Non-Hermitian degeneracies in the Lieb-Liniger model and exotic quantum holonomy

2016-08-12 updated; 2016-08-06 updated

Transcript

1. 1 / 25 Non-Hermitian degeneracies in the Lieb-Liniger model and exotic quantum holonomy Atushi Tanaka (Tokyo Metropolitan University, Japan) (joint work with Nobuhiro Yonezawa and Taksu Cheon) 2016-08-10 PHHQP16 @ Maskawa Hall, Kyoto Univ.
2. Introduction 2 / 25 Exceptional Points (EPs) in non-Hermitian quantum theory An EP is a branch point of eigenenergies. The parametric evolution around an EP induces a permutation of eigenenergies. Ref. Heiss and Steeb (1991).
3. Introduction 3 / 25 Adiabatic state flip around an EP? The parametric evolution around an EP also induces a permutations of eigenspaces. EP Decay due to non-unitary evol. Is such a permutation relevant with time evolution? — The decay prevents the permutation in the adiabatic limit. Ref. Uzdin et al. 2011, Berry and Uzdin 2011, Graefe et al. 2013.
4. Introduction 4 / 25 Exotic Quantum Holonomy in Hermitian quantum theory An adiabatic cycle may induce a permutation of eigenspaces, which is called exotic quantum holonomy. Ref. Cheon 1998; AT and Miyamoto 2007.
5. Introduction 5 / 25 Exotic Quantum Holonomy in terms of Exceptional Points The unitary cycles that exhibit EQH may “enclose” EPs: S.W. Kim, T. Cheon and AT, PLA 374 1958 (2010) . . . reported at PHHQP XI AT, S.W. Kim and T. Cheon, PRE 89 042904 (2014) . . . N-level systems including NEP case (N > 2) EP Previous examples for the “EP interpretation” are kicked tops, which are few-level periodically driven (Floquet) systems.
6. Introduction 6 / 25 Aim of this talk Another example of the interplay between the Exceptional Points and Exotic Quantum Holonomy in a many-body system is shown. . . . Lieb and Liniger’s model, which describes Bosons in a 1D space. Refs. Yonezawa, AT and Cheon, PRA 87 062113 (2013). AT, Yonezawa and Cheon, JPA 46 315302 (2013).
7. Introduction 7 / 25 Outline Introduction Hermitian side — Exotic Quantum Holonomy in Lieb-Liniger’s model Non-hermitian side — EPs in LL model (cf. Ushveridze 1988) Interplay between EPs and EQH Summary
8. Hermitian side — Exotic Quantum Holonomy in LL 8 / 25 The Lieb-Liniger model Bosons are confined in a one-dimensional periodic space: H(g) = − 1 2 N j=1 ∂2 ∂x2 j +g (i,j) δ(xi −xj), where N is the number of Bosons. g is the interaction strength (two-body, contact interaction). Lieb and Liniger (1963).
9. Hermitian side — Exotic Quantum Holonomy in LL 9 / 25 An adiabatic cycle C for the Lieb-Liniger model The coupling strength g is adiabatically varied to form a “closed” path C: 1. C starts from g = 0 (free Bosons). 2. g is increased to +∞ (Tonks-Girardeau regime). 3. g is suddenly flipped to −∞ (super-Tonks-Girardeau regime). 4. g is increased to 0 (free Bosons). C is relevant to a recent experiment of super-Tonks-Girardeau gas (with finite T and uncertain N) [Haller et al., Science (2009)].
10. Hermitian side — Exotic Quantum Holonomy in LL 10 / 25 Exotic quantum holonomy in the LL model (N = 2) The unitary and adiabatic cycle C shifts the eigenenergies of the LL model. From Yonezawa, AT and Cheon, PRA 87 062113 (2013).
11. Hermitian side — Exotic Quantum Holonomy in LL 11 / 25 Exotic quantum holonomy in the LL model (cont.) N = 2 N = 3 N = 4 From Yonezawa, AT and Cheon, PRA 87 062113 (2013).
12. Non-hermitian side — EPs in LL model 12 / 25 Introduction Hermitian side — Exotic Quantum Holonomy in Lieb-Liniger’s model Non-hermitian side — EPs in LL model (cf. Ushveridze 1988) Interplay between EPs and EQH Summary
13. Non-hermitian side — EPs in LL model 13 / 25 Non-Hermitian Lieb-Liniger model The coupling strength g is complexified, which includes effects of inelastic collisions (D¨ urr et al., PRA 2009): H(g) = − 1 2 N j=1 ∂2 ∂x2 j +g (i,j) δ(xi −xj). In the following, we examine the N = 2 case.
14. Non-hermitian side — EPs in LL model 14 / 25 “Energy landscape” in complex g-plane ReE0(g) (ground) 2 1 0 1 4 2 0 Re g Im g ReE2(g) (1st) 2 1 0 1 4 2 0 Re g Im g ReE4(g) (2nd) 2 1 0 1 4 2 0 Re g Im g Note: We restrict the case that the total momentum is zero.
