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Complex entanglement entropy for complex conformal field theory

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Pith's one-line read Complex entanglement entropy confirms complex CFT scaling

desk verdict A solid numerical demonstration that biorthogonal entanglement entropy extracts the complex central charge of the non-Hermitian Potts model, but the universal claim is undercut by an unexplained OBC deviation and a fragile three-point fit. read the letter →

arxiv 2502.02001 v2 pith:EAABJ6G4 submitted 2025-02-04 cond-mat.stat-mech hep-thquant-ph

Haruki Shimizu , Kohei Kawabata This is my paper · ORCID
classification cond-mat.stat-mechhep-thquant-ph
keywords complexconformalfieldtheoryentanglemententropynon-Hermitianquantumsystemsfive-statePottsmodeldensitymatrixrenormalizationgroupcentralchargepseudo-entropynonunitarycriticalphenomena
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

The paper argues that a complex-valued generalization of entanglement entropy can characterize conformal field theory (CFT) with complex central charges, which describe non-Hermitian critical systems. It constructs non-Hermitian reduced density matrices from the right and left ground states and defines the entropy through the complex logarithm. For the critical non-Hermitian five-state Potts model, density-matrix-renormalization-group calculations yield the universal CFT scaling $S = \frac{c}{3} \log\left(\frac{L}{\pi} \sin\frac{\pi l}{L}\right) + S_0$ with complex $c$. The fitted central charge approaches the CFT prediction $c \approx 1.13755 - 0.02107i$, and the paper reads this as evidence that complex CFT controls the model's critical behavior.

What carries the argument

The central object is the non-Hermitian (biorthogonal) reduced density matrix $\hat{\rho}^{RL}_A = \mathrm{tr}_B\left(|\psi\rangle\langle\langle\psi| / \langle\langle\psi|\psi\rangle\right)$, built from a right ground state $|\psi\rangle$ and a left ground state $|\psi\rangle\rangle$ of a non-Hermitian Hamiltonian. The complex entanglement entropy is $S = -\mathrm{tr}_A\left(\hat{\rho}^{RL}_A \log \hat{\rho}^{RL}_A\right)$ with the logarithm branch $\arg z \in (-\pi, \pi]$. This object carries the argument because it is the quantity whose finite-size scaling reproduces the CFT formula, and it is the ingredient that makes the entropy complex rather than real.

What would settle it

An exact or high-precision computation for larger systems, say $L = 32$ with bond dimension $D = 600$, that shows the fitted complex central charge moving away from $c \approx 1.13755 - 0.02107i$, or an imaginary part of $S(L, L/2)$ that does not grow as $\frac{\mathrm{Im}\, c}{3} \log\left(\frac{L}{\pi}\right)$, would disprove the claim. The open-boundary result already provides a partial negative control: a robust theory would need to explain why the same fitting protocol gives $\mathrm{Re}\, c \approx 1.4415$ there.

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Extended reading notes

Core claim

Conformal field theory's entanglement scaling formula extends without modification to complex central charges, provided the reduced density matrix is built biorthogonally from both right and left eigenstates rather than from a single Hermitian density matrix. The resulting entropy is complex, and its real and imaginary parts both follow the universal finite-size scaling predicted by CFT under periodic boundary conditions. Numerical fitting for the non-Hermitian five-state Potts model gives $c \approx 1.1385 - 0.0205i$ at $L = 20$, $D = 400$, close to the analytic $c \approx 1.13755 - 0.02107i$, and closer to theory than the central charge extracted from eigenenergy scaling in earlier work. The authors take this as validation that complex entanglement entropy is a meaningful probe of complex CFT and of open quantum many-body criticality.

Load-bearing premise

The argument assumes that the entropy computed from paired right and left ground-state wave functions obeys the same CFT scaling formula as a Hermitian system's entropy in the middle of the chain, so fitting only the three central subsystem sizes yields an unbiased estimate of the complex central charge.

