- Email: [email protected]
2 anisotropic antiferromagnet in a staggered magnetic field
Journal of Magnetism and Magnetic Materials 323 (2011) 1064–1067
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Thermal transport in the one-dimensional spin-1/2 anisotropic antiferromagnet in a staggered magnetic field L.S. Lima Department of Physics, University of Kaiserslautern, D-67663 Kaiserslautern, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 October 2010 Received in revised form 24 November 2010 Available online 7 December 2010
We study the thermal transport in the one-dimensional spin-1/2 Heisenberg antiferromagnet in a staggered magnetic field. The thermal conductivity was calculated using bosonization and the Kubo linear response formalism in order to determine the thermal Drude weight Dth(T). & 2010 Elsevier B.V. All rights reserved.
Keywords: One-dimensional Thermal transport
1. Introduction Measurements of the thermal conductivity are an efficient experimental tool in magnetic materials, since heat transport is sensitive to weak disorder and influenced by phase transitions [1]. As pointed out by several authors [2,3], there is an experimental evidence that spin excitations may contribute significantly to the heat current in low-dimensional spin systems and thus a lot of attention has been focused on the possibility of magnetic heat transport in one-dimensional systems, with ballistic heat transport are good candidates for technological applications, e.g., for carrying heat in electronic devices. In this paper we study the thermal conductivity in the anisotropic one-dimensional spin-1/2 Heisenberg antiferromagnet in a staggered magnetic field using bosonization, mean-field theory and the Kubo formalism of transport with the objective of to verify the influence of the external staggered magnetic field in the thermal conductivity. The model Hamiltonian is given by H¼
N N X X ½JðSxl Sxlþ 1 þ Syl Sylþ 1 Þ þ Jz Szl Szlþ 1 þ h ð1Þl Szl , l
ð1Þ
l
where h is the module of the external magnetic field. A systematic study of coupled spin-1/2 antiferromagnetic chains in an effective staggered field was performed by [4]. The mechanisms generating the staggered fields in real magnets was discussed in [5–8]. Many works have been published focusing on the importance of this study [9,10]. All materials studied so far are highly one-dimensional or quasi-one-dimensional.
Tel.: + 49 631 205 3021; fax: + 49 631 205 3907.
E-mail address: [email protected] 0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.12.004
We make the Jordan–Wigner transformation: Pl1 y o n Slþ ¼ cly eip m ¼ 1 cm cm , n Pl1 y o ip c c m ¼ 1 m m cl , S l ¼ e 1 Szl ¼ cly cl , 2
ð2Þ
and the transformed Hamiltonian has the form H¼
N X J l
1 1 1 ½cly cl þ 1 þ clyþ 1 cl þ Jz nl nl þ 1 þ hð1Þl nl , 2 2 2 2
ð3Þ
where nl ¼ cly cl is the number operator. We outline briefly the definitions of the thermal current and the thermal conductivity within the linear response theory. The thermal conductivity is defined as the response of energy-current density je to a thermal gradient je ¼ krT. Writing X hl,l þ 1 , ð4Þ H¼ l
