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Interaction-Induced Second-Order Skin Effect

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read By adding interactions to a non-Hermitian square lattice of bosons, the paper shows that bound boson pairs localize at the corners in a way that scales with system size—a second-order skin effect driven entirely by interactions.

desk verdict Interaction-induced corner skin modes for doublons are convincingly demonstrated, but the second-order classification needs a stronger scaling check and a less heuristic bulk-delocalization argument. read the letter →

arxiv 2501.06816 v2 pith:GMIPIXG6 submitted 2025-01-12 quant-ph cond-mat.mes-hall

Wen-Zheng Ling , Zhao-Fan Cai , Tao Liu This is my paper · ORCID
classification quant-phcond-mat.mes-hall
keywords non-Hermitianskineffecthigher-ordermany-bodyinteractionsdoublonsBose-Hubbardmodelnonreciprocalhoppingcornermodespoint-gaptopology
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to show that many-body interactions, rather than single-particle band engineering, can produce a second-order non-Hermitian skin effect. The authors study a two-dimensional square lattice of bosons with nonreciprocal hopping, onsite interactions, and two-boson pairing hopping, and work in the two-boson sector. They find that bound boson pairs, called doublons, form in-gap states that localize at the lattice corners while the bulk doublon states stay extended, and that the number of corner modes grows linearly with the linear size of the lattice. That linear scaling is the defining signature of a second-order skin effect, so the paper concludes that interactions alone can drive this higher-order phenomenon.

What carries the argument

The engine of the argument is the effective doublon Hamiltonian obtained by quasi-degenerate second-order perturbation theory. In the strong-interaction limit, two bosons bind into a doublon that acts as a single quasiparticle with effective symmetric hopping $J^2/U$ along x, alternating effective unidirectional hopping $t^2/U$ along y, and a direct pairing-hopping term $P$ that staggers the x hoppings into an SSH pattern. That effective model supplies two ingredients: SSH-like edge states localized at the right boundary when the x chain is open, and destructive interference of the alternating y-hopping in the bulk that leaves only the boundary hopping unidirectional. The point-gap topology of the edge states, quantified by the winding number $W=2$, then drives the corner localization when both directions are open.

What would settle it

For a range of interaction strengths and hopping parameters, compute the inverse participation ratio of every doublon eigenstate under open boundary conditions in both directions, and plot the corner-mode count $N_c$ versus $L_y$ at fixed $L_x$. If a macroscopic fraction of the bulk doublon states also localizes at an edge, or if $N_c$ grows with the area $L_x L_y$ instead of linearly with $L_y$, then the corner modes are part of a first-order skin effect and the second-order claim fails.

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Extended reading notes

Core claim

The central claim is that interactions alone induce a second-order non-Hermitian skin effect in the two-boson sector of this non-Hermitian Bose-Hubbard model, even though the single-boson model exhibits no such effect. In the strong-interaction limit, second-order perturbation theory maps the two-boson sector onto an effective single-doublon model: the doublon hops symmetrically along x with amplitude $J^2/U$, experiences an additional staggered pairing-hopping term $P$, and undergoes alternating unidirectional hopping along y. In the x direction this effective lattice is an SSH-like chain whose topologically protected edge states sit at the right boundary; along the y direction the alternating unidirectional hopping cancels in the bulk through destructive interference but remains unidirectional at the boundary, pushing those edge states into the top-right corner. The corner modes carry a nonzero many-body winding number ($W=2$), survive a compensating potential that eliminates Tamm-Shockley edge states, remain localized under disorder, and scale in number with the lattice length, which the authors identify as the second-order skin effect.

Load-bearing premise

The second-order classification stands on the assumption that bulk doublon states remain extended, and do not also pile up at a boundary, once both directions are open; the paper argues this with a destructive-interference picture and shows density plots for chosen parameters, but does not give a general proof.

Editorial extensions

If this is right

  • The corner skin modes appear for both $P>0$ and $P<-2J^2/U$, and their number $N_c$ grows linearly with the lattice length $L_y$, distinguishing the second-order effect from a first-order volume skin effect.
  • The in-gap corner states survive a compensating potential that removes Tamm-Shockley edge states, showing that the effect originates from interaction-driven topological edge states rather than trivial surface defects.
  • The corner modes remain localized under strong disorder in the hopping amplitudes, consistent with the topological point-gap origin claimed by the authors.
  • The same mechanism produces corner skin modes in the three-excitation subspace, indicating that the effect is not limited to the two-boson sector.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The effective doublon Hamiltonian is a single-particle model, so a classical platform that emulates the doublon hopping, such as an electrical circuit network, could in principle display the same corner skin modes without needing true two-particle quantum dynamics.
  • The design principle implied by the paper is modular: any one-dimensional chain with interaction-induced edge states, when coupled by boundary-only nonreciprocal hopping, should give corner skin modes. Testing this on other interacting lattice models would show whether the phenomenon is generic rather than specific to this Bose-Hubbard realization.
  • A stress test beyond the paper's parameter sets is to leave the strong-interaction limit: if the corner modes persist at moderate $U$, the mechanism is robust; if they vanish exactly where the perturbative doublon picture breaks down, the effect hinges on the doublon approximation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies a two-dimensional non-Hermitian Bose-Hubbard model with nonreciprocal single-boson hopping along y, reciprocal hopping along x, two-boson pairing hopping, and onsite interactions. It claims that, in the two-boson sector, interactions create a band of doublon states with in-gap edge states under x-OBC; when y-OBC is also imposed, these edge states become corner-localized doublon skin modes. The authors derive an effective doublon Hamiltonian by second-order perturbation theory, solve the 1D SSH-like edge problem analytically, obtain the condition P>0 or P<-2J^2/U for edge states, and compute a many-body winding number W=2. They also argue that the corner modes are robust to disorder and propose an ultracold-atom realization. The central claim is an interaction-induced second-order non-Hermitian skin effect whose corner-mode count grows linearly with the linear system size.

