- Email: [email protected]
Spin gap in generalized spin models with dimerization and the model for CaV4O9
PHYSICA ELSEVIER
Physica C 263 (1996) 114-117
Spin gap in generalized spin models with dimerization and the model for C a V 4 0 9 Nobuyuki Katoh *, Masatoshi Imada Institutefor Solid State Physics, University of Tokyo 7-22-1, Roppongi, Minato-ku, Tokyo 106, Japan
Abstract
Mechanisms of spin-gap formation in two-dimensional Mott insulators are investigated. The origin of the spin gap is analyzed as a dimer gap in generalized Heisenberg models with dimerization, while it is interpreted as the plaquette singlet gap in CaV409.
We have investigated the generalized Heisenberg model with dimerized antiferromagnetic exchange [1]. The Hamiltonian is written as ~ " = J Y ' ~ ' ( ( 1 + 6)Sr_ ~ ' s r + (1 -- 6 ) S r" Sr+~) r
+J'Esr.s.
,
(1)
r
where the summation ~ represents the sum over all the lattice points on a square lattice, while E' is over all the points with even x-coordinates. In this model, there are two parameters. One is the dimer parameter 6 and the other is the uniform interchain coupling J'. The Hamiltonian (1) is investigated in the region of 0_< 6_< 1, while in the other region, 6 > 1, a modified Hamiltonian is employed in the form defined by ,,~ = J Y'~'(2Sr_ , "Sr it- (1 -- 6 ) S r " Sr+y.) r -I- J ' E ' S r . r
Sr+ ~ .
" Corresponding author. Fax: +81 3 3402 8174; e-mail; [email protected].
(2)
Then, our model includes several seemingly different models. In the region of J ' = 0 and 0 < 8 < l, the model is reduced to the S = 7 l dimerized antiferromagnetic Heisenberg (DAFH) chain, while it is reduced to the S = ~1 ladder model at 6 = 1 [2]. In the 6 >> 1 region, the bond J(1 - 6) becomes ferromagnetic. The model is mapped to the S = 1 AFH chain in the limit of 6 ~ oc. The spin gap As(L) is defined as the energy difference between the ground state and the lowest triplet state. The spin gap has been calculated by quantum Monte Carlo (QMC) simulation and exact diagonalization (ED). The sizes we have taken are L X L--- 4 X 4 ~ 10 X 10 for QMC, while the size is up to 24 sites for ED. Figs. l(a) and (b) show the spin gap as a function of 6 and J ' / J . The spin gap is a continuous and smooth function in the parameter space of 6 and J ' / J which connects the S = 7 1 D A F H chain, the 1 S = 1 AFH chain and S = ~ ladder model. Therefore, the origin of the spin gap in these models must be essentially the same and is understood as the dimer gap in a unified way. Recently, the temperature dependence of the uniform magnetic susceptibility X and nuclear magnetic
0921-4534/96/$15.00 © 1996 Elsevier Science B.V. All fights reserved SSDI 0921-4534(95)00816-0
115
N. Katoh, M. Imada / Physica C 263 (1996) 114-117
1 From these facts, we take the S = ~AFH model with the network shown in Fig. 2(b) as a starting point to discuss the Mott insulating phase. The Hamiltonian is given as , ¢ Y f = J ~ ( i , j ) S i • Sj, where (i, j ) denotes the nearest-neighbor bonds. We first calculate the spin gap by QMC to check whether our model has a spin gap or not. Fig. 3 shows that the spin gap extrapolated to the thermodynamic limit is estimated to be (0.11 + 0.03)J. We also calculate the temperature dependence of X, which is shown in Fig. 3. This also suggests the
relaxation time T 1 of CaV409 have indicated the existence of a spin gap [3]. We have considered the possible mechanism of the spin gap in CaV409 [4]. In this paper, after reviewing our recent studies, we present several new results. Fig. 2(a) shows the crystal structure of VO 5 pyramids in CaV409. A V atom can be treated as the 1 localized spin with S = i because the valence of the V atom is 4 + . The antiferromagnetic exchange coupling is expected to be dominant between the adjacent d, orbitals on the nearest-neighbor V atoms.
(a)
I
S=1 AFH Haldanes,stem
00
[
I
~o~
A.