the current operator corresponding to the local energy density, hl,l + 1 is obtained from the equation of continuity je,l þ 1 je,l ¼ i½H,hl,l þ 1 ðtÞ,
ð5Þ P where the total thermal current je is given by: je ¼ l je,l , and hl,l + 1(t)¼eiHthl,l + 1e iHt and je,l ¼ i[hl 1,l,hl,l + 1]. From Eqs. (3)–(5) we obtained the following expression to the thermal current je: N 2 iX J y 1 ½cly cl þ 1 clyþ 1 cl ½cl1 cl þ 1 clyþ 1 cl1 þ JJ z nl1 je ¼ 2 l 2 2 ) 1 hJð1Þl y y þ ½cl1 cl cly cl1 : þ JJz ½cl1 cl cly cl1 nl þ 1 ð6Þ 2 2
L.S. Lima / Journal of Magnetism and Magnetic Materials 323 (2011) 1064–1067
In the Kubo linear response theory, the thermal conductivity at a finite frequency o, is computed from current–current correlation function [11]: Z Z b b 1 iot kðo,TÞ ¼ dte dt/je je ðt þ itÞS: ð7Þ L 0 0 Here / S denotes the thermodynamic average and L is the system size. The real part of kðo,TÞ can be decomposed into
kðo,TÞ ¼ Dth ðTÞdðoÞ þ kreg ðo,TÞ,
ð8Þ
where Dth(T) is the thermal Drude weight. If Dth(T) vanishes, the thermal conductivity of the system is expected to be completely governed by the regular contribution kreg ðo,TÞ. A non-zero Dth(T) implies ballistic transport. If the total thermal current je is conserved, it is time independent and we can write [12]
pb2
kðo,TÞ ¼
LkB
1065
given by the Bethe Ansatz by K¼
p , v ¼ ðJ p=2Þsing=g, pg
ð12Þ
where the anisotropy is parameterized by Jz ¼ cosg. fðxÞ and f~ ðxÞ are written in terms of the boson operators bk and y bk as [15] ( sffiffiffiffiffi ) X 2px 1 n0 þ sinðkxÞ½bk þ byk , fðxÞ ¼ 2i l nk k40 pffiffiffiffi
f~ ðxÞ ¼ 2f0 K þ
( sffiffiffiffiffi ) 1 2 cosðkxÞ½bk þ byk , nk k40 X
ð13Þ
where bk j0S ¼ 0,
/j2e SdðoÞ,
ð9Þ
where we take kB ¼1 and b ¼ 1=T. Spin-1/2 one-dimensional systems show ballistic heat transport at all temperatures, since in this case the energy current commutes with the Heisenberg Hamiltonian [je,H]¼0 [13].
2. Thermal transport
ð14Þ
pffiffiffiffi ð15Þ K n0 j0S ¼ Sz j0S: P P y y y n0 ¼ k : ck ck :¼ k ðck ck /0jck ck j0SÞ measures the number of P z particles on the ground state, and Sz ¼ N j ¼ 1 Sj . Therefore, we have * pffiffiffi pffiffiffiffi 2px n0 /ei K fðxÞ S ¼ 0 exp i K l (sffiffiffiffiffi )) + pffiffiffiffi X 1 y ð16Þ þ2 K sinðkxÞ½bk þbk 0 : nk k40
2.1. Low-temperature limit Bosonization of the Hamiltonian equation (1), in the gapless regime, 0 rJz r1, and in half filling, leads to the low energy effective Hamiltonian H ¼ H0 þ Hu þHu þ Hbc , H0 ¼
v 2
Z
l
eA eB ¼ eA þ B e1=2½A,B ,
pffiffiffi i K fðxÞ
/e
dx½P2 þð@x fðxÞÞ2 , Z
pffiffiffiffi dxsinð K fðxÞÞ,
l
0
Z
l
pffiffiffiffiffiffiffi dxcosð 4K fðxÞÞ,
0
Hbc ¼ 2pvl þ
Z
l 0
dxð@x fR Þ2 ð@x fL Þ2 2pvl
Z 0
l
dx½ð@x fR Þ4 þ ð@x fL Þ4 , ð10Þ
2
where 2jCj ¼ dð0Þ,l ¼ Na. N is the number of sites on the lattice and a is the lattice space. The amplitudes l, l þ , l are given by [14] " #2K2 1 G 1 þ 2K2 K GðKÞsinðp=KÞ l¼ , ð11Þ pffiffiffiffi K pGð2KÞ 2 pG 1þ 2K2
