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Eldad Bettelheim

Publications and source records attributed to Eldad Bettelheim.

35 records · Page 2Linked to original sources
arXiv2013

Coherence Factors Beyond the BCS Expressions -- A Derivation

We present a derivation of a previously announced result for matrix elements between exact eigenstates of the pairing Hamiltnonian. Our results, which generalize the well known BCS (Bardeen-Cooper-Schrieffer) expressions for what is known as 'coherence factors', are derived based on the Slavnov formula for overlaps between Bethe-ansatz states, thus making use of the known connection between the exact diagonalization of the BCS Hamiltonian, due to Richardson, and the algebraic Bethe ansatz. The resulting formula has a compact form after a suitable parameterization of the Energy plane. Although we apply our method here to the pairing Hamiltonian, it may be adjusted to study what is termed the 'Sutherland limit' for exactly solvable models, namely where a macroscopic number of rapidities form a large string.

cond-mat.supr-con↗
arXiv2013

Coherence Factors Beyond the BCS Result

The dynamics of BCS (Bardeen-Cooper-Schrieffer) superconductors is fairly well understood due to the availability of a mean field solution for the pairing Hamiltonian, a solution which gives the quantum state of superconductor as a state of almost-free fermions interacting only with a condensate. As a result, transition probabilities may be computed, and expressed in terms of matrix elements of electron creation and annihilation operators between approximate eigenstates. These matrix elements are also called 'coherent factors'. Mean-field theory is however not sufficient to describe all eigenstates of a superconductor, a deficiency which is hardly important in (or very close) to equilibrium, but one that becomes relevant in certain out of equilibrium situations. We report here on a computation of matrix elements (coherence factors) for the pairing Hamiltonian between any 'two-arc' eigenstates in the thermodynamic limit.

cond-mat.supr-con↗
arXiv2012

Quantum ripples over a semi-classical shock

The evolution of an initially smooth spatial inhomogeneity in the density of a one-dimensional Fermi gas is well described by classical mechanics. The classical evolution leads to the formation of a shock wave: the density develops kinks in its coordinate dependence. We show that quantum corrections to the shock wave produce density ripples which run off the kinks. Despite their quantum origin, the amplitude and period of the ripples are expressed only in terms of classical objects derived from a smooth density profile.

cond-mat.stat-mech↗
arXiv2011

Exact expectation values within Richardson's approach for the pairing Hamiltonian in a macroscopic system

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-field method, which is adequate to study equilibrium properties and a variety of nonequilibrium effects. Nevertheless, in certain nonequilibrium situations, even in a macroscopic rather than a microscopic superconductor, the application of mean-field theory may not be valid. In such cases, one may resort to the full solution of the Hamiltonian, as given by Richardson in the 1960s. The relevance of Richardson's solution to macroscopic nonequilibrium 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 Richardson's 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 (e.g., for the electron occupation or pairing correlations) to cases where the spectrum of excitations diverges from BCS theory (e.g., in cases where the spectrum exhibits multiple gaps).

cond-mat.supr-con↗
arXiv2011

Fermi distribution of semicalssical non-eqilibrium Fermi states

When a classical device suddenly perturbs a degenerate Fermi gas a semiclassical non-equilibrium Fermi state arises. 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 Airy functions regardless of the details of the perturbation. In this letter we also give a general discussion of coherent Fermi states.

cond-mat.quant-gas↗
arXiv2011

Gradient Catastrophe and Fermi Edge Resonances in Fermi Gas

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.

cond-mat.mes-hall↗
arXiv2010

Multi-condensate states in BCS superconductors

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 $Δ$). 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.