DAFH c h a i n spin-Peierls
system
r
1.0
2.0 10/3 J'/J (J'=l)
2.0 cO
1.0
0.5
"1""
0.0
~
i
I
I
i
0.3
0.0
I
I
1
0.7 J'/J (J=l)
(b)
2.O . . . . . . . . . . . . . . . . . .
1.5
,
~
oo
........
0
~ ' ' ~ ~"" 1.0 2.0 oo 6 Fig. 1. (a) The spin gap as a function of ~ and J ' / J . (b) The &dependenceof the spin gap for several choices of J ' / J . 0.0 0.0
, 0
0.5
N. Katoh,M. lmada/ Physica C 263 (1996) 114-117
116 (a)
(b)
l) I I
NIcff
Nl):ff
;I
Nl)Off
I
(I
NI
I "°
- -
jr!
(b) --j, Os
in
1/2
Fig. 4. (a) The lattice structure in PE. (b) A plaquette with frustration.
Fig. 2. (a) The lattice structure of a VO5 pyramid in CaV409.(b) The network of the S = 7I AFH model. In the QMC calculation, we take 20 sites as a cell as specified by the wavy-line square.
existence of the spin gap in our model. T o make a more quantitative comparison, J in this c o m p o u n d is estimated at high temperatures. In Fig. 3, the fitting
0 . 1 5 ,,,,,,.
,
.
- 41- <3- ..... ,
our model (QMC) | - s q u a r e lattice AFH ( Q M C ) | 4-site p l a q u e t t e | 4-site.plaquette w l i h f r u s t r a t i o n (J'=O.SJ)|
data [3]
--experimental
[ .
0.10
of experimental data is shown. F r o m the comparison of X obtained from Q M C with the experimental data for CaV409 [3], J in CaV409 is estimated to be around 1 4 0 - 1 6 0 K. The spin gap appears to be larger than the Q M C data, as is clearly seen in the deviation at low temperatures. These quantitative aspects will be discussed below. Next, the perturbation expansion (PE) is carried out to elucidate the origin of the spin gap. The four-site plaquette is treated as the unperturbed H a m i l t o n i a n , while the other bonds are taken as the perturbation, as is shown in Fig. 4(a). W i t h i n the second-order PE, the ground state energy per site E(G2) and the energy difference b e t w e e n the ground state and the triplet state A(2)(k) are obtained as ,g)=-
.
0.05 ti
0.00 ~/L . . . . . . . 0.0 0.5
~02
J
1+--')-576
T
(3)
,
)5...+
....-,....- "
o.,, o.+ o.,o.++o++,,,+ o:, o+ o+ ,.o ,.o ,.+ ,'.2 ,,. y t+, .... 1.0 1.5 T/ J
A(se)(k) = J + 3 J ' ( c o s 1
4 108 2.0
Fig. 3. Temperature dependence of the uniform magnetic susceptibility. Solid circles represent the susceptibility of our model for 2 x 2 cells, while open circles show that for the conventional square-lattice AFH model for 8 × 8 lattices. Fiuing of experimental data [3] to solid circles is shown as a bold line. The high-temperature parts are fitted to give the same coefficient for the Tand T-2 terms. The thin solid curve is the susceptibility for a single plaquette, while the bold broken curve is for a single plaquette with J" = 0.5J. The inset shows the size dependence of the spin gap in our model.
kxa+cos
kya)
31 j , 2 864 J
j,2 J (cos 2kxa +cos 2kya)
1 j,2 . . . . cos k~a cos kya. 9 J
(4)
A n interesting point is that the spin gap survives ( 0 . 2 0 5 J ) e v e n if the value of J ' is equal to J at k = (Tr, 7r). In the PE in term of J / J ' , the ground state e n e r g y per site g:(c2) is obtained as --
f:(2)
_jr
8
1 + --4 ")-77 "
(5)
N. Katoh, M. Imada / Physica C 263 (1996) 114-117
(a)
(b)
DRaDIRlglii DmO mO Ui DRSIIRglImE lDDlailSa Fig. 5. (a) The one-dimensional four-site plaquette model. (b) The lattice structure of a possible quasi-one-dimensional four-siteplaquette system of transition metal oxides.
