[Paper Review] Inner Skin Effects on Non-Hermitian Topological Fractals

Sourav Manna, Bitan Roy|arXiv (Cornell University)|Feb 15, 2022

Topological Materials and Phenomena4 citations

TL;DR

This paper proposes the inner skin effect (ISE) in non-Hermitian (NH) topological fractals, demonstrating that periodic boundary conditions induce macroscopic state localization at inner boundaries of Sierpinski carpet lattices—previously inaccessible under open boundaries. The ISE manifests as first-order (edge) and second-order (corner) localization for both charged and neutral Majorana fermions, with intrinsic ISE tied to nontrivial NH topological invariants, offering a pathway for experimental observation in photonic, phononic, and topolectric circuits.

ABSTRACT

Non-Hermitian (NH) crystals, quasicrystals and amorphous network display an accumulation of a macroscopic number of states near one of its specific interfaces with vacuum, such as edge, surface, hinge or corner. This phenomenon is known as the NH skin effect, which can only be observed with open boundary condition. In this regard self-similar fractals, manifesting inner boundaries in the interior of the system, harbor a novel phenomenon, the \emph{inner skin effect} (ISE). Then the NH skin effect appears at the inner boundaries of the fractal lattice with periodic boundary condition. We showcase this observation by implementing prominent models for NH insulators and superconductors on representative planar Sierpinski carpet fractal lattices. They accommodate both first-order and second-order ISEs at inner edges and corners, respectively, for charged as well as neutral Majorana fermions. Furthermore, over extended parameter regimes ISEs are tied with nontrivial bulk topological invariants, yielding intrinsic ISEs. With the recent success in engineering NH topological phases on highly tunable metamaterial platforms, such as photonic and phononic lattices, as well as topolectric circuits, the proposed ISEs can be observed experimentally at least on fractal metamaterials with periodic boundary condition.

Motivation & Objective

• To investigate non-Hermitian (NH) skin effects in fractal lattices, which possess intrinsic inner boundaries due to self-similarity.
• To demonstrate that the NH skin effect can emerge at inner edges and corners of fractals under periodic boundary conditions (PBC), a phenomenon absent in conventional crystals.
• To establish connections between the inner skin effect (ISE) and nontrivial bulk topological invariants, such as the NH Bott index and NH quadrupole moment.
• To explore the feasibility of experimentally observing ISE in tunable metamaterial platforms like photonic, phononic, and topolectric circuits.

Proposed method

• Numerical diagonalization of NH Hamiltonians on Sierpinski carpet fractal lattices with both open and periodic boundary conditions.
• Construction of real-space Hamiltonians from momentum-space counterparts by preserving discrete symmetries (e.g., reflection, rotation) and replacing Bloch terms with equivalent real-space hopping terms.
• Incorporation of exponentially decaying finite-range hoppings to ensure full connectivity across the fractal lattice while preserving topological features.
• Computation of NH topological invariants: NH Bott index $B_{\rm NH}$ for first-order ISE and NH quadrupole moment $Q^{\rm NH}_{xy}$ for second-order ISE on half-filled ground states.
• Analysis of eigenvector localization via left/right eigenvector weight distributions to identify skin effect at outer vs. inner boundaries.
• Comparison of NH skin effects on Sierpinski carpet (fractal), square lattice (crystalline), Ammann-Beenker quasicrystal, and amorphous networks to isolate fractal-specific phenomena.

Experimental results

Research questions

• RQ1 Can non-Hermitian skin effects emerge at inner boundaries of fractal lattices under periodic boundary conditions, where they are absent in conventional crystals?
• RQ2 How do first-order and second-order inner skin effects manifest in Sierpinski carpet fractals for both charged and neutral Majorana fermions?
• RQ3 To what extent are the inner skin effects correlated with nontrivial bulk topological invariants such as the NH Bott index and NH quadrupole moment?
• RQ4 What is the role of fractal geometry in enabling novel topological phenomena inaccessible in periodic or quasicrystalline systems?

Key findings

• First-order inner skin effect is observed at inner edges of Sierpinski carpet fractals under PBC, with a macroscopic number of eigenvectors localized at inner boundaries, even when the outer system shows no skin effect under PBC.
• Second-order inner skin effect emerges at inner corners under PBC in both directions, with eigenvector weight concentrated at the innermost corner, distinct from outer corner localization under OBC.
• For small non-Hermitian coupling, the first-order ISE is intrinsically tied to a nontrivial NH Bott index $B_{\rm NH} = -1$, confirming a topological origin of the effect.
• The ISE persists over extended parameter regimes and is robust against variations in non-Hermitian coupling strength, indicating a stable topological phase.
• The inner skin effect is absent in square lattices, quasicrystals (e.g., Ammann-Beenker), and amorphous networks under PBC, highlighting its unique dependence on fractal geometry.
• The results suggest experimental feasibility in existing NH metamaterial platforms such as photonic, phononic, and topolectric circuits, where periodic boundary conditions can be engineered.