Mean-field theory for extended Bose-Hubbard model with hard-core ...

Mean-field theory for extended Bose-Hubbard model
with hard-core bosons
Mathias May, Nicolas
Gheeraert, Shai Chester,
Sebastian Eggert, Axel
Pelster

Outline
◮ Introduction
◮ Experiment
◮ Hamiltonian
◮ Periodic patterns
◮ Main part
◮ Mean-field (MF) approximation method
◮ MF-Hamiltonian for unit cell in a periodic pattern (dependent on
mean-field parameters)
◮ General calculation method for any pattern
◮ energy eigenvalues
◮ mean-field parameters
◮ Results (quadratic lattice, triangular lattice)
◮ Outlook
◮ Completing mean-field for triangular lattice
◮ Negative hopping J
◮ Other lattices
◮ Quantum corrections

Experiment
T ≈ 0

Hamiltonian
Extended Bose-Hubbard model
Ĥ = −J ∑ ↠i âj + U 2
∑
i ˆn i (ˆn i − 1) − µ ∑ i ˆn i + V ∑
2 ˆn i ˆn j
Hopping parameter J
J = J (i, j) = − ´ [
]
d 3 x w ∗ (x − x i ) − 2 △ + 2m Vext (x) w (x − x j )
On-site interaction U
U = U (i) = 4πa2
m
Next neighbour interaction V
´
d 3 x |w (x − x i )| 4
V = V (i, j) = ´ d 3 x ´ d 3 x ′ V dipol (x, x ′ ) |w (x − x i )| 2 |w (x ′ − x j )| 2

Feshbach resonance
( )
a (B) = a bg 1 − ∆
B−B 0
Hamiltonian for hard-core bosons:
Ĥ = −J ∑ ↠i âj + V 2
Hard-core limit:
a → ∞ ⇒ U → ∞ ⇒ n ∈ {0, 1}
∑
ˆn i ˆn j − µ ∑ i ˆn i

Patterns
not frustrated
frustrated
Connectivity matrices:
(
0 4
N =
4 0
)
N =
⎛
⎝ 3 0 3
0 3 3
3 3 0
⎞
⎠

Outline
◮ Introduction
◮ Experiment
◮ Hamiltonian
◮ Periodic patterns
◮ Main part
◮ Mean-field (MF) approximation method
◮ MF-Hamiltonian for unit cell in a periodic pattern (dependent on
mean-field parameters)
◮ General calculation method for any pattern
◮ energy eigenvalues
◮ mean-field parameters
◮ Results (quadratic lattice, triangular lattice)
◮ Outlook
◮ Completing mean-field for triangular lattice
◮ Negative hopping J
◮ Other lattices
◮ Quantum corrections

Mean-field approximation
Problem: bilocal terms in Hamiltonian
Mean-field approximation:
â i = 〈â
〈 i 〉
〉
+ δâ i
δâ † i δâ j ≈ 0 ∀i, j
â † i
= â † i
+ δâ † i
ˆn i = 〈ˆn i 〉 + δˆn i δˆn i δˆn j ≈ 0 ∀i, j
Operators appearing in the Hamiltonian:
â † i âj
≈
〈 〉
〈 〉
â † i
â j + 〈â j 〉 â † i
− â † i
〈â j 〉
= ψ ∗ i â j + ψ j â † i
− ψ ∗ i ψ j
ψ i := 〈â i 〉
ϱ i := 〈ˆn i 〉
ˆn i ˆn j ≈ 〈ˆn i 〉 ˆn j + 〈ˆn j 〉 ˆn i − 〈ˆn i 〉 〈ˆn j 〉

Mean-field Hamiltonian
Hamiltonian for the whole lattice
Ĥ
MF
≈
∑ i
ĥ MF ,i
Ψ i := ∑ j∈NN i
ψ j
)
ĥ MF ,i := −J
(â i Ψ ∗ i + â † i Ψ i − ψi ∗ Ψ i
R i := ∑ j∈NN i
ϱ j
+ V 2 (2ˆn i R i − ϱ i R i )
−µˆn i

Mean-field Hamiltonian
Hamiltonian for periodic patterns
For periodic patterns:
Hamiltonian reduces to a sum over the finite number of sites in
the unit cell (UC).
ĥ MF ,UC = ( JΨ − V 2 R) − J ∑ X
(
)
â X Ψ ∗ X + â † X Ψ X + 2 ∑ X ˆn ( V
X R )
2 X − µ 2
X ∈ {A, B, . . .}, Ψ := ∑ X ψ∗ X Ψ X , R := ∑ X ϱ X R X

Usual calculation method
◮ Finding Hamiltonian for a special case (pattern)
◮ Defining a basis for the pattern
◮ Finding matrix representations for operators on this basis
◮ For n sites in the unit cell the size of the matrix is 2 n !
◮ Combining them to the matrix representation of the
Hamiltonian
)
◮ Solving the eigenvalue equation det
(Ĥ − E = 0
◮ roots of 2 n -order polynomial !
◮ no general solution for the energies E to expect !

