Abstract
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.
Highlights
The spectral gap is one of the most important properties of a quantum many-body system
The behaviour of the spectral gap is intimately related to the phase diagram of a quantum many-body system, with quantum phase transitions occurring at critical points where the gap vanishes
The lowtemperature physics of the system are governed by the spectral gap: gapped systems exhibit “non-critical” behaviour, with low-energy excitations that behave as massive particles, preventing long-range correlations [MH06]; gapless systems exhibit “critical” behaviour, with low-energy excitations that behave as massless particles, allowing long-range correlations
Summary
The spectral gap is one of the most important properties of a quantum many-body system. In the related setting of quantum field theory, determining if Yang-Mills theory is gapped is one of the Millennium Prize Problems [JW00], and is closely related to one of the most important open problems in high-energy physics: explaining the phenomenon of quark confinement All of these problems are specific cases of the general spectral gap problem: given a quantum many-body Hamiltonian, is the system it describes gapped or gapless? Undecidability of other physical quantities has been shown in many-body systems for the much easier cases where either the many-body lattice structure or the translational-invariance is removed [Gu+09; Llo[93]; PER89] Proving such impossibility theorems is highlighted as one of the main open problems in mathematical physics in the list published by the International Association of Mathematical Physics in the late 90’s, edited by Aizenman [Aiz98]. Can be seen as a major contribution to this
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