The time- and frequency dependent energy fluctuations in the Heisenberg chain are studied by means of a continued fraction representation. In a broad wave vector and temperature range, the energy fluctuations are found to display dominant oscillatory behavior.

Raedt, Bart De,

Raedt, Hans De,

English

University of Groningen Digital Archive

ENERGY FLUCTUATIONS IN ONE DIMENSIONAL CLASSICAL MAGNETS 
HANS DE RAEDT 
Departement Natuurkunde,  Universitaire Instelling Antwerperg B-2610 Wilrijk, Belgium 
and BART DE RAEDT 
Institut fi~r Theoretische Physik,  Universitiit des Saarlandes, D-6600 Saarbriicken, Fed Rep. Germany 
The time- and frequency dependent energy fluctuations in the Heisenberg chain are studied by means of a  continued fraction 
representation. In a broad wave vector and temperature range, the energy fluctuations are found to display dominant oscillatory 
behavior. 
It is well known that the time evolution of the 
energy density of the Heisenberg chain in the hy- 
drodynamic  region  is  governed  by  the  diffusion 
equation.  In  particular,  the  diffusion constant  at 
infinite temperature  has  been  calculated by com- 
puter simulation [ I, 2]. 
The  aim of this paper  is  to  investigate the dy- 
namic properties  of the energy fluctuations at low 
temperature  and  to  shed  light  on  the  question 
whether the diffusive nature of the energy fluctua- 
tions  in  the  hydrodynamic region  persists  in  the 
wave vector range  where  hydrodynamic theory is 
not valid. 
Here,  we will confine ourselves  to  the  analytic 
results obtained by means of a  continued fraction 
representation  for  the  energy-energy  correlation 
functions  for  the  classical  Heisenberg  chain.  The 
Hamiltonian is given by 
N 
H  =  -J  E  Si°Si+l,  (1) 
i=l 
and the energy density Eq  is given by the Fourier 
transformed local energy 
J 
En =  -'2  Sn)(Sn+I  -t-  S n_l).  (2) 
In  order  to  study  the  time-dependent  energy- 
energy  correlation  functions  we  write  its  Fourier 
transform  in  the  exact  continued  fraction  repre- 
sentation [3] and we truncate this expansion follow- 
ing the method given in ref. [4]. Application of this 
method  requires  the  knowledge  of  the  frequency 
moments 
<[[...[e  q,n]  .... nl, e,]>  (3) 
~((E_qEq)  --  ~E)2~q, 0)  ' 
where the number of commutators equals 2n, and 
because  of  the  nature  of  the  spin  commutation 
relations evaluation results in complicated expres- 
sions. 
In  order  to  evaluate  the  commutators  analyti- 
cally, we have  developed an  interactive computer 
program. This program takes the various symmetry 
properties  of  the  Hamiltonian  into  account  and 
writes  the  final results  in  terms  of eigenvalues of 
the transfer operator  of the Hamiltonian (1).  The 
final results for the second and fourth moment are 
(J)q  =  (12/fl2)y~u/  (I  -  3y~ +  2y2),  (4) 
=  8u  y~ -  4y~y2  +  3y~ + y~(4 +  Sy2)u 
(~4)q  f12  1 -- 3Yl  2 +  2y  2  ' 
(5) 
where Yl  =  coth fl  -  1/fl,  Y2 =  1 -  3y~/fl  [5],  u 
1  -  cos q and fl denotes the inverse temperature 
in units of JS(S  +  1).  Note that these expressions 
do not depend on the sign of J. To our knowledge, 
the general result eq.  (5)  is new. For infinite tem- 
perature,  our expressions reduce to those given in 
ref.  [1]. We  were  unable  to  calculate higher  mo- 
