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Register a new userRandom Walks on a Fluctuating Lattice: A Renormalization Group Approach Applied in One Dimension
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We study the problem of a random walk on a lattice in which bonds connecting nearest neighbor sites open and close randomly in time, a situation often encountered in fluctuating media. We present a simple renormalization group technique to solve for the effective diffusive behavior at long times. For one-dimensional lattices we obtain better quantitative agreement with simulation data than earlier effective medium results. Our technique works in principle in any dimension, although the amount of computation required rises with dimensionality of the lattice.
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We study a spin-1/2-particle moving on a one dimensional lattice subject to disorder induced by a random, space-dependent quantum coin. The discrete time evolution is given by a family of random unitary quantum walk operators, where the shift operation is assumed to be deterministic. Each coin is an independent identically distributed random variable with values in the group of two dimensional unitary matrices. We derive sufficient conditions on the probability distribution of the coins such that the system exhibits dynamical localization. Put differently, the tunneling probability between two lattice sites decays rapidly for almost all choices of random coins and after arbitrary many time steps with increasing distance. Our findings imply that this effect takes place if the coin is chosen at random from the Haar measure, or some measure continuous with respect to it, but also for a class of discrete probability measures which support consists of two coins, one of them being the Hadamard coin.
We study a mixture of ultracold spin-half fermionic and spin-one bosonic atoms in a shallow optical lattice where the bosons are coupled to the fermions via both density-density and spin-spin interactions. We consider the parameter regime where the bosons are in a superfluid ground state, integrate them out, and obtain an effective action for the fermions. We carry out a renormalization group analysis of this effective fermionic action at low temperatures, show that the presence of the spinor bosons may lead to a separation of Fermi surfaces of the spin-up and spin-down fermions, and investigate the parameter range where this phenomenon occurs. We also calculate the susceptibilities corresponding to the possible superfluid instabilities of the fermions and obtain their possible broken-symmetry ground states at low temperatures and weak interactions.
We present a real space renormalization-group map for probabilities of random walks on a hierarchical lattice. From this, we study the asymptotic behavior of the end-to-end distance of a weakly self- avoiding random walk (SARW) that penalizes the (self-)intersection of two random walks in dimension four on the hierarchical lattice.
The non-perturbative renormalization-group approach is extended to lattice models, considering as an example a $phi^4$ theory defined on a $d$-dimensional hypercubic lattice. Within a simple approximation for the effective action, we solve the flow equations and obtain the renormalized dispersion $eps(q)$ over the whole Brillouin zone of the reciprocal lattice. In the long-distance limit, where the lattice does not matter any more, we reproduce the usual flow equations of the continuum model. We show how the numerical solution of the flow equations can be simplified by expanding the dispersion in a finite number of circular harmonics.
Exact results are obtained for random walks on finite lattice tubes with a single source and absorbing lattice sites at the ends. Explicit formulae are derived for the absorption probabilities at the ends and for the expectations that a random walk will visit a particular lattice site before being absorbed. Results are obtained for lattice tubes of arbitrary size and each of the regular lattice types; square, triangular and honeycomb. The results include an adjustable parameter to model the effects of strain, such as surface curvature, on the surface diffusion. Results for the triangular lattice tubes and the honeycomb lattice tubes model diffusion of adatoms on single walled zig-zag carbon nano-tubes with open ends.
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