🔬 mesoscale physics

Spectral-topology-induced criticality in non-Hermitian fermionic metals

This paper establishes a unified framework for non-Hermitian quantum criticality by introducing a symmetry-protected dynamical topological index derived from Morse theory, which links spectral topology to emergent imaginary Fermi surfaces and universal many-body phenomena like logarithmic entanglement scaling in non-equilibrium steady states.

Original authors: Ayan Banerjee, Julius T. Gohsrich, Flore K. Kunst

Published 2026-07-07

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CC BY 4.0

Original authors: Ayan Banerjee, Julius T. Gohsrich, Flore K. Kunst

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the rules of quantum physics are slightly "leaky." In our everyday world, things usually stay put or fade away slowly. But in the strange realm of non-Hermitian systems (which the authors study), energy can be gained or lost, like a sound that gets louder or quieter as it travels. These systems are "out of equilibrium," meaning they aren't resting peacefully; they are constantly being pushed and pulled by their environment.

The authors of this paper ask a big question: When these systems are in this chaotic, "leaky" state, how do we find the "critical points" where the material changes its fundamental nature?

Here is the story of their discovery, broken down into simple concepts:

1. The Map of a Leaky World (The Complex Spectrum)

In normal quantum physics, energy levels are like rungs on a ladder—they are real, solid numbers. But in this "leaky" world, energy levels are like coordinates on a map with both a real part (left-right) and an imaginary part (up-down).

The Analogy: Imagine a rollercoaster track that exists not just in a 2D plane, but in a 3D space where the "height" represents how fast the ride is speeding up or slowing down (gaining or losing energy).
The authors realized that the shape of this track (the "spectrum") holds the secret to the system's behavior.

2. The "Topological Index" (Counting the Hills and Valleys)

The team invented a new way to count things, which they call a Dynamical Topological Index.

The Analogy: Imagine you are hiking on a circular mountain trail (the "Brillouin Zone"). You look at the "imaginary height" of the trail. Sometimes the trail goes up (gaining energy), and sometimes it goes down (losing energy).
The authors found that if you count the hills (peaks) that are above the zero line and subtract the valleys (troughs) that are above the zero line, you get a special number.
This number is robust. Just like you can't turn a mountain into a valley without digging a massive hole (a "phase transition"), you can't change this number just by slightly bending the track. It is a topological fingerprint.

3. The "Imaginary Fermi Surface" (The New Ocean Level)

In normal metals, electrons fill up energy levels like water filling a glass until it hits a "Fermi surface" (the water line). In this leaky world, the authors discovered a new kind of water line: the Imaginary Fermi Surface.

The Analogy: Think of the system as a giant ocean. Some waves are growing (gaining energy), and some are dying out (losing energy). The "Imaginary Fermi Surface" is the exact horizon line where the waves stop growing and start dying.
The system naturally settles into a state where it only keeps the waves that are growing. The "Fermi points" are the specific locations on the horizon where the growing waves touch the dying waves.

4. The Connection: Hills Create Waves

The paper's biggest "Aha!" moment is linking the hills on the map (the topological index) to the waves on the ocean (the Fermi points).

The Claim: Every time your topological index goes up by 1, it forces the creation of two new "Fermi points" on the horizon.
Why it matters: These points are special. They are "gapless," meaning they allow energy to flow freely without resistance. They act like open doors in a wall.

5. The Result: A New Kind of Metal

Because of these open doors, the material behaves like a metal even though it's in a chaotic, non-equilibrium state.

Entanglement (The Invisible String): In quantum physics, particles can be "entangled," meaning they are linked across distances. The authors found that the amount of this "invisible string" (entanglement) grows logarithmically. The topological index they invented directly tells you exactly how strong this connection is.
Currents (The Flow): Because the system is asymmetric (some waves grow, some die), it creates a persistent flow of particles, like a river that never stops, even without a pump.
Correlations (The Ripples): If you drop a stone in this "ocean," the ripples don't just fade away; they travel in a specific pattern determined by the number of Fermi points. The paper shows that changing the shape of the track (the spectrum) changes the pattern of these ripples.

Summary

The authors have built a bridge between mathematical shapes (topology) and physical behavior (criticality) in a chaotic world.

The Input: A complex, leaky energy map.
The Tool: A new counting method (Topological Index) that counts the imbalance of hills and valleys.
The Output: This count predicts exactly how many "open doors" (Fermi points) exist, which in turn dictates how much the particles are linked (entanglement), how much current flows, and how ripples move through the system.

They have shown that even in a world that is constantly changing and losing energy, there is a hidden, unchangeable geometric order that controls how the material behaves. This order is the key to understanding "quantum criticality" in these new, non-equilibrium systems.

Technical Summary: Spectral-topology-induced criticality in non-Hermitian fermionic metals

Problem Statement
The paper addresses a fundamental challenge in non-Hermitian physics: defining and understanding quantum criticality in systems that are intrinsically out-of-equilibrium. Unlike Hermitian systems where criticality is governed by equilibrium ground states and real spectra, non-Hermitian systems (found in photonic, electric circuit, and open many-body platforms) exhibit complex spectra, non-unitary dynamics, and exceptional points. A central difficulty lies in determining how the complex single-particle spectrum governs universal many-body behaviors, such as entanglement scaling and transport, when the system is driven by gain and loss. The authors seek a topological principle that captures this criticality without relying on equilibrium assumptions.

