arXiv:cond-mat/9804223 · Physical review. B, Condensed matter

Critical behavior of three-dimensional magnets with complicated ordering from three-loop renormalization-group expansions

Aleksandr Ivanovich Sokolov, K. B. Varnashev

Published Apr 1, 1999• 17 citations

Abstract

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.

DOI: 10.1103/physrevb.59.8363

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arXiv:cond-mat/9804223 · Physical review. B, Condensed matter

Critical behavior of three-dimensional magnets with complicated ordering from three-loop renormalization-group expansions

Aleksandr Ivanovich Sokolov, K. B. Varnashev

Published Apr 1, 1999• 17 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physrevb.59.8363

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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.

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Critical thermodynamics of three-dimensional MN-component field model with cubic anisotropy from higher-loop RG expansions

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

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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.

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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.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.

arXiv:hep-th/9803264Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsMar 1, 1995156 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 $N$.

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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

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

2N

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_c^C=1.447 \pm 0.020

is obtained from resumming the known five-loop $\ve$-series for N_c^R. Since N_c^C < 2, the critical thermodynamics of the model relevant to the phase transitions in real substances should be governed by the complex cubic fixed point with a new set of critical exponents: $\gm=1.404(25)$,

ν=0.715(10)

, $\et=0.0343(20)$ for N=2 and $\gm=1.390(25)$,

ν=0.702(10)

, $\et=0.0345(15)$ for N=3.

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.

{\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.

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>{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.