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Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory

Tien-Tsan Shieh

joint work with

I-Liang Chern and Chiu-Fen Chou

National Center of Theoretical Science

December 19, 2015

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 1 / 41

Outline

Introduction to Spin-1 Bose-Einstein condensate Thomas-Fermi Approximation of the spin-1 BEC Γ-convergence result of the spin-1 BEC

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 2 / 41

The Bose-Einstein condensation (BEC)

In 1925 Einstein and Bose predicted a new state of matter for very dilute Boson gas which tend to occupy the state of the lowest energy at very low temperature and behave as a coherent matter wave.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 3 / 41

Realization of BEC

BECs were realized in lab by E. Cornell, W. Ketterle and C. Wieman (1995).

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 4 / 41

The mean field model for BEC

N particle system: wave function ΨN(x1, · · · , xN, t), Hamiltonian: HN =

N

• j=1
• − 2

2Ma ∇2

j + V (xj)

• +
• 1≤j<k≤N

Vint(xj − xk), Ultracold and dilute gases, the mean field approximation: Vint(xj − xk) ≈ gδ(xj − xk) Hartree ansatz: all boson particles are in the same quantum state ΨN(x1, · · · , xN, t) =

N

• j=1

ψ(xj, t).

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 5 / 41

The Gross-Pitaeviskii equation

Hamiltonian: H = 2 2Ma |∇ψ|2 + V (x)|ψ|2 + β 2 |ψ|4 , β = gN Energy E[ψ] =

• H dx.

Gross-Pitaevskii equation: i∂tψ = δE/δψ∗. i∂tψ = − 2 2Ma ∇2ψ + V (x)ψ + β|ψ|2ψ ψ wave function V (x) trap potential: V (x) = 1

2

3

i=1 ω2 i x2 i .

Interaction: repulsive if β > 0, attractive if β < 0.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 6 / 41

One-, multi-component and spinor BECs

One-component BECs: atoms with a single quantum state are trapped. E.g. Using magnetic trap Two-component BECs: mixture of two different species of bosons. E.g. two isotopes of the same elements, or two different elements Spinor BECs: mixture of different hyperfine states of the same isotopes. E.g. Spin-1 atoms using optical trap. There are 3 hyperfine states mF = 1, 0, −1

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 7 / 41

Spinor BECs

Spin-1 atom has 3 hyperfine states: mF = 1, 0, −1. Vector order parameter Ψ = (ψ1, ψ0, ψ−1). Associate with a spinor Ψ, the spin vector F = Ψ†FΨ ∈ R3, which is just like a magnetic dipole moment. F = (Fx, Fy, Fz) is the spin-1 Pauli operator:

Fx = 1 √ 2   1 1 1 1   , Fy = i √ 2   −1 1 −1 1   , Fz =   1 −1   .

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 8 / 41

G-P equation for spin-1 BECs

Hamiltonian: H = 2 2Ma |∇Ψ|2 + V (x)|Ψ|2 + cn 2 |Ψ|4 + cs 2 |Ψ†FΨ|2 |Ψ|2 · |Ψ|2: spin-independent interaction |Ψ†FΨ|2: spin-spin interaction (spin-exchange). The total energy E[Ψ] =

• H dx.

The G-P equation i∂tΨ = δE δΨ†

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 9 / 41

Physical parameters

H = 2 2Ma |∇Ψ|2

• Hkin

+ V (x)|Ψ|2

• Hpot

+ cn 2 |Ψ|4

Hn

+ cs 2 |Ψ†FΨ|2

• Hs

interaction > 0 < 0 cn spin-independent repulsive attractive cs spin-exchange antiferromagnetic ferromagnetic cn cs

87Rb

7.793

• 0.0361

ferromagnetic

23Na

15.587 0.4871 anti-ferromagnetic

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 10 / 41

Spinor BEC in uniform magnetic field

Under an uniform magnetic field, we have to consider additional Zeeman shift energy in the Hamiltonian. Hamiltonian H = Hkin + Hpot + Hn + Hs + HZee Zeeman shift energy: Suppose magnetic field B ˆ z, HZee =

1

• j=−1

Ej(B)nj = q(n1 + n−1) + p(n1 − n−1) + E0n where nj = |ψj|2 and p = 1 2(E−1 − E1) ≈ −µBB 2 q = 1 2(E−1 + E1 − 2E0) ≈ µ2

