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Undecidability of the spectral gap

机译：光谱间隙的不确定性

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摘要

The spectral gap-the energy difference between the ground state and first excited state of a system-is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the Yang-Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum many-body 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 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'. 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.
机译：光谱间隙-系统基态与第一激发态之间的能量差-是量子多体物理学的核心。许多具有挑战性的开放性问题，例如Haldane猜想，存在空位拓扑自旋液相的问题以及Yang-Mills间隙猜想，都与光谱间隙有关。这些和其他问题是一般谱隙问题的特例：给定量子多体系统的哈密顿量，它是有间隙的还是无间隙的？在这里，我们证明这是一个无法确定的问题。具体而言，我们在具有平移不变的最近邻相互作用的二维晶格上构造量子自旋系统族，对此，谱隙问题是不确定的。该结果扩展到其他低能量特性的不确定性，例如存在代数衰减的基态相关性。该证明将哈密顿复杂性技术与非周期性平铺相结合，以构造哈密顿，其基态编码量子相位估计算法的演化，然后跟随通用图灵机。光谱间隙取决于相应“停止问题”的结果。我们的结果表明，没有算法可以确定任意模型是有间隙的还是无间隙的，并且存在模型中是否存在谱隙与数学公理无关。

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  • 来源
    《Nature》 |2015年第7581期|207-211|共5页
  • 作者

    Cubitt Toby S.; Perez-Garcia David; Wolf Michael M.;

  • 作者单位

    UCL, Dept Comp Sci, London WC1E 6BT, England|Univ Cambridge, Ctr Math Sci, DAMTP, Cambridge CB3 0WA, England;

    Univ Complutense Madrid, Fac CC Matemat, Dept Anal Matemat, E-28040 Madrid, Spain|Univ Complutense Madrid, Fac CC Matemat, IMI, E-28040 Madrid, Spain|ICMAT, Madrid 28049, Spain;

    Tech Univ Munich, Dept Math, D-85748 Garching, Germany;

  • 收录信息 美国《科学引文索引》(SCI);美国《工程索引》(EI);美国《生物学医学文摘》(MEDLINE);美国《化学文摘》(CA);
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