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Spin correlations in 2 + ε dimensions - ordered and disordered phase
Volume 67A, number 2
SPIN CORRELATIONS
PHYSICS LETTERS
24 July 1978
IN 2 + E DIMENSIONS - ORDERED AND DISORDERED
PHASE
T. NATTERMANN Sektion Physik, Karl-Marx-Universitl, 701 Leipzig, DDR Received 23 February 1978
Closed form expressions for the correlation functions of fixed length spins in 2 and 2 + E dimensions below and above the critical temperature are obtained by a skeleton graph approach.
In 1975 Polyakov [l] and Migdal [2] have developed techniques to treat n-component classical fixed length spins in 2 + E dimensions. Their work has been generalized by Brezin et al. [3] using field-theoretic methods. Recently, Nelson and Pelcovits [4] used momentum shell recursion relations to calculate various thermodynamic quantities. Their results complete calculations in 2 ,dimensions by Pokrovskii and Uimin [ 51 and Khoklachev [6] . In this note we compute closed form expressions for the correlation functions in an external field both below and above the critical temperature. To this aim we use a skeleton graph approach very similar to that in d = $ - E dimensions [7]. The results are valid to O(e) and to O(n”). In particular we are able to treat also the high temperature phase which was considered so far only for the large-n case [8] . The resulting expressions improve considerably those found by Patashinskli and Pokrovskii [9] in 1973. Following refs. [l-4] we consider an n-component isotropic spin model (n > 2) in d = 2 t E dimensions with the reduced hamiltonian: a/T=1
ddx{(Js2/T)+($,n)2
- (s/T)Hn),
(1)
tudinal part with respect to the external field, n(x) = (&w
, fi
o),
X = T,‘Js2
(3)
(rr is a (n - 1) component vector) we may integrate over the o-tield in the partition function. This leads to the hamiltonian of the non-linear u-model [3,4] : d G+
BI(A)/T = j
{(a,qW-l(a,JZ2~2 (4)
- p ln(1
- hn2) - 2hh-l
JiXZ},
where p = AdKd/d,K;l = 2d-1nd/2y(d/2) and h = = H/Js. A denotes a cut-off which will be chosen later to be unity. In order to find the critical behaviour of the model we have to eliminate the fllctuations of n(x) = J ddq n4 ei@ with momenta A < 4 < A from the partition function. From eq. (1) it is obvious that this procedure effects in a renormalization of the only two parameters J and s, according to J(x) = g1 6% A) J(A),
4% = g26i, O(A),
which leads to a renormalization tor (t = h&):
of the bare propaga-
G,(k, h; A) + G,(k, h; X) = t(??ikni _ k)qo(x) (5) = t/(glk2
(2) where J, s and H denote the exchange coupling, the spin length and an external field, respectively. Decomposing the unit vector n(x) into a transverse and longi144
+ h$).
To leading order in e we find from a skeleton graph calculation in 2 t E dimensions the following equations for gl, g2 : E dgl (X)/dX’
= -tgT2
X2/(X2 + k* + hgi’gz’),
(6)
Volume 67A, number 2
E dg2(x)/d%
PHYSICS LETTERS
= +(n - 1) f&-‘g;’
x2/(x2 + hg&$). (7)
For k = 0 these equations are equivalent to-those found in ref. [4]. From eq. (6) follows, that for A 2 k*(k, h), where k*2 = k2 + hg;‘(k*)g;‘(k*),
(8)
gl(x) changes into gl(k*) if we neglect contributions smaller by a factor E. Hence
n-
h/m6 = {
1 t (r/rc - l)m-1/P}2/‘(r/r,)-2’“,
p=,(,yn-l
’
(9)
6 - 4(n - 2) + n - 3 (n- 1)e n- 1 ’
in agreement with ref. [3]. The free energy density is f= -G,(O, h)h2/r2. For the longitudinal correlation function we write
G&k,h) = r((ok = (t2/4&)
gi(k) = (1 - r/tc + (t/t& ke)*i, Al=-
24 July 1978
$
(u,$)(u_~
- (u_~)))
d dx eikx0t2(x)n2(0)),,
and therefore
Using the zero field correlation length 6 = (r/r,)l/e X I1 - r/rCI-l/E, and relative magnetization u = (1 - r/rc)(n-N2( n- 2, below rc the result for the transverse correlation function can be written as [3] : G,(k, h) = u2EdF,(&, hoid/r) >
(10)
with F,(x,y)
or
= r,z-2(x,
z~(x,Y)=x~+
x = tk,
(12)
r(5k
y)(l + zE(x,y))ll(n-2),
rcy(1
tz~(x,y))-(n-3)/2(n-2)
=
t2k*2, (11)
Contrary to the situation above r, or .in anisotropic systems, scaling holds in the whole low temperature phase. However, there is a crossover between the hydrodynamic region z’(x, y) -%1 and the critical region z’(x,y) Z+1. In the latter case we get z = (r,y)11Xho-1(x2/(r,y)2/“h), where G(r2) is a solution of the equation Ah = 2 + [(n - 3)/2(n - 2)] E.
