{"id":"b9496067-ba8c-43ee-bcc1-a918ac69e05e","arxiv_id":"2412.11623","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Interactions expand the localization critical point of a nonreciprocal quasicrystal into an intermediate mobility edge phase, and bound boson pairs acquire a tunable non-Hermitian skin effect.","lead":"This paper studies two interacting bosons in a non-Hermitian quasiperiodic lattice and finds that interactions turn the sharp single-particle localization transition into a wider intermediate phase where localized boson pairs coexist with extended states. It also shows that these boson pairs, called doublons, can pile up at the edges under open boundaries, with the pile-up direction controlled by the hopping imbalance.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The intermediate mobility-edge phase is diagnosed only at L=89 with right-eigenvector IPRs; finite-size scaling and biorthogonal checks are needed to rule out a finite-size crossover.","rationale":"The reader's weakest assumption targets precisely the lack of finite-size scaling and the use of right-eigenvector IPRs in a non-Hermitian system; my stress-test identifies the same load-bearing point. The central claim requires that the apparent coexistence of localized doublons and extended states persists as L -> infinity, and that this coexistence is not an artifact of IPR normalization in a nonorthogonal eigenbasis. Since all phase diagrams in Figs. 3, 4, 6, and 7 are generated at L = 89 with no scaling analysis, this is the least secure link in the argument. I do not see an internal algebraic error in the doublon effective model (Eq. 15) or in the large-U NHSE discussion: the second-order hopping amplitudes scale as J^2/U, the transition condition Eq. (16) follows from the standard AAH criterion for that effective model, and Fig. 8 provides direct spectral confirmation of the doublon subspace. Thus the doublon skin-effect part of the paper has independent analytic support. The moderate-U intermediate phase, however, rests entirely on exact diagonalization at one system size. A finite-size scaling run would either confirm the phase boundaries as thermodynamic features or expose them as finite-size crossovers. Before such a check is reported, the CONDITIONAL verdict is appropriate, but I would not escalate to rejection because the effective-model support and the qualitative spectra are consistent with the claimed intermediate phase.","tokens_in":19234,"tokens_out":16052,"duration_ms":162693,"concrete_test":"For representative points in each claimed phase of the UHM and AHM (e.g., UHM with J = 1: (U,V) = (2,0.25) extended, (2,0.75) intermediate, (2,1.25) localized), recompute IPRmax, IPRmin, and zeta at Fibonacci sizes L = 89, 144, 233, 377 using the same approximant for alpha. Require IPRmin ~ L^-2 and zeta -> finite constant in the intermediate phase, IPRmax -> 0 in the extended phase, and IPRmin -> constant in the localized phase. Also repeat the intermediate-phase points with biorthogonal IPRs (left-right eigenvector overlaps) to check that the coexistence is not a right-eigenvector normalization artifact. If the phase boundaries drift by more than the L = 89 finite-size spread, or if zeta scales toward -infinity in the nominally intermediate phase, the central claim is not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that for U != 0 the noninteracting critical point at |V| = max(J_L, J_R) expands into a finite intermediate phase with coexisting extended states and localized doublons (Figs. 3 and 6), and that doublons show switchable NHSE under OBC (Sec. IV). The phase labels rest on IPRmax > 0, IPRmin -> 0, and zeta finite, computed at L = 89 only (Sec. II.B, Figs. 3 and 6), while the paper invokes L -> infinity in Sec. II.B. No finite-size scaling or threshold analysis is presented. This matters because the localized sector contains at most L doublon states in a Fock space of dimension D ~ L^2, so a single localized state can make IPRmax > 0 even if the localized fraction vanishes in the thermodynamic limit; whether the apparent coexistence is a true phase or a finite-size crossover is not established. In addition, IPRs are computed from right eigenvectors with standard normalization, and nonorthogonality in the non-Hermitian regime can distort IPR-based phase boundaries. If finite-size or normalization effects produce the apparent IPRmax > 0 / IPRmin -> 0 signature, the interaction-expanded intermediate phase and the winding-number characterization built on it (Sec. III) would not survive.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19506,"tokens_out":9091,"duration_ms":86638,"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":[{"comment":"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.","section":"II.B; Figs. 3 and 6"},{"comment":"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.","section":"II.B, Eq. (4)"},{"comment":"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.","section":"III, Eq. (10); Figs. 4 and 7"}],"minor_comments":[{"comment":"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.","section":"General"},{"comment":"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.","section":"Figs. 3(d) and 6(d)"},{"comment":"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)'.","section":"II.A, after Eq. (3)"},{"comment":"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.","section":"IV, Eq. (15)"},{"comment":"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.","section":"V, Conclusion"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of cond-mat.dis-nn, and the effective-doublon part in Sec. IV is solid. The main risk is the thermodynamic status of the intermediate phase; the finite-size scaling request in Major Comment 1 is the key condition for acceptance. The base-energy choice for omega_2 should also be checked for circularity before the topological claims are emphasized further."