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Possible permutation symmetry in two dimensional Heisenberg Model
Solid State Communications, Printed in Great Britain.
Possible
Vol.
Permutation
72,
No. 6,
Symmetry
pp.
517-521,
1989.
0038-1098/89$3.00+.00 Pergamon Press
in Two Dimensional
Heisenberg
plc
Model
Riichiro Saito Department of Physics, Faculty of Science, University of Tokyo 113 Bunkyo-ku, Tokyo Japan (Received 10 August 1989 by T. Tsuzuki) A method for obtaining the ground state of the two dimensional Heisenberg Model is proposed in terms of permutation operations. First a minimal basis set of spin functions for the ground state is prepared for a given Hamiltonian using the high symmetry of the spin functions. The symmetry of the spin functions can be expressed graphically by permutations commutable with the Hamiltonian. We then solve the Hamiltonian using the minimal basis set whose dimension is much smaller than that of the subspace of singlet states. An analytical solution for the finite system of N=lO is given.
ALTHOUGH CHALLENGING ATTEMPTS to ob t&n the exact solution of the two dimensional antiferromagnetic (2D-AF) Heisenberg model have been performed for long times, the exact solution has not yet obtained except for finite number of spins. The rigorous ground state of the unfrustrated spin l/2 antiferromagnetic lattice lies in the spin singlet subspace in which the total spin Stit is zero ‘1’). However the number of basis spin-functions for St, = 0 accerelatingly increases with increasing number of spins N. Up to now the large& N that has been solved exactly is only 263). In spite of such difficulties, the character of the ground state in the spin l/2 2DAF lattice is now generally understood to have a staggered magnetization smaller than that in the NC1 state because of fluctuations but long range order exists. Anderson proposed an idea of the resonating valence bond (RVB) in which low energy states are given by the spin functions expressed as possible combinations of the products of singlet bonds consisting of nearest neighbor sites’). The RVB spin functions describe well the fluctuations of the low energy states by using singlet functions. However trial functions considering only nearest neighhour singlet pairs do not cover completely the space of St, = 0, and thus it becomes inaccurate with incressing N. Liang et al. have improved the trial functions to take account of long distance singlet pairs, weighted by functions of distance of the pairs’). They obtain lower energies using a variational principle. Recently Oguchi and Kitatani showed that the set of non crossed singlet bonds gives a complete basis set for the singlet (Se& = 0) subspace for N < 16, and asserted that this is very plausible true for all Ns). Thus the exact solution for any finite N may be obtained by diagonalizing the complete set of S, = 0 non crossed spin functions. The idea of non crossed diagrams was proposed first by Rumer’) and Pauling*) for the electronic structure of molecules. Very recently Nakagawa et al. have proposed a new trial function for the ground state by introducing a small number of variational parameters as
weights for topologically inequivalent singlet pai&). The optimized trial state lies very close to the exact ground state (1 - (@~,a1@_~) < lo-’ ) even for N = 26 for which there are only 3 variational parameters. Their results are very striking in the sense that the number of variational parameters does not increase much though they have calculated huge numbers of wave functions. Their surprising result should be clarified rigorously in order to understand the symmetry of the ground state of the 2D Heisenberg model. I. In this communication, a new method is proposed; for obtaining the symmetries possessed by the ground state. After taking account of the symmetry of the 2D AF system, we get a surprisingly small number of basis functions which are linearly independent. First, we define the Hamiltonian of the antiferromagnetic Heisenberg model of spin l/2 in the two dimensional square lattice, divided into two sublattices A and B (the bipartite lattice), as H = J
C Si*Sj,
J > 0,
where summation tween the A and stant. The total It is known that the transposition
is over the nearest neighbour spins beB sublattices and J is a coupling connumber of spins N is finite and even. Si. Sj and !j(fij - $) in which Pij is between
tors for any spin functions@. - Cij
i and j, are identical
opera
Taking the energy origin as
i, in the unit of J/2, Eq.(l) is expressed as follows,
ET=
C
.Pij.
