{"id":"bd03f1a2-21d7-484d-986c-9273e41e17f9","arxiv_id":"2511.22669","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A large-cell CTMRG algorithm computes iPEPS energy variances accurately and efficiently, enabling zero-variance energy extrapolation and a steady-state quality measure for open quantum systems.","lead":"This paper introduces a more accurate way to calculate the energy variance of infinite tensor-network wavefunctions, a number that tells how close a state is to an exact ground state. It also helps judge the quality of steady states of open quantum systems and locate first-order phase transitions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Open-system ϵ estimate uses L=12 cell despite known non-vanishing disconnected tail; transition location lacks controlled uncertainty.","rationale":"The reader's CONDITIONAL verdict is appropriate. The paper delivers a valuable practical method for energy variance with iPEPS, and the Hermitian benchmarks (Heisenberg vs QMC, free fermions vs exact, Shastry-Sutherland vs other methods) provide strong empirical support. The main weakness is indeed the open-system application: the L=12 cell is used despite the manuscript's own footnote admitting the disconnected part of the Liouvillian correlator does not vanish. This is not merely a disagreement with consensus; it is an internal inconsistency between the stated limitation and the claimed reliability of the transition location. A concrete fix would be an L-extrapolation or disconnected-part subtraction, plus reported uncertainties. Since this concerns the second application rather than the core algorithm, conditional acceptance remains appropriate. I agree with the reader's overall assessment, though my emphasis is more on the open-system finite-cell error than on the general CTMRG two-sweep premise.","tokens_in":21005,"tokens_out":9330,"duration_ms":82618,"concrete_test":"Recompute ϵ for the dissipative Ising model near h_x/γ≈7.2 at fixed bond dimension (e.g., D=4) for cells L=12, 16, 20 (and if feasible L=24), with sufficiently large χ so that χ-convergence is established at each L. Subtract the disconnected contribution c = ⟨⟨L†(∞)⟩⟩⟨⟨L0⟩⟩ estimated from the large-distance plateau of ⟨⟨L†(r)L0⟩⟩, and check whether the intensive ϵ(L) becomes L-independent; alternatively, extrapolate ϵ(L) to L→∞. If the intersection of the two phase curves shifts by more than ~0.1 in h_x/γ, the quoted location is not reliable. Also report the magnitude of ⟨⟨L⟩⟩ to quantify the offset.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim for the open-system extension — that the LC-CTMRG approach computes ⟨⟨L†L⟩⟩ reliably and locates the dissipative Ising transition at h_x/γ ≈ 7.2 — is weakened by an uncontrolled finite-cell error. Section IV and Fig. 7 use a single cell size L=12, but footnote [80] explicitly states that ⟨⟨L†(r)L0⟩⟩ does not vanish at large distances because ⟨⟨L⟩⟩ ≠ 0 for the approximate steady state. Consequently, the disconnected part contributes an L-dependent offset to the sum over the cell; without subtracting this contribution or extrapolating in L, the computed ϵ is not an intensive, converged quantity. No L-convergence data, no disconnected-part subtraction, and no uncertainty estimate are provided for the transition location, and the resulting h_x/γ ≈ 7.2 lies outside the previously reported range ~6–7. The Hermitian variance benchmarks are independent and appear sound, but the open-system demonstration is quantitatively uncontrolled. Because the title and abstract explicitly claim the computation of ⟨⟨L†L⟩⟩ and the location of first-order transitions, this limitation undermines a significant part of the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces an LC-CTMRG approach for computing the energy variance of iPEPS. The unit cell is enlarged to L×L, a Hamiltonian term H0 is applied at the center to form |Ψ′⟩=H0|Ψ⟩, the overlap ⟨Ψ|Ψ′⟩ is contracted with CTMRG, and all two-point Hamiltonian correlators in the cell are evaluated. The method is benchmarked on the Heisenberg model (zero-variance extrapolated energy matching QMC), on free fermions with a staggered potential (matching exact results), and on the Shastry-Sutherland model. The same contraction is used to compute ϵ=⟨⟨L†L⟩⟩ for an open system, and the dissipative Ising model is used to locate a first-order transition at h_x/γ≈7.2.","tokens_in":21298,"tokens_out":7293,"duration_ms":74847,"significance":"If the claims hold, this is a practical and accurate tool for a notoriously difficult iPEPS observable. The Hermitian benchmarks are convincing: the variance is computed directly from the tensors and the Hamiltonian, with no fitted input, and the Heisenberg zero-variance energy agrees with QMC to about 6×10⁻⁶. The χ-convergence improvement over the H-environment