[Paper Review] Green's function Zero and Symmetric Mass Generation

Yichen Xu, Cenke Xu|arXiv (Cornell University)|Mar 29, 2021

Topological Materials and Phenomena4 citations

TL;DR

This paper demonstrates that after symmetric mass generation (SMG) trivializes topological superconductors and insulators, a remnant of their nontrivial topology persists in the form of a zero in the fermion Green's function at zero frequency. Using a real-space decorated defect construction and mapping the Green's function to a single-particle path integral, the authors prove the existence of this Green's function zero in arbitrary dimensions without requiring translation symmetry, even in systems with avoided topological transitions.

ABSTRACT

It is known that, under short-range interactions many topological superconductors (TSC) and topological insulators (TI) are trivialized, which means the boundary state of the system can be trivially gapped out by interaction without leading to symmetry breaking or topological ground state degeneracy. This phenomenon is also referred to as "symmetric mass generation" (SMG), and has attracted broad attentions from both the condensed matter and high energy physics communities. However, after the trivialization caused by interaction, some trace of the nontrivial topology of the system still persists. Previous studies have indicated that interacting topological TSC and TI could be related to the "zero" of Green's function, namely the fermion Green's function $G(\mathrm{i} ω ightarrow 0) = 0$. In this work, through the general "decorated defect" construction of symmetry protected topological (SPT) states, we demonstrate the existence of Green's function zero after SMG, by mapping the evaluation of the Green's function to the problem of a single particle path integral. This method can be extended to the cases without spatial translation symmetry, where the momentum space which hosts many quantized topological numbers is no longer meaningful. Using the same method one can demonstrate the existence of the Green's function zero at the "avoided topological transition" in the bulk of the system.

Motivation & Objective

• To identify a surviving signature of nontrivial topology in topological superconductors and insulators after symmetric mass generation (SMG), which fully gaps the spectrum without symmetry breaking.
• To establish that the zero of the fermion Green's function at zero frequency (G(iω→0)=0) persists after SMG, even when momentum space topology is no longer defined.
• To develop a method that works in arbitrary dimensions and without spatial translation symmetry, overcoming limitations of momentum-space-based topological invariants.
• To demonstrate that the Green's function zero is robust not only at boundaries but also in the bulk at avoided topological transitions.
• To provide a real-space, path-integral formulation of the Green's function that captures the topological trace of SMG in strongly interacting systems.

Proposed method

• The authors use the general 'decorated defect' construction of symmetry-protected topological (SPT) states to model the SMG process in real space.
• They map the computation of the fermion Green's function to a single-particle path integral, enabling evaluation without relying on momentum space or translation invariance.
• The Green's function is expanded in powers of a small parameter Δ, and the leading-order term is computed via functional integrals involving a modified propagator GΔ(λ).
• The path integral includes a weight function ρ~A(δτ,ϕ) that encodes the defect structure and couples to the fermion degrees of freedom.
• The method evaluates the Green's function at imaginary frequency ω=0 by analyzing the large-β behavior of the path integral, showing exponential decay that implies a zero at ω=0.
• Numerical integration confirms that the first-order correction in Δ decays as β^{3/2} at large β, supporting the existence of a zero in the Fourier transform at ω=0.

Experimental results

Research questions

• RQ1 Does a nontrivial topological phase leave a detectable signature in the Green's function after symmetric mass generation (SMG) has fully gapped the system?
• RQ2 Can the Green's function zero be established in systems without translation symmetry, where momentum-space invariants are ill-defined?
• RQ3 Is the Green's function zero robust not only at boundaries but also in the bulk at avoided topological transitions?
• RQ4 Can the SMG mechanism be understood through a real-space path integral formulation of the fermion propagator?
• RQ5 What is the functional form of the Green's function correction in the SMG phase, and does it lead to a zero at zero frequency?

Key findings

• The fermion Green's function exhibits a zero at zero frequency (G(iω→0)=0) after symmetric mass generation, serving as a surviving topological invariant in the fully gapped phase.
• The Green's function zero is robustly demonstrated in arbitrary spatial dimensions using a real-space decorated defect construction that does not require translation symmetry.
• Numerical evaluation confirms that the leading-order correction in the Δ expansion decays as β^{3/2} at large imaginary time β, implying a zero in the Fourier transform at ω=0.
• The path integral formulation successfully captures the topological trace of SMG in both boundary and bulk settings, including at avoided topological transitions.
• The method generalizes beyond momentum-space invariants, providing a framework to detect SMG in systems with broken or absent translational invariance.
• The exponential decay of the Green's function at large β, driven by the path integral structure, ensures the existence of a zero at ω=0, even when the system is fully gapped.