We present real-space renormalization-group (RG) calculations of the critical properties of the random-field Ising model on a cubic lattice in three dimensions. We calculate the RG flows in a two-parameter truncation of the Hamiltonian space. As predicted, the transition at finite randomness is controlled by a zero-temperature, disordered critical fixed point, and we exhibit the universal crossover trajectory from the pure Ising critical point. We extract scaling fields and critical exponents, and study the distribution of barrier heights between states as a function of length scale.We present real-space renormalization-group (RG) calculations of the critical properties of the random-field Ising model on a cubic lattice in three dimensions. We calculate the RG flows in a two-parameter truncation of the Hamiltonian space. As predicted, the transition at finite randomness is controlled by a zero-temperature, disordered critical fixed point, and we exhibit the universal crossover trajectory from the pure Ising critical point. We extract scaling fields and critical exponents, and study the distribution of barrier heights between states as a function of length scale.

arXiv:cond-mat/9304047Physical review. B, Condensed matterDec 1, 199319 citations

Critical behavior of two-dimensional cubic and MN models in the five-loop renormalization group approximation

Pasquale Calabrese, E. V. Orlov, D. V. Pakhnin +1 more

The critical thermodynamics of the two-dimensional $N$ -vector cubic and $MN$ models is studied within the field-theoretical renormalization group (RG) approach. The $\ensuremath{\beta}$ functions and critical exponents are calculated in the five-loop approximation and the RG series obtained are resummed using the Borel-Leroy transformation combined with the generalized Pad\'e approximant and conformal mapping techniques. For the cubic model, the RG flows for various $N$ are investigated. For $N=2$ it is found that the continuous line of fixed points running from the $XY$ fixed point to the Ising one is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both $\ensuremath{\beta}$ functions closer to each another. For the cubic model with $N\ensuremath{\geqslant}3$, the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under $N>2$ is an artifact of the perturbative analysis. For the quenched dilute $O(M)$ models ( $MN$ models with $N=0$ ) the results are compatible with a stable pure fixed point for $M\ensuremath{\geqslant}1$. For the $MN$ model with $M,N\ensuremath{\geqslant}2$ all the nonperturbative results are reproduced. In addition a new stable fixed point is found for moderate values of $M$ and $N$ .The critical thermodynamics of the two-dimensional $N$-vector cubic and $MN$ models is studied within the field-theoretical renormalization group (RG) approach. The $\ensuremath{\beta}$ functions and critical exponents are calculated in the five-loop approximation and the RG series obtained are resummed using the Borel-Leroy transformation combined with the generalized Pad\'e approximant and conformal mapping techniques. For the cubic model, the RG flows for various $N$ are investigated. For $N=2$ it is found that the continuous line of fixed points running from the $XY$ fixed point to the Ising one is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both $\ensuremath{\beta}$ functions closer to each another. For the cubic model with $N\ensuremath{\geqslant}3$, the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under $N>2$ is an artifact of the perturbative analysis. For the quenched dilute $O(M)$ models ($MN$ models with $N=0$) the results are compatible with a stable pure fixed point for $M\ensuremath{\geqslant}1$. For the $MN$ model with $M,N\ensuremath{\geqslant}2$ all the nonperturbative results are reproduced. In addition a new stable fixed point is found for moderate values of $M$ and $N$.

arXiv:cond-mat/0405432Physical Review BSep 29, 200434 citations

Renormalization group methods

Kenneth Geddes Wilson

Advances in MathematicsMay 1, 197574 citations

Numerical renormalization group methods

Martin Andersson

Jan 1, 19990 citations

Five-loop renormalization-group expansions for the three-dimensionaln-vector cubic model and critical exponents for impure Ising systems

