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Magnetization and specific heat of classical ferromagnetic chains in a field: Crossover between two regimes
Volume 103A, number 9
PHYSICS LETTERS
30 July 1984
MAGNETIZATION AND SPECIFIC HEAT OF CLASSICAL FERROMAGNETIC CHAINS IN A FIELD: CROSSOVER BETWEEN TWO REGIMES U. BALUCANI Istituto di Elettronica Quantistica del CNR, Florence, Italy V. TOGNETTI 1 and R. VAIA Dipartimento di Fisiea dell'Univers#d, Florence, Italy Received 17 April 1984 Revised manuscript received 5 June 1984
A theoretical approach, previously introduced to treat the statistical mechanics of a classical ferromagnetic chain in a field, has been extended in a systematic way to give the crossover from low-field to spin-waveregime. The results are in quantitative agreement with numerical transfer matrix data.
One-dimensional magnetic systems have received great attention in the last years [ 1]. Their most interesting feature is the lack of long range order ((S z) = O) for finite temperatures in the absence of a magnetic field. The usual many-body perturbative approach in terms of interacting spin waves was proved to be valid when the ordering effect of the applied magnetic field H overcomes the thermal disorder [2]. On the other hand, for vanishing magnetic fields, the presence of unpleasant divergencies points out the necessity of non-perturbative methods to calculate both static and dynamical quantities. In the classical case, the data obtained numerically by transfer-matrix techniques [2,3] give a test for the validity of any theory of the thermodynamical properties. For the isotropic Heisenberg classical chain, we have previously shown [4] that a consistent scheme can be set up to calculate the partition function in order to reproduce, at least at low temperatures, the two limiting regimes, i.e. both the standard spinwaves and the H = 0 case where exact analytical expressions are available [5]. In particular, a zero magnetization for vanishing field was obtained for finite
temperatures, just taking into full account the anharmonic terms. In this paper we present quantitative results for the magnetization and specific heat of the classical Heisenberg magnetic chain after having improved the previous formalism introducing a simpler and more extendible mathematical framework. The present approach permits, in principle, to build up a rigorous expansion which interpolates both the limiting situations and whose results can be compared with transfer matrix data. Moreover, the correct vanishing magnetization for H ~ 0 is analytically obtained also in the presence of a small exchange anisotropy. Let us consider the following hamfltonian N-1 j=l
[ i~ u ( S } S+m
-
÷ s j - s j +÷ , ) ÷ s j s jz+ l zl
N
- gUB~l.~ S[,
(1)
]°= l
where N classical "spin" vectors Sj interact through the J > 0 exchange integrals and can be parametrized by [41
i Also with Unit~ di Firenze del GNSM-CNR, Italy.
0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North.Holland Physics Publishing Division)
435
S1 = 2S[vi(1 - vi)]l/2exp(±i¢]), S]z = S(1 - 2v/). For/l = 1 the isotropic case is recovered. In the following we shall use normalized units for magnetic field and temperature
h = glaBH/JS,
T* = [3*-1 = kB T/JS 2"
The partition function Z N turns out to be [4]
Z N = exp~*[N(1 + h ) 1
30 July 1984
PHYSICS LETTERS
Volume 103A, number 9
1])
1
X f dv 1 ... f dVNexp {-2~* [(1 +h) v 1 - VN] } 0 0 N-1
X ~ A (vi, vi+1), i=1
(2)
with
From eqs. (4), (5) one can see that at low temperatures ~* >> 1) the negative exponential in sinh x can be neglected for any value of the magnetic field, indicating that the "kinematic" interaction is significant only at higher temperatures (where it gives rise to activation terms like e-l/T*). Furthermore, the expansion of R (h, la, VN_ 1) in terms of powers of VN_ 1 can be performed both for small anharmonicity (spin wave regime, SW) and in the case of low field (LF, h < 1) and small anisotropy ((1 -/~2) ,~ 1). Such an approximation permits one to compute iteratively the preceding integrals, and finally to get an expression for Z N that describes the crossover between the SW and LF regimes. By considering an approximation involving up to the nth power of v, one deals with successive integrals like 1
A (v', v) = exp [-2[3*(2 + h - 2ev') v] 21r
n
f dvA(v',v)exp(2[3*~v) 0
21t × f 0 x
k~=Oakv k
n
dTexp{4[3*~[vv'(1 - ev)(1 -
ev')l t/2
cos-t}.
