L=S and the order of the approximant considered. Fig. (1) shows the numerical results for one dimension as a function of L=S and of the approximant’s order (F n ; F n1 ). As t=J gets larger, the spin gap deviates considerably from the linear behavior and there is no indication at all suggesting a critical value for t=J where the gap would vanish. Since the excitation spectra are real and positive everywhere in the Brillouin zone, the system will be in a quantum disordered Kondo spin liquid state for finite values of the coupling strength t=J. Thus, starting from the limit t=J ¼ 0 with localized Kondo spin singlets on each site, we see that any finite coupling strength delocalizes the local Kondo singlets, reducing the magnitude of the gap but not closing it completely. Moreover, there are no qualitative changes on the behavior of D=J for all studied cases. Thereby, the structural aperiodicity (in the approximant considered) just renormalizes the behavior of the spin gap energy for one dimension. Besides, the above behavior of the energy gap clearly suggests that the ground state of this
D. Reyes et al. / Journal of Magnetism and Magnetic Materials 441 (2017) 85–87
Kondo necklace model with structural aperiodicity exchange modulation belongs to the same universality class as the periodic one. 4. Conclusions In this work we have studied the influence of a structural aperiodicity on the one dimensional Kondo necklace model at zero temperature. We have investigated the influence of the ratio L=S and of the order of the approximant on the spin gap energy. We concluded that this structural aperiodicity just renormalizes the behavior of the spin gap energy then any finite coupling strength delocalizes the local Kondo singlets, reducing it but not closing the spin gap completely. Acknowledgments We would like to thank the Programa Nacional de Innovación para la Competitividad y Productividad of the Peruvian Agency Innovate Peru for financial support under contract number N° 457PNICP-ECIP-2015. Daniel Reyes would like to thank the Condensed Matter Research Group of San Marcos University for the hospitality and the stimulating intellectual atmosphere. References [1] A. Jagannathan, A. Szallas, Stefan Wessel, Michel Duneau, Phys. Rev. B 75 (2007) 212407. [2] Stefan Wessel, Igor Milat, Phys. Rev. B 71 (2005) 104427. [3] A. Jagannathan, Phys. Rev. B 71 (2005) 115101. [4] Stefan Wessel, Anuradha Jagannathan, Stephan Haas, Phys. Rev. Lett. 90 (2003) 177205. [5] A. Jagannathan, H.J. Schulz, Phys. Rev. B 55 (1997) 8045. [6] D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53 (1984) 1951. [7] Kazuo Hida, Phys. Rev. Lett. 93 (2004) 037205; J. Phys. Soc. Jpn. 73 (2004) 2296; J. Phys. Soc. Jpn. 74 (2005) 57.
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[8] Daniel Reyes, C.V. Landauro, Journal of Magnetism and Magnetic Materials 322 (2010) 3298. [9] For a pedagogic textbook on the physics of quasicrystals see C. Janot, Quasicrystals: A Primer, Second Edition (Clarendon Press, Oxford, 1994); for a textbook on the mathematics of quasicrystals see M. Senechal, Quasicrystals and Geometry (Cambridge University Press, Cambridge, 1995). [10] T. Janssen, G. Chapuis, M. de Boissieu, Aperiodic Crystals, Oxford University Press, 2007. [11] Kazuhiko Deguchi, Shuya Matsukawa1, Noriaki K. Sato1, Taisuke Hattori, Kenji Ishida, Hiroyuki Takakura, Tsutomu Ishimasa, Nat. Mater. 11 (2012) 1013. [12] T. Watanuki, S. Kashimoto, T. Ishimasa, A. Machida, S. Yamamoto, Y. Tanaka, M. Mizumaki, N. Kawamura, S. Watanabe, Solid State Commun. 