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Absence of vector chiral order in the two-dimensional antiferromagnetic Heisenberg model
Volume 143, number 1,2
PHYSICS LETTERS A
1 January 1990
ABSENCE OF VECTOR CHIRAL ORDER IN THE TWO-DIMENSIONAL ANTIFERROMAGNETIC HEISENBERG MODEL Tohru KAWARABAYASHI and Masuo SUZUKI Department ofPhysics, Faculty of Science, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan Received 31 October 1989; accepted for publication 31 October 1989 Communicated by A.A. Maradudin
It is proved using Bogoliubov’s inequality that no vector chiral order appears at finite temperatures in the two-dimensional antiferromagnetic Heisenberg model with short-range interaction.
It is one of the challenging problems to study chiral orders in two-dimensional frustrated quantum spin systems, particularly for the purpose of clarifyingthe mechanism of high-Ta superconductivity [1]. There are two kinds of chiral order, namely the vector chiral order [2—121and scalar chiral order [13]. In the present paper, we discuss the vector chiral order defined by QIJ,kx(SIXS,+SJXSk+SkXSI) at the three lattice points i, j and k, while the scalar chiral order is defined by X~Jk=S,(SJXSk). Since Villain [2] pointed out two-fold degeneracy in fully frustrated spin systems, there have been reported many numerical calculations and approximate theories [2—12] about the vector chiral order in frustrated spin systems. We consider here the two-dimensional square lattice and triangt~larlattice. For quantum spin systems, the normalized chirality operator
where ~, is a modular factor and Q~is the z-component of the chirality operator defined on the plaquette shown in fig. 1. It is explicitly given by
r
n~=
~
—
A
(n. \ W.,l,I+U.I+U+V
(S~XS1+u+Si+u
(SIXSJ+SJXSk+SkXSI)
(1)
~=
(~,~) ,
(2)
(3)
where u (v) is a unit vector of the x-direction (y-direction) and r1 is the position vector ofthe site i. Lattice spacing is taken as a unit length. (ii) For the triangular lattice, we have ~tr~
~
~
(Qi,i+a.i+a+b_Qi.i+a+b.i+b)Z,
b
(4)
denote unit vectors shown in fig. 2.
Thewe Hamiltonian consider here is of short range. Thus assume thetoHamiltonian
is defined [8,91in each plaquette on the triangular lattice, following definitions for classical spins [2—7]. We consider th~z-component of this operator and define a chiral order parameter for each lattice as follows. (i) For the square lattice, the total chiral order Q~ • is defined by
~ ,Q~, , =exp(i,cr1),
XSi+u+v
+Sj+U+V xS1+,~+S1~,, xS1)z,
where a and 3
+n ~4+U,i+U+V,l+V
I~V
.
I
I~U+V
l+U
Fig. 1. A typical plaquette to define the chirality operator the square lattice.
0375-9601/90/S 03.50 © Elsevier Science Publishers B.V. (North-Holland)
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Volume 143, number 1,2
i÷b
PHYSICS LETTERS A
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1 January 1990
in the case of the square lattice, and A(k)= —2 ~ exp(—ik•~)S(S~+a—Sj~_a+S~+b (12)
S1_b+Sj_a_bSj+a+b)
Fig. 2. A typical plaquette to define the combined chirality oper0~.in the triangular lattice. ator
in the case of the triangular lattice. The direct calculation shows that i[C(k),A(k)]=Q (13) and
Ho=—~J~S
(5) with the couplings which satisfy the following condition, 1.S~,
Jo~ ~ IJ~Ijr~—r~I2
(6)
t]
[ C(k), II], C(k) =A~+2~ J,~{1—cos[k.
(k,—~)]}
X(S~SJ+SfSJ)
(14)
J
The present argument is similar to that of Mermin and Wagner [14]. Bogoliubov’s inequality for quanturn systems states that
for any k, where ~ is the chiral operator defined previously. It follows from eq. (14) that
~/3 <[C,
H], Ci]> ~ V( IA~I+2J 2jk12) , (15) 0s where Vis the volume ofthe system and s2=S(S+ 1) and ~ <~> / V is the chiral order parameter per site.
