Generalized $η$-pairing approach to interacting non-Hermitian systems in arbitrary dimensions

This paper establishes a generalized $\eta$-pairing framework for interacting non-Hermitian Hubbard models in arbitrary dimensions, revealing novel phenomena such as asymmetric eigenoperators, spatially modulated states, and anomalous skin effects that unify and extend traditional symmetry concepts beyond Hermitian limits.

The Gist

Imagine a vast, bustling city made of tiny particles called electrons. In the world of standard physics (which we'll call "Hermitian physics"), this city follows strict, fair rules: energy is conserved, and if you swap two people, the laws of physics look the same. In this city, there's a special dance called Eta-Pairing.

Think of Eta-Pairing as a magical dance move where two electrons jump together, holding hands, and form a perfect pair. In the standard city, these pairs are like a well-organized marching band: they spread out evenly across the whole city, moving in perfect unison. They are stable, predictable, and symmetrical.

But what happens if the city becomes "Non-Hermitian"?

In this paper, the author, Kai Lieta, explores a chaotic, "non-Hermitian" version of this city. Here, the rules are skewed. Energy can leak out, or the "wind" (hopping amplitudes) might blow harder in one direction than the other. It's like a city where the streets are one-way, or where the ground is slippery on one side and sticky on the other.

The author asks: Can our magical dancing pairs (Eta-Pairing) still exist in this chaotic city? And if they do, how do they behave?

The answer is a resounding YES, but they behave in ways that would be impossible in the normal city. Here are the three biggest surprises, explained with simple analogies:

1. The "Broken Mirror" (Asymmetry)

In the normal city, if you have a dance move, its "mirror image" (doing the move in reverse) is also a valid dance move.

• The New Twist: In the chaotic city, the author discovers that the dance move and its mirror image are not the same. The "forward" dance might work perfectly, but the "backward" dance might be impossible, or it might look completely different.
• The Analogy: Imagine a dance where stepping forward is easy, but stepping backward is impossible because the floor is tilted. The "forward" dancer and the "backward" dancer are no longer reflections of each other; they are two different species of dancers.

2. The "Crowded Corner" (Skin Effect)

In the normal city, the dancing pairs spread out evenly, like a crowd at a festival.

• The New Twist: In the chaotic city, the pairs don't spread out. Instead, they all get pushed to the edges or corners of the city.
• The Analogy: Imagine a crowd of people in a hallway. In a normal hallway, they spread out evenly. But in this "Non-Hermitian" hallway, there's a strong wind blowing from one end. The crowd gets swept away and piles up tightly against the exit door. This is called the Skin Effect. The author shows that even when these particles are interacting and dancing together, they still get swept to the walls, creating a "skin" of particles on the boundary.

3. The "Lost Symmetry" (Broken Rules)

In the normal city, the dance has a beautiful symmetry called SU(2) Pseudospin. It's like a rule that says, "No matter how you rotate the dance, it looks the same."

• The New Twist: The author finds that in the chaotic city, even if the dance moves exist, this beautiful symmetry can be broken. The dance might work, but the "rotation rule" no longer applies.
• The Analogy: Imagine a spinning top. In a normal room, it spins perfectly symmetrically. In this chaotic room, the top might still spin (the dance exists), but it wobbles so much that it no longer looks the same from every angle. The "perfect symmetry" is gone, even though the spinning continues.

The "Universal Translator"

The most brilliant part of this paper is that the author didn't just find these weird phenomena; he built a general theory (a "Universal Translator") that explains them all.

He created a set of mathematical rules (Theorems) that act like a recipe. If you know how the "wind" (hopping) behaves in your chaotic city, this recipe tells you:

• Will the pairs form?
• Will they pile up at the edge?
• Will the symmetry break?

He tested this recipe on different "cities" (lattices):

• 1D City (The Hatano-Nelson-Hubbard model): He showed pairs piling up at opposite ends of a line.
• 2D City: He showed pairs piling up in the corners of a square.
• General City: He built a flexible model that works on any shape of city, proving these weird effects happen everywhere, not just in simple lines.

Why Does This Matter?

For decades, physicists have struggled to understand how particles behave when they interact in these "leaky" or "biased" non-Hermitian systems. Most previous theories only worked for simple, one-dimensional cases.

