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Two-body interaction induced phase transitions and intermediate phases in nonreciprocal non-Hermitian quasicrystals

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Pith's one-line read Two interacting bosons on a nonreciprocal quasiperiodic lattice turn the single-particle localization critical point into a finite intermediate mobility-edge phase, and under open boundaries the resulting doublons skin-localize to an edge…

desk verdict Clean doublon effective model and a plausible but under-supported intermediate-phase claim; worth refereeing with requests for scaling and biorthogonal checks. read the letter →

arxiv 2412.11623 v3 pith:FNQHZWGO submitted 2024-12-16 cond-mat.dis-nn cond-mat.quant-gascond-mat.stat-mechquant-ph

Yalun Zhang , Longwen Zhou This is my paper · ORCID
classification cond-mat.dis-nncond-mat.quant-gascond-mat.stat-mechquant-ph
keywords non-HermitianquasicrystalsAubry-André-HarpermodeltwointeractingbosonsBose-Hubbardinteractionmobilityedgedoublonsskineffectlocalizationtransition
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

Two interacting bosons on a one-dimensional Aubry-André-Harper lattice with asymmetric, direction-biased hopping can turn a single sharp localization critical point into a finite intermediate phase. In that phase, delocalized two-boson states coexist with spatially localized bosonic pairs, called doublons, that form only because of the onsite Hubbard interaction. The paper argues that this occurs in both a unidirectional-hopping limit and the general asymmetric-hopping model, and that the boundaries of the phase carry topological signatures in a pair of winding numbers. Under open boundaries, the doublons also show a non-Hermitian skin effect, condensing at the right or left edge according to which hopping amplitude is larger. If correct, this offers a concrete route to interaction-generated mobility edges and switchable edge condensation in non-Hermitian quasicrystals.

What carries the argument

The argument runs on two objects. The first is the inverse-participation-ratio family $\mathrm{IPR}_{\max}$, $\mathrm{IPR}_{\min}$, and $\zeta = \log_{10}(\mathrm{IPR}_{\mathrm{ave}} \cdot \mathrm{NPR}_{\mathrm{ave}})$, which distinguishes extended, intermediate, and localized phases from exact diagonalization of the two-boson Fock space. The second is an effective doublon Hamiltonian produced by second-order degenerate perturbation theory, in which a bound boson pair hops nonreciprocally with amplitudes $J_L^2/U$ and $J_R^2/U$ on the same quasiperiodic lattice. A pair of winding numbers $(\omega_1, \omega_2)$, taken at two base energies, marks the two phase boundaries by jumping from zero to nonzero values.

What would settle it

Compute $\mathrm{IPR}_{\min}$ and $\zeta$ at the same parameter points inside the claimed intermediate phase (e.g., where $\zeta$ is finite in Fig. 3(d) or Fig. 6(d)) for $L = 144, 233, 377$; if $\mathrm{IPR}_{\min}$ does not approach zero or $\zeta$ does not remain finite as $L$ grows, the intermediate mobility edge is a finite-size artifact. A complementary check is to recompute the IPRs with biorthogonal eigenvectors and see whether the localized-extended coexistence persists.

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Extended reading notes

Core claim

The paper's central claim is that the noninteracting triple critical point at $|V| = \max(J_L, J_R)$, where the spectrum, the eigenstates, and the winding number all change at once, is not robust against two-boson interactions. For $U \neq 0$, the real-to-complex spectral transition stays at the noninteracting position, but the full delocalization transition moves toward stronger hopping asymmetry, leaving an intermediate mobility-edge phase in which localized doublons with real energies coexist with extended states with complex energies. The localization diagnostics are $\mathrm{IPR}_{\max}$, $\mathrm{IPR}_{\min}$, and the $\zeta$ function: in the claimed intermediate phase $\mathrm{IPR}_{\max} > 0$, $\mathrm{IPR}_{\min} \to 0$, and $\zeta$ stays finite. In the strong-interaction regime, the paper derives an effective single-doublon Hamiltonian with nonreciprocal hopping amplitudes $J_L^2/U$ and $J_R^2/U$, yielding a doublon-level transition at $|V| = \max(J_L^2, J_R^2)/U$. Under open boundary conditions, doublons skin-localize to the right edge when $J_L^2 < J_R^2$ and to the left edge when $J_L^2 > J_R^2$, so the hopping imbalance selects the direction of doublon condensation.