This expression for the HamiItonidn is very useful for consideration of the permutation symmetry of the Hamilt* nian. We adopt a periodic boundary condition. It should be mentioned that boundary condition affects the ground state energy l& for a given N. Thus we must adopt a consistent boundary condition when discussing the N de517
518
POSSIBLE PERMUTATION
SYMMETRY
pendence of E0 etc. In Fig. la, we number the sites from 1 to 10 for ten spins (N=10) on a two dimensional square lattice, where the A and B sublattices correspond to odd and even numbers respectively. The periodic boundary condition or the connectivity of Pij in Fig. la can be graphically represented by the solid lines in Fig. lb. We can see that the Hamiltonian represented by Fig. lb has the highest symmetry in the case of N=10. Following Oguchi and Kitatani 6), we now prepare a complete basis set {/i} of singlet spin (Stot/= 0) functions using non crossed singlet bonds, in which all singlet wavefunctions are expressed as N/2 products of the singlet pair by t¢/2
f~ = [I[i,j],
NI
(k = 1,...,no - (~ + 1)I~f).
(3)
Here [i, j] is a singlet spin function, that is, [i, j] = (aifljfliaj). The number no of the wavefunctions fk's is shown to be identical with the dimension of Stot = 07). In Fig.2 six inequivalent, no crossed patterns are shown in the case of N=10. By rotating each pattern, we can get the topologically equivalent bases. The total number of such basis set no is 42 for N=10. One notes here that the completeness of the non crossed singlet bonds in the singlet space has not yet been proved rigorously, though it has been checked to be true for small N without any exception. We shall assume the completeness of fi in the following discussion.
a)
IN TWO DIMENSIONAL
2-1. ~1 (2)
HEISENBERG MODEL
2-2. f2 (5) 10
6~5
Vol. 72, No.
2-3, f3 (10) 1
10
1
6~5
2-4. f4 (10)
87/~5~43
2-5. A (10)
87~6~
2-6. /6 (5)
5 /43
87~~5~43
Fig. 2: Non crossed singlet pair for N=10. Six inequivalent patterns are shown in Fig. 2-1 to 2-6. The number for each figure shows the number of rotationally equivalent pattern and total number no is 42. Very recently a rigorous proof has been completed by the author which will be reported elsewhere 1°). Using the complete set of the non crossed bonds, we can express the ground state of Eq.(2) as n0
~ = ~ bifi,
(4)
where the coefficients bi's are uniquely determined for the non-degenerate ground state because of the linear independence of {fl}. Next we consider permutations {T} commutable with the Hamiltonian which change the N/2 spins in the A sublattice into the N/2 spins in the B sublattice and vice versa. When we operate T on the ground state, we obtain
Tq~g = ~ bjTfi = ~ bjTj, f,.
N=IO 10
1
It is important to note here that any permutation operator P commutes with S 2. Thus the last term of Eq. (5) is uniquely determined since Tfj belongs in the subspace of St~ = 0. If the product of any two singlet bonds in Tfi crosses each other, that is [a, c][b,d] for a < b < c < d, we can expand it into two non crossed diagrarns using the following decomposition formula, b, c][b, d l = In, b][c, d] + b, dl[b, el.
6
5
Fig. 1: (a) The boundary condition of a Hamiltonian for N=10 which can be graphically shown in Fig. 1-(b).
(5)
(6)
This prccedure may be continued until no product of any pairs crosses each other. After carrying this out a finite number of times, we get integers for the coefficients of ~i's. Further, TCg should be the ground state since IT, H] = 0 and the ground state is non-degenerate. If Cg is the ground state of unfrustrated spin 1/2 AF lattice, the wavefunction obeys Marshall's sign rule 1) when ~g is written in Sz representation. Using the Marshall's sign rule for q~, we get another equation for the above T's 11). That is, T ¢ , = (-1)I¢/2¢g. (7)
Vol. 72, No. 6
POSSIBLE PERMUTATION
SYMMETRY IN TWO DIMENSIONAL HEISENBERG MODEL
Here we use the fact that any non crossed bonds consist of the singlet pairs between A and B sites. Combining Eqs. (5) and (7), we finally obtain a set of simultaneous equations for the hi's, no
bi = (-1)Ul2y]~Tjlbj,
(i= 1,...,no).
