Lieb-Liniger delta Bose gas

introduction

• N bosons interacting on the line $[0,L]$ of length L via the delta function potential
  
• one-dimensional Bose gas
  
• 1963 Lieb and Liniger solved by Bethe ansatz
  

 

Hamiltonian

• quantum mechanical Hamiltonian

\[H=-\sum_{j=1}^{N}\frac{\partial^2}{\partial x_j^2}+2c\sum_{1\leq i<j\leq N}^{N}\delta(x_i-x_j)\]
 

wave function

• $\psi(x_1, x_2, \dots, x_j, \dots,x_N)$
• $\psi(x_1, \dots, x_N) = \sum_P a(P)\exp \left( i \sum_{j=1}^N k_{Pj} x_j\right)$

$$ a(P) = \prod\nolimits_{1\leq i<j \leq N}\left(1+\frac{ic}{k_{Pi} -k_{Pj}}\right) \ . $$

two-body scattering term

• \(s_{ab}=k_a-k_b+ic\)
  

Bethe-ansatz equation

\[\exp(ik_jL)=\prod_{l=1}^{N}\frac{k_j-k_l+ic}{k_j-k_l-ic}\]

 

energy spectrum

• energy of a Bethe state

\[E=\sum_{j=1}^{N}k_j^2\]

 

 

computational resource

• Day 5 - Yang-Baxter, Delta Bosons, Contact Terms
  • Bose-Einstein Condensation and BAE exercise .pdf
  • Bose-Einstein Condensation and BAE solution .nb

encyclopedia

• http://en.wikipedia.org/wiki/Lieb-Liniger_model

articles

• Dorlas, T. C. “Orthogonality and Completeness of the Bethe Ansatz Eigenstates of the Nonlinear Schroedinger Model.” Communications in Mathematical Physics 154, no. 2 (June 1, 1993): 347–76. doi:10.1007/BF02097001.
• Yang, C. N., and C. P. Yang. “Thermodynamics of a One‐Dimensional System of Bosons with Repulsive Delta‐Function Interaction.” Journal of Mathematical Physics 10, no. 7 (July 1, 1969): 1115–22. doi:[10.1063/1.1664947 http://dx.doi.org/10.1063/1.1664947].
• C.N. Yang Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312-1315
• Elliott H. Lieb and Werner Liniger Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State, 1963