1-s2 0-0038109881900491-main

Prévia do material em texto

~Solid State pp.983-986. Communications, Vol.40, 
Pergamon Press Ltd. 1981. Printed in Great Britain. 
0038-i098181/470983-04502.00/0 
TRANSVERSE SPIN CORRELATION FUNCTION OF THE ONE-DLMENSIONAL SPIN-I/2 XY MODEL 
Takashi Tonegawa 
Department of Physics, Faculty of Science, Kobe University, Rokkodai, Kobe 657, Japan 
(Received 31 August 1981 by J. Kanamori) 
The transverse spin pair correlation function 9 x = = is 
calculated exactly in the thermodynamic limit of the system described by 
the one-dimensional, isotropic, spin-i/2, XY Hamiltonian 
N 
H 2J ~ xx yy = _ (S~S~+ I + SzS~+ I) • 
It is found that at absolute zero temperature (T =0), the correlation 
function pX for n ~_ 0 is given by 
x i [212P p-if 4j 2 ~2p-2j 
x I [212p+Ip[ 4j 2 ]2p+l-mj 
P2p+l 4 [ j=i[ j - j = -+ -- ~J I I /44-~_1J if n = 2p+l, 
where the plus sign applies when J is positive and the minus sign ap- 
plies when J is negative. From these the asymptotic behavior as n÷ ~ of 
10nXl at T=0 is derived to be 10xl ~,a//nn with a=0.147088 ---. For fi- 
nite temperatures, pX n is calculated numerically. By using the results 
for 0 x, the transverse inverse correlation length and the wavenumber 
dependent transverse spin pair correlation function are also calculated 
exactly. 
There has been a considerable amount of 
success in treating exactly the thermodynamics 
of one-dimensional spin systems I . In this paper 
we consider the system described by the one- 
dimensional, isotropic, spin-i/2, XY Hamiltonian 
N 
H 2J [ xx yy = _ ( s ~ s z + 1 + s ~ s ~ + 1) , 
~.=1 
(i) 
where J represents the exchange constant between 
nearest neighboring spins and S~ (~ = x, y) is 
equal to 1/2 times the ~-component of the Pauli 
spin operator for the £th spin. Throughout the 
paper we are confined ourselves to the thermo- 
dynamic limit (N +~). Exact results for various 
physical quantities of the system described by 
the Hamiltonian (i) have been obtained by Lieh, 
Schultz and Mattis 2 and by Katsura 3. The pur- 
pose of this paper is to calculate exactly the 
transverse spin pair correlation function 
X 0 n = . (2) 
m m*n m m-n 
Using the exact results for pn x, we also calcu- 
late the transverse inverse correlation length 
and the wavenumber dependent transverse spin 
pair correlation function (the Fourier transform 
of pX). 
Lieb, Schultz and Mattis 2 have shown that 
px for n-~0 is given by 
I Rp2/4 if n = 2p 
X = 
o n (3 ) 
t RpRp+i/4 if n = 2p+l , 
where R 0=I and R for p>l is the pxp determi- 
nant P = 
I G_I G_3 G_ (2p-l) I 
G I G_ I G-(2p-3) 
Rp . . . . (4) 
°2 3 °2p5 G I L 
The quantity G2£+I in Eq. (4) is 
_ ~ [~ cos{ (2£+i)k} 
G2Z+I - ~ JO exp(-2JB cos k) + I 
d k , (5 ) 
where B = (kBT)-I. We note that in this paper, 
the lattice constant is taken to be unity. It 
has also been shown by Lieb, Schultz and Mattis 2 
that the longitudinal spin pair correlation 
function is given by 
z = = 
Pn m m-~n m m-n 
= I ~P'O/4 
t- G-(2p+I)G2p+I/4 
if n = 2p , 
if n = 2p+l. 
