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Magnetic excitations in the stripe phase of the Hubbard model
Physica C 460–462 (2007) 1165–1166 www.elsevier.com/locate/physc
Magnetic excitations in the stripe phase of the Hubbard model G. Seibold b
a,*
, J. Lorenzana
b
a Institut fu¨r Physik, BTU Cottbus, P.O. Box 101344, 03013 Cottbus, Germany Center for Statistical Mechanics and Complexity, INFM, Dipartimento di Fisica, Universita` di Roma La Sapienza, P. Aldo Moro 2, 00185 Roma, Italy
Available online 14 April 2007
Abstract We calculate the frequency and momentum dependence of magnetic excitations as arising from striped ground states. Our investigations are based on the time-dependent Gutzwiller approximation of the two-dimensional Hubbard model and allow for quantitative predictions for the doping dependence of the magnetic susceptibility in La-based cuprate superconductors. Ó 2007 Elsevier B.V. All rights reserved. PACS: 74.72.h; 71.10.w; 71.28.+d; 74.25.Ha Keywords: Magnetic excitations; Stripes; Cuprate superconductors
Since the discovery of stripes in codoped lanthanum based cuprates [1] the question whether electronically ordered states and its associated dynamics may determine the physics in high-temperature superconductors has been intensively debated. Such self-organized electronic structures should have a characteristic influence on the spectrum of magnetic fluctuations which can be probed by inelastic neutron scattering (INS). Most interestingly it has turned out recently that the spin excitations in at least two different cuprate families, namely LCO [2] and YBCO [3], show large similarities over a wide energy and momentum range. Here, we are going to show that the magnetic excitation spectra can be understood on the basis of an electronically ordered stripe state in these materials. Our investigations are based on the extended one-band Hubbard model (onsite repulsion U/t = 7.5, t 0 /t = 0.2, t = 342 meV; parameters derived in Ref. [4]) which is treated within an unrestricted Gutzwiller approximation supplemented by RPA-type fluctuations. As shown in Ref. [5] this approach leads to striped ground states which doping dependence reproduces the experimentally found
*
Corresponding author. E-mail address: [email protected] (G. Seibold).
0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.416
incommensurability in LCO compounds [7]. Due to the arguments presented in Refs. [6,4] we restrict to bond centered (BC) stripes but results for site centered structures are qualitatively similar. Fig. 1 shows the dispersion of magnetic excitations for BC stripes oriented along the y-axis for doping x = 0.08, 0.125, 0.15. The lower (acoustic) branch perpendicular to the stripes shows the correct Goldstone-like behavior going to zero frequency at the ordering wave-vector Qs ¼ ðp pd ; pÞ. d denotes the distance between charge stripes. Starting from Qs one observes two branches of spin-waves where the one dispersing towards smaller qx rapidly loses intensity. The other one remains very intense up to the antiferromagnetic ordering vector QAF where it continuously connects to the dispersive excitations along the stripes. The resulting saddle-point feature at QAF has a pronounced peak structure in the momentum integrated structure factor at an energy xres which is additionally amplified by the ‘roton-type’ structure for x P 1/8 along the stripe. Fig. 2 displays the doping dependence of xres which is compared with available INS data on LCO compounds [2,8,9]. Basically xres is determined by the magnetic interaction across the stripes mediated by the charge carriers inside the domain wall. Therefore, it is only weakly doping
200 100
Energy [meV]
(π,π)
(0,π) 250 200 150 100 50 0
x=0.125, d=4 200 100
(π,0)
Energy [meV]
250 200 150 100 50 0
x=0.08, d=6
(π,0)
(π,π)
250 200 150 100 50 0
200 100
(π,π)
150
100
Ref. 3 Ref. 8 Ref. 9
50
0 0.05
PS
0.1
0.15
0.2
Doping Fig. 2. Doping dependence of the saddle-point energy xres compared with INS data from Refs. [2,8,9]. The dashed line indicates the expected behavior for phase separation (PS) between stripes and homogeneously doped regions for x > 0.125.