Since our results are valid only for small r = x/yllxh = k/hlIxh, we find for the expansion of 9: @= 4n(l -+(n
mp~-p+R)-l ah,
hk#O’O
y = ho.$d/r = h[@lvrfl-l.
1 = ,2#2 + @h,
@;P,Y)= n_l
- 3) rc9: m Go), do = l/(1 +r2).
In the special case k = 0 one obtains from eqs. (lo), (11) and the relation G,(k = 0) = (m/h)r, m = (u(x)) X $i, for the equation of state:
hk denotes the Fourier transform of an inhomogeneous external field. In general, I’(x, u, y) is not known and we are unable to calculate F,, . However in the special case of zero external momentum we can use the result I’(u,u,y)=(l tz+,y))-@. We note that G,,(k,h) exhibits then scaling behaviour only if the upper integration limit tends to infinity. Then we get
~$(O,Y) =f@- 1)jd u d+v+y)r+,f4,y) zo
(13)
= rc azi2+e(1 + Z;)-(n -3)/b-2), where a = (n - l)r,/4, z. = ~(0, y).The reason for using the square of I’ and z. as a low momentum cutoff in eq. (13) follows from the approximate calculation of F,,(O, y) (see ref. [lo] for a discussion). For k = 0 one finds from eqs. (lo)-(12) the results for the transverse and longitudinal susceptibility given in ref. [4]. For k f 0 the result is new. So far all expressions are restricted to the low temperature phase r < r,. Considering the region r > t, one has to take into account, that the n-fields acquire 145
PHYSICS LETTERS
Volume 67A, number 2
a mass f-l (8) into
which changes our low momentum
z *2 = z-2 + k2 t &;‘(~*)g;‘(~*).
cut-off
(14)
For h = k = 0 we find from the equality of the longitudinal and transverse propagator above t,: -g =
(JJ
pc- 1)--lk(1 - Cr)l/E= E(1 -a)+
(15)
From eq. (15) we obtain the results of Polyakov [I] and of Bardeen et al. [8] in the limit E + 0 and n -+ m, respectively. Since (1 -,)-1/e > 1 we get 1 - t/tC + (t/t,_) ff*E > 0 for all t, i.e. we have a mechanism to continue the gj factors into the high temperature phase. In particular, we get for the renormahzed coupling
&(k,
h) = X/ [ 1 - t/tc t (t/t& z*‘] .
Since perturbation theory requires hren Q 1 this formula is only valid not too close to the critical point. For the scaling function we get F&y)
= I;-2(x,y)(?(x,y)
+ t,u(zc(X,Y)
_ 1)-6+3)/w-2)
(16) = 52 x*2.
Inside the critical region z e, Ze S 1, FL changes smoothly into FL as expected from scaling theory. We note further, that for n > 2 it is easy to perform the limit e + 0 (d = 2) in the results. For n = 2 the transition temperature becomes infinity. Following Brezin et al. [3] we write for
G,(k, h) = tdje e-t/EFl(ktllE, he-t/ht2/E). Then from eqs. (5)-(9)
we get:
F,(x,y)
= z-~(x,~)
zQ,r)
=x2 tyeZE(XGY2E.
eZE(X~Y)IE, (17)
Finally, we consider the limit E + 0, n = 2. However, there is strong indication, that our model does not de-
146
scribe the plane rotator in this limit since we neglected vortex configurations [ 121. Formally these can be taken into account by substituting J by Jp, (i.e. t + t/ps, h + h/p,), where p,(t) denotes the transverse stiffness. Then we get
G(k,h) = t/p,(t)k*2-q, (18) k*2 = k2 + (h/p,)k*Q/2,
q = v(t) = t/p&t).