~ exp(2[3* q/v')
(3)
A perturbative parameter e (e = 1 in the real system) has been introduced to measure the effect of the anharmonic interactions. The integral in the variable vN is straightforward: 1
ak v'k ,
(6)
where qJ' = #2/(2 + h - ~). In the thermodynamic limit this leads to a continued fraction for the q/s, converging to the value = 1'- ~(K - h),
(7)
with
K = [h(h + 4) + 4(1 - At2)] 1/2.
f d vN A (vN_ 1, vN) exp (2 [3*vN ) 0
The coefficients a~ depend linearly on the ak : n
= exp[-(fl*/e)(1 +h - 2eVN_l) ]
aj : k~=OMlk (T*, h, U) ak,
(7')
X sinh[[3*(1 + h ) e - l R ( h , / & vU_l) ] X [/~*(1 +h)R(h,la, VN_I)] -1,
(4)
with
R(h, la, VN_l) = [ 1 - 4
-u s +4(11 +h)2
436
Z N "" (exp [[3"(1 + h)] h) N.
1 + h - / a 2 eVN 1 (I + h) 2 -
1/2
and in the thermodynamic limit the leading contribution to Z N is given by the Nth power of the maximum eigenvalue X = X(T*, h,/a) of the (n + 1) X (n + 1) matrix M:
(5)
(8)
The ruth column of M is associated with the coefficients of the expansion in powers of v' of the general integral
1
Jm(V') = f
d v A ( v ' , v)exp(2/3*~Ov) v m
o rl
"" exp(2~*~ v') k~__OMk m v 'k.
(9)
Analytical expressions for the integrals Jm can easily be obtained from the low temperature result j0(/f) = ~(C 1 2 + B) -1/2 exp ~e-1 [ - B + (C 2 + B E)l/2] }, (lO)
with B =#*0a2/~ - 2ev'),
30 July 1984
PHYSICS LETTERS
Volume 103A, number 9
C - 2~*p[ev'(1 - ev')] 1/2,
noting that Jm (v') = ( - ~blbB) m Jo(v').
(11)
The expansion to be performed in such expressions involves halfinteger powers of c 2 +a 2 - (~*u2/~) 2
× [1-4(~//a2)~ev'+4(ff2/p4)52e2v'2],
(12)
where we have set ~ = 1 - ff -_ ] ( K _ h) and 52 = 1 - / a 2, which are small parameters in the LF (~ ~ (h + 52) 1/2) and low anisotropy situation. Note that 52 can be negative in the case of planar exchange anisotropy. By referring to the parameters e (to be considered small in the SW regime), ~ and 52, rather to v', one gets better control on the accuracy of the expansion and has the possibility of constructing "mixed" approximations. For the sake of simplicity, we shall consider in the following a vanishing anisotropy (6 2 - 0). Furthermore, we shall indicate an approximation procedure correct up to terms en and ~rn as a (en, ~m)-theory: in such a theory, only terms of order equal or higher than the product e n+l ~m+l can be disregarded. One can easily see that: (a) a theory correct up to terms of order e n requires expansions up to the 2nth power of v'; (b) a theory correct up to terms of order ~n requires expansions up to the nth power of v'. The natural interpolating theories will then be (e n, ~2n) and (e n , ~2n+l)-theories, respectively involving expansions up to v 2n and v 2n+l . We can equivalently say that the coefficients a2n and a2n+l are of order
en~ 2n and en+l~ 2n+l. As a consequence, one needs a lower accuracy in the expansion procedure of the integrals Jm, because in eq. (6) they already are multiplied by a factor a m . The next columns of the matrix M will then be computed with decreasing accuracy. This "graded" structure of M makes possible a computation of the maximum eigenvalue ~ by means of increasing order contributions, which are obtained as solutions of linear equations (in principle, ~ is a solution of det(M - 20 = 0, a secular equation of degree equal to the order of M). Indeed, suppose we know a (e n - l , ~2(n-1)) value of ?~0 for ~. The following contribution x is of order en~ 2 n - l , and will be recovered in the (e n, ~2n-1)-theory. Putting ?~= ?'0 +x, only the x in the first column of M - 2~survives, because in the o t h er columns quantities of order en~ 2n-1 can be neglected: then the secular equation det (M - ?0 = 0 becomes linear in the unknown variable x. In a similar way, one gets the next en~ 2n correction, and thus, the (e n, ~2n)-value for ~,. Analytical solutions for ?, can easily be obtained in the first lower order theories, but this iterative procedure is very useful also in the numerical computations. In the (e 0, ~3)-theory (LF and low anisotropy limit) we obtain the analytical result ? ~ = ~ T * [ 1 - / ~ * h - 5 (1f l • - 1 ) 5 2].