(8)
j=l
Here again we have used the linear independence of {fi} which requires that the coefficients of fi in Eqs. (5) and (7) to be identical. Eliminating some of the bi's using Eq. (8), we get the remaining linearly independent bi's, the number of which corresponds to the dimension of the eigenvalue problem for the ground stgte. Thus we can solve for the eigenvalue of the ground state using a much smaller dimension than no. Let us summerize our procedure; [1] find T's such that [T, H] = 0 where T is a permutation between the A and B sublattices, [2] for each T, expand Tf~ in fl and obtain the set of T/~'s, and [3] obtain the linearly dependent equations for the b~'s for all T's and eliminate the linearly dependent coefficients. Next we show as an example the case for N=10. In Fig. 3 we show the possible 120 T's which satisfy the above conditions. In the figure closed loops and solid lines represent cyclic permutations and transpositions, respectively. For example, Fig. 3-a and 3-i correspond to the permutations, Ts-'=
( 1 2 3 4 5 6 7 8 9 1 0 ) 2 3 4 5 6 7 8 9 10 1
and
(9) T3-t=
( 1 2 3 4 5 6 7 8 9 1 0 ) 2 1 4 3 1 0 7 6 9 8 5
'
respectively. All these permutations are permutations between odd (A sublattice) and even (B) numbers and it is easy to check [T, HI = 0 for each T. These commutable permutations are classified as the classes of the symmetry group S~v, that is the partition $ of N which is listed in the figure. An important fact is that these T's form a subgroup RN of S~v when combined with other permutations commutable with the Hamiltouian, which change sites in the A sublattice with sites in the A sublattice and similarly for the B sublttice. We found 120 permutations for T(A --* A, B ~ B) which can be expressed as the products of T(A ~ B, B ~ A). Thus for T(A --* A, B ---* B), the same procedure as above exists if we change Eq.(7) for the following;
T(A ..-+A, B -+ B)¢g = eg.
(10)
Moreover a product of any two T's in R~v, TaTs, satisfies the commutation law,
[T1T2, H] = TI[T,,H] + [T,,H]T2 = 0,
since these axe linear operators. Thus it is sufficient to check for the linear dependent equations for the generating elements of R,v only. Now let us consider the linear dependent equations in the case of N--10. If we adopt a permutation of Fig. 3-a, that is a cyclic permutation or rotation, each fi changes into a rotationally equivalent pattern fT~. Thus we can define fi's such that all the coefficients b~'s for the rorationally equivalent patterns are equal. As a result, of 42 b~'s only 6 are independent, denoted as bl to be, corresponding to the 6 inequivalent patterns shown in Fig. 2-1 to 2-6, respectively. Next we will carry out the same procedure for T3-t. After some calculations of T/~'s using the decomposition formula of Eq.(6), we obtain three linear dependent equations, b2=b4,
barbs,
(11)
and the product is the permutation of either T ( A --.* B, B ~ A) or T ( A -'-* A, B ..~ B). Thus the 240 T's form a subgroup. If we obtain the linearly dependent equations of Eq.(7) for T1 and T2, we do not get any other independent equations from the product of T1T2
b1+ba=b2+2b5
(12)
Since T3-, and Ta-t are generating elements of R~, no more linear independent equations can be obtained from the remaining 118 T's. Eliminating three of the b'~ from Eq. (12), we get the minimum number of bases for the ground state, which is three for N=10. The reduction of the dimension from no is very effective for large N if we take a suitable representation for the Hamiltonian. Finally the ground state energy is calculated using the Hamiltonian of Eq.(2) instead of Eq.(7), Heg
'
519
E0¢g.
=
(13)
Selecting bl, b2, and b5 as the independent coefficients, we obtain the following equations by the same procedure of Eq.(8), E0 = Eoz = Eoy =
-20 -8 -6
+ + +
20x + 12z + 5z +
60y 24y , 22y
(14)
where z = b~/ba, and y = bs/ba. Eq.(14) is not a normal seqular equation. In fact we do not use any overlap integrals of spin functions. After eliminating x and y, we obtain an equation of E0. Eoa - 14Eo~ - 16Eo + 320 = 0.
(15)
This has only one real solution which is given analytically as
14 Eo = 3
.
~[~'~-.
t arcttm(I $7 J ~ ) x
' * v o l cos(
3
3
)
=-4.6001...,
(16)
which is checked to be identical to the numerical result from diagonalizing a 42 x 42 matrix. The other two imaginary eigenvalues give non-physical solutions. It is noted here that E0 is irrational. If we get (no - 1) independent equations from Eq.(8), we can solve them to obtain the ground state with a rational value of E0, indicating that the 120 T's as above are insufficient to define the ground state. For the other possible linear T's, linear combination of permutations may give a independent equations for bi's. However we can prove that there are no W's such that [7'1, H] # 0, ITs, H] # 0 and
520
POSSIBLE PERMUTATION SYMMETRY IN TWO DIMENSIONAL HEISENBERG MODEL
= [lO1
3-b. T3-b (10)
3-~. T3-~ (2)
3-c. T3-c (10)
9
= [6,2,21
3-f. T3-/ (10)
3-e. T3-, (10) 10~-.l
8 11 76
/~43
= [,t,4,21
3-i. T3-i (10) 10 1
2
~ 5
~3 4
7
3-m. T3-m (5)
10.
