{"id":"83d843dd-fb21-4176-95ba-d3c9e1137466","arxiv_id":"2608.03024","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"In the hole-doped t-J Kitaev model, ferromagnetic Kitaev coupling produces triplet p-wave superconductivity that coexists with ferromagnetism, and antiferromagnetic coupling drives a doping-dependent transition from spin-selective triplet to d+id singlet pairing.","lead":"This paper uses large-scale variational Monte Carlo simulations to map what happens when holes are added to the Kitaev quantum spin liquid on a honeycomb lattice. It finds that ferromagnetic Kitaev couplings yield a triplet p-wave superconductor coexisting with ferromagnetism, while antiferromagnetic couplings switch the dominant pairing symmetry as doping grows.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The phase diagram rests on the unverified assumption that no charge-ordered or enlarged-unit-cell state outside the translationally invariant ansatz is lower in energy; the paper's own limitation statement concedes this.","rationale":"The reader's weakest_assumption identifies exactly this issue: the variational pair-product wavefunction is restricted to translationally invariant F_IJ and local Jastrow factors, so competing states such as charge order or enlarged-unit-cell magnetic or pairing states are not represented. The paper's strongest claim—a first beyond-mean-field phase diagram with robust triplet superconductivity—would be invalidated if such a state were lower in energy in any doping window. I agree that this is the most load-bearing concern. The alternative concern about the absence of thermodynamic extrapolation or superfluid stiffness is real but secondary: even perfect extrapolation within the chosen ansatz would not rule out a missing competing state. The paper itself concedes the limitation in its Summary and Discussion, which strengthens the case that this is a genuine unresolved condition rather than a speculative objection. A concrete supercell calculation directly tests whether the variational space is sufficient at the doping values where the phase boundaries are drawn. Since the concern is already the basis for the reader's CONDITIONAL verdict, my recommendation is to keep that verdict rather than change it.","tokens_in":17633,"tokens_out":3634,"duration_ms":41654,"concrete_test":"At δ≈0.19 for K/t=-1 and δ≈0.17 for K/t=+1 on the L=12 cluster, re-run the mVMC optimization with a 2×1 and a 2×2 supercell, allowing F_IJ and v_ij to vary independently on sublattice sites and adding a staggered site-dependent potential (i.e., a Jastrow factor with Fourier components at stripe wavevectors). If the optimized energy is lower than the reported translationally invariant state by more than the statistical resolution, or if the long-range pairing plateau in Eq. (7) disappears, the superconducting phase is an artifact of the restricted ansatz. If no energy gain is found and the pairing plateaus persist, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—robust p-wave superconductivity in the doped Kitaev spin liquid—depends on the variational ansatz of Eq. (2) being able to represent the true ground state at every doping. Both F_IJ and v_ij are constrained to be translationally invariant, and the Jastrow factor is a density-density term only; there are no variational parameters that can describe a charge-density wave, a stripe, or a magnetic state with an enlarged unit cell. The selection protocol in Supplemental Sec. B retains the lowest energy among initial states derived from the KQSL simple/symmetric representations and the mean-field pSC2/d+id states, but all of these share the same translational and local-Jastrow structure. If a period-2 charge-ordered or stripe state is lower in energy near the claimed superconducting maxima (δ~0.19–0.22 for K/t=-1; δ~0.13–0.19 for K/t=+1), the reported phase diagram and the 'robust superconductivity' headline would be wrong. This is not a purely hypothetical risk: DMRG studies of doped Kitaev-type models report charge-density-wave and pairing-competition tendencies (Refs. [39], [41], [44]), and the authors themselves state in the Summary that 'allowing competing magnetic and charge orders in enlarged-unit-cell wave functions' is needed for future work. The half-filling benchmark does not constrain the doped ansatz space, because the exact KQSL representation is special to the undoped point. Thus the concern is load-bearing and unresolved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies hole doping of the Kitaev honeycomb model using the many-variable variational Monte Carlo (mVMC) method with a projected pair-product (Pfaffian) wave function. The authors consider the t-J-type Kitaev model on isotropic clusters with L=8, 10, and 12, and report two main phase diagrams: for ferromagnetic Kitaev interaction (K/t=-1), triplet p-wave superconductivity coexists with ferromagnetism at low-to-intermediate doping and is suppressed near full spin polarization; for antiferromagnetic interaction (K/t=+1), the dominant pairing changes from