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Dynamical correlation functions of one-dimensional Kondo insulators
ELSEVIER
Physica B 206 & 207 (1995) 813-815
Dynamical correlation functions of one-dimensional Kondo insulators Tomotoshi Nishino a'*, Kazuo Ueda b "Department of Physics, Tohoku University, Sendai 980-77, Japan blnstitute for Solid State Physics, University of Tokyo, Roppongi 7-22-1. Tokyo 106, Japan
Abstract
Dynamical correlation functions of the one-dimensional Kondo lattice model (1D KLM) are calculated as a reference for dynamic properties of Kondo insulators. In the strong coupling limit of the 1D KLM, the antiferromagnetic dynamic spin susceptibility consists of a peak at the lowest spin excitation energy As and multiple peaks in the higher energy region. The optical conductivity has a dominant peak just above the charge excitation energy A. The ratio A~/As is larger than 3/2, and is a useful quantity to measure the electron correlation in a Kondo insulator.
I. Introduction
Kondo insulators have attracted much interest because of the unusual opening of a gap at low temperatures. A recent experiment [1] on dynamic responses of a Kondo insulator shows that the charge gap Ac is greater than the spin gap A . In this paper we examine the excitation structure of the one-dimensional (1D) Kondo lattice model (KLM) at half-filling, which is a theoretical prototype for the Kondo insulators, and show the overall structure of optical conductivity and dynamic spin susceptibility. The KLM consists of a conduction band and local f-electrons. In one dimension, the Hamiltonian is given by H = - t ~'~ (c*,~,+,~+h.c.)+1,,Y.s~,.S~,
(1)
where S c is the spin of the conduction electrons 2~T, c* (~r/2). .ciT, ' S~' is the spin of the f-electrons and ~ is the Pauli matrix. The ~" -r'r ' f i t. ( c r l 2 ) . . f , ,
* Corresponding author.
f-electron occupation f ~ , f , , + f ~ f , ~ is always one. The exchange coupling is assumed to be antiferromagnetic (JK > 0). The KLM on bipartite lattices has SO(4) symmetry [2]; the pseudospin I, whose components are I z = 1/2 Ei~(c~,,ci~ +f*~,,f~ - 1), I + = E ~ ( - 1 ) i ( c ~ , c ~ - f ~ t f ~ $ ) , and 1 - = (I+)*, is a good quantum number of the KLM as the total spin S =
z, (sT + s~) is. The SO(4) symmetry imposes a selection rule on dynamic responses of the 1D-KLM. For example, only those excited states In) with S = 1 and I = 0 contribute to the dynamic spin susceptibility
Im X~(q, w)= 7r E [(~lS~lg)l~8(o~ - E,, + E~)
(2)
n
at half-filling, since the ground state [g) is shown to have S = I = 0 [3] and the Fourier transform of the spin S o commutes with the pseudospin 1. To the optical conductivity "n"
Im cr(~o) = ~-,~_, I ( n l ) l g ) l ~ ( ~
- Eo + E , ) ,
(3)
only those excited states with S = 0 and I = 1 contribute, where ] is the current operator ] = h.c.) [4]. i ~i~(Ci~Ci * .1~
0921-4526/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0921-4526(94)00593-1
- -
T. Nishimo, K. Ueda
814
/
Physica B 206 & 207 (1995) 813-815
2. Excitations in the strong coupling limit
complicated, and detailed calculations will be discussed elsewhere.
