arXiv:hep-th/9803264 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical exponents for a three-dimensional O(n)-symmetric model withn>3

S. A. Antonenko, Aleksandr Ivanovich Sokolov

Published Mar 1, 1995• 156 citations

Abstract

Critical exponents for the three-dimensional O(n)-symmetric model with n>3 are estimated on the basis of six-loop renormalization-group (RG) expansions. A simple Pad\'e-Borel technique is used for the resummation of the RG series and the Pad\'e approximants [L/1] are shown to give rather good numerical results for all calculated quantities. For large n, the fixed point location ${\mathit{g}}_{\mathit{c}}$ and the critical exponents are also determined directly from six-loop expansions without addressing the resummation procedure. An analysis of the numbers obtained shows that resummation becomes unnecessary when n exceeds 28 provided an accuracy of about 0.01 is adopted as satisfactory for ${\mathit{g}}_{\mathit{c}}$ and the critical exponents. Further, results of the calculations performed are used to estimate the numerical accuracy of the 1/n expansion. The same value n=28 is shown to play the role of the lower boundary of the domain where this approximation provides high-precision estimates for the critical exponents.

DOI: 10.1103/physreve.51.1894

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Critical exponents for a three-dimensional impure Ising model in the five-loop approximation

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arXiv:hep-th/9803264 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical exponents for a three-dimensional O(n)-symmetric model withn>3

S. A. Antonenko, Aleksandr Ivanovich Sokolov

Published Mar 1, 1995• 156 citations

arXiv HTML PDF

Abstract

Critical exponents for the three-dimensional O(n)-symmetric model with n>3 are estimated on the basis of six-loop renormalization-group (RG) expansions. A simple Pad'e-Borel technique is used for the resummation of the RG series and the Pad'e approximants [L/1] are shown to give rather good numerical results for all calculated quantities. For large n, the fixed point location

{\mathit{g}}_{\mathit{c}}

and the critical exponents are also determined directly from six-loop expansions without addressing the resummation procedure. An analysis of the numbers obtained shows that resummation becomes unnecessary when n exceeds 28 provided an accuracy of about 0.01 is adopted as satisfactory for

{\mathit{g}}_{\mathit{c}}

and the critical exponents. Further, results of the calculations performed are used to estimate the numerical accuracy of the 1/n expansion. The same value n=28 is shown to play the role of the lower boundary of the domain where this approximation provides high-precision estimates for the critical exponents.

DOI: 10.1103/physreve.51.1894

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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.

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Critical exponents for a three-dimensional impure Ising model in the five-loop approximation

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The renormalization-group functions governing the critical behavior of the three-dimensional weakly-disordered Ising model are calculated in the five-loop approximation. The random fixed point location and critical exponents for impure Ising systems are estimated by means of the Padé-Borel-Leroy resummation of the renormalization-group expansions derived. The asymptotic critical exponents are found to be γ=1.325 ± 0.003, η=0.025 ± 0.01, ν= 0.671 ± 0.005, α=−0.0125 ± 0.008, β=0.344 ± 0.006, while for the correction-to-scaling exponent, a less accurate estimate ω=0.32 ± 0.06 is obtained.The renormalization-group functions governing the critical behavior of the three-dimensional weakly-disordered Ising model are calculated in the five-loop approximation. The random fixed point location and critical exponents for impure Ising systems are estimated by means of the Padé-Borel-Leroy resummation of the renormalization-group expansions derived. The asymptotic critical exponents are found to be γ=1.325 ± 0.003, η=0.025 ± 0.01, ν= 0.671 ± 0.005, α=−0.0125 ± 0.008, β=0.344 ± 0.006, while for the correction-to-scaling exponent, a less accurate estimate ω=0.32 ± 0.06 is obtained.

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The critical behavior of a model describing phase transitions in 3D antiferromagnets with $2N$ -component real order parameters is studied within the renormalization-group (RG) approach. The RG functions are calculated in the three-loop order and resummed by the generalized Pad\'e-Borel procedure preserving the specific symmetry properties of the model. An anisotropic stable fixed point is found to exist in the RG flow diagram for $N>~2$ and lies near the Bose fixed point; corresponding critical exponents are close to those of the $\mathrm{XY}$ model. The accuracy of the results obtained is discussed and estimated.The critical behavior of a model describing phase transitions in 3D antiferromagnets with $2N$-component real order parameters is studied within the renormalization-group (RG) approach. The RG functions are calculated in the three-loop order and resummed by the generalized Pad\'e-Borel procedure preserving the specific symmetry properties of the model. An anisotropic stable fixed point is found to exist in the RG flow diagram for $N>~2$ and lies near the Bose fixed point; corresponding critical exponents are close to those of the $\mathrm{XY}$ model. The accuracy of the results obtained is discussed and estimated.

