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On the generalization of the exponential basis for tensor network representations of long-range interactions in two and three dimensions

Li, Zhendong and O'Rourke, Matthew J. and Chan, Garnet Kin-Lic (2019) On the generalization of the exponential basis for tensor network representations of long-range interactions in two and three dimensions. Physical Review B, 100 (15). Art. No. 155121. ISSN 2469-9950. https://resolver.caltech.edu/CaltechAUTHORS:20190801-151930232

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Abstract

In one dimension (1D), a general decaying long-range interaction can be fit to a sum of exponential interactions e^(−λrij) with varying exponents λ, each of which can be represented by a simple matrix product operator with bond dimension D=3. Using this technique, efficient and accurate simulations of 1D quantum systems with long-range interactions can be performed using matrix product states. However, the extension of this construction to higher dimensions is not obvious. We report how to generalize the exponential basis to two and three dimensions by defining the basis functions as the Green's functions of the discretized Helmholtz equation for different Helmholtz parameters λ, a construction which is valid for lattices of any spatial dimension. Compact tensor network representations can then be found for the discretized Green's functions, by expressing them as correlation functions of auxiliary fermionic fields with nearest-neighbor interactions via Grassmann Gaussian integration. Interestingly, this analytic construction in three dimensions yields a D=4 tensor network representation of correlation functions which (asymptotically) decay as the inverse distance (r^(−1)_(ij)), thus generating the (screened) Coulomb potential on a cubic lattice. These techniques will be useful in tensor network simulations of realistic materials.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1103/PhysRevB.100.155121DOIArticle
https://arxiv.org/abs/1907.06018arXivDiscussion Paper
ORCID:
AuthorORCID
Li, Zhendong0000-0002-0683-6293
O'Rourke, Matthew J.0000-0002-5779-2577
Chan, Garnet Kin-Lic0000-0001-8009-6038
Additional Information:© 2019 American Physical Society. Received 16 July 2019; revised manuscript received 20 September 2019; published 14 October 2019. This work was supported by the U. S. National Science Foundation via Grant No. 1665333. Z.L. is supported by the Simons Foundation via the Simons Collaboration on the Many-Electron Problem. M.J.O. is supported by a U. S. National Science Foundation Graduate Research Fellowship under Grant No. DEG-1745301. G.K.-L.C. is the recipient of a Simons Investigator in Physics.
Funders:
Funding AgencyGrant Number
NSFCHE-1665333
Simons FoundationUNSPECIFIED
NSF Graduate Research FellowshipDEG-1745301
Issue or Number:15
Record Number:CaltechAUTHORS:20190801-151930232
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20190801-151930232
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:97609
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:01 Aug 2019 22:25
Last Modified:14 Oct 2019 21:19

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