BB2

4Ehfs

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 11 / 41

Gauge invariants and conservation laws

Energy E[Ψ] =

• (Hkin + Hpot + Hn + Hs + HZee) dx

Gauge invariant: energy is invariant under transform Ψ → eiφRz(α)Ψ This leads to two conservation laws:

◮ Total number of atoms

• (|ψ1|2 + |ψ0|2 + |ψ−1|2) dx = N

◮ Total magnetization

• (|ψ1|2 − |ψ−1|2) dx = M

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 12 / 41

The ground state problem

min E[Ψ] subject to

• n(x) dx = N,
• m(x) dx = M.

E[Ψ] =

• H dx

H = Hkin + Hpot + Hn + Hs + HZee nj = |ψj|2, n = n1 + n0 + n−1 m = n1 − n−1 Set uj = |ψj|, j = 1, 0, −1.

E[u] =

• R3

2 2Ma

1

• j=−1

|∇uj|2 + cn 2 |u|4 + V (x)|u|2 +cs 2

• 2u2

0(u1 − sgn(cs) u−1)2 + (u2 1 − u2 −1)2

+

• q(u2

1 + u2 −1)

• dx + E0N + pM.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 13 / 41

The goal of this project

Show the phase separation do occur in the Ground states Give a complete phase diagram Characterize the patterns of the Ground states The problem could be formulated as a problem in calculus of variation.

inf

• R3

2 2Ma

1

• j=−1

|∇uj|2 + cn 2 |u|4 + V (x)|u|2 +cs 2

• 2u2

0(u1 − sgn(cs) u−1)2 + (u2 1 − u2 −1)2

+

• p(u2

1 − u2 −1) + q(u2 1 + u2 −1)

• dx
• subject to the constrains
• R3 u2

1 + u2 0 + u2 −1 dx = N,

• R3 u2

1 − u2 −1 dx = M.

In particular, we are interesting in the case that ǫ =

2 2Mi ≪ 1.

The problem becomes a singular perturbation problem.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 14 / 41

The corresponding nonlinear eigenvalue problem

The corresponding Euler-Lagrange equation is (µ + λ)u1 =

• − 2

2m∆ + V (x) + q + c0n

• u1 + c2(n1 + n0 − n−1)u1 + c2u∗

−1u0 2

µu0 =

• − 2

2m∆ + V (x) + c0n

• u0 + 2c2(n1 − n−1)u0 + c2u1u−1u∗

(µ − λ)u−1 =

• − 2

2m∆ + V (x) + q + c0n

• u−1 + c2(n−1 + n0 − n1)u−1 + c2u∗

1 u2

We denote that n1 = |u1|2, n0 = |u0|2, n−1 = |u−1|2.

Here, µ and λ are the two Lagrange multipliers corresponding to the two constraints.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 15 / 41

A brief survey on the gradient theory of phase transition

Potential W : Rn → R+.

• Ω

ǫ2 2 |∇u|2 + W (u(x)) dx inf

• Ω
• ǫ|∇u|2 + 1

ǫ W (u)

• : u ∈ W 1,2(Ω),
• Ω

u(x) dx = m

• ,

Scalar case (n = 1): Modica[87], Sternberg[88], Kohn and Sternberg[89]. Vector case (n > 1): Sternberg[91], Fonseca and Tartar[89]. More general perturbation: Owen[88], Owen and Sternberg[92], Ishige[94]. Dirichlet boundary condition: Owen, Rubinstein and Sternberg[90], Ishige[96]. Higer dimension transition: Lin, Pan and Wang[12]

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 16 / 41

The Thomas-Fermi Approximation

Ignoring the kinetic energy, we consider the problem inf

• R3

cn 2 |u|4 + V (x)|u|2 +cs 2

• 2u2

0(u1 − sgn(cs) u−1)2 + (u2 1 − u2 −1)2

+

• p(u2

1 − u2 −1) + q(u2 1 + u2 −1)

• dx
• subject to the constrains
• R3 u2

1 + u2 0 + u2 −1 dx = N,

• R3 u2

1 − u2 −1 dx = M.

