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 $N$ states on each site. The Hamiltonian of this model is

H=-J\left(\sum _{\langle i,j\rangle }(Z_{i}^{\dagger }Z_{j}+Z_{i}Z_{j}^{\dagger })+g\sum _{j}(X_{j}+X_{j}^{\dagger })\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 ^[2] of Kosterlitz–Thouless transition, whose physical nature is still largely unknown.

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:^[3]

{\begin{aligned}{\tilde {X_{j}}}&=Z_{j}^{\dagger }Z_{j+1}\\{\tilde {Z}}_{j}^{\dagger }{\tilde {Z}}_{j+1}&=X_{j+1}\end{aligned}}

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 $H=-Jg\sum _{j}({\tilde {Z}}_{j}^{\dagger }{\tilde {Z}}_{j+1}+g^{-1}{\tilde {X}}_{j}^{\dagger }+{\textrm {h.c.}})$ . 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.

Phase transition

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 $g_{1}<1$ and $g_{2}=1/g_{1}>1$ . Strong evidences have been found that these phase transitions should be a class of generalizations^[2] of Kosterlitz–Thouless transition. The KT transition predicts that the free energy has an essential singularity that goes like $e^{-{\tfrac {c}{\sqrt {|g-g_{c}|}}}}$ , 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^[4] of these phase transitions are still not clear.

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].
1 2 Bingnan Zhang (2020). "Perturbative study of the one-dimensional quantum clock model". Phys. Rev. E. 102 (4): 042110. arXiv:2006.11361. Bibcode:2020PhRvE.102d2110Z. doi:10.1103/PhysRevE.102.042110. PMID 33212691. S2CID 219966942.
↑ Radicevic, Djordje (2018). "Spin Structures and Exact Dualities in Low Dimensions". arXiv:1809.07757 [hep-th].
↑ Martin B. Einhorn, Robert Savit, Eliezer Rabinovici (1980). "A physical picture for the phase transitions in Zn symmetric models". Nuclear Physics B. 170 (1): 16-31. Bibcode:1980NuPhB.170...16E. doi:10.1016/0550-3213(80)90473-3. hdl:2027.42/23169.{{cite journal}}: CS1 maint: multiple names: authors list (link)

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

[bingnan-2] 1 2 Bingnan Zhang (2020). "Perturbative study of the one-dimensional quantum clock model". Phys. Rev. E. 102 (4): 042110. arXiv:2006.11361. Bibcode:2020PhRvE.102d2110Z. doi:10.1103/PhysRevE.102.042110. PMID 33212691. S2CID 219966942.

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

[eliezer-4] Martin B. Einhorn, Robert Savit, Eliezer Rabinovici (1980). "A physical picture for the phase transitions in Zn symmetric models". Nuclear Physics B. 170 (1): 16-31. Bibcode:1980NuPhB.170...16E. doi:10.1016/0550-3213(80)90473-3. hdl:2027.42/23169.{{cite journal}}: CS1 maint: multiple names: authors list (link)

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