Meeting on Large Quantum Systems,
Warwick University, Coventry,
June 11-15,
2007

Stefan
ADAMS

Probabilistic approaches to Bose-Einstein condensation

Interacting many-particle systems of Bosons have been studied widely in the physics and mathematics literature. They pose hard mathematical problems, which

have been attacked by a variety of methods. We outline how a stochastic analysis of interacting Brownian motions via large deviations techniques leads to effective descriptions of large systems of Bosons and how it raises new mathematical questions. The starting point are systems of Brownian bridges under certain initial distributions, which are linked via Fenyman-Kac formulae to traces of quantum mechanical density matrices. We study the large particle limits of random measures like the mean of occupation or the mean of path measures. We will outline how the specific symmetry for systems of Bosons can be implemented in the stochastic model. With this approach we can prove the Gross-Pitaevskii formula, a variational formula for the ground state of Bosons. Furthermore, we will discuss an empirical path measure phase transition and its interpretation as Bose-Einstein condensation.

Walter ASCHBACHER

Landauer-Büttiker formulas in systems of independent fermions

Using the scattering approach to the construction of non-equilibrium steady states in the sense of Ruelle, we establish Landauer-Büttiker type formulas in systems of independent fermions under general conditions. This is a joint work with V. Jaksic, Y. Pautrat, and C.-A. Pillet.

Laurent BRUNEAU

Repeated Interaction Systems II: Random

(Part I given by Marco Merkli)

In this second talk on repeated interaction systems, we consider the case where the various interactions are not identical but taken at random. This randomness may have various origins: the interaction time, the elements of the chain, the coupling between the system S and the various elements. In a first part we show how to extend the approach of identical interactions to this situation, and reduce the study of the large time limits to an infinite product of iid random matrices satisfying two basic properties: for some suitable norm they are contractions and they have a common invariant vector. We then study the convergence of such infinite products of random matrices, and show how they translate into the large time behaviour of repeated interaction systems (existence of an asymptotic state and its thermodynamical properties).

Horia CORNEAN

Adiabatically switched-on electrical bias in continuous systems, and the Landauer-Buttiker formula

This is joint work with P. Duclos, G. Nenciu and R. Purice. Consider a three dimensional system which looks like a cross-connected pipe system, i.e. a small sample coupled to a finite number of leads. We investigate the current which runs through this system, in the linear response regime, when we adiabatically turn on an electrical bias between leads. The main technical tool is the use of the finite volume regularization, which allows us to define the current coming out of a lead as the time derivative of its charge. We then prove that in virtually all physically interesting situations, the conductivity tensor is given by a Landauer-Buttiker type formula. Finally we discuss the connection with the Jaksic-Pillet-Ruelle approach.

Tony DORLAS

The HSW theorem for a chennal with long-term memory

Patrik FERRARI

A stochastic growth of an interface on a plane: fluctuations for the KPZ universality class

For some classes of growth models universal behaviors appear for large growth time t. We consider a model in the Kardar-Parisi-Zhang universality class in 1+1 dimensions. For large t, the surface is described by the Airy_2 process or Airy_1 process depending on classes of initial conditions. The transition between the two processes has been recently analyzed too.

Ostap HRYNIV

Birth of the critical droplet and related topics

Phase transition in various models of statistical mechanics often leads to phase coexistence. We shall describe typical configurations in the 2D Ising ferromagnet at the edge of the phase coexistence region, and compare our results with the "birth of the giant component" phenomenon in the random graphs theory.

(Partially based on a joint work with D.Ioffe and R.Kotecky.)

Marco MERKLI

Repeated Interaction Systems I: Deterministic

(Part II given by Laurent Bruneau)

The goal of this talk is to explain what repeated interaction systems are and why they are important, to set up a suitable mathematical framework, and to present results on the dynamics of such systems. In this talk we focus on deterministic systems, while a second talk is devoted to random repeated interaction systems.

A quantum system S interacts successively with elements E of a chain of identical independent quantum subsystems. Each interaction lasts for a duration T and is governed by a fixed coupling between S and E. We show that the system approaches a "repeated interaction asymptotic state" in the limit of large times. Our approach is constructive, it allows us to examine the thermodynamic properties of the asymptotic state, and to analyze it by rigorous perturbation theory (in the interaction strength). Our method takes advantage of the structural properties of the systems in question and involves a generalization of some recent ideas from the analysis of open quantum systems far from equilibrium.

Bruno NACHTERGAELE

A multi-dimensional Lieb-Schultz-Mattis theorem

For a large class of finite-range quantum spin models with half-integer spins, we prove that uniqueness of the ground state implies the existence of a low-lying excited state. For systems of linear size L, of arbitrary finite dimension, we obtain an upper bound on the excitation energy (i.e., the gap above the ground state) of the form C log(L)/L. This result is a multi-dimensional version of the Lieb-Schultz-Mattis theorem and provides a rigorous proof of a recent result by Hastings. We will discuss the main ideas in our proof.

