Enhancement of charge correlations and real-space topological marker on an interacting non-Hermitian Su-Schrieffer-Heeger model

This paper investigates the interacting non-Hermitian Su-Schrieffer-Heeger model to demonstrate that a real-space topological marker robustly identifies topological phases and their breakdown into charge density waves, while revealing that non-Hermiticity significantly amplifies interaction-induced charge correlations, particularly near exceptional points under open boundary conditions.

The Gist

Imagine a crowded dance floor where the dancers are electrons. In a normal, stable world (what physicists call a "Hermitian" system), these dancers follow strict, predictable rules: if you push one, they push back equally. But in this paper, the authors explore a weird, "non-Hermitian" world. Here, the dance floor is slightly tilted, and the rules are lopsided. If a dancer moves left, it's easy; if they try to move right, it's much harder. This creates a "one-way street" effect for the electrons.

The researchers are studying a specific dance pattern called the SSH model (named after Su, Schrieffer, and Heeger). Think of this as a line of dancers holding hands in pairs. Sometimes the pairs hold hands tightly (strong bond), and sometimes loosely (weak bond). This alternating pattern creates a special "topological" state—a hidden order that makes the dancers at the very ends of the line behave differently than those in the middle, almost like they are wearing invisible "topological hats" that protect them.

The Twist: Adding "Pushiness" (Interactions)
In the real world, electrons don't just dance alone; they push and pull on each other. This is called "interaction." The paper asks: What happens to our special topological dance when the electrons start pushing each other around, especially in this weird, one-way street world?

They found three main things:

1. The "Topological Marker" is a Reliable Compass:
   To figure out if the dancers are in a topological state or a normal state, the authors used a special tool called a "real-space topological marker." Imagine this as a GPS tracker that looks at the dancers' positions right where they are, rather than trying to predict the whole crowd's movement from far away.

   • The Claim: Even when the electrons start pushing each other hard, this GPS tracker works perfectly. It correctly identifies the "Topological" phases (where the edge dancers are special) and tells you exactly when the system breaks down into a chaotic mess.

2. The "Charge Density Wave" (CDW) is the Villain:
   As the electrons push each other harder (increasing the interaction strength), they eventually stop dancing in their topological pattern. Instead, they get stuck in a rigid, alternating pattern of "heavy" and "light" spots, like a checkerboard of crowded and empty seats. This is called a Charge Density Wave (CDW).

   • The Claim: This rigid CDW pattern destroys the topological protection. Once the electrons lock into this checkerboard pattern, the "topological hats" disappear, and the special edge behavior is lost. The topological marker drops to zero, signaling the end of the special phase.

3. The "One-Way Street" Makes Things Worse (The Skin Effect):
   This is the most surprising part. The authors compared two scenarios:

   • Scenario A (Periodic Boundary): The dance floor is a circle. The dancers can go around forever.
   • Scenario B (Open Boundary): The dance floor is a straight line with walls at the ends.
   • The Claim: In the "Open Boundary" (straight line) scenario, the one-way street rules cause a massive buildup of dancers near the walls (a phenomenon called the Non-Hermitian Skin Effect). When the system gets close to a critical tipping point (called an "Exceptional Point"), this buildup acts like a megaphone. It amplifies the electrons' tendency to push each other into that rigid checkerboard pattern.
   • The Metaphor: In the circular dance floor, the pushiness is mild. But in the straight line, the "walls" and the "one-way rules" pile the dancers up so tightly that they are forced into the rigid checkerboard pattern much more easily and violently. The "Exceptional Point" is like a singularity where the music changes pitch so drastically that the dancers lose their rhythm and freeze into place.

Summary of the Findings:

• Robustness: The special topological order is surprisingly tough against electron pushing, until the pushing gets too strong.
• The Breakdown: Once the pushing gets strong enough to create a "checkerboard" (CDW) pattern, the topological magic vanishes.
• The Amplifier: If you put this system in a straight line (Open Boundary) instead of a circle, the "one-way" nature of the world makes the electrons pile up at the edges. This pile-up makes them much more likely to freeze into that checkerboard pattern, destroying the topological state faster than in a circular setup.

The paper essentially maps out exactly where the "special topological dance" ends and the "rigid checkerboard freeze" begins, showing that the shape of the room (the boundary conditions) and the one-way nature of the rules play a huge role in how quickly the system loses its special properties.

Technical Summary

Technical Summary: Enhancement of Charge Correlations and Real-Space Topological Marker on an Interacting Non-Hermitian Su-Schrieffer-Heeger Model

Problem Statement
The paper investigates the interplay between topology and charge ordering in one-dimensional interacting non-Hermitian systems. While non-Hermitian topological phases have been extensively studied in single-particle regimes, their behavior in the presence of interactions remains an emerging field. Specifically, the authors address how nearest-neighbor interactions affect the topological phases of the Su-Schrieffer-Heeger (SSH) model with non-reciprocal hopping. A central challenge is determining whether topological invariants, such as winding numbers or real-space markers, remain robust diagnostics in the presence of both non-Hermiticity (which introduces complex eigenenergies and the skin effect) and many-body interactions (which can induce charge density waves, CDW). The study also explores how boundary conditions (Periodic vs. Open) and the proximity to exceptional points (EPs) influence these competing phases.

