Bose–Hubbard model

The Bose–Hubbard model gives a description of the physics of interacting bosons on a lattice. It is closely related to the Hubbard model which originated in solid-state physics as an approximate description of superconducting systems and the motion of electrons between the atoms of a crystalline solid. The model was first introduced by Gersch and Knollman ^[1] in 1963; the term 'Bose' in its name refers to the fact that the particles in the system are bosonic.

The Bose–Hubbard model can be used to model systems such as bosonic atoms in an optical lattice. Furthermore, it can also be generalized and applied to Bose–Fermi mixtures, in which case the corresponding Hamiltonian is called the Bose–Fermi–Hubbard Hamiltonian.

The Hamiltonian

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

${\displaystyle H=-t\sum _{\left\langle i,j\right\rangle }{\hat {b}}_{i}^{\dagger }{\hat {b}}_{j}+{\frac {U}{2}}\sum _{i}{\hat {n}}_{i}\left({\hat {n}}_{i}-1\right)-\mu \sum _{i}{\hat {n}}_{i}}$.

Here ${\displaystyle \left\langle i,j\right\rangle }$ denotes summation over all neighboring lattice sites $i$ and $j$, while ${\displaystyle {\hat {b}}_{i}^{\dagger }}$ and ${\displaystyle {\hat {b}}_{i}^{}}$ are bosonic creation and annihilation operators such that ${\displaystyle {\hat {n}}_{i}={\hat {b}}_{i}^{\dagger }{\hat {b}}_{i}}$ gives the number of particles on site $i$. The model is parametrized by the hopping amplitude $t$ describing the mobility of bosons in the lattice, the on-site interaction $U$ which can be attractive (${\displaystyle U<0}$) or repulsive (${\displaystyle U>0}$), and the chemical potential $\mu$.

The dimension of the Hilbert space of the Bose–Hubbard model is given by ${\displaystyle D_{b}={\frac {(N_{b}+L-1)!}{N_{b}!(L-1)!}}}$. At fixed ${\displaystyle N_{b}}$ or $L$, ${\displaystyle D_{b}}$ grows polynomially, but at a fixed density of ${\displaystyle n_{b}}$ bosons per site, it grows exponentially as ${\displaystyle D_{b}\sim \left((1+n_{b})\left(1+{\frac {1}{n_{b}}}\right)^{n_{b}}\right)^{L}}$. The model can also be considered with spinless fermionic atoms, then it is called Fermi–Hubbard Model. In this case ${\displaystyle D_{f}={\frac {L!}{N_{f}!(L-N_{f})!}}}$, which grows polynomially at fixed ${\displaystyle N_{f}}$ but exponentially as ${\displaystyle D_{f}\sim \left({\frac {(1-n_{f})^{-(1-n_{f})}}{{n_{f}}^{n_{f}}}}\right)^{L}}$ at fixed density ${\displaystyle n_{f}}$ fermions per site. The different results stem from different statistics of fermions and bosons.

Analogous Hamiltonians may be formulated to describe mixtures of different atom species (Bose–Fermi mixtures being a prominent example). Then 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.

Phase diagram

At zero temperature, the Bose–Hubbard model (in the absence of disorder) is in either a Mott insulating (MI) state at small ${\displaystyle t/U}$, or in a superfluid (SF) state at large ${\displaystyle t/U}$.^[2] 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. In the presence of disorder, a third, ‘‘Bose glass’’ phase exists. 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.

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, lack of lattice dynamics, present in electronic systems provides very good conditions for experimental study of this model.^[4][5]

The Hamiltonian in second quantization formalism describing a gas of ultracold atoms in the optical lattice potential is of the form