When J' is equal to J, ~ 2 ) = - 0 . 5 3 7 J obtained from Eq. (3) is lower than ~ 2 ) = - 0 . 4 6 9 J in Eq. (5). The above results imply that the origin of the spin gap is qualitatively understood from the four-site plaquette singlet ( = resonating valence bond state). In estimating the spin gap quantitatively, the next-nearest-neighbor frustrate interaction J" may have a secondary effect [5]. Fig. 3 shows the temperature dependences of X for a plaquette with and without intraplaquette frustration J" illustrated in Fig. 4(b). The spin gap for the single plaquette is always J when the intraplaquette frustration is included [5]. It is expected that the interplaquette frustration J" may effectively reduce the amplitude of the interplaquette bond J' and hence approaches the system more or less to a single plaquette. This increases the spin gap from 0.11J towards ~ J. This similarity of overall behavior of X between the
117
experiment and a single plaquette seen in Fig. 4(b) might result from this frustration effect. Another possible origin of the large spin gap in the experiment might be the temperature dependence of the orbital degeneracy. The exchange J can sensitively increase with decreasing temperature if the occupied orbital is temperature dependent. In order to investigate the mechanism of the spin gap in more detail from the viewpoint of the plaquette singlet formation, we investigate a one-dimensional analog of this model. The lattice structure we treat is shown in Fig. 5(a). The value of the spin gap is estimated to be ~ 0.60J by ED, while it is 0.561J by PE. This one-dimensional model may be not a toy model but a relevant model in some transition metal oxide compounds if a lattice structure such as that in Fig. 5(b) is realized. In summary, we have studied a possible mechanism of the spin-gap formation in the Mott insulating phase. We found that the spin gaps of the spin-Peierls system, the ladder system and the Haldane system are identified as the dimer gap, while that of the C a V 4 0 9 system is due to the plaquette singlet gap.
References [1] N. Katoh and M. Imada, J. Phys. Soc. Jpn. 63 (1994) 4529; ibid. 64 (1995) 1437. [2] T. Barnes, E. Dagouo, J. Riera and E.S. Swanson, Phys. Rev. B 47 (1993) 3196; M. Troyer, H. Tsunetsugu and D. Wiirtz, Phys. Rev. B 50 (1994) 13515. [3] S. Taniguchi et al., J. Phys. Soc. Jpn. 64 (1995) 2758. [4] N. Katoh and M. lmada, J. Phys. Soc. Jpn., to be published. [5] K. Ueda, H. Kontani, M. Sigrist and P.A. Lee, private communication.
Physica C 263 (1996) 114-117
Spin gap in generalized spin models with dimerization and the model for C a V 4 0 9 Nobuyuki Katoh *, Masatoshi Imada Institutefor Solid State Physics, University of Tokyo 7-22-1, Roppongi, Minato-ku, Tokyo 106, Japan
Abstract
Mechanisms of spin-gap formation in two-dimensional Mott insulators are investigated. The origin of the spin gap is analyzed as a dimer gap in generalized Heisenberg models with dimerization, while it is interpreted as the plaquette singlet gap in CaV409.
We have investigated the generalized Heisenberg model with dimerized antiferromagnetic exchange [1]. The Hamiltonian is written as ~ " = J Y ' ~ ' ( ( 1 + 6)Sr_ ~ ' s r + (1 -- 6 ) S r" Sr+~) r
+J'Esr.s.
,
(1)
r
where the summation ~ represents the sum over all the lattice points on a square lattice, while E' is over all the points with even x-coordinates. In this model, there are two parameters. One is the dimer parameter 6 and the other is the uniform interchain coupling J'. The Hamiltonian (1) is investigated in the region of 0_< 6_< 1, while in the other region, 6 > 1, a modified Hamiltonian is employed in the form defined by ,,~ = J Y'~'(2Sr_ , "Sr it- (1 -- 6 ) S r " Sr+y.) r -I- J ' E ' S r . r
Sr+ ~ .
" Corresponding author. Fax: +81 3 3402 8174; e-mail; [email protected].
