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Parafermions and Topological States

Following on from the initial paper defining D_\mathcal{F}, the second paper investigated interaction distance in models of topological phases of matter. The most interesting result of this study was that the ground states of topological models alternate between maximally interacting and completely free, depending on their topological symmetry index {\mathbb{Z}_N }. In some cases of models where the ground state was identified to have \mathcal{D}_{\mathcal{F}} = 0, the corresponding free model was given.

The paper investigated flat entanglement spectra which are physically realised by parafermion chains at their fixed point f=0, where they can be expressed as

\displaystyle H_{\mathbb{Z}_{N}} = -\sum_{j}\tau_{j}^{\dagger}\tau_{j+1} + h.c.,

and \tau_j is a clock operator on site j. For example, for N=3 we have \tau_j = {\rm diag}(1,\exp(i2\pi/3),\exp(-i2\pi/3)). The entanglement spectra of the ground states of these models are N-fold degenerate and denoted as:

\displaystyle \bar{\rho}(N) = \begin{pmatrix} \frac{1}{N} & \cdots & 0 \\ \vdots & \ddots & \\ 0 & & \frac{1}{N}\end{pmatrix} \rightarrow \bar{\rho_{i}}(N) = \frac{1}{N}.

It is clear that the entanglement energies are degenerate and equal to -ln(\bar{\rho_{i}}) = -ln(\frac{1}{N}). The goal is to find the free spectrum that optimally models these flat spectra.

Let n be the greatest integer such that 2^{n} \leq N. The flat probability entanglement spectrum then consists of 2^{n} values, N entries of \frac{1}{N}, and padded out with 0's. Adding these 0's to the entanglement spectrum is viable as it leaves the entropy invariant, corresponding to an additional infinite energy level, as explained here. We conjecture that the optimal free spectrum, \sigma_{ansatz} is of the form:

\displaystyle \sigma_{ansatz} \simeq {\rm diag}(N^{-1}, \cdots, N^{-1}, p, \cdots, p),

where there are 2^{n} entries of both N^{-1} and p. Enforcing normalisation Tr(\sigma_{ansatz}) = 1 leads to p = 2^{-n} - N^{-1}. This leads to an upper bound for D_{\mathcal{F}}(\bar{\rho}(N)), shown in the figure below:

\displaystyle D_{\mathcal{F}}(\bar{\rho}(N)) \leq 3 - \frac{N}{2^{n}} - \frac{2^{n+1}}{N} \qquad \qquad \qquad (1)

It is found that the numerically computed values of D_{\mathcal{F}} for these flat, padded spectra are in remarkable agreement with this upper bound, again shown in the figure below.

From this agreement, the paper suggests that the upper bound of Eq.(1) is the exact maximum of D_{\mathcal{F}}(\bar{\rho}), it is shown to have the value D_{\mathcal{F}}^{max} = 3-2\sqrt{2}. By exhaustive numerical maximisation of D_{\mathcal{F}} of random spectra, \rho, a value for the interaction distance above this upper bound has not been found, suggesting that this is the upper bound of D_{\mathcal{F}} for any state.

Interestingly, as seen in the figure, \mathbb{Z}_{N=2^{n}} chains have D_{\mathcal{F}} = 0, implying that they can be expressed in terms of free fermions. For example, in the case of \mathbb{Z}_{4} the ground state is expressed in terms of the Gaussian ground state of \mathbb{Z}_{2} \times \mathbb{Z}_{2} with the help of the rotation \mathcal{U} = \otimes_{j}U_{j}, where the local unitaries, U_{j} are acting on each site j of the site. Using this relation, the parent free fermion Hamiltonian, H^{free}_{\mathbb{Z}_{4}}, can be found, which has the same ground state as H_{\mathbb{Z}_{4}}:

H^{free}_{\mathbb{Z}_{4}} = \mathcal{U}H_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}\mathcal{U}^{\dagger}

This can be generalised to all \mathbb{Z}_{2^{n}} models with D_{\mathcal{F}} = 0. These results can also be extended to other topological states such as string-net models.