Opening workshop

The first of these workshops will take place April 11-14, 2016 at Wuppertal University.

Correlations in Integrable Quantum Many-Body Systems

This workshop will focus on the lattice realization of quantum field theories with non-compact degrees of freedom and on functional equations for the calculation of spectra and correlation functions of integrable lattice models.

Abstracts

Form factors for staircase models

P. Dorey, University of Durham

Off-diagonal correlations and Bose-Einstein condensation in coupled chains of interacting bosons

S. Eggert, TU Kaiserslautern

The emergence of new properties from low-dimensional building blocks is a universal theme in different areas in physics. The investigation of transitions between isolated and coupled low-dimensional systems promises to reveal new phenomena and exotic phases. Interacting 1D bosons, which are coupled in a two-dimensional array, are maybe the most fundamental example of a system which illustrates the concept of a dimensional phase transition. However, recent experiments using ultracold gases have shown a surprising discrepancy between theory and experiment. We discuss how the off-diagonal correlations in isolated chains can be used to predict the nature of the phase transition to a Bose-Einstein condensate. Comparison with large scale quantum Monte Carlo simulations show surprising discrepancies at low filling, which demonstrate that a careful analysis of different non-commuting limiting cases is necessary.

Non-Linear Integral Equations with a singular kernel

Y. Ikhlef, Université Pierre et Marie Curie

Non-Linear Integral Equations (NLIEs) were introduced in the 1990s as a powerful tool to study the scaling limit of integrable lattice models. Generically, the scaling theory describing the critical point of a spin chain is a Conformal Field Theory (CFT) with a discrete spectrum of scaling dimensions, which can be extracted from the NLIEs by a well-controlled procedure. In this talk, I will present a simple integrable spin-chain model whose Bethe Ansatz equations are governed by a singular kernel: I will explain how to derive the NLIEs in this situation, and how to treat the singularity to obtain the CFT spectrum. It turns out that the corresponding CFT is the SL(2, R)/U (1) “black hole” WZW model, a toy model of CFT with non-compact target space.

Non-compact a_2^(2) and a_3^(2) spin chains and their physical applications

J.L. Jacobsen, ENS Paris

We describe how non-compact continuum limits arise from aN-1(2) spin chains based on the second baxterisation of the so(N) Birman-Murakami-Wenzl algebra. These chains can be physically realised, for N=3, as the Izergin-Korepin 19-vertex model or the dilute O(n) loop model, and for N=4 as two coupled antiferromagnetic Potts models. The non-compact physics arises in the so-called regime III, which for N=3 contains a theta point of polymers (n → 0 limit) with self-attraction.

The continuum limit of these models is identified by a series of arguments, including level-rank duality, RSOS restrictions, and numerical resolution of the Bethe Ansatz equations. For N=3, the corresponding conformal field theory turns out to be the Euclidean black hole sigma model. In particular, we show that the discrete states in the black hole model emerge from the non-compact continuum upon changing the twist of the spin chains. Some implications of non-compactness for the logarithmic behaviour of observables in the polymer problem are given.

Multi-state extension of the asymmetric simple exclusion process

C. Matsui, Tokyo University

There are few far-from-equilibrium systems which are analytically solvable. One of those examples is the asymmetric simple exclusion process (ASEP). The ASEP is an integrable two-state stochastic process in one dimension. The integrability of the model lies in the Uq(sl2)-invariance of the bulk part. We consider the multi-state extension of the ASEP based on the fact that the Markov matrix of this process satisfies the Temperley–Lieb algebra. Besides the construction of steady states, we derive the exact expressions of particle-density profiles and currents on the steady states under the closed boundary condition. Although strong restrictions are imposed on hopping rates to keep integrability, we show that they are simplified in the limit q to 0.

On quantum loop algebras: q-oscillator vs. prefundamental representations

Kh. Nirov, University of Wuppertal and INR Moscow

Modern approaches to quantum integrable systems are based on the notion of quantum groups. Here, the choice of a representation in the auxiliary space defines an integrability object, and by a representation in the quantum space one fixes a model subject to consideration. The functional relations between integrability objects follow from the characteristics of the representations of the quantum group. We discuss various representations of quantum loop algebras giving rise to different integrability objects and functional relations. Specifically, we give a comparative analysis of q-oscillator and prefundamental representations of the corresponding Borel subalgebras.