19~"~~
3
3-o. T3-o (5) 10 1 2 8:~6
~43
8-
3
9
l i \ ~ ! 2 4 3 1 0 7 .8. ~ ~ J k ~ 5 ~ 4 3 7 -
3
3-1l. T3_n (5) 23
3-p. T3-p (5) 10 1 89\ ~ 2 76 ~ 5
.2
3-h. T3-h (10)
~
3 8
)~= [2,2,2,2,2] 3-e. T3-t (5) %2
8
No. 6
3-k. T3-1, (10) 10 1
3 8
8} ~
9
3-j. T3-j (10) 10 -1
8
43 8
3-g. T3-9 (10)
9
87~5
9
3 8~6 ~ 4 5
72,
3-d. T3-a (2)
~
8
Vol.
89. 1 0 ~ 3
3-q. T3-q (1) 10
1
3 ~43
87T ~ 5
24
Fig. 3:120 commutable permutations with Hamiltonian for N=10 are graphically shown from 3-a to 3-% in which closed loops with arrow and solid lines represent cyclic permutations and transposition, respectively. These permutations are classified by the partition X of N. [TI+T2, H] = 0. Thus to obtain the ground state exactly, non-linear T's must be considered. Finally we remark on some points. [1] For a small number of N, we can get a rational E0 directly from Eq.(8) which is not suitable as an example for the present paper. [2] The number of T's depends on N and the boundary condition of the Hamiltonian. We have calculated the ground state for different Hamiltonians for different N's and we very often get only the trivial T's
corresponding to the rotation. However we can find a large number of T's even for large N if we select a suitable Hamiltonian. It is found that, for a systematic boundary condition of the Hamiltonian, a unit cell of 2 x N/2 lattice gives a large number of T's. Detailed rules for finding the boundary conditions will be reported elsewhere. [3] If we apply this method to a one-dimensional system, the commutable T's with Hamiltonian are only the rotational ones. In order to consider the relationship to the
Vol. 72, No. 6
POSSIBLE PERMUTATION SYMMETRY IN TWO DIMENSIONAL HEISENBERG MODEL
exact solution, we must use other non-linear operators. [4] For the dimensions higher than two, this method will be more effective, since there exist a greater symmetry in the unit cell In the highest dimension for given N, the Hamiltonian may be given by //=
E P,j.
(17)
i
In conclusiOn we have proposed a new method for obtaining a minimal basis set for the ground state of the spin 1/2.antiferromagnetic Heisenberg model using the linear operators T's which are commutable permutations with Harniltonian. The method is applicable for any T's which reflect the symmetry of the system.
Acknowledgements - The author thank Dr. Sunji Tsuneyuld for valuable discussion, Professor Hiroshi Kamimura for the disucussion concerning on the high temperature superconductivity, and Dr. Y. K. Ko for critical reading of this paper. The author would like also to thank Professors Y. Natsume and T. Oguchi for sending preprints before publications.
REFERENCES I. W. Marshall: Proc. Roy. Soc. (London)A232, 48 (1955). 2. E. Lieb and D. Mattis: J. Math. Phys. 3, 749
(1962). 3. S. Nakagawa: Master Thesis, Chiba University (1989) in japanese.
4. P.W. Anderson: Mater. Res. Bull. 8, 153 (1973). 5. S. Liang, B. Doucot, and P.W. Anderson: Phys. Rev. 61,356 (1988). related papers therein. 6. T. Oguchi and H. Kitatani: J. Phys. Soc. Japan, 58, 1403 (1989).
521
7. G. Ruiner, E. Teller and H. Weyl: Nachr. Gott., Math-physik. Klasse, 499 (1932). 8. L. Pauling: J. Chem. Phys. 1, 280 (1933). 9. S. Nakagawa, T. Hamncla, J. Ka~e, and Y. Natsume: preprint. 10. R. Saito: preprint 11. see Eq.(A-2) of Ref. 1. We can apply Eq.(A-2) to any dimension if the system is defined in spin 1/2 bipartite lattice.
