symmetry breaking the optical lattice

outline

1. spontaneous

symmetry breaking the optical lattice

2.

3. orbital dance

- 1 -

Spontaneous Symmetry

Breaking

Example: Ising model

H = J X

i

z(i) (i+1)

z + h X

i

z(i)

The ground state of the system depends on external field

|Gi = |upi = | "" . . . "i, for h < 0

|Gi = |downi = | ## . . . #i, for h > 0

|Gi = |upi = | "" . . . "i, for h < 0

|Gi = |downi = | ## . . . #i, for h > 0

What will happen when the external field vanishes?

COMMON VIEW

the ground state is degenerate; the system has to decide and in given realization one of these two states is chosen

SPONTANEOUS SYMMETRY BREAKING

Z²

in

HALF-TRUTH

Example: Ising model

H = J X

i

z(i) (i+1)

z + h X

i

z(i)

FIRST OBSERVATION

any superposition of these two states can be chosen

|Gi = sin ✓|upi + cos ✓e^i'|downi U(1) ⇥

U(1)

COMMON VIEW

the ground state is degenerate; the system has to decide and in given realization one of these two states is chosen

SPONTANEOUS SYMMETRY BREAKING

Z²

in

HA LF -TR UTH

Example: Ising model

H = J X

i

z(i) (i+1)

z + h X

i

z(i)

FIRST OBSERVATION

any superposition of these two states can be chosen

|Gi = sin ✓|upi + cos ✓e^i'|downi U(1) ⇥

U(1)

SECOND OBSERVATION

Hamiltonian has additional symmetry of flipping all spins

[S, H] = 0

Common eigenvectors are:

|±i = 1

p2 (|upi ± |downi) S|±i = ±1|±i

in

Example: Ising model

H = J X

i

z(i) (i+1)

z + h X

i

z(i)

[S, H] = 0 |±i = 1

p2 (|upi ± |downi) S|±i = ±1|±i

decoherence-imposed selection

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)

… In this way symmetry operator is broken

|upi |downi

U (1) ⇥ U(1) ! Z²

or

in

- 2 -

The 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

Energy

s

px py

dxx dxy dyy

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

1D chain

Energy

s

px py

dxx

dyy

dxy

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

Energy

s

px py

dxx dyy

dxy

V_x < V_y

V_xk_x² = V_yk_y²

harmonic approximation

k_x > k_y

2D optical lattice

- 3 -

Orbital Dance

p-orbital physics

H =ˆ X

i

H(i)ˆ X

hiji

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

H(i) =ˆ U_xx

2 nˆ_x(i)(ˆn_x(i) 1) + U_yy

2 nˆ_y(i)(ˆn_y(i) 1) + U_xy

2

⇥4ˆn_x(i)ˆn_y(i) + ˆa^†_x(i)²ˆa_y(i)² + ˆa^†_y(i)²ˆa_x(i)²⇤ local Hamiltonian

Nˆ_y = X

i

ˆ n_y(i) Nˆ_x = X

i

ˆ n_x(i)

additional symmetry of the system

S = exp(i⇡ ˆ ˆ N

)

hH, ˆˆ N_x + ˆN_yi

= 0

total number of particles is conserved

role of contact interactions

|n, mi = (ˆa^†x)ⁿ(ˆa^†_y)^m|⌦i

local Fock basis

|0, 1i

|1, 1i

|1, 0i

|2, 0i

|0, 2i

energy of local states

Energy

?

role of contact interactions

|n, mi = (ˆa^†x)ⁿ(ˆa^†_y)^m|⌦i

Local Fock basis

|0, 1i |1, 0i |0, 2i |2, 0i |1, 1i

Energy of local states

Energy

H(i) =ˆ U_xx

2 nˆ_x(i)(ˆn_x(i) 1) + U_yy

2 nˆ_y(i)(ˆn_y(i) 1) + U_xy

2

⇥4ˆn_x(i)ˆn_y(i) + ˆa^†_x(i)²aˆ_y(i)² + ˆa^†_y(i)²ˆa_x(i)²⇤

in

|0, 2i |2, 0i

U_xx = U_yy = 3U_xy

role of contact interactions

|n, mi = (ˆa^†x)ⁿ(ˆa^†_y)^m|⌦i

local Fock basis

|0, 1i

|1, 1i

|1, 0i

|2, 0i

|0, 2i

energy of local states

Energy

H(i) =ˆ U_xx

2 nˆ_x(i)(ˆn_x(i) 1) + U_yy

2 nˆ_y(i)(ˆn_y(i) 1) + U_xy

2

⇥4ˆn_x(i)ˆn_y(i) + ˆa^†_x(i)²aˆ_y(i)² + ˆa^†_y(i)²ˆa_x(i)²⇤

|0, 2i

|2, 0i +

U_yy > U_xx > 3U_xy

p-orbital physics

|G^oddi |G^eveni

two-fold degeneracy

of the many-body ground state

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

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

conclusions

in asymmetric lattices it is possible to obtain


degeneracy between the single-particle energies in given orbital (p-orbital)

this degeneracy is lifted by an anharmonicity when contact interactions are taken into account

the degeneracy between orbitals

is dynamically restored due to tunneling

BUT

In this region an additional symmetry of the system is spontaneously broken

The state which breaks the time-reversal symmetry becomes the true ground state of the system