Symmetry-Fractionalized Skin Effects in Non-Hermitian Luttinger Liquids

This paper extends the concept of symmetry-fractionalization to non-Hermitian systems by demonstrating that skin effects associated with different symmetry sectors, such as spin and charge, exhibit emergent decoupling in one-dimensional gapless systems, a phenomenon verified numerically for the Hubbard model and theoretically constructed for an interaction-enabled $E_8$ skin effect.

The Gist

Imagine a crowded hallway where people are trying to walk from one end to the other. In a normal, "fair" world (what physicists call a Hermitian system), everyone moves at the same speed, and if you look at the crowd, the people are spread out evenly.

But now, imagine the hallway has a strange, invisible wind blowing only in one direction. This is the Non-Hermitian Skin Effect. In this scenario, almost everyone gets blown into a massive pile-up at one specific wall, leaving the rest of the hallway empty. It's a chaotic, one-sided crowd.

This paper asks a fascinating question: What happens if we add "social rules" (interactions) to this windy hallway? Specifically, what if the people in the crowd are not just individuals, but groups with different identities, like "Charge" (how much they weigh) and "Spin" (which way they are facing)?

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

1. The Magic of "Fractionalization" (Splitting the Crowd)

In the world of quantum physics, there's a famous trick called Luttinger Liquid theory. It says that in a one-dimensional line (like our hallway), a particle isn't just one thing. It's actually two separate things traveling together:

• The Charge: How much "stuff" the particle has.
• The Spin: The particle's internal direction.

Usually, these two travel together. But in this paper, the authors show that in a non-Hermitian system (the windy hallway), these two identities can split apart.

The Analogy: Imagine a group of dancers. In a normal room, they dance in pairs. But in this windy room, the "Charge" dancers get blown to the Left Wall, while the "Spin" dancers get blown to the Right Wall. They have completely separated! The paper calls this Symmetry-Fractionalized Skin Effect.

2. The "Imaginary Wind" (Gauge Fields)

How do they control this? They use something called an Imaginary Gauge Field.

• Think of this as a "wind" that only blows on specific types of people.
• You can set a wind that only pushes the "Charge" dancers.
• You can set a different wind that only pushes the "Spin" dancers.

The authors discovered that if the hallway is perfectly straight (linear), these two winds don't interfere with each other. You can push the charges to the left and the spins to the right simultaneously, and they stay perfectly separated.

3. The "Traffic Jam" (Interactions)

The paper also looks at what happens when the dancers bump into each other (interactions).

• Strong Repulsion (Pushing away): If the dancers hate each other, they form a rigid line. In this state, the "Charge" dancers get stuck in a traffic jam (a "gap") and stop moving. Only the "Spin" dancers can still get blown to the wall.
• Strong Attraction (Hugging): If the dancers love each other, they clump together. Now the "Spin" dancers get stuck, and only the "Charge" dancers get blown to the wall.

This means scientists can tune the system. By changing how much the particles like or dislike each other, they can choose whether the "Charge" piles up, the "Spin" piles up, or both pile up independently.

4. The "Ghost Dance" (The E8 Effect)

The most mind-blowing part of the paper is the final example. The authors built a theoretical model with 11 different types of particles.

• In a normal world, you can't have a skin effect for a group of 11 particles unless they are all independent.
• But here, the interactions are so strong and complex that they create a brand new, unified "super-identity" called E8.
• The Analogy: Imagine 11 different musical instruments playing. Normally, they just make noise. But if they play a specific, complex song together (interact), they suddenly sound like a single, perfect, 8-dimensional chord that has never existed before.
• This "E8 Skin Effect" is a pile-up of particles that cannot exist without these complex interactions. It's a phenomenon that is impossible for simple, non-interacting particles.

Why Does This Matter?

This isn't just math for math's sake.

