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

Transcription

1 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 Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December viascience) Γ-convergence 19, Theory / 41

2 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 Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December viascience) Γ-convergence 19, Theory / 41

3 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. ien-tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December viascience) Γ-convergence 19, Theory / 41

4 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 Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December viascience) Γ-convergence 19, Theory / 41

5 The mean field model for BEC N particle system: wave function Ψ N (x 1,, x N, t), Hamiltonian: H N = N j=1 ) ( 2 2 j + V (x j ) + V int (x j x k ), 2M a 1 j<k N Ultracold and dilute gases, the mean field approximation: V int (x j x k ) gδ(x j x k ) Hartree ansatz: all boson particles are in the same quantum state Ψ N (x 1,, x N, t) = N ψ(x j, t). j=1 Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December viascience) Γ-convergence 19, Theory / 41

6 The Gross-Pitaeviskii equation Hamiltonian: H = 2 2M a ψ 2 + V (x) ψ 2 + β 2 ψ 4, β = gn Energy E[ψ] = H dx. Gross-Pitaevskii equation: i t ψ = δe/δψ. ψ wave function i t ψ = 2 2M a 2 ψ + V (x)ψ + β ψ 2 ψ V (x) trap potential: V (x) = i=1 ω2 i x 2 i. Interaction: repulsive if β > 0, attractive if β < 0. Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December viascience) Γ-convergence 19, Theory / 41

7 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 m F = 1, 0, 1 Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December viascience) Γ-convergence 19, Theory / 41

8 Spinor BECs Spin-1 atom has 3 hyperfine states: m F = 1, 0, 1. Vector order parameter Ψ = (ψ 1, ψ 0, ψ 1 ). Associate with a spinor Ψ, the spin vector F = Ψ FΨ R 3, which is just like a magnetic dipole moment. F = (F x, F y, F z ) is the spin-1 Pauli operator: F x = , F y = i , F z = Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December viascience) Γ-convergence 19, Theory / 41

9 G-P equation for spin-1 BECs Hamiltonian: H = 2 2M a Ψ 2 + V (x) Ψ 2 + c n 2 Ψ 4 + c s 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 t Ψ = δe δψ Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December viascience) Γ-convergence 19, Theory / 41

10 Physical parameters 2 H = Ψ 2 + V (x) Ψ 2 2M } a }{{}{{} H kin H pot + c n 2 Ψ 4 + c s }{{} 2 Ψ FΨ 2 }{{} H n H s interaction > 0 < 0 c n spin-independent repulsive attractive c s spin-exchange antiferromagnetic ferromagnetic c n c s 87 Rb ferromagnetic 23 Na anti-ferromagnetic Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

11 Spinor BEC in uniform magnetic field Under an uniform magnetic field, we have to consider additional Zeeman shift energy in the Hamiltonian. Hamiltonian H = H kin + H pot + H n + H s + H Zee Zeeman shift energy: Suppose magnetic field Bẑ, H Zee = where n j = ψ j 2 and 1 E j (B)n j j= 1 = q(n 1 + n 1 ) + p(n 1 n 1 ) + E 0 n p = 1 2 (E 1 E 1 ) µ BB 2 q = 1 2 (E 1 + E 1 2E 0 ) µ2 B B2 4E hfs Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

12 Gauge invariants and conservation laws Energy E[Ψ] = (H kin + H pot + H n + H s + H Zee ) dx Gauge invariant: energy is invariant under transform Ψ e iφ R z (α)ψ This leads to two conservation laws: Total number of atoms ( ψ ψ ψ 1 2 ) dx = N Total magnetization ( ψ 1 2 ψ 1 2 ) dx = M Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

13 The ground state problem min E[Ψ] subject to n(x) dx = N, m(x) dx = M. E[Ψ] = H dx H = H kin + H pot + H n + H s + H Zee n j = ψ j 2, n = n 1 + n 0 + n 1 m = n 1 n 1 Set u j = ψ j, j = 1, 0, 1. E[u] = R3 2 1 u j 2 + c n 2M a 2 u 4 + V (x) u 2 j= 1 + c s [ 2u (u 1 sgn(c s ) u 1 ) 2 + (u1 2 u 1) 2 2] + [ q(u1 2 + u 1) 2 ] dx + E 0 N + pm. Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

