Extracting Boundary Conformal Data from Periodic Non-Hermitian Critical Chains

Dongchang Liu; Haruki Shimizu; Kohei Kawabata; Yifan Liu

REVIEW 3 major objections 3 minor 3 cited by

Extracting Boundary Conformal Data from Periodic Non-Hermitian Critical Chains

T0 review · 3 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Boundary conformal data—including the Affleck–Ludwig g-factor—can be extracted from periodic non-Hermitian critical chains alone, using paired overlaps whose dual implements the nonunitary CFT's bilinear pairing; the resulting coefficients

desk verdict A genuinely new periodic-chain route to nonunitary boundary data with strong Yang-Lee and fixed/free Potts numerics, held back only by an unproven—though plausible—identification of the lattice dual pairing with the CFT bilinear pairing. read the letter →

arxiv 2606.16785 v2 pith:YHZNTLCX submitted 2026-06-15 cond-mat.stat-mech cond-mat.str-elhep-th

Yifan Liu , Haruki Shimizu , Dongchang Liu , Kohei Kawabata This is my paper

classification cond-mat.stat-mechcond-mat.str-elhep-th MSC 81T4082B2082B27 PACS 11.25.Hf05.30.-d

keywords non-HermitiancriticalityboundaryconformalfieldtheoryAffleck-Ludwigg-factorYang-LeeCFTfive-statePottsmodelprojectedoverlapsKramers-Wannierdualitynonunitary

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 introduces a method for extracting universal boundary conformal data—the Affleck–Ludwig g-factor and a full set of sector-resolved boundary coefficients—directly from periodic non-Hermitian chains at criticality, without engineering any open boundary. The central formula states that a projected overlap of a short-range-entangled preparation state with a low-energy periodic eigenstate decays as exp(-αL) times a universal coefficient G_{a,i} = (B_i^a)^2; the non-Hermitian dual preparation is chosen to match the bilinear pairing of the infrared nonunitary CFT. Using this, the authors recover the Yang-Lee boundary g-factor and discover a negative excited-to-ground sector ratio, a sign that is impossible in unitary theories and serves as a crisp microscopic signature of nonunitarity. For the genuinely complex fixed points of the non-Hermitian five-state Potts chain, they extract intrinsically complex boundary coefficients and reproduce the exact Kramers-Wannier duality relation G_free = 5 G_fixed. If correct, this provides a route to nonunitary BCFT data using only bulk knowledge, bypassing the problematic open-boundary physics of non-Hermitian systems.

What carries the argument

Projected-partition-function spectroscopy: for a periodic bulk-critical Hamiltonian H, one takes a short-range-entangled state |Φ⟩ representing a boundary condition a and a compatible dual ⟨eΦ| defined either by η-pseudo-Hermiticity (Eq. 10) or by a complex-symmetric bilinear pairing (Eq. 12). The paired amplitude Z_{Φ,i}(L) = ⟨eΦ|Ψ_i,R⟩⟨Ψ_i,L|Φ⟩ / ⟨Ψ_i,L|Ψ_i,R⟩ projects onto the low-energy sector i via the biorthogonal projector; in the scaling limit it gives exp(-α_Φ L) G_{a,i} with G_{a,i} = (B_i^a)^2. The dual pairing is the load-bearing piece: it implements the BPZ bilinear form of the nonunitary theory, which can make the coefficients negative or complex.

What would settle it

Compute the projected overlap ratio R_{Φ,1} = Z_{Φ,1}/Z_{Φ,ϕ} in the Yang-Lee chain at larger L with the η-inserted dual; if the negative sign flips or the magnitude drifts from (1+√5)/2 beyond the reported sub-percent precision, the claim of a universal negative sector ratio fails. For the Potts case, re-derive the analytic continuation with the opposite γ branch and compare the fixed and free coefficients; only one branch can match them, and a mismatch would signal that the method selects the wrong branch or that the dual-pairing assumption is incorrect.

