arXiv subjects

P. B. Wiegmann

Publications and source records attributed to P. B. Wiegmann.

At least 19 recordsLinked to original sources

arXiv2024

Chern-Simons modification of Fluid Mechanics

We show that the hydrodynamics of a perfect fluid admits a natural modification that incorporates a chiral gravitational anomaly (also known as a mixed gauge-gravity anomaly) alongside the chiral current anomaly. This modification introduces in parallel a gravitational and gauge Chern-Simons terms in a manner analogous to the Jackiw-Pi Chern-Simons modification of gravity, and features an axion in fluid mechanics. As a result, flows of a spinless perfect fluid acquire spin through motion, with the spin being equal to the fluid helicity. Furthermore, spacetime curvature induces an analogue of the Mathisson-Papapetrou force and a geometric counterpart of the chiral magnetic effect.

hep-th↗

arXiv2024

Multivalued Wess-Zumino-Novikov Functional and Chiral Anomaly in Hydrodynamics

We present a hydrodynamic framework derived from the action of a perfect fluid, modified by the hydrodynamic analog of Novikov's multivalued functional. This modification introduces spin degrees of freedom into the fluid. The structure closely resembles the Abelian version of the Wess-Zumino functional, commonly applied in field theories with chiral anomalies. The deformation incorporates the transport properties of Weyl fermions and exhibits the chiral anomaly in the case of a charged fluid. It is also consistent with Onsager's semiclassical quantization of circulation. Additionally, we discuss the hydrodynamic analog of instantons and related topological invariants.

hep-th↗

arXiv2022

Hamilton Principle for Chiral Anomalies in Hydrodynamics

We developed the spacetime-covariant Hamilton principle for barotropic flows of a perfect fluid in the external axial-vector potential conjugate to the helicity current. Such flows carry helicity, a chiral imbalance, controlled by the axial potential. The interest in such a setting is motivated by the recent observation that the axial-current anomaly of quantum field theories with Dirac fermions appears as a kinematic property of classical hydrodynamics. Especially interesting effects occur under the simultaneous actions of the electromagnetic field and the axial-vector potential. With the help of the Hamilton principle, we obtain the extension of the Euler equations by the axial potential and derive anomalies in the divergence of the axial and vector current. Our approach provides a hydrodynamic expression for vector and axial currents and lays down a platform for studying flows with a chiral imbalance and their anomalies.

hep-th↗

arXiv2022

Chiral Anomaly in Euler Fluid and Beltrami Flow

We show that the chiral anomaly of quantum field theories with Dirac fermions subject to an axial background field is an inherent property of kinematics of a perfect classical fluid. Celebrated Beltrami flows (stationary solutions of Euler equations with extensive helicity) exhibit the chiral anomaly equivalent to that known for Dirac fermions. A prominent effect of the chiral anomaly is the transport electric current at equilibrium. We show that it is also a property of Beltrami flows.

hep-th↗

arXiv2021

Axial-Current Anomaly in Euler Fluid

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 $\partial_μj_{A}^μ= 2\,\bf E\!\cdot\! \bf B$ similar to that of the axial current of a quantum field theory with Dirac fermions such as QED.

hep-th↗

arXiv2013

Hydrodynamics of Euler incompressible fluid and the Fractional Quantum Hall Effect

We show that the Fractional Quantum Hall Effect can be phenomenologically described as a special flow of a quantum incompressible Euler liquid. This flow consists of a large number of vortices of the same chirality. In this approach each vortex is identified with an electron while the fluid is neutral. We show that the Laughlin wave function emerges as a stationary flow of the system of vortices in quantum fluid dynamics. Subtle features of FQHE such as effects of Lorentz shear stress, the spectral function, the Hall current in a modulated electric fields, etc., naturally follow from the hydrodynamics approach. In the paper we develop the hydrodynamics of the vortex liquid, and able consistently quantize it. As a demonstration of the efficiency of the hydrodynamics we discuss some new results for FQHE in a non-uniform magnetic field and a curved space.

cond-mat.str-el↗

arXiv2008

Comment on "Superinsulator and Quantum Synchronization"