Possible
Vol.
Permutation
72,
No. 6,
Symmetry
pp.
517-521,
1989.
0038-1098/89$3.00+.00 Pergamon Press
in Two Dimensional
Heisenberg
plc
Model
Riichiro Saito Department of Physics, Faculty of Science, University of Tokyo 113 Bunkyo-ku, Tokyo Japan (Received 10 August 1989 by T. Tsuzuki) A method for obtaining the ground state of the two dimensional Heisenberg Model is proposed in terms of permutation operations. First a minimal basis set of spin functions for the ground state is prepared for a given Hamiltonian using the high symmetry of the spin functions. The symmetry of the spin functions can be expressed graphically by permutations commutable with the Hamiltonian. We then solve the Hamiltonian using the minimal basis set whose dimension is much smaller than that of the subspace of singlet states. An analytical solution for the finite system of N=lO is given.
ALTHOUGH CHALLENGING ATTEMPTS to ob t&n the exact solution of the two dimensional antiferromagnetic (2D-AF) Heisenberg model have been performed for long times, the exact solution has not yet obtained except for finite number of spins. The rigorous ground state of the unfrustrated spin l/2 antiferromagnetic lattice lies in the spin singlet subspace in which the total spin Stit is zero ‘1’). However the number of basis spin-functions for St, = 0 accerelatingly increases with increasing number of spins N. Up to now the large& N that has been solved exactly is only 263). In spite of such difficulties, the character of the ground state in the spin l/2 2DAF lattice is now generally understood to have a staggered magnetization smaller than that in the NC1 state because of fluctuations but long range order exists. Anderson proposed an idea of the resonating valence bond (RVB) in which low energy states are given by the spin functions expressed as possible combinations of the products of singlet bonds consisting of nearest neighbor sites’). The RVB spin functions describe well the fluctuations of the low energy states by using singlet functions. However trial functions considering only nearest neighhour singlet pairs do not cover completely the space of St, = 0, and thus it becomes inaccurate with incressing N. Liang et al. have improved the trial functions to take account of long distance singlet pairs, weighted by functions of distance of the pairs’). They obtain lower energies using a variational principle. Recently Oguchi and Kitatani showed that the set of non crossed singlet bonds gives a complete basis set for the singlet (Se& = 0) subspace for N < 16, and asserted that this is very plausible true for all Ns). Thus the exact solution for any finite N may be obtained by diagonalizing the complete set of S, = 0 non crossed spin functions. The idea of non crossed diagrams was proposed first by Rumer’) and Pauling*) for the electronic structure of molecules. Very recently Nakagawa et al. have proposed a new trial function for the ground state by introducing a small number of variational parameters as
weights for topologically inequivalent singlet pai&). The optimized trial state lies very close to the exact ground state (1 - (@~,a1@_~) < lo-’ ) even for N = 26 for which there are only 3 variational parameters. Their results are very striking in the sense that the number of variational parameters does not increase much though they have calculated huge numbers of wave functions. Their surprising result should be clarified rigorously in order to understand the symmetry of the ground state of the 2D Heisenberg model. I. In this communication, a new method is proposed; for obtaining the symmetries possessed by the ground state. After taking account of the symmetry of the 2D AF system, we get a surprisingly small number of basis functions which are linearly independent. First, we define the Hamiltonian of the antiferromagnetic Heisenberg model of spin l/2 in the two dimensional square lattice, divided into two sublattices A and B (the bipartite lattice), as H = J
C Si*Sj,
J > 0,
is over the nearest neighbour spins beB sublattices and J is a coupling connumber of spins N is finite and even. Si. Sj and !j(fij - $) in which Pij is between
tors for any spin functions@. - Cij
i and j, are identical
opera
Taking the energy origin as
i, in the unit of J/2, Eq.(l) is expressed as follows,
ET=
C
.Pij.
