Random-field systems

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

In the next paper [160], Villain discussed the model in which the local impurities are to some extent treated in the same fashion as in the random field Ising model, and concluded, in agreement with earlier predictions for RFIM [165], that the commensurate, ordered phase is always unstable, so that the C-IC transition is destroyed by impurities as well. The argument of Villain, though presented only for the special case of 7 = 0, suggests that at finite temperatures the effects of impurities should be even stronger, due to the presence of strong statistical fluctuations in two-dimensional systems which further destabilize the commensurate phase. 

In this paper we have derived expressions for the environment-induced correction to the Berry phase, for a spin coupled to an environment. On one hand, we presented a simple quantum-mechanical derivation for the case when the environment is treated as a separate quantum system. On the other hand, we analyzed the case of a spin subject to a random classical field. The quantum-mechanical derivation provides a result which is insensitive to the antisymmetric part of the random-field correlations. In other words, the results for the Lamb shift and the Berry phase are insensitive to whether the different-time values of the random-field operator commute with each other or not. This observation gives rise to the expectation that for a random classical field, with the same noise power, one should obtain the same result. For the quantities at hand, our analysis outlined above involving classical randomly fluctuating fields has confirmed this expectation. 

The first two of the above equations represent Langevin equation for the Rouse chain in presence of extra random force 4>f. One can note that dynamics of a polymer chain in random fields was studied extensively (Baumgarter and Muthukumar 1996 Ebert et al. 1996), as a possible mode of motion of a macromolecule in entangled system. Note also that the two top equation... 

Depending on the considered system, additional terms must be added to the micromagnetic equation. In lowest order, DM interactions amount to a random field Zj(Z),Jjyex - Dlj Xey)/2 where the summation (or integration) over j includes all atomic neighbors the resulting structure may be called a spin colloid. ... 

Statistical mechanical manipulations of the functional integral representation of Q are necessary for inhomogeneous systems (Helfand, 1975c Hong and Noolandi, 1981). Minimization of the free energy fixes the equilibrium spatial distribution of polymer and solvent. Edwards random field technique (1965) leads to... 

This chapter is organized as follows. In Section II, we show how quantum chaos systems can be controlled under the optimal fields obtained by OCT. The examples are a random matrix system and a quantum kicked rotor. (The former is considered as a strong-chaos-limit case, and the latter is considered as mixed regular-chaotic cases.) In Section III, a coarse-grained Rabi state is introduced to analyze the controlled dynamics in quantum chaos systems. We numerically obtain a smooth transition between time-dependent states, which justifies the use of such a picture. In Section IV, we derive an analytic expression for the optimal field under the assumption of the CG Rabi state, and we numerically show that the field can really steer an initial state to a target state in random matrix systems. Finally, we summarize the chapter and discuss further aspects of controlling quantum chaos. 

The random matrix was first introduced by E. P. Wigner as a model to mimic unknown interactions in nuclei, and it has been studied to describe statistical natures of spectral fluctuations in quantum chaos systems [17]. Here, we introduce a random matrix system driven by a time-dependent external field E(t), which is considered as a model of highly excited atoms or molecules under an electromagnetic field. We write the Hamiltonian... 

One sees that the ZBR scheme is effective enough for random matrix systems that is, the optimal fields can be obtained even for this type of complicated problem of multilevel-multilevel transitions. However, it seems that the further analysis is difficult because the power spectra for the optimal fields, Figs, lb and 2b, are very complex that is, they contain many frequency components. ... 

Figure 1. Optimal control between Gaussian random vectors in a 64 x 64 random matrix system by the Zhu-Botina-Rabitz scheme with T — 20 and a = 1. a) The optimal field after 100 iterations b) its power spectrum (c) the optimal evolution of the squared overlap with the target (([)(r) (py) as well as its magnified values near the target time in the inset (d) the convergence behavior of the overlap Jq (solid curve) and the functional J (dashed curve) versus the number of iteration steps.

In Section II.A, we have already obtained the optimal field e t) by the numerical calculation for the random matrix systems, Eq. (6). However, only the overlap between the time-evolving controlled state < )(t)) and the target state (pj) was shown there. In this section, we show the overlaps between the time-dependent states defined by Eq. (15) and < )(t)), and we find a smooth transition picture. 

Figures, (a) The final overlap 7o = ((t)(r) (p ) and ( ) the averaged field amplitude 8 for a 64 X 64 random matrix system are shown as a function of the target time T. Crosses (x) represent the numerical results by the Zhu-Botina-Rabitz scheme. Solid curves represent our analytic results under the assumption of the CG Rabi state.

We next examine when and how the analytic optimal field works for a random matrix system (256 x 256 GOE random matrix). Figure 9 demonstrates the coarse-grained Rabi oscillation induced by the analytic field, Eq. (45), with k = 3, where smooth oscillations of ((j)o(f) (t)(f))p and (Xo(0l4 (0)P observed. The initial and the target states are both Gaussian random vectors with 256 elements. This result shows that the field actually produces the CG Rabi oscillation in the random matrix system. 

We have numerically diagonalized small spin systems containing up to 5 by 5 spins subjected to a small random field hf j flatly distributed in the interval (—(5/2, (5/2). We see that indeed the gap closes rather fast away from the special Jx = Jy point (Fig. 3) but remains significant near Jz = Jx point where it clearly has a much weaker size dependence. Interestingly, the gap between the lowest 2ra states and the rest of the spectrum expected in the limits Jz 33> Jx or Jz -C Jx appears only at Jx/ Jz > jc with a practically size independent jc 1.2. We also see that the condition l 1 eliminates all low lying states in the Jz lowest excited state in l 1 sector... 

Figure 4. Ground state splitting by random field in 5-direction for 5x5 and 4x4 systems. The random field acted on each spin and was randomly distributed in the interval (—0.05,0.05). Note that the effect of the random field in z-direction becomes larger for Jx 1.2 the difference Ei — Eq is difficult to resolve numerically.

Domain structures in CSB systems experiencing a random-field t5qie disorder stabilize in size. Many theoretical studies of such systems use approaches based on equilibrium statistical mechanics. Such systems are parameterized by the physical dimension of the system d, a disorder strength parameter w, the volume proportion of the impurities p, and the dimension n of the broken spin symmetry. There are two paradigms of the low temperature behavior of these systems. 

Equations (D.6) and (D.7) are Gilbert s equation in spherical polar coordinates. To obtain the Gilbert-Langevin equation in such coordinates we augment the field components //, and with random field terms and h. By graphical comparison of the Cartesian and spherical, polar coordinate systems, we find these to be... 

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