ments because of lack of sufficient computer time. 
Truncating the continued fraction at the corre- 
sponding level, we have [4] 
SEE(q, w)/SEE(q,  O) =  ((Wa)q  -- (w2)2q) 
X  ((034)q/(Oj2)q) 1/2 
/[  (02( 0)2  --  <0j4>q/<O.)2>q) 2 
"1- (03')q(O) 2  --  (032)q) 2 
/(w2)q],  (6) 
Journal of Magnetism and Magnetic Materials  15-18 (1980)  1045-1046 ©North Holland  1045 1046  H. De Raedt, B. De Raedt  / Energy fluctuations in classical  spin chains 
o 
--  T=I 
T=2 
'L-  -.  T= 3 
-3c,  B  L  -  q:TC/,6 
% 
r  "\- 
"%_ 
L  t  L  I  ~  ~  I  I 
0  2  ~  5  8 
W 
Fig. 1. The normalized energy-energy structure factor for q = 
*r/16 and some temperatures.  The propagating  mode has  the 
largest  spectral  weight if  the  wave  vector  is  larger  than  the 
inverse correlation length. 
I 
G  4 0  ~  T  =.05 
,s,,  _  .....  T=2 
~--  ..............  T=.3 
3 
"  3 0  I[[I  t 
--~  -  I  I  q  :  31X/16 
tt3 
20  i  / 
...............  k  tl  ~, 
I  ..................  /  il  '..  \ 
..............  "/  S//  ",'\ 
~  o  ~.  7\ 
..... 
I 
~.  ~  _  1  L  J- 
0  0 /.  08  12  16 
(J.) 
Fig. 2. The normalized energy-energy structure factor for q =, 
3,r/16 and a set of temperatures. 
with 
S~E(q, 0) ---- (1  -- 3y  2 +  2),2)(1  +  cos q)/6. 
In the limit  T --, O, we obtain 
(7) 
SEE(q, 0)  ----- T2(I  +  COS q)/2,  (8) 
(~O2)q  ----- 8  sin  2 q/2,  (9) 
(o~4)q  -  96 sin  4 q/2,  (10) 
From  eq.  (8),  it  follows  that  the  probability  for 
energy  excitations  to  occur  vanishes  for  T--> 0. 
From  eqs.  (9)  and  (10)  we  conclude  that,  in  con- 
trast with the spin excitation spectrum, the normal- 
ized excitation spectrum for T  =  0  cannot be writ- 
ten  as  a  sum  of  two  delta  functions  because  this 
requires  (o~4)q  =  (~02)~. 
In the limit T ~  00  and q ~  0, eq. (6) reduces to 
a  Lorentzian centered  around ¢o =  0. The width  is 
determined by DEq  2. The energy diffusion constant 
D E =  0.821dlS is larger than the one obtained by a 
Gaussian  approximation  [1]  but  is  in  serious  dis- 
agreement  with  the  result  D E =  (3_  1)IJIS  ob- 
tained  by  computer  simulation  [1,  2].  This  can  be 
due to the fact that the approximations that lead to 
eq. (6) are invalid at infinite temperature.  In figs.  1, 
2  some  typical  lineshapes  for  the  energy  density 
fluctuations  are  depicted.  It is  clear  that  for small 
wave vectors and low temperatures,  energy fluctua- 
tions are  characterized by a  dominant propagating 
mode  and  a  relaxation  mode.  The  spectral  weight 
of  the  propagating  part  decreases  with  increasing 
temperature.  In general, we find that the propagat- 
ing  mode  disappears  if  the  inverse  correlation 
length  K =  -  In Y t  is  larger  than  the  wave vector. 
It is  remarkable  that  the  three  pole  expansion  for 
the  energy-energy  correlation  function  does  not 
lead to the criterion q >>  ~1/2 which seems to be the 
typical  criterion  for  the  three  pole  approximation 
of the spin-spin  correlation function [4, 5]. 
References 
[1] D. L. Huber, J. S. Semura and C. G. W/ndsor, Phys. Rev. 
186 (1969) 534. 
[2] N. A. Lurie, D.  L. Huber and M.  Blume, Phys. Rev. B 9 
(1974) 2171. 
[3] M. Mori, Prog. Theor. Phys. 34 (1965) 399. 
[4] H. De Raedt and B. De Raedt, Phys. Rev. B 15 (1977) 5379. 
[5] H. Tomita and H. Mashiyama, Prog. Theor. Phys. 48 (1972) 
1133. 