(2)
Then, our model includes several seemingly different models. In the region of J ' = 0 and 0 < 8 < l, the model is reduced to the S = 7 l dimerized antiferromagnetic Heisenberg (DAFH) chain, while it is reduced to the S = ~1 ladder model at 6 = 1 [2]. In the 6 >> 1 region, the bond J(1 - 6) becomes ferromagnetic. The model is mapped to the S = 1 AFH chain in the limit of 6 ~ oc. The spin gap As(L) is defined as the energy difference between the ground state and the lowest triplet state. The spin gap has been calculated by quantum Monte Carlo (QMC) simulation and exact diagonalization (ED). The sizes we have taken are L X L--- 4 X 4 ~ 10 X 10 for QMC, while the size is up to 24 sites for ED. Figs. l(a) and (b) show the spin gap as a function of 6 and J ' / J . The spin gap is a continuous and smooth function in the parameter space of 6 and J ' / J which connects the S = 7 1 D A F H chain, the 1 S = 1 AFH chain and S = ~ ladder model. Therefore, the origin of the spin gap in these models must be essentially the same and is understood as the dimer gap in a unified way. Recently, the temperature dependence of the uniform magnetic susceptibility X and nuclear magnetic
0921-4534/96/$15.00 © 1996 Elsevier Science B.V. All fights reserved SSDI 0921-4534(95)00816-0
115
N. Katoh, M. Imada / Physica C 263 (1996) 114-117
1 From these facts, we take the S = ~AFH model with the network shown in Fig. 2(b) as a starting point to discuss the Mott insulating phase. The Hamiltonian is given as , ¢ Y f = J ~ ( i , j ) S i • Sj, where (i, j ) denotes the nearest-neighbor bonds. We first calculate the spin gap by QMC to check whether our model has a spin gap or not. Fig. 3 shows that the spin gap extrapolated to the thermodynamic limit is estimated to be (0.11 + 0.03)J. We also calculate the temperature dependence of X, which is shown in Fig. 3. This also suggests the
relaxation time T 1 of CaV409 have indicated the existence of a spin gap [3]. We have considered the possible mechanism of the spin gap in CaV409 [4]. In this paper, after reviewing our recent studies, we present several new results. Fig. 2(a) shows the crystal structure of VO 5 pyramids in CaV409. A V atom can be treated as the 1 localized spin with S = i because the valence of the V atom is 4 + . The antiferromagnetic exchange coupling is expected to be dominant between the adjacent d, orbitals on the nearest-neighbor V atoms.
(a)
I
S=1 AFH Haldanes,stem
00
[
I
~o~
A.
DAFH c h a i n spin-Peierls
system
r
1.0
2.0 10/3 J'/J (J'=l)
2.0 cO
1.0
0.5
"1""
0.0
~
i
I
I
i
0.3
0.0
I
I
1
0.7 J'/J (J=l)
(b)
2.O . . . . . . . . . . . . . . . . . .
1.5
,
~
oo
........
0
~ ' ' ~ ~"" 1.0 2.0 oo 6 Fig. 1. (a) The spin gap as a function of ~ and J ' / J . (b) The &dependenceof the spin gap for several choices of J ' / J . 0.0 0.0
, 0
0.5
N. Katoh,M. lmada/ Physica C 263 (1996) 114-117
116 (a)
(b)
l) I I
NIcff
Nl):ff
;I
Nl)Off
I
(I
NI
I "°
- -
jr!
(b) --j, Os
in
1/2
Fig. 4. (a) The lattice structure in PE. (b) A plaquette with frustration.
Fig. 2. (a) The lattice structure of a VO5 pyramid in CaV409.(b) The network of the S = 7I AFH model. In the QMC calculation, we take 20 sites as a cell as specified by the wavy-line square.
existence of the spin gap in our model. T o make a more quantitative comparison, J in this c o m p o u n d is estimated at high temperatures. In Fig. 3, the fitting
0 . 1 5 ,,,,,,.
,
.
- 41- <3- ..... ,
our model (QMC) | - s q u a r e lattice AFH ( Q M C ) | 4-site p l a q u e t t e | 4-site.plaquette w l i h f r u s t r a t i o n (J'=O.SJ)|
data [3]
--experimental
[ .
0.10
of experimental data is shown. F r o m the comparison of X obtained from Q M C with the experimental data for CaV409 [3], J in CaV409 is estimated to be around 1 4 0 - 1 6 0 K. The spin gap appears to be larger than the Q M C data, as is clearly seen in the deviation at low temperatures. These quantitative aspects will be discussed below. Next, the perturbation expansion (PE) is carried out to elucidate the origin of the spin gap. The four-site plaquette is treated as the unperturbed H a m i l t o n i a n , while the other bonds are taken as the perturbation, as is shown in Fig. 4(a). W i t h i n the second-order PE, the ground state energy per site E(G2) and the energy difference b e t w e e n the ground state and the triplet state A(2)(k) are obtained as ,g)=-
.
