Within a real-space renormalization-group (RG) framework, we study the criticality of the Z(4) ferromagnet on the square lattice. The phase diagram (exhibiting ferromagnetic, paramagnetic, and nematiclike phase) recovers all the available exact results, and possibly is of high precision everywhere. In particular, we establish the main asymptotic behaviors (bifurcation and Ising regions). In addition, we develop an operational procedure (break-collapse method) which considerably simplifies the exact calculation of arbitrary Z(4) two-terminal clusters (commonly appearing in RG approaches).Within a real-space renormalization-group (RG) framework, we study the criticality of the Z(4) ferromagnet on the square lattice. The phase diagram (exhibiting ferromagnetic, paramagnetic, and nematiclike phase) recovers all the available exact results, and possibly is of high precision everywhere. In particular, we establish the main asymptotic behaviors (bifurcation and Ising regions). In addition, we develop an operational procedure (break-collapse method) which considerably simplifies the exact calculation of arbitrary Z(4) two-terminal clusters (commonly appearing in RG approaches).

Physical review. B, Condensed matterNov 1, 198521 citations

Precise estimate of correlation length exponents from simple real-space renormalization group analysis

Aleksander Kubica, Beni Yoshida

Bulletin of the American Physical SocietyMar 6, 20140 citations

Phase diagram of the Z(4) ferromagnet in an anisotropic square lattice

Constantino Tsallis, Jean Souletie

Within a real space renormalisation group (RG) scheme, the authors study the criticality of the ferromagnetic Z(4) model on an anisotropic square lattice. They use an RG cluster which has already proved to be very efficient for the Potts model on the same lattice. The establishment of the RG recurrence relations is greatly simplified through the break-collapse method. The phase diagram (exhibiting ferromagnetic, paramagnetic and nematic-like phases) recovers all the available exact results, and is believed to be of high precision everywhere. If the model is alternatively thought of as being associated with a particular hierarchical lattice rather than with the square lattice, then it is exact everywhere.Within a real space renormalisation group (RG) scheme, the authors study the criticality of the ferromagnetic Z(4) model on an anisotropic square lattice. They use an RG cluster which has already proved to be very efficient for the Potts model on the same lattice. The establishment of the RG recurrence relations is greatly simplified through the break-collapse method. The phase diagram (exhibiting ferromagnetic, paramagnetic and nematic-like phases) recovers all the available exact results, and is believed to be of high precision everywhere. If the model is alternatively thought of as being associated with a particular hierarchical lattice rather than with the square lattice, then it is exact everywhere.

Journal of Physics A Mathematical and GeneralJun 21, 19865 citations

Lifshitz-point correlation length exponents from the large-n expansion

Mykola A. Shpot, Yu. M. Pis’mak

arXiv:1202.2464Nuclear Physics BApr 16, 201210 citations

Proof of the exact correlation-length exponent $ν= \frac{\sqrt[3]{2}}{2}$ and algebraic relations for $β$ and $γ$ in the 3D Ising modelProof of the exact correlation-length exponent $ν= \frac{\sqrt[3]{2}}{2}$ and algebraic relations for $β$ and $γ$ in the 3D Ising model

Aleksandr Liivand

Based on our previous results regarding critical exponent estimates for the 3D Ising model and scaling relations [1], we have proven that the exact value of the correlation-length exponent is $\frac{\sqrt[3]{2}}{2}$ . This is in very close agreement with the most precise numerical estimates currently available. Using a group theoretical approach and some propositions of constructive geometry, we derived an exact construction of algebraic expressions for the critical exponents $γ$ and $β$ . We proved that these expressions depend solely on the exact value of $ν\,$ .Based on our previous results regarding critical exponent estimates for the 3D Ising model and scaling relations [1], we have proven that the exact value of the correlation-length exponent is $\frac{\sqrt[3]{2}}{2}$. This is in very close agreement with the most precise numerical estimates currently available. Using a group theoretical approach and some propositions of constructive geometry, we derived an exact construction of algebraic expressions for the critical exponents $γ$ and $β$. We proved that these expressions depend solely on the exact value of $ν\,$.

arXiv:2610.17797Oct 13, 20260 citations

Proof of the exact correlation-length exponent

ν= \frac{\sqrt[3]{2}}{2}

and algebraic relations for

β

and

γ

in the 3D Ising model

Based on our previous results regarding critical exponent estimates for the 3D Ising model and scaling relations [1], we have proven that the exact value of the correlation-length exponent is

\frac{\sqrt[3]{2}}{2}

. This is in very close agreement with the most precise numerical estimates currently available. Using a group theoretical approach and some propositions of constructive geometry, we derived an exact construction of algebraic expressions for the critical exponents

γ

and

β

. We proved that these expressions depend solely on the exact value of

ν\,