lþ ¼
1 pK , tan 2p 2K2
l ¼
3K 3 1 G 1 G 2K2 2K2 : 12pK G 3 G3 K 2K2
ð½A,B ¼ const:Þ,
ð17Þ
and the properties of the ground states of the Luttinger–Liquid Hamiltonian operators we have " ð2pix=lÞSz
S¼e
0
Hu ¼ 2hjCj2
Hu ¼ l
Using the Campbell–Baker–Hausdorff formula
lim e-0
#K
e p :
ð18Þ
2 sin l x
From the expression above, we have then Z l pffiffiffiffi /HuS ¼ 2hjCj2 dx/sinð K fðxÞÞS 0 Z pffiffiffi pffiffiffi 1 l ¼ 2jCj2 dx½/ei K fðxÞ S/ei K fðxÞ S 2i 0 8 " #K 9 Z l < = 2px z e 2 S lim p dx sin ¼ 2jCj : ; e-0 2 sin x l 0 l Z l Z l K p dx dx sin x , K l 0 0 x where in the point x0 A f0,lg the field fðxÞ result # rffiffiffiffi" pffiffiffi! 1 p 2 arccos fðx0 Þ ¼ ¼ 0: K 2 4
ð19Þ
ð20Þ
Therefore, the contribution of the Hu term in Eq. (10) for the energy is very small for K 4 2 can be neglected, being relevant for 1o K o 2. For the Hu term, we can make the same analyse where we have
2K2
Hu and Hbc are the leading irrelevant perturbations due to umklapp scattering and band curvature, respectively, that are small in the low energy limit. The bosonic field f ¼ fR þ fL and its conjugate ~ ~ ¼ f f , obey the canonical momentum P ¼ @x fðxÞ, where f R L commutation relation ½fðxÞ, PðxuÞ ¼ idðxxuÞ. Where the fields fR and fL are associated with the fermion densities moving to the right and left, respectively, in Fermi point kF. K and v are the Luttinger parameter and the spin velocity, respectively, and are
/Hu Sp
Z
l
pffiffiffiffiffiffiffi 1 dx/cosð 4K fðxÞÞS ¼
2
0
8 <
Z
l
dx½/ei
0
" 4p x z e dx cos S lim p : l e-0 2 sin l x 0 Z l 4K Z l p dx dx sin x , 4K l 0 0 x Z
¼
l
pffiffiffiffiffi 4KfðxÞ
Sþ /ei
pffiffiffiffiffi 4KfðxÞ
S
#4K 9 = ;
ð21Þ
1066
L.S. Lima / Journal of Magnetism and Magnetic Materials 323 (2011) 1064–1067
which is an irrelevant perturbation in the range of the Luttinger parameter 1 oK o 2. K ¼1 corresponds to the isotropic point (Jz ¼1) and K ¼2 corresponds to the free fermion point (Jz ¼0). Using Eq. (6) we obtain the energy-current operator, je, for the model above as Z je dx½vðv þ hÞð@x fR Þ2 vðvhÞð@x fL Þ2 : ð22Þ 2
For determine the Drude weight Dth ðTÞ ¼ pb /j2e S=N we have to evaluate the two-point correlation function /je ðx, tÞje ð0,0ÞS, where t is the time variable [16]. We change to coordinates z ¼ vt þ ix and z ¼ vtix and we obtain " # 1 v2 ðv þhÞ2 v2 ðvhÞ2 þ , ð23Þ /je ðt,xÞje ð0,0ÞS ¼ 2 8p ðvt þixÞ4 ðvtixÞ4 where we use the averages [15] /JL ðt,xÞJL ð0,0ÞS ¼
/JR ðt,xÞJR ð0,0ÞS ¼
1 8p2 ðvtixÞ2
,
1 8p2 ðvt þixÞ2
ð24Þ
,
Fig. 1. Behavior of the Drude weight Dth(T) for different values of magnetic field h. Dashed lines show the exact result for the low-temperature limit and solid lines are results obtained by Jordan–Wigner transformation and mean-field treatment of the interaction term and of the magnetic field term. The value of the anisotropy constant used in the calculations was Jz ¼ cosp=6. We consider also in our calculations J¼ 1.
