arXiv:2005.04548

Stability of the Spectral Gap for Lattice Fermions

Tohru Koma

Published May 10, 2020• 2 citations

Abstract

It is widely believed that the spectral gap above the Fermi sea ground state of lattice free fermions is stable against generic weak interactions. Quite recently, Hastings presented a new interesting idea to prove the stability for Majorana free fermions. He ingeniously used Majorana fermions so that the unperturbed Hamiltonian of the free fermion part is mapped into a frustration-free form. In this paper, we refine Hastings's argument, and give a complete proof of the stability of the spectral gap above the Fermi sea ground state against generic weak interactions for generic lattice free fermions.

DOI: 10.48550/arxiv.2005.04548

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arXiv:2005.04548

Stability of the Spectral Gap for Lattice Fermions

Tohru Koma

Published May 10, 2020• 2 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2005.04548

Related papers

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Φ_h

be a free fermionic interaction, possibly including pairing terms, whose infinite-volume Majorana coefficient operator

h

has no spectrum in

(-g,g)

, and let

\mathcal V

be an even self-adjoint many-body interaction. For sufficiently small

|λ|

, we construct pure ground states

ω_λ

of the interacting bulk system

Φ_λ=Φ_h+λ\mathcal V

and quasi-local automorphisms

β_{0,λ}

such that

ω_λ=ω_0\circβ_{0,λ}

, where

ω_0

is the free ground state of

Φ_h

. Each

ω_λ

has a GNS Hamiltonian with gap at least

g-C_*|λ| \|\mathcal V\|_*

, where

\|\cdot\|_*

is the relevant interaction norm. Our result allows polynomial decay of both the free coefficients and the many-body interaction. For two-site interactions on a uniformly discrete point set

Γ\subset\mathbb{R}^d

, a sufficient condition is decay as

(1+|x-y|)^{-\varpi}

with

\varpi>2d

, for both the hopping and pairing coefficients and the interaction terms. The spectral-flow generator preserves the spatial exponent of the polynomial interaction norm, and its automorphisms satisfy locality bounds at that exponent. If

h

and

\mathcal V

have stretched-exponential decay, so does the spectral-flow generator. The proof constructs the spectral flow and establishes the gap through an excitation-number comparison, using polynomial propagation bounds for evolved interactions proved in a companion paper.

\mathbb{Z}_2

index for the Majorana edge zero modes.