14 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. { 2 1 inf u j 2 + cn R 3 2M a 2 u 4 + V (x) u 2 + cs 2 j= 1 [ 2u 2 0 (u 1 sgn(c s) u 1) 2 + (u 2 1 u 2 1) 2] + [ p(u1 2 u 1) 2 + q(u1 2 + u 1) 2 ] } dx subject to the constrains R 3 u u u 2 1 dx = N, R 3 u 2 1 u 2 1 dx = M. In particular, we are interesting in the case that ɛ = 2 2M i 1. The problem becomes a singular perturbation problem. Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

15 The corresponding nonlinear eigenvalue problem The corresponding Euler-Lagrange equation is [ ] (µ + λ)u 1 = 2 + V (x) + q + c0n u 2m 1 + c 2(n 1 + n 0 n 1)u 1 + c 2u 1u 2 0 [ ] µu 0 = 2 + V (x) + c0n u 2m 0 + 2c 2(n 1 n 1)u 0 + c 2u 1u 1u0 [ ] (µ λ)u 1 = 2 + V (x) + q + c0n u 2m 1 + c 2(n 1 + n 0 n 1)u 1 + c 2u1 u0 2 We denote that n 1 = u 1 2, n 0 = u 0 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 Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

16 A brief survey on the gradient theory of phase transition Potential W : R n R +. { inf Ω Ω (ɛ u 2 + 1ɛ W (u) ) ɛ 2 2 u 2 + W (u(x)) dx } : u W 1,2 (Ω), u(x) dx = m, Ω 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 Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

17 The Thomas-Fermi Approximation Ignoring the kinetic energy, we consider the problem { c n inf R 3 2 u 4 + V (x) u 2 + c s [ 2u (u 1 sgn(c s ) u 1 ) 2 + (u1 2 u 1) 2 2] + [ p(u1 2 u 1) 2 + q(u1 2 + u 1) 2 ] } dx subject to the constrains R 3 u u u 2 1 dx = N, R 3 u 2 1 u 2 1 dx = M. In order to make the problem simple, we consider the case when trap potential is { 0 on Ω, V (x) = + on R n \Ω. Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

18 The Phase Diagram Minimize c n Ω 2 u 4 + cs [ 2u 2 0 (u 1 sgn(c s) u 1) 2 + (u1 2 u 1) 2 2] + [ p(u1 2 u 1) 2 + q(u1 2 + u 1) 2 ] dx 2 subject to the constrains Ω u2 1 + u0 2 + u 1 2 dx = N, Ω u2 1 u 1 2 dx = M. Anti-ferromagnetic (c s > 0) When q > q 2, we have NS + MS state. (0, u 0, 0) + (u 1, 0, 0) When q 1 < q < q 2, we have NS + 2C state. (0, u 0, 0) + (u 1, 0, u 1 ) When q < q 1, we have 2C state. (u 1, 0, u 1 ) Ferromagetic (c s < 0) When q > 0, we have 3C state. (u 1, u 0, u 1 ) When q < 0, we have MS + MS state. (u 1, 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 Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

19 Phase Diagram for anti-ferromagnetic spin 1 BEC (c s > 0) Minimize c n Ω 2 u 4 + cs [ 2u 2 0 (u 1 sgn(c s) u 1) 2 + (u1 2 u 1) 2 2] + [ p(u1 2 u 1) 2 + q(u1 2 + u 1) 2 ] dx 2 subject to the constrains Ω u2 1 + u0 2 + u 1 2 dx = N, Ω u2 1 u 1 2 dx = M. Anti-Ferromagnetic c s > 0 Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

20 Phase Diagram for ferromagnetic spin-1 BEC (c s < 0) Minimize c n Ω 2 u 4 + cs [ 2u 2 0 (u 1 sgn(c s) u 1) 2 + (u1 2 u 1) 2 2] + [ p(u1 2 u 1) 2 + q(u1 2 + u 1) 2 ] dx 2 subject to the constrains Ω u2 1 + u0 2 + u 1 2 dx = N, Ω u2 1 u 1 2 dx = M. Ferromagnetic c s < 0 Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

