The discrete rotational symmetry can be extended to the discrete translational symmetry in the whole phase space. For such purpose, the displacement operator in phase space is defined by D ^ ^ ( ξ ξ ) = exp ⁡ ⁡ [ ( ξ ξ a ^ ^ † † − − ξ ξ ∗ ∗ a ^ ^ ) / 2 λ λ ] which has the property D ^ ^ † † ( ξ ξ ) a ^ ^ D ^ ^ ( ξ ξ ) = a ^ ^ + ξ ξ , where ξ ξ is a complex number corresponding to the displacement vector in phase space. The system has discrete translational symmetry if the Hamiltonian commutates with translational operator [ H ^ ^ , D ^ ^ † † ( ξ ξ ) ] = 0 , i.e., H ^ ^ = D ^ ^ † † ( ξ ξ ) H ^ ^ D ^ ^ ( ξ ξ ) → → H ( a ^ ^ , a ^ ^ † † ) = H ( D ^ ^ † † ( ξ ξ ) a ^ ^ D ^ ^ ( ξ ξ ) , D ^ ^ † † a ^ ^ † † D ^ ^ ( ξ ξ ) ) = H ( a ^ ^ + ξ ξ , a ^ ^ † † + ξ ξ ∗ ∗ ) . If there exist two elementary displacements D ^ ^ ( ξ ξ 1 ) and D ^ ^ ( ξ ξ 2 ) that satisfy the above condition simultaneously, the phase space Hamiltonian possesses 2D lattice symmetry in phase space. However, the two displacement operators are not commutative in general [ D ^ ^ ( ξ ξ 1 ) , D ^ ^ ( ξ ξ 2 ) ] ≠ ≠ 0 . In the non-commutative phase space, the concept of a "point" is meaningless. Instead, a coherent state | α α ⟩ ⟩ is defined as the eigenstate of the lowering operator via a ^ ^ | α α ⟩ ⟩ = α α | α α ⟩ ⟩ . The displacement operator displaces the coherent state with an additional phase, i.e., D ^ ^ ( ξ ξ ) | α α ⟩ ⟩ = e i I m ( ξ ξ α α ∗ ∗ ) | α α + ξ ξ ⟩ ⟩ . A coherent state that is moved along a closed path, e.g., a triangle with three edges given by ( ξ ξ 1 , ξ ξ 2 , − − ξ ξ 1 − − ξ ξ 2 ) in phase space, acquires a geometric phase factor D ^ ^ [ − − ξ ξ 1 − − ξ ξ 2 ] D ^ ^ ( ξ ξ 2 ) D ^ ^ ( ξ ξ 1 ) | α α ⟩ ⟩ = e i S λ λ | α α ⟩ ⟩ , where S = 1 2 I m ( ξ ξ 2 ξ ξ 1 ∗ ∗ ) is the enclosed area. This geometric phase is analogous to the Aharonov–Bohm phase of charged particle in a magnetic field. If the magnetic unit cell and the lattice unit cell are commensurable, namely, there exist two integers r and s such that [ D ^ ^ r ( ξ ξ 1 ) , D ^ ^ s ( ξ ξ 2 ) ] = 0 , one can calculate the band structure defined in a 2D Brillouin. For example, the spectrum of a square phase space lattice Hamiltonian H ^ ^ = cos ⁡ ⁡ x ^ ^ + cos ⁡ ⁡ p ^ ^ displays Hofstadter's butterfly band structure that describes the hopping of charged particles between tight-binding lattice sites in a magnetic field. In this case, the eigenstates are called 2D lattice phase space crystals.