arXiv:cond-mat/0405432 · Physical Review B

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

Pasquale Calabrese, E. V. Orlov, D. V. Pakhnin, Aleksandr Ivanovich Sokolov

Published Sep 29, 2004• 34 citations

Abstract

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 $N$.

DOI: 10.1103/physrevb.70.094425

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arXiv:cond-mat/0405432 · Physical Review B

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

Pasquale Calabrese, E. V. Orlov, D. V. Pakhnin, Aleksandr Ivanovich Sokolov

Published Sep 29, 2004• 34 citations

arXiv HTML PDF

Abstract

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

DOI: 10.1103/physrevb.70.094425

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Critical thermodynamics of two-dimensional systems in the five-loop renormalization-group approximation

E. V. Orlov, Aleksandr Ivanovich Sokolov

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Physics of the Solid StateNov 1, 200020 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}

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

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 RG functions of the 2D

n

-vector

ϕ^4

model are calculated in the five-loop approximation. Perturbative series for the

β

function and critical exponents are resummed by the Pade-Borel and Pade-Borel-Leroy techniques, resummation procedures are optimized and an accuracy of the numerical results is estimated. In the Ising case

n = 1

as well as in the others (

n = 0

n = -1

n = 2, 3,...32

) an account for the five-loop term is found to shift the Wilson fixed point location only briefly, leaving it outside the segment formed by the results of the corresponding lattice calculations; even error bars of the RG and lattice estimates do not overlap in the most cases studied. This is argued to reflect the influence of the singular (non-analytical) contribution to the

β

function that can not be found perturbatively. The evaluation of the critical exponents for

n = 1

n = 0

and

n = -1

in the five-loop approximation and comparison of the numbers obtained with their known exact counterparts confirm the conclusion that non-analytical contributions are visible in two dimensions.

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.