Quantum clock model

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.…

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. When N>4 , strong evidences have been found that the phase transitions exhibited in these models should be certain generalizations of Kosterlitz–Thouless transition, whose physical nature is still largely unknown. \begin{align}\tilde{X_j} &= Z^\dagger_j Z_{j+1} \\ \tilde{Z}^\dagger_j \tilde{Z}_{j+1} &= X_{j+1} \end{align} Then, in terms of the newly defined clock matrices with tildes, which obey the same algebraic relations as the original clock matrices, the Hamiltonian is simply H = -Jg \sum_j ( \tilde{Z}^\dagger_j \tilde{Z}_{j+1} + g^{-1}\tilde{X}^\dagger_{j} + \textrm{h.c.} ) . This indicates that the model with coupling parameter g is dual to the model with coupling parameter g^{-1} , and establishes a duality between the ordered phase and the disordered phase. Note that there are some subtle considerations at the boundaries of the one dimensional chain; as a result of these, the degeneracy and \mathbb{Z}_N symmetry properties of phases are changed under the Kramers–Wannier duality. A more careful analysis involves coupling the theory to a \mathbb{Z}_N gauge field; fixing the gauge reproduces the results of the Kramers Wannier transformation. For N=2,3,4 , there is a unique phase transition from the ordered phase to the disordered phase at g=1 . The model is said to be "self-dual" because Kramers–Wannier transformation transforms the Hamiltonian to itself. For N>4 , there are two phase transition points at g_1 and g_2=1/g_1>1 . Strong evidences have been found that these phase transitions should be a class of generalizations of Kosterlitz–Thouless transition. The KT transition predicts that the free energy has an essential singularity that goes like e^{-\tfrac{c}{\sqrt }} , while perturbative study found that the essential singularity behaves as e^{-\tfrac{c}{|g-g_c|^\sigma}} where \sigma goes from 0.2 to 0.5 as N increases from 5 to 9 . The physical pictures of these phase transitions are still not clear.

References

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