518
POSSIBLE PERMUTATION
SYMMETRY
pendence of E0 etc. In Fig. la, we number the sites from 1 to 10 for ten spins (N=10) on a two dimensional square lattice, where the A and B sublattices correspond to odd and even numbers respectively. The periodic boundary condition or the connectivity of Pij in Fig. la can be graphically represented by the solid lines in Fig. lb. We can see that the Hamiltonian represented by Fig. lb has the highest symmetry in the case of N=10. Following Oguchi and Kitatani 6), we now prepare a complete basis set {/i} of singlet spin (Stot/= 0) functions using non crossed singlet bonds, in which all singlet wavefunctions are expressed as N/2 products of the singlet pair by t¢/2
f~ = [I[i,j],
NI
(k = 1,...,no - (~ + 1)I~f).
(3)
Here [i, j] is a singlet spin function, that is, [i, j] = (aifljfliaj). The number no of the wavefunctions fk's is shown to be identical with the dimension of Stot = 07). In Fig.2 six inequivalent, no crossed patterns are shown in the case of N=10. By rotating each pattern, we can get the topologically equivalent bases. The total number of such basis set no is 42 for N=10. One notes here that the completeness of the non crossed singlet bonds in the singlet space has not yet been proved rigorously, though it has been checked to be true for small N without any exception. We shall assume the completeness of fi in the following discussion.
a)
IN TWO DIMENSIONAL
2-1. ~1 (2)
HEISENBERG MODEL
2-2. f2 (5) 10
6~5
Vol. 72, No.
2-3, f3 (10) 1
10
1
6~5
2-4. f4 (10)
87/~5~43
2-5. A (10)
87~6~
2-6. /6 (5)
5 /43
87~~5~43
Fig. 2: Non crossed singlet pair for N=10. Six inequivalent patterns are shown in Fig. 2-1 to 2-6. The number for each figure shows the number of rotationally equivalent pattern and total number no is 42. Very recently a rigorous proof has been completed by the author which will be reported elsewhere 1°). Using the complete set of the non crossed bonds, we can express the ground state of Eq.(2) as n0
~ = ~ bifi,
(4)
where the coefficients bi's are uniquely determined for the non-degenerate ground state because of the linear independence of {fl}. Next we consider permutations {T} commutable with the Hamiltonian which change the N/2 spins in the A sublattice into the N/2 spins in the B sublattice and vice versa. When we operate T on the ground state, we obtain
Tq~g = ~ bjTfi = ~ bjTj, f,.
N=IO 10
1
It is important to note here that any permutation operator P commutes with S 2. Thus the last term of Eq. (5) is uniquely determined since Tfj belongs in the subspace of St~ = 0. If the product of any two singlet bonds in Tfi crosses each other, that is [a, c][b,d] for a < b < c < d, we can expand it into two non crossed diagrarns using the following decomposition formula, b, c][b, d l = In, b][c, d] + b, dl[b, el.
6
5
Fig. 1: (a) The boundary condition of a Hamiltonian for N=10 which can be graphically shown in Fig. 1-(b).
(5)
(6)
This prccedure may be continued until no product of any pairs crosses each other. After carrying this out a finite number of times, we get integers for the coefficients of ~i's. Further, TCg should be the ground state since IT, H] = 0 and the ground state is non-degenerate. If Cg is the ground state of unfrustrated spin 1/2 AF lattice, the wavefunction obeys Marshall's sign rule 1) when ~g is written in Sz representation. Using the Marshall's sign rule for q~, we get another equation for the above T's 11). That is, T ¢ , = (-1)I¢/2¢g. (7)
Vol. 72, No. 6
POSSIBLE PERMUTATION
SYMMETRY IN TWO DIMENSIONAL HEISENBERG MODEL
Here we use the fact that any non crossed bonds consist of the singlet pairs between A and B sites. Combining Eqs. (5) and (7), we finally obtain a set of simultaneous equations for the hi's, no
bi = (-1)Ul2y]~Tjlbj,
(i= 1,...,no).
