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Error estimates for extrapolations with matrix-product states

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Project
ERQUAF (Entanglement and Renormalisation for Quantum Fields)
Abstract
We introduce an error measure for matrix-product states without requiring the relatively costly two-site density-matrix renormalization group (2DMRG). This error measure is based on an approximation of the full variance <psi broken vertical bar(H) over cap - E)(2) broken vertical bar psi >. When applied to a series of matrix-product states at different bond dimensions obtained from a single-site density-matrix renormalization group (1DMRG) calculation, it allows for the extrapolation of observables towards the zero-error case representing the exact ground state of the system. The calculation of the error measure is split into a sequential part of cost equivalent to two calculations of <psi broken vertical bar(H) over cap broken vertical bar psi > and a trivially parallelized part scaling like a single operator application in 2DMRG. The reliability of this error measure is demonstrated by four examples: the L = 30, S = 1/2 Heisenberg chain, the L = 50 Hubbard chain, an electronic model with long-range Coulomb-like interactions, and the Hubbard model on a cylinder with a size of 10 x 4. Extrapolation in this error measure is shown to be on par with extrapolation in the 2DMRG truncation error or the full variance <psi broken vertical bar(H) over cap - E)(2) broken vertical bar psi > at a fraction of the computational effort.
Keywords
RENORMALIZATION-GROUP

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Citation

Please use this url to cite or link to this publication:

Chicago
Hubig, C, Jutho Haegeman, and U Schollwöck. 2018. “Error Estimates for Extrapolations with Matrix-product States.” Physical Review B 97 (4).
APA
Hubig, C., Haegeman, J., & Schollwöck, U. (2018). Error estimates for extrapolations with matrix-product states. PHYSICAL REVIEW B, 97(4).
Vancouver
1.
Hubig C, Haegeman J, Schollwöck U. Error estimates for extrapolations with matrix-product states. PHYSICAL REVIEW B. 2018;97(4).
MLA
Hubig, C, Jutho Haegeman, and U Schollwöck. “Error Estimates for Extrapolations with Matrix-product States.” PHYSICAL REVIEW B 97.4 (2018): n. pag. Print.
@article{8556681,
  abstract     = {We introduce an error measure for matrix-product states without requiring the relatively costly two-site density-matrix renormalization group (2DMRG). This error measure is based on an approximation of the full variance {\textlangle}psi broken vertical bar(H) over cap - E)(2) broken vertical bar psi {\textrangle}. When applied to a series of matrix-product states at different bond dimensions obtained from a single-site density-matrix renormalization group (1DMRG) calculation, it allows for the extrapolation of observables towards the zero-error case representing the exact ground state of the system. The calculation of the error measure is split into a sequential part of cost equivalent to two calculations of {\textlangle}psi broken vertical bar(H) over cap broken vertical bar psi {\textrangle} and a trivially parallelized part scaling like a single operator application in 2DMRG. The reliability of this error measure is demonstrated by four examples: the L = 30, S = 1/2 Heisenberg chain, the L = 50 Hubbard chain, an electronic model with long-range Coulomb-like interactions, and the Hubbard model on a cylinder with a size of 10 x 4. Extrapolation in this error measure is shown to be on par with extrapolation in the 2DMRG truncation error or the full variance {\textlangle}psi broken vertical bar(H) over cap - E)(2) broken vertical bar psi {\textrangle} at a fraction of the computational effort.},
  articleno    = {045125},
  author       = {Hubig, C and Haegeman, Jutho and Schollw{\"o}ck, U},
  issn         = {2469-9950},
  journal      = {PHYSICAL REVIEW B},
  keyword      = {RENORMALIZATION-GROUP},
  language     = {eng},
  number       = {4},
  pages        = {9},
  title        = {Error estimates for extrapolations with matrix-product states},
  url          = {http://dx.doi.org/10.1103/physrevb.97.045125},
  volume       = {97},
  year         = {2018},
}

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