Continuum Landau surface states in a non-Hermitian Weyl semimetal

This paper reveals that in non-Hermitian Weyl semimetals, the non-Hermitian chiral anomaly inflow is mediated by continuum Landau modes—special eigenstates with both spatial localization and a continuous spectrum—resulting in a unique volume-scaling of surface modes that contradicts traditional area-based expectations and can be probed via metamaterials.

The Gist

Imagine a bustling city where the laws of physics are slightly "glitched." In our normal world, energy is conserved, and things behave predictably. But in this paper, the researchers are exploring a Non-Hermitian world—a place where energy can leak in or out, like a city with open doors where people can mysteriously appear or disappear.

Here is the story of what happens when you put a magnetic field into this glitchy city, explained through simple analogies.

1. The Setup: The Glitchy City (The Non-Hermitian Weyl Semimetal)

Think of the material they are studying as a 3D grid of tiny houses (atoms). In a normal city (a standard metal), if you put a magnetic field on the surface, you get "traffic lanes" called Landau levels. These are like distinct, separate bus routes. Each route can only carry one specific type of bus, and the number of routes depends on how big the surface of the city is.

But in this "glitchy" city (the Non-Hermitian Weyl Semimetal), the rules are different. The "traffic" here is weird. The researchers found that when they apply a magnetic field, the surface doesn't just get a few bus routes. Instead, it gets a Continuum of Landau Modes (CLMs).

2. The Discovery: The "Infinite Bus Stop" (Continuum Landau Modes)

In a normal city, a bus stop is either a specific, fixed location (a bound state) or a highway where cars zoom by freely (a free state). You can't have a bus stop that is both a fixed location and part of a continuous highway.

The CLM is the impossible bus stop.
Imagine a bus stop that is physically stuck to a specific spot on the ground (localized), but the buses arriving there can have any speed or color (a continuous spectrum). It's like a magical bus stop where an infinite number of different buses can park at the exact same spot without crashing into each other.

In this paper, the researchers show that on the surface of their glitchy material, these "impossible bus stops" appear when a magnetic field is turned on. They are stuck to the surface, but they form a continuous, flowing river of energy.

3. The Big Surprise: The Volume vs. Surface Area Trick

This is the most mind-bending part of the paper.

• The Old Rule: In normal physics, if you want to know how many surface states (bus stops) you have, you look at the surface area. A bigger city surface means more bus stops. It's like painting a wall: the amount of paint you need depends on the wall's size, not the room's volume.
• The New Rule: In this glitchy world, the number of these special "impossible bus stops" depends on the volume of the entire city, not just the surface.

The Analogy:
Imagine you are building a wall of bricks.

• Normal Physics: If you double the size of the wall (surface area), you get twice as many bricks.
• This Paper's Physics: If you double the size of the wall, you get four times as many bricks because the "depth" of the wall (the volume) is also pushing more bricks out to the surface.

Why? Because these "impossible bus stops" are so crowded. In a normal system, only one bus can fit at a specific spot. In this glitchy system, because of the "glitch" (non-Hermiticity), you can stack many different buses at the same spot. As the city gets deeper (larger volume), more and more of these stacked buses are forced to the surface.

4. The "One-Way Street" Effect (The Chiral Anomaly)

The paper also talks about a "chiral anomaly." Think of this as a one-way street that only allows traffic to flow in one direction.

• In a normal city, traffic flows both ways.
• In this glitchy city, the magnetic field creates a situation where traffic flows only down the stairs on the bottom floor, but never up on the top floor.
• Because the number of "impossible bus stops" depends on the volume, the amount of traffic flowing down the stairs is massive—much more than anyone expected. It's as if the depth of the building is pumping extra cars onto the one-way street.

5. How Do We See This? (The Experiment)

The researchers suggest we can't just look at this with our eyes; we need to build a model using metamaterials (artificial materials designed to bend waves).

• Imagine building a giant 3D maze of acoustic tubes (sound) or fiber optics (light).
• You send a sound wave or light beam into the maze.
• Without the magnetic field: The sound bounces around or leaks out in a predictable way.
• With the magnetic field: The sound suddenly gets "trapped" in those "impossible bus stops" on the surface. It forms a distinct, Gaussian-shaped (bell-curve) pattern that is much brighter and more intense than before.

Summary

This paper is about discovering a new type of "traffic jam" on the surface of a weird, glitchy material.

1. The Glitch: The material allows energy to leak, breaking normal rules.
2. The Phenomenon: A magnetic field creates "Continuum Landau Modes"—special states that are stuck in place but have a continuous range of energies.
3. The Shock: The number of these states doesn't grow with the surface size; it grows with the total volume of the material.
4. The Result: This creates a massive, unexpected flow of particles (current) along the surface, a phenomenon that was previously thought impossible.

It's like realizing that if you make a building taller, you don't just get more rooms; you suddenly get a super-highway of people appearing on the roof, and the number of people depends on how deep the building is, not just how wide the roof is.

Technical Summary

1. Problem Statement

The paper addresses the manifestation of non-Hermitian (NH) anomalies in three-dimensional topological phases, specifically within a Non-Hermitian Weyl Semimetal (NHWSM).