21 Key idea H TF (u) := 1 2 u2 1 + u u α 2 [ 2u 2 0 (u 1 sgn(α) u 1 ) 2 + (u 2 1 u 2 1) 2] + q(u u 2 1) Try to find suitable choice of β 1, β 2, β 3 such that W (u) := H TF (u) + β 1 (u1 2 + u0 2 + u 1) 2 + β 2 (u1 2 u 1) 2 + β 3 is a double-well potential, that is W (a) = W (b) = 0, W (u) > 0 for u R 3 +\{a, b} that satisfying the constrains. Main trick: to complete a perfect square. x 4 2ax 2 + a 2 = (x 2 a) 2 Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

22 Anti-ferromagnetic: c s > 0, q > q 2 implies NS + MS state Theorem For 0 < α 1 and q > q 2 := ( ) 1 ( ( ) ) 1 n + (α + 1) 1/2 1 m., (α + 1) 1/2 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 where U Ω is a measurable set of size u = a χ U + b χ Ω\U and and U = (α + 1) 1/2 m n + ((α + 1) 1/2 1)m Ω A a = ( (α + 1), 0, 0), and b = (0, A, 0) 1/2 A = ( ( ) ) n + (α + 1) 1/2 1 m. Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

23 Sketch proof: (α > 0, q > q 2 ) (I) 2H TF = (u u u 2 1) 2 + α(u 2 1 u 2 1) 2 + 2αu 2 0(u 1 u 1) 2 + 2q(u u 2 1) = (1 + α)u1 4 + u0 4 + (1 + α)u (1 α)u1u [ + 2u0 2 (u1 2 + u 1) 2 + α(u 1 u 1) 2] + 2q(u1 2 + u 1) 2 = ( (1 + α) 1/2 u u α + 2u 2 0 [ ( 1 + α (1 + α) 1/2) u ) 2 (1 + α) 1/2 u α 1 + α u q(u1 2 + u 1) 2 ( 1 + α 1 α ) ] u 2 1 2αu 1u (1 + α) 1/2 1 Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

24 Sketch proof: (α > 0, q > q 2 ) (II) 2W = W (u) := H TF (u) + β1 2 (u2 1 + u u 2 1) + β2 2 (u2 1 u 2 1) + A2 2, ( (α + 1) 1/2 u1 2 + u α ) 2 (1+α) 1/2 u2 1 A ( ( + 2u0 2 (α + 1) (α + 1) 1/2) 1/2 α u1 ((α + 1) (α + 1) 1/2 u 1 1/2 ) 2α + 4 2(α + 1)1/2 + (α + 1) (α + 1) 1/2 u2 1u α α + 1 u4 1 ( + β 1 + β 2 + 2q + 2A(α + 1) 1/2) u1 2 + (β 1 + 2A)u0 2 ( + β 1 β 2 + 2q + 2A 1 α ) u 1. 2 (α + 1) 1/2 We choose β 1 and β 2 to satisfy { β1 + β 2 + 2q + 2A(α + 1) 1/2 = 0 β 1 + 2A = 0, β 1 β 2 + 2q + 2A 1 α ( ) 1 = 4A (α + 1) 1/2 (α + 1) 1 + 4q. 1/2 Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41 ) 2

25 Sketch proof: (α > 0, q > q 2 ) (III) 2W = ( (α + 1) 1/2 u1 2 + u α ) 2 (1 + α) 1/2 u2 1 A ( ( + 2u0 2 (α + 1) (α + 1) 1/2) 1/2 α u1 ((α + 1) (α + 1) 1/2 u 1 1/2 ) + 2α + 4 2(α + 1)1/2 (α + 1) (α + 1) 1/2 u2 0u α α + 1 u (q q 2) u 2 1. Therefore, we have ( ) A a =, 0, 0 (α + 1) 1/2 and b = (0, A, 0) ) 2 such that W (a) = W (b) = 0. Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

26 Sketch proof: (α > 0, q > q 2 ) (IV) ( ) A a =, 0, 0 (α + 1) 1/2 u = a χ U + b χ Ω\U and b = (0, A, 0). for some measurable set U Ω. We determine the size of U by using the constraints of the total mass and total magnetization u1 2 + u0 2 + u 1 2 dx = N, u1 2 u 1 2 dx = M and find { Thus, we have Ω A U + A( Ω U ) (α+1) 1/2 = N A U (α+1) 1/2 = M. A = U = ( ) n + ((α + 1) 1/2 1)m (α + 1) 1/2 m n + ((α + 1) 1/2 1)m Ω. Ω Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

