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arXiv:cond-mat/0207187 · Physical review. B, Condensed matter

Chiral critical behavior of frustrated spin systems in two dimensions from five-loop renormalization-group expansions

Pasquale Calabrese, E. V. Orlov, Pietro Parruccini, Aleksandr Ivanovich Sokolov

Published Jan 16, 2003• 23 citations
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Abstract

We analyze the critical behavior of two-dimensional N-vector spin systems with noncollinear order within the five-loop renormalization-group (RG) approximation. The structure of the RG flow is studied for different N leading to the conclusion that the chiral fixed point governing the critical behavior of physical systems with $N=2$ and $N=3$ does not coincide with that given by the $1/N$ expansion. We show that the stable chiral fixed point for $N<~{N}^{*},$ including $N=2$ and $N=3,$ turns out to be a focus. We give a complete characterization of the critical behavior controlled by this fixed point, also evaluating the subleading crossover exponents. The spiral-like approach of the chiral fixed point is argued to give rise to unusual crossover and near-critical regimes that may imitate varying critical exponents seen in numerous physical and computer experiments.

DOI
10.1103/physrevb.67.024413

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We analyze the critical behavior of two-dimensional N-vector spin systems with noncollinear order within the five-loop renormalization-group (RG) approximation. The structure of the RG flow is studied for different N leading to the conclusion that the chiral fixed point governing the critical behavior of physical systems with N=2N=2N=2 and N=3N=3N=3 does not coincide with that given by the 1/N1/N1/N expansion. We show that the stable chiral fixed point for N< N∗,N<~{N}^{*},N< N∗, including N=2N=2N=2 and N=3,N=3,N=3, turns out to be a focus. We give a complete characterization of the critical behavior controlled by this fixed point, also evaluating the subleading crossover exponents. The spiral-like approach of the chiral fixed point is argued to give rise to unusual crossover and near-critical regimes that may imitate varying critical exponents seen in numerous physical and computer experiments.
The fixed point that governs the critical behavior of magnets described by the N-vector chiral model under the physical values of N (N=2,3)(N=2,3)(N=2,3) is shown to be a stable focus both in two and three dimensions. Robust evidence in favor of this conclusion is obtained within the five-loop and six-loop renormalization-group analysis in fixed dimension. The spirallike approach of the chiral fixed point results in unusual crossover and near-critical regimes that may imitate varying critical exponents seen in 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>Nc1N>{N}_{c1}N>Nc1​ and N<Nc2N<{N}_{c2}N<Nc2​ from the region Nc1>N>Nc2{N}_{c1}>N>{N}_{c2}Nc1​>N>Nc2​ where such transitions are first order. Our calculations yield Nc1=6.4(4){N}_{c1}=6.4(4)Nc1​=6.4(4) and Nc2=5.7(3).{N}_{c2}=5.7(3).Nc2​=5.7(3). For N>Nc1N>{N}_{c1}N>Nc1​ the structure of RG flows is identical to that given by the $\ensuremath{\epsilon}$ and 1/N1/N1/N expansions with the chiral fixed point being a stable node. For N<Nc2N<{N}_{c2}N<Nc2​ 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=2N=2N=2 and N=3,N=3,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.
Critical behavior associated with the helical or noncollinear ordering for isotropic n >3-component spins is studied near d =2 dimensions based on an O ( n )× O (2) nonlinear σ model and the renormalization-group ε= d -2 expansion. A chiral fixed point is found for general n characterized by the critical exponents that are consistent with the previous 1/ n expansion calculation. For n =3, it is argued that because of the nontrivial global topology of the order-parameter space, ∝ 1 ( S O (3))= Z 2 , and the possible existence of a finite-temperature topological transition at d =2, the standard ε= d -2 expansion technique is not applicable.
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 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.