Lieb-Liniger anholonomy 1. Introduction 1-1. topology of delta potential Dirichlet condition = + Berry phase (TC, Phys. Lett. A 248, 285 (1998)) 1-2. delta potential in quantum well Does N body delta potential system have Anholonomy ? Berry phase 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 Quantum many body system on circle Bosonic system E. H. Lieb et al., Phys. Rev. 130 (1963) periodic boundary conditions 2-2 Connection to field theory N particle state Basis: Linear combination: Commutation relation: Vaccum: Non-Linear-Schrodinger Equation Eigenstate Heisenberg rep. 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 Ground state for : Super Tonks Girardeau state Continuous transition 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 5.1 N=2 (0,0) (-1,1) (-2,2) (-3,3) (0,1) (-1,2) (-2,3) 5.2 N=3 (0,0,0) (-2,0,2) (-4,0,4) (0,0,1) (-2,0,3) (-4,0,5) (-1,0,1) (-3,0,3) (-5,0,5) 6. Conclusion Quasi-momenta: Difference of quasi-momenta: Initial state Final state cf. Berry phase New example in Many body system Anholonomy 3-2 Limit of g to + is real when g > 0. if x is real. 2-2 Limit of g to 0 2-2 Connection to field theory 4.3 N=4 (0,0,0,0) (-2,-1,1,2) (-4,-2,2,4) (-1,-1,1,1) (-4,-1,1,4) (-7,-2,2,7) (-1,0,0,1) (-4,-2,2,4) (-7,-3,3,7)