Theory of Metastable States in Many-Body Quantum Systems

We present a mathematical theory of metastable pure states in closed many-body quantum systems with finite-dimensional Hilbert space. Given a Hamiltonian, a pure state is defined to be metastable when all sufficiently local operators either stabilize the state or raise its average energy. We prove that short-range-entangled metastable states are necessarily eigenstates (scars) of a perturbatively close Hamiltonian. Given any metastable eigenstate of a Hamiltonian, in the presence of perturbations, we prove the presence of prethermal behavior: Local correlation functions decay at a rate bounded by a timescale nonperturbatively long in the inverse metastability radius, rather than Fermi’s golden rule. Inspired by this general theory, we prove that the lifetime of the false vacuum in certain <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>d</a:mi></a:math>-dimensional quantum models grows at least as fast as <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>exp</c:mi><c:mo stretchy="false">(</c:mo><c:msup><c:mi>ε</c:mi><c:mrow><c:mo>−</c:mo><c:mi>d</c:mi></c:mrow></c:msup><c:mo stretchy="false">)</c:mo></c:math>, where <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>ε</g:mi><g:mo stretchy="false">→</g:mo><g:mn>0</g:mn></g:math> is the relative energy density of the false vacuum; this lower bound matches, for the first time, explicit calculations using quantum field theory. We identify metastable states at finite energy density in the PXP model, along with exponentially many metastable states in “helical” spin chains and the two-dimensional Ising model. Our inherently quantum formalism reveals precise connections between many problems, including prethermalization, robust quantum scars, and quantum nucleation theory, and applies to systems without known semiclassical and/or field-theoretic limits.