Complex Wannier centers and drifting Wannier functions in non-Hermitian Hamiltonians

This paper introduces complex Wannier centers derived from nonunitary Wilson loops in non-Hermitian systems to explain directional drift in Wannier functions, establishes symmetry-protected bulk-boundary correspondences linking these centers to filling anomalies and edge mode gain/loss, and proposes a photonic waveguide implementation for experimental verification.

The Gist

The Big Picture: A World Where Energy Leaks

Imagine you are studying a crystal, like a perfect grid of atoms. In the "normal" world of physics (called Hermitian), energy is perfectly conserved. If you drop a ball, it bounces, but it never gains or loses energy on its own. In this world, the "center" of an electron's wave packet (a Wannier function) sits still or moves predictably, like a car on a highway with a steady speed.

But this paper explores a different world: Non-Hermitian physics. Think of this as a world where the crystal is "leaky." Some parts of the crystal absorb energy (loss), while others pump energy in (gain). It's like a city where some buildings are on fire (losing energy) and others have giant generators (gaining energy).

In this leaky world, the usual rules of physics break down. The authors discovered something strange: the "centers" of these electron waves don't just sit at a specific point; they can exist in a complex plane. This sounds like math jargon, but here is the simple version:

The Main Discovery: The "Ghost" Center

In normal physics, an electron's center is at a real location, say, "3 meters to the right."
In this new physics, the center can be at "3 meters to the right plus a ghostly imaginary shift."

The authors call this a Complex Wannier Center.

• The Real part is the normal position (where the electron is).
• The Imaginary part is a hidden "momentum kick."

The Analogy: The Leaky Boat
Imagine a boat (the electron) floating on a river.

• Normal Physics: The boat is perfectly balanced. If you push it, it moves, but if you let go, it stays put.
• This Paper's Physics: The boat has a hole in the bottom (energy loss) on the left side and a water jet (energy gain) on the right side.
  • Even if the boat looks like it's sitting in the middle of the river, the imbalance of water leaking out one side and spraying in the other creates a current.
  • The "Imaginary part" of the center is like a hidden compass needle pointing in the direction of this current.
  • Result: The boat doesn't just sit there; it drifts continuously in one direction, even without an engine. This is called Drifting Wannier Functions.

How They Found It: The "Magic Loop"

To find these drifting centers, the scientists used a tool called a Wilson Loop.

• The Analogy: Imagine walking around a giant circular track (the crystal). In a normal world, if you walk a full circle, you end up exactly where you started with the same energy.
• The Twist: In this leaky world, if you walk the full circle, you might end up with more energy (amplified) or less energy (attenuated) than when you started.
• The authors realized that this "gain or loss" accumulated over the loop is the Imaginary Part of the center. It tells you exactly how fast and in which direction the electron wave will drift.

The Rules of the Game: Symmetry and "Signatures"

The paper also asks: "Can we control this drift? Can we stop it?"
They found that Symmetries (rules about how the crystal looks when flipped or rotated) act like traffic cops.

1. The "Krein Signature" (The ID Card):
   Imagine every electron wave has an ID card with a sign: +1 or -1. This is called a Krein Signature.

   • If two waves have the same sign, they are "friendly." They can't swap places or drift apart easily. They stay stuck in a real position.
   • If two waves have opposite signs, they are "rivals." If they meet, they can collide and suddenly split into a pair: one drifting left, one drifting right. This is a Krein Collision.

2. The "Bulk-Boundary Correspondence" (The Edge Effect):
   This is the most exciting part. The authors found that by looking at the "drift" inside the material (the bulk), you can predict what happens at the edge.

   • The Analogy: Imagine a crowd of people in a stadium (the bulk). If the crowd is shifting to the left (drifting), it means people will pile up at the left wall (the edge).
   • The paper shows that if the "Complex Centers" inside the crystal have a specific pattern, it guarantees that edge states (waves stuck to the surface) will exist.
   • Furthermore, the imaginary part tells you if these edge waves will grow (amplify) or shrink (die out). This is huge for designing lasers or sensors that need to be super sensitive.

