Abstract: We give a new proof for the area law for general 1D gapped systems, which exponentially improves Hastings' famous result [1]. Specifically, we show that for a chain of d-dimensional spins, governed by a 1D local Hamiltonian with a spectral gap ε > 0, the entanglement entropy of the ground state with respect to any cut in the chain is upper bounded by O(log^3 d/ε ). Our approach uses the framework of Refs. [2, 3] to construct a Chebyshev-based AGSP (Approximate Ground Space Projection) with favorable factors. However, our construction uses the Hamiltonian directly, instead of using the Detectability lemma, which allows us to work with general (frustrated) Hamiltonians, as well as slightly improving the 1/ε dependence of the bound in Ref. [3]. To achieve that, we establish a new, “random-walk like”, bound on the entanglement rank of an arbitrary power of a 1D Hamiltonian, which might be of independent interest: ER(H^ℓ) ≤ (ℓd)O(√ℓ). Finally, treating d as a constant, our AGSP shows that the ground state is well approximated by a matrix product state with a sublinear bond dimension B = ε ^O(log^(3/4) n/ε^(1/4)). Using this in conjunction with known dynamical programing algorithms, yields an algorithm for a 1=poly(n) approximation of the ground energy with a subexponential running time T ≤ exp (εO(log^(3/4) n/ε^(1/4))).

ID: CaltechAUTHORS:20140130-142058060

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Abstract: Inspired by quantum information theory, we look for representations of the braid groups B_n on V^(⊗(n+m−2)) for some fixed vector space V such that each braid generator σ_i, i = 1, ..., n−1, acts on m consecutive tensor factors from i through i +m−1. The braid relation for m = 2 is essentially the Yang-Baxter equation, and the cases for m > 2 are called generalized Yang-Baxter equations. We observe that certain objects in ribbon fusion categories naturally give rise to such representations for the case m = 3. Examples are given from the Ising theory (or the closely related SU(2)_2), SO(N)_2 for N odd, and SU(3)_3. The solution from the Jones-Kauffman theory at a 6th root of unity, which is closely related to SO(3)_2 or SU(2)_4, is explicitly described in the end.

ID: CaltechAUTHORS:20120713-102318475

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