Quantum quenches and excited state correlations in the XXZ spin chain

B. Pozsgay, Hungarian Academy of Sciences

In this talk we will discuss non-equilibrium situations of the XXZ chain, in particular time evolution from simple product states such as the N'eel state or the dimerized state. The focus will be on calculating the long-time limit of local observables, which can be performed using the Quench Action method. Two very important ingredients for this method are the overlaps with the initial state and the calculation of short range correlators in arbitrary excited states. We will discuss the latter topic in detail, and present a conjectured formula which calculates the correlation functions in arbitrary excited states of the finite XXZ chain. Our result builds on the theory of factorization of correlation functions, and it calculates the physical part of the construction using a finite set of Bethe roots. In the thermodynamic limit the formula leads to TBA-like sets of linear equations which can be solved effectively for arbitrary Bethe root distributions.

Integrability and the Conformal Bootstrap

V. Schomerus, DESY Theory

The conformal bootstrap programme relies on the expansion of 4-point functions into kinematically determined conformal blocks. I will explain that conformal blocks of scalar 4-point functions in a d-dimensional conformal field theory can mapped to eigenfunctions of a 2-particle hyperbolic Calogero-Sutherland Hamiltonian. The link makes considerable mathematical developments in integrability and the modern theory of special functions available for conformal field theory.

Fermionic basis and reflection relations

F. Smirnov, Université Pierre et Marie Curie

In this talk I shall discuss the CFT limit of the fermionic basis and its connection with the reflection relations. Emphasis will be done on including the descendants created by the local integrals of motion.

Spinon expansion of correlation functions of the spin 1/2 XXZ model in massive regime

J. Suzuki, Shizuoka University

The recent advance on the form factor expansion approach to correlation functions of the spin 1/2 XXZ model in massive regime will be reviewed. We put emphasis on the advantage in starting from the finite temperature problem with a non-vanishing magnetic field. This yields relatively simple expressions, without complicate multiple contour integrals. The comparison with other numerical approaches will be briefly commented. The talk is based on a collaboration with M. Dugave, F. Göhmann and K. K. Kozlowski.

Heisenberg spin chains by separation of variables: recent advances

V. Terras, Université Paris Sud

During the last decades, important progresses have been made concerning the computa- tion of form factors and correlation functions of simple models solvable by algebraic Bethe Ansatz (ABA) such as the XXZ spin-1/2 chain or 1D Bose gas with periodic boundary conditions. However, the generalization of these results to more complicated models or different types of integrable boundary conditions is for the moment limited by the range of applicability of ABA or by some difficulties of the method. In this talk, we discuss the solution of Heisenberg spin chains (XXX, XXZ or XYZ) in the framework of a complementary approach, Sklyanin’s quantum Separation of Variables approach. This enables us notably to consider for these models various types of boundary conditions (quasi-periodic, open…) not directly solvable by Bethe ansatz. More precisely, we discuss in this framework some new results and open problems concerning the description of the spectrum by means of solutions of a functional T- Q equation (or equivalently in terms of Bethe-type equations). We also discuss the problem of the computation of the eigenstates scalar products and of the form factors of local operators.

Magneto-thermal transport in the s=1/2 Heisenberg model - revisited

X. Zotos, University of Heraklion

I will discuss recent results on the evaluation of spin and thermal Drude weight in the spin-1/2 easy-axis Heisenberg chain and an application in far-out of equilibrium transport.

Preliminary Schedule

11 - 14 April 2016
Time Monday Tuesday Wednesday Thursday
11:00 - 12:00 J. Suzuki F. Smirnov J. Jacobsen Kh. Nirov
14:00 - 15:00 B. Pozsgay V. Schomerus S. Eggert X. Zotos
15:30 - 16:30 V. Terras J. Ikhlef C. Matsui P. Dorey
19:00 joint dinner

Participation

If you are interested in participating in this workshop, please send an e-mail message to any of the PIs.

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Second Workshop

The second workshop will take place September 05-08, 2017 at the Institute of Theoretical Physics, Leibniz Universität Hannover.

Correlations in Integrable Quantum Many-Body Systems

The second workshop on Correlations in Integrable Quantum Many-Body Systems will focus on applications of exactly solvable models to

  • cold gases
  • anyons and topological matter
  • non-equilibrium, dynamics and transport
  • form factors on higher rank (super-)spin chains

Invited Speakers (Abstracts)

Chains of anyons: from their structure to integrability

Eddy Ardonne

Stockholm University

In this talk, I will give an introduction to anyonic chains, starting from their mathematical formulation, using so-called anyon models, or more precisely, modular tensor categories. The main ingredients are the F-symbols, which can be obtained using quantum groups. The physics of anyonic chains will be discussed, focussing on the rich phase diagrams, and in particular, the integrability at special points.

Quantum quenches near criticality

Gesualdo Delfino

SISSA

We present the theory of quantum quenches in near-critical one-dimensional systems [1,2]. Aspects that are discussed include role of interaction, role of integrability, appearance of timescales, analytic determination of one-point functions and their long time behavior.