and the currents JL, JR, given by i JL ¼ pffiffiffiffiffiffi @z fL , 2p i JR ¼ pffiffiffiffiffiffi @z fR : 2p
ð25Þ
Before performing the space integration the imaginary time direction is compactified by mapping the plane z into strip z using zðzÞ ¼ expð2pz=bÞ leading to the replacement
vb pðvt 7 ixÞ vt 7 ixsin : ð26Þ vb p We have then 9 8 > > > > > > = < 2 2 2 2 b v ðv þ hÞ v ðvhÞ Dth ðTÞ ¼ dx þ
4
4 >,
> 8p 1 > vb vb > ixÞp p > > ; : sin ðvt þ sin ðvtixÞ vb vb 2
Z
1
p
¼
p2 ðv2 þh2 Þ 4vb
2
ðu þ 1Þ
2
Z
p
1 1
1w2 ðu þiwÞ4
dw,
ð27Þ
where we finally find Dth ðTÞ ¼
p2 3v
ðv2 þh2 ÞT:
ð28Þ
¨ When h¼0 this result is equal to Klumper’s and Sakai’s analytic expression [17] for the low-temperature limit of the XXZ model with the spin velocity given by v ¼ ðJ p=2Þsing=g. This result is more generally valid for models with the continuum limit given by the Luttinger model. 2.2. Mean-field theory When Jz ¼0 we have that the Hamiltonian in the momentum space P y is given by H ¼ k ½ek cky ck þ hck ck þ p , with the dispersion ek given by ek ¼ Jcosk. A non-zero value of Jz leads to a four-fermion interaction term in the Hamiltonian that can be treated approximately by Hartree– Fock (for details, see, e.g., Ref. [18]), resulting in a renormalization of ek P to e~ k , where HMF ¼ k e~ k cky ck . At zero temperature and h ¼ 0 the spinon dispersion of the MF theory simplifies to e~ k ¼ Jð1 þ 2=pÞcosðkÞ which compare reasonably well with the exact spinon dispersion by des Cloizeaux and Person, ek ¼ ðpJ=2ÞcosðkÞ. Where the MF ground-state energy is E0 ð0Þ 0:420, which agrees reasonably well with the Bethe
Ansatz result E0 ¼ 1=4lnð2Þ 0:443. Therefore, the MF theory is not completely accurate at low temperature hence it generates a small error in the ground state energy in T¼0. From Eqs. (6) and (9), we obtained the thermal conductivity directly for the case 0 rJz r1, which is known to exhibit gapless spinon-like excitations, as Dth ðTÞ ¼
J4 16T 2
Z 0
L
sin2 ð2kÞ dk
, 2 e~ k cosh 2T
ð29Þ
where L p is a lattice cutoff. Results for Dth(T) for the XXZ model with staggered magnetic field using bosonization and mean-field theory are shown in Fig. 1. We obtained a small variation of the thermal Drude weight with the increase of h. At low temperature, the results are obtained exactly by Eq. (28). The mean-field theory produces a qualitatively picture for the temperature dependence of Dth(T). At low temperature both the slope of Dth ðTÞ T and the position of the maximum are well predicted. However, at high temperatures the results describe only qualitatively due to neglect of many-particle excitations in the mean-field approximation that will generate deviations from the correct value in this range of T. A treatment that also includes the many-particle excitations should generate a more quantitative description in the limit of large T. In conclusion, we have calculated the thermal Drude weight for the one-dimensional Heisenberg antiferromagnet with a staggered magnetic field using bosonization, mean-field theory and the Kubo formalism of transport. We obtained a small influence of the staggered magnetic field on thermal conductivity and that this increases with the increase of the intensity of h. In h¼ 0, our theory ¨ reduces to the result of Klumper and Sakai [17], in low temperature. Using the mean-field approximation, our results also agree with the results of Heidrich-Meisner et al. [16] in h¼0. We obtain a ballistic behavior for the thermal conductivity in T a 0 for all values of magnetic field.
Acknowledgment This work was supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPQ).
L.S. Lima / Journal of Magnetism and Magnetic Materials 323 (2011) 1064–1067
References [1] A.V. Sologubenko, T. Lorentz, H.R. Ott, A. Freimuth, J. Low Temp. Phys. 147 (2007) 387. [2] M. Hofmann, T. Lorenz, K. Berggold, M. Gruninger, A. Freimuth, G.S. Uhrig, E. Bruck, Phys. Rev. B 67 (2003) 184502 references there in. [3] A. Sanchez, S. Paschen, B. Wand, H. Aoki, A. Ochiai, F. Steglich, J. Magn. Magn. Mater. 272–276 (Suppl 1) (2004) E703–E705. [4] M. Sato, M. Oshikawa, Phys. Rev. B 69 (2004) 054406. [5] M. Oshikawa, I. Affleck, Phys. Rev. Lett. 79 (1997) 2883. [6] Y.J. Wang, F.H.L. Essler, M. Fabrizio, A.A. Nersesyan, Phys. Rev. B 66 (2002) 024412. [7] A. Zheludev, et al., Phys. Rev. Lett. 80 (1998) 3630.