(13)
By eq. (8) one correctly arrives at (S z) = 0 + O(h).
0.8
h = 0.01
0.6-
~.~" ?
0.4.
"?':"-.
0.2
~
0
0
0".1
~ "'.
0~2
"'"
T*
Fig. 1. Magnetization of the classicalferromagnetic chain in a field h = 0.01 as a function of temperature. The full line is the theoretical result at order e 2 ,/~4, the dashed line is the result of the lower-order (e, f2) approximation, and the dotted
line derives from a Hartree-Fock perturbative approach. Open circles represent the numerical transfer-matrix data. 437
PHYSICS LETTERS
Volume 103A, number 9 C ~'~.
h -
ly. The results of the simpler (e, ~2)-theory are also reported, showing the degree of improvement obtained by raising the order of the approximation. It is worthwhile to note that our treatment is essentially valid down to very low fields, where interacting spin-wave theories fail, and at the same time to very low temperatures, where standard transfer matrix techniques meet problems of numerical accuracy. In this region, our results approach those obtained numerically within the continuum approximation [7].
0.01
o o
1.06
o
lO,
"
1.02
1.00
..
/
o
~ -"~
o:1
o12
T*
Fig. 2. Magnetic specific heat of the classical ferromagnetic chain in a field h = 0.01 as a function of temperature. All curves and data have the same meaning as in fig. 1.
Another analytical solution can straightforwardly be obtained in the (e, ~2)-theory, which in the SW case accounts for the perturbative H a r t r e e - F o c k correction. Both these limiting results were obtained by a different method in ref. [4]. The theoretical results obtained in the (e 2, ~4) approximation at a field h = 0.01 for the magnetization and the specific heat are reported in figs. 1 and 2, respectively, together with numerical transfer matrix data [6] obtained with the method of refs. [2,8]. The comparison is quite satisfactory, even in the temperature region where the H a r t r e e - F o c k perturbative approach fails complete-
438
30 July 1984
We wish to thank Dr. M.G. Pini for having furnished unpublished transfer matrix data.
References [1 ] J. Bernasconi and T. Schneider, eds., Physics in one dimension (Springer, Berlin, 1981). [2] U. Balucani, M.G. Pini, V. Tognetti and A. Rettori, Phys. Rev. B26 (1982) 4974. [3] M. Blume, P. Heller and N.A. Lurie, Phys. Rev. B11 (1975) 4483. [4] U. Balucani, M.G. Pini, A. Rettori and V. Tognetti, Phys. Rev. B29 (1984) 1517. [5] M.E. Fisher, Am. J. Phys. 32 (1967) 343. [6] M.G. Pini, private communication. [7] A.R. McGurn and D.J. Scalapino, Phys. Rev. Bll (1975) 2552. [8] V. Tognetti, A. Rettori, M.G. Pini, J.M. Loveluck, U. Balucani and E. Balcar, J. Phys. C16 (1983) 5641.
