Lieb-Liniger模型とanholonomy
阪市大数研：
米澤 信拓
首都大学東京： 田中 篤司
高知工科大学： 全
卓樹
1. Introduction
1-1. topology of delta potential
𝜙 +0 =
𝜙′ +0 −𝜙′ −0 ℏ2
2𝑚𝑔
Dirichlet condition
Parameter space of 𝑔
=
+
Adiabatic cycle of 𝑔
𝑔 = +∞
Start:
𝑔 = +0
Goal:
𝑔 = −0
𝑔 = −∞
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
𝑚=ℏ=1
Bosonic system
periodic boundary conditions
𝜓 2𝜋, 𝑥2 , 𝑥3 , ⋯ , 𝑥𝑁
𝜕1 𝜓 2𝜋, 𝑥2 , 𝑥3 , ⋯ , 𝑥𝑁
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
𝑥1 < 𝑥2 < ⋯ < 𝑥𝑁
3-2. Two Bethe equation
discontinuous at 𝑡 = ±∞
𝑛𝑖
∞
continuous at
discontinuous at
𝑛𝑖
0
continuous at
discontinuous at
Parameter space of 𝑔
We need two chart at least.
4. anholonomy of spectrum
4-1 Super Tonks Girardeau state
Ground state for
: Tonks Girardeau state
Continuous transition
: Super Tonks Girardeau state
Ground state for
Adiabatic cycle of 𝑔
𝑔 = +∞
Start:
𝑔 = +0
Experiment
Goal:
𝑔 = −0
E. Haller et. al.,
Science 325 (2009) 1224
𝑔 = −∞
4-2. calculation of anholonomy
Im
𝑥
1
−)
Re
4-3. Calculation of anholonomy
𝑔 = +0
𝑔>0
𝑘𝑖 is real
𝑔 = ±∞
𝑔<0
𝑘𝑖 → 𝑘𝑖 (−∞)
𝑔 = −0
4-4. summary
Total
5. Example
5.1 N=2
E
(-3,3)
3
(-2,3)
2
(-2,2)
(-1,2)
1
(-1,1)
(0,1)
(0,0)
2
−2
(𝑔
g = ±∞)
1
−1
(𝑔
g = −1)
1
00
(𝑔
g =00)
(0,0)
11
(𝑔
g =11)
2
(𝑔
g = ±∞)
x
5.2 N=3
(-5,0,5)
E
5
(-4,0,5)
4
(-4,0,4)
(-3,0,3) 3
(-2,0,3)
2
(-2,0,2)
(-1,0,1)
1
(0,0,1)
(0,0,0)
2
−2
(𝑔 = ±∞)
g
1
−1
(𝑔 = −1)
g
1
0
0
(𝑔 = 0)
g 0
11
(𝑔 = 1)
g 1
2
2
(𝑔 = ±∞)
g
x
6. Conclusion
𝑔 = +∞
We propose a cycle of 𝑔: 𝒞
Start:
𝑔 = +0
Goal: 𝑔 = −0
𝑔 = −∞
Parameter space of 𝑔
Quasi-momenta:
Initial state
≠
Difference of quasi-momenta:
cf. Berry phase
Final state
Anholonomy
New example in Many body system

Further Quantum Theory, MT2014, Problem sheet 1