Energy Fluctuations in One Dimensional Classical Magnets

The time- and frequency dependent energy fluctuations in the Heisenberg chain are studied by means of a continued fraction representation. In a broad wave vector and temperature range, the energy fluctuations are found to display dominant oscillatory behavior

Raedt, Hans De

Raedt, Bart De

NARCIS 

ARTS repository - University of Groningen

   University of GroningenEnergy Fluctuations in One Dimensional Classical MagnetsRaedt, Hans De; Raedt, Bart DePublished in:Journal of Magnetism and Magnetic MaterialsDOI:10.1016/0304-8853(80)90880-XIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1980Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Raedt, H. D., & Raedt, B. D. (1980). Energy Fluctuations in One Dimensional Classical Magnets. Journal ofMagnetism and Magnetic Materials, 15(2). https://doi.org/10.1016/0304-8853(80)90880-XCopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 08-06-2022ENERGY FLUCTUATIONS IN ONE DIMENSIONAL CLASSICAL MAGNETS HANS DE RAEDT Departement Natuurkunde, Universitaire Instelling Antwerperg B-2610 Wilrijk, Belgium and BART DE RAEDT Institut fi~r Theoretische Physik, Universitiit des Saarlandes, D-6600 Saarbriicken, Fed Rep. Germany The time- and frequency dependent energy fluctuations in the Heisenberg chain are studied by means of a continued fraction representation. In a broad wave vector and temperature range, the energy fluctuations are found to display dominant oscillatory behavior. It is well known that the time evolution of the energy density of the Heisenberg chain in the hy- drodynamic region is governed by the diffusion equation. In particular, the diffusion constant at infinite temperature has been calculated by com- puter simulation [ I, 2]. The aim of this paper is to investigate the dy- namic properties of the energy fluctuations at low temperature and to shed light on the question whether the diffusive nature of the energy fluctua- tions in the hydrodynamic region persists in the wave vector range where hydrodynamic theory is not valid. Here, we will confine ourselves to the analytic results obtained by means of a continued fraction representation for the energy-energy correlation functions for the classical Heisenberg chain. The Hamiltonian is given by N H = - J  E Si°Si+l, (1)  i = l  and the energy density Eq is given by the Fourier transformed local energy J En = - ' 2  Sn)(Sn+I -t- S n_l).  (2)  In order to study the time-dependent energy- energy correlation functions we write its Fourier transform in the exact continued fraction repre- sentation [3] and we truncate this expansion follow- ing the method given in ref. [4]. Application of this method requires the knowledge of the frequency moments < [ [ . . . [ e  q,n] . . . .  nl, e,]> (3) ~ ( ( E _ q E q )  - -  ~E)2~q,  0) ' where the number of commutators equals 2n, and because of the nature of the spin commutation relations evaluation results in complicated expres- sions. In order to evaluate the commutators analyti- cally, we have developed an interactive computer program. This program takes the various symmetry properties of the Hamiltonian into account and writes the final results in terms of eigenvalues of the transfer operator of the Hamiltonian (1). The final results for the second and fourth moment are (J)q = (12 / f l 2 ) y~u /  (I  - 3y~ + 2y2), (4) = 8u y~ - 4y~y2 + 3y~ + y~(4 + Sy2)u (~4)q f12 1 -- 3Yl 2 + 2y 2 ' (5) where Yl = coth fl - 1/ f l ,  Y2 = 1 - 3y~/ f l  [5], u 1 - cos q and fl denotes the inverse temperature in units of J S ( S  + 1). Note that these expressions do not depend on the sign of J.  To our knowledge, the general result eq. (5) is new. For infinite tem- perature, our expressions reduce to those given in ref. [1]. We were unable to calculate higher mo- ments because of lack of sufficient computer time. Truncating the continued fraction at the corre- sponding level, we have [4] SEE(q, w)/SEE(q ,  O) = ((Wa)q -- (w2)2q) X ((034)q/(Oj2)q) 1/2 / [  (02( 0)2 - -  <0j4>q/<O.)2>q) 2 "1- (03')q(O) 2 - -  (032)q) 2 / ( w 2 ) q ] ,  (6) Journal of Magnetism and Magnetic Materials 15-18 (1980) 1045-1046 ©North Holland 1045 1046 H. De Raedt, B. De Raedt / Energy fluctuations in classical spin chains o -- T=I T=2 'L- -. T= 3 -3c, B L - q : T C / , 6  % r " \ -  " % _  L t L I ~ ~ I I 0 2 ~ 5 8 W Fig. 1. The normalized energy-energy structure factor for q = *r/16 and some temperatures. The propagating mode has the largest spectral weight if the wave vector is larger than the inverse correlation length. I G 4 0 ~ T = . 0 5  , s , ,  _ . . . . .  T = 2  ~-- . . . . . . . . . . . . . .  T = . 3  3 " 3 0 I [ [ I  t --~ - I I q : 31X/16 tt3 20 i / ............... k t l  ~, I .................. / il '.. \ .............. "/ S// ",'\ ~ o ~ .  7 \  ..... I ~ .  ~ _ 1 L J -  0 0 /. 08 12 16 (J.) Fig. 2. The normalized energy-energy structure factor for q =, 3,r/16 and a set of temperatures. with S~E(q, 0) ---- (1 -- 3y 2 + 2),2)(1 + cos q) /6 .  In  the limit T --, O, we obtain  (7) SEE(q, 0) ----- T2(I + COS q) /2 ,  (8) ( ~ O 2 ) q  ----- 8 sin 2 q /2 ,  (9) (o~4)q - 96 sin 4 q/2 ,  (10) F r o m  eq. (8), it follows that  the probabil i ty for energy excitations to occur  vanishes for T--> 0. F r o m  eqs. (9) and (10) we conc lude  that, in con-  trast with the spin excitation spectrum, the normal-  ized excitation spect rum for T = 0 canno t  be writ- ten as a sum of two delta funct ions because this requires (o~4)q = (~02)~. In  the limit T ~ 00 and  q ~ 0, eq. (6) reduces to a Lorentz ian  centered a round  ¢o = 0. The  width is de termined by DEq 2. The energy diffusion cons tant  D E = 0.821dlS is larger than the one obta ined by  a Gauss ian  approximat ion  [1] bu t  is in serious dis- agreement  with the result D E = ( 3 _  1)IJIS ob-  tained by computer  s imulat ion [1, 2]. This can be due to the fact that  the approximat ions  that  lead to eq. (6) are invalid at infinite temperature.  In  figs. 1, 2 some typical lineshapes for the energy density f luctuations are depicted. It  is clear that  for small wave vectors and low temperatures,  energy fluctua- tions are characterized by a d o m i n a n t  propagat ing  mode  and  a relaxation mode.  The spectral weight of the propagat ing par t  decreases with increasing temperature.  In general, we f ind that  the propagat-  ing mode  disappears if the inverse correlat ion length K = - In Y t is larger than  the wave vector.  It  is remarkable  that  the three pole expansion for the ene rgy-ene rgy  correlat ion funct ion does no t  lead to the criterion q >> ~1/2 which seems to be the typical criterion for the three pole approximat ion  of  the sp in-sp in  correlat ion funct ion [4, 5]. References [1] D. L. Huber, J. S. Semura and C. G. W/ndsor, Phys. Rev. 186 (1969) 534. [2] N. A. Lurie, D. L. Huber and M. Blume, Phys. Rev. B 9 (1974) 2171. [3] M. Mori, Prog. Theor. Phys. 34 (1965) 399. [4] H. De Raedt and B. De Raedt, Phys. Rev. B 15 (1977) 5379. [5] H. Tomita and H. Mashiyama, Prog. Theor. Phys. 48 (1972) 1133. 

