Odd $q$-state clock spin-glass models in three dimensions, asymmetric phase diagrams, and multiple algebraically ordered phases

Distinctive orderings and phase diagram structures are found, from renormalization-group theory, for odd $q$-state clock spin-glass models in $d=3$ dimensions. These models exhibit asymmetric phase diagrams, as is also the case for quantum Heisenberg spin-glass models. No finite-temperature spin-glass phase occurs. For all odd $q\ge 5$, algebraically ordered antiferromagnetic phases occur. One such phase is dominant and occurs for all $q\ge 5$. Other such phases occupy small low-temperature portions of the phase diagrams and occur for $5\le q\le 15$. All algebraically ordered phases have the same structure, determined by an attractive finite-temperature sink fixed point where a dominant and a subdominant pair states have the only nonzero Boltzmann weights. The phase transition critical exponents quickly saturate to the high $q$ value.

Figure 1

(a) Migdal-Kadanoff approximate renormalization-group transformation for the $d=3$ cubic lattice with the length-rescaling factor of $b=3$. Bond moving is followed by decimation. (b) Exact renormalization-group transformation for the equivalent $d=3$ hierarchical lattice with the length-rescaling factor of $b=3$. (c) Pairwise applications of the quenched probability convolution of Eq. (5), leading to the exact transformation in (b).

Figure 2

Calculated phase diagrams of the odd $q$-state clock spin-glass models on the hierarchical lattice with $d=3$ dimensions. These phase diagrams do not have ferromagnetic-antiferromagnetic symmetry, i.e., they are not left-right symmetric with respect to the $p=0.5$ line. The phase diagrams do not have a spin-glass phase, but show a multiplicity of algebraically ordered phases on the antiferromagnetic side. The phase diagrams show true reentrance (disordered-ordered-disordered) as temperature is lowered at fixed antiferromagnetic bond concentration $p$, on both the ferromagnetic and antiferromagnetic sides of the phase diagram. The phase diagrams also show lateral, true double reentrance (ferromagnetic-disordered-ferromagnetic-disordered) as the antiferromagnetic bond concentration $p$ is increased at fixed temperature, only on the ferromagnetic side. No antiferromagnetic ordering occurs for the lowest model, $q=3$. Algebraically ordered antiferromagnetic phases occur for all higher $q\ge 5$ models. In these cases, the phase boundary between the dominant antiferromagnetic algebraically ordered phase and the disordered phase is slightly asymmetric with the phase boundary between the ferromagnetic and disordered phases. To make this slight asymmetry evident, the latter boundary is also shown (dashed) reflected about the $p=0.5$ line. The lower temperature details of these phase diagrams are shown in Fig. 3.

Figure 3

Lower-temperature details of the phase diagrams shown in Fig. 2.

Figure 4

Evolution of the quenched probability distribution under successive renormalization-group transformations. The case of $q=9$, starting with the initial condition temperature $1/J=4$ and antiferromagnetic bond concentration $p=0.8$ is shown here. For $q=9$, the generalized interaction potential unavoidably generated by the renormalization-group transformation is determined by four interaction constants (see Table 1). The renormalization-group transformation gives the evolution, under scale change, of the correlated quenched probability distribution $P(V_0,V_1,V_2,V_3,V_4)$. Shown in this figure are the projections $P_0\left(V_0\right)=\int d{V}_{1}d{V}_{2}d{V}_{3}d{V}_{4}P(V_0,V_1,V_2,V_3,V_4)$ and similarly for $P_1\left(V_1\right),P_2\left(V_2\right),P_3\left(V_3\right),$ and $P_4\left(V_4\right)$. Each row corresponds to another renormalization-group step $k$, as marked on the figure. It is seen here that in four renormalization-group transformations, the renormalized system essentially reaches the critical phase sink described in Sec. IV: The most misaligned pair state is dominant with Boltzmann weight $e^{V(8\pi /9)}=1$ and the next-most misaligned pair state is also present but less dominant with $e^{V(6\pi /9)}=1/3$. The other two less misaligned pair states and the aligned pair state have zero Boltzmann weight at the sink.

Figure 5

Top: Critical temperatures $1/J_C$ of the ferromagnetic (circles) and antiferromagnetic (asterisks) $q$-state clock models in $d=3$. Bottom: Critical exponents $y_T$ of the ferromagnetic (circles) and antiferromagnetic (asterisks) $q$-state clock models in $d=3$. In both panels, the values exactly coincide for even $q$, due to the ferromagnetic-antiferromagnetic symmetry that is present for even $q$ but absent for odd $q$.