Load-bearing premise

The central claim rests on classifying states by inverse participation ratios at a single lattice size $L=89$, with no finite-size scaling; if those labels shift as the lattice grows, or are distorted by the nonorthogonality of right eigenvectors, the intermediate phase could disappear.

Editorial extensions

If this is right

  • At any nonzero $U$, the single-particle critical point of the noninteracting model is replaced by a finite intermediate phase, so the triple transition is split.
  • The spectral and localization transitions decouple: the spectrum becomes complex at the old critical value, while full delocalization is pushed to larger hopping imbalance.
  • In the strong-interaction limit, doublons localize at $|V| = \max(J_L^2, J_R^2)/U$, a threshold that differs from the single-particle one by a factor set by the interaction.
  • Under open boundaries, the doublon skin effect is switchable: crossing $J_L = J_R$ reverses the edge on which doublons condense, which acts as a doublon pump.
  • The localized-to-intermediate and intermediate-to-extended transitions are topological, signalled by the quantized changes of $\omega_1$ and $\omega_2$.

Reading between the lines

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

  • A dynamical quench crossing $J_L = J_R$ should shuttle the doublon cloud from one edge to the other; this is a directly testable consequence of the open-boundary skin-effect direction rule, though the paper does not simulate the quench.
  • Because the phase labels rely on right-eigenvector IPRs, recomputing them with biorthogonal participation ratios would show whether the intermediate phase survives the nonorthogonality of non-Hermitian eigenstates.
  • The same effective-Hamiltonian logic for three bosons would predict triplon skin effects and intermediate phases with thresholds set by higher powers of $J/U$; the paper reports preliminary three-boson evidence in that direction.
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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 two interacting bosons in a one-dimensional Aubry-André-Harper lattice with nonreciprocal hopping (Eq. (1)). It claims that the Hubbard interaction expands the noninteracting critical point |V| = max(J_L, J_R) into an intermediate mobility-edge phase in which extended two-boson states coexist with localized doublons, that the relevant transitions carry winding-number signatures, and that in the strong-interaction limit the doublons obey an effective nonreciprocal AAH model whose skin-effect direction is switchable by tuning the hopping imbalance. Numerical support comes from exact diagonalization of the two-boson Fock space, mostly at L = 89, using IPR-based diagnostics, plus a second-order effective doublon Hamiltonian in Sec. IV.

Significance. If the intermediate-phase claim survives the thermodynamic limit, the paper would provide a useful few-body example in which interactions convert a single-particle critical point into a coexistence region, and the doublon skin-effect prediction in Sec. IV is a clean, experimentally meaningful statement. The effective doublon derivation leading to Eq. (15) is a notable strength: it is transparent, parameter-light, and explicitly benchmarked against exact spectra in Fig. 8. The paper also states a falsifiable transition condition for doublon localization, Eq. (16). However, the central phase diagrams rest on L = 89 IPR data without finite-size scaling, so the thermodynamic status of the intermediate phase is not yet established.