University of Groningen Research Database

  
 University of Groningen
Energy Fluctuations in One Dimensional Classical Magnets
de Raedt, Hans; Raedt, Bart De
Published in:
Journal of Magnetism and Magnetic Materials
DOI:
10.1016/0304-8853(80)90880-X
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
it. Please check the document version below.
Document Version
Publisher's PDF, also known as Version of record
Publication date:
1980
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Raedt, H. D., & Raedt, B. D. (1980). Energy Fluctuations in One Dimensional Classical Magnets. Journal of
Magnetism and Magnetic Materials, 15(2). DOI: 10.1016/0304-8853(80)90880-X
Copyright
Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the
author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).
Take-down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the
number of authors shown on this cover page is limited to 10 maximum.
Download date: 11-02-2018
ENERGY FLUCTUATIONS IN ONE DIMENSIONAL CLASSICAL MAGNETS 
HANS DE RAEDT 
Departement Natuurkunde, Universitaire Instelling Antwerperg B-2610 Wilrijk, Belgium 
and BART DE RAEDT 
Institut fi~r Theoretische Physik, Universitiit des Saarlandes, D-6600 Saarbriicken, Fed Rep. Germany 
The time- and frequency dependent energy fluctuations in the Heisenberg chain are studied by means of a continued fraction 
representation. I  a broad wave vector and temperature ange, the energy fluctuations are found to display dominant oscillatory 
behavior. 
It is well known that the time evolution of the 
energy density of the Heisenberg chain in the hy- 
drodynamic region is governed by the diffusion 
equation. In particular, the diffusion constant at 
infinite temperature has been calculated by com- 
puter simulation [ I, 2]. 
The aim of this paper is to investigate the dy- 
namic properties of the energy fluctuations at low 
temperature and to shed light on the question 
whether the diffusive nature of the energy fluctua- 
tions in the hydrodynamic region persists in the 
wave vector range where hydrodynamic theory is 
not valid. 
Here, we will confine ourselves to the analytic 
results obtained by means of a continued fraction 
representation for the energy-energy correlation 
functions for the classical Heisenberg chain. The 
Hamiltonian is given by 
N 
H = - J  E Si°Si+l, (1) 
i= l  
and the energy density Eq is given by the Fourier 
transformed local energy 
J 
En = - '2  Sn)(Sn+I -t- S n_l). (2) 
In order to study the time-dependent energy- 
energy correlation functions we write its Fourier 
transform in the exact continued fraction repre- 
sentation [3] and we truncate this expansion follow- 
ing the method given in ref. [4]. Application of this 
method requires the knowledge of the frequency 
moments 
<[[ . . . [e  q,n] . . . .  nl, e,]> (3) 
~((E_qEq)  --  ~E)2~q, 0) ' 
where the number of commutators equals 2n, and 
because of the nature of the spin commutation 
relations evaluation results in complicated expres- 
sions. 
In order to evaluate the commutators analyti- 
cally, we have developed an interactive computer 
program. This program takes the various symmetry 
properties of the Hamiltonian into account and 
writes the final results in terms of eigenvalues of 
the transfer operator of the Hamiltonian (1). The 
final results for the second and fourth moment are 
(J)q = (12/ f l2)y~u/  (I - 3y~ + 2y2), (4) 
= 8u y~ - 4y~y2 + 3y~ + y~(4 + Sy2)u 
(~4)q f12 1 -- 3Yl 2 + 2y 2 ' 
(5) 
where Yl = coth fl - 1/f l ,  Y2 = 1 - 3y~/fl  [5], u 
