Exact ground state spin configurations for 2D and 3D lattices with nearest neighbor bilinear exchange

~

Pergamon

Solid State Communications, Voi. 96, No. 1 !, pp. 853-858, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/95 $9.50+.00 0038.1098(9b')00588-9

E X A C T G R O U N D STATE SPIN C O N F I G U R A T I O N S F O R 2D AND 3D L A T T I C E S WITH N E A R E S T N E I G H B O R B I L I N E A R E X C H A N G E E. Belorizky Laboratoire de Spectrom~trie Physique (UA 08), B.P. 87, 38402 Saint Martin d'H~res, France

(Received 4 July 1995 by P. Burlet)

A systematic research of the bilinear exchange Hamiltonians providing an exactly known ground state spin configuration is performed for two and three dimensional lattices. Apart from the well known ferromagnetic Heisenberg and ferro or antiferromagnetic Ising systems, particular attention is paid to the anisotropic Heisenberg model ,Y'eij= - Jx Six Sjx - Jy Siy Sjy - Jz Siz Sjz for a bond along the z axis. It is shown that, under special conditions, this pair interaction leads to a two sublattice antiferromagnetic configuration which is an eigenstate of the system at T -- 0 K, i.e. without spin deviation. All the possible structures displaying this remarkable property are derived.

Keywords : A. magnetically ordered materials, C. crystal structure and symmetry, D. exchange and superexchange.

1. INTRODUCTION parallel to Oz and the same ground state energy Eo as above,

There are very few two and three dimensional magnetic systems for which the ground state is exactly known. If we

while for J < 0 the ground state is a two sublattice

restrict the problem to bilinear exchange interactions with p

antiferromagnet with one sublattice having up spins in the z

equivalent nearest neighbors at each magnetic site, we have the N Heisenberg model H = - 2j- ~ Jij Si. ~ , which for i'm j [

direction and the other down spins in the same direction with Eo = p JNS 2 • Both Ising ground configurations have a double degeneracy according to time reversal symmetry. Several studies 1,2,3 have been performed on the xyz chain N H = - 1 ~ (Jx Six Sjx + Jy Sly Sjy + Jz Siz Sjz) leading to an 2 i=l exact value of the ground state energy but the corresponding eigenstate is generally not a simple spin configuration. This

Jij = J > 0 leads to a ferromagnetic ground state with a total spin St = NS and an energy Eo = - p JNS 2 with a degeneracy 2NS+I. All the spins are parallel in an arbitrary direction. In presence of an external magnetic field in the z direction, the degeneracy is lifted by the Zeeman energy g~tBHStz.

includes the xy model when Jx = Jy = J and Jz = 0. We also mention the Potts model 4 Hij = - 1 j ~ ~ (~i, oj) between nearest neighbors with oi i,j , o j = + 1, which is equivalent to the Ising model for

(Obviously the ferromagnetic Heisenberg ground state is not restricted to n.n. exchange interactions). Similarly, the Ising model H = - 21-~ . Jij Siz Sj leads for ij Jij = J > 0 to a ferromagnetic ground state with all the spins

S = 1/2 spins. Indeed ~ (oi, o i) = 1 + 4 Siz Sjz 853

m

854

EXACT GROUND STATE SPIN CONFIGURATIONS

In contrast to the above cases, for an Heisenberg antiferromagnet it is well known that the simple Ndel classical state described by two sublattices with Siz = S and Sjz = - S, is not an eigenstate of the Hamiltonian. The classical energy is higher than the approximate quantum energy ground state 5. The anisotropic exchange Heisenberg model in which the

Vol. 96, No. i I

We consider two spins S1 = $2 = 1/2 with their bond along the z direction and we denote by I+> or I-> the states 1:t:1/2>. A stable configuration of the spins corresponds to orientations 71 (01, and 72(02, ~02) of each spin.This