211 (2015) 19. [13] S. Matsukawa, K. Tanaka, M. Nakayama, S. Kunikata, K. Deguchi, K. Imura, T. Ishimasa, N.K. Sato, Acta Physica Polonica A 126 (2014) 527. [14] V.R. Shaginyan, A.Z. Msezane, K.G. Popov, G.S. Japaridze, V.A. Khodel, Phys. Rev. B 87 (2013) 245122; V.R. Shaginyan, M. Ya Amusia, A.Z. Msezane, K.G. Popov, Phys. Rep. 492 (2010) 31. [15] Shinji Watanabe, Kazumasa Miyake, Journal of the Physical Society of Japan 82 (2013) 083704, 85, 063703 (2016).. [16] Tetsu Watanuki, Shiro Kashimoto, Daichi Kawana, Teruo Yamazaki, Akihiko Machida, Yukinori Tanaka, Taku J. Sato, Phys. Rev. B 86 (2012) 094201. [17] S. Doniach, Phys. B 91 (1977) 231. [18] R. Jullien, J.N. Fields, S. Doniach, Phys. Rev. B 16 (1977) 4889. [19] R.T. Scalettar, D.J. Scalapino, R.L. Sugar, Phys. Rev. B 31 (1985) 7316. [20] D. Reyes, M.A. Continentino, Phys. Rev. B 76 (2007) 075114. [21] Wolfram Brenig, Phys. Rev. B 73 (2006) 104450. [22] Guang-Ming Zhang, Gu. Qiang, Yu. Lu, Phys. Rev. B 62 (2000) 69. [23] Daniel Reyes, Mucio Continentino, JMMM 320 (2008) 461–463. [24] A.M. Tsvelik, I.A. Zaliznyak, Phys. Rev. B 94 (2016) 075152. [25] A. Saguia, M.S. Sarandy, Phys. Rev. A 67 (2003) 012315. [26] D. Reyes, M.A. Continentino, Physica B 403 (2008) 829–830. [27] D. Reyes, M.A. Continentino, J. Phys.: Condens. Matter 19 (2007) 406203. [28] D. Reyes, Han-Ting Wang, M.A. Continentino, JMMM 321 (2009) 348. [29] H. Rezania, A. Langari, P. Thalmeier, Phys. Rev. B 79 (2009) 094401. [30] A. Saguia, T.G. Rappoport, B. Boechat, M.A. Continentino, Physica A 344 (2004) 644–648. [31] S. Sachdev, R.N. Bhatt, Phys. Rev. B 41 (1990) 9323. [32] Daniel Reyes, Mucio A. Continentino, Christopher Thomas, Claudine Lacroix, Ann. Phys. 373 (2016) 257.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Review articles
Kondo necklace model in approximants of Fibonacci chains Daniel Reyes a,⇑, H. Tarazona b, G. Cuba-Supanta b, C.V. Landauro b, R. Espinoza b, J. Quispe-Marcatoma b a b
Instituto Militar de Engenharia, Praça General Tibúrcio, 80, 22290-270 Praia Vermelha, Rio de Janeiro, Brazil Faculty of Physical Sciences, National University of San Marcos, P.O.-Box 14-0149, Lima 14, Peru
a r t i c l e
i n f o
Article history: Received 30 January 2017 Accepted 11 May 2017 Available online 19 May 2017
a b s t r a c t The low energy behavior of the one dimensional Kondo necklace model with structural aperiodicity is studied using a representation for the localized and conduction electron spins, in terms of local Kondo singlet and triplet operators at zero temperature. A decoupling scheme on the double time Green’s functions is used to find the dispersion relation for the excitations of the system. We determine the dependence between the structural aperiodicity modulation and the spin gap in a Fibonacci approximant chain at zero temperature and in the paramagnetic side of the phase diagram. Ó 2017 Elsevier B.V. All rights reserved.
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . Bond operator mean-field approximation Analysis of results . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction In the last years many studies on the low energy properties of quantum spin systems with modulated spatial structure have been attracting wide interest due to the huge sensitivity to structural details in such systems [1–5]. Even, in real quasicrystalline systems [6] with structural [1,2] and chemical [8,7] aperiodicity, the physical properties strongly differ from their related crystalline phases [9,10]. Among these systems, it has been recently observed a quantum criticality heavy fermion like behavior on the Yb15Al34Au51 icosahedral quasicrystal compound and its approximant crystal Yb14Al35Au51 [11–14]. Besides several authors have claimed that this behavior is due to strong correlation of the critical Yb-valence fluctuation and small Brillouin zone reflecting the large unit cell [15,16]. The heavy fermion behavior is mainly due to the competition of two effects: the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between the magnetic ions, which favors long-range magnetic ⇑ Corresponding author. E-mail address: [email protected] (D. Reyes). http://dx.doi.org/10.1016/j.jmmm.2017.05.030 0304-8853/Ó 2017 Elsevier B.V. All rights reserved.
