Abstract
We define and calculate versions of complexity for free fermionic quantum field theories in 1+1 and 3+1 dimensions, adopting Nielsen's geodesic perspective in the space of circuits. We do this both by discretizing and identifying appropriate classes of Bogoliubov-Valatin transformations, and also directly in the continuum by defining squeezing operators and their generalizations. As a closely related problem, we consider cMERA tensor networks for fermions: viewing them as paths in circuit space, we compute their path lengths. Certain ambiguities that arise in some of these results because of cut-off dependence are discussed.
Highlights
The formation of black holes via gravitational collapse in anti–de Sitter space is expected to be dual to thermalization in the dual conformal field theory
This leads one to think of a thermal Conformal Field Theory (CFT) state as a gravitational configuration that can be approximated by an eternal black hole
One needs to have a CFT explanation for the fact that the Einstein-Rosen bridge that shows up inside the horizon of an eternal black hole is a time-dependent configuration, and that its “size” increases towards the future. It has recently been proposed by Susskind and others [1] that the quantity one should compare against the size of the Einstein-Rosen wormhole is the “complexity” of the CFT state: the idea being that the complexity of a state can increase even after it has thermalized in some appropriate sense
Summary
The formation of black holes via gravitational collapse in anti–de Sitter space is expected to be dual to thermalization in the dual conformal field theory. By working with a class of Gaussian wave functions that interpolated between the decoupled reference state and the true ground state (chosen as the target state) they were able to define complexity for the state by minimizing the Nielsen-like path length in the space of unitaries that did such an interpolation. Since when the cutoff is finite the target state for the cMERA is not quite the target state of the previous paragraph (even though they both tend to the ground state as Λ → ∞), it is not possible to meaningfully compare the lengths of the paths at finite cutoff by looking at their leading divergences This leads us to a discussion of the meaning of a cutoff dependent quantity like the complexity, and to speculations on the possibility that subleading terms in complexity could be of physical interest, in analogy with (holographic) entanglement entropy. Various Appendices contain some review material as well as technical details
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