Homogeneous pattern
(ĥMF ,UC
)
B
hard-core basis: B = {|0〉 , |1〉}
( (
JΨ −
V
=
R) −JΨ ∗ 2 ( A
−JΨ A JΨ −
V
R) + 2 ( V
R )
2 2 A − µ 2
)
E ±A = ( JΨ − V R) + ( V
R )
√ (
2 2 A − µ V ±A R )
2 2 A − µ 2
2 + (J |ΨA |) 2

2 sites in the unit cell
hard-core basis: B = {|0, 0〉 , |1, 0〉 , |0, 1〉 , |1, 1〉}
E ±A ,± B
=
(JΨ − V 2 R )
( V
+
2 R A − µ )
√ ( V
± A
2 2 R A − µ ) 2
+ (J |Ψ A |) 2
2
( V
+
2 R B − µ )
√ ( V
± B
2 2 R B − µ ) 2
+ (J |Ψ B |) 2
2

+ ∑ X
Energies
General formula for the energies
E ±A =
E ±A ,± B
=
(JΨ − V ) ( V
2 R +
2 R A − µ )
√ ( V
± A
2 2 R A − µ ) 2
+ (J |Ψ A |) 2
2
(JΨ − V 2 R )
( V
+
2 R A − µ )
√ ( V
± A
2 2 R A − µ ) 2
+ (J |Ψ A |) 2
2
( V
+
2 R B − µ )
√ ( V
± B
2 2 R B − µ ) 2
+ (J |Ψ B |) 2
2
E ±A ,± B ,... =
(JΨ − V 2 R )
( V
2 R X − µ 2
)
+ ∑ X
(± X )
√ ( V
2 R X − µ 2
) 2
+ (J |Ψ X |) 2

Finding the mean-field parameters
leading to:
∂ ϱX E = 0 ∀X , ∂ ψX E = 0 ∀X
(
ϱX − 1 )
2 =
1
2 (± X )
√
(
V
ψ X = − 1 2 (± X ) √(
V
( V 2 R X − µ 2 )
2 R X − µ 2 ) ,
2 +(J|Ψ X |) 2
JΨ X
2 R X − µ 2 ) 2 +(J|Ψ X |) 2

Resulting formulas
Relation between ϱ and ψ independent of other sites
(
ϱX − 1 ) 2
2 + |ψX | 2 = 1 4
|ψ X | = 0 ⇒ ϱ X ∈ {0, 1}
no hopping ⇒ bosons localized in valleys
⇒ expectation value for density: 0 (absent) or 1 (present)

Resulting formulas
Relation between ϱ’s and ψ’s of different sites
⎛
⎜
⎝
N X1 ,X 1
N X1 ,X 2
. . .
N X2 ,X 1
N X2 ,X 2
. . .
.
.
⎞ ⎛
⎟ ⎜
. ⎠ ⎝ . .
} {{ }
reduced connectivity matrix
ψ X1
ψ X2
.
Ψ X
η X := ⇓ ψ X
⎞ ⎛
⎟
⎠ = ⎜
⎝
η X1 0 . . .
0 η X2 . . .
.
.
⎞ ⎛
⎟ ⎜
. ⎠ ⎝ . .
ψ X1
ψ X2
.
⎞
⎟
⎠
⎛
⎜
⎝
ϱ X1
ϱ X2
.
⎛
⎞
⎛
⎟
⎠ = J ⎜
⎝
⎜
⎝
η X1 0 . . .
0 η X2 . . .
.
.
reduced connectivity matrix ⎞−1
{ }} {
⎞ ⎛
⎞
N X1 ,X ⎟
. ⎠ + V 1
N X1 ,X 2
. . .
N X2 ,X
⎜ 1
N X2 ,X 2
. . .
⎟
2 ⎝
. ⎠
. . . .
. . ⎟
⎠
⎛
×
⎜
⎝
1
2
Jη X1 + µ 2 − V 2
∑
j N Y j X 1
(01) Yj
1
2
Jη X2 + µ 2 − V 2
∑
j N Y j X 2
(01) Yj
.
⎞
⎟
⎠
X i ∈ M ψ≠0 ∀i Y j ∈ M ψ=0 ∀j ϱ Yj = (01) Yj ∈ {0, 1}

Resulting formulas
Energies
( (
E ϱ JV A , µ ) (
, ϱ JV
V B , µ )
)
, . . . , η
V
A , η B , . . .
=
V
1 ∑ ∑
N XY ϱ Y ϱ X − µ ∑
ϱ X
2
V
X ∈M ψ=0 Y ∈M ψ=0 X ∈M ψ=0
} {{ }
=: E ψ=0
V
+ 1 ∑ ∑
N XY ϱ Y ϱ X
2
X ∈M ψ=0 Y ∈M ψ≠0
} {{ }
=: E mix
V
− 1 ∑
( J
ϱ X
2
V η X + µ )
V
X ∈M ψ≠0
} {{ }
=: E ψ≠0
V

