1. The Hamiltonian

The physics of this model is given by the Bose–Hubbard Hamiltonian:

[math]\displaystyle{ H = -t \sum_{ \left\langle i, j \right\rangle } \hat{b}^{\dagger}_i \hat{b}_j + \frac{U}{2} \sum_{i} \hat{n}_i \left( \hat{n}_i - 1 \right) - \mu \sum_i \hat{n}_i }[/math].

Here, [math]\displaystyle{ \left\langle i, j \right\rangle }[/math] denotes summation over all neighboring lattice sites [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math], while [math]\displaystyle{ \hat{b}^{\dagger}_i }[/math] and [math]\displaystyle{ \hat{b}^{}_i }[/math] are bosonic creation and annihilation operators such that [math]\displaystyle{ \hat{n}_i = \hat{b}^{\dagger}_i \hat{b}_i }[/math] gives the number of particles on site [math]\displaystyle{ i }[/math]. The model is parametrized by the hopping amplitude [math]\displaystyle{ t }[/math] describing the mobility of bosons in the lattice, the on-site interaction [math]\displaystyle{ U }[/math] which can be attractive ([math]\displaystyle{ U \lt 0 }[/math]) or repulsive ([math]\displaystyle{ U \gt 0 }[/math]), and the chemical potential [math]\displaystyle{ \mu }[/math], which essentially sets the total number of particles. If unspecified, typically the phrase 'Bose–Hubbard model' refers to the case where the on-site interaction is repulsive.

This Hamiltonian has a global [math]\displaystyle{ U(1) }[/math] symmetry, which means that it is invariant (i.e. its physical properties are unchanged) by the transformation [math]\displaystyle{ \hat{b}_i \rightarrow e^{i \theta} \hat{b}_i }[/math]. In a superfluid phase, this symmetry is spontaneously broken.

Hilbert Space

The dimension of the Hilbert space of the Bose–Hubbard model is given by [math]\displaystyle{ D_{b}= (N_{b}+L-1)!/N_{b}!(L-1)! }[/math], where [math]\displaystyle{ N_{b} }[/math] is the total number of particles, while [math]\displaystyle{ L }[/math] denotes the total number of lattice sites. At fixed [math]\displaystyle{ N_b }[/math] or [math]\displaystyle{ L }[/math], the Hilbert space dimension [math]\displaystyle{ D_b }[/math] grows polynomially, but at a fixed density of [math]\displaystyle{ n_b }[/math] bosons per site, it grows exponentially as [math]\displaystyle{ D_b \sim \left[ (1 + n_b) \left( 1 + \frac{1}{n_b} \right)^{n_b} \right]^L }[/math]. Analogous Hamiltonians may be formulated to describe spinless fermions (the Fermi-Hubbard model) or mixtures of different atom species (Bose–Fermi mixtures, for example). In the case of a mixture, the Hilbert space is simply the tensor product of the Hilbert spaces of the individual species. Typically additional terms need to be included to model interaction between species.

2. Phase Diagram

At zero temperature, the Bose–Hubbard model (in the absence of disorder) is in either a Mott insulating state at small [math]\displaystyle{ t / U }[/math], or in a superfluid state at large [math]\displaystyle{ t / U }[/math].^[1] The Mott insulating phases are characterized by integer boson densities, by the existence of an energy gap for particle-hole excitations, and by zero compressibility. The superfluid is characterized by long-range phase coherence, a spontaneous breaking of the Hamiltonian's continuous [math]\displaystyle{ U(1) }[/math] symmetry, a non-zero compressibility and superfluid susceptibility. At non-zero temperature, in certain parameter regimes there will also be a regular fluid phase which does not break the [math]\displaystyle{ U(1) }[/math] symmetry and does not display phase coherence. Both of these phases have been experimentally observed in ultracold atomic gases.^[2]

In the presence of disorder, a third, "Bose glass" phase exists.^[3] The Bose glass is a Griffiths phase, and can be thought of as a Mott insulator containing rare 'puddles' of superfluid. These superfluid pools are not connected to each other, so the system remains insulating, but their presence significantly changes the thermodynamics of the model. The Bose glass phase is characterized by a finite compressibility, the absence of a gap, and by an infinite superfluid susceptibility.^[3] It is insulating despite the absence of a gap, as low tunneling prevents the generation of excitations which, although close in energy, are spatially separated. The Bose glass has been shown to have a non-zero Edwards-Anderson order parameter^[4][5] and has been suggested to display replica symmetry breaking,^[6] however this has not been proven.