The Real-World Test: A Photonic Ladder

Finally, the authors didn't just do math; they proposed a way to build this in a lab.

• The Setup: Imagine a ladder made of optical fibers (light guides).
• The Trick: They suggest putting tiny "sinks" (to absorb light) on some rungs and "sources" (to amplify light) on others.
• The Result: If you shine a laser into this ladder, the light shouldn't just bounce back and forth. It should flow in one direction, like water down a slide, because of the "Complex Centers" they designed.

Summary for the Everyday Reader

1. The Problem: In systems with gain and loss (like lasers or open biological systems), electrons don't behave normally. They drift.
2. The Discovery: The authors found a mathematical way to describe this drift using "Complex Centers." The "Imaginary" part of the center is actually the speed and direction of the drift.
3. The Control: By arranging the crystal's symmetry, you can force these drifts to happen or stop them.
4. The Application: This allows scientists to design materials where light or electricity flows in only one direction without resistance, or where edge waves can be made to amplify (brighten) or dampen (dim) on command.

In a nutshell: They found the "hidden compass" inside leaky crystals that tells you exactly which way the energy will flow, and they figured out how to build a machine that uses this to create one-way traffic for light.

Technical Summary

1. Problem Statement

The extension of topological band theory to non-Hermitian (NH) systems has largely focused on Hamiltonians with point gaps, which exhibit phenomena like the non-Hermitian skin effect (NHSE) and exceptional points. However, line-gapped NH Hamiltonians (where the spectrum has a gap along a line in the complex plane, often allowing for real spectra) remain less explored.

In Hermitian systems, the Wilson loop (the holonomy of the Berry connection) is unitary, and its eigenvalues encode the Wannier centers (WCs), which determine polarization and bulk-boundary correspondence. In NH systems, the biorthogonal nature of the eigenbasis introduces a gauge freedom that renders Wilson loops non-unitary. The physical consequences of this non-unitarity, particularly regarding the geometric interpretation of Wannier functions and their dynamics, have been unclear. The paper addresses:

• What is the physical meaning of non-unitary Wilson loops?
• How do symmetries constrain the spectrum of these loops?
• Can a bulk-boundary correspondence (BBC) be established for line-gapped NH systems using these concepts?

2. Methodology

The authors employ the framework of biorthogonal quantum mechanics, utilizing right ($|\psi^R\rangle$) and left ($|\psi^L\rangle$) eigenvectors to define a biorthonormal basis.

• Complex Wannier Centers: They define the Wilson loop eigenvalues $\lambda$ as $\lambda = e^{-2\pi i z}$, where $z = \nu + i\kappa$ is a complex Wannier center.
  • $\nu$ (real part): Represents the spatial position within the unit cell (standard polarization).
  • $\kappa$ (imaginary part): Quantifies the deviation from unitarity and encodes accumulated amplification/attenuation over the Brillouin zone (BZ).
• Projected Position Operator: The authors distinguish between two definitions of Wannier functions:
  1. Bloch-Fourier: Equal-weight superposition of Bloch states (inequivalent in NH systems).
  2. Projected-Position: Eigenstates of the projected position operator $X_P = PXP$. They argue this is the physically relevant definition for NH systems.
• Symmetry Analysis: They analyze how pseudo-Hermiticity (pH) and crystalline symmetries (specifically Inversion and Pseudo-Inversion) constrain the Wilson loop spectrum. They utilize Krein signatures (indefinite inner products defined by the pH metric) to classify stability.
• Symmetry Ramification: They demonstrate that Hermitian symmetries "ramify" into distinct non-Hermitian branches:
  • Similarity-type: $\mathcal{H}(k) = S \mathcal{H}(-k) S^{-1}$ (acts within right/left spaces).
  • Conjugation-type: $\mathcal{H}^\dagger(k) = S \mathcal{H}(-k) S^{-1}$ (relates right to left spaces).
• Dynamical Simulations: They perform time-evolution simulations of Wannier functions to verify theoretical predictions regarding drift.

3. Key Contributions

A. Complex Wannier Centers and Drifting Dynamics

The central finding is that the imaginary part $\kappa$ of a complex Wannier center corresponds to a linear phase gradient in the Wannier function's envelope.

