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Spin dynamics of cuprate superconductors: Exact results from numerical continued fraction expansions
Physica C 162-164 (1989) 207-208 North-Holland
SPIN DYNAMICS OF C U P R A T E SUPERCONDUCTORS: CONTINUED FRACTION EXPANSIONS
EXACT RESULTS F R O M NUMERICAL
C.-X CHEN and H.-B. SCHUTTLER Center for Simulational Physics, University of Georgia, Athens, Georgia 30602, USA We present the results for the T = 0 dyaamical structure factor of finite-sized 2D square--lattice spin-l/2 antiferromagnetic Heisenberg systems obtained by numerical continued fraction expansion techniques. Our results are compared to a recently proposed Schwinger boson mean-field theory. The coexistence of superconductivity and strong antiferromagnetic correlations in the high-Tc cuprate materials 1 has renewed our theoretical interest in the two-dimensional (2D) spin-l/2 Heisenberg and strong coupling Hubbard models. Here we report the exact numerical results for the dynamical structure factor of the 2D spin-l/2 Heisenberg antiferromagnet on finite--sized lattices at zero temperature. 2 For dusters with up to 16 lattice sites, we have applied a Mori-type 3 continued fraction expansion (CFE) technique 4-6 to calculate real-frequency spectra from Lanczos--generated groundstate wavefunctions. We compare our results to the SU(2) Schwinger boson mean-field theory which was recently proposed by Arovas and Auerbach (AA) 7 and is here applied to the finite--sized systems solved numerically. The Hamiltonian is in the following assumed to be of the form H =
J l. m
(l,m) with ~l denotingthe spin-i/2vectoroperatorat siteI.
S(q,ca)= NL1 • I dteiwt-iq(l-m)[SIZ)(t)Sm(Z)(0)], l,m where stZ)(t) [ is the spin z-component at site I after time--evolution in real-time t and [...] denotes the thermal average at temperature T. For finite systems, each S(q, ca) can be written as a superposition of 5--function terms of the form Sv(q)6(ca-Wv(q) ) with discrete excitation energies car(q) and intensities Sv(q). These frequency 5--functions are in the following Fig. 1 displayed as Lorentzian peaks with a finite width (FWHM) ~ = 0.03. (a)
....
, ....
The unitsof energyand lengthare chosensuch thatthe exchange constant J = 1 and lattice constant a = 1, respectively. The dynamical spin--spin structure factor S(q,w) at wavevector q and frequency cais given by
0921-4534/89/$03.50 @ Elsevier Science Publishers B.V. (North-Holland)
, ....
,
....
,.
16 sites ,o-
/
to-
~/
sites
8
~/ 4 ~ites
5 -
0
The (l,m)--summationruns once over allnearest neighborbonds on a square--shapedlatticeclusterwith periodicboundary conditionsand N L = L X L sites.
, ....
.
0
1
.
.
.
2
.
I
3
. . . .
I . . . .
4
I.
5
Fig. 1 S(q,ca) for lattice sizes N L = 4,8,16 at q = (%~r). Exact (full) and mean-field (dashed) curves have been ulotted for each N L and q. In most cases, mean-field and exact curves cannot be discerned on the scale of this plot. To get exact results for S(q, oJ) at T = 0, we need to solve first for the groundstate of H using a modified
208
C.-X. Chertand H.-B. Schuttler / Spin dynamics of cuprate superconductors
Lanczos algorithm. From the groundstate wavefunction, continued fraction coefficients of (an,bn) were generated. Following ref. (7b), we compare our results to a renormalized mean-field structure factor, S(q,w) = (2/3)sMF(q, oJ) where sMF(q,w) denotes the "bare" mean-field result, given by eq. (6) of ref. (7b). The normalization factor 2/3 was introduced ad hoc in ref. (Tb) so that S(q,w) obeys the susceptibility sum rule Zq ~ ~ r S(q,w) = 1/4. The comparison between MF and exact results is shown in Fig. 1. Clearly, there is excellent agreement between both sets of data. In fact, most of the main excitation peak shown in Fig. 1 at the deviations between MF and exact results are so small that they cannot be discerned on the scale of the plot. We have compared our exact diagonalization data and the MC data of Reger and YoungS for the integrated spectral intensity; i.e., the static structure factor Ss(q) = ~ ~ rw S(q,w), evaluated at q = (r,r) with the mean-field results. Again, we find remarkably close agreement between all three sets of data. Consistent with the MC data, s the exact results and analytical arguments, 9 the mean-field results for Ss(%~)/N L fall on a straight line when plotted vs. the
We have also studied the effects of dopant induced holes on the antiferromagnetic background in single-- and multi-orbital Hubbard model. The results are reported in a separate paper in these proceedings.i 0 We would like to thank Drs. M. H. Lee and E. Y. Loh for useful discussions concerning the CFE method. This work was supported by the Advanced Computa- tional Methods Center and from the Office of the Vice President for Research at the University of Georgia. REFERENCES 1. For a review see: R. J. Birgeneau and G. Shirane, in "Physical Properties of High Temperature Superconductors," D. M. Ginsberg Ed., World Scientific Publishing, Feb. 1989 (in press). 2. C.X. Chen and H. B. Schiittler, Phys. Rev. B 40 (in press). 3. H. Mori, Prog. Theor. Phys. 34, 399 (1965); ibid. 33, 423 (1965).