cond-mat.supr-con↗
arXiv2007

Asymptotic eigenvalue distribution of large Toeplitz matrices

We study the asymptotic eigenvalue distribution of Toeplitz matrices generated by a singular symbol. It has been conjectured by Widom that, for a generic symbol, the eigenvalues converge to the image of the symbol. In this paper we ask how the eigenvalues converge to the image. For a given Toeplitz matrix $T_n(a)$ of size $n$, we take the standard approach of looking at $\det(ζ-T_n(a))$, of which the asymptotic information is given by the Fisher-Hartwig theorem. For a symbol with single jump, we obtain the distribution of eigenvalues as an expansion involving $1/n$ and $\log n/n$. To demonstrate the validity of our result we compare our result against the numerics using a pure Fisher-Hartwig symbol.

math-ph↗
arXiv2006

Nonlinear Dynamics of Quantum Systems and Soliton Theory

We show that space-time evolution of one-dimensional fermionic systems is described by nonlinear equations of soliton theory. We identify a space-time dependence of a matrix element of fermionic systems related to the {\it Orthogonality Catastrophe} or {boundary states} with the $τ$-function of the modified KP-hierarchy. The established relation allows to apply the apparatus of soliton theory to the study of non-linear aspects of quantum dynamics. We also describe a {\it bosonization in momentum space} - a representation of a fermion operator by a Bose field in the presence of a boundary state.

nlin.SI↗
arXiv2006

Quantum Shock Waves - the case for non-linear effects in dynamics of electronic liquids

Using the Calogero model as an example, we show that the transport in interacting non-dissipative electronic systems is essentially non-linear. Non-linear effects are due to the curvature of the electronic spectrum near the Fermi energy. As is typical for non-linear systems, propagating wave packets are unstable. At finite time shock wave singularities develop, the wave packet collapses, and oscillatory features arise. They evolve into regularly structured localized pulses carrying a fractionally quantized charge - {\it soliton trains}. We briefly discuss perspectives of observation of Quantum Shock Waves in edge states of Fractional Quantum Hall Effect and a direct measurement of the fractional charge.

cond-mat.str-el↗
arXiv2006

Singular limit of Hele-Shaw flow and dispersive regularization of shock waves

We study a family of solutions to the Saffman-Taylor problem with zero surface tension at a critical regime. In this regime, the interface develops a thin singular finger. The flow of an isolated finger is given by the Whitham equations for the KdV integrable hierarchy. We show that the flow describing bubble break-off is identical to the Gurevich-Pitaevsky solution for regularization of shock waves in dispersive media. The method provides a scheme for the continuation of the flow through singularites.

nlin.SI↗
arXiv2006

Bubble break-off in Hele-Shaw flows : Singularities and integrable structures

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 Painlevé II equation at the break-off.

nlin.SI↗
arXiv2004

Field Theory Analysis of Laplacian Growth Models

We consider Laplacian growth problems using a field theory approach. In particular we consider the Saffman-Taylor (ST) problem. The idealized settings of the problem, with vanishing surface tension between the bubble and the surrounding fluid, is singular due to the formation of cusps after a finite time (for generic initial conditions). A natural regularization of the cusp, is the addition of surface tension, but this complicates the mathematical description of the problem a great deal. We discuss a different method of regularization which arises from the relation of the ST problem with integrable systems and matrix models.

cond-mat.soft↗
arXiv2002

Viscous fingering and a shape of an electronic droplet in the Quantum Hall regime

We show that the semiclassical dynamics of an electronic droplet confined in the plane in a quantizing inhomogeneous magnetic field in the regime when 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.

cond-mat↗
arXiv2000

"Quantum phase transitions" in classical nonequilibrium processes

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.

cond-mat↗
arXiv1999

The Importance of Being Discrete - Life Always Wins on the Surface

Many systems in chemistry, biology, finance and social sciences present emerging features which 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 quite 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 the population (respectively the vanishing of the invested capital or of the concentration of a chemical substance, etc), the microscopic granularity insures the emergence of macroscopic localized sub-populations with collective adaptive properties which allow their survival and development. In particular it is found that in 2 dimensions "life" (the localized proliferating phase) always prevails.

adap-org↗