Ouyang, Y. orcid.org/0000-0003-1115-0074 (2019) Computing spectral bounds of the Heisenberg ferromagnet from geometric considerations. Journal of Mathematical Physics, 60 (7). ISSN 0022-2488
Abstract
We give a polynomial-time algorithm for computing upper bounds on some of the smaller energy eigenvalues in a spin-1/2 ferromagnetic Heisenberg model with any graph G for the underlying interactions. An important ingredient is the connection between Heisenberg models and the symmetric products of G. Our algorithms for computing upper bounds are based on generalized diameters of graphs. Computing the upper bounds amounts to solving the minimum assignment problem on G, which has well-known polynomial-time algorithms from the field of combinatorial optimization. We also study the possibility of computing the lower bounds on some of the smaller energy eigenvalues of Heisenberg models. This amounts to estimating the isoperimetric inequalities of the symmetric product of graphs. By using connections with discrete Sobolev inequalities, we show that this can be performed by considering just the vertex-induced subgraphs of G. If our conjecture for a polynomial time approximation algorithm to solve the edge-isoperimetric problem holds, then our proposed method of estimating the energy eigenvalues via approximating the edge-isoperimetric properties of vertex-induced subgraphs will yield a polynomial time algorithm for estimating the smaller energy eigenvalues of the Heisenberg ferromagnet.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2019 The Author(s). Reproduced in accordance with the publisher's self-archiving policy. |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > Department of Physics and Astronomy (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 11 Oct 2019 08:18 |
Last Modified: | 24 May 2023 08:58 |
Status: | Published |
Publisher: | AIP Publishing |
Refereed: | Yes |
Identification Number: | 10.1063/1.5084136 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:152018 |