1 - cos q and fl denotes the inverse temperature 
in units of J S (S  + 1). Note that these expressions 
do not depend on the sign of J. To our knowledge, 
the general result eq. (5) is new. For infinite tem- 
perature, our expressions reduce to those given in 
ref. [1]. We were unable to calculate higher mo- 
ments because of lack of sufficient computer time. 
Truncating the continued fraction at the corre- 
sponding level, we have [4] 
SEE(q, w)/SEE(q, O) = ((Wa)q -- (w2)2q) 
X ((034)q/(Oj2)q) 1/2 
/ [  (02( 0)2 - -  <0j4>q/<O.)2>q) 2 
"1- (03')q(O) 2 -- (032)q) 2 
/ (w2)q] ,  (6) 
Journal of Magnetism and Magnetic Materials 15-18 (1980) 1045-1046 ©North Holland 1045 
1046 H. De Raedt, B. De Raedt / Energy fluctuations in classical spin chains 
o 
-- T=I  
T=2 
'L- -. T= 3 
-3c, B L - q:TC/ ,6  
% 
r " \ -  
"%_  
L t L I ~ ~ I I 
0 2 ~ 5 8 
W 
Fig. 1. The normalized energy-energy structure factor for q = 
*r/16 and some temperatures. The propagating mode has the 
largest spectral weight if the wave vector is larger than the 
inverse correlation length. 
I 
G 4 0 ~ T =.05  
, s , ,  _ . . . . .  T=2 
~-- . . . . . . . . . . . . . .  T=.3  
3 
" 3 0 I [ [ I  t 
--~ - I I q : 31X/16 
tt3 
20 i / 
............... k t l  ~, 
I .................. / il '. \ 
.............. "/ S// ",'\ 
~ o ~ .  7 \  
..... 
I 
~.  ~ _ 1 L J -  
0 0 /. 08  12 16 
(J.) 
Fig. 2. The normalized energy-energy structure factor for q =, 
3,r/16 and a set of temperatures. 
with 
S~E(q, 0) ---- (1 -- 3y 2 + 2),2)(1 + cos q)/6. 
In the limit T --, O, we obtain 
(7) 
SEE(q, 0) ----- T2(I + COS q)/2,  (8) 
(~O2)q  ----- 8 sin 2 q/2,  (9) 
(o~4)q - 96 sin 4 q/2, (10) 
From eq. (8), it follows that the probabil ity for 
energy excitations to occur vanishes for T--> 0. 
F rom eqs. (9) and (10) we conclude that, in con- 
trast with the spin excitation spectrum, the normal- 
ized excitation spectrum for T = 0 cannot be writ- 
ten as a sum of two delta functions because this 
requires (o~4)q = (~02)~. 
In the limit T ~ 00 and q ~ 0, eq. (6) reduces to 
a Lorentzian centered around ¢o = 0. The width is 
determined by DEq 2. The energy diffusion constant 
D E = 0.821dlS is larger than the one obtained by a 
Gaussian approximation [1] but is in serious dis- 
agreement with the result D E = (3_  1)IJ IS ob- 
tained by computer simulation [1, 2]. This can be 
due to the fact that the approximations that lead to 
eq. (6) are invalid at infinite temperature. In figs. 1, 
2 some typical lineshapes for the energy density 
fluctuations are depicted. It is clear that for small 
wave vectors and low temperatures, energy fluctua- 
tions are characterized by a dominant  propagating 
mode and a relaxation mode. The spectral weight 
of the propagating part decreases with increasing 
temperature. In general, we find that the propagat- 
ing mode disappears if the inverse correlation 
length K = - In Y t is larger than the wave vector. 
It is remarkable that the three pole expansion for 
the energy-energy correlation function does not 
lead to the criterion q >> ~1/2 which seems to be the 
typical criterion for the three pole approximation 
of the spin-spin correlation function [4, 5]. 
References 
[1] D. L. Huber, J. S. Semura nd C. G. W/ndsor, Phys. Rev. 
186 (1969) 534. 
[2] N. A. Lurie, D. L. Huber and M. Blume, Phys. Rev. B 9 
(1974) 2171. 
[3] M. Mori, Prog. Theor. Phys. 34 (1965) 399. 
[4] H. De Raedt and B. De Raedt, Phys. Rev. B 15 (1977) 5379. 
[5] H. Tomita and H. Mashiyama, Prog. Theor. Phys. 48 (1972) 
1133. 