major comments (3)
  1. [II.B; Figs. 3 and 6] Section II.B defines the intermediate-phase criteria (IPRmax > 0, IPRmin -> 0, zeta finite) as statements in the L -> infinity limit, but Figs. 3 and 6 show only L = 89. This is load-bearing because the localized sector contains at most L doublon states in a Fock space of dimension D ~ L^2, so a single localized state already makes IPRmax > 0 without implying a finite density of localized states. Moreover, for a subextensive localized fraction one has zeta = log10(IPRave * NPRave) ~ -log L, so the 'finite zeta' signature at L = 89 may be a finite-size artifact. I request a scaling analysis of IPRmax, IPRmin and zeta for several system sizes at fixed parameters, together with a statement of whether the intermediate region persists in the thermodynamic limit or collapses onto the U = 0 critical line.
  2. [II.B, Eq. (4)] The IPR is computed from right eigenvectors with the standard normalization, as the paper explicitly notes. In non-Hermitian systems, the nonorthogonality of the right eigenbasis can distort IPR values and phase boundaries. Since IPRmax and IPRmin are the primary phase labels in Figs. 3 and 6, please provide a biorthogonal IPR (or at least a comparison with left-eigenvector IPR) and show that the apparent coexistence of extended and localized states is not a normalization artifact.
  3. [III, Eq. (10); Figs. 4 and 7] The second winding number omega_2 uses a base energy E_B2 defined as the real part of the energy of the eigenstate with the largest IPR. Because the same largest-IPR criterion is used to identify the intermediate-to-extended boundary, omega_2 is not an independent topological witness; the paper itself notes that the integer values depend on the system size and on base-point encirclement. Please either define E_B2 from a parameter-independent spectral feature and provide scaling of omega_2, or restrict the conclusions to 'topological signatures' rather than a topological characterization of the transitions.
minor comments (5)
  1. [General] Please add an availability statement for code and data; the exact-diagonalization data behind Figs. 3 and 6 would allow the requested finite-size checks to be reproduced by other groups.
  2. [Figs. 3(d) and 6(d)] The plotted zeta values are shifted and rescaled as (zeta - min)/max; the raw zeta scale should be stated in the caption so that the reader can judge what 'finite' means in the intermediate phase.
  3. [II.A, after Eq. (3)] There is a typographical spacing issue in 'When rho_Im = 0(rho_Im > 0)' that should be corrected to 'When rho_Im = 0 (rho_Im > 0)'.
  4. [IV, Eq. (15)] The second-order derivation of the effective doublon Hamiltonian is compressed; even though Refs. [119,120] are cited, a few lines showing the intermediate |1_l,1_m> states and the energy denominators would make the result easier to verify.
  5. [V, Conclusion] The statement about preliminary three-boson calculations is not supported by any data in the paper; either add supporting results or label the claim explicitly as speculation.

Circularity Check

1 steps flagged · score 2.0 of 10

The winding number omega_2 is defined from the largest-IPR eigenstate, so its transition boundary is built from the same localization diagnostic it is said to confirm; the central intermediate-phase and doublon claims rest on independent exact-diagonalization and effective-model comparisons and are not circular.

  1. self definitional [Sec. II.B, Eq. (10); Sec. III.A, Fig. 4; Sec. III.B, Fig. 7]
    "In numerical calculations, we choose E_B^1 = 0 and set E_B^2 as the real part of energy of the eigenstate of H with the largest IPR. ... Meanwhile, the quantized jump of omega_2 from zero to a finite integer corresponds to the delocalization transition, after which all the eigenstates become extended."

    The base energy E_B^2 is defined from the eigenstate with the largest IPR, so the zero-to-nonzero transition of omega_2 is tied to whether that maximal-IPR state is localized (real energy, outside the complex spectral loop) or extended (complex energy, with Re(E) inside the loop). The boundary where omega_2 changes is therefore the same boundary set by IPR_max, merely repackaged as a winding number. Presenting omega_2 as a topological signature of the delocalization transition does not provide an independent check of the IPR_max-based phase boundary. This circularity is local to the topological characterization, not to the central intermediate-phase claim, which is supported by IPR_max/IPR_min/zeta and by the separately derived effective doublon Hamiltonian.

full rationale

The paper's central physical claims are not circular. The intermediate mobility-edge phase is diagnosed from IPR_max, IPR_min and zeta computed by exact diagonalization of the microscopic two-boson Hamiltonian, and the doublon sector is independently described by a second-order effective Hamiltonian (Eq. 15) whose parameters (J_L^2/U, J_R^2/U, 2V) are derived from the microscopic model rather than fitted. The effective doublon model is then compared directly to the full Hamiltonian spectra in Fig. 8, providing a non-circular consistency check. There is no load-bearing self-citation: prior work by the same author is cited for context and for well-known single-particle nonreciprocal AAH results, not to justify the new phase. The one genuine circular step is the definition of the winding number omega_2: its base energy E_B^2 is chosen as the real part of the energy of the eigenstate with the largest IPR, so the omega_2 transition is constructed to track the IPR_max boundary. The paper itself softens this by noting that the winding numbers provide 'topological signatures ... rather than topological characterizations of the phases' and that their integer values depend on system size. Because the intermediate-phase claim and the doublon skin-effect claim do not rest on omega_2, the overall circularity is mild and localized to the topological-diagnostic framing.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The model parameters V, J_L, J_R, U and alpha are physical inputs, not fitted parameters. The only ad hoc parameter in the analysis is the diagnostic base energy E_B2, which is chosen from eigenstate data. No new particles, forces, or conserved quantities are introduced.

