Highq-state clock spin glasses in three dimensions and the Lyapunov exponents of chaotic phases and chaotic phase boundaries

Abstract

Spin-glass phases and phase transitions for q-state clock models and their q → ∞ limit the XY model, in spatial dimension d = 3, are studied by a detailed renormalization-group study that is exact for the d = 3 hierarchical lattice and approximate for the cubic lattice. In addition to the now well-established chaotic rescaling behavior of the spin-glass phase, each of the two types of spin-glass phase boundaries displays, under renormalization-group trajectories, their own distinctive chaotic behavior. These chaotic renormalization-group trajectories subdivide into two categories, namely as strong-coupling chaos (in the spin-glass phase and, distinctly, on the spinglass-ferromagnetic phase boundary) and as intermediate-coupling chaos (on the spinglass-paramagnetic phase boundary). We thus characterize each different phase and phase boundary exhibiting chaos by its distinct Lyapunov exponent, which we calculate. We show that, under renormalization-group, chaotic trajectories and fixed distributions are mechanistically and quantitatively equivalent. The phase diagrams of arbitrary even q-state clock spin-glass models in d = 3 are calculated. These models, for all noninfinite q, have a finite-temperature spin-glass phase. Furthermore, the spin-glass phases exhibit a universal ordering behavior, independent of q. The spin-glass phases and the spinglass-paramagnetic phase boundaries exhibit universal fixed distributions, chaotic trajectories and Lyapunov exponents. In the XY model limit, our calculations indicate a zero-temperature spin-glass phase.

Phase transitions between different spin-glass phases and between different chaoses in quenched random chiral systems

“…Similar previous studies, on other spin-glass systems, are reported in Refs. [12,13,[40][41][42][43][44][45][46][47]. Figure 1 shows a calculated sequence of phase diagram cross sections for the left-chiral (c = 0) (top row) and quenched random left-and right-chiral (c = 0.5) (bottom row) systems with, in both cases, quenched random ferromagnetic and antiferromagnetic interactions.…”

“…Recent studies on ferromagneticantiferromagnetic clock spin glasses are reported in Refs. [12][13][14].…”

“…Another distinctive mechanism, that of chaos under scale change [2][3][4] or, equivalently [13], chaos under spatial translation, occurs within the spin-glass phase and differently at the spin-glass phase boundary [13] in systems with competing ferromagnetic and antiferromagnetic interactions [2][3][4]13,46, and, more recently, with competing left-and right-chiral interactions [5,6]. Originally found in hierarchical systems [2][3][4], scaling or, equivalently, translation spin-glass chaos is now well accepted for real d = 3 lattices and experimental systems [2][3][4]13,46,.…”

“…The traditional ferromagnetic-antiferromagnetic spin-glass phase does not occur and the antiferromagnetic phase is a critical phase lacking conventional long-range order [12]. On the other hand, the even-q clock spins can achieve complete antiferromagnetic pairing and exhibit the conventional antiferromagnetic long-range order and the traditional ferromagnetic-antiferromagnetic spin-glass phase [13]. Thus, the current study is on the random chiral clock model with an even number of states (q = 4), which supports the ferromagnetic-antiferromagnetic usual spin-glass phase [13], as well as, as we see below, with added phase diagram complexity, the chiral spin-glass phase and two * nihatberker@khas.edu.tr other new spin-glass phases.…”

“…On the other hand, the even-q clock spins can achieve complete antiferromagnetic pairing and exhibit the conventional antiferromagnetic long-range order and the traditional ferromagnetic-antiferromagnetic spin-glass phase [13]. Thus, the current study is on the random chiral clock model with an even number of states (q = 4), which supports the ferromagnetic-antiferromagnetic usual spin-glass phase [13], as well as, as we see below, with added phase diagram complexity, the chiral spin-glass phase and two * nihatberker@khas.edu.tr other new spin-glass phases. A double spin-glass model is constructed, including competing quenched random leftright chiral and ferromagnetic-antiferromagnetic interactions and solved in three dimensions by renormalization-group theory.…”