Dynamical correlation functions stand for the relation between the ground state and excited states. We start analyzing eigenstates of the 1D-KLM at halffilling in the strong coupling limit (JK >>t). The atomic limit (t = 0) is the simplest case. The ground state is nothing but a singlet sea Is 1,s 2, . . . SL), t t where s i is a local singlet (c~t f*~ - c,~ fir )/V2[0) and L is the system size. The lowest spin excitation is a spin-flip from singlet s i to triplet t,; the spin gap As is JK. The next lowest excitation is a pair creation of doublets: one is a 'holon' h~" expressed by f/*~10) and # t ~t the other is a 'doublon' dj- - o " by ci~ cj~ l j - ~ l 1 0 ~). A m o n g linear combinations of these two-doublet states = E
"L - I ") ~ij
I. . . . d7 . . . . . hj . . . . )
3. Dynamic correlation functions In the strong coupling limit, structures of dynamic correlation functions are easily determined. As examples, we calculate antiferromagnetic ( A F ) spin susceptibility Im X~z(~r, to) and optical conductivity Im g(to). In the following, we use approximate wave-functions in the first-order perturbation. Only spin excitations with m o m e n t u m zr contribute to Xzz(Tr, to). Within the perturbation, the lowest spin excitation and two-doublet states specified by ~mr~, = cos p(m - n) give the formula
(4)
i~j,cf
64t4~-
16t27r
X,z(Tr, to) = ---_-7_7-_,6(to - A,) + .--;5-7. ~" sin2p w h e r e the dots ' . . . ' denote the singlet sea and ~ r ' = - o ' , those states that satisfy the c o n d i t i o n
are spin excited states with S = 1 and I = 0. The states that satisfy ~,
= ~ , ~ = - ~ 0 ~ = - - ~ m~
9JKL p
9J~
× (5(oo - ~JK -- 2t COS p ) .
(8)
Fig. l(a) shows numerically calculated Im Xzz(~r, to) of the 8-site 1D-KLM at half-filling. The function consists of an isolated spin excitation peak at A s = 7.3 ( < Jx =
(6)
Im Z= (zr, co) Jk=8
are charge excited states with S = 0 and I = 1. Thus the charge gap A in this limit is 3JK/2. Introduction of finite t lifts the degeneracy in excitations. The first order perturbation in t gives the amplitudes ~p~' in Eq. (4) as
,L - -
PBC
" --
APBC .
&
~tm$n : eipm +iqn
-
-
eipn +iqm
(7)
for both spin and charge excitations. The momenta p and q are 2~r/L times half the integer if the periodic boundary conditions (PBC) are imposed, and are 2~r/ L times the integer if the antiperiodic boundary conditions ( A P B C ) are imposed. Excitation energy is given by 3JK/2 -- t COS p -- t COS q. Second order perturbations in t take account of virtual processes leading to a correction of the order of a t2/JK . Polarization processes s~, s~+~ h i , dT~ lower the ground state energy by 2t2/3JK per site. Now we consider the lowest spin excitations. The degeneracy in this subspace is lifted by the processes t~, s~+ ~---~hi t , d J+ ~~ s~, ti + 1- Therefore the spin excitation energy is JK +4t2/3JK +4t2( cos P - - 1)/J~, where p is the m o m e n t u m of the whole system. The spin gap is A = J - 20t2/3JK [3]. The pair-doublet creation processes in the two-doublet states of Eq. (4) are rather
i
i
8
2
CO
16
Im X.. (n:, c0) ~ F ~ C --AI~C Jk=2
0
i
,
2
4
i
(0
6
Fig. 1. ]m X~,(~', ~) of the half-filled ] D K L M at (a) JK = 8 and (b) JK = 2.
T. Nishimo, K. Ueda / Physica B 206 & 207 (1995) 813-815
8) and multiple peaks around 3JK/2 as we expect; the perturbation results are shown by the triangle marks. The Im Xzz(Tr, to) at JK = 2 (Fig. l(b)) also consists of a peak at As and multiple peaks in the higher energy side. In the weak JK region, the spin gap As is known to scale as exp ( - 2~rt/vJ~) [3]. Charge excitations with zero momentum contribute to the optical conductivity or(to). In contrast to the Im X~(zr, to), the optical conductivity in the framework of the first-order perturbation is just a delta-function Im tr(to) ~/~(to - 3JK/2), because corresponding excited states (~. =eipn+i(Tr-P)ra--eipm+i(~r-P)n) have the same energy. We get the same result for the system with open boundary conditions (OBC). Fig. 2 shows numerically calculated Im g(to) of the 8-site 1D-KLM at half-filling. The OBC is assumed. Compared to Im X~(~r, to) in Fig. 1, the structure of the Im tr(to) is quite simple. It has a dominant peak just above the charge excitation energy A , where the peak width decreases with the Kondo coupling JK. In
Jk=1
]J33CT(03)
i Jk~ CO Ii
contrast to the spin gap d,, the charge gap Ac is nearly proportional to JK in the weak JK region; the ratio Ac/A ~ diverges in the weak coupling limit JK---~0 [5].
