Abstract The behavior of
b=2 b = 2 real-space renormalization group (RSRG) maps like the majority rule and the decimation map was examined by numerically applying RSRG steps to critical
q=2,3,4 q = 2 , 3 , 4 Potts spin configurations. While the majority rule is generally believed to work well, a more thorough investigation of the action of the map has yet to be considered in the literature. When fixing the size of the renormalized lattice
Lg L g and allowing the source configuration size
L0 L 0 to vary, we observed that the RG flow of the spin and energy correlation under the majority rule map appear to converge to a nontrivial model-dependent curve. We denote this property as “faithfulness”, because it implies that some information remains preserved by RSRG maps that fall under this class. Furthermore, we show that
b=2 b = 2 weighted majority-like RSRG maps acting on the
q=2 q = 2 Potts model can be divided into two categories, maps that behave like decimation and maps that behave like the majority rule.
We employ the Multiscale Entanglement Renormalization Ansatz (MERA) tensor network to investigate a critical line of continuous quantum phase transitions of the
Z3 chiral clock model. This critical line is believed to be described by a slow renormalization group flow from the 3-state Potts fixed point to another fixed point that features anisotropic scaling of space and time. We use the variational principle to construct a MERA representation of the model's ground state, from which we obtain the ground state energy and the set of scaling operators and their scaling dimensions. These scaling dimensions determine the critical exponents of the model, and we study these critical exponents and other scaling data as a function of the model's chiral parameter. We find a set of effective scaling data that smoothly varies starting from the Potts data as the chiral parameter is increased. Within the context of our approach, we discuss how this result may nevertheless be consistent with the two fixed point hypothesis provided the renormalization group flow is sufficiently slow. Our findings demonstrate MERA's effectiveness in capturing the complex low-energy physics of the chiral clock model and in extracting field theory data for an anisotropic continuum theory.
The real-space renormalization group theory using the first order cumulant is applied to the q -state antiferromagnetic Potts model on 2-d (dimensional) and 3-d lattice. The special features of this paper are that the Potts states are represented by q unit vectors in q -1 dimensional space and the two-sublattice structure is preserved after the renormalization transformation. The critical exponents obtained for q =2 in 3-d are in fairly good agreement with the reliable values and those for q =3 and 4 are obtained for the first time.