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SUMMARY:Undecidability of the spectral gap - Toby Cubitt (University Colle
 ge London)
DTSTART:20170216T150000Z
DTEND:20170216T160000Z
UID:TALK69729@talks.cam.ac.uk
CONTACT:Dr Hansen
DESCRIPTION:The spectral gap - the difference between the smallest and\nse
 cond-smallest eigenvalue of a quantum many-body Hamiltonian - is of\ncentr
 al importance to quantum many-body physics. It determines the phase\ndiagr
 am at low temperature\, with quantum phase transitions occurring when\nthe
  gap vanishes. Some of the most challenging and long-standing open\nproble
 ms in theoretical physics concern the spectral gap\, such as the\nfamous H
 aldane conjecture\, or the infamous Yang-Mills gap conjecture (one\nof the
  Millennium Prize problems). These problems - and many others - are\nall p
 articular cases of the general spectral gap problem: Given a quantum\nmany
 -body Hamiltonian\, is the system it describes gapped or gapless?\n\nWe pr
 ove that this problem is undecidable (in the Goedel and Turing\nsense). Ou
 r results also extend to many other important zero-temperature\nproperties
  of quantum many-body systems\, such as correlation functions.\n\nThe proo
 f is by reduction from the Halting problem. But the construction\nis compl
 ex and draws on a wide variety of techniques\, ranging from\nspectral theo
 ry\, Hamiltonian complexity theory\, quantum algorithms\, and\nnew results
  on aperiodic tilings.\n\nI will explain the result\, sketch the technique
 s involved in the proof at\nan accessible level\, discuss the striking imp
 lications this may have for\nphysics\, and outline some interesting comput
 ability questions related to\nthis problem that remain open.\n\nBased on t
 he following papers:\n\nUndecidability of the Spectral Gap\nToby Cubitt\, 
 David Perez-Garcia and Michael Wolf\nNature\, 528\, p207-211\, (2015)\narX
 iv:1502.04135[quant-ph]\n\nUndecidability of the Spectral Gap (full versio
 n\, 143 pages)\nToby Cubitt\, David Perez-Garcia and Michael Wolf\narXiv:1
 502.04573[quant-ph]\n
LOCATION:MR 14\, CMS
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