Possible phases
ϱ
ψ
Mott ∀X , Y ϱ X = ϱ Y ∀X ψ X = 0
Density wave ∃X , Y ϱ X ≠ ϱ Y ∀X ψ X = 0
Superfluid ∀X , Y ϱ X = ϱ Y ∀X , Y ψ X = ψ Y ≠ 0
Supersolid ∃X , Y ϱ X ≠ ϱ Y ∃X , Y ψ X ≠ ψ Y ≠ 0

Quadratic lattice
(V = 1)
Μ
5 MOTT INSULATOR
4
SUPERFLUID:ΡA⩵ΡB0,Ψ0
3
2
DENSITYWAVE:ΡA⩵1,ΡB⩵0,Ψ⩵0
1
0
1 MOTT INSULATOR
0.0 0.1 0.2 0.3 0.4 0.5 0.6
J

Triangular lattice
(V = 1)
Μ
Mott Insulator:ΡA⩵ΡB⩵ΡC⩵1,Ψ⩵0
6
Superfluid:ΡA⩵ΡB⩵ΡC≠0,Ψ≠0
4
Density Wave:ΡA⩵ΡB⩵1,ΡC⩵0,Ψ⩵0
Supersolid: 0≠ΡA⩵ΡB≠ΡC≠0,
0≠ΨA⩵Ψb≠Ψc≠0
2
Density Wave:ΡA⩵ΡB⩵0,ΡC⩵1,Ψ⩵0
0.05 0.10 0.15 0.20 0.25 J
Mott Insulator:ΡA⩵ΡB⩵ΡC⩵0,Ψ⩵0
X.-F. Zhang, R. Dillenschneider, Y. Yu and S.
Eggert, Phys. Rev. B 84, 174515 (2011)
G. Murthy, D. Arovas, A. Auerbach, Phys. Rev.
B 55, 3104 (1996)

Outline
◮ Introduction
◮ Experiment
◮ Hamiltonian
◮ Periodic patterns
◮ Main part
◮ Mean-field (MF) approximation method
◮ MF-Hamiltonian for unit cell in a periodic pattern (dependent on
mean-field parameters)
◮ General calculation method for any pattern
◮ energy eigenvalues
◮ mean-field parameters
◮ Results (quadratic lattice, triangular lattice)
◮ Outlook
◮ Completing mean-field for triangular lattice
◮ Negative hopping J
◮ Other lattices
◮ Quantum corrections

Negative hopping J
Quadratic lattice
(V = 1)
Μ
5 MOTT INSULATOR
4
3
2
1
0
DENSITYWAVE:ΡA⩵1,ΡB⩵0,Ψ⩵0
SUPERFLUID:ΡA⩵ΡB0,Ψ0
SF1:
|ψ A | = |ψ B |
signψ A = signψ B
SF2:
|ψ A | = |ψ B |
signψ A = −signψ B
1 MOTT INSULATOR
0.0 0.1 0.2 0.3 0.4 0.5 0.6
J
A. Eckardt, C. Weiss, and M. Holthaus, Phys. Rev. Lett., 95, 260404 (2005)
A. Zenesini, H.Lignier, C. Sias, O. Morsch, D. Ciampini, E. Arimondo, Phys. Rev. Lett., 102, 100403
(2009)

Other lattices
Kagome, 1D Stripe, decorated triangular, ...
G.-B. Jo, J. Guzman, C. K. Thomas, P. Hosur, A. Vishwanath and D. M. Stamper-Kurn, PRL 108,
045305 (2012)

Quantum corrections
inspired by variational perturbation theory
(H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets,
World Scientific Publishing Company, 5th edition (2009))
)
Ĥ (ξ) = Ĥ MF + ξ
(ĤBH − Ĥ MF
◮ Free energy:
F (ξ) = − 1 β ln Z (ξ), Z (ξ) = Tr (
e −βĤ(ξ) )
◮ Expansion with respect to ξ: 〉
F (ξ) = F MF + ξ
〈Ĥ − ĤMF
MF
+ . . .,
〈Ô〉
MF
)
:= 1
−βĤMF
Z MF
Tr
(Ôe
◮ Truncation after n-th order depends artificially on ϱ X , ψ X for ξ = 1:
F (n) (ξ = 1) = F (n) (ϱ X , ψ X )
◮ Principle of minimal sensitivity:
∂ ϱX F (n) = 0, ∂ ψX F (n) = 0
◮ Quantum corrections:
n = 0: mean-field
n = 1: first quantum corrections
n = 2: ...
F. E. A. dos Santos and A. Pelster, Phys. Rev. A 79, 013614 (2009)