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
• In 1963, Lieb and Liniger solved exactly a one dimensional model of bosons interacting by a repulsive \delta-potential and calculated the ground state in the thermodynamic limit

 

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

• Tracy, Craig A., and Harold Widom. “On the Ground State Energy of the Delta-Function Bose Gas.” arXiv:1601.04677 [math-Ph], January 18, 2016. http://arxiv.org/abs/1601.04677.
• Zill, J. C., T. M. Wright, K. V. Kheruntsyan, T. Gasenzer, and M. J. Davis. “A Coordinate Bethe Ansatz Approach to the Calculation of Equilibrium and Nonequilibrium Correlations of the One-Dimensional Bose Gas.” arXiv:1601.00434 [cond-Mat, Physics:hep-Th], January 4, 2016. http://arxiv.org/abs/1601.00434.
• Veksler, Hagar, and Shmuel Fishman. “A Generalized Lieb-Liniger Model.” arXiv:1508.02011 [cond-Mat, Physics:math-Ph], August 9, 2015. http://arxiv.org/abs/1508.02011.
• Flassig, Daniel, Andre Franca, and Alexander Pritzel. “Large-N Ground State of the Lieb-Liniger Model and Yang-Mills Theory on a Two-Sphere.” arXiv:1508.01515 [cond-Mat, Physics:hep-Th], August 6, 2015. http://arxiv.org/abs/1508.01515.
• 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