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A hierarchy of topological tensor network states
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View: Figures


Image of FIG. 1.
FIG. 1.

A graph edge representing both the Hopf algebra H (via L ±) and X = (H op)* (via T ±). These different H- and X-module structures are related via and , respectively.

Image of FIG. 2.
FIG. 2.

(a) The minimal graph Γ1 which carries a representation of D(H) on . (b) The graph Γ2 associated with . (c) Graph used in the proof of Theorem 1.

Image of FIG. 3.
FIG. 3.

Degenerate vertices and faces with spikes and loops.

Image of FIG. 4.
FIG. 4.

Decorated face.

Image of FIG. 5.
FIG. 5.

Proper faces (white) vs. boundary “faces” F M (light red). Upon continuous deformation of those pieces of the boundary ∂M which do not coincide with graph edges (dashed gray) the vertices marked in gray on the left and on the right are identified with each other, respectively. The boundary faces F M are precisely those which are completed to proper faces by this process. Furthermore, both the boundary edges E M (red) and interior edges (black) are shown.

Image of FIG. 6.
FIG. 6.

A region R whose boundary ∂R never crosses any interior edges cleanly partitions the edges into the set E R (black, red) which belongs to R and the rest (gray). The internal boundary (∂R)° consists of those pieces which do not coincide with the boundary of the ambient surface M and gives rise to the internal boundary faces (light red, middle).

Image of FIG. 7.
FIG. 7.

Diagram encoding the tensor trace . While the interior degrees of freedom {x e } ⊂ H and {f p } ⊂ X are only shown partially the boundary degrees of freedom {y e } ⊂ H and {g q } ⊂ X are labelled in such a way that the ordering of the boundary is evident.

Image of FIG. 8.
FIG. 8.

Hierarchy of Hopf tensor network states for before the partial collapse. Subalgebras and are ordered by dimension, not necessarily by inclusion. Surviving equivalence classes of quantum states share the same value of the topological entanglement entropy γ, see Sec. V for a detailed discussion.

Image of FIG. 9.
FIG. 9.

Hopf tensor network state on a small graph Γ embedded in S 2. The outer circle is identified with the north pole of S 2.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: A hierarchy of topological tensor network states