1. Monte Carlo Renormalization Group studies of the Baxter-Wu model in the presence of an external magnetic field
An application of Monte Carlo Renormalization Group for calculating critical points and exponents
Ioannis N. Velonakis
University of Athens, Department of Physics, Section of Solid State Physics, GR Athens, Greece

2. Abstract
In the present work we use the well known Monte Carlo Renormalization Group method to study the phase transition of the Baxter-Wu model in the presence of a negative external magnetic field. We find that the critical exponent ν is equal to 0.8, while the exponent γ is Our results are consistent to all scaling equalities, including the hyperscaling relation, within statistical errors. The validity of two usual order-parameters is also checked.

3. Baxter-Wu model Hamiltonian:
2-d model which involves only triple-spin interactions.
For zero external magnetic field it has 4 ground states with energy E = −2JN, a ferromagnetic (M = +NμB) and three ferrimagnetic ones (M = −3 −1 NμB).
For negative external magnetic field it has 4 ground states with energy E = −2JN − 3 −1 μBNΗc(T), a ferromagnetic (M = − NμB) and three ferrimagnetic ones (M = −3 −1 NμB).
For H = 0 and T = Tc , where kBTc = 2.269J, the model undergoes critical phase transition from ferrimagnetism to paramagnetism and vice versa.
For H = 0 and T < Tc , where kBTc = 2.269J, the model undergoes 1st-order phase transition from ferrimagnetism to paramagnetism and vice versa.
For H = Hc(T) < 0, the model undergoes phase transition from ferrimagnetism to phase (−−−), which is probably critical.

4. Baxter-Wu model Physical Importance- Applications (1):
Α simple, toy model of ferrimagnetism in two dimensions (most real ferrimagnetic materials are 3-d)
A simple 2-d model for describing critical behavior, whose critical temperature and critical exponents can found with analytic methods at zero external magnetic field.
It belongs to the Universality Class of q = 4 and d = 2 Potts models but no logarithmic correction-to-scaling terms appear when calculating critical exponents.
Triple interactions appear when quarx form hadrons but the corresponding phase transition does not belong to the above universality class.

5. Baxter-Wu model Physical Importance- Applications (2): :
Up to our knowledge, only three 2-dimensional real systems have been studied whose critical behavior seems to belong to the same universality class. These are the chemisorbed overlayer p (2×2) Oxygen on Ni(111) (Roelofs et al, 1981), ], the adsorption system O/Ru(0001) at ¼ monolayer (Piercy and Pfnür, 1987) and the (2×2)-2H structure on Ni(111) (Schwenger et al, 1994). The Hamiltonian of the system in the last case is that of the following lattice gas where (Chin and Landau, 1987)
Here μ is the chemical potential, εt is the energy of triple interactions, εb is the binding energy of heteroatms in adsorption, but

6. Baxter-Wu model Order- parameter for phase transitions:
For H = 0 it is a linear combination of the lattice’s mean magnetization (Novotny and Landau, 1981) or the rms average of the magnetizations of the three sublattices or, equivalently, the average of their absolute values (Santos and Figueiredo, 2001).
Tsai, Wang and Landau (2006, 2007a, 2007b) have defined a “vector” order parameter P similar to that used for the q = 3 Potts model. The parameter P is unity for the three ferrimagnetic states while it is zero everywhere else. The susceptibility is treated not only as the derivative of the mean magnetization but also as the derivative of the order-parameter P.

7. Methods of calculating critical points and the corresponding critical exponents from Monte Carlo data
Finite-Size Scaling (FSS)
Monte Carlo Renormalization Group (MCRG)
It is based on Phenomenological Renormalization Theory of Nightingale (1978).
It can give all critical exponents directly.
It requires considerable computational effort to give results of high quality
It is based on Renormalization Group Theory of Wilson (1972, 1974) and Fisher (1974).
It was introduced by Ma (1976) and was improved by Swendsen (1979, 1982).
It requires less computational effort to give results of high quality.
It suffers from uncontrollable sources of errors.

