Ferromagnetic and Spin-Glass Finite-Temperature Order but no Antiferromagnetic Order in the d=1 Ising Model with Long-Range Power-Law Interactions

TL;DR

This study investigates the phase transitions and ordering phenomena in the one-dimensional Ising model with long-range power-law interactions, revealing ferromagnetic order without antiferromagnetic order and complex spin-glass behavior.

Contribution

It provides a comprehensive renormalization-group analysis of the d=1 Ising model with long-range interactions, detailing phase boundaries, critical exponents, and the absence of antiferromagnetic order.

Findings

Ferromagnetic phase exists for 0.74<a<2 at finite temperatures.

Phase transition changes from second to first order at a=2.

Spin-glass phase exhibits chaotic interactions without antiferromagnetic order.

Abstract

The d=1 Ising ferromagnet and spin glass with long-range power-law interactions J r^-a is studied for all interaction range exponents a by a renormalization-group transformation that simultaneously projects local ferromagnetism and antiferromagnetism. In the ferromagnetic case, J>0, a finite-temperature ferromagnetic phase occurs for interaction range 0.74<a<2. The second-order phase transition temperature monotonically decreases between these two limits. At a=2, the phase transition becomes first order, as predicted by rigorous results. For a>2, the phase transition temperature discontinuously drops to zero and for a>2 there is no ordered phase above zero temperature, also as predicted by rigorous results. At the other end, on approaching a=0.74 from above, namely increasing the range of the interaction, the phase transition temperature diverges to infinity, meaning that, at all non-infinite temperatures, the system is ferromagnetically ordered. Thus, the equivalent-neighbor interactions regime is entered before (a > 0) the neighbors become equivalent, namely before the interactions become equal for all separations. The critical exponents alpha,beta, gamma,delta,eta,nu are calculated, from a large recursion matrix, varying as function of a. For antiferromagnetic J<0, all triplets of spins at all ranges have competing interactions and this highly frustrated system does not have an ordered phase. In the spin-glass system, where all couplings for all separations are randomly ferromagnetic or antiferromagnetic (with probability p), a finite-temperatures spin-glass phase is obtained in the absence of antiferromagnetic phase. In the spin-glass phase, the signature chaotic behavior under scale change occurs in a richer version than previously: In the long-range interaction of this system, the interactions at every separation become chaotic, yielding a piecewise chaotic interaction function.