- Email: [email protected]
Order parameters for various Ising lattices with competing ±J interactions
Journal of Magnetism and Magnetic Materials 140-144 (1995) 1793-1794
~
Journalof renalnetlsm magnetic ~ 4 materials
ELSEVIER
Order parameters for various Ising lattices with competing + J interactions E.E. Vogel *, S. Contreras, W. Lebrecht, J. Cartes Departamento de Ciencias Fisicas, Universidad de La Frontera, Casilla 54-D, Temuco, Chile
Abstract Two recently introduced order parameters p and h are studied for Ising lattices with competing ferromagnetic and antiferromagnetic interactions. Square, triangular and honeycomb lattices are considered. The results are compared with the traditional order parameters q (Edwards-Anderson) and C R (rigidity), respectively. The advantages of the new magnitudes are brought out.
In a recent article [1], we introduced two order parameters to characterize the properties of the ground level of Ising lattices with competing ferromagnetic (F) and antiferromagnetic (AF) exchange interactions. Let us consider a two-dimensional lattice with N spins, each possessing two possible orientations ( _ 1). The number of nearest-neighbor interactions or bonds is B, whose value depends on the geometry of the lattice [2]. The Ising Hamiltonian for such a system is: B
n = EJijSiSj, i,j
(1)
where the sums extend over all bonds, S i and Sj represent Ising spins at sites i and j, respectively. The bond between a pair of spins is represented by Jij and can be either - 1 (F) or + 1 (AF). This Hamiltonian is invariant with respect to the inversion of all spins, which allows one to deal with only half of the configuration space. Periodic boundary conditions are assumed. We have solved this Hamiltonian for triangular lattices (TL), square lattices (SL) and honeycomb lattices (HL), for different values of N (size) distributed in different arrays (shape). Equal amounts of F and AF bonds randomly distributed for a given array lead to a particular sample. Due to fluctuations from sample to sample, results given below are obtained after averaging over 500 samples for each array to get general properties as N increases. Each state is an ordered collection of spin orientations. The ground level energy Eg can be solved by different methods that also allow us to find the degeneracy of the ground manifold W [1].
* Corresponding author. Fax: +56-45-251847; 252547; email: [email protected].
+56-45-
The spin site correlation parameter p is defined as
Ei
P=N
Es; a
iv)w ,
(2)
where I [ means absolute value, while div represents an integer division, so the result inside { } can be either 0 or 1. The index a runs over all the ground states. This parameter is more drastic than the q introduced by Edwards and Anderson [3], and other related magnitudes derived from it [4]. The parameter q can be calculated by means of the following expression:
% <_~ rTsTs? NW(W+I) '
q=2
(3)
where c~ and /3 play the role of initial and final states within the ground manifold. Each pair of states is counted once and the case a =/3 is allowed. Thus q is restricted to the interval [ - 1, 1]. It can be proven that I q l >--p. The bond correlation parameter h has been defined as [1]: '
h = "~ .,<~j
~
~-
div W,
(4)
where the first sum extends over the B pairs of nearest neighbors. The range for h is [0, 1]. However, it oscillates around 0.50 for SLs and TLs [1], while it stays close to 0.75 for HLs [5]. In this way, h represents the fraction of bonds that are not frustrated when scanning through all the states of the ground manifold. A similar parameter Ca, called rigidity, has been previously defined [6]. It represents the fraction of those bonds that are never frustrated plus those that always frustrate
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 8 8 5 3 ( 9 4 ) 0 1 1 8 3 - 4
E.E. Vogel et al. /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1793-1794
1794 1.o.
~G01.~tIR
[,,ATT]( ~
1.0
0.9-
0.9
o.5-
0.8
tion. Similar calculations were performed for SL where the descent is more pronounced than the one shown in Fig. 1
*
O.7
[1].
