271 research outputs found
Introduction to the Bethe ansatz I
The Bethe ansatz for the one-dimensional s=1/2 Heisenberg ferromagnet is
introduced at an elementary level. The presentation follows Bethe's original
work very closely. A detailed description and a complete classification of all
two-magnon scattering states and two-magnon bound states are given for finite
and infinite chains. The paper is designed as a tutorial for beginning graduate
students. It includes 10 problems for further study.Comment: 8 pages, 4 figure
Computational probes of collective excitations in low-dimensional magnetism
The investigation of the dynamics of quantum many-body systems is a concerted
effort involving computational studies of mathematical models and experimental
studies of material samples. Some commonalities of the two tracks of
investigation are discussed in the context of the quantum spin dynamics of
low-dimensional magnetic systems, in particular spin chains. The study of
quantum fluctuations in such systems at equilibrium amounts to exploring the
spectrum of collective excitations and the rate at which they are excited from
the ground state by dynamical variables of interest. The exact results obtained
via Bethe ansatz or algebraic analysis (quantum groups) for a select class of
completely integrable models can be used as benchmarks for numerical studies of
nonintegrable models, for which computational access to the spectrum of
collective excitations is limited.Comment: 22 pages. Talk given at the 7th Summer School on Neutron Scattering,
Zuoz Switzerland, August 199
Quasiparticles in the XXZ model
The coordinate Bethe ansatz solutions of the XXZ model for a one-dimensional
spin-1/2 chain are analyzed with focus on the statistical properties of the
constituent quasiparticles. Emphasis is given to the special cases known as XX,
XXX, and Ising models, where considerable simplifications occur. The XXZ
spectrum can be generated from separate pseudovacua as configurations of sets
of quasiparticles with different exclusion statistics. These sets are
complementary in the sense that the pseudovacuum of one set contains the
maximum number of particles from the other set. The Bethe ansatz string
solutions of the XXX model evolve differently in the planar and axial regimes.
In the Ising limit they become ferromagnetic domains with integer-valued
exclusion statistics. In the XX limit they brake apart into hard-core bosons
with (effectively) fermionic statistics. Two sets of quasiparticles with spin
1/2 and fractional statistics are distinguished, where one set (spinons)
generates the XXZ spectrum from the unique, critical ground state realized in
the planar regime, and the other set (solitons) generates the same spectrum
from the twofold, antiferromagnetically ordered ground state realized in the
axial regime. In the Ising limit, the solitons become antiferromagnetic domain
walls.Comment: 6 figure
Introduction to the Bethe Ansatz III
Having introduced the magnon in part I and the spinon in part II as the
relevant quasi-particles for the interpretation of the spectrum of low-lying
excitations in the one-dimensional (1D) s=1/2 Heisenberg ferromagnet and
antiferromagnet, respectively, we now study the low-lying excitations of the
Heisenberg antiferromagnet in a magnetic field and interpret these collective
states as composites of quasi-particles from a different species. We employ the
Bethe ansatz to calculate matrix elements and show how the results of such a
calculation can be used to predict lineshapes for neutron scattering
experiments on quasi-1D antiferromagnetic compounds. The paper is designed as a
tutorial for beginning graduate students. It includes 11 problems for further
study.Comment: 11 page
Transition rates via Bethe ansatz for the spin-1/2 Heisenberg chain
We use the exact determinantal representation derived by Kitanine, Maillet,
and Terras for matrix elements of local spin operators between Bethe wave
functions of the one-dimensional s=1/2 Heisenberg model to calculate and
numerically evaluate transition rates pertaining to dynamic spin structure
factors. For real solutions z_1,...,z_r of the Bethe ansatz equations, the size
of the determinants is of order r x r. We present applications to the
zero-temperature spin fluctuations parallel and perpendicular to an external
magnetic field.Comment: 4 pages, 4 figures and LaTeX-svjour clas
Monodisperse hard rods in external potentials
We consider linear arrays of cells of volume populated by
monodisperse rods of size , , subject
to hardcore exclusion interaction. Each rod experiences a position-dependent
external potential. In one application we also examine effects of contact
forces between rods. We employ two distinct methods of exact analysis with
complementary strengths and different limits of spatial resolution to calculate
