Abstract
We apply the recently developed notion of complexity for field theory to a quantum quench through a critical point in 1+1 dimensions. We begin with a toy model consisting of a quantum harmonic oscillator, and show that complexity exhibits universal scalings in both the slow and fast quench regimes. We then generalize our results to a one-dimensional harmonic chain, and show that preservation of these scaling behaviors in free field theory depends on the choice of norm. Applying our setup to the case of two oscillators, we quantify the complexity of purification associated with a subregion, and demonstrate that complexity is capable of probing features to which the entanglement entropy is insensitive. We find that the complexity of subregions is subadditive, and comment on potential implications for holography.
Highlights
Introduction.—Among the most exciting developments in theoretical physics is the confluence of ideas from quantum many-body systems, quantum information theory, and gravitational physics
We find that the complexity of subregions is subadditive, and comment on potential implications for holography
Insights from black hole physics [11,12,13,14] suggest that certain codimension-0 and codimension-1 surfaces may play an important role in reconstructing bulk spacetime in holography, since these capture information beyond that which is accessible to the aforementioned codimension-2 surfaces—that is, beyond entanglement entropy
Summary
Studies have mainly focused on the scalings of a restricted set of one- and two-point functions, and recently on entanglement [51]. Ω0 is a free parameter, but will gain an interpretation as the dimensionless reference-state frequency below This profile has the property that the system is initially gapped at t 1⁄4 −∞, but becomes gapless at t 1⁄4 0, corresponding to oscillator excitations above the ground state (3) as the system evolves via Eq (2). In this case, the function fðtÞ can be written explicitly in terms of hypergeometric functions; see Ref. The corresponding wave function is given by ψðxþ; x−; tÞ 1⁄4 ψ0ðxþ; tÞψ0ðx−; tÞ; ð8Þ with ψ0 given by Eq (3) Note that this construction naturally generalizes to an N-oscillator harmonic chain, which we will consider after introducing complexity below. Our reference state will be provided by the ground state of our time-dependent Hamiltonian (6) at t 1⁄4 −∞, ψ
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