<[C, Al>
H], C~1>
2
<[C,
(7)
where fl=(k 8T)~, H is a Hermitian matrix (H=Ht), and C and A are arbitrary matrices of the
same size. The angular brackets denote a thermal average: ~Tr[Bexp(—/3H)]/Trexp(—flH) (8) and the square brackets denote a commutator. One of the keypoints in our argument is to consider the following Hamiltonian with a super-effective field A [121 conjugate to the chiral order ~:
Substituting eqs. (13) and (14) into Bogoliubov’s inequality (7), and using eq. (15), we obtain ~fl 2q2 ~ V(lA~I+2Jos2IkI2)~ V
(16)
Taking the summation over k’s, and taking the limit V—+ ~, we obtain
H=H 0+H’,
H’=—A~,
(9)
where H0 is the Heisenberg Hamiltonian (5), and Q is the chiral order parameter, given by ~ (~tr) in the case of the square lattice (triangular lattice). We choose C in eq. (7) as C(k) = ~ exp(ik-r~)S~, (10) i
where k is the wave vector restricted to the first Brilbum zone. Furthermore we choose A in eq. (7) as A(k) = —2 ~ exp( —ikr1)~jSJ 3’ _SY _SJ! x (SJ÷~ +S- i—u J+v J—v) 18
(11)
flM~ JkdkIA~I+~js2IkI2,
(17)
where M is some finite constant. It is clear that for finite temperatures ~ has to go to zero as A—~0. Thus we have shown that no chiral order exists at finite temperatures in this system. It is easy to extend the proof to the case in which the y- or z-component of Q1•J,k is an order parameter. The present proof is also valid even if the Hamiltonian has anisotropy in the y-direction. The present authors would like to thank N. Kawashima for stimulating discussion. This study is partially financed by the Research Fund of the Min-
Volume 143, number 1,2
PHYSICS LETTERS A
istry of Education, Science and Culture.
References El] J.A. Bednorz and K.A. Muller, Z. Phys. B 64 (1986) 189. [2] J. Villain, J. Phys. C 10 (1977)1717, 4793; G. Forgacs, Phys. Rev. B 22 (1980) 4473; D.H. Lee, R.G. Caflish, J.D. Joannopoulos and F.Y. Wu, Phys. Rev. B 29(1984)2680. [3] S. teitel and C. Jayaprakash, Phys. Rev. B 27 (1983) 598. [4]S. Miyashita and H. Shiba, J. Phys. Soc. Japan 53 (1984) 1145. [5] N. Kawashima and M. Suzuki, J. Phys. Soc. Japan 58 (1989) 3123. [6] D.H. Lee, J.D. Joannopoulos, J.W. Negele and D.P. ~ Phys. Rev. Lett. 52 (1984) 433; Phys. Rev. B 33 (1986) 450. [7] B. Berge, H.T. Diep, A. Ghazali and P. Lallemand, Phys. Rev.B34 (1986)3177.
1 January 1990
[8]S. Fujiki and D.D. Betts, Can. J. Phys. 64 (1986) 876; 65 (1987) 76; S. Fujiki, Can.J. Phys. 65 (1987) 489. [9] 5. Fujiki268. and D.D. Betts, Prog. Theor. Phys. Suppl. 87 (1986) [10] H. Nishimori and H. Nakanishi, J. Phys. Soc. Japan 57 (1988) 626; S. Miyashita, J. Phys. Soc. Japan 57 (1988) 1934. [11] F. Matsubara and S. Inawashiro, Solid State Commun. 67 (1988) 229. [12JM. Suzuki, J. Phys. Soc. Japan 57 (1988) 2310. [13] W.G. Wen, F. Wilczek and A. Zee, Phys. Rev. B 39 (1989) 11413; Y.H. Chen, F. Wilczek, E. Witten and B.!. Halperin, Int. J. Mod. Phys. B 3 (1989) 1001. [14] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133; P.C. Hohenberg, Phys. Rev. 158 (1967) 383.