(0,π)
x=0.149, d=4
(π,0)
Resonance energy [meV]
G. Seibold, J. Lorenzana / Physica C 460–462 (2007) 1165–1166
Energy [meV]
1166
(0,π)
Fig. 1. Dispersion of magnetic excitations perpendicular (p,p) ! (0,p) and parallel (p,p) ! (p,0) to the stripe orientation. Doping x and distance d between charge stripes are indicated in the panels. The dispersion is obtained from an intensity plot of the imaginary part of the transverse magnetic susceptibility.
dependent up to x = 0.125 below which the charge concentration m of the stripes is constant m = 1/2. For x > 0.125 the separation between stripes stays at 4 unit cells and therefore additional charge carriers are doped into the domain walls leading to a significant increase in xres. The three available data points [2,8,9] seem to support this behavior in favor of a phase separation scenario where stripes stay half-filled for x > 0.125 but are spatially segregated from homogeneously doped regions. However, more detailed experimental investigations are required since the points at x = 0.07 and x = 0.1 are obtained from an extrapolation of low energy data.
At low doping the high energy excitations (>xres) are almost isotropic although the underlying ground state breaks C4 symmetry. As a result constant energy cuts of v00 (q,x) show almost isotropic ring shaped intensity distributions at high energies around QAF [6,4] in agreement with experiment [2,3]. At larger doping we find a substantial anisotropy in the spectra which, however, should be compensated for by the strong stripe fluctuations expected at these concentrations. Obviously at larger doping the long-range ordered ground state should be replaced by some kind of a ‘stripe nematic’ [10] which also would lead to a ‘spin-gap’ in the excitation spectrum as observed for optimally doped cuprates. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
J.M. Tranquada et al., Nature 375 (1995) 56. J.M. Tranquada et al., Nature 429 (2004) 534. S.M. Hayden et al., Nature 429 (2004) 531. G. Seibold, J. Lorenzana, Phys. Rev. B 73 (2006) 144515. G. Seibold, J. Lorenzana, Phys. Rev. B 69 (2004) 134513. G. Seibold, J. Lorenzana, Phys. Rev. Lett. 94 (2005) 107006. K. Yamada et al., Phys. Rev. B 57 (1998) 6165. H. Hiraka et al., J. Phys. Soc. Jpn. 70 (2001) 853. N.B. Christensen et al., Phys. Rev. Lett. 93 (2004) 147002. S.A. Kivelson et al., Rev. Mod. Phys. 75 (2003) 1201.
Magnetic excitations in the stripe phase of the Hubbard model G. Seibold b
a,*
, J. Lorenzana
b
a Institut fu¨r Physik, BTU Cottbus, P.O. Box 101344, 03013 Cottbus, Germany Center for Statistical Mechanics and Complexity, INFM, Dipartimento di Fisica, Universita` di Roma La Sapienza, P. Aldo Moro 2, 00185 Roma, Italy
Available online 14 April 2007
Abstract We calculate the frequency and momentum dependence of magnetic excitations as arising from striped ground states. Our investigations are based on the time-dependent Gutzwiller approximation of the two-dimensional Hubbard model and allow for quantitative predictions for the doping dependence of the magnetic susceptibility in La-based cuprate superconductors. Ó 2007 Elsevier B.V. All rights reserved. PACS: 74.72.h; 71.10.w; 71.28.+d; 74.25.Ha Keywords: Magnetic excitations; Stripes; Cuprate superconductors
Since the discovery of stripes in codoped lanthanum based cuprates [1] the question whether electronically ordered states and its associated dynamics may determine the physics in high-temperature superconductors has been intensively debated. Such self-organized electronic structures should have a characteristic influence on the spectrum of magnetic fluctuations which can be probed by inelastic neutron scattering (INS). Most interestingly it has turned out recently that the spin excitations in at least two different cuprate families, namely LCO [2] and YBCO [3], show large similarities over a wide energy and momentum range. Here, we are going to show that the magnetic excitation spectra can be understood on the basis of an electronically ordered stripe state in these materials. Our investigations are based on the extended one-band Hubbard model (onsite repulsion U/t = 7.5, t 0 /t = 0.2, t = 342 meV; parameters derived in Ref. [4]) which is treated within an unrestricted Gutzwiller approximation supplemented by RPA-type fluctuations. As shown in Ref. [5] this approach leads to striped ground states which doping dependence reproduces the experimentally found
*
Corresponding author. E-mail address: [email protected] (G. Seibold).