0.05 ti
0.00 ~/L . . . . . . . 0.0 0.5
~02
J
1+--')-576
T
(3)
,
)5...+
....-,....- "
o.,, o.+ o.,o.++o++,,,+ o:, o+ o+ ,.o ,.o ,.+ ,'.2 ,,. y t+, .... 1.0 1.5 T/ J
A(se)(k) = J + 3 J ' ( c o s 1
4 108 2.0
Fig. 3. Temperature dependence of the uniform magnetic susceptibility. Solid circles represent the susceptibility of our model for 2 x 2 cells, while open circles show that for the conventional square-lattice AFH model for 8 × 8 lattices. Fiuing of experimental data [3] to solid circles is shown as a bold line. The high-temperature parts are fitted to give the same coefficient for the Tand T-2 terms. The thin solid curve is the susceptibility for a single plaquette, while the bold broken curve is for a single plaquette with J" = 0.5J. The inset shows the size dependence of the spin gap in our model.
kxa+cos
kya)
31 j , 2 864 J
j,2 J (cos 2kxa +cos 2kya)
1 j,2 . . . . cos k~a cos kya. 9 J
(4)
A n interesting point is that the spin gap survives ( 0 . 2 0 5 J ) e v e n if the value of J ' is equal to J at k = (Tr, 7r). In the PE in term of J / J ' , the ground state e n e r g y per site g:(c2) is obtained as --
f:(2)
_jr
8
1 + --4 ")-77 "
(5)
N. Katoh, M. Imada / Physica C 263 (1996) 114-117
(a)
(b)
DRaDIRlglii DmO mO Ui DRSIIRglImE lDDlailSa Fig. 5. (a) The one-dimensional four-site plaquette model. (b) The lattice structure of a possible quasi-one-dimensional four-siteplaquette system of transition metal oxides.
When J' is equal to J, ~ 2 ) = - 0 . 5 3 7 J obtained from Eq. (3) is lower than ~ 2 ) = - 0 . 4 6 9 J in Eq. (5). The above results imply that the origin of the spin gap is qualitatively understood from the four-site plaquette singlet ( = resonating valence bond state). In estimating the spin gap quantitatively, the next-nearest-neighbor frustrate interaction J" may have a secondary effect [5]. Fig. 3 shows the temperature dependences of X for a plaquette with and without intraplaquette frustration J" illustrated in Fig. 4(b). The spin gap for the single plaquette is always J when the intraplaquette frustration is included [5]. It is expected that the interplaquette frustration J" may effectively reduce the amplitude of the interplaquette bond J' and hence approaches the system more or less to a single plaquette. This increases the spin gap from 0.11J towards ~ J. This similarity of overall behavior of X between the
117
experiment and a single plaquette seen in Fig. 4(b) might result from this frustration effect. Another possible origin of the large spin gap in the experiment might be the temperature dependence of the orbital degeneracy. The exchange J can sensitively increase with decreasing temperature if the occupied orbital is temperature dependent. In order to investigate the mechanism of the spin gap in more detail from the viewpoint of the plaquette singlet formation, we investigate a one-dimensional analog of this model. The lattice structure we treat is shown in Fig. 5(a). The value of the spin gap is estimated to be ~ 0.60J by ED, while it is 0.561J by PE. This one-dimensional model may be not a toy model but a relevant model in some transition metal oxide compounds if a lattice structure such as that in Fig. 5(b) is realized. In summary, we have studied a possible mechanism of the spin-gap formation in the Mott insulating phase. We found that the spin gaps of the spin-Peierls system, the ladder system and the Haldane system are identified as the dimer gap, while that of the C a V 4 0 9 system is due to the plaquette singlet gap.
References [1] N. Katoh and M. Imada, J. Phys. Soc. Jpn. 63 (1994) 4529; ibid. 64 (1995) 1437. [2] T. Barnes, E. Dagouo, J. Riera and E.S. Swanson, Phys. Rev. B 47 (1993) 3196; M. Troyer, H. Tsunetsugu and D. Wiirtz, Phys. Rev. B 50 (1994) 13515. [3] S. Taniguchi et al., J. Phys. Soc. Jpn. 64 (1995) 2758. [4] N. Katoh and M. lmada, J. Phys. Soc. Jpn., to be published. [5] K. Ueda, H. Kontani, M. Sigrist and P.A. Lee, private communication.