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Dependence of the optical conductivity on the magnetic order
" hysica C 235-240 (1994) 2291-2292
PHYSICA
North-Holland
Dependence of the Optical Conductivity on the Magnetic Order Henk Eskes a and Andrzej M. Ole§b a Max-Planck-Institut fiir FestkSrperforschung, Heisenbergstra6e 1, D-70569 Stuttgart, Federal Republic of Germany b Institute of Physics, Jagellonian University, Reymonta 4, PL-30059 KrakSw, Poland * We derive optical sum rules for the upper and lower bands of the large-U Hubbard model. The weight of the upper band is directly related to the local magnetic order. In spite of considerable progress both in experimental and in theoretical studies of the excitation spectra of strongly correlated materials, there is little understanding of the observed dramatic transfers of spectral weight under doping, found recently in the optical conductivity of the cuprate materials [1]. The weight around 2-3 eV disappears very rapidly as soon as holes or electrons are introduced in the insulating parent rr, aterihis, and reappears in the low- and mid-infrared regime. Similar results are found for La2NiO4 compounds [2], which suggests this phenomenon is characteristic of doped charge-transfer (MottHubbard) insulators in general. It is therefore important to find the physical reasons of these rapid weight transfers. We will start from the Hubbard model,
H = V +T = UE
ni,Tni,l- t Z i
aL'ai+',a'(1)
i,6,a
with ni,a = h it, h a l , a , and i = 1 ,..., Na. Assuming t << U, the upper (UHB) and lower (LHB) Hubbard subbands can be studied separately by a transformation to new fermion operators c~,a = e-Sa~,oe s, where S is determined by the requirement that the transformed Hamiltonian commutes with the potential energy part. The transformation S is described in Ref. [3], and the first order term leads to the strongcoupling model (t-J plus 3-site hopping). Because the number of doubly occupied ci,a sites is now a good quantum number one can decompose any operator into parts which create different *A.M.O. acknowledges the support by the C o m m i t t e e of Scientific Research (KBN) Project 2 P302 93 055 05.
amounts of doubly occupied sites. For instance ci,o;v -- ci,ofii,o, where fii,o = c ,aci,a, causes a transition to the UHB (energy is increased by U). With this one can derive sum rules for the UHB and LHB separately [4]. The finite-frequency conductivity per site is [5],
1
j2
trz(w) = wNa E I(flJ~10) 6@ - E! + E0), (2) : where 10) is the ground-state wavefunction and j~ is the paramagnetic current operator for the field in x-direction. We consider the UHB by restricting j~ to j~.;u, which creates one doubly occupied site. Transforming to ci,~ fermions and integrating over w one finds the sum rule of the UHB, W UHB = t2
i,$,61,a
x
t ( c i+~,~i,~ci+~,~
t t - ci+~,~ci,oci,~ci+~
, ,~),
(a)
with the &translations restricting the hopping to be in the x-direction. Following a similar procedure, the total sum rule W is found, 1
W -
zN(T) +
2t 2
z U N--'----~
x
i,6,6' ,a
with 7~ being the kinetic energy of the ci,a fermions and z the number of nearest neighbors. This determines as well the sum rule for the LHB [6]. (..) is the expectation value in the groundstate of the strong-coupling model. In one dimension (1D) for large U the expectation value is a product of a charge and a spin part [7]. In this case both terms (6. = 6'~
0921-4534/94/$07.00 © 1994 - l-lsevicr Science B.V. All rights reserved. SSDI 0921-4534(94)01711-5
2292
H. Eskes, A.M. Oie,~/Physica C 235-240 (1994) 2291-2292
(a) o
t
I o & o
OjxO
l
(b) t l t o .
.
.
.
.
.
.
.
t l
lit
Y
1.2
T -~ x
1.0
U-5t total
j
0.8
Figure 1. a): Excitations to the UHB only occur if the two spins are antiparallel, b): A hole removes two spin bonds in the x-direction.