Robert SEIRINGER

The Free Energy of Dilute Quantum Gases

We discuss bounds on the free energy of homogeneous Bose and Fermi gases at non-zero temperature. In the dilute regime, the leading order correction compared to an ideal quantum gas depends on the particle interaction only through the scattering length. Moreover, in the Bose case it depends non-trivially on temperature through the critical density for Bose-Einstein condensation. Some of the key ingredients in the proof are the use of coherent states to deal with the condensation, as well as new correlation estimates relying on the monotonicity of the relative entropy under partial traces.

Valeriy SLASTIKOV

Fast rotating Bose-Einstein condensates: variational approach.

András SUTO

Remarks on the variational quasiparticle theory for bosons

Forian THEIL

Towards a Mathematical Justification of Kinetic Theory

Jakob YNGVASON

Rotating Bose Gases

The talk will give a brief overview of the mathematical physics of rotating Bose gases and present some recent results on gases in rapid rotation.

Valentin ZAGREBNOV

About Variational Principle for a Pair Hamiltonian Boson Model

We give a two parameter variational formula for the grand-canonical pressure of the Pair Boson Hamiltonian model. By using the Approximating Hamiltonian Method we provide a rigorous proof of this variational principle. The Euler-Lagrange equations allows a detailed study of Bose-Einstein condensation. In particular, this concerns the sequence of phase transitions in the Pair-Boson (BCS) model and the conditions for the coexistence of the generalized Bose-condensation and the condensation of boson pairs.

Miloš ZAHRADNIK

Combinatorial approach to cluster (Mayer) expansions

I will present a selfcontained introduction to this method, which is really a "theory of general exponential function of many variables". Applications range from the elementary expansion of log(1+x), or more generally of det(J-A), to complicated, possibly also non absolutely convergent series for logarithms of polynomials of many variables. These describe partition functions of some spin models with continuous symmetry (perturbed zero mass gaussians).

Most of the presented material is well known to specialists and many crucial features of these expansions, like the marvellous simplifications obtained after the cancellations of terms indexed by different graphs (especially for the hard repulsion case), positivity of the final coefficients (if they are written properly) or K-P type bounds (which I will present as equations) were stressed recently by other authors. I will mention also interesting relations to exactly computable models (determinants of lattice Laplacians, Kramers - Wannier- Onsager solution) through a suitable, (possibly) new method of resummation of the cluster expansion terms. The method is especially useful for models where incompatibility of polymers is determined by the nonempty intersection of their "supports", and polymers itself are given as closed paths (possibly with "decorations" influencing the compatibility) in d-dimensional lattice, d>2.

Probabilistic approaches to Bose-Einstein condensation

Interacting many-particle systems of Bosons have been studied widely in the physics and mathematics literature. They pose hard mathematical problems, which

have been attacked by a variety of methods. We outline how a stochastic analysis of interacting Brownian motions via large deviations techniques leads to effective descriptions of large systems of Bosons and how it raises new mathematical questions. The starting point are systems of Brownian bridges under certain initial distributions, which are linked via Fenyman-Kac formulae to traces of quantum mechanical density matrices. We study the large particle limits of random measures like the mean of occupation or the mean of path measures. We will outline how the specific symmetry for systems of Bosons can be implemented in the stochastic model. With this approach we can prove the Gross-Pitaevskii formula, a variational formula for the ground state of Bosons. Furthermore, we will discuss an empirical path measure phase transition and its interpretation as Bose-Einstein condensation.

Walter ASCHBACHER

Landauer-Büttiker formulas in systems of independent fermions

Using the scattering approach to the construction of non-equilibrium steady states in the sense of Ruelle, we establish Landauer-Büttiker type formulas in systems of independent fermions under general conditions. This is a joint work with V. Jaksic, Y. Pautrat, and C.-A. Pillet.

Laurent BRUNEAU

Repeated Interaction Systems II: Random

(Part I given by Marco Merkli)

In this second talk on repeated interaction systems, we consider the case where the various interactions are not identical but taken at random. This randomness may have various origins: the interaction time, the elements of the chain, the coupling between the system S and the various elements. In a first part we show how to extend the approach of identical interactions to this situation, and reduce the study of the large time limits to an infinite product of iid random matrices satisfying two basic properties: for some suitable norm they are contractions and they have a common invariant vector. We then study the convergence of such infinite products of random matrices, and show how they translate into the large time behaviour of repeated interaction systems (existence of an asymptotic state and its thermodynamical properties).

Horia CORNEAN

Adiabatically switched-on electrical bias in continuous systems, and the Landauer-Buttiker formula

This is joint work with P. Duclos, G. Nenciu and R. Purice. Consider a three dimensional system which looks like a cross-connected pipe system, i.e. a small sample coupled to a finite number of leads. We investigate the current which runs through this system, in the linear response regime, when we adiabatically turn on an electrical bias between leads. The main technical tool is the use of the finite volume regularization, which allows us to define the current coming out of a lead as the time derivative of its charge. We then prove that in virtually all physically interesting situations, the conductivity tensor is given by a Landauer-Buttiker type formula. Finally we discuss the connection with the Jaksic-Pillet-Ruelle approach.

