computing ground states of spin-2 bose-einstein ... · introduction models: scalar bose{einstein...

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




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)


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).


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’;


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)


Introduction


� 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


� 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` .


Introduction


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.


Introduction


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 .


Introduction


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.


Introduction


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






Single Mode Approximation (SMA)


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.





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.





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)



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)



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)



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 .



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).



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)




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:


g0 , ξ

g−1, ξ

g−2)T φg(x) =: φg ξg. (55)



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))


Numerical methods and results

Outline

1 Introduction






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).



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


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.



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)



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)



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...


Numerical 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).



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)



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


(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


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)



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.



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.


Conclusion and remarks

Outline

1 Introduction






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.



Sichuan University and ChengDu

Thank you for attention !


Proof of Lemma 3

Outline

5 Proof of Lemma 3


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}.


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).


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