Correlation transitions in the Ising chain with competing short-range and long-range mirror interactions

István Daruka; Zsolt Gulácsi

doi:10.1103/PhysRevE.58.5403

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Correlation transitions in the Ising chain with competing short-range and long-range mirror interactions

István Daruka¹ and Zsolt Gulácsi²

¹Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556
²Department of Theoretical Physics, Lajos Kossuth University, H-4010 Debrecen, Hungary

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Phys. Rev. E 58, 5403 – Published 1 November, 1998

DOI: https://doi.org/10.1103/PhysRevE.58.5403

Abstract

We present exact four-spin- and string-correlation functions derived for an Ising chain with competing geometrical nearest-neighbor short-range and mirror-image-type long-range interactions. Unusual $T \neq 0$ correlation effects were found, indicating qualitative changes in the behavior of the system in different T domains. These domains are not separated by a sharp, traditional phase transition, but are delimited by a temperature $T_{i}$ at which the exact connected four-spin-correlation function vanishes. At the same $T_{i}$ the pair-correlation function changes its character: The functional form of the correlation length and the nature of the long-range decay are modified.

References (26)

S. R. White, Phys. Rev. B 53, 52 (1996); F. D. M. Haldane, Phys. Lett. 93A, 464 (1983).
S. P. Strong and A. J. Millis, Phys. Rev. Lett. 69, 2419 (1992); I. Affleck (unpublished); J. Zang, A. R. Bishop, and D. Schmeltzer, Phys. Rev. B 52, 6723 (1995).
S. Chakraverty, Phys. Rev. Lett. 77, 4446 (1996); E. Dagotto and T. M. Rice, Science 271, 618 (1996).
A. Kolezhuk, R. Roth, and U. Schollwöck, Phys. Rev. Lett. 77, 5142 (1996).
Z. Wang, Phys. Rev. Lett. 78, 126 (1997).
R. Chitra, S. Pati, H. R. Krishnamurthy, D. Sen, and S. Ramasesha, Phys. Rev. B 52, 6581 (1995).
K. Hida, Phys. Rev. B 45, 2207 (1992); ibid.P. J. M. Bastiaansen and H. J. F. Knops, 53, 126 (1996).
M. Azuma (unpublished).
U. Schollwöck, T. Jolicoeur, and T. Garel, Phys. Rev. B 53, 3304 (1996).
Y. Muraoka, K. Oda, J. W. Tucker, and T. Idogaki, J. Phys. A 29, 949 (1996).
M. Kardar and M. Kaufman, Phys. Rev. Lett. 51, 1210 (1983); ibid.P. Bak and R. Bruinsma, 49, 249 (1982).
U. Löw, V. J. Emery, K. Fabritius, and S. A. Kivelson, Phys. Rev. Lett. 72, 1918 (1994).
M. Sobkowicz and B. Chakraborty, J. Stat. Phys. 83, 739 (1996).
P. J. M. Bastiansen and H. J. F. Knops, Phys. Rev. B 53, 126 (1996).
M. Sahimi and D. Stauffer, Phys. Rev. Lett. 71, 4271 (1993).
W. Jezewski, J. Stat. Phys. 82, 1099 (1996).
M. Mito et al., J. Phys. Soc. Jpn. 64, 4402 (1995).
W. Selke, Phys. Rep. 170, 213 (1988).
In this case, i is the ladder step position, while $α$ is the chain index within the ladder.
M. Hunyadi and Zs. Gulácsi, Phys. Rev. B 53, 2326 (1996).
B. D. Simons and B. L. Altshuler, Phys. Rev. B 50, 1102 (1994).
F. D. M. Haldane, Phys. Rev. Lett. 60, 635 (1988); ibid.B. S. Shastry, 60, 639 (1988).
R. J. de Boer et al., in Thinking about Biology, edited by F. J. Varela and W. D. Stein (Addison-Wesley, New York, 1992).
That is when $Γ^{4}$ or $L_{n}$ cannot be expressed as a product of $Γ^{2}$ terms.
A. K. Kolezhuk and H. J. Mikeska, Phys. Rev. B 53, R8848 (1996).
For all $S_{i, α}$ with the same $α,$ $a^{'} = ψ (+ 1),$ $b^{'} = τ ψ (- 1),$ $ψ (x) = [0.5 \exp {(2 xJ}_{1} β)] \sinh [{4 (J}_{3} {+ x J}_{2}) β],$ $ρ_{5 / 6} {= t}_{1 / 2} / λ_{1}^{2}, τ {= + 1, t}_{1 / 2} = 0.5 ({(a}^{'} {+ b}^{'}) + (+ / -) {{(a}^{'} - b^{'})^{2} + 4 τ \sinh [{2 (J}_{2} {+ J}_{3}) β] \sinh [{2 (J}_{3} - J_{2}) β]}^{1 / 2}) .$ Alternating sequence of subsequent spin positions on different sides of O gives the same results as before with $τ = - 1.$ For a symmetric adjacent spin sequence $(O$ in the middle) $a^{'} = φ [1] {= a, b}^{'} = \exp (- J_{1} β), b = φ [- 1],$ $t_{1 / 2} = 0.5 {(a - b) + (+ / -) [{(a + b)}^{2} - 4]^{1 / 2}},$ and $ρ_{5 / 6} {= t}_{1 / 2} / λ_{1} .$

[c1] S. R. White, Phys. Rev. B 53, 52 (1996); F. D. M. Haldane, Phys. Lett. 93A, 464 (1983).

