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Маслов ИНТРОДУЦТИОН ТО ПХЫСИЦС ОФ СЕЦОНД-ОРДЕР МАГНЕТИЦ 2015

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This is the sum of diagonal elements of 2 ×2 matrix. In other words Q =Tr (AN ). So, knowing the matrix A, we should find the matrix AN

and add up its diagonal elements. Taking into account that the trace of square n-by-n matrix A is Tr (Ak )= ∑λik , where λi are the eigenvalues

of A, the Q can described as the eigenvalues sum, i.e., as the sum of two elements. Let us find them

−

T −λ

= 0;

det

−

−λ

λ2 −λe

−λe

−

+ e T −e−

T = 0;

−

λ2 −λeT

eT + e T

+ e T

−e

= 0;

(23.9)

−

− 2λeT cosh

+ e T

−e

= 0;

cosh2

−

= eT cosh

−1

+ e

1,2

Let us

analyze

the

latter

expression. Since

0 <1−e− T <1

( J > 0,

−e

−4J

0 <T < +∞) and

cosh

≥ 0 , therefore 1

< cosh

. So, the

term

under

the

square

root

positive.

Next,

cosh2

−1+ e−

< cosh2

, therefore the both λ

and λ

are pos-

N 1, for the Q we ob-

itive. Considering that λ1 ≡ λ+

> λ2

≡ λ−

and

tain

Q = λN

+ λN = λN

λ2

(23.10)

≈ λN .

Therefore the Helmholtz free energy is equal to

F = −T ln Q = −TN ln l1 =

−

= −TN ln eT

cosh

−1

+ e

= (23.11)

−

= −NJ −TN ln cosh

cosh

−1+ e

T .

Let us find the magnetization. Recall that the expression for the differ-

ential of free energy is

H = Hez ,

then

M = Mez

= −MdH.

well. So, the magnetization can be described as

M = −

dF = −µB dF .

(23.12)

Nµ2

V dH

Since M

x , see Eq. (12.6), the order parameter takes the form

dF .

x = −

(23.13)

N dh

Considering the Eq. (23.11), let us calculate it

cosh

sinh

TeT

sinh

−

d (ln l )

cosh

−1

+ e

x =T

−4J

cosh

−1+ e T

+ e

−

sinh

cosh

−1

+ e

−

2 h

−1+ e

−

cosh

−1

cosh

Finally, for the order parameter we obtain

sinh

x =

(23.14)

sinh

+ e

−

Let us analyze the resulting expression.

1. Without the interactions between the magnetic moments ( J = 0 )

	h
we can rewrite the formula for the order parameter as	x = tanh	.

	T

This result we already know. The Curie law follows from it.

2. If J ≠ 0 and h →0 the order parameter is equal to zero at any temperature T > 0 .

Therefore the ferromagnetic transition in the one-dimensional Ising model with interactions only between the nearest neighbors is absent! In other words the one-dimensional chain of magnetic moments with interactions only between the nearest neighbors is paramagnet at any temperature T ≠ 0 and exchange integral J ≠ 0 . So, the mean-field theory gives a qualitatively wrong result. The reason is the small number of the nearest neighbors in the chain z = 2 , so, the fluctuations are very high. Finally, note that if we apply long-range exchange potential between magnetic moments in the one-dimensional Ising model, the ferromagnetic phase transition will be occur.

24. Short-range and long-range orders. Correlations. Fluctuations

In the Ising model, the order parameter is defined as a thermodynamic average of projection of the dimensionless atomic magnetic moment

σi = µi µB on z-axis, σiz T (see § 13). In the case of ferromagnetic ordering the value of σiz T ≡ x does not depend on i, vanishing at T > θ and being nonzero at T < θ. As for the average value of the product

σiz σjz T , in the mean field approximation (that is, neglecting fluctua-

							, so that
tions), we had σiz σjz = σiz		σjz	T + σiz	T σjz − σiz T		σjz T	, so that

				0,	T > θ;	(24.1)
σiz σjz		= σiz	σjz	=	, T < θ	(24.1)
	T		T	T x2	, T < θ

for any pair (i, j ≠ i) of magnetic moments.