Our result for the free energy density,f= -@;n x h4)1/(4 -o)t-1 ) is in general agreement with Zittartz [ 1 l] (but not with ref. [4]) who predicts transitions at q(t,) = 2,4. However, taking vortex excitations into account, there is a transition at to = $ps(tO) < t, due to a jump in ps which cannot be calculated in our framework. In this case our results are restricted to the low temperature phase. References [l] A.M. Polyakov, Phys. Lett. 59B (1975) 79. [ 21 A.A. Migdal, Zh. Eksp. Teor. Fiz. 69 (1975) 810,1457;
- l)Il(“-%,,
?(X, y) = (1 - a)-21’ t X2
24 July 1978
EngI. transl. Sov. Phys. JETP 42 (1976) 413,743. [3] E. Brezin and J. Zinn-Justin, Phys. Rev. B14 (1976) 3110; Phys. Rev. Lett. 36 (1976) 691. [4] R.A. Pelcovits and D.R. Nelson, Phys. Lett. 57A (1976) 23; D.R. Nelson and R.A. Pelcovits, Phys. Rev. B16 (1977) 2191. [S] V.L. Pokrovskii and G.V. Uimin, Phys. Lett. 45A (1973) 467. [6] S.B. Khokhlachev, Zh. Eksp. Teor. Fiz. 70 (1976) 265; EngI. transl. Sov. Phys. JETP 43 (1977) 137. [ 7) T. Nattermann and S. Trimper, Phys. Lett. SOA (1974) 307. [ 81 W.A. Bardeen, B.W. Lee and R.E. Shrock, Phys. Rev. D14 (1976) 985. [9] A.S. Patashinskii and V.L. Pokrovskii, Zh. Eksp. Teor. Fiz. 64 (1973) 1445. [lo] B. Roulet, J. Gavoret and P. Nozieres, Phys. Rev. 178 (1969) 1072. [ll] J. Zittartz, Z. Phys. 23B (1976) 55,63. [12] J.M. Kosterlitz, J. Phys. C7 (1974) 1046; J.V. Jose, L.P. Kadanoff, S. Kirkpatrick and D.R. Nelson, Phys. Rev. B16 (1977) 1217.
SPIN CORRELATIONS
PHYSICS LETTERS
24 July 1978
IN 2 + E DIMENSIONS - ORDERED AND DISORDERED
PHASE
T. NATTERMANN Sektion Physik, Karl-Marx-Universitl, 701 Leipzig, DDR Received 23 February 1978
Closed form expressions for the correlation functions of fixed length spins in 2 and 2 + E dimensions below and above the critical temperature are obtained by a skeleton graph approach.
In 1975 Polyakov [l] and Migdal [2] have developed techniques to treat n-component classical fixed length spins in 2 + E dimensions. Their work has been generalized by Brezin et al. [3] using field-theoretic methods. Recently, Nelson and Pelcovits [4] used momentum shell recursion relations to calculate various thermodynamic quantities. Their results complete calculations in 2 ,dimensions by Pokrovskii and Uimin [ 51 and Khoklachev [6] . In this note we compute closed form expressions for the correlation functions in an external field both below and above the critical temperature. To this aim we use a skeleton graph approach very similar to that in d = $ - E dimensions [7]. The results are valid to O(e) and to O(n”). In particular we are able to treat also the high temperature phase which was considered so far only for the large-n case [8] . The resulting expressions improve considerably those found by Patashinskli and Pokrovskii [9] in 1973. Following refs. [l-4] we consider an n-component isotropic spin model (n > 2) in d = 2 t E dimensions with the reduced hamiltonian: a/T=1
ddx{(Js2/T)+($,n)2
- (s/T)Hn),
(1)
tudinal part with respect to the external field, n(x) = (&w
, fi
o),
X = T,‘Js2
(3)
(rr is a (n - 1) component vector) we may integrate over the o-tield in the partition function. This leads to the hamiltonian of the non-linear u-model [3,4] : d G+
BI(A)/T = j
{(a,qW-l(a,JZ2~2 (4)
- p ln(1
- hn2) - 2hh-l
JiXZ},
where p = AdKd/d,K;l = 2d-1nd/2y(d/2) and h = = H/Js. A denotes a cut-off which will be chosen later to be unity. In order to find the critical behaviour of the model we have to eliminate the fllctuations of n(x) = J ddq n4 ei@ with momenta A < 4 < A from the partition function. From eq. (1) it is obvious that this procedure effects in a renormalization of the only two parameters J and s, according to J(x) = g1 6% A) J(A),
4% = g26i, O(A),
which leads to a renormalization tor (t = h&):
of the bare propaga-
G,(k, h; A) + G,(k, h; X) = t(??ikni _ k)qo(x) (5) = t/(glk2
(2) where J, s and H denote the exchange coupling, the spin length and an external field, respectively. Decomposing the unit vector n(x) into a transverse and longi144
+ h$).
