Undecidability of the Spectral Gap in One Dimension

TL;DR

This paper proves that determining the spectral gap in one-dimensional quantum spin systems is undecidable, showing that even in 1D, the problem remains as complex and intractable as in higher dimensions, with implications for understanding quantum physics.

Contribution

The paper constructs 1D spin chains with translationally-invariant interactions demonstrating undecidability of the spectral gap, extending the undecidability result to one-dimensional systems.

Findings

Spectral gap undecidability applies to 1D systems.

Existence of 1D systems with uncomputably large constant gap.

Transition from gapped to gapless behavior at uncomputably large sizes.

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.