Bloch electrons in a magnetic field

A complete set of basis functions for the expansion of the wavefunction of a Bloch electron in a uniform magnetic field is derived. In the empty lattice limit this set gives the appropriate Landau free-electron wavefunctions, contrary to the Roth functions which in that limit are plane waves.     

The Bethe ansatz equations for reflecting magnons

We derive the Bethe ansatz equations on the half line for particles interacting through factorized S-matrices invariant relative to the centrally extended su(2|2) Lie superalgebra and su(1|2) open boundaries. These equations may be of relevance for the study of the spectrum of open strings on AdS₅×S⁵ background attached to Y=0 giant graviton branes. A one-dimensional spin chain Hamiltonian associated to this system is also derived.     

Generalized Bethe ansatz equations for Hofstadter problem

The problem of diagonalization of the quantum mechanical Hamiltonian, governing dynamics of an electron on a two-dimensional triangular or square lattice in external uniform magnetic field, applied perpendicularly to the lattice plane, the flux through lattice cell, divided by the elementary quantum flux, being a rational number, is reduced to the generalized Bethe ansatz like equations on the high genus algebraic curve. Our formulae for the trigonometric case, where the genus of the curve vanishes, contain as a particular case a recent result of Wiegmann and Zabrodin.Supported by the Russian Academy of Sciences and Academy of Finland     

Bethe ansatz equations for the brokenZ N -symmetric model

We obtain the Bethe ansatz equations for the brokenZ _N -symmetric model by constructing a functional relation of the transfer matrix ofL-operators. This model is an elliptic off-critical extension of the Fateev-Zamolodchikov model. We calculate the free energy of this model on the basis of the string hypothesis.     

Pseudo-differential equations,and the Bethe ansatz for the classical Lie algebras

The correspondence between ordinary differential equations and Bethe ansatz equations for integrable lattice models in their continuum limits is generalised to vertex models related to classical simple Lie algebras. New families of pseudo-differential equations are proposed, and a link between specific generalised eigenvalue problems for these equations and the Bethe ansatz is deduced. The pseudo-differential operators resemble in form the Miura-transformed Lax operators studied in work on generalised KdV equations, classical W-algebras and, more recently, in the context of the geometric Langlands correspondence. Negative-dimension and boundary-condition dualities are also observed.     

Bloch electrons in a magnetic field: Rigorous justification of the Peierls-Onsager effective Hamiltonian

We present a rigorous justification of the Peierls-Onsager one-band effective Hamiltonian, starting from the one-particle Hamiltonian for Bloch electrons in a homogeneous magnetic field.     

Analysis of the Bethe ansatz equations of the XXZ model

We analyse the Bethe ansatz equations of the XXZ model in the antiferromagnetic region, without assuming a priori the existence of strings. Excited states are described by a finite number of parameters. These parameters satisfy a closed system of equations, which we obtain by eliminating the parameters of the vacuum from the original Bethe ansatz equations. Strings are only particular solutions of these equations.     

Bloch electrons in a magnetic field: Topology of quantum states and transport

Quantum states and Hall conductances of electrons in n-type heterojunctions and holes in p-type heterojunctions in a field of a lateral superlattice and a perpendicular magnetic field were studied. It is shown that the energy spectrum of magnetic subbands in a periodic potential without inversion center is not symmetric about the reversal of the quasi-momentum sign. The properties of wave functions and the related topological invariants determining the Hall conductance were examined. The method of calculating the magnetic Bloch states of holes was developed on the basis of the Luttinger Hamiltonian, allowing the spin and spin-orbit interactions to be taken into account in this problem. The Hall conductance quantization law was determined for 2D holes in a periodic superlattice potential.     

Bloch electrons in constant electric field

The motion of a conduction electron of a crystal in a constant electric field is studied. It is shown that the modulus of the wave function inp-representation is well approximated by a periodic function for times smaller than several hundred periods.On leave from Department of Physics, Luminy, Université d'Aix-Marseille II, and Centre de Physique Théorique, CNRS, Marseille, France     

Explicit solutions to some nonlinear physical models by a two-step ansatz

Explicit solutions are derived for some nonlinear physical model equations by using a delicate way of two-step ansatz method.     

