Abstract: We revisit the problem of the plateau transition in the integer quantum Hall eï¬ect. Here we
develop a novel analytical approach for this transition, and for other 2D disordered systems, based
on the theory of âconformal restrictionâ. This is a mathematical theory that was recently developed
within the context of the âSchramm-Loewner Evolutionâ [SLE] which describes the âstochastic geometryâ of fractal curves and other stochastic geometrical fractal objects in two-dimensional space.
Observables elucidating the connection with the plateau transition include the so-called point-contact
conductances [PCCs] between points on the boundary of the sample, described within the language
of the Chalker-Coddington network model for the transition. We show that the disorder-averaged
PCCs are characterized by a classical probability distribution for certain geometric objects in the
plane (which we call pictures), occuring with positive statistical weights, that satisfy the crucial
so-called ârestriction propertyâ with respect to changes in the shape of the sample with absorbing
boundaries - physically, these are boundaries connected to ideal leads. At the transition point these
geometrical objects (pictures) become fractals. Upon combining this ârestriction propertyâ with the
expected conformal invariance at the transition point, we employ the mathematical theory of âconformal restriction measuresâ to relate the disorder-averaged PCCs to correlation functions of (Virasoro)
primary operators in a conformal ï¬eld theory (of central charge c = 0). We show how this can be
used to calculate these functions in a number of geometries with various boundary conditions. Since
our results employ only the conformal restriction property, they are equally applicable to a number
of other critical disordered electronic systems in two spatial dimension, including for example the
spin quantum Hall eï¬ect, the thermal metal phase in symmetry class D, and classical diï¬usion in
two dimensions in a perpendicular magnetic ï¬eld. For most of these systems we also predict exact
values of critical exponents related to the spatial behavior of various disorder-averaged PCCs
Abstract: A smooth spatial disturbance of the Fermi surface in a Fermi gas inevitably becomes sharp. This phenomenon, called {\it the gradient catastrophe}, causes the breakdown of a Fermi sea to disconnected parts with multiple Fermi points. We study how the gradient catastrophe effects probing the Fermi system via a Fermi edge singularity measurement. We show that the gradient catastrophe transforms the single-peaked Fermi-edge singularity of the tunneling (or absorption) spectrum to a set of multiple asymmetric singular resonances. Also we gave a mathematical formulation of FES as a matrix Riemann-Hilbert problem.
Notes: Accepted for publication in Phys. Rev. Lett.
Abstract: When a classical instrument suddenly perturbs a degenerate Fermi gas semiclassical non-
equilibrium Fermi states arise. Semiclassical Fermi states are characterized by a Fermi energy or
Fermi momentum that slowly depends on space or/and time. We show that the Fermi distribution of a
semiclassical Fermi state has a universal nature. It is described by the Airy function regardless of the details
of the perturbation. We also give a general discussion of coherent Fermi states.
Abstract: We formulate the problem of the Fermi Edge Singularity in non-equilibrium states of a Fermi gas as a matrix Riemann-Hilbert problem with an integrable kernel. This formulation is the most suitable for studying the singular behavior at each edge of non-equilibrium Fermi states by means of the method of steepest descent, and also reveals the integrable structure of the problem. We supplement this result by extending the familiar approach to the problem of the Fermi Edge Singularity via the bosonic representation of the electronic operators to non-equilibrium settings. It provides a compact way to extract the leading asymptotes.
Abstract: BCS superconductivity is explained by a simple Hamiltonian describing an attractive pairing
interaction between pairs of electrons. The Hamiltonian may be treated using a mean ï¬eld method,
which is adequate to study equilibrium properties and a variety of non-equilibrium eï¬ects.
Nevertheless, in certain non-equilibrium situations, even in a macroscopic, rather than a microscopic,
superconductor, the application of mean ï¬eld may not be valid. In such cases, one may resort to
the full solution of the Hamiltonian, as given by Richardson in the 60âs. The relevance of Richardsonâs
solution to macroscopic non-equilibrium superconductors was pointed out recently based on the
existence of quantum instabilities out of equilibrium. It is then of interest to obtain analytical
expressions for expectation values between exact eigenvalues of the pairing Hamiltonian within the
Richardson approach for macroscopic systems. We undertake this task in the current paper. It
should be noted that Richardsonâs approach yields the full set of eigenvalues of the Hamiltonian,
while BCS theory yields only a subset. The results obtained here, then, generalize the familiar BCS
expressions for, e.g., the electron occupation or pairing correlations, to cases where the spectrum
of excitations diverges from BCS theory, for example in cases where the spectrum exhibits multiple
gaps.
