Free-Fermion entanglement and orthogonal polynomials

Nicolas Crampé, Rafael I. Nepomechie, Luc Vinet

Research output: Contribution to journalArticle

Abstract

We present a simple construction for a tridiagonal matrix T that commutes with the hopping matrix for the entanglement Hamiltonian of open finite free-Fermion chains associated with families of discrete orthogonal polynomials. It is based on the notion of algebraic Heun operator attached to bispectral problems, and the parallel between entanglement studies and the theory of time and band limiting. As examples, we consider Fermionic chains related to the Chebychev, Krawtchouk and dual Hahn polynomials. For the former case, which corresponds to a homogeneous chain, the outcome of our construction coincides with a recent result of Eisler and Peschel; the latter cases yield commuting operators for particular inhomogeneous chains. Since T is tridiagonal and non-degenerate, it can be readily diagonalized numerically, which in turn can be used to calculate the spectrum of , and therefore the entanglement entropy.

Original languageEnglish (US)
Article number093101
JournalJournal of Statistical Mechanics: Theory and Experiment
Volume2019
Issue number9
DOIs
StatePublished - Sep 2 2019
Externally publishedYes

Fingerprint

Entanglement
Orthogonal Polynomials
Fermions
polynomials
fermions
Tridiagonal matrix
Discrete Orthogonal Polynomials
Hahn Polynomials
operators
matrices
Commute
Operator
Limiting
Entropy
entropy
Calculate
Polynomials
Commuting

Keywords

  • entanglement entropies
  • entanglement in extended quantum systems
  • ladders
  • planes
  • solvable lattice models
  • spin chains

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Free-Fermion entanglement and orthogonal polynomials. / Crampé, Nicolas; Nepomechie, Rafael I.; Vinet, Luc.

In: Journal of Statistical Mechanics: Theory and Experiment, Vol. 2019, No. 9, 093101, 02.09.2019.

Research output: Contribution to journalArticle

@article{178c5598a2264be2b561ec3eb26f1ed8,
title = "Free-Fermion entanglement and orthogonal polynomials",
abstract = "We present a simple construction for a tridiagonal matrix T that commutes with the hopping matrix for the entanglement Hamiltonian of open finite free-Fermion chains associated with families of discrete orthogonal polynomials. It is based on the notion of algebraic Heun operator attached to bispectral problems, and the parallel between entanglement studies and the theory of time and band limiting. As examples, we consider Fermionic chains related to the Chebychev, Krawtchouk and dual Hahn polynomials. For the former case, which corresponds to a homogeneous chain, the outcome of our construction coincides with a recent result of Eisler and Peschel; the latter cases yield commuting operators for particular inhomogeneous chains. Since T is tridiagonal and non-degenerate, it can be readily diagonalized numerically, which in turn can be used to calculate the spectrum of , and therefore the entanglement entropy.",
keywords = "entanglement entropies, entanglement in extended quantum systems, ladders, planes, solvable lattice models, spin chains",
author = "Nicolas Cramp{\'e} and Nepomechie, {Rafael I.} and Luc Vinet",
year = "2019",
month = "9",
day = "2",
doi = "10.1088/1742-5468/ab3787",
language = "English (US)",
volume = "2019",
journal = "Journal of Statistical Mechanics: Theory and Experiment",
issn = "1742-5468",
publisher = "IOP Publishing Ltd.",
number = "9",

}

TY - JOUR

T1 - Free-Fermion entanglement and orthogonal polynomials

AU - Crampé, Nicolas

AU - Nepomechie, Rafael I.

AU - Vinet, Luc

PY - 2019/9/2

Y1 - 2019/9/2

N2 - We present a simple construction for a tridiagonal matrix T that commutes with the hopping matrix for the entanglement Hamiltonian of open finite free-Fermion chains associated with families of discrete orthogonal polynomials. It is based on the notion of algebraic Heun operator attached to bispectral problems, and the parallel between entanglement studies and the theory of time and band limiting. As examples, we consider Fermionic chains related to the Chebychev, Krawtchouk and dual Hahn polynomials. For the former case, which corresponds to a homogeneous chain, the outcome of our construction coincides with a recent result of Eisler and Peschel; the latter cases yield commuting operators for particular inhomogeneous chains. Since T is tridiagonal and non-degenerate, it can be readily diagonalized numerically, which in turn can be used to calculate the spectrum of , and therefore the entanglement entropy.

AB - We present a simple construction for a tridiagonal matrix T that commutes with the hopping matrix for the entanglement Hamiltonian of open finite free-Fermion chains associated with families of discrete orthogonal polynomials. It is based on the notion of algebraic Heun operator attached to bispectral problems, and the parallel between entanglement studies and the theory of time and band limiting. As examples, we consider Fermionic chains related to the Chebychev, Krawtchouk and dual Hahn polynomials. For the former case, which corresponds to a homogeneous chain, the outcome of our construction coincides with a recent result of Eisler and Peschel; the latter cases yield commuting operators for particular inhomogeneous chains. Since T is tridiagonal and non-degenerate, it can be readily diagonalized numerically, which in turn can be used to calculate the spectrum of , and therefore the entanglement entropy.

KW - entanglement entropies

KW - entanglement in extended quantum systems

KW - ladders

KW - planes

KW - solvable lattice models

KW - spin chains

UR - http://www.scopus.com/inward/record.url?scp=85072040722&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072040722&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/ab3787

DO - 10.1088/1742-5468/ab3787

M3 - Article

AN - SCOPUS:85072040722

VL - 2019

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

SN - 1742-5468

IS - 9

M1 - 093101

ER -