References: [1] G. Delfino, Quantum quenches with integrable pre-quench dynamics, J. Phys. A 47 (2014) 402001 [2] G. Delfino and J. Viti, On the theory of quantum quenches in near-critical systems, J. Phys. A 50 (2017) 084004

Full counting statistics in the spin-1/2 Heisenberg XXZ chain

Fabian Essler

University of Oxford

The spin-1/2 Heisenberg chain exhibits a quantum critical regime characterized by quasi long-range magnetic order at zero temperature. The strength of quantum fluctuations in the ground state can be quantified by determining the probability distributions of the components of the (staggered) subsystem magnetization. Some of these are shown to exhibit scaling and the corresponding universal scaling functions are determined by free fermion methods and by exploiting a relation with the boundary sine-Gordon model.

Tan’s contact for one-dimensional Bose and Fermi gases

Anna Minguzzi

Université Grenoble Alpes

A universal decay power-law of the large-momentum tails of the momentum distribution, fixed by Tan’s contact coefficients, constitutes a direct signature of strong correlations in a short-range interacting quantum gas. We derive the Tan’s contact of one-dimensional harmonically trapped gases, both in the case of a multicomponent Fermi gas and of a Bose gas. In the case of a multicomponent mixture, we find a direct correspondence between the value of the Tan’s contact and the symmetry of the state. We show that a local density approximation (LDA) on the Bethe-Ansatz equation of state for the homogeneous gas is in excellent agreement with the results for the harmonically confined gas and predicts a scaling behavior of the Tan’s contact. This provides useful analytical expressions for the dependence on the number of particles, number of components and on interaction strength. Based on work with J. Decamp, J. Jünemann, G. Lang, M. Albert, M. Rizzi, and P. Vignolo

Quantum loop algebras and highest ℓ-weight representations: General linear case

Khazretali Nirov

Bergische Universität Wuppertal and Institute for Nuclear Research / Moscow

Universal integrability objects are determined by the choice of representations of the quantum group in the auxiliary space. We consider various representations of Borel subalgebras of quantum loop algebras to define such objects. We are especially interested in the highest ℓ-weight representations. Here we compare the q-oscillator and prefundamental representations and argue that the latter can be obtained by tensoring the former. We also discuss how relations between the corresponding highest ℓ-weights reproduce functional relations between integrability objects.

Universal relations for quantum gases in one dimension

Ovidiu Patu

Institute for Space Science / Bucharest

For the most general case of an one-dimensional multi-component system (bosons, fermions or an arbitrary mixture) interacting through a delta function potential and subjected to an external potential we show that the large momentum distribution of these systems behaves like nσ∼Cσ/k4 with Cσ the contact of species σ which can be computed from the thermodynamic properties of the system (derivatives of appropriate thermodynamic potentials with respect to scattering lengths). We obtain short distance expansions for the Green's function and pair distribution function and show that the coefficients of these expansions can be expressed in terms of the density, kinetic energy and contact. In addition we derive universal thermodynamic identities for homogeneous and inhomogeneous systems connecting the pressure, total energy, trapping energy and contact. Based on work with A. Klümper.

Integrable models: their Bethe vectors, scalar products and form factors

Eric Ragoucy

Laboratoire de Physique Théorique / Annecy

We apply the nested algebraic Bethe ansatz to integrable models. We present some explicit representations for the Bethe vectors and their scalar products, in the framework of periodic generalized models, that encompass all integrable spin chain models with (twisted) periodic boundary conditions. We review what has been (or can be) done, depending on the algebra which underlies the model (Yangian, super-Yangian or quantum group). Starting from these formulas we present some general methods that allow to deduce the form factors of the models. They are of two types: the twisted scalar product and the zero modes method.

Spin Drude weight in the XXZ chain at finite temperatures

Kazumitsu Sakai

Tokyo University of Science

In this talk, I will present our recent results on spin transport properties in the spin-1/2 XXZ spin chain at finite temperatures. Based on the functional relations among the row-to-row transfer matrices (T-systems) and their certain combinations (Y-system), the spin Drude weight and its size dependence are evaluated. This talk is based on joint work with Andreas Klümper.