Abstract: A BCS (Bardeen-Cooper-Schrieffer) superconductor, which is placed out of equilibrium, can develop quantum instabilities, which manifest themselves in oscillations of the superconductor's order parameter (pairing amplitude $\Delta$). These instabilities are a manifestations of the Cooper instability. Inelastic collisions are essential in resolving those instabilities. Incorporating the quantum instabilities and collisions in a unified approach based on Richardson's exact solution of the pairing Hamiltonian, we find that a BCS superconductor may end up in a state in which the spectrum has more than one gap.
Abstract: We develop a hydrodynamic description of the classical Calogero-Sutherland liquid: a Calogero-Sutherland model with an infinite number of particles and a non-vanishing density of particles. The hydrodynamic equations, being written for the density and velocity fields of the liquid, are shown to be a bidirectional analog of the Benjamin-Ono equation. The latter is known to describe internal waves of deep stratified fluids. We show that the bidirectional Benjamin-Ono equation appears as a real reduction of the modified KP hierarchy. We derive the chiral nonlinear equation which appears as a chiral reduction of the bidirectional equation. The conventional Benjamin-Ono equation is a degeneration of the chiral nonlinear equation at large density. We construct multi-phase solutions of the bidirectional Benjamin-Ono equations and of the chiral nonlinear equations.
Abstract: Most mature B lymphocytes express one BCR L chain, kappa or lambda, but recent work has shown that there are exceptions in that some B lymphocytes express both kappa and lambda and some even bear two different kappa L chains. Using the anti-DNA H chain-transgenic mouse, 56R, we find that B cells with pre-existing autoreactivity are especially subject to L chain inclusion. Specifically, we show that isotypic and allelic inclusion enables autoreactive B cells to bypass central tolerance giving rise to B cells that retain dangerous features. One receptor in dual receptor B cells is an editor L chain, i.e., neutralizes or alters self-reactivity of the 56R H chain transgene. We compare the 56R mouse when on the C57/BL/6 background, a strain prone to autoimmunity, with that of 56R when on the BALB/c background, a strain that resists autoimmunity. In the B6.56R mouse, polyreactive B cells with dual L chain move to the follicular B cell compartment. Their localization in the follicular compartment may explain the ease with which B cells in the B6.56R differentiate into autoantibody-producing plasma cells. Likewise, in the BALB/c.56R mouse, dual L chain B cells are found in the follicular B cell compartment. Yet, the lack of autoantibody-producing plasma cells in the BALB/c.56R suggests that postfollicular tolerance checkpoints are intact. The J kappa usage in dual kappa L chain B cells suggests increased receptor editing activity and is consistent with the expected distribution of J kappa genes in our computational model for random selection of J kappa.
Abstract: We present new nonlinear differential equations for spacetime correlation functions of Fermi gas in one spatial dimension. The correlation functions we consider describe non-stationary processes out of equilibrium. The equations we obtain are integrable equations. They generalize known nonlinear differential equations for correlation functions at equilibrium [1-4] and provide vital tools for studying non-equilibrium dynamics of electronic systems. The method we developed is based only on Wick's theorem and the hydrodynamic description of the Fermi gas. Differential equations appear directly in bilinear form.
Abstract: In many situations a BCS-type superconductor will develop an imbalance between the populations of the holelike and electronlike spectral branches. This imbalance suppresses the gap. It has been noted by Gal'perin et al. [Sov. Phys. JETP 54, 1126 (1981)] that at large imbalance, when the gap is substantially suppressed, an instability develops. The analytic treatment of the system beyond the instability point is complicated by the fact that the Boltzmann approach breaks down. We study the short-time behavior following the instability, in the collisionless regime, using methods developed by Yuzbashyan et al. [J. Phys. A 38, 7831 (2005); Phys. Rev. B 72, 220503 (R) (2005)].
