Computing spectral bounds of the Heisenberg ferromagnet from geometric considerations
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.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Jul 01, 2019
- Source ID
- 10.1063/1.5084136
Entities
People
- Yingkai Ouyang
Organizations
- Air Force Office of Scientific Research
- Engineering and Physical Sciences Research Council
- Ministry of Education
- National Research Foundation
- National University of Singapore
- Singapore University of Technology and Design
- University of Sheffield