Kosterlitz-Thouless

The roughening transition itself is believed to be of the Kosterlitz-Thouless type , according to which one should expect the following properties ... 

The above results show that the APB undergoes a roughening transition at T j 1490 K, which is below the bulk first-order transition, and this transition is in agreement with the Kosterlitz-Thouless theory. In brief, the APB is flat below T j and, above T j, it develops transverse fluctuations and wanders through the lattice. 

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. 

Kosterlitz-Thouless and commensurate-incommensurate transitions in the triangular lattice gas... 

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... 

The critical behavior is, however, the same there is a Kosterlitz-Thouless (KT) transition at the phase boundary Ku between a disorder dominated, pinned and a free, unpinned phase which terminates in the fixed point K = 6/p2. One can derive an implicit equation for Ku by combining (23a) and (23b) to a differential equation... 

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. 

Fig. 3 SPA-LEED results for NiAl(lOO) Intensity variation of the normalized (00) spot (full circles) and the background intensity (open squares) with temperature recorded for in-phase conditions (Eq = 95 eV). The full lines are guidelines for the eye the dashed line indicates a result for the situation of surface roughening of the Kosterlitz-Thouless type, (from ref [37]).

D. The Kosterlitz-Thouless-Halperin-Nelson-Young Theory of Two-Dimensional Melting... 

While for q = 2 this is still identical to the Ising model and for q = 3 identical to the standard Potts model, a different behavior results for q > 4. In particular, the exponents for q = 4 can be non-universal for variants of this model (Knops, 1980). For q > 4 there occurs a Kosterlitz-Thouless phase transition to a floating phase at higher temperatures, where the correlation function decays algebraically, and a IC-transition to a commensurate phase with ordered structure at lower temperatures (Elitzur et al., 1979 Jose et al., 1977). 

Of course, the approximations made by the spin wave theory are reasonable at very low temperatures only, and thus it is plausible that this line of critical temperatures terminates at a transition point 7 Kt, the Kosterlitz-Thouless (1973) transition, while for T > 7kt one has a correlation function that decays exponentially at large distances. This behavior is recognized when singular spin configurations called vortices (fig. 33 Kawabata and Binder, 1977) are included in the treatment (Berezinskii, 1971, 1972). Because 0(x) is a multivalued function it is possible that a line integral such... 

For an XT-model at dimensionalities d > dt this coefficient vanishes at Tc with a power law S a ]f 2jff T,v. Although in d = 2 the magnetization (cos0(x)) = 0, the stiffness S is non-zero in the spin wave regime in fact, comparison of eqs. (160) and (168) suggests S = J, independent of T. The Kosterlitz-Thouless (1973) theory implies that S is reduced from J at non-zero temperatures due to vortex-antivortex pairs, and that the equation that yields the transition temperature rather is... 

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