[Paper Review] Stability of the Spectral Gap for Lattice Fermions

Tohru Koma|arXiv (Cornell University)|May 10, 2020

Spectral Theory in Mathematical Physics12 references4 citations

TL;DR

This paper provides a complete proof of the stability of the spectral gap above the Fermi sea ground state for generic lattice fermions under weak, short-range interactions. By refining Hastings' approach using Majorana fermion transformations, the authors map the unperturbed Hamiltonian into a frustration-free form, enabling a relative bound on the perturbation that ensures the spectral gap remains open in the thermodynamic limit, even without translational invariance.

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.

Motivation & Objective

• To establish the stability of the spectral gap above the Fermi sea ground state for generic lattice fermions under weak interactions.
• To extend and complete Hastings' recent argument by providing a rigorous proof without assuming translational invariance.
• To demonstrate that the spectral gap remains open in the thermodynamic limit for short-range, even-parity interactions.
• To develop a framework using Majorana fermion mappings to achieve a frustration-free representation of the unperturbed Hamiltonian.
• To derive a relative bound on the perturbation that ensures the gap does not close under weak interactions.

Proposed method

• Transform the original fermion system into a Majorana fermion representation to map the unperturbed Hamiltonian into a frustration-free form.
• Apply a unitary transformation to the full Hamiltonian such that all local interactions annihilate the unperturbed ground state.
• Use Lieb-Robinson bounds and spatial decay estimates to control the propagation of perturbations in the system.
• Establish a relative bound on the perturbation using exponential decay of interaction matrix elements and spectral gap estimates.
• Employ technical estimates from Appendices A–I to control the decay of correlation functions and the convergence of perturbative series.
• Introduce a cutoff parameter $ K_h > 1 $ to suppress high-body interactions when the interaction range is not bounded.

Experimental results

Research questions

• RQ1 Does the spectral gap above the Fermi sea remain open under generic weak interactions in lattice fermion systems?
• RQ2 Can the stability of the spectral gap be rigorously proven without assuming translational invariance?
• RQ3 Is Hastings' Majorana fermion approach sufficient and generalizable to all lattice fermion systems with a spectral gap?
• RQ4 How do short-range, even-parity interactions affect the stability of the Fermi sea ground state?
• RQ5 What conditions ensure that the perturbation does not close the spectral gap in the thermodynamic limit?

Key findings

• The spectral gap above the Fermi sea remains open for generic lattice fermions under weak, short-range interactions, even without translational invariance.
• The proof establishes a relative bound on the perturbation that ensures the gap does not close, with the bound depending on exponential decay of interaction matrix elements.
• The transformation to Majorana fermions allows the unperturbed Hamiltonian to be cast in a frustration-free form, simplifying the analysis of the ground state.
• The interaction terms are shown to decay exponentially with distance, satisfying the condition $ \sum_{X\ni x,y} \|V_X\| \leq C_V e^{-m_V \text{dist}(x,y)} $.
• The use of a cutoff $ K_h > 1 $ ensures convergence of the perturbative expansion even when many-body interactions are present.
• The final bound on the deviation of the perturbed dynamics from the unperturbed one decays sub-exponentially with distance, confirming the stability of the gap.