207 research outputs found
The Theorem of Jentzsch--Szeg\H{o} on an analytic curve. Application to the irreducibility of truncations of power series
The theorem of Jentzsch--Szeg\H{o} describes the limit measure of a sequence
of discrete measures associated to the zeroes of a sequence of polynomials in
one variable. Following the presentation of this result by Andrievskii and
Blatt in their book, we extend this theorem to compact Riemann surfaces, then
to analytic curves over an ultrametric field. The particular case of the
projective line over an ultrametric field gives as corollaries information
about the irreducibility of the truncations of a power series in one variable.Comment: 16 pages; the application to irreducibility and the final example
have been correcte
Quantum Clifford-Hopf Algebras for Even Dimensions
In this paper we study the quantum Clifford-Hopf algebras
for even dimensions and obtain their intertwiner matrices, which are
elliptic solutions to the Yang- Baxter equation. In the trigonometric limit of
these new algebras we find the possibility to connect with extended
supersymmetry. We also analyze the corresponding spin chain hamiltonian, which
leads to Suzuki's generalized model.Comment: 12 pages, LaTeX, IMAFF-12/93 (final version to be published, 2
uuencoded figures added
Continued Fractions and Fermionic Representations for Characters of M(p,p') minimal models
We present fermionic sum representations of the characters
of the minimal models for all relatively prime
integers for some allowed values of and . Our starting point is
binomial (q-binomial) identities derived from a truncation of the state
counting equations of the XXZ spin chain of anisotropy
. We use the Takahashi-Suzuki method to express
the allowed values of (and ) in terms of the continued fraction
decomposition of (and ) where stands for
the fractional part of These values are, in fact, the dimensions of the
hermitian irreducible representations of (and )
with (and We also establish the duality relation and discuss the action of the Andrews-Bailey transformation in the
space of minimal models. Many new identities of the Rogers-Ramanujan type are
presented.Comment: Several references, one further explicit result and several
discussion remarks adde
Singular responses of spin-incoherent Luttinger liquids
When a local potential changes abruptly in time, an electron gas shifts to a
new state which at long times is orthogonal to the one in the absence of the
local potential. This is known as Anderson's orthogonality catastrophe and it
is relevant for the so-called X-ray edge or Fermi edge singularity, and for
tunneling into an interacting one dimensional system of fermions. It often
happens that the finite frequency response of the photon absorption or the
tunneling density of states exhibits a singular behavior as a function of
frequency: where is a
threshold frequency and is an exponent characterizing the singular
response. In this paper singular responses of spin-incoherent Luttinger liquids
are reviewed. Such responses most often do not fall into the familiar form
above, but instead typically exhibit logarithmic corrections and display a much
higher universality in terms of the microscopic interactions in the theory.
Specific predictions are made, the current experimental situation is
summarized, and key outstanding theoretical issues related to spin-incoherent
Luttinger liquids are highlighted.Comment: 21 pages, 3 figures. Invited Topical Review Articl
Exceptional structure of the dilute A model: E and E Rogers--Ramanujan identities
The dilute A lattice model in regime 2 is in the universality class of
the Ising model in a magnetic field. Here we establish directly the existence
of an E structure in the dilute A model in this regime by expressing
the 1-dimensional configuration sums in terms of fermionic sums which
explicitly involve the E root system. In the thermodynamic limit, these
polynomial identities yield a proof of the E Rogers--Ramanujan identity
recently conjectured by Kedem {\em et al}.