Let us look on this issue from somewhat different point of view. To gain insight into the behavior of σiz σjz T at T < θ, consider for clarity

sake the limiting case T = 0. In this limit, the state of the system is doubly degenerate (see § 11), all magnetic moments being directed either

“up” or		“down”, so that				for any i and hence
					σiz T = +1 or –1
		= 1 for any (i, j), in accordance with the mean field result,
σiz σjz	T
Eq. (24.1). In terms of				probabilities P↑↑ , P↑↓ ,		P↓↑ , P↓↓ of magnetic
moments		orientations		we	have either P↑↑ =1,	P↓↓ = P↑↓ =P↓↑ = 0 or
P↓↓ =1,	P↑↑ = P↑↓ =P↓↑ = 0 , and the value of
			= (+1)(+1)P↑↑		+ (−1)(−1)P↓↓ + (+1)(−1)P↑↓ + (−1)(+1)P↓↑
σiz σjz
			T

equals to 1 in both cases. The physical reason is that at T < θ the value

of		is not only nonzero but always equals to the value of
	σjz T		σiz T .

If the value of one physical quantity ( σjz T in our case) depends on the value of another physical quantity ( σiz T in our case), those quantities

are said to be correlated. So, in the ferromagnetic state there is an intimate correlation between the orientations of different magnetic mo-

ments. It is worth noting that below the Curie temperature σiz σjz T ≠ 0

even if the distance Ri − R j between i-th and j-th moments greatly exceeds the interatomic spacing. Hence, there is a long-range order in ori-

entation of magnetic moments. Since the value of		for the
	σiz σjz T

nearest moments is also nonzero, there is a short-range order as well. Here is a close analogy with arrangement of atoms in the crystalline lattice.

Let us turn to the case T > θ. Now σiz T = 0 for any i because of equal probabilities of “up” and “down” orientation of every magnetic moment, P↑ = P↓ = 12 (no ferromagnetic ordering). As for a pair of dis-

tant moments, in the absence of long-range order four types of their mutual orientation can occur with equal probabilities 1/4, so that

σiz σjz T = (+1)(+1) 14 + (−1)(−1) 14 + (+1)(−1) 14 + (−1)(+1) 14 = 0

for any (i, j ≠ i), in accordance with the mean field result, Eq.(24.1). It

may appear at first sight that at T > θ there are no correlations between the nearest moments as well. Note however that once in a ferromagnet the exchange energy Jij for the nearest neighbor atoms is positive, it is energetically favorable for two nearest magnetic moments to be oriented in one direction. Hence, we should have

P↑↑ > 14 , P↓↓ > 14 , P↑↓ < 14 , P↓↑ < 14 ,

that is, σiz σjz T ≠ 0 . In other words, one might expect that at T > θ, in

spite of the absence of long-range ferromagnetic order, there are nevertheless short-range ferromagnetic correlations which the mean field approximation fails to describe. Below we shall see that this is so indeed.

25. Correlation function of a ferromagnet in the Ising model

Now our purpose is to study ferromagnetic correlations in the Ising model beyond the mean field approximation. Let us consider two mag-

netic moments, µ1 and µ2 (not necessarily nearest neighbors) situated at atoms “1” and “2” with coordinates R1 and R2 respectively. The

thermodynamic average σ1z σ2 z T is called the correlation function. In

a crystal with regular atomic arrangement it depends on a relative position of two atoms:

				(25.1)
g(R1	− R2 ) = σ1z σ2z .			(25.1)
			T
In the mean field approximation we had
		0, T > θ;		(25.2)
g(R1	− R2 ) =		, T < θ	(25.2)
		x2	, T < θ
		is the order parameter, see § 11.
for any R1 ≠ R2 , where x = σiz		is the order parameter, see § 11.
	T

