Non-Hermitian Delocalization Realizes Random Dirac Criticality in One Dimension

This paper demonstrates that non-Hermitian delocalized states in one dimension under periodic boundary conditions intrinsically realize the universality class of random Dirac criticality, emerging generically from spectral topology rather than fine-tuning.

The Gist

Imagine a crowded hallway where people (representing electrons) are trying to walk from one end to the other. In a normal, "Hermitian" world (the physics of our everyday reality), if you throw enough random obstacles (disorder) into the hallway, the people will get stuck. They will huddle in one spot and refuse to move. This is called Anderson localization. In a one-dimensional hallway, it is almost impossible for them to ever break free and walk freely again.

Now, imagine a "Non-Hermitian" world. This is a slightly magical version of physics where the rules are different. In this world, the hallway has a strange property: it pushes people in one direction more than the other. This is called the Non-Hermitian Skin Effect. Instead of getting stuck in a pile, the people are swept along the walls, accumulating at the edges. They seem to "delocalize" (spread out) and move freely, even with all the obstacles.

The Big Discovery
The authors of this paper asked a simple question: If these people are moving freely in this magical hallway, are they just wandering aimlessly, or is there a hidden order to their movement?

Their answer is surprising: They are not just moving freely; they are in a state of "Criticality."

Think of "Criticality" like a tightrope walker. They aren't stuck on the ground (localized), and they aren't flying freely in the sky (fully extended). They are balancing perfectly on the edge. In normal physics, finding a tightrope walker requires perfect, fine-tuned conditions (like adjusting the wind speed to the exact millimeter). But in this Non-Hermitian world, the tightrope walker appears automatically and generically. You don't need to tune anything; the very shape of the energy landscape forces them to be in this critical state.

The "Loop" and the "Map"
To prove this, the authors used a clever trick called "Hermitization." Imagine you have a complex, twisted map of the magical hallway (the Non-Hermitian system). They unfolded this map into a flat, standard map (a Hermitian system).

They found that the "loop" of energy states in the magical hallway corresponds exactly to the Topological Anderson Transition on the flat map.

• The Analogy: Imagine the magical hallway is a rollercoaster loop. The people riding the loop are the "delocalized states." The authors showed that riding this loop is mathematically identical to standing exactly on the edge of a cliff on a normal map. You are neither falling off (localized) nor standing safely on solid ground (extended); you are in a precarious, critical balance.

The "Fingerprint" of Criticality
How do we know they are in this critical state? The authors looked at how the people (waves) correlate with each other over distance.

• Normal Localized People: Their connection drops off exponentially fast. If you are 10 steps away, you have almost no connection to someone 1 step away. It's like a light turning off instantly.
• Normal Extended People: Their connection is uniform everywhere.
• The Critical People (This Paper): Their connection drops off in a very specific, slow way: like $1/x^{1.5}$ (a power law).

The authors calculated this mathematically and confirmed it with computer simulations. They found that the "fingerprint" of these Non-Hermitian states matches the famous "Random Dirac Fermion" model. This is a specific type of critical behavior usually only seen in very special, fine-tuned 2D systems, but here, it appears naturally in 1D Non-Hermitian systems.

Why This Matters (According to the Paper)
The paper claims that Non-Hermitian systems provide a universal mechanism to create these critical states.

• In the old world (Hermitian), you had to build a machine with perfect precision to get this critical behavior.
• In this new world (Non-Hermitian), the "skin effect" (the tendency to pile up at the edges) automatically creates a spectral loop. This loop is the critical state.

Summary in a Nutshell
The paper reveals that in one-dimensional systems with non-reciprocal (one-way) interactions and disorder, the "free-moving" states are not truly free. Instead, they are intrinsically critical. They behave like a specific type of quantum tightrope walker (Random Dirac criticality) that emerges naturally due to the topology of the system's energy spectrum, without needing any fine-tuning. This unifies the understanding of why these states delocalize and reveals their hidden, critical nature.

Technical Summary

Technical Summary: Non-Hermitian Delocalization Realizes Random Dirac Criticality in One Dimension

Problem Statement
In one-dimensional (1D) Hermitian systems, disorder typically induces Anderson localization, preventing the existence of extended states. While critical states (delocalized but not fully extended) exist in 1D Hermitian systems, they are generally restricted to finely tuned transition points, such as topological Anderson transitions (TAT). Conversely, non-Hermitian systems, particularly those exhibiting the non-Hermitian skin effect (NHSE), can evade Anderson localization and support delocalized states even in 1D. However, the fundamental nature of these non-Hermitian delocalized states—specifically whether they represent a generic critical phase or a distinct universality class—remains an open question. This work addresses the characterization of non-Hermitian delocalized states under periodic boundary conditions (PBC) and their relationship to random Dirac criticality.