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know: the doublon effective model is a clean, checkable result, and the intermediate mobility-edge phase is plausible but rests on L=89 IPR diagnostics with no scaling. The paper derives an effective doublon Hamiltonian in the strong-U limit, with hopping amplitudes J_L^2/U and J_R^2/U, and verifies it against exact diagonalization spectra. That derivation is the strongest part; it also gives a tunable non-Hermitian skin effect for doublons, with direction controlled by the hopping imbalance, which is a concrete prediction. The winding-number pair (omega1, omega2) is a reasonable way to assign topological signatures to the transitions, though omega2's base energy is chosen from the state with largest IPR, which is somewhat circular.\n\nThe soft spot is the center of the paper. The interaction-expanded intermediate phase is identified via IPRmax > 0, IPRmin -> 0, and finite zeta, all at L=89, with no finite-size scaling. In a two-boson Fock space of dimension ~L^2, even a single localized doublon makes IPRmax > 0, so the coexistence signature could be a finite-size crossover. Right-eigenvector IPRs in non-Hermitian systems can also be distorted by nonorthogonality; a biorthogonal check would help. None of this kills the effective-model result, but it means the headline claim about a true intermediate phase is not yet established.\n\nThe paper also cites Refs [113,114] for related interaction-induced mobility edges but does not explicitly say what is new relative to them. That should be clarified.\n\nVerdict: worth a serious referee. The effective doublon Hamiltonian is a solid contribution, and the question of whether interactions expand critical points into intermediate phases in non-Hermitian quasicrystals is worth settling. I would send it to review with a request for finite-size scaling, biorthogonal IPR, and sharper positioning against Qian et al. and Longhi. A full many-body calculation is not needed; the two-boson setting is already at the edge of exact diagonalization, and the effective model is the useful part. People working on non-Hermitian localization or interacting bosons in quasicrystals will get value from this paper, especially from the doublon physics.","headline":"Clean doublon effective model and a plausible but under-supported intermediate-phase claim; worth refereeing with requests for scaling and biorthogonal checks.","tokens_in":20010,"tokens_out":2540,"would_cite":true,"duration_ms":22775,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["non-Hermitian quasicrystals","Aubry-André-Harper model","two interacting bosons","Bose-Hubbard interaction","mobility edge","doublons","non-Hermitian skin effect","localization transition"],"falsifier":"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.","tokens_in":19019,"feed_emoji":"⚛️","tokens_out":14827,"duration_ms":119940,"temperature":0.7,"pith_summary":"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.","feed_headline":"Interactions expand a single critical point into a finite phase","feed_subtitle":"Localized doublons can coexist with extended states; hopping imbalance flips where they cluster.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $\\omega_2$."],"supporting_citations":[{"why":"Defines the noninteracting nonreciprocal AAH model and establishes its triple transition at $|V|=\\max(J_L,J_R)$, the reference point that interactions are claimed to enlarge into an intermediate phase.","marker":"[49, 62]"},{"why":"Introduces the $\\zeta$ function that the paper uses as the smoking-gun diagnostic for an intermediate mobility-edge phase.","marker":"[118]"},{"why":"Supplies the second-order degenerate perturbation theory used to derive the effective doublon Hamiltonian with hopping amplitudes $J_L^2/U$ and $J_R^2/U$.","marker":"[119, 120]"}],"fun_headline_variants":["Interactions split a triple point into a mobility-edge window","Doublons localize, singles roam: non-Hermitian quasicrystal phases","Hopping imbalance picks the edge where boson pairs condense","Interactions enlarge localization into an intermediate doublon phase","Two-body interactions create a finite mobility-edge phase"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Interactions split a triple point into a mobility-edge window","Doublons localize, singles roam: non-Hermitian quasicrystal phases","Hopping imbalance picks the edge where boson pairs condense","Interactions enlarge localization into an intermediate doublon phase","Two-body interactions create a finite mobility-edge phase"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000224,"raw_usage":{"total_tokens":1507,"prompt_tokens":1035,"completion_tokens":472,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":651,"completion_tokens_details":{"reasoning_tokens":386}},"tokens_in":651,"tokens_out":472,"duration_ms":4738,"temperature":1.0,"reasoning_tokens":386,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:45:45.761466+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}