🔬 mesoscale physics

Tunneling density of states in Luttinger Liquid in proximity to a superconductor: Effect of non-local interaction

This paper demonstrates that non-local density-density interactions can further amplify the tunneling density of states (TDOS) enhancement near a superconductor-Luttinger liquid junction, while revealing that the spatial dependencies of TDOS and the induced pair potential are distinct, indicating that TDOS enhancement cannot be directly attributed to the proximity-induced pair potential.

Original authors: Amulya Ratnakar, Sourin Das

Published 2026-05-01

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CC BY 4.0

Original authors: Amulya Ratnakar, Sourin Das

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where electricity doesn't flow like water in a pipe, but like a crowded dance floor where everyone is holding hands. In this "Luttinger Liquid" (a special state of matter found in thin wires or the edges of quantum materials), the dancers are so sensitive to each other that if one person moves, everyone else has to adjust. Usually, if you try to sneak a new dancer (an electron) onto this floor, the crowd pushes back, making it very hard to get in. This is called "suppression."

However, physicists have discovered two ways to make it easier for new dancers to join, and this paper explores what happens when you combine these two methods.

The Two Magic Tricks

1. The Superconductor Neighbor (The "Super-Partner")
Imagine placing a superconductor (a material where electricity flows with zero resistance) right next to our crowded dance floor. The superconductor acts like a magical mirror. When a dancer (electron) hits the edge, the superconductor reflects them back not as a dancer, but as a "hole" (a missing dancer). This reflection creates a special rhythm that actually helps new dancers join the floor near the edge. This is known as the "proximity effect," and it usually leads to an enhancement (an increase) in the number of dancers you can fit in.

2. The Non-Local Whisper (The "Long-Distance Chat")
In a previous study, the authors found that if the dancers on the floor can "whisper" to each other from far away (non-local interactions), rather than just bumping into their immediate neighbors, it can also create a zone where it's easier to add new dancers.

The Big Discovery: A Tug-of-War

The main goal of this paper was to see what happens when you use both tricks at once: a superconductor neighbor and the long-distance whispers.

The authors found a surprising "tug-of-war":

The Conflict: The conditions that make the "long-distance whispers" work best to help dancers join are exactly the opposite of the conditions that make the "superconductor mirror" work best.
The Result: You cannot have the maximum boost from both at the same time. If you tune the system to get the best help from the whispers, the superconductor actually starts to push dancers away (suppression). If you tune it for the superconductor, the whispers stop helping. They are "mutually exclusive."

Think of it like trying to tune a radio. You can tune it to hear the "Whisper Station" clearly, or the "Super-Mirror Station" clearly, but you can't have both stations playing at their loudest volume simultaneously. The paper proves this using a mathematical "symmetry," showing that the physics of the two scenarios are like mirror images of each other.

The Mystery of the Decay (The "Fading Echo")

The paper also looked at how far these effects reach.

The Expectation: You might think that if the superconductor helps the dancers, the "help" (the density of dancers) and the "reason for the help" (the superconductor's influence) should fade away at the exact same rate as you move away from the edge.
The Reality: The paper shows they fade at different speeds. The "help" (the increased number of dancers) lasts for a much longer distance than the "influence" (the superconductor's direct pull).
The Analogy: Imagine a lighthouse (the superconductor) shining on a foggy sea. The light (the influence) might fade quickly as you move away from the lighthouse. However, the waves created by the light hitting the water (the effect on the dancers) might keep rolling out much further. The paper confirms that even with the "long-distance whispers," the waves travel further than the light itself.

Summary of Findings

Enhancement is Possible: Even with these complex interactions, it is still possible to create a zone where it's easier to add electrons, but only in specific "weak interaction" settings.
No Double Boost: You cannot combine the non-local interaction boost with the superconductor boost to get a super-enhancement. They cancel each other out in terms of parameter space.
Different Decay Rates: The "enhancement" of the electron density persists further away from the junction than the "pair potential" (the superconductor's direct influence). This proves that the increase in electrons isn't just a simple copy of the superconductor's effect; it's a more complex result of the system's internal dynamics.

In short, the paper maps out the rules of a complex quantum dance floor, showing that while you can make it easier for new dancers to join using specific tricks, you can't simply stack all the tricks together to get a super-dance floor. The physics of the "whispers" and the "mirror" are fundamentally at odds with each other.

1. Problem Statement

The paper investigates the interplay between non-local density-density interactions and superconducting proximity effects in a Luttinger Liquid (LL) system. Specifically, it examines a junction where the edge states of two Fractional Quantum Hall (FQH) systems (or integer QH systems) are coupled to a superconductor (SC).

Context: It is well-established that a single LL in proximity to a superconductor exhibits an enhancement in the Tunneling Density of States (TDOS) near the junction due to Andreev reflection. Conversely, recent studies (e.g., Ref. 49) showed that non-local interactions between two LLs can also induce TDOS enhancement at a junction with current-conserving (normal) boundary conditions.
The Gap: The authors aim to determine if these two mechanisms (superconducting proximity and non-local interactions) can synergistically enhance TDOS further, or if they compete. They also seek to understand the stability of the resulting fixed points and the spatial decay profiles of the induced pair correlations versus the TDOS.

2. Methodology

The authors employ bosonization and renormalization group (RG) techniques to model the system.

