2D optical lattice

2D optical lattice

V (x, y) = V_x sin²(k_xx) + V_y sin²(k_yy)

Square lattice

V_x = V_y k_x = k_y

V (x, y) = V_x sin²(k_xx) + V_y sin²(k_yy)

1D chain

V_x < V_y k_x = k_y

2D optical lattice

V (x, y) = V_x sin²(k_xx) + V_y sin²(k_yy)

1D chain

V_x < V_y

V_xk_x² = V_yk_y² k_x > k_y

2D optical lattice

- 2 -

optical chain

in p-band

H =ˆ X

i

H(i)ˆ X

hiji

⇥t_xaˆ^†_x(i)ˆa_x(j) + t_yˆa^†_y(i)ˆa_y(j) + h.c.⇤

t_x t_y

p-orbital physics

x

|t ^x | > |t ^y |

H =ˆ X

i

H(i)ˆ X

hiji

⇥t_xaˆ^†_x(i)ˆa_x(j) + t_yˆa^†_y(i)ˆa_y(j) + h.c.⇤

p-orbital physics

local Hamiltonian

U_yy > U_xx > 3U_xy

H =ˆ X

i

H(i)ˆ X

hiji

⇥t_xaˆ^†_x(i)ˆa_x(j) + t_yˆa^†_y(i)ˆa_y(j) + h.c.⇤

p-orbital physics

local Hamiltonian

additional symmetry of the system

S = exp(i⇡ ˆ ˆ N

)

hH, ˆˆ N_x + ˆN_yi

= 0

total number of particles is conserved

Nˆ_y = X

i

ˆ

n_y(i) Nˆ_x = X

i

ˆ

n_x(i)

p-orbital physics

h_x = hG|ˆa^†x(i)ˆa^†_x(i + 1)|Gi h_y = hG|ˆa^†y(i)ˆa^†_y(i + 1)|Gi

|G^oddi |G^eveni

two-fold degeneracy

of the many-body ground state

region of restored degeneracy

|G^oddi |G^eveni

two-fold degeneracy

of the many-body ground state

cos ✓

|Gi = + sin ✓e^i'

many-body ground state in the thermodynamic limit

Chosen ground state should be as close to the product state as possible

Einselec(on*principle*

W.*H.*Żurek,*Rev.*Mod.*Phys.*75,*715*(2003)

S(✓, ') = X

i

i log _i

|G_±i = |G^oddi ± i|G^eveni p2

entanglement entropy for single lattice site

eigenvalues of the single-site reduced density matrix

|G_±i = |G^oddi ± i|G^eveni p2

properties of the ground-state

local quasi-angular momentum operator

staggered angular momentum operator

Lˆ_z(j) = i ⇥ ˆ

a^†_x(j)ˆa_y(j) ˆa^†_y(j)ˆa_x(j)⇤

Lˆ^z = X

j

( 1)^jLˆ_z(j)

non-trivial correlations

C^↵ (j) = C ^↵(j) C^↵ (j) = hˆa^†↵(j)ˆa (j)i 6= 0

properties of the ground-state

positive staggered angular momentum negative staggered angular momentum

time-reversal symmetry is BROKEN

…thank you

for your attention…