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Antiferromagnetict riangular Ising model
Volume 58A, number 7
PHYSICS LETTERS
18 October 1976
ANTIFERROMAGNETICT RIANGULAR ISING MODEL M. SCHICK, J.S. WALKER and M. WORTIS * Department of Physics, University of Washington, Seattle, Wa. 98195, USA Received 7 July 1976 We calculate the magnetic field dependent transition temperature of the triangular Ising lattice with nearest neighbor antiferromagnetic interactions employing renormalization group techniques. Our result agrees well with Monte Carlo calculations.
The spin ~ Ising model on a triangular lattice with nearest neighbor antiferromagnetic interactions ~
K
(1)
is experimentally relevant because of its application (as a lattice-gas model) to phase transitions in monolayers adsorbed on hexagonal substrates [1,2]. The triangular lattice may be decomposed into three interpenetrating sublattices A, B, C which only contain spins which are second-nearest neighbors of one another. Theoretical interest attaches to the possibility of a transition from the paramagnetic phase, in which all sublattice magnetizations are equal, to an antiferromagnetic phase characterized by two equal sublattice magnetizations which are unequal to the third. As this third sublattice can be either the A, B or C sublattice, there are three possible ordered phases and the transition to the disordered phase corresponds to the special tricritical point of the three-state Potts model [3,4]. The magnetic field dependence of this transition temperature T~(H),has attracted considerable attention. Parity requires Tc(H) = Tc(~H).It is easy to thow from ground state energy considerations that the antiferromagnetic phase is unstable to ferromagnetic alignment if H/K I >6. Furthermore at H 0, the problem is exactly soluble [5—i11 and there is no phase transition at any finite temperature. The absence of a phase transition at H 0 is associated with a finite entropy per spin at zero temperature [7]. Fluctuations associated with the large degenerate manifold suffice to suppress ordering. For non-zero H, this *
Permanent address: Department of Physics, University of Illinois at Urbana-Champaign, Urbana, ill. U.S.A.
~
A
‘~..J
B
C A
B
A A
A
A
B
A
B
C
c
B C
B
A
B C
A B
c
B C
A~(
B<
c A
B
A A
c
~
B
B
~—1
A C
B
_—~1 ~.—1
Fig. 1. Division of lattice into sublattices and clusters.
choice of
th.~
degeneracy is absent and Monte Carlo results [12] confirm the supposition of a phase boundary at finite Tc(H). All previous analytic calculations of the phase boundary have been of the mean-field type and lead to finite values of Tc(O) in contradiction to the exact result [13]. The reason for this is clear; mean-field theory neglects the fluctuations which supress the transition at H 0. We report below a renormalization group calculation of Tc(H) which yields Tc(O) = 0 and rather good agreement with the Monte Carlo results over the full range of H. We employ the position space renormalization group techniques of Niemeyer and van Leeuwen [141in the cluster approximation. it is necessary to subdivide the lattice As intonoted A, B,above C sublattices. We assign all spins to clusters which contain three spins from
479
Volume 58A, number 7
PHYSICS LETTERS
is \/~for our0.956. renormalization group transformation, we find ~T= This corresponds to a specific
_________
I
~
.4
-
.2
-
18 October 1976
sults for the three state Potts model yield [151 a
With two irrelevant eigenvalues, the domain of the Potts fixed point is a surface and it is the intersection
I.0~.
0.8
-
heat of 0.05±0.1. this exponent surface with a 2(1 the—y~) P 0ofplane = which defines is shown TJH) for inthe fig.Hamiltonian 2.= Also shown eq.—0.093. are (1). the points ThisSeries intersection andreerror bars calculated in ref. [121. The agreement is gratifying. In addition to the three state Potts transition, a system governed by the Hamiltonian of eq. (2) exhibits other phenomena of current interest including the
0.4 ~ 0.2 00
_______
I
___________________
2
3 4I H/lKI
5I
~
6
Fig. 2. Phase diagram for triangular antiferromagnet from this work (solid line). Data points are Monte Carlo results.
a given sublattice as shown in fig. 1. Our approximation is to treat only three inter-penetrating clusters, one from each sublattice, or nine spins in all. We assign the spin of the cluster according to simple majority rule. The specification of periodic boundary conditions completes the definition of our renormalization group operation and associated recursion relations. Although we start with only the physical interactions of eq. (1), our renormalization group operation will, in general, introduce an interaction P among three spins which are all nearest neighbors of one another. Thus we are led to consider the following generalization of eq. (1)
—h
=
K ~ S1S1 + H ~ S, + P
~ SiSISk
.