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[8] S. Maslov, A. Zheludev, Phys. Rev. B 57 (1998) 68. [9] A.V. Chubukov, S. Sachdev, J. Ye, Phys. Rev. B 49 (1994) 11919. [10] E. Ercolessi, G. Morandi, P. Pieri, M. Roncaglia, Phys. Rev. B 62 (2000) 14860. [11] G.D. Mahan, Many Particle Physics, Plenum Press, New York, 1990. [12] E. Orignac, R. Chitra, R. Citro, Phys. Rev. B 67 (2003) 134426. [13] X. Zotos, F. Naef, P. Prelovsek, Phys. Rev. B 55 (1997) 11029. [14] S. Lukyanov, Nucl. Phys. B 522 (1998) 533. [15] D. Se´ne´chal, arXiv:cond-mat/9908262. [16] F. Heidrich-Meisner, A. Honecker, D.C. Cabra, W. Brenig, Phys. Rev. B 66 (2002) 140406(R). ¨ [17] A. Klumper, K. Sakai, J. Phys. A 35 (2002) 2173. [18] W. Brenig, Phys. Rev. B 56 (1997) 2551.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Thermal transport in the one-dimensional spin-1/2 anisotropic antiferromagnet in a staggered magnetic field L.S. Lima Department of Physics, University of Kaiserslautern, D-67663 Kaiserslautern, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 October 2010 Received in revised form 24 November 2010 Available online 7 December 2010
We study the thermal transport in the one-dimensional spin-1/2 Heisenberg antiferromagnet in a staggered magnetic field. The thermal conductivity was calculated using bosonization and the Kubo linear response formalism in order to determine the thermal Drude weight Dth(T). & 2010 Elsevier B.V. All rights reserved.
Keywords: One-dimensional Thermal transport
1. Introduction Measurements of the thermal conductivity are an efficient experimental tool in magnetic materials, since heat transport is sensitive to weak disorder and influenced by phase transitions [1]. As pointed out by several authors [2,3], there is an experimental evidence that spin excitations may contribute significantly to the heat current in low-dimensional spin systems and thus a lot of attention has been focused on the possibility of magnetic heat transport in one-dimensional systems, with ballistic heat transport are good candidates for technological applications, e.g., for carrying heat in electronic devices. In this paper we study the thermal conductivity in the anisotropic one-dimensional spin-1/2 Heisenberg antiferromagnet in a staggered magnetic field using bosonization, mean-field theory and the Kubo formalism of transport with the objective of to verify the influence of the external staggered magnetic field in the thermal conductivity. The model Hamiltonian is given by H¼
N N X X ½JðSxl Sxlþ 1 þ Syl Sylþ 1 Þ þ Jz Szl Szlþ 1 þ h ð1Þl Szl , l
ð1Þ
l
where h is the module of the external magnetic field. A systematic study of coupled spin-1/2 antiferromagnetic chains in an effective staggered field was performed by [4]. The mechanisms generating the staggered fields in real magnets was discussed in [5–8]. Many works have been published focusing on the importance of this study [9,10]. All materials studied so far are highly one-dimensional or quasi-one-dimensional.
Tel.: + 49 631 205 3021; fax: + 49 631 205 3907.
