Bose-Hubbard and extended Bose-Hubbard models

Consider a lattice with bosons on each site, where operators are defined as the following:

• $\hat{a}_i, \hat{a}^{\dagger}_i$ - boson annihilation and creation operators for site $i$,
• $\hat{n}_{i}$ - boson occupation number operator.

Bose-Hubbard model includes three terms:

• chemical potential term $\mu$, which tells the price of adding more particles to the system,
• hopping amplitude $t$, which defines how likely the bosons are hopping between the nearest neighbour lattice sites
• onsite interaction potential $U$, which prevents bosons from occupying the same lattice site

Bose-Hubbard hamiltonian:

$H = -t\sum_{\left<i, j\right>}\left(\hat{a}_i^\dagger\hat{a}_j + \hat{a}_i\hat{a}_j^{\dagger}\right) + \frac{U}{2}\sum_i\hat{n}_i(\hat{n}_i - 1) - \mu \sum_i \hat{n}_i$

If we include repulsion between nearest neighbours, we will end up with extended Bose-Hubbard model, which demonstrates several new phases even in one-dimensional case. Besides Mott insulator and superfluid phases present in the original Bose-Hubbard model, you can observe density wave, supersolid and Haldane insulator phases in 1D extended Bose-Hubbard model.

Extended Bose-Hubbard hamiltonian:

$H = -t\sum_{\left<i, j\right>}\left(\hat{a}_i^\dagger\hat{a}_j + \hat{a}_i\hat{a}_j^{\dagger}\right) + \frac{U}{2}\sum_i\hat{n}_i(\hat{n}_i - 1) - V\sum n_i n_j - \mu \sum_i \hat{n}_i$

DMRG Calculations and Correlators

Using DMRG and Julia ITensor package, we can compute groundstates of the $N$-site 1D Bose-Hubbard and extended Bose-Hubbard models in MPS form. After that we can calculate correlation matrices and different order parameters, including:

• $\Gamma_{i,j} = \left< \hat{a}_i^\dagger \hat{a}_j\right>$ - correlation matrix for creation and annihilation operators

• $n_{i,j} = \left< \hat{n}_i \hat{n}_j\right>$ - correlation matrix for occuptaion numbers

• $n_{i} = \left< n_i\right>$ - occupation number of sites

• $\Gamma(r) = \Gamma_{N/2, N/2+r}$ - correlation function

• $\Gamma(r) \sim e^{-r/\xi}, \; \Gamma \sim r^{-K/2}$ - correlation lengths $\xi$ and $K$ from fits to exponential and polynomial decay correspondingly

• $S_{\pi} = \frac{1}{N^2}\sum_{i,j} (-1)^{|i-j|} \left< \hat{n}_i \hat{n}_j\right>$ - structure factor

Phase diagram of extended Bose-Hubbard model. Upper the white stripped line: correlator is taken to be |n - 1.0| to detect Mott insulator with average filling equal to one; lower the white stripped line: correlator is taken to be |n - 0.5| to detect density wave phase with average filling equal to one half.

Slight change in average occupation number in the Mott insulator region poses a question whether this is a numerical error or a new phase. If you think for a while about the possible source of a change in average occupation number on a level $10^{-2}-10^{-1}$ in a chain of size $N=51$, you might end up with the idea that there is some contribution from the edge states.

Dependence of different correlators on chemical potential, crossing MI, SF, DW and HI phases.

Haldane insulator phase is detected by appearance of edge states in the region, where Mott insulator is expected. Left: Mott insulator occupation numbers; right: Haldane insulator occupation numbers.

Mott insulator is characterized by exponential decay of correlation function $\Gamma(r) \sim e^{-r/\xi}$, while superfluid phase demonstrates polynomial decay $\Gamma(r) \sim r^{-K/2}$. Correlation lengths $\xi, K$ can be extracted by fitting $\Gamma(r)$ with exponent and polynomial.

Left: polynomial decay of correlation function manifests superfluid phase with long-range correlations; right: Mott insulator phase with exponential decay of correlation function and short-range correlations.

We can also plot phase diagrams using correlation lenghts as order parameters

Bose-Hubbard phase diagram. Left: exponential correlation length; right: polynomial correlation length. Both order parameters show that there are two different phases.

Extended Bose-Hubbard phase diagram with exponential correlation length as order parameter.