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Ground-state entropy of the q-state Potts antiferromagnet in a magnetic field
Volume 95A, number 5
PHYSICS LETTERS
2 May 1983
GROUND-STATE ENTROPY OF THE q-STATE POTTS ANTIFERROMAGNET IN A M A G N E T I C F I E L D D.S. HAJDUKOVIC and S. MILO~EVIC
Department o f Physics and Meteorology, Faculty o f Natural and Mathematical Sciences, P.O. Box 550, Belgrade, Yugoslavia Received 5 January 1983
We present a simple method to obtain reliable ground-state entropies of the q-state Potts antiferromagnet in an external magnetic field. As an example, the ground-state entropy for the triangular lattice is established for all q. In the particular case q = 2, our method gives results which coincide with the first-order approximation obtained by the corner transfer matrix method.
It is known that Potts antiferromagnets have highly degenerate ground states accompanied with nonzero residual entropies per site of the lattice. However, except for the one-dimensional lattice, the exact ground-state entropy is known only in a few cases. So, for q t> 3 there are only three exact resuits [ 1 - 3 ] . These are the residual entropies, in zero field, for the square lattice (q = 3), for the triangular lattice (q = 4), and for the Kagom6 lattice (q = 3). The situation for q = 2 (the Ising model) is not much better. The triangular lattice, with q = 2, is a rare example when the exact residual entropy is known in both cases - in zero field [4] and in a non-zero external field [5]. In this paper, in the case of the triangular lattice, as an example, we are going to present a simple method to obtain approximate, and reliable, residual entropies of a Potts antiferromagnet in an external field for all q. To introduce the q-state Potts antiferromagnetic model we attach to every lattice site i a spin variable a i that takes values in the set {0 .... , q - 1} and define the hamiltonian by N ~=e~Skr(Oi, ~ij)
oj)--H~
i =1
is applied to the zero-spin state (a i = 0). If we introduce new variables: n(0) the number of terms with 6kr(a i, a/) = 0 in the first sum o f ( l ) , and n, the number of spins in spin states different from 0, the hamiltonian (1) can be written in the form = N (z e]2 - H) - en (0) + n n ,
where z is the coordination number of the lattice. As we are interested in the minimum value of ~ , we will be concerned only with those arrangements of spins that give the largest possible value of n(0), for a given value o f n . For every lattice there exists an interval n E [0, N/r], where N/r is the maximum of the possible number of spins in states different from 0 which can be distributed on the lattice so that they are not neighbouring one another. For loosely packed lattices (e.g. the square lattice) it is evident that r = 2. For close-packed lattices r > 2 (for example, the triangular lattice has r = 3). Thus, for a given n in the interval [O,N/r] the maximum value of n(0) is n(0) = zn, with the corresponding minimum value of 9~ given by E 0 (n) = N ( z e/2 - H ) - (ze - H ) n .
~k,(Oi, 0),
e>0.
(1)
The first sum is over nearest-neighbour sites (//3, and ~kr(Oi, a/) is a Kronecker delta. The external field H 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
(2)
(3)
As we can see from (3), the ground-state configuration when H dominates (i.e. H > ze) has all spins in the spin state 0 and has an energy E0(0 ) = N ( z e / 2 - H). This state is nondegenerate. However, when H 249
Volume 95A, number 5
PHYSICS LETTERS
is reduced to the value H c =ze,
(4)
the ground-state energy E0(0 ) will be the same for all n E [O,N/r]. It means that for Potts antiferromagnets, H c is the maximum critical field, in which the ground state is highly degenerate. We will find an expression for the residual entropy associated with this field in the case of a triangular lattice. A triangular lattice with N = L X L spins can be conceived as a set of L parallel chains. In the first step we ignore interactions between the chains. Then, if on the pth chain (p = 1,2, ,., L) there are np spins in spin states different from 0, the number of configurations in which they are not neighbouring one another is
2 May 1983
ground-state degeneracy (for given np and np+l) of the system of two chains. This number can be straightforwardly found. Indeed, np elements of the second set and L - 2np - np+ 1 elements of the third set can be arranged in