E-mail address: [email protected] 0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.12.004
We make the Jordan–Wigner transformation: Pl1 y o n Slþ ¼ cly eip m ¼ 1 cm cm , n Pl1 y o ip c c m ¼ 1 m m cl , S l ¼ e 1 Szl ¼ cly cl , 2
ð2Þ
and the transformed Hamiltonian has the form H¼
N X J l
1 1 1 ½cly cl þ 1 þ clyþ 1 cl þ Jz nl nl þ 1 þ hð1Þl nl , 2 2 2 2
ð3Þ
where nl ¼ cly cl is the number operator. We outline briefly the definitions of the thermal current and the thermal conductivity within the linear response theory. The thermal conductivity is defined as the response of energy-current density je to a thermal gradient je ¼ krT. Writing X hl,l þ 1 , ð4Þ H¼ l
the current operator corresponding to the local energy density, hl,l + 1 is obtained from the equation of continuity je,l þ 1 je,l ¼ i½H,hl,l þ 1 ðtÞ,
ð5Þ P where the total thermal current je is given by: je ¼ l je,l , and hl,l + 1(t)¼eiHthl,l + 1e iHt and je,l ¼ i[hl 1,l,hl,l + 1]. From Eqs. (3)–(5) we obtained the following expression to the thermal current je: N 2 iX J y 1 ½cly cl þ 1 clyþ 1 cl ½cl1 cl þ 1 clyþ 1 cl1 þ JJ z nl1 je ¼ 2 l 2 2 ) 1 hJð1Þl y y þ ½cl1 cl cly cl1 : þ JJz ½cl1 cl cly cl1 nl þ 1 ð6Þ 2 2
L.S. Lima / Journal of Magnetism and Magnetic Materials 323 (2011) 1064–1067
In the Kubo linear response theory, the thermal conductivity at a finite frequency o, is computed from current–current correlation function [11]: Z Z b b 1 iot kðo,TÞ ¼ dte dt/je je ðt þ itÞS: ð7Þ L 0 0 Here / S denotes the thermodynamic average and L is the system size. The real part of kðo,TÞ can be decomposed into
kðo,TÞ ¼ Dth ðTÞdðoÞ þ kreg ðo,TÞ,
ð8Þ
where Dth(T) is the thermal Drude weight. If Dth(T) vanishes, the thermal conductivity of the system is expected to be completely governed by the regular contribution kreg ðo,TÞ. A non-zero Dth(T) implies ballistic transport. If the total thermal current je is conserved, it is time independent and we can write [12]
pb2
kðo,TÞ ¼
LkB
1065
given by the Bethe Ansatz by K¼
p , v ¼ ðJ p=2Þsing=g, pg
ð12Þ
where the anisotropy is parameterized by Jz ¼ cosg. fðxÞ and f~ ðxÞ are written in terms of the boson operators bk and y bk as [15] ( sffiffiffiffiffi ) X 2px 1 n0 þ sinðkxÞ½bk þ byk , fðxÞ ¼ 2i l nk k40 pffiffiffiffi
f~ ðxÞ ¼ 2f0 K þ
( sffiffiffiffiffi ) 1 2 cosðkxÞ½bk þ byk , nk k40 X
ð13Þ
where bk j0S ¼ 0,
/j2e SdðoÞ,
ð9Þ
where we take kB ¼1 and b ¼ 1=T. Spin-1/2 one-dimensional systems show ballistic heat transport at all temperatures, since in this case the energy current commutes with the Heisenberg Hamiltonian [je,H]¼0 [13].
2. Thermal transport
ð14Þ
pffiffiffiffi ð15Þ K n0 j0S ¼ Sz j0S: P P y y y n0 ¼ k : ck ck :¼ k ðck ck /0jck ck j0SÞ measures the number of P z particles on the ground state, and Sz ¼ N j ¼ 1 Sj . Therefore, we have * pffiffiffi pffiffiffiffi 2px n0 /ei K fðxÞ S ¼ 0 exp i K l (sffiffiffiffiffi )) + pffiffiffiffi X 1 y ð16Þ þ2 K sinðkxÞ½bk þbk 0 : nk k40
2.1. Low-temperature limit Bosonization of the Hamiltonian equation (1), in the gapless regime, 0 rJz r1, and in half filling, leads to the low energy effective Hamiltonian H ¼ H0 þ Hu þHu þ Hbc , H0 ¼
v 2
Z
l
eA eB ¼ eA þ B e1=2½A,B ,
pffiffiffi i K fðxÞ
/e
dx½P2 þð@x fðxÞÞ2 , Z
pffiffiffiffi dxsinð K fðxÞÞ,
l
0
Z
l