(2)
This Hamiltonian is symmetric under the simultaneous reflection ofH and P. Our recursion relations exhibit a fixed point which Potts+ transition at 1’ governs 0.893, the exp(H* 6p*_~6K*) expH* expP’ = 2.48. = By0,symmetry there is an equivalent fixed point whose coordinates are obtained from the above by reflection ofP and H. Following the standard procedure [14] of linearizing the recursion relations about the fixed point, we find that the Potts fixed point is characterized by two irrelevant eigenvalues and one relevant one, A. If we express this eigenvalue in the usual form A = b ~‘T where b is the dilation factor which
480
=
Baxter—Wu transition [16—191. The description of these phenomena provided by our recursion relations will be published elsewhere. We wish to acknowledge very helpful conversations with colleagues and friends G. Golner, E. Riedel and K. Subbarao.
References [1] CE. Campbell and M. Schick, Phys. Rev. A5 (1972) 1919. 12] M. Bretz eta!., Phys. Rev. A8 (1973) 1589. 131 RB. Potts, Proc. Camb. Phil. Soc. 48 (1952) 106. [4] S. Alexander, Phys. Lett. 54A (1975) 353. [5] R.M.F. Houtappel, Physica 16 (1950) 425. 161 K. Husimi and I. Syozi, Prog. Theor. Phys. 5 (1950) 177, 341. [7] G. Wannier, Phys. Rev. 79 (1950) 357. [8] G.F. Newell, Phys. R~v.79 (1950) 876. [91 RB. Potts, Phys. Rev. 88 (1952) 352. [10] R.N.V. Temperley, Phys. Rev. 103 (1956) 1. [11] T.P. Eggarter, Ph~s.Rev. B12 (1975) 1933. [12] B.D. Metcalf, Phys. Lett. 45A (1973) 1. [13] Burley, Proc. Phys. Soc. 85 (1965)Physica 1163. 71 [14] D.M. Th. Niemeyer and J.M.J. van Leeuwen, (1974) 17. [15] J.P. Straley and M.E. Fisher, J. Phys. A6 (1973) 1310. Note however the valuea 0.26 obtained by D. Kim and R.I. Joseph, J. Phys. A8 (1975) 891. [16] R.J. Baxter and F.Y. Wu, Phys. Rev. Letters 31(1973) 1924. [171 R.J. Baxter, M.F. Sykes and M.G. Watts, J. Phys. A8 (1975) 245. [18] D. Imbro and P.C. Hemmer, Phys. Lett. 57A (1976) 297. [19] M.P.M. den Nijs et al., preprint.
PHYSICS LETTERS
18 October 1976
ANTIFERROMAGNETICT RIANGULAR ISING MODEL M. SCHICK, J.S. WALKER and M. WORTIS * Department of Physics, University of Washington, Seattle, Wa. 98195, USA Received 7 July 1976 We calculate the magnetic field dependent transition temperature of the triangular Ising lattice with nearest neighbor antiferromagnetic interactions employing renormalization group techniques. Our result agrees well with Monte Carlo calculations.
The spin ~ Ising model on a triangular lattice with nearest neighbor antiferromagnetic interactions ~
K
(1)
is experimentally relevant because of its application (as a lattice-gas model) to phase transitions in monolayers adsorbed on hexagonal substrates [1,2]. The triangular lattice may be decomposed into three interpenetrating sublattices A, B, C which only contain spins which are second-nearest neighbors of one another. Theoretical interest attaches to the possibility of a transition from the paramagnetic phase, in which all sublattice magnetizations are equal, to an antiferromagnetic phase characterized by two equal sublattice magnetizations which are unequal to the third. As this third sublattice can be either the A, B or C sublattice, there are three possible ordered phases and the transition to the disordered phase corresponds to the special tricritical point of the three-state Potts model [3,4]. The magnetic field dependence of this transition temperature T~(H),has attracted considerable attention. Parity requires Tc(H) = Tc(~H).It is easy to thow from ground state energy considerations that the antiferromagnetic phase is unstable to ferromagnetic alignment if H/K I >6. Furthermore at H 0, the problem is exactly soluble [5—i11 and there is no phase transition at any finite temperature. The absence of a phase transition at H 0 is associated with a finite entropy per spin at zero temperature [7]. Fluctuations associated with the large degenerate manifold suffice to suppress ordering. For non-zero H, this *
Permanent address: Department of Physics, University of Illinois at Urbana-Champaign, Urbana, ill. U.S.A.
