arXiv subjects

Paul Wiegmann

Publications and source records attributed to Paul Wiegmann.

14 recordsLinked to original sources

arXiv2025

Fluid dynamics as intersection problem

We formulate the equations of fluid dynamics as an intersection-theoretic problem on an infinite-dimensional symplectic manifold naturally associated with spacetime. This perspective separates the structures determined by the equation of state and the spacetime geometry from the differential-topological data of spacetime. It leads to a geometric derivation of the covariant formulation of hydrodynamics due to Lichnerowicz and Carter, clarifies the role of the canonical velocity and hydrodynamic invariants, including the asymptotic Hopf invariant and the Ertel invariant, and yields a generalized Kelvin circulation theorem. We also explain the relation between the canonical velocity, the four-velocity, and other choices of hydrodynamic frame. In addition, we identify a five-dimensional geometric origin of the formalism underlying covariant hydrodynamics. The formalism extends naturally to fluids with additional degrees of freedom, including multicomponent fluids, charged fluids, and superfluids, and incorporates the chiral anomaly and Onsager quantization. It also suggests a possible bridge between hydrodynamics, Poisson sigma models, and topological field theories. We further argue that the same intersection-theoretic viewpoint applies to self-dual fields, including chiral bosons in 1+1 dimensions, tensor fields of the (2,0) theory in 1+5 dimensions, and the self-dual four-form field of type-IIB supergravity in 1+9 dimensions.

hep-th↗

arXiv2025

Finite-gap potentials as a semiclassical limit of the thermodynamic Bethe Ansatz

We show that the semiclassical limit of thermodynamic Bethe Ansatz equations naturally reconstructs the algebro-geometric spectra of finite-gap periodic potentials. This correspondence is illustrated using the traveling-wave (snoidal) solution of the defocusing modified Korteweg--de Vries equation. In this framework, the Bethe-root distribution of the associated quantum field theory yields an Abelian differential of the second kind on the elliptic Riemann surface specified by the spectral endpoints, a structure central to the algebro-geometric theory of solitons. The semiclassical parameter is identified with the large-rank limit of the internal symmetry group ($O(2N)$) of the underlying quantum field theory (the Gross-Neveu model with a chemical potential). Our analysis indicates that the analytic structure of the spectrum is dictated solely by the Dynkin diagram ($D_N$) and its large-rank limit ($D_\infty$), independently of the particular integrable model used to realize it.

hep-th↗

arXiv2024

Peierls Transition in Gross-Neveu Model from Bethe Ansatz

The two-dimensional Gross-Neveu model is anticipated to undergo a crystalline phase transition at high baryon charge densities. This conclusion is drawn from the mean-field approximation, which closely resembles models of Peierls instability. We demonstrate that this transition indeed occurs when both the rank of the symmetry group and the dimension of the particle representation contributing to the baryon density are large (the large-N limit). We derive this result through the exact solution of the model, developing the large-N limit of the Bethe Ansatz. Our analytical construction of the large-N solution of the Bethe Ansatz equations aligns perfectly with the periodic (finite-gap) solution of the Korteweg-de Vries (KdV) of the mean-field analysis.

hep-th↗

arXiv2019

Edge wave and boundary layer of vortex matter

We show that a 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 sign-like 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 the 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 odd viscosity of the 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.

physics.flu-dyn↗

arXiv2016

Quantum Hall effect and Quillen metric

We study the generating functional, the adiabatic curvature and the adiabatic phase for the integer quantum Hall effect (QHE) on a compact Riemann surface. For the generating functional we derive its asymptotic expansion for the large flux of the magnetic field, i.e., for the large degree $k$ of the positive Hermitian line bundle $L^k$. The expansion consists of the anomalous and exact terms. The anomalous terms are the leading terms of the expansion. This part is responsible for the quantization of the adiabatic transport coefficients in QHE. We then identify the non-local (anomalous) part of the expansion with the Quillen metric on the determinant line bundle, and the subleading exact part with the asymptotics of the regularized spectral determinant of the Laplacian for the line bundle $L^k$, at large $k$. Finally, we show how the generating functional of the integer QHE is related to the gauge and gravitational (2+1)d Chern-Simons functionals. We observe the relation between the Bismut-Gillet-Soulé curvature formula for the Quillen metric and the adiabatic curvature for the electromagnetic and geometric adiabatic transport of the integer Quantum Hall state. Then we relate the adiabatic phase in QHE to the eta invariant and show that the geometric part of the adiabatic phase is given by the Chern-Simons functional.

hep-th↗

arXiv2016

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.

cond-mat.str-el↗

arXiv2013

Anomalous Hydrodynamics of Two-Dimensional Vortex Fluid

Turbulent flows of incompressible liquid in two dimensions are comprised of dense systems of vortices. Such system of vortices can be treated as a fluid and itself could be described in terms of hydrodynamics. We develop the hydrodynamics of the vortex fluid. This hydrodynamics captures characteristics of fluid flows averaged over fast circulations in the inter-vortex space. The hydrodynamics of the vortex fluid features the anomalous stress absent in Euler's hydrodynamics. The anomalous stress yields a number of interesting effects. Some of them are: a deflection of stream lines, a correction to the Bernoulli law, accumulation of vortices in regions with high curvature in the curved space. The origin of the anomalous stresses is a divergence of inter-vortex interactions at the micro-scale which manifest at the macro-scale. We obtain the hydrodynamics of the vortex fluid from the Kirchhoff equations for dynamics of point-like vortices

physics.flu-dyn↗

arXiv2010

Weak solution of the Hele-Shaw problem: shocks and viscous fingering

In Hele-Shaw flows, boundaries between fluids develop unstable viscous fingers. At vanishing surface tension, the fingers further evolve to cusp-like singularities. We show that the problem admits a {\it weak solution} where shock fronts triggered by a singularity propagate together with a fluid. Shocks form a growing, branching tree of a mass deficit, and a line distribution of vorticity where pressure and velocity of the fluid have finite discontinuities. Imposing that the flow remain curl-free at macroscale determines the shock graph structure. We present a self-similar solution describing shocks emerging from a generic (2,3)-cusp singularity -- an elementary branching event.

cond-mat.soft↗

arXiv2010

Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painleve I transcendent

In Hele-Shaw flows at vanishing surface tension, the boundary of a viscous fluid develops cusp-like singularities. In recent papers [1, 2] we have showed that singularities trigger viscous shocks propagating through the viscous fluid. Here we show that the weak solution of the Hele-Shaw problem describing viscous shocks is equivalent to a semiclassical approximation of a special real solution of the Painleve I equation. We argue that the Painleve I equation provides an integrable deformation of the Hele-Shaw problem which describes flow passing through singularities. In this interpretation shocks appear as Stokes level-lines of the Painleve linear problem.

nlin.SI↗

arXiv2008

Shocks and finite-time singularities in Hele-Shaw flow

Hele-Shaw flow at vanishing surface tension is ill-defined. In finite time, the flow develops cusp-like singularities. We show that the ill-defined problem admits a weak {\it dispersive} solution when singularities give rise to a graph of shock waves propagating in the viscous fluid. The graph of shocks grows and branches. Velocity and pressure jump across the shock. We formulate a few simple physical principles which single out the dispersive solution and interpret shocks as lines of decompressed fluid. We also formulate the dispersive weak solution in algebro-geometrical terms as an evolution of the Krichever-Boutroux complex curve. We study in detail the most generic (2,3) cusp singularity, which gives rise to an elementary branching event. This solution is self-similar and expressed in terms of elliptic functions.

nlin.SI↗

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↗