Axial-Current Anomaly in Euler Fluids.

We argue that a close analog of the axial-current anomaly of quantum field theories with fermions occurs in the classical Euler fluid. The conservation of the axial current (closely related to the helicity of inviscid barotropic flow) is anomalously broken by the external electromagnetic field as ∂_{µ}j_{A}^{µ}=2E·B, similar to that of the axial current of a quantum field theory with Dirac fermions, such as QED.

Edge Wave and Boundary Layer of Vortex Matter.

We show that vortex matter, that is, a dense assembly of vortices in an incompressible two-dimensional flow, such as a fast rotating superfluid or turbulent flows with signlike eddies, exhibits (i) a boundary layer of vorticity (vorticity layer) and (ii) a nonlinear wave localized within the vorticity layer, the edge wave. Both are solely an effect of the topological nature of vortices. Both are lost if vortex matter is approximated as a continuous vorticity patch. The edge wave is governed by the integrable Benjamin-Davis-Ono equation, exhibiting solitons with a quantized total vorticity. Quantized solitons reveal the topological nature of the vortices through their dynamics. The edge wave and the vorticity layer are due to the odd viscosity of vortex matter. We also identify the dynamics with the action of the Virasoro-Bott group of diffeomorphisms of the circle, where odd viscosity parametrizes the central extension. Our edge wave is a hydrodynamic analog of the edge states of the fractional quantum Hall effect.

Inner Nonlinear Waves and Inelastic Light Scattering of Fractional Quantum Hall States as Evidence of the Gravitational Anomaly.

We develop the quantum hydrodynamics of inner waves in the bulk of fractional quantum Hall states. We show that the inelastic light scattering by inner waves is a sole effect of the gravitational anomaly. We obtain the formula for the oscillator strength or mean energy of optical absorption expressed solely in terms of an independently measurable static structure factor. The formula does not explicitly depend on a model interaction potential.

Emergent Conformal Symmetry and Geometric Transport Properties of Quantum Hall States on Singular Surfaces.

We study quantum Hall states on surfaces with conical singularities. We show that the electronic fluid at the cone tip possesses an intrinsic angular momentum, which is due solely to the gravitational anomaly. We also show that quantum Hall states behave as conformal primaries near singular points, with a conformal dimension equal to the angular momentum. Finally, we argue that the gravitational anomaly and conformal dimension determine the fine structure of the electronic density at the conical point. The singularities emerge as quasiparticles with spin and exchange statistics arising from adiabatically braiding conical singularities. Thus, the gravitational anomaly, which appears as a finite size correction on smooth surfaces, dominates geometric transport on singular surfaces.

Geometric Adiabatic Transport in Quantum Hall States.

We argue that in addition to the Hall conductance and the nondissipative component of the viscous tensor, there exists a third independent transport coefficient, which is precisely quantized. It takes constant values along quantum Hall plateaus. We show that the new coefficient is the Chern number of a vector bundle over moduli space of surfaces of genus 2 or higher and therefore cannot change continuously along the plateau. As such, it does not transpire on a sphere or a torus. In the linear response theory, this coefficient determines intensive forces exerted on electronic fluid by adiabatic deformations of geometry and represents the effect of the gravitational anomaly. We also present the method of computing the transport coefficients for quantum Hall states.

Fractional quantum Hall effect in a curved space: gravitational anomaly and electromagnetic response.

We develop a general method to compute correlation functions of fractional quantum Hall (FQH) states on a curved space. In a curved space, local transformation properties of FQH states are examined through local geometric variations, which are essentially governed by the gravitational anomaly. Furthermore, we show that the electromagnetic response of FQH states is related to the gravitational response (a response to curvature). Thus, the gravitational anomaly is also seen in the structure factor and the Hall conductance in flat space. The method is based on an iteration of a Ward identity obtained for FQH states.

Nonlinear hydrodynamics and fractionally quantized solitons at the fractional quantum Hall edge.

We argue that the dynamics of fractional quantum Hall (FQH) edge states is essentially nonlinear and that it features fractionally quantized solitons with charges -νe propagating along the edge. The observation of solitons would be direct evidence of fractional charges. We show that the nonlinear dynamics of the Laughlin's FQH state is governed by the quantum Benjamin-Ono equation. Nonlinear dynamics of gapless edge states is determined by gapped modes in the bulk of FQH liquid and is traced to the double boundary layer "overshoot" of FQH states. The dipole moment of the layer η=1-ν/4π is obtained in paper. Quantum hydrodynamics of FQH liquid is outlined.

Gradient catastrophe and Fermi-edge resonances in Fermi gas.

Any smooth spatial disturbance of a degenerate Fermi gas inevitably becomes sharp. This phenomenon, called the gradient catastrophe, causes the breakdown of a Fermi sea to multiconnected components characterized by multiple Fermi points. We argue that the gradient catastrophe can be probed through a Fermi-edge singularity measurement. In the regime of the gradient catastrophe the Fermi-edge singularity problem becomes a nonequilibrium and nonstationary phenomenon. We show that the gradient catastrophe transforms the single-peaked Fermi-edge singularity of the tunneling (or absorption) spectrum to a sequence of multiple asymmetric singular resonances. An extension of the bosonic representation of the electronic operator to nonequilibrium states captures the singular behavior of the resonances.

Nonlinear quantum shock waves in fractional quantum Hall edge states.

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.

Orthogonality catastrophe and shock waves in a nonequilibrium Fermi gas.

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.

Harmonic measure of critical curves.

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 < or = 1, scaling exponents of the harmonic measure have been computed by Duplantier [Phys. Rev. Lett. 84, 1363 (2000)10.1103/PhysRevLett.84.1363] 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 < or = 1.

Stochastic Loewner evolution for conformal field theories with lie group symmetries.

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.

Viscous fingering and the shape of an electronic droplet in the quantum Hall regime.

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.

Chiral Nonlinear sigma models as models for topological superconductivity.

We study the mechanism of topological superconductivity in a hierarchical chain of chiral nonlinear sigma models (models of current algebra) in one, two, and three spatial dimensions. The models illustrate how the 1D Fröhlich's ideal conductivity extends to a genuine superconductivity in dimensions higher than one. The mechanism is based on the fact that a pointlike topological soliton carries an electric charge. We discuss a flux quantization mechanism and show that it is essentially a generalization of the persistent current phenomenon, known in quantum wires. We also discuss why the superconducting state is stable in the presence of a weak disorder.