This paper is a blueprint. It provides the first rigorous, mathematically perfect framework to understand interacting particles in any dimension, even in messy, irregular shapes. It reveals that the "Skin Effect" (particles piling up at the edge) isn't just a single-particle trick; it's a fundamental property of interacting quantum matter in these weird systems.

In short: The author took a complex, chaotic quantum world, found a hidden order within the chaos, and showed us that even in a broken, asymmetric universe, particles can still dance together—but they will dance right up against the walls.

Technical Summary

Here is a detailed technical summary of the paper "Generalized $\eta$-pairing approach to interacting non-Hermitian systems in arbitrary dimensions" by Kai Lieta.

1. Problem Statement

The paper addresses the lack of a rigorous analytical framework for interacting non-Hermitian many-body systems, particularly in arbitrary spatial dimensions and without bulk translation symmetry.

• Context: While the non-Hermitian skin effect (anomalous boundary localization) is well-understood for non-interacting single-particle systems via non-Bloch band theory, its fate in interacting systems remains unclear. Previous rigorous results were limited to 1D systems solvable by the Bethe ansatz.
• Gap: There is no general theory to construct exact eigenstates for interacting non-Hermitian Hubbard models or to understand how interactions and non-Hermiticity (specifically asymmetric hoppings) affect phenomena like the skin effect, symmetries, and off-diagonal long-range order (ODLRO).
• Goal: To generalize C. N. Yang's seminal $\eta$-pairing theory (originally for Hermitian systems) to non-Hermitian Hubbard models, thereby providing exact solutions and revealing novel non-Hermitian phenomena.

2. Methodology

The authors develop a generalized $\eta$-pairing theory based on rigorous algebraic derivations rather than numerical simulations or Bethe ansatz techniques.

• Model: They consider a generic non-Hermitian Hubbard model on an arbitrary lattice $\Lambda$ with complex hopping amplitudes $t_{ij}$, on-site interactions $U_i$, and chemical potentials $\mu_i$.
• Operator Definition: They define the $\eta$-raising operator as $\eta^\dagger = \sum_{j \in \Lambda} \omega_j c^\dagger_{j\uparrow} c^\dagger_{j\downarrow}$, where the pairing amplitude $\omega_j$ is allowed to be site-dependent and complex.
• Eigenoperator Condition: They derive the necessary and sufficient conditions for $\eta^\dagger$ (and its lowering counterpart $\eta'$) to be an eigenoperator of the Hamiltonian $H$ (i.e., $[H, \eta^\dagger] = \lambda \eta^\dagger$). This leads to a set of coupled equations relating the hopping amplitudes $t_{ij}$ and the pairing amplitudes $\omega_j$.
• Symmetry Analysis: The authors re-examine the symmetry groups generated by spin and $\eta$-pairing (pseudospin) operators, specifically investigating the $SO(4)$ symmetry and various Particle-Hole (PH) symmetries in the non-Hermitian context.
• Model Construction: They apply the theory to specific models:
  1. Hatano-Nelson-Hubbard (HNH) models in 1D and 2D (with asymmetric hoppings).
  2. A general two-sublattice model defined on arbitrary lattices to realize all possible exotic phenomena.

3. Key Contributions and Novel Phenomena

The paper establishes that non-Hermiticity fundamentally alters the properties of $\eta$-pairing, leading to five distinct phenomena (labeled a–e) that have no Hermitian analogs. These are shown to be interconnected via a hierarchy of implications: (e) $\Rightarrow$ (d) $\Rightarrow$ (a) $\Leftrightarrow$ (b) $\Leftrightarrow$ (c).

