Relaxation and Diffusion for the Kicked Rotor

The dynamics of the kicked-rotor, that is a paradigm for a mixed system, where the motion in some parts of phase space is chaotic and in other parts is regular is studied statistically. The evolution (Frobenius-Perron) operator of phase space densities in the chaotic component is calculated in presence of noise, and the limit of vanishing noise is taken is taken in the end of calculation. The relaxation rates (related to the Ruelle resonances) to the invariant equilibrium density are calculated analytically within an approximation that improves with increasing stochasticity. The results are tested numerically. The global picture of relaxation to the equilibrium density in the chaotic component when the system is bounded and of diffusive behavior when it is unbounded is presented

Exact Eigenfunctions of a Chaotic System

The interest in the properties of quantum systems, whose classical dynamics are chaotic, derives from their abundance in nature. The spectrum of such systems can be related, in the semiclassical approximation (SCA), to the unstable classical periodic orbits, through Gutzwiller's trace formula. The class of systems studied in this work, tiling billiards on the pseudo-sphere, is special in this correspondence being exact, via Selberg's trace formula. In this work, an exact expression for Green's function (GF) and the eigenfunctions (EF) of tiling billiards on the pseudo-sphere, whose classical dynamics are chaotic, is derived. GF is shown to be equal to the quotient of two infinite sums over periodic orbits, where the denominator is the spectral determinant. Such a result is known to be true for typical chaotic systems, in the leading SCA. From the exact expression for GF, individual EF can be identified. In order to obtain a SCA by finite series for the infinite sums encountered, resummation by analytic continuation in $\hbar$ was performed. The result is similar to known results for EF of typical chaotic systems. The lowest EF of the Hamiltonian were calculated with the help of the resulting formulae, and compared with exact numerical results. A search for scars with the help of analytical and numerical methods failed to find evidence for their existence.Comment: 53 pages LaTeX, 10 Postscript figure

Double Exchange in a Magnetically Frustrated System

This work examines the magnetic order and spin dynamics of a double-exchange model with competing ferromagnetic and antiferromagnetic Heisenberg interactions between the local moments. The Heisenberg interactions are periodically arranged in a Villain configuration in two dimensions with nearest-neighbor, ferromagnetic coupling $J$ and antiferromagnetic coupling $-\eta J$. This model is solved at zero temperature by performing a $1/\sqrt{S}$ expansion in the rotated reference frame of each local moment. When $\eta$ exceeds a critical value, the ground state is a magnetically frustrated, canted antiferromagnet. With increasing hopping energy $t$ or magnetic field $B$, the local moments become aligned and the ferromagnetic phase is stabilized above critical values of $t$ or $B$. In the canted phase, a charge-density wave forms because the electrons prefer to sit on lines of sites that are coupled ferromagnetically. Due to a change in the topology of the Fermi surface from closed to open, phase separation occurs in a narrow range of parameters in the canted phase. In zero field, the long-wavelength spin waves are isotropic in the region of phase separation. Whereas the average spin-wave stiffness in the canted phase increases with $t$ or $\eta$, it exhibits a more complicated dependence on field. This work strongly suggests that the jump in the spin-wave stiffness observed in Pr$_{1-x}$Ca$_x$MnO$_3$ with $0.3 \le x \le 0.4$ at a field of 3 T is caused by the delocalization of the electrons rather than by the alignment of the antiferromagnetic regions.Comment: 28 pages, 12 figure

Localization of eigenstates in a modified Tomonaga-Luttinger model

We study the localization in the Hilbert space of a modified Tomonaga-Luttinger model. For the standard version of this model, the states are found to be extended in the basis of Slater determinants, representing the eigenstates of the non-interacting system. The linear dispersion which leads to the fact that these eigenstates are extended in the modified model is replaced by one with random level spacings modeling the complicated one-particle spectra of realistic models. The localization properties of the eigenstates are studied. The interactions are simplified and an effective one-dimensional Lloyd model is obtained. The effects of many-body energy correlations are studied numerically. The eigenstates of the system are found to be localized in Fock space for any strength of the interactions, but the localization is not exponential.Comment: 19 pages, 7 figure

Stable Quantum Resonances in Atom Optics

A theory for stabilization of quantum resonances by a mechanism similar to one leading to classical resonances in nonlinear systems is presented. It explains recent surprising experimental results, obtained for cold Cesium atoms when driven in the presence of gravity, and leads to further predictions. The theory makes use of invariance properties of the system, that are similar to those of solids, allowing for separation into independent kicked rotor problems. The analysis relies on a fictitious classical limit where the small parameter is {\em not} Planck's constant, but rather the detuning from the frequency that is resonant in absence of gravity.Comment: 5 pages, 3 figure

On the Spectrum of the Resonant Quantum Kicked Rotor

It is proven that none of the bands in the quasi-energy spectrum of the Quantum Kicked Rotor is flat at any primitive resonance of any order. Perturbative estimates of bandwidths at small kick strength are established for the case of primitive resonances of prime order. Different bands scale with different powers of the kick strength, due to degeneracies in the spectrum of the free rotor.Comment: Description of related published work has been expanded in the Introductio

Faster Methods for Contracting Infinite 2D Tensor Networks

We revisit the corner transfer matrix renormalization group (CTMRG) method of Nishino and Okunishi for contracting two-dimensional (2D) tensor networks and demonstrate that its performance can be substantially improved by determining the tensors using an eigenvalue solver as opposed to the power method used in CTMRG. We also generalize the variational uniform matrix product state (VUMPS) ansatz for diagonalizing 1D quantum Hamiltonians to the case of 2D transfer matrices and discuss similarities with the corner methods. These two new algorithms will be crucial to improving the performance of variational infinite projected entangled pair state (PEPS) methods.Comment: 20 pages, 5 figures, V. Zauner-Stauber previously also published under the name V. Zaune