3. Mean-Field Theory

The phases of the clean Bose–Hubbard model can be described using a mean-field Hamiltonian:^[7][math]\displaystyle{ \begin{align} H_{\textrm{MF}}&=\sum_i \left[ -\mu \hat{n}_i +\frac{U}{2} \hat{n}_i(\hat{n}_i-1)-zt(\psi^{*} \hat{b}_i +\psi\hat{b}^{\dagger}_i)+zt\psi^{*}\psi \right] \end{align} }[/math]where [math]\displaystyle{ z }[/math] is the lattice co-ordination number. This can be obtained from the full Bose-Hubbard Hamiltonian by setting [math]\displaystyle{ \hat{b}_{i} \rightarrow \psi+\delta \hat{b} }[/math] where [math]\displaystyle{ \psi=\langle \hat{b}_{i} \rangle }[/math], neglecting terms quadratic in [math]\displaystyle{ \delta \hat{b}_{i} }[/math] (which we assume to be infinitesimal) and relabelling [math]\displaystyle{ \delta \hat{b}_{i} \rightarrow \hat{b}_{i} }[/math]. Because this decoupling breaks the [math]\displaystyle{ U(1) }[/math] symmetry of the initial Hamiltonian for all non-zero values of [math]\displaystyle{ \psi }[/math], this parameter acts as a superfluid order parameter. For simplicity, this decoupling assumes [math]\displaystyle{ \psi }[/math] to be the same on every site - this precludes exotic phases such as supersolids or other inhomogeneous phases. (Other decouplings are of course possible if one wishes to allow for such phases.)

We can obtain the phase diagram by calculating the energy of this mean-field Hamiltonian using second-order perturbation theory and finding the condition for which [math]\displaystyle{ \psi \neq 0 }[/math]. To do this, we first write the Hamiltonian as a site-local piece plus a perturbation:[math]\displaystyle{ H_{\textrm{MF}}=\sum_{i}\left[ h^{(0)}_{i}-zt(\psi^{*} \hat{b}_i +\psi\hat{b}^{\dagger}_i) \right] \quad \textrm{with} \quad h^{(0)}_i=-\mu \hat{n}_i +\frac{U}{2} \hat{n}_i(\hat{n}_i-1)+zt\psi^{*}\psi }[/math]where the bilinear terms [math]\displaystyle{ \psi^{*}\hat{b}_i }[/math] and its conjugate are treated as the perturbation, as we assume the order parameter [math]\displaystyle{ \psi }[/math] to be small near the phase transition. The local term is diagonal in the Fock basis, giving the zeroth-order energy contribution:[math]\displaystyle{ E^{(0)}_m=-\mu m + \frac{U}{2}m(m-1)+zt |\psi|^{2} }[/math]where [math]\displaystyle{ m }[/math] is an integer that labels the filling of the Fock state. The perturbative piece can be treated with second-order perturbation theory, which leads to:[math]\displaystyle{ E^{(2)}_n=zt |\psi|^2 \sum_{n \neq m} \frac{|\langle m|(\hat{b}_{i}^{\dagger}+\hat{b}_{i})|n \rangle|^2}{E^{(0)}_n-E^{(0)}_m} =-(zt)^2 |\psi|^{2} \left( \frac{m}{U(m-1)-\mu} + \frac{m+1}{\mu-U m} \right). }[/math]We can then express the energy as a series expansion in even powers of the order parameter (also known as the Landau formalism):[math]\displaystyle{ E=\text{constant} + R |\psi|^2 + W |\psi|^4 +... }[/math]After doing so, the condition for the mean-field, second-order phase transition between the Mott insulator and the superfluid phase is given by:[math]\displaystyle{ r=\frac{R}{zt}=1+zt \left( \frac{m}{U(m-1)-\mu} + \frac{m+1}{\mu-U m} \right)=0 }[/math]where the integer [math]\displaystyle{ m }[/math] describes the filling of the [math]\displaystyle{ m^{th} }[/math] Mott insulating lobe. Plotting the line [math]\displaystyle{ r=0 }[/math] for different integer values of [math]\displaystyle{ m }[/math] will generate the boundary of the different Mott lobes, as shown in the phase diagram.^[3]

4. Implementation in Optical Lattices

Ultracold atoms in optical lattices are considered a standard realization of the Bose–Hubbard model. The ability to tune parameters of the model using simple experimental techniques and the lack of the lattice dynamics which are present in solid-state electronic systems mean that ultracold atoms offer a very clean, controllable realisation of the Bose–Hubbard model.^[8][9] The biggest downside with optical lattice technology is the trap lifetime, with atoms typically only being trapped for a few tens of seconds.

To see why ultracold atoms offer such a convenient realisation of Bose-Hubbard physics, we can derive the Bose-Hubbard Hamiltonian starting from the second quantized Hamiltonian which describes a gas of ultracold atoms in the optical lattice potential. This Hamiltonian is given by the expression:

