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Absence of magnetic phase transition in one- and two-dimensional s-d interaction models
Volume 50A, number 2
PHYSICS LETTERS
18 November 1974
ABSENCE OF MAGNETIC PHASE TRANSITION IN ONE- AND TWO-DIMENSIONAL s-d INTERACfION MODELS M. Van den BERGH and G. VERTOGEN Institute for Theoretical Physics, University of Groningen, the Netherlands Received 1 October 1974 By means of the Bogoliubov inequality it is proved that the s-d interaction between conduction electrons and localized magnetic moments does not give rise to a spontaneous magnetic ordering in one- or two-dimensional systems at finite temperatures.
An important part in the theory of magnetism in metals is played by the concept of an indirect exchange interaction between the localized magnetic moments. This effective interaction is brought about by the s-d or s-f interaction of the localized magnetic moments with the conduction electrons. A well-known expression for the indirect exchange interaction is the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [1], which is derived by second order perturbation theory. Although the validity of the RKKY expression is highly questionable because of the appearance of divergences in higher order terms of the perturbation expansion [21,the itinerancy of the conduction electrons undoubtedly induces some sort of effective long-range interaction between the localized magnetic moments. Recently interest has arisen in the possible existence of two-dimensional or even one-dimensional magnetic systems. Therefore it is worthwile to investigate whether the s-d interaction can provide a possible mechanism for the appearance of a spontaneous magnetic ordering at fmite temperatures in one or two dimensions. Starting with 1966 several papers have been published concerning the absence of magnetic ordering in various models. These papers were all based on an inequality due to Bogoliubov. Mermin and Wagner [3] proved the absence of ferromagnetism and antiferromagnetism in one- and two-dimensional Heisenberg models. Wegner [4] showed the absence of ferromagnetism in one- and two-dimensional systems consisting of locally interacting electrons and nuclei under the condition [C,V1 = 0, where V describes the interaction of the electrons with the nuclei and among themselves, while C = E~exp (ik’ r1) (SI + iS3f) with r1 and S1 denoting the position and spin of electron 1. As shown by Walker and Ruygrok [5] the results of Wegner can be extended to systems with non-local potentials, which do not satisfy Wegner’s criterion. In this paper we demonstrate the absence of ferromagnetism and antiferromagnetism in one- and two-dimensional s-d models. The s-d interaction also does not satisfy Wegner’s criterion as can be seen easily. Because of the fact that we study a system composed of two subsystems the proof is slightly more complicated due to the fact that we have to show that both the conduction electron system and the system of localized magnetic moments do not exhibit a magnetic phase transition. The s-d Hamiltonian reads Hs..d
=
kCkoCk0
—
~
J(k—k’)exp [i(k—k’)~]
Nk,k’,n
X ~
(1)
where N is the total number of lattice points, R~represents the position of the localized magnetic moments S,~, the summation index n runs over the sites of the Bravais lattice with the usual periodic boundary conditions, J(k—k’) is the exchange integral between a conduction electron and a d-(or f-) core spin, k is the wave vector of a conduction electron and the summation over k extends over the complete conduction band. Changing to a local representation by means of ~ we obtain
4
85
Volume 50A, number 2
PHYSICS LETFERS
2I~T(Ri_R,n)40cm0
Hsd
l,m,u
18 November 1974
J(R1 _Rm)[(c~ci+—c~_c1_)S,~ + C,~Cl_S~+c~_ci+S~1,
—~
(2)
1,m
With T(R, Rm)= T(Rm R1) =(l/N) ~k~k exp [ik’(Ri_Rm)J and J(Ri_Rm)=J(Rm _R1)(l/N) ~kJ(k) X exp [ik’(R, R,~)].We now add to the s-d Hamitonian Zeeman R1)pe and p 3e and j3 arethe defined byterm pe =H~ 13e5e —h and~p =exp (35,(—iq. where X [13~S7~ +i3Sfl withthe S7~ ~ (c’÷moment c1÷—c~_c1_); denote respectively magnetic of the‘ electron and the localized spin. Taking S7~= c~÷ c 1_ and S7 = c1÷we easily verify that [57÷, S7._] = 2~ll.S~and [S7~ S7’~]= ± S7... The Fourier components of the appearing angular momentum operators S7 and S~are given by —
—
—
,
st:) (k)
exp [—ik’R1]~
=
(a = +,
—,
~‘
z).
k
Because we are interested in both the magnetization of the electron gas and the system of localized magnetic moments we consider the following two quantities exp(—iq.R1)
E13e
(Sf~)+l3(S~)],
(3a)
a=~ ~exp(_iq~R1)[I3e(Sfz>_I3(S~)].
(3b)
It is clear that a = = 0 imply the absence of magnetization of both the electron system and the localized magnetic moment system. In order to estimate a and ä~we make use of the Bogoliubov inequality 2([C,H],C’~])~ ~<{A,A~}>. (4) 2kBT([C,A]) First we take the following operators C and A
C=S~(—k)+S_(—k),
A =i3~S~(k+q)+f3S÷(k+q).