Watch

Extended reading notes

Core claim

The paper claims that the finite-size scaling of paired projected overlaps Z_{Φ,i}(L) = ⟨eΦ|Ψ_i,R⟩⟨Ψ_i,L|Φ⟩ / ⟨Ψ_i,L|Ψ_i,R⟩ yields, up to a nonuniversal exponential factor, the squared boundary-state coefficients G_{a,i} = (B_i^a)^2, where B_i^a is the coefficient of the Ishibashi state |i⟩⟩ in the Cardy boundary state |a⟩. In nonunitary and complex fixed-point theories, the usual Hermitian conjugation is replaced by the infrared CFT's bilinear (BPZ) pairing, so the extracted coefficients may be negative or complex. The method is demonstrated for the Yang-Lee CFT, where the excited-to-ground ratio equals -(1+√5)/2, a sign impossible in unitary theories, and for the non-Hermitian five-state P

Load-bearing premise

The dual preparation chosen by Eq. (10) or (12) exactly reproduces the BPZ bilinear pairing of the infrared nonunitary CFT; if this lattice-to-infrared identification fails, the extracted G_{a,i}—especially their signs and phases—are not the true BCFT coefficients, and the paper verifies it only by numerical agreement for Yang-Lee and fixed/free Potts, with visible mismatches for k=3,4.

Editorial extensions

If this is right

Boundary universal data (g-factors, boundary coefficients, signs and phases) become accessible in non-Hermitian systems without engineering open boundaries, avoiding the skin effect and other open-boundary pathologies.
A negative excited-to-ground sector ratio is a universal signature of nonunitarity: no unitary CFT can produce it, so observing a similar ratio in another critical non-Hermitian chain would indicate a nonunitary infrared fixed point.
For complex fixed points, the method resolves intrinsically complex boundary coefficients and verifies Kramers-Wannier duality; the finite-size fits select the correct analytic-continuation branch of the boundary state.
The sector-resolved nature of the extracted G_{a,i} enables consistency checks beyond a single g-factor, including first-excited-multiplet projections that constrain which boundary conditions actually describe the lattice model.
Applying the method to other non-Hermitian critical models could map out boundary RG flows without ever needing a microscopic open-boundary Hamiltonian.

Reading between the lines

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

The same paired-overlap principle could be extended to extract bulk-to-boundary OPE coefficients and mixed-boundary amplitudes, or to diagnose boundary RG flow by tracking how different preparations approach fixed points.
The visible ground- and excited-sector discrepancies for the k=3,4 Potts preparations suggest those intermediate blob boundaries may actually flow to the free or fixed boundary condition in the infrared; our method provides a direct test by comparing full sector-resolved coefficients across preparations.
If the negative Yang-Lee ratio persists in larger systems and in other real nonunitary CFTs, it could serve as a 'nonunitarity detector' analogous to how entanglement scaling detects complex criticality.
A natural stress test would be to apply the method to a non-Hermitian chain with an exactly known boundary-state expansion (e.g., a free-fermion model with an explicit open-boundary solution) to isolate whether the η-pairing or the analytic continuation is the source of any residual mismatch.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know this paper for two things. First, it gives a concrete way to extract BCFT boundary coefficients from periodic non-Hermitian critical chains, sidestepping the unreliable open-boundary engineering that plagues this field. Second, it actually works in the benchmarks: the Yang-Lee g-factor and the negative excited-to-ground ratio match the BCFT predictions to four digits, and the complex Potts fixed/free coefficients agree with the analytically continued 2BTL values at the subpercent level. That is real evidence, not noise.

The new ingredient is the use of a left dual (Eqs. 10 and 12) chosen to match the infrared CFT's bilinear pairing—eta for pseudo-Hermitian Yang-Lee, simple transpose for the complex-symmetric Potts chain. This is a sensible extension of the Hermitian overlap method, and the negative ratio in a nonunitary theory is a nice, clean signature that would be impossible in a unitary setting. The paper is also honest about its blemishes: it reports the k=2,3,4 Potts discrepancies, shows the excited-sector ratio for k=2 is incompatible with the 2BTL value, and puts the disagreement on the analytic-continuation assignment rather than hiding it.

