Nematic ordering in the Heisenberg spin-glass system in three dimensions

Nematic ordering, where the spins globally align along a spontaneously chosen axis irrespective of direction, occurs in spin-glass systems of classical Heisenberg spins in $d=3$. In this system where the nearest-neighbor interactions are quenched randomly ferromagnetic or antiferromagnetic, instead of the locally randomly ordered spin-glass phase, the system orders globally as a nematic phase. This nematic ordering of the Heisenberg spin-glass system is dramatically different from the spin-glass ordering of the Ising spin-glass system. The system is solved exactly on a hierarchical lattice and, equivalently, Migdal-Kadanoff approximately on a cubic lattice. The global phase diagram is calculated, exhibiting this nematic phase, and ferromagnetic, antiferromagnetic, disordered phases. The nematic phase of the classical Heisenberg spin-glass system is also found in other dimensions $d>2$: We calculate nematic transition temperatures in 24 different dimensions in $2<d\le 4$.

Figure 1

Construction of the $d=3$ hierarchical model used in this study. A hierarchical model is constructed by repeatedly self-imbedding a graph into each of its bonds. This hierarchical model is the $d=3,b=3$ generalization of the original $d=2,b=2$ hierarchical model introduced in Fig. 2(c) of Ref. [24] in 1979 and is a member of the most used family of hierarchical models, namely the so-called “diamond” family. For example, this model family has been used [25] in the determination of the spin-glass lower-critical dimension by solving in 23 sequenced dimensions $d$. The exact solution of a hierarchical model proceeds in the opposite direction of its construction [24, 26, 27].

Figure 2

Calculated phase diagram of the classical Heisenberg spin-glass system in $d=3$. The phase diagram shows no spin-glass phase, but at low temperatures an extended nematic phase where the spins globally align along a spontaneously chosen axis irrespective of direction. This spontaneous nematic axis is spontaneously chosen isotropically in the three-dimensional spin space of the model. This characteristic low-temperature high-quenched-disorder phase is also distinctive in its low-temperature penetration into the ferromagnetic and antiferromagnetic phases, in contrast to the spin-glass phase.

Figure 3

Low-temperature portion of the calculated phase diagram of the classical Heisenberg spin-glass system in $d=3$. As more fully seen here, the phase diagram shows no spin-glass phase, but at low temperatures an extended nematic phase where the spins globally align along a spontaneously chosen axis irrespective of direction. The nematic phase distinctively penetrates the ferromagnetic and antiferromagnetic phases to its mid-temperature range, where the boundary drops to zero temperature at very low quenched randomness ($p=0.05,0.95$). Thus, the nematic phase covers an unusually broad range between $p=0.05$ and 0.95, where it robustly stretches to finite temperatures.

Figure 4

The fixed-point exponentiated potential $u\left(\gamma \right)$ at the renormalization-group sink of the nematic phase. The neighboring spins align (nearest-neighbor angle $\gamma =0)$ or antialign ($\gamma =\pi$), creating the nematic phase with a global spontaneous alignment axis along which both spin directions occur. All points in the nematic phase flow, under repeated renormalization-group transformations, to this sink which epitomizes the ordering of this phase. This potential function, in terms of the nearest-neighbor angle $\gamma$, is reconstructed from the Fourier-Legendre coefficients at the renormalization-group sink.

Figure 5

The fixed-point exponentiated potential $u\left(\gamma \right)$ of the sink of the nematic phase of the $d=3$ classical Heisenberg spin-glass system. This potential function, in terms of the spherical coordinate angles $\theta$ and $\varphi$ of one spin with respect to the other, is reconstructed from the Fourier-Legendre coefficients at the sink.

Figure 6

Calculated transition temperatures for the nematic phase (at $p=0.5$), squares and right vertical scale, and for the ferromagnetic phase (at $p=0$), circles and left vertical scale, for 24 different dimensions in $2<d\le 4$. No nematic (or ferromagnetic [23]) phase occurs in $d=2$, which is expected.