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 the density matrix renormalization group—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-neighbor 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 it also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with a constant spectral gap and nondegenerate classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behavior with dense spectrum.
- Received 22 October 2019
- Revised 24 April 2020
- Accepted 27 May 2020
DOI:https://doi.org/10.1103/PhysRevX.10.031038
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
The spectral gap—the energy difference between the lowest and next-lowest state of a quantum system—is one of the fundamental quantities that determine the system’s properties at low temperature. Determining whether a system is gapped or gapless is therefore one of the most important questions in understanding the physics of a quantum system. In 2015, some of us proved mathematically that, in general, this problem is impossible to solve in quantum systems with two or more dimensions. But that leads us to wonder if it is easier to solve in 1D. Here, we prove that, surprisingly, this is not the case: The 1D spectral gap problem is also undecidable.
Usually, 1D quantum systems are much easier to solve than those with more dimensions. Many important 1D systems were solved with pen and paper in the 1930s, and many more can now be solved numerically with modern computers. In contrast, some 2D models have resisted full solution for over 30 years, even using the world’s biggest supercomputers.
There are many good numerical algorithms for computing properties of 1D quantum systems, so we analyze how they would fare if applied to the models involved in our 1D undecidability proof. We find that these algorithms would predict a classical gapped system, whereas the real behavior of the model becomes apparent only at system sizes too large to be accessible.
Despite the widespread success of numerical algorithms for 1D quantum systems, predicting their elementary properties is, in principle, no easier than in higher dimensions. While at first discouraging, this also implies that there is richer physics in 1D quantum systems than hitherto predicted.
