Computing ground states of spin-2 Bose-Einsteincondensates by the normalized gradient flow
Qinglin TANG
School of Mathematics, SiChuan University, China
Joint work with: Weizhu BAO and Yongjun YUAN
Modeling and Simulation for Quantum Condensation, Fluids and Information
18/11/2019-22/11/2019, IMS, NUS, Singapore
Outline
1 Introduction
2 SMA and GS in spatial-uniform system
3 Numerical methods and results
4 Conclusion and remarks
Introduction
Outline
1 Introduction
2 SMA and GS in spatial-uniform system
3 Numerical methods and results
4 Conclusion and remarks
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 3 / 41
Introduction
Models: scalar Bose–Einstein condensate
� Early experiments, bosons magnetically trapped → direction of atomic spinswere polarised & spin integral freedom is frozen → BECs of these system welldescribed by a single wave function ψ(x, t)
Mean-field approximation
I Gross-Pitaevskii equation (GPE)ab
i∂tψ(x, t) = −1
2
δEδψ
=
[−1
2∇2 + V (x) + β|ψ|2
]ψ(x, t), (1)
Energy : E(ψ) =
∫Rd
[1
2|∇ψ|2 + V |ψ|2 +
1
2β|ψ|4
]dx. (2)
aL. Pitaevskii & S. Stringari, Bose-Einstein Condensation, Oxford, 03’.bC. Pethick & H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University, 01’.
ψ : complex-valued wave function, V (x): trapping potential.β : short-range interaction ( > 0: repulsive, < 0: attractive)
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 4 / 41
Introduction
Important quantities
Mass : N (ψ) =
∫Rd|ψ(x, t)|2dx = 1, (3)
Energy : E(ψ) =
∫Rd
[1
2|∇ψ|2 + V (x)|ψ|2 +
β
2|ψ|4
]dx. (4)
Ground states φg(x): non-convex minimization problem
φg(x) = arg minφ∈SE(φ), with S = {φ | N (φ) = 1, E(φ) <∞}. (5)
Problem of interest: (non-)existence, phase diagram, numerics, etc
I Well-studied: X. Antoine, W. Bao, C. Besse, Y. Cai, I. Danaila, K. Burnett, E. Cances, C. M. Dion,
Q. Du, R. Duboscq, M. Edwards, , D. L. Feder, F. Hecht, P. Kazemi, B. I. Schneider, J. Shen, Z. Wen,
H. Wang, X. Wu, S. K. Adhikari, M. L. Chiofalo, M. P. Tosi, R. J. Dodd, etc...
Gradient flow with discrete normalization (GFDN): Bao, Du, 04’, etc
∂tφ = −1
2
δE(φ)
δφ=
[1
2∇2 − V (x)− β|φ|2
]φ(x, t), tn−1 ≤ t < tn, (6)
φ(x, tn) = σnφ(x, t−n ). (7)
σn = 1/‖φ(x, t−n )‖, φ(x, t−n ) = limt→t−n
φ(x, t).
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 5 / 41
Introduction
Important quantities
Mass : N (ψ) =
∫Rd|ψ(x, t)|2dx = 1, (3)
Energy : E(ψ) =
∫Rd
[1
2|∇ψ|2 + V (x)|ψ|2 +
β
2|ψ|4
]dx. (4)
Ground states φg(x): non-convex minimization problem
φg(x) = arg minφ∈SE(φ), with S = {φ | N (φ) = 1, E(φ) <∞}. (5)
Problem of interest: (non-)existence, phase diagram, numerics, etc
I Well-studied: X. Antoine, W. Bao, C. Besse, Y. Cai, I. Danaila, K. Burnett, E. Cances, C. M. Dion,
Q. Du, R. Duboscq, M. Edwards, , D. L. Feder, F. Hecht, P. Kazemi, B. I. Schneider, J. Shen, Z. Wen,
H. Wang, X. Wu, S. K. Adhikari, M. L. Chiofalo, M. P. Tosi, R. J. Dodd, etc...
Gradient flow with discrete normalization (GFDN): Bao, Du, 04’, etc
∂tφ = −1
2
δE(φ)
δφ=
[1
2∇2 − V (x)− β|φ|2
]φ(x, t), tn−1 ≤ t < tn, (6)
φ(x, tn) = σnφ(x, t−n ). (7)
σn = 1/‖φ(x, t−n )‖, φ(x, t−n ) = limt→t−n
φ(x, t).
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 5 / 41
Introduction
Spinor Bose–Einstein condensates2
I Confined in optical traps ⇒ spin integral freedom is released ⇒ atomic spinscan change due to interparticle interaction.
I Spin-F BECs: vector wave function Ψ = (ψF , · · · , ψ−F )T , 2F + 1 coupled GPEs.
I Experiments on spin-1, 2, 3 BECs1
I Spin-1 BEC: W. Bao, I.-L. Chern, F. Lim, Y. Zhang, H. Wang, etc
Coupled Gross-Pitaevskii Equations (CGPE)
i∂tψ±1 =(−∇2/2 + V (x) + β0ρ± β1Fz
)ψ±1 + β1F∓ψ0, (8)
i∂tψ0 =(−∇2/2 + V (x) + β0ρ
)ψ0 + β1
(F+ψ1 + F−ψ−1
). (9)
F+ = F̄− = (ψ̄1ψ0 + ψ̄0ψ−1)/√
2, Fz = |ψ1|2 − |ψ−1|2, ρ =2∑
`=−1
|ψ`|2, (10)
I Spin vector: F = (Fx, Fy, Fz)T , F± = Fx ± iFy.
I β0, β1: consts. represent spin-independent/spin-exchange interaction.
1J. Stenger et al, 98’, T. Schmaljohann et al, 04’, B. Pasquiou et al, 11’, etc2Y. Kawaguchi & M. Ueda, Spinor Bose-Einstein condensates, Phys. Rep. 12’;
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 6 / 41
Introduction
Spinor Bose–Einstein condensates2
I Confined in optical traps ⇒ spin integral freedom is released ⇒ atomic spinscan change due to interparticle interaction.
I Spin-F BECs: vector wave function Ψ = (ψF , · · · , ψ−F )T , 2F + 1 coupled GPEs.
