Undecidability of the Spectral Gap in One Dimension

Authors: Johannes Bausch, Toby S Cubitt, Angelo Lucia and David Pérez-García (ICMAT-UCM)

Source: Physical Review X

Date of publication: August 2020



One of the main steps in the scientific method for experimental sciences is the need to make predictions for a given hypothetical explanation of a phenomenon, so that such predictions can be tested again in experiments. At the level of quantum systems, this is the problem of, starting from a description of the microscopic interactions of a system, derive the measurable global properties of lowest energy state in the ideal scenario when the number of elementary constituents of the system grows to infinity.

In this article, the authors show that this problem is impossible to solve, even in the simplest possible case of a one-dimensional system. Impossible here means that there exists no algorithm, or mathematical reasoning, no matter how complex, that can solve it.

This result is especially unexpected since there was a general consensus in condensed matter physics that one-dimensional systems were “easy”. And indeed, there were several evidences in this direction. For instance, for a given fixed number of particles and assuming a control on the energy gap between the lowest and second-lowest energy levels of the system, it was shown recently that there are even efficient algorithms that solve the problem. It is also known since the 60s that temperature cannot create phase transitions in 1D systems (as opposed to the 2D case) and it is also well known that some of the most exotic quantum properties of nature, like topological order (whose discovery was awarded with the Nobel prize on 2016), cannot exist in one-dimensional systems.

The result proven in this paper builds on the techniques developed in 2015 by the authors to show the analogue result for 2D systems. On top of that, the authors introduce a new extra key ingredient, also quite unexpected. This is the existence of quantum interactions that, despite having a local nature (that is, particles interact only with their nearest neighbors) create in 1D patterns with arbitrary long periods. This is a new, purely quantum, property, since it is well known that this property cannot hold for classical systems in 1D. Indeed, that it holds for classical systems in 2D was a breakthrough results in the context of tiling problems in the 60s.

The result proven in this paper, despite being a no-go result, has a beautiful positive side. It implies the existence of 1D quantum systems that display a new quantum effect named “size-driven transition”: the system behaves as an insulator for all system sizes below a critical threshold and, from this threshold on, the system switches dramatically its behavior to become a superconductor. Moreover, this critical threshold can be uncomputably large.