4.
M.H. Lee, Phys. Rev. Lett. 49, 1072 (1982); Phys. Rev. B 26, 1072 (1982); J. Math. Phys. 24,
5.
E.R. Gagliano and C. A. Balseiro, Phys. Rev. Lett. 59, 2999 (1987); Phys. Rev. B 38, i1766
2512
inverse linear system size 1/L = NL1/2 , as shown in Fig. 2. The mean-field value for the staggered magnetization m (MF) = 0..306, in excellent agreement with the MC value3 m (MC) = 0.30 • 0.02.
0.15
O +
Monte Carlo m e a n field
o
exact ~b
~:Z
6. 7.
Q 8.
d7
0.10
¢ 0.05
0.00
9.
+ ...I
.... 0.1
I .... 0.R
1 .... 0.3
I .... 0.4
1 .... 0.5
0.6
t/L Fig. 2 The static structure factor Ss(q)/N L vs. 1/L.
10.
(1983).
(1988). E.Y. Loh and D. Campbell, Synth. Metals 27, 499 (1988). Ca) D. Arovas and A. Auerbach, Phys. Rev. B 38, 316 (1988); (b) A. Auerbach and D. Arovas, Phys. Lett. 61, 617 (1988). Ca) J. D. Reger and A. P. Young, Phys. Rev. B 37, 5978 (1988); (b) J. D. Reger, J. A. Riera, and A. P. Young, (unpublished). D.A. Huse, Phys. Rev. B 37, 2380 (1988); D. A. Huse and V. Elser,Phys. Rev. Lett. 80, 2531 (1988). H.B. Schiittler, A. Fedro, and C. X. Chen, these proceedings.
SPIN DYNAMICS OF C U P R A T E SUPERCONDUCTORS: CONTINUED FRACTION EXPANSIONS
EXACT RESULTS F R O M NUMERICAL
C.-X CHEN and H.-B. SCHUTTLER Center for Simulational Physics, University of Georgia, Athens, Georgia 30602, USA We present the results for the T = 0 dyaamical structure factor of finite-sized 2D square--lattice spin-l/2 antiferromagnetic Heisenberg systems obtained by numerical continued fraction expansion techniques. Our results are compared to a recently proposed Schwinger boson mean-field theory. The coexistence of superconductivity and strong antiferromagnetic correlations in the high-Tc cuprate materials 1 has renewed our theoretical interest in the two-dimensional (2D) spin-l/2 Heisenberg and strong coupling Hubbard models. Here we report the exact numerical results for the dynamical structure factor of the 2D spin-l/2 Heisenberg antiferromagnet on finite--sized lattices at zero temperature. 2 For dusters with up to 16 lattice sites, we have applied a Mori-type 3 continued fraction expansion (CFE) technique 4-6 to calculate real-frequency spectra from Lanczos--generated groundstate wavefunctions. We compare our results to the SU(2) Schwinger boson mean-field theory which was recently proposed by Arovas and Auerbach (AA) 7 and is here applied to the finite--sized systems solved numerically. The Hamiltonian is in the following assumed to be of the form H =
J l. m
(l,m) with ~l denotingthe spin-i/2vectoroperatorat siteI.
S(q,ca)= NL1 • I dteiwt-iq(l-m)[SIZ)(t)Sm(Z)(0)], l,m where stZ)(t) [ is the spin z-component at site I after time--evolution in real-time t and [...] denotes the thermal average at temperature T. For finite systems, each S(q, ca) can be written as a superposition of 5--function terms of the form Sv(q)6(ca-Wv(q) ) with discrete excitation energies car(q) and intensities Sv(q). These frequency 5--functions are in the following Fig. 1 displayed as Lorentzian peaks with a finite width (FWHM) ~ = 0.03. (a)
....
, ....
