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arxiv: 2202.00029 · v2 · submitted 2022-01-31 · ❄️ cond-mat.mes-hall

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

Amulya Ratnakar , Sourin Das This is my paper

Pith reviewed 2026-05-24 12:58 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords Luttinger liquidtunneling density of statesnon-local interactionsuperconducting proximityquantum Hall edgeboundary conditions
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The pith

Non-local interactions can further amplify the tunneling density of states enhancement near a superconductor-Luttinger liquid junction.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the combined effect of non-local density-density interactions and superconducting correlations on the tunneling density of states in a Luttinger liquid wire at the junction with a superconductor. It shows algebraically, in the weak-interaction limit, that suitable non-local interaction parameters increase the TDOS enhancement that the superconducting boundary condition already produces. The analysis establishes that the parameter regions yielding TDOS enhancement under superconducting boundary conditions and under non-superconducting charge-conserving boundary conditions are completely disjoint, a separation that follows from a symmetry relating the two sectors of the theory. The spatial power-law exponents that govern how TDOS and the proximity-induced pair potential vary with distance from the junction are shown to be distinct functions of the local and non-local interaction strengths, implying that the TDOS increase cannot be traced directly to the pair potential.

Core claim

In a Luttinger liquid with both local and non-local interactions connected to a superconductor, the tunneling density of states near the junction is enhanced by the superconducting boundary condition and this enhancement can be amplified further by appropriate non-local parameters in the weak-interaction limit; the regions of such enhancement for superconducting versus charge-conserving boundary conditions are mutually exclusive because of a symmetry between the sectors; and the distance dependence of TDOS and of the induced pair potential are governed by distinct functions of the interaction parameters.

What carries the argument

Bosonized Luttinger liquid with local plus non-local density-density interactions subject to superconducting boundary conditions at the junction, together with the symmetry that maps the superconducting sector onto the non-superconducting sector.

If this is right

  • The TDOS enhancement produced by the superconducting junction grows when non-local interaction parameters are chosen appropriately in the weak-interaction regime.
  • The set of interaction parameters that produce enhanced TDOS under superconducting boundary conditions is disjoint from the set that produces enhancement under non-superconducting charge-conserving boundary conditions.
  • The power-law exponent for the spatial variation of TDOS differs from the exponent for the spatial variation of the proximity-induced pair potential.
  • The TDOS enhancement therefore cannot be attributed directly to the proximity-induced pair potential.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Spatial scans of both TDOS and pair potential on the same device could distinguish interaction-driven amplification from pair-potential effects.
  • The mutual exclusivity of the two enhancement regimes suggests that changing boundary conditions would switch the system between enhanced and non-enhanced TDOS without an intermediate regime.
  • The algebraic structure that separates the sectors may extend to other boundary-condition problems in one-dimensional wires with long-range interactions.

Load-bearing premise

The bosonized or renormalization-group treatment of the Luttinger liquid that includes both local and non-local interactions remains valid all the way to the junction and captures the interplay without higher-order corrections that would mix the superconducting and charge sectors.

What would settle it

A measurement of tunneling density of states and induced pair potential at several distances from the junction that checks whether their spatial power-law exponents follow the distinct interaction-parameter dependence predicted by the theory.

Figures

Figures reproduced from arXiv: 2202.00029 by Amulya Ratnakar, Sourin Das.

Figure 1
Figure 1. Figure 1: Figure a) and b) corresponds to the physical setting of the LL QWs. Schematic figure shows the junction of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: fig. 2. We allow for repulsive density-density interaction [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: The schematic figure a) shows two QH states, with [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The schematic figure a) and b) shows the unfolded version of the junction of SC with the edge states of two FQH [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The Schematic picture on the top shows the unfolded version of the direct Andreev reflection [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The Schematic picture on the left shows the unfolded version of the direct Andreev reflection [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The Schematic picture on the top shows the unfolded version of the Cross Andreev reflection [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The Schematic picture on the left shows the unfolded version of the Cross Andreev reflection [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The Schematic picture on the top shows the unfolded version of the junction of [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The Schematic picture on the left shows the unfolded version of the Cross Andreev reflection [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The schematic figure on the top shows the unfolded version of the junction of bilayer QH edge states with unequal [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The schematic figure on the left shows the unfolded version of the junction of bilayer QH edge states with equal [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
read the original abstract