(6) 
Using the spin pair correlation functions 
discussed above, we define the transverse and 
longitudinal inverse correlation lengths as 
: - ~ ~n lp~+ ir:>~l , (7 ) KX n ''l~ 1 
983 
984 THE ON~-DIMENSIONAL SPIN-I/2 XY MODEL Vo[. 40, No. 
(8) 
We also define the wavenumber dependent spin 
pair correlation functions as 
transverse spin pair correlation function Sx(k) 
at T = 0 in the case of J >0 is asymptotically 
given as 
Sx(k) - a ~ . (16) 
S (k) = ~ exp(ikn) O~ (==x, z) . (9) 
It is noted that Sx(k) and Sz(k ) are proportion- 
al, respectively, to the transverse and longi- 
tudinal components of the cross section for mag- 
netic scattering of neutrons in the quasielastic 
approximation. 
Now, let us consider the limit of absolute 
zero temperature (T =0). In this limit, G2£+I 
is calculated to be 
2 
G2£+I = ± (-i)£ Tr(2£+l) ' (i0) 
where the plus sign is taken when J is positive 
(ferromagnetic) and the minus sign is taken when 
J is negative (antiferromagnetic). We can show 
that this result for G2£+I together with Eqs. 
(3) and (4) lead to 
x i [_212P p-l( 4j2 ~2p-2j 
P2p = 4" ~Tr) -IT l ~ J ' 
j = l ~ . j - ; 
(z l ) 
For finite temperatures, Pn, K= and S=(k) 
(= = x, z) can be calculated numerically with the 
use of the values of G2Z+I obtained from Eq. (5) 
by performing the numerical integration. It is 
noted that even when T>0, the correlation func- 
tion p~ in the case of J0. Therefore, S x(k) 
in the former case is given by Sx(k+#) [or 
Sx(k-~)] in the latter case. On the other 
hand, all of Kx, p~, mz and Sz(k) do not depend 
on the sign of J. 
A part of the numerical results obtained in 
the case of J >0 is summarized in Figs. i-3: 
The n dependence of P~ is plotted in Fig. i for 
representative values of kBT/J , the temperature 
dependence of ~x and Kz is plotted in Fig. 2 
and the k dependence of Sx(k ) is plotged in 
Fig. 3 for representative values of kBT/J. It 
can be seen from Fig. 2 that ~x holds when 
T>0. The numerical result shows that the ratio 
KZ/KX approaches to 4 as T tends to 0. The 
wavenumber dependent longitudinal spin pair cor- 
' " " " # . . . . I . . . . 
I x = + ! 2p+l~gvrT_~; (12) 
P2p+l - 4 j=l 
In Eq. (12), the plus sign applies when J is 
positive and the minus sign applies when J is 
negative. From Eqs. (ii) and (12) the asymptot- 
ic behavior as n+~ of Ip~I at r =0 can be de- 
rived to be 
Ip~l ~ a/,a'n (~3) 
with 
[{3 ~ 1 
a = exp - £n2 + ~ + 
+ ~ ~(2j-l)/(j 22j-i)}] 
j=2 
= 0.147088°.. , (14) 
y being Euler's constant and ~(j) the Riemann 
zeta function defined by ~(j) = ~ £-J. A der- 
£=I 
ivation of Eq. (13) is given in the Appendix. 
The fact that the leadin$ term of the asymptotic 
expansion as n-~ of IpX| at T =0 is proportion- 
al to i/~n has already been found by McCoy 4, who 
has employed another method. 
From Eqs. (7), (ll) and (12) we can obtain 
at T=05 
[2j~ 4j2 I=0 
K x = - £n ~ "= 4j 2-I] (15) 
Furthermore, using Eqs. (9) and (13), we can 
show that when k + 0, the wavenumber dependent 
10 "1 
P~ 
:sT/J =0.00 
i 0.01 
I .10 
t o.5o 
i 0 - ~ ~ J 
0 50 100 n 150 
Fig. i. Transverse spin pair correlation func- 
x lion O n as a function of n in the case 
of J > 0. Labels on the individual 
curves denote the values of kBT/J. The 
arrow indicated the value, 0.25, of P~- 
Note that p~ in the case of J 0. 
