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: 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: 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.
Abstract: 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.
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.
Abstract: In Hele-Shaw flows, a boundary of a viscous fluid develops unstable fingering patterns. At vanishing surface tension, fingers evolve to cusp-like singularities preventing a smooth flow. We show that the Hele-Shaw problem admits a weak solution where a singularity triggers viscous shocks. Shocks form a growing, branching tree of a line distribution of vorticity where pressure has a finite discontinuity. A condition that the flow remains curl-free at a macroscale uniquely 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 of a branching shock graph.
Abstract: 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.
Abstract: Family members play an important role in the physical and mental recovery of soldiers returning from Operation Iraqi Freedom (OIF) and Operation Enduring Freedom (OEF). Posttraumatic stress disorder (PTSD) has been associated with strained marital and family relations and parenting difficulties, and many veterans with PTSD experience difficulty finding and maintaining employment. Family members who assist with the veteran's recovery also experience significant strain and may have to leave employment to care for the veteran. Our objective was to identify appropriate assessment measures for examining the well-being of spouses assisting with veterans' recovery and to identify opportunities for supporting veterans' spouses. We used a combination of expert panel input and qualitative methods (focus group interviews) to develop a battery of instruments for use in future research with OIF/OEF family members to examine well-being. Research is needed to elucidate and refine the special needs and issues surrounding PTSD in current and future OIF/OEF veterans and their families. This study provides a first step toward understanding appropriate measures. Expert panel methods and focus group interviews yielded valuable input on the domains and measures that should be included in the assessment battery as well as opportunities for assisting spouses.
Abstract: A Comment on the Letter by M. V. Fistul, V. M. Vinokur, and T. I. Baturina, [Phys. Rev. Lett. 100, 086805 (2008)]. The authors of the Letter offer a Reply.
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: Hele-Shaw flow at vanishing surface tension is ill-defined. in finite time, the flow develops cusp-like singularities. We show that this ill-defined problem admits a weak dispersive solution when singularities give rise to a graph of shock waves propagating into the Viscous fluid. The graph of shocks grows and branches. Velocity and pressure have finite discontinuities 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. Published by Elsevier B.V.
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: 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 and provide vital tools to study 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: 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: 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: 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: 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: 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 discuss the 1/N expansion of the free energy of N logarithmically interacting charges in the plane in an external field. For some particular values of the inverse temperature beta, this system is equivalent to the eigenvalue version of certain random matrix models, where it is referred to as the `Dyson gas' of eigenvalues. To find the free energy at large N and the structure of 1/N-corrections, we first use the effective action approach and then confirm the results by solving the loop equation. The results obtained give some new representations of the mathematical objects related to the Dirichlet boundary value problem, complex analysis and spectral geometry of exterior domains. They also suggest interesting links with bosonic field theory on Riemann surfaces, gravitational anomalies and topological field theories.
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: 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: 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: Collective field theory for the Calogero model represents particles with fractional statistics in terms of hydrodynamic modes-density and velocity fields. We show that the quantum hydrodynamics of this model can be written as a single evolution equation on a real holomorphic Bose field-the quantum integrable Benjamin-Ono equation. It renders tools of integrable systems to studies of nonlinear dynamics of 1D quantum liquids.
Abstract: We show that unstable fingering patterns of two-dimensional flows of viscous fluids with open boundary are described by a dispersionless limit of the Korteweg-de Vries hierarchy. In this framework, the fingering instability is linked to a known instability leading to regularized shock solutions for nonlinear waves, in dispersive media. The integrable structure of the flow suggests a dispersive regularization of the finite-time singularities.
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 Laplacian growth (the Hele-Shaw problem) of multiply-connected domains in the case of zero surface tension is proven to be equivalent to an integrable system of Whitham equations known in soliton theory. The Whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations. Through this connection the Laplacian,growth is understood as a flow in the moduli space of Riemann surfaces. (C) 2004 Elsevier B.V. All rights reserved.
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 compute the large scale (macroscopic) correlations in ensembles of normal random matrices with a general non-Gaussian measure and in ensembles of general non-Hermitian matrices with a class of non-Gaussian measures. In both cases, the eigenvalues are complex and in the large N limit they occupy a domain in the complex plane. For the case when the support of eigenvalues is a connected compact domain, we compute two-, three- and four-point connected correlation functions in the first non-vanishing order in 1/N, in a manner that the algorithm of computing higher correlations becomes clear. The correlation functions are expressed through the solution of the Dirichlet boundary problem in the domain complementary to the support of eigenvalues.