A recent study have shown that it is possible to have enhancement, in contrast to an expected suppression, in tunneling density of states (TDOS) in a Luttinger liquid (LL) which is solely driven by the non-local density-density interactions. Also, it is well known that a LL in proximity to a superconductor (SC) shows enhancement in TDOS in the vicinity of the junction in the zero energy limit. In this paper, we study the interplay of nonlocal density-density interaction and superconducting correlations in the TDOS in the vicinity of the SC-LL junction, where the LL maybe realized on the edge of an integer or a fractional quantum Hall state. We show that the TDOS enhancement in a LL quantum wire (QW) due to superconducting correlation at the junction can be farther amplified in presence of appropriate choice of non-local interaction parameters, which is demonstrated through a neat algebraic expression in the weak interaction limit. We also show that, in the full parameter regime comprising both the local and the non-local interaction, the region of enhanced TDOS for LL junction with "superconducting" boundary condition and that of "non-superconducting charge conserving" boundary condition (discussed in Phys. Rev. B 104, 045402 (2021)) are mutually exclusive. We show that this fact can be understood in terms of symmetry relation established between the superconducting and non-superconducting sectors of the theory. We compare the dependence of the proximity induced pair potential and TDOS as a function of distance $x$ from the junction. We demonstrate that the dependence of the spatial power law exponent for the `TDOS(x)/TDOS(x -> 0)' and the `pair-potential(x)/pair-potential(x -> 0)' are distinct function of the various local and non-local interaction parameters, which implies that the TDOS enhancement can not be directly attributed to the proximity induced pair potential in the LL.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript examines the tunneling density of states (TDOS) near a superconductor-Luttinger liquid (SC-LL) junction, including both local and non-local density-density interactions. It claims that suitable non-local interaction parameters further amplify the TDOS enhancement (already present from SC correlations) in the weak-interaction limit, shown via an algebraic expression. It also asserts that the regions of enhanced TDOS under superconducting versus non-superconducting charge-conserving boundary conditions are mutually exclusive throughout the full parameter space, owing to a symmetry relation between the sectors, and that the spatial power-law exponents of TDOS(x) and the proximity-induced pair potential differ as functions of the interaction parameters.

Significance. If the central claims hold, the work clarifies how non-local interactions can be used to tune proximity-induced effects in LL wires (including quantum-Hall edges), and the symmetry-based mutual exclusivity plus the distinct spatial scaling provide a diagnostic separating TDOS enhancement from simple pair-potential leakage. The algebraic weak-limit result is a concrete, checkable prediction.

major comments (2)
  1. [symmetry relation and full-parameter-regime statement] The mutual-exclusivity claim and the symmetry relation between superconducting and charge-conserving sectors are load-bearing for the main conclusions, yet the manuscript supplies no explicit bound on higher-order operator mixing generated by the non-local density-density terms at the junction; the bosonized treatment is assumed to remain valid without such corrections (see the paragraph containing the symmetry argument and the comparison of spatial exponents).
  2. [weak-interaction algebraic expression] The algebraic demonstration of further TDOS amplification is restricted to the weak-interaction limit; it is unclear whether the same non-local parameters continue to produce net enhancement once local interactions are treated non-perturbatively or when the boundary condition is imposed exactly (the abstract states the result holds for the chosen SC boundary condition but does not show the intermediate steps that would confirm absence of mixing).
minor comments (2)
  1. [Abstract] Abstract contains a grammatical error: 'A recent study have shown' should read 'has shown'.
  2. [Abstract] The phrase 'farther amplified' is imprecise; 'further amplified' is the conventional usage.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the key claims regarding non-local interactions, symmetry relations, and spatial exponents in the TDOS at SC-LL junctions. We address the major comments point by point below, providing clarifications on the scope of our bosonized treatment and the exact nature of the symmetry argument.

read point-by-point responses

  1. Referee: [symmetry relation and full-parameter-regime statement] The mutual-exclusivity claim and the symmetry relation between superconducting and charge-conserving sectors are load-bearing for the main conclusions, yet the manuscript supplies no explicit bound on higher-order operator mixing generated by the non-local density-density terms at the junction; the bosonized treatment is assumed to remain valid without such corrections (see the paragraph containing the symmetry argument and the comparison of spatial exponents).

