Informal workshop on “ One-Dimensional Quantum Gases ”

Aug 21-23, 2012

Institute for Advanced Study, Tsinghua University

地点：清华大学高等研究院 科学馆 322报告厅

This workshop is very informal. It contains two pedagogical lectures given by Xiwen Guan, two research seminars and two sessions of informal discussions.

Aug 21

10:00-11:30 Pedagogical Lecture I: Xiwen Guan

(1hour lecture+30minute discussion)

3:00-4:30 Pedagogical Lecture II: Xiwen Guan

(1hour lecture+30minute discussion)

In 1931 Hans Bethe introduced a particular form of wavefunction (Bethe ansatz) to obtain the energy eigenspectrum of the one dimensional (1D) Heisenberg spin chain. Since the pioneering work in the 60's, 70's and 80's of Lieb, Yang, Baxter, Faddeev and many others, the study of Bethe ansatz solvable models has flourished into a major activity. For such exactly solved models, the energy eigenspectrum of the model Hamiltonian can be obtained exactly in terms of the Bethe ansatz equations, from which physical properties can be derived via mathematical analysis.

Lecture I: I will introduce Bethe's hypothesis and Yang and Yang's grand canonical description of the finite temperature problem through one simple but notable Bethe ansatz integrable model - the Lieb-Liniger delta-function interacting Bose gas. I will demonstrate how to calculate ground state properties and the thermodynamics of the Lieb–Liniger bosons in analytic fashion in the framework of the Bethe ansatz.

Lecture II: In 1967, Yang first time proved that Bethe's hypothesis still works in the complicated many-body problem -delta-function interacting fermion problem. Right at the heart is the so-called Yang-Baxter equation (YBE). I will present a thorough study of Yang's seminal work on the delta-function interacting fermion problem and demonstrate how fermions behave ``harmonioursly"— they can cooperatively share the space where they live and collectively respond to external influences such as magnetic field and temperature.

Aug 22

10:00-11:00 Seminar I: Georgy Shlyapnikov

Title: One-dimensional bosons with a large spin

Abstract: We use the saddle point approximation and show that for a negative pairing interaction the ground state of a 1D gas of bosons with a large spin is the so-called polar phase in which one has a (quasi)condensate of singlet bosonic pairs with an algebraic order. Since in 1D one needs a finite energy to destroy a pair, the spectrum of spin excitations has a gap. hence, in the absence of magnetic field there is only one gapless mode corresponding to phase fluctuations of the pair quasicondensate. Once the field exceeds the gap, there emerges a condensate of unpaired bosons with spins aligned along the field. the corresponding quantum transition is of the commensurate-incommensurate type.

3:00-5:00 Informal Discussions I

Topic: One-dimensional quantum gases with spin-orbit coupling

3:00-3:45

Xiaoling Cui (IASTU): One-dimensional fermions with spin-orbit coupling

3:45-4:30

Junpeng Cao (IOP): Exact solvable models with spin-orbit coupling

4:30-5:00 discussion

Aug 23

10:00-11:00 Seminar II: Xiwen Guan

Title: Quantum Critical Matter in one dimension

Abstract:

It has long been appreciated that exactly solved mathematical models describing the statistical mechanics of interacting particles have played a key role in the development of formerly unrelated areas of mathematics and theoretical physics, such as the study of knots, links and braids, quantum groups, conformal field theory and condensed matter physics. However, over the past few years striking experimental achievements in trapping and cooling atoms in one-dimensional optical waveguides have provided remarkable realisations of exactly solved models in the lab. More generally the study of cold atomic matter provides a unique environment to explore novel quantum many-body effects like quantum liquids, quantum correlations and quantum criticality. In this talk I will describe some of these fundamental mathematical models and their relevance to recent and future experiments on such exotic many-body physics, such as spin-charge separation, Fulde-Ferrel-Larkin-Ovchinnikov pairing and quantum criticality.

3:00-5:00 Informal Discussions II

Topic: One-dimensional quantum gases with resonant interactions

3:00-3:45

Ran Qi (IASTU): Tonk and super-Tonk gases across a narrow resonance

3:45-4:30

Xiaoling Cui (IASTU): Ferromagnetism in one-dimension Fermi gases

4:30-5:00 discussion