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: We study how the solution of the two-dimensional Dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem. We show that the Hadamard formula for the variation of the Dirichlet Green function under deformations of the domain reveals an integrable structure. The independent variables corresponding to the infinite set of commuting flows are identified with harmonic moments of the domain. The solution to the Dirichlet boundary problem is expressed through the tau-function of the dispersionless Toda hierarchy. We also discuss a degenerate case of the Dirichlet problem on the plane with a gap. In this case the tau-function is identical to the partition function of the planar large N limit of the Hermitian one-matrix model.
Abstract: 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 Frohlich'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 on 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. Vile also discuss why the superconducting state is stable in the presence of a weak disorder.
Abstract: We discuss the origin of the associativity (WDVV) equations in the context of quasiclassical or Whitham hierarchies. The associativity equations are shown to be encoded in the dispersionless limit of the Hirota equations for KP and Toda hierarchies. We show, therefore:, that any tau-function of dispersionless KP or Toda hierarchy provides a solution to associativity equations. In general, they depend on infinitely many variables. We also discuss the particular solution to the dispersionless Toda hierarchy that describes conformal mappings and construct a family of new solutions to the WDVV equations depending on finite number of variables. (C) 2001 Published by Elsevier Science B.V.
Abstract: Solitons of a nonlinear field interacting with fermions often acquire a fermionic number or an electric charge if fermions carry a charge. We establish a correspondence between charge and statistics (or spin) of solitons showing 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.
Abstract: We review the concept of $\tau$-function for simple analytic curves. The $\tau$-function gives a formal solution to the 2D inverse potential problem and appears as the $\tau$-function of the integrable hierarchy which describes conformal maps of simply-connected domains bounded by analytic curves to the unit disk. The $\tau$-function also emerges in the context of topological gravity and enjoys an interpretation as a large $N$ limit of the normal matrix model.
Abstract: We trace the origin of theta-terms in nonlinear sigma-models as a nonperturbative anomaly of current algebras. The nonlinear 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 space-time 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 B-terms and quantum numbers of solitons, (C) 2000 Elsevier Science B.V. All rights reserved.
Abstract: We establish the equivalence of 2D contour dynamics to the dispersionless limit of the integrable Toda hierarchy constrained by a string equation. Remarkably, the same hierarchy underlies 2D quantum gravity.
Abstract: 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 tau-function for analytic curves.
Abstract: I review a recent progress towards solution of the so-called almost Mathieu equation, 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.
Abstract: There is a class of electronic liquids in dimensions greater than 1 that shows 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 1, we first review the Frohlich-Peierls mechanism of ideal conductivity in one dimension and then extend the physics and the methods into two spatial dimensions. [S0163-1829(99)05224-8].
Abstract: We study algebro-geometric (finite-gap) and elliptic solutions of fully discretized KP or 2D Toda equations. In bilinear form they are Hirota's difference equation for tau-functions. Starting from a given algebraic curve, we express the tau-function and the Baker-Akhiezer function in terms of the Riemann theta function. We show that the elliptic solutions, when the tau-function is an elliptic polynomial, form a subclass of the general algebro-geometric solutions. We construct the algebraic curves of the elliptic solutions. The evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the Ruijsenaars-Schneider many body system. The zeros obey equations which have the form of nested Bethe-ansatz equations, known from integrable quantum field theories. We discuss the Lax representation and the action-angle-type Variables for the many body system, We also discuss elliptic solutions to discrete analogues of KdV, sine-Gordon and 1D Toda equations and describe the loci of the zeros.
Abstract: We present numerical evidence that solutions of the Bethe anstaz 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 Aux 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 non-interacting 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. (C) 1998 Elsevier Science B.V.
Abstract: We present asymptotically exact wave functions of an incommensurate Harper equation-one-dimensional Schrodinger 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.
Abstract: We compute the order parameter 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 degrees 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.
Abstract: The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to the Bethe ansatz are studied. The nested Bethe ansatz equations for A(k-1)-type models appear as discrete time equations of motions for zeros of classical tau-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained. Difference equations for eigenvalues of the Q-operators which generalize Baxter's three-term T-Q-relation are derived.
Abstract: We show that the set of transfer matrices of an arbitrary fusion type for an integrable quantum model obeys 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.
