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Renormalization group and critical exponents for a Gaussian spin model with long range interactions
Volume 4 1 A, number 3
PHYSICS LETTERS
RENORMALIZATION
25 September 1972
GROUP AND CRITICAL EXPONENTS FOR A GAUSSIAN SPIN
MODEL WITH LONG RANGE INTERACTIONS Th. NIEMEIJER and J.M.J. Van LEEUWEN Laboratorium
voor Technische Natuurkunde,
Technische Hogeschool,
Delft, The Netherlands
Received 20 July 1972 For Gaussian spin models with long range interactions a renormalization group of transformations is constructed explicitly showing an (improper) fixed point. Correct critical exponents result from the fmed point eigenvalue spectrum, as long as the phase transition is non-classical.
The one-dimensional, translationally invariant, Gaussian spin model defined by the Hamiltonian:
4j
= qe2
C
J2i,2j-k
:N 2
J2i2k+lx2k+1,21tlJ21+1,2j
1
(‘)
(1) and was discussed by Joyce [l] for Jti a li-jl-(l+u) and This system is thermodynamically stable and shows a ferromagnetic phase transition when 0 < u < 1(h = 0); its critical exponents are
hi = h.
o
%
Pr
6
77
aJ
0
h 1
3
2-o
+-$
(6)
(A’is the inverse of J2i+l,2fil), U; = qU2im The scalefactor 4 will be fured later. Under this last position and scale transformation the correlation function behaves as g2i_2j=(U2jU2f)=~2(Uj~~>=q-2q~_j
*
(2)
In view of the ideas recently expounded by Wilson [2] , a renormalization group of transformations is constructed for (1) by first integrating over the odd spin variables. The free energy f(J, h) (always per spin) of the system defined by (1) can then be expressed as
f(J, h) = 1If,,&& h) +f(s, h’)+ln41
h:=4-l
(3)
where fodd(J, h) is the free energy of the system consisting of the odd spins only and f(J’, h’) is the free energy of the system described by
(4)
(7)
Eqs. (5) and (6) define a renormalization group of equations for a Hamiltonian of type (1). Introducing fourier transforms of the field hi,
susceptibility
1
-1
,$
Jtj exp t -W-131
,
N) and magnetization m(k) = $?$~~~=ark~gously h’(k) = c N/2 h!exp(-ikj) (k = 4nn/N, n = 1, .. . . @) etc, one de&.: f&m (5) and (6) the transformation equations
211
Volume 4 1A, number 3
X’W =w
[X(W)
PHYSICS LETTERS
+ X(~+wN
(8)
25 September 1972
Af(e, m) (E = (T--T,)/T,) one immediately derives [3] , assuming thatfhas a singular part, that
and m(k) = q,[Vz(k/2)+m(n+k/2)]
.
(9)
In order that (8) has a fixed point in the hyperplane of critical points (determined by J(0) = 0 or x(O) = -) corresponding to interactions with the same strength of the long range part of (I), 9 has to be chosen as q2 = 2l-O. This follows from the requirement that both x(k) and x’(k) start out as Ak-O. The fixed point x*(k) can be represented by X*(k) = I= IF
IO-1 cos (Ik) .
.( 101
> r.. Since this series is conditionally convergent, (10) is only formal. In fact (8) allows no solution x*(k) > 0 (as one should require for the susceptibility). For m*(k) the trivial solution m*(k) = 0 is allowed. To find the critical exponents one has to determine the bigen-values hT and h, of the linearized transformations (here they already are linear) at the futed point
X’(k) = hTX(‘%
m’(k) = h,m(k)
.
(11)
From (8), (9) and (10) one derives for the complete spectrum of the linearized equations hT = 21-rs-n, A, = 2(1-o)/2-n’, n, n’ = 0, 1,2, . . . . Thus there is only one XT > 1 and one h, > 1, namely A$ = 2l-O and AZ = 2(lPu)12. If one writes the free energy as a generalized harmonic functionf(XaTe, A”mm) =
212
“m =lnX~/ln
2= a(l-0)
(12)
these lead to the values of (Y,p, y and 6 as listed in [2] for 1 < u < 1. The solutionq2 = 2l-u also leads to the correct values of v and n since from the transformation law (7) it follows that g(r) = g( l)/rl-” in the fixed point. For 1 < u < : the phase transition 1s classical as can be seen from (2) and the free energy can actually be expanded round the critical point. The expressions (11) do not distinguish between the cases u > 1 and u < 4. One can resolve this by stating that the linearized renormalization equations lead to the correct critical exponents when the free energy has a singular part. The two-dimensional system behaves in a similar way. The authors want to thank Prof. B. Widom for his stimulating lectures on the Wilson theory and one of us (J.M.J. van Leeuwen) is indebted to Prof. M.S. Green for explaining his ideas on the renormalization group.
References [l] G.S. Joyce, Phys. Rev. 146 (1966) 349. [2] K.G. Wilson, Phys. Rev. B4 (1971) 3174,3184. [3] H.E. Stanley, Introduction to phase transitions and critical phenomena (Oxford University Press, 197 1).