Vol. 40, No. II THE ONE-DI.~NSIONAL 
51 , , i 
K 
4 Kz 
Kx 
O ~ 2 3 
ksT/J 
Fig. 2. Transverse inverse correlation length K x 
and longitudinal inverse correlation 
length 0. Note thatthe following, we assume that p-> 2. Let us 
now introduce the Riemann zeta function ~(x), 
3 
2 
1 
0 
~- I ~ kBT/J=0"05 
// t I o3o 
i I jo. o 
-0.2 -0.1 0.0 0.1 0.2 k/T[ 
Fig. 3. Wavenumber dependent transverse spin pair correlation function 
Sx(k) as a function of k/~ in the case of J •0. Labels on the 
individual solid curves denote the values of kBT/J. The dashed 
curves represent the asymptotic behavior as k ÷0 of Sx(k) at 
T =0, which is given by Eq. (16). Note that Sx(k) in the case 
of J 0• 
986 Th~ ON~-DIMENSIONAL SPIN-I/2 XY MODEL Vol. 40, 
the digemma function ~(x) and the polygamma 
function @(i)(x) (i=l, 2, 3,--.). Then, we 
have the relations 7 
~ i~(2i_l ) + ~(2i-2)(D)% f 22i_i) 
- i= 2 (2i'2){ j/(i . 
No. 11 
(A5) 
p~l j-l = Y + ~(p), 
j=l 
p-l l ~(i-1) j-i = ~(i) - (-i) i (p) (i-1)----~. j=l 
(i=2, 3, 4,--') , 
(A2) 
(A3) 
where y is Euler's constant. We substitute 
Eqs. (A2) and (A3) into Eq. (AI) and use the 
relation S 
When x ÷~, the digamma function ~(x) and the 
polygamma function ~(i)(x) are given by 7 
~(x) = Znx + O(x -1) , (A6) 
~(i)(x) = (-i) i-I (i-i)[ x -i + O~x -i-l) 
(i =I, 2, 3,-'') . (A7) 
Thus, from Eqs. (A5), (A6) and (AT), we obtain 
in the limit of p+~ 
~(2i)l[i 22i) -- £n(~/2) 
i=l 
to find 
£nP2p - £n4 - Y - ~(p) 
- 2p ~ @(2i-l)(p)/{i22i (2i-i)!} 
i--i 
(A4) 
x ~ - Izn(2p) + Zna, (AS) £n p 2p 
where a is given by Eq. (14). Equation (AS) re- 
duces immediately to Eq. (13). Starting with 
Eq. (12), we can, of course, derive Eq. (13) in 
a similar way. 
REFERENCES 
i. See, for example, C.J. Thompson, in Phase 
Transitions and Critical Phenomena, edited by 
C. Domb and M.S. Green (Academic, London, 
1972), Vol. I, p. 177. 
2. E. Lieb, T. Schultz and D. Mattis, Ann. Phys. 
(N.Y.) 16, 407 (1961). 
3. S. Katsura, Phys. Rev. 127, 1508 (1962). 
&. B.M. McCoy, Phys. Rev. 173, 531 (1968). 
5. The relation j~l{4J2/(4j2-1)} : 7/2 is obtain- 
ed by substituting x =7/2 into the infinite 
product representation of sinx/x: sinx/x = 
j~l{l- (x/=j) 2} (see Ref. 7). Equation (A4) 
can be readily derived from the above rela- 
tion. 
6. S. Katsura, T. Horiguchi and M. Suzuki, Phys- 
ica, 46, 67 (1970). 
7. M. Abramowitz and I.A. Stegun, Handbook of 
Mathematical Functions (Dover, New York, 
1970).