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arxiv: 2606.32035 · v1 · pith:55LLT3FUnew · submitted 2026-06-30 · ❄️ cond-mat.stat-mech · hep-th

Exactly solvable non-unitary conformal interfaces in unitary CFTs

Qicheng Tang , Zixia Wei , Xueda Wen This is my paper

Pith reviewed 2026-07-01 02:24 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech hep-th
keywords non-unitary interfacesconformal interfacesboundary CFTCardy's conditionentanglement scalingquantum quenchesanalytic continuationbiorthogonal bases
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The pith

Analytic continuation of unitary interface scattering data produces exactly solvable non-unitary conformal interfaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs non-unitary interfaces directly on the lattice by analytically continuing the scattering data of known unitary conformal interfaces, producing an SL(2,C)-parametrized family that remains exactly conformal and solvable while breaking probability-current conservation. Exploiting the lattice-continuum correspondence, the authors derive the corresponding boundary states in the folded picture, showing that the Hilbert space requires independent incoming and outgoing states specified by a pair of dual biorthogonal bases. This setup yields a non-unitary generalization of Cardy's condition. The interfaces exhibit logarithmic entanglement scaling with a generally complex effective central charge; the SU(1,1) subclass keeps the central charge real but unbounded as transmission grows. The work also analyzes global quenches initialized with such non-unitary boundaries and relates their effective temperature to the dimension of the associated boundary-condition-changing operators.

Core claim

By analytically continuing the scattering data of known exact unitary conformal interfaces on the lattice, we obtain an SL(2,C)-parametrized family of non-unitary interfaces that remain exactly conformal and solvable. This construction determines consistent non-unitary boundary and interface CFT descriptions via biorthogonal bases in the closed-string channel and produces a non-unitary generalization of Cardy's condition, with the interfaces showing logarithmic entanglement governed by a generally complex effective central charge.

What carries the argument

The SL(2,C)-parametrized family obtained from analytic continuation of scattering data of unitary interfaces, together with the requirement of independent incoming and outgoing boundary states from dual biorthogonal bases.

Load-bearing premise

Analytic continuation of the scattering data from known unitary conformal interfaces preserves exact conformality and solvability on the lattice.

What would settle it

A direct lattice calculation showing that the continued scattering parameters violate the conformal crossing relations or produce non-conformal finite-size scaling would disprove that the family remains exactly conformal.

Figures

Figures reproduced from arXiv: 2606.32035 by Qicheng Tang, Xueda Wen, Zixia Wei.

Figure 1
Figure 1. Figure 1: FIG. 1. Half-chain entanglement entropy (EE) [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
read the original abstract

We construct directly on the lattice a class of non-unitary interfaces that are both exactly conformal and exactly solvable, and establish their corresponding boundary and interface conformal field theory (CFT) descriptions. The construction is obtained by analytically continuing the scattering data of known exact unitary conformal interfaces on the lattice, yielding an $SL(2,\mathbb C)$-parametrized family, which is non-compact and breaks probability-current conservation. Exploiting the exact lattice-continuum correspondence, we derive the conformal boundary states in the folded picture. We show that a proper definition of the Hilbert space in the closed-string channel requires the incoming and outgoing boundary states to be specified independently by boundary data associated with a pair of dual biorthogonal bases, in close analogy with the right and left eigenvectors of a non-Hermitian Hamiltonian. This requirement determines a consistent CFT construction of non-unitary boundaries and interfaces, and leads to a non-unitary generalization of the conventional Cardy's condition for unitary boundary CFT. Beyond their formal construction, these non-unitary interfaces are shown to exhibit logarithmic entanglement scaling governed by an effective central charge that is generally complex. For the $SU(1,1)$ subclass, the effective central charge remains real but grows without bound as the transmission coefficient increases. This result is demonstrated through analytical and numerical lattice calculations, as well as an interface CFT analysis in the unfolded picture. Finally, we present a general CFT analysis of a class of global quantum quenches whose initial states are prepared with non-unitary boundaries. We relate their effective temperature to the conformal dimension of the boundary-condition-changing operators associated with non-unitary boundary conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper constructs a class of non-unitary conformal interfaces directly on the lattice by analytic continuation of the scattering data of known unitary exact conformal interfaces, yielding an SL(2,C)-parametrized family. It derives the corresponding boundary and interface CFT descriptions in the folded picture using independent incoming/outgoing boundary states from dual biorthogonal bases, generalizes Cardy's condition to the non-unitary setting, demonstrates logarithmic entanglement entropy scaling with a generally complex effective central charge (real and unbounded for the SU(1,1) subclass), and analyzes global quantum quenches initialized with non-unitary boundaries, relating effective temperature to boundary-condition-changing operator dimensions.

Significance. If the central claims hold, the work supplies an exactly solvable lattice realization of non-unitary conformal interfaces together with a consistent CFT framework that extends Cardy-type conditions and yields concrete predictions for entanglement and quench dynamics. The explicit use of lattice-continuum correspondence and the biorthogonal construction are notable strengths that could enable further analytic and numerical studies in non-unitary settings.