I Experiments on spin-1, 2, 3 BECs1
I Spin-1 BEC: W. Bao, I.-L. Chern, F. Lim, Y. Zhang, H. Wang, etc
Coupled Gross-Pitaevskii Equations (CGPE)
i∂tψ±1 =(−∇2/2 + V (x) + β0ρ± β1Fz
)ψ±1 + β1F∓ψ0, (8)
i∂tψ0 =(−∇2/2 + V (x) + β0ρ
)ψ0 + β1
(F+ψ1 + F−ψ−1
). (9)
F+ = F̄− = (ψ̄1ψ0 + ψ̄0ψ−1)/√
2, Fz = |ψ1|2 − |ψ−1|2, ρ =2∑
`=−1
|ψ`|2, (10)
I Spin vector: F = (Fx, Fy, Fz)T , F± = Fx ± iFy.
I β0, β1: consts. represent spin-independent/spin-exchange interaction.
1J. Stenger et al, 98’, T. Schmaljohann et al, 04’, B. Pasquiou et al, 11’, etc2Y. Kawaguchi & M. Ueda, Spinor Bose-Einstein condensates, Phys. Rep. 12’;
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 6 / 41
Introduction
Spin-1 Bose–Einstein condensates
Total Energy
E(Ψ(·, t)) =
∫Rd
[ 1∑`=−1
(1
2|∇ψ`|2 + V (x)|ψ`|2
)+β0
2ρ2 +
β1
2|F |2
]dx. (11)�� ��|F |2 = |F+|2 + |Fz|2
I Mass conservation:
N (t) = N (Ψ(·, t)) =1∑
`=−1
∫Rd|ψ`(x, t)|2dx ≡ N (t = 0), t ≥ 0. (12)
I Magnetization conservation (−1 ≤M ≤ 1):
M(Ψ(·, t)) :=1∑
`=−1
∫Rd`|ψ`(x, t)|2dx ≡M(Ψ(·, 0)) =: M. (13)
Ground states: Φg(x) = (φg1, φg0, φ
g−1)T
Φg = arg minΦ∈SE(Φ), with (14)
S ={
Φ = (φ1, φ0, φ−1)T | N (Φ) = 1, M(Φ) = M, E(Φ) <∞}. (15)
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 7 / 41
Introduction
Spin-1 Bose–Einstein condensates
� Classification of GS according to β1 (or |F+|):
I Ferromagnetic Phase: β1 < 0, |F+(Φg)| =√
1−M2 > 0. (M2 6= 1)
I Anti-Ferromagnetic Phase: β1 > 0, |F+(Φg)| = 0.
� Single Mode Approximation (SMA) and Vanishing phenomena3:
Φgsma = (ξg1 , ξg0 , ξ
g−1)T φg(x) =: φg ξg, (16)
ξgj : real constants. φg(x): GS of specific single-component GPE.
I Ferromagnetic Phase: SMA valid for M ∈ [−1, 1].
I Anti-Ferromagnetic Phase: SMA valid for M = 0.
I Anti-Ferromagnetic Phase: M 6= 0, SMA invalid, but φg0 ≡ 0 ! Reduce totwo-component GPEs on (φ1, φ−1).
Question: How about spin-2 BEC?
3L. Lin, I.-L. Chern, Discrete Cont. Dyn-B., 14’Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 8 / 41
Introduction
Spin-1 Bose–Einstein condensates
� Classification of GS according to β1 (or |F+|):
I Ferromagnetic Phase: β1 < 0, |F+(Φg)| =√
1−M2 > 0. (M2 6= 1)
I Anti-Ferromagnetic Phase: β1 > 0, |F+(Φg)| = 0.
� Single Mode Approximation (SMA) and Vanishing phenomena3:
Φgsma = (ξg1 , ξg0 , ξ
g−1)T φg(x) =: φg ξg, (16)
ξgj : real constants. φg(x): GS of specific single-component GPE.
I Ferromagnetic Phase: SMA valid for M ∈ [−1, 1].
I Anti-Ferromagnetic Phase: SMA valid for M = 0.
I Anti-Ferromagnetic Phase: M 6= 0, SMA invalid, but φg0 ≡ 0 ! Reduce totwo-component GPEs on (φ1, φ−1).
Question: How about spin-2 BEC?
3L. Lin, I.-L. Chern, Discrete Cont. Dyn-B., 14’Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 8 / 41
Introduction
Spin-1 Bose–Einstein condensates
� Classification of GS according to β1 (or |F+|):
I Ferromagnetic Phase: β1 < 0, |F+(Φg)| =√
1−M2 > 0. (M2 6= 1)
I Anti-Ferromagnetic Phase: β1 > 0, |F+(Φg)| = 0.
� Single Mode Approximation (SMA) and Vanishing phenomena3:
Φgsma = (ξg1 , ξg0 , ξ
g−1)T φg(x) =: φg ξg, (16)
ξgj : real constants. φg(x): GS of specific single-component GPE.
I Ferromagnetic Phase: SMA valid for M ∈ [−1, 1].
I Anti-Ferromagnetic Phase: SMA valid for M = 0.
I Anti-Ferromagnetic Phase: M 6= 0, SMA invalid, but φg0 ≡ 0 ! Reduce totwo-component GPEs on (φ1, φ−1).
Question: How about spin-2 BEC?
3L. Lin, I.-L. Chern, Discrete Cont. Dyn-B., 14’Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 8 / 41
Introduction
Spin-1 Bose–Einstein condensates
� Numerics and key points: W. Bao & F. Lim, SISC, 08’.
Gradient flow with discrete normalization
Step1: evolve Gradient flow tn−1 ≤ t < tn
∂tφ±1 =(∇2/2− V (x)− β0ρ∓ β1Fz
)ψ±1 − β1F∓φ0/
√2, (17)
∂tφ0 =(∇2/2− V (x)− β0ρ
)ψ0 − β1
(F+ψ1 + F−φ−1
)/√
2. (18)
Step2: projection back to S := {Φ| N (Φ) = 1, M(Φ) = M}
φ`(x, tn) = σn` φ`(x, t−n ), ` = −1, 0, 1. (19)
KEY Point: find the third condition for the projection constants: σn` .
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 9 / 41
Introduction
Spin-2 Bose–Einstein condensates
Coupled Gross-Pitaevskii Equations (CGPE)
i∂tψ±2 = (H + β0ρ± 2β1Fz)ψ±2 + β1F∓ψ±1 + β2A00ψ̄∓2/√
5, (20)
i∂tψ±1 = (H + β0ρ± β1Fz)ψ±1 + β1(√
6F∓ψ0/2 + F±ψ±2)− β2A00ψ̄∓1/√
5, (21)
i∂tψ0 = (H + β0ρ)ψ0 +√
6β1
(F+ψ1 + F−ψ−1
)/2 + β2A00ψ̄0/
√5. (22)
H = −1
2∇2
+ V, F+ = F̄− = 2(ψ̄2ψ1 + ψ̄−1ψ−2
)+√
6(ψ̄1ψ0 + ψ̄0ψ−1
), (23)
Fz =2∑
`=−2
`|ψ`|2, ρ =2∑
`=−2
|ψ`|2, A00 =1√
5
[2ψ2ψ−2 − 2ψ1ψ−1 + ψ
20
], (24)
I Spin-singlet pair: A00.
