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In Kitaev's notes, he reviewed the toric code model. Consider on square lattice the Hamiltonian $H=-J_e \sum_s A_s-J_m \sum_p B_p,\ A_s=\prod_{j\in vertices} \sigma_j^x,\ B_p=\prod_{j\in plaquettes} \sigma_j^z$. Now add two kinds of perturbation to the Hamiltonian: $H_1=-\sum_j h_x \sigma_j^x$ and $H_2=-\sum_j h_z\sigma_j^z$. When $h_x\gg J_e, J_m$ and $h_z=0$, if $h_x$ increases the vortices condense. When $h_z\gg J_e, J_m$ and $h_x=0$, increase of $h_z$ condenses the charges.

Then he comments that "the high-field phase is just a paramagnet, so one can continuously rotate the field between the $x$- and $z$- direction without inducing a phase transition. Thus, the charge and vortex condensates are actually the same phase."

I can understand the literal meaning of the argument but I'm unable to really accept it. For example, we know that putting a toric code model on a cylinder, condensation of charges on one boundary and vortices on the other gives ground degeneracy $GSD=1$, while condensation of charges (or vortices) on both boundaries gives $GSD=2$. The difference should thus be physically detectable.(See reference.)

Did I understand anything wrong? Does a generalization of the argument indicate that, for a 2D topological system described by a modular tensor category, the condensations of different condensable algebras (or called Lagrangian subsets) in the MTC will give the same phase?

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    $\begingroup$ The argument works fine for PBC, but as you observe it might give problems for OBC -- this is, the phase is the same in the bulk, but not at the boundary. This is not surprising, since depending on how you choose the boundary, it can either condense e or m particles (smooth/rough boundaries), and thus breaks e/m duality. Nevertheless, I would argue that this is the same bulk phase, since we can interpolate between the two phases without closing a gap in the bulk. (You can create similar models even in 1D which are in the same bulk phase but are in different phases once you put them on OBC.) $\endgroup$ – Norbert Schuch Jan 12 '16 at 19:15
  • $\begingroup$ Happy it helped; I have turned it into an answer. $\endgroup$ – Norbert Schuch Jan 12 '16 at 20:25
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The argument works fine for PBC, but as you observe it might give problems for OBC -- this is, the phase is the same in the bulk, but not at the boundary. This is not surprising, since depending on how you choose the boundary, it can either condense e or m particles (smooth/rough boundaries), and thus breaks e/m duality. Nevertheless, I would argue that this is the same bulk phase, since we can interpolate between the two phases without closing a gap in the bulk. (You can create similar models even in 1D which are in the same bulk phase but are in different phases once you put them on OBC.)

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  • $\begingroup$ BTW I think similar problems will also arise when we consider the two e-condensed and m-condensed phases separated by a domain wall. $\endgroup$ – Zhuxi Luo Jan 12 '16 at 20:30
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I'm just going to put a slightly more general remark, essentially the same as the answer by Norbert. The condensation of Lagrangian subalgebra (it is not just a set, really an algebra satisfying several conditions) in a MTC always leads to a trivial phase, namely, a state that can be continuously connected to a product state (I'm ignoring a subtlety of chiral central charge being multiples of $8$). This does not contradict the GSD of toric code on cylinder, because there you are looking at the domain walls between the original MTCs and the phases after condensation. Although the condensed phase is the same topologically, the interface between the condensed phase and the original ones are certainly different because different types of anyons condense.

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The other two guys had made a clear argument on this issue. condensation of Lagrangian subsets always give rise to trivial phase in the bulk. however, based on the way u condensed the anyons, there could be sth different on the boundary. The toric code lattice model is pretty subtle, it has either rough/smooth boundary so you can only condense either e/m on the edge.

another related example is the double semion model. one can condense the semion+anitsemion pair to confine the other anyons. there are two ways to do so, with a phase \pi in the Higgs term or not.(see http://arxiv.org/abs/1509.00355 equation 6-8). two distinct ways of condensation give the same bulk, but different boundary. The semionic excitations on the boundary can transform either as time reversal singlets or as time reversal (Kramers) doublets.

As u further comment "I think similar problems will also arise when we consider the two e-condensed and m-condensed phases separated by a domain wall". Yes, like in double semion problem I mentioned above, if we have a boundary between two distinct anyon condensation phases, and them together with the original double semion in a tri-junction on sphere(see Fig 5 in http://arxiv.org/pdf/1509.00355v1.pdf), there would be some degeneracy protected by time reversal symmetry.

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