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Finite temperature dynamics of the spin-half Heisenberg chain
ELSEVIER
Physica B 241 243 (1998) 563 565
Finite temperature dynamics of the spin-half Heisenberg chain O . A . S t a r y k h a'*, A . W . S a n d v i k b, R . R . P . S i n g h c Department o f Physics, Universi O, o[ Florida, Gainesville, FL 32611, USA h Department (?fPhysics, UniversiO~ (?['lllinois, Urbana, IL 61801. USA Department ~/" Physics. Universi O, o[' Cafilbrnia, Daris, CA 95616, USA
Abstract
We study the dynamic susceptibility z(q, to) of the S = ~ Heisenberg chain. Closed form analytic expression for z(q, ~)) at low T ~ J (J is exchange constant) is derived, which contains subleading logarithmic corrections due to umklapp scattering processes between left and right moving excitations. These corrections lead to noticeable deviations from quantum-critical scaling, and can be observed in NMR and neutron scattering experiments. At higher temperatures, we extend the recursion method to finite T using high-temperature expansions, and also carry out quantum Monte-Carlo simulations. We find a gradual transfer of spectral weight from diffusive (q ~ 0) modes at high T ~>J to propagating antiferromagnetic (q ~ ~) excitations at intermediate T ~ J. 1998 Elsevier Science B.V. All rights reserved.
Keywords: Heisenberg chain; Spin dynamics; Quantum critical scaling
The linear chain S = ½ Heisenberg antiferromagnet exhibits critical spin correlations at T = 0 in a wide range of microscopic exchange parameters J(n) describing interaction between spins separated by distance ha. This makes quasi-lD antiferromagnets model systems for studying quantum critical (QC) behavior [1], i.e. low temperature (T) behavior of a system with critical correlations in the ground state. Highly increased precision of NMR [2] and neutron scattering experiments [3] provide the possibility of detailed comparison of experimental data with theory.
*Corresponding author. [email protected].
Fax:
(352) 392-0524: e-mail:
We recently performed detailed studies of the dynamics of spin-half Heisenberg chain with nearest-neighbor interactions [4,5]. (a) T ~ J regime: Low-energy long-distance behavior of spin correlations is described by the continuum bosonized Hamiltonian H =~a~dx (.f'((~xC~)2_b p 2 ( g / a 2 ) c o s ( x / ~ O ) ) [6]. The cosine term, which describes umklapp scattering of spinons, produces multiplicative logarithmic corrections to the spin correlation function at T = 0 [7]: ( S(x)S(O)) = ( - 1)XD{x/[ln(x/xo)]I/x. At finite T, however, effect of umklapp scattering is two-fold: (i) it produces multiplicative logarithmic correction lnx/in~o/T) to the staggered spin susceptibility and structure factor [8,4], (ii) it leads to additive T-dependent logarithmic correction to the
0921-4526/98/$19.00 ~' 1998 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 7 ) 0 0 6 4 5 - 5
O.A. Starykh et al. / Physica B 241 243 (1998) 563 565
564
scaling dimension of the spin o p e r a t o r [4],
4.0
As a result the dynamic staggered susceptibility
22A-3/2D
z(q, co) =
re--T- sin (2rtA)
(
....
3.0
To) U21F2(l _ 2 A )
ln~-
,
/I,,
I
F(A - i(co - cq/4~T)) x F(1 - A - i(co - cq/4rtT))
~
c
sin (2~A)
In
i
(2)
ceases to be a universal function of cq/T and oJ/T [4]. In collaboration with M. T a k i g a w a we have found [9] impressive agreement of N M R relaxation rates 1/T1 and 1/T2c, calculated from Eq. (2) with experimental data on Sr2CuO3 [2]. Parameters of the fit To/J = 4.5 and D = 0.062 are in good agreement with our q u a n t u m M o n t e Carlo ( Q M C ) calculations [4,5]. F o r neutron scattering, our theory predicts subleading logarithmic corrections to the T-dependence of the spin correlation length ~, and deviation from Q C scaling of the local susceptibility (g(co) = ~_' .~(dq/2n) z(q, co)) -
~
%
F(A - i(co + cq/47t T)) x F(1 - A - i(co + cq/4nT))
Z"(co) - -
iI// 2.0
tI ~' ,
1.0 ' i
T=O.01, Eq.(3) T=O.01, Eq.(4) .... T/J=O.1, Eq.(3) T/J=0.1, Eq.(4) ----- T/J=O.25, Eq.(3) T/J=0.25, Eq.(4)
t
//
L - - ~
°°oo
t
21o
4.o
61o
lo.o
Fig. 1. Imaginary part of local susceptibility versus various temperatures.