method is dramatic and reproducible from Figs. 3(a) and 6. The free-fermion and Shastry-Sutherland results strengthen the case. However, the open-system part, which is also advertised in the title and abstract, is not yet controlled enough to support the transition-location claim.","major_comments":[{"comment":"The open-system computation of ϵ is uncontrolled. Footnote [80] states that ⟨⟨L†(r)L0⟩⟩ does not vanish at large distances because ⟨⟨L⟩⟩≠0 for the approximate steady state. Hence the sum over an L=12 cell contains an L-dependent disconnected contribution. No subtraction of the disconnected part and no L-extrapolation are provided, so ϵ is not an intensive, converged quantity. This offset can shift the intersection in Fig. 7 and likely explains why h_x/γ≈7.2 lies outside the previously reported range ~6–7. The abstract's claim that the transition is located at ~7.2 is therefore not supported by the data as presented.","section":"Sec. IV, Eq. (7), Fig. 7, footnote [80]"},{"comment":"The transition estimate has no uncertainty or convergence analysis. Only L=12 is used; no χ-convergence or sweep-convergence data are shown. The D-trend (6.2 for D=1, 6.5 for D=2, 7.2 for D=3,4) has not converged, and the intersection procedure using lowest-ϵ selection may systematically favor the more converged branch. Please provide error bars, a D→∞ extrapolation or at least a discussion of the D-dependence, and an explicit criterion for the intersection.","section":"Sec. IV, Fig. 7"},{"comment":"The central numerical assumption is that the CTMRG contraction of the overlap after two sweeps faithfully captures the long-distance correlations introduced by H0. No a priori error bound is given, and Fig. 5 shows that small χ can produce an artificial drift with L. For the energy variance this is mitigated by the benchmarks, but the same check is entirely absent for ϵ in Sec. IV. A concrete protocol (e.g., require stability under increasing L and χ) should be stated for both applications.","section":"Sec. II.C, Fig. 5"}],"minor_comments":[{"comment":"The statement that two sweeps are sufficient is not demonstrated. A small convergence plot in the number of sweeps would help, especially for the open-system case.","section":"Sec. II.C"},{"comment":"The normalization of |ρ_s⟩⟩ in the inner product should be specified. If Tr ρ_s=1 but the vector norm is not 1, the value of ϵ changes accordingly.","section":"Sec. IV, Eq. (7)"},{"comment":"The extrapolated free-fermion energies are plotted without error bars. Please provide uncertainties for the zero-variance extrapolated values.","section":"Sec. III.B, Fig. 4"},{"comment":"The text says 'sufficiently large L and χ' but does not list the values used. A table with L, χ, and sweep counts for each D and Δ would improve reproducibility.","section":"Sec. III.B"},{"comment":"References [56,57] appear to be missing journal, volume, and page details; please update.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The Hermitian core of the paper is strong and publishable. The main issue is the open-system section: the finite-cell offset admitted in footnote [80] is a genuine uncontrolled error for ϵ, and the transition estimate has no uncertainty. If the authors can add L-convergence data, subtract the disconnected contribution, and give honest error bars for h_x/γ, this would be an accept-level paper. If they cannot, they should downgrade the open-system claims to a preliminary illustration and adjust the title and abstract accordingly. The self-citations are relevant and not excessive."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core of this paper is the LC-CTMRG scheme for computing the energy variance (and more generally sums of two-point Hamiltonian correlators) with iPEPS. That part is a real step forward. The idea of enlarging the unit cell and contracting the overlap with a Hamiltonian insertion using standard CTMRG is clean, borrows from the group's earlier structure-factor work, and—as the benchmarks show—converges in χ much faster than the older H-environment approach. The Heisenberg extrapolation to zero variance gives -0.669436(21), essentially on top of QMC; the free-fermion results match exact energies; the Shastry-Sutherland comparison makes the χ-convergence advantage concrete. I believe the variance computation works as advertised, and this should make zero-variance extrapolation practical for larger D than before. The paper also honestly flags the small-χ drift with L in Fig. 5 and tells readers to monitor it.