19
PHYSICS LETTERS A
1 January 1990
ABSENCE OF VECTOR CHIRAL ORDER IN THE TWO-DIMENSIONAL ANTIFERROMAGNETIC HEISENBERG MODEL Tohru KAWARABAYASHI and Masuo SUZUKI Department ofPhysics, Faculty of Science, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan Received 31 October 1989; accepted for publication 31 October 1989 Communicated by A.A. Maradudin
It is proved using Bogoliubov’s inequality that no vector chiral order appears at finite temperatures in the two-dimensional antiferromagnetic Heisenberg model with short-range interaction.
It is one of the challenging problems to study chiral orders in two-dimensional frustrated quantum spin systems, particularly for the purpose of clarifyingthe mechanism of high-Ta superconductivity [1]. There are two kinds of chiral order, namely the vector chiral order [2—121and scalar chiral order [13]. In the present paper, we discuss the vector chiral order defined by QIJ,kx(SIXS,+SJXSk+SkXSI) at the three lattice points i, j and k, while the scalar chiral order is defined by X~Jk=S,(SJXSk). Since Villain [2] pointed out two-fold degeneracy in fully frustrated spin systems, there have been reported many numerical calculations and approximate theories [2—12] about the vector chiral order in frustrated spin systems. We consider here the two-dimensional square lattice and triangt~larlattice. For quantum spin systems, the normalized chirality operator
where ~, is a modular factor and Q~is the z-component of the chirality operator defined on the plaquette shown in fig. 1. It is explicitly given by
r
n~=
~
—
A
(n. \ W.,l,I+U.I+U+V
(S~XS1+u+Si+u
(SIXSJ+SJXSk+SkXSI)
(1)
~=
(~,~) ,
(2)
(3)
where u (v) is a unit vector of the x-direction (y-direction) and r1 is the position vector ofthe site i. Lattice spacing is taken as a unit length. (ii) For the triangular lattice, we have ~tr~
~
~
(Qi,i+a.i+a+b_Qi.i+a+b.i+b)Z,
b
(4)
denote unit vectors shown in fig. 2.
Thewe Hamiltonian consider here is of short range. Thus assume thetoHamiltonian
is defined [8,91in each plaquette on the triangular lattice, following definitions for classical spins [2—7]. We consider th~z-component of this operator and define a chiral order parameter for each lattice as follows. (i) For the square lattice, the total chiral order Q~ • is defined by
~ ,Q~, , =exp(i,cr1),
XSi+u+v
+Sj+U+V xS1+,~+S1~,, xS1)z,
where a and 3
+n ~4+U,i+U+V,l+V
I~V
.
I
I~U+V
l+U
Fig. 1. A typical plaquette to define the chirality operator the square lattice.
0375-9601/90/S 03.50 © Elsevier Science Publishers B.V. (North-Holland)
Qf
in
17
Volume 143, number 1,2
i÷b
PHYSICS LETTERS A
i÷a+b
1 January 1990
in the case of the square lattice, and A(k)= —2 ~ exp(—ik•~)S(S~+a—Sj~_a+S~+b (12)
S1_b+Sj_a_bSj+a+b)
Fig. 2. A typical plaquette to define the combined chirality oper0~.in the triangular lattice. ator
in the case of the triangular lattice. The direct calculation shows that i[C(k),A(k)]=Q (13) and
Ho=—~J~S
(5) with the couplings which satisfy the following condition, 1.S~,
Jo~ ~ IJ~Ijr~—r~I2
(6)
t]
[ C(k), II], C(k) =A~+2~ J,~{1—cos[k.
(k,—~)]}
X(S~SJ+SfSJ)
(14)
J
The present argument is similar to that of Mermin and Wagner [14]. Bogoliubov’s inequality for quanturn systems states that
for any k, where ~ is the chiral operator defined previously. It follows from eq. (14) that
~/3
H], Ci]> ~ V( IA~I+2J 2jk12) , (15) 0s where Vis the volume ofthe system and s2=S(S+ 1) and ~ <~> / V is the chiral order parameter per site.