free parameters (1)
  • E_B2, second winding base energy = Re(E) of the eigenstate with largest IPR at each parameter point
    Chosen in Eq. (10) to make the winding number omega_2 vanish in the intermediate phase and become nonzero in the extended phase. Because it is defined from the same eigenstate data used to identify phases, the resulting transition boundary is not an independent check.
assumptions (4)
  • domain assumption Noninteracting non-Hermitian AAH phase boundary at |V| = max(J_L, J_R)
    Adopted from Refs. [49,62] as the baseline against which interaction effects are measured; not rederived in this paper.
  • domain assumption Second-order degenerate perturbation theory is sufficient for doublons when U >> (J_L, J_R, V)
    Used to derive Eq. (15); higher-order processes could renormalize the effective hopping and alter the threshold Eq. (16).
  • ad hoc to paper IPR-based criteria (IPRmax, IPRmin, zeta) at L=89 represent the thermodynamic limit
    The paper states L tending to infinity limits in Sec. II.B but only shows L=89 data with no finite-size scaling (Figs. 3 and 6).
  • domain assumption Right-eigenvector IPR with standard normalization is a valid localization measure for non-Hermitian systems
    Stated in Sec. II.B; nonorthogonality of right eigenvectors can distort IPR values in non-Hermitian systems.

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Cite this review

Pith. "Pith review of Two-body interaction induced phase transitions and intermediate phases in nonreciprocal non-Hermitian quasicrystals." pith.science (2026). https://pith.science/paper/FNQHZWGO

@misc{pith2026241211623,
  author       = {Pith},
  title        = {Pith review of: Two-body interaction induced phase transitions and intermediate phases in nonreciprocal non-Hermitian quasicrystals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FNQHZWGO}},
  note         = {Machine review of arXiv:2412.11623}
}
read the original abstract

Non-Hermitian phenomena, such as exceptional points, non-Hermitian skin effects, and topologically nontrivial phases have attracted continued attention. In this work, we reveal how interactions and nonreciprocal hopping could collectively influence the behavior of two interacting bosons on quasiperiodic lattices. Focusing on the Bose-Hubbard model with Aubry-Andr\'e-Harper quasiperiodic modulations and hopping asymmetry, we discover that interactions could enlarge the localization transition point of the noninteracting system into an intermediate mobility edge phase, in which localized doublons formed by bosonic pairs can coexist with delocalized states. Under the open boundary condition, the bosonic doublons could further show non-Hermitian skin effects, realizing doublon condensation at the edges, and their direction of skin-localization can be flexibly tuned by the hopping parameters. A framework is developed to characterize the spectral, localization, and topological transitions accompanying these phenomena. Our work advances the understanding of localization and topological phases in non-Hermitian systems, particularly in relation to multiparticle interactions.

Figures

Figures reproduced from arXiv: 2412.11623 by the authors.

Figure 1
Figure 1. FIG. 1. A sketch of the model [in (a)] and its single-particle [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Energy spectra of the UHM under the PBC. The [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Spectral and localization transitions of the UHM vs [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Energy spectra of the AHM under the PBC. Other [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Spectral and localization transitions of the AHM [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Energy spectra, IPRs and probability distributions of the bosonic doublons. The length of lattice [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Phase diagram of the doublon effective Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Topological doublon edge states induced by the spatially modulated interactions

    cond-mat.str-el 2025-02 conditional novelty 5.0 of 10

    Spatial modulation of interactions in 1D two-particle systems generates topological doublon edge states via an effective Aubry-André-Harper model.

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