0.6 0.4 0.2
and , = -8~) in W UHB are explicitly proportional to the nearest-neighbor spin order (holes are squeezed out) ITyrUHB,ID 412 [ 2 . sin(2~rn) sin2(~'n)] =
2
j
(5) (..)H is the expectation value for a Heisenberg spin chain and n is the density. This demonstrates the explicit dependence of the weight of the UHB on the local spin order, which is easily understood (see Fig.l): the current can only generate a doubly occupied site if (/) there are two neighboring spins and (ii) these form a singlet. Using (S~. S i + l - 1 / 4 ) , = - l n 2 we obtain the curve labeled "UHB" in Fig.2. A similar calculation gives the total and LHB weight (other two curves). Taking U/t = 5 we realize a similar ratio of U and the bandwidth as that in the high-To cuprates. The total weight is almost constant close to half filling. For larger U/t the analytic expressions reproduce very well the numerical data. The weight of the upper band vanishes rapidly moving away from n - 1, as experimentally observed. The figure shows that higher order terms enhance this weight-transfer process. We note that the weight distribution over the different bands is very similar in 2D. The fast reduction of the UHB follows from the dilution of the spin order. First of all doping reduces the number of spin bonds (Fig.l). Furthermore (for D>I) holes weaken the local spin order. Thirdly the kinetic three-site hopping term (~= = -6~) reduces further W UHB. In ID the above formula gives }vUHB(z) __~ (1 - 3x)WUHB(0), where x = 1 - n is the doping percentage. The weight of the UHB is largest at half-filling, where the local spin order has the strongest antiferromagnetic tendency. There are two ways of
0.0 0.0
m
0.2
0.4
0.6
•
0.8
1.0
n
Figure 2. The integrated conductivities: total, UHB and LHB versus occupation n in 1D for U = 5, ~ = 1. The perturbation results (lines) are compared with numerical diagonalization data for a 10-site ring (dots). reducing (Si" Si+6): (i) by doping, and (ii) by an external magnetic field. The first could be useful ~o detect changes of the local spin order. For instance, if the doped holes form large ferromagnetic polarons, the UHB will disappear extremely rapidly when holes are introduced. The second might be used to monitor the field dependence of the local spin order by measuring the optical conductivity in a magnetic field. REFERENCES
1. S. L. Cooper et ai., Phys. Rev. B, 41 (1990) 11605; S. Uchida et al., Phys. Rev. B, 43 (1991) 7942. 2. T. Ido et al., Phys. Rev. B, 44 (1991) 12094. 3. A. B. Harris and R. V. Lange, Phys. Rev., 157 (1967) 295. 4. H. Eskes, A. M. Ole§, M. B. J. Meinders and W. Stephan, to be published. 5. W. Kohn, Phys. Rev., 133 (1964) A171; B. S. Shastry and B. Sutherland, Phys. Rev. Lett., 65 (1990) 243. 6. W. Stephan and P. Horsch, Int. J. Mod. Phys. B, 6 (1992) 589. 7. M. Ogata ~nd H. Shiba, Phys. Rev. B, 41 (1990) 2326.
PHYSICA
North-Holland
Dependence of the Optical Conductivity on the Magnetic Order Henk Eskes a and Andrzej M. Ole§b a Max-Planck-Institut fiir FestkSrperforschung, Heisenbergstra6e 1, D-70569 Stuttgart, Federal Republic of Germany b Institute of Physics, Jagellonian University, Reymonta 4, PL-30059 KrakSw, Poland * We derive optical sum rules for the upper and lower bands of the large-U Hubbard model. The weight of the upper band is directly related to the local magnetic order. In spite of considerable progress both in experimental and in theoretical studies of the excitation spectra of strongly correlated materials, there is little understanding of the observed dramatic transfers of spectral weight under doping, found recently in the optical conductivity of the cuprate materials [1]. The weight around 2-3 eV disappears very rapidly as soon as holes or electrons are introduced in the insulating parent rr, aterihis, and reappears in the low- and mid-infrared regime. Similar results are found for La2NiO4 compounds [2], which suggests this phenomenon is characteristic of doped charge-transfer (MottHubbard) insulators in general. It is therefore important to find the physical reasons of these rapid weight transfers. We will start from the Hubbard model,
H = V +T = UE
ni,Tni,l- t Z i
aL'ai+',a'(1)
i,6,a
with ni,a = h it, h a l , a , and i = 1 ,..., Na. Assuming t << U, the upper (UHB) and lower (LHB) Hubbard subbands can be studied separately by a transformation to new fermion operators c~,a = e-Sa~,oe s, where S is determined by the requirement that the transformed Hamiltonian commutes with the potential energy part. The transformation S is described in Ref. [3], and the first order term leads to the strongcoupling model (t-J plus 3-site hopping). Because the number of doubly occupied ci,a sites is now a good quantum number one can decompose any operator into parts which create different *A.M.O. acknowledges the support by the C o m m i t t e e of Scientific Research (KBN) Project 2 P302 93 055 05.
