Monte Carlo study of the Baxter-Wu model

Monte Carlo study of the Baxter-Wu model Nir Schreiber and Dr. Joan Adler Monte Carlo study of the Baxter-Wu model p.1/40

Outline Theory of phase transitions, Monte Carlo simulations and finite size scaling Landau-Wang algorithm Results Summary Monte Carlo study of the Baxter-Wu model p.2/40

Phase transitions A physical system is said to exhibit a phase transition when its order parameter changes from zero in the disordered phase to a nonzero value in the ordered phase. This change occurs at the critical point. The order parameter can be a scalar function of external variables such as magnetic field and temperature (magnetic transition) or a multicomponent quantity. When the change in the order parameter is discontinuous the system undergoes a first order transition and when it is continuous the transition is second order. Monte Carlo study of the Baxter-Wu model p.3/40

Ising models Introduced by W. Lenz (1920) to explain magnetism Defined by the Hamiltonian H = J i,j σ iσ j If J is positive (negative) the spin interaction in the Hamiltonian above is said to be F erromagnetic (Antif erromagnetic) A net spontaneous magnetization occurs at the critical temperature, T c, in the absence of an external field and the system exhibits a magnetic phase transition at T c Ising (1925) showed that spontaneous magnetization cannot occur in d=1 Onsager (1944) exactly solved the 2d model on the square lattice Monte Carlo study of the Baxter-Wu model p.4/40

Critical exponents The thermodynamic functions are expected to have power law behavior near the critical point (t (T T c )/T c ): M t β t 0 χ t γ t 0 C t α t 0 ξ t ν t 0 where α, β, γ and ν are called critical exponents The following scaling relations between the exponents hold: α + 2β + γ = 2 (Rushbrooke equality), and the hyperscaling relation dν = 2 α (d is the dimensionality) Monte Carlo study of the Baxter-Wu model p.5/40

The hyper scaling relation is easily seen from the equation for the specific heat. Substituting it in the other two, and manipulating Monte Carlo study of the Baxter-Wu model p.6/40 them, yields the Rushbrooke equality The Scaling Hypothesis The power law behavior results from the scaling hypothesis which assumes homogeneity of the free energy near T c. The singular part of the latter has the form f s (t, h) = l d φ ± (tl y t, hl y h ) where l is a scaling parameter. Set t l y t = 1, define y h /y t, 2 α d/y t, and identify 1/y t with ν to obtain: C = 2 f s / t 2 h=0 t dν 2 = t α M = f s / h h=0 t dν = t β χ = 2 f s / h 2 h=0 t dν 2 = t γ

Monte Carlo (MC) Simulations In order to calculate the partition function accurately for a finite lattice with N spins and Q states per spin we need to count Q N configurations. Even for relatively small lattices this is far beyond the capability of a powerful PC, and even of a supercomputer We consider, instead, a sequence of equilibrium configurations satisfying the detailed balance condition p i T ij = p j T ji where p i is the probability to be in a state i and T ij is the transition probability between states i and j. In order that the configurations will not be highly correlated, we choose the typical time scale 1 MCS (Monte Carlo Step per Spin) to be a random path through the lattice, consisting of N single spin flips Monte Carlo study of the Baxter-Wu model p.7/40

Monte Carlo (MC) Simulations The simulation stops when a large number of uncorrelated states are generated. Near T c, the simulation takes a "very" long time since configurations in the vicinity of a phase transition are highly correlated even for large time scales. This is better known as "critical slowing down" Metropolis MC We choose configurations with probability proportional to the Boltzmann factor E/kT. It can be shown that the choice T ij = min [1, ] e (E j E i )/kt satisfies detailed balance Monte Carlo study of the Baxter-Wu model p.8/40

Finite Size Scaling-second order transitions On finite lattices we no longer have singularities at T c and the lattice linear dimension, L, must also be considered. The correlation length at T c has a cutoff at the linear endpoint of the lattice, so that the free energy scales with L as f L = L (2 α)/ν φ(tl 1/ν, hl /ν ) Appropriate differentiation yields the following scaling forms at t = 0: M(0) L β/ν χ(0) L γ/ν C(0) L α/ν The theory is valid below an upper critical dimension, d u. In higher dimensions, the correlations are much smaller than L and no significant finite-size effects should be Monte Carlo study of the Baxter-Wu model p.9/40 seen

The reweighting method The energy distribution at an inverse temperature β P β (E) = g(e)e βe /Z β can be approximated by the energy histogram H(E) P β H(E)/M where M is the number of measurements made. Manipulating these two equations gives the reweighted distribution (histogram) at another temperature β P β (E) H(E)e (β β)e / E H(E)e (β β)e The last formula can be used to calculate averages at the new temperature f(e) β = f(e)p β (E) Monte Carlo study of the Baxter-Wu model p.10/40