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85 86 86 87 87 87
order, and the Kondo effect which tends to screen the moments and produces a nonmagnetic ground state. These effects are contained in the Kondo necklace model [17] which neglects charge fluctuations and considers only spin fluctuations. This model was previously used in one dimension, at zero and at finite temperature. These works showed that the pure Kondo necklace model (KNM) in a chain lattice at zero [18] and finite temperature [19], does not present a long-range magnetic order. On the other hand, at higher dimensions, it is found that the model presents a quantum critical point (QCP) where the nonmagnetic gapped phase goes to zero and at the same time appears a magnetic gapless phase [20–22]. This model was also employed for studying these systems under a magnetic field [23,24], considering thermal and magnetic entanglement [25], dimensional crossover [26,27], anisotropy [28–30] and for studying the role of an aperiodic exchange modulation in heavy fermions materials [8]. Based on a previous work [8] we introduce in this paper a structural aperiodicity, employing the Fibonacci sequence, in the one dimensional KNM for studying its effects on the quantum critical behavior spin gap. The Fibonacci chain is an example of a one dimensional quasicrystal. It can be constructed from two difference segments, one long and one short, which are denoted by L
86
D. Reyes et al. / Journal of Magnetism and Magnetic Materials 441 (2017) 85–87
and S, respectively. The constructing rule is to replace S with L, and L with LS such that we obtain the sequence LSLLSLSL . . .. The KNM replaces the hopping term of the conduction electrons by an XY interaction among the conduction electron spins [17] and is given by,
H¼t
X X ðrix rjx þ riy rjy Þ þ J i Si ri ; hi;ji
20 15
where we use the triplet operator Fourier transformation qffiffiffiffiffiffiP 1 ikrði;lÞ t a ði; lÞ ¼ NN ; N s is the number of localized moments k t a ðkÞe s per unit cell (approximant) and N is the number of unit cells. The singlet operators have been condensed, hsyi i ¼ hsi i ¼ s, following the standard bond operator mean-field approximation [20,8]. Also c ¼ x; y; Kk ¼ x0 þ 2Dk , kðkÞ ¼ F n cosðkLÞ þ F n1 cosðkSÞ; Dk ¼ ts2 kðkÞ=4, x0 ¼ 4J þ l , Ns ¼ F nþ1 ¼ F n þ F n1 where F n is the number of L distance’s and F n1 is the number of S distance’s between nearestneighbor sites, and L ¼ F n0 b; S ¼ F n0 1 b, have been defined in function of parameter n0 which generate Fibonacci numbers F n0 and pffiffiffi F n0 1 . In the limit n0 ! 1 L=S ¼ ð 5 þ 1Þ=2 (golden mean). The parameter b ¼ 1=ðF n F n0 þ F n1 F n01 Þ is chosen such that lattice parameter is kept to be unity and n is the approximant order. The wave-vectors k are taken in the first Brillouin zone and the lattice spacing was assumed to be unity. The mean-field Hamiltonian Eq. (2) can be diagonalized using the Green’s functions method [20,32] to obtain the thermal averages of the singlet and triplet correlation functions and the spectrum of excitations in the paramagnetic phase [8]. The energies of these modes are given by x0 ¼ 4J þ l , which is the z-branch dispersionless spectrum of the longitudinal spin triplet states, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and xk ¼ K2k ð2Dk Þ2 ¼ x0 1 þ ts2 kðkÞ=x0 that corresponds to the excitation spectrum of the transverse spin triplet states for both branches xx ¼ xy . Following the mean-field approximation, the parameters introduced can then be obtained by minimization of the ground state energy of Eq. (2), deriving the following saddle-point equations:
2ð2 s2 Þ ¼ I1 ðyÞ þ I2 ðyÞ 1 3 J l ¼ I2 ðyÞ I1 ðyÞ; 2y t 4 with
1 Ns p
Z p
dk pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 þ ykðkÞ 0 Z p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 dk 1 þ ykðkÞ; I2 ðyÞ ¼ Ns p 0
I1 ðyÞ ¼
ð3Þ
ð4Þ
Δ/J
3
4
5
(2,1)
(8,5)
Δ/J
15 10
0
L/S=F2 /F1 F3 /F2 F4 /F3 F6 /F5
(5,3)
5 0
ð2Þ
2
20
2. Bond operator mean-field approximation
k
1
0
where ri and Si are independent sets of spin-1/2 Pauli operators, representing the conduction electron spin and localized spin operators, respectively. The sum hi; ji denotes summation over nearestneighbor sites. The first term mimics electron propagations and the second term represents the magnetic interaction between conduction electrons and localized spins Si via the coupling Ji .
X 3 J þl Hmf ¼ NN s s2 J þ ls2 l þ t yk;z tyk;z 4 4 k i Xh þ Kk tyk;c tyk;c þ Dk t yk;c t yk;c þ tk;c tk;c
0
Zhang et. al.