PHYSICS LETTERS
RENORMALIZATION
25 September 1972
GROUP AND CRITICAL EXPONENTS FOR A GAUSSIAN SPIN
MODEL WITH LONG RANGE INTERACTIONS Th. NIEMEIJER and J.M.J. Van LEEUWEN Laboratorium
voor Technische Natuurkunde,
Technische Hogeschool,
Delft, The Netherlands
Received 20 July 1972 For Gaussian spin models with long range interactions a renormalization group of transformations is constructed explicitly showing an (improper) fixed point. Correct critical exponents result from the fmed point eigenvalue spectrum, as long as the phase transition is non-classical.
The one-dimensional, translationally invariant, Gaussian spin model defined by the Hamiltonian:
4j
= qe2
C
J2i,2j-k
:N 2
J2i2k+lx2k+1,21tlJ21+1,2j
1
(‘)
(1) and was discussed by Joyce [l] for Jti a li-jl-(l+u) and This system is thermodynamically stable and shows a ferromagnetic phase transition when 0 < u < 1(h = 0); its critical exponents are
hi = h.
o
%
Pr
6
77
aJ
0
h 1
3
2-o
+-$
(6)
(A’is the inverse of J2i+l,2fil), U; = qU2im The scalefactor 4 will be fured later. Under this last position and scale transformation the correlation function behaves as g2i_2j=(U2jU2f)=~2(Uj~~>=q-2q~_j
*
(2)
In view of the ideas recently expounded by Wilson [2] , a renormalization group of transformations is constructed for (1) by first integrating over the odd spin variables. The free energy f(J, h) (always per spin) of the system defined by (1) can then be expressed as
f(J, h) = 1If,,&& h) +f(s, h’)+ln41
h:=4-l
(3)
where fodd(J, h) is the free energy of the system consisting of the odd spins only and f(J’, h’) is the free energy of the system described by
(4)
(7)
Eqs. (5) and (6) define a renormalization group of equations for a Hamiltonian of type (1). Introducing fourier transforms of the field hi,
susceptibility
1
-1
,$
Jtj exp t -W-131
,
N) and magnetization m(k) = $?$~~~=ark~gously h’(k) = c N/2 h!exp(-ikj) (k = 4nn/N, n = 1, .. . . @) etc, one de&.: f&m (5) and (6) the transformation equations
211
Volume 4 1A, number 3
X’W =w
[X(W)
PHYSICS LETTERS
+ X(~+wN
(8)
25 September 1972
Af(e, m) (E = (T--T,)/T,) one immediately derives [3] , assuming thatfhas a singular part, that
and m(k) = q,[Vz(k/2)+m(n+k/2)]
.
(9)
In order that (8) has a fixed point in the hyperplane of critical points (determined by J(0) = 0 or x(O) = -) corresponding to interactions with the same strength of the long range part of (I), 9 has to be chosen as q2 = 2l-O. This follows from the requirement that both x(k) and x’(k) start out as Ak-O. The fixed point x*(k) can be represented by X*(k) = I= IF
IO-1 cos (Ik) .
.( 101
> r.. Since this series is conditionally convergent, (10) is only formal. In fact (8) allows no solution x*(k) > 0 (as one should require for the susceptibility). For m*(k) the trivial solution m*(k) = 0 is allowed. To find the critical exponents one has to determine the bigen-values hT and h, of the linearized transformations (here they already are linear) at the futed point
X’(k) = hTX(‘%
m’(k) = h,m(k)
.
(11)
From (8), (9) and (10) one derives for the complete spectrum of the linearized equations hT = 21-rs-n, A, = 2(1-o)/2-n’, n, n’ = 0, 1,2, . . . . Thus there is only one XT > 1 and one h, > 1, namely A$ = 2l-O and AZ = 2(lPu)12. If one writes the free energy as a generalized harmonic functionf(XaTe, A”mm) =
212
“m =lnX~/ln
2= a(l-0)
(12)
these lead to the values of (Y,p, y and 6 as listed in [2] for 1 < u < 1. The solutionq2 = 2l-u also leads to the correct values of v and n since from the transformation law (7) it follows that g(r) = g( l)/rl-” in the fixed point. For 1 < u < : the phase transition 1s classical as can be seen from (2) and the free energy can actually be expanded round the critical point. The expressions (11) do not distinguish between the cases u > 1 and u < 4. One can resolve this by stating that the linearized renormalization equations lead to the correct critical exponents when the free energy has a singular part. The two-dimensional system behaves in a similar way. The authors want to thank Prof. B. Widom for his stimulating lectures on the Wilson theory and one of us (J.M.J. van Leeuwen) is indebted to Prof. M.S. Green for explaining his ideas on the renormalization group.
References [l] G.S. Joyce, Phys. Rev. 146 (1966) 349. [2] K.G. Wilson, Phys. Rev. B4 (1971) 3174,3184. [3] H.E. Stanley, Introduction to phase transitions and critical phenomena (Oxford University Press, 197 1).