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Undecidability of the spectral gap

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If you have a question about this talk, please contact Mahdi Godazgar.

The spectral gap—the difference in energy between the ground state and the first excited state—is of central importance to quantum many-body physics. Some of the most challenging and long-standing open problems in theoretical physics concern the spectral gap, such as the famous Haldane conjecture, or the infamous Yang-Mills gap conjecture (one of the Millennium Prize problems). These problems—and many others— are all particular cases of the general spectral gap problem: Given a quantum many-body Hamiltonian, is the system it describes gapped or gapless?

We prove that this problem is undecidable (in exactly the same sense as the Halting Problem was proven to be undecidable by Turing). This also implies that the spectral gap of certain quantum many-body Hamiltonians is not determined by the axioms of mathematics (in much the same sense as Goedel’s incompleteness theorem implies that certain theorems are mathematically unprovable). The results also extend to many other important low-temperature properties of quantum many-body systems, such correlation functions.

The proof is complex and draws on a wide variety of techniques, ranging from mathematical physics to theoretical computer science, from Hamiltonian complexity theory, quantum algorithms and quantum computing to fractal tilings. I will explain the result, sketch the techniques involved in the proof at an accessible level, and discuss the striking implications this may have both for theoretical physics, and for physics more generally (which, after all, happens in the laboratory not in Hilbert space!).

This talk is part of the Theoretical Physics Colloquium series.

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