profiles of pressure and density on mesoscopic and microscopic length scales at
thermal equilibrium. One method uses density functionals and the other
statistically interacting vacancy particles. The applications worked out
include gravity, power-law traps, and hard walls. We identify oscillations in
the profiles on a microscopic length scale and show how they are systematically
averaged out on a well-defined mesoscopic length scale to establish full
consistency between the two approaches. The continuum limit, realized as
, at nonzero and finite , connects our highest-resolution results with known exact results
for monodisperse rods in a continuum. We also compare the pressure profiles
obtained from density functionals with the average microscopic pressure
profiles derived from the pair distribution function.Comment: 17 pages, 14 figure
Spinon excitations in the XX chain: spectra, transition rates, observability
The exact one-to-one mapping between (spinless) Jordan-Wigner lattice
fermions and (spin-1/2) spinons is established for all eigenstates of the
one-dimensional s = 1=2 XX model on a lattice with an even or odd number N of
lattice sites and periodic boundary conditions. Exact product formulas for the
transition rates derived via Bethe ansatz are used to calculate asymptotic
expressions of the 2-spinon and 4-spinon parts (for large even N) as well as of
the 1-spinon and 3-spinon parts (for large odd N) of the dynamic spin structure
factors. The observability of these spectral contributions is assessed for
finite and infinite N.Comment: 19 pages, 10 figure
Disks in a narrow channel jammed by gravity and centrifuge: profiles of pressure, mass density and entropy density
This work investigates jammed granular matter under conditions that produce
heterogeneous mass distributions on a mesoscopic scale. We consider a system of
identical disks that are confined to a narrow channel, open at one end and
closed off at the other end. The disks are jammed by the local pressure in a
gravitational field or centrifuge. All surfaces are hard and frictionless. We
calculate the profiles of pressure, mass density, and entropy density on a
mesoscopic length scale under the assumption that the jammed states are
produced by random agitations of uniform intensity along the channel. These
profiles exhibit trends and features governed by the balancing of
position-dependent forces and potential energies. The analysis employs a method
of configurational statistics that uses interlinking two-disk tiles as the
fundamental degrees of freedom. Configurational statistics weighs the
probabilities of tiles according to competing potential energies associated
with gravity and centrifugation. Amendments account for the effects of the
marginal stability of some tiles due to competing forces.Comment: 20 pages, 10 figure
Statistically interacting vacancy particles
The equilibrium statistical mechanics of one-dimensional lattice gases with
interactions of arbitrary range and shape between first-neighbor atoms is
solved exactly on the basis of statistically interacting vacancy particles. Two
sets of vacancy particles are considered. In one set all vacancies are of
one-cell size. In the other set the sizes of vacancy particles match the
separation between atoms. Explicit expressions are obtained for the Gibbs free
energy and the distribution of spaces between atoms at thermal equilibrium.
Applications to various types of interaction potentials are discussed,
including long-range potentials that give rise to phase transitions. Extensions
to hard rod systems are straightforward and are shown to agree with existing
results for lattice models and their continuum limits.Comment: 10 pages, 5 figure
On the theory of microwave absorption by the spin-1/2 Heisenberg-Ising magnet
We analyze the problem of microwave absorption by the Heisenberg-Ising magnet
in terms of shifted moments of the imaginary part of the dynamical
susceptibility. When both, the Zeeman field and the wave vector of the incident
microwave, are parallel to the anisotropy axis, the first four moments
determine the shift of the resonance frequency and the line width in a
situation where the frequency is varied for fixed Zeeman field. For the
one-dimensional model we can calculate the moments exactly. This provides exact
data for the resonance shift and the line width at arbitrary temperatures and
magnetic fields. In current ESR experiments the Zeeman field is varied for
fixed frequency. We show how in this situation the moments give perturbative
results for the resonance shift and for the integrated intensity at small
anisotropy as well as an explicit formula connecting the line width with the
anisotropy parameter in the high-temperature limit.Comment: 4 page
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