27 Anti-ferromagnetic: c s > 0, q < q 1 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 q 1 defined by q 1 = ( n + n 2 + αm 2 ), n = N Ω, m = M Ω, such that for q < q 1, the minimizer is a constant state ( ) n + m n m u =, 0,. 2 2 Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

28 Anti-ferro: c s > 0, q 1 < q < q 2 implies NS + 2C state Theorem Suppose M > 0. If α > 0 and q 1 < q < q 2, a global minimizer of the variational problem have the form where u = a χ U + b χ Ω\U A + B A B a = (, 0, ) and b = (0, 4 A αb 2, 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 ( 2q 2 x 3 + 2qn q 2) x 2 αm 2 = 0. Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

29 Ferrmomagnetic: c s < 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 χ U + b χ Ω\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 Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

30 Ferromagnetic: c s < 0, q > 0 implies 3C state Theorem For α < 0 and q > 0, the variational problem has a unique global minimizer u = (u 1, u 0, u 1 ) where u 1 = q + b [ n + 1 ( )] 1/2 q 2q α 2 b2 2q [ q 2 b 2 u 0 = 2q 2 n + q2 + b 2 ( 1 q 2q 2 α 2 b2 2q u 1 = q b [ n + 1 ( )] 1/2 q 2q α 2 b2 2q )] 1/2 The value b is the unique root in ( q 2 + 2αqn, q) of the cubic equation b 3 (q 2 + 2αqn)b + 2αq 2 m = 0. Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

31 Beyond the Thomas-Fermi approximation There are too many solutions for the Thomas-Fermi approximation, which are not reveal the real BEC system. inf W (u) dx subject to the constraints of total mass and total magnetization. Ω Putting back the kenetic energy, we consider the problem inf ɛ 2 u 2 + W (u) dx Ω subject to the constraints of total mass and total magnetization. Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

32 Γ-convergence Define the functional G ɛ : ( L 2 (Ω) ) 3 R+ by Ω ɛ u W (u) dx for u = (u1, u0, u 1) ( H 1 (Ω; R) ) 3 ɛ, uk 0 a.e. o Ω u1 2 + u u 1 2 dx = N, G ɛ(u) = Ω u1 2 u 1 2 dx = M, + otherwise. Its (L 2 ) Γ limit is the functional G 0 : ( L 2 (Ω) ) 3 R+ given by 2g(a, b) Per Ω (u = a) for u = (u 1, u 0, u 1) (BV (Ω; R)) 3 +2g(0, a) H 2 ({x Ω : u(x) = a}) u = aχ U + bχ Ω\U G 0(u) = +2g(0, b) H 2 ({x Ω : u(x) = b}) for some U Ω s.t. U / Ω = r, + otherwise. where { 1 g(v, u) = inf W (γ(t)) γ (t) dt : γ : [0, 1] R 3 + Lipchitz continuous, 0 γ(0) = v, γ(1) = u}. Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

33 Γ-convergent result Theorem The sequence {G ɛ } Γ-convergence to G 0 in L 2 (Ω)-topology. i.e. 1 (Lower-semicontinuity) For any sequence {u ɛ } converging to u 0 in (L 2 (Ω)) 3, we have G 0 (u 0 ) lim inf ɛ 0 G ɛ(u ɛ ). 2 (Recovery sequence) For any v 0 (L 2 (Ω)) 3, there exists a sequence {v ɛ } converging to v 0 in (L 2 (Ω)) 3 such that Corollary G 0 (v 0 ) = lim ɛ 0 G ɛ (v ɛ ). Suppose u ɛ is a global minimizers of G ɛ. If the sequence {u ɛ } converges to some u 0 in L 2 (Ω). then u 0 is a global minimizer of G 0. G 0(u 0) lim inf G ɛ(u ɛ) lim G ɛ(v ɛ) = G 0(v 0) Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