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

where ${\displaystyle V_{latt}}$ is the optical lattice potential, $g$ is the (contact) interaction amplitude, and $\mu$ is the chemical potential. The standard tight binding approximation results in the substitution ${\displaystyle {\hat {\psi }}({\vec {r}})=\sum \limits _{i}w_{i}^{\alpha }({\vec {r}})b_{i}^{\alpha }}$ which gives the Bose-Hubbard Hamiltonian if one restricts physics to the lowest band ($\alpha =0$) and the interactions are local also at the level of the discrete mode: ${\displaystyle \int w_{i}^{\alpha }({\vec {r}})w_{j}^{\beta }({\vec {r}})w_{k}^{\gamma }(r)w_{l}^{\delta }({\vec {r}})\,{\rm {d}}^{3}r=0}$ except for case ${\displaystyle i=j=k=l\wedge \alpha =\beta =\gamma =\delta =0}$. Here, ${\displaystyle w_{i}^{\alpha }({\vec {r}})}$ is a Wannier function for a particle in an optical lattice potential localized around site i of the lattice and for $\alpha$th Bloch band.^[6]

Subtleties and approximations

The tight-binding approximation simplifies significantly the second quantization Hamiltonian, introducing several limitations in 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 $U_{n}$ close, but not equal to U.^[6]
• When considering (fast) lattice dynamics, additional terms should be added to the Bose-Hubbard hamiltonian, so that the time-dependent Schrödinger equation was obeyed in the (time-dependent) Wannier function basis. They come from dependence on time of Wannier functions.^[7][8] Otherwise the dynamics of the lattice may be incorporated by making the key parameters of the model: U and J time-dependent and be linked to the instantaneous values of the optical potential most prominently its depth.

Experimental results

Quantum phase transitions in the Bose–Hubbard model were experimentally observed by Greiner et al.^[9] in Germany. Density dependent interaction parameters ${\displaystyle U_{n}}$ were observed by I.Bloch's group ^[10]

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.^[11]

Numerical simulation

In the calculation of low energy states the term proportional to ${\displaystyle n^{2}U}$ means that large occupation of a single site is improbable, allowing for truncation of local Hilbert space to states containing at most ${\displaystyle d<\infty }$ particles. Then the local Hilbert space dimension is ${\displaystyle d+1.}$ The dimension of the full Hilbert space grows exponentially with the number of sites in the lattice, therefore computer simulations 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. For the numerical simulation of this model, an algorithm of exact diagonalization is presented in this paper.^[12]

One-dimensional lattices may be treated by 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.

Higher dimensions are significantly more difficult due to the quick growth of entanglement.^[13]

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

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 density-density interaction ${\displaystyle Vn_{i}n_{j}}$
• long-range dipolar interaction ^[14]
• systems with interaction-induced tunneling terms ${\displaystyle a_{i}^{\dagger }(n_{i}+n_{j})a_{j}}$ ^[15]
• internal spin structure of atoms, for example due to trapping entire degenerate manifold of hyperfine spin states (for F=1 i leads to the spin-1 Bose–Hubbard model) ^[16]
• situation where the gas feel presence of an additional potential for example for disordered systems.^[17] 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: ${\displaystyle H_{I}=V_{d}\sum \limits _{i}\cos(ki+\varphi ){\hat {n}}_{i}}$