*
0.6
HLs were also considered where the descent is even more evident [5]. The conclusion may be that p tends to zero in the thermodynamic limit (N--* oo). However, we realize that this conclusion depends on two facts that will not be included here. On the one hand, is the definition of ergodic valley: we have found ways of optimizing the value of p according to this fact. On the other hand, the descent diminishes when the coordination number is increased. This lead us to think that two-dimensional systems with interactions beyond nearest neighbors, or systems with higher dimensionality, or both, will finally stabilize the value of p, and therefore of q. This is what the initial result actually showed [7,8]. In Fig. 2 we present the variations of h and C R with size for the case of SL. These parameters do not require any ergodic considerations for their calculation. The first striking feature is that both h and C R are constant, independent of size or shape. Evidently, h < C a as can be expected from Eq. (5) suggesting that the new parameter is more strict than the old one. The difference remains constant, at a value close to 0.03 for the range reported here. The advantage of h over C R is that the former is based on the bonds that are never frustrated when scanning the ground manifold and hence propagate information without distortion through the lattice. This makes h a good independent variable for studying percolation in these systems. We have done initial calculations of this property for SLs which show very interesting behavior [7]. A comparison of h with C R for HLs shows similar properties to those presented in Fig. 2. Namely, the difference C R - h remains constant, at the same value 0.03, for the same range as the one considered for SLs. Acknowledgements: The authors thank Fondecyt (Contract 1930385) and the Direcci6n de Investigaci6n Universidad de La Frontera ( # 9408) for partial support.
o.7~
*
0.6
*
*
=
*
*
*
o.5
0.5
0.3
0.3
0.4
0.2
****.
q
0.1
*****
P
1()
'
0.0
'
20.2
20.1 2'0
3'0
'
400.0
t[
Fig. 1. Comparison of p and q as a function of size N for different triangular lattices in arrays close to square shape. when scanning the ground manifold. We can calculate it by means of
CR=h+~.i~
~
IS'SJ2+J'Jl
divW,
(5)
where the symbols are the same as those used in previous equations. We have calculated these two pairs of order parameters for TLs, SLs and HLs as a function of size, staying close to shapes that resemble square arrays. We present a comparison of p and q for TLs in Fig. 1, and of h and C a for SLs in Fig. 2. Here, we comment on the general trends of these parameters for the different lattices. The results for p and q in Fig. 1 show that they decrease slowly with size. Both parameters tend to follow the same variations with size, so they represent essentially the same information about the system. However, the effort required to calculate p is negligible compared to that for q. In fact, the latter requires storing all the states to perform the calculations for q after the diagonalization. In contrast, p does not require storing states and it can be calculated during the diagonalization in a very straightforward way. Despite this advantage, p still requires a decision on the ergodic valley (half the configuration space) over which it will be calculated. This applies to q in the same way. We have used the criterion of choosing one spin strongly attached to its neighbors (surrounded by the maximum number of non-frustrated bonds), assigning an arbitrary orientation to it, and letting the rest of the spins follow from the diagonalization that considers this restricL0 -
sqffARIg I£I~CES
1.0
0.9 ~
0.9
o.8 ~
o.a
0.7 ~ O,6 -
0.7 ÷ n ÷o
O.6
=+ +oo* ~* =*
0.5~ •
o.,~
~i
o
o
0.5 o.4
0.3 ~
0.3
0.2 -
o.,
~
0.2
;;;;;
~
Ol
0.0 'lb'2b'3'o'4b'sb'6b'700.0 N
Fig. 2. Comparison of h and C R as a function of size N for different square lattices in arrays close to square shape.
References
[1] E.E. Vogel, J. Cartes, S. Contreras, W. Lebrecht and J. Villegas, Phys. Rev. B 49 (1994) 6018. [2] W. Lebrecht, and E.E. Vogel, Pro. 2nd Latin-American Workshop on Magnetism, Magnetic Materials and their Applications, eds. J.L. Morfin-L6pez and J.M. Sfinchez (Plenum, New York, 1994). [3] S.F. Edwards and P.W. Anderson, J. Phys. F 5 (1975) 965. [4] K. Binder and A.P. Young, Rev. Mod. Phys. 58 (1986) 801. [5] W. Lebrecht, and E.E. Vogel, Phys. Rev. B (1994) submitted. [6] F. Barahona, R. Maynard, R. Rammal and J.P. Uhry, J. Phys. A 15 (1982) 673. [7] E.E. Vogel, S. Contreras and J. Cartes, Proc. 2nd LatinAmerican Workshop on Magnetism, Magnetic Materials and their Applications, eds. J.L. Mor~n-L6pez and J.M. S~nchez (Plenum, New York, 1994). [8] F. Nieto, A.J. Ramirez and E.E. Vogel, Proc. VIII Taller Sur de Fisica del S61ido (Universidad de Concepci6n, Chile, 1994).
