Accurate calculations of the thermodynamic properties of a system near its critical point are now possible using finite-lattice approximations to renormalization groups. In order to investigate some features of such approximations, we introduce a linear-renormalization-group transformation appropriate to a system of continuous spins like the Gaussian model of Berlin and Kac. We solve the renormalization-group equations for the Gaussian model exactly, and then study in detail a finite-lattice approximation to the renormalization group for the three-dimensional case. The renormalization-group transformation contains a parameter $b$ . The exact transformation has fixed points only for $b={2}^{\ensuremath{-}\frac{5}{2}}$. The approximate transformation has fixed points for a range of values of $b$ around ${2}^{\ensuremath{-}\frac{5}{2}}$. Eigenvalues of the transformation, which determine critical exponents, depend on the value of $b$ used in the calculation. We study the problem of identifying the correct value of $b$ .Accurate calculations of the thermodynamic properties of a system near its critical point are now possible using finite-lattice approximations to renormalization groups. In order to investigate some features of such approximations, we introduce a linear-renormalization-group transformation appropriate to a system of continuous spins like the Gaussian model of Berlin and Kac. We solve the renormalization-group equations for the Gaussian model exactly, and then study in detail a finite-lattice approximation to the renormalization group for the three-dimensional case. The renormalization-group transformation contains a parameter $b$. The exact transformation has fixed points only for $b={2}^{\ensuremath{-}\frac{5}{2}}$. The approximate transformation has fixed points for a range of values of $b$ around ${2}^{\ensuremath{-}\frac{5}{2}}$. Eigenvalues of the transformation, which determine critical exponents, depend on the value of $b$ used in the calculation. We study the problem of identifying the correct value of $b$.

Physical review. B, Solid stateMay 1, 1975114 citations

Renormalization Group on the Lattice

Andreas Wipf

Lecture notes in physicsOct 26, 20120 citations

Nonperturbative renormalization group approach to quantum XY spin models

Adam Rançon

We present a lattice nonperturbative renormalization group (NPRG) approach to quantum $XY$ spin models by using a mapping onto hardcore bosons. The NPRG takes as initial condition of the renormalization group flow the (local) limit of decoupled sites, allowing us to take into account the hardcore constraint exactly. The initial condition of the flow is equivalent to the large $S$ classical results of the corresponding spin system. Furthermore, the hardcore constraint is conserved along the RG flow, and we can describe both local and long-distance fluctuations in a nontrivial way. We discuss a simple approximation scheme, and solve the corresponding flow equations. We compute both the zero-temperature thermodynamics and the finite temperature phase diagram on the square and cubic lattices. The NPRG allows us to recover the correct critical physics at finite temperature in two and three dimensions. The results compare well with numerical simulations.We present a lattice nonperturbative renormalization group (NPRG) approach to quantum $XY$ spin models by using a mapping onto hardcore bosons. The NPRG takes as initial condition of the renormalization group flow the (local) limit of decoupled sites, allowing us to take into account the hardcore constraint exactly. The initial condition of the flow is equivalent to the large $S$ classical results of the corresponding spin system. Furthermore, the hardcore constraint is conserved along the RG flow, and we can describe both local and long-distance fluctuations in a nontrivial way. We discuss a simple approximation scheme, and solve the corresponding flow equations. We compute both the zero-temperature thermodynamics and the finite temperature phase diagram on the square and cubic lattices. The NPRG allows us to recover the correct critical physics at finite temperature in two and three dimensions. The results compare well with numerical simulations.

arXiv:1404.3425Physical Review BJun 23, 201423 citations

Phenomenological renormalization group methods

J. A. Plascak, W. Figueiredo, B. C. S. Grandi

Some renormalization group approaches have been proposed during the last few years which are close in spirit to the Nightingale phenomenological procedure. In essence, by exploiting the finite size scaling hypothesis, the approximate critical behavior of model on infinite lattice is obtained through the exact computation of some thermal quantities of the model on finite clusters. In this work some of these methods are reviewed, namely the mean field renormalization group, the effective field renormalization group and the finite size scaling renormalization group procedures. Although special emphasis is given to the mean field renormalization group (since it has been, up to now, much more used to study a wide variety of different systems) a discussion of their potentialities and interrelations to other methods is also presented.Some renormalization group approaches have been proposed during the last few years which are close in spirit to the Nightingale phenomenological procedure. In essence, by exploiting the finite size scaling hypothesis, the approximate critical behavior of model on infinite lattice is obtained through the exact computation of some thermal quantities of the model on finite clusters. In this work some of these methods are reviewed, namely the mean field renormalization group, the effective field renormalization group and the finite size scaling renormalization group procedures. Although special emphasis is given to the mean field renormalization group (since it has been, up to now, much more used to study a wide variety of different systems) a discussion of their potentialities and interrelations to other methods is also presented.

arXiv:cond-mat/9910458Brazilian Journal of PhysicsSep 1, 199943 citations

The Renormalization Group: Fundamental Aspects

Lothar Schäfer

Springer eBooksJan 1, 19990 citations

b

. The exact transformation has fixed points only for $b={2}^{\ensuremath{-}\frac{5}{2}}$. The approximate transformation has fixed points for a range of values of

b

around ${2}^{\ensuremath{-}\frac{5}{2}}$. Eigenvalues of the transformation, which determine critical exponents, depend on the value of

b

used in the calculation. We study the problem of identifying the correct value of

b

We present a lattice nonperturbative renormalization group (NPRG) approach to quantum

XY

spin models by using a mapping onto hardcore bosons. The NPRG takes as initial condition of the renormalization group flow the (local) limit of decoupled sites, allowing us to take into account the hardcore constraint exactly. The initial condition of the flow is equivalent to the large

S

classical results of the corresponding spin system. Furthermore, the hardcore constraint is conserved along the RG flow, and we can describe both local and long-distance fluctuations in a nontrivial way. We discuss a simple approximation scheme, and solve the corresponding flow equations. We compute both the zero-temperature thermodynamics and the finite temperature phase diagram on the square and cubic lattices. The NPRG allows us to recover the correct critical physics at finite temperature in two and three dimensions. The results compare well with numerical simulations.