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Cubbit, Toby S. and Pérez García, David and Wolf, Michael M. (2015) Undecidability of the spectral gap. Nature, 528 (7581). pp. 207211. ISSN 00280836

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Official URL: http://www.nature.com/nature/journal/v528/n7581/full/nature16059.html
URL  URL Type 

https://arxiv.org/abs/1502.04135  Organisation 
https://arxiv.org/abs/1502.04573  Organisation 
Abstract
The spectral gapthe energy difference between the ground state and first excited state of a systemis central to quantum manybody physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the YangMills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum manybody system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a twodimensional lattice with translationally invariant, nearestneighbour interactions, for which the spectral gap problem is undecidable. This result extends to undecidability of other lowenergy properties, such as the existence of algebraically decaying groundstate correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phaseestimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding 'halting problem'. Our 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.
Item Type:  Article 

Additional Information:  Supplementary material: http://eprints.sim.ucm.es/38062/ 
Uncontrolled Keywords:  Complexity; Quantum; Antiferromagnet; Lattice; State 
Subjects:  Sciences > Physics > Quantum theory 
ID Code:  34985 
Deposited On:  18 Jan 2016 13:27 
Last Modified:  10 Jun 2016 08:36 
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