PHYSICS LETTERS
30 July 1984
MAGNETIZATION AND SPECIFIC HEAT OF CLASSICAL FERROMAGNETIC CHAINS IN A FIELD: CROSSOVER BETWEEN TWO REGIMES U. BALUCANI Istituto di Elettronica Quantistica del CNR, Florence, Italy V. TOGNETTI 1 and R. VAIA Dipartimento di Fisiea dell'Univers#d, Florence, Italy Received 17 April 1984 Revised manuscript received 5 June 1984
A theoretical approach, previously introduced to treat the statistical mechanics of a classical ferromagnetic chain in a field, has been extended in a systematic way to give the crossover from low-field to spin-waveregime. The results are in quantitative agreement with numerical transfer matrix data.
One-dimensional magnetic systems have received great attention in the last years [ 1]. Their most interesting feature is the lack of long range order ((S z) = O) for finite temperatures in the absence of a magnetic field. The usual many-body perturbative approach in terms of interacting spin waves was proved to be valid when the ordering effect of the applied magnetic field H overcomes the thermal disorder [2]. On the other hand, for vanishing magnetic fields, the presence of unpleasant divergencies points out the necessity of non-perturbative methods to calculate both static and dynamical quantities. In the classical case, the data obtained numerically by transfer-matrix techniques [2,3] give a test for the validity of any theory of the thermodynamical properties. For the isotropic Heisenberg classical chain, we have previously shown [4] that a consistent scheme can be set up to calculate the partition function in order to reproduce, at least at low temperatures, the two limiting regimes, i.e. both the standard spinwaves and the H = 0 case where exact analytical expressions are available [5]. In particular, a zero magnetization for vanishing field was obtained for finite
temperatures, just taking into full account the anharmonic terms. In this paper we present quantitative results for the magnetization and specific heat of the classical Heisenberg magnetic chain after having improved the previous formalism introducing a simpler and more extendible mathematical framework. The present approach permits, in principle, to build up a rigorous expansion which interpolates both the limiting situations and whose results can be compared with transfer matrix data. Moreover, the correct vanishing magnetization for H ~ 0 is analytically obtained also in the presence of a small exchange anisotropy. Let us consider the following hamfltonian N-1 j=l
[ i~ u ( S } S+m
-
÷ s j - s j +÷ , ) ÷ s j s jz+ l zl
N
- gUB~l.~ S[,
(1)
]°= l
where N classical "spin" vectors Sj interact through the J > 0 exchange integrals and can be parametrized by [41
i Also with Unit~ di Firenze del GNSM-CNR, Italy.
0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North.Holland Physics Publishing Division)
435
S1 = 2S[vi(1 - vi)]l/2exp(±i¢]), S]z = S(1 - 2v/). For/l = 1 the isotropic case is recovered. In the following we shall use normalized units for magnetic field and temperature
h = glaBH/JS,
T* = [3*-1 = kB T/JS 2"
The partition function Z N turns out to be [4]
Z N = exp~*[N(1 + h ) 1
30 July 1984
PHYSICS LETTERS
Volume 103A, number 9
1])
1
X f dv 1 ... f dVNexp {-2~* [(1 +h) v 1 - VN] } 0 0 N-1
X ~ A (vi, vi+1), i=1
(2)
with
From eqs. (4), (5) one can see that at low temperatures ~* >> 1) the negative exponential in sinh x can be neglected for any value of the magnetic field, indicating that the "kinematic" interaction is significant only at higher temperatures (where it gives rise to activation terms like e-l/T*). Furthermore, the expansion of R (h, la, VN_ 1) in terms of powers of VN_ 1 can be performed both for small anharmonicity (spin wave regime, SW) and in the case of low field (LF, h < 1) and small anisotropy ((1 -/~2) ,~ 1). Such an approximation permits one to compute iteratively the preceding integrals, and finally to get an expression for Z N that describes the crossover between the SW and LF regimes. By considering an approximation involving up to the nth power of v, one deals with successive integrals like 1
A (v', v) = exp [-2[3*(2 + h - 2ev') v] 21r
n
f dvA(v',v)exp(2[3*~v) 0
21t × f 0 x
k~=Oakv k
n
dTexp{4[3*~[vv'(1 - ev)(1 -
ev')l t/2
cos-t}.