pffiffiffiffiffiffiffi dxcosð 4K fðxÞÞ,
0
Hbc ¼ 2pvl þ
Z
l 0
dxð@x fR Þ2 ð@x fL Þ2 2pvl
Z 0
l
dx½ð@x fR Þ4 þ ð@x fL Þ4 , ð10Þ
2
where 2jCj ¼ dð0Þ,l ¼ Na. N is the number of sites on the lattice and a is the lattice space. The amplitudes l, l þ , l are given by [14] " #2K2 1 G 1 þ 2K2 K GðKÞsinðp=KÞ l¼ , ð11Þ pffiffiffiffi K pGð2KÞ 2 pG 1þ 2K2
lþ ¼
1 pK , tan 2p 2K2
l ¼
3K 3 1 G 1 G 2K2 2K2 : 12pK G 3 G3 K 2K2
ð½A,B ¼ const:Þ,
ð17Þ
and the properties of the ground states of the Luttinger–Liquid Hamiltonian operators we have " ð2pix=lÞSz
S¼e
0
Hu ¼ 2hjCj2
Hu ¼ l
Using the Campbell–Baker–Hausdorff formula
lim e-0
#K
e p :
ð18Þ
2 sin l x
From the expression above, we have then Z l pffiffiffiffi /HuS ¼ 2hjCj2 dx/sinð K fðxÞÞS 0 Z pffiffiffi pffiffiffi 1 l ¼ 2jCj2 dx½/ei K fðxÞ S/ei K fðxÞ S 2i 0 8 " #K 9 Z l < = 2px z e 2 S lim p dx sin ¼ 2jCj : ; e-0 2 sin x l 0 l Z l Z l K p dx dx sin x , K l 0 0 x where in the point x0 A f0,lg the field fðxÞ result # rffiffiffiffi" pffiffiffi! 1 p 2 arccos fðx0 Þ ¼ ¼ 0: K 2 4
ð19Þ
ð20Þ
Therefore, the contribution of the Hu term in Eq. (10) for the energy is very small for K 4 2 can be neglected, being relevant for 1o K o 2. For the Hu term, we can make the same analyse where we have
2K2
Hu and Hbc are the leading irrelevant perturbations due to umklapp scattering and band curvature, respectively, that are small in the low energy limit. The bosonic field f ¼ fR þ fL and its conjugate ~ ~ ¼ f f , obey the canonical momentum P ¼ @x fðxÞ, where f R L commutation relation ½fðxÞ, PðxuÞ ¼ idðxxuÞ. Where the fields fR and fL are associated with the fermion densities moving to the right and left, respectively, in Fermi point kF. K and v are the Luttinger parameter and the spin velocity, respectively, and are
/Hu Sp
Z
l
pffiffiffiffiffiffiffi 1 dx/cosð 4K fðxÞÞS ¼
2
0
8 <
Z
l
dx½/ei
0
" 4p x z e dx cos S lim p : l e-0 2 sin l x 0 Z l 4K Z l p dx dx sin x , 4K l 0 0 x Z
¼
l
pffiffiffiffiffi 4KfðxÞ
Sþ /ei
pffiffiffiffiffi 4KfðxÞ
S
#4K 9 = ;
ð21Þ
1066
L.S. Lima / Journal of Magnetism and Magnetic Materials 323 (2011) 1064–1067
which is an irrelevant perturbation in the range of the Luttinger parameter 1 oK o 2. K ¼1 corresponds to the isotropic point (Jz ¼1) and K ¼2 corresponds to the free fermion point (Jz ¼0). Using Eq. (6) we obtain the energy-current operator, je, for the model above as Z je dx½vðv þ hÞð@x fR Þ2 vðvhÞð@x fL Þ2 : ð22Þ 2
For determine the Drude weight Dth ðTÞ ¼ pb /j2e S=N we have to evaluate the two-point correlation function /je ðx, tÞje ð0,0ÞS, where t is the time variable [16]. We change to coordinates z ¼ vt þ ix and z ¼ vtix and we obtain " # 1 v2 ðv þhÞ2 v2 ðvhÞ2 þ , ð23Þ /je ðt,xÞje ð0,0ÞS ¼ 2 8p ðvt þixÞ4 ðvtixÞ4 where we use the averages [15] /JL ðt,xÞJL ð0,0ÞS ¼
/JR ðt,xÞJR ð0,0ÞS ¼
1 8p2 ðvtixÞ2
,
1 8p2 ðvt þixÞ2
ð24Þ
,
Fig. 1. Behavior of the Drude weight Dth(T) for different values of magnetic field h. Dashed lines show the exact result for the low-temperature limit and solid lines are results obtained by Jordan–Wigner transformation and mean-field treatment of the interaction term and of the magnetic field term. The value of the anisotropy constant used in the calculations was Jz ¼ cosp=6. We consider also in our calculations J¼ 1.