To begin with consider the case T > θ where we expect the qualitative modifications to the mean field approximation. Note that for a sys-

		N
tem of N magnetic moments the average σ1z σ2z T		is taken over 2

configurations. In one half of those configurations the moment µ1 is

directed “up” (that is, the eigenvalue σ1z of the operator σ1z equals to +1), while in another half it is directed “down” (σ1z = –1), and hence the

	can be represented as
average σ1z σ2z T	can be represented as
				1			1		,	(25.3)
	σ1z σ2z	T	=	2	σ1z σ2z +	+	2	σ1z σ2z −	,	(25.3)

where the symbols ...+ and ...− stand for the thermodynamic averag-

es over 2N-1 configurations with the fixed values of σ1z = +1 and σ1z = –1 respectively. Since in zero magnetic field the Hamiltonian and the value

	are invariant with the direction of the z axis (as z → –z,
of σ1z σ2z T	are invariant with the direction of the z axis (as z → –z,
σiz → –σiz for every i), we have						− , that is
σiz → –σiz for every i), we have			σ1z σ2z	+	= σ1z σ2z	− , that is
						(25.4)
	g(R1	− R2 ) = σ1z σ2z .				(25.4)
					+

By virtue of the fact that σ1z = +1 in all configurations involved, from Eq. (25.4) one has


				Tr+ σ2z exp(−H / T )
g(R1	− R2 ) =	σ2z +	=			,	(25.5)
					/ T )
				Tr+ exp(−H

where the symbol Tr+ implies summation over the diagonal matrix elements calculated for the states with σ1z = +1 only.

Let us now rewrite the Hamiltonian of the Ising model, Eq. (10.7), separating out the terms that account for interactions of the magnetic moment µ2 with all other moments:

		1	∑	/		(25.6)
H	= −∑J2 j σ2z σjz −		∑		Jij σiz σjz	(25.6)
	j≠2	2 i, j≠2

(the external magnetic field H = 0). Next we apply the mean field approximation to the first part of Eq. (25.6), leaving the second one intact:

−

H = −∑J2 j

σ2z

σjz

+ σjz

σ2z

− σ2z

σjz

j≠2

(25.7)

∑

−

Jij σiz σjz .

2 i, j≠2

In doing so, we neglect only the interactions between the fluctuations of the magnetic moment µ2 and fluctuations of other moments, while the

interactions between the fluctuations of all remaining moments are tak-

en	into	account without any approximations. Let us write
	(2)
H	= H	+ ∆H , where

	(2)				(25.8)
	H	= −∑J2 j σjz		σ2z	(25.8)
		j≠2		+
		j≠2

and ∆H	is the rest of H . Considering that			H does not contain the op-
			in Eq. (25.5) we find
erator σ2 z , for the average		σ2z +	in Eq. (25.5) we find

			(2)
		Tr+ σ2z exp(−H			/ T )Tr+ exp(−∆H / T )
σ2z +	=							=
σ2z +	=		(2)					=
		Tr+ exp(−H		/ T )Tr+ exp(−∆H / T )				(25.9)
					(2)		/ T )	(25.9)
					(2)
		=	Tr+ σ2z exp(−H
		=			(2)	/ T )
					(2)
			Tr+ exp(−H

Designating

	,	(25.10)
h′ = ∑J2 j σjz
j≠2	+

(2)

we see that H can be viewed as the Hamiltonian of the single magnetic moment µ2 placed in an effective magnetic field H / = H / ez where H / =h′/ µB , see Eq. (25.10). By analogy with an isolated magnet-

ic moment in the field H = Hez , see § 4, we come up with the expression

h′

∑J2 j σjz

j≠2

σ2z = tanh

= tanh

(25.11)

Because

the

averages

are

taken

over configurations

with

σjz

σ1z = +1, one has

(25.12)

σjz

= σ1z σjz +

Recalling

the

definition

of the

correlation

function

g(R1 − R2 ) ,

see

Eq. (25.4), and taking into account Eqs. (25.5), (25.11) and (25.12), we arrive at the following expression:

			1
g(R1	− R2 ) = tanh		1	∑J (Rj − R2 )g(R1	− Rj ) .	(25.13)
g(R1	− R2 ) = tanh			∑J (Rj − R2 )g(R1	− Rj ) .	(25.13)
		T		j≠2

Writing down similar expressions for g(R1 − Rj ) at j ≠ 2, we obtain a

set of N coupled equations for correlation functions g(R1 − Ri ) with

i = 2, 3, ..., N .

