Kosterlitz-Thouless transition

We review Monte Carlo calculations of phase transitions and ordering behavior in lattice gas models of adsorbed layers on surfaces. The technical aspects of Monte Carlo methods are briefly summarized and results for a wide variety of models are described. Included are calculations of internal energies and order parameters for these models as a function of temperature and coverage along with adsorption isotherms and dynamic quantities such as self-diffusion constants. We also show results which are applicable to the interpretation of experimental data on physical systems such as H on Pd(lOO) and H on Fe(110). Other studies which are presented address fundamental theoretical questions about the nature of phase transitions in a two-dimensional geometry such as the existence of Kosterlitz-Thouless transitions or the nature of dynamic critical exponents. Lastly, we briefly mention multilayer adsorption and wetting phenomena and touch on the kinetics of domain growth at surfaces. 

It is not clear whether in the centered rectangular lattice gas of section 3.2 such a Kosterlitz-Thouless transition occurs, or whether the disordered phase extends, though being incommensurate, down to the commensurate (3x1) phase (then this transition is believed to belong to a new chiral universality class ), or whether there is another disorder line for (3 x 1) correlations. However, Kosterlitz-Thouless type transitions have been found for various two-dimensional models the XY ferromagnet , the Coulomb gas . ... 

In the flnite size scaling analysis Eqs. (36)-(38) need modiHcation at Kosterlitz-Thouless transitions the argument (1 — T/T )L or alternatively must be replaced by L/ itself since 1/v = 0, and Eq. (56) must be... 

A connection between Manning counterion condensation [31] and Kosterlitz-Thouless transition [58a] was conjectured by Mohanty, and supported by phenomenological arguments [58b]. An explicit calculation using nonlinear PB confirmed the hypothesis [59]. 

The impact of this is tremendous. No long-range order (LRO) can exist at finite temperature in one dimension no crystals, no magnets, no superconductors. Only special transitions are possible in two dimensions. The Ising model (n = 1 component) is an example [7]. The Kosterlitz-Thouless transition [8], without LRO, is another case for d = 2 and n = 2, discussed in Section V.C. The thermal fluctuations are very destructive in lower dimensions. Quantum fluctuations (i.e., those associated with the dynamics of a system) also tend to suppress LRO and can sometimes destroy it even at 0 K when the Mermin-Wagner theorem does not apply. Such is the case of the quantum spin- antiferromagnetic models [9] in one dimension. 

There is, however, another type of transition possible in two dimensions, a transition between states without LRO. This is the Kosterlitz-Thouless transition [8] mentioned in Sections II and V.B.l. It is relevant to superconductivity, commensurate-incommensurate transitions [61], planar magnetism, the electron gas system, and to many other systems in two dimensions. It involves vortices (thus the requirement of a two-component order parameter) characterized by a winding number q = (1/2-rr) dr V0, in which 0 is the phase of the order parameter (see also Ref. 4), the amplitude being fixed. These free vortices have an energy [see Eq. (28)] given by... 

A. Alavi, Molecular Dynamics Simulation of Methane Adsorbed in MgO Evidence for a Kosterlitz-Thouless Transition, Mol. Phys. 71 (1990) 1173-1191 Evidence for a Kosterlitz-Thouless Transition in a Simulation of CD4 Adsorbed on MgO, Phys. Rev. Lett. 64 (1990), 2289-2292. A. Alavi and I. R. McDonald, Molecular Dynamics Simulation of Argon Physisorbed on Magnesium Oxide, Mol. Phys. 69 (1990) 703-713. 

Park JM, Lubensky TO (1995) Topological defects on fluctuating surfaces general properties and the Kosterlitz-Thouless transition. Phys Rev E 53(3) 2648-2664... 

T.J. Sluckin and A. Poniewierski, Novel surface phase transition in nematic liquid crystals Wetting and the Kosterlitz-Thouless transition, Phys. Rev. Lett. 

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