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Power law violation of the area law in critical spin chains (joint work with Peter W. Shor)

August 11, 2014 - 2:00pm
Ramis Movassagh
Department of Mathematics, Northeastern and MIT

The sub-volume scaling of the entanglement entropy with the system's size, n, has been a subject of vigorous study in the last decade. Hastings proved “the area law” for gapped one dimensional systems and it is largely believed that, in quantum critical systems, the area law would be violated by at most a factor of log(n).

In this work, we generalize the spin-1 model of Bravyi et al [*] to all integer spin-s chains, whereby we introduce a class of exactly solvable models that exhibit signatures of criticality. The proposed Hamiltonian is local and translationally invariant. We prove that it is frustration free and has a unique ground state. Moreover, we prove that the energy gap scales as poly(1/n). The upper bound we prove is O(n^-2) which also improves the one given in [*]  This rules out the possibility of this class of models being described by a conformal field theory. We analytically show that the Schmidt rank grows exponentially with n and that the half-chain entanglement entropy to the leading order scales as \sqrt{n}. Geometrically, the ground state is seen as a uniform superposition of all s-colored Motzkin walks. Our techniques for obtaining the asymptotic form of the entanglement entropy, the gap upper bound and the self-contained expositions of the combinatorial techniques, more akin to lattice paths, may be of independent interest. 

[*]  Criticality without Frustration for Quantum Spin-1 Chains

Sergey Bravyi, Libor Caha, Ramis Movassagh, Daniel Nagaj, and Peter W. Shor

Phys. Rev. Lett. 109, 207202 

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