Quantum clock model

Article summary

The quantum clock model is a quantum lattice model. It is a generalisation of the transverse-field Ising model . It is defined on a lattice with N states on each site. The Hamiltonian of this model is : H = -J \left( \sum_{ \langle i, j \rangle} (Z^\dagger_i Z_j + Z_i Z^\dagger_j ) + g \sum_j (X_j + X^\dagger_j) \right) Here, the subscripts refer to lattice sites, and the sum \sum_{\langle i, j \rangle} is done over pairs of nearest neighbour sites i and j . The clock matrices X_j and Z_j are N \times N generalisations of the Pauli matrices satisfying : Z_j X_k = e^{\frac{2\pi i }{N}\delta_{j,k}} X_k Z_j and X_j^N = Z_j^N = 1 where \delta_{j,k} is 1 if j and k are the same site and zero otherwise. J is a prefactor with dimensions of energy, and g is another coupling coefficient that determines the relative strength of the external field compared to the nearest neighbor interaction. The model obeys a global \mathbb{Z}_N symmetry, which is generated by the unitary operator U_X = \prod_j X_j where the product is over every site of the lattice. In other words, U_X commutes with the Hamiltonian. When N=2 the quantum clock model is identical to the transverse-field Ising model. When N=3 the quantum clock model is equivalent to the quantum three-state Potts model. When N=4 , the model is again equivalent to the Ising model.…

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