Fig: Scheme of the hamiltonian construction.
the authors prove that deciding whetherthe Hamiltonian of a quantum many-body system
is gapped or gapless is an undecidable problem.
They construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable. This result extends to undecidability of other low-energy properties, such as the existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding ‘halting problem’. Their result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.
MathQI Seminar: ¡¡¡new schedule!!!
From Feb 1st on it's being hold on Wednesdays at 12:30.
New positions in the group: PhD, Postdoc and Senior Postdoc.
Topics: mathematical problems arising in quantum many body systems, mainly in the context of Tensor Network States.
More info and contact details: see Job Offers.
Undecidability of the spectral gap in the media
10th Feb 2016 12:30 MathQI Seminar.
Andrea Coser: "Entanglement negativity in Quantum Field Theory".
21st Jan 2016 09:45 MathQI Seminar.
Carlos Fernández: "The structure of MPS".