Morning lectures
Ed Corrigan
(University of York, Gran Bretaña)
An introduction to integrable classical and quantum field theory in two dimensions
The main example will be the sine-Gordon model, which will be used to illustrate features that are key to the study of many other models.
For example, the classical and quantum spectrum, conservation laws, scattering (including a brief look at the Yang-Baxter equation) and bound states.
If time allows there will be a brief exploration of other topics especially the influence of boundaries.
Presentación de Prof Ed Corrigan
Andreas Klümper
(Bergische Universität Wuppertal, Alemania)
Thermodynamics of spin chains
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Thermodynamical Bethe Ansatz (TBA)
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Lattice Path Integral Formulation and Quantum Transfer Matrix (QTM)
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Fusion Algebra: T- and Y-Systems
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Non-Linear Integral Equations
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Higher Rank Models and Continuum Systems
The main topic of my lectures will be the finite temperature physics of
integrable 1d quantum systems.
The discussion will be rather comprehensive and detailed for the case of the spin-1/2 Heisenberg chain, but not restricted to this.
In the
first lecture I will discuss the combinatorial TBA method
introduced by Yang and Yang for the single component Bose gas and generalized
by Gaudin and Takahashi to the Heisenberg chain.
Lectures 2 and 3 are devoted
to an algebraic approach to the thermodynamical properties of integrable
quantum chains.
The finite temperature systems are mapped to classical models on 2d lattices.
The partition function is obtained from just the largest eigenvalue of the column-to-column transfer matrix, also called the
quantum transfer matrix which acts in an infinite dimensional space.
A hierarchy of transfer matrices is derived by the fusion method, and algebraic relations of various type are established.
Lecture 4: By use of the so-called T- and
Y-systems
(i) the TBA equations are derived rigorously,
(ii) alternative, especially finite sets of non-linear integral equations are derived,
(iii) correlation lengths/mass gaps are calculated.
In
Lecture 5 generalizations to higher rank models and continuum limits like Bose gases are discussed.
Vladimir Korepin
(State University of New York at Stony Brook, USA)
Quantum non-linear Schrödinger equation
Short communications
- Primitivo Acosta-Humánez (Universidad del Norte)
This talk will be concerning to a Differential Galois (Picard-Vessiot) Theory point of view of the Supersymmetric Quantum Mechanics.
The main object is the non-relativistic stationary Schrödinger equation, where are introduced the concepts of Algebraic Spectrum and Hamiltonian Algebrization.
Using the Kovacic's Algorithm and the Hamiltonian Algebrization are analyzed Darboux transformations, Crum iterations and supersymmetric quantum mechanics, including their Algebrized Versions from a Galoisian approach.
In particular are obtained the ground state, eigenvalues, eigenfunctions, the differential Galois groups and eigenrings of some Schrödinger equations with potentials such as exactly solvable, quasi-exactly solvable and shape invariant potentials.
Finally is introduced one methodology to find Algebraically Solvable and Algebraically Quasi-Solvable Potentials.
It consists in to apply the Hamiltonian Algebrization, as inverse process, over families of second order linear differential equations integrables in the Picard-Vessiot sense for a set of parameters, in particular, involving orthogonal polynomials and special functions.
Presentación de Primitivo
- Andrés Ángel (Universidad de Los Andes)
In this talk I will report on work of Givental and Coates interpreting intersection numbers of moduli spaces of stable curves as Gromov-Witten invariants with values in complex cobordisms.
This interpretation allows us to see the problem as a quantum version of the Hirzebruch-Riemann-Roch theory corresponding to the Chern-Dold character from the cobordisms to cohomology.
- David Blázquez Sanz (Universidad Sergio Arboleda)
We study a Sturm-Liouville type eigenvalue problem for second-order
differential equations on the infinite interval. Here the
eigenfunctions are nonzero solutions exponentially decaying at
infinity. We prove that at any discrete eigenvalue the differential
equations are integrable in the setting of differential Galois theory
under general assumptions. Our result is illustrated with three
examples for a stationary Schrödinger equation having a generalized
Hulthen potential; a linear stability equation for a traveling front
in the Allen-Cahn equation; and an eigenvalue problem related to the
Lame equation. This talk is based in a collaborative research with K.
Yagasaki, from Niigata University.
Presentación de David
- Alexander Cardona (Universidad de Los Andes)
En esta charla describiremos los sistemas integrables clásicos en términos de estructuras de Dirac torcidas sobre variedades, también describiremos un modelo geométrico de cuantización para las álgebras de Poisson de funciones admisibles asociadas con tales tipos de estructura.
- Otaivin Martínez (Universidad Industrial de Santander)
La idea de esta charla es exponer, a modo de sketch, la demostración del teorema de Levinson, el cual asegura que al menos un tercio de los ceros no triviales de la función zeta de Riemann tiene parte real 1/2.
Además de mostrar las mejoras que se han logrado para el estimativo, como lo es el teorema de Young:
Al menos el 40% de los ceros no triviales tiene parte real 1/2.