1. New Materials: It helps us understand how to build new materials where we can control exactly where electrons go, even if they are interacting strongly.
2. Quantum Computers: It suggests ways to protect quantum information. If you can separate "Charge" and "Spin" and push them to different walls, you might be able to store data in a way that is harder to mess up.
3. Real Experiments: The authors suggest this could be tested with ultracold atoms (like Strontium atoms) in a lab. These atoms have many internal states, making them perfect "dancers" to test these theories.

The Bottom Line

The paper shows that when you mix non-Hermitian physics (systems that lose energy or have one-way winds) with strong interactions (particles bumping into each other), you get a beautiful new phenomenon. The "skin effect" (the pile-up at the wall) doesn't just happen to the whole crowd; it splits apart. Different parts of the particle's identity get blown to different walls, and with enough complexity, you can create entirely new types of "ghostly" pile-ups that nature didn't show us before.

It's like discovering that in a hurricane, your shoes fly left, your hat flies right, and your soul flies up, and you can control exactly where each piece goes!

Technical Summary

1. Problem Statement

The paper addresses a fundamental gap in the understanding of Non-Hermitian Skin Effects (NHSE) in interacting many-body systems.

• Context: In non-Hermitian systems, an extensive number of eigenstates accumulate at the boundaries under open boundary conditions (OBC), a phenomenon known as NHSE. While well-understood in single-particle (free fermion) models and classified via topological invariants, the behavior of NHSE in strongly correlated, interacting 1D systems remains unclear.
• The Gap: Existing frameworks often treat NHSE as a single-particle spectral instability. It is unknown how non-reciprocity (the driver of NHSE) interacts with many-body correlations, specifically how boundary localization reorganizes when multiple symmetry sectors (e.g., spin and charge) are present and coupled.
• Specific Question: Can the low-energy physics of interacting non-Hermitian 1D fermions be described by a symmetry-resolved framework where different symmetry sectors exhibit independent or coupled skin effects?

2. Methodology

The authors develop a theoretical framework combining bosonization, conformal field theory (CFT), and numerical exact diagonalization (ED).

• Theoretical Framework (Bosonization):

  • They map $N$-flavor interacting fermions in 1D to a bosonic field theory (Luttinger Liquid theory).
  • They model non-reciprocal hopping by coupling the system to constant imaginary background gauge fields ($H_\mu$).
  • They analyze the system using the affine Lie algebra structure. By decomposing the symmetry group $U(N)$ into subalgebras (e.g., $G$ and its complement $G'$), they derive analytical expressions for the ground-state density profiles.
  • Key Derivation: They show that for a purely linear dispersion, the expectation value of the density $\langle \rho_h(x) \rangle$ in a symmetry sector $G'$ depends on the product of gauge fields and Luttinger parameters. If the sectors are decoupled (linear dispersion), skin effects in distinct symmetry sectors do not mix.

• Analytical Models:

  • Hatano-Nelson-Hubbard Model: A 1D model with non-reciprocal hopping ($t_R, t_L$) and on-site interaction $U$. The non-reciprocity is parameterized by imaginary gauge fields $h$ (charge/U(1)) and $H$ (spin/SU(2)).
  • $E_8$ Skin Effect Model: A theoretical construction using 11 fermionic flavors to realize an emergent $E_8$ symmetry sector, which is gapped out from the rest of the spectrum to isolate a strongly correlated skin effect.

• Numerical Validation:

  • The authors perform Exact Diagonalization (ED) on the Hatano-Nelson-Hubbard chain (up to $L=16$ sites) using the EDITH software.
  • They calculate the ground-state density ($\langle n_j \rangle$) and spin ($\langle S^z_j \rangle$) profiles.
  • They quantify localization using the normalized mean center of mass (mcom) to distinguish between delocalized (bulk-like) and localized (skin) states.

3. Key Contributions

A. Symmetry-Fractionalized Skin Effects

The primary contribution is the demonstration that in 1D interacting systems, the NHSE can be fractionalized into distinct symmetry sectors.