I Spin vector: F = (Fx, Fy, Fz)T , F± = Fx ± iFy.
I β1: consts. represents spin-exchange interaction.
I β0, β2: consts. represents spin-independent interaction.
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 10 / 41
Introduction
Spin-2 Bose–Einstein condensates
Total Energy
E(Ψ(·, t)) =
∫Rd
[ 2∑`=−2
(1
2|∇ψ`|2 + V (x)|ψ`|2
)+β0
2ρ2 +
β1
2|F |2 +
β2
2|A00|2
]dx. (25)
�� ��|F |2 = |F+|2 + |Fz|2I Mass conservation:
N (t) = N (Ψ(·, t)) =2∑
`=−2
∫Rd|ψ`(x, t)|2dx ≡ N (t = 0), t ≥ 0. (26)
I Magnetization conservation (−2 ≤M ≤ 2):
M(Ψ(·, t)) :=2∑
`=−2
∫Rdl|ψ`(x, t)|2dx ≡M(Ψ(·, 0)) =: M. (27)
I Remark: Consider only M ∈ [0, 2).1). M = ±2 ⇒ reduce to single comp. GPE on ψ±2, consider M 6= ±2.
2). Φg = (φ2, φ1, φ0, φ−1, φ−2)T GS corrsp. M
⇐⇒ Φ̃g = (φ−2, φ−1, φ0, φ1, φ2)T GS corrsp. −M .
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 11 / 41
Introduction
Spin-2 Bose–Einstein condensates
Ground states: Φg(x) = (φg2, φg1, φ
g0, φ
g−1, φ
g−2)T
Φg = arg minΦ∈SE(Φ), with (28)
S ={
Φ = (φ2, · · · , φ−2)T | N (Φ) = 1, M(Φ) = M, E(Φ) <∞}. (29)
E(Φ) =
∫Rd
[ 2∑`=−2
(1
2|∇φ`|2 + V (x)|φ`|2
)+β0
2ρ2 +
β1
2
(|F+|2 + |Fz|2
)+β2
2|A00|2
]dx
� Classification4 of Ground states according to (|F+|, |A00|):
I Ferromagnetic Phase: |A00(Φg)| = 0, |F+(Φg)| > 0.
I Nematic Phase: |A00(Φg)| > 0, |F+(Φg)| = 0.
I Cyclic Phase: |A00(Φg)| = 0, |F+(Φg)| = 0.
4C.V. Cilbanu, S.-K. Yip & T.-L. Ho, PRA, 00’; M. Ueda & M. Koashi, PRA, 02’; H. Saito & M. Ueda,PRA, 05.
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 12 / 41
Introduction
Spin-2 Bose–Einstein condensates
Ground states: Φg(x) = (φg2, φg1, φ
g0, φ
g−1, φ
g−2)T
Φg = arg minΦ∈SE(Φ), with (28)
S ={
Φ = (φ2, · · · , φ−2)T | N (Φ) = 1, M(Φ) = M, E(Φ) <∞}. (29)
E(Φ) =
∫Rd
[ 2∑`=−2
(1
2|∇φ`|2 + V (x)|φ`|2
)+β0
2ρ2 +
β1
2
(|F+|2 + |Fz|2
)+β2
2|A00|2
]dx
� Classification4 of Ground states according to (|F+|, |A00|):
I Ferromagnetic Phase: |A00(Φg)| = 0, |F+(Φg)| > 0.
I Nematic Phase: |A00(Φg)| > 0, |F+(Φg)| = 0.
I Cyclic Phase: |A00(Φg)| = 0, |F+(Φg)| = 0.
4C.V. Cilbanu, S.-K. Yip & T.-L. Ho, PRA, 00’; M. Ueda & M. Koashi, PRA, 02’; H. Saito & M. Ueda,PRA, 05.
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 12 / 41
Introduction
Spin-2 Bose–Einstein condensates
I GS phase diagram5 if V (x) ≡ 0 ( bounded domain with periodic B.C.), only mass constrain.
nematic
cyclic
0
Γ fn
Γ fc
β2
ferromagnetic
β1
Γun c
�� ��Γfn = {(β1, β2)|β1 < 0, β2 = 20β1}.
Problem of interest (W. Bao & Y. Cai, Review Article, CICP, 18’.)
I Existence & (non)-uniqueness of GS?
I Phase diagram of GS when V (x) 6≡ 0? Validity of SMA? What are ξg & φg?
I Numerics: H. Wang, JCP 14’, PGF with CNFD. GFDN applicable? −→ YES!
5Y. Kawaguchi & M. Ueda, Phys. Rep., 12’.Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 13 / 41
SMA and GS in spatial-uniform system
Outline
1 Introduction
2 SMA and GS in spatial-uniform system
3 Numerical methods and results
4 Conclusion and remarks
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 14 / 41
SMA and GS in spatial-uniform system
Single Mode Approximation (SMA)
Φgsma = (ξg2 , ξg1 , ξ
g0 , ξ
g−1, ξ
g−2)T φg(x) =: φg ξg, (30)
• ξg ∈ SC =:{ξ ∈ C5 |
∑2`=−2 |ξ`|2 = 1,
∑2`=−2 `|ξ`|2 = M
}: complex const. vector.
• φg ∈ S1 =:{φ|∫Rd |φ(x)|2dx = 1
}: GS of single-component GPE.
I The total energy:
E(Φgsma) =
∫Rd
[1
2|∇φg|2 + V |φg|2 + EU (ξg) |φg|4
]dx =: Esma(φg, ξg).
Function EU (ξ) reads as:
EU (ξ)| = 1
2
(β1|F+(ξ)|2 + β2|A00(ξ)|2 + β0 + β1M
2
). (31)I EU (ξ) : energy for a spatial-uniform (V (x) ≡ 0) system on bounded domain D with
periodic B.C. (assume |D| = 1)
E(Ψ) =
∫D
[ 2∑`=−2
(1
2|∇ψ`|2 + V (x)|ψ`|2
)+β0
2ρ2 +
β1
2|F |2 +
β2
2|A00|2
]dx.
with|F |2 = |F+|2 + |Fz |2 = M2.