The soft spot is the load-bearing assumption that the lattice eta/transpose pairing really implements the BPZ bilinear pairing of the infrared CFT. That is asserted in Appendix A, not derived, and the paper's only validation is numerical agreement in two models. The stress-test note is right that the quantities most sensitive to this—the sign of the Yang-Lee ratio and the complex phases of the Potts coefficients—are exactly the ones that would change if the pairing were wrong. I don't think this sinks the paper, because the Yang-Lee result is robust across three different preparations and the fixed/free Potts agreement is too sharp to be accidental. But the method can't yet tell you, a priori, whether a given preparation's dual is the correct one; it can only show you after the fact that some choices match and others do not. Also minor: the Potts error bars in Table II are propagated fit uncertainties only; systematic ansatz variation is shown in the supplement but not folded into the quoted errors.

Who gets value: people working on non-Hermitian criticality, boundary RG, and complex CFT. It deserves a serious referee—the central idea is novel, the numerical benchmarks are strong, and the limitations are disclosed. I would engage with it, probably after asking the authors to either prove or substantially justify the pairing identification, and to show at least one more nonunitary benchmark where the dual is independently known.

Referee Report

3 major / 3 minor

Summary. The paper proposes a periodic-chain projected-partition-function spectroscopy to extract boundary conformal data from non-Hermitian critical chains. For a bulk-critical periodic Hamiltonian and a short-range-entangled preparation |Φ⟩, the authors define the paired projected amplitude Z_{Φ,i}(L) in Eq. (2) and argue that it scales as e^{-α_Φ L}(G_{a(Φ),i}+o(1)), where G_{a,i}=(B_i^a)^2 is a boundary-state coefficient. The dual preparation is chosen through an η-pseudo-Hermitian pairing, Eq. (10), or through a complex-symmetric bilinear form, Eq. (12). The method is tested on a PT-symmetric Yang-Lee realization, where the g-factor and a negative excited-to-ground ratio are recovered, and on the non-Hermitian five-state Potts chain, where complex fixed/free boundary coefficients are extracted and the Kramers-Wannier duality G_free/G_fixed=5 is reproduced. The paper also includes Hermitian benchmarks and openly reports discrepancies for the k=2,3,4 intermediate Potts preparations.

Significance. If the central claim holds, the paper establishes a genuinely new route to nonunitary boundary universal data without engineering open-boundary Hamiltonians, which is especially valuable in non-Hermitian systems where open boundaries are subtle. The Yang-Lee negative projected ratio and the complex Potts coefficients are concrete, falsifiable predictions that go beyond standard unitary BCFT. Strengths of the paper include the transparent finite-size extrapolation protocols, the independent Hermitian benchmarks in Supplemental Sec. B, the explicit disclosure of the k=2,3,4 mismatches in Tables II and S3, and the use of independent analytically continued 2BTL predictions for comparison. The main risk is that the microscopic-to-infrared identification of the bilinear pairing is assumed rather than derived, and the quantities that most depend on it—signs and complex phases—are exactly the headline results.

major comments (3)

