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| The '''Bose–Hubbard model''' gives an approximate description of the physics of interacting [[boson]]s on a [[Lattice model (physics)|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 name Bose refers to the fact that the particles in the system are [[boson]]ic; the model was first introduced by Gersch H., Knollman G <ref>{{cite doi | 10.1103/PhysRev.129.959}}</ref> in 1963, The Bose–Hubbard model can be used to study systems such as bosonic atoms on an [[optical lattice]]. In contrast, the Hubbard model applies to [[fermions|fermionic]] particles such as electrons, rather than bosons. Furthermore, it can also be generalized and applied to Bose–Fermi mixtures, in which case the corresponding [[Hamiltonian (quantum mechanics)|Hamiltonian]] is called the Bose–Fermi–Hubbard Hamiltonian.
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| == The hamiltonian ==
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| <!-- Deleted image removed: [[Image:Bose Hubbard phase diagram.jpg|thumb|right|350px|The zero-temperature phase diagram of the Bose–Hubbard model. MI = Mott insulator, SF = superfluid.]] -->
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| The physics of this model is given by the Bose–Hubbard Hamiltonian:
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| <math> H = -t \sum_{ \left\langle i, j \right\rangle } b^{\dagger}_i b_j + \frac{U}{2} \sum_{i} \hat{n}_i \left( \hat{n}_i - 1 \right) - \mu \sum_i \hat{n}_i </math>.
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| Here i is summed over all lattice sites, and <math>\left\langle i, j \right\rangle</math> denotes summation over all neighboring sites i and j.. <math>b^{\dagger}_i</math> and <math>b^{}_i</math> are bosonic [[creation and annihilation operators]]. <math>\hat{n}_i = b^{\dagger}_i b_i</math> gives the number of particles on site i. Parameter <math>t</math> is the hopping matrix element, signifying mobility of bosons in the lattice. Parameter <math>U</math> describes the on-site interaction, if <math>U > 0</math> it describes repulsive interaction, if <math>U < 0</math> then the interaction is attractive. <math>\mu </math> is the [[chemical potential]].
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| The dimension of the [[Hilbert space]] of the Bose–Hubbard model grows exponentially with respect to the number of particles ''N'' and lattice sites L. It is given by:
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| <math> D_{b}= \frac{(N_{b}+L-1)!}{N_{b}!(L-1)!} </math>
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| while that of Fermi–Hubbard Model is given by:
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| <math> D_{f}= \frac{L!}{N_{f}!(L-N_{f})!}. </math> The different results stem from [[Spin-statistics theorem|different statistics of fermions and bosons]].
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| For the Bose–Fermi mixtures, the corresponding Hilbert space of the Bose–Fermi–Hubbard model is simply the tensor product of Hilbert spaces of the bosonic model and the fermionic model. | |
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| == Phase diagram==
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| At zero temperature, the Bose–Hubbard model (in the absence of disorder) is in either a [[Mott insulators|Mott insulating]] (MI) state at small <math>t / U </math>, or in a [[superfluid]] (SF) state at large <math> t / U </math>.<ref>{{cite doi | 10.1103/PhysRevB.58.R14741}}</ref> The Mott insulating phases are characterized by integer boson densities, by the existence of an [[band gap|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|superfluid susceptibility]].<ref>{{cite journal |doi=10.1103/PhysRevB.40.546 |title=Boson localization and the superfluid-insulator transition |year=1989 |last1=Fisher |first1=Matthew P. A. |last2=Grinstein |first2=G. |last3=Fisher |first3=Daniel S. |journal=Physical Review B |volume=40 |pages=546–70|bibcode = 1989PhRvB..40..546F }},</ref> It is insulating despite the presence of a gap, as low tunneling prevents the generation of excitations which, although close in energy, are spatially separated.
