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Variational ansatz for the superfluid Mott-insulator transition in optical lattices

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Abstract

We develop a variational wave function for the ground state of a one-dimensional bosonic lattice gas. The variational theory is initally developed for the quantum rotor model and later on extended to the Bose-Hubbard model. This theory is compared with quasi-exact numerical results obtained by Density Matrix Renormalization Group (DMRG) studies and with results from other analytical approximations. Our approach accurately gives local properties for strong and weak interactions, and it also describes the crossover from the superfluid phase to the Mott-insulator phase.

©2004 Optical Society of America

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Figures (5)

Fig. 1.
Fig. 1. Instead of working directly with the population of each well, nk , we can use other quantum numbers, wk , defined by the relation nk =wk -wk -1+, and which behave like a set of chemical potentials acting on the barriers that connect neigboring sites.
Fig. 2.
Fig. 2. Estimates for (a) energy energy per lattice site and (b) density fluctuations of the quantum rotor Hamiltonian (6) obtained with the variational method (solid), and perturbative calculations for U≪J (dashed) and UJ (dots).
Fig. 3.
Fig. 3. (a) The ground state energy per site, ε, (b) nearest neighbor correlation, c 1=〈aj+1aj 〉, and (c) variance of the number of atoms per site, σ 2=〈(nj -)2〉. Plots (b) and (c) use a log-log scale. The results of the DMRG (solid line) are obtained on a system with 128 sites, a maximum occupation number of 9 bosons per site and a reduced space of states of about 200 states. The estimates from the variational theory are plotted using dashed lines. The vertical lines mark the location of the phase transition according to [11]. The mean occupation numbers are denoted with circles (=1), diamonds (=2) and boxes (=3).
Fig. 4.
Fig. 4. (a) The ground state energy per site, ε, (b) nearest neighbor correlation, c 1=〈aj+1aj 〉, and (c) variance of the number of atoms per site, σ 2=〈(nj -)2〉. Plot (b) and (c) are in log-log scale. Using filling factor n̄=1, we show results from the variational model for the Bose-Hubbard model using phase coherent states (solid), the quantum rotor model (dashed), the Gutzwiller ansatz for the Bose-Hubbard Hamiltonian (dots) and DMRG (circles). Vertical dash-dot lines mark the location of the phase transition according to [11].
Fig. 5.
Fig. 5. The energy of the product ansatz contains a contribution from each connection, εest , plus the interaction between neighbouring connections, Δεest . In Fig. (a) we show that Δεest (dash) is actually negative, and improves the estimate εest moving it towards the exact value, εDMRG (circles). Everything has been computed for =1. In Fig. (b) we show that the correction Δεest does not change much for large lattices.

Equations (39)

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H B H = j = 1 M [ J ( a j + 1 a j + a j a j + 1 ) + U 2 a j a j a j a j ] U 2 M n ¯ ( n ¯ 1 ) ,
ψ = n c n n = n c n n 1 n M ,
a l a j ψ = n ¯ ( n ¯ + 1 ) 𝓟 A l + A j n + Δ l j
Δ l j = n c n ( n ¯ + 1 ) ( n j n ¯ ) + n ( n l n ) 2 n ¯ ( n ¯ + 1 ) n ,
Δ l j n ¯ 2 + ( n ¯ + 1 ) 2 2 n ¯ ( n ¯ + 1 ) σ l ,
H Q R = 𝓟 j [ 2 ρ J ( A j + 1 + A j + A j + A j + 1 ) + U 2 ( A j z ) 2 ]
ε 1 M H Q R .
n ϕ = e i n · ϕ ( 2 π ) M 2 , ϕ [ π , π ] M .
H Q R = j [ 2 ρ J cos ( ϕ j ϕ j + 1 ) U 2 2 ϕ j 2 ] ,
ψ = ( 2 π ) M 2 e i n ¯ Σ ϕ k Ψ ( ϕ ) | ϕ d M ϕ .
Ψ ( ϕ ) = j = 1 M h ( ϕ j ϕ j + 1 ) .
n k = w k w k 1 + n ¯ .
ψ = h ˜ ( M 1 ) = w h ˜ w 1 h ˜ w M 1 w 1 w M 1 ,
h ˜ m = h ( ξ ) e i m ξ d ξ .
H Q R = k = 1 M 1 [ 2 ρ J ( k + + k ) + U 2 ( k z k 1 z ) 2 ] ,
ε [ h ˜ ] 4 ρ J Re Σ + + U ( Σ z ) 2 U Σ z 2 ,
2 ρ J ( h ˜ j + 1 + h ˜ j 1 ) + U j 2 h ˜ j = ε est h ˜ j ,
[ U 2 2 ξ 2 2 ρ J cos ( ξ ) ] h ( ξ ) = ε est h ( ξ ) .
σ j 2 = ( a j a j n ¯ ) 2 = 2 ( Σ z ) 2 ,
a j + 1 a j = ρ j ρ γ 1 ,
a j + l a j = ρ k = j j + l 1 k = ρ ( γ 1 ) l ,
H Q R j [ U 2 2 ϕ j 2 ρ J ( ϕ j ϕ j + 1 ) 2 ] .
E g 2 M π 2 ρ J U .
σ 1 π 8 J ρ U .
n | ϕ = e i n ϕ n ! .
H coh t = J i , j [ 2 ( n ¯ + 1 ) cos ( ϕ i ϕ j ) i e i ( ϕ i ϕ j ) ϕ j ] + U 2 j ( 2 ϕ j 2 ) .
O n = k = 1 M n k ! n .
H coh = O H B H O 1
= J i , j ( A i z + n ¯ ) A i + A j + U 2 j ( A j z ) 2 .
H coh = H 1 + H 2 ,
H 1 = j [ J n ¯ Σ x + i J j z j y + U ( j z ) 2 ] ,
H 2 = j [ J ( j 1 z j + j + 1 z j ) + U j z j + 1 z ] ,
ε 0 = min ψ 0 ψ H B H ψ ψ 2 = min χ 0 χ O 2 H coh χ χ O 2 χ ,
ε 0 ε est + 1 N h ˜ M O 2 H 2 h ˜ M h ˜ M O 2 h ˜ M ε est + Δ ε est .
[ U 2 ξ 2 2 J ( n ¯ + 1 ) cos ( ξ ) 2 J sin ( ξ ) ξ ] h = ε est h ,
a j + 1 a j = J ε var ,
σ 2 = U ε var n ¯ 2 ,
ϕ = h ˜ 1 h ˜ k 1 A h ˜ k B h ˜ k + 1 h ˜ k + 2 h ˜ M ,
a k + Δ a k ψ = u t O ( H O ) k 1 ( H O ) Δ ( H O ) M k + 1 u u t O ( H O ) M u .
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