• (a) Non-Hermitian Conjugacy Failure: The Hermitian conjugate of an $\eta$-pairing eigenoperator is not necessarily an eigenoperator. In Hermitian systems, if $\eta^\dagger$ is an eigenoperator, so is $\eta$. In non-Hermitian systems with asymmetric hoppings, $\eta^\dagger$ and $\eta$ are distinct, and $\eta$ may not even exist as an eigenoperator.
• (b) Spatially Modulated Pairing Amplitude: The on-site pairing amplitude $|\omega_j|$ can be site-dependent (spatially modulated). In Hermitian systems, $|\omega_j|$ is constant. This modulation leads to anomalous localization of the eigenstates.
• (c) Breakdown of $SU(2)$ Pseudospin Symmetry: Even if the Hamiltonian commutes with the $\eta$-pairing operators, the $SU(2)$ pseudospin symmetry (which requires $\eta^\dagger$ and $\eta$ to form an angular momentum algebra) may be absent if hoppings are asymmetric.
• (d) Independent Eigenvalues: The eigenvalues $\lambda$ (for $\eta^\dagger$) and $\lambda'$ (for $\eta'$) can be independent constants, whereas in Hermitian systems, $\lambda = \lambda'$.
• (e) Non-Uniqueness: The $\eta$-raising or -lowering eigenoperators may not be unique (unlike in Hermitian systems where they are unique up to a constant factor).

New Concept: The authors introduce "non-Hermitian angular-momentum operators," where $\eta^\dagger$ and $\eta'$ satisfy angular momentum commutation relations but are not Hermitian conjugates of each other.

4. Key Results

A. Anomalous Localization (Skin Effect)

• Mechanism: The generalized theory reveals that $\eta$-pairing provides a new mechanism for the skin effect in interacting systems.
• 1D HNH Model: The right ($\eta^\dagger|0\rangle$) and left ($(\eta')^\dagger|0\rangle$) two-particle $\eta$-pairing eigenstates are exponentially localized at opposite boundaries of the chain.
• 2D HNH Model: The eigenstates exhibit first-order (localized at edges) or second-order (localized at corners) skin effects, depending on the anisotropy of the asymmetric hoppings.
• General Lattice: In the general two-sublattice model, eigenstates can be localized in arbitrary subregions (not just boundaries) if the lattice is partitioned such that hopping exists only between specific sublattices.

B. Symmetry Unification

• $SO(4)$ Symmetry: The authors clarify that the true physical symmetry of the Hubbard model is always $SO(4)$, regardless of whether the number of lattice sites $N_\Lambda$ is even or odd (resolving previous ambiguities regarding $SU(2) \times SU(2)$ vs. $SO(4)$).
• Equivalence of Symmetries: They prove that in the context of the Hubbard model, the $SU(2)$ pseudospin symmetry, the $SO(4)$ symmetry, and various discrete Particle-Hole (PH) symmetries (including $Z_2$, $Z_4$, and general $Z_{2k}$) are equivalent. If the Hamiltonian possesses one, it possesses all.
• Shiba Transformation: They clarify that the $Z_2$ Shiba transformation (a partial PH transformation) does not belong to the $SO(4)$ symmetry but is related to it via the same underlying algebraic constraints.

C. Off-Diagonal Long-Range Order (ODLRO)

• The paper derives explicit expressions for the ODLRO in non-Hermitian $\eta$-pairing states.
• Unlike the simple function of filling fraction in Hermitian systems, the ODLRO in non-Hermitian systems depends on the ratio of pairing amplitudes on different sublattices ($c = |\omega_A|^2 / |\omega_B|^2$) and exhibits complex scaling behavior in the thermodynamic limit.

5. Significance

• Theoretical Framework: This work provides the first rigorous, general analytical framework for interacting non-Hermitian many-body systems in arbitrary dimensions, bypassing the limitations of the Bethe ansatz.
• New Mechanism for Skin Effects: It identifies $\eta$-pairing as a distinct mechanism for the non-Hermitian skin effect in interacting systems, demonstrating that interactions do not destroy but can even enhance or modify boundary localization.
• Symmetry Clarification: It resolves long-standing ambiguities regarding the symmetry groups of the Hubbard model and unifies various symmetries under a single algebraic structure.
• Broad Applicability: The general two-sublattice model constructed can describe a wide range of systems, including topological insulators (Haldane model, SSH model, higher-order topological insulators) and various lattice geometries (square, triangular, honeycomb), revealing $\eta$-pairing structures in systems previously thought to lack them.
• Experimental Relevance: The findings offer concrete predictions for cold atom experiments and photonic systems where non-Hermitian Hubbard models can be engineered, particularly regarding the observation of interacting skin effects and symmetry-breaking phenomena.

In summary, the paper fundamentally extends the concept of $\eta$-pairing into the non-Hermitian regime, uncovering a rich landscape of novel quantum phenomena driven by the interplay of interactions and non-Hermiticity.