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| ==Implementation in Optical Lattices==
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| Ultracold atoms in [[optical lattice]]s 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.<ref>{{cite doi| 10.1103/PhysRevLett.81.3108}}</ref><ref>{{cite doi | 10.1016/j.aop.2004.09.010 }}</ref>
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| The hamiltonian in [[Second quantization]] formalism describing a gas of ultracold atoms in the optical lattice potential is of the form :
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| <math> H= \int d^3\vec r \hat\psi^\dagger(\vec r) \left ( -\frac{\hbar^2}{2m} \nabla^2 +V_{latt.}(x) \right) \hat\psi(\vec r)
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| + \frac{g}{2}\hat \psi^\dagger(\vec r)\hat\psi^\dagger(\vec r)\hat\psi(\vec r)\hat\psi(\vec r) - \mu \psi^\dagger(\vec r)\hat\psi(\vec r)
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| </math>
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| where, <math> V_{latt} </math> is the optical lattice potential, g is interaction amplitude (here contact interaction is assumed), <math>\mu</math> is a chemical potential. Standard [[Tight binding| tight binding approximation]] (see this article for details) <math>\hat\psi(\vec r) = \sum\limits_i w_i^\alpha (\vec r) b_i^\alpha</math> yields the Bose-Hubbard hamiltonians if one assumes additionally
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| that <math> \int w_i^\alpha(\vec r)w_j^\beta(\vec r)w_k^\alpha(\gamma r)w_l^\delta(\vec r) d^3\vec r=0</math> except for case <math>i=j=k=l , \alpha=\beta=\gamma=\delta=0</math>. Here <math>w_i^\alpha(\vec r) </math> is a [[Wannier function]] for a particle in a optical lattice potential localized around site i of the lattice and for <math>\alpha</math>th [[Bloch band]].<ref name="Luhmann"/>
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| ===Subtleties & approximations===
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| The tight-binding approximation simplifies significantly the second quantization hamltonian, introducing several limitations in the same time:
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| * Parameters U and J may in fact depend on density, as neglected terms are in fact not exactly zero; instead of one parameter U, the interaction energy of n particles may be described by <math>U_n</math> close, but not equal to U <ref name="Luhmann">{{cite doi | 10.1088/1367-2630/14/3/033021}}</ref>
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| * When considering fast lattice dynamics, additional terms should be added to the Bose-Hubbard hamiltonian, so that the [[Schrödinger equation|Time-dependent Schrödinger equation]] was obeyed. They come from dependence on time of Wannier functions.<ref name="Sakmann2011">{{cite doi | 10.1088/1367-2630/13/4/043003 }}</ref><ref>{{cite doi | 10.1103/PhysRevLett.110.065301}}</ref>
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| ==Experimental results==
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| Quantum phase transitions in the Bose–Hubbard model were experimentally observed by Greiner et al.<ref>{{cite journal |doi=10.1038/415039a |title=Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms |year=2002 |last1=Greiner |first1=Markus |last2=Mandel |first2=Olaf |last3=Esslinger |first3=Tilman |last4=Hänsch |first4=Theodor W. |last5=Bloch |first5=Immanuel |journal=Nature |volume=415 |issue=6867 |pages=39–44 |pmid=11780110}}</ref> in Germany. Density dependent interaction parameters <math>U_n </math> were observed by [[Immanuel Bloch|I.Bloch's]] group <ref>{{cite doi|10.1038/nature09036}}</ref>
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| ==Further applications of the model==
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| 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.<ref>{{cite journal |doi=10.1088/1751-8113/40/28/S11 |title=Transport and entanglement generation in the Bose–Hubbard model |year=2007 |last1=Romero-Isart |first1=O |last2=Eckert |first2=K |last3=Rodó |first3=C |last4=Sanpera |first4=A |journal=Journal of Physics A: Mathematical and Theoretical |volume=40 |issue=28 |pages=8019–31|arxiv = quant-ph/0703177 |bibcode = 2007JPhA...40.8019R }}</ref>
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| == Numerical Simulation ==
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| In the calculation of low energy states the term proportional to <math>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>d <\infty</math> particles. Then the local Hilbert space dimension is <math>d+1.</math> The dimension of full Hilbert space grows exponentially with number of sites in the lattice, therefore computer simulations are limited to 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 numerical simulation of this model, an algorithm of exact diagonalization is presented in this paper.<ref>{{cite journal |doi=10.1088/0143-0807/31/3/016 |title=Exact diagonalization: The Bose–Hubbard model as an example |year=2010 |last1=Zhang |first1=J M |last2=Dong |first2=R X |journal=European Journal of Physics |volume=31 |issue=3 |pages=591–602|arxiv = 1102.4006 |bibcode = 2010EJPh...31..591Z }}</ref>
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| 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 ground state of the hamiltonian for systems of thousands particles on thousands of lattice sites, and simulate its dynamics governed by [[Schrödinger equation|Time-dependent Schrödinger equation]].
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| Higher dimensions are significantly more difficult to quick growth of [[entanglement]].<ref>{{cite doi | 10.1103/RevModPhys.82.277 }}</ref>
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| 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.
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| ==Generalizations==
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| The Bose-Hubbard-like hamiltonians may be derived for:
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| * systems with density-density interaction <math> V n_i n_j </math>
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| * long-range dipolar interaction <ref>{{cite doi | 10.1103/PhysRevLett.88.170406 }}</ref>
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| * systems with interaction-induced tunneling terms <math> a_i^\dagger ( n_i + n_j ) a_j </math> <ref>{{cite doi|10.1103/PhysRevLett.108.115301}}</ref>
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| * internal spin structure (spin-1 Bose–Hubbard model) <ref>{{cite doi | 10.1103/PhysRevA.70.043628 }}</ref>
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| * disordered systems <ref>{{Cite doi|10.1103/PhysRevB.80.214519}}</ref>
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| ==See also==
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| *[[Hubbard model]]
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| *[[Jaynes-Cummings-Hubbard model]]
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| == References ==
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| {{reflist}}
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| {{DEFAULTSORT:Bose-Hubbard model}}
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| [[Category:Condensed matter physics]]
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| [[Category:Quantum Lattice models]]
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