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Critical Exponents for the Model with Unique Stable Fixed Point From Three-Loop RG Expansions

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The critical behavior of a model describing phase transitions in cubic and tetragonal anti-ferromagnets with 2N-component (N>1) real order parameters as well as the structural transition in NbO 2 crystal is studied within the field-theoretical renormalization-group (RG) approach in three and (4-∊)-dimensions. Perturbative expansions for RG functions are calculated up to three-loop order and resummed, in 3D, by means of the generalized Padé–Borel procedure which is shown to preserve the specific symmetry properties of the model. It is found that a stable fixed point does exist in the three-dimensional RG flow diagram for N>1, in accordance with predictions obtained earlier within the ∊-expansion. Fixed-point coordinates and critical-exponent values are presented for physically interesting cases N=2 and N=3. In both cases critical exponents are found to be numerically close to those of the 3DXY model. The analysis of the results given by the ∊-expansion and by the RG approach in three dimensions is performed resulting in a conclusion that the latter provides much more accurate numerical estimates.The critical behavior of a model describing phase transitions in cubic and tetragonal anti-ferromagnets with 2N-component (N>1) real order parameters as well as the structural transition in NbO 2 crystal is studied within the field-theoretical renormalization-group (RG) approach in three and (4-∊)-dimensions. Perturbative expansions for RG functions are calculated up to three-loop order and resummed, in 3D, by means of the generalized Padé–Borel procedure which is shown to preserve the specific symmetry properties of the model. It is found that a stable fixed point does exist in the three-dimensional RG flow diagram for N>1, in accordance with predictions obtained earlier within the ∊-expansion. Fixed-point coordinates and critical-exponent values are presented for physically interesting cases N=2 and N=3. In both cases critical exponents are found to be numerically close to those of the 3DXY model. The analysis of the results given by the ∊-expansion and by the RG approach in three dimensions is performed resulting in a conclusion that the latter provides much more accurate numerical estimates.

International Journal of Modern Physics BMay 30, 199816 citations

Stability of a cubic fixed point in three dimensions: Critical exponents for genericN

K. B. Varnashev

The detailed analysis of the global structure of the renormalization-group (RG) flow diagram for a model with isotropic and cubic interactions is carried out in the framework of the massive field theory directly in three dimensions (3D) within an assumption of isotropic exchange. Perturbative expansions for RG functions are calculated for arbitrary N up to four-loop order and resummed by means of the generalized Pad\'e-Borel-Leroy technique. Coordinates and stability matrix eigenvalues for the cubic fixed point are found under the optimal value of the transformation parameter. Critical dimensionality of the model is proved to be equal to ${N}_{c}=2.89\ifmmode\pm\else\textpm\fi{}0.02$ that agrees well with the estimate obtained on the basis of the five-loop $\ensuremath{\varepsilon}$ expansion [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995)] resummed by the above method. As a consequence, the cubic fixed point should be stable in 3D for $N>~3,$ and the critical exponents controlling phase transitions in three-dimensional magnets should belong to the cubic universality class. The critical behavior of the random Ising model being the nontrivial particular case of the cubic model when $N=0$ is also investigated. For all physical quantities of interest the most accurate numerical estimates with their error bounds are obtained. The results achieved in the work are discussed along with the predictions given by other theoretical approaches and experimental data.The detailed analysis of the global structure of the renormalization-group (RG) flow diagram for a model with isotropic and cubic interactions is carried out in the framework of the massive field theory directly in three dimensions (3D) within an assumption of isotropic exchange. Perturbative expansions for RG functions are calculated for arbitrary N up to four-loop order and resummed by means of the generalized Pad\'e-Borel-Leroy technique. Coordinates and stability matrix eigenvalues for the cubic fixed point are found under the optimal value of the transformation parameter. Critical dimensionality of the model is proved to be equal to ${N}_{c}=2.89\ifmmode\pm\else\textpm\fi{}0.02$ that agrees well with the estimate obtained on the basis of the five-loop $\ensuremath{\varepsilon}$ expansion [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995)] resummed by the above method. As a consequence, the cubic fixed point should be stable in 3D for $N>~3,$ and the critical exponents controlling phase transitions in three-dimensional magnets should belong to the cubic universality class. The critical behavior of the random Ising model being the nontrivial particular case of the cubic model when $N=0$ is also investigated. For all physical quantities of interest the most accurate numerical estimates with their error bounds are obtained. The results achieved in the work are discussed along with the predictions given by other theoretical approaches and experimental data.

