TY - JOUR

T1 - Zero-temperature complex replica zeros of the ±j Ising spin glass on mean-field systems and beyond

AU - Obuchi, Tomoyuki

AU - Kabashima, Yoshiyuki

AU - Nishimori, Hidetoshi

AU - Ohzeki, Masayuki

N1 - Funding Information:
This work was supported by the Japan Society for the Promotion of Science (JSPS) Research Fellowship for Young Scientists program, CREST(JST) , the 21th Century COE Program ‘Nanometer-Scale Quantum Physics’ and the Global COE Program ‘Nanoscience and Quantum Physics’ at Tokyo Institute of Technology, and by the Grant-in-Aid for Scientific Research on the Priority Area “Deepening and Expansion of Statistical Mechanical Informatics” by the Ministry of Education, Culture, Sports, Science and Technology.

PY - 2011/1

Y1 - 2011/1

N2 - Zeros of the moment of the partition function [Zn]J with respect to complex n are investigated in the zero-temperature limit β→∞, n→0 keeping y=βn≈O(1). We numerically investigate the zeros of the ±J Ising spin-glass models on several Cayley trees and hierarchical lattices and compare those results. In both lattices, the calculations are carried out with feasible computational costs by using recursion relations originated from the structures of those lattices. The results for Cayley trees show that a sequence of the zeros approaches the real axis of y implying that a certain type of analyticity breaking actually occurs, although it is irrelevant for any known replica symmetry breaking. The result of hierarchical lattices also shows the presence of analyticity breaking, even in the two dimensional case in which there is no finite-temperature spin-glass transition, which implies the existence of the zero-temperature phase transition in the system. A notable tendency of hierarchical lattices is that the zeros spread in a wide region of the complex y plane in comparison with the case of Cayley trees, which may reflect the difference between the mean-field and finite-dimensional systems.

AB - Zeros of the moment of the partition function [Zn]J with respect to complex n are investigated in the zero-temperature limit β→∞, n→0 keeping y=βn≈O(1). We numerically investigate the zeros of the ±J Ising spin-glass models on several Cayley trees and hierarchical lattices and compare those results. In both lattices, the calculations are carried out with feasible computational costs by using recursion relations originated from the structures of those lattices. The results for Cayley trees show that a sequence of the zeros approaches the real axis of y implying that a certain type of analyticity breaking actually occurs, although it is irrelevant for any known replica symmetry breaking. The result of hierarchical lattices also shows the presence of analyticity breaking, even in the two dimensional case in which there is no finite-temperature spin-glass transition, which implies the existence of the zero-temperature phase transition in the system. A notable tendency of hierarchical lattices is that the zeros spread in a wide region of the complex y plane in comparison with the case of Cayley trees, which may reflect the difference between the mean-field and finite-dimensional systems.

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U2 - 10.1016/j.physe.2010.07.052

DO - 10.1016/j.physe.2010.07.052

M3 - Article

AN - SCOPUS:78651080658

VL - 43

SP - 786

EP - 789

JO - Physica E: Low-Dimensional Systems and Nanostructures

JF - Physica E: Low-Dimensional Systems and Nanostructures

SN - 1386-9477

IS - 3

ER -