1. Anomalous Scaling in the Conserved
KPZ Equation and Discrete Growth Models
C.K. Lee, S.H.Yook, Yup Kim

2. Abstract Kyung-Hee Univ.
Recently anomalous scaling relation a+z=4-3d for the conserved KPZ equations with conserved or nonconserved noises are suggested by Janssen [1]. The estimated d by two-loop RG calculation is very small. Through the two discrete stochastic growth models [2,3] which are believed to follow the conserved KPZ equations with conserved or nonconserved noises, we test whether there exists a consistent correction in the scaling relation a+z=4.

3. I Continuum equations [4-7]
Kyung-Hee Univ.
Introduction
I Continuum equations [4-7]
(1)
where
and
for nonconserved noise
(2)
for conserved noise
I Scaling relations [8,9]
Surface width
(3)
Height-height correlation function

4. I One - loop Renormalization Group(RG) results
Kyung-Hee Univ.
I One - loop Renormalization Group(RG) results
· nonconserved noise
(4)
· conserved noise
(5)
d : substrate dimension
I Janssen’s two - loop RG results [1]
(6)
where dc is upper critical dimension.
for nonconserved noise
(7)
for conserved noise

5. Relations used to determine d
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Motivation
From the theoretical point of view it should be very interesting to find whether the scaling relation (6) is exact or not. So the motivation of our study is to compare Janssen’s correction[eq.(6),(7)] with the values of exponents in the CRSOS model and the model suggested by Krug.
Relations used to determine d

6. Models I Conserved noise Kyung-Hee Univ. I Noncolserved noise
Conserved Restricted Solid-on-Solid (CRSOS) Model[2,10].
(1) A site is selected randomly in a d-dimensional substrate
(2) If the restricted solid-on-solid (RSOS) condition[8] on the neighboring heights = 0,1,2,3,……,N is obeyed after a particle is deposited at , where N is a preassigned restriction parameter, then a growth is permitted by increasing the height
(3) If the RSOS condition is not obeyed at the position , the dropped particle is allowed to hop to the nearest site where the RSOS condition is satisfied. If there is more than one neighboring site at the same distance that satisfies the RSOS condition, one of them is chosen randomly.
I Conserved noise
Model suggested by Krug [3,12]
(1) A site is chosen at random.
(2) It is checked whether the height at any one of the neighboring sites exceeds , by at least one lattice spacing.
(3) If so, the particle at is regarded as immobile and a new site is chosen
(4) Otherwise, the particle at is moved to a randomly chosen neighbor site

7. Kyung-Hee Univ.
Results for CRSOS Model(d=2) [2]

8. Kyung-Hee Univ.
Results for Krug’s Model

9. Kyung-Hee Univ.

10. Higher dimensions I Nonconserved noise I Conserved case
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Higher dimensions
I Nonconserved noise
I Conserved case
In d = 2, W 2 exhibits logarithmic behavior (a =b = 0 ) [12].

11. Summary and Discussions
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Summary and Discussions
We have found that there is a consistent trend of nonzero d. The estimated d ’s are
For CRSOS model
For Krug’s model
The estimated values are somewhat larger than Janssen’s result [eq.(6)]. One of the possible explanations for the larger value of d is due to the fact that eq. (6) is a result of an asymptotic expansion.

12. References Kyung-Hee Univ.
[1] H.K.Janssen, Phys. Rev. Lett. 78, 1082 (1997).
[2] Yup Kim and J.M.Kim, Phys. Rev. E 55, 3977(1997); S.H.Yook,J.M.Kim, and Yup Kim, Phys. Rev. E 56, 4085 (1997)
[3] J. Krug, Adv. Phys. 46,139 (1997)
[4] Z.W.Lai and Das Sarma, Phys. Rev. Lett (1991) ; L.H. Tang and T. Nattermann Phys. Rev. Lett. 66, 2899 (1991)
[5] J. Villain, J. Phys. I (France) (1991)
[6] T. Sun, H.Guo, and M. Grant, Phys. Rev. A 40,6763 (1989)
[7] Z. Rácz, M. Siegert, D. Liu, and M. Plischke, Phys. Rev. A 43, 5275 (1991)
[8] F. Famil and T. Vicsek, J. Phys. A 18, L75 (1985)
[9] J. Krug and H. Spohn in Solids Far From Equilibrium: Growth, Morphology and Defects, edited by C. Godreche (Cambridge University Press, New York, 1991)
[10] Yup Kim, D.K.Park, and J.M.Kim, J.Phys.A 27, L553(1994)
[11] J.M.Kim and J.M. Kosterlitz, Phys. Rev. Lett. 62, 2289 (1989)
[12] In-mook Kim, J.Yang, and J.M.Kim, J. Kor. Phys. Soc. 31, 898 (1997)