(q --1)np+np+l ( L - np - np +1) np ways in a small chain with L - np - np+ 1 elements. Every configuration of the small chain can be completed by spins from the first set in
(L - np - np+l ) np+l ways. Hence, the ground-state degeneracy of the system of interacting chains is a sum of the terms
-
(5)
The ground-state degeneracy for the isolated pth chain is a sum of the terms (5) over all possible values np = 0, 1, ..., L/2. Similarly, in the case of I, non-interacting chains the ground-state degeneracy is a sum of the products -
np
]" (7)
We can consider (7) as product of two factors, the factor (5) for the isolated pth chain, and a factor
(6)
over all possible sets (nl, n2, ..., np ..... nL) - {rip). In the product (6) every chain is represented by a factor of the same form [given by (5)]. Now, we will try to write a similar product in the case of L interacting chains. To this end, let us consider two interacting chains, e.g. the pth chain and (p + 1)th chain. Every spin in a spin state different from 0, in the pth chain, must be accompanied with two neighbouring spins in the zero-spin state in the (p + 1)th chain. It means that the (p + 1)th chain can be imagined as a union of three sets of elements. The first set consists of np+ 1 spins in spin states different from 0, which cannot be nearest neighbours. The second set includes np couples of spins. Spins in each couple are nearest neighbours, and every couple must be considered as one element in all permutations. The third set is formed o f L - 2n_ - n p + 1 spins in the zero-spin state. The number o~ways of possible coexistence of these three sets, on the (p + 1)th chain, is equal to the 250
rip+I
C(p + 1) = (q - 1)np +I
L
p =1
np
(L-np-np+l)(L-n ×
np
p - np+l) np+ 1
(8)
for the (p + 1)th chain, which takes into account the influence of the pth chain. Now, in the case of L interacting chains we assume we can write a product similar to (6), but with factor (8) instead of factor (5) for each chain. The ground-state degeneracy can then be approximated by a sum of terms L
[-I C(p + 1),
p=l
(9)
over all possible sets (np). However, in order to find out the corresponding ground-state entropy per site in the thermodynamic limit, we have merely to find the corresponding maximum term. The latter occurs when
np=-xL,
p = 1,2,...,L,
(10)
Volume 95A, number 5
PHYSICS LETTERS
where x is the smallest positive root, in the interval (0, 1/r), of the equation
2 May 1983
When we know the solution of (11) we can obtain the maximum of (9), and thereby the ground-state entropy per site
pies OHc= 0.45464, OHc = 0.53374 and trHe = 0.59298, obtained for q = 3 , 4 and 5, respectively. To complete our discussion, we observe that for fields smaller than (4), it is evident that ground state occurs for n = N/r. Each of the n spins can be found in q - 1 different states. Thus, the exact residual entropy per site in this case is
OHc = ln[(1
O(O,ze) = (1/r)ln(q
[(1 - 2x)/(1 - 3x)] 6 = (q - 1)(1
-x)/x.
- - x ) - l ( 1 -- 2x)3/(1 -- 3x)2].
(11)
(12)
In the particular case q = 2 our result eric = 0.33305, determined by (11) and (12), is the same as the first-order approximation obtained by Baxter and Tsang [6], who used the comer transfer matrix method and who termed their first-order calculation as the Kramers-Wannier approximation. At the same time, this result differs by less than 0.06% from Baxter's exact result [5]. The virtue of our approach is that it can straightforwardly be generalized to other lattices, producing results which are of a comparable accuracy as the results obtained by sophisticated numerical methods (e.g. the Monte Carlo method). This point will be elaborated on in a future publication. Here we present only a few additional residual entro-
- 1),
(13)
and therefore it follows that for the Potts model all fields in the interval (0, z e] are critical fields. We acknowledge a stimulating discussion with Professor H. Capel.
References [1] [2] 13] [4] [5 ] [6]
E.H. Lieb, Phys. Rev. Lett. 18 (1967) 692. E.H. Lieb, Phys. Rev. 162 (1967) 162. R.J. Baxter, J. Math. Phys. 11 (1970) 784. G.H. Wannier, Phys. Rev. 79 (1950) 357. R.J. Baxter, J. Phys. A13 (1980) L61. R.J. Baxter and S.K. Tsang, J. Phys. A13 (1980) 1023.
