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Optical spectral weight anomalies and strong correlation
Physica C 460–462 (2007) 1045–1046 www.elsevier.com/locate/physc
Optical spectral weight anomalies and strong correlation A. Toschi
a,c,* ,
M. Capone
b,c
, M. Ortolani
d,e
, P. Calvani
c,e
, S. Lupi
c,e
, C. Castellani
c
a
Max Planck Institut fu¨r Festko¨rperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany INFM-SMC and Instituto dei Sistemi Complessi, CNR, Via dei Taurini 19, 00185 Roma, Italy c Dip. di Fisica, Universita` di Roma ‘‘La Sapienza’’, Piazzale Aldo Moro 5, 00185 Roma, Italy Berliner Elektronenspeicherring-Gesellschaft fu¨r Synchrotronstrahlung, A. Einstein Strasse 15, D-12489 Berlin, Germany e CNR-INFM ‘‘Coherentia’’, Universita` di Roma ‘‘La Sapienza’’, Piazzale A. Moro 2, 00185 Roma, Italy b
d
Available online 28 March 2007
Abstract The anomalous behavior observed in the optical spectral weight (W) of the cuprates provides valuable information about the physics of these compounds. Both the doping and the temperature dependences of W are hardly explained through conventional estimates based on the f-sum rule. By computing the optical conductivity of the doped Hubbard model with the Dynamical Mean Field Theory, we point out that the strong correlation plays a key role in determining the basic features of the observed anomalies: the proximity to a Mott insulating phase accounts simultaneously for the strong temperature dependence of W and for its zero temperature value. Ó 2007 Elsevier B.V. All rights reserved. PACS: 71.10.Fd; 71.10.w; 74.25.q Keywords: Optical spectral weight; Strong correlations; Sum rules; Hubbard model
One of the most surprising features that has emerged from infrared (IR) spectroscopy experiments [1,2] in the high-temperature superconducting cuprates is the sizable R Xc redistribution of the spectral weight W ðT Þ ¼ dx rðx; T Þ on energy scales Xc much larger (up to 1– 0 1.5 eV) than those expected for a conventional superconductor (less than 100 meV). More specifically, if we focus on the normal state (T > Tc), a relevant enhancement of W(T) when the temperature is decreased from T = 300 K to T = Tc has been found. The low-temperature behavior of W(T) is generally quadratic (W(T) = W0 BT2), although indications of a linear behavior in some samples have been reported [3]. On the other hand, a general consensus has been reached on the doping dependence of the low-temperature spectral weight in the low-temperature
*
Corresponding author. Address: Max Planck Institut fu¨r Festko¨rperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany. E-mail address: [email protected] (A. Toschi). 0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.211
limit (W0), which is monotonically increasing with the doping level x. 1. Optical spectral weight and strong correlation The temperature dependence of W is not surprising by itself, at least for cut-offs Xc ’ 1 1.5 eV, for which the integral contains all the contribution from a single band. In the case of an uncorrelated single-band system W naturally acquires a T-dependence, being related to the kinetic energy of the system. However, these general considerations cannot account for the extremely large size of the overall variation of W(T) between T = 300 K and T = 0 observed in the cuprates. As pointed out in Refs. [2,4,5], the T-dependence of W in La2xSrxCuO4 (LSCO) could be reproduced only by assuming a very narrow band (t = 20 meV, i.e., at least an order of magnitude smaller than reasonable estimates for cuprates). This fact, together with the increase of W0 with the doping level, calls for the inclusion of strong-correlation
1046
A. Toschi et al. / Physica C 460–462 (2007) 1045–1046
500 IR DMFT U=0
300
20
-1 B (e V )
400
W0 (meV)
IR DMFT U=0
25
200 100
15 10 5
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0 0
x
effects. We account for those correlation effects by studying, by means of Dynamical Mean Field Theory (DMFT), the single-band Hubbard model X y X X H ¼ t cir cjr þ U ni" ni# l nir ; ð1Þ i
0.1
0.15
0.2
0.25
0.3
x
Fig. 1. W(T = 0) for Xc = 1.5D as a function of doping for the Hubbard model (squares) compared with the IR spectroscopy data in LSCO (circles).