(5)
After some calculations we find <[C,HSd],C])=~
{exp[ik•(Rj—R~)] _1}T(Ri_Rm)](c~_Cm_+c,~+ci+>+
1, m
—2 ~ {exp [ik’(R
1 —R,~)]—1}J(R1 _Rm){2(S7zS~>+ 2(s~~ Sf> + (S7S~>+ (S~+S~},
(6)
1, m
([C,H~],C~])2hNa,
([C,A]>—2Na.
Since each lattice point is a center of symmetry the factors {exp [ik’ (R1 — Rm)] —1 } can be replaced by ~cos[k’ (R1 — Rm)] 1 }. After some calculations we find 2(a+b) (7) I([C,H5d],C~])I~MkI with a = 2 E 21T(R 2J(R,).Obviously it is essential that both a and b andthat b = 4{S~(5~ + 1)+S(S + w l)} ~1IR,I converge. It11R11 is easy to 1)I show ~k({A, A~}>
IaI
(1 dimension),
Repeating the whole procedure with 86
IaI
Volume 50A, number 2
C=Sej_k) +S_ (—k),
PHYSICS LETTERS
18 November 1974
A =j3~S÷e(k+q) — (3S÷(k+q)
we obtain for sufficiently small fields IâI
I~I
(9) (2 dimensions).
(lOa,b)
The conclusion is obvious: both a and~tend to zero in the limit h -÷0,i.e. there is no spontaneous magnetization in one and two dimensions for s-d interaction models at any non-zero temperature provided the energy band structure and the exchange integrals are such that both quantitiesa and b converge. Finally it should be remarked that we can easily incorporate into our s-d model the interactions considered by the previous authors. Our conclusion, however, still holds: no spontaneous magnetization will occur at finite temperatures.
References [1] M.A. Ruderman and C. Kittel, Phys. Rev. 96(1954)99; T. Kasuya, Progr. Theor. Phys. 16 (1956) 45; K. Yosida, Phys. Rev. 106 (1957) 893. [2] G. Vertogen, Phys. Stat. So!. 25 (1968) 721. [3] N.D. Mermin and H. Wagner, Phys. Rev. Letters 17 (1966) 1133. [4] F. Wegner, Phys. Letters 24 (1967) 131. [5] M.B. Walker and Th.W. Ruygrok, PISyS. Rev. 171 (2) (1968) 513.
87
PHYSICS LETTERS
18 November 1974
ABSENCE OF MAGNETIC PHASE TRANSITION IN ONE- AND TWO-DIMENSIONAL s-d INTERACfION MODELS M. Van den BERGH and G. VERTOGEN Institute for Theoretical Physics, University of Groningen, the Netherlands Received 1 October 1974 By means of the Bogoliubov inequality it is proved that the s-d interaction between conduction electrons and localized magnetic moments does not give rise to a spontaneous magnetic ordering in one- or two-dimensional systems at finite temperatures.
An important part in the theory of magnetism in metals is played by the concept of an indirect exchange interaction between the localized magnetic moments. This effective interaction is brought about by the s-d or s-f interaction of the localized magnetic moments with the conduction electrons. A well-known expression for the indirect exchange interaction is the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [1], which is derived by second order perturbation theory. Although the validity of the RKKY expression is highly questionable because of the appearance of divergences in higher order terms of the perturbation expansion [21,the itinerancy of the conduction electrons undoubtedly induces some sort of effective long-range interaction between the localized magnetic moments. Recently interest has arisen in the possible existence of two-dimensional or even one-dimensional magnetic systems. Therefore it is worthwile to investigate whether the s-d interaction can provide a possible mechanism for the appearance of a spontaneous magnetic ordering at fmite temperatures in one or two dimensions. Starting with 1966 several papers have been published concerning the absence of magnetic ordering in various models. These papers were all based on an inequality due to Bogoliubov. Mermin and Wagner [3] proved the absence of ferromagnetism and antiferromagnetism in one- and two-dimensional Heisenberg models. Wegner [4] showed the absence of ferromagnetism in one- and two-dimensional systems consisting of locally interacting electrons and nuclei under the condition [C,V1 = 0, where V describes the interaction of the electrons with the nuclei and among themselves, while C = E~exp (ik’ r1) (SI + iS3f) with r1 and S1 denoting the position and spin of electron 1. As shown by Walker and Ruygrok [5] the results of Wegner can be extended to systems with non-local potentials, which do not satisfy Wegner’s criterion. In this paper we demonstrate the absence of ferromagnetism and antiferromagnetism in one- and two-dimensional s-d models. The s-d interaction also does not satisfy Wegner’s criterion as can be seen easily. Because of the fact that we study a system composed of two subsystems the proof is slightly more complicated due to the fact that we have to show that both the conduction electron system and the system of localized magnetic moments do not exhibit a magnetic phase transition. The s-d Hamiltonian reads Hs..d