251
PHYSICS LETTERS
2 May 1983
GROUND-STATE ENTROPY OF THE q-STATE POTTS ANTIFERROMAGNET IN A M A G N E T I C F I E L D D.S. HAJDUKOVIC and S. MILO~EVIC
Department o f Physics and Meteorology, Faculty o f Natural and Mathematical Sciences, P.O. Box 550, Belgrade, Yugoslavia Received 5 January 1983
We present a simple method to obtain reliable ground-state entropies of the q-state Potts antiferromagnet in an external magnetic field. As an example, the ground-state entropy for the triangular lattice is established for all q. In the particular case q = 2, our method gives results which coincide with the first-order approximation obtained by the corner transfer matrix method.
It is known that Potts antiferromagnets have highly degenerate ground states accompanied with nonzero residual entropies per site of the lattice. However, except for the one-dimensional lattice, the exact ground-state entropy is known only in a few cases. So, for q t> 3 there are only three exact resuits [ 1 - 3 ] . These are the residual entropies, in zero field, for the square lattice (q = 3), for the triangular lattice (q = 4), and for the Kagom6 lattice (q = 3). The situation for q = 2 (the Ising model) is not much better. The triangular lattice, with q = 2, is a rare example when the exact residual entropy is known in both cases - in zero field [4] and in a non-zero external field [5]. In this paper, in the case of the triangular lattice, as an example, we are going to present a simple method to obtain approximate, and reliable, residual entropies of a Potts antiferromagnet in an external field for all q. To introduce the q-state Potts antiferromagnetic model we attach to every lattice site i a spin variable a i that takes values in the set {0 .... , q - 1} and define the hamiltonian by N ~=e~Skr(Oi, ~ij)
oj)--H~
i =1
is applied to the zero-spin state (a i = 0). If we introduce new variables: n(0) the number of terms with 6kr(a i, a/) = 0 in the first sum o f ( l ) , and n, the number of spins in spin states different from 0, the hamiltonian (1) can be written in the form = N (z e]2 - H) - en (0) + n n ,
where z is the coordination number of the lattice. As we are interested in the minimum value of ~ , we will be concerned only with those arrangements of spins that give the largest possible value of n(0), for a given value o f n . For every lattice there exists an interval n E [0, N/r], where N/r is the maximum of the possible number of spins in states different from 0 which can be distributed on the lattice so that they are not neighbouring one another. For loosely packed lattices (e.g. the square lattice) it is evident that r = 2. For close-packed lattices r > 2 (for example, the triangular lattice has r = 3). Thus, for a given n in the interval [O,N/r] the maximum value of n(0) is n(0) = zn, with the corresponding minimum value of 9~ given by E 0 (n) = N ( z e/2 - H ) - (ze - H ) n .
~k,(Oi, 0),
e>0.
(1)
The first sum is over nearest-neighbour sites (//3, and ~kr(Oi, a/) is a Kronecker delta. The external field H 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
(2)
(3)
As we can see from (3), the ground-state configuration when H dominates (i.e. H > ze) has all spins in the spin state 0 and has an energy E0(0 ) = N ( z e / 2 - H). This state is nondegenerate. However, when H 249
Volume 95A, number 5
PHYSICS LETTERS
is reduced to the value H c =ze,
(4)
the ground-state energy E0(0 ) will be the same for all n E [O,N/r]. It means that for Potts antiferromagnets, H c is the maximum critical field, in which the ground state is highly degenerate. We will find an expression for the residual entropy associated with this field in the case of a triangular lattice. A triangular lattice with N = L X L spins can be conceived as a set of L parallel chains. In the first step we ignore interactions between the chains. Then, if on the pth chain (p = 1,2, ,., L) there are np spins in spin states different from 0, the number of configurations in which they are not neighbouring one another is
2 May 1983
ground-state degeneracy (for given np and np+l) of the system of two chains. This number can be straightforwardly found. Indeed, np elements of the second set and L - 2np - np+ 1 elements of the third set can be arranged in