[c2] S. P. Strong and A. J. Millis, Phys. Rev. Lett. 69, 2419 (1992); I. Affleck (unpublished); J. Zang, A. R. Bishop, and D. Schmeltzer, Phys. Rev. B 52, 6723 (1995).

[c3] S. Chakraverty, Phys. Rev. Lett. 77, 4446 (1996); E. Dagotto and T. M. Rice, Science 271, 618 (1996).

[c4] A. Kolezhuk, R. Roth, and U. Schollwöck, Phys. Rev. Lett. 77, 5142 (1996).

[c5] Z. Wang, Phys. Rev. Lett. 78, 126 (1997).

[c6] R. Chitra, S. Pati, H. R. Krishnamurthy, D. Sen, and S. Ramasesha, Phys. Rev. B 52, 6581 (1995).

[c7] K. Hida, Phys. Rev. B 45, 2207 (1992); ibid.P. J. M. Bastiaansen and H. J. F. Knops, 53, 126 (1996).

[c8] M. Azuma (unpublished).

[c9] U. Schollwöck, T. Jolicoeur, and T. Garel, Phys. Rev. B 53, 3304 (1996).

[c10] Y. Muraoka, K. Oda, J. W. Tucker, and T. Idogaki, J. Phys. A 29, 949 (1996).

[c11] M. Kardar and M. Kaufman, Phys. Rev. Lett. 51, 1210 (1983); ibid.P. Bak and R. Bruinsma, 49, 249 (1982).

[c12] U. Löw, V. J. Emery, K. Fabritius, and S. A. Kivelson, Phys. Rev. Lett. 72, 1918 (1994).

[c13] M. Sobkowicz and B. Chakraborty, J. Stat. Phys. 83, 739 (1996).

[c14] P. J. M. Bastiansen and H. J. F. Knops, Phys. Rev. B 53, 126 (1996).

[c15] M. Sahimi and D. Stauffer, Phys. Rev. Lett. 71, 4271 (1993).

[c16] W. Jezewski, J. Stat. Phys. 82, 1099 (1996).

[c17] M. Mito et al., J. Phys. Soc. Jpn. 64, 4402 (1995).

[c18] W. Selke, Phys. Rep. 170, 213 (1988).

[c19] In this case, i is the ladder step position, while $α$ is the chain index within the ladder.

[c20] M. Hunyadi and Zs. Gulácsi, Phys. Rev. B 53, 2326 (1996).

[c21] B. D. Simons and B. L. Altshuler, Phys. Rev. B 50, 1102 (1994).

[c22] F. D. M. Haldane, Phys. Rev. Lett. 60, 635 (1988); ibid.B. S. Shastry, 60, 639 (1988).

[c23] R. J. de Boer et al., in Thinking about Biology, edited by F. J. Varela and W. D. Stein (Addison-Wesley, New York, 1992).

[c24] That is when $Γ^{4}$ or $L_{n}$ cannot be expressed as a product of $Γ^{2}$ terms.

[c25] A. K. Kolezhuk and H. J. Mikeska, Phys. Rev. B 53, R8848 (1996).

[c26] For all $S_{i, α}$ with the same $α,$ $a^{'} = ψ (+ 1),$ $b^{'} = τ ψ (- 1),$ $ψ (x) = [0.5 \exp {(2 xJ}_{1} β)] \sinh [{4 (J}_{3} {+ x J}_{2}) β],$ $ρ_{5 / 6} {= t}_{1 / 2} / λ_{1}^{2}, τ {= + 1, t}_{1 / 2} = 0.5 ({(a}^{'} {+ b}^{'}) + (+ / -) {{(a}^{'} - b^{'})^{2} + 4 τ \sinh [{2 (J}_{2} {+ J}_{3}) β] \sinh [{2 (J}_{3} - J_{2}) β]}^{1 / 2}) .$ Alternating sequence of subsequent spin positions on different sides of O gives the same results as before with $τ = - 1.$ For a symmetric adjacent spin sequence $(O$ in the middle) $a^{'} = φ [1] {= a, b}^{'} = \exp (- J_{1} β), b = φ [- 1],$ $t_{1 / 2} = 0.5 {(a - b) + (+ / -) [{(a + b)}^{2} - 4]^{1 / 2}},$ and $ρ_{5 / 6} {= t}_{1 / 2} / λ_{1} .$

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Correlation transitions in the Ising chain with competing short-range and long-range mirror interactions

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