Finite temperature phase diagrams

G. Schmid, S. Todo, M. Troyer, and A. Dorneich (2002) Finite-temperature phase diagram of hard-core bosons in two dimensions. Phys. Rev. Lett. 88, p. 167208... 

Up to now, we have concentrated on the physics at zero kelvin. In this section, we extend the studies to finite temperatures and discuss finite temperature phase diagrams. The physics at finite temperatures is dominated by thermal fluctuations between low lying excited states of the system. These fluctuations can include spin fluctuations, fluctuations between different valence states, or fluctuations between different orbitally ordered states, if present. Such fluctuations can be addressed througih a so-called alloy analogy. If there is a timescale that is slow compared to the motion of the valence electrons, and on which the configurations persist between the system fluctuations, one can replace the temporal average over all fluctuations by an ensemble average over all possible (spatially... 

Figure 2 Schematic phase diagram close to a quantum critical point for systems having an ordered phase at nonzero temperature. The solid line is the finite-temperature phase boundary while the dashed lines are crossover lines separating different regions within the disordered phase. QCP denotes the quantum critical point.

Phase transitions in binary systems, nomially measured at constant pressure and composition, usually do not take place entirely at a single temperature, but rather extend over a finite but nonzero temperature range. Figure A2.5.3 shows a temperature-mole fraction T, x) phase diagram for one of the simplest of such examples, vaporization of an ideal liquid mixture to an ideal gas mixture, all at a fixed pressure, (e.g. 1 atm). Because there is an additional composition variable, the sample path shown in tlie figure is not only at constant pressure, but also at a constant total mole fraction, here chosen to be v = 1/2. 

In this section we review several studies of phase transitions in adsorbed layers. Phase transitions in adsorbed (2D) fluids and in adsorbed layers of molecules are studied with a combination of path integral Monte Carlo, Gibbs ensemble Monte Carlo (GEMC), and finite size scaling techniques. Phase diagrams of fluids with internal quantum states are analyzed. Adsorbed layers of H2 molecules at a full monolayer coverage in the /3 X /3 structure have a higher transition temperature to the disordered phase compared to the system with the heavier D2 molecules this effect is... 

FIG. 14 Phase diagram of the quantum APR model in the Q -T plane. The solid curve shows the line of continuous phase transitions from an ordered phase at low temperatures and small rotational constants to a disordered phase according to the mean-field approximation. The symbols show the transitions found by the finite-size scaling analysis of the path integral Monte Carlo data. The dashed line connecting these data is for visual help only. (Reprinted with permission from Ref. 328, Fig. 2. 1997, American Physical Society.)... 

However, the domain of the QCD phase diagram where neutron star conditions are met is not yet accessible to Lattice QCD studies and theoretical approaches have to rely on nonperturbative QCD modeling. The class of models closest to QCD are Dyson-Schwinger equation (DSE) approaches which have been extended recently to finite temperatures and densities [11-13], Within simple, infrared-dominant DSE models early studies of quark stars [14] and diquark condensation [15] have been performed. 

Fig. 18. Phase diagram of the centered rectangular lattice gas model with ==0, 3/4 2 = V3> vJ

A particular complex problem has been the modelling of Si/W(l 10) Amar et have included pairwise interactions up to the sixth nearest neighbor shell, as estimated experimentally from field-ion microscopic studies The predicted phase diagram (Fig. 30) exhibits (5 x 1), (6 x 1) and p(2 x 1) commensurate phases, as well as a broad regime of an incommensurate phase. In contrast to the ANNNI model the present model does seem to have a finite-temperature Lifshitz point, where the incommensurate, commensurate... 

Fig. 30. Phase diagram of a model for Si/W(110) in the temperature versus 9 plane. Experimentally determined interactions J Jj,are used. Full dots are from Monte Carlo calculations, while triangles are based on transfer matrix finite size scaling using strip widths of 8 and 12. The point labelled L indicates approximate location of Lifshitz point. The dotted line indicates the transition region between the (5 x l)and(6 x 1) phases. (From...

The influence of a commensurate lattice potential on a free density wave is considered in section 5. The full finite temperature renormalization group flow equation for this sine-Gordon type model are derived and resulting phase diagram is discussed. Furthermore a qualitative picture of the combined effect of disorder and a commensurate lattice potential at zero temperature is presented in section 6, including the phase diagram. 

This procedure may be a good approximation at zero temperature, but if one considerers finite temperatures this does not hold anymore, since the extension in r-direction is now finite. As a result, there is a region ir/L < A < 7t/At where fluctuations are mainly one-dimensional and purely thermal. This region was disregarded in previous treatments. As we will see, fluctuations from this region have an important influence on the overall phase diagram. 

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