Fractal Dimensions of Self-Avoiding Walks and Ising High-Temperature Graphs in 3D Conformal Bootstrap

• Published in: Journal of statistical physics 2016-12, Vol.165 (6), p.1006-1035
• Main Authors: Shimada, Hirohiko, Hikami, Shinobu
• Format: Article
• Language: English
• Subjects: Fractals; Graphs; Ising model; Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Supersymmetry; Theoretical; Three dimensional models
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-016-1658-x

The fractal dimensions of polymer chains and high-temperature graphs in the Ising model both in three dimension are determined using the conformal bootstrap applied for the continuation of the O ( N ) models from N = 1 (Ising model) to N = 0 (polymer). Even for non-integer N , the O ( N ) sum rule allows one to study the unitarity bound formally defined from the positivity, which may be violated in a non-unitary CFT. This unitarity bound of the scaling dimension for the O ( N )-symmetric-tensor develops a kink as a function of the fundamental field as in the case of the energy operator dimension in the Z 2 (Ising) sum rule. Although this kink structure becomes less pronounced as N tends to zero, we found instead an emerging asymmetric minimum in the current central charge C J . Despite the non-unitarity of the O ( N ) model at non-integer N , we find the C J -kink along the unitarity bound lies very close to the location of the infrared (IR) O ( N ) CFT estimated by other methods. It is pointed out that certain level degeneracies at the IR CFT should induce these singular shapes of the unitarity bounds. As an application to the quantum and classical spin systems, we also predict critical exponents associated with the N = 1 supersymmetry, which could be relevant for locating the corresponding fixed point in the phase diagram.