The reweighting method The reweighting method can also be used to determine a first order transition point. Finite size scaling theory of first order transitions tells us that if there is a ratio, r, between the number of ordered and disordered states, at the transition point, the energy distribution is a sum of 2 weighted Gaussians of the form P L (E) e Ld (E Eo) 2 2kT 2 C centered at the ordered state energy E and at the disordered state energy E +, such that P L (E ) = rp L (E + ) Monte Carlo study of the Baxter-Wu model p.11/40

The reweighting method The reweighting method applied for the ferromagnetic q = 5 Potts model defined by the Hamiltonian H = J i,j δ σi σ j where σ i = 1, 2,..., 5 and J > 0. 10 (a) Equal height distribution at T Cmax 50 (b) Reweighted distribution r=5 Energy distribution 8 6 4 2 Energy distribution 40 30 20 10 0-2 -1 0 Energy [J] 0-3 -2 E - -1 E + 0 Energy [J] Monte Carlo study of the Baxter-Wu model p.12/40

Landau-Wang (LW) Sampling The transition probability between energy level E i and E j is proportional to the ratio between the "densities" (or more precisely the number of states with a given energy) of the two states The energy space is divided into segments and a random walk is carried out independently on each segment until the system attains equilibrium The random walk is done iteratively and at each iteration the densities of states are multiplied by a "modification factor", f. From H(E) g(e)e βe it is clear that f is proportional to the Boltzmann factor at an inverse temperature β The simulation converges to the true value when f = f final is approximately 1 or when the system is heated to an "infinite" temperature Monte Carlo study of the Baxter-Wu model p.13/40

LW flow-chart Set all density of states entries, {g(e)}, to unity Choose initial modification factor f 0 = 2.7182... Perform a random walk with transition probability P Ei Ej = min On each MC sweep [ 1, g(ei) ] g(ej) Modify a given density of states log (g(e)) log (g(e)) + log(f) Accumulate the histogram Check the histogram every 10,000 sweeps: If H(E) H E reduce f a to f a+1 = fa 1/2 (a=0,1,2,..) and set the histogram to zero Repeat above procedure from the third ( ) step until f a 1 Monte Carlo study of the Baxter-Wu model p.14/40

Calculation of thermodynamic functions From the calculation of the density of states we can compute the partition function Z β = E g(e) exp( βe) and its first and second moments which yield the Internal Energy U E β = E Eg(E) exp( βe)/z β and the Specific Heat C = U/ T = ( E 2 β E 2 β)/kt 2 Monte Carlo study of the Baxter-Wu model p.15/40

Analytic results The partition function for the 2d Ising model on the square lattice with length L can be written as a low temperature expansion Z N = e 2NK N n=0 g n x 2n where N = L L is the number of spins, K is the dimensionless inverse temperature and x = e 2K is the low temperature variable. Using the extension of Onsager s solution to give the exact expression for the partition function on a finite lattice (Kaufman 1949) one can look at successive powers of x and extract the density of states coefficients in the expansion above (Beale 1995) Monte Carlo study of the Baxter-Wu model p.16/40

Analytic results Exact (Beale) and simulated densities (this work) for 32x32 Ising model on the square lattice 800 Exact Simulation 600 Ln[g(E)] 400 0.2 Ln(g sim )-Ln(g exact ) 0.1 200 0-0.1 0-0.2-2 -1.5-1 -0.5 0-2 -1 0 1 2 E/N [J] Monte Carlo study of the Baxter-Wu model p.17/40

Analytic results The error in the simulated densities depends on many factors such as the size of the system, the "flatness" of histogram (in our simulation H(E) H / H 0.05 for any energy E) and, of course, on the modification factor. The error, hence, can be no smaller than log(f final ) 0.0001. 620.5 620 Log(g) 619.5 619 618.5 618-0.1-0.08-0.06-0.04-0.02 E [J] Monte Carlo study of the Baxter-Wu model p.18/40

The pure Baxter-Wu (BW) model A three body interaction model on a triangular lattice with the Hamiltonian H = J i,j,k σ i σ j σ k where i, j and k are the vertices of a triangle. σ i σ j σ k It was solved exactly by R.J. Baxter and F.Y.Wu (1974) and was found to exhibit a second order transition with the specific heat exponent α = 2/3, and the correlation length exponent ν = 2/3 (c.f. for Ising model α = 0 and ν = 1). The critical temperature is the same as for the Ising model T c k/j = 2/ln( 2 + 1). Monte Carlo study of the Baxter-Wu model p.19/40