8. Applying MCRG method to the Baxter-Wu model
Perform a Monte Carlo simulation (e.g. using the Metropolis Algorithm) at temperature T and calculate measurable quantities like the internal energy per bond u.
Take each of the states generated by the simulation and block them (majority rule blocking scheme).
Calculate the internal energy u΄ for the blocked system, averaged over all of these blocked states.
Vary the temperature T using the Single Histogram Method until we find the point at which u = u΄. This is the critical temperature of the system.
The method can be varied to give the critical external magnetic field.

9. The majority rule for the Baxter-Wu model
Choose each spin “block” to be made up of seven spins all from the same sublattice (next-nearest-neighbor spins) with the spins arranged in the shape of a star (Novotny and Landau,1982).

10. MCRG results from internal energy for varying temperature and constant external magnetic fieldH
Tc1 (21×18)
δTc1( 21×18)
ν1 (21×18)
δν1(21×18)
Tc2(9×6)
δTc2(9×6)
ν2(9×6)
δν2(9×6)
+0.0
2.2657
0.0001
0.67
0.04
2.2673
0.6650
0.0009
−0.1
2.3231
0.0002
0.82
2.325
0.002
−0.2
2.3599
0.802
0.006
2.3625
0.762
−0.4
2.407
0.003
0.9
0.2
2.4047
−0.5
2.4276
0.0006
0.80
0.02
2.4296
0.771
0.009
−0.6
2.4424
0.7993
0.0004
2.4433
0.782
0.001
−0.8
2.4622
0.8041
0.0007
2.4632
0.7889
0.0003
−1.0
2.4712
0.8062
2.4726
0.7917
−1.5
2.4551
0.8067
0.0005
2.4577
0.7931
−2.0
2.3874
0.8021
2.3910
0.7900
−2.5
2.2763
0.7908
2.2803
0.7841
−3.0
2.1007
0.7737
2.1044
0.7781
−3.5
1.8879
0.7863
1.8908
0.7808
−4.0
1.5943
0.7719
1.5963
0.7783
−4.5
1.2799
0.745
1.2811
0.777
−5.0
0.8864
0.81
0.8870
−5.5
0.4194
0.7372
0.4195
0.7840
−5.8
0.1698
−5.9
0.0840
0.6486
0.7020

11. MCRG results from P order parameter for varying external magnetic field and constant temperature
H1 (21×18)
δHc1( 21×18)
r1 (42×18)
δr1(21×18)
Hc2(9×6)
δHc2(9×6)
r2(9×6)
δr2(9×6)
2.2692
0.0001
1.82
0.01
1.790
0.005
2.426
−0.5492
0.0003
0.449
0.002
−0.5462
0.0002
0.4633
0.0008
2.471
−1.1693
0.0004
0.429
0.003
−1.1514
0.411
0.001
2.457
−1.4335
0.385
−1.4502
0.3535
0.0005
2.389
−1.9557
0.3878
0.0009
−1.9650
0.3622
2.275
−2.4752
0.398
−2.4817
0.3741
0.0007
2.111
−2.9907
0.409
−2.9957
0.3866
1.883
−3.4894
0.426
−3.4931
0.4039
1.591
−3.9920
0.437
−3.9949
0.416
1.230
−4.4915
0.448
0.004
−4.4936
0.425
0.850
−5.0220
0.453
−5.0234
0.430
0.4195
−5.5041
0.45
−5.5048

12. MCRG results from magnetic order parameter for varying external magnetic field and constant temperature T
H1 (21×18)
δHc1( 21×18)
r1 (42×18)
δr1(21×18)
Hc2(9×6)
δHc2(9×6)
r2(9×6)
δr2(9×6)
2.2692
0.0001
1.765
0.006
1.773
2.426 -
2.471
2.457
2.389
2.275
−2.5577
−0.0188
0.0008
−2.5222
0.1465
0.0005
2.111
−3.0220
0.383
0.003
−3.0126
0.3532
1.883
−3.4993
0.563
0.001
−3.4991
0.4572
0.0006
1.591
−3.9931
0.634
0.007
−3.9962
0.505
1.230
−4.4899
0.661
−4.4933
0.526
0.850
−5.0204
0.665
−5.0230
0.532
0.4195
−5.5033
0.667
0.002
−5.5046
0.534