~
Journalof renalnetlsm magnetic ~ 4 materials
ELSEVIER
Order parameters for various Ising lattices with competing + J interactions E.E. Vogel *, S. Contreras, W. Lebrecht, J. Cartes Departamento de Ciencias Fisicas, Universidad de La Frontera, Casilla 54-D, Temuco, Chile
Abstract Two recently introduced order parameters p and h are studied for Ising lattices with competing ferromagnetic and antiferromagnetic interactions. Square, triangular and honeycomb lattices are considered. The results are compared with the traditional order parameters q (Edwards-Anderson) and C R (rigidity), respectively. The advantages of the new magnitudes are brought out.
In a recent article [1], we introduced two order parameters to characterize the properties of the ground level of Ising lattices with competing ferromagnetic (F) and antiferromagnetic (AF) exchange interactions. Let us consider a two-dimensional lattice with N spins, each possessing two possible orientations ( _ 1). The number of nearest-neighbor interactions or bonds is B, whose value depends on the geometry of the lattice [2]. The Ising Hamiltonian for such a system is: B
n = EJijSiSj, i,j
(1)
where the sums extend over all bonds, S i and Sj represent Ising spins at sites i and j, respectively. The bond between a pair of spins is represented by Jij and can be either - 1 (F) or + 1 (AF). This Hamiltonian is invariant with respect to the inversion of all spins, which allows one to deal with only half of the configuration space. Periodic boundary conditions are assumed. We have solved this Hamiltonian for triangular lattices (TL), square lattices (SL) and honeycomb lattices (HL), for different values of N (size) distributed in different arrays (shape). Equal amounts of F and AF bonds randomly distributed for a given array lead to a particular sample. Due to fluctuations from sample to sample, results given below are obtained after averaging over 500 samples for each array to get general properties as N increases. Each state is an ordered collection of spin orientations. The ground level energy Eg can be solved by different methods that also allow us to find the degeneracy of the ground manifold W [1].
* Corresponding author. Fax: +56-45-251847; 252547; email: [email protected].
+56-45-
The spin site correlation parameter p is defined as
Ei
P=N
Es; a
iv)w ,
(2)
where I [ means absolute value, while div represents an integer division, so the result inside { } can be either 0 or 1. The index a runs over all the ground states. This parameter is more drastic than the q introduced by Edwards and Anderson [3], and other related magnitudes derived from it [4]. The parameter q can be calculated by means of the following expression:
% <_~ rTsTs? NW(W+I) '
q=2
(3)
where c~ and /3 play the role of initial and final states within the ground manifold. Each pair of states is counted once and the case a =/3 is allowed. Thus q is restricted to the interval [ - 1, 1]. It can be proven that I q l >--p. The bond correlation parameter h has been defined as [1]: '
h = "~ .,<~j
~
~-
div W,
(4)
where the first sum extends over the B pairs of nearest neighbors. The range for h is [0, 1]. However, it oscillates around 0.50 for SLs and TLs [1], while it stays close to 0.75 for HLs [5]. In this way, h represents the fraction of bonds that are not frustrated when scanning through all the states of the ground manifold. A similar parameter Ca, called rigidity, has been previously defined [6]. It represents the fraction of those bonds that are never frustrated plus those that always frustrate
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 8 8 5 3 ( 9 4 ) 0 1 1 8 3 - 4
E.E. Vogel et al. /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1793-1794
1794 1.o.
~G01.~tIR
[,,ATT]( ~
1.0
0.9-
0.9
o.5-
0.8
tion. Similar calculations were performed for SL where the descent is more pronounced than the one shown in Fig. 1
*
O.7
[1].
