Hamiltonians phase transitions

Phase transitions in adsorbed layers often take place at low temperatures where quantum effects are important. A method suitable for the study of phase transitions in such systems is PIMC (see Sec. IV D). Next we study the gas-liquid transition of a model fluid with internal quantum states. The model [193,293-300] is intended to mimic an adsorbate in the limit of strong binding and small corrugation. No attempt is made to model any real adsorbate realistically. Despite the crudeness of the model, it has been shown by various previous investigations [193,297-300] that it captures the essential features also observed in real adsorbates. For example, the quite complex phase diagram of the model is in qualitative agreement with that of real substances. The Hamiltonian is given by... 

Freezing transitions have been examined in recent years by density functional methods [306-313]. Here we review the results [298] of a modification of the Ramakrishnan-Yussouff theory to the model fluid with Hamiltonian (Eq. (25)) a related study of phase transitions in a system of hard discs in two dimensions with Ising internal states which couple anti-ferromagnetically to their neighbors is shown in Ref. 304. First, a combined... 

The other class of phenomenological approaches subsumes the random surface theories (Sec. B). These reduce the system to a set of internal surfaces, supposedly filled with amphiphiles, which can be described by an effective interface Hamiltonian. The internal surfaces represent either bilayers or monolayers—bilayers in binary amphiphile—water mixtures, and monolayers in ternary mixtures, where the monolayers are assumed to separate oil domains from water domains. Random surface theories have been formulated on lattices and in the continuum. In the latter case, they are an interesting application of the membrane theories which are studied in many areas of physics, from general statistical field theory to elementary particle physics [26]. Random surface theories for amphiphilic systems have been used to calculate shapes and distributions of vesicles, and phase transitions [27-31]. 

The theoretical foundation for describing critical phenomena in confined systems is the finite-size scaling approach [64], by which the dependence of physical quantities on system size is investigated. On the basis of the Ising Hamiltonian and finite-size scaling theory, Fisher and Nakanishi computed the critical temperature of a fluid confined between parallel plates of distance D [66]. The critical temperature refers to, e.g., a liquid/vapor phase transition. Alternatively, the demixing phase transition of an initially miscible Kquid/Kquid mixture could be considered. Fisher and Nakashini foimd that compared with free space, the critical temperature is shifted by an amoimt... 

Then, there are model Hamiltonians. Effectively a model Hamiltonian includes only some effects, in order to focus on those effects. It is generally simpler than the true full Coulomb Hamiltonian, but is made that way to focus on a particular aspect, be it magnetization, Coulomb interaction, diffusion, phase transitions, etc. A good example is the set of model Hamiltonians used to describe the IETS experiment and (more generally) vibronic and vibrational effects in transport junctions. Special models are also used to deal with chirality in molecular transport junctions [42, 43], as well as optical excitation, Raman excitation [44], spin dynamics, and other aspects that go well beyond the simple transport phenomena associated with these systems. 

An inference of fundamental importance follows from Eqs. (2.3.9) and (2.3.11) When long axes of nonpolar molecules deviate from the surface-normal direction slightly enough, their azimuthal orientational behavior is accounted for by much the same Hamiltonian as that for a two-dimensional dipole system. Indeed, at sin<9 1 the main nonlocal contribution to Eq. (2.3.9) is provided by a term quadratic in which contains the interaction tensor V 2 (r) of much the same structure as dipole-dipole interaction tensor 2B3 > 0, B4 < 0, only differing in values 2B3 and B4. For dipole-dipole interactions, 2B3 = D = flic (p is the dipole moment) and B4 = -3D, whereas, e.g., purely quadrupole-quadrupole interactions are characterized by 2B3 = 3U, B4 = - SU (see Table 2.2). Evidently, it is for this reason that the dipole model applied to the system CO/NaCl(100), with rather small values 0(6 25°), provided an adequate picture for the ground-state orientational structure.81 A contradiction arose only in the estimation of the temperature Tc of the observed orientational phase transition For the experimental value Tc = 25 K to be reproduced, the dipole moment should have been set n = 1.3D, which is ten times as large as the corresponding value n in a gas phase. Section 2.4 will be devoted to a detailed consideration of orientational states and excitation spectra of a model system on a square lattice described by relations (2.3.9)-(2.3.11). 