91)

means that a measure of the components St.~I o r S2.U2 will give the value + 1/2 with certainty. Taking the z direction as quantification axis, the state of the pair is 12 :

nearest neighbor interaction between two spins is : ,~f'~ij= - [J//Siz Sjz + J.L (Six Sjx + Siy Sjy)]

(l)

~ 1,2)= (cos-~1-1+, + ei¢, sin--~l- I-,} x

for a bond along the z axis has been the subject of extensive studies concerning the ground state 6,7,8 (T -- 0 K) and the thermodynamic properties9.10 (T ~ 0). A striking feature was

= CLC21++> + ei(~+~a}sls21-->+ei~acl s21-->+ei~slc21-+>

(2)

the discovery of an exact ground state antiferromagnetic configuration for a simple cubic lattice when J//-- - J.l. = J > 0, corresponding to the Z5 configuration in the notations of Luttinger and Tisza 11 displayed in fig. 3. The energy is

with ci = cos 2

' si=sin~

, ( i -= 1,2 ", 0 < 0 i < ~

,"

Eo = - 3 NS2J. In this case the classical and quantum energies are equal and there is no spin deviations for each sublattice at T = 0 K : = 5: S. Apart from the Ising model it is the

Taking '~lt'~12= " (Jx Slx S2x + Jy Sly S2y + Jz Slz S2z),

(3)

only three dimensional two sublattice antiferromagnet for which the ground state is exactly known. An Hamiltonian o f the form (1) is the most general interaction between two

which is the most general form of ,.cf~12for a C 2 , C2h, C2v, D2 or D2h symmetry of the bond, we must have

Kramers doublets (pseudo spin S = 1/2) if the bond has a D 4 , C 4 v , D2d, C3v, D3 , D 3 d , D 6 , C6v, D3h, D 6 , C6v, D3h, D6, D6h symmetry. The expression (1) may be used for

,~f4~12V(1,2) = ~, V(1,2)

(4)

The following system is obtained :

spins S > 1/2, but then biquadratic terms are also allowed. Similarly for lower symmetries of the bond other bilinear

l [ ( j , _ jx) ei(tpt+q~2)sis2_ Jz CLC2]= ~' ClC2

terms are present. 41--[(Jy - Jx)CLC2-Jz ei(q~"l~2)$1s2] = ~" ei(~'+~2) SIS2 The aim of this paper is to investigate in a systematic way the possibility of finding other structures with the same

- l[(Jx + Jy)e itpl SIC2- Jz ei92 c1s2] = ~, ei92clS2

property. Three steps are necessary : - l [ ( J x + Jy)eiq~2ClS2 - Jz eig, sic2] = ~,eiq~,slc2 (i)

between a pair of spins 1 and 2 with a bond along the z

The solutions are given in Table 1. The conditions for

axis, we must find the stable spin configurations (i.e. the

which the stable spin pair configurations correspond to the

eigenstates for which both spins have a fixed direction). (ii)

(5)

For a general form ,~t4~12of the exchange Hamiltonian

Checking that the obtained spin configuration corresponds to the ground state of the pair,

(iii) Building two or three dimensional structures from a given pair satisfying to (i) and (ii) with a Bravais lattice. This will ensure that any magnetic configuration obtained in this way will he the ground state of the whole system ~ = 21- .~. ,~ij. l,J

2. STABLE STATES FOR A PAIR

ground state are given in the last column. I corresponds to the ferromagnetic Heisenberg coupling ; II is the case discussed in the introduction with Ht2 given by equation (1) with J//= - J.L = J > 0. III and VII are similar but with J//= J < 0 and J_L = J in one direction normal to the bond and J.I. = - J in the other. XI, IV and VIII are similar to III, VII and VIII respectively but with IJ//I # IJ.l.I. VI, X, XII correspond to an Ising ferromagnetic (a) or antiferromagnetic (b) coupfing in the x, y, z directions and to the "xy" coupling (c) in the three planes normal to these directions. For the latter case, we note that the energy k = 0 is never the ground state, which explains why the xy model is out of the scope of this study. The other

