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Lieb-Robinson bounds, automorphic equivalence and LPPL for long-range interacting fermions

Stefan Teufel, Tom Wessel

TL;DR

The work develops Lieb-Robinson bounds for lattice fermions with polynomially decaying long-range interactions, achieving a linear light cone through an iteration-based refinement of finite-range bounds. These LR bounds underpin the locality of the quasi-local inverse Liouvillian and enable automorphic equivalence of gapped ground states via a spectral flow generated by a localized generator. Consequently, the authors establish the LPPL principle for these long-range fermionic systems and discuss extensions and limitations related to spin systems and the conditional expectation technique. The results provide a framework for adiabatic-type theorems and stability analyses in long-range interacting many-body systems, with explicit attention to how locality propagates through repeated inverse-Liouvillian applications and perturbations.

Abstract

We prove a Lieb-Robinson bound for lattice fermion models with polynomially decaying interactions, which can be used to show the locality of the quasi-local inverse Liouvillian. This allows us to prove automorphic equivalence and the local perturbations perturb locally (LPPL) principle for these systems. The proof of the Lieb-Robinson bound is based on the work of Else et al. (2020), and our results also apply to spin systems. We explain why some newer Lieb-Robinson bounds for long-range spin systems cannot be used to prove the locality of the quasi-local inverse Liouvillian, and in some cases may not even hold for fermionic systems.

Lieb-Robinson bounds, automorphic equivalence and LPPL for long-range interacting fermions

TL;DR

The work develops Lieb-Robinson bounds for lattice fermions with polynomially decaying long-range interactions, achieving a linear light cone through an iteration-based refinement of finite-range bounds. These LR bounds underpin the locality of the quasi-local inverse Liouvillian and enable automorphic equivalence of gapped ground states via a spectral flow generated by a localized generator. Consequently, the authors establish the LPPL principle for these long-range fermionic systems and discuss extensions and limitations related to spin systems and the conditional expectation technique. The results provide a framework for adiabatic-type theorems and stability analyses in long-range interacting many-body systems, with explicit attention to how locality propagates through repeated inverse-Liouvillian applications and perturbations.

Abstract

We prove a Lieb-Robinson bound for lattice fermion models with polynomially decaying interactions, which can be used to show the locality of the quasi-local inverse Liouvillian. This allows us to prove automorphic equivalence and the local perturbations perturb locally (LPPL) principle for these systems. The proof of the Lieb-Robinson bound is based on the work of Else et al. (2020), and our results also apply to spin systems. We explain why some newer Lieb-Robinson bounds for long-range spin systems cannot be used to prove the locality of the quasi-local inverse Liouvillian, and in some cases may not even hold for fermionic systems.

Paper Structure

This paper contains 16 sections, 17 theorems, 125 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mathematical framework
  3. Lattice and operator algebras
  4. Operator Algebra for fermions
  5. Interactions and operator-families
  6. Improved Lieb-Robinson bounds for time-dependent long-range interactions
  7. Automorphic equivalence
  8. Local perturbations perturb locally
  9. Comments on spin systems and the conditional expectation
  10. Mathematical framework
  11. Conditional expectation to improve Lieb-Robinson bounds in spin systems
  12. Difference in lattice fermions
  13. Proof of the finite-range Lieb-Robinson bound
  14. Proof of the range-splitting lemma
  15. Proof of the iteratively improved Lieb-Robinson bound
  16. ...and 1 more sections

Key Result

proposition 1

Let $\Ldim\in{\N}_+$, $\Aconst>0$, $\alpha>\Ldim$ and $\Lat\in\graphs$. Moreover, let $I\subset{\R}$ be an interval, $\Phi\in\interactions{\Lambda}(I)$ be a time-dependent interaction and $H_0$ be an on-site Hamiltonian. For all $X$, $Y\subset \Lambda$, $A\in \alg_X$, $B\in\alg_Y$, with $A$ or $B$ e where and $\velocity = 2 \e^{} \, \norm{F_\alpha}_{\Lat} \, \norm{\Phi}_{\alpha}$.

Figures (3)

  • Figure 1: Situation after applying \ref{['lem:splitting-LR-bound']} once, as in \ref{['thm:LR-bound-with-r^1-p']} or after applying \ref{['lem:LR-bound-iteration-step']} once. The Hamiltonian is split into a short- and long-range part. Terms are considered short-range, if their diameter is smaller than $R'=r^\sigma$, where $r=\Ldist{A,B}$. Depicted are some terms appearing in \ref{['eq:lem-splitting-LR-bound']}: $\uts<>{A}[<r^\sigma]$ is the Heisenberg time evolution of $A$ according to the Hamiltonian $H_{<R'}$. $\Phi(Z_1)$ and $\Phi(Z_2)$ are examples of different long-range contributions, i.e. $\Ldiam{Z_1},\Ldiam{Z_2}\geq R'=r^\sigma$, appearing in \ref{['eq:lem-splitting-LR-bound']}.
  • Figure 2: Second step in the iteration from EMN2020.
  • Figure 3: Second step in our iteration after applying \ref{['lem:splitting-LR-bound']} with a twist.

Theorems & Definitions (26)

  • proposition 1
  • lemma 1
  • lemma 2
  • proof : Proof of \ref{['lem:LR-bound-iteration-step']}
  • theorem 1
  • corollary 1
  • theorem 2
  • proposition 2
  • theorem 3
  • theorem 4
  • ...and 16 more