XY universality class

The hexatic B-S transition is expected to belong to the XY universality class/ but high resolution calorimetric investigations appear to indicate that this may not be the case. Tilted analogues of hexatic B, namely Sp and Sj, have also been identified. ... 

Critical Behavior of the Three-Dimensional XY Universality Class. 

This fixed point is isotropic and belongs to the so-called inverted XY universality class. Inverted refers to the inversion of the high and low temperature sides of the tran-... 

The order parameter has two components and the SmA-SmB transition is expected to belong to the XY universality class. 

The SmA liquid crystalline phase results from the development of a one-dimensional density wave in the orientationally ordered nematic phase. The smectic wave vector q is parallel to the nematic director (along the z-axis) and the SmA order parameter i/r= i/r e is introduced by P( ) = Po[1+R6V ]- Thus the order parameter has a magnitude and a phase. This led de Gennes to point out the analogy with superfluid helium and the normal-superconductor transition in metals [7, 59]. This would than place the N-SmA transition in the three-dimensional XY universality class. However, there are two important sources of deviations from isotropic 3D-XY behavior. The first one is crossover from second-order to first-order behavior via a tricritical point due to coupling between the smectic order parameter y/ and the nematic order parameter Q. The second source of deviation from isotropic 3D-XY behavior arises from the coupling between director fluctuations and the smectic order parameter, which is intrinsically anisotropic [60-62]. 

With a complex scalar order parameter, the SmA-SmC transition is expected [131] to belong to the 3D XY universality class, which does not have the critical exponent P equal to 1/2. Nevertheless, experiments show [132] that it is surprisingly well described by mean field theory, although with an unusually large sixth order term. 

The SmA-N transition is often encountered in liquid crystals and has been extensively studied both theoretically and experimentally. Based on the analogy with the superconductor to normal metal transition, de Gennes [19,21] classified the SmA-N transition in the isotropic three-dimensional XY universality class. Actually, however, deviation from the isotropic 3D-XY behaviour occurs to give the cross-over from second order to first order behaviour via a tricritical point due to coupling between smectic and nematic order parameters. This type of deviation has been predicted for the... 

Figures 6-9 illustrate the use of these finite size scaling relations for the square lattice gas with repulsion between both nearest and next nearest neighbors. In Fig. 6 the raw data of Fig. 5 are replotted in scaled form, as suggested by Eq. (37). Note that neither = TJcc) nor the critical exponents are known in beforehand - the phase transition of the (2x1) phase falls in the universality class of the XY model with uniaxial anisotropy which has nonuniversal exponents depending on R. Clearly, it is desirable to estimate without being biased by the choice of the critical exponents. This is possible...

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 this case, there is a symmetry against the change of sign of 0 (fig. 10 shows that this simply corresponds to an interchange of sublattices) and hence a term 03 cannot occur. The fourth order term, however, now contains two cubic invariants rather than a single term [(02)2 =

rotational invariance in the order parameter space (0i, 02), since all directions in the (01, 02) plane are equivalent. No such rotational symmetry applies to the (2x1) structure, of course. So the expansion eq. (26) results, which defines the universality class of the X Y model with cubic anisotropy . Of course, in this approach not much can he said on the phenomenological coefficients r, u, u, R in eq. (26). 

At this point, we mention a further consequence of the universality principle alluded to above. For each universality class (such as that of the Ising model or that of the XY model, etc.) not just the critical exponents are universal, but also the scaling function F(H), apart from non-universal scale factors for the occurring variables (a factor for H we have expressed via the ratio C/B in eq. (84), for instance). A necessary implication then is the universality of certain critical amplitude ratios, where all scale factors for the variables of interest cancel out. In particular, ratios of critical amplitudes of corresponding quantities above and below Tc, A+j A [eq. (7)], C+jC [eq. (6)] and f+/ [eq. (38)] are universal (Privman et al., 1991). A further relation exists between the amplitude D and B and C 1 Writing M H -> oo) = XHl/ cf. eqs. (87) and (91), the universality of M(H) states that X is universal. But since 0 = B tfM H) = B t PXH = B] SC S H] X, a comparison with eq. (45) yields... 

Universality class and critical exponents Ising XY with cubic anisotropy 3-state Potts 4-state Potts... 

In the vicinity of fluctuation dominated phase transitions, the temperature dependence of thermodynamic parameters such as the specific heat at constant pressure, Cp=e , and the order parameter, y/ e, are all related to through a free energy density giving rise to scaling relations. For example a=2-vdmdp=(d-2) v/2 [2]. Despite the variety of their continuous broken symmetries, most liquid crystal phase transitions are expected to fall in the 3D-XY (helium) universality class with, a=-0.01, V-0.67 and y8-0.33. 

From (5.5.20) and (5.5.21) it is seen that Beet, where = 2vj — v, and D oc fK If V = 2, B should be finite at the transition temperature. However, experimentally, it appears that B at the transition is almost vanishingly small within experimental limits. Few measurements are available on D to draw any definite conclusions. In any case, as pointed out earlier, the exponents are neither universal nor do they agree with the predictions of any of the theoretical models. Vithana et a/. have suggested that the widely differing values of the exponents for the different compounds may be a consequence of the fact that one is measuring effective values associated with crossover effects between the XY class and a tricritical point. A further complication is that the experiments of Evans-Lutterodt et appear to indicate that the occurrence of different... 

However, the data recorded for different materials do not seem to correspond to a universal behavior of the transition. For example, in the case of compounds with large nematic ranges, the critical exponents are close to an xy, helium-like behavior with a=-0.007, V = v =0.67, and 7= 1.32 [40], but the question of the anisotropy is not explained [35,41], Other proposals have been made to consider that the transition belongs to an anisotropic class with v =2 [42], or... 

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