In order to make the problem simple, we consider the case when trap potential is V (x) =

• n Ω,

+∞

• n Rn\Ω.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 17 / 41

The Phase Diagram

Minimize

• Ω

cn 2 |u|4 + cs 2

• 2u2

0(u1 − sgn(cs) u−1)2 + (u2 1 − u2 −1)2

+

• p(u2

1 − u2 −1) + q(u2 1 + u2 −1)

• dx

subject to the constrains

• Ω u2

1 + u2 0 + u2 −1 dx = N,

• Ω u2

1 − u2 −1 dx = M.

Anti-ferromagnetic (cs > 0) When q > q2, we have NS + MS state.

(0, u0, 0) + (u1, 0, 0)

When q1 < q < q2, we have NS + 2C state.

(0, u0, 0) + (u1, 0, u−1)

When q < q1, we have 2C state.

(u1, 0, u−1)

Ferromagetic (cs < 0) When q > 0, we have 3C state.

(u1, u0, u−1)

When q < 0, we have MS + MS state.

(u1, 0, 0) + (0, 0, u−1)

Note: Here, the notation NS + MS means that there is a measurable set U ⊂ Ω such that u(x) = a for x ∈ U and u(x) = b for x ∈ Ω \ U.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 18 / 41

Phase Diagram for anti-ferromagnetic spin 1 BEC (cs > 0)

Minimize

• Ω

cn 2 |u|4 + cs 2

• 2u2

0(u1 − sgn(cs) u−1)2 + (u2 1 − u2 −1)2

+

• p(u2

1 − u2 −1) + q(u2 1 + u2 −1)

• dx

subject to the constrains

• Ω u2

1 + u2 0 + u2 −1 dx = N,

• Ω u2

1 − u2 −1 dx = M.

Anti-Ferromagnetic cs > 0

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 19 / 41

Phase Diagram for ferromagnetic spin-1 BEC (cs < 0)

Minimize

• Ω

cn 2 |u|4 + cs 2

• 2u2

0(u1 − sgn(cs) u−1)2 + (u2 1 − u2 −1)2

+

• p(u2

1 − u2 −1) + q(u2 1 + u2 −1)

• dx

subject to the constrains

• Ω u2

1 + u2 0 + u2 −1 dx = N,

• Ω u2

1 − u2 −1 dx = M.

Ferromagnetic cs < 0

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 20 / 41

Key idea

HTF(u) := 1 2|u2

1 + u2 0 + u2 −1|2

+ α 2

• 2u2

0(u1 − sgn(α) u−1)2 + (u2 1 − u2 −1)2

+ q(u2

1 + u2 −1)

Try to find suitable choice of β1, β2, β3 such that W (u) := HTF(u) + β1(u2

1 + u2 0 + u2 −1) + β2(u2 1 − u2 −1) + β3

is a double-well potential, that is W (a) = W (b) = 0, W (u) > 0 for u ∈ R3

+\{a, b} that satisfying the constrains.

Main trick: to complete a perfect square. x4 − 2ax2 + a2 = (x2 − a)2

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 21 / 41

Anti-ferromagnetic: cs > 0, q > q2 implies NS + MS state

Theorem

For 0 < α ≤ 1 and q > q2 :=

• 1 −

1 (α + 1)1/2 n +

• (α + 1)1/2 − 1
• m
• .,

the global minimizer of the variational problem of TF approximation with finite domain Ω subject to the constraints of total mass and total magnetization takes the form u = a χU + b χΩ\U where U ⊂ Ω is a measurable set of size |U| = (α + 1)1/2m n + ((α + 1)1/2 − 1)m |Ω| and a = (

• A

(α + 1)1/2 , 0, 0), and b = (0, √ A, 0) and A =

• n +
• (α + 1)1/2 − 1
• m
• .

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 22 / 41

Sketch proof: (α > 0, q > q2) (I)

2HTF = (u2

1 + u2 0 + u2 −1)2 + α(u2 1 − u2 −1)2 + 2αu2 0(u1 − u−1)2 + 2q(u2 1 + u2 −1)

= (1 + α)u4

1 + u4 0 + (1 + α)u4 −1 + 2(1 − α)u2 1u2 −1

+ 2u2

• (u2

1 + u2 −1) + α(u1 − u−1)2

+ 2q(u2

1 + u2 −1)

=

• (1 + α)1/2u2

1 + u2 0 +

1 − α (1 + α)1/2 u2

−1

2 + 4α 1 + αu4

−1 + 2q(u2 1 + u2 −1)