13. Phase diagram of the Baxter-Wu model for H < 0
Hc (FSS)
Tc (FSS)
δTc (FSS)
Tc (MCRG)
δTc (MCRG)
+0.0
2.270
0.001
2.2673
0.0001
−0.1
2.324
0.002
2.325
−0.2
2.358
2.3625
−0.4
2.410
2.4047
0.0002
−0.5
2.426
2.4296
0.0009
−0.6
2.440
2.4433
−0.8
2.461
2.4632
−1.0
2.471
2.4726
−1.5
2.457
2.4577
0.0005
−2.0
2.389
2.3910
−2.5
2.275
0.003
2.2803
−3.0
2.111
0.007
2.1044
−3.5
1.883
1.8908
−4.0
1.591
0.008
1.5963
−4.5
1.230
1.2811
−5.0
0.85
0.02
0.8870
Polynomial Fit of degree = 8, R = 1

14. Comparing the results FSS method MCRG method
Baxter-Wu model undergoes a 2nd order phase transition along the phase boundary Hc = Hc(T) which begins from the point (−6J, 0) and ends to the well-known critical point (0, Tc).
The zero-field Baxter-Wu critical point is a tetracritical point.
The line Hc = Hc(T) < 0 separates the region of the three coexisting ferrimagnetic phases and the region of the ordered (− − −) phase (tricritical line). So, its critical points should be in a different universality class from the tetracritical point.
The corresponding critical exponents form a new universality class, with values ν ≈ 1.00 ± 0.02, β ≈ 0.75 ± 0.05, γ ≈ 0.40 ± 0.05, α ≈ 0.50 ± 0.01.
The phase transition line and other results of FSS method are valid.
The corresponding critical exponents form a new universality class, with values ν ≈ 0.80 ± 0.02 and
The critical exponents are ν ≈ 0.80 ± 0.02, β ≈ 0.64 ± 0.05, γ ≈ 0.32 ± 0.05, α ≈ 0.40 ± 0.01.

15. Possible choices for critical exponents
The difference in ν is due to FSS effects
The critical exponents are ν ≈ 0.80 ± 0.02, β ≈ 0.64 ± 0.05, γ ≈ 0.32 ± 0.05, α ≈ 0.40 ± 0.01.
All critical exponent inequalities are held as equalities, as expected, including hyperscaling relation and Josephson relation.
The difference in ν is a characteristic of the phase transition (following Binder et al., 1985)
yT* = 1.00 ± 0.02, ω* = −0.50 ± 0.02, d = d* = 2.
The critical exponents are ν ≈ 0.80 ± 0.02, β ≈ 0.85 ± 0.05, γ ≈ 0.30 ± 0.05 and α ≈ 0, while in this case FSS gives α ≈ 0.50 ± 0.01 (the case is rejected !!!).

16. Order-parameter of the phase transition for negative external magnetic field
The P order parameter (Tsai, Wang, Landau, 2006, 2007a, 2007b) is clearly a better choice than the magnetic order parameter.
The gap that appears near tetracritical point in the calculation of magnetization critical exponents is probably due to poor order parameter choice.

17. Conclusions
Baxter-Wu model undergoes a 2nd order phase transition along the phase boundary Hc = Hc(T) which begins from the point (−6J, 0) and ends to the well-known critical point (0, Tc).
The zero-field Baxter-Wu critical point is a tetracritical point.
The line Hc = Hc(T) < 0 separates the region of the three coexisting ferrimagnetic phases and the region of the ordered (− − −) phase (tricritical line). So, its critical points should be in a different universality class from the tetracritical point.
The critical exponents are ν ≈ 0.80 ± 0.02, β ≈ 0.64 ± 0.05, γ ≈ 0.32 ± 0.05, α ≈ 0.40 ± 0.01.
All critical exponent inequalities are held as equalities, as expected, including hyperscaling relation and Josephson relation.
The P order parameter (Tsai, Wang, Landau, 2006, 2007a, 2007b) is clearly a better choice than the magnetic order parameter.

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20. Acknowledgments State Scholarships Foundation (I. K. Y.), Greece.
This work was partly supported by the Special Account for Research Grants of the University of Athens under Grant no. 70/40/7677.
Ass. Professors S. S. Martinos, A. Malakis, I. Hadjiagapiou
Professor M. A. Novotny, for his advice.
My colleagues to Solid State Physics Session, Department of Physics, University of Athens

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