We address generic behavior of quantum dislocations in almost ideal crystals. It is proven that the combination of arbitrary small Peierls potential and Coulomb-type elastic interaction between dislocation kinks prevents quantum roughening of dislocations. Thermally created kinks induce classical roughening which leads to softening of crystal shear modulus at temperatures comparable to the kink energy. This effect is discussed in the context of the shear modulus softening observed by Day & Beamish in solid 4 He.

We introduce a new method to analysis the many-body problem with disorder. The method is an extension of the real space renormalization group based on the operator product expansion. We consider the problem in the presence of interaction, large elastic mean free path, and finite temperatures. As a result scaling is stopped either by temperature or the length scale set by the diverging many-body length scale (superconductivity). Due to disorder a su-

We solve the quantum mechanical problem for electrons confined to two coupled metallic rings which form an '8' structure. The solution is achieved by gluing the wavefunction at the interface of the two rings using a modification of Dirac's constrained method. The Heisenberg equation of motion for the wavefunction is a fermionic and bosonic mixture, suggesting that it is impossible to solve the problem for two coupled rings by standard boundary conditions.As an explicit demonstration of our method we present an exact solution for the persistent current on coupled rings in the presence of two external magnetic fluxes. For large coupled rings with equal fluxes we find that the persistent current in the two coupled rings is equal to that in a single ring. For opposite fluxes the energy has a chaotic structure. For both cases the periodicity is h/e.In order to compare our theory to the experimental situations we consider two rings with a finite width in the ballistic regime in the presence of 2K F impurity scattering.

A new method for computing the spin Hall conductivity for a two dimensional electron gas in the presence of the spin Orbit interaction is presented. The spin current is computed using the Many-Body wave function which is degenerated at zero momentum The degeneracy at K = 0 gives rise to non-commuting Cartesian coordinates. The non-commuting Cartesian coordinate are a result of an effective Aharonov-Bohm vortex at K = 0. An explicit calculation for the Rashba model is presented. The conductivity is determined by the linear response theory which has two parts :A-a static spin Hall conductivity which is determined by the non-commuting coordinates and has the value |e| 4π . B-a time dependent conductivity which renormalizes the static conductivity. The value of this renormalization depends on inelastic time scattering ,spin Orbit polarization energy and Zeeman energy. As a result the spin Hall conductivity vary between |e| 4π and |e| 8π . In the absence of a Zeeman field we find that the long time behavior is given by the renormalized conductivity |e| 8π . For relative small magnetic field the Zeeman field allows to probe continuously the spin Hall conductivity from the static unrenormalized value |e| 4π to the fully renormalized value |e| 8π . When the Zeeman energy exceeds the Fermi energy only one Fermi Dirac band is occupied and as a result the static Hall conductivity is half the static spin Hall conductivity.We compute the uniform magnetization without the Zeeman field and show that the spin current is covariantly conserved and satisfies effectively the continuity equation.The effect of a time reversal scattering potential due to a single impurity in the Rashba model causes the the spin Hall current to decrease with the size of the system.

A renormalization-group transformation for quantum statistics is developed and applied to the P4 model. We find that quantum fluctuations at T =0 and thermal fluctuations at T AO restore the symmetry giving rise to a ferroelectric-paraelectric transition. The renormalized mass (the inverse dielectric susceptibility) and the coupling constant become temperature dependent. The renormalization constants and the Wilson functions are given by the calculation at T =0. The inverse susceptibility for n =1 and d =3 (n being the number of components of the order parameter and d the dimension) is given by X ' -X~'&Ilogxq f! 'I (qmf refers to the quantum-mean-field susceptibility in the paraelectric phase). For materials with T, =0 we find X '& -T and X ' -T IlogT IThe purpose of this Communication is to show how one can construct a renormalization-group (RG) theory for quantum statistics. As a specific example we investigate the case of quantum ferroelectrics.A common feature of studies of phase transitions is the use of classical statistical mechanics. In these cases the classical RG treats the gradient term which is the origin of the thermal fluctuations. It is known that these fluctuations change the result predicted by the Landau theory. On the other hand the presence of the dynamic momenta which do not commute with the coordinates may suppress the phase transition due to the nonvanishing zero-point motion of the particle. Quantum effects' 6 can no longer be neglected if the model parameters become small. In those cases it is necessary to reformulate the RG and to treat in addition to the gradient fluctuations also the fluctuations produced by the time derivatives.For quantum ferroelectrics the experimental results in Ref. 3 indicate that the inverse dielectric susceptibility is given by X ' -T~, where y changes from 2 in the quantum regime to 1 in the classical regime.In the first part of this Communication we will show how one can make use of the RG scheme developed in quantum field theory and often used in phase transitions. 7 Our main purpose will be to show the changes which we have to perform in order to use it for fluctuations at finite temperature.As a specific example we consider the @4 model in a Euclidean field representation at finite temperature T = I/p, d d 1+ (~@-)2 + 2@2+ )~@ 4 2 r)r 2 2 4T his action is the continuous and simplified version of the model introduced by Bilz et al. 4 in ferroelectrics.In order to formulate the RG theory we have to compute the N-point vertex functions I"'~'. In our case the temperature enters as a finite length P = I/T in the fourth dimension. Therefore, I are functions of a continuous momentum variable K' [in (d -I) space dimensions] and a discrete variable (2~/P)'n', n =0, +I, +2, . . . .We derive the RG equations for a massless field (we work at the critical point) at a fixed momentum In doing this we obtain a renormalization Z& for the field @(q). In order to obtain the behavior in the critical region we introduce the renormalized mass m2 A 0 (which is the quantum-mean...