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 15 / 41
SMA and GS in spatial-uniform system
Single Mode Approximation (SMA)
Φgsma = (ξg2 , ξg1 , ξ
g0 , ξ
g−1, ξ
g−2)T φg(x) =: φg ξg, (30)
• ξg ∈ SC =:{ξ ∈ C5 |
∑2`=−2 |ξ`|2 = 1,
∑2`=−2 `|ξ`|2 = M
}: complex const. vector.
• φg ∈ S1 =:{φ|∫Rd |φ(x)|2dx = 1
}: GS of single-component GPE.
I The total energy:
E(Φgsma) =
∫Rd
[1
2|∇φg|2 + V |φg|2 + EU (ξg) |φg|4
]dx =: Esma(φg, ξg).
Function EU (ξ) reads as:
EU (ξ)| = 1
2
(β1|F+(ξ)|2 + β2|A00(ξ)|2 + β0 + β1M
2
). (31)
I EU (ξ) : energy for a spatial-uniform (V (x) ≡ 0) system on bounded domain D with
periodic B.C. (assume |D| = 1)
E(Ψ) =
∫D
[ 2∑`=−2
(1
2|∇ψ`|2 + V (x)|ψ`|2
)+β0
2ρ2 +
β1
2|F |2 +
β2
2|A00|2
]dx.
with|F |2 = |F+|2 + |Fz |2 = M2.
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 15 / 41
SMA and GS in spatial-uniform system
Single Mode Approximation (SMA)
Φgsma = (ξg2 , ξg1 , ξ
g0 , ξ
g−1, ξ
g−2)T φg(x) =: φg ξg, (30)
• ξg ∈ SC =:{ξ ∈ C5 |
∑2`=−2 |ξ`|2 = 1,
∑2`=−2 `|ξ`|2 = M
}: complex const. vector.
• φg ∈ S1 =:{φ|∫Rd |φ(x)|2dx = 1
}: GS of single-component GPE.
I The total energy:
E(Φgsma) =
∫Rd
[1
2|∇φg|2 + V |φg|2 + EU (ξg) |φg|4
]dx =: Esma(φg, ξg).
Function EU (ξ) reads as:
EU (ξ)| = 1
2
(β1|F+(ξ)|2 + β2|A00(ξ)|2 + β0 + β1M
2
). (31)
I EU (ξ) : energy for a spatial-uniform (V (x) ≡ 0) system on bounded domain D with
periodic B.C. (assume |D| = 1)
E(Ψ) =
∫D
[ 2∑`=−2
(1
2|∇ψ`|2 + V (x)|ψ`|2
)+β0
2ρ2 +
β1
2|F |2 +
β2
2|A00|2
]dx.
with|F |2 = |F+|2 + |Fz |2 = M2.
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 15 / 41
SMA and GS in spatial-uniform system
Single Mode Approximation (SMA)
Φgsma = (ξg2 , ξg1 , ξ
g0 , ξ
g−1, ξ
g−2)T φg(x) =: φg ξg, (32)
Φgsma = arg minΦsma∈S
E(Φsma) = arg minφ∈S1
{∫Rd
[1
2|∇φ|2 + V |φ|2 +
[minξ∈SC
EU (ξ)
]|φ|4
]dx
}.
m
Pro 1 : ξg = arg minξ∈SC
EU (ξ) = arg minξ∈SC
{β1|F+(ξ)|2 + β2|A00(ξ)|2 + β1M
2 + β0
},(33)
Pro 2 : φg = arg minφ∈S1
{∫Rd
[1
2|∇φ|2 + V |φ|2 + βgξ |φ|
4
]dx
}. (34)
βgξ = EU (ξg), S1 =:
{φ|∫Rd|φ(x)|2dx = 1
}, (35)
SC =:
{ξ ∈ C5 |
2∑`=−2
|ξ`|2 = 1,
2∑`=−2
`|ξ`|2 = M
}, (36)
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 16 / 41
SMA and GS in spatial-uniform system
Single Mode Approximation (SMA)
Φgsma = (ξg2 , ξg1 , ξ
g0 , ξ
g−1, ξ
g−2)T φg(x) =: φg ξg, (32)
Φgsma = arg minΦsma∈S
E(Φsma) = arg minφ∈S1
{∫Rd
[1
2|∇φ|2 + V |φ|2 +
[minξ∈SC
EU (ξ)
]|φ|4
]dx
}.
m
Pro 1 : ξg = arg minξ∈SC
EU (ξ) = arg minξ∈SC
{β1|F+(ξ)|2 + β2|A00(ξ)|2 + β1M
2 + β0
},(33)
Pro 2 : φg = arg minφ∈S1
{∫Rd
[1
2|∇φ|2 + V |φ|2 + βgξ |φ|
4
]dx
}. (34)
βgξ = EU (ξg), S1 =:
{φ|∫Rd|φ(x)|2dx = 1
}, (35)
SC =:
{ξ ∈ C5 |
2∑`=−2
|ξ`|2 = 1,
2∑`=−2
`|ξ`|2 = M
}, (36)
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 16 / 41
SMA and GS in spatial-uniform system
GS in spatial-uniform system: EU = β1|F+|2 + β2|A00|2 + β0 + β1M2.
Reduce from SC into SR = R5 ∩ SC ={ξ ∈ R5 |
∑2`=−2 |ξ`|2 = 1,
∑2`=−2 `|ξ`|2 = M
}.
Pro 1 : ξg = arg minξ∈SC
EU (ξ) = arg minξ∈SR
EU (ξ). (37)
{|ξ2|2 + |ξ−2|2 + |ξ0|2 + |ξ1|2 + |ξ−1|2 = 1,
2(|ξ2|2 − |ξ−2|2) + |ξ1|2 − |ξ−1|2 = M,∀M ∈ [0, 2). (38)
Lemma 1
If ξ ∈ R5, then system (38) has real solution if and only if
F 2+(ξ) + 20A2
00(ξ) ≤ 4−M2 (39)
{F+(ξ) = 2
(ξ̄2ξ1 + ξ̄−1ξ−2
)+√
6(ξ̄1ξ0 + ξ̄0ξ−1
),
A00(ξ) = (2ξ2ξ−2 − 2ξ1ξ−1 + ξ20)/√
5.(40)
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 17 / 41
SMA and GS in spatial-uniform system
GS in spatial-uniform system: EU = β1|F+|2 + β2|A00|2 + β0 + β1M2.
