Abstract
We investigate the ground-state properties of two lines with “on-site” repulsion on disordered Cayley tree and (Berker) hierarchical lattices, in connection with the question of multiple “pure states” for the corresponding one-line problem. Exact recursion relations for the distribution of ground-state energies and of the overlaps are derived. Based on a numerical study of the recursion relations, we establish that the total interaction energy on average is asymptotically proportional to the width δ of the ground-state energy fluctuation of a single line for both weak and strong (i.e., hard-core) repulsion. When the lengtht of the lines is finite, there is a finite probability of ordert −a for (nearly) degenerate, nonoverlapping one-line ground-state configurations, in which case the interaction energy vanishes. We show thata=ω (δ∼t ω) on hierarchical lattices. Monte Carlo transfer matrix calculation on a (1+1)-dimensional model yields the same scaling for the interaction energy but ana different from ω=1/3. Finitelength scalings of the distribution of the interaction energy and of the overlap are also discussed.
Similar content being viewed by others
References
D. R. Nelson,Phys. Rev. Lett. 60:1973 (1988); D. R. Nelson and P. Le Doussal,Phys. Rev. B 42:10113 (1990).
T. Nattermann and R. Lipowsky,Phys. Rev. Lett. 61:2508 (1988).
T. Natterman, M. Feigelman, and I. Lyuksyutov,Z. Phys. B 84:353 (1991).
D. S. Fisher, M. P. A. Fisher, and D. A. Huse,Phys. Rev. B 43:130 (1991); D. J. Bishop, P. L. Gammel, D. A. Huse, and C. A. Murray,Science 255:165 (1992); and references therein.
B. Derrida and H. Spohn,J. Stat. Phys. 51:817 (1988).
M. Kardar and Y.-C. Zhang,Phys. Rev. Lett. 58:2087 (1987).
M. Kadar,Nucl. Phys. B 290:582 (1987).
G. Parisi,J. Phys. (Paris)51:1595 (1990).
M. Mézard,J. Phys. (Paris)51:1831 (1990).
D. S. Fisher and D. A. Huse,Phys. Rev. B. 43:10728 (1991).
L.-H. Tang and I. F. Lyuksyutov,Phys. Rev. Lett. 71:2745 (1993).
L. Balents and M. Kardar,Europhys. Lett. 23:503 (1993).
T. Hwa and T. Nattermann, to be published.
B. Derrida and R. B. Griffiths,Europhys. Lett. 8:111 (1989).
A. N. Berker and S. Ostlund,J. Phys. C 12:4961 (1981).
J. Cook and B. Derrida,J. Stat. Phys. 57:89 (1989).
T. Halpin-Healy,Phys. Rev. A 42:711 (1990).
L.-H. Tang, J. Kertész, and D. E. Wolf,J. Phys. A 24:L1193 (1991).
D. A. Huse, C. L. Henley, and D. S. Fisher,Phys. Rev. Lett. 55:2924 (1985).
E. Medina, T. Hwa, M. Kardar, and Y.-C. Zhang,Phys. Rev. A 39:3053 (1989)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tang, LH. Two repulsive lines on disordered lattices. J Stat Phys 77, 581–606 (1994). https://doi.org/10.1007/BF02179451
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02179451