    Authors: Within our model the non-local density-density interactions enter the bulk Hamiltonian as quadratic terms in the bosonic fields and do not generate additional relevant operators at the junction that would invalidate the bosonization or induce mixing between sectors. The symmetry mapping between the superconducting and charge-conserving boundary-condition sectors follows directly from the structure of the interaction Hamiltonian and the imposed boundary conditions; it holds exactly for arbitrary values of the local and non-local couplings. We will add an explicit paragraph in the revised manuscript deriving the symmetry and stating why higher-order mixing is absent in this quadratic theory. revision: yes

  2. Referee: [weak-interaction algebraic expression] The algebraic demonstration of further TDOS amplification is restricted to the weak-interaction limit; it is unclear whether the same non-local parameters continue to produce net enhancement once local interactions are treated non-perturbatively or when the boundary condition is imposed exactly (the abstract states the result holds for the chosen SC boundary condition but does not show the intermediate steps that would confirm absence of mixing).

    Authors: The algebraic amplification result is derived and presented strictly in the weak-interaction limit, as stated in the abstract and the body; we do not claim non-perturbative enhancement. The mutual-exclusivity statement, however, rests on the exact symmetry relation that is independent of perturbation theory and holds when the boundary condition is imposed exactly within the bosonized theory. We will include the intermediate algebraic steps of the symmetry derivation in the revision and add a clarifying sentence that the weak-limit result is a concrete prediction while the exclusivity is non-perturbative. revision: partial

Circularity Check

1 steps flagged

Minor self-citation to prior non-SC boundary condition work; derivation otherwise independent within LL model

specific steps
  1. self citation load bearing [Abstract]
    "We also show that, in the full parameter regime comprising both the local and the non-local interaction, the region of enhanced TDOS for LL junction with superconducting boundary condition and that of non-superconducting charge conserving boundary condition (discussed in Phys. Rev. B 104, 045402 (2021)) are mutually exclusive. We show that this fact can be understood in terms of symmetry relation established between the superconducting and non-superconducting sectors of the theory."

    The mutual-exclusivity claim for the full parameter regime is justified by reference to the 2021 paper (likely same-author prior work) for the non-SC sector; while the symmetry relation is asserted here, the load-bearing comparison between SC and non-SC regions reduces to that external result rather than being independently re-derived from first principles in the present manuscript.

full rationale

The paper derives an algebraic expression for TDOS amplification in the weak-interaction limit directly from the bosonized Hamiltonian with local and non-local terms plus the superconducting boundary condition. The mutual-exclusivity statement references the 2021 paper only for the contrasting non-SC case and invokes a symmetry relation established in the present theory; this citation supplies context rather than defining or forcing the new results. No fitted parameters are relabeled as predictions, no ansatz is smuggled via citation, and no self-definitional loop appears in the claimed expressions. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The calculation assumes the standard Luttinger-liquid bosonization framework plus a specific non-local interaction kernel whose form is not derived here. No new particles or forces are introduced; the interaction parameters are treated as tunable inputs rather than fitted constants.

free parameters (1)
  • non-local interaction strength parameters
    Chosen to demonstrate amplification; their specific values are not derived from microscopic physics within the paper.
axioms (2)
  • domain assumption Luttinger liquid description remains valid near the junction when both local and non-local density-density interactions are present
    Invoked to justify the algebraic treatment and boundary-condition analysis.
  • domain assumption Superconducting and charge-conserving boundary conditions can be imposed independently without mixing higher-order terms
    Required for the symmetry relation that produces mutual exclusion of enhanced-TDOS regions.

pith-pipeline@v0.9.0 · 5899 in / 1597 out tokens · 24879 ms · 2026-05-24T12:58:39.809312+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 1 internal anchor

  1. [1]

    The incident incoming electron at the junction reflects back as the outgoing electron in the same LL QW

    Disconnected Normal (DN2) fixed point, where the QWs are disconnected from each other and also with the SC. The incident incoming electron at the junction reflects back as the outgoing electron in the same LL QW

    work page

  2. [2]

    Fully transmitting charge conserving fixed point, where the QWs are strongly coupled to each other at the junction but disconnected from the super- conductor, such that an incoming electron along the wire 1 is perfectly transmitted in the wire 2 and vice-versa

    work page

  3. [3]