(q --1)np+np+l ( L - np - np +1) np ways in a small chain with L - np - np+ 1 elements. Every configuration of the small chain can be completed by spins from the first set in
(L - np - np+l ) np+l ways. Hence, the ground-state degeneracy of the system of interacting chains is a sum of the terms
-
(5)
The ground-state degeneracy for the isolated pth chain is a sum of the terms (5) over all possible values np = 0, 1, ..., L/2. Similarly, in the case of I, non-interacting chains the ground-state degeneracy is a sum of the products -
np
]" (7)
We can consider (7) as product of two factors, the factor (5) for the isolated pth chain, and a factor
(6)
over all possible sets (nl, n2, ..., np ..... nL) - {rip). In the product (6) every chain is represented by a factor of the same form [given by (5)]. Now, we will try to write a similar product in the case of L interacting chains. To this end, let us consider two interacting chains, e.g. the pth chain and (p + 1)th chain. Every spin in a spin state different from 0, in the pth chain, must be accompanied with two neighbouring spins in the zero-spin state in the (p + 1)th chain. It means that the (p + 1)th chain can be imagined as a union of three sets of elements. The first set consists of np+ 1 spins in spin states different from 0, which cannot be nearest neighbours. The second set includes np couples of spins. Spins in each couple are nearest neighbours, and every couple must be considered as one element in all permutations. The third set is formed o f L - 2n_ - n p + 1 spins in the zero-spin state. The number o~ways of possible coexistence of these three sets, on the (p + 1)th chain, is equal to the 250
rip+I
C(p + 1) = (q - 1)np +I
L
p =1
np
(L-np-np+l)(L-n ×
np
p - np+l) np+ 1
(8)
for the (p + 1)th chain, which takes into account the influence of the pth chain. Now, in the case of L interacting chains we assume we can write a product similar to (6), but with factor (8) instead of factor (5) for each chain. The ground-state degeneracy can then be approximated by a sum of terms L
[-I C(p + 1),
p=l
(9)
over all possible sets (np). However, in order to find out the corresponding ground-state entropy per site in the thermodynamic limit, we have merely to find the corresponding maximum term. The latter occurs when
np=-xL,
p = 1,2,...,L,
(10)
Volume 95A, number 5
PHYSICS LETTERS
where x is the smallest positive root, in the interval (0, 1/r), of the equation
2 May 1983
When we know the solution of (11) we can obtain the maximum of (9), and thereby the ground-state entropy per site
pies OHc= 0.45464, OHc = 0.53374 and trHe = 0.59298, obtained for q = 3 , 4 and 5, respectively. To complete our discussion, we observe that for fields smaller than (4), it is evident that ground state occurs for n = N/r. Each of the n spins can be found in q - 1 different states. Thus, the exact residual entropy per site in this case is
OHc = ln[(1
O(O,ze) = (1/r)ln(q
[(1 - 2x)/(1 - 3x)] 6 = (q - 1)(1
-x)/x.
- - x ) - l ( 1 -- 2x)3/(1 -- 3x)2].
(11)
(12)
In the particular case q = 2 our result eric = 0.33305, determined by (11) and (12), is the same as the first-order approximation obtained by Baxter and Tsang [6], who used the comer transfer matrix method and who termed their first-order calculation as the Kramers-Wannier approximation. At the same time, this result differs by less than 0.06% from Baxter's exact result [5]. The virtue of our approach is that it can straightforwardly be generalized to other lattices, producing results which are of a comparable accuracy as the results obtained by sophisticated numerical methods (e.g. the Monte Carlo method). This point will be elaborated on in a future publication. Here we present only a few additional residual entro-
- 1),
(13)
and therefore it follows that for the Potts model all fields in the interval (0, z e] are critical fields. We acknowledge a stimulating discussion with Professor H. Capel.
References [1] [2] 13] [4] [5 ] [6]
E.H. Lieb, Phys. Rev. Lett. 18 (1967) 692. E.H. Lieb, Phys. Rev. 162 (1967) 162. R.J. Baxter, J. Math. Phys. 11 (1970) 784. G.H. Wannier, Phys. Rev. 79 (1950) 357. R.J. Baxter, J. Phys. A13 (1980) L61. R.J. Baxter and S.K. Tsang, J. Phys. A13 (1980) 1023.
251