Abstract: We consider critical curves-conformally invariant curves-that appear at critical points of two-dimensional statistical mechanical systems. We show how to describe these curves in terms of the Coulomb gas formalism of conformal field theory (CFT). We also provide links between this description and the stochastic (Schramm-) Loewner evolution (SLE). The connection appears in the long-time limit of stochastic evolution of various SLE observables related to CFT primary fields. We show how the multifractal spectrum of harmonic measure and other fractal characteristics of critical curves can be obtained.
Abstract: We show that spacetime evolution of one-dimensional fermionic systems is described by nonlinear equations of soliton theory. We identify a spacetime dependence of a matrix element of fermionic systems related to the orthogonality catastrophe or boundary states with the t-function of the modified KP-hierarchy. The established relation allows us to apply the apparatus of soliton theory to the study of nonlinear aspects of quantum dynamics. We also describe a bosonization in momentum space a representation of a fermion operator by a Bose field in the presence of a boundary state.
Abstract: Using the Calogero model as an example, we show that the transport in interacting nondissipative electronic systems is essentially nonlinear and unstable. Nonlinear effects are due to the curvature of the electronic spectrum near the Fermi energy. As is typical for nonlinear systems, a propagating semiclassical wave packet develops a shock wave at a finite time. A wave packet collapses into oscillatory features which further evolve into regularly structured localized pulses carrying a fractionally quantized charge. The Calogero model can be used to describe fractional quantum Hall edge states. We discuss perspectives of observation of quantum shock waves and a direct measurement of the fractional charge in fractional quantum Hall edge states.
Abstract: A semiclassical wave packet propagating in a dissipationless Fermi gas inevitably enters a "gradient catastrophe" regime, where an initially smooth front develops large gradients and undergoes a dramatic shock-wave phenomenon. The nonlinear effects in electronic transport are due to the curvature of the electronic spectrum at the Fermi surface. They can be probed by a sudden switching of a local potential. In equilibrium, this process produces a large number of particle-hole pairs, a phenomenon closely related to the orthogonality catastrophe. We study a generalization of this phenomenon to the nonequilibrium regime and show how the orthogonality catastrophe cures the gradient catastrophe, by providing a dispersive regularization mechanism.
Abstract: Bubbles of inviscid fluid surrounded by a viscous fluid in a Hele-Shaw cell can merge and break off. During the process of break-off, a thinning neck pinches off to a universal self-similar singularity. We describe this process and reveal its integrable structure: it is a solution of the dispersionless limit of the AKNS hierarchy. The singular break-off patterns are universal, not sensitive to details of the process and can be seen experimentally. We briefly discuss the dispersive regularization of the Hele-Shaw problem and the emergence of the Painlee II equation at the break-off. (c) 2006 Elsevier B.V. Al rights reserved.
Abstract: We show that singularities developed in the Hele-Shaw problem have a structure identical to shock waves in dissipativeless dispersive media. We propose an experimental setup where the cell is permeable to a nonviscous fluid and study continuation of the flow through singularities. We show that a singular flow in this nontraditional cell is described by the Whitham equations identical to Gurevich-Pitaevski solution for a regularization of shock waves in Korteveg-de Vriez equation. This solution describes regularization of singularities through creation of disconnected bubbles.
Abstract: In general or normal random matrix ensembles, the support of eigenvalues of large size matrices is a planar domain (or several domains) with a sharp boundary. This domain evolves under a change of parameters of the potential and of the size of matrices. The boundary of the support of eigenvalues is a real section of a complex curve. Algebro-geometrical properties of this curve encode physical properties of random matrix ensembles. This curve can be treated as a limit of a spectral curve which is canonically defined for models of finite matrices. We interpret the evolution of the eigenvalue distribution as a growth problem, and describe the growth in terms of evolution of the spectral curve. We discuss algebro-geometrical properties of the spectral curve and describe the wave functions (normalized characteristic polynomials) in terms of differentials on the curve. General formulae and emergence of the spectral curve are illustrated by three meaningful examples. (C) 2004 Elsevier B.V. All rights reserved.
Abstract: The stochastic Loewner evolution is a recent tool in the study of two-dimensional critical systems. We extend this approach to the case of critical systems with continuous symmetries, such as SU(2) Wess-Zumino-Witten models, where domain walls carry an additional spin-1/2 degree of freedom.