[Appendix A / Eqs. (10)-(12)] The identification of the lattice dual with the BPZ pairing is the central premise of Eq. (3), but it is assumed rather than derived. Appendix A postulates in Eq. (A5) that |Φ⟩ and ⟨Φ̃| realize regularized boundary states in the same CFT, and then Eqs. (10) and (12) are asserted to supply the correct dual. No argument is given that the η-inner product or the complex-symmetric transpose reproduces the BPZ bilinear pairing, including its normalization and sign. The numerical agreement in Yang-Lee and in the fixed/free Potts sectors is encouraging, but it involves only one real nonunitary fixed point with all preparations flowing to the same minimal-g boundary condition. Because the headline outputs—the negative Yang-Lee ratio and the complex phases of the Potts coefficients—are precisely pairing-dependent, the manuscript should either provide a derivation from the microscopic-to-continuum
[Tables II and S3] The intermediate preparations do not match the analytically continued 2BTL predictions. In the ground sector, k=3 and k=4 deviate by about 6% in Table II; in the excited sector, the k=2 extrapolated ratio 1.842(6)+0.025(6)i is incompatible with the 2BTL value 1.4464+0.1256i in Table S3. The manuscript attributes this to a limitation of the analytic-continuation assignment rather than to the extraction. This may be correct, but it means that the method does not, by itself, identify when the infrared pairing has been implemented correctly for an arbitrary preparation. The fixed/free agreement and Eq. (17) are strong, but the general claim that the full family in Eq. (16) yields reliably extracted boundary data is not yet supported.
[Tables I, II, and S3; Figs. S2-S3] The quoted numerical uncertainties are propagated fitting uncertainties only; the deviation between the two fit ansatzes shown in Figs. S2 and S3 (1/L versus 1/L+1/L^2) is not propagated into the quoted error bars. For several Potts intercepts, the ansatz shift is comparable to or larger than the stated uncertainty. Since Eq. (A.12) contains many possible correction powers and the accessible L range is small, the systematic error from the fit-form choice should be either quantified in the quoted values or used to define the central estimates; otherwise the stated precision overstates the reliability of the extracted coefficients.

minor comments (3)

[Main text vs. Sec. D] The main text quotes the Potts fixed-point coupling as λ_c=0.079+0.060i, while Supplemental Sec. D states λ=0.0788+0.0603i. These should be aligned.
[Appendix C, Eq. (C15)] The branch selection from the multivalued relation (C14) is central to the analytic continuation. The statement 'continuously connected to the Potts blob boundary conditions' should be made more explicit, e.g., by specifying the continuation path or the choice of logarithm branch in the complex plane.
[Summary and discussion] The possible generalized g-theorem for complex BCFT, Eq. (18), is framed as speculation and supported only by the fitted magnitudes. This is clearly labeled, but it might be separated more explicitly from the extracted data to avoid giving it the same status as the fixed/free benchmark.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the universal coefficients are extrapolated intercepts compared against independent BCFT/2BTL data; the lattice-to-CFT pairing is an explicit ansatz, not a fitted input.

full rationale

The central extraction is not circular by construction. Eq. (2) defines Z_{Phi,i}(L) from fixed microscopic preparation states and the biorthogonal eigenstates; Eq. (3) is the scaling form whose intercept gives G_{a,i}. The universal coefficient is not used as input: Eqs. (7)-(8) fit only nonuniversal extensive and 1/L corrections, while the target G is the extrapolated constant, and ratios cancel the nuisance terms. The dual preparations in Eqs. (10) and (12) are prescribed by lattice symmetries (eta-pseudo-Hermiticity for Yang-Lee and complex-symmetric transpose for the Potts chain), not chosen to reproduce the BCFT coefficients being tested. Appendix A does rest on the Calabrese-Cardy regularized-boundary-state ansatz and assumes that the lattice dual realizes the BPZ pairing (Eq. A5); this is a genuine unproven load-bearing premise. But it is a hypothesis with falsifiable content: the extracted Yang-Lee value g_1 and negative ratio, and the fixed/free Potts complex coefficients, agree with independent Cardy and analytically continued 2BTL predictions, while the k=2 excited sector and k=3,4 sectors show discrepancies that the paper reports rather than hides. Thus the method could have failed, and did partially fail for intermediate sectors; it is not an identity between input and output. Self-citations enter only in finite-size-correction power counting (Ref. [48]) and in a speculative RG-flow interpretation of the k=3,4 mismatches (Ref. [43]); neither is a uniqueness theorem and neither is needed to obtain the headline fixed/free and Yang-Lee results. The unproven pairing identification is a correctness risk, not a circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central method relies on a handful of input assumptions, the regularized-boundary-state picture, the pairing correspondence, and the analytic continuation, but introduces no new physical entities. All fitted quantities are nuisance finite-size coefficients, not the target universal data.