=
kCkoCk0
—
~
J(k—k’)exp [i(k—k’)~]
Nk,k’,n
X ~
(1)
where N is the total number of lattice points, R~represents the position of the localized magnetic moments S,~, the summation index n runs over the sites of the Bravais lattice with the usual periodic boundary conditions, J(k—k’) is the exchange integral between a conduction electron and a d-(or f-) core spin, k is the wave vector of a conduction electron and the summation over k extends over the complete conduction band. Changing to a local representation by means of ~ we obtain
4
85
Volume 50A, number 2
PHYSICS LETFERS
2I~T(Ri_R,n)40cm0
Hsd
l,m,u
18 November 1974
J(R1 _Rm)[(c~ci+—c~_c1_)S,~ + C,~Cl_S~+c~_ci+S~1,
—~
(2)
1,m
With T(R, Rm)= T(Rm R1) =(l/N) ~k~k exp [ik’(Ri_Rm)J and J(Ri_Rm)=J(Rm _R1)(l/N) ~kJ(k) X exp [ik’(R, R,~)].We now add to the s-d Hamitonian Zeeman R1)pe and p 3e and j3 arethe defined byterm pe =H~ 13e5e —h and~p =exp (35,(—iq. where X [13~S7~ +i3Sfl withthe S7~ ~ (c’÷moment c1÷—c~_c1_); denote respectively magnetic of the‘ electron and the localized spin. Taking S7~= c~÷ c 1_ and S7 = c1÷we easily verify that [57÷, S7._] = 2~ll.S~and [S7~ S7’~]= ± S7... The Fourier components of the appearing angular momentum operators S7 and S~are given by —
—
—
,
st:) (k)
exp [—ik’R1]~
=
(a = +,
—,
~‘
z).
k
Because we are interested in both the magnetization of the electron gas and the system of localized magnetic moments we consider the following two quantities exp(—iq.R1)
E13e
(Sf~)+l3(S~)],
(3a)
a=~ ~exp(_iq~R1)[I3e(Sfz>_I3(S~)].
(3b)
It is clear that a = = 0 imply the absence of magnetization of both the electron system and the localized magnetic moment system. In order to estimate a and ä~we make use of the Bogoliubov inequality 2([C,H],C’~])~ ~<{A,A~}>. (4) 2kBT([C,A]) First we take the following operators C and A
C=S~(—k)+S_(—k),
A =i3~S~(k+q)+f3S÷(k+q).
(5)
After some calculations we find <[C,HSd],C])=~
{exp[ik•(Rj—R~)] _1}T(Ri_Rm)](c~_Cm_+c,~+ci+>+
1, m
—2 ~ {exp [ik’(R
1 —R,~)]—1}J(R1 _Rm){2(S7zS~>+ 2(s~~ Sf> + (S7S~>+ (S~+S~},
(6)
1, m
([C,H~],C~])2hNa,
([C,A]>—2Na.
Since each lattice point is a center of symmetry the factors {exp [ik’ (R1 — Rm)] —1 } can be replaced by ~cos[k’ (R1 — Rm)] 1 }. After some calculations we find 2(a+b) (7) I([C,H5d],C~])I~MkI with a = 2 E 21T(R 2J(R,).Obviously it is essential that both a and b andthat b = 4{S~(5~ + 1)+S(S + w l)} ~1IR,I converge. It11R11 is easy to 1)I show ~k({A, A~}>
IaI
(1 dimension),
Repeating the whole procedure with 86
IaI
Volume 50A, number 2
C=Sej_k) +S_ (—k),
PHYSICS LETTERS
18 November 1974
A =j3~S÷e(k+q) — (3S÷(k+q)
we obtain for sufficiently small fields IâI
I~I
(9) (2 dimensions).
(lOa,b)
The conclusion is obvious: both a and~tend to zero in the limit h -÷0,i.e. there is no spontaneous magnetization in one and two dimensions for s-d interaction models at any non-zero temperature provided the energy band structure and the exchange integrals are such that both quantitiesa and b converge. Finally it should be remarked that we can easily incorporate into our s-d model the interactions considered by the previous authors. Our conclusion, however, still holds: no spontaneous magnetization will occur at finite temperatures.
References [1] M.A. Ruderman and C. Kittel, Phys. Rev. 96(1954)99; T. Kasuya, Progr. Theor. Phys. 16 (1956) 45; K. Yosida, Phys. Rev. 106 (1957) 893. [2] G. Vertogen, Phys. Stat. So!. 25 (1968) 721. [3] N.D. Mermin and H. Wagner, Phys. Rev. Letters 17 (1966) 1133. [4] F. Wegner, Phys. Letters 24 (1967) 131. [5] M.B. Walker and Th.W. Ruygrok, PISyS. Rev. 171 (2) (1968) 513.
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