The pure Baxter-Wu (BW) model It has four ground states: one f erromagnetic state (upper left) and three f errimagnetic states Monte Carlo study of the Baxter-Wu model p.20/40

The pure Baxter-Wu (BW) model Near the transition point (T k/j(l=150)=2.271) domains of different ground states can be seen. Monte Carlo study of the Baxter-Wu model p.21/40

The pure Baxter-Wu (BW) model Calculation of thermodynamic functions (54 54 lattice). -0.5 (a) -1.5-2 (d) U [J] -1-1.5-2.5-3 -3.5 F [J] -2 1.5 2 2.5 3 100 (b) -4 1 1.5 2 2.5 3 3.5 4 2 1.5 (c) C [k B ] 50 1 0.5 S [k B ] 0 1.5 2 2.5 3 T [J/k B ] 0 1.5 2 2.5 3 T [J/k B ] Monte Carlo study of the Baxter-Wu model p.22/40

The pure Baxter-Wu (BW) model The specific heat maximum scales perfectly well according to the 2nd order hypothesis C max (L) L α/ν, with α/ν = 1. 250 200 C max 150 100 50 0 0 50 100 L Monte Carlo study of the Baxter-Wu model p.23/40

The pure Baxter-Wu (BW) model Two pronounced peaks at the "ordered" ferromagnetic and "disordered" ferrimagnetic states energies E and E + Energy distribution 10 5 L 21 24 27 30 36 54 120 Energy distribution 10 8 6 4 2 L=54 exact BW at T c BW BW at T Cmax BW Ising at T Cmax Ising Ising at T Cmax 0-2 -1.75-1.5-1.25-1 -0.75-0.5 E/N [J] 0-2 -1.75-1.5-1.25-1 Eenergy [J] Monte Carlo study of the Baxter-Wu model p.24/40

The pure Baxter-Wu (BW) model "Time" evolution of The Internal Energy show long range fluctuations around E and E + (L=150, T k/j = 2.271) Monte Carlo study of the Baxter-Wu model p.25/40

The pure Baxter-Wu (BW) model The doubly peaked distribution is a finite size effect E ± (L) = U c E 0± L φ ±/ν ; U c /J = 2 with the scaling ratio φ + /ν = 0.531 ± 0.048-0.8-1 ln(e + ) -1.2-1.4-1.6-1.8 3 3.2 3.4 3.6 3.8 4 4.2 ln(l) Monte Carlo study of the Baxter-Wu model p.26/40

The pure Baxter-Wu (BW) model The condition for extrema of the energy distribution P (E) is satisfied by d (lng(e)) /de = 1/kT Two local solutions in [E, E + ], at T Cmax, f 1 (E) = E/kT Cmax + C 1 f 2 (E) = E/kT Cmax + C 2, such that f 1 (E) is tangent to lng(e) at E and E + respectively, and f 2 (E) is tangent to lng(e) at the minimum of the distribution between the peaks, give P (E ) = P (E + ) = e C 1 Monte Carlo study of the Baxter-Wu model p.27/40

The pure Baxter-Wu (BW) model The Binder parameter B = 1 E 4 /3 E 2 2 has an inverse peak, usually seen in first order transitions, which tends to the limit B min = 1 2(E 4 + E 4 +)/3(E 2 + E 2 +) 2 0.67 0.665 0.655 Binder parameter 0.66 0.655 0.65 B min =0.6583 L 60 66 72 78 111 120 B min 0.65 0.645 y=0.6583-52.628x 0.645 0.64 2.2 2.25 2.3 2.35 Temperature [J/k B ] 0.64 0 0.0001 0.0002 0.0003 0.0004 Monte Carlo L -2 study of the Baxter-Wu model p.28/40

The pure Baxter-Wu (BW) model Scaling of the Binder parameter minimum temperature T Bmin, with the inverse volume 2.278 T c sim =2.2696±0.0004 T c exact =2.26918... 2.276 T Bmin 2.274 2.272 2.27 2.268 0 0.0001 0.0002 0.0003 0.0004 L -2 Monte Carlo study of the Baxter-Wu model p.29/40

The pure Baxter-Wu (BW) model A new exponent θ B B min = 2/3 B 0 L θ B/ν θ B = α C max = C 0 L α/ν 1000 C max and (2/3-B min ) -1 100 10 C max B min 1 10 100 L Monte Carlo study of the Baxter-Wu model p.30/40