Significance. If fully established, the result would be a genuinely new mechanism for higher-order NHSE: many-body interactions, rather than single-particle nonreciprocity, produce corner skin modes in the doublon sector. The paper has clear strengths: the effective doublon Hamiltonian is derived from the original model without fitting and matches exact diagonalization of the full Hamiltonian in Fig. 9; the analytic edge-state condition is explicit and also tested for the negative-P regime in Appendix D; the winding number is computed directly from the interacting Hamiltonian; and the disorder robustness is demonstrated numerically. The main gap is that the scaling evidence that distinguishes a second-order effect from a first-order skin effect is incomplete, and the bulk-delocalization assumption that is load-bearing for the order classification is supported only heuristically.

major comments (3)
  1. [Sec. III.A and Fig. 3(b)] The central scaling claim is not established by the presented data. Figure 3(b) plots N_c versus L_y at fixed L_x, and the text states that a first-order skin effect would scale as L_x L_y 'regardless of variations in Ly.' At fixed L_x, a first-order skin effect along y also produces a number of localized states proportional to L_y, so the plot cannot distinguish N_c ~ L_y (second order) from N_c ~ L_x L_y (first order along y). The effective model in Sec. III.B predicts one edge state per x-chain, hence N_c ~ L_y independent of L_x, which is a direct and testable consequence. Please add the N_c versus L_x dependence at fixed L_y, or otherwise report the full (L_x, L_y) scaling and compare with the first-order prediction.
  2. [Sec. III.A and Sec. III.B] The classification as a second-order skin effect assumes that bulk doublon states remain extended and do not accumulate at a boundary. The paper supports this only by the heuristic destructive-interference argument in Sec. III.B and by selected density plots in Figs. 2(i-l) and 10(g,h); no quantitative measure of localization is given. If a macroscopic fraction of bulk doublon states localized along the top or bottom edge, the corner modes would be part of a first-order skin effect rather than a genuine second-order one. Please compute the inverse participation ratio or participation entropy for the doublon bulk eigenstates as a function of L_x and L_y, and show that the number of non-extended bulk states is O(1), or provide a generalized-Brillouin-zone argument for the effective Hamiltonian in Eq. (8) that proves the absence of bulk skin accumulation.
  3. [Sec. III.A, Fig. 3(b)] The definition of the integer N_c used in the scaling plot should be stated precisely. It is not clear from the text whether N_c counts all in-gap states in a chosen energy window, all states with density above a corner threshold, or something else. Without a reproducible counting criterion, the scaling plot in Fig. 3(b) cannot be independently verified, and this is directly related to the load-bearing claim of linear scaling.
minor comments (5)
  1. [Sec. III.A] The statement that 'the single-boson case ... exhibits no second-order skin effects [see Appendix B]' is not supported by Appendix B, which instead computes the skin corner weight for the two-particle model with U=0 or P=0. Please either add an explicit single-boson analysis or correct the cited appendix.
  2. [Sec. III.B, around Eq. (19)] The wording 'it is disappears when P>0 or P<-2J^2/U' is ungrammatical and the logic of the sentence is hard to parse. Please rephrase to state explicitly which boundary state exists for which parameter range, e.g., 'the higher-energy boundary state is absent, and the lower-energy state is localized, when ...'.
  3. [Sec. III.B, Eq. (8)] The notation for the effective Hamiltonian would be clearer if the authors stated explicitly that the sum over x runs over sites in the effective lattice and that the boundary modification of U_eff applies only under x-OBC; the transition from Eq. (8) to Eq. (9) is currently abrupt.
  4. [Eq. (4)] The many-body winding number is stated to be W=2, but the text gives no numerical illustration of the determinant loop or its winding. Please clarify which subspace (full two-boson Hilbert space or doublon subspace) is used in the determinant, and show the loop or provide the numerical values used.
  5. [References] Reference [88] and reference [109] appear to be the same paper (Brighi and Nunnenkamp, Phys. Rev. A 110, L020201 (2024)). Please deduplicate or cite different sources if both entries were intended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the effective-model derivation, winding-number computation, and scaling check are self-contained.