spin-dependent triplet p-wave at low doping to singlet d+id at intermediate doping. They also report a nonmonotonic dependence of the triplet correlation on |K|/t at fixed low doping. The half-filling wave function is benchmarked against exact Majorana and exact-diagonalization results to roughly 10^-5 per site up to L=20.","tokens_in":17938,"tokens_out":6592,"duration_ms":65086,"significance":"If the reported phase diagrams are correct, the paper provides the first beyond-mean-field variational phase diagram for the hole-doped t-J-type Kitaev model on isotropic two-dimensional clusters, including a coexistence regime of triplet superconductivity and ferromagnetism that has not been identified in earlier slave-boson mean-field studies. The strengths of the paper are the exact representation of the Kitaev quantum spin liquid at half filling, systematic use of three cluster sizes, multiple initial states for the optimization, and absence of any parameter fitted to reproduce the reported pairing correlations. The main significance is limited by the variational ansatz, which is restricted to translationally invariant pairing amplitudes and a local density-density Jastrow factor; the paper's own summary acknowledges that enlarged-unit-cell and charge-order states are not included. The central claims therefore hold only within this variational class unless additional tests are provided.","major_comments":[{"comment":"The variational wave function (2) imposes translational invariance on both F_IJ and v_ij, and the Jastrow factor is restricted to a density-density form. Consequently, charge-density-wave, stripe, and enlarged-unit-cell magnetic states are not representable. The authors themselves state in the Summary that 'allowing competing magnetic and charge orders in enlarged-unit-cell wave functions' is future work. Given that DMRG studies (Refs. [39], [41], [44]) report charge-density-wave and competing pairing tendencies in doped Kitaev systems, the absence of these variational degrees of freedom is load-bearing for the central claim that the retained state is the ground state. Please either extend the ansatz (e.g., allow period-2 modulations of F_IJ and v_ij) or explicitly restrict the conclusions to the chosen variational class.","section":"Model and method, Eq. (2); Summary and discussion"},{"comment":"The long-range average P^eta_alpha in Eq. (7) is an average of Re D^eta_alpha(r) over distances d >= L/2, not an extrapolated order parameter. The plateau in the distance profile is used to infer long-range order, but no finite-size scaling analysis is presented: Figures 2(a) and 3(a) show P for L=8, 10, and 12 without an extrapolation or a statement of how P would behave in the thermodynamic limit. Because the pair-product ansatz can produce long-range pairing correlations by construction, the plateau in the optimized variational state does not by itself establish that the Hamiltonian ground state has off-diagonal long-range order. I request a finite-size scaling of P (e.g., versus 1/L) or an independent cross-check (e.g., DMRG on one doping) before the word 'robust' is used.","section":"Results, Eq. (7)"},{"comment":"For the antiferromagnetic Kitaev case, the low-doping triplet regime is obtained from initial states generated from the 'simple' representation of the KQSL, and the paper states that the spin-dependent triplet pairing 'reflects the gap structure of the simple representation' (Table SI). Because the optimization is initialized from this state, the final symmetry may be inherited from the initial condition rather than selected by energy. To establish that this spin- and form-factor-selective triplet pairing is an intrinsic low-doping phase, the authors should show that independent initializations (e.g., from the symmetric representation or from states with reversed spin-channel weights) converge to the same energy and pairing correlations. Without this test, the claimed new phase is not fully supported.","section":"Supplemental Sec. B and Table SI"},{"comment":"The gauge-pinning penalty parameter lambda in Eq. (A14) enters the construction of the KQSL and pSC2 initial states, and the channel decomposition in Table SI depends on lambda (e.g., the 'simple' entries contain (3K/4+2lambda)^2 and 16 lambda^2). No numerical value of lambda is reported, and no sensitivity to lambda is discussed. Since the selection of the lowest-energy state is performed among these initial conditions, the chosen lambda could influence which basin of attraction the optimization reaches, especially in the antiferromagnetic case. Please specify lambda (and the procedure for fixing it) and show that the phase diagram is stable under reasonable variations of lambda.","section":"Supplemental Sec. A, Eq. (A14); Supplemental Sec. B"}],"minor_comments":[{"comment":"The dashed line is labeled '(1-delta)/2' but the text calls it the fully polarized value; please clarify that this is the full-polarization value per site in the presence of holes. Several data points near delta=0.12, 0.2, and 0.3 deviate strongly from