Landau rule

The first Landau rule states that for a second-order transition to be possible there should occur just a single star of q in the description of the ordered phase. Now the order parameter components [Pg.152]

If Sc undergoes a transition directly to the nematic phase, 9 is generally found to be temperature independent and usually about 45°. According to the Landau rules, the C-N transition can be continuous, but when fluctuations are taken into account it is predicted to be of first order. Experimentally, only first order C-N transitions have been observed. Some compounds exhibit transitions from Sg to the isotropic phase. Interestingly, a slight increase of 9 with increasing temperature has been reported for two such compounds. ... 

In the time-dependent perturbation theory [Landau and Lifshitz 1981] the transition probability from the state 1 to 2 is related with the perturbation by the golden rule,... 

To obtain we may try to elaborate several rules, taking into account what is already known in the literature relative to the construction of the Landau Hamiltonian [25] or at the level of quantum field theory [17,22,26]. 

In principle, we already have in our disposal the SRPA formalism for description of the collective motion in space of collective variables. Indeed, Eqs. (11), (12), (18), and (19) deliver one-body operators and strength matrices we need for the separable expansion of the two-body interaction. The number K of the collective variables qk(t) and pk(t) and separable terms depends on how precisely we want to describe the collecive motion (see discussion in Section 4). For K = 1, SRPA converges to the sum rule approach with a one collective mode [6]. For K > 1, we have a system of K coupled oscillators and SRPA is reduced to the local RPA [6,24] suitable for a rough description of several modes and or main gross-structure efects. However, SRPA is still not ready to describe the Landau fragmentation. For this aim, we should consider the detailed Iph space. This will be done in the next subsection. 

In this regard, we should notice that the time evolution of a quantum system is ruled by two different types of eigenvalues corresponding to the wave function and the statistical descriptions. On the one hand, we have the eigenenergies of the Hamiltonian within the wave function description. On the other hand, we have the eigenvalues of the Landau-von Neumann superoperator in the Liouville formulation of quantum mechanics. These quantum Liouvillian eigenvalues j are related to the Bohr frequencies according to... 

This expression is nothing but the Bohr-Sommerfeld quantization rule (see, e.g., Landau and Lifshitz [1981]). In the metastable potential of Figure 3.3 there are also imaginary-time periodic orbits satisfying (3.41) that develop between the turning points inside the classically forbidden region. It is these trajectories that are responsible for tunneling [Levit et... 

Although the shortest way to the tunneling gap 8 is the solution of Landau and Lifshits [27], here we consider the problem from a different perspective. Like in the theory of electric circuits, instead of a detailed consideration of each particle, one can apply some simple rules that provide enough equations to solve the problem. One is the junction rule. It is based upon the probability conservation law for a stationary state, PiQ, t). At any point Q in the domain of 77(2, t), the probability density, I PiQ, t) 2 remains constant, dl P(Q. f)P/df = 0. Consider the part of a vibronic state that is located in a potential well. In this region, the probability density, P(Q, t) 2, looks like an octopus with its tentacles extended into the restricted areas under the barriers.2 If we construct a closed surface S around the body of the octopus , then, due to conservation of probability density, the total flux of probability through the surface S must be equal to zero,... 

For two terms of the same symmetry, this condition is always satisfied at low velocity as, r /v and the noncrossing rule implies that w 0. Terms of different symmetry may cross, but the most efficient nonadiabatic coupling need not be localized at the crossing point, though this is usually the case. If v is a small parameter, a perturbation treatment of equation (2) can be attempted. It was shown by Landau [10] that for noncrossing terms the matrix elements Nkm are exponentially small, that is, Nkm exp (—a>kmTkm). As for the preexponential factor Bkm, more detailed treatment shows two possibilities Either Bkm depends on velocity v and is small for low v, or it is independent of v. Only the first case, which is realized for terms of different symmetries, can be treated by the usual perturbation method. Thus, to first... 

Next, I would like to discuss intricacy of CP violation. In gauge theories one may compute the fundamental CP asymmetry, namely the difference of baryon numbers between particle and antiparticle decays using perturbation theory. It is thus given by an interference term of, for instance, the tree and the one-loop contributions. A convenient tool of computing the interference term is the Landau-Cutkovsky rule [19]. The result is like... 

The universal Eq. (2) [Kofman 2000 Kofman 2001 (a)] has much broader applicability than its Golden-Rule counterpart, Rqr = 27t Gn(ujn), obtained by standard perturbation theory [Landau 1977] lin(l) may decrease with the modulation period r, thus obeying the QZE, but only if r is shorter than the memory (correlation) time rc (the time over which G(L) changes). Equivalently, the rate 1/r and the corresponding spectral width of Ft (uj) must exceed the width of the coupling spectrum Gn(oj). The opposite, AZE-like, behavior, i.e., an increase of lln t) with 1/r, obtains for lyfuj) with a narrow (but non-negligible) spectral width compared to l/rc. 

This overlap measures the rate of tunnel transitions that entails the detailed balance condition for the SETs. The transition dynamics fall into the generic class of two-level-band systems [Akulin 2004], According to the common quantum-mechanical rules [Landau 1977] we identify the following cases ... 

In the strong field problem, much effort and attention has been given to situations where the quantum defects m are nearly integral, i.e. the solutions are quasihydrogenic. An example is the spectrum of Li, which was investigated in considerable detail [569]. In the quasi-Landau resonance range, it had been noticed quite early that the spacings observed in photoabsorption experiments do not follow the usual rule but... 

Ao and Rammer [166] obtained the same result (and more) on the basis of a fully quantum mechanical treatment. Frauenfelder and Wolynes [78] derived it from simple physical arguments. Equation (9.98) predicts a quasiadiabatic result, = h k/ v 1 and the Golden Rule result, Pk = k/ v, in the opposite limit, which is qualitatively similar to the Landau-Zener behavior of the transition probability but the implications are different. Equation (9.98) is the result of multiple nonadiabatic crossings of the delta sink although it does not depend on details of the stochastic process Xj- t). This can be understood from the following consideration. For each moment of time, the fast coordinate has a Gaussian distribution, p Xf, t) = (xy — Xj, transition region, the fast coordinate crosses it very frequently and thus forms an effective sink for the slow coordinate. 

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