We show that the "theory" of "superinsulating" state presented by V.Vinokur et al (Nature vol. 452, p. 613, 2008) and Fistul et al (Phys. Rev. Lett. vol. 100, 086805, 2008) is essentially incorrect due to a sequence of errors in the theoretical analysis of the standard model of Josephson arrays which properties have been carefully studied and described in the literature. The line of calculations suggested in these articles lead to unphysical results. In particular, the calculations predict a direct current flowing through a capacitor. Moreover, this current may flow even in the absence of voltage - a sort of supercurrent flowing in the "superinsulating" state. We also question that the theoretical model employed in these works, even if treated correctly, is applicable to the analysis of experimental data on homogeneously disordered superconductive films.

cond-mat.supr-con↗

arXiv2001

Geometrical phases and quantum numbers of solitons in nonlinear sigma-models

Solitons of a nonlinear field interacting with fermions often acquire a fermionic number or an electric charge if fermions carry a charge. We show how the same mechanism (chiral anomaly) gives solitons statistical and rotational properties of fermions. These properties are encoded in a geometrical phase, i.e., an imaginary part of a Euclidian action for a nonlinear sigma-model. In the most interesting cases the geometrical phase is non-perturbative and has a form of an integer-valued theta-term.

hep-th↗

arXiv2000

Chiral non-linear sigma-models as models for topological superconductivity

We study the mechanism of topological superconductivity in a hierarchical chain of chiral non-linear sigma-models (models of current algebra) in one, two, and three spatial dimensions. The models have roots in the 1D Peierls-Frohlich model and illustrate how the 1D Frohlich's ideal conductivity extends to a genuine superconductivity in dimensions higher than one. The mechanism is based on the fact that a point-like 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.

hep-th↗

arXiv2000

Conformal maps and dispersionless integrable hierarchies

We show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2D Toda integrable hierarchy. The maps are determined by a particular solution to the hierarchy singled out by the conditions known as "string equations". The same hierarchy locally solves the 2D inverse potential problem, i.e. reconstruction of the domain out of a set of its harmonic moments. This is the same solution which is known to describe 2D gravity coupled to c=1 matter. We also introduce a concept of the $τ$-function for analytic curves.

hep-th↗

arXiv1999

Theta-terms in nonlinear sigma-models

We trace the origin of theta-terms in non-linear sigma-models as a nonperturbative anomaly of current algebras. The non-linear sigma-models emerge as a low energy limit of fermionic sigma-models. The latter describe Dirac fermions coupled to chiral bosonic fields. We discuss the geometric phases in three hierarchies of fermionic sigma-models in spacetime dimension (d+1) with chiral bosonic fields taking values on d-, d+1-, and d+2-dimensional spheres. The geometric phases in the first two hierarchies are theta-terms. We emphasize a relation between theta-terms and quantum numbers of solitons.

hep-th↗

arXiv1999

On the singular spectrum of the Almost Mathieu operator. Arithmetics and Cantor spectra of integrable models

I review a recent progress towards solution of the Almost Mathieu equation (A.G. Abanov, J.C. Talstra, P.B. Wiegmann, Nucl. Phys. B 525, 571, 1998), known also as Harper's equation or Azbel-Hofstadter problem. The spectrum of this equation is known to be a pure singular continuum with a rich hierarchical structure. Few years ago it has been found that the almost Mathieu operator is integrable. An asymptotic solution of this operator became possible due analysis the Bethe Ansatz equations.

hep-th↗

arXiv1998

Topological Electronic Liquids: Electronic Physics of One Dimension Beyond the One Dimension

There is a class of electronic liquids in dimensions greater than one, which show all essential properties of one dimensional electronic physics. These are topological liquids - correlated electronic systems with a spectral flow. Compressible topological electronic liquids are superfluids. In this paper we present a study of a conventional model of a topological superfluid in two spatial dimensions. This model is thought to be relevant to a doped Mott insulator. We show how the spectral flow leads to the superfluid hydrodynamics and how the Orthogonality Catastrophe affects off-diagonal matrix elements. We also compute the major electronic correlation functions. Among them are the spectral function, the pair wave function and various tunneling amplitudes. To compute correlation functions we develop a method of current algebra - an extension of the bosonization technique of one spatial dimension. In order to emphasize a similarity between electronic liquids in one dimension and topological liquids in dimensions greater than one, we first review the Frohlich-Peierls mechanism of ideal conductivity in one dimension and then extend the physics and the methods into two spatial dimension.

