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Physics of the Solid State

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

E. V. Orlov, Aleksandr Ivanovich Sokolov

Published Nov 1, 2000• 20 citations
Publisher

Abstract

The paper is devoted to the calculation of renormalization-group (RG) functions in the O ( n )-symmetry two-dimensional model of the λϕ 4 type in the five-loop approximation and to an analysis of the critical behavior of systems described by this model. Five-loop expansions for the β function and the critical indices are determined in bulk theory. They are summed up using the Padé-Borel and Padé-Borel-Le Roy methods, making it possible to optimize the summation procedure and to estimate the accuracy of the obtained numerical values. It is shown that in the Ising ( n =1) case, as well as in other cases, the inclusion of the five-loop contribution to the β function displaces the coordinate of the Wilson fixed point only insignificantly, leaving it outside the interval formed by the results of computations on lattices; even “spreads” of the error in the renormalization group and lattice estimates do not overlap. This discrepancy is attributed to the effect of the nonanalytic com-ponent of the β function, which cannot be determined in perturbation theory. A computation of critical indices proves that, although the inclusion of the five-loop terms in the corresponding RG expansion slightly improves the concordance with the exact results, the nonanalytic contributions are apparently also significant in this case.

DOI
10.1134/1.1324056

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arXiv:cond-mat/0510088•Physics of the Solid State•Nov 1, 2005•14 citations
The paper is devoted to the calculation of renormalization-group (RG) functions in the O ( n )-symmetry two-dimensional model of the λϕ 4 type in the five-loop approximation and to an analysis of the critical behavior of systems described by this model. Five-loop expansions for the β function and the critical indices are determined in bulk theory. They are summed up using the Padé-Borel and Padé-Borel-Le Roy methods, making it possible to optimize the summation procedure and to estimate the accuracy of the obtained numerical values. It is shown that in the Ising ( n =1) case, as well as in other cases, the inclusion of the five-loop contribution to the β function displaces the coordinate of the Wilson fixed point only insignificantly, leaving it outside the interval formed by the results of computations on lattices; even “spreads” of the error in the renormalization group and lattice estimates do not overlap. This discrepancy is attributed to the effect of the nonanalytic com-ponent of the β function, which cannot be determined in perturbation theory. A computation of critical indices proves that, although the inclusion of the five-loop terms in the corresponding RG expansion slightly improves the concordance with the exact results, the nonanalytic contributions are apparently also significant in this case.
The RG functions of the 2D nnn-vector ϕ4ϕ^4ϕ4 model are calculated in the five-loop approximation. Perturbative series for the βββ 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=1n = 1n=1 as well as in the others (n=0n = 0n=0, n=−1n = -1n=−1, n=2,3,...32n = 2, 3,...32n=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 βββ function that can not be found perturbatively. The evaluation of the critical exponents for n=1n = 1n=1, n=0n = 0n=0 and n=−1n = -1n=−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.
The critical behavior of the two-dimensional N-vector cubic model is studied within the field-theoretical renormalization-group (RG) approach. The beta-functions and critical exponents are calculated in the five-loop approximation, RG series obtained are resummed using Pade-Borel-Leroy and conformal mapping techniques. It is found that for N = 2 the continuous line of fixed points is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both beta-functions closer to each another. For N > 2 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. In the case N = 0 the results obtained are compatible with the conclusion that the impure critical behavior is controlled by the Ising fixed point.
The critical thermodynamics of the two-dimensional NNN-vector cubic and MNMNMN 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 NNN are investigated. For N=2N=2N=2 it is found that the continuous line of fixed points running from the XYXYXY 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>2N>2N>2 is an artifact of the perturbative analysis. For the quenched dilute O(M)O(M)O(M) models (MNMNMN models with N=0N=0N=0) the results are compatible with a stable pure fixed point for $M\ensuremath{\geqslant}1$. For the MNMNMN 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 MMM and NNN.
The critical behavior of the two-dimensional N -vector cubic model is studied within the field-theoretical renormalization-group (RG) approach.The β functions and critical exponents are calculated in the five-loop approximation, RG series obtained are resummed using Pad é-Borel-Leroy and conformal mapping techniques.It is found that for N = 2 the continuous line of fixed points is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both β functions closer to each other.For N 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.In the case N = 0 the results obtained are compatible with the conclusion that the impure critical behavior is controlled by the Ising fixed point.
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 pseudo-ε-expansions for the coordinate of the fixed point g * , the critical exponents, and the sextic effective coupling constant g 6 are determined for the two-dimensional Ising model on the basis of the five-loop renormalization group series. It is found that the pseudoe-expansions for the coordinate of the fixed point g * , the inverse exponent γ −1 , and the constant g 6 possess a remarkable property, namely, the higher terms of these series are so small that reliable numerical results can be obtained without invoking Borel summation.