~
A
‘~..J
B
C A
B
A A
A
A
B
A
B
C
c
B C
B
A
B C
A B
c
B C
A~(
B<
c A
B
A A
c
~
B
B
~—1
A C
B
_—~1 ~.—1
Fig. 1. Division of lattice into sublattices and clusters.
choice of
th.~
degeneracy is absent and Monte Carlo results [12] confirm the supposition of a phase boundary at finite Tc(H). All previous analytic calculations of the phase boundary have been of the mean-field type and lead to finite values of Tc(O) in contradiction to the exact result [13]. The reason for this is clear; mean-field theory neglects the fluctuations which supress the transition at H 0. We report below a renormalization group calculation of Tc(H) which yields Tc(O) = 0 and rather good agreement with the Monte Carlo results over the full range of H. We employ the position space renormalization group techniques of Niemeyer and van Leeuwen [141in the cluster approximation. it is necessary to subdivide the lattice As intonoted A, B,above C sublattices. We assign all spins to clusters which contain three spins from
479
Volume 58A, number 7
PHYSICS LETTERS
is \/~for our0.956. renormalization group transformation, we find ~T= This corresponds to a specific
_________
I
~
.4
-
.2
-
18 October 1976
sults for the three state Potts model yield [151 a
With two irrelevant eigenvalues, the domain of the Potts fixed point is a surface and it is the intersection
I.0~.
0.8
-
heat of 0.05±0.1. this exponent surface with a 2(1 the—y~) P 0ofplane = which defines is shown TJH) for inthe fig.Hamiltonian 2.= Also shown eq.—0.093. are (1). the points ThisSeries intersection andreerror bars calculated in ref. [121. The agreement is gratifying. In addition to the three state Potts transition, a system governed by the Hamiltonian of eq. (2) exhibits other phenomena of current interest including the
0.4 ~ 0.2 00
_______
I
___________________
2
3 4I H/lKI
5I
~
6
Fig. 2. Phase diagram for triangular antiferromagnet from this work (solid line). Data points are Monte Carlo results.
a given sublattice as shown in fig. 1. Our approximation is to treat only three inter-penetrating clusters, one from each sublattice, or nine spins in all. We assign the spin of the cluster according to simple majority rule. The specification of periodic boundary conditions completes the definition of our renormalization group operation and associated recursion relations. Although we start with only the physical interactions of eq. (1), our renormalization group operation will, in general, introduce an interaction P among three spins which are all nearest neighbors of one another. Thus we are led to consider the following generalization of eq. (1)
—h
=
K ~ S1S1 + H ~ S, + P
~ SiSISk
.
(2)
This Hamiltonian is symmetric under the simultaneous reflection ofH and P. Our recursion relations exhibit a fixed point which Potts+ transition at 1’ governs 0.893, the exp(H* 6p*_~6K*) expH* expP’ = 2.48. = By0,symmetry there is an equivalent fixed point whose coordinates are obtained from the above by reflection ofP and H. Following the standard procedure [14] of linearizing the recursion relations about the fixed point, we find that the Potts fixed point is characterized by two irrelevant eigenvalues and one relevant one, A. If we express this eigenvalue in the usual form A = b ~‘T where b is the dilation factor which
480
=
Baxter—Wu transition [16—191. The description of these phenomena provided by our recursion relations will be published elsewhere. We wish to acknowledge very helpful conversations with colleagues and friends G. Golner, E. Riedel and K. Subbarao.
References [1] CE. Campbell and M. Schick, Phys. Rev. A5 (1972) 1919. 12] M. Bretz eta!., Phys. Rev. A8 (1973) 1589. 131 RB. Potts, Proc. Camb. Phil. Soc. 48 (1952) 106. [4] S. Alexander, Phys. Lett. 54A (1975) 353. [5] R.M.F. Houtappel, Physica 16 (1950) 425. 161 K. Husimi and I. Syozi, Prog. Theor. Phys. 5 (1950) 177, 341. [7] G. Wannier, Phys. Rev. 79 (1950) 357. [8] G.F. Newell, Phys. R~v.79 (1950) 876. [91 RB. Potts, Phys. Rev. 88 (1952) 352. [10] R.N.V. Temperley, Phys. Rev. 103 (1956) 1. [11] T.P. Eggarter, Ph~s.Rev. B12 (1975) 1933. [12] B.D. Metcalf, Phys. Lett. 45A (1973) 1. [13] Burley, Proc. Phys. Soc. 85 (1965)Physica 1163. 71 [14] D.M. Th. Niemeyer and J.M.J. van Leeuwen, (1974) 17. [15] J.P. Straley and M.E. Fisher, J. Phys. A6 (1973) 1310. Note however the valuea 0.26 obtained by D. Kim and R.I. Joseph, J. Phys. A8 (1975) 891. [16] R.J. Baxter and F.Y. Wu, Phys. Rev. Letters 31(1973) 1924. [171 R.J. Baxter, M.F. Sykes and M.G. Watts, J. Phys. A8 (1975) 245. [18] D. Imbro and P.C. Hemmer, Phys. Lett. 57A (1976) 297. [19] M.P.M. den Nijs et al., preprint.