- Servio Pérez (Universidad del Cauca)
El método de matriz densidad y Grupo de renormalización (DMRG) se ha convertido en un poderoso método numérico que se puede aplicar a sistemas de baja dimensionalidad, fermiónicos y bosónicos fuertemente correlacionados.
Este permite un cálculo muy preciso de las propiedades estáticas, dinámicas y termodinámicas.
Su campo de aplicación se ha extendido más allá de la Materia Condensada, y es
utilizado con éxito en mecánica estadística y también en física de altas Energías.
En cada caso se revisan los aspectos principales del método y se comentan algunas de
las aplicaciones más relevantes, a fin de dar una visión general sobre el alcance y las
posibilidades de DMRG y mencionar las extensiones más importantes del método;
como el cálculo de las propiedades dinámicas, la aplicación a los sistemas clásicos, la
inclusión de la temperatura, fonónes y desorden, teoría de campos, propiedades que
dependen del tiempo y el cálculo ab initio de los estadoséelectrónicos en las moléculas.
Presentación de Servio (pdf)
(pptx)
- Otto Rendón (Universidad de Carabobo, Venezuela)
We have studied scattering of two identical particles [1, 2].
The device consists of one input lead and output leads attached to a quantum dot.
The movement of two fermions is without interaction in the leads but exist repulsion U when the dot is doubly occupied.
In this device, one may avoid both single and triple processes with putting certain values in the leads energy.
Within this model We have done formal derivations for transition amplitudes, in terms of the T-matrix, to second orders in the coupling to the dot
Vcoupling
and consider a finite lead bandwidth, V.
In fact, we put
Vcoupling << V.
At T = 0 K, the devices filters singlet entangled pairs if U ≠ 0.
However, here the postselection is not used.
Moreover, resonance structure for the singlet transition amplitude is studied as function of energy difference between the input lead and the dot single-particle state.
In U = 0, there isn't tunneling and the two-fermion scattering matrix tend to the obtained by custom scattering matrix theory for noninteracting electrons.
The first part of this work has been done in collaboration with Gladys León and Ernesto Medina D.
[1] Oliver W. D., F. Yamaguchi, and Y. Yamamoto, Phys. Rev. Lett. vol. 63, 037901 (2002).
[2] G. León, O. Rendón, H. M. Pastawski, V. Mujica, and E. Medina, Europhys. Lett. vol. 66, 624 (2004).
[3] M. Büttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B vol. 31, 6207(1985).
Presentación de Otto
- Juliana Restrepo (Universidad Antonio Nariño, sede Medellin)
In this seminar, I will address a few fundamental questions related to the decoherence and relaxation of quantum systems.
In particular I will present the study of the full evolution of a two level system or qubit coupled to a bath composed of non-interacting and interacting electrons and I will focus on the case where the bath exhibits long range order.
The interest of this problem is related to quantum computation because in order to construct an efficient quantum computer one needs to understand the effect of the surroundings on the basic building block, i.e., a quantum bit or qubit.
Alternatively, these problems are also interesting from the viewpoint of using the two level system as a probe of the physics of the bath.
Presentación de Juliana
- Andrés Reyes (Universidad de Los Andes)
In this talk I will present an approach to entanglement which is based on the Gelfand-Naimark-Segal (GNS) construction.
The conventional approach to the emergence of mixed from pure states based on taking partial traces is replaced by the more general notion of the restriction of a state to a subalgebra.
For bipartite systems of nonidentical particles, this approach reproduces the standard results.
But it also very naturally overcomes the limitations of the usual treatment of systems of identical particles.
The content of this talk is based on joint work with A.P. Balchandran, T.R. Govindarajanand A.R de Queiroz.
Presentación de Andrés
- Miguel Rodríguez (Universidad Pedagógica Nacional)
La charla se centrará en como la métrica de Gödel genera líneas de mundo cerradas, siendo estas las causantes de un universo retro causal teóricamente; siendo una solución exacta a la ecuación de campo de Einstein, además seria interesante el mostrar el comportamiento de la misma y que implicaciones físicas conllevaría un universo al cual solo seria aplicarle la propiedad de rotación a la materia.
- Gabriel Téllez (Universidad de Los Andes)
The two-dimensional one-component plasma (2dOCP) is a system of
N
mobile particles of the same charge
q on a surface with a
neutralising background.
The Boltzmann factor of the 2dOCP at temperature
T can be expressed as a Vandermonde determinant to the
power
Γ = q2/(kBT). Recent advances in the theory of
symmetric and anti-symmetric Jack polymonials provide an efficient way
to expand this power of the Vandermonde in their monomial basis,
allowing the computation of several thermodynamic and structural
properties of the 2dOCP for
N values up to 14 and
Γ equal to
4, 6 and 8.
In this work, we present some applications of this
formalism.
Presentación de Gabriel
- Emilio Torres (Universidad de Los Andes)
In this talk, I'll adress the relation between coulomb gases, and in particular the two component plasma (a system with two kind of charges of equal magnitude and opposite sign), and some classically integrable field theories, such as the sine-Gordon model.
For the case of the TCP, I'll show how to transform the classical expression of the partition function to an expression involving the sine-Gordon action and I'll show some results derived from an explicit calculation of the correlation functions of this theory.
Presentación de Emilio
