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Stable Unitary Integrators for the Numerical Implementation of Continuous Unitary Transformations

Savitz, Samuel and Refael, Gil (2017) Stable Unitary Integrators for the Numerical Implementation of Continuous Unitary Transformations. Physical Review B, 96 (11). Art. No. 115129. ISSN 2469-9950. https://resolver.caltech.edu/CaltechAUTHORS:20171004-144154017

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Abstract

The technique of continuous unitary transformations has recently been used to provide physical insight into a diverse array of quantum mechanical systems. However, the question of how to best numerically implement the flow equations has received little attention. The most immediately apparent approach, using standard Runge-Kutta numerical integration algorithms, suffers from both severe inefficiency due to stiffness and the loss of unitarity. After reviewing the formalism of continuous unitary transformations and Wegner's original choice for the infinitesimal generator of the flow, we present a number of approaches to resolving these issues including a choice of generator which induces what we call the “uniform tangent decay flow” and three numerical integrators specifically designed to perform continuous unitary transformations efficiently while preserving the unitarity of flow. We conclude by applying one of the flow algorithms to a simple calculation that visually demonstrates the many-body localization transition.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1103/PhysRevB.96.115129DOIArticle
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.96.115129PublisherArticle
http://arxiv.org/abs/1707.03407arXivDiscussion Paper
ORCID:
AuthorORCID
Savitz, Samuel0000-0003-2112-3758
Additional Information:© 2017 American Physical Society. Received 31 July 2017; published 18 September 2017. This work was supported by the Institute of Quantum Information and Matter, a National Science Foundation frontier center partially funded by the Gordon and Betty Moore Foundation. G.R. acknowledges the generous support of the Packard Foundation and the National Science Foundation through award DMR-1410435. Thanks to E. van Nieuwenburg, S. Kehrein, M. Bintz, C. White, A. Bourzutschky, and P. Titum for many fruitful discussions. The numerical CUT flows were implemented using double-precision floating-point matrices calculated by the open-source linear algebra library armadillo [62]. Special thanks to C. Peng for GPU-accelerating the flows in Sec. V using the open-source linear algebra library arrayfire [63] and thereby helping to test their performance.
Group:IQIM, Institute for Quantum Information and Matter
Funders:
Funding AgencyGrant Number
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
Gordon and Betty Moore FoundationUNSPECIFIED
David and Lucile Packard FoundationUNSPECIFIED
NSFDMR-1410435
Issue or Number:11
Record Number:CaltechAUTHORS:20171004-144154017
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20171004-144154017
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:82076
Collection:CaltechAUTHORS
Deposited By: Joy Painter
Deposited On:05 Oct 2017 16:56
Last Modified:03 Oct 2019 18:50

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