Abstract: Fractal geometry of critical curves appearing in 2D critical systems is characterized by their harmonic measure. For systems described by conformal field theories with central charge c <= 1, scaling exponents of the harmonic measure have been computed by Duplantier [Phys. Rev. Lett. 84, 1363 ( 2000)] by relating the problem to boundary two-dimensional gravity. We present a simple argument connecting the harmonic measure of critical curves to operators obtained by fusion of primary fields and compute characteristics of the fractal geometry by means of regular methods of conformal field theory. The method is not limited to theories with c <= 1.
Abstract: We continue the analysis of the spectral curve of the normal random matrix ensemble, introduced in an earlier paper. Evolution of the full quantum curve is given in terms of compatibility equations of independent flows. The semiclassical limit of these flows is expressed through canonical differential forms of the spectral curve. We also prove that the semiclassical limit of the evolution equations is equivalent to Whitham hierarchy. (C) 2004 Elsevier B.V. All rights reserved.
Abstract: We show that the semiclassical dynamics of an electronic droplet, confined in a plane in a quantizing inhomogeneous magnetic field in the regime where the electrostatic interaction is negligible, is similar to viscous (Saffman-Taylor) fingering on the interface between two fluids with different viscosities confined in a Hele-Shaw cell. Both phenomena are described by the same equations with scales differing by a factor of up to 10(-9) . We also report the quasiclassical wave function of the droplet in an inhomogeneous magnetic field.
Abstract: Evolution of a system of diffusing and proliferating mortal reactants is analyzed in the presence of randomly moving catalysts. While the continuum description of the problem predicts reactant extinction as the average growth rate becomes negative, growth rate fluctuations induced by the discrete nature of the agents are shown to allow for an active phase, where reactants proliferate as their spatial configuration adapts to the fluctuations of the catalyst density. The model is explored by employing field theoretical techniques, numerical simulations, and strong. coupling analysis. For d less than or equal to2, the system is shown to exhibits an active phase at any growth rate, while for d>2 a kinetic phase transition is predicted. The applicability of this model as a prototype for a host of phenomena that exhibit self-organization is discussed.
Abstract: Diffusion-limited reaction of the Lotka-Volterra type is analyzed taking into account the discrete nature of the reactants. In the continuum approximation, the dynamics is dominated by an elliptic fixed point. This fixed point becomes unstable due to discretization effects, a scenario similar to quantum phase transitions. As a result, the long-time asymptotic behavior of the system changes and the dynamics flows into a limit cycle. The results are verified by numerical simulations. (C) 2001 Elsevier Science B.V. All rights reserved.
Abstract: Many systems in chemistry, biology, finance, and social sciences present emerging features that are not easy to guess from the elementary interactions of their microscopic individual components. In the past, the macroscopic behavior of such systems was modeled by assuming that the collective dynamics of microscopic components can be effectively described collectively by equations acting on spatially continuous density distributions. It turns out that, to the contrary, taking into account the actual individual/discrete character of the microscopic components of these systems is crucial for explaining their macroscopic behavior. In fact, we find that in conditions in which the continuum approach would predict the extinction of all of the population (respectively the vanishing of the invested capital or the concentration of a chemical substance, etc.), the microscopic granularity insures the emergence of macroscopic localized subpopulations with collective adaptive properties that allow their survival and development. In particular it is found that in two dimensions "life" (the localized proliferating phase) always prevails.
Abstract: Hen-egg-white lysozyme was exponentially inactivated by the azide radical. The inactivation yield was pH dependent with a maximum near 5.5. Similar pH effect was found in the protein for the intramolecular one-electron reaction between the indolyl radical and tyrosine. Remarkably, an analogous pH dependence was observed for the catalytic activity of the hen-egg white lysozyme. This was interpreted as representing the dependence of the catalytic rate on two ionizations of two amino acids (Asp52 and Glu35). The similarities between the apparent different phenomena lead us to suggest that it is reasonable to argue that at least two proton equilibria may perturb the intrinsic pH-rate profiles for the tyrOH/trp(.) electron transfer process and radiation inactivation. In other words, the pH dependencies that we observe affected by either the protein structure or its ionization.