°.° I
0.4
at
[3=1.0 [3=1.5 -[3=2.0
~
(3)
ea/T
[3--0.0 [3---O.5
F(l - 4A)
~ . [ F(2A - [i(eo/(2~T))]) "~ x z l m / F ]- . . . . . J. \ ( -- 2A -- [i(~/(2xT))])J
810
co/T
/f r~
As T ~ 0, it factorizes into a nonuniversal amplitude and a universal function [6] of ¢o/T Z"(co) =
In
x~
tanh
~
.
0.2
~
(4)
These expressions contain no contributions from q ~ 0 mode, as well as from the interior of the Brillouin zone, which is important at higher T. Deviation from Q C scaling is illustrated in Fig. 1 for a set of temperatures. (b) T ~ J regime: In this regime the spin correlation length becomes of order the lattice spacing and hence the c o n t i n u u m a p p r o x i m a t i o n begins to fail. This temperature region is also of experimental and
o.o
0
1
2
~
3
co/3
Fig. 2. Local structure factor So((o) versus ~,J for various
fi = J/T. theoretical significance, containing the crossover from diffusive high-T behavior to the l o w - T one d o m i n a t e d by elementary excitations. To study the spin dynamics we employed [5] (i) high-temperature
O.A. Starykh et al. /Physica B 241-243 (1998) 563-565
expansion (HTE) technique combined with continued fraction method [10], (ii) "stochastic series expansion" Q M C technique combined with maximum-entropy method for analytic continuation to real frequencies. Details of both techniques can be found in Ref. [5]. We found that HTE results agree well with QMC calculations for 0.5 ~< T/J <~ ~ , and describe a gradual transfer of the spectral weight from diffusive modes with q ~ 0 at high T (T >> J) to antiferromagnetic propagating q ~ rt magnon-like modes at intermediate T (T ~< J). HTE results for the local (on-site) dynamic structure factor So(~O)at various temperatures, illustrating this behavior, are summarized in Fig. 2. We found that NMR rate I/T1 reaches its approximately T-independent (up to logarithmic corrections discussed above) lowtemperature value at T/J ~ 0.25. We would like to thank M. Takigawa for many clarifying discussions and fruitful collaboration on comparison of NMR data with present theory.
565
References [1] S. Chakravarty, B.I. Halperin, D.R. Nelson, Phys. Rev. B 39 (1989) 2344. [2] M. Takigawa, N. Motoyama, It. Eisaki, S. Uchida, Phys. Rev. Lett. 76 (1996) 4612. [3] D.C. Dender et al., Phys. Rev. B 53 (1996) 2583. [4] O.A. Starykh, R.R.P. Singh~ A.W. Sandvik, Phys. Rev. Lett. 78 (1997) 539. [5] O.A. Starykh, A.W. Sandvik, R.R.P. Singh, Phys. Rev. B 55 (1997) 14953. [6] H.J. Schulz, Phys. Rev. B 34 (1986) 6372. [7] 1. Affleck et al., J. Phys. A 22 (1989) 511. [8] S. Sachdev, Phys. Rev. B 50 i1994) 13006. I-9] M. Takigawa, O.A. Starykh, A.W. Sandvik, R.R.P. Singh, cond-mat/9706177; Phys. Rev. B 56 (1997) 13681. [10] V.S. Viswanath, G. Miiller, Recursion Method Applications to Many-Body Dynamics, Lecture Notes in Physics, Vol. 23, Springer, New York~ 1994.