+ 2u2

• 1 + α − (1 + α)1/2

u2

1 +

• 1 + α −

1 − α (1 + α)1/2

• u2

−1 − 2αu1u−1

• Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science)

Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 23 / 41

Sketch proof: (α > 0, q > q2) (II)

W (u) := HTF(u) + β1 2 (u2

1 + u2 0 + u2 −1) + β2

2 (u2

1 − u2 −1) + A2

2 , 2W =

• (α + 1)1/2u2

1 + u2 0 +

1 − α (1+α)1/2 u2

−1 − A

2 + 2u2

• (α + 1) − (α + 1)1/21/2

u1 − α ((α + 1) − (α + 1)1/2)1/2 u−1 2 + 2α + 4 − 2(α + 1)1/2 (α + 1) − (α + 1)1/2 u2

−1u2 0 +

4α α + 1 u4

−1

+

• β1 + β2 + 2q + 2A(α + 1)1/2

u2

1 + (β1 + 2A)u2

+

• β1 − β2 + 2q + 2A

1 − α (α + 1)1/2

• u2

−1.

We choose β1 and β2 to satisfy

• β1 + β2 + 2q + 2A(α + 1)1/2

= β1 + 2A = , β1 − β2 + 2q + 2A 1 − α (α + 1)1/2 = 4A

• 1

(α + 1)1/2 − 1

• + 4q.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 24 / 41

Sketch proof: (α > 0, q > q2) (III)

2W =

• (α + 1)1/2u2

1 + u2 0 +

1 − α (1 + α)1/2 u2

−1 − A

2 + 2u2

• (α + 1) − (α + 1)1/21/2

u1 − α ((α + 1) − (α + 1)1/2)1/2 u−1 2 + 2α + 4 − 2(α + 1)1/2 (α + 1) − (α + 1)1/2 u2

0u2 −1 +

4α α + 1 u4

−1 + 4 (q − q2) u2 −1.

Therefore, we have a =

• A

(α + 1)1/2 , 0, 0

• and

b = (0, √ A, 0) such that W (a) = W (b) = 0.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 25 / 41

Sketch proof: (α > 0, q > q2) (IV)

a =

• A

(α + 1)1/2 , 0, 0

• and

b = (0, √ A, 0). u = a χU + b χΩ\U for some measurable set U ⊂ Ω. We determine the size of U by using the constraints of the total mass and total magnetization

• Ω

u2

1 + u2 0 + u2 −1 dx = N,

• Ω

u2

1 − u2 −1 dx = M

and find

• A

(α+1)1/2 |U| + A(|Ω| − |U|)

= N

A (α+1)1/2 |U|

= M. Thus, we have A =

• n + ((α + 1)1/2 − 1)m
• |U| =

(α + 1)1/2m n + ((α + 1)1/2 − 1)m |Ω|.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 26 / 41

Anti-ferromagnetic: cs > 0, q < q1 implies 2C state

Theorem

Consider the variational problem of TF approximation in a finite domain Ω subject to the constraints of total mass and total magnetization. For 0 < α ≤ 1, there is a function q1 defined by q1 =

• −n +
• n2 + αm2
• , n = N

|Ω|, m = M |Ω|, such that for q < q1, the minimizer is a constant state u =

• n + m

2 , 0,

• n − m

2

• .

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 27 / 41

Anti-ferro: cs > 0, q1 < q < q2 implies NS + 2C state

Theorem

Suppose M > 0. If α > 0 and q1 < q < q2, a global minimizer of the variational problem have the form u = a χU + b χΩ\U where a = (

• A + B

2 , 0,

• A − B

2 ) and b = (0,

4

• A2 + αB2, 0)

and two values A and B are given by A = n + (x − 1)q and B = m x

and the relative size of a measurable set U, denoted by x := |U|/|Ω|, satisfies the equation 2q2x3 +

• 2qn − q2

x2 − αm2 = 0.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 28 / 41

Ferrmomagnetic: cs < 0, q < 0 implies MS + MS state

Theorem

For α < 0 and q > 0, a global minimizer of the variational problem have the form u = a χU + b χΩ\U where U is a measurable set of size |U| = 1 2

• 1 + m

n

• |Ω|

and a = (√n, 0, 0) and b = (0, 0, √n).