    Disconnected Andreev(A2) fixed point, where the QWs are disconnected with each other but are strongly coupled to the superconductor, such that the incident incoming electron current at the junc- tion reflects as the outgoing hole current in the same LL wire

    work page

  4. [4]

    The incoming electron current along wire 1 is perfectly transmit- ted as a hole current in wire 2 and vice-versa55–60

    Cross Andreev reflection (CA2) fixed point, where the QWs are connected to the superconductor and strongly coupled to each other also. The incoming electron current along wire 1 is perfectly transmit- ted as a hole current in wire 2 and vice-versa55–60. Here, we primarily focus on the Andreev fixed point A2 and the cross Andreev reflection CA2 fixed points, wh...

    work page

  5. [5]

    L" and “R

    α is the interaction betweenρiR/L and ρiL/R, 2) β is the interaction between ρ1R/L and ρ2R/L, and 3) γ is the inter- action between ρ1R/L and ρ2L/R. The parameters, α, β, γ will symbolize the interaction strength in this article unless otherwise mentioned. The dashed black line at the junction denotes the direct Andreev Reflection. The dashed red line at t...

    work page

  6. [6]

    Before we go further, we first discuss the TDOS in the limitx→∞

    Here, we mainly focused on TDOS at the junction and its relative evolution at finite distancex away from the junction. Before we go further, we first discuss the TDOS in the limitx→∞ . The energy power law of the TDOS in this limit is denoted by∆∞, such that ∆∞ i = 1 2νi ( 1− β√ (1− β)2− (α− γ)2 + 1 + β√ (1 + β)2− (α + γ)2 ( . (19) ∆∞ i does not depend on t...

    work page

  7. [7]

    Even the highly suppressedν = 1/3 edge shows the TDOS enhancement, although in the strong interaction regime

    TDOS can show enhancement in the vicinity of the superconducting junction in the presence of non- local interactions. Even the highly suppressedν = 1/3 edge shows the TDOS enhancement, although in the strong interaction regime. This should be contrasted with the current conserving BC studied in Ref.49, where TDOS enhancement was not pos- sible ν = 1/3 QH ...

    work page

  8. [8]

    If we only consider the Andreev type insta- bilities, then we can have simultaneous TDOS en- hancement and stability of the fixed point at the junction (see fig

    Simultaneous TDOS enhancement and stability of the junction fixed point is impossible for a super- conducting junction against all physically relevant perturbations that can be switched on at the junc- tion. If we only consider the Andreev type insta- bilities, then we can have simultaneous TDOS en- hancement and stability of the fixed point at the junction...

    work page

  9. [9]

    A LL QW in proximity to a superconductor shows TDOS enhancement at the junction. In the weakα limit (α << 1), upon introducing non-local inter- action, specifically, increasingβ while keeping γ at γ = 0, can further boost the existing enhancement in TDOS for finiteα

    work page

  10. [10]

    There exist a symmetry relation between the fixed point corresponding to current conserving bound- ary condition and superconducting boundary con- dition. As a result of which, upon the introduction of superconducting correlations at the junction, the interaction parameter regime in which TDOS is enhanced at the junction for the fixed point cor- responding ...

    work page

  11. [11]

    Exploring the fractional quantum Hall effect with electron tunneling

    In general, the TDOS does not follow spatial power law behavior at finite distance away from the junc- tion. Hence, we can identify a quantity "relative TDOS", defined as ρ(x, E)/ρ(x −→ 0, E), which shows a pure spatial power law dependence in the E−→0 limit. We observed that, in the interaction parameter regime in which TDOS is less suppressed atthejunctio...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.87 2019

  12. [12]

    Hence, the induced pair amplitude, Re [Fij(x)], is given by Re [Fij(x)] ∼ ( 1 2πδ ) 2∏ k=1 (2π L )Λ0 ijk (δ)Λ1 ijk × ( δ2 + 4x2) Λ2 ijk+Λ3 ijk 2 cos π 2 ( Λ3 ijk− Λ2 ijk ) (B4) Then, the relative pair correlation function is given by Re [Fij(x)] Re [Fij(x→ 0)]∼ (δ2 + 4x2 δ2 )∑2 k=1 Λ2 ijk+Λ3 ijk 2 . (B5) Appendix C: Stability analysis against Andreev type...

    work page