~ exp(2[3* q/v')
(3)
A perturbative parameter e (e = 1 in the real system) has been introduced to measure the effect of the anharmonic interactions. The integral in the variable vN is straightforward: 1
ak v'k ,
(6)
where qJ' = #2/(2 + h - ~). In the thermodynamic limit this leads to a continued fraction for the q/s, converging to the value = 1'- ~(K - h),
(7)
with
K = [h(h + 4) + 4(1 - At2)] 1/2.
f d vN A (vN_ 1, vN) exp (2 [3*vN ) 0
The coefficients a~ depend linearly on the ak : n
= exp[-(fl*/e)(1 +h - 2eVN_l) ]
aj : k~=OMlk (T*, h, U) ak,
(7')
X sinh[[3*(1 + h ) e - l R ( h , / & vU_l) ] X [/~*(1 +h)R(h,la, VN_I)] -1,
(4)
with
R(h, la, VN_l) = [ 1 - 4
-u s +4(11 +h)2
436
Z N "" (exp [[3"(1 + h)] h) N.
1 + h - / a 2 eVN 1 (I + h) 2 -
1/2
and in the thermodynamic limit the leading contribution to Z N is given by the Nth power of the maximum eigenvalue X = X(T*, h,/a) of the (n + 1) X (n + 1) matrix M:
(5)
(8)
The ruth column of M is associated with the coefficients of the expansion in powers of v' of the general integral
1
Jm(V') = f
d v A ( v ' , v)exp(2/3*~Ov) v m
o rl
"" exp(2~*~ v') k~__OMk m v 'k.
(9)
Analytical expressions for the integrals Jm can easily be obtained from the low temperature result j0(/f) = ~(C 1 2 + B) -1/2 exp ~e-1 [ - B + (C 2 + B E)l/2] }, (lO)
with B =#*0a2/~ - 2ev'),
30 July 1984
PHYSICS LETTERS
Volume 103A, number 9
C - 2~*p[ev'(1 - ev')] 1/2,
noting that Jm (v') = ( - ~blbB) m Jo(v').
(11)
The expansion to be performed in such expressions involves halfinteger powers of c 2 +a 2 - (~*u2/~) 2
× [1-4(~//a2)~ev'+4(ff2/p4)52e2v'2],
(12)
where we have set ~ = 1 - ff -_ ] ( K _ h) and 52 = 1 - / a 2, which are small parameters in the LF (~ ~ (h + 52) 1/2) and low anisotropy situation. Note that 52 can be negative in the case of planar exchange anisotropy. By referring to the parameters e (to be considered small in the SW regime), ~ and 52, rather to v', one gets better control on the accuracy of the expansion and has the possibility of constructing "mixed" approximations. For the sake of simplicity, we shall consider in the following a vanishing anisotropy (6 2 - 0). Furthermore, we shall indicate an approximation procedure correct up to terms en and ~rn as a (en, ~m)-theory: in such a theory, only terms of order equal or higher than the product e n+l ~m+l can be disregarded. One can easily see that: (a) a theory correct up to terms of order e n requires expansions up to the 2nth power of v'; (b) a theory correct up to terms of order ~n requires expansions up to the nth power of v'. The natural interpolating theories will then be (e n, ~2n) and (e n , ~2n+l)-theories, respectively involving expansions up to v 2n and v 2n+l . We can equivalently say that the coefficients a2n and a2n+l are of order
en~ 2n and en+l~ 2n+l. As a consequence, one needs a lower accuracy in the expansion procedure of the integrals Jm, because in eq. (6) they already are multiplied by a factor a m . The next columns of the matrix M will then be computed with decreasing accuracy. This "graded" structure of M makes possible a computation of the maximum eigenvalue ~ by means of increasing order contributions, which are obtained as solutions of linear equations (in principle, ~ is a solution of det(M - 20 = 0, a secular equation of degree equal to the order of M). Indeed, suppose we know a (e n - l , ~2(n-1)) value of ?~0 for ~. The following contribution x is of order en~ 2 n - l , and will be recovered in the (e n, ~2n-1)-theory. Putting ?~= ?'0 +x, only the x in the first column of M - 2~survives, because in the o t h er columns quantities of order en~ 2n-1 can be neglected: then the secular equation det (M - ?0 = 0 becomes linear in the unknown variable x. In a similar way, one gets the next en~ 2n correction, and thus, the (e n, ~2n)-value for ~,. Analytical solutions for ?, can easily be obtained in the first lower order theories, but this iterative procedure is very useful also in the numerical computations. In the (e 0, ~3)-theory (LF and low anisotropy limit) we obtain the analytical result ? ~ = ~ T * [ 1 - / ~ * h - 5 (1f l • - 1 ) 5 2].