this line; the text explains this, but the figure would benefit from marking those points.","section":"Figure 2(b) and text"},{"comment":"The phase boundaries are estimated from L=12 results, but no criterion (e.g., crossing of P or M values) is specified. Please state the criterion used to draw the schematic boundaries.","section":"Figure 1(a)"},{"comment":"The sentence 'A few points near delta=0.12, 0.2, and 0.3 in Fig. 2(b) show strongly reduced magnetization' refers to competing nearly unpolarized states; this competition is size dependent and could affect the phase diagram. Consider presenting these points with distinct markers or discussing the energy differences.","section":"Results, second paragraph"},{"comment":"The triplet pairing operator in Eq. (4) includes a 1/sqrt(2) normalization, while Table SI defines Delta^T_rho without the sqrt(2) normalization. Please state this convention explicitly near Eq. (4) to avoid confusion in comparing bare amplitudes with correlation functions.","section":"Eq. (4) and Table SI"},{"comment":"The term 'robust triplet p-wave superconductivity' is used in the Abstract and Summary. Given the finite-size and ansatz caveats, consider replacing 'robust' with 'variational evidence for' or a similar formulation unless the requested finite-size and enlarged-unit-cell tests are provided.","section":"Abstract and Summary"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious variational study with a strong half-filling benchmark and no fitted pairing parameters. The main risk is not internal inconsistency but the restricted variational manifold, which the authors themselves acknowledge. I would not reject, but the requested additions (enlarged-unit-cell or tempered claims, finite-size scaling, specification of lambda, and an initialization test for the AFM triplet regime) are substantial enough for a major revision. The manuscript may be suitable for a rapid-communication journal after these issues are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new result here is the spin- and form-factor-selective triplet pairing in the AFM low-doping regime and the nonmonotonic dependence of triplet correlations on the Kitaev coupling. Neither was in the earlier mean-field phase diagrams, and the selective pattern (P^T_py, P^up_px, P^down_py enhanced while their partners stay small) is specific enough that it would be hard to produce accidentally. That is the part worth reading.\n\nThe paper does several things properly. The half-filling benchmark against Majorana and exact-diagonalization energies to 10^-5 per site is solid, and it is the right check for the claim that the pair-product ansatz can represent the KQSL. The use of three cluster sizes (L=8,10,12) with consistent pairing plateaus in the FM low-doping regime is a meaningful step beyond earlier ladder DMRG results. The comparisons to mean-field [34-36] and to the DMRG literature are fair and not self-serving, and no free parameter was tuned to produce the reported pairing.\n\nThe soft spots are the ones the stress-test note flags, and the authors themselves concede the main one. The variational ansatz is translationally invariant and has only a density-density Jastrow, so charge order, stripes, or enlarged-unit-cell magnetic states are excluded by construction. The authors state in the Summary that allowing such competing orders in enlarged-unit-cell wave functions is needed for future work. That is an honest limitation, but it means the 'robust superconductivity' headline rests on the lowest energy within a restricted variational class, not on a demonstrated thermodynamic phase. The pairing evidence is also finite-cluster correlation plateaus rather than extrapolated order parameters or superfluid stiffness. A few magnetization dips in Fig. 2(b) are size dependent and explicitly unresolved, and the phase boundaries in Fig. 1(a) come from L=12 only. These are standard limits for mVMC on 2D clusters, not signs of sloppiness, but they should temper the abstract's 'robust' language.\n\nThe central qualitative picture—triplet p-wave coexisting with ferromagnetism before full polarization, suppression near the fully polarized state, and an AFM transition from spin-selective triplet to d+id—is plausible and reasonably supported by the size trends. I would not bet the house on the exact phase boundaries, but I would be surprised if the main features were artifacts of the ansatz class alone.