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 29 / 41

Ferromagnetic: cs < 0, q > 0 implies 3C state

Theorem

For α < 0 and q > 0, the variational problem has a unique global minimizer u = (u1, u0, u−1) where u1 = q + b 2q

• n + 1

α q 2 − b2 2q 1/2 u0 = q2 − b2 2q2 n + q2 + b2 2q2 1 α q 2 − b2 2q 1/2 u−1 = q − b 2q

• n + 1

α q 2 − b2 2q 1/2 The value b is the unique root in (

• q2 + 2αqn, q) of the cubic equation

b3 − (q2 + 2αqn)b + 2αq2m = 0.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 30 / 41

Beyond the Thomas-Fermi approximation

There are too many solutions for the Thomas-Fermi approximation, which are not reveal the real BEC system. inf

• Ω

W (u) dx subject to the constraints of total mass and total magnetization. Putting back the kenetic energy, we consider the problem inf

• Ω

ǫ2|∇u|2 + W (u) dx subject to the constraints of total mass and total magnetization.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 31 / 41

Γ-convergence

Define the functional Gǫ :

• L2(Ω)

3 → R+ by Gǫ(u) =           

• Ω ǫ|∇u|2 + 1

ǫW (u) dx

for u = (u1, u0, u−1) ∈

• H1(Ω; R)

3 , uk ≥ 0 a.e. on

• Ω |u1|2 + |u0|2 + |u−1|2 dx = N,
• Ω |u1|2 − |u−1|2 dx = M,

+∞

• therwise.

Its (L2) − Γ limit is the functional G0 :

• L2(Ω)

3 → R+ given by G0(u) =            2g(a, b) PerΩ(u = a) for u = (u1, u0, u−1) ∈ (BV (Ω; R))3 +2g(0, a) H2({x ∈ ∂Ω : u(x) = a}) u = aχU + bχΩ\U +2g(0, b) H2({x ∈ ∂Ω : u(x) = b}) for some U ⊂ Ω s.t. |U|/|Ω| = r, +∞

• therwise.

where g(v, u) = inf 1

• W (γ(t)) |γ′(t)| dt : γ : [0, 1] → R3

+ Lipchitz continuous,

γ(0) = v, γ(1) = u} .

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 32 / 41

Γ-convergent result

Theorem

The sequence {Gǫ} Γ-convergence to G0 in L2(Ω)-topology. i.e.

1 (Lower-semicontinuity) For any sequence {uǫ} converging to u0 in

(L2(Ω))3, we have G0(u0) ≤ lim inf

ǫ→0 Gǫ(uǫ).

2 (Recovery sequence) For any v0 ∈ (L2(Ω))3, there exists a sequence

{vǫ} converging to v0 in (L2(Ω))3 such that G0(v0) = lim

ǫ→0 Gǫ(vǫ).

Corollary

Suppose uǫ is a global minimizers of Gǫ. If the sequence {uǫ} converges to some u0 in L2(Ω). then u0 is a global minimizer of G0.

G0(u0) ≤ lim inf Gǫ(uǫ) ≤ lim Gǫ(vǫ) = G0(v0)

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 33 / 41

Key idea in the proof of Γ-convergence result

Theorem

1 (Lower-semicontinuity) For any sequence {uǫ} converging to u0 in

(L2(Ω))3, we have G0(u0) ≤ lim inf

ǫ→0 Gǫ(uǫ).

2 (Recovery sequence) For any v0 ∈ (L2(Ω))3, there exists a sequence

{vǫ} converging to v0 in (L2(Ω))3 such that G0(v0) = lim

ǫ→0 Gǫ(vǫ).

Key idea by using simple example: Fǫ(uǫ) =

• Ω

ǫ|∇uǫ|2 + 1 ǫ (u2

ǫ − 1)2 dx ≥ 2

• Ω

|∇uǫ||u2

ǫ − 1| dx

≈2Hn−1({Jump set}) 1

−1

1 − u2 dx = 4 3Hn−1(Jump set)

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 34 / 41

wǫ(x) =

• 1 − ζ

db(x) ǫγ ζ d(x) ǫγ

• η

d(x) ǫ

• +
• 1 − ζ

d(x) ǫγ

• v0(x)
• + ζ

db(x) ǫγ 1 − ζ d(x) ǫγ

• ˜

η db(x) ǫ

• .

