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arXiv:cond-mat/0304154 · Physical review. B, Condensed matter

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

Pasquale Calabrese, Pietro Parruccini, Aleksandr Ivanovich Sokolov

Published Sep 17, 2003• 30 citations
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Abstract

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.

DOI
10.1103/physrevb.68.094415

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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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ 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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ 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.
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.
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 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.
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The critical behavior of a model describing phase transitions in 3D antiferromagnets with 2N2N2N-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> 2N>~2N> 2 and lies near the Bose fixed point; corresponding critical exponents are close to those of the XY\mathrm{XY}XY model. The accuracy of the results obtained is discussed and estimated.
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.