full rationale

The derivation chain is self-contained. The effective doublon Hamiltonian (Eq. 8) is obtained by quasi-degenerate second-order perturbation theory (Eq. 7, Appendix C) directly from the original Hamiltonian (Eq. 1), with no fitted parameters; its accuracy is cross-checked against the full two-boson spectrum in Fig. 9. The existence and localization of the 1D edge states are derived analytically via Eqs. (10)-(18), and the nonzero winding number W=2 is computed from the interacting Hamiltonian via Eq. (4), not imported from prior work. The linear scaling of N_c with L_y (Fig. 3b) is a numerical observation, not a refit of a parameter used to produce the model; the corner-mode count is read off from exact diagonalization. Self-citations (refs. 32, 35, 36, 79) appear only as background context. The skeptical concern that Fig. 3(b) varies L_y at fixed L_x, so an O(L_x L_y) first-order skin effect would also be linear in L_y, is an evidential limitation of the scaling test, not a circular reduction: nothing in the paper's construction forces the corner count by definition, and the analytic one-edge-state-per-chain picture could in principle be falsified by a future N_c versus L_x scan. No circular step is exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The model parameters J, t, U, P are Hamiltonian inputs, not fitted quantities; the central claim holds over a range (P>0 or P<-2J^2/U). The effective doublon Hamiltonian and the boundary-state condition are derived from these inputs. The main unproven load-bearing premise is the bulk delocalization via destructive interference, listed as an axiom. No new entities are introduced.

assumptions (4)
  • standard math Quasi-degenerate second-order perturbation theory is valid for the parameter regime considered (|U| >> J, t, P).
    Used to derive the effective doublon Hamiltonian Heff in Eq. (8) and Appendix C. The numerical comparison in Fig. 9 supports it at U/J=8, but P/U=0.5 is not asymptotically small.
  • domain assumption The effective doublon Hamiltonian captures the full two-boson physics; the coupling to scattering states is negligible.
    Second-order perturbation theory projects onto the doublon subspace; the agreement in Fig. 9 under PBC and OBC supports this.
  • ad hoc to paper Bulk doublon states remain extended due to destructive interference of alternating unidirectional y-hopping; no bulk skin accumulation occurs.
    Stated heuristically in Sec. III.A and III.B without a rigorous derivation; verified numerically for specific parameters (Fig. 2 i-l) and essential for classifying the effect as second-order rather than first-order.
  • standard math The many-body winding number W computed with twisted boundary conditions (Eq. 4) characterizes the point-gap topology of the in-gap edge states.
    Follows refs 76, 78, 80; used to establish the topological origin of corner modes.

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Pith. "Pith review of Interaction-Induced Second-Order Skin Effect." pith.science (2026). https://pith.science/paper/GMIPIXG6

@misc{pith2026250106816,
  author       = {Pith},
  title        = {Pith review of: Interaction-Induced Second-Order Skin Effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GMIPIXG6}},
  note         = {Machine review of arXiv:2501.06816}
}
abstract

In contrast to the conventional (first-order) non-Hermitian skin effect (NHSE) in a $d$-dimensional system with linear size $L$, the $n$th-order (higher-order) NHSE is characterized by skin modes localized at lower-dimensional boundaries of dimension $(d-n)$. The total number of these modes scales linearly with the system size $L$. Significant progress has been made in understanding higher-order NHSE in non-interacting systems. In this work, we demonstrate the many-body interaction induced second-order skin effect in a two-dimensional non-Hermitian bosonic system. Specifically, we construct a square lattice that incorporates nonreciprocal single-boson hopping, onsite many-body interactions and two-boson pairing hopping. In the absence of interactions, no second-order NHSE is observed. However, with the inclusion of interactions, we identify interaction-induced skin modes for in-gap doublon states (i.e., bound pairs of bosons) localized at the corners of the lattice, while the bulk doublon states remain extended. These corner-localized skin modes arise from the interplay between interaction-induced edge states, localized along one-dimensional boundaries, and the nonreciprocal hopping along these boundaries. Furthermore, the number of corner skin modes scales linearly with the system size, confirming the presence of second-order NHSE in this interacting system. Our findings introduce a novel approach to realizing higher-order skin effects by leveraging interactions.

Figures

Figures reproduced from arXiv: 2501.06816 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic of the square lattice model in the presence [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a-d) Complex eigenenergy spectrum of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Complex eigenenergy spectrum of in-gap corner [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Complex eigenenergy spectrum of in-gap corner [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Real part and (b) absolute value of eigenenergies [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Complex eigenenergy spectrum of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Implementation of asymmetric hopping in optical lattices. (a) Schematic diagram of the experimental setup for a [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The complex eigenenergy spectrum of doublon bands, [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. (a-d) Complex eigenenergy spectrum of [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. (a) Complex eigenenergy spectrum of in-gap corner [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. (a) Complex eigenenergy spectrum of [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.