On the interacting Majorana chain

Dirk Schuricht

Utrecht University

We study the effect of interactions on Kitaev's toy model for Majorana wires. We demonstrate that even though strong repulsive interaction eventually drive the system into a Mott insulating state the competition between the (trivial) band-insulator and the (trivial) Mott insulator leads to an interjacent topological insulating state for arbitrary strong interactions. We show that the exact ground states can be obtained analytically even in the presence of interactions when the chemical potential is tuned to a particular function of the other parameters. The ground states obtained are two-fold degenerate and differ in fermion parity, as is the case with the Kitaev/Majorana chain in a topological phase. We prove that the ground state is unique in each fermion parity sector and that there exists an energy gap. Furthermore, we investigate the effect of disorder in the chemical potential. We find that, like the non-interacting system, moderate disorder supports the topological phase, while at large disorder strengths the system becomes trivial. Based on work with Niklas Gergs, Fabian Hassler, Hosho Katsura, Masahiro Takahashi

Form factors of the monodromy matrix entries in the models with gl(2|1) symmetry

Nikita Slavnov

Steklov Mathematical Institute

We apply the nested algebraic Bethe ansatz to the models with gl(2|1) symmetry. We obtain explicit representations for the Bethe vectors scalar products. In some particular cases we find determinant formulas for the scalar products. Starting from these formulas and using the zero modes method we obtain compact determinant representations for the form factors of the monodromy matrix entries. The latter, in their turn, lead us to determinant formulas for form factors of local operators.

New approach to computation of correlation functions for XXX model

Fedor Smirnov

Université Pierre et Marie Curie

It has been shown that using the fermionic basis the correlation functions for XXZ model on a cylinder are expressed in universal form for arbitrary Matsubara data. We use this arbitrariness in order to fix the coefficients in the expansion of operators in the fermionic basis. For XXX case this allows to compute the correlation functions up to 11 sites. The dependence of coefficients on the lattice spacing exhibits remarkable regularity.

The static and the dynamical form factor expansion approach to quantum correlations

Junji Suzuki

Shizuoka University

We discuss a quantitative analysis on the correlation functions of spin 1/2 XXZ model based on the quantum transfer matrix. The static and the dynamical aspects of the correlations will be discussed within the framework of form factor expansions. The content of the talk is based on collaborations with M. Dugave, F Göhmann, A. Klümper and K. K. Kozlowski.

SU(2) -symmetric lattice spin model with Majorana fermion excitations

Alexei Tsvelik

Brookhaven National Lab

We have constructed a 2D lattice model of spins 1/2 interacting with nearest neighbor 2- and 3-spin interactions which have gapped bulk and gapless chiral excitations on the edges. The bulk excitations are 2D Majorana fermions and 1D solitons. The 2D propagating Majorana's are bound states of fractionalized 1D solitons. The model has two phases – Abelian and non-Abelian topological ones.

Dynamics of observables in out-of-equilibrium many-body quantum systems: the Loschmidt echo

Eric Vernier

SISSA

While much progress has been made over the last years in understanding the relaxation mechanisms taking place in the non-equilibrium dynamics of quantum many-body systems, very few analytical results exist concerning the full time evolution of physical observables, however urgently called for by the ongoing advances in cold-atomic experiments as well as by the recently emerged subject of dynamical phase transitions. A reason for this fact is that despite the existence of prototypical integrable models, the time dynamics involves contributions of arbitrarily excited eigenstates of the Hamiltonian which render exact calculations prohibitively difficult. In this seminar I will present a method to tackle the exact-time dynamics of quantum observables in a prototypical interacting integrable quantum many-body system, the Heisenberg XXZ spin chain, starting with a simple observable that is the Loschmidt echo (or quantum fidelity). The latter measures the overlap between the system's state at a given time and its initial state, and has attracted a renewed interest recently in the context of dynamical phase transitions, which it signals through its non-analyticities as a function of the time. Using a reformulation of the problem in terms of an auxiliary boundary quantum transfer matrix, the Loschmidt echo is written as the solution of a set of Non Linear Integral Equations, which allows for its exact determination at arbitrarily large time. This method overcomes the time limitations experienced by numerical approaches, and allows to for an analytic approach to dynamical transitions. I will further discuss perspectives concerning the classification of “integrable initial states” allowing for such exact computations, as well as the extension of this method to the study of more general physical observables. This is based on L Piroli, B Pozsgay, E Vernier, J. Stat. Mech., 023106 (2017), as well as some ongoing work.

Preliminary Schedule

05 - 08 September 2017
Time Tuesday Wednesday Thursday Friday
10:00 - 11:00 - Minguzzi Sakai Ragoucy
11:30 - 12:30 Suzuki Patu Vernier Slavnov
14:30 - 15:30 Smirnov Essler Schuricht -
16:00 - 17:00 Nirov Delfino Ardonne -
17:00 - 18:00 Posters Posters Tsvelik -
Dinner
workshops.txt · Last modified: 2017/08/26 10:16 by admin
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