Methodology
The authors employ Exact Diagonalization (ED) to solve the interacting non-Hermitian SSH Hamiltonian for finite system sizes (up to $N=24$ sites). The model includes intra-cell and inter-cell hopping with non-reciprocity parameter $\delta$, and nearest-neighbor density-density interactions $V$.

Key methodological components include:

1. Ground State Selection: Since the Hamiltonian is non-Hermitian, the ground state is defined as the eigenpair (left and right eigenvectors) with the smallest real part of the eigenvalue. If degenerate, the state with the smallest imaginary part is selected.
2. Real-Space Topological Marker: To diagnose topology without relying on translational invariance (crucial for Open Boundary Conditions and interacting systems), the authors utilize a real-space topological marker $W$. This marker is constructed using the projector $\hat{P}$ onto the ground state eigenpair, incorporating chiral symmetry $\hat{S}$ and the position operator $\hat{X}$. The calculation explicitly uses a biorthogonal convention ($\eta = L$, utilizing both left and right eigenvectors) to ensure physical consistency.
3. Charge Correlation Analysis: The emergence of the CDW phase is tracked via the charge structure factor $S_{cdw}(q)$ and its finite-size scaling. The authors analyze the staggered charge correlations at wavevectors $q=\pi$ (for PBC) and $q=\pi - 2\pi/N$ (for OBC).
4. Phase Boundary Determination: Phase transitions are identified through the behavior of the many-body energy gap (real and imaginary parts), the topological marker, and finite-size scaling analysis of the CDW order parameter, assuming the $(1+1)$D Ising universality class for the CDW transition.

Key Contributions and Results
The study maps out the phase diagrams for both Periodic Boundary Conditions (PBC) and Open Boundary Conditions (OBC), revealing a competition between topological phases and the CDW phase.

1. Phase Diagrams and Competing Orders:

   • PBC: The system exhibits three distinct regimes for weak to intermediate interactions: Topo-I (winding number $W=1$), Topo-II ($W=1/2$, intrinsic to non-Hermitian systems), and a Trivial phase ($W=0$). As interaction strength $V$ increases, the system transitions into a CDW phase. The topological marker vanishes in the CDW phase, indicating that charge ordering breaks the chiral symmetry protecting the topological phases.
   • OBC: The phase diagram differs significantly due to the non-Hermitian skin effect. A competition is observed primarily between the Topo-I and CDW phases. The Topo-II phase is more strongly suppressed by interactions under OBC compared to PBC.

2. Robustness of the Topological Marker:
   The real-space topological marker $W$ is shown to be a robust diagnostic for non-Hermitian topological phases even in the presence of interactions. It consistently signals the breakdown of topological order at the onset of the CDW phase. The authors emphasize that using only right eigenvectors ($\eta=R$) fails to reproduce the correct phase diagram and yields unphysical artifacts (such as asymmetric density profiles under PBC), validating the necessity of the biorthogonal ($\eta=L$) approach.

3. Enhancement of Charge Correlations near Exceptional Points:
   A significant finding is that non-Hermiticity enhances interaction effects, particularly under OBC. As the system approaches the exceptional line ($\delta \approx t_1$), the staggered charge correlations are pronouncedly amplified. This enhancement arises from the accumulation of low-energy states near the exceptional points, which increases the density of states and promotes electronic instabilities.

   • Under PBC, this enhancement is modest.
   • Under OBC, the structure factor increases sharply near the EP, separating two distinct charge-ordered regimes: one dominated by $q=\pi$ and another by nearby finite-size wavevectors.

4. Comparison with Mean-Field Theory:
   The authors note that mean-field treatments fail to quantitatively reproduce the full phase diagram, particularly regarding the suppression of the Topo-II phase and the specific nature of interaction-driven transitions. The ED results reveal that the Topo-II to CDW transition is first-order, whereas other transitions remain second-order.

Significance and Claims
The paper claims to provide a comprehensive characterization of interacting non-Hermitian topological phases using a real-space marker that does not rely on translational invariance. The primary significance lies in demonstrating that:

• Interactions can enlarge the region of topological phases (specifically Topo-I) by suppressing the non-Hermitian Topo-II phase more strongly than in the Hermitian case.
• Non-Hermiticity acts as a catalyst for interaction effects; the proximity to exceptional points under OBC significantly enhances charge correlations, potentially driving the system into a CDW state more readily than in Hermitian counterparts.
• The biorthogonal real-space topological marker is essential for correctly diagnosing topology in interacting non-Hermitian systems, as single-eigenvector approaches lead to incorrect phase boundaries and unphysical observables.

The work establishes a framework for studying topology in open quantum systems with interactions, highlighting the delicate balance between topological protection and symmetry-breaking charge ordering. The authors suggest future directions involving steady-state formulations and the investigation of other symmetries (time-reversal, particle-hole) using entanglement measures, but do not propose specific experimental realizations beyond the theoretical model derived from Lindblad dynamics.