arXiv:cond-mat/9909087Physical review. B, Condensed matterJun 1, 200047 citations

Critical thermodynamics of a three-dimensional chiral model for N>3

Pasquale Calabrese, Pietro Parruccini, Aleksandr Ivanovich Sokolov

The critical behavior of the three-dimensional N-vector chiral model is studied for arbitrary N. The known six-loop renormalization-group (RG) expansions are resummed using the Borel transformation combined with the conformal mapping and Pad\'e approximant techniques. Analyzing the fixed-point location and the structure of RG flows, it is found that two marginal values of N exist which separate domains of continuous chiral phase transitions $N>{N}_{c1}$ and $N<{N}_{c2}$ from the region ${N}_{c1}>N>{N}_{c2}$ where such transitions are first order. Our calculations yield ${N}_{c1}=6.4(4)$ and ${N}_{c2}=5.7(3).$ For $N>{N}_{c1}$ the structure of RG flows is identical to that given by the $\ensuremath{\epsilon}$ and $1/N$ expansions with the chiral fixed point being a stable node. For $N<{N}_{c2}$ the chiral fixed point turns out to be a focus having no generic relation to the stable fixed point seen at small $\ensuremath{\epsilon}$ and large N. In this domain, containing the physical values $N=2$ and $N=3,$ phase trajectories approach the fixed point in a spiral-like manner giving rise to unusual crossover regimes which may imitate varying (scattered) critical exponents seen in numerous physical and computer experiments.The critical behavior of the three-dimensional N-vector chiral model is studied for arbitrary N. The known six-loop renormalization-group (RG) expansions are resummed using the Borel transformation combined with the conformal mapping and Pad\'e approximant techniques. Analyzing the fixed-point location and the structure of RG flows, it is found that two marginal values of N exist which separate domains of continuous chiral phase transitions $N>{N}_{c1}$ and $N<{N}_{c2}$ from the region ${N}_{c1}>N>{N}_{c2}$ where such transitions are first order. Our calculations yield ${N}_{c1}=6.4(4)$ and ${N}_{c2}=5.7(3).$ For $N>{N}_{c1}$ the structure of RG flows is identical to that given by the $\ensuremath{\epsilon}$ and $1/N$ expansions with the chiral fixed point being a stable node. For $N<{N}_{c2}$ the chiral fixed point turns out to be a focus having no generic relation to the stable fixed point seen at small $\ensuremath{\epsilon}$ and large N. In this domain, containing the physical values $N=2$ and $N=3,$ phase trajectories approach the fixed point in a spiral-like manner giving rise to unusual crossover regimes which may imitate varying (scattered) critical exponents seen in numerous physical and computer experiments.

arXiv:cond-mat/0304154Physical review. B, Condensed matterSep 17, 200330 citations

{\mathit{g}}_{\mathit{c}}

{\mathit{g}}_{\mathit{c}}

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 critical behavior of a model describing phase transitions in 3D antiferromagnets with

2N

-component real order parameters is studied within the renormalization-group (RG) approach. The RG functions are calculated in the three-loop order and resummed by the generalized Pad\'e-Borel procedure preserving the specific symmetry properties of the model. An anisotropic stable fixed point is found to exist in the RG flow diagram for

N>~2

and lies near the Bose fixed point; corresponding critical exponents are close to those of the

\mathrm{XY}

model. The accuracy of the results obtained is discussed and estimated.

N>~3,

and the critical exponents controlling phase transitions in three-dimensional magnets should belong to the cubic universality class. The critical behavior of the random Ising model being the nontrivial particular case of the cubic model when

N=0

is also investigated. For all physical quantities of interest the most accurate numerical estimates with their error bounds are obtained. The results achieved in the work are discussed along with the predictions given by other theoretical approaches and experimental data.

N>{N}_{c1}

and

N<{N}_{c2}

from the region

{N}_{c1}>N>{N}_{c2}

where such transitions are first order. Our calculations yield

{N}_{c1}=6.4(4)

and

{N}_{c2}=5.7(3).

For

N>{N}_{c1}

the structure of RG flows is identical to that given by the $\ensuremath{\epsilon}$ and

1/N

expansions with the chiral fixed point being a stable node. For

N<{N}_{c2}

the chiral fixed point turns out to be a focus having no generic relation to the stable fixed point seen at small $\ensuremath{\epsilon}$ and large N. In this domain, containing the physical values

N=2

and

N=3,

phase trajectories approach the fixed point in a spiral-like manner giving rise to unusual crossover regimes which may imitate varying (scattered) critical exponents seen in numerous physical and computer experiments.