279 research outputs found
Polynomial-time complexity for instances of the endomorphism problem in free groups
We say the endomorphism problem is solvable for an element W in a free group F if it can be decided effectively whether, given U in F, there is an endomorphism Φ of F sending W to U. This work analyzes an approach due to C. Edmunds and improved by C. Sims. Here we prove that the approach provides an efficient algorithm for solving the endomorphism problem when W is a two- generator word. We show that when W is a two-generator word this algorithm solves the problem in time polynomial in the length of U. This result gives a polynomial-time algorithm for solving, in free groups, two-variable equations in which all the variables occur on one side of the equality and all the constants on the other side
Quantum-field-theoretical techniques for stochastic representation of quantum problems
We describe quantum-field-theoretical (QFT) techniques for mapping quantum
problems onto c-number stochastic problems. This approach yields results which
are identical to phase-space techniques [C.W. Gardiner, {\em Quantum Noise}
(1991)] when the latter result in a Fokker-Planck equation for a corresponding
pseudo-probability distribution. If phase-space techniques do not result in a
Fokker-Planck equation and hence fail to produce a stochastic representation,
the QFT techniques nevertheless yield stochastic difference equations in
discretised time
Projective Ribbon Permutation Statistics: a Remnant of non-Abelian Braiding in Higher Dimensions
In a recent paper, Teo and Kane proposed a 3D model in which the defects
support Majorana fermion zero modes. They argued that exchanging and twisting
these defects would implement a set R of unitary transformations on the zero
mode Hilbert space which is a 'ghostly' recollection of the action of the braid
group on Ising anyons in 2D. In this paper, we find the group T_{2n} which
governs the statistics of these defects by analyzing the topology of the space
K_{2n} of configurations of 2n defects in a slowly spatially-varying gapped
free fermion Hamiltonian: T_{2n}\equiv {\pi_1}(K_{2n})$. We find that the group
T_{2n}= Z \times T^r_{2n}, where the 'ribbon permutation group' T^r_{2n} is a
mild enhancement of the permutation group S_{2n}: T^r_{2n} \equiv \Z_2 \times
E((Z_2)^{2n}\rtimes S_{2n}). Here, E((Z_2)^{2n}\rtimes S_{2n}) is the 'even
part' of (Z_2)^{2n} \rtimes S_{2n}, namely those elements for which the total
parity of the element in (Z_2)^{2n} added to the parity of the permutation is
even. Surprisingly, R is only a projective representation of T_{2n}, a
possibility proposed by Wilczek. Thus, Teo and Kane's defects realize
`Projective Ribbon Permutation Statistics', which we show to be consistent with
locality. We extend this phenomenon to other dimensions, co-dimensions, and
symmetry classes. Since it is an essential input for our calculation, we review
the topological classification of gapped free fermion systems and its relation
to Bott periodicity.Comment: Missing figures added. Fixed some typos. Added a paragraph to the
conclusio
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