(13)
By eq. (8) one correctly arrives at (S z) = 0 + O(h).
0.8
h = 0.01
0.6-
~.~" ?
0.4.
"?':"-.
0.2
~
0
0
0".1
~ "'.
0~2
"'"
T*
Fig. 1. Magnetization of the classicalferromagnetic chain in a field h = 0.01 as a function of temperature. The full line is the theoretical result at order e 2 ,/~4, the dashed line is the result of the lower-order (e, f2) approximation, and the dotted
line derives from a Hartree-Fock perturbative approach. Open circles represent the numerical transfer-matrix data. 437
PHYSICS LETTERS
Volume 103A, number 9 C ~'~.
h -
ly. The results of the simpler (e, ~2)-theory are also reported, showing the degree of improvement obtained by raising the order of the approximation. It is worthwhile to note that our treatment is essentially valid down to very low fields, where interacting spin-wave theories fail, and at the same time to very low temperatures, where standard transfer matrix techniques meet problems of numerical accuracy. In this region, our results approach those obtained numerically within the continuum approximation [7].
0.01
o o
1.06
o
lO,
"
1.02
1.00
..
/
o
~ -"~
o:1
o12
T*
Fig. 2. Magnetic specific heat of the classical ferromagnetic chain in a field h = 0.01 as a function of temperature. All curves and data have the same meaning as in fig. 1.
Another analytical solution can straightforwardly be obtained in the (e, ~2)-theory, which in the SW case accounts for the perturbative H a r t r e e - F o c k correction. Both these limiting results were obtained by a different method in ref. [4]. The theoretical results obtained in the (e 2, ~4) approximation at a field h = 0.01 for the magnetization and the specific heat are reported in figs. 1 and 2, respectively, together with numerical transfer matrix data [6] obtained with the method of refs. [2,8]. The comparison is quite satisfactory, even in the temperature region where the H a r t r e e - F o c k perturbative approach fails complete-
438
30 July 1984
We wish to thank Dr. M.G. Pini for having furnished unpublished transfer matrix data.
References [1 ] J. Bernasconi and T. Schneider, eds., Physics in one dimension (Springer, Berlin, 1981). [2] U. Balucani, M.G. Pini, V. Tognetti and A. Rettori, Phys. Rev. B26 (1982) 4974. [3] M. Blume, P. Heller and N.A. Lurie, Phys. Rev. B11 (1975) 4483. [4] U. Balucani, M.G. Pini, A. Rettori and V. Tognetti, Phys. Rev. B29 (1984) 1517. [5] M.E. Fisher, Am. J. Phys. 32 (1967) 343. [6] M.G. Pini, private communication. [7] A.R. McGurn and D.J. Scalapino, Phys. Rev. Bll (1975) 2552. [8] V. Tognetti, A. Rettori, M.G. Pini, J.M. Loveluck, U. Balucani and E. Balcar, J. Phys. C16 (1983) 5641.