26,616 research outputs found
How many double squares can a string contain?
Counting the types of squares rather than their occurrences, we consider the
problem of bounding the number of distinct squares in a string. Fraenkel and
Simpson showed in 1998 that a string of length n contains at most 2n distinct
squares. Ilie presented in 2007 an asymptotic upper bound of 2n - Theta(log n).
We show that a string of length n contains at most 5n/3 distinct squares. This
new upper bound is obtained by investigating the combinatorial structure of
double squares and showing that a string of length n contains at most 2n/3
double squares. In addition, the established structural properties provide a
novel proof of Fraenkel and Simpson's result.Comment: 29 pages, 20 figure
Time reversal invariant gapped boundaries of the double semion state
The boundary of a fractionalized topological phase can be gapped by
condensing a proper set of bosonic quasiparticles. Interestingly, in the
presence of a global symmetry, such a boundary can have different symmetry
transformation properties. Here we present an explicit example of this kind, in
the double semion state with time reversal symmetry. We find two distinct cases
where the semionic excitations on the boundary can transform either as time
reversal singlets or as time reversal (Kramers) doublets, depending on the
coherent phase factor of the Bose condensate. The existence of these two
possibilities are demonstrated using both field theory argument and exactly
solvable lattice models. Furthermore, we study the domain walls between these
two types of gapped boundaries and find that the application of time reversal
symmetry tunnels a semion between them.Comment: 11 pages, 8 figure
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