1 Introduction 1 1 topology of delta potential

1. Introduction

1 -1. topology of delta potential Dirichlet condition = + Berry phase

1 -2. delta potential in quantum well (TC, Phys. Lett. A 248, 285 (1998)) Berry phase Does N body delta potential system have “Anholonomy” ? Quantum holonomy or Anholonomy

1 -3. Plan • • • 1. Introduction 2. Lieb-Liniger model 3. Bethe equation 4. anholonomy of spectrum 5. Example 6. Conclusion

2. Lieb-Liniger model

2 -1. Definition E. H. Lieb et al. , Phys. Rev. 130 (1963) 1605. Quantum many body system on circle Bosonic system periodic boundary conditions

2 -2 Connection to field theory Commutation relation: Vaccum: N particle state Basis: Linear combination: Eigenstate Heisenberg rep. Non-Linear-Schrodinger Equation

3. Bethe equation

3 -1. Bethe equation

3 -2. Two Bethe equation continuous at discontinuous at We need two chart at least.

4. anholonomy of spectrum

4 -1 Super Tonks Girardeau state Ground state for : Tonks Girardeau state Continuous transition Ground state for : Super Tonks Girardeau state Experiment E. Haller et. al. , Science 325 (2009) 1224

4 -2. calculation of anholonomy 1

4 -3. Calculation of anholonomy

4 -4. summary Total

5. Example

6. Conclusion Quasi-momenta: ≠ Difference of quasi-momenta: Initial state Final state cf. Berry phase Anholonomy New example in Many body system