To leading order in e we find from a skeleton graph calculation in 2 t E dimensions the following equations for gl, g2 : E dgl (X)/dX’
= -tgT2
X2/(X2 + k* + hgi’gz’),
(6)
Volume 67A, number 2
E dg2(x)/d%
PHYSICS LETTERS
= +(n - 1) f&-‘g;’
x2/(x2 + hg&$). (7)
For k = 0 these equations are equivalent to-those found in ref. [4]. From eq. (6) follows, that for A 2 k*(k, h), where k*2 = k2 + hg;‘(k*)g;‘(k*),
(8)
gl(x) changes into gl(k*) if we neglect contributions smaller by a factor E. Hence
n-
h/m6 = {
1 t (r/rc - l)m-1/P}2/‘(r/r,)-2’“,
p=,(,yn-l
’
(9)
6 - 4(n - 2) + n - 3 (n- 1)e n- 1 ’
in agreement with ref. [3]. The free energy density is f= -G,(O, h)h2/r2. For the longitudinal correlation function we write
G&k,h) = r((ok = (t2/4&)
gi(k) = (1 - r/tc + (t/t& ke)*i, Al=-
24 July 1978
$
(u,$)(u_~
- (u_~)))
d dx eikx0t2(x)n2(0)),,
and therefore
Using the zero field correlation length 6 = (r/r,)l/e X I1 - r/rCI-l/E, and relative magnetization u = (1 - r/rc)(n-N2( n- 2, below rc the result for the transverse correlation function can be written as [3] : G,(k, h) = u2EdF,(&, hoid/r) >
(10)
with F,(x,y)
or
= r,z-2(x,
z~(x,Y)=x~+
x = tk,
(12)
r(5k
y)(l + zE(x,y))ll(n-2),
rcy(1
tz~(x,y))-(n-3)/2(n-2)
=
t2k*2, (11)
Contrary to the situation above r, or .in anisotropic systems, scaling holds in the whole low temperature phase. However, there is a crossover between the hydrodynamic region z’(x, y) -%1 and the critical region z’(x,y) Z+1. In the latter case we get z = (r,y)11Xho-1(x2/(r,y)2/“h), where G(r2) is a solution of the equation Ah = 2 + [(n - 3)/2(n - 2)] E.
Since our results are valid only for small r = x/yllxh = k/hlIxh, we find for the expansion of 9: @= 4n(l -+(n
mp~-p+R)-l ah,
hk#O’O
y = ho.$d/r = h[@lvrfl-l.
1 = ,2#2 + @h,
@;P,Y)= n_l
- 3) rc9: m Go), do = l/(1 +r2).
In the special case k = 0 one obtains from eqs. (lo), (11) and the relation G,(k = 0) = (m/h)r, m = (u(x)) X $i, for the equation of state:
hk denotes the Fourier transform of an inhomogeneous external field. In general, I’(x, u, y) is not known and we are unable to calculate F,, . However in the special case of zero external momentum we can use the result I’(u,u,y)=(l tz+,y))-@. We note that G,,(k,h) exhibits then scaling behaviour only if the upper integration limit tends to infinity. Then we get
~$(O,Y) =f@- 1)jd u d+v+y)r+,f4,y) zo
(13)
= rc azi2+e(1 + Z;)-(n -3)/b-2), where a = (n - l)r,/4, z. = ~(0, y).The reason for using the square of I’ and z. as a low momentum cutoff in eq. (13) follows from the approximate calculation of F,,(O, y) (see ref. [lo] for a discussion). For k = 0 one finds from eqs. (lo)-(12) the results for the transverse and longitudinal susceptibility given in ref. [4]. For k f 0 the result is new. So far all expressions are restricted to the low temperature phase r < r,. Considering the region r > t, one has to take into account, that the n-fields acquire 145
PHYSICS LETTERS
Volume 67A, number 2
a mass f-l (8) into
which changes our low momentum
z *2 = z-2 + k2 t &;‘(~*)g;‘(~*).
