Groupe de travail en matière condensée

The notion of operator resonances was introduced by Al.~Zamolodchikov (1991) within the framework of the conformal field theory. The resonances are related to logarithmic divergences of the perturbation integrals, and manifest themselves as poles in correlation functions as functions of conformal dimensions. I show that the higher integral of motion in the Liouville theory discovered by Al.~Zamolodchikov (2003) can be interpreted as resonance identities in the conformal perturbation expansion from the free field theory, and that the construction can be generalized to the sinh-Gordon theory, which is, in turn, a perturbation of the Liouville theory. I compare the results with the identities obtained (in collaboration with Y.~Pugai) for exact form factors of local operators.

Based on recently completed PhD research, we study the quantum nonlinear Schrödinger (QNLS) model, which describes the Lieb-Liniger system of quantum particles. We briefly review the two main methods used to study this model. In the quantum inverse scattering method (QISM), the algebraic Bethe ansatz expresses the QNLS wavefunction as a product of bosonic particle creation operators acting on the pseudovacuum. On the other hand, in the approach using the degenerate affine Hecke algebra (dAHA) of type A, Gutkin’s propagation operator (the intertwiner for two dAHA representations) produces a non-symmetric function which, upon symmetrization, also produces the QNLS wavefunction. We present an alternative, recursive, way of constructing this non-symmetric function, using a product of particle creation operators, thus explicitly connecting the QISM and the dAHA approach. Furthermore, some of the commutation relations encoded in the Yang-Baxter equation for the QNLS monodromy matrix are generalized to the non-symmetric case.

I will summarize recent results aimed to the construction of a S-matrix theory that interpolates between the exact S-matrix of the superstring world-sheet theory on AdS_(5) x S(5) and the S-matrix corresponding to its Pohlmeyer reduction. The construction makes use of the R-matrix of the q-deformed Hubbard model which depends on two coupling constants. It provides an explicit realization of the conjectured equivalence of the two theories, and realizes at the quantum level the fact that the two theories correspond to different limits of a one-parameter family of symplectic structures of the same classical integrable system.

Several combinatorial problems in physics, mathematics and computer science lead to a natural generalization of the partitions of integers? these are called higher-dimensional partitions and were first introduced by MacMahon. Two-dimensional or plane partitions have a nice generating function like the one due to Euler for usual partitions. It was shown in 1967 by Atkin et. al. that a similar generating function guessed by MacMahon for dimensions > 2 was wrong. One needs to take recourse to exact enumeration. It was also shown by Atkin et. al. that in order to enumerate partitions of a positive integer N in any dimension, one needs to compute (N-1) independent numbers. This could be, say, the partitions of N in dimensions 1,…, (N-1). We improve on the 1967 result by showing that one only needs [(N-1)/2] independent numbers to obtain the result. This reduction is summarized by a simple transform which leads to a new combinatorial problem that we hope can be used for direct enumerations. As a by-product, we are able to compute partitions of all integers <=26 in any dimension — this would have been impossible without the reduction. We will also comment on the asymptotics of these partitions.

Based on arXiv:1203.4419

The calculation of scalar products between Bethe vectors is an important topic of quantum integrable models. In the case when both vectors are on-shell, the scalar product is the norm-squared of the Bethe eigenstate. More generally, off-shell scalar products (in which one vector is not an eigenvector of the transfer matrix) are important to the study of correlation functions.

In the case of models based on the SU(2)-invariant R-matrix, it is known from the work of Slavnov that off-shell scalar products have a determinant expression. A difficult unsolved problem is the generalization of Slavnov’s formula to models based on SU(n). In this talk I will describe my recent work on the SU(3) problem, where by sending some of the variables in the scalar product to infinity, it factorizes into a product of two determinants.

Les systèmes magnétiques frustrés présentent fréquemment de nouvelles  phases de la matière qui ne peuvent pas être simplement décrites par la théorie de Landau. Depuis une quinzaine d’année, la glace de spin en est devenu un exemple typique, avec un état fondamental macroscopiquement dégénéré aux corrélations dipolaires entre spins, appelé phase de Coulomb.

Dans un premier temps, nous présenterons brièvement les phénomènes critiques pouvant émerger dans cette phase sous perturbations extérieures, tels que la transition de Kasteleyn sous champ [1] ou de KDP [2]. Ensuite, nous nous intéresserons au couplage des spins du réseau — qui peuvent être décrit par un modèle de boucles [3] — avec des électrons itinérants [4]. En l’occurrence quel type d’ordre magnétique apparaît en fonction du dopage d’électrons, et comment peut-on utiliser ces électrons itinérants pour donner corps à ces degrés de libertés unidimensionnels émergents ?

[1] Jaubert, Chalker, Holdsworth, Moessner, Phys. Rev. Lett. (2008)
[2] Jaubert, Chalker, Holdsworth, Moessner, Phys. Rev. Lett. (2010)
[3] Jaubert, Haque, Moessner, Phys. Rev. Lett. (2011)
[4] Jaubert, Pitaecki, Haque & Moessner, Phys. Rev. B (2012)

Quantum kinetic theory involves the study of the dynamics of operators and correlation functions in non-equilibrium states. However, even at weak coupling, non-equilibrium dynamics is often not amenable to usual perturbative analysis. Semi-classical kinetic equations (like the Boltzmann equation and DMFT) are usually uncontrolled, but intuitively justified approximations, which work well in practice. We will explore a derivation of quantum kinetic theory from holography. We will see that in the holographic approach, it is possible to have systematic perturbative expansions even at non-equilibrium. Also, holography suggests some novel results in quantum kinetic theory. We will argue that field theory and holography complement each other well, so that by mining information from non-equilibrium processes we can know the underlying microscopic dynamics exactly, at least in the strong-coupling limit. We will discuss some possible applications of this novel approach in modelling strongly correlated electron systems which may be successful phenomenologically (if not quantitatively).