Physica B 241 243 (1998) 563 565
Finite temperature dynamics of the spin-half Heisenberg chain O . A . S t a r y k h a'*, A . W . S a n d v i k b, R . R . P . S i n g h c Department o f Physics, Universi O, o[ Florida, Gainesville, FL 32611, USA h Department (?fPhysics, UniversiO~ (?['lllinois, Urbana, IL 61801. USA Department ~/" Physics. Universi O, o[' Cafilbrnia, Daris, CA 95616, USA
Abstract
We study the dynamic susceptibility z(q, to) of the S = ~ Heisenberg chain. Closed form analytic expression for z(q, ~)) at low T ~ J (J is exchange constant) is derived, which contains subleading logarithmic corrections due to umklapp scattering processes between left and right moving excitations. These corrections lead to noticeable deviations from quantum-critical scaling, and can be observed in NMR and neutron scattering experiments. At higher temperatures, we extend the recursion method to finite T using high-temperature expansions, and also carry out quantum Monte-Carlo simulations. We find a gradual transfer of spectral weight from diffusive (q ~ 0) modes at high T ~>J to propagating antiferromagnetic (q ~ ~) excitations at intermediate T ~ J. 1998 Elsevier Science B.V. All rights reserved.
Keywords: Heisenberg chain; Spin dynamics; Quantum critical scaling
The linear chain S = ½ Heisenberg antiferromagnet exhibits critical spin correlations at T = 0 in a wide range of microscopic exchange parameters J(n) describing interaction between spins separated by distance ha. This makes quasi-lD antiferromagnets model systems for studying quantum critical (QC) behavior [1], i.e. low temperature (T) behavior of a system with critical correlations in the ground state. Highly increased precision of NMR [2] and neutron scattering experiments [3] provide the possibility of detailed comparison of experimental data with theory.
*Corresponding author. [email protected].
Fax:
(352) 392-0524: e-mail:
We recently performed detailed studies of the dynamics of spin-half Heisenberg chain with nearest-neighbor interactions [4,5]. (a) T ~ J regime: Low-energy long-distance behavior of spin correlations is described by the continuum bosonized Hamiltonian H =~a~dx (.f'((~xC~)2_b p 2 ( g / a 2 ) c o s ( x / ~ O ) ) [6]. The cosine term, which describes umklapp scattering of spinons, produces multiplicative logarithmic corrections to the spin correlation function at T = 0 [7]: ( S(x)S(O)) = ( - 1)XD{x/[ln(x/xo)]I/x. At finite T, however, effect of umklapp scattering is two-fold: (i) it produces multiplicative logarithmic correction lnx/in~o/T) to the staggered spin susceptibility and structure factor [8,4], (ii) it leads to additive T-dependent logarithmic correction to the
0921-4526/98/$19.00 ~' 1998 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 7 ) 0 0 6 4 5 - 5
O.A. Starykh et al. / Physica B 241 243 (1998) 563 565
564
scaling dimension of the spin o p e r a t o r [4],
4.0
As a result the dynamic staggered susceptibility
22A-3/2D
z(q, co) =
re--T- sin (2rtA)
(
....
3.0
To) U21F2(l _ 2 A )
ln~-
,
/I,,
I
F(A - i(co - cq/4~T)) x F(1 - A - i(co - cq/4rtT))
~
c
sin (2~A)
In
i
(2)
ceases to be a universal function of cq/T and oJ/T [4]. In collaboration with M. T a k i g a w a we have found [9] impressive agreement of N M R relaxation rates 1/T1 and 1/T2c, calculated from Eq. (2) with experimental data on Sr2CuO3 [2]. Parameters of the fit To/J = 4.5 and D = 0.062 are in good agreement with our q u a n t u m M o n t e Carlo ( Q M C ) calculations [4,5]. F o r neutron scattering, our theory predicts subleading logarithmic corrections to the T-dependence of the spin correlation length ~, and deviation from Q C scaling of the local susceptibility (g(co) = ~_' .~(dq/2n) z(q, co)) -
~
%
F(A - i(co + cq/47t T)) x F(1 - A - i(co + cq/4nT))
Z"(co) - -
iI// 2.0
tI ~' ,
1.0 ' i
T=O.01, Eq.(3) T=O.01, Eq.(4) .... T/J=O.1, Eq.(3) T/J=0.1, Eq.(4) ----- T/J=O.25, Eq.(3) T/J=0.25, Eq.(4)
t
//
L - - ~
°°oo
t
21o
4.o
61o
lo.o
Fig. 1. Imaginary part of local susceptibility versus various temperatures.