Vol. 96, No. 11

EXACT GROUND STATE SPIN CONFIGURATIONS

855

O.A z

O

1I

1

1

x

IIa (0 arbitrary

I

(0, ~arbitrary)

IIb XI

IIc XI

~P2=CPl+ g) ¢2 2.,q~

l

I

2

1 q~l

x

~Pl x

IIIb VIIb XIIb

( ~ + ~ 1 = 2 g) IIIa

llIc VIIc XIIb

.Y.* 1

°

V

IV

.y..

Y..~

°°

y•

°

VIII

"

IX

Fig. 1 : Allowed ground state spin structures for a pair. A

,y

II3

II2

Ht

(Ol~arbitrary)

II4,III 3 V,VII3,IX

IIs,I I h V,VII 4,IX

~z a/~

a ~y/~

V HI 1

VII 1

III2 VII 2

HI 5,IV VII 5,VI11

III6,IV VII6,VIII

Fig. 2 : Exact ground state spin configurations for 2D lattices. (The states obtained by time reversal symmetry are not shown).

III7,IV,VIII

856

EXACT GROUND STATE SPIN CONFIGURATIONS

Vol. 96, No. 11

Table 1 : Exact spin 1/2 configurations for a pair

Exchange parameters Jx = Jy = Jz = J

Spin configuration -j/4

Ground state J>0

el=e2;o 1=~2 01=02 ;02=01+~

b

Jx=Jy=-Jz=-J

-j/4

C

J>0

e 1 =02=~

a

III

01 = 02 = 0

el = 0 2 = ~ / 2 Jy=Jz =-Jx=J

J/4

; 01+02 =0,2~

J<0

Ol = 0

e2=

Ol=X

02=0 01 = 02 = 0

IV

Jy = Jz = J ~ Jx

-Jx/4 e 1 = 0 2 = ~ / 2

Jx>0;Jx>lJI

; 01 =02 = ~ 01 = 0 0 2 = n

V

Jy=-Jz =-J~Jx

Jx/4

01=02=n/2

Jx<0;-Jx>lJI

; 01=~

a

Jy=Jz=0

C

Jy=Jz=J

d

- J y = J z =J

- Jx/4

identical to IV

Jx>0

Jx/4

identical to V

Jx
0

identical to IV

never

0

identical to V

never

Jx~0

b VI

Jx=0 Jx=0

a

VII b

01 = 02 = ~ / 2 ; 01 + 02 = ~,3n Jx=Jz = -Jy=J

j/4

C

VIII

02=0

el =n

Jx = Jz = J ~ Jy

J<0

el = 0 e2 = 02=0

i - jy/4 01 = 02 = n / 2 ; 01 = 02 = n / 2 , 3 n / 2

Jy>0,ly>lJI

01 = ~ / 2 , 02 = 3 n / 2

IX

Jx = - Jz = - J ~ Jy

jy/4

e 1 = e2 = ~/2;

Jy<0,-Jy
cases are obtained for particular values of the exchange parameters. All the ground state stable configurations of a pair are displayed in Fig. 1. The results are valid for S > 1/2 but in Table 1 the values J/4 of ~, must be replaced by J S 2.

3.

2D A N D 3D S T R U C T U R E S

In two dimensions, for having the same distance and symmetry between nearest neighbors, only the square,

