Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the asymptotic scaling of the von Neumann entropy in quasifree fermionic right mover/left mover systems

Walter H. Aschbacher

TL;DR

This work establishes a rigorous, algebraic framework for analyzing the asymptotic scaling of von Neumann entropy in translation-invariant quasifree fermionic right/left mover systems, modeled on the CAR algebra over the infinite 1D lattice. By tying the local reduced density matrices to a 2-point operator and a Majorana correlation matrix, and employing Bogoliubov transformations and Toeplitz operator techniques, it proves that the entropy density of large subsystems generically remains nonzero and encodes a two-source thermal mixture from the left and right reservoirs. A precise momentum-space formula for the limiting entropy density is derived, valid across nonequilibrium steady states, thermal states, and ground states, with a clear criterion for vanishing entropy density. The results link nonequilibrium transport data to entropic measures, and are underpinned by Szegő-type limit theorems for block Toeplitz operators, enabling a rigorous asymptotic analysis of entanglement-like quantities in open quantum systems.

Abstract

For the general class of quasifree fermionic right mover/left mover systems over the infinitely extended two-sided discrete line introduced in [8] within the algebraic framework of quantum statistical mechanics, we study the von Neumann entropy of a contiguous subsystem of finite length in interaction with its environment. In particular, under the assumption of spatial translation invariance, we analyze the asymptotic behavior of the von Neumann entropy for large subsystem lengths and prove that its leading order density is, in general, nonvanishing and displays the signature of a mixture of the independent thermal species underlying the right mover/left mover system. As special cases, the formalism covers so-called nonequilibrium steady states, thermal equilibrium states, and ground states. Moreover, for general Fermi functions, we derive a necessary and sufficient criterion for the von Neumann entropy density to vanish.

On the asymptotic scaling of the von Neumann entropy in quasifree fermionic right mover/left mover systems

TL;DR

This work establishes a rigorous, algebraic framework for analyzing the asymptotic scaling of von Neumann entropy in translation-invariant quasifree fermionic right/left mover systems, modeled on the CAR algebra over the infinite 1D lattice. By tying the local reduced density matrices to a 2-point operator and a Majorana correlation matrix, and employing Bogoliubov transformations and Toeplitz operator techniques, it proves that the entropy density of large subsystems generically remains nonzero and encodes a two-source thermal mixture from the left and right reservoirs. A precise momentum-space formula for the limiting entropy density is derived, valid across nonequilibrium steady states, thermal states, and ground states, with a clear criterion for vanishing entropy density. The results link nonequilibrium transport data to entropic measures, and are underpinned by Szegő-type limit theorems for block Toeplitz operators, enabling a rigorous asymptotic analysis of entanglement-like quantities in open quantum systems.

Abstract

For the general class of quasifree fermionic right mover/left mover systems over the infinitely extended two-sided discrete line introduced in [8] within the algebraic framework of quantum statistical mechanics, we study the von Neumann entropy of a contiguous subsystem of finite length in interaction with its environment. In particular, under the assumption of spatial translation invariance, we analyze the asymptotic behavior of the von Neumann entropy for large subsystem lengths and prove that its leading order density is, in general, nonvanishing and displays the signature of a mixture of the independent thermal species underlying the right mover/left mover system. As special cases, the formalism covers so-called nonequilibrium steady states, thermal equilibrium states, and ground states. Moreover, for general Fermi functions, we derive a necessary and sufficient criterion for the von Neumann entropy density to vanish.

Paper Structure

This paper contains 8 sections, 14 theorems, 18 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Infinite fermionic systems
  3. Reduced density matrix
  4. Von Neumann entropy
  5. Translation invariance
  6. R/L mover entropy asymptotics
  7. Proofs
  8. Toeplitz operators

Key Result

Proposition 10

Let $\Lambda\in{\rm Fin}({\mathord{\mathbb Z}})$ and set ${\mathfrak B}_\Lambda:=\{a_x\,|\,x\in\Lambda\}$. Then, the local observable algebra, defined by A_Λ := ⋂_ B: $\text{C}^\ast$ - subalgebra of A B_Λ⊆ B B, is a unital $\text{C}^\ast$ - subalgebras of ${\mathfrak A}$ which equals the set o

Figures (7)

  • Figure 1: All the pairings for $n=1$, $n=2$, and $n=3$. The total number of intersections $\#$ in each graph relates to the sign of the permutation $\pi$ as ${\rm sgn}(\pi)=(-1)^\#$.
  • Figure 2: The functions $\widetilde{\ell}$ (dashed line) and $\widetilde{\eta}$ (continuous line) which are the extensions by zero to the whole of ${\mathord{\mathbb R}}$ of $\ell$ and $\eta$ from \ref{['Shan']} and \ref{['eta']}.
  • Figure 3: The bijection $\kappa$ and the surjection $\widetilde{\kappa}$.
  • Figure 4: The function \ref{['E-XY']} for $c_{2,1}=1/50$ and $c_{3,0}=1/2$. We have ${\rm a}=\arccos(625/1248)\approx1.05$ and ${\rm b}=E({\rm e}^{{\rm i} {\rm a}})=\sqrt{1871/39}/200\approx0.03$.
  • Figure 5: Some of the configurations in the proof of Lemma \ref{['lem:StdCtg']}\ref{['StdCtg-a']}.
  • ...and 2 more figures

Theorems & Definitions (33)

  • Definition 1: Observables
  • Definition 4: 2-point operator
  • Definition 6: Quasifree state
  • Definition 7: Fermi family
  • Example 8
  • Proposition 10: Local observable algebras
  • Lemma 11: Matrix units
  • Proposition 12: Reduced density matrix
  • Lemma 15: Factorization
  • Definition 17: Bogoliubov transformations
  • ...and 23 more