Tony DORLAS

The HSW theorem for a chennal with long-term memory

Patrik FERRARI

A stochastic growth of an interface on a plane: fluctuations for the KPZ universality class

For some classes of growth models universal behaviors appear for large growth time t. We consider a model in the Kardar-Parisi-Zhang universality class in 1+1 dimensions. For large t, the surface is described by the Airy_2 process or Airy_1 process depending on classes of initial conditions. The transition between the two processes has been recently analyzed too.

Ostap HRYNIV

Birth of the critical droplet and related topics

Phase transition in various models of statistical mechanics often leads to phase coexistence. We shall describe typical configurations in the 2D Ising ferromagnet at the edge of the phase coexistence region, and compare our results with the "birth of the giant component" phenomenon in the random graphs theory.

(Partially based on a joint work with D.Ioffe and R.Kotecky.)

Marco MERKLI

Repeated Interaction Systems I: Deterministic

(Part II given by Laurent Bruneau)

The goal of this talk is to explain what repeated interaction systems are and why they are important, to set up a suitable mathematical framework, and to present results on the dynamics of such systems. In this talk we focus on deterministic systems, while a second talk is devoted to random repeated interaction systems.

A quantum system S interacts successively with elements E of a chain of identical independent quantum subsystems. Each interaction lasts for a duration T and is governed by a fixed coupling between S and E. We show that the system approaches a "repeated interaction asymptotic state" in the limit of large times. Our approach is constructive, it allows us to examine the thermodynamic properties of the asymptotic state, and to analyze it by rigorous perturbation theory (in the interaction strength). Our method takes advantage of the structural properties of the systems in question and involves a generalization of some recent ideas from the analysis of open quantum systems far from equilibrium.

Bruno NACHTERGAELE

A multi-dimensional Lieb-Schultz-Mattis theorem

For a large class of finite-range quantum spin models with half-integer spins, we prove that uniqueness of the ground state implies the existence of a low-lying excited state. For systems of linear size L, of arbitrary finite dimension, we obtain an upper bound on the excitation energy (i.e., the gap above the ground state) of the form C log(L)/L. This result is a multi-dimensional version of the Lieb-Schultz-Mattis theorem and provides a rigorous proof of a recent result by Hastings. We will discuss the main ideas in our proof.

Robert SEIRINGER

The Free Energy of Dilute Quantum Gases

We discuss bounds on the free energy of homogeneous Bose and Fermi gases at non-zero temperature. In the dilute regime, the leading order correction compared to an ideal quantum gas depends on the particle interaction only through the scattering length. Moreover, in the Bose case it depends non-trivially on temperature through the critical density for Bose-Einstein condensation. Some of the key ingredients in the proof are the use of coherent states to deal with the condensation, as well as new correlation estimates relying on the monotonicity of the relative entropy under partial traces.

Valeriy SLASTIKOV

Fast rotating Bose-Einstein condensates: variational approach.

András SUTO

Remarks on the variational quasiparticle theory for bosons

Forian THEIL

Towards a Mathematical Justification of Kinetic Theory

Jakob YNGVASON

Rotating Bose Gases

The talk will give a brief overview of the mathematical physics of rotating Bose gases and present some recent results on gases in rapid rotation.

Valentin ZAGREBNOV

About Variational Principle for a Pair Hamiltonian Boson Model

We give a two parameter variational formula for the grand-canonical pressure of the Pair Boson Hamiltonian model. By using the Approximating Hamiltonian Method we provide a rigorous proof of this variational principle. The Euler-Lagrange equations allows a detailed study of Bose-Einstein condensation. In particular, this concerns the sequence of phase transitions in the Pair-Boson (BCS) model and the conditions for the coexistence of the generalized Bose-condensation and the condensation of boson pairs.

Miloš ZAHRADNIK

Combinatorial approach to cluster (Mayer) expansions

I will present a selfcontained introduction to this method, which is really a "theory of general exponential function of many variables". Applications range from the elementary expansion of log(1+x), or more generally of det(J-A), to complicated, possibly also non absolutely convergent series for logarithms of polynomials of many variables. These describe partition functions of some spin models with continuous symmetry (perturbed zero mass gaussians).

Most of the presented material is well known to specialists and many crucial features of these expansions, like the marvellous simplifications obtained after the cancellations of terms indexed by different graphs (especially for the hard repulsion case), positivity of the final coefficients (if they are written properly) or K-P type bounds (which I will present as equations) were stressed recently by other authors. I will mention also interesting relations to exactly computable models (determinants of lattice Laplacians, Kramers - Wannier- Onsager solution) through a suitable, (possibly) new method of resummation of the cluster expansion terms. The method is especially useful for models where incompatibility of polymers is determined by the nonempty intersection of their "supports", and polymers itself are given as closed paths (possibly with "decorations" influencing the compatibility) in d-dimensional lattice, d>2.