5
ð1Þ
(3,2)
0.5 0.0
10
i
Considering the bond-operator representation [31] of the Hamiltonian in Eq. (1) the resulting effective Hamiltonian Hmf in one dimension with only quadratic operators can be written by,
1.0
1
2
3
4
5
t/J
0
1
2
3
4
5
t/J
Fig. 1. Spin-gap as a function of t=J for different approximants (2/1), (3/2), (5/3) and (8/5). Four ratios L=S are considered: F 2 =F 1 (cross lines), F 3 =F 2 (dashed lines), F 4 =F 3 (square lines), and F 6 =F 5 (dots lines). Also for L ¼ S we obtain the original spin gap (solid line) in the KNM at zero temperature [22].
where y ¼ ts2 =Jx0 . An equation for y can be obtained:
y¼2
t I1 ðyÞ 1 : J 2
ð5Þ
3. Analysis of results We will now obtain the numerical solutions to the zero temperature self-consistent Eqs. (3) using Eq. (5). In this paramagnetic case, the z-polarized branch of excitations has a dispersionless value xz ðkÞ ¼ x0 and the other two branches show a dispersion which has a minimum at the AF reciprocal vector k ¼ Q . The minimum value of the excitations defines the spin gap as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 D 1 l u t u kðpÞ; þ ¼ t1 þ 1 4 J J J þ lJ 4
ð6Þ
The spin gap D=J defines the energy scale for the Kondo singlet phase. In Fig. 1 for L ¼ S we obtain the original spin gap (solid line) in the KNM at zero temperature [22]. The analysis of the spin gap is important because the vanishing of this and the appearance of soft modes define the transition from the disordered Kondo spin liquid to the AF phase at a critical value ðJ=t ¼ ðJ=tÞc Þ. From Eq. (6) we can see that the influence of the structural aperiodicity on the model is controlled by the ratio L=S and the order of the approximant considered. Fig. (1) shows the numerical results for one dimension as a function of L=S and of the approximant’s order (F n ; F n1 ). As t=J gets larger, the spin gap deviates considerably from the linear behavior and there is no indication at all suggesting a critical value for t=J where the gap would vanish. Since the excitation spectra are real and positive everywhere in the Brillouin zone, the system will be in a quantum disordered Kondo spin liquid state for finite values of the coupling strength t=J. Thus, starting from the limit t=J ¼ 0 with localized Kondo spin singlets on each site, we see that any finite coupling strength delocalizes the local Kondo singlets, reducing the magnitude of the gap but not closing it completely. Moreover, there are no qualitative changes on the behavior of D=J for all studied cases. Thereby, the structural aperiodicity (in the approximant considered) just renormalizes the behavior of the spin gap energy for one dimension. Besides, the above behavior of the energy gap clearly suggests that the ground state of this
D. Reyes et al. / Journal of Magnetism and Magnetic Materials 441 (2017) 85–87
Kondo necklace model with structural aperiodicity exchange modulation belongs to the same universality class as the periodic one. 4. Conclusions In this work we have studied the influence of a structural aperiodicity on the one dimensional Kondo necklace model at zero temperature. We have investigated the influence of the ratio L=S and of the order of the approximant on the spin gap energy. We concluded that this structural aperiodicity just renormalizes the behavior of the spin gap energy then any finite coupling strength delocalizes the local Kondo singlets, reducing it but not closing the spin gap completely. Acknowledgments We would like to thank the Programa Nacional de Innovación para la Competitividad y Productividad of the Peruvian Agency Innovate Peru for financial support under contract number N° 457PNICP-ECIP-2015. Daniel Reyes would like to thank the Condensed Matter Research Group of San Marcos University for the hospitality and the stimulating intellectual atmosphere. References [1] A. Jagannathan, A. Szallas, Stefan Wessel, Michel Duneau, Phys. Rev. B 75 (2007) 212407. [2] Stefan Wessel, Igor Milat, Phys. Rev. B 71 (2005) 104427. [3] A. Jagannathan, Phys. Rev. B 71 (2005) 115101. [4] Stefan Wessel, Anuradha Jagannathan, Stephan Haas, Phys. Rev. Lett. 90 (2003) 177205. [5] A. Jagannathan, H.J. Schulz, Phys. Rev. B 55 (1997) 8045. [6] D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53 (1984) 1951. [7] Kazuo Hida, Phys. Rev. Lett. 93 (2004) 037205; J. Phys. Soc. Jpn. 73 (2004) 2296; J. Phys. Soc. Jpn. 74 (2005) 57.
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