\n\nThe soft spot is exactly where the stress-test note lands: the open-quantum-system section. Using L=12 while footnote [80] concedes that the disconnected part of ⟨⟨L†(r)L0⟩⟩ does not vanish, because ⟨⟨L⟩⟩ is nonzero for the approximate steady state, means the computed ϵ has an uncontrolled L-dependent offset. No L-convergence data, no subtraction, no uncertainty estimate on the transition location. The claim h_x/γ≈7.2 for the dissipative Ising first-order transition therefore is not quantitatively controlled. That doesn't sink the main methodological contribution—the same contraction scheme is still the right way to evaluate the object—but the open-system demonstration is at the level of a proof of principle, not a reliable phase-boundary estimate.\n\nOne small thing: no code or data is provided, which would help others adopt the method. I would not call that a flaw in the paper itself, just a practical wish.\n\nWho is this for? People working with iPEPS on 2D ground states will benefit most; open-system tensor-network practitioners will read the last section with appropriate caution. The paper deserves a serious referee—it is important enough, the Hermitian part is solid, and the authors show clear thinking and honest reporting of the limitation. I would send it to peer review and ask for the open-system systematics to be addressed (extrapolation in L or subtraction of the disconnected contribution) before publication. My verdict is close to the reader's: conditional on that revision.","headline":"A genuinely useful method for computing energy variances with iPEPS, with solid Hermitian benchmarks; the open-system application is a suggestive demo with an uncontrolled finite-cell error that should be fixed before publication.","tokens_in":21785,"tokens_out":1486,"would_cite":true,"duration_ms":17127,"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":"Contracting a large cell of iPEPS tensors with CTMRG computes the energy variance accurately at small boundary dimension χ, enabling systematic zero-variance extrapolation that matches quantum Monte Carlo results.","keywords":["iPEPS","energy variance","CTMRG","zero-variance extrapolation","open quantum systems","Liouvillian","dissipative Ising model","Shastry-Sutherland model"],"falsifier":"Compute the variance for the Heisenberg model at D=5 with the LC-CTMRG method at χ=40, L=40, and compare it against the value obtained at χ=600 with the previous H-environment method; if the two disagree by more than the claimed uncertainty, the central claim fails. Alternatively, check whether the connected correlator ⟨H(r)H0⟩−⟨H(r)⟩⟨H0⟩ decays to zero within numerical tolerance for r≈L/2; if it does not, the large-cell sum carries an unresolved additive error that grows with L.","tokens_in":20900,"feed_emoji":"🧮","tokens_out":7301,"duration_ms":57611,"temperature":0.7,"pith_summary":"This paper addresses a standing problem in infinite projected entangled-pair state (iPEPS) simulations: computing the energy variance Var(E)=⟨H²⟩−⟨H⟩², which is zero only for an exact eigenstate and therefore a natural target for ground-state energy extrapolation. The authors claim that the variance can be computed accurately by enlarging the unit cell to size L×L, applying one Hamiltonian term at the center to create a modified state, and contracting the resulting overlap with the standard corner-transfer-matrix renormalization group (CTMRG) method. Because the CTMRG projectors are computed from the overlap itself, the inserted term is naturally accounted for, so the required boundary dimension χ is far smaller than in previous H-environment schemes. This makes variance-based zero-energy extrapolation practical at large bond dimension D, and the benchmarks — the Heisenberg model, free fermions with a staggered potential, and the Shastry-Sutherland model — land on exact or quantum Monte Carlo results. The same machinery computes ⟨⟨L†L⟩⟩ for open quantum systems, giving a measure of closeness of an iPEPS steady state and a tool for locating first-order dissipative phase transitions, demonstrated on the dissipative quantum Ising model.","feed_headline":"Energy variance from iPEPS at small χ matches QMC","feed_subtitle":"A large-cell CTMRG contraction makes zero-variance extrapolation practical and also locates dissipative phase transitions.","key_machinery":"The key object is the large-cell overlap contraction: the iPEPS unit cell is enlarged to L×L, a Hamiltonian term H0 is applied at the center to form |Ψ′⟩ with two modified tensors and an enlarged bond between them, and CTMRG is used to contract the tensor network of the overlap ⟨Ψ|Ψ′⟩, sweeping from the center outward (two sweeps suffice). The CTMRG projectors are computed from the overlap network itself, so they automatically incorporate the perturbation H0, which is why the method converges at much smaller χ than the previous H-environment approach. Summing the resulting local expectation values ⟨Ψ|Hb|Ψ′⟩/⟨Ψ|Ψ′⟩ over all bonds in the cell, weighted by the central expectation value, gives t","core_discovery":"The central discovery is that