A staggered fermion chain with supersymmetry on open intervals

A strongly-interacting fermion chain with supersymmetry on the lattice and open boundary conditions is analysed. The local coupling constants of the model are staggered, and the properties of the ground states as a function of the staggering parameter are examined. In particular, a connection between certain ground-state components and solutions of non-linear recursion relations associated with the Painlev\'e VI equation is conjectured. Moreover, various local occupation probabilities in the ground state have the so-called scale-free property, and allow for an exact resummation in the limit of infinite system size.Comment: 21 pages, no figures; v2: typos correcte

Sinai model in presence of dilute absorbers

We study the Sinai model for the diffusion of a particle in a one dimension random potential in presence of a small concentration $\rho$ of perfect absorbers using the asymptotically exact real space renormalization method. We compute the survival probability, the averaged diffusion front and return probability, the two particle meeting probability, the distribution of total distance traveled before absorption and the averaged Green's function of the associated Schrodinger operator. Our work confirms some recent results of Texier and Hagendorf obtained by Dyson-Schmidt methods, and extends them to other observables and in presence of a drift. In particular the power law density of states is found to hold in all cases. Irrespective of the drift, the asymptotic rescaled diffusion front of surviving particles is found to be a symmetric step distribution, uniform for $|x| < {1/2} \xi(t)$, where $\xi(t)$ is a new, survival length scale ($\xi(t)=T \ln t/\sqrt{\rho}$ in the absence of drift). Survival outside this sharp region is found to decay with a larger exponent, continuously varying with the rescaled distance $x/\xi(t)$. A simple physical picture based on a saddle point is given, and universality is discussed.Comment: 21 pages, 2 figure

One-dimensional classical diffusion in a random force field with weakly concentrated absorbers

A one-dimensional model of classical diffusion in a random force field with a weak concentration $\rho$ of absorbers is studied. The force field is taken as a Gaussian white noise with \mean{\phi(x)}=0 and \mean{\phi(x)\phi(x')}=g \delta(x-x'). Our analysis relies on the relation between the Fokker-Planck operator and a quantum Hamiltonian in which absorption leads to breaking of supersymmetry. Using a Lifshits argument, it is shown that the average return probability is a power law \smean{P(x,t|x,0)}\sim{}t^{-\sqrt{2\rho/g}} (to be compared with the usual Lifshits exponential decay $\exp{-(\rho^2t)^{1/3}}$ in the absence of the random force field). The localisation properties of the underlying quantum Hamiltonian are discussed as well.Comment: 6 pages, LaTeX, 5 eps figure

Random RNA under tension

The Laessig-Wiese (LW) field theory for the freezing transition of random RNA secondary structures is generalized to the situation of an external force. We find a second-order phase transition at a critical applied force f = f_c. For f f_c, the extension L as a function of pulling force f scales as (f-f_c)^(1/gamma-1). The exponent gamma is calculated in an epsilon-expansion: At 1-loop order gamma = epsilon/2 = 1/2, equivalent to the disorder-free case. 2-loop results yielding gamma = 0.6 are briefly mentioned. Using a locking argument, we speculate that this result extends to the strong-disorder phase.Comment: 6 pages, 10 figures. v2: corrected typos, discussion on locking argument improve

Fluctuation force exerted by a planar self-avoiding polymer

Using results from Schramm Loewner evolution (SLE), we give the expression of the fluctuation-induced force exerted by a polymer on a small impenetrable disk, in various 2-dimensional domain geometries. We generalize to two polymers and examine whether the fluctuation force can trap the object into a stable equilibrium. We compute the force exerted on objects at the domain boundary, and the force mediated by the polymer between such objects. The results can straightforwardly be extended to any SLE interface, including Ising, percolation, and loop-erased random walks. Some are relevant for extremal value statistics.Comment: 7 pages, 22 figure

Ordered spectral statistics in 1D disordered supersymmetric quantum mechanics and Sinai diffusion with dilute absorbers

