### 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 language | English (US) |
---|---|

Article number | 093101 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Volume | 2019 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2 2019 |

Externally published | Yes |

### Fingerprint

### 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

*Journal of Statistical Mechanics: Theory and Experiment*,

*2019*(9), [093101]. https://doi.org/10.1088/1742-5468/ab3787

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

Research output: Contribution to journal › Article

*Journal of Statistical Mechanics: Theory and Experiment*, vol. 2019, no. 9, 093101. https://doi.org/10.1088/1742-5468/ab3787

}

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 -