This is an elementary introduction to cluster algebras, invented by Fomin and Zelevinsky around 2000, in relation to questions on total positivity, such as “characterize minimally the set of nxn matrices with only non-negative minors”. Starting from there, we show the connection to the theory of networks, give general definitions and a brief overview of the ever-expanding field of applications of cluster algebras. We then concentrate on an important application: the Q- and T-systems arising from integrable quantum spin chains and their Bethe Ansatz solution. We show how admissible initial data for such discrete evolution equations form a set of clusters in a cluster algebra, and how the general solutions may be expressed as statistical path or network models. Time permitting, we will explore the non-commutative versions of the above, some of which appear in the context of non-commutative Donaldson-Thomas invariant theory.

We consider square lattice bond percolation on an infinite strip of width L, with boundary conditions that allow the bond clusters to attach to the boundary. We refer to clusters that touch both boundaries of the strip as percolating clusters. By means of analytic transfer-matrix methods including the qKZ equation, we compute the *exact finite size* expression for the number of percolating clusters passing between two chosen points. Note that this quantity (which we denote by F) is a discretely holomorphic observable, which becomes a holomorphic field in the continuous limit.

As an application, using the exact equivalence of bond percolation with the Chalker-Coddington model for the quantum spin Hall effect [Cardy,Gruzberg et al], we can interpret F as the spin current between the two boundaries. Our result is then a lattice realisation of Cardy’s CFT computation of this spin current.

In contrast to higher dimensional systems where pair collisions provide finite relaxation rate and lifetime, the situation in one dimension is peculiar. A one-dimensional electron gas requires three-particle collisions for finite relaxation due to constraints imposed by the conservation laws. At zero temperature the fastest relaxation is provided by the interbranch processes which enable energy exchange between counterpropagating particles. At sufficiently high temperatures the leading mechanism is due to the intrabranch scattering of comoving electrons. We derive the corresponding relaxation rates that strongly depend whether one considers screened or unscreneed Coulomb interaction. The abovementioned relaxation processes are responsible for interaction-induced modifications of electrical and thermal conductance in quantum wires. Our approach is based on the Boltzmann equation that is beyond the Luttinger-liquid theory.

After a brief review on the Casimir effect (theory and experiments), we will show the Casimir force between two finite-width mirrors at finite temperature, working in a simplified model in 1+1 dimensions. The mirrors, considered as dissipative media, are modeled by a continuous set of harmonic oscillators which in turn are coupled to an external environment at thermal equilibrium. The calculation of the Casimir force is performed in the framework of the theory of quantum open systems. It is shown that the Casimir interaction has two different contributions: the usual radiation pressure from vacuum, which is obtained for ideal mirrors without dissipation or losses, and a Langevin force associated with the noise induced by the interaction between dielectric atoms in the slabs and the thermal bath. Both contributions to the Casimir force are needed in order to reproduce the analogous of Lifshitz formula in 1+1 dimensions. We also discuss the relation between the electromagnetic properties of the mirrors and the spectral density of the environment.

I talk about the dynamical behavior of non-Abelian line defects characterized by the non-Abelian fundamental groups. Non-Abelian defects appear as disclinations in biaxial nematic liquid crystals and quantized vortices in Bose-Einstein condensates with spin degrees of freedom. First, I explain how non-Abelian defects become possible in such a system. Next, I talk about their collisional properties. Unlike Abelian defect, they neither reconnect nor pass through each other, but create a new defect between them. Such a dynamical change of defects is expected to cause drastic change of dynamics and statistics of systems which is concerned in many non-Abelian line defects. For example, in phase-ordering dynamics, a large-scale networking structure can be seen as a stable state.

Over the last few years, integrability has become a method of choice for the calculation of equilibrium dynamical correlation functions of systems such as spin chains and interacting atomic gases, its main strength being its ability to go beyond commonly-used low-energy effective theories. After a brief review, this talk will present some new results on simple and more elaborate observables, with a discussion of their relevance to new types of experiments such as resonant inelastic x-ray scattering, and of their relation to new developments in Luttinger liquid theory.

For out-of-equilibrium dynamics, most importantly following a sudden change in one of the global parameters of a system (quantum quench), integrability will be shown to be able to offer reliable results on (re)equilibration phenomena with potential applications in quantum dots, quantum magnets and atomic gases, this approach having the unique advantage of being applicable to arbitrary time scales.

Spin-ice materials are frustrated magnets that have the Pauling’s zero temperature entropy of water ice. This type of materials provide a variety of novel states, one of the most surprising one is the emergence of de-confined magnetic monopoles as thermal excitations. I will introduce a general 2d vertex model to study this kind of system numerically. It is a generalization of the extensively studied six and eight vertex models, which are integrable. After reviewing the equilibrium properties of the model I will discuss its ordering dynamics after a quench from a disordered state into an ordered state. The geometry of the model gives rise to unusual growing structures.