Vol. 96, No. 11

EXACT GROUND STATE SPIN CONFIGURATIONS

a

b

857

-Jy/4

identical to VIII

Jy/4

identical to IX

Jy
Jy>0

lx=h=0 jy,e

C

Jx = Jz = J Jy = 0

0

identical to VIII

never

d

- Jx = Jz = J Jy = 0

0

identical to IX

never

Jx = Jy ~ Jz = J

- j/4

le 1 = e 2 = 0

XI

J>0;J>lJxl

e I =e2=x

-j/4

identical to XI

J>0

J x = J y = 0 Jz = J¢0 b

J/4

XII c

Jx=Jy *0

d

Jx=-Jy¢O

Jz=J =0 Jz=J=O

!e 1 = 0 e 2 = n

J<0

el=re e2=0

0

e l=e2=0,n

never

0

!Cell=° °2=~t =n 02=0

never

hexagonal and rectangular centered (for a < f-3-b < 3a) lattices are concerned. The correspondintg ground state spin configurations are displayed in Fig. 2. The solution I gives the ferromagnetic Heisenberg structure for all lattices with a ground state energy Eo = - 2 NJ S2. The solution II provides a

four sublattices antiferromagnetic spin configuration for the square lattice (II1) or a two sublattices antiferromagnetie structure for the square and rectangular centered lattices (II2 to II5) with Eo = - 2 NJ S2. The Figs. II2,3,4 are particular cases of II1. The solutions III and VII lead to a two sublattices

¢ Z1

Z8

Z2

1 and8 2 and 7 3 and 6 4and5

Z5

(~., g, v) (-~., -It, v) (-~,, It, - v ) (~., -g, - v )

4~ (XYZ) 5

Fig. 3 : Exact ground state spin configurations for a 3D simple cubic lattice (The states obtained by time reversal symmetry are not shown). For the XYZ configuration, the orientation of the various spins are given by their director cosines ~, g , v.

858

EXACT GROUND STATE SPIN CONFIGURATIONS

Vol. 96, No. 1 i

antiferromagnet with the spins along the primitive vectors of a

Eo = 3 JN S2. But in contrast to the (XYZ)5 case these

square lattice (1 and 2) or perpendicular to the square or

solutions cannot be linearly combined.

rectangular centered lattices (3 and 4) and to a ferromagnetic structure with the spins perpendicular to the three kinds of

4. C O N C L U S I O N

lattice planes (5,6,7), with Eo = 2 NJ S2. The solutions IV We have shown that in two dimensions, there are several

and VIII provide ferromagnetic structures identical to II15,6,7 with Eo = - 2 NJx S2 and Eo = - 2 NJy S2 respectively. Finally

exact ground state spin configurations, for various set of the

V and IX give the same antiferromagnetic structure for the

exchange

square and rectangular centered lattices as 114,5 with Eo = 2 NJx S2 or 2 NJy S2.

antiferromagnetic structures and for square, hexagonal and rectangular centered lattices. In three dimensions only the

In three dimensions, apart from the trivial case I of the

antiferromagnetic ground state. The configurations Z 5 ,

constants,

corresponding

to

ferro

or

simple cubic lattice can accommodate with an exact Heisenberg ferromagnet, a ground stable structure is available

(XYZ)5 correspond to a situation where JII = " J.t. = J > 0 and

only if there is some degeneracy in a given solution for a pair.

is allowed in many cases with a threefold, fourfold or sixfold

The solution II leads to the previously discussed Z5

equivalent ones correspond to JII = J < 0 and J.l. = J in one

structure and equivalently to the X5 and Y5 structures where

direction normal to the bond and J.l. = - J in the other. These

symmetry of each bond. The configurations Z2, Z8 and the

the spins are along the x or y axis of a simple cubic lattice. But

situations can occur if the bond is a twofold symmetry axis,

there is also an exact four sublattices antiferromagnetic

but not for a higher symmetry.

structure denoted (XYZ)5 (see Fig. 3) obtained by linear superposition of the structures X S , YS, Z5 , with Eo = - 3 JN S2. The solutions III and VIII provide the

Our systematic derivation of these structures was performed with spins 1/2 but we checked that all the results

magnetic structures Z2 and Z8 respectively or the equivalent

are valid for any value of S. Surprisingly there are many solutions to the problem.

X2, Y2, X8, Y8 structures for a simple cubic lattice with

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