4. C o n c l u s i o n
The AF dynamic spin susceptibility Im Xzz0r, to) of 1D-KLM at half-filling consists of a peak at the spin excitation energy to = As and multiple peaks in the higher energy side above A~. The optical conductivity Im or(w) has a dominant peak just above the charge excitation energy Ac. The ratio Ac/A s diverges in the limit J-->0, and is a useful quantity to measure the strength of electron correlation in a Kondo insulator.
Acknowledgement
T. Nishino thanks the Kasuya Foundation. The computations were done on the SX-3 at Osaka University and SX-3 at the Institute for Molecular Science, Okazaki National Research Institutes.
References
Jk=2 Jk=4
' 4
815
' 12
Fig. 2. Optical conductivity Im tr(w) of the 8-site 1D KLM at half-filling.
[1] B. Bucher and Z. Schlesinger, Phys. Rev. Lett. 24 (1994) 522. [2] S.C. Zhang, Phys. Rev. Lett. 65 (1990) 120. [3] H. Tsunetsugu, Y. Hatsugai, K. Ueda and M. Sigrist, Phys. Rev. B 46 (1992) 3175. [4] J. Wagner, W. Hanke and D.J. Scalapino, Phys. Rev. B 43 (1991) 10517. [5] T. Nishino and K. Ueda, Phys. Rev. B 47 (1993) 12451.
Physica B 206 & 207 (1995) 813-815
Dynamical correlation functions of one-dimensional Kondo insulators Tomotoshi Nishino a'*, Kazuo Ueda b "Department of Physics, Tohoku University, Sendai 980-77, Japan blnstitute for Solid State Physics, University of Tokyo, Roppongi 7-22-1. Tokyo 106, Japan
Abstract
Dynamical correlation functions of the one-dimensional Kondo lattice model (1D KLM) are calculated as a reference for dynamic properties of Kondo insulators. In the strong coupling limit of the 1D KLM, the antiferromagnetic dynamic spin susceptibility consists of a peak at the lowest spin excitation energy As and multiple peaks in the higher energy region. The optical conductivity has a dominant peak just above the charge excitation energy A. The ratio A~/As is larger than 3/2, and is a useful quantity to measure the electron correlation in a Kondo insulator.
I. Introduction
Kondo insulators have attracted much interest because of the unusual opening of a gap at low temperatures. A recent experiment [1] on dynamic responses of a Kondo insulator shows that the charge gap Ac is greater than the spin gap A . In this paper we examine the excitation structure of the one-dimensional (1D) Kondo lattice model (KLM) at half-filling, which is a theoretical prototype for the Kondo insulators, and show the overall structure of optical conductivity and dynamic spin susceptibility. The KLM consists of a conduction band and local f-electrons. In one dimension, the Hamiltonian is given by H = - t ~'~ (c*,~,+,~+h.c.)+1,,Y.s~,.S~,
(1)
where S c is the spin of the conduction electrons 2~T, c* (~r/2). .ciT, ' S~' is the spin of the f-electrons and ~ is the Pauli matrix. The ~" -r'r ' f i t. ( c r l 2 ) . . f , ,
* Corresponding author.