D. V. Pakhnin, Aleksandr Ivanovich Sokolov

The renormalization-group (RG) functions for the three-dimensional n-vector cubic model are calculated in the five-loop approximation. High-precision numerical estimates for the asymptotic critical exponents of the three-dimensional impure Ising systems are extracted from the five-loop RG series by means of the Pad\'e-Borel-Leroy resummation under $n=0.$ These exponents are found to be $\ensuremath{\gamma}=1.325\ifmmode\pm\else\textpm\fi{}0.003,$ $\ensuremath{\eta}=0.025\ifmmode\pm\else\textpm\fi{}0.01,$ $\ensuremath{\nu}=0.671\ifmmode\pm\else\textpm\fi{}0.005,$ $\ensuremath{\alpha}=\ensuremath{-}0.0125\ifmmode\pm\else\textpm\fi{}0.008,$ and $\ensuremath{\beta}=0.344\ifmmode\pm\else\textpm\fi{}0.006.$ For the correction-to-scaling exponent, the less accurate estimate $\ensuremath{\omega}=0.32\ifmmode\pm\else\textpm\fi{}0.06$ is obtained.The renormalization-group (RG) functions for the three-dimensional n-vector cubic model are calculated in the five-loop approximation. High-precision numerical estimates for the asymptotic critical exponents of the three-dimensional impure Ising systems are extracted from the five-loop RG series by means of the Pad\'e-Borel-Leroy resummation under $n=0.$ These exponents are found to be $\ensuremath{\gamma}=1.325\ifmmode\pm\else\textpm\fi{}0.003,$ $\ensuremath{\eta}=0.025\ifmmode\pm\else\textpm\fi{}0.01,$ $\ensuremath{\nu}=0.671\ifmmode\pm\else\textpm\fi{}0.005,$ $\ensuremath{\alpha}=\ensuremath{-}0.0125\ifmmode\pm\else\textpm\fi{}0.008,$ and $\ensuremath{\beta}=0.344\ifmmode\pm\else\textpm\fi{}0.006.$ For the correction-to-scaling exponent, the less accurate estimate $\ensuremath{\omega}=0.32\ifmmode\pm\else\textpm\fi{}0.06$ is obtained.

arXiv:cond-mat/9912071Physical review. B, Condensed matterJun 1, 200060 citations

The critical thermodynamics of the two-dimensional

N

-vector cubic and

MN

models is studied within the field-theoretical renormalization group (RG) approach. The $\ensuremath{\beta}$ functions and critical exponents are calculated in the five-loop approximation and the RG series obtained are resummed using the Borel-Leroy transformation combined with the generalized Pad\'e approximant and conformal mapping techniques. For the cubic model, the RG flows for various

N

are investigated. For

N=2

it is found that the continuous line of fixed points running from the

XY

fixed point to the Ising one is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both $\ensuremath{\beta}$ functions closer to each another. For the cubic model with $N\ensuremath{\geqslant}3$, the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under

N>2

is an artifact of the perturbative analysis. For the quenched dilute

O(M)

models (

MN

models with

N=0

) the results are compatible with a stable pure fixed point for $M\ensuremath{\geqslant}1$. For the

MN

model with $M,N\ensuremath{\geqslant}2$ all the nonperturbative results are reproduced. In addition a new stable fixed point is found for moderate values of

M

and

N

n=0.

These exponents are found to be $\ensuremath{\gamma}=1.325\ifmmode\pm\else\textpm\fi{}0.003,$ $\ensuremath{\eta}=0.025\ifmmode\pm\else\textpm\fi{}0.01,$ $\ensuremath{\nu}=0.671\ifmmode\pm\else\textpm\fi{}0.005,$ $\ensuremath{\alpha}=\ensuremath{-}0.0125\ifmmode\pm\else\textpm\fi{}0.008,$ and $\ensuremath{\beta}=0.344\ifmmode\pm\else\textpm\fi{}0.006.$ For the correction-to-scaling exponent, the less accurate estimate $\ensuremath{\omega}=0.32\ifmmode\pm\else\textpm\fi{}0.06$ is obtained.