arXiv:cond-mat/0205046 · Physical review. B, Condensed matter

Chiral phase transitions: Focus driven critical behavior in systems with planar and vector ordering

Pasquale Calabrese, Pietro Parruccini, Aleksandr Ivanovich Sokolov

Published Nov 22, 2002• 56 citations

Abstract

The fixed point that governs the critical behavior of magnets described by the N-vector chiral model under the physical values of N $(N=2,3)$ 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.

DOI: 10.1103/physrevb.66.180403

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arXiv:cond-mat/0205046 · Physical review. B, Condensed matter

Chiral phase transitions: Focus driven critical behavior in systems with planar and vector ordering

Pasquale Calabrese, Pietro Parruccini, Aleksandr Ivanovich Sokolov

Published Nov 22, 2002• 56 citations

arXiv HTML PDF

Abstract

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)

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.

DOI: 10.1103/physrevb.66.180403

Related papers

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

Pasquale Calabrese, E. V. Orlov, Pietro Parruccini +1 more

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.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.

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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

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Relevance of the fixed dimension perturbative approach to frustrated magnets in two and three dimensions

Bertrand Delamotte, M. Dudka, Yurij V. Holovatch +1 more

We show that the critical behavior of two- and three-dimensional frustrated magnets cannot reliably be described from the known five- and six-loop perturbative renormalization-group results. Our conclusions are based on a careful reanalysis of the resummed perturbative series obtained within the zero-momentum massive scheme. In three dimensions, the critical exponents for XY and Heisenberg spins display strong dependences on the parameters of the resummation procedure and on the loop order. This behavior strongly suggests that the fixed points found are in fact spurious. In two dimensions, we find, as in the $O(N)$ case, that there is apparent convergence of the critical exponents but toward erroneous values. As a consequence, the interesting question of the description of the crossover/transition induced by ${Z}_{2}$ topological defects in two-dimensional frustrated Heisenberg spins remains open.We show that the critical behavior of two- and three-dimensional frustrated magnets cannot reliably be described from the known five- and six-loop perturbative renormalization-group results. Our conclusions are based on a careful reanalysis of the resummed perturbative series obtained within the zero-momentum massive scheme. In three dimensions, the critical exponents for XY and Heisenberg spins display strong dependences on the parameters of the resummation procedure and on the loop order. This behavior strongly suggests that the fixed points found are in fact spurious. In two dimensions, we find, as in the $O(N)$ case, that there is apparent convergence of the critical exponents but toward erroneous values. As a consequence, the interesting question of the description of the crossover/transition induced by ${Z}_{2}$ topological defects in two-dimensional frustrated Heisenberg spins remains open.

arXiv:1009.1492Physical Review BSep 30, 201043 citations

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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.

arXiv:cond-mat/9909087Physical review. B, Condensed matterJun 1, 200047 citations

Chiral Criticality near Two Dimensions

Hikaru Kawamura

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.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.

Journal of the Physical Society of JapanJun 15, 199124 citations

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

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.

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 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.

O(N)

case, that there is apparent convergence of the critical exponents but toward erroneous values. As a consequence, the interesting question of the description of the crossover/transition induced by

{Z}_{2}

topological defects in two-dimensional frustrated Heisenberg spins remains open.

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.