The system of nonlinear equations (25.13) has no exact analytical solution. Let us find an approximate expression for g(R1 − R2 ) in the case that the distance R1 − R2 between the magnetic moments greatly ex-

ceeds the interatomic spacing a. In this case, at least one of two strong inequalities, R1 − Rj >> a and Rj − R2 >> a , should hold for any j, so

that either g(R1 − Rj ) <<1 (because there is no long range magnetic or-

der at T > θ) or J (Rj − R2 ) is exponentially small (see § 8), or combina-

tion of both. As a consequence, the argument of hyperbolic tangent in the right-hand side of Eq. (25.13) is much less than unity, and one can restrict himself to the first term in a Taylor series tanh(z) = z + … :

		1
g(R1	− R2 ) =		∑J (Rj − R2 )g(R1	− Rj ) .	(25.14)
g(R1	− R2 ) =	T	∑J (Rj − R2 )g(R1	− Rj ) .	(25.14)
		T	j≠2
Denoting R1 − R2 = R and Rj − R2 = R′, we have
		1
	g(R) =		∑J (R′)g(R − R′) .		(25.15)
	g(R) =	T	∑J (R′)g(R − R′) .		(25.15)
		T	R′≠0

It only remains for us to supplement Eq. (25.15) with a boundary condition at R = 0 . Since g(R = 0) = σiz σiz T =1 for any T, one can approximately write

	δ	1	δ	δ δ
	g(R) =		∑J (R′)g(R − R′) + δRδ,0		,	(25.16)
	g(R) =	T	∑J (R′)g(R − R′) + δRδ,0		,	(25.16)
		T	Rδ′≠0
where δδ	= 1 and 0 at R =		0 and R ≠ 0 respectively.
R,0

Expanding the functions g(R) and J (R) into the Fourier series,

	1
g(R) =	1	∑g			(q)exp(iqR);
g(R) =	N	∑g			(q)exp(iqR);
	N		q
	1
J (R) =	1	∑J			(q)exp(iqR),
J (R) =	N	∑J			(q)exp(iqR),
	N		q
from Eq.(25.16) we have				1
				1
g(q) =1			+			J	(q)g(q) ,
g(q) =1			+	T		J	(q)g(q) ,
				T

whence it follows that at T > θ

g(q) = 1−β1J (q) ,

(25.17)

(25.18)

(25.19)

where β = 1/T. We also present without proof the expression for g(q) at T < θ:

g(qδ) = Nx2δδ	+	1− x2	δ	,	(25.20)

q,0		1−β(1− x2 )J (q)

where x is the temperature dependent order parameter. One can see that at T → θ (that is, at x → 0) Eq. (25.20) transforms into Eq. (25.19).

With g(q) at hand, it	is straightforward to calculate g(R) , see
Eq. (25.17). However first we should find				J (q) . Since we are interested
mainly in behavior of g(R)			>> a ,
	at	R		the major contribution to the

sum over q in Eq. (25.17) comes from the region of small q. Expanding

J (q) in powers of qR and restricting ourselves to quadratic terms we have

				(−iqR)2	. (25.21)
J (q) = ∑J (R)exp(−iqR) = ∑J (R) 1			−iqR +	2	. (25.21)
R	≠0	R≠0		2
Inasmuch as	∑J (R)	is, by definition, the Curie temperature θ (recall
	R≠0

that we set kB = 1), and the second term in Eq. (25.21) equals to zero from the symmetry considerations, we arrive at

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