On the Geodesic Nature of Wegner’s Flow

Abstract

Wegner's method of flow equations offers a useful tool for diagonalizing a given Hamiltonian and is widely used in various branches of quantum physics. Here, generalizing this method, a condition is derived, under which the corresponding flow of a quantum state becomes geodesic in a submanifold of the projective Hilbert space, independently of specific initial conditions. This implies the geometric optimality of the present method as an algorithm of generating stationary states. The result is illustrated by analyzing some physical examples.

Quantifying Unitary Flow Efficiency and Entanglement for Many-Body Localization

“…The WWF ensures V (β) → 0 monotonically; in particular, the decay of off-diagonal elements of H(β) scales exponentially with [β] = energy −2 [28,40]. Moreover, the WWF induces a geodesic flow with respect to the Fubini-Study metric given appropriate constraints on H 0 and {ψ 0 } [28].…”

“…The WWF ensures V (β) → 0 monotonically; in particular, the decay of off-diagonal elements of H(β) scales exponentially with [β] = energy −2 [28,40]. Moreover, the WWF induces a geodesic flow with respect to the Fubini-Study metric given appropriate constraints on H 0 and {ψ 0 } [28]. Numerical studies have demonstrated the WWF generally produces more quasilocal (in the sense of support on the lattice) integrals of motion in the MBL phase than other flow equation methodologies [18,41].…”

“…However, this connection between bulk geometry and boundary entanglement is less clear in a generic tensor network construction. Beyond the tensor network description, the WWF has strong ties to quantum geometry and geodesicity in the projective Hilbert space [28].…”

Stable unitary integrators for the numerical implementation of continuous unitary transformations

“…The flows reviewed in her paper are summarized in table I. In conjunction with equation (7), the polar forms of these generators make it apparent why they must induce evolution towards a diagonalized Hamiltonian. * The function atan2 has the minor advantage over the usual inverse tangent that there is no ±π ambiguity.…”

“…They have also been analyzed in terms of geodesic flows. 7 Interest in CUTs has increased recently due to the discovery that they may provide physical insights when applied to important many-body problems such as the poorly understood phenomenon of many-body localization (MBL). 8,9 The study of MBL has exploded since Basko, Aleiner, and Altshuler published their seminal diagrammatic analysis of the phenomenon in 2006 10 .…”