Lemma 2
For ∀ ξ ∈ SC , we have
|F+(ξ)|2 + 20 |A00(ξ)|2 ≤ 4−M2. (41)
By Lemma 1, ∃ ζR ∈ SR s.t.
F+(ζR) = |F+(ξ)|, A00(ζR) = |A00(ξ)|.
Hence, EU (ζR) = EU (ξ), i.e, the spatial-uniform system has real GS if the GS exists.
Lemma 3
minξ∈SC
EU (ξ) = minξ∈SR
EU (ξ) ⇐⇒ min|F+|2+20|A00|2≤4−M2
EU (F+, A00)
⇐⇒ min|F+|2+20|A00|2≤4−M2
(β1|F+|2 + β2|A00|2
). (42)
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 18 / 41
SMA and GS in spatial-uniform system
GS of the spatial-uniform system: ξg
Lemma 4
If β1 < 0 & β2 > 20β1, EU (F+, A00) attains minimum at
(F+, A00) = (√
4−M2, 0), (43)
i.e., the GS is ferromagnetic. Moreover, for ∀M ∈ [0, 2), ξg reads as
ξg =
(m4
1
16,m3
1m2
8,
√6m2
1m22
16,m1m
32
8,m4
2
16
)T, (44)
with m1 =√
2 +M and m2 =√
2−M .
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 19 / 41
SMA and GS in spatial-uniform system
GS of the spatial-uniform system (cont1.)
Lemma 4If β1 < 0 & β2 < 20β1, EU (F+, A00) attains minimum at
(F+, A00) = (0,√
4−M2/2√
5), (45)
i.e., the GS is nematic. Moreover,
I For ∀ 0 < M < 2, ξg reads as:
ξg =(√
2 +M/2, 0, 0, 0,√
2−M/2)T
. (46)
I For M = 0, ξg are not unique and reads as
Type1 : ξg = (γ1 cos θ, γ1 sin θ, γ, −γ1 sin θ, γ1 cos θ)T , (47)
Type2 : ξg =(
cos θ/√
2, sin θ/√
2, 0, sin θ/√
2, − cos θ/√
2)T
, (48)
∀ |γ| ≤ 1, γ1 =√
(1− γ2)/2, θ ∈ [0, 2π).
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 20 / 41
SMA and GS in spatial-uniform system
GS of the spatial-uniform system (cont2.)
Lemma 4If β1 > 0 & β2 > 0, EU (F+, A00) attains minimum at
(F+, A00) = (0, 0), (49)
i.e., the GS is cyclic. Moreover,
I For ∀M ∈ [0, 1], ξg reads as: m3 =√
1 +M and m4 =√
1−M .
Type1 : ξg =(m2
1/4, 0,√
2m1m2/4, 0, m22/4)T
, (50)
Type2 : ξg =(√
3m3m4/4, m23/2, −
√2m1m2/4, m
24/2,
√3m3m4/4
)T, (51)
Type3 :
ξg0 = − 3
√6
8M sin2 θ cos θ ∓−
√2
8
(2 cot(2θ) + cot θ
)g(θ),
ξg1 = 34M sin3 θ + 1
4m2
2 sin θ ±√
34g(θ), ξg−1 = sin θ − ξg1 ,
ξg2 = 18
(3M sin2 θ + 2m2
1
)cos θ ∓
√3
8g(θ) tan θ, ξg−2 = ξg2 − cos θ,
(52)
with g(θ) =√(m1m2 − 3M2 sin2 θ
)sin2 θ cos2 θ, θ ∈ (0, 2π) s.t. | sin θ| ≤ min
{m1m2√
3M, 1
}&| sin θ| 6= 0, 1.
I For M ∈ (1, 2), ξg are not unique and reads as Type 3. (52)
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 21 / 41
SMA and GS in spatial-uniform system
Single Mode Approximation (SMA)
I GS energy in the spatial-uniform system:
βgξ(M) =: EU (ξg) =β0
2+
2β1, β1 < 0 & β2 > 20β1, Ferromag.,
β210
+(20β1−β2)M2
40, β2 < 0 & β2 < 20β1, Nematic,
β1M2
2, β1 > 0 & β2 > 0, Cyclic.
(53)
I Solving the ground state φg of the following single component GPE:
i∂tψ =
[−1
2∇2 + V (x) + βgξ(M)|ψ|2
]ψ, with
∫Rd|ψ|2dx = 1. (54)
I The SMA read as:
Φgsma = (ξg2 , ξg1 , ξ
g0 , ξ
g−1, ξ
g−2)T φg(x) =: φg ξg. (55)
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 22 / 41
SMA and GS in spatial-uniform system
A Summarise on GS in spatial uniform system
I GS phase diagram of spatial uniform system with both mass & magnetisation
conservation constrains is same as the one with only mass conservation constrain.
nematic
cyclic
0
Γ fn
Γ fc
β2
ferromagnetic
β1
Γun c
GS is unique for Ferrormagnetic
phase, Nematic with M 6= 0.
GS is not-unique for: Cyclic phase,
Nematic with M = 0.
Different with V (x) 6≡ 0, e.g., GS is
unique for Cyclic phase with M 6= 0.
I V (x) 6≡ 0 : the SMA is not always valid, e.g, Nematic phase: Φg = (φg2, 0, 0, 0, φg−2)T .
i). -5 0 5
0
0.5
1
1.5M = 0
φ0
g/φ
2
g
φ-2
g/φ
2
g
φ1
g/φ
2
g
φ-1
g/φ
2
g
ii). -5 0 5
0
0.5
1
1.5
2M = 0.5
φ0
g/φ
2
g
φ-2
g/φ
2
g
φ1
g/φ
2
g
φ-1
g/φ
2
g
iii). -5 0 5
0
0.5
1
1.5M = 1.5
φ0
g/φ
2
g
φ-2
g/φ
2
g
φ1
g/φ
2
g
φ-1
g/φ
2
g
Figure: SMA valid for M = 0 (i)). Invalid for M 6= 0 (ii)&iii))
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 23 / 41
SMA and GS in spatial-uniform system
A Summarise on GS in spatial uniform system
I GS phase diagram of spatial uniform system with both mass & magnetisation
conservation constrains is same as the one with only mass conservation constrain.
nematic
cyclic
0
Γ fn
Γ fc
β2
ferromagnetic
β1
Γun c
GS is unique for Ferrormagnetic
phase, Nematic with M 6= 0.
GS is not-unique for: Cyclic phase,
Nematic with M = 0.