*
0.6
HLs were also considered where the descent is even more evident [5]. The conclusion may be that p tends to zero in the thermodynamic limit (N--* oo). However, we realize that this conclusion depends on two facts that will not be included here. On the one hand, is the definition of ergodic valley: we have found ways of optimizing the value of p according to this fact. On the other hand, the descent diminishes when the coordination number is increased. This lead us to think that two-dimensional systems with interactions beyond nearest neighbors, or systems with higher dimensionality, or both, will finally stabilize the value of p, and therefore of q. This is what the initial result actually showed [7,8]. In Fig. 2 we present the variations of h and C R with size for the case of SL. These parameters do not require any ergodic considerations for their calculation. The first striking feature is that both h and C R are constant, independent of size or shape. Evidently, h < C a as can be expected from Eq. (5) suggesting that the new parameter is more strict than the old one. The difference remains constant, at a value close to 0.03 for the range reported here. The advantage of h over C R is that the former is based on the bonds that are never frustrated when scanning the ground manifold and hence propagate information without distortion through the lattice. This makes h a good independent variable for studying percolation in these systems. We have done initial calculations of this property for SLs which show very interesting behavior [7]. A comparison of h with C R for HLs shows similar properties to those presented in Fig. 2. Namely, the difference C R - h remains constant, at the same value 0.03, for the same range as the one considered for SLs. Acknowledgements: The authors thank Fondecyt (Contract 1930385) and the Direcci6n de Investigaci6n Universidad de La Frontera ( # 9408) for partial support.
o.7~
*
0.6
*
*
=
*
*
*
o.5
0.5
0.3
0.3
0.4
0.2
****.
q
0.1
*****
P
1()
'
0.0
'
20.2
20.1 2'0
3'0
'
400.0
t[
Fig. 1. Comparison of p and q as a function of size N for different triangular lattices in arrays close to square shape. when scanning the ground manifold. We can calculate it by means of
CR=h+~.i~
~
IS'SJ2+J'Jl
divW,
(5)
where the symbols are the same as those used in previous equations. We have calculated these two pairs of order parameters for TLs, SLs and HLs as a function of size, staying close to shapes that resemble square arrays. We present a comparison of p and q for TLs in Fig. 1, and of h and C a for SLs in Fig. 2. Here, we comment on the general trends of these parameters for the different lattices. The results for p and q in Fig. 1 show that they decrease slowly with size. Both parameters tend to follow the same variations with size, so they represent essentially the same information about the system. However, the effort required to calculate p is negligible compared to that for q. In fact, the latter requires storing all the states to perform the calculations for q after the diagonalization. In contrast, p does not require storing states and it can be calculated during the diagonalization in a very straightforward way. Despite this advantage, p still requires a decision on the ergodic valley (half the configuration space) over which it will be calculated. This applies to q in the same way. We have used the criterion of choosing one spin strongly attached to its neighbors (surrounded by the maximum number of non-frustrated bonds), assigning an arbitrary orientation to it, and letting the rest of the spins follow from the diagonalization that considers this restricL0 -
sqffARIg I£I~CES
1.0
0.9 ~
0.9
o.8 ~
o.a
0.7 ~ O,6 -
0.7 ÷ n ÷o
O.6
=+ +oo* ~* =*
0.5~ •
o.,~
~i
o
o
0.5 o.4
0.3 ~
0.3
0.2 -
o.,
~
0.2
;;;;;
~
Ol
0.0 'lb'2b'3'o'4b'sb'6b'700.0 N
Fig. 2. Comparison of h and C R as a function of size N for different square lattices in arrays close to square shape.
References
[1] E.E. Vogel, J. Cartes, S. Contreras, W. Lebrecht and J. Villegas, Phys. Rev. B 49 (1994) 6018. [2] W. Lebrecht, and E.E. Vogel, Pro. 2nd Latin-American Workshop on Magnetism, Magnetic Materials and their Applications, eds. J.L. Morfin-L6pez and J.M. Sfinchez (Plenum, New York, 1994). [3] S.F. Edwards and P.W. Anderson, J. Phys. F 5 (1975) 965. [4] K. Binder and A.P. Young, Rev. Mod. Phys. 58 (1986) 801. [5] W. Lebrecht, and E.E. Vogel, Phys. Rev. B (1994) submitted. [6] F. Barahona, R. Maynard, R. Rammal and J.P. Uhry, J. Phys. A 15 (1982) 673. [7] E.E. Vogel, S. Contreras and J. Cartes, Proc. 2nd LatinAmerican Workshop on Magnetism, Magnetic Materials and their Applications, eds. J.L. Mor~n-L6pez and J.M. S~nchez (Plenum, New York, 1994). [8] F. Nieto, A.J. Ramirez and E.E. Vogel, Proc. VIII Taller Sur de Fisica del S61ido (Universidad de Concepci6n, Chile, 1994).