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arXiv:2110.10148 · Physical review. B./Physical review. B

Quantifying unitary flow efficiency and entanglement for many-body localization

Gregory A. Hamilton, Bryan K. Clark

Published Feb 14, 2023• 3 citations
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Abstract

We probe the bulk geometry of the Wegner Wilson Flow (WWF) in the context of many-body localization by addressing efficiency and bulk entanglement growth measures through approximating upper bounds on the boundary entanglement entropy. We connect these upper bounds to the Fubini-Study metric and clarify how a central quantity, the information fluctuation complexity, distinguishes bulk unitary rotation from entanglement production. We also give a short new proof of the small incremental entangling theorem in the absence of ancillas, achieving a dimensionindependent, universal factor of $c=2$.

DOI
10.1103/physrevb.107.064203

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We probe the bulk geometry of the Wegner Wilson Flow (WWF) in the context of many-body localization by addressing efficiency and bulk entanglement growth measures through approximating upper bounds on the boundary entanglement entropy. We connect these upper bounds to the Fubini-Study metric and clarify how a central quantity, the information fluctuation complexity, distinguishes bulk unitary rotation from entanglement production. We also give a short new proof of the small incremental entangling theorem in the absence of ancillas, achieving a dimensionindependent, universal factor of c=2c=2c=2.
The entanglement entropy in one-dimensional critical systems with boundaries has been associated with the noninteger ground-state degeneracy. This quantity, being a characteristic of boundary fixed points, decreases under renormalization group flow, as predicted by the g theorem. Here, using conformal field theory methods, we exactly calculate the entanglement entropy in the boundary Ising universality class. Our expression can be separated into the well-known bulk term and a boundary entanglement term, displaying a universal flow between two boundary conditions, in accordance with the g theorem. These results are obtained within the replica trick approach, where we show that the associated twist field, a central object generating the geometry of an nnn-sheeted Riemann surface, can be bosonized, giving simple analytic access to multiple quantities of interest. We argue that our result applies to other models falling into the same universality class. This includes the vicinity of the quantum critical point of the two-channel Kondo model, allowing one to track in real space the presence of a region containing one-half of a qubit with entropy 12log(2)\frac{1}{2}log(2)21​log(2), associated with a free local Majorana fermion.
We prove novel far-from-equilibrum bounds on conformal field theories, upper bounding the growth rate of entanglement entropy in spatially uniform states. When equal-time correlators and Wilson loops are computed by geodesics and world sheets, we also bound the growth rate of these. Example applications include lower bounds on thermalization times or the time it takes to achieve deconfinement. Our bounds are proven for holographic conformal field theories at strong coupling and large--NNN, but we provide evidence that they are valid independent of these assumptions. In two dimensions, our results prove a conjectured bound on entanglement growth by Liu and Suh for a large class of states. We also derive bounds on spatial derivatives of correlation measures. From a gravitational perspective, our results constitute new lower bounds on the mass of asymptotically AdS spacetimes with planar symmetry, strengthening the positive mass theorem for these spacetimes. We also derive novel relations in AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT relating various geometric features directly to entanglement entropy derivatives. For example, we show that conformal field theory entanglement growth corresponds to bulk matter falling deeper into the bulk.
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