See also

• Hubbard model
• Jaynes-Cummings-Hubbard model

References

1. Gersch, H.; Knollman, G. (1963). "Quantum Cell Model for Bosons". Physical Review. 129 (2): 959. Bibcode:1963PhRv..129..959G. doi:10.1103/PhysRev.129.959.
2. Kühner, T.; Monien, H. (1998). "Phases of the one-dimensional Bose-Hubbard model". Physical Review B. 58 (22): R14741. arXiv:cond-mat/9712307. Bibcode:1998PhRvB..5814741K. doi:10.1103/PhysRevB.58.R14741.
3. Fisher, Matthew P. A.; Grinstein, G.; Fisher, Daniel S. (1989). "Boson localization and the superfluid-insulator transition". Physical Review B. 40: 546–70. Bibcode:1989PhRvB..40..546F. doi:10.1103/PhysRevB.40.546.,
4. Jaksch, D.; Bruder, C.; Cirac, J.; Gardiner, C.; Zoller, P. (1998). "Cold Bosonic Atoms in Optical Lattices". Physical Review Letters. 81 (15): 3108. arXiv:cond-mat/9805329. Bibcode:1998PhRvL..81.3108J. doi:10.1103/PhysRevLett.81.3108.
5. Jaksch, D.; Zoller, P. (2005). "The cold atom Hubbard toolbox". Annals of Physics. 315: 52. arXiv:cond-mat/0410614. Bibcode:2005AnPhy.315...52J. doi:10.1016/j.aop.2004.09.010.
6. Lühmann, D. S. R.; Jürgensen, O.; Sengstock, K. (2012). "Multi-orbital and density-induced tunneling of bosons in optical lattices". New Journal of Physics. 14 (3): 033021. arXiv:1108.3013. Bibcode:2012NJPh...14c3021L. doi:10.1088/1367-2630/14/3/033021.
7. Sakmann, K.; Streltsov, A. I.; Alon, O. E.; Cederbaum, L. S. (2011). "Optimal time-dependent lattice models for nonequilibrium dynamics". New Journal of Physics. 13 (4): 043003. arXiv:1006.3530. Bibcode:2011NJPh...13d3003S. doi:10.1088/1367-2630/13/4/043003.
8. Łącki, M.; Zakrzewski, J. (2013). "Fast Dynamics for Atoms in Optical Lattices". Physical Review Letters. 110 (6). arXiv:1210.7957. Bibcode:2013PhRvL.110f5301L. doi:10.1103/PhysRevLett.110.065301.
9. Greiner, Markus; Mandel, Olaf; Esslinger, Tilman; Hänsch, Theodor W.; Bloch, Immanuel (2002). "Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms". Nature. 415 (6867): 39–44. doi:10.1038/415039a. PMID 11780110.
10. Will, S.; Best, T.; Schneider, U.; Hackermüller, L.; Lühmann, D. S. R.; Bloch, I. (2010). "Time-resolved observation of coherent multi-body interactions in quantum phase revivals". Nature. 465 (7295): 197–201. Bibcode:2010Natur.465..197W. doi:10.1038/nature09036. PMID 20463733.
11. Romero-Isart, O; Eckert, K; Rodó, C; Sanpera, A (2007). "Transport and entanglement generation in the Bose–Hubbard model". Journal of Physics A: Mathematical and Theoretical. 40 (28): 8019–31. arXiv:quant-ph/0703177. Bibcode:2007JPhA...40.8019R. doi:10.1088/1751-8113/40/28/S11.
12. Zhang, J M; Dong, R X (2010). "Exact diagonalization: The Bose–Hubbard model as an example". European Journal of Physics. 31 (3): 591–602. arXiv:1102.4006. Bibcode:2010EJPh...31..591Z. doi:10.1088/0143-0807/31/3/016.
13. Eisert, J.; Cramer, M.; Plenio, M. B. (2010). "Colloquium: Area laws for the entanglement entropy". Reviews of Modern Physics. 82: 277. arXiv:0808.3773. Bibcode:2010RvMP...82..277E. doi:10.1103/RevModPhys.82.277.
14. Góral, K.; Santos, L.; Lewenstein, M. (2002). "Quantum Phases of Dipolar Bosons in Optical Lattices". Physical Review Letters. 88 (17). arXiv:cond-mat/0112363. Bibcode:2002PhRvL..88q0406G. doi:10.1103/PhysRevLett.88.170406.
15. Sowiński, T.; Dutta, O.; Hauke, P.; Tagliacozzo, L.; Lewenstein, M. (2012). "Dipolar Molecules in Optical Lattices". Physical Review Letters. 108 (11). arXiv:1109.4782. Bibcode:2012PhRvL.108k5301S. doi:10.1103/PhysRevLett.108.115301.
16. Tsuchiya, S.; Kurihara, S.; Kimura, T. (2004). "Superfluid–Mott insulator transition of spin-1 bosons in an optical lattice". Physical Review A. 70 (4). arXiv:cond-mat/0209676. Bibcode:2004PhRvA..70d3628T. doi:10.1103/PhysRevA.70.043628.
17. Gurarie, V.; Pollet, L.; Prokof’Ev, N. V.; Svistunov, B. V.; Troyer, M. (2009). "Phase diagram of the disordered Bose-Hubbard model". Physical Review B. 80 (21). arXiv:0909.4593. Bibcode:2009PhRvB..80u4519G. doi:10.1103/PhysRevB.80.214519.