0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.416
incommensurability in LCO compounds [7]. Due to the arguments presented in Refs. [6,4] we restrict to bond centered (BC) stripes but results for site centered structures are qualitatively similar. Fig. 1 shows the dispersion of magnetic excitations for BC stripes oriented along the y-axis for doping x = 0.08, 0.125, 0.15. The lower (acoustic) branch perpendicular to the stripes shows the correct Goldstone-like behavior going to zero frequency at the ordering wave-vector Qs ¼ ðp pd ; pÞ. d denotes the distance between charge stripes. Starting from Qs one observes two branches of spin-waves where the one dispersing towards smaller qx rapidly loses intensity. The other one remains very intense up to the antiferromagnetic ordering vector QAF where it continuously connects to the dispersive excitations along the stripes. The resulting saddle-point feature at QAF has a pronounced peak structure in the momentum integrated structure factor at an energy xres which is additionally amplified by the ‘roton-type’ structure for x P 1/8 along the stripe. Fig. 2 displays the doping dependence of xres which is compared with available INS data on LCO compounds [2,8,9]. Basically xres is determined by the magnetic interaction across the stripes mediated by the charge carriers inside the domain wall. Therefore, it is only weakly doping
200 100
Energy [meV]
(π,π)
(0,π) 250 200 150 100 50 0
x=0.125, d=4 200 100
(π,0)
Energy [meV]
250 200 150 100 50 0
x=0.08, d=6
(π,0)
(π,π)
250 200 150 100 50 0
200 100
(π,π)
150
100
Ref. 3 Ref. 8 Ref. 9
50
0 0.05
PS
0.1
0.15
0.2
Doping Fig. 2. Doping dependence of the saddle-point energy xres compared with INS data from Refs. [2,8,9]. The dashed line indicates the expected behavior for phase separation (PS) between stripes and homogeneously doped regions for x > 0.125.
(0,π)
x=0.149, d=4
(π,0)
Resonance energy [meV]
G. Seibold, J. Lorenzana / Physica C 460–462 (2007) 1165–1166
Energy [meV]
1166
(0,π)
Fig. 1. Dispersion of magnetic excitations perpendicular (p,p) ! (0,p) and parallel (p,p) ! (p,0) to the stripe orientation. Doping x and distance d between charge stripes are indicated in the panels. The dispersion is obtained from an intensity plot of the imaginary part of the transverse magnetic susceptibility.
dependent up to x = 0.125 below which the charge concentration m of the stripes is constant m = 1/2. For x > 0.125 the separation between stripes stays at 4 unit cells and therefore additional charge carriers are doped into the domain walls leading to a significant increase in xres. The three available data points [2,8,9] seem to support this behavior in favor of a phase separation scenario where stripes stay half-filled for x > 0.125 but are spatially segregated from homogeneously doped regions. However, more detailed experimental investigations are required since the points at x = 0.07 and x = 0.1 are obtained from an extrapolation of low energy data.
At low doping the high energy excitations (>xres) are almost isotropic although the underlying ground state breaks C4 symmetry. As a result constant energy cuts of v00 (q,x) show almost isotropic ring shaped intensity distributions at high energies around QAF [6,4] in agreement with experiment [2,3]. At larger doping we find a substantial anisotropy in the spectra which, however, should be compensated for by the strong stripe fluctuations expected at these concentrations. Obviously at larger doping the long-range ordered ground state should be replaced by some kind of a ‘stripe nematic’ [10] which also would lead to a ‘spin-gap’ in the excitation spectrum as observed for optimally doped cuprates. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
J.M. Tranquada et al., Nature 375 (1995) 56. J.M. Tranquada et al., Nature 429 (2004) 534. S.M. Hayden et al., Nature 429 (2004) 531. G. Seibold, J. Lorenzana, Phys. Rev. B 73 (2006) 144515. G. Seibold, J. Lorenzana, Phys. Rev. B 69 (2004) 134513. G. Seibold, J. Lorenzana, Phys. Rev. Lett. 94 (2005) 107006. K. Yamada et al., Phys. Rev. B 57 (1998) 6165. H. Hiraka et al., J. Phys. Soc. Jpn. 70 (2001) 853. N.B. Christensen et al., Phys. Rev. Lett. 93 (2004) 147002. S.A. Kivelson et al., Rev. Mod. Phys. 75 (2003) 1201.