Critical behaviour of the continuous n-component Potts model

Zia, R. K. P.; Wallace, D. J.

doi:10.1088/0305-4470/8/9/019

Cited by 188 publications

(108 citation statements)

References 24 publications

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“…On the contrary, exact two-dimensional results, numerical simulations and RG analysis suggest that for small n, the phase transition is of the second order [9,23].…”

Section: Introductionmentioning

confidence: 93%

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Critical behaviour of theO(n)-ϕ⁴model with an antisymmetric tensor order parameter

Antonov¹,

Kompaniets²,

Lebedev³

2013

J. Phys. A: Math. Theor.

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Abstract. Critical behaviour of the O(n)-symmetric φ4 -model with an antisymmetric tensor order parameter is studied by means of the field-theoretic renormalization group (RG) in the leading order of the ε = 4 − d-expansion (one-loop approximation). For n = 2 and 3 the model is equivalent to the scalar and the O(3)-symmetric vector models, for n ≥ 4 it involves two independent interaction terms and two coupling constants. It is shown that for n > 4 the RG equations have no infrared (IR) attractive fixed points and their solutions (RG flows) leave the stability region of the model. This means that fluctuations of the order parameter change the nature of the phase transition from the second-order type (suggested by the mean-field theory) to the first-order one. For n = 4, the IR attractive fixed point exists and the IR behaviour is non-universal: if the coupling constants belong to the basin of attraction for the IR point, the phase transition is of the second order and the IR critical scaling regime realizes. The corresponding critical exponents ν and η are presented in the order ε and ε 2 , respectively. Otherwise the RG flows pass outside the stability region and the first-order transition takes place.

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“…On the contrary, exact two-dimensional results, numerical simulations and RG analysis suggest that for small n, the phase transition is of the second order [9,23].…”

Section: Introductionmentioning

confidence: 93%

“…The corresponding RG equations can have several fixed points with different attractive properties [9,10,13,14,15,16,17,19]. This can lead to a very complicated pattern of the corresponding RG flows (solutions of the RG equations for the invariant charges) in the space of model parameters.…”

Section: Introductionmentioning

confidence: 99%

Critical behaviour of theO(n)-ϕ⁴model with an antisymmetric tensor order parameter

Antonov¹,

Kompaniets²,

Lebedev³

2013

J. Phys. A: Math. Theor.

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“…defines the simplex representation 16 that we will use to describe the observables measured in the simulations. In order to investigate the possible presence of ͑spurious͒ ferromagnetic effects we have carefully checked the value of the magnetization, looking for possible signs of spontaneous ferromagnetic ordering.…”

Section: ͑1͒mentioning

confidence: 99%

Spin glass phase in the four-state three-dimensional Potts model

et al. 2009

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We perform numerical simulations, including parallel tempering, a four-state Potts glass model with binary random quenched couplings using the JANUS application-oriented computer. We find and characterize a glassy transition, estimating the critical temperature and the value of the critical exponents. Nevertheless, the extrapolation to infinite volume is hampered by strong scaling corrections. We show that there is no ferromagnetic transition in a large temperature range around the glassy critical temperature. We also compare our results with those obtained recently on the "random permutation" Potts glass.

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“…will start by rewriting the Hamiltonian (3) for the case of uniaxial symmetry, introducing in addition a single-particle anisotropy term: (6) where N is the total number of spins and z is the coordination number. lt can be seen that there is no information about the transverse coupling in this mean-field Hamiltonian, which for D =O would be exactly the same as in the symmetric case J 1 = J 2·…”

Section: Mean-field Theorymentioning

confidence: 99%

Phase transitions in asymmetric Potts models: Breakdown of the classical mean-field picture

Gusmao

1987

Phys. Rev. B

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It is shown that mean-field theory fails to give a correct qualitative picture of the thermodynamic behavior of the q-state Potts model when the exchange interaction is anisotropic in spin space. The correct picture is recovered either by introducing a single-particle anisotropy or by taking correlations into account via a Bethe-Peierls approximation. This analysis helps the interpretation of previous renormalization-group results for asymmetric Potts models. I. INTRODUCfiONThe Potts model 1 of a ferromagnet has been extensively studied 2 either in its original lattice formulation or in the continuum version 3 -6 as a Euclidean f{J 3 field theory. Extensions of the model have also been introduced 7 -9 that allow for anisotropy in spin space. These are used, for example, to describe structural phase transitions in perovskites. 7 • 8 In a recent paper, Barbosa, Gusmão, and Theumann 9 discussed the phase transitions in the continuum version of the q-state Potts model with symmetry breaking, using a form of the renormalization group (RG) suitable for studying the crossover behavior when some components of the order-parameter field remain massive through the transitiono 10 The purpose of this work is to complement the analysis of Ref. 9, where the interpretation of the RG results was based on a mean-field theory (MFT) which, as I shall discuss below, is not appropriate for the anisotropic case. In particular, the MFT predicts the existence of a disordered phase with zero magnetization even when the exchange anisotropy favors only one of the q states against ali the otherso The existence of such a phase is obviously not expected on physical groundso This failure can be explained by the fact that an asymmetry in the exchange interaction manifests itself through correlations, which are neglected in MFTo lndeed, when a single-particle (crystal-field) anisotropy is introduced (Seco 111) it appears as an effective magnetic field in the Landau free energy and, consequently, the disordered phase is not present. I also show that in the case of purely exchange asymmetry the introduction of correlations through a simple Bethe-Peierls approximation 11 (Seco IV) results in the absence of a disordered phase. This precludes a paramagnetic-to-ferromagnetic second-order phase transition, although a first-order transition between a weakly and a strongly magnetized phase is not ruled outo 8 It also explains the absence of a nontrivial fixed point of the RG for this case, 9 since the mean-field minimum at zero-order parameter, around which the perturbation expansion is performed, is no longer a minimum when the asymmetric interaction is taken into accounto A controversial point concerning the continuation of the results for small noninteger q that is important to understand the role of the RG fixed point and criticai exponents is also addressed (Seco V). THE MODEL AND RG RESULTSThe Hamiltonian of the q-state Potts model in a lattice with only nearest-neighbor interactions is usually 2 written aswhere 8u 1 u 1 is the Kronecker 8 func...

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Critical behaviour of the continuous n-component Potts model

Cited by 188 publications

References 24 publications

Critical behaviour of theO(n)-ϕ⁴model with an antisymmetric tensor order parameter

Critical behaviour of theO(n)-ϕ⁴model with an antisymmetric tensor order parameter

Spin glass phase in the four-state three-dimensional Potts model

Phase transitions in asymmetric Potts models: Breakdown of the classical mean-field picture

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Critical behaviour of the continuous n-component Potts model

Cited by 188 publications

References 24 publications

Critical behaviour of theO(n)-ϕ4model with an antisymmetric tensor order parameter

Critical behaviour of theO(n)-ϕ4model with an antisymmetric tensor order parameter

Spin glass phase in the four-state three-dimensional Potts model

Phase transitions in asymmetric Potts models: Breakdown of the classical mean-field picture

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Product

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Critical behaviour of theO(n)-ϕ⁴model with an antisymmetric tensor order parameter

Critical behaviour of theO(n)-ϕ⁴model with an antisymmetric tensor order parameter