Abstract
The spectral gap problem - determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of low-energy excitations - pervades quantum many-body physics. Recently, this important problem was shown to be undecidable for quantum spin systems in two (or more) spatial dimensions: there exists no algorithm that determines in general whether a system is gapped or gapless, a result which has many unexpected consequences for the physics of such systems. However, there are many indications that one dimensional spin systems are simpler than their higher-dimensional counterparts: for example, they cannot have thermal phase transitions or topological order, and there exist highly-effective numerical algorithms such as DMRG - and even provably polynomial-time ones - for gapped 1D systems, exploiting the fact that such systems obey an entropy area-law. Furthermore, the spectral gap undecidability construction crucially relied on aperiodic tilings, which are not possible in 1D. So does the spectral gap problem become decidable in 1D? In this paper we prove this is not the case, by constructing a family of 1D spin chains with translationally-invariant nearest neighbour interactions for which no algorithm can determine the presence of a spectral gap. This not only proves that the spectral gap of 1D systems is just as intractable as in higher dimensions, but also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with constant spectral gap and non-degenerate classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behaviour with dense spectrum.
Highlights
One-dimensional spin chains are an important and widely studied class of quantum many-body systems
Satisfiability and tiling problems are nondeterministic polynomial time hard [21] and undecidable [22] in two dimensions and higher. Despite these indications that one-dimensional systems appear qualitatively easier to analyze than their higherdimensional counterparts, we show in this paper that the spectral gap problem is undecidable, even in 1D
Despite the indications that 1D spin chains are simpler systems than higher-dimensional lattice models, we show that the spectral gap problem is undecidable even in dimension one, settling one of the big open questions left in Ref. [19]
Summary
One-dimensional spin chains are an important and widely studied class of quantum many-body systems. For the simplest class of spin chains—qubit chains with translationally invariant nearest-neighbor interactions—the spectral gap problem has been completely solved when the system is frustration-free [15]. Contrast this tractability with the situation in 2D and higher, where even simple theoretical models such as the 2D Fermi-Hubbard model (believed to underlie high-temperature superconductivity) cannot be reliably solved numerically even for moderately large system sizes [16,17], the entropy area law remains an unproven conjecture [18], and the spectral gap problem—i.e., the question of the existence of a spectral gap above the ground state in the thermodynamic limit—is undecidable [19,20]. FHNg is gapped if there exists γ > 0 such that for all N ∈ N, HN have a nondegenerate ground state and a spectral gap ΔðHNÞ > γ where ΔðHNÞ is the difference in energy between the (unique) ground state and the first excited state [23] (see Fig. 1)
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