On the Bethe ansatz for random directed polymers

We show that the inclusion of the (gapless) center-of-mass motion together with a functional integral representation of the Bethe wave function allows one to predict exactly the critical exponents for random directed polymers in (1+1) dimensions. The corresponding amplitudes are computed; they compare satisfactorily with existing numerical data. Within a replica-symmetric theory, we find that the Green function of the polymer has the form recently proposed by Parisi.     

Numerical solution to the Bloch equations:paramagnetic solutions under wideband continuous radio frequency irradiation in a pulsed magnetic field

A novel nuclear magnetic resonance(NMR) experimental scheme,called wideband continuous wave NMR(WB-CW-NMR),is presented in this article.This experimental scheme has promising applications in pulsed magnetic fields,and can dramatically improve the utilization of the pulsed field.The feasibility of WB-CW-NMR scheme is verified by numerically solving modified Bloch equations.In the numerical simulation,the applied magnetic field is a pulsed magnetic field up to 80 T,and the wideband continuous radio frequency(RF) excitation is a band-limited(0.68-3.40 GHz) white noise.Furthermore,the influences of some experimental parameters,such as relaxation time,applied magnetic field strength and wideband continuous RF power,on the WB-CW-NMR signal are analyzed briefly.Finally,a multi-channel system framework for transmitting and receiving ultra wideband signals is proposed,and the basic requirements of this experimental system are discussed.Meanwhile,the amplitude of the NMR signal,the level of noise and RF interference in WB-CW-NMR experiments are estimated,and a preliminary adaptive cancellation plan is given for detecting WB-CW-NMR signal from large background interference.     

Bethe ansatz calculations for the eight-vertex model on a finite strip

Bethe ansatz equations for the eigenvalues of the transfer matrix of the eight-vertex model are solved numerically to yield mass gap data on infinitely long strips of up to 512 sites in width. The finite-size corrections, at criticality, to the free energy per site and polarization gap are found to be in agreement with recent studies of theXXZ spin chain. The leading corrections to the finite-size scaling estimates of the critical line and thermal exponent are also found, providing an explanation of the poor convergence seen in earlier studies. Away from criticality, the linear scaling fields are derived exactly in the full parameter space of the spin system, allowing a thorough test of a recently proposed method of extracting linear scaling fields and related exponents from finite lattice data.     

Analytic Bethe ansatz for fundamental representations of Yangians

We study the analytic Bethe, ansatz in solvable vertex models associated with the YangianY(X _r ) or its quantum affine analogueU _q (X _r ⁽¹⁾ ) forX _r =B _r ,C _r andD _r . Eigenvalue formulas are proposed for the transfer matrices related to all the fundamental representations ofY(X _r ). Under the Bethe ansatz equation, we explicitly prove that they are pole-free, a crucial property in the ansatz. Conjectures are also given on higher representation cases by applying theT-system, the transfer matrix functional relations proposed recently. The eigenvalues are neatly described in terms of Yangian analogues of the semi-standard Young tableaux.     

Effective hamiltonian of Bloch electrons in a homogeneous electrical field

In order to construct a band mechanics of Bloch electrons in a homogeneous electrical field E with the interband interaction taken into account, a method of determining the exact single-band Hamiltonian $$H_q = \varepsilon _q^F (\kappa ) + Fi\frac{\partial }{{\partial \kappa }}$$ is proposed, where ε _q ^F (κ) is the renormalized (effective) electron dispersion law for R = 0 and the q-th Bloch band,F= ¦e¦·E. The function ε _q ^F (κ) is expressed in terms of the interband element coordinates as well as in terms of periodic solutions of the system of ordinary differential equations which degenerateinto a common Riccati equation in a two-band approximation. A solution of the system and the form of ε _q ^F (κ), in agreement with the Wanhier result, is found in the quasiclassical approximation.