[math]\displaystyle{ H= \int {\rm d}^3 r \! \left[ \hat\psi^\dagger(\vec r) \left ( -\frac{\hbar^2}{2m} \nabla^2 +V_{\rm latt.}(\vec r) \right) \hat\psi(\vec r) + \frac{g}{2}\hat \psi^\dagger(\vec r)\hat\psi^\dagger(\vec r)\hat\psi(\vec r)\hat\psi(\vec r) - \mu \hat{\psi}^\dagger(\vec r)\hat\psi(\vec r)\right] }[/math],

where [math]\displaystyle{ V_{latt} }[/math] is the optical lattice potential, [math]\displaystyle{ g }[/math] is the (contact) interaction amplitude, and [math]\displaystyle{ \mu }[/math] is the chemical potential. The tight binding approximation results in the substitution [math]\displaystyle{ \hat\psi(\vec r) = \sum\limits_i w_i^\alpha (\vec r) b_i^\alpha }[/math] which leads to the Bose-Hubbard Hamiltonian if one restricts the physics to the lowest band ([math]\displaystyle{ \alpha=0 }[/math]) and the interactions are local at the level of the discrete mode. Mathematically, this can be stated as the requirement that [math]\displaystyle{ \int w_i^\alpha(\vec r)w_j^\beta(\vec r)w_k^\gamma(r)w_l^\delta(\vec r)\,{\rm d}^3 r=0 }[/math] except for case [math]\displaystyle{ i=j=k=l \wedge \alpha=\beta=\gamma=\delta=0 }[/math]. Here, [math]\displaystyle{ w_i^\alpha(\vec r) }[/math] is a Wannier function for a particle in an optical lattice potential localized around site [math]\displaystyle{ i }[/math] of the lattice and for the [math]\displaystyle{ \alpha }[/math]th Bloch band.^[10]

Subtleties and Approximations

The tight-binding approximation significantly simplifies the second quantized Hamiltonian, though it introduces several limitations at the same time:

• For single-site states with several particles in a single state, the interactions may couple to higher Bloch bands, which contradicts base assumptions. Still, a single band model is able to address low-energy physics of such setting but with parameters U and J becoming in fact density-dependent. Instead of one parameter U, the interaction energy of n particles may be described by [math]\displaystyle{ U_n }[/math] close, but not equal to U.^[10]
• When considering (fast) lattice dynamics, additional terms should be added to the Bose-Hubbard Hamiltonian, so that the time-dependent Schrödinger equation is obeyed in the (time-dependent) Wannier function basis. They come from the time dependence of Wannier functions.^[11][12] Otherwise, the dynamics of the lattice may be incorporated by making the key parameters of the model time-dependent, varying with the instantaneous value of the optical potential.

5. Experimental Results

Quantum phase transitions in the Bose–Hubbard model were experimentally observed by Greiner et al.,^[2] and density dependent interaction parameters [math]\displaystyle{ U_n }[/math] were observed by I.Bloch's group.^[13] Single-atom resolution imaging of the Bose–Hubbard model has been possible since 2009 using quantum gas microscopes.^[14][15][16]

6. Further Applications of the Model

The Bose–Hubbard model is also of interest to those working in the field of quantum computation and quantum information. Entanglement of ultra-cold atoms can be studied using this model.^[17]

7. Numerical Simulation

In the calculation of low energy states the term proportional to [math]\displaystyle{ n^2 U }[/math] means that large occupation of a single site is improbable, allowing for truncation of local Hilbert space to states containing at most [math]\displaystyle{ d \lt \infty }[/math] particles. Then the local Hilbert space dimension is [math]\displaystyle{ d+1. }[/math] The dimension of the full Hilbert space grows exponentially with the number of sites in the lattice, therefore exact computer simulations of the entire Hilbert space are limited to the study of systems of 15-20 particles in 15-20 lattice sites. Experimental systems contain several millions lattice sites, with average filling above unity.

One-dimensional lattices may be studied using density matrix renormalization group (DMRG) and related techniques such as time-evolving block decimation (TEBD). This includes to calculate the ground state of the Hamiltonian for systems of thousands of particles on thousands of lattice sites, and simulate its dynamics governed by the Time-dependent Schrödinger equation. Recently, two dimensional lattices have also been studied using Projected Entangled Pair States which is a generalization of Matrix Product States in higher dimensions, both for the ground state ^[18] as well as finite temperature.^[19]

Higher dimensions are significantly more difficult due to the quick growth of entanglement.^[20]

All dimensions may be treated by Quantum Monte Carlo algorithms, which provide a way to study properties of thermal states of the Hamiltonian, and in particular the ground state.

8. Generalizations

Bose-Hubbard-like Hamiltonians may be derived for different physical systems containing ultracold atom gas in the periodic potential. They include, but are not limited to:

• systems with longer-ranged density-density interactions of the form [math]\displaystyle{ V n_i n_j }[/math], which may stabilise a supersolid phase for certain parameter values
• dimerised magnets, where spin-1/2 electrons are bound together in pairs called dimers which have bosonic excitation statistics and are described by a hard-core Bose–Hubbard model
• long-range dipolar interaction ^[21]
• systems with interaction-induced tunneling terms [math]\displaystyle{ a_i^\dagger ( n_i + n_j ) a_j }[/math] ^[22]
• internal spin structure of atoms, for example due to trapping entire degenerate manifold of hyperfine spin states (for F=1 it leads to the spin-1 Bose–Hubbard model)^[23]
• situations where the gas experiences an additional potential—for example, in disordered systems.^[24] The disorder might be realised by a speckle pattern, or using a second incommensurate, weaker optical lattice. In the latter case inclusion of the disorder amounts to including extra term of the form: [math]\displaystyle{ H_I=V_d \sum\limits_i \cos(k i+\varphi) \hat{n}_i }[/math]

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