• Context: In Hermitian systems, topological surface states (like Fermi arcs) are linked to quantum anomalies (e.g., the chiral anomaly). In NH systems, recent theories suggested a generalized NH chiral anomaly affecting surface states.
• The Gap: Previous studies assumed that NH surface states would follow standard area-scaling laws (similar to Hermitian Landau levels) and possess a degeneracy structure similar to Hermitian chiral fermions. However, the specific nature of surface states in NH systems under magnetic fields, particularly their spectral properties and scaling behavior with system size, remained unexplored.
• Core Question: How do NH anomalies manifest in the surface states of a 3D NHWSM under a magnetic field, and what are the unique scaling properties of these states?

2. Methodology

The authors employed a combination of analytical field theory, tight-binding modeling, and numerical simulations.

• Model Construction:
  • They constructed a 3D tight-binding model on a cubic lattice with two sublattices ($A$ and $B$).
  • The model includes Hermitian reciprocal hoppings ($x, z$ directions), Hermitian non-reciprocal hoppings ($x$ direction, breaking time-reversal symmetry), and non-Hermitian one-way hoppings ($y$ direction).
  • A uniform magnetic field ($B$) is applied along the $+z$ direction via a vector potential $A = Bxy$, modifying the phases of the $y$-hoppings.
• Boundary Conditions:
  • Periodic Boundary Conditions (PBC) in the $y$-direction and Open Boundary Conditions (OBC) in $x$ and $z$ to isolate surface states.
  • Full OBC (all directions) to study the interplay between the NH Skin Effect (NHSE) and magnetic fields.
• Analysis Techniques:
  • Spectral Analysis: Calculated complex energy spectra to identify point gaps and surface modes.
  • Localization Metrics: Used the Participation Ratio (PR) along the $z$-direction ($z$-PR) to distinguish between bulk skin modes, extended surface states, and localized states.
  • Mode Counting: Calculated the mode imbalance ($\Delta n = n_- - n_+$) between the top ($+z$) and bottom ($-z$) surfaces to quantify the chiral magnetic effect.
  • Transmission Simulations: Modeled experimental transmission spectra using a source-probe setup to predict observable signatures in metamaterials.

3. Key Contributions & Results

A. Discovery of Continuum Landau Modes (CLMs)

The most significant finding is the emergence of Continuum Landau Modes (CLMs) on the $-z$ surface under a magnetic field.

• Nature of CLMs: Unlike standard Hermitian Landau levels (discrete, orthogonal, extended over the surface) or bound states, CLMs are spatially localized (Gaussian wavepackets) yet possess a continuous energy spectrum.
• Violation of Dichotomy: CLMs violate the standard quantum mechanical distinction between bound states (discrete spectrum) and free states (continuous spectrum).
• Localization: They are localized on the $-z$ surface with a spatial width scaling as $L \propto B^{-1/2}$. On the $+z$ surface, these modes are non-normalizable.

B. Volume-Scaling of Surface States

The paper overturns the expectation that surface state density scales with surface area.

• Hermitian Expectation: In Hermitian Weyl semimetals, the number of surface modes scales with the cross-sectional area ($L_x L_y$).
• NH Discovery: The number of CLMs induced by the NH anomaly scales linearly with the sample volume ($L_x L_y L_z$).
  • Mechanism: In the NHWSM, for a given transverse momentum $k_y$ and center position $x_0$, there is a continuum of energies (multiplicity) rather than a single discrete state. This extra multiplicity allows the number of states to scale with the system thickness ($L_z$).
  • Formula: The mode imbalance is derived as $\Delta n \sim \frac{B L_x L_y L_z}{2\pi}$.

C. Interplay with the Non-Hermitian Skin Effect (NHSE)

• Without Magnetic Field ($B=0$): Under full OBC, the NHSE causes all eigenstates to collapse into skin modes localized on the $\pm y$ boundaries, destroying the Fermi arc surface states.
• With Magnetic Field ($B \neq 0$): The magnetic field suppresses the NHSE dominance and causes the re-emergence of CLMs on the $-z$ surface. The system transitions from skin modes to a mix of skin modes and CLMs.

D. Experimental Probing via Metamaterials

The authors propose experimental signatures using classical wave platforms (photonic, acoustic, or electrical circuits):

• Transmittance: A strong transmission peak appears inside the point gap when $B \neq 0$, corresponding to CLM excitation.
• Spatial Profile: The field intensity on the surface transitions from an exponential decay (characteristic of ballistic Fermi arcs at $B=0$) to a Gaussian profile (characteristic of CLMs at $B \neq 0$).
• Chiral Magnetic Effect: The difference in transmission between the top and bottom halves of the sample increases linearly with both the magnetic field $B$ and the sample thickness $L_z$.

4. Significance

• Theoretical Breakthrough: This work identifies a unique class of eigenstates (CLMs) that bridge the gap between bound and free states, fundamentally altering the understanding of surface physics in non-Hermitian systems.
• Redefining Anomaly Inflow: It demonstrates that the NH chiral anomaly inflow is mediated by CLMs and results in a volume-scaling of surface states, a phenomenon not predicted by earlier field theories which assumed area-scaling.
• Experimental Relevance: The findings provide concrete, measurable signatures (Gaussian spatial profiles, volume-dependent transmission) for detecting NH anomalies in existing metamaterial platforms, such as fiber loops, gyromagnetic crystals, and topolectrical circuits.
• Generalizability: The mechanism suggests that unusual multiplicity in the spectrum can lead to macroscopic scaling effects in topological systems, opening new avenues for designing topological devices with enhanced sensitivity to system size.