amounts of doubly occupied sites. For instance ci,o;v -- ci,ofii,o, where fii,o = c ,aci,a, causes a transition to the UHB (energy is increased by U). With this one can derive sum rules for the UHB and LHB separately [4]. The finite-frequency conductivity per site is [5],
1
j2
trz(w) = wNa E I(flJ~10) 6@ - E! + E0), (2) : where 10) is the ground-state wavefunction and j~ is the paramagnetic current operator for the field in x-direction. We consider the UHB by restricting j~ to j~.;u, which creates one doubly occupied site. Transforming to ci,~ fermions and integrating over w one finds the sum rule of the UHB, W UHB = t2
i,$,61,a
x
t ( c i+~,~i,~ci+~,~
t t - ci+~,~ci,oci,~ci+~
, ,~),
(a)
with the &translations restricting the hopping to be in the x-direction. Following a similar procedure, the total sum rule W is found, 1
W -
zN(T) +
2t 2
z U N--'----~
x
i,6,6' ,a
with 7~ being the kinetic energy of the ci,a fermions and z the number of nearest neighbors. This determines as well the sum rule for the LHB [6]. (..) is the expectation value in the groundstate of the strong-coupling model. In one dimension (1D) for large U the expectation value is a product of a charge and a spin part [7]. In this case both terms (6. = 6'~
0921-4534/94/$07.00 © 1994 - l-lsevicr Science B.V. All rights reserved. SSDI 0921-4534(94)01711-5
2292
H. Eskes, A.M. Oie,~/Physica C 235-240 (1994) 2291-2292
(a) o
t
I o & o
OjxO
l
(b) t l t o .
.
.
.
.
.
.
.
t l
lit
Y
1.2
T -~ x
1.0
U-5t total
j
0.8
Figure 1. a): Excitations to the UHB only occur if the two spins are antiparallel, b): A hole removes two spin bonds in the x-direction.
0.6 0.4 0.2
and , = -8~) in W UHB are explicitly proportional to the nearest-neighbor spin order (holes are squeezed out) ITyrUHB,ID 412 [ 2 . sin(2~rn) sin2(~'n)] =
2
j
(5) (..)H is the expectation value for a Heisenberg spin chain and n is the density. This demonstrates the explicit dependence of the weight of the UHB on the local spin order, which is easily understood (see Fig.l): the current can only generate a doubly occupied site if (/) there are two neighboring spins and (ii) these form a singlet. Using (S~. S i + l - 1 / 4 ) , = - l n 2 we obtain the curve labeled "UHB" in Fig.2. A similar calculation gives the total and LHB weight (other two curves). Taking U/t = 5 we realize a similar ratio of U and the bandwidth as that in the high-To cuprates. The total weight is almost constant close to half filling. For larger U/t the analytic expressions reproduce very well the numerical data. The weight of the upper band vanishes rapidly moving away from n - 1, as experimentally observed. The figure shows that higher order terms enhance this weight-transfer process. We note that the weight distribution over the different bands is very similar in 2D. The fast reduction of the UHB follows from the dilution of the spin order. First of all doping reduces the number of spin bonds (Fig.l). Furthermore (for D>I) holes weaken the local spin order. Thirdly the kinetic three-site hopping term (~= = -6~) reduces further W UHB. In ID the above formula gives }vUHB(z) __~ (1 - 3x)WUHB(0), where x = 1 - n is the doping percentage. The weight of the UHB is largest at half-filling, where the local spin order has the strongest antiferromagnetic tendency. There are two ways of
0.0 0.0
m
0.2
0.4
0.6
•
0.8
1.0
n
Figure 2. The integrated conductivities: total, UHB and LHB versus occupation n in 1D for U = 5, ~ = 1. The perturbation results (lines) are compared with numerical diagonalization data for a 10-site ring (dots). reducing (Si" Si+6): (i) by doping, and (ii) by an external magnetic field. The first could be useful ~o detect changes of the local spin order. For instance, if the doped holes form large ferromagnetic polarons, the UHB will disappear extremely rapidly when holes are introduced. The second might be used to monitor the field dependence of the local spin order by measuring the optical conductivity in a magnetic field. REFERENCES
1. S. L. Cooper et ai., Phys. Rev. B, 41 (1990) 11605; S. Uchida et al., Phys. Rev. B, 43 (1991) 7942. 2. T. Ido et al., Phys. Rev. B, 44 (1991) 12094. 3. A. B. Harris and R. V. Lange, Phys. Rev., 157 (1967) 295. 4. H. Eskes, A. M. Ole§, M. B. J. Meinders and W. Stephan, to be published. 5. W. Kohn, Phys. Rev., 133 (1964) A171; B. S. Shastry and B. Sutherland, Phys. Rev. Lett., 65 (1990) 243. 6. W. Stephan and P. Horsch, Int. J. Mod. Phys. B, 6 (1992) 589. 7. M. Ogata ~nd H. Shiba, Phys. Rev. B, 41 (1990) 2326.