<[C, Al>
H], C~1>
2
<[C,
(7)
where fl=(k 8T)~, H is a Hermitian matrix (H=Ht), and C and A are arbitrary matrices of the
same size. The angular brackets denote a thermal average: ~Tr[Bexp(—/3H)]/Trexp(—flH) (8) and the square brackets denote a commutator. One of the keypoints in our argument is to consider the following Hamiltonian with a super-effective field A [121 conjugate to the chiral order ~:
Substituting eqs. (13) and (14) into Bogoliubov’s inequality (7), and using eq. (15), we obtain ~fl
(16)
Taking the summation over k’s, and taking the limit V—+ ~, we obtain
H=H 0+H’,
H’=—A~,
(9)
where H0 is the Heisenberg Hamiltonian (5), and Q is the chiral order parameter, given by ~ (~tr) in the case of the square lattice (triangular lattice). We choose C in eq. (7) as C(k) = ~ exp(ik-r~)S~, (10) i
where k is the wave vector restricted to the first Brilbum zone. Furthermore we choose A in eq. (7) as A(k) = —2 ~ exp( —ikr1)~jSJ 3’ _SY _SJ! x (SJ÷~ +S- i—u J+v J—v) 18
(11)
flM~ JkdkIA~I+~js2IkI2,
(17)
where M is some finite constant. It is clear that for finite temperatures ~ has to go to zero as A—~0. Thus we have shown that no chiral order exists at finite temperatures in this system. It is easy to extend the proof to the case in which the y- or z-component of Q1•J,k is an order parameter. The present proof is also valid even if the Hamiltonian has anisotropy in the y-direction. The present authors would like to thank N. Kawashima for stimulating discussion. This study is partially financed by the Research Fund of the Min-
Volume 143, number 1,2
PHYSICS LETTERS A
istry of Education, Science and Culture.
References El] J.A. Bednorz and K.A. Muller, Z. Phys. B 64 (1986) 189. [2] J. Villain, J. Phys. C 10 (1977)1717, 4793; G. Forgacs, Phys. Rev. B 22 (1980) 4473; D.H. Lee, R.G. Caflish, J.D. Joannopoulos and F.Y. Wu, Phys. Rev. B 29(1984)2680. [3] S. teitel and C. Jayaprakash, Phys. Rev. B 27 (1983) 598. [4]S. Miyashita and H. Shiba, J. Phys. Soc. Japan 53 (1984) 1145. [5] N. Kawashima and M. Suzuki, J. Phys. Soc. Japan 58 (1989) 3123. [6] D.H. Lee, J.D. Joannopoulos, J.W. Negele and D.P. ~ Phys. Rev. Lett. 52 (1984) 433; Phys. Rev. B 33 (1986) 450. [7] B. Berge, H.T. Diep, A. Ghazali and P. Lallemand, Phys. Rev.B34 (1986)3177.
1 January 1990
[8]S. Fujiki and D.D. Betts, Can. J. Phys. 64 (1986) 876; 65 (1987) 76; S. Fujiki, Can.J. Phys. 65 (1987) 489. [9] 5. Fujiki268. and D.D. Betts, Prog. Theor. Phys. Suppl. 87 (1986) [10] H. Nishimori and H. Nakanishi, J. Phys. Soc. Japan 57 (1988) 626; S. Miyashita, J. Phys. Soc. Japan 57 (1988) 1934. [11] F. Matsubara and S. Inawashiro, Solid State Commun. 67 (1988) 229. [12JM. Suzuki, J. Phys. Soc. Japan 57 (1988) 2310. [13] W.G. Wen, F. Wilczek and A. Zee, Phys. Rev. B 39 (1989) 11413; Y.H. Chen, F. Wilczek, E. Witten and B.!. Halperin, Int. J. Mod. Phys. B 3 (1989) 1001. [14] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133; P.C. Hohenberg, Phys. Rev. 158 (1967) 383.
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