Methodology
The authors develop a framework rooted in algebraic topology, specifically Morse theory, to analyze the complex single-particle spectrum of non-Hermitian fermionic systems.

Dynamical Topological Index ( $\nu_{dyn}$ ): The core methodological innovation is the definition of a symmetry-protected dynamical topological index derived directly from the imaginary part of the complex energy spectrum, $f(k) = \text{Im } E(k)$ .
- For a time-reversal symmetric, single-band model in one dimension, $f(k)$ is an odd, periodic Morse function.
- The index is defined as the net imbalance of extrema of $f(k)$ above zero:
  $\nu_{dyn} = - \sum_{k_c: f'(k_c)=0, f(k_c)>0} \text{sign}[f''(k)|_{k=k_c}]$
- This index counts the difference between the number of maxima and minima in the upper half of the complex energy plane. The authors prove that due to the periodicity and symmetry of the spectrum, extrema in the upper and lower half-planes are paired, and the index is a robust topological invariant that changes only when the spectrum crosses zero via non-simple zeros or exceptional points.
Non-Equilibrium Steady State (NESS) Construction: The authors construct the NESS for a non-interacting fermionic system by considering the long-time limit of non-unitary time evolution. In this regime, the system asymptotically relaxes into a state populated by single-particle modes with the largest (positive) imaginary energies. This selection process dynamically generates an "imaginary Fermi surface" defined by $\text{Im } E(k) = 0$ .
Many-Body Observables: The framework connects the single-particle spectral topology to many-body observables by analyzing the NESS. The authors compute:
- Entanglement Entropy: Using Peschel's prescription on the correlation matrix of the NESS.
- Transport: Calculating persistent currents based on the occupation of chiral modes.
- Correlations: Analyzing the decay and oscillation of two-point correlation functions.

Key Contributions and Results

Topological Origin of Criticality: The paper establishes that non-Hermitian quantum criticality in non-interacting systems is controlled by the dynamical topological index $\nu_{dyn}$ . Critical points in the spectrum are identified as topological defects whose stability is protected under continuous deformations, linking spectral geometry directly to many-body physics.
Emergent Imaginary Fermi Surface: The authors demonstrate that the NESS is characterized by an emergent imaginary Fermi surface. The Fermi points ( $k_F$ ) are the momenta where the highest-occupied modes touch the boundary $\text{Im } E(k) = 0$ .
Quantitative Relations:
- Fermi Points: The number of Fermi points $n_F$ is strictly determined by the dynamical topological index: $n_F = 2\nu_{dyn}$ . This relation arises because each unit of $\nu_{dyn}$ forces the complex dispersion to cross the zero axis twice, creating a pair of symmetry-related Fermi points.
- Effective Central Charge: The logarithmic scaling of entanglement entropy in the NESS, $S_\ell \sim \frac{c_{eff}}{3} \log \ell$ , is governed by an effective central charge $c_{eff}$ that equals the dynamical topological index ( $c_{eff} = \nu_{dyn}$ ).
- Transport and Correlations: The system exhibits persistent currents and algebraic (power-law) correlations ( $\propto 1/d$ ) characteristic of a gapless critical phase. The number of Fermi points dictates the interference patterns in correlations.
Violation of Nielsen-Ninomiya Theorem: While the Nielsen-Ninomiya theorem typically forbids net chirality in lattice fermions, the authors show that non-unitary dynamics dynamically evade this constraint. The system selectively amplifies chiral modes, resulting in a symmetry-protected fermionic chiral metal with a non-zero net chirality stabilized by the spectral topology rather than band topology alone.
Phase Transitions: Topological phase transitions occur when the spectral topology changes (e.g., an extremum crosses the zero axis or an exceptional point is encountered). These transitions manifest as non-analyticities in the persistent current and changes in the entanglement scaling and correlation patterns.

Significance and Claims
The paper claims to establish a unified framework for non-Hermitian quantum matter that connects spectral topology to Morse theory. By identifying critical points with topological defects, the work reveals a topological foundation for non-equilibrium quantum criticality.

The authors assert that their approach provides a natural language for connecting spectral geometry to emergent scale invariance, entanglement, and transport in non-Hermitian fermionic systems. They posit that the dynamical topological index serves as a robust, topologically protected measure of chiral asymmetry and entanglement scaling. The work suggests that non-Hermitian criticality is not merely a breakdown of Hermitian physics but a distinct phenomenon governed by the interplay of gain-and-loss-selected steady states and spectral winding.

The paper concludes by noting that while formulated for one-dimensional systems, the framework offers a pathway to explore higher-dimensional extensions, interactions, and experimental realizations in cold-atom and photonic platforms where non-Hermitian control is achievable. The authors frame their contribution as a step toward bridging modern mathematical structures (Morse theory) with physically realizable non-Hermitian dynamics.