free parameters (4)

alpha_Phi (per-preparation extensive exponent) = fitted; not tabulated
Nonuniversal extensive term in Eq. (3)/(7) removed by fitting -log Z vs L.
beta_Phi and B_Phi (Yang-Lee 1/L and 1/L^2 amplitudes) = fitted; not tabulated
Finite-size correction coefficients in the Yang-Lee ansatz, Eqs. (A.14)-(A.15).
A_k (Potts 1/L amplitude) = fitted; not tabulated
Finite-size correction coefficient in the complex Potts fit, Eq. (A.16).
h_c^infinity extrapolation coefficients (A,B,C in Eq. C.1) = e.g., h_c^inf=0.8834983(1) at lambda=3; 1.55872195(5) at lambda=4
Thermodynamic critical field used as input for Yang-Lee projected overlaps; obtained by fitting finite-size exceptional points with assumed scaling powers.

assumptions (5)

domain assumption Short-range-entangled preparation states flow to smeared Cardy boundary states of the infrared CFT (Calabrese-Cardy regularized-boundary-state ansatz), also in non-Hermitian systems.
Used in Appendix A, Eq. (A4)-(A5), to identify lattice overlaps with the boundary cylinder amplitude.
ad hoc to paper The lattice bilinear pairing encoded by eta (pseudo-Hermitian) or by the complex-symmetric transpose S_sigma reproduces the BPZ pairing of the nonunitary CFT; therefore the left dual in Eqs. (10)/(12) is the correct dual.
Eqs. (10)-(12) define the dual preparation; no derivation from microscopic dynamics is given, only consistency checks.
domain assumption The biorthogonal projector onto low-energy periodic eigenstates corresponds to the CFT projector onto a single conformal sector i.
Appendix A, Eq. (A6); essential for Z_Phi,i to be a sector-resolved boundary amplitude.
domain assumption The finite-size correction powers appearing in the fits (1/L, 1/L^2, L^-2.4, etc.) are the leading corrections from boundary and bulk irrelevant operators.
SM Sec. A power counting; used in all fits and in Eq. (C.1) for h_c extrapolation.
domain assumption The analytic continuation of the two-boundary Temperley-Lieb loop model to Q=5 with branch gamma = -i log phi describes the complex fixed point of the non-Hermitian five-state Potts chain.
Appendix C; taken from Refs [26,30-32]; central to the Potts comparison.

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Pith. "Pith review of Extracting Boundary Conformal Data from Periodic Non-Hermitian Critical Chains." pith.science (2026). https://pith.science/paper/YHZNTLCX

@misc{pith2026260616785,
  author       = {Pith},
  title        = {Pith review of: Extracting Boundary Conformal Data from Periodic Non-Hermitian Critical Chains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YHZNTLCX}},
  note         = {Machine review of arXiv:2606.16785}
}

abstract

Boundary conformal field theory (BCFT) contains universal data that are usually accessed microscopically by imposing spatial boundaries on the lattice. In non-Hermitian many-body systems, however, changing boundary conditions can qualitatively reorganize spectra and eigenstates, and their boundary criticality remains elusive. Here, we introduce a periodic-chain spectroscopy to extract universal boundary quantities, such as the Affleck-Ludwig $g$-factor, directly from non-Hermitian bulk-critical quantum chains, avoiding the need to engineer microscopic open boundaries and circumventing subtle boundary effects in non-Hermitian systems. We illustrate our method with a $\mathcal{PT}$-symmetric Ising realization of the real nonunitary Yang-Lee CFT and reveal a universal negative excited-to-ground ratio, which has no counterpart in unitary critical theories and provides a microscopic signature of nonunitarity. For the genuinely complex fixed points of the non-Hermitian five-state Potts chain, we extract intrinsically complex boundary coefficients, verify the exact Kramers-Wannier duality relation, and select the consistent analytic-continuation branch of the boundary states. Our results establish a route to nonunitary BCFT universal data using only knowledge of the bulk critical system, opening a window into non-Hermitian boundary criticality.

Figures

Figures reproduced from arXiv: 2606.16785 by the authors.

**Figure 2.** FIG. 2. Finite-size extraction of complex Potts projected [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Forward citations

Cited by 3 Pith papers

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Reviewed August 3, 2026 · model on record in the stance chip above.

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