Now we consider thermodynamic properties of the system described by the Hamiltonian (2.4.5) it is a generalized Hamiltonian of the isotropic Ashkin-Teller model100,101 expressed in terms of interactions between pairs of spins lattice site nm of a square lattice. Hamiltonian (2.4.5) differs from the known one in that it includes not only the contribution from the four-spin interaction (the term with the coefficient J3), but also the anisotropic contribution (the term with the coefficient J2) which accounts for cross interactions of spins a m and s m between neighboring lattice sites. This term is so structured that it vanishes if there are no fluctuation interactions between cr- and s-subsystems. As a result, with sufficiently small coefficients J2, we arrive at a typical phase diagram of the isotropic Ashkin-Teller model,101 102 limited by the plausible values of coefficients in Eq. (2.4.6). At J, > J3, the phase transition line... 

We now turn to the potential (4) for nonequilibrium phase transition. We separate the Hamiltonian density H into a quadratic part Ho and a perturbation part Hp ... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

For 7=1 the Hamiltonian reduces to the Ising model and for y = 0 to the XY model. For the pure homogeneous case, /,, +i = / and = B, the system exhibits a quantum phase transition at a dimensionless coupling constant... 

The structural phase transition for a quantum paraelectric was handled by the following model Hamiltonian [2] ... 

The ground-state energy terms of this form contribute to the potential energy of the Hamiltonian from Eq. 6, where they act as effective ferroelectric interactions between neighboring PO4 dipoles, thus being responsible for phase transition in KDP and DKDP. In the case of KDP (H atom and Ro = 2.50 A) calculated values of parameters h and I are = llOmeV and = 0.22 A, while in the case of DKDP (D atom and Ro-o = 2.52 A), they are = 58 meV and P = I = 0.22 A. 

The aim of this work is to elucidate these problems. To this end, we calculate the effective spin Hamiltonian of the 5f2—5f2 superexchange interaction between the neighboring U4+ ions in the cubic crystal lattice of UO2 and we calculate T5 <%> eg, rs f2g(l) ancl r5 f2g(2) linear vibronic coupling constants. These data are then used to draw a more definite conclusion about the driving force of the phase transition and especially about the actual mechanism of the spin and orbital ordering in U02. 

The fact that the phase transition in UO2 has the first-order character and the ordered magnetic moment of 1.74 /ab is considerably lower than the paramagnetic moment (about 3 /ab) is qualitatively consistent with the ratio of the strength of the bilinear and biquadratic parts of the effective spin Hamiltonian (8) of the 5f2-5f2... 

The salient feature of the normal coordinate description is that there is no coupling between the various normal modes, so the Hamiltonian can be written as a sum of Hamiltonians for each normal mode. The reaction coordinate is defined, like in gas-phase transition-state theory, as the normal mode for which the associated frequency is imaginary. The Hamiltonian for the activated complex may be written as... 

The motion in the reaction coordinate Q is, like in gas-phase transition-state theory, described as a free translational motion in a very narrow range of the reaction coordinate at the transition state, that is, for Q = 0 hence the subscript trans on the Hamiltonian. The potential may be considered to be constant and with zero slope in the direction of the reaction coordinate (that is, zero force in that direction) at the transition state. The central assumption in the theory is now that the flow about the transition state is given solely by the free motion at the transition state with no recrossings. So when we associate a free translational motion with that coordinate, it does not mean that the interaction potential energy is independent of the reaction coordinate, but rather that it has been set to its value at the transition state, Q j = 0, because we only consider the motion at that point. The Hamiltonian HXlans accordingly only depends on Px, as for a free translational motion, so... 

In order to understand the problem of finding TS with three or more DOFs, it is useful to address the question of dimensionalities, in configuration and phase space. In classical, Hamiltonian dynamics, transition states are grounded on the idea that certain surfaces (more precisely, certain manifolds) act as barriers in phase space. It is possible to devise barriers in phase space, since in phase space, in contrast to configuration space, two trajectories never cross [uniqueness of solutions of ODEs, see Eq. (4)]. In order to construct a barrier in phase space, the first step is to construct a manifold if that is made of a set of trajectories [8]. 

The CITE system will be considered as a crystal containing a lattice (or one of the sublattices) of JT ions (structural units). A typical Hamiltonian should describe the JT effect at each of the corresponding centers plus the elastic energy related to the appearance of the homogeneous strain as a result of the structural phase transition. 

In this Hamiltonian k is the wave vector of the phonons, y is the phonon mode branch, go and Vmk are the electron-strain and the electron-phonon interaction constants. It is important to remind that as it was noted for the first time by Kanamori [3], the electron interaction with the homogeneous strain U should be considered separately from the electron-phonon interaction as that type of strain can not be represented by phonons. The introduction of the last ones depends upon the Born-Karman conditions that are changing at the structural phase transition. 

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