First-Order Transitions for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>n</mml:mi></mml:math>-Vector Models in Two and More Dimensions: Rigorous Proof

Enter, Aernout van; Shlosman, Senya

doi:10.1103/physrevlett.89.285702

Cited by 79 publications

(98 citation statements)

References 15 publications

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“…For general interaction potentials, entropy and elasticity are no longer strictly linked and order-disorder transitions, which can then take place as a function of temperature or of density, might realize other melting scenarios [27]. Theoretical, computational and experimental research on more complex microscopic models will build on the hard-disk solution obtained in this work.…”

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confidence: 99%

Two-Step Melting in Two Dimensions: First-Order Liquid-Hexatic Transition

2011

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Melting in two spatial dimensions, as realized in thin films or at interfaces, represents one of the most fascinating phase transitions in nature, but it remains poorly understood. Even for the fundamental hard-disk model, the melting mechanism has not been agreed on after fifty years of studies. A recent Monte Carlo algorithm allows us to thermalize systems large enough to access the thermodynamic regime. We show that melting in hard disks proceeds in two steps with a liquid phase, a hexatic phase, and a solid. The hexatic-solid transition is continuous while, surprisingly, the liquid-hexatic transition is of first-order. This melting scenario solves one of the fundamental statistical-physics models, which is at the root of a large body of theoretical, computational and experimental research. Generic two-dimensional particle systems cannot crystallize at finite temperature [1][2][3] because of the importance of fluctuations, yet they may form solids [4]. This paradox has provided the motivation for elucidating the fundamental melting transition in two spatial dimensions. A crystal is characterized by particle positions which fluctuate about the sites of an infinite regular lattice. It has long-range positional order. Bond orientations are also the same throughout the lattice. A crystal thus possesses long-range orientational order. The positional correlations of a two-dimensional solid decay to zero as a power law at large distances. Because of the absence of a scale, one speaks of "quasi-long range" order. In a two-dimensional solid, the lattice distortions preserve long-range orientational order [5], while in a liquid, both the positional and the orientational correlations decay exponentially.Besides the solid and the liquid, a third phase, called "hexatic", has been discussed but never clearly identified in particle systems. The hexatic phase is characterized by exponential positional but quasi-long range orientational correlations. It has long been discussed whether the melting transition follows a one-step first-order scenario between the liquid and the solid (without the hexatic) as in three spatial dimensions Two-dimensional melting was discovered [4] in the simplest particle system, the hard-disk model. Hard disks (of radius σ) are structureless and all configurations of nonoverlapping disks have zero potential energy. Two isolated disks only feel the hard-core repulsion, but the other disks mediate an entropic "depletion" interaction (see, e.g., [19]). Phase transitions result from an "order from disorder" phenomenon: At high density, ordered configurations can allow for larger local fluctuations, thus higher entropy, than the disordered liquid. For hard disks, no difference exists between the liquid and the gas. At fixed density η, the phase diagram is independent of temperature T = 1/k B β, and the pressure is proportional to T , as discovered by D. Bernoulli in 1738. Even for this basic model, the nature of the melting transition has not been agreed on.The hard-disk model has been simulated with the...

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confidence: 99%

Two-Step Melting in Two Dimensions: First-Order Liquid-Hexatic Transition

2011

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“…. , t K be an ordering of all sites of T. Then we have 16) where the first sum runs over collections of pairs (x j , y j ), j = 1, . .…”

Section: Technical Lemmasmentioning

confidence: 99%

“…In addition, for all shift-ergodic Gibbs states µ ∈ G βt , we have We remark that the existence of a first-order transition in energy density has been a matter of some controversy in the physics literature; see [16,17] for more discussion and relevant references. The proof of Theorem 4.4 is fairly technical and it is therefore deferred to Sect.…”

Section: Nonlinear Vector Modelsmentioning

confidence: 99%

Forbidden Gap Argument for Phase Transitions Proved by Means of Chessboard Estimates

Biskup¹,

Kotecký

2006

Commun. Math. Phys.

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Chessboard estimates are one of the standard tools for proving phase coexistence in spin systems of physical interest. In this note we show that the method not only produces a point in the phase diagram where more than one Gibbs states coexist, but that it can also be used to rule out the existence of shift-ergodic states that differ significantly from those proved to exist. For models depending on a parameter (say, the temperature), this shows that the values of the conjugate thermodynamic quantity (the energy) inside the "transitional gap" are forbidden in all shift-ergodic Gibbs states. We point out several models where our result provides useful additional information concerning the set of possible thermodynamic equilibria.

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“…Above this value the transition becomes of first order, a result which does not violate Mermin-Wagner-Hohenberg theorem, since the correlation length is finite at the transition. For finite value of n, the question of the nature of the transition at high k is still a challenging problem, although there is a rigorous proof that the transition becomes of first-order for large enough values of k for arbitrary n 2 [22,23]. This observation suggests inspecting the effect of a higher value of k for the O(2) model as well.…”

Section: Definition Of the Model And Of The Observablesmentioning

confidence: 99%

Nematic phase transitions in two-dimensional systems

Berche¹,

Paredes²

2005

Condens. Matter Phys.

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Simulations of nematic-isotropic transition of liquid crystals in two dimensions are performed using an O(2) vector model characterized by non linear nearest neighbour spin interaction governed by the fourth Legendre polynomial P 4 . The system is studied through standard Finite-Size Scaling and conformal rescaling of density profiles or correlation functions. The low temperature limit is discussed in the spin wave approximation and confirms the numerical results, while the value of the correlation function exponent at the deconfining transition seems controversial. : 05.40.+j, 64.60.Fr, 75.10.Hk Key words: liquid crystal, orientational transition, nematic phase, topological transition PACS Ordering in two dimensionsIn the context of phase transitions, two-dimensional models exhibit a very rich variety of typical behaviours, ranging from conventional temperature-driven second order phase transitions (e.g. Ising model) to first-order ones (e.g. q > 4-state Potts model), with specific properties of models having continuous global symmetry which may present defect-mediated topological phase transitions (e.g. XY model) or even no transition at all (e.g. Heisenberg model). Models of nematic-isotropic orientational phase transitions belong to this latter category of systems displaying a continuous symmetry. Ordering in low dimensional systems is likely to be frustrated by the strength of fluctuations. On qualitative grounds, let us consider for instance *

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First-Order Transitions forn-Vector Models in Two and More Dimensions: Rigorous Proof

Cited by 79 publications

References 15 publications

Two-Step Melting in Two Dimensions: First-Order Liquid-Hexatic Transition

Two-Step Melting in Two Dimensions: First-Order Liquid-Hexatic Transition

Forbidden Gap Argument for Phase Transitions Proved by Means of Chessboard Estimates

Nematic phase transitions in two-dimensional systems

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