University of Groningen

   University of GroningenEnergy Fluctuations in One Dimensional Classical MagnetsRaedt, Hans De; Raedt, Bart DePublished in:Journal of Magnetism and Magnetic MaterialsDOI:10.1016/0304-8853(80)90880-XIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1980Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Raedt, H. D., & Raedt, B. D. (1980). Energy Fluctuations in One Dimensional Classical Magnets. Journal ofMagnetism and Magnetic Materials, 15(2). https://doi.org/10.1016/0304-8853(80)90880-XCopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019ENERGY FLUCTUATIONS IN ONE DIMENSIONAL CLASSICAL MAGNETS HANS DE RAEDT Departement Natuurkunde, Universitaire Instelling Antwerperg B-2610 Wilrijk, Belgium and BART DE RAEDT Institut fi~r Theoretische Physik, Universitiit des Saarlandes, D-6600 Saarbriicken, Fed Rep. Germany The time- and frequency dependent energy fluctuations in the Heisenberg chain are studied by means of a continued fraction representation. I  a broad wave vector and temperature ange, the energy fluctuations are found to display dominant oscillatory behavior. It is well known that the time evolution of the energy density of the Heisenberg chain in the hy- drodynamic region is governed by the diffusion equation. In particular, the diffusion constant at infinite temperature has been calculated by com- puter simulation [ I, 2]. The aim of this paper is to investigate the dy- namic properties of the energy fluctuations at low temperature and to shed light on the question whether the diffusive nature of the energy fluctua- tions in the hydrodynamic region persists in the wave vector range where hydrodynamic theory is not valid. Here, we will confine ourselves to the analytic results obtained by means of a continued fraction representation for the energy-energy correlation functions for the classical Heisenberg chain. The Hamiltonian is given by N H = - J  E Si°Si+l, (1) i= l  and the energy density Eq is given by the Fourier transformed local energy J En = - '2  Sn)(Sn+I -t- S n_l). (2) In order to study the time-dependent energy- energy correlation functions we write its Fourier transform in the exact continued fraction repre- sentation [3] and we truncate this expansion follow- ing the method given in ref. [4]. Application of this method requires the knowledge of the frequency moments <[[ . . . [e  q,n] . . . .  nl, e,]> (3) ~((E_qEq)  --  ~E)2~q, 0) ' where the number of commutators equals 2n, and because of the nature of the spin commutation relations evaluation results in complicated expres- sions. In order to evaluate the commutators analyti- cally, we have developed an interactive computer program. This program takes the various symmetry properties of the Hamiltonian into account and writes the final results in terms of eigenvalues of the transfer operator of the Hamiltonian (1). The final results for the second and fourth moment are (J)q = (12/ f l2)y~u/  (I - 3y~ + 2y2), (4) = 8u y~ - 4y~y2 + 3y~ + y~(4 + Sy2)u (~4)q f12 1 -- 3Yl 2 + 2y 2 ' (5) where Yl = coth fl - 1/f l ,  Y2 = 1 - 3y~/fl  [5], u 1 - cos q and fl denotes the inverse temperature in units of J S (S  + 1). Note that these expressions do not depend on the sign of J. To our knowledge, the general result eq. (5) is new. For infinite tem- perature, our expressions reduce to those given in ref. [1]. We were unable to calculate higher mo- ments because of lack of sufficient computer time. Truncating the continued fraction at the corre- sponding level, we have [4] SEE(q, w)/SEE(q, O) = ((Wa)q -- (w2)2q) X ((034)q/(Oj2)q) 1/2 / [  (02( 0)2 - -  <0j4>q/<O.)2>q) 2 "1- (03')q(O) 2 -- (032)q) 2 / (w2)q] ,  (6) Journal of Magnetism and Magnetic Materials 15-18 (1980) 1045-1046 ©North Holland 1045 1046 H. De Raedt, B. De Raedt / Energy fluctuations in classical spin chains o -- T=I  T=2 'L- -. T= 3 -3c, B L - q:TC/ ,6  % r " \ -  "%_  L t L I ~ ~ I I 0 2 ~ 5 8 W Fig. 1. The normalized energy-energy structure factor for q = *r/16 and some temperatures. The propagating mode has the largest spectral weight if the wave vector is larger than the inverse correlation length. I G 4 0 ~ T =.05  , s , ,  _ . . . . .  T=2 ~-- . . . . . . . . . . . . . .  T=.3  3 " 3 0 I [ [ I  t --~ - I I q : 31X/16 tt3 20 i / ............... k t l  ~, I .................. / il '. \ .............. "/ S// ",'\ ~ o ~ .  7 \  ..... I ~.  ~ _ 1 L J -  0 0 /. 08  12 16 (J.) Fig. 2. The normalized energy-energy structure factor for q =, 3,r/16 and a set of temperatures. with S~E(q, 0) ---- (1 -- 3y 2 + 2),2)(1 + cos q)/6. In the limit T --, O, we obtain (7) SEE(q, 0) ----- T2(I + COS q)/2,  (8) (~O2)q  ----- 8 sin 2 q/2,  (9) (o~4)q - 96 sin 4 q/2, (10) From eq. (8), it follows that the probabil ity for energy excitations to occur vanishes for T--> 0. F rom eqs. (9) and (10) we conclude that, in con- trast with the spin excitation spectrum, the normal- ized excitation spectrum for T = 0 cannot be writ- ten as a sum of two delta functions because this requires (o~4)q = (~02)~. In the limit T ~ 00 and q ~ 0, eq. (6) reduces to a Lorentzian centered around ¢o = 0. The width is determined by DEq 2. The energy diffusion constant D E = 0.821dlS is larger than the one obtained by a Gaussian approximation [1] but is in serious dis- agreement with the result D E = (3_  1)IJ IS ob- tained by computer simulation [1, 2]. This can be due to the fact that the approximations that lead to eq. (6) are invalid at infinite temperature. In figs. 1, 2 some typical lineshapes for the energy density fluctuations are depicted. It is clear that for small wave vectors and low temperatures, energy fluctua- tions are characterized by a dominant  propagating mode and a relaxation mode. The spectral weight of the propagating part decreases with increasing temperature. In general, we find that the propagat- ing mode disappears if the inverse correlation length K = - In Y t is larger than the wave vector. It is remarkable that the three pole expansion for the energy-energy correlation function does not lead to the criterion q >> ~1/2 which seems to be the typical criterion for the three pole approximation of the spin-spin correlation function [4, 5]. References [1] D. L. Huber, J. S. Semura nd C. G. W/ndsor, Phys. Rev. 186 (1969) 534. [2] N. A. Lurie, D. L. Huber and M. Blume, Phys. Rev. B 9 (1974) 2171. [3] M. Mori, Prog. Theor. Phys. 34 (1965) 399. [4] H. De Raedt and B. De Raedt, Phys. Rev. B 15 (1977) 5379. [5] H. Tomita and H. Mashiyama, Prog. Theor. Phys. 48 (1972) 1133. 