Model System: A junction of two chiral edge states from QH layers (filling fractions $\nu_1, \nu_2$ $ν_{1}, ν_{2}$ ) coupled to a superconductor. The system includes:
- Intralayer interactions ( $\alpha$ ): Between counter-propagating modes in the same layer.
- Interlayer interactions ( $\beta, \gamma$ ): Non-local interactions between co-propagating ( $\beta$ ) and counter-propagating ( $\gamma$ ) modes of different layers.
Bosonization: The fermionic fields are mapped to bosonic fields. The interacting Hamiltonian is diagonalized using Bogoliubov transformations to define new bosonic modes ( $\tilde{\phi}$ ).
Boundary Conditions (Fixed Points): The study focuses on two superconducting fixed points at the junction ( $x=0$ $x = 0$ ):
1. Disconnected Andreev ( $A_2$ ): Each wire reflects an electron as a hole within the same wire.
2. Cross Andreev Reflection ( $CA_2$ ): An electron in wire 1 is transmitted as a hole in wire 2 (and vice versa).
Calculations:
- TDOS: Calculated via the Green's function of the fermionic fields. The energy scaling dimension ( $\Delta^0$ ) determines enhancement ( $\Delta^0 < 1$ ) or suppression ( $\Delta^0 > 1$ ).
- Stability: Analyzed by calculating the scaling dimensions of perturbation operators (Andreev reflection, cross Andreev reflection, electron tunneling, and quasiparticle backscattering). A fixed point is stable if all relevant perturbations have scaling dimensions $> 1$ .
- Spatial Dependence: The authors derive the spatial power-law exponents for the "relative TDOS" ( $\rho(x)/\rho(0)$ ) and the "relative pair amplitude" ( $F(x)/F(0)$ ) to compare their decay profiles.

3. Key Contributions and Results

A. Mutual Exclusivity of Enhancement Regimes

The most significant finding is that the parameter regimes allowing for TDOS enhancement in current-conserving (normal) junctions and superconducting junctions are mutually exclusive.

Symmetry Relation: The authors establish a symmetry relation between the scaling dimensions of the superconducting fixed points ( $A_2, CA_2$ ) and the normal fixed points ( $S_1, S_2$ ) under the exchange of interaction parameters ( $\alpha \to -\alpha$ , $\gamma \to -\gamma$ ).
Consequence: If non-local interactions enhance TDOS in a normal junction, introducing superconducting correlations in the same parameter regime leads to suppression of TDOS. Enhancement in the superconducting case requires a different set of interaction parameters (often involving strong $\gamma$ or specific $\alpha, \beta$ combinations) that would suppress TDOS in the normal case.

B. Stability vs. Enhancement Trade-off

Simultaneous Stability and Enhancement: The authors find that for a superconducting junction, it is impossible to achieve simultaneous TDOS enhancement and stability against all physically relevant perturbations (including quasiparticle backscattering and electron tunneling).
Contrast with Normal Junctions: In contrast, previous work (Ref. 49) showed that for current-conserving junctions, simultaneous enhancement and stability are possible.
Specific Cases:
- For $\nu_1 = \nu_2 = 1/3$ , the $A_2$ fixed point shows TDOS enhancement in the strong interaction limit but is unstable against quasiparticle backscattering.
- For $\nu_1 = 1, \nu_2 = 1/3$ , similar instability issues arise for the $CA_2$ fixed point.

C. Distinct Spatial Decay Profiles

The paper clarifies a common misconception regarding the proximity effect:

Decoupling of TDOS and Pair Amplitude: The spatial decay of the enhanced TDOS is not identical to the decay of the induced pair correlation function (pair amplitude).
Power Laws: The relative TDOS decays as $x^{\varsigma}$ , while the relative pair amplitude decays as $x^{\Lambda}$ . The authors prove that $\varsigma \neq \Lambda$ even in the presence of non-local interactions.
Implication: The enhancement of TDOS persists over longer distances than the induced pair amplitude. Therefore, TDOS enhancement cannot be solely attributed to the finite density of Cooper pairs in the LL; it is a distinct consequence of the boundary fixed point and the specific Bogoliubov modes of the system.

D. Effect of Non-Local Interactions

Weak Interaction Limit: In the weak interaction limit, introducing non-local interactions (specifically increasing $\beta$ while keeping $\gamma=0$ ) can further amplify the existing TDOS enhancement caused by local interactions ( $\alpha$ ).
Strong Interaction Limit: In the strong interaction regime, non-local interactions allow for TDOS enhancement even in highly suppressed edges (e.g., $\nu=1/3$ ), which was not possible in normal junctions under similar conditions.

4. Significance and Conclusion

Theoretical Insight: The work provides a rigorous symmetry-based understanding of why superconducting and normal junctions exhibit opposite behaviors regarding TDOS enhancement in the presence of non-local interactions. It highlights that the "superconducting proximity effect" in LLs is more complex than a simple amplification of pair correlations.
Experimental Relevance: The proposed setup (bilayer FQH systems coupled to a superconductor) is experimentally feasible with current technology (e.g., GaAs quantum wells with interlayer distances $\sim 30$ nm and superconducting coherence lengths $\sim 38-230$ nm).
Implications for Majorana/Parafermions: Since such junctions are candidates for hosting Majorana or parafermion zero modes, understanding the stability and TDOS profiles is crucial for identifying these exotic states. The finding that stability and enhancement are mutually exclusive suggests that experimental signatures of these states must be sought in specific, carefully tuned interaction regimes.

In summary, the paper demonstrates that while non-local interactions can enhance TDOS in superconducting LL junctions, this enhancement comes at the cost of fixed-point stability and occurs in parameter regimes distinct from those of normal junctions. Furthermore, the spatial decay of TDOS is fundamentally distinct from the decay of the induced superconducting order parameter.