and the currents JL, JR, given by i JL ¼ pffiffiffiffiffiffi @z fL , 2p i JR ¼ pffiffiffiffiffiffi @z fR : 2p
ð25Þ
Before performing the space integration the imaginary time direction is compactified by mapping the plane z into strip z using zðzÞ ¼ expð2pz=bÞ leading to the replacement
vb pðvt 7 ixÞ vt 7 ixsin : ð26Þ vb p We have then 9 8 > > > > > > = < 2 2 2 2 b v ðv þ hÞ v ðvhÞ Dth ðTÞ ¼ dx þ
4
4 >,
> 8p 1 > vb vb > ixÞp p > > ; : sin ðvt þ sin ðvtixÞ vb vb 2
Z
1
p
¼
p2 ðv2 þh2 Þ 4vb
2
ðu þ 1Þ
2
Z
p
1 1
1w2 ðu þiwÞ4
dw,
ð27Þ
where we finally find Dth ðTÞ ¼
p2 3v
ðv2 þh2 ÞT:
ð28Þ
¨ When h¼0 this result is equal to Klumper’s and Sakai’s analytic expression [17] for the low-temperature limit of the XXZ model with the spin velocity given by v ¼ ðJ p=2Þsing=g. This result is more generally valid for models with the continuum limit given by the Luttinger model. 2.2. Mean-field theory When Jz ¼0 we have that the Hamiltonian in the momentum space P y is given by H ¼ k ½ek cky ck þ hck ck þ p , with the dispersion ek given by ek ¼ Jcosk. A non-zero value of Jz leads to a four-fermion interaction term in the Hamiltonian that can be treated approximately by Hartree– Fock (for details, see, e.g., Ref. [18]), resulting in a renormalization of ek P to e~ k , where HMF ¼ k e~ k cky ck . At zero temperature and h ¼ 0 the spinon dispersion of the MF theory simplifies to e~ k ¼ Jð1 þ 2=pÞcosðkÞ which compare reasonably well with the exact spinon dispersion by des Cloizeaux and Person, ek ¼ ðpJ=2ÞcosðkÞ. Where the MF ground-state energy is E0 ð0Þ 0:420, which agrees reasonably well with the Bethe
Ansatz result E0 ¼ 1=4lnð2Þ 0:443. Therefore, the MF theory is not completely accurate at low temperature hence it generates a small error in the ground state energy in T¼0. From Eqs. (6) and (9), we obtained the thermal conductivity directly for the case 0 rJz r1, which is known to exhibit gapless spinon-like excitations, as Dth ðTÞ ¼
J4 16T 2
Z 0
L
sin2 ð2kÞ dk
, 2 e~ k cosh 2T
ð29Þ
where L p is a lattice cutoff. Results for Dth(T) for the XXZ model with staggered magnetic field using bosonization and mean-field theory are shown in Fig. 1. We obtained a small variation of the thermal Drude weight with the increase of h. At low temperature, the results are obtained exactly by Eq. (28). The mean-field theory produces a qualitatively picture for the temperature dependence of Dth(T). At low temperature both the slope of Dth ðTÞ T and the position of the maximum are well predicted. However, at high temperatures the results describe only qualitatively due to neglect of many-particle excitations in the mean-field approximation that will generate deviations from the correct value in this range of T. A treatment that also includes the many-particle excitations should generate a more quantitative description in the limit of large T. In conclusion, we have calculated the thermal Drude weight for the one-dimensional Heisenberg antiferromagnet with a staggered magnetic field using bosonization, mean-field theory and the Kubo formalism of transport. We obtained a small influence of the staggered magnetic field on thermal conductivity and that this increases with the increase of the intensity of h. In h¼ 0, our theory ¨ reduces to the result of Klumper and Sakai [17], in low temperature. Using the mean-field approximation, our results also agree with the results of Heidrich-Meisner et al. [16] in h¼0. We obtain a ballistic behavior for the thermal conductivity in T a 0 for all values of magnetic field.
Acknowledgment This work was supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPQ).
L.S. Lima / Journal of Magnetism and Magnetic Materials 323 (2011) 1064–1067
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