cut-off
(14)
For h = k = 0 we find from the equality of the longitudinal and transverse propagator above t,: -g =
(JJ
pc- 1)--lk(1 - Cr)l/E= E(1 -a)+
(15)
From eq. (15) we obtain the results of Polyakov [I] and of Bardeen et al. [8] in the limit E + 0 and n -+ m, respectively. Since (1 -,)-1/e > 1 we get 1 - t/tC + (t/t,_) ff*E > 0 for all t, i.e. we have a mechanism to continue the gj factors into the high temperature phase. In particular, we get for the renormahzed coupling
&(k,
h) = X/ [ 1 - t/tc t (t/t& z*‘] .
Since perturbation theory requires hren Q 1 this formula is only valid not too close to the critical point. For the scaling function we get F&y)
= I;-2(x,y)(?(x,y)
+ t,u(zc(X,Y)
_ 1)-6+3)/w-2)
(16) = 52 x*2.
Inside the critical region z e, Ze S 1, FL changes smoothly into FL as expected from scaling theory. We note further, that for n > 2 it is easy to perform the limit e + 0 (d = 2) in the results. For n = 2 the transition temperature becomes infinity. Following Brezin et al. [3] we write for
G,(k, h) = tdje e-t/EFl(ktllE, he-t/ht2/E). Then from eqs. (5)-(9)
we get:
F,(x,y)
= z-~(x,~)
zQ,r)
=x2 tyeZE(XGY2E.
eZE(X~Y)IE, (17)
Finally, we consider the limit E + 0, n = 2. However, there is strong indication, that our model does not de-
146
scribe the plane rotator in this limit since we neglected vortex configurations [ 121. Formally these can be taken into account by substituting J by Jp, (i.e. t + t/ps, h + h/p,), where p,(t) denotes the transverse stiffness. Then we get
G(k,h) = t/p,(t)k*2-q, (18) k*2 = k2 + (h/p,)k*Q/2,
q = v(t) = t/p&t).
Our result for the free energy density,f= -@;n x h4)1/(4 -o)t-1 ) is in general agreement with Zittartz [ 1 l] (but not with ref. [4]) who predicts transitions at q(t,) = 2,4. However, taking vortex excitations into account, there is a transition at to = $ps(tO) < t, due to a jump in ps which cannot be calculated in our framework. In this case our results are restricted to the low temperature phase. References [l] A.M. Polyakov, Phys. Lett. 59B (1975) 79. [ 21 A.A. Migdal, Zh. Eksp. Teor. Fiz. 69 (1975) 810,1457;
- l)Il(“-%,,
?(X, y) = (1 - a)-21’ t X2
24 July 1978
EngI. transl. Sov. Phys. JETP 42 (1976) 413,743. [3] E. Brezin and J. Zinn-Justin, Phys. Rev. B14 (1976) 3110; Phys. Rev. Lett. 36 (1976) 691. [4] R.A. Pelcovits and D.R. Nelson, Phys. Lett. 57A (1976) 23; D.R. Nelson and R.A. Pelcovits, Phys. Rev. B16 (1977) 2191. [S] V.L. Pokrovskii and G.V. Uimin, Phys. Lett. 45A (1973) 467. [6] S.B. Khokhlachev, Zh. Eksp. Teor. Fiz. 70 (1976) 265; EngI. transl. Sov. Phys. JETP 43 (1977) 137. [ 7) T. Nattermann and S. Trimper, Phys. Lett. SOA (1974) 307. [ 81 W.A. Bardeen, B.W. Lee and R.E. Shrock, Phys. Rev. D14 (1976) 985. [9] A.S. Patashinskii and V.L. Pokrovskii, Zh. Eksp. Teor. Fiz. 64 (1973) 1445. [lo] B. Roulet, J. Gavoret and P. Nozieres, Phys. Rev. 178 (1969) 1072. [ll] J. Zittartz, Z. Phys. 23B (1976) 55,63. [12] J.M. Kosterlitz, J. Phys. C7 (1974) 1046; J.V. Jose, L.P. Kadanoff, S. Kirkpatrick and D.R. Nelson, Phys. Rev. B16 (1977) 1217.