f-electron occupation f ~ , f , , + f ~ f , ~ is always one. The exchange coupling is assumed to be antiferromagnetic (JK > 0). The KLM on bipartite lattices has SO(4) symmetry [2]; the pseudospin I, whose components are I z = 1/2 Ei~(c~,,ci~ +f*~,,f~ - 1), I + = E ~ ( - 1 ) i ( c ~ , c ~ - f ~ t f ~ $ ) , and 1 - = (I+)*, is a good quantum number of the KLM as the total spin S =
z, (sT + s~) is. The SO(4) symmetry imposes a selection rule on dynamic responses of the 1D-KLM. For example, only those excited states In) with S = 1 and I = 0 contribute to the dynamic spin susceptibility
Im X~(q, w)= 7r E [(~lS~lg)l~8(o~ - E,, + E~)
(2)
n
at half-filling, since the ground state [g) is shown to have S = I = 0 [3] and the Fourier transform of the spin S o commutes with the pseudospin 1. To the optical conductivity "n"
Im cr(~o) = ~-,~_, I ( n l ) l g ) l ~ ( ~
- Eo + E , ) ,
(3)
only those excited states with S = 0 and I = 1 contribute, where ] is the current operator ] = h.c.) [4]. i ~i~(Ci~Ci * .1~
0921-4526/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0921-4526(94)00593-1
- -
T. Nishimo, K. Ueda
814
/
Physica B 206 & 207 (1995) 813-815
2. Excitations in the strong coupling limit
complicated, and detailed calculations will be discussed elsewhere.
Dynamical correlation functions stand for the relation between the ground state and excited states. We start analyzing eigenstates of the 1D-KLM at halffilling in the strong coupling limit (JK >>t). The atomic limit (t = 0) is the simplest case. The ground state is nothing but a singlet sea Is 1,s 2, . . . SL), t t where s i is a local singlet (c~t f*~ - c,~ fir )/V2[0) and L is the system size. The lowest spin excitation is a spin-flip from singlet s i to triplet t,; the spin gap As is JK. The next lowest excitation is a pair creation of doublets: one is a 'holon' h~" expressed by f/*~10) and # t ~t the other is a 'doublon' dj- - o " by ci~ cj~ l j - ~ l 1 0 ~). A m o n g linear combinations of these two-doublet states = E
"L - I ") ~ij
I. . . . d7 . . . . . hj . . . . )
3. Dynamic correlation functions In the strong coupling limit, structures of dynamic correlation functions are easily determined. As examples, we calculate antiferromagnetic ( A F ) spin susceptibility Im X~z(~r, to) and optical conductivity Im g(to). In the following, we use approximate wave-functions in the first-order perturbation. Only spin excitations with m o m e n t u m zr contribute to Xzz(Tr, to). Within the perturbation, the lowest spin excitation and two-doublet states specified by ~mr~, = cos p(m - n) give the formula
(4)
i~j,cf
64t4~-
16t27r
X,z(Tr, to) = ---_-7_7-_,6(to - A,) + .--;5-7. ~" sin2p w h e r e the dots ' . . . ' denote the singlet sea and ~ r ' = - o ' , those states that satisfy the c o n d i t i o n
are spin excited states with S = 1 and I = 0. The states that satisfy ~,
= ~ , ~ = - ~ 0 ~ = - - ~ m~
9JKL p
9J~
× (5(oo - ~JK -- 2t COS p ) .
(8)
Fig. l(a) shows numerically calculated Im Xzz(~r, to) of the 8-site 1D-KLM at half-filling. The function consists of an isolated spin excitation peak at A s = 7.3 ( < Jx =
(6)
Im Z= (zr, co) Jk=8
are charge excited states with S = 0 and I = 1. Thus the charge gap A in this limit is 3JK/2. Introduction of finite t lifts the degeneracy in excitations. The first order perturbation in t gives the amplitudes ~p~' in Eq. (4) as
,L - -
PBC
" --
APBC .