The polynomial identities also apply to regime 3, which is obtained by
transforming the modular parameter by . In this case we find an
A_1\times\mbox{E}_7 structure and prove a Rogers--Ramanujan identity of
A_1\times\mbox{E}_7 type. Finally, in the critical limit, we give
some intriguing expressions for the number of -step paths on the A
Dynkin diagram with tadpoles in terms of the E Cartan matrix. All our
findings confirm the E and E structure of the dilute A model found
recently by means of the thermodynamic Bethe Ansatz.Comment: 9 pages, 1 postscript figur
Symmetry-protected phases for measurement-based quantum computation
Ground states of spin lattices can serve as a resource for measurement-based
quantum computation. Ideally, the ability to perform quantum gates via
measurements on such states would be insensitive to small variations in the
Hamiltonian. Here, we describe a class of symmetry-protected topological orders
in one-dimensional systems, any one of which ensures the perfect operation of
the identity gate. As a result, measurement-based quantum gates can be a robust
property of an entire phase in a quantum spin lattice, when protected by an
appropriate symmetry.Comment: 5 pages, 1 figure, comments welcome; v2 fixed minor typographic
errors; v3 published versio
Local height probabilities in a composite Andrews-Baxter-Forrester model
We study the local height probabilities in a composite height model, derived
from the restricted solid-on-solid model introduced by Andrews, Baxter and
Forrester, and their connection with conformal field theory characters. The
obtained conformal field theories also describe the critical behavior of the
model at two different critical points. In addition, at criticality, the model
is equivalent to a one-dimensional chain of anyons, subject to competing two-
and three-body interactions. The anyonic-chain interpretation provided the
original motivation to introduce the composite height model, and by obtaining
the critical behaviour of the composite height model, the critical behaviour of
the anyonic chains is established as well. Depending on the overall sign of the
hamiltonian, this critical behaviour is described by a diagonal coset-model,
generalizing the minimal models for one sign, and by Fateev-Zamolodchikov
parafermions for the other.Comment: 34 pages, 5 figures; v2: expanded introduction, references added and
other minor change
Motivic Serre invariants, ramification, and the analytic Milnor fiber
We show how formal and rigid geometry can be used in the theory of complex
singularities, and in particular in the study of the Milnor fibration and the
motivic zeta function. We introduce the so-called analytic Milnor fiber
associated to the germ of a morphism f from a smooth complex algebraic variety
X to the affine line. This analytic Milnor fiber is a smooth rigid variety over
the field of Laurent series C((t)). Its etale cohomology coincides with the
singular cohomology of the classical topological Milnor fiber of f; the
monodromy transformation is given by the Galois action. Moreover, the points on
the analytic Milnor fiber are closely related to the motivic zeta function of
f, and the arc space of X.
We show how the motivic zeta function can be recovered as some kind of Weil
zeta function of the formal completion of X along the special fiber of f, and
we establish a corresponding Grothendieck trace formula, which relates, in
particular, the rational points on the analytic Milnor fiber over finite
extensions of C((t)), to the Galois action on its etale cohomology.
The general observation is that the arithmetic properties of the analytic
Milnor fiber reflect the structure of the singularity of the germ f.Comment: Some minor errors corrected. The original publication is available at
http://www.springerlink.co
Fermionic representations for characters of M(3,t), M(4,5), M(5,6) and M(6,7) minimal models and related Rogers-Ramanujan type and dilogarithm identities
Characters and linear combinations of characters that admit a fermionic sum
representation as well as a factorized form are considered for some minimal
Virasoro models. As a consequence, various Rogers-Ramanujan type identities are
obtained. Dilogarithm identities producing corresponding effective central
charges and secondary effective central charges are derived. Several ways of
constructing more general fermionic representations are discussed.Comment: 14 pages, LaTex; minor correction
Superposition rules for higher-order systems and their applications
Superposition rules form a class of functions that describe general solutions
of systems of first-order ordinary differential equations in terms of generic
families of particular solutions and certain constants. In this work we extend
this notion and other related ones to systems of higher-order differential
equations and analyse their properties. Several results concerning the
existence of various types of superposition rules for higher-order systems are
proved and illustrated with examples extracted from the physics and mathematics
literature. In particular, two new superposition rules for second- and
third-order Kummer--Schwarz equations are derived.Comment: (v2) 33 pages, some typos corrected, added some references and minor
commentarie
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