   University of GroningenEnergy Fluctuations in One Dimensional Classical MagnetsRaedt, Hans De; Raedt, Bart DePublished in:Journal of Magnetism and Magnetic MaterialsDOI:10.1016/0304-8853(80)90880-XIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1980Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Raedt, H. D., & Raedt, B. D. (1980). Energy Fluctuations in One Dimensional Classical Magnets. Journal ofMagnetism and Magnetic Materials, 15(2). https://doi.org/10.1016/0304-8853(80)90880-XCopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 27-04-2023ENERGY FLUCTUATIONS IN ONE DIMENSIONAL CLASSICAL MAGNETS HANS DE RAEDT Departement Natuurkunde, Universitaire Instelling Antwerperg B-2610 Wilrijk, Belgium and BART DE RAEDT Institut fi~r Theoretische Physik, Universitiit des Saarlandes, D-6600 Saarbriicken, Fed Rep. Germany The time- and frequency dependent energy fluctuations in the Heisenberg chain are studied by means of a continued fraction representation. In a broad wave vector and temperature range, the energy fluctuations are found to display dominant oscillatory behavior. It is well known that the time evolution of the energy density of the Heisenberg chain in the hy- drodynamic region is governed by the diffusion equation. In particular, the diffusion constant at infinite temperature has been calculated by com- puter simulation [ I, 2]. The aim of this paper is to investigate the dy- namic properties of the energy fluctuations at low temperature and to shed light on the question whether the diffusive nature of the energy fluctua- tions in the hydrodynamic region persists in the wave vector range where hydrodynamic theory is not valid. Here, we will confine ourselves to the analytic results obtained by means of a continued fraction representation for the energy-energy correlation functions for the classical Heisenberg chain. The Hamiltonian is given by N H = - J  E Si°Si+l, (1)  i = l  and the energy density Eq is given by the Fourier transformed local energy J En = - ' 2  Sn)(Sn+I -t- S n_l).  (2)  In order to study the time-dependent energy- energy correlation functions we write its Fourier transform in the exact continued fraction repre- sentation [3] and we truncate this expansion follow- ing the method given in ref. [4]. Application of this method requires the knowledge of the frequency moments < [ [ . . . [ e  q,n] . . . .  nl, e,]> (3) ~ ( ( E _ q E q )  - -  ~E)2~q,  0) ' where the number of commutators equals 2n, and because of the nature of the spin commutation relations evaluation results in complicated expres- sions. In order to evaluate the commutators analyti- cally, we have developed an interactive computer program. This program takes the various symmetry properties of the Hamiltonian into account and writes the final results in terms of eigenvalues of the transfer operator of the Hamiltonian (1). The final results for the second and fourth moment are (J)q = (12 / f l 2 ) y~u /  (I  - 3y~ + 2y2), (4) = 8u y~ - 4y~y2 + 3y~ + y~(4 + Sy2)u (~4)q f12 1 -- 3Yl 2 + 2y 2 ' (5) where Yl = coth fl - 1/ f l ,  Y2 = 1 - 3y~/ f l  [5], u 1 - cos q and fl denotes the inverse temperature in units of J S ( S  + 1). Note that these expressions do not depend on the sign of J.  To our knowledge, the general result eq. (5) is new. For infinite tem- perature, our expressions reduce to those given in ref. [1]. We were unable to calculate higher mo- ments because of lack of sufficient computer time. Truncating the continued fraction at the corre- sponding level, we have [4] SEE(q, w)/SEE(q ,  O) = ((Wa)q -- (w2)2q) X ((034)q/(Oj2)q) 1/2 / [  (02( 0)2 - -  <0j4>q/<O.)2>q) 2 "1- (03')q(O) 2 - -  (032)q) 2 / ( w 2 ) q ] ,  (6) Journal of Magnetism and Magnetic Materials 15-18 (1980) 1045-1046 ©North Holland 1045 1046 H. De Raedt, B. De Raedt / Energy fluctuations in classical spin chains o -- T=I T=2 'L- -. T= 3 -3c, B L - q : T C / , 6  % r " \ -  " % _  L t L I ~ ~ I I 0 2 ~ 5 8 W Fig. 1. The normalized energy-energy structure factor for q = *r/16 and some temperatures. The propagating mode has the largest spectral weight if the wave vector is larger than the inverse correlation length. I G 4 0 ~ T = . 0 5  , s , ,  _ . . . . .  T = 2  ~-- . . . . . . . . . . . . . .  T = . 3  3 " 3 0 I [ [ I  t --~ - I I q : 31X/16 tt3 20 i / ............... k t l  ~, I .................. / il '.. \ .............. "/ S// ",'\ ~ o ~ .  7 \  ..... I ~ .  ~ _ 1 L J -  0 0 /. 08 12 16 (J.) Fig. 2. The normalized energy-energy structure factor for q =, 3,r/16 and a set of temperatures. with S~E(q, 0) ---- (1 -- 3y 2 + 2),2)(1 + cos q) /6 .  In  the limit T --, O, we obtain  (7) SEE(q, 0) ----- T2(I + COS q) /2 ,  (8) ( ~ O 2 ) q  ----- 8 sin 2 q /2 ,  (9) (o~4)q - 96 sin 4 q/2 ,  (10) F r o m  eq. (8), it follows that  the probabil i ty for energy excitations to occur  vanishes for T--> 0. F r o m  eqs. (9) and (10) we conc lude  that, in con-  trast with the spin excitation spectrum, the normal-  ized excitation spect rum for T = 0 canno t  be writ- ten as a sum of two delta funct ions because this requires (o~4)q = (~02)~. In  the limit T ~ 00 and  q ~ 0, eq. (6) reduces to a Lorentz ian  centered a round  ¢o = 0. The  width is de termined by DEq 2. The energy diffusion cons tant  D E = 0.821dlS is larger than the one obta ined by  a Gauss ian  approximat ion  [1] bu t  is in serious dis- agreement  with the result D E = ( 3 _  1)IJIS ob-  tained by computer  s imulat ion [1, 2]. This can be due to the fact that  the approximat ions  that  lead to eq. (6) are invalid at infinite temperature.  In  figs. 1, 2 some typical lineshapes for the energy density f luctuations are depicted. It  is clear that  for small wave vectors and low temperatures,  energy fluctua- tions are characterized by a d o m i n a n t  propagat ing  mode  and  a relaxation mode.  The spectral weight of the propagat ing par t  decreases with increasing temperature.  In general, we f ind that  the propagat-  ing mode  disappears if the inverse correlat ion length K = - In Y t is larger than  the wave vector.  It  is remarkable  that  the three pole expansion for the ene rgy-ene rgy  correlat ion funct ion does no t  lead to the criterion q >> ~1/2 which seems to be the typical criterion for the three pole approximat ion  of  the sp in-sp in  correlat ion funct ion [4, 5]. References [1] D. L. Huber, J. S. Semura and C. G. W/ndsor, Phys. Rev. 186 (1969) 534. [2] N. A. Lurie, D. L. Huber and M. Blume, Phys. Rev. B 9 (1974) 2171. [3] M. Mori, Prog. Theor. Phys. 34 (1965) 399. [4] H. De Raedt and B. De Raedt, Phys. Rev. B 15 (1977) 5379. [5] H. Tomita and H. Mashiyama, Prog. Theor. Phys. 48 (1972) 1133. 

