# Quantum clock model

The **quantum clock model** is a quantum lattice model.^{[1]} It is a generalisation of the transverse-field Ising model . It is defined on a lattice with [math]\displaystyle{ N }[/math] states on each site. The Hamiltonian of this model is

- [math]\displaystyle{ 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) }[/math]

Here, the subscripts refer to lattice sites, and the sum [math]\displaystyle{ \sum_{\langle i, j \rangle} }[/math] is done over pairs of nearest neighbour sites [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math]. The clock matrices [math]\displaystyle{ X_j }[/math] and [math]\displaystyle{ Z_j }[/math] are [math]\displaystyle{ N \times N }[/math] generalisations of the Pauli matrices satisfying

- [math]\displaystyle{ Z_j X_k = e^{\frac{2\pi i }{N}\delta_{j,k}} X_k Z_j }[/math] and [math]\displaystyle{ X_j^N = Z_j^N = 1 }[/math]

where [math]\displaystyle{ \delta_{j,k} }[/math] is 1 if [math]\displaystyle{ j }[/math] and [math]\displaystyle{ k }[/math] are the same site and zero otherwise. [math]\displaystyle{ J }[/math] is a prefactor with dimensions of energy, and [math]\displaystyle{ g }[/math] is another coupling coefficient that determines the relative strength of the external field compared to the nearest neighbour interaction.

The model obeys a global [math]\displaystyle{ \mathbb{Z}_N }[/math] symmetry, which is generated by the unitary operator [math]\displaystyle{ U_X = \prod_j X_j }[/math] where the product is over every site of the lattice. In other words, [math]\displaystyle{ U_X }[/math] commutes with the Hamiltonian.

When [math]\displaystyle{ N=2 }[/math] the quantum clock model is identical to the transverse-field Ising model. When [math]\displaystyle{ N=3 }[/math] the quantum clock model is equivalent to the quantum three-state Potts model.

## One-dimensional model

There are various analytical methods that can be used to study the quantum clock model specifically in one dimension.

### Kramers–Wannier duality

A nonlocal mapping of clock matrices known as the Kramers–Wannier duality transformation can be done as follows:^{[2]}
[math]\displaystyle{ \begin{align}\tilde{X_j} &= Z^\dagger_j Z_{j+1} \\
\tilde{Z}^\dagger_j \tilde{Z}_{j+1} &= X_{j+1} \end{align}
}[/math]
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 [math]\displaystyle{ H = -Jg \sum_j ( \tilde{Z}^\dagger_j \tilde{Z}_{j+1} + g^{-1}\tilde{X}^\dagger_{j} + \textrm{h.c.} ) }[/math]. This indicates that the model with coupling parameter [math]\displaystyle{ g }[/math] is dual to the model with coupling parameter [math]\displaystyle{ g^{-1} }[/math], 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 [math]\displaystyle{ \mathbb{Z}_N }[/math] symmetry properties of phases are changed under the Kramers–Wannier duality. A more careful analysis involves coupling the theory to a [math]\displaystyle{ \mathbb{Z}_N }[/math] gauge field; fixing the gauge reproduces the results of the Kramers Wannier transformation.

### Phase transition

When [math]\displaystyle{ g =1 }[/math], the quantum clock model is said to be "self-dual" because Kramers–Wannier transformation transforms the Hamiltonian to itself. This self-dual theory generally represents a phase transition between the ordered phase at [math]\displaystyle{ 0 \lt g \lt 1 }[/math] and the disordered phase at [math]\displaystyle{ 1 \lt g }[/math].

### Jordan–Wigner transformation

Another nonlocal mapping known as the Jordan Wigner transformation can be used to express the theory in terms of parafermions.

## References

- ↑ Radicevic, Djordje (2018). "Spin Structures and Exact Dualities in Low Dimensions". arXiv:1809.07757 [hep-th].
- ↑ Radicevic, Djordje (2018). "Spin Structures and Exact Dualities in Low Dimensions". arXiv:1809.07757 [hep-th].

Original source: https://en.wikipedia.org/wiki/ Quantum clock model.
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