cond-mat.str-el↗

arXiv1998

Tunneling in the topological mechanism of superconductivity

We compute the two-particle matrix element and Josephson tunneling amplitude in a two-dimensional model of topological superconductivity which captures the physics of the doped Mott insulator. The hydrodynamics of topological electronic liquid consists of the compressible charge sector and the incompressible chiral topological spin liquid. We show that ground states differing by an odd number of particles are orthogonal and insertion of two extra electrons is followed by the emission of soft modes of the transversal spin current. The orthogonality catastrophe makes the physics of superconductivity drastically different from the BCS-theory but similar to the physics of one-dimensional electronic liquids. The wave function of a pair is dressed by soft modes. As a result the two particle matrix element forms a complex d-wave representation (i.e., changes sign under $90^o$ degree rotation), although the gap in the electronic spectrum has no nodes. In contrast to the BCS-theory the tunneling amplitude has an asymmetric broad peak (much bigger than the gap) around the Fermi surface. We develop an operator algebra, that allows one to compute other correlation functions.

cond-mat.str-el↗

arXiv1998

Asymptotically exact solutions of Harper equation

We present asymptotically exact solutions of an incommensurate Harper equation---one-dimensional Schroedinger equation of one particle on a lattice in a cosine potential. The wave functions can be written as an infinite product of string polynomials. The roots of these polynomials are solutions of Bethe equations. They are classified according to the string hypothesis. The string hypothesis gives asymptotically exact values of roots and reveals the hierarchical structure of the spectrum of the Harper equation.

cond-mat↗

arXiv1997

Hierarchical Structure of Azbel-Hofstader Problem: Strings and loose ends of Bethe Ansatz

We present numerical evidence that solutions of the Bethe Ansatz equations for a Bloch particle in an incommensurate magnetic field (Azbel-Hofstadter or AH model), consist of complexes-"strings". String solutions are well-known from integrable field theories. They become asymptotically exact in the thermodynamic limit. The string solutions for the AH model are exact in the incommensurate limit, where the flux through the unit cell is an irrational number in units of the elementary flux quantum. We introduce the notion of the integral spectral flow and conjecture a hierarchical tree for the problem. The hierarchical tree describes the topology of the singular continuous spectrum of the problem. We show that the string content of a state is determined uniquely by the rate of the spectral flow (Hall conductance) along the tree. We identify the Hall conductances with the set of Takahashi-Suzuki numbers (the set of dimensions of the irreducible representations of $U_q(sl_2)$ with definite parity). In this paper we consider the approximation of noninteracting strings. It provides the gap distribution function, the mean scaling dimension for the bandwidths and gives a very good approximation for some wave functions which even captures their multifractal properties. However, it misses the multifractal character of the spectrum.

cond-mat↗

arXiv1997

Fusion rules for Quantum Transfer Matrices as a Dynamical System on Grassmann Manifolds

We show that the set of transfer matrices of an arbitrary fusion type for an integrable quantum model obey these bilinear functional relations, which are identified with an integrable dynamical system on a Grassmann manifold (higher Hirota equation). The bilinear relations were previously known for a particular class of transfer matrices corresponding to rectangular Young diagrams. We extend this result for general Young diagrams. A general solution of the bilinear equations is presented.

solv-int↗

arXiv1997

Tunneling and orthogonality catastrophe in the topological mechanism of superconductivity

We compute the angular dependence of the order parameter and tunneling amplitude in a model exhibiting topological superconductivity and sketch its derivation as a model of a doped Mott insulator. We show that ground states differing by an odd number of particles are orthogonal and the order parameter is in the d-representation, although the gap in the electronic spectrum has no nodes. We also develop an operator algebra, that allowes one to compute off-diagonal correlation functions.

cond-mat.supr-con↗