major comments (3)
  1. [Lattice construction and analytic continuation (Section 2)] The central claim that analytic continuation of unitary lattice scattering data produces an SL(2,C) family that remains exactly conformal and exactly solvable on the lattice is load-bearing, yet the manuscript invokes the lattice-continuum correspondence after continuation without an independent verification that the continued interface operator still commutes with the transfer matrix (or satisfies the algebraic relations enforcing conformal invariance). This verification is required to substantiate the 'exactly conformal and exactly solvable' assertion in the abstract and the subsequent CFT analysis.
  2. [Folded-picture CFT construction (Section 3)] The biorthogonal Hilbert-space definition in the closed-string channel, which requires independent specification of incoming and outgoing boundary states, is introduced to enable the non-unitary generalization of Cardy's condition, but the manuscript does not provide an explicit check that this construction preserves the modular invariance or consistency conditions needed for a well-defined CFT partition function on the torus or cylinder. This step is load-bearing for the claimed non-unitary Cardy generalization.
  3. [Entanglement and interface CFT analysis (Section 5)] §5 (entanglement scaling) and the interface CFT analysis: the reported unbounded growth of the real effective central charge for the SU(1,1) subclass is shown to increase with the transmission coefficient, which is a free parameter of the SL(2,C) family; the growth is therefore a direct parametric dependence rather than an independent prediction, which limits the strength of the claim that the interfaces 'exhibit logarithmic entanglement scaling governed by an effective central charge'.
minor comments (2)
  1. [Section 3] Notation for the biorthogonal bases and the distinction between left and right eigenvectors should be clarified with explicit definitions early in the folded-picture section to avoid ambiguity when comparing to the unitary case.
  2. [Numerical results (Section 5)] The numerical lattice calculations supporting the entanglement scaling would benefit from additional details on system sizes, boundary conditions, and error estimates to allow independent reproduction.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [Lattice construction and analytic continuation (Section 2)] The central claim that analytic continuation of unitary lattice scattering data produces an SL(2,C) family that remains exactly conformal and exactly solvable on the lattice is load-bearing, yet the manuscript invokes the lattice-continuum correspondence after continuation without an independent verification that the continued interface operator still commutes with the transfer matrix (or satisfies the algebraic relations enforcing conformal invariance). This verification is required to substantiate the 'exactly conformal and exactly solvable' assertion in the abstract and the subsequent CFT analysis.

    Authors: We agree that an explicit post-continuation verification strengthens the presentation. The relevant commutation relations are algebraic identities in the scattering parameters that do not rely on unitarity and therefore continue analytically. In the revised manuscript we have added a direct check in Section 2 confirming that the continued interface operator commutes with the transfer matrix for generic SL(2,C) parameters. revision: yes

  2. Referee: [Folded-picture CFT construction (Section 3)] The biorthogonal Hilbert-space definition in the closed-string channel, which requires independent specification of incoming and outgoing boundary states, is introduced to enable the non-unitary generalization of Cardy's condition, but the manuscript does not provide an explicit check that this construction preserves the modular invariance or consistency conditions needed for a well-defined CFT partition function on the torus or cylinder. This step is load-bearing for the claimed non-unitary Cardy generalization.

    Authors: We acknowledge the value of an explicit consistency check. The biorthogonal construction ensures that the sewing conditions and the resulting partition functions remain well-defined by direct analogy with the left/right eigenvector structure of non-Hermitian transfer matrices. In the revised manuscript we have added a short verification that the cylinder partition function obtained from the dual bases satisfies the generalized Cardy condition and is consistent under modular transformations. revision: yes

  3. Referee: [Entanglement and interface CFT analysis (Section 5)] §5 (entanglement scaling) and the interface CFT analysis: the reported unbounded growth of the real effective central charge for the SU(1,1) subclass is shown to increase with the transmission coefficient, which is a free parameter of the SL(2,C) family; the growth is therefore a direct parametric dependence rather than an independent prediction, which limits the strength of the claim that the interfaces 'exhibit logarithmic entanglement scaling governed by an effective central charge'.

    Authors: The referee correctly observes the parametric dependence on the transmission coefficient. This dependence is, however, a central and non-trivial feature of the construction: it shows that the SU(1,1) subclass realizes arbitrarily large real effective central charges. The logarithmic scaling itself is established independently by three routes (exact lattice computation, analytic continuation of the entanglement formula, and the unfolded interface CFT analysis). We therefore maintain that the claim is substantiated and that the unbounded growth does not weaken but rather illustrates the result. No revision is required. revision: no

Circularity Check

0 steps flagged

No significant circularity; construction and derived properties are independent

full rationale

The paper's derivation starts from an explicit lattice construction obtained by analytic continuation of scattering data from known unitary conformal interfaces, producing an SL(2,C)-parametrized family. It then exploits the lattice-continuum correspondence to derive boundary states and a non-unitary generalization of Cardy's condition, with properties such as logarithmic entanglement and effective central charge growth demonstrated through separate analytical/numerical lattice calculations and unfolded CFT analysis. No quoted step reduces a claimed prediction or result to an input parameter or self-citation by construction; the transmission coefficient is an explicit free parameter of the family, and its relation to c_eff is computed rather than tautological. The work is self-contained against external benchmarks with no load-bearing self-citation chains.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The construction adds the analytic-continuation procedure and the biorthogonal-basis definition as new elements; it rests on the assumption that the bulk remains unitary and that the lattice-continuum map survives continuation.

free parameters (2)
  • SL(2,C) parameters
    The family is explicitly parametrized by SL(2,C) elements that label the non-unitary interfaces.
  • transmission coefficient
    Effective central charge is reported to grow without bound as this coefficient increases in the SU(1,1) subclass.
axioms (2)
  • domain assumption The bulk theory remains a unitary CFT after interface continuation
    The paper starts from unitary CFTs and continues only the interface data.
  • ad hoc to paper Lattice-continuum correspondence holds for the analytically continued non-unitary interfaces
    Exploiting the exact lattice-continuum correspondence is invoked to derive the CFT boundary states.

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discussion (0)

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Reference graph

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