Bose–Hubbard model: Difference between revisions

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→‎The Hamiltonian: deleted word "regular" which doesn't mean anything specific here and may fool readers into thinking it's a technical term
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m (→‎The Hamiltonian: deleted word "regular" which doesn't mean anything specific here and may fool readers into thinking it's a technical term)
<math> 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>\left\langle i, j \right\rangle</math> denotes summation over all neighboring lattice sites <math>i</math> and <math>j</math>, while <math>\hat{b}^{\dagger}_i</math> and <math>\hat{b}^{}_i</math> are regular bosonic [[creation and annihilation operators]] such that <math>\hat{n}_i = \hat{b}^{\dagger}_i \hat{b}_i</math> gives the number of particles on site <math>i</math>. The model is parametrized by the hopping amplitude <math>t</math> describing the mobility of bosons in the lattice, the on-site interaction <math>U</math> which can be attractive (<math>U < 0</math>) or repulsive (<math>U > 0</math>), and the [[chemical potential]] <math>\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>U(1)</math> symmetry, which means that it is invariant (i.e. its physical properties are unchanged) by the transformation <math>\hat{b}_i \rightarrow e^{i \theta} \hat{b}_i</math>. In a [[Superfluidity|superfluid]] phase, this symmetry is [[Spontaneous symmetry breaking|spontaneously broken]].