RENORMALIZATION GROUP ANALYSIS OF THE ISING MODEL ON TWO-DIMENSIONAL QUASI-LATTICES

1988 ◽  
Vol 02 (01) ◽  
pp. 13-35 ◽  
Author(s):  
HIDEAKI AOYAMA ◽  
TAKASHI ODAGAKI
Keyword(s):  
Phase Transition ◽  
Renormalization Group ◽  
Critical Temperature ◽  
Ising Model ◽  
Group Analysis ◽  
Critical Index ◽  
Two Dimensional ◽  
Renormalization Group Analysis

We present a renormalization group analysis of the two-dimensional Ising model on both the Kite-Dart and Rhombi Penrose lattices in which spins are located at the vertices of the lattice. We demonstrate the existence of the phase transition and obtain the critical temperature and the thermal critical index. The thermal critical index is found to be close to unity.

scholarly journals DOMAIN WALL RENORMALIZATION GROUP ANALYSIS OF TWO-DIMENSIONAL ISING MODEL

2009 ◽  
Vol 23 (18) ◽  
pp. 3739-3751 ◽  
Author(s):  
KEN-ICHI AOKI ◽  
TAMAO KOBAYASHI ◽  
HIROSHI TOMITA
Keyword(s):  
Renormalization Group ◽  
Ising Model ◽  
Domain Wall ◽  
Domain Walls ◽  
Group Analysis ◽  
Coarse Graining ◽  
Group Method ◽  
Two Dimensional ◽  
Renormalization Group Analysis ◽  
Renormalization Transformation

Using a recently proposed new renormalization group method (tensor renormalization group), we analyze the Ising model on the two-dimensional square lattice. For the lowest-order approximation with two-domain wall states, it realizes the idea of coarse graining of domain walls. We write down explicit analytic renormalization transformation and prove that the picture of the coarse graining of the physical domain walls does hold for all physical renormalization group flows. We solve it to get the fixed point structure and obtain the critical exponents and the critical temperature. These results are very near to the exact values. We also briefly report the improvement using four-domain wall states.

RENORMALIZATION GROUP ANALYSIS OF A TWO-DIMENSIONAL INTERACTING ELECTRON SYSTEM

2001 ◽  
Vol 16 (11) ◽  
pp. 1889-1898
Author(s):  
WALTER METZNER
Keyword(s):  
Renormalization Group ◽  
Group Analysis ◽  
Interaction Strength ◽  
Electron System ◽  
Flow Equation ◽  
Group Method ◽  
Model Parameters ◽  
Two Dimensional ◽  
Renormalization Group Analysis ◽  
Effective Interactions

We describe a Wick ordered functional renormalization group method for interacting Fermi systems, where the complete flow from the bare action of the microscopic model to the effective low-energy action is obtained from a differential flow equation. We apply this renormalization group approach to a prototypical two-dimensional lattice electron system, the Hubbard model on a square lattice. The flow equation for the effective interactions is evaluated numerically on 1-loop level. The effective interactions diverge at a finite energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signalled by the diverging interactions we compute the flow of the singlet superconducting susceptibilities for various pairing symmetries and also charge and spin density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength and band-filling) we find antiferromagnetic order or d-wave superconductivity as leading symmetry breaking instability.

A renormalization group analysis of lattice models of two-dimensional membranes

1989 ◽  
Vol 55 (1-2) ◽  
pp. 29-85 ◽  
Author(s):  
Jan Ambj�rn ◽  
Bergfinnur Durhuus ◽  
J�rg Fr�hlich ◽  
Th�rdur J�nsson
Keyword(s):  
Renormalization Group ◽  
Group Analysis ◽  
Lattice Models ◽  
Two Dimensional ◽  
Renormalization Group Analysis

scholarly journals Plastic Yielding as a Phase Transition

1999 ◽  
Vol 66 (2) ◽  
pp. 289-298 ◽  
Author(s):  
M. Ortiz
Keyword(s):  
Phase Transition ◽  
Shear Stress ◽  
Critical Temperature ◽  
Ising Model ◽  
Yield Point ◽  
Slip Plane ◽  
Dislocation Loop ◽  
Lattice Gas ◽  
Two Dimensional ◽  
Statistical Mechanical

A statistical mechanical theory of forest hardening is developed in which yielding arises as a phase transition. For simplicity, we consider the case of a single dislocation loop moving on a slip plane through randomly distributed forest dislocations, which we treat as point obstacles. The occurrence of slip at the sites occupied by these obstacles is assumed to require the expenditure of a certain amount of work commensurate with the strength of the obstacle. The case of obstacles of infinite strength is treated in detail. We show that the behavior of the dislocation loop as it sweeps the slip plane under the action of a resolved shear stress is identical to that of a lattice gas, or, equivalently, to that of the two-dimensional spin-1/2 Ising model. In particular, there exists a critical temperature Tc below which the system exhibits a yield point, i.e., the slip strain increases sharply when the applied resolved shear stress attains a critical value. Above the critical temperature the yield point disappears and the slip strain depends continuously on the applied stress. The critical exponents, which describe the behavior of the system near the critical temperature, coincide with those of the two-dimensional spin-1/2 Ising model.

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