hijir
0.05
i;r
cir ðcyir Þ
where are annihilation (creation) operators for fermions of spin r on site i, and nir ¼ cyir cir . We assume [4] that t = 300 meV and U = 12t, so that the antiferromagnetic (AF) superexchange J = 4t2/U ’ 100 meV as in neutron scattering data. 2. DMFT results and experimental data Here we show our DMFT results for the spectral weight at low temperature (W0 in Fig. 1) and the coefficient B controlling the quadratic T-dependence (Fig. 2, obtained from a T2 fit of the data). In both cases we also compare with the experimental data for LSCO [4]. We see, then, how the inclusion of correlation – without adjusting any model parameter – already captures the main features of the phenomenology, remarkably improving any single-particle theory. More specifically, the proximity to a Mott insulating phase can qualitatively explain both (i) the magnitude and the doping dependence of the spectral weight (The latter behavior would be the opposite in the U = 0 case, see Fig. 1), and (ii) the amplitude of the temperature variation (B is one order of magnitude larger than the U = 0 results, see Fig. 2). From a more quantitative point of view we observe that both the W0 and the coefficient B differs at most a factor two from the experiments, except for the very low-doping regime. We emphasize again that this agree-
Fig. 2. Coefficient B of the T2-fit for the spectral weight in the Hubbard model (squares) and in LSCO (circles).
ment is obtained in a model with no adjustable parameters. Due to the neglect of the non-local AF interactions, the DMFT yields probably a too narrow quasiparticle peak near the Fermi level, leading to an overestimate of the coefficient B. Including short-range correlations as done in cellular-DMFT [6] can improve the estimate of W and its T- dependence [7]. 3. Conclusions We show that strong electron correlation is a necessary ingredient to describe the basic features of the spectral weight behavior in the cuprates. The inclusion of other effects, like more realistic descriptions of the band structure and the interactions, and non-local AF fluctuations, beyond DMFT is necessary to quantitatively describe the experimental spectra in detail. Strong-correlation effects should also be taken into account in the superconducting phase, where some effective low-energy description, like that of the BCS–BE crossover [8], can account only partially of the IR-spectroscopy data. References [1] See, e.g., H.J.A. Molegraaf et al., Science 295 (2002) 2239; A.F. Santander-Syro et al., Phys. Rev. B 70 (2004) 134504. [2] M. Ortolani, P. Calvani, S. Lupi, Phys. Rev. Lett. 94 (2005) 067002. [3] N. Bontemps et al., Ann. Phys. 321 (2006) 1547. [4] A. Toschi et al., Phys. Rev. Lett. 95 (2005) 097002. [5] L. Benfatto, G.S. Sharapov, Low Temp. Phys. 32 (2006) 533. [6] M. Civelli et al., Phys. Rev. Lett. 95 (2005) 106402. [7] F. Carbone et al., Phys. Rev. B 74 (2006) 064510; T.A. Maier et al., Phys. Rev. Lett. 92 (2004) 027005. [8] A. Toschi, M. Capone, C. Castellani, Phys. Rev. B 72 (2005) 235118.
Optical spectral weight anomalies and strong correlation A. Toschi
a,c,* ,
M. Capone
b,c
, M. Ortolani
d,e
, P. Calvani
c,e
, S. Lupi
c,e
, C. Castellani
c
a
Max Planck Institut fu¨r Festko¨rperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany INFM-SMC and Instituto dei Sistemi Complessi, CNR, Via dei Taurini 19, 00185 Roma, Italy c Dip. di Fisica, Universita` di Roma ‘‘La Sapienza’’, Piazzale Aldo Moro 5, 00185 Roma, Italy Berliner Elektronenspeicherring-Gesellschaft fu¨r Synchrotronstrahlung, A. Einstein Strasse 15, D-12489 Berlin, Germany e CNR-INFM ‘‘Coherentia’’, Universita` di Roma ‘‘La Sapienza’’, Piazzale A. Moro 2, 00185 Roma, Italy b
d
Available online 28 March 2007
Abstract The anomalous behavior observed in the optical spectral weight (W) of the cuprates provides valuable information about the physics of these compounds. Both the doping and the temperature dependences of W are hardly explained through conventional estimates based on the f-sum rule. By computing the optical conductivity of the doped Hubbard model with the Dynamical Mean Field Theory, we point out that the strong correlation plays a key role in determining the basic features of the observed anomalies: the proximity to a Mott insulating phase accounts simultaneously for the strong temperature dependence of W and for its zero temperature value. Ó 2007 Elsevier B.V. All rights reserved. PACS: 71.10.Fd; 71.10.w; 74.25.q Keywords: Optical spectral weight; Strong correlations; Sum rules; Hubbard model
One of the most surprising features that has emerged from infrared (IR) spectroscopy experiments [1,2] in the high-temperature superconducting cuprates is the sizable R Xc redistribution of the spectral weight W ðT Þ ¼ dx rðx; T Þ on energy scales Xc much larger (up to 1– 0 1.5 eV) than those expected for a conventional superconductor (less than 100 meV). More specifically, if we focus on the normal state (T > Tc), a relevant enhancement of W(T) when the temperature is decreased from T = 300 K to T = Tc has been found. The low-temperature behavior of W(T) is generally quadratic (W(T) = W0 BT2), although indications of a linear behavior in some samples have been reported [3]. On the other hand, a general consensus has been reached on the doping dependence of the low-temperature spectral weight in the low-temperature
