Quasiparticle density of states in dirty high-T_c superconductors

We study the density of quasiparticle states of dirty d-wave superconductors. We show the existence of singular corrections to the density of states due to quantum interference effects. We then argue that the density of states actually vanishes in the localized phase as $|E|$ or $E^2$ depending on whether time reversal is a good symmetry or not. We verify this result for systems without time reversal symmetry in one dimension using supersymmetry techniques. This simple, instructive calculation also provides the exact universal scaling function for the density of states for the crossover from ballistic to localized behaviour in one dimension. Above two dimensions, we argue that in contrast to the conventional Anderson localization transition, the density of states has critical singularities which we calculate in a $2+\epsilon$ expansion. We discuss consequences of our results for various experiments on dirty high-$T_c$ materials

Localization-delocalization transition of disordered d-wave superconductors in class CI

A lattice model for disordered d-wave superconductors in class CI is reconsidered. Near the band-center, the lattice model can be described by Dirac fermions with several species, each of which yields WZW term for an effective action of the Goldstone mode. The WZW terms cancel out each other because of the four-fold symmetry of the model, which suggests that the quasiparticle states are localized. If the lattice model has, however, symmetry breaking terms which generate mass for any species of the Dirac fermions, remaining WZW term which avoids the cancellation can derive the system to a delocalized strong-coupling fixed point.Comment: 4 pages, revte

Statistical mechanics of RNA folding: a lattice approach

We propose a lattice model for RNA based on a self-interacting two-tolerant trail. Self-avoidance and elements of tertiary structure are taken into account. We investigate a simple version of the model in which the native state of RNA consists of just one hairpin. Using exact arguments and Monte Carlo simulations we determine the phase diagram for this case. We show that the denaturation transition is first order and can either occur directly or through an intermediate molten phase.Comment: 8 pages, 9 figure

Superconducting ``metals'' and ``insulators''

We propose a characterization of zero temperature phases in disordered superconductors on the basis of the nature of quasiparticle transport. In three dimensional systems, there are two distinct phases in close analogy to the distinction between normal metals and insulators: the superconducting "metal" with delocalized quasiparticle excitations and the superconducting "insulator" with localized quasiparticles. We describe experimental realizations of either phase, and study their general properties theoretically. We suggest experiments where it should be possible to tune from one superconducting phase to the other, thereby probing a novel "metal-insulator" transition inside a superconductor. We point out various implications of our results for the phase transitions where the superconductor is destroyed at zero temperature to form either a normal metal or a normal insulator.Comment: 18 page

Quasiparticle localization in superconductors with spin-orbit scattering

We develop a theory of quasiparticle localization in superconductors in situations without spin rotation invariance. We discuss the existence, and properties of superconducting phases with localized/delocalized quasiparticle excitations in such systems in various dimensionalities. Implications for a variety of experimental systems, and to the properties of random Ising models in two dimensions, are briefly discussed.Comment: 10 page

Griffiths effects and quantum critical points in dirty superconductors without spin-rotation invariance: One-dimensional examples

We introduce a strong-disorder renormalization group (RG) approach suitable for investigating the quasiparticle excitations of disordered superconductors in which the quasiparticle spin is not conserved. We analyze one-dimensional models with this RG and with elementary transfer matrix methods. We find that such models with broken spin rotation invariance {\it generically} lie in one of two topologically distinct localized phases. Close enough to the critical point separating the two phases, the system has a power-law divergent low-energy density of states (with a non-universal continuously varying power-law) in either phase, due to quantum Griffiths singularities. This critical point belongs to the same infinite-disorder universality class as the one dimensional particle-hole symmetric Anderson localization problem, while the Griffiths phases in the vicinity of the transition are controlled by lines of strong (but not infinite) disorder fixed points terminating in the critical point.Comment: 14 pages (two-column PRB format), 9 eps figure

Weak localization of disordered quasiparticles in the mixed superconducting state

Starting from a random matrix model, we construct the low-energy effective field theory for the noninteracting gas of quasiparticles of a disordered superconductor in the mixed state. The theory is a nonlinear sigma model, with the order parameter field being a supermatrix whose form is determined solely on symmetry grounds. The weak localization correction to the field-axis thermal conductivity is computed for a dilute array of s-wave vortices near the lower critical field H_c1. We propose that weak localization effects, cut off at low temperatures by the Zeeman splitting, are responsible for the field dependence of the thermal conductivity seen in recent high-T_c experiments by Aubin et al.Comment: RevTex, 8 pages, 1 eps figure, typos correcte

Localization and delocalization in dirty superconducting wires

We present Fokker-Planck equations that describe transport of heat and spin in dirty unconventional superconducting quantum wires. Four symmetry classes are distinguished, depending on the presence or absence of time-reversal and spin rotation invariance. In the absence of spin-rotation symmetry, heat transport is anomalous in that the mean conductance decays like $1/\sqrt{L}$ instead of exponentially fast for large enough length $L$ of the wire. The Fokker-Planck equations in the presence of time-reversal symmetry are solved exactly and the mean conductance for quasiparticle transport is calculated for the crossover from the diffusive to the localized regime.Comment: 4 pages, RevTe

Statistical mechanics of secondary structures formed by random RNA sequences

The formation of secondary structures by a random RNA sequence is studied as a model system for the sequence-structure problem omnipresent in biopolymers. Several toy energy models are introduced to allow detailed analytical and numerical studies. First, a two-replica calculation is performed. By mapping the two-replica problem to the denaturation of a single homogeneous RNA in 6-dimensional embedding space, we show that sequence disorder is perturbatively irrelevant, i.e., an RNA molecule with weak sequence disorder is in a molten phase where many secondary structures with comparable total energy coexist. A numerical study of various models at high temperature reproduces behaviors characteristic of the molten phase. On the other hand, a scaling argument based on the extremal statistics of rare regions can be constructed to show that the low temperature phase is unstable to sequence disorder. We performed a detailed numerical study of the low temperature phase using the droplet theory as a guide, and characterized the statistics of large-scale, low-energy excitations of the secondary structures from the ground state structure. We find the excitation energy to grow very slowly (i.e., logarithmically) with the length scale of the excitation, suggesting the existence of a marginal glass phase. The transition between the low temperature glass phase and the high temperature molten phase is also characterized numerically. It is revealed by a change in the coefficient of the logarithmic excitation energy, from being disorder dominated to entropy dominated.Comment: 24 pages, 16 figure

Compositionality, stochasticity and cooperativity in dynamic models of gene regulation

We present an approach for constructing dynamic models for the simulation of gene regulatory networks from simple computational elements. Each element is called a ``gene gate'' and defines an input/output-relationship corresponding to the binding and production of transcription factors. The proposed reaction kinetics of the gene gates can be mapped onto stochastic processes and the standard ode-description. While the ode-approach requires fixing the system's topology before its correct implementation, expressing them in stochastic pi-calculus leads to a fully compositional scheme: network elements become autonomous and only the input/output relationships fix their wiring. The modularity of our approach allows to pass easily from a basic first-level description to refined models which capture more details of the biological system. As an illustrative application we present the stochastic repressilator, an artificial cellular clock, which oscillates readily without any cooperative effects.Comment: 15 pages, 8 figures. Accepted by the HFSP journal (13/09/07