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 35 / 41

Compactness Result

Theorem

For any family {uǫ} with an uniformly bounded energy Gǫ(uǫ) ≤ C < +∞, then there exists a subsequence uǫj converges to some u0 in (L2(Ω))3.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 36 / 41

The limiting problem == Minimal interface problem

Theorem

Suppose uǫ is a minimizer of the variational problem inf

u∈H1(Ω:R3

+)

• Ω |u1|2+|u0|2+|u−1|2 dx=N
• Ω |u1|2−|u−1|2 dx=M
• Ω

ǫ|∇u|2 + 1 ǫ W (u) dx and uǫj → u0 in (L2(Ω))3 for some subsequence ǫj → 0. Then u0 solves the minimization problem inf

u∈A {2g(a, b) PerΩ(u = a)

+ 2g(0, a) H2({x ∈ ∂Ω : u(x) = a}) +2g(0, b) H2({x ∈ ∂Ω : u(x) = b})

• among the admissible set

A = {u ∈ BV (Ω : {a, b}) ∩ L2(Ω; R3

+) :

• Ω

|u1|2 + |u0|2 + |u−1|2 dx = N

• Ω

|u1|2 − |u−1|2 dx = M}.

Here, PerΩ(u) :=

• Ω |∇u| and |∇u| is the total variation measure.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 37 / 41

Construction of local minimizers from the limiting problem

Theorem

Suppose u0 ∈ (L2(Ω))3 is an isolated L2-local minimizer of the variational problem inf

u∈BV (Ω:{a,b})∩L2(Ω;R3

+)

• Ω |u1|2+|u0|2+|u−1|2 dx=N
• Ω |u1|2−|u−1|2 dx=M

G0(u) Then there exists a sequence {uǫ} ⊂ (L2(Ω))3 such that each uǫ is a local minimizer of the perturbed variational problem inf

u∈H1(Ω:R3

+)

• Ω |u1|2+|u0|2+|u−1|2 dx=N
• Ω |u1|2−|u−1|2 dx=M
• Ω

ǫ|∇u|2 + 1 ǫ W (u) dx and uǫ → u0 in (L2(Ω))3 as ǫ → 0.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 38 / 41

Criticality of the interface

Theorem

Let u0 be a critical point of G0 such that ∂U ∩ Ω is of class C2 with mean curvature H : ∂U ∩ Ω → R. Then for any C 2-vector field X : Ω → R3 which is a tangential vector field on the boundary ∂Ω and satisfies the condition

• ∂A∩Ω

X · n dH2(x) = 0 where n is an outward unit normal vector of ∂A ∩ Ω, we have 0 = −g(a, b)

• ∂U∩Ω

H(x)(X · n) dH2(x) +

• ∂(∂U∩∂Ω)

[g(a, b)(ν · t) + g(0, a) − g(0, b)] (t · X) dH1(x) = 0. Here, n : ∂U ∩ Ω → Sn−1 is a normal unit vector to the interface ∂U ∩ Ω; ν : ∂(∂U ∩ Ω) → Sn−1 is an outward unit tangential vector to the interface ∂U ∩ Ω and normal to ∂(∂U ∩ Ω); t : ∂(∂U ∩ ∂Ω) → Sn−1 is the outward unit tangential vector to ∂U ∩ ∂Ω and normal to ∂(∂U ∩ ∂Ω). The corresponding Euler-Lagrange equation is H(x) = Const. for x ∈ ∂U ∩ Ω, g(a, b)(ν · t) + g(0, a) − g(0, b) = 0 for x ∈ ∂(∂U ∩ Ω).

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 39 / 41

H(x) = Const. for x ∈ ∂U ∩ Ω, g(a, b)(ν · t) + g(0, a) − g(0, b) = 0 for x ∈ ∂(∂U ∩ Ω).

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 40 / 41

Summary

Show the phase separation do occur in the Ground states Give a complete phase diagram for the TF approximation Anti-Ferromagnetic Ferromagnetic Characterize the patterns of the Ground states Minimal Interface Problem

inf

u∈BV (Ω:{a,b})∩L2(Ω;R3

+) PerΩ(u)

subject to

Ω

|u1|2 + |u0|2 + |u−1|2 dx = N,

• Ω

|u1|2 − |u−1|2 dx = M.

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 41 / 41

Thank you for your attention!

Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory December 19, 2015 42 / 41