• Decoupling: In the linear dispersion limit, the charge ($U(1)$) and spin ($SU(2)$) sectors decouple. Non-reciprocity applied to one sector (e.g., charge) induces a skin effect only in that sector, leaving the other sector (spin) delocalized.
• Tunability: The localization direction and presence can be independently tuned by the imaginary gauge fields ($h$ for charge, $H$ for spin) and interaction strength ($U$).

B. Role of Interactions and Gapping

The paper elucidates how interactions control the skin effect:

• Gap Induction: Interactions can open energy gaps in specific sectors (e.g., Mott insulating behavior). If a sector is gapped, its skin effect is suppressed.
• Phase Diagram: They identify regimes where:
  • Only charge skin effects exist (strong repulsive $U$, half-filling).
  • Only spin skin effects exist (strong attractive $U$).
  • Both exist simultaneously (weak interactions).
  • Neither exists (gapped sectors).

C. Interaction-Enabled $E_8$ Skin Effect

The authors construct a novel model where the skin effect arises purely from interactions, with no free-fermion counterpart.

• By utilizing the conformal embedding $U(11)_1 = (E_8)_1 \otimes U(3)_1$, they gap out the $U(3)$ sector using backscattering interactions.
• The remaining low-energy theory possesses only $E_8$ symmetry. Coupling this sector to an imaginary gauge field generates a skin effect localized to the $E_8$ charges, a phenomenon impossible in non-interacting fermion systems.

D. Breakdown of Decoupling

The study clarifies that the strict decoupling of symmetry sectors is a feature of the linear dispersion limit. Away from half-filling or with band curvature, irrelevant operators (e.g., $\partial_x \phi_c (\partial_x \phi_\sigma)^2$) mix the sectors. This allows a skin effect in one sector (e.g., spin) to induce a secondary, weaker localization in the coupled sector (e.g., charge), even without a direct gauge field.

4. Results

• Analytical vs. Numerical Agreement: The analytical predictions derived from the bosonized Luttinger Liquid theory match the ED numerical results with high precision across weak and strong coupling regimes. The spatial profiles of charge and spin accumulation follow the predicted logarithmic forms.
• Phase Diagrams:
  • Half-filling: Clear separation of phases. Strong repulsion ($U>0$) gaps the charge sector, suppressing charge skin effects but allowing spin skin effects. Strong attraction ($U<0$) gaps the spin sector, suppressing spin skin effects but allowing charge skin effects.
  • Below Half-filling: Particle-hole symmetry is broken. The mixing terms become relevant, leading to "double-sided" skin effects where a gauge field in the spin sector induces a non-trivial charge profile, though this effect diminishes in the thermodynamic limit.
• $E_8$ Realization: The theoretical construction confirms that strongly correlated systems can host skin effects associated with exceptional Lie groups ($E_8$), expanding the classification of non-Hermitian phases beyond unitary symmetries.

5. Significance

• Theoretical Advancement: This work provides the first universal low-energy framework for NHSE in interacting 1D systems, bridging the gap between non-Hermitian topology and Luttinger liquid theory. It establishes that symmetry resolution is the key organizing principle for many-body skin effects.
• New Phenomena: It predicts "interaction-enabled" skin effects (like the $E_8$ case) that cannot be understood via single-particle physics, suggesting that interactions can fundamentally alter non-Hermitian localization.
• Experimental Relevance: The authors propose that these effects are observable in ultracold atomic gases (specifically alkaline-earth atoms like $^{87}\text{Sr}$) with large internal symmetries. By engineering controlled dissipation (loss) that couples selectively to specific internal states (simulating imaginary gauge fields), experimentalists could observe the fractionalization of skin effects into spin and charge components.
• Future Directions: The framework suggests extensions to higher dimensions (where sectors do not decouple naturally but can be gapped) and to open quantum systems described by Lindblad equations, offering a roadmap for studying non-equilibrium steady states in correlated matter.

In summary, the paper demonstrates that in 1D, interactions do not destroy the skin effect but rather fractionalize it, allowing for a rich phase diagram where boundary localization can be selectively controlled, suppressed, or generated purely through many-body correlations.