• Mechanism: A complex WC implies a non-uniform momentum-weight distribution $\|w_k\|^2$ in the projected-position Wannier function.
• Result: This asymmetry creates an effective momentum $k_{\text{eff}} \propto \kappa$, causing the Wannier wave packet to drift directionally over time.
• Significance: This establishes a direct link between the geometric property of the Wilson loop (non-unitarity) and real-space dynamical transport (non-reciprocity).

B. Pseudo-Hermiticity and Krein Signatures

For pseudo-Hermitian Hamiltonians ($H^\dagger = \eta H \eta^{-1}$), the Wilson loop becomes pseudo-unitary.

• Spectral Constraints: The eigenvalues of the Wilson loop are either unimodular ($|\lambda|=1$, real $\kappa=0$) or form inverse-conjugate pairs $(\lambda, 1/\lambda^*)$, corresponding to complex-conjugate WCs $(\nu+i\kappa, \nu-i\kappa)$.
• Krein Stability: The authors introduce Krein signatures based on the projected metric operator. Real Wannier centers carry well-defined signatures. Transitions to complex conjugate pairs (instability) can only occur via Krein collisions between states of opposite signature.
• Global Reciprocity: While individual channels may drift (non-reciprocal), the sum of imaginary parts $\sum \kappa_n = 0$ is enforced globally by pseudo-unitarity, ensuring no net amplification/attenuation in the total system.

C. Symmetry Ramification and Bulk-Boundary Correspondence (BBC)

The paper analyzes the interplay of Inversion (IS) and Pseudo-Inversion (pIS) symmetries.

• Commuting Case: If IS and pIS commute, the system is constrained such that non-trivial Krein collisions are forbidden; WCs remain real.
• Anticommuting Case: If IS and pIS anticommute, they generate a Clifford algebra structure. This allows for Krein collisions at high-symmetry points.
  • BBC: The real part of the WC determines the existence of a filling anomaly (and thus edge states). The imaginary part $\kappa$ predicts the dynamical stability of these edge modes (whether they experience gain or loss).
  • This provides a robust BBC for line-gapped NH systems, linking bulk topological invariants (complex WCs) to boundary physics.

4. Results

• Analytical Derivation: Derived the relationship between the imaginary part of the Berry connection and the drift velocity of Wannier functions.
• Numerical Verification:
  • Simulated a two-band model with alternating gain/loss. Confirmed that non-zero $\kappa$ leads to directional drift of wave packets, while $\kappa=0$ (trivial phase) results in stationary packets.
  • Verified that the momentum-weight distribution $\|w_k\|^2$ is asymmetric in the non-Hermitian regime, matching the analytical prediction derived from $\Gamma(k)$.
• Model (80): Demonstrated a specific model where anticommuting symmetries lead to a phase transition where edge states emerge (filling anomaly) and acquire imaginary energies (gain/loss) precisely when bulk Wannier centers undergo a Krein collision and split into complex conjugates.
• Photonic Implementation: Proposed a photonic waveguide array (ladder geometry with $s$ and $d$ orbital modes) to experimentally realize the model and observe the predicted drift and edge mode behavior.

5. Significance

• New Topological Invariants: Introduces complex Wannier centers as a fundamental topological invariant for line-gapped non-Hermitian systems, extending the concept of polarization beyond the Hermitian limit.
• Dynamical Topology: Connects static band geometry (Wilson loops) to non-reciprocal transport (drifting wave packets), offering a new mechanism for unidirectional transport that does not rely on the skin effect.
• Refined Classification: The concept of symmetry ramification and Krein signatures provides a more nuanced classification of NH topological phases, distinguishing between phases that are topologically distinct in Hermitian limits but behave differently in NH regimes.
• Experimental Relevance: The proposal for a photonic implementation offers a concrete path to experimentally verify these theoretical predictions, potentially leading to new types of non-reciprocal photonic devices and topological solitons.

In summary, this work bridges the gap between the geometric structure of non-Hermitian bands and their physical dynamics, establishing a rigorous framework for understanding how complex topology manifests as directional drift and edge-state instability in open quantum systems.