*
Corresponding author. Address: Max Planck Institut fu¨r Festko¨rperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany. E-mail address: [email protected] (A. Toschi). 0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.211
limit (W0), which is monotonically increasing with the doping level x. 1. Optical spectral weight and strong correlation The temperature dependence of W is not surprising by itself, at least for cut-offs Xc ’ 1 1.5 eV, for which the integral contains all the contribution from a single band. In the case of an uncorrelated single-band system W naturally acquires a T-dependence, being related to the kinetic energy of the system. However, these general considerations cannot account for the extremely large size of the overall variation of W(T) between T = 300 K and T = 0 observed in the cuprates. As pointed out in Refs. [2,4,5], the T-dependence of W in La2xSrxCuO4 (LSCO) could be reproduced only by assuming a very narrow band (t = 20 meV, i.e., at least an order of magnitude smaller than reasonable estimates for cuprates). This fact, together with the increase of W0 with the doping level, calls for the inclusion of strong-correlation
1046
A. Toschi et al. / Physica C 460–462 (2007) 1045–1046
500 IR DMFT U=0
300
20
-1 B (e V )
400
W0 (meV)
IR DMFT U=0
25
200 100
15 10 5
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0 0
x
effects. We account for those correlation effects by studying, by means of Dynamical Mean Field Theory (DMFT), the single-band Hubbard model X y X X H ¼ t cir cjr þ U ni" ni# l nir ; ð1Þ i
0.1
0.15
0.2
0.25
0.3
x
Fig. 1. W(T = 0) for Xc = 1.5D as a function of doping for the Hubbard model (squares) compared with the IR spectroscopy data in LSCO (circles).
hijir
0.05
i;r
cir ðcyir Þ
where are annihilation (creation) operators for fermions of spin r on site i, and nir ¼ cyir cir . We assume [4] that t = 300 meV and U = 12t, so that the antiferromagnetic (AF) superexchange J = 4t2/U ’ 100 meV as in neutron scattering data. 2. DMFT results and experimental data Here we show our DMFT results for the spectral weight at low temperature (W0 in Fig. 1) and the coefficient B controlling the quadratic T-dependence (Fig. 2, obtained from a T2 fit of the data). In both cases we also compare with the experimental data for LSCO [4]. We see, then, how the inclusion of correlation – without adjusting any model parameter – already captures the main features of the phenomenology, remarkably improving any single-particle theory. More specifically, the proximity to a Mott insulating phase can qualitatively explain both (i) the magnitude and the doping dependence of the spectral weight (The latter behavior would be the opposite in the U = 0 case, see Fig. 1), and (ii) the amplitude of the temperature variation (B is one order of magnitude larger than the U = 0 results, see Fig. 2). From a more quantitative point of view we observe that both the W0 and the coefficient B differs at most a factor two from the experiments, except for the very low-doping regime. We emphasize again that this agree-
Fig. 2. Coefficient B of the T2-fit for the spectral weight in the Hubbard model (squares) and in LSCO (circles).
ment is obtained in a model with no adjustable parameters. Due to the neglect of the non-local AF interactions, the DMFT yields probably a too narrow quasiparticle peak near the Fermi level, leading to an overestimate of the coefficient B. Including short-range correlations as done in cellular-DMFT [6] can improve the estimate of W and its T- dependence [7]. 3. Conclusions We show that strong electron correlation is a necessary ingredient to describe the basic features of the spectral weight behavior in the cuprates. The inclusion of other effects, like more realistic descriptions of the band structure and the interactions, and non-local AF fluctuations, beyond DMFT is necessary to quantitatively describe the experimental spectra in detail. Strong-correlation effects should also be taken into account in the superconducting phase, where some effective low-energy description, like that of the BCS–BE crossover [8], can account only partially of the IR-spectroscopy data. References [1] See, e.g., H.J.A. Molegraaf et al., Science 295 (2002) 2239; A.F. Santander-Syro et al., Phys. Rev. B 70 (2004) 134504. [2] M. Ortolani, P. Calvani, S. Lupi, Phys. Rev. Lett. 94 (2005) 067002. [3] N. Bontemps et al., Ann. Phys. 321 (2006) 1547. [4] A. Toschi et al., Phys. Rev. Lett. 95 (2005) 097002. [5] L. Benfatto, G.S. Sharapov, Low Temp. Phys. 32 (2006) 533. [6] M. Civelli et al., Phys. Rev. Lett. 95 (2005) 106402. [7] F. Carbone et al., Phys. Rev. B 74 (2006) 064510; T.A. Maier et al., Phys. Rev. Lett. 92 (2004) 027005. [8] A. Toschi, M. Capone, C. Castellani, Phys. Rev. B 72 (2005) 235118.