   University of GroningenEnergy Fluctuations in One Dimensional Classical MagnetsRaedt, Hans De; Raedt, Bart DePublished in:Journal of Magnetism and Magnetic MaterialsDOI:10.1016/0304-8853(80)90880-XIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1980Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Raedt, H. D., & Raedt, B. D. (1980). Energy Fluctuations in One Dimensional Classical Magnets. Journal ofMagnetism and Magnetic Materials, 15(2). https://doi.org/10.1016/0304-8853(80)90880-XCopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 31-10-2023ENERGY FLUCTUATIONS IN ONE DIMENSIONAL CLASSICAL MAGNETS HANS DE RAEDT Departement Natuurkunde, Universitaire Instelling Antwerperg B-2610 Wilrijk, Belgium and BART DE RAEDT Institut fi~r Theoretische Physik, Universitiit des Saarlandes, D-6600 Saarbriicken, Fed Rep. Germany The time- and frequency dependent energy fluctuations in the Heisenberg chain are studied by means of a continued fraction representation. In a broad wave vector and temperature range, the energy fluctuations are found to display dominant oscillatory behavior. It is well known that the time evolution of the energy density of the Heisenberg chain in the hy- drodynamic region is governed by the diffusion equation. In particular, the diffusion constant at infinite temperature has been calculated by com- puter simulation [ I, 2]. The aim of this paper is to investigate the dy- namic properties of the energy fluctuations at low temperature and to shed light on the question whether the diffusive nature of the energy fluctua- tions in the hydrodynamic region persists in the wave vector range where hydrodynamic theory is not valid. Here, we will confine ourselves to the analytic results obtained by means of a continued fraction representation for the energy-energy correlation functions for the classical Heisenberg chain. The Hamiltonian is given by N H = - J  E Si°Si+l, (1)  i = l  and the energy density Eq is given by the Fourier transformed local energy J En = - ' 2  Sn)(Sn+I -t- S n_l).  (2)  In order to study the time-dependent energy- energy correlation functions we write its Fourier transform in the exact continued fraction repre- sentation [3] and we truncate this expansion follow- ing the method given in ref. [4]. Application of this method requires the knowledge of the frequency moments < [ [ . . . [ e  q,n] . . . .  nl, e,]> (3) ~ ( ( E _ q E q )  - -  ~E)2~q,  0) ' where the number of commutators equals 2n, and because of the nature of the spin commutation relations evaluation results in complicated expres- sions. In order to evaluate the commutators analyti- cally, we have developed an interactive computer program. This program takes the various symmetry properties of the Hamiltonian into account and writes the final results in terms of eigenvalues of the transfer operator of the Hamiltonian (1). The final results for the second and fourth moment are (J)q = (12 / f l 2 ) y~u /  (I  - 3y~ + 2y2), (4) = 8u y~ - 4y~y2 + 3y~ + y~(4 + Sy2)u (~4)q f12 1 -- 3Yl 2 + 2y 2 ' (5) where Yl = coth fl - 1/ f l ,  Y2 = 1 - 3y~/ f l  [5], u 1 - cos q and fl denotes the inverse temperature in units of J S ( S  + 1). Note that these expressions do not depend on the sign of J.  To our knowledge, the general result eq. (5) is new. For infinite tem- perature, our expressions reduce to those given in ref. [1]. We were unable to calculate higher mo- ments because of lack of sufficient computer time. Truncating the continued fraction at the corre- sponding level, we have [4] SEE(q, w)/SEE(q ,  O) = ((Wa)q -- (w2)2q) X ((034)q/(Oj2)q) 1/2 / [  (02( 0)2 - -  <0j4>q/<O.)2>q) 2 "1- (03')q(O) 2 - -  (032)q) 2 / ( w 2 ) q ] ,  (6) Journal of Magnetism and Magnetic Materials 15-18 (1980) 1045-1046 ©North Holland 1045 1046 H. De Raedt, B. De Raedt / Energy fluctuations in classical spin chains o -- T=I T=2 'L- -. T= 3 -3c, B L - q : T C / , 6  % r " \ -  " % _  L t L I ~ ~ I I 0 2 ~ 5 8 W Fig. 1. The normalized energy-energy structure factor for q = *r/16 and some temperatures. The propagating mode has the largest spectral weight if the wave vector is larger than the inverse correlation length. I G 4 0 ~ T = . 0 5  , s , ,  _ . . . . .  T = 2  ~-- . . . . . . . . . . . . . .  T = . 3  3 " 3 0 I [ [ I  t --~ - I I q : 31X/16 tt3 20 i / ............... k t l  ~, I .................. / il '.. \ .............. "/ S// ",'\ ~ o ~ .  7 \  ..... I ~ .  ~ _ 1 L J -  0 0 /. 08 12 16 (J.) Fig. 2. The normalized energy-energy structure factor for q =, 3,r/16 and a set of temperatures. with S~E(q, 0) ---- (1 -- 3y 2 + 2),2)(1 + cos q) /6 .  In  the limit T --, O, we obtain  (7) SEE(q, 0) ----- T2(I + COS q) /2 ,  (8) ( ~ O 2 ) q  ----- 8 sin 2 q /2 ,  (9) (o~4)q - 96 sin 4 q/2 ,  (10) F r o m  eq. (8), it follows that  the probabil i ty for energy excitations to occur  vanishes for T--> 0. F r o m  eqs. (9) and (10) we conc lude  that, in con-  trast with the spin excitation spectrum, the normal-  ized excitation spect rum for T = 0 canno t  be writ- ten as a sum of two delta funct ions because this requires (o~4)q = (~02)~. In  the limit T ~ 00 and  q ~ 0, eq. (6) reduces to a Lorentz ian  centered a round  ¢o = 0. The  width is de termined by DEq 2. The energy diffusion cons tant  D E = 0.821dlS is larger than the one obta ined by  a Gauss ian  approximat ion  [1] bu t  is in serious dis- agreement  with the result D E = ( 3 _  1)IJIS ob-  tained by computer  s imulat ion [1, 2]. This can be due to the fact that  the approximat ions  that  lead to eq. (6) are invalid at infinite temperature.  In  figs. 1, 2 some typical lineshapes for the energy density f luctuations are depicted. It  is clear that  for small wave vectors and low temperatures,  energy fluctua- tions are characterized by a d o m i n a n t  propagat ing  mode  and  a relaxation mode.  The spectral weight of the propagat ing par t  decreases with increasing temperature.  In general, we f ind that  the propagat-  ing mode  disappears if the inverse correlat ion length K = - In Y t is larger than  the wave vector.  It  is remarkable  that  the three pole expansion for the ene rgy-ene rgy  correlat ion funct ion does no t  lead to the criterion q >> ~1/2 which seems to be the typical criterion for the three pole approximat ion  of  the sp in-sp in  correlat ion funct ion [4, 5]. References [1] D. L. Huber, J. S. Semura and C. G. W/ndsor, Phys. Rev. 186 (1969) 534. [2] N. A. Lurie, D. L. Huber and M. Blume, Phys. Rev. B 9 (1974) 2171. [3] M. Mori, Prog. Theor. Phys. 34 (1965) 399. [4] H. De Raedt and B. De Raedt, Phys. Rev. B 15 (1977) 5379. [5] H. Tomita and H. Mashiyama, Prog. Theor. Phys. 48 (1972) 1133. 

Energy Fluctuations in One Dimensional Classical Magnets

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