&
~tm$n : eipm +iqn
-
-
eipn +iqm
(7)
for both spin and charge excitations. The momenta p and q are 2~r/L times half the integer if the periodic boundary conditions (PBC) are imposed, and are 2~r/ L times the integer if the antiperiodic boundary conditions ( A P B C ) are imposed. Excitation energy is given by 3JK/2 -- t COS p -- t COS q. Second order perturbations in t take account of virtual processes leading to a correction of the order of a t2/JK . Polarization processes s~, s~+~ h i , dT~ lower the ground state energy by 2t2/3JK per site. Now we consider the lowest spin excitations. The degeneracy in this subspace is lifted by the processes t~, s~+ ~---~hi t , d J+ ~~ s~, ti + 1- Therefore the spin excitation energy is JK +4t2/3JK +4t2( cos P - - 1)/J~, where p is the m o m e n t u m of the whole system. The spin gap is A = J - 20t2/3JK [3]. The pair-doublet creation processes in the two-doublet states of Eq. (4) are rather
i
i
8
2
CO
16
Im X.. (n:, c0) ~ F ~ C --AI~C Jk=2
0
i
,
2
4
i
(0
6
Fig. 1. ]m X~,(~', ~) of the half-filled ] D K L M at (a) JK = 8 and (b) JK = 2.
T. Nishimo, K. Ueda / Physica B 206 & 207 (1995) 813-815
8) and multiple peaks around 3JK/2 as we expect; the perturbation results are shown by the triangle marks. The Im Xzz(Tr, to) at JK = 2 (Fig. l(b)) also consists of a peak at As and multiple peaks in the higher energy side. In the weak JK region, the spin gap As is known to scale as exp ( - 2~rt/vJ~) [3]. Charge excitations with zero momentum contribute to the optical conductivity or(to). In contrast to the Im X~(zr, to), the optical conductivity in the framework of the first-order perturbation is just a delta-function Im tr(to) ~/~(to - 3JK/2), because corresponding excited states (~. =eipn+i(Tr-P)ra--eipm+i(~r-P)n) have the same energy. We get the same result for the system with open boundary conditions (OBC). Fig. 2 shows numerically calculated Im g(to) of the 8-site 1D-KLM at half-filling. The OBC is assumed. Compared to Im X~(~r, to) in Fig. 1, the structure of the Im tr(to) is quite simple. It has a dominant peak just above the charge excitation energy A , where the peak width decreases with the Kondo coupling JK. In
Jk=1
]J33CT(03)
i Jk~ CO Ii
contrast to the spin gap d,, the charge gap Ac is nearly proportional to JK in the weak JK region; the ratio Ac/A ~ diverges in the weak coupling limit JK---~0 [5].
4. C o n c l u s i o n
The AF dynamic spin susceptibility Im Xzz0r, to) of 1D-KLM at half-filling consists of a peak at the spin excitation energy to = As and multiple peaks in the higher energy side above A~. The optical conductivity Im or(w) has a dominant peak just above the charge excitation energy Ac. The ratio Ac/A s diverges in the limit J-->0, and is a useful quantity to measure the strength of electron correlation in a Kondo insulator.
Acknowledgement
T. Nishino thanks the Kasuya Foundation. The computations were done on the SX-3 at Osaka University and SX-3 at the Institute for Molecular Science, Okazaki National Research Institutes.
References
Jk=2 Jk=4
' 4
815
' 12
Fig. 2. Optical conductivity Im tr(w) of the 8-site 1D KLM at half-filling.
[1] B. Bucher and Z. Schlesinger, Phys. Rev. Lett. 24 (1994) 522. [2] S.C. Zhang, Phys. Rev. Lett. 65 (1990) 120. [3] H. Tsunetsugu, Y. Hatsugai, K. Ueda and M. Sigrist, Phys. Rev. B 46 (1992) 3175. [4] J. Wagner, W. Hanke and D.J. Scalapino, Phys. Rev. B 43 (1991) 10517. [5] T. Nishino and K. Ueda, Phys. Rev. B 47 (1993) 12451.