Different with V (x) 6≡ 0, e.g., GS is
unique for Cyclic phase with M 6= 0.
I V (x) 6≡ 0 : the SMA is not always valid, e.g, Nematic phase: Φg = (φg2, 0, 0, 0, φg−2)T .
i). -5 0 5
0
0.5
1
1.5M = 0
φ0
g/φ
2
g
φ-2
g/φ
2
g
φ1
g/φ
2
g
φ-1
g/φ
2
g
ii). -5 0 5
0
0.5
1
1.5
2M = 0.5
φ0
g/φ
2
g
φ-2
g/φ
2
g
φ1
g/φ
2
g
φ-1
g/φ
2
g
iii). -5 0 5
0
0.5
1
1.5M = 1.5
φ0
g/φ
2
g
φ-2
g/φ
2
g
φ1
g/φ
2
g
φ-1
g/φ
2
g
Figure: SMA valid for M = 0 (i)). Invalid for M 6= 0 (ii)&iii))
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 23 / 41
Numerical methods and results
Outline
1 Introduction
2 SMA and GS in spatial-uniform system
3 Numerical methods and results
4 Conclusion and remarks
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 24 / 41
Numerical methods and results
Gradient flow with discrete nomalization (GFDN)
I Denote Φ(x, t) = (φ−2, φ−1, φ0, φ1, φ2)T
∂tφ`(x, t) =(∇2/2− V (x)− a`(Φ)
)φ` − f`(Φ), t ∈ [tn−1, tn], (56)
φ`(x, tn) = σn` φ`(x, t−n ), ` = −2,−1, 0, 1, 2. (57)
I Here, φ`(x, t−n ) = limt→t−n
φ`(x, t).
σn` (` = −2,−1, 0, 1, 2): projection constants to be determined.
I Mass and Magnetization conservation laws lead to:
2∑`=−2
(σn` )2‖φ`(·, t−n )‖2 = 1,
2∑`=−2
`(σn` )2‖φ`(·, t−n )‖2 = M. (58)
Three more constrains are needed to determine all σn` (` = −2,−1, 0, 1, 2).
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 25 / 41
Numerical methods and results
Gradient flow with discrete nomalization (GFDN)
I Denote Φ(x, t) = (φ−2, φ−1, φ0, φ1, φ2)T
∂tφ`(x, t) =(∇2/2− V (x)− a`(Φ)
)φ` − f`(Φ), t ∈ [tn−1, tn], (56)
φ`(x, tn) = σn` φ`(x, t−n ), ` = −2,−1, 0, 1, 2. (57)
I Here, φ`(x, t−n ) = limt→t−n
φ`(x, t).
σn` (` = −2,−1, 0, 1, 2): projection constants to be determined.
I Mass and Magnetization conservation laws lead to:
2∑`=−2
(σn` )2‖φ`(·, t−n )‖2 = 1,
2∑`=−2
`(σn` )2‖φ`(·, t−n )‖2 = M. (58)
Three more constrains are needed to determine all σn` (` = −2,−1, 0, 1, 2).
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 25 / 41
Numerical methods and results
Continuous normalized gradient flow (CNGF) 6
∂tφ` =(∇2/2− V (x)− a`(Φ)
)φ` − f`(Φ) +
[µ(t) + ` λ(t)
]φ` =: (HΦ)` (59)
I µ(t) & λ(t) are chosen s.t. the CNGF (59) satisfies:
(1). conserve mass & magnetization. (2). diminishing the energy
µ(t) =R(t)D(t)−M(t)F(t)
R(t)N (t)−M2(t), λ(t) =
N (t)F(t)−M(t)D(t)
R(t)N (t)−M2(t), (60)
D(t) =
2∑`=−2
∫Rdφ̄`(HΦ
)3−` dx, F(t) =
2∑`=−2
∫Rd
` φ̄`(HΦ
)3−` dx, (61)
R(t) =
2∑`=−2
`2‖φ`‖2, N (t) =
2∑`=−2
‖φ`‖2, M(t) =
2∑`=−2
`‖φ`‖2 (62)
I CNGF (59) satisfies (1) & (2), i.e.
N (Φ(·, t)) ≡ 1, M(Φ(·, t)) ≡M, E(Φ(·, t)) ≤ E(Φ(·, s)) for ∀ t ≥ s ≥ 0. (63)
6H. Wang, A projection gradiant method for computing GS of spin-2 BECs, JCP, 14’.Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 26 / 41
Numerical methods and results
Continuous normalized gradient flow (CNGF) 6
∂tφ` =(∇2/2− V (x)− a`(Φ)
)φ` − f`(Φ) +
[µ(t) + ` λ(t)
]φ` =: (HΦ)` (59)
I µ(t) & λ(t) are chosen s.t. the CNGF (59) satisfies:
(1). conserve mass & magnetization. (2). diminishing the energy
µ(t) =R(t)D(t)−M(t)F(t)
R(t)N (t)−M2(t), λ(t) =
N (t)F(t)−M(t)D(t)
R(t)N (t)−M2(t), (60)
D(t) =
2∑`=−2
∫Rdφ̄`(HΦ
)3−` dx, F(t) =
2∑`=−2
∫Rd
` φ̄`(HΦ
)3−` dx, (61)
R(t) =
2∑`=−2
`2‖φ`‖2, N (t) =
2∑`=−2
‖φ`‖2, M(t) =
2∑`=−2
`‖φ`‖2 (62)
I CNGF (59) satisfies (1) & (2), i.e.
N (Φ(·, t)) ≡ 1, M(Φ(·, t)) ≡M, E(Φ(·, t)) ≤ E(Φ(·, s)) for ∀ t ≥ s ≥ 0. (63)
6H. Wang, A projection gradiant method for computing GS of spin-2 BECs, JCP, 14’.Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 26 / 41
Numerical methods and results
Additional constrains on σn` (` = −2,−1, 0, 1, 2)
I GFDN (56) can be viewed as applying a time-splitting scheme to the CNGF (59):
∂tφ` =(∇2/2− V (x)− a`(Φ)
)φ` − f`(Φ) (64)
∂tφ` =[µ(t) + ` λ(t)
]φ`. (65)
I Solving ODEs (65), we have
φ`(x, tn) = φ`(x, tn−1) exp
(∫ tn
tn−1
[µ(s) + `λ(s)] ds
)=: φ`(x, tn−1)σ̃`(tn), (66)
I Projection constant σn` is an approximation of σ̃`(tn).