(8)
j=l
Here again we have used the linear independence of {fi} which requires that the coefficients of fi in Eqs. (5) and (7) to be identical. Eliminating some of the bi's using Eq. (8), we get the remaining linearly independent bi's, the number of which corresponds to the dimension of the eigenvalue problem for the ground stgte. Thus we can solve for the eigenvalue of the ground state using a much smaller dimension than no. Let us summerize our procedure; [1] find T's such that [T, H] = 0 where T is a permutation between the A and B sublattices, [2] for each T, expand Tf~ in fl and obtain the set of T/~'s, and [3] obtain the linearly dependent equations for the b~'s for all T's and eliminate the linearly dependent coefficients. Next we show as an example the case for N=10. In Fig. 3 we show the possible 120 T's which satisfy the above conditions. In the figure closed loops and solid lines represent cyclic permutations and transpositions, respectively. For example, Fig. 3-a and 3-i correspond to the permutations, Ts-'=
( 1 2 3 4 5 6 7 8 9 1 0 ) 2 3 4 5 6 7 8 9 10 1
and
(9) T3-t=
( 1 2 3 4 5 6 7 8 9 1 0 ) 2 1 4 3 1 0 7 6 9 8 5
'
respectively. All these permutations are permutations between odd (A sublattice) and even (B) numbers and it is easy to check [T, HI = 0 for each T. These commutable permutations are classified as the classes of the symmetry group S~v, that is the partition $ of N which is listed in the figure. An important fact is that these T's form a subgroup RN of S~v when combined with other permutations commutable with the Hamiltouian, which change sites in the A sublattice with sites in the A sublattice and similarly for the B sublttice. We found 120 permutations for T(A --* A, B ~ B) which can be expressed as the products of T(A ~ B, B ~ A). Thus for T(A --* A, B ---* B), the same procedure as above exists if we change Eq.(7) for the following;
T(A ..-+A, B -+ B)¢g = eg.
(10)
Moreover a product of any two T's in R~v, TaTs, satisfies the commutation law,
[T1T2, H] = TI[T,,H] + [T,,H]T2 = 0,
since these axe linear operators. Thus it is sufficient to check for the linear dependent equations for the generating elements of R,v only. Now let us consider the linear dependent equations in the case of N--10. If we adopt a permutation of Fig. 3-a, that is a cyclic permutation or rotation, each fi changes into a rotationally equivalent pattern fT~. Thus we can define fi's such that all the coefficients b~'s for the rorationally equivalent patterns are equal. As a result, of 42 b~'s only 6 are independent, denoted as bl to be, corresponding to the 6 inequivalent patterns shown in Fig. 2-1 to 2-6, respectively. Next we will carry out the same procedure for T3-t. After some calculations of T/~'s using the decomposition formula of Eq.(6), we obtain three linear dependent equations, b2=b4,
barbs,
(11)
and the product is the permutation of either T ( A --.* B, B ~ A) or T ( A -'-* A, B ..~ B). Thus the 240 T's form a subgroup. If we obtain the linearly dependent equations of Eq.(7) for T1 and T2, we do not get any other independent equations from the product of T1T2
b1+ba=b2+2b5
(12)
Since T3-, and Ta-t are generating elements of R~, no more linear independent equations can be obtained from the remaining 118 T's. Eliminating three of the b'~ from Eq. (12), we get the minimum number of bases for the ground state, which is three for N=10. The reduction of the dimension from no is very effective for large N if we take a suitable representation for the Hamiltonian. Finally the ground state energy is calculated using the Hamiltonian of Eq.(2) instead of Eq.(7), Heg
'
519
E0¢g.
=
(13)
Selecting bl, b2, and b5 as the independent coefficients, we obtain the following equations by the same procedure of Eq.(8), E0 = Eoz = Eoy =
-20 -8 -6
+ + +
20x + 12z + 5z +
60y 24y , 22y
(14)
where z = b~/ba, and y = bs/ba. Eq.(14) is not a normal seqular equation. In fact we do not use any overlap integrals of spin functions. After eliminating x and y, we obtain an equation of E0. Eoa - 14Eo~ - 16Eo + 320 = 0.
(15)
This has only one real solution which is given analytically as
14 Eo = 3
.
~[~'~-.
t arcttm(I $7 J ~ ) x
' * v o l cos(
3
3
)
=-4.6001...,
(16)
which is checked to be identical to the numerical result from diagonalizing a 42 x 42 matrix. The other two imaginary eigenvalues give non-physical solutions. It is noted here that E0 is irrational. If we get (no - 1) independent equations from Eq.(8), we can solve them to obtain the ground state with a rational value of E0, indicating that the 120 T's as above are insufficient to define the ground state. For the other possible linear T's, linear combination of permutations may give a independent equations for bi's. However we can prove that there are no W's such that [7'1, H] # 0, ITs, H] # 0 and
520
POSSIBLE PERMUTATION SYMMETRY IN TWO DIMENSIONAL HEISENBERG MODEL
= [lO1
3-b. T3-b (10)
3-~. T3-~ (2)
3-c. T3-c (10)
9
= [6,2,21
3-f. T3-/ (10)
3-e. T3-, (10) 10~-.l
8 11 76
/~43
= [,t,4,21
3-i. T3-i (10) 10 1
2
~ 5
~3 4
7
3-m. T3-m (5)
10.
