Metropolis-type evolution rules for surface growth models with the global constraints on one and two dimensional substrates Yup Kim, H. B. Heo, S. Y. Yoon KHU

1. Motivation In equilibrium state, Normal restricted solid-on-solid model : Edward-Wilkinson universality class Two-particle correlated surface growth - Yup Kim, T. S. Kim and H. Park, PRE 66, 046123 (2002) Dimer-type surface growth J. D. Noh, H. Park, D. Kim and M. den Nijs, PRE 64, 046131 (2001) - J. D. Noh, H. Park and M. den Nijs, PRL 84, 3891 (2000) Self-flattening surface growth -Yup Kim, S. Y. Yoon and H. Park, PRE(RC) 66, 040602(R) (2002)

nh : the number of columns with height h 2 Partition function, nh : the number of columns with height h Steady state or Saturation regime , 1. Normal RSOS (z =1) Normal Random Walk (1D) (EW) 2. Two-particle correlated (dimer-type) growth (z = -1) nh=even number, Even-Visiting Random Walk (1D)

? ? 3. Self-flattening surface growth (z = 0) Self-attracting random walk (1D) Phase diagram (1D) z = 0 z = 1 z =-1 Normal Random Walk Even-Visiting ? Self-attracting z Phase diagram (2D) z = 0 z = 1 z =-1 Normal Random Walk Even-Visiting ? Self-attracting z

2. Generalized Model Evaluate the weight 4 2. Generalized Model ( nh : the number of sites which have the same height h ) Evaluate the weight in a given height configuration Choose a column randomly. Decide the deposition (the evaporation) attempt with probability p (1-p) Calculate for the new configuration from the decided deposition (evaporation) process Acceptance parameter P is defined by

If P  1 , then new configuration is accepted unconditionally. 5 If P  1 , then new configuration is accepted unconditionally. If P < 1 , then new configuration is accepted only when P  R. where R is generated random number 0< R < 1 (Metropolis algorithm) Any new configuration is rejected if it would result in violating the RSOS contraint ( a primitive lattice vector in the i – th direction ) P  R hmax hmin p =1/2 L = 10 z = 0.5 n+2 = 1 n+1 = 3 n 0 = 2 n-1 = 2 n-2 = 2 w n´ +2 = 2 n´ +1 = 2 n´ 0 = 2 n´ -1 = 2 n´ -2 = 2 w´

 Equilibrium model (1D, p=1/2) 6 3. Simulation Results  Equilibrium model (1D, p=1/2)

7 0.22 0.33 1.1 0.9  0.19  0.22 1/4   0.33  0.34 1/2  (L) -1 -0.5 0.5 1 1.5 z

 Equilibrium model (2D, p=1/2) 7  Equilibrium model (2D, p=1/2) z -1 -0.5 0.5 1 1.5 a 0.176 0.175 0.179

Scaling Collapse to in 2D equilibrium state. 7 Scaling Collapse to in 2D equilibrium state. , Z = 2.5

Phase diagram in equilibrium (1D) 7 Phase diagram in equilibrium (1D) z =-1 -1/2 z = 0 1/2 z = 1 3/2 z = 0.9 z = 1.1 2-particle corr. growth Self-flattening surface growth Normal RSOS Phase diagram in equilibrium (2D) z z =-1 z = -0.5 z = 0 z = 0.5 z = 1 z = 1.5 Even-Visiting Random Walk Self-attracting Random Walk Normal Random Walk

Growing (eroding) phase (1D, p=1(0) ) 9 Growing (eroding) phase (1D, p=1(0) ) z  0 : Normal RSOS model (Kardar-Parisi-Zhang universality class) p  (L) 1.5  0.52 0.5  0.51  0.49

10 z  0 Normal RSOS Model (KPZ)

11 z  0 z=-0.5 p=1 L=128

 Equilibrium model (1D, p=1/2) 12 4. Conclusion  Equilibrium model (1D, p=1/2) z -1 -1/2 1/2 0.9 1 1.1 3/2 2-particle corr. growth (EVRW) Self-flattening surface growth (SATW) Normal RSOS (Normal RW) Growing (eroding) phase (1D, p = 1(0) ) 1. z  0 : Normal RSOS model (KPZ universality class) 2. z  0 : Groove phase ( = 1) Phase transition at z=0 (?)

Scaling Collapse to in 1D equilibrium state. ( = 1/3 , z = 1.5) 12-1 Scaling Collapse to in 1D equilibrium state. ( = 1/3 , z = 1.5)

Monomer & Extremal & Dimer & 2-site 7 Monomer & Extremal & Dimer & 2-site 0.175 0.162 Dimer Monomer Slope a Model Extremal 0.174 2-site