34 Key idea in the proof of Γ-convergence result Theorem 1 (Lower-semicontinuity) For any sequence {u ɛ } converging to u 0 in (L 2 (Ω)) 3, we have G 0 (u 0 ) lim inf ɛ 0 G ɛ(u ɛ ). 2 (Recovery sequence) For any v 0 (L 2 (Ω)) 3, there exists a sequence {v ɛ } converging to v 0 in (L 2 (Ω)) 3 such that Key idea by using simple example: G 0 (v 0 ) = lim ɛ 0 G ɛ (v ɛ ). F ɛ (u ɛ ) = ɛ u ɛ Ω ɛ (u2 ɛ 1) 2 dx 2 2H n 1 ({Jump set}) 1 1 Ω u ɛ u 2 ɛ 1 dx 1 u 2 dx = 4 3 Hn 1 (Jump set) Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

35 [ w ɛ(x) = 1 ζ ( db (x) ɛ γ ( db (x) + ζ ɛ γ )] { ζ ) [ 1 ζ ( d(x) ɛ γ ( d(x) ɛ γ ) η )] η ( d(x) ɛ ( db (x) ) [ + 1 ζ ɛ ). ( d(x) ɛ γ )] } v 0(x) Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

36 Compactness Result Theorem For any family {u ɛ } with an uniformly bounded energy G ɛ (u ɛ ) C < +, then there exists a subsequence u ɛj converges to some u 0 in (L 2 (Ω)) 3. Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

37 The limiting problem == Minimal interface problem Theorem Suppose u ɛ is a minimizer of the variational problem inf u H 1 (Ω:R 3 + ) Ω u u u 1 2 dx=n Ω u 1 2 u 1 2 dx=m and u ɛj u 0 in (L 2 (Ω)) 3 for some subsequence ɛ j 0. Then u 0 solves the minimization problem among the admissible set inf {2g(a, b) Per Ω(u = a) u A Ω ɛ u W (u) dx ɛ + 2g(0, a) H 2 ({x Ω : u(x) = a}) } +2g(0, b) H 2 ({x Ω : u(x) = b}) A = {u BV (Ω : {a, b}) L 2 (Ω; R 3 + ) : u u u 1 2 dx = N u 1 2 u 1 2 dx = M}. Ω Ω Here, Per Ω (u) := u and u is the total variation measure. Ω Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

38 Construction of local minimizers from the limiting problem Theorem Suppose u 0 (L 2 (Ω)) 3 is an isolated L 2 -local minimizer of the variational problem inf G 0 (u) u BV (Ω:{a,b}) L 2 (Ω;R 3 + ) Ω u u u 1 2 dx=n Ω u 1 2 u 1 2 dx=m Then there exists a sequence {u ɛ } (L 2 (Ω)) 3 such that each u ɛ is a local minimizer of the perturbed variational problem inf ɛ u W (u) dx u H 1 (Ω:R 3 + ) ɛ Ω u u u 1 2 dx=n Ω u 1 2 u 1 2 dx=m Ω and u ɛ u 0 in (L 2 (Ω)) 3 as ɛ 0. Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

39 Criticality of the interface Theorem Let u 0 be a critical point of G 0 such that U Ω is of class C 2 with mean curvature H : U Ω R. Then for any C 2 -vector field X : Ω R 3 which is a tangential vector field on the boundary Ω and satisfies the condition X n dh 2 (x) = 0 where n is an outward unit normal vector of A Ω, A Ω we have 0 = g(a, b) + ( U Ω) U Ω H(x)(X n) dh 2 (x) [g(a, b)(ν t) + g(0, a) g(0, b)] (t X) dh 1 (x) = 0. Here, n : U Ω S n 1 is a normal unit vector to the interface U Ω; ν : ( U Ω) S n 1 is an outward unit tangential vector to the interface U Ω and normal to ( U Ω); t : ( U Ω) S n 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 Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

40 { 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 Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

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 subject to inf Per Ω (u) u BV (Ω:{a,b}) L 2 (Ω;R 3 + ) u u u 1 2 dx = N, Ω Ω u 1 2 u 1 2 dx = M. Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41

42 Thank you for your attention! Tien-Tsan Shieh joint work with I-Liang Ground Chern State andpatterns Chiu-Fen of Chou Spin-1(National Bose-Einstein Center condensation of Theoretical December via19, Science) Γ-convergence Theory / 41