19~"~~
3
3-o. T3-o (5) 10 1 2 8:~6
~43
8-
3
9
l i \ ~ ! 2 4 3 1 0 7 .8. ~ ~ J k ~ 5 ~ 4 3 7 -
3
3-1l. T3_n (5) 23
3-p. T3-p (5) 10 1 89\ ~ 2 76 ~ 5
.2
3-h. T3-h (10)
~
3 8
)~= [2,2,2,2,2] 3-e. T3-t (5) %2
8
No. 6
3-k. T3-1, (10) 10 1
3 8
8} ~
9
3-j. T3-j (10) 10 -1
8
43 8
3-g. T3-9 (10)
9
87~5
9
3 8~6 ~ 4 5
72,
3-d. T3-a (2)
~
8
Vol.
89. 1 0 ~ 3
3-q. T3-q (1) 10
1
3 ~43
87T ~ 5
24
Fig. 3:120 commutable permutations with Hamiltonian for N=10 are graphically shown from 3-a to 3-% in which closed loops with arrow and solid lines represent cyclic permutations and transposition, respectively. These permutations are classified by the partition X of N. [TI+T2, H] = 0. Thus to obtain the ground state exactly, non-linear T's must be considered. Finally we remark on some points. [1] For a small number of N, we can get a rational E0 directly from Eq.(8) which is not suitable as an example for the present paper. [2] The number of T's depends on N and the boundary condition of the Hamiltonian. We have calculated the ground state for different Hamiltonians for different N's and we very often get only the trivial T's
corresponding to the rotation. However we can find a large number of T's even for large N if we select a suitable Hamiltonian. It is found that, for a systematic boundary condition of the Hamiltonian, a unit cell of 2 x N/2 lattice gives a large number of T's. Detailed rules for finding the boundary conditions will be reported elsewhere. [3] If we apply this method to a one-dimensional system, the commutable T's with Hamiltonian are only the rotational ones. In order to consider the relationship to the
Vol. 72, No. 6
POSSIBLE PERMUTATION SYMMETRY IN TWO DIMENSIONAL HEISENBERG MODEL
exact solution, we must use other non-linear operators. [4] For the dimensions higher than two, this method will be more effective, since there exist a greater symmetry in the unit cell In the highest dimension for given N, the Hamiltonian may be given by //=
E P,j.
(17)
i
In conclusiOn we have proposed a new method for obtaining a minimal basis set for the ground state of the spin 1/2.antiferromagnetic Heisenberg model using the linear operators T's which are commutable permutations with Harniltonian. The method is applicable for any T's which reflect the symmetry of the system.
Acknowledgements - The author thank Dr. Sunji Tsuneyuld for valuable discussion, Professor Hiroshi Kamimura for the disucussion concerning on the high temperature superconductivity, and Dr. Y. K. Ko for critical reading of this paper. The author would like also to thank Professors Y. Natsume and T. Oguchi for sending preprints before publications.
REFERENCES I. W. Marshall: Proc. Roy. Soc. (London)A232, 48 (1955). 2. E. Lieb and D. Mattis: J. Math. Phys. 3, 749
(1962). 3. S. Nakagawa: Master Thesis, Chiba University (1989) in japanese.
4. P.W. Anderson: Mater. Res. Bull. 8, 153 (1973). 5. S. Liang, B. Doucot, and P.W. Anderson: Phys. Rev. 61,356 (1988). related papers therein. 6. T. Oguchi and H. Kitatani: J. Phys. Soc. Japan, 58, 1403 (1989).
521
7. G. Ruiner, E. Teller and H. Weyl: Nachr. Gott., Math-physik. Klasse, 499 (1932). 8. L. Pauling: J. Chem. Phys. 1, 280 (1933). 9. S. Nakagawa, T. Hamncla, J. Ka~e, and Y. Natsume: preprint. 10. R. Saito: preprint 11. see Eq.(A-2) of Ref. 1. We can apply Eq.(A-2) to any dimension if the system is defined in spin 1/2 bipartite lattice.