The unitsof energyand lengthare chosensuch thatthe exchange constant J = 1 and lattice constant a = 1, respectively. The dynamical spin--spin structure factor S(q,w) at wavevector q and frequency cais given by
0921-4534/89/$03.50 @ Elsevier Science Publishers B.V. (North-Holland)
, ....
,
....
,.
16 sites ,o-
/
to-
~/
sites
8
~/ 4 ~ites
5 -
0
The (l,m)--summationruns once over allnearest neighborbonds on a square--shapedlatticeclusterwith periodicboundary conditionsand N L = L X L sites.
, ....
.
0
1
.
.
.
2
.
I
3
. . . .
I . . . .
4
I.
5
Fig. 1 S(q,ca) for lattice sizes N L = 4,8,16 at q = (%~r). Exact (full) and mean-field (dashed) curves have been ulotted for each N L and q. In most cases, mean-field and exact curves cannot be discerned on the scale of this plot. To get exact results for S(q, oJ) at T = 0, we need to solve first for the groundstate of H using a modified
208
C.-X. Chertand H.-B. Schuttler / Spin dynamics of cuprate superconductors
Lanczos algorithm. From the groundstate wavefunction, continued fraction coefficients of (an,bn) were generated. Following ref. (7b), we compare our results to a renormalized mean-field structure factor, S(q,w) = (2/3)sMF(q, oJ) where sMF(q,w) denotes the "bare" mean-field result, given by eq. (6) of ref. (7b). The normalization factor 2/3 was introduced ad hoc in ref. (Tb) so that S(q,w) obeys the susceptibility sum rule Zq ~ ~ r S(q,w) = 1/4. The comparison between MF and exact results is shown in Fig. 1. Clearly, there is excellent agreement between both sets of data. In fact, most of the main excitation peak shown in Fig. 1 at the deviations between MF and exact results are so small that they cannot be discerned on the scale of the plot. We have compared our exact diagonalization data and the MC data of Reger and YoungS for the integrated spectral intensity; i.e., the static structure factor Ss(q) = ~ ~ rw S(q,w), evaluated at q = (r,r) with the mean-field results. Again, we find remarkably close agreement between all three sets of data. Consistent with the MC data, s the exact results and analytical arguments, 9 the mean-field results for Ss(%~)/N L fall on a straight line when plotted vs. the
We have also studied the effects of dopant induced holes on the antiferromagnetic background in single-- and multi-orbital Hubbard model. The results are reported in a separate paper in these proceedings.i 0 We would like to thank Drs. M. H. Lee and E. Y. Loh for useful discussions concerning the CFE method. This work was supported by the Advanced Computa- tional Methods Center and from the Office of the Vice President for Research at the University of Georgia. REFERENCES 1. For a review see: R. J. Birgeneau and G. Shirane, in "Physical Properties of High Temperature Superconductors," D. M. Ginsberg Ed., World Scientific Publishing, Feb. 1989 (in press). 2. C.X. Chen and H. B. Schiittler, Phys. Rev. B 40 (in press). 3. H. Mori, Prog. Theor. Phys. 34, 399 (1965); ibid. 33, 423 (1965).
4.
M.H. Lee, Phys. Rev. Lett. 49, 1072 (1982); Phys. Rev. B 26, 1072 (1982); J. Math. Phys. 24,
5.
E.R. Gagliano and C. A. Balseiro, Phys. Rev. Lett. 59, 2999 (1987); Phys. Rev. B 38, i1766
2512
inverse linear system size 1/L = NL1/2 , as shown in Fig. 2. The mean-field value for the staggered magnetization m (MF) = 0..306, in excellent agreement with the MC value3 m (MC) = 0.30 • 0.02.
0.15
O +
Monte Carlo m e a n field
o
exact ~b
~:Z
6. 7.
Q 8.
d7
0.10
¢ 0.05
0.00
9.
+ ...I
.... 0.1
I .... 0.R
1 .... 0.3
I .... 0.4
1 .... 0.5
0.6
t/L Fig. 2 The static structure factor Ss(q)/N L vs. 1/L.
10.
(1983).
(1988). E.Y. Loh and D. Campbell, Synth. Metals 27, 499 (1988). Ca) D. Arovas and A. Auerbach, Phys. Rev. B 38, 316 (1988); (b) A. Auerbach and D. Arovas, Phys. Lett. 61, 617 (1988). Ca) J. D. Reger and A. P. Young, Phys. Rev. B 37, 5978 (1988); (b) J. D. Reger, J. A. Riera, and A. P. Young, (unpublished). D.A. Huse, Phys. Rev. B 37, 2380 (1988); D. A. Huse and V. Elser,Phys. Rev. Lett. 80, 2531 (1988). H.B. Schiittler, A. Fedro, and C. X. Chen, these proceedings.