Abstract: We discuss an interrelation between quantum integrable models and classical soliton equations with discretized time. It appeared that spectral characteristics of quantum integrable systems may be obtained from entirely classical set up. Namely, the eigenvalues of the quantum transfer matrix and the scattering S-matrix itself are identified with a certain tau-functions of the discrete Liouville equation. The Bethe ansatz equations are obtained as dynamics of zeros. For comparison we also present the Bethe ansatz equations for elliptic solutions of the classical discrete Sine-Gordon equation. The paper is based on the recent study of classical integrable structures in quantum integrable systems.(1)
Abstract: 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 allows one to compute off-diagonal correlation functions.
Abstract: There exists strong experimental evidence for the dimensional crossover from two to three dimensions as La2-xSrxCuO4 compounds are overdoped. In this paper we describe the dimensional crossover of the layered correlated metal in the gauge theory framework. In particular, we obtain the anomalous exponent 3/2 for the temperature dependence of resistivity observed in overdoped La2-xSrxCuO4.
Abstract: A class of second order difference (discrete) operators with a partial algebraization of the spectrum is introduced. The eigenfuncions of the algebraized part of the spectrum are polynoms (discrete polynoms). Such difference operators can be constructed by means of U-q(sl(2)), the quantum deformation of the sl(2) algebra. The roots of the polynoms determine the spectrum and obey the Bethe ansatz equations. A particular case of difference equations for q-hypergeometric and Askey-Wilson polynoms is discussed. Applications to the problem of Bloch electrons in a magnetic field are outlined.
Abstract: We present a new approach to the problem of Bloch electrons in a magnetic field (sometimes called the Asbel-Hofstadter problem), by making explicit a natural relation between the group of magnetic translations and the quantum group U(q)(2l2). The approach allows us to express the ''mid''-band spectrum of the model and the Bloch wave function as solutions of the Bethe ansatz equations typical for completely integrable quantum systems. The zero-mode wave functions are found explicitly in terms of q-deformed classical orthogonal polynomials. In this paper we present a solution for the isotropic problem. We also present a class of solvable quasiperiodic equations related to U(q)(sl2).
Abstract: We present an exact and explicit solution of the principal chiral field in two dimensions for a group manifold with infinitely large rank. The energy of the ground state is explicitly found for the external Noether fields of arbitrary magnitude. The exact Gell-Mann-Low function exhibits asymptotic freedom behaviour at large value of the field in agreement with perturbative calculations. Coefficients of the perturbative expansion in the renormalized charge are calculated. They grow factorially with the order showing the presence of renormalons. At small field we found an inverse logarithmic singularity in the ground-state energy at the mass gap which indicates that at N = infinity the spectrum of the theory contains extended objects rather than pointlike particles.
Abstract: We present the exact and explicit solution of the principle chiral field model in two dimensions for infinitely large rank group. In particular we show that in the large-N limit the spectrum of the theory does not contain pointlike particles. The energy of the ground state as a function of external ''Noether'' field and the beta function are explicitly found. The nonperturbative threshold behavior near the mass gap m is f(h) similar to (h - m)/ln(h - m), exhibiting a similarity with the 1D bosonic string theory.
Abstract: We present a new approach to the problem of Bloch electrons in a magnetic field, by making explicit a natural relation between magnetic translations and the quantum group U(q)(sl2). The approach allows us to express the spectrum and the Bloch function as solutions of the Bethe-ansatz equations typical for completely integrable quantum systems.
Abstract: We present a theory of high energy large shift Raman scattering in Mott insulators and show that it provides an instrument for direct measurements of local chirality and anomalous terms in the electronic current algebra. On this basis we argue that the electric-dipole-forbidden electronic transition at energy just below the charge-transfer gap recently observed in inelastic light scattering in insulating cuprates can be interpreted as a zero mode bound state in the chiral spin liquid.
Abstract: A novel mechanism of superconductivity is presented, exploring the possibility that a sufficiently strong interaction in electronic systems gives rise to a topological order.
Abstract: We show that the gauge theory of the two-dimensional doped Mott insulator predicts an anomalous anisotropic negative contribution to the magnetoresistance, which dominates at low temperatures. This anomalous contribution to the magnetoresistivity falls rapidly with increasing temperature, so the total magnetoresistivity is positive at room temperatures. This effect is due to orbital motion and is highly anisotropic with respect to the orientation of the magnetic field. We discuss possible applications of these results to the high-T(c) materials.
Abstract: We show that the Hall coefficient R(H) of a doped two-dimensional Mott insulator is temperature dependent in the state where long-range magnetic order vanishes. The Hall number n(H) = R(H)-1 is linear in the doping density at high temperatures and decreases monotonically with decreasing temperature. At low temperatures R(H) diverges, indicating the possible onset of a phase with spontaneous chirality.