15. Non-hermitian side — EPs in LL model 15 / 25 EPs in the non-Hermitian Lieb-Liniger model 2 1 0 1 4 2 0 Re g Im g ReE0(g) 2 1 0 1 4 2 0 Re g Im g ReE2(g) 3 2 1 1 2 Re g 14 12 10 8 6 4 2 Im g EPs in g-plane Ref. A. G. Ushveridze, JPA 21 955 (1988).
16. Non-hermitian side — EPs in LL model 16 / 25 Encircling several EPs C +C0: trivial C +C2: permutation of two levels (E0,E2) → (E2,E0) C +C4: permutation of three levels (E0,E2,E4) → (E2,E4,E0) 2 1 0 1 4 2 0 Re g Im g Ref. A. G. Ushveridze, JPA 21 955 (1988).
17. Non-hermitian side — EPs in LL model 17 / 25 C ≡ encirclement of an infinite number of EPs 3 2 1 1 2 Re g 14 12 10 8 6 4 2 Im g When a path encircles all the relevant EPs, the path emulates the permutation induced by C, i.e., (E0,E2,E4,...) → (E2,E4,E6,...).
18. Interplay between EPs and EQH 18 / 25 Introduction Hermitian side — Exotic Quantum Holonomy in Lieb-Liniger’s model Non-hermitian side — EPs in LL model (cf. Ushveridze 1988) Interplay between EPs and EQH Summary
19. Interplay between EPs and EQH 19 / 25 How we characterize the permutation of eigenspaces? Compare the initial and final states of an adiabatic cycle C through the overlap integrals, which form a matrix M(C): Mmn(C) ∼ m|n(C) where |n(C) is the final state corresponding to the initial state |n , and the dynamical phase is excluded from RHS.
20. Interplay between EPs and EQH 20 / 25 Gauge covariant expression of M(C) This is an “extension” of Berry’s formula for the geometric phase: M(C) = exp → −i C A(g)dg exp i C AD(g)dg , where Amn(g) ≡ i m(g)| ∂|n(g) ∂g , a non-Abelian gauge connection and AD mn (g) ≡ δmnAnn(g) the diagonal part of A. Fujikawa 2007; (for EQH) Cheon and AT 2009.
21. Interplay between EPs and EQH 21 / 25 Evaluation of M(C) along a complex contour Evaluate the anti-path-ordered exponential M(C) = exp → −i C A(g)dg , where the parallel transport condition Ann = 0 is imposed for all n. C is deformed to a complex contour. The integral may be decomposed into the contributions from EPs.
22. Interplay between EPs and EQH 22 / 25 Singularity (“pole”) of A(g) around an EP g(2) Let us examine around the EP g(2) that involves the ground and first excited eigenenergies. The gauge connection A(g) diverges at g(2): A(g(2) + ) = i R(2) 0 0 0 +O( −1 2 ), where R(2) ≡ − 1 4 0 −i i 0 .
23. Interplay between EPs and EQH 23 / 25 Contribution from the exceptional point g(2) g(2) is “enclosed” by C(2) ≡ g(2) + eiθ|0 ≤ θ ≤ 2π : exp → −i C(2) A(g)dg = P → 1−i C(2) A(g)dg − 1 2 C(2) A(g1)dg1 C(2) A(g2)dg2 +... → exp −i2πR(2) 0 0 1 as → 0, where exp −i2πR(2) = 0 −1 1 0 . . . . C(2) permutates the ground and first excited eigenspaces accompanying a off-diagonal phase factor.
24. Summary 24 / 25 Summary The interplay between EPs and exotic quantum holonomy in the Lieb-Liniger model is explained. 1. The adiabatic cycle C connecting free Bosons (g = 0), the Tonks-Girardeau gas (g = ∞), the super-Tonks-Girardeau gas (g = −∞), and free Bosons induces exotic quantum holonomy. [Yonezawa, AT and Cheon, PRA 87 062113 (2013)] 2. 2EPs of the N = 2 LL models are shown. [old result, Ushveridze, JPA 21 955 (1988)] 3. The permutation between eigenspaces is described in terms of non-Abelian gauge connection. In N = 2 case, M(C) can be decomposed into contributions from 2EPs, which are “poles” of the non-Abelian gauge connection. [AT, Yonezawa and Cheon, JPA 46 315302 (2013)] This result is an extension of the previous works on few level Floquet systems [Kim, Cheon and AT 2010; AT, Kim and Cheon 2014].
25. Summary 25 / 25 A digression: Another interplay between nHQM and EQH Recently, there appears another parallelism between non-Hermitian quantum theory and exotic quantum holonomy. This stems from Simon’s formulation of geometric phase in terms of fiber bundles (1983). Mehri-Dehnavi and Mostafazadeh’s formulation on geometric phase for non-Hermitian systems involves a covering space structure, which can be regarded as a fiber bundle with a discrete structure group. [JMP 49, 082105 (2008)] It is shown that the counterpart of Simon’s formulation for exotic quantum holonomy involves a covering space structure [AT and Cheon, PLA 379 1693 (2015)]
26. Appendix 1 / 1 How about N > 2? N = 2 has only a single “family” whose total momentum is zero. N = 3 has an infinite number of such families.