Some results on the ordered statistics of eigenvalues for one-dimensional random Schr\"odinger Hamiltonians are reviewed. In the case of supersymmetric quantum mechanics with disorder, the existence of low energy delocalized states induces eigenvalue correlations and makes the ordered statistics problem nontrivial. The resulting distributions are used to analyze the problem of classical diffusion in a random force field (Sinai problem) in the presence of weakly concentrated absorbers. It is shown that the slowly decaying averaged return probability of the Sinai problem, \mean{P(x,t|x,0)}\sim \ln^{-2}t, is converted into a power law decay, \mean{P(x,t|x,0)}\sim t^{-\sqrt{2\rho/g}}, where $g$ is the strength of the random force field and $\rho$ the density of absorbers.Comment: 10 pages ; LaTeX ; 4 pdf figures ; Proceedings of the meeting "Fundations and Applications of non-equilibrium statistical mechanics", Nordita, Stockholm, october 2011 ; v2: appendix added ; v3: figure 2.left adde

Ground-state properties of a supersymmetric fermion chain

We analyze the ground state of a strongly interacting fermion chain with a supersymmetry. We conjecture a number of exact results, such as a hidden duality between weak and strong couplings. By exploiting a scale free property of the perturbative expansions, we find exact expressions for the order parameters, yielding the critical exponents. We show that the ground state of this fermion chain and another model in the same universality class, the XYZ chain along a line of couplings, are both written in terms of the same polynomials. We demonstrate this explicitly for up to N = 24 sites, and provide consistency checks for large N. These polynomials satisfy a recursion relation related to the Painlev\'e VI differential equation, and using a scale-free property of these polynomials, we derive a simple and exact formula for their limit as N goes to infinity.Comment: v2: added more information on scaling function, fixed typo

Influence of slim rod material properties to the Siemens feed rod and the float zone process

The identification and understanding of material properties influencing the float zone process is important to crystallize high purity silicon for high efficiency solar cells. Also the knowledge of minimal requirements to crystallize monocrystalline silicon with the float zone process is of interest from an economic point of view. In the present study, feed rods for the float zone process composed of a central slim rod and the deposited silicon from the Siemens process are investigated. Previous studies have shown that the slim rod has a significant impact on the purity and suitability for further crystallization processes. In particular, contaminations like substitutional carbon and the presence of precipitates as well as the formation of oxide layers play an important role and are investigated in detail. For this purpose different slim rod materials were used in deposition and float zone crystallization experiments. Samples were prepared by cross sectioning and core drilling of Siemens rods, which were recrystallized with the float zone process. Recrystallized drilled cores are analyzed with FT-IR spectrometry concerning the carbon and oxygen content. To estimate the grain growth behavior on the slim rod surface in dependence of the used slim rod material, EBSD mappings inside a SEM are performed on squared and circular slim rods. TEM analysis was used to investigate the presence of an oxide layer at the interface between slim rod and deposited polycrystalline silicon. Additionally the influence of a nitrogen-containing gas atmosphere during the slim rod pulling is investigated by IR microscopy and ToF-SIMS regarding Si3N4 precipitation

Structural and chemical investigations of adapted Siemens feed rods for an optimized float zone process

The optimization of the float zone process for industrial application is a promising way to crystallize high purity silicon for high efficiency solar cells with reduced process costs. We investigated two differently produced Siemens rods which should be used as feed material for the float zone process. The aim is to identify and to improve material properties of the feed rods which have a high impact to the float zone process. We show here microstructural and chemical analysis comparing feed rods manufactured under standard conditions and under float zone adapted conditions. To resolve the growth behavior of the grains SEM/EBSD mappings are performed at different positions. TEM analyses are used to investigate the interface region between the mono- and the multicrystalline silicon within the Siemens feed rod. Additionally, drilled cores are cut out from the feed rods containing the region of the slim rod. Afterwards, the drilled cores are crystallized with the float zone process. Finally, carbon and oxygen measurements with FT-IR spectrometry on different positions of the crystallized drilled cores of the Siemens feed rods show the influence of the slim rod material to the float zone process

Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices

The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localisation; its interest in this particular context is that it provides a reasonable measure of the localisation length. The Lyapunov exponent also features prominently in the theory of products of random matrices pioneered by Furstenberg. After a brief historical survey, we describe some recent work that exploits the close connections between these topics. We review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case. Finally, we point out some limitations of the Lyapunov exponent as a means of studying localisation properties.Comment: LaTeX, 23 pages, 3 pdf figures ; review for a special issue on "Lyapunov analysis" ; v2 : typo corrected in eq.(3) & minor change