Chromatic zeros on the limit G_∞^(p,ℓ) of the family G_m^(p,ℓ) of hierarchical graphs

We calculate the continuous accumulation set B_q(p,ℓ) of zeros of the chromatic polynomial P(G_m^(p,ℓ),q) in the limit m→∞, on a family of graphs G_m^(p,ℓ) defined such that G_m^(p,ℓ) is obtained from G_m−1^(p,ℓ) by replacing each edge (i.e., bond) on G_m^(p,ℓ) by p paths each of length ℓ edges, starting with the tree graph T₂. Our method uses the property that the chromatic polynomial P(G,q) of a graph G is equal to the v=−1 evaluation of the partition function of the q-state Potts model, together with (i) the property that Z(G_m^(p,ℓ),q,v) can be expressed via an exact closed-form real-space renormalization (RG) group transformation in terms of Z(G_m−1^(p,ℓ),q,v^′), where v^′=F_(p,ℓ),q(v) is a rational function of v and q and (ii) B_q(p,ℓ)(v) is the locus in the complex q-plane that separates regions of different asymptotic behavior of the m-fold iterated RG transformation F_(p,ℓ),q(v) in the m→∞ limit. Thus, our results involve calculations of region diagrams in the complex q-plane showing the type of behavior that occurs in the m→∞ limit of the m-fold iterated RG transformation mapping F_(p,ℓ),q(v) starting with the initial value v=v₀=−1. Calculations are presented of the maximal point q_c(G_∞^(p,ℓ)) at which the locus B_q crosses the real-q axis, as well as several other points at which, depending on p and ℓ, the locus B_q crosses this axis. We give explicit results for a variety of (p,ℓ) cases and observe a number of interesting features. Among our new results is the finding that for (p,ℓ) families with even p and ℓ (denoted (p_even,ℓ_even)), the locus B_q(p,ℓ) contains an infinite sequence of crossings on the real-q axis. We also find the first cases for nonrandom graphs where the leftmost point, denoted q_L(p,ℓ), at which B_q(p,ℓ) crosses the real-q axis, is negative, namely cases where p>ℓ. Calculations of the ground-state degeneracy of the Potts antiferromagnet at q_c(G_∞^(p,ℓ)) are presented. This work extends a previous study with R. Roeder of the (p,ℓ)=(2,2) case to higher p and ℓ values.