°.° I
0.4
at
[3=1.0 [3=1.5 -[3=2.0
~
(3)
ea/T
[3--0.0 [3---O.5
F(l - 4A)
~ . [ F(2A - [i(eo/(2~T))]) "~ x z l m / F ]- . . . . . J. \ ( -- 2A -- [i(~/(2xT))])J
810
co/T
/f r~
As T ~ 0, it factorizes into a nonuniversal amplitude and a universal function [6] of ¢o/T Z"(co) =
In
x~
tanh
~
.
0.2
~
(4)
These expressions contain no contributions from q ~ 0 mode, as well as from the interior of the Brillouin zone, which is important at higher T. Deviation from Q C scaling is illustrated in Fig. 1 for a set of temperatures. (b) T ~ J regime: In this regime the spin correlation length becomes of order the lattice spacing and hence the c o n t i n u u m a p p r o x i m a t i o n begins to fail. This temperature region is also of experimental and
o.o
0
1
2
~
3
co/3
Fig. 2. Local structure factor So((o) versus ~,J for various
fi = J/T. theoretical significance, containing the crossover from diffusive high-T behavior to the l o w - T one d o m i n a t e d by elementary excitations. To study the spin dynamics we employed [5] (i) high-temperature
O.A. Starykh et al. /Physica B 241-243 (1998) 563-565
expansion (HTE) technique combined with continued fraction method [10], (ii) "stochastic series expansion" Q M C technique combined with maximum-entropy method for analytic continuation to real frequencies. Details of both techniques can be found in Ref. [5]. We found that HTE results agree well with QMC calculations for 0.5 ~< T/J <~ ~ , and describe a gradual transfer of the spectral weight from diffusive modes with q ~ 0 at high T (T >> J) to antiferromagnetic propagating q ~ rt magnon-like modes at intermediate T (T ~< J). HTE results for the local (on-site) dynamic structure factor So(~O)at various temperatures, illustrating this behavior, are summarized in Fig. 2. We found that NMR rate I/T1 reaches its approximately T-independent (up to logarithmic corrections discussed above) lowtemperature value at T/J ~ 0.25. We would like to thank M. Takigawa for many clarifying discussions and fruitful collaboration on comparison of NMR data with present theory.
565
References [1] S. Chakravarty, B.I. Halperin, D.R. Nelson, Phys. Rev. B 39 (1989) 2344. [2] M. Takigawa, N. Motoyama, It. Eisaki, S. Uchida, Phys. Rev. Lett. 76 (1996) 4612. [3] D.C. Dender et al., Phys. Rev. B 53 (1996) 2583. [4] O.A. Starykh, R.R.P. Singh~ A.W. Sandvik, Phys. Rev. Lett. 78 (1997) 539. [5] O.A. Starykh, A.W. Sandvik, R.R.P. Singh, Phys. Rev. B 55 (1997) 14953. [6] H.J. Schulz, Phys. Rev. B 34 (1986) 6372. [7] 1. Affleck et al., J. Phys. A 22 (1989) 511. [8] S. Sachdev, Phys. Rev. B 50 i1994) 13006. I-9] M. Takigawa, O.A. Starykh, A.W. Sandvik, R.R.P. Singh, cond-mat/9706177; Phys. Rev. B 56 (1997) 13681. [10] V.S. Viswanath, G. Miiller, Recursion Method Applications to Many-Body Dynamics, Lecture Notes in Physics, Vol. 23, Springer, New York~ 1994.