Methodology
The authors employ a combination of analytical field theory and numerical simulations to investigate the Hatano–Nelson (HN) model with disorder. The core methodological innovation is the application of the Hermitization technique to link non-Hermitian spectral topology to Hermitian topological phases.

1. Hermitization Mapping: The non-Hermitian Hamiltonian $H_{HN}$ is mapped to a Hermitian doubled Hamiltonian $\tilde{H}$ defined as:
   $$\tilde{H} = \begin{pmatrix} 0 & E - H_{HN} \\ E^* - H_{HN}^\dagger & 0 \end{pmatrix}$$
   This mapping establishes a correspondence between the spectral winding of $H_{HN}$ in the complex energy plane and the topological invariant (winding number) of $\tilde{H}$ in the chiral AIII class.
2. Long-Wavelength Expansion: The authors perform a long-wavelength expansion of the Hermitized Hamiltonian $\tilde{H}$ around the gapless point. This yields an effective Dirac Hamiltonian with a random mass term:
   $$\tilde{H}_{eff} = (\eta_x \sigma_x + \eta_y \sigma_y)(-i\partial_x) + m(x)\sigma_x$$
   where $m(x)$ represents the disorder potential.
3. Liouville Field Theory: Using the Kogan–Mudry–Tsvelik approach, the statistical properties of the wavefunctions are analyzed by mapping the problem to a Liouville-type field theory. This allows for the analytical derivation of wavefunction correlation functions.
4. Numerical Verification: Large-scale numerical simulations (up to $L=3000$ and $5 \times 10^5$ disorder realizations) are conducted on the lattice HN model to verify the scaling of participation ratios (PR) and wavefunction correlations.

Key Contributions and Results

• Generic Criticality of NHSE States: The paper demonstrates that delocalized states under PBC in non-Hermitian disordered systems are not merely extended but are intrinsically critical. They belong to the universality class of 1D random Dirac fermions.
• Spectral Winding and TAT Correspondence: A direct link is established between the spectral winding number of the non-Hermitian system and the topological Anderson transition of the Hermitized system. The "loop states" (delocalized modes forming the loop in the complex spectrum) correspond precisely to the critical states at the TAT of $\tilde{H}$.
• Universal Algebraic Correlations: The central result is the derivation of the universal spatial correlation for these critical states. The disorder-averaged correlation function of the wavefunction moments scales algebraically:
  $$\langle |\psi(x)\psi(0)|^q \rangle_{dis} \sim x^{-3/2} \quad (x \gg 1)$$
  This $x^{-3/2}$ decay is a hallmark of 1D Dirac criticality and contrasts sharply with the exponential decay of localized states or the multifractal scaling seen in 2D random Dirac systems.
• Participation Ratio Behavior: The study resolves an apparent contradiction regarding the participation ratio (PR). While the localization length diverges (suggesting delocalization), the PR remains size-independent ($PR \sim 1$) for large systems. This is explained by the $x^{-3/2}$ correlation scaling, which implies that the wavefunction intensity is not uniformly distributed but exhibits critical fluctuations that prevent the PR from growing with system size as in extended states.
• Robustness and Universality: The findings are shown to be universal across different non-Hermitian scenarios, including:
  • Hatano–Nelson models with complex-valued disorder.
  • Two-band models with local gain and loss.
  • Random hopping models where non-reciprocity arises purely from bond disorder (including the "erratic skin effect").

Significance
The paper claims to provide a unified physical picture of non-Hermitian delocalization in one dimension. Its primary significance lies in identifying a mechanism by which non-Hermiticity promotes criticality generically, without the need for fine-tuning required in Hermitian systems.

• Theoretical Unification: By linking spectral winding to topological Anderson transitions via Hermitization, the work unifies the understanding of non-Hermitian skin effects and critical phenomena. It identifies the loop states in the complex spectrum as Dirac critical states.
• New Route to Criticality: The results establish non-Hermitian systems as a natural platform for realizing the criticality of Dirac fermions in 1D. Unlike Hermitian systems where such criticality is restricted to specific transition points, non-Hermitian spectral topology ensures that these critical states emerge generically along spectral loops.
• Resolution of Localization Debate: The work clarifies the nature of "delocalized" states in 1D non-Hermitian systems, redefining them as critical states with universal algebraic correlations rather than conventional extended states.

The authors conclude that these findings offer a distinct class of Dirac critical states accessible through non-Hermitian topology, providing a robust theoretical framework for understanding disordered non-Hermitian systems.