I Relation between σ̃`(tn)
σ̃2(tn) σ̃−2(tn) = (σ̃0(tn) )2, σ̃1(tn) σ̃−1(tn) = (σ̃0(tn) )2, σ̃2(tn) σ̃0(tn) = (σ̃1(tn) )2.
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 27 / 41
Numerical methods and results
Additional constrains on σn` (` = −2,−1, 0, 1, 2)
I GFDN (56) can be viewed as applying a time-splitting scheme to the CNGF (59):
∂tφ` =(∇2/2− V (x)− a`(Φ)
)φ` − f`(Φ) (64)
∂tφ` =[µ(t) + ` λ(t)
]φ`. (65)
I Solving ODEs (65), we have
φ`(x, tn) = φ`(x, tn−1) exp
(∫ tn
tn−1
[µ(s) + `λ(s)] ds
)=: φ`(x, tn−1)σ̃`(tn), (66)
I Projection constant σn` is an approximation of σ̃`(tn).
I Relation between σ̃`(tn)
σ̃2(tn) σ̃−2(tn) = (σ̃0(tn) )2, σ̃1(tn) σ̃−1(tn) = (σ̃0(tn) )2, σ̃2(tn) σ̃0(tn) = (σ̃1(tn) )2.
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 27 / 41
Numerical methods and results
Additional constrains on σn` (` = −2,−1, 0, 1, 2)
I GFDN (56) can be viewed as applying a time-splitting scheme to the CNGF (59):
∂tφ` =(∇2/2− V (x)− a`(Φ)
)φ` − f`(Φ) (64)
∂tφ` =[µ(t) + ` λ(t)
]φ`. (65)
I Solving ODEs (65), we have
φ`(x, tn) = φ`(x, tn−1) exp
(∫ tn
tn−1
[µ(s) + `λ(s)] ds
)=: φ`(x, tn−1)σ̃`(tn), (66)
I Projection constant σn` is an approximation of σ̃`(tn).
I Relation between σ̃`(tn)
σ̃2(tn) σ̃−2(tn) = (σ̃0(tn) )2, σ̃1(tn) σ̃−1(tn) = (σ̃0(tn) )2, σ̃2(tn) σ̃0(tn) = (σ̃1(tn) )2.
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 27 / 41
Numerical methods and results
Projection constants: σn` (` = −2,−1, 0, 1, 2)
Constrains on σn`
σn2 σn−2 = (σn0 )2, σn1 σ
n−1 = (σn0 )2, σn2 σ
n0 = (σn1 )2, (67)
2∑`=−2
(σn` )2‖φ`(·, t−n )‖2 = 1,
2∑`=−2
`(σn` )2‖φ`(·, t−n )‖2 = M. (68)
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 28 / 41
Numerical methods and results
Lemma 6: formula of σn` , ∀M ∈ [0, 2) & sufficiently small time step ∆t
(a). If M = 1 & ‖φ2(·, t−n )‖ = 0, then for ` = −2,−1, 2
σn1 = 1/‖φ1(·, t−n )‖, σn0 ∈ R+, σn` =(σn1)`/(σn0)`−1
. (69)
(b). If M = 0 & ‖φ−2(·, t−n )‖ = ‖φ−1(·, t−n )‖ = 0, then for ` = −2,−1, 2
σn0 = 1/‖φ0(·, t−n )‖, σn1 ∈ R+, σn` =(σn1)`/(σn0)`−1
. (70)
(c). Otherwise, we have for ` = −2,−1, 1, 2
σn0 =1√∑2
`=−2 λ`n‖φ`(·, t−n )‖2
, σn` =√λ`nσ
n0 . (71)
where λn is the unique positive solution of the following equation:
(2−M)‖φ2(·, t−n )‖2λ4n + (1−M)‖φ1(·, t−n )‖2λ3
n −M‖φ0(·, t−n )‖2λ2n
−(1 +M)‖φ−1(·, t−n )‖2λn − (2 +M)‖φ−2(·, t−n )‖2 = 0 (72)
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 29 / 41
Numerical methods and results
Full discretization of the GFDN
� Time discretization: Back-Euler (BE) scheme
for Gradient Flowφ∗` − φn`
∆t=
(∇2/2− V (x)− a`(Φn)
)φ∗` − f`(Φn), (73)
φn+1` = σnl φ
∗` , l = 2, 1, 0,−1,−2. (74)
for Projected Gradient Flowφ∗` − φn`
∆t=
(∇2/2− V (x)− a`(Φn)
)φ∗` − f`(Φn) + [µ(Φn) + `λ(Φn)]φ∗n, (75)
φn+1` = σn` φ
∗` , ` = 2, 1, 0,−1,−2. (76)
� Spatial discretization: FDM, Fourier spectral methods...
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 30 / 41
Numerical Results
Numerical methods and results
Phases diagram of GS in spatial non-uniform sys. V (x) = |x|2/2.
I Three phases: ferrormagnetic, nematic, cyclic. The boundaries are:
Γfn ={
(β1, β2) | β1 < 0, β2 = 20β1
}, Γfc =
{(β1, β2) | β1 = 0, β2 > 0
}, (77)
Γunc ={
(β1, β2) | β2 = 0, β1 > 0}, Γnnc =
{(β1, β2) | β1 > 0, β2 = fb(β1)
}. (78)
fb(β1) = (0.0054β31 − 0.468β2
1 + 15.535β1)/1000. (79)
nematic
cyclic
0
Γ fn
Γ fc
β2
ferromagnetic
β1
Γun c
0
Γ fc
Γ fn
ferromagnetic Γnn c
β2
β1
nematic
cyclic
Figure: V (x) = 0 (left) & V (x) 6≡ 0 (right). And boundary of the three phases (blue lines).
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 32 / 41
Numerical methods and results
Masses of the GS, Wave function Φg = (φ2, φ1, φ0, φ−1, φ−2)
Figure: Masses N` = ‖φ`‖2 W.R.T. different β1 & β2 for M = 0.5
I For M ∈ (0, 2), the GS are in the form of:
Ferromagnetc : ΦFg =
(φF2 , φ
F1 , φ
F0 , φ
F−1, φ
F−2
), all φF` > 0 (80)
Nematic : ΦNg =
(φN2 , 0, 0, 0, φN−2
), all φN` > 0 (81)
Cyclic : ΦCg =
(φC2 , 0, 0, φC−1, 0
), all φC` > 0 (82)
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 33 / 41
Numerical methods and results
Validity of SMA in spatial non-uniform system: Φ = ξgφg(x)
� The validity regimes of SMA are:
I i). ∀ M ∈ [0, 2) : whole regime of ferromagnetic phase −→ proved7.