the energy variance of an iPEPS can be obtained by evaluating the correlator between Hamiltonian terms in a large cell using standard CTMRG, rather than by building specialized H-environments. The calculation exploits the identity ⟨Ψ|HbH0|Ψ⟩/⟨Ψ|Ψ⟩ = [⟨Ψ|Hb|Ψ′⟩/⟨Ψ|Ψ′⟩]·[⟨Ψ|Ψ′⟩/⟨Ψ|Ψ⟩], where |Ψ′⟩=H0|Ψ⟩ has a Hamiltonian term inserted at the center. The first factor is a local expectation value in the overlap network, and the second is the central bond energy. Contracting the overlap ⟨Ψ|Ψ′⟩ with CTMRG, starting from the center and sweeping twice, yields accurate correlators at much smaller boundary dimension χ than previous methods; for the Heisenberg model at D=4,","pith_inferences":["Because the CTMRG projectors adapt to the inserted operator, the same large-cell contraction should generalize to other non-local quantities, such as higher moments ⟨H^k⟩ or real-frequency spectral functions, at similar cost savings — an extension the paper does not pursue.","The paper's observation of a small-χ drift with L suggests a practical per-run error diagnostic: monitoring the large-distance limit of the connected correlator ⟨H(r)H0⟩−⟨H(r)⟩⟨H0⟩, which should vanish, would give a concrete error bar; the paper notes the drift but does not propose such a diagnostic.","The footnote that the disconnected part of ⟨⟨L†(r)L0⟩⟩ does not vanish for approximate steady states implies that the reported ϵ values at L=12 contain a systematic offset; a connected correlator or an L-extrapolation of ϵ could sharpen the transition estimate, and the intersection method may then shift slightly from the quoted h_x/γ≈7.2.","The success of variance extrapolation on the Shastry-Sutherland model indicates the method transfers to frustrated magnets, where accurate variational energies are otherwise hard to obtain."],"forward_implications":["Zero-variance extrapolation becomes a practical tool for iPEPS at large bond dimensions, removing the need for uncontrolled 1/D extrapolation; the Heisenberg energy extrapolates to −0.669436(21), in agreement with quantum Monte Carlo.","The method works for both gapless and gapped systems, as the free-fermion benchmark reaches the exact ground-state energy for two staggered potentials.","A single large-cell calculation yields variance data for all smaller cell sizes, making convergence checks in L inexpensive; however, the paper notes that small χ can cause an artificial drift at large L and must be monitored.","For open quantum systems, computing ϵ=⟨⟨L†L⟩⟩ provides both a steady-state quality check and a way to locate first-order dissipative phase transitions by intersecting ϵ curves from the two metastable phases; the dissipative Ising transition shifts to h_x/γ≈7.2 with increasing bond dimension."],"fun_headline_variants":["CTMRG yields accurate iPEPS energy variances at small chi","Zero-variance extrapolation for iPEPS made practical with CTMRG","iPEPS variance via CTMRG: beats QMC at low boundary dimension","Dissipative phase transitions located by iPEPS energy variance"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that a standard CTMRG contraction of the large-cell overlap ⟨Ψ|Ψ′⟩, using only two sweeps, faithfully captures the long-distance correlations introduced by the single inserted Hamiltonian term, so that summing the correlators over the L×L cell converges to the true variance with controlled error — an assumption the paper supports empirically but does not prove; for the open-system extension, the premise is that L=12 is sufficient, even though the p","fun_headline_variants_meta":{"raw":{"variants":["CTMRG yields accurate iPEPS energy variances at small chi","Zero-variance extrapolation for iPEPS made practical with CTMRG","iPEPS variance via CTMRG: beats QMC at low boundary dimension","Dissipative phase transitions located by iPEPS energy variance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00139,"raw_usage":{"total_tokens":5485,"prompt_tokens":791,"completion_tokens":4694,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":535,"completion_tokens_details":{"reasoning_tokens":4615}},"tokens_in":535,"tokens_out":4694,"duration_ms":29002,"temperature":1.0,"reasoning_tokens":4615,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T19:43:02.226281+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the variance for the Heisenberg model at D=5 with the LC-CTMRG method at χ=40, L=40, and compare it against the value obtained at χ=600 with the previous H-environment method; if the two disagree by more than the claimed uncertainty, the central claim fails. Alternatively, check whether the connected correlator ⟨H(r)H0⟩−⟨H(r)⟩⟨H0⟩ decays to zero within numerical tolerance for r≈L/2; if it does not, the large-cell sum carries an unresolved additive error that grows with L.","supporting_citations":[],"review_version":1}