The pure Baxter-Wu (BW) model Large corrections to scaling can mislead... T Bmin = T c + L 1/ν ( a + bl θ 1 +... ) ; θ 1 0.25 1/ν + θ 1 1.50 + 0.25 d = 2 1000 t Bmin t Cmax Inverse distances 100 10 10 100 L Monte Carlo study of the Baxter-Wu model p.31/40

Quenched site dilute disorder Consider a binary alloy of atoms A, which are magnetic, and B which are non magnetic, with a concentration, x, of magnetic atoms, A x B 1 x The site disorder is described by uncorrelated variables n i which take the values 0 and 1 such that their configurational average is n i c = x Quenching means that configurational averages... c are independent of thermal averages and have the canonical probability P (n i ) = xδ(n i 1) + (1 x)δ(n i ) The critical behavior will change if the specific heat exponent α in the undiluted (pure) system is positive (Harris 1974) Monte Carlo study of the Baxter-Wu model p.32/40

The Dilute Ferromagnetic Ising model (DFI) Described by the Hamiltonian H DFI = J i,j n i n j σ i σ j The transition remains second order for increasing dilution There is a controversy about the nature of the critical exponents : while Shalev (1984), Shankar (1987) and Ludwig (1988) argue that the magnetization and susceptibility have the same exponents as in the pure case and only display logarithmic corrections, Heuer (1991) claims that the susceptibility exponent γ continuously increases with dilution, and changes from γ 1.74 (γ = 7/4 exactly) at x = 1 to γ 2.16 at x = 0.7 Monte Carlo study of the Baxter-Wu model p.33/40

The Dilute Ferromagnetic Ising model (DFI) Small energy fluctuations for x 0.8 make it hard to reliably determine the critical temperature Temperature [J/k B ] 3 2.5 2 1.5 1 0.5 Phase diagram in the T-p plane The solid line represents the critical line T c (p) 1 0.95 0.9 0.8 T c (p)={tanh -1 [e -1.45(p-p c ) ]} -1 MCRG L=192 p c =0.593 C [k B ] Variation of the specific heat with temperature 10 8 6 4 2 L=22 p 1 0.95 0.9 0.8 0.75 0 0.5 0.6 0.7 0.8 0.9 1 Concentration x 0 1 1.5 2 2.5 3 Temperature [J/k B ] Monte Carlo study of the Baxter-Wu model p.34/40

The Dilute Ferromagnetic BW model (DFBW) The dilute BW model is defined by the Hamiltonian H DFBW = J i,j,k n i n j n k σ i σ j σ k Normalized temperature 1 0.8 0.6 0.4 0.2 L 24 T(1)=1 X c =0.755±0.020 0 0.7 0.75 0.8 0.85 0.9 0.95 1 Concentration Normalized temperature 1 0.5 L 18 30 linear fit to the L=24 data 33 0 0.7 0.75 0.8 0.85 0.9 0.95 1 Concentration Monte Carlo study of the Baxter-Wu model p.35/40

The Dilute Ferromagnetic BW model (DFBW) The energy distribution becomes sharper when the concentration is reduced. It may indicate that magnetic fluctuations become larger, in contrary to the reduction in energy fluctuations. 20 Energy distribution 15 10 5 A T c k B /J A AAAA A AAA A AAAA x 2.29164 1 2.10953 0.97 1.93379 0.95 1.74926 0.85 AA A AAA A A A AA A A AA A A AA A A AAA A A A A A AA A A A A AAA A A AA A A AAA A AA A AAAAA A AAA A 0 A AAAA -2-1.5-1 -0.5 0 E/N [J] Monte Carlo study of the Baxter-Wu model p.36/40

The Dilute Ferromagnetic BW model (DFBW) The specific heat exponent appears to be close to zero(!) at a concentration of x 0.9. This may indicate an "Ising like" logarithmic singularity at the infinite lattice transition point. 1000 x=1 x~0.9 100 C max 10 1 10 100 L Monte Carlo study of the Baxter-Wu model p.37/40

Summary The LW sampling is a very accurate algorithm It can be parallelized to reduce the simulation time to that required for calculating energy densities at low temperatures, unlike usual Monte Carlo methods which require multiple runs at different temperatures The pure Baxter-Wu model is strongly influenced by finite size effects and corrections to scaling Monte Carlo study of the Baxter-Wu model p.38/40

Summary The peaks of the energy distribution appear to scale with exponents φ + and φ The field B min scales well according to a second order finite size scaling theory The dilute model shows clearly a second order behavior as expected The change in the specific heat exponent is qualitatively confirmed, in agreement with previous results (Landau and Novotny 1981) and with the Harris argument Monte Carlo study of the Baxter-Wu model p.39/40

Summary Is θ B universal? Monte Carlo study of the Baxter-Wu model p.40/40