I ii). M = 0 : whole regime of nematic and cyclic phase −→ proved for cyclic8.
I iii) For M 6= 0 : SMA invalid, 5-comp. system reduce to 2-comp. sys., i.e.:
Nematic : (φ2, φ−2)T , Cyclic : (φ2, φ−1).
−→ proved partially for nematic, while not yet for cyclic8.
7Y. Cai, in preparation, 2018Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 34 / 41
Numerical methods and results
(Non)-Uniqueness of GS in spatial non-uniform system
� The GS are not unique at8:
I i) M ∈ [0, 2) : along the boundary Γfn, Γfc and Γnc.
I ii) M = 0, the whole regime of nematic and cyclic phases.
� ∀M ∈ (0, 2), GS are unique for all the three phases
proved9 in the regime
• Ferromagnetic Phase: β0 + 4β1 ≥ 0.
• Nematic Phase: β0 + β1/5 ≥ 0.
� Cyclic phases, different for V (x) = 0 & V (x) 6≡ 0 :
• GS not-unique for V (x) = 0.
• Phase-shift in the diagram
0
Γ fc
Γ fn
ferromagnetic Γnn c
β2
β1
nematic
cyclic
8Y. Cai, in preparation, 2018Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 35 / 41
Numerical methods and results
Example of Non-uniqueness of GS V (x) = x2/2, β0 = 100
I (a) Γfn with M = 0.5: β1 = −1, β2 = −20.
I (b) Nematic phase with M = 0: β1 = 1, β2 = −2,M = 0.
(a). -8 0 8x
0
0.15
0.3
M=0.5
φ0
φ2
φ−2
φ1
φ−1
-8 0 8x
0
0.15
0.3
M=0.5
(b). -8 0 8x-0.2
-0.1
0
0.1
0.2
M=0
-8 0 8x
-0.1
0
0.1
0.2
M=0
Figure: Plot of the wave function of the GS φ` (` = 2, 1, 0,−1,−2)
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 36 / 41
Numerical methods and results
Numerical results in 2D (with optical lattice potential)
Optical lattice potential: V (x) = 12 (x2 + y2) + 10
[sin2(πx2 ) + sin2(πy2 )
]
I Case 1. Ferromagnetic interaction, β0 = 100, β1 = −1 and β2 = −5;
I Case 2. Nematic interaction, β0 = 100, β1 = −1 and β2 = −25;
I Case 3. Cyclic interaction, β0 = 100, β1 = 10 and β2 = 2.
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 37 / 41
Numerical methods and results
Numerical results in 2D (with optical lattice potential)
a).
b).
c).
Figure: M = 1. Contour plots of the components of the ground states φg` (from left to right,` = 2, 1, 0,−1,−2): a). Ferromagnetic phase. b). Nematic phase. c). Cyclic phase.
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 38 / 41
Conclusion and remarks
Outline
1 Introduction
2 SMA and GS in spatial-uniform system
3 Numerical methods and results
4 Conclusion and remarks
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 39 / 41
Conclusion and remarks
Summary
I GS of spatial-uniform system. Also of the non-uniform case if SMA if valid.
I Additional conditions for projection constant, extend GFDN.
I Numerically characterising properties of GS: phase diagram of GS, validityregime of SMA, vanishing component.
Remarks: Theoretical Justification on
I Validity of SMA, vanishing phenomena for Nematic/Cyclic phase.
I Shifted boundary between Cyclic phase v.s. Nematic phase.
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 40 / 41
Conclusion and remarks
Sichuan University and ChengDu
Thank you for attention !
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 41 / 41
Proof of Lemma 3
Outline
5 Proof of Lemma 3
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 42 / 41
Proof of Lemma 3
I (1) If A00(ξ) = 0. Notice that
ξ ∈ S =⇒ ζ = (|ξ2|, |ξ1|, |ξ0|, |ξ−1|, |ξ−2|)T ∈ S, (83)
hence system (38) is fulfilled for ζ and by lemma 2.1, we have
|F+(ξ)|2 + 20|A00(ξ)|2 = |F+(ξ)|2 ≤ F 2+(ζ) ≤ F 2
+(ζ) + 20A200(ζ) ≤ 4−M2. (84)
I (2) If A00(ξ) 6= 0. Notice that
maxξ∈S
{|F+(ξ)|2 + 20|A00(ξ)|2 +M2} = max
ξ∈S
{|F+(ξ)|2 + 20|A00(ξ)|2 + F 2
z (ξ)}
≤ maxζ∈S1
{|F+(ζ)|2 + 20|A00(ζ)|2 + F 2
z (ζ)}
(85)
to prove the inequality (41), we only need to show
maxζ∈S1
{|F+(ζ)|2 + 20|A00(ζ)|2 + F 2
z (ζ)}≤ 4. (86)
�� ��S1 ={ζ ∈ C5 |
∑2`=−2 |ζl|
2 = 1}, S =
{ζ ∈ C5 |
∑2`=−2 |ζl|
2 = 1,∑2`=−2 l|ζl|
2 = M}.
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 43 / 41
Proof of Lemma 3
I To this end, we consider the auxiliary minimization problem
ζg := arg minζ∈S1F(ζ) = arg min
ζ∈S1
{−[|F+(ζ)|2 + 20|A00(ζ)|2 + F 2
z (ζ)]},(87)
= −maxζ∈S1
{|F+(ζ)|2 + 20|A00(ζ)|2 + F 2
z (ζ)}. (88)
I ζg satisfies the Euler-Lagrange equation associated with problem (87)
∇ζ̄ F(ζ) = λζ ζ (89)
where λζ ∈ R is the Lagrange multiplier.
I Denoting ηg = (ζg−2,−ζg−1, ζ
g0 ,−ζ
g1 , ζ
g2 )T , we have{
ζ̄g · ∇ζ̄ F(ζg) = λζg
ηg · ∇ζ̄ F(ζg) = λζg ηg · ζ=⇒
{−2(|F+|2 + 20|A00|2 + F 2
z ) = λζg
(λζg + 8)δ = 0,(90)
I Therefore, we have{λζg = −8
|F+(ζg)|2 + 20|A00(ζg)|2 + F 2z (ζg) = 4,
=⇒ F(ζg) = −4. (91)
Noticing (88), we have (86), hence we prove the inequality (41).
Q. Tang (Sch. of Math., SCU, China) GS computation of Spin 2 BEC IMS, 19/Nov/2019, Singapore 44 / 41
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