\n\nBottom line: this is a competent, honest paper that gives the community a concrete beyond-mean-field phase diagram to argue against. It deserves a serious referee, not a desk rejection. My own verdict would be conditional: recommend publication after the authors soften the 'robust' claim, state the restricted-ansatz caveat in the abstract, and ideally add a short discussion of what enlarged-unit-cell states might change.","headline":"A well-executed mVMC study with genuinely new pairing-selection results, though the 'robust superconductivity' claim outruns the finite-cluster variational evidence; still deserves a serious referee.","tokens_in":18468,"tokens_out":1443,"would_cite":true,"duration_ms":17162,"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":"Hole doping a Kitaev quantum spin liquid produces ferromagnetic triplet p-wave superconductivity, with singlet d+id pairing appearing for antiferromagnetic Kitaev coupling.","keywords":["Kitaev quantum spin liquid","doped Kitaev model","triplet superconductivity","p-wave pairing","d+id pairing","variational Monte Carlo","ferromagnetism","honeycomb lattice"],"falsifier":"On an $L=12$ cluster at $K/t=-1$ and $\\delta\\simeq 0.2$, compute the variational energy of a charge-density-wave or stripe ansatz with a doubled unit cell; if its energy is below the reported triplet $p$-wave state, the central claim that triplet $p$-wave superconductivity is the ground state in that window collapses, whereas observing the $p_y$ triplet correlation plateau to decay with system size would falsify long-range pairing order.","tokens_in":17430,"feed_emoji":"🧲","tokens_out":6540,"duration_ms":60698,"temperature":0.7,"pith_summary":"The paper asks what happens when holes are doped into a Kitaev quantum spin liquid on the honeycomb lattice, and answers with a variational Monte Carlo phase diagram on two-dimensional clusters of up to 288 sites. For ferromagnetic Kitaev coupling, it finds robust triplet $p$-wave superconducting correlations coexisting with ferromagnetism at low to intermediate doping, disappearing only as the system approaches full spin polarization. For antiferromagnetic coupling, the dominant pairing shifts from a spin-dependent triplet $p$-wave channel at low doping to singlet $d+id$ pairing at intermediate doping. The study also shows that, at fixed low doping, triplet pairing is strongest at intermediate Kitaev coupling, slightly below full polarization. If correct, this provides the first beyond-mean-field evidence for unconventional superconductivity in carrier-doped Kitaev materials and a microscopic route to ferromagnetic-superconducting coexistence outside $f$-electron systems.","feed_headline":"Triplet superconductivity emerges in doped Kitaev spin liquid","feed_subtitle":"Variational Monte Carlo maps how doping stabilizes p-wave pairing alongside ferromagnetism.","key_machinery":"The machinery is the many-variable variational Monte Carlo method applied to a projected pair-product (Pfaffian) wave function, $|\\Psi\\rangle = P_J P_G^\\infty |\\phi_{\\mathrm{pair}}\\rangle$, with pair matrix $F_{IJ}$ and Jastrow factors $v_{ij}$ optimized by stochastic reconfiguration. The key identity is that the Kitaev quantum spin liquid at half filling is exactly a Gutzwiller-projected BCS state, so the same variational family can represent both the spin liquid and doped superconducting phases. Superconductivity is diagnosed by equal-time pairing correlations in singlet and triplet channels, decomposed into four honeycomb form factors ($s/f$, $d_{xy}/p_x$, $d_{x^2-y^2}/p_y$, $d+id/p+ip$), with the long-range average $P^\\eta_\\alpha$ over distances $d \\ge L/2$ used as the order parameter.","core_discovery":"The central discovery is that the hole-doped $t$-$J$-type Kitaev model, studied with a projected pair-product wave function that exactly represents the undoped Kitaev quantum spin liquid, supports spin-triplet superconducting order. For $K<0$, the long-range-averaged $p_y$ triplet pairing correlation rises with doping, peaks around $\\delta\\simeq 0.2$, and coexists with a finite uniform magnetization; both are suppressed as the magnetization saturates to the fully polarized value. For $K>0$, a spin- and form-factor-selective triplet $p$-wave state at low doping—where only specific equal-spin and $S^z=0$ components are enhanced—gives way to a singlet $d+id$ state at intermediate doping, with negligible magnetization in the singlet regime. At fixed $\\delta\\simeq 0.11$ and $K<0$, the triplet correlation peaks at $|K|/t\\simeq 0.5$, nonmonotonically, while the magnetization decreases monotonically. The paper interprets the suppression of triplet pairing at full polarization as evidence that spin fluctuations are needed to mediate the pairing.","pith_inferences":["A natural next test is to enlarge the variational unit cell and allow charge-density-wave or stripe orders; if such a state wins near $\\delta\\simeq 0.2$, the reported phase boundaries would shift, but the qualitative coexistence scenario may survive.","The spin- and form-factor-selective triplet pairing at low doping suggests a pairing state that breaks lattice rotational symmetry; determining its topological invariant (e.g., via Chern number or edge states) would settle whether it is a chiral topological superconductor.","For experiments, the nonmonotonic coupling dependence implies that stronger Kitaev character is not always better for triplet pairing; samples with intermediate $K$ may be the best targets for detecting triplet order.","Extending the method to multi-orbital Hubbard models from ab initio calculations could reveal whether the predicted phases survive in real materials beyond the $t$-$J$ approximation."],"forward_implications":["For ferromagnetic Kitaev coupling, triplet $p$-wave superconductivity and ferromagnetism coexist over a low-to-intermediate doping window, with the superconducting correlation vanishing as magnetization saturates.","For antiferromagnetic coupling, doping drives a transition from spin-dependent triplet $p$-wave to singlet $d+id$ pairing, and the singlet regime has no spontaneous magnetization.","Triplet pairing strength depends nonmonotonically on $|K|/t$ at fixed low doping, peaking around $|K|/t\\simeq 0.5$, while magnetization decreases monotonically.","These results give experimental guidance: carrier-doped Kitaev candidate materials such as heterostructures of $\\alpha$-RuCl$_3$ with graphene or graphite may realize triplet superconductivity coexisting with ferromagnetism.","The doped Kitaev model offers a microscopic realization of spin-triplet superconductor / ferromagnet coexistence previously studied mainly in uranium-based $f$-electron compounds."],"supporting_citations":[{"why":"Defines the exactly solvable Kitaev model whose ground state is the quantum spin liquid being doped.","marker":"[1]"},{"why":"Shows the Kitaev quantum spin liquid can be represented as a projected BCS (slave-fermion) state, the starting point of the variational wave function.","marker":"[19]"},{"why":"Introduces the variational Monte Carlo method combined with quantum-number projection and multi-variable optimization that the study uses.","marker":"[45]"},{"why":"Provides the open-source software used to optimize the variational wave functions.","marker":"[46]"},{"why":"Supplies the pair-product/Pfaffian wave function framework and its optimization context.","marker":"[47]"},{"why":"Earlier mean-field prediction of topological superconductivity from the doped Kitaev-Heisenberg model, a baseline the paper's phase diagram is compared with.","marker":"[34]"},{"why":"Mean-field global phase diagram used as the reference for the pSC2 and d+id states and for the width of superconducting regimes.","marker":"[36]"},{"why":"DMRG evidence for kinetic ferromagnetism in the hole-doped Kitaev spin liquid, consistent with the extended ferromagnetic regime reported here.","marker":"[40]"},{"why":"Review of uranium-based ferromagnetic superconductors that motivates the analogy of coexisting triplet superconductivity and ferromagnetism.","marker":"[49]"}],"fun_headline_variants":["Doped Kitaev spin liquid yields triplet superconductivity","Triplet p-wave pairing shows up in doped Kitaev magnet","Kitaev doping turns spin fluctuations into triplet pairs","Ferromagnetic Kitaev phase hosts robust p-wave superconductivity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results assume that the optimized variational state, restricted to translationally invariant pair-product amplitudes and local density-density Jastrow factors, is the true ground state at every doping; a lower-energy state outside this ansatz class—such as a charge-ordered or enlarged-unit-cell magnetic or pairing state—would change the phase diagram.","fun_headline_variants_meta":{"raw":{"variants":["Doped Kitaev spin liquid yields triplet superconductivity","Triplet p-wave pairing shows up in doped Kitaev magnet","Kitaev doping turns spin fluctuations into triplet pairs","Ferromagnetic Kitaev phase hosts robust p-wave superconductivity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0002,"raw_usage":{"total_tokens":1388,"prompt_tokens":971,"completion_tokens":417,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":587,"completion_tokens_details":{"reasoning_tokens":352}},"tokens_in":587,"tokens_out":417,"duration_ms":5016,"temperature":1.0,"reasoning_tokens":352,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T04:15:31.231541+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On an $L=12$ cluster at $K/t=-1$ and $\\delta\\simeq 0.2$, compute the variational energy of a charge-density-wave or stripe ansatz with a doubled unit cell; if its energy is below the reported triplet $p$-wave state, the central claim that triplet $p$-wave superconductivity is the ground state in that window collapses, whereas observing the $p_y$ triplet correlation plateau to decay with system size would falsify long-range pairing order.","supporting_citations":[{"cited_title":"Tahara and M","cited_arxiv_id":null,"evidence_quote":"Introduces the variational Monte Carlo method combined with quantum-number projection and multi-variable optimization that the study uses."},{"cited_title":"Misawa, K","cited_arxiv_id":null,"evidence_quote":"Provides the open-source software used to optimize the variational wave functions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"DMRG evidence for kinetic ferromagnetism in the hole-doped Kitaev spin liquid, consistent with the extended ferromagnetic regime reported here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Review of uranium-based ferromagnetic superconductors that motivates the analogy of coexisting triplet superconductivity and ferromagnetism."}],"review_version":1}