Editorial extensions

If this is right

  • The universal formula $S = \frac{c}{3} \log\left(\frac{L}{\pi} \sin\frac{\pi l}{L}\right) + S_0$ can be used to extract complex central charges directly from entanglement data.
  • Complex entanglement entropy is a numerical probe of complex CFT that is more precise than finite-size eigenenergy scaling, because it avoids an extra energy-scale fit.
  • The same construction connects to pseudo-entropy in high-energy physics, so CFT results for complex entropy may transfer to that setting.
  • Real-valued alternatives based on Hermitian reduced density matrices or SVD fail to capture the complex central charge, indicating that biorthogonality is essential.
  • Open boundary conditions do not obey the same scaling, signaling boundary effects specific to non-Hermitian systems that require separate treatment.

Reading between the lines

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

  • If the scaling law holds for other $Q > 4$ Potts models or complexified Kondo couplings, complex entanglement entropy would offer a general route to determining complex fixed points from lattice data.
  • The open-boundary deviation suggests a boundary-condition-dependent effective description, possibly connected to the non-Hermitian skin effect, which could be tested by studying how the fitted $c$ depends on boundary coupling.
  • A direct extension would be to compute the complex Rényi entropy and check whether its $n$-dependence matches the CFT prediction, giving an independent falsifier beyond von Neumann entropy.
  • The Petermann factor analysis implies that the finite-size spectral gap at the complex fixed point is what makes the CFT scaling visible; systems with exceptional points may require a different scaling ansatz.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper introduces a complex-valued entanglement entropy constructed from the biorthogonal (right/left) ground states of non-Hermitian Hamiltonians and applies it to the critical non-Hermitian five-state Potts model, which is believed to be described by complex conformal field theory with central charge c = 1.13755 - 0.02107i. Using finite-system DMRG, the authors compute the entropy as a function of subsystem size for periodic and open boundary conditions. By fitting the universal CFT scaling form to three central points, they extract complex central charges that approach the theoretical value for PBC with increasing system size (e.g., c = 1.1385 - 0.0205i at L=20, D=400). The OBC data, however, yield a significantly different real part (Re c ≈ 1.4415) and a positive imaginary part (Im c ≈ 0.05), which the paper attributes to non-Hermitian boundary effects without quantitative analysis. The paper also compares alternative entropy definitions and examines the Petermann factor to contrast with c = -2 nonunitary CFT.

Significance. If the PBC agreement is taken at face value, the result would be a useful numerical demonstration that biorthogonal entanglement entropy can serve as a probe of complex CFT, complementing previous energy-based extractions of complex central charges. The paper's strengths are that the fitted central charge is compared with an independent theoretical prediction (Eq. 9) from the literature, that bond-dimension convergence is checked for the PBC and OBC data, and that alternative definitions (Sabs, SVD entropy) are analyzed to show the specificity of the biorthogonal construction. However, the central claim of universality is weakened by the unexplained OBC failure, by the lack of error bars and of a full finite-size scaling analysis, and by the absence of any analytic derivation of the assumed CFT form. The paper is therefore best viewed as a promising numerical study that needs additional systematic checks before the universality claim can be considered established.

major comments (3)
  1. [IV.B, Eq. (30), Fig. 1, Table I] The fit protocol used throughout the paper is a least-squares fit of Eq. (30) (or (31)) to only the three central values of l for each L, with no reported error bars or residuals. This leaves one degree of freedom (three data points, two fit parameters) and makes the extracted central charge highly sensitive to boundary distortions: Fig. 1(b) shows clear deviations for l near 1 and L-1, and Table I shows that the L=4 system yields Im c = +0.0104, opposite in sign to the CFT value -0.0211. The paper attributes these deviations to boundary effects but provides no quantitative model for them. To support the universality claim, the authors should either fit the full l-dependence with an explicit boundary-correction term or demonstrate that the three-point estimate converges with L in a controlled way, e.g., by a finite-size extrapolation.
  2. [IV.C, Eq. (31), Fig. 3] The open-boundary results in §IV.C are in sharp conflict with the CFT prediction: the fit gives Re c ≈ 1.4415 and Im c ≈ 0.05, whereas Eq. (9) predicts Re c = 1.1376 and Im c = -0.0211. The authors invoke 'subtle boundary effects' and the non-Hermitian skin effect, but no quantitative boundary-effect analysis is presented. Since Eq. (31) is the standard CFT formula for open boundaries, the observed failure means that the paper's central claim — that the universal scaling S = (c/3) log[...] holds for complex CFT — is not supported by the OBC geometry. The manuscript needs either a demonstration that the OBC deviations vanish as L increases, or a concrete model of the non-Hermitian boundary contributions that reproduces the fitted values; without this, the PBC success cannot be unambiguously separated from boundary artifacts.
  3. [III, Eq. (17), and IV.B] Equation (30) is posited as the CFT scaling law for the biorthogonal entanglement entropy, but no derivation is given for why the non-Hermitian reduced density matrix (15) should obey this form, nor why the central charge should appear as a complex coefficient in the logarithm. The numerical test is therefore a check that the coefficient extracted from data matches Eq. (9) under the assumption that the l-dependence is log[(L/π) sin(πl/L)]. The paper should explicitly state that this is an assumed functional form rather than a derived result, and ideally should provide a derivation or a more direct test, for example by verifying the predicted l-dependence over the full range after subtracting a modeled boundary term. This issue is load-bearing because the universality claim rests entirely on this assumed form.
minor comments (4)
  1. [Abstract and Section I] The paper uses 'we demonstrate' in the abstract and introduction, but the evidence is numerical and conditional on the assumed CFT form; a more cautious phrasing such as 'we provide numerical evidence' would better match the content.
  2. [IV.B] The statement that the L=20, D=300 imaginary part is a coincidence due to competition between finite-size and bond-dimension effects is informal; a quantitative estimate of both corrections would be more persuasive.
  3. [IV.A and Figs. 1(c)-(d), 3(c)-(d)] The DMRG convergence analysis reports the difference between successive bond dimensions, but not the truncation error or the overlap between right and left ground states; specifying these would strengthen the reliability argument.
  4. [IV.D, Fig. 4] The comparison with Sabs and Ssvd is presented without a quantitative explanation of why the differences appear where they do; a few sentences on the origin of these deviations would improve the discussion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the fitted central charge is tested against an independent analytic prediction, and the acknowledged OBC discrepancy is a limitation, not a circular step.

full rationale

The paper's derivation chain does not reduce to its own inputs. The central charge is not defined in terms of the entanglement entropy; it is an independent input from Eq. (9), quoted from the earlier complex-CFT analysis of the Potts model by Gorbenko, Rychkov, and Zan (Ref. [48], not the present authors). The paper fits Eq. (30) to three central values of the numerically computed biorthogonal entanglement entropy and finds c = 1.1385 - 0.0205i at L = 20, D = 400, then compares with Eq. (9). This is a standard consistency test: a parameter is fitted to data and checked against an external prediction. The biorthogonal reduced density matrix (Eq. 15) and the generalized entropy (Eq. 17) are constructed from right and left eigenstates and do not encode the target value of c. The alternative entropies S_abs and S_svd in Sec. IV D are also defined independently, and comparing them with S_cpx strengthens rather than presupposes the claim. The one genuine weakness noted in the manuscript is the open-boundary-condition result in Sec. IV C, where the same fit yields Re c ≈ 1.4415 and an imaginary part inconsistent with complex CFT; the paper attributes this to non-Hermitian boundary and skin effects without a quantitative model. That is an acknowledged limitation of the evidence, not a circularity, because the OBC data are not used to define or force the periodic-boundary conclusion. There is no load-bearing self-citation chain, no uniqueness theorem imported from the authors' own work, and no fitted quantity is renamed as a prediction. Accordingly, the circularity score is 0.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central claim rests on assumed CFT scaling, externally supplied critical parameters, and a convention for the complex logarithm. The fitted central charge and offset are the only numerical parameters extracted in the paper.

free parameters (3)
  • complex central charge c = 1.1385 - 0.0205i (L=20, D=400, PBC)
    Fitted to the three central points of the complex entanglement entropy vs subsystem size, used to compare with theory, not derived.
  • constant offset S0 = not reported
    The intercept in the CFT scaling formula; fitted simultaneously with c.
  • branch cut convention for log(z) = arg z in (-pi, pi]
    Chosen by hand to define the complex logarithm; different branch choices would alter the imaginary part of the entropy.
assumptions (3)
  • domain assumption The non-Hermitian Potts model with parameters in Eq. (8) is at a complex fixed point described by complex CFT with central charge Eq. (9).
    The paper takes the critical parameters from Ref. [57] and the CFT value from Ref. [48] as given inputs.
  • domain assumption The CFT scaling form S = (c/3) log[...] + S0 applies to the biorthogonal entanglement entropy with complex c.
    The paper assumes this functional form to extract c, rather than deriving it from first principles.
  • domain assumption The ground state is the eigenstate with minimum real part of the complex eigenenergy and biorthogonal normalization <psi|psi> != 0 holds.
    Selection of the target state and well-definedness of the non-Hermitian reduced density matrix; the paper notes this can fail at exceptional points but argues the finite-size Potts model avoids it.

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Cite this review

Pith. "Pith review of Complex entanglement entropy for complex conformal field theory." pith.science (2026). https://pith.science/paper/EAABJ6G4

@misc{pith2026250202001,
  author       = {Pith},
  title        = {Pith review of: Complex entanglement entropy for complex conformal field theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EAABJ6G4}},
  note         = {Machine review of arXiv:2502.02001}
}
read the original abstract

Conformal field theory underlies critical ground states of quantum many-body systems. While conventional conformal field theory is associated with positive central charges, nonunitary conformal field theory with complex-valued central charges has recently been recognized as physically relevant. Here, we demonstrate that complex-valued entanglement entropy characterizes complex conformal field theory and critical phenomena of open quantum many-body systems. This is based on non-Hermitian reduced density matrices constructed from the combination of right and left ground states. Applying the density matrix renormalization group to non-Hermitian systems, we numerically calculate the complex entanglement entropy of the non-Hermitian five-state Potts model, thereby confirming the scaling behavior predicted by complex conformal field theory.

Figures

Figures reproduced from arXiv: 2502.02001 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Real and (b) imaginary parts of complex entangle [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Imaginary part of complex entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Real and (b) imaginary parts of complex entangle [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Petermann factor [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 4
Figure 4. Figure 4: FIG. 4. (a), (c) Real and (b), (d) imaginary parts of com [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Complex Conformal Manifolds

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    Complexified exactly-marginal couplings produce solvable complex CFTs, with the Ising defect verified numerically in non-Hermitian chains.

  2. Extracting Boundary Conformal Data from Periodic Non-Hermitian Critical Chains

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    Periodic-chain projected overlaps with paired left duals extract universal boundary CFT coefficients, including a negative Yang-Lee ratio and complex Potts boundary data.

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    Tensor network simulations of the non-Hermitian 5-state Potts model reveal the predicted spiral RG flow and a boundary spectrum matching complex conformal field theory.

Reference graph

Works this paper leans on

104 extracted references · 59 canonical work pages · cited by 3 Pith papers

  1. [1]

    M. A. Nielsen and I. L. Chuang, Quantum Computa- tion and Quantum Information(Cambridge University Press, Cambridge, England, 2000)

    2000

  2. [2]

    Calabrese and J

    P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 2005, P04010 (2005); Time Dependence of Correlation Functions Following a Quantum Quench, Phys. Rev. Lett. 96, 136801 (2006)

    2005

  3. [3]

    Kim and D

    H. Kim and D. A. Huse, Ballistic Spreading of Entangle- ment in a Diffusive Nonintegrable System, Phys. Rev. Lett. 111, 127205 (2013)

    2013

  4. [4]

    Nahum, J

    A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum Entanglement Growth under Random Unitary Dynam- ics, Phys. Rev. X 7, 031016 (2017)

    2017

  5. [5]

    Osterloh, L

    A. Osterloh, L. Amico, G. Falci, and R. Fazio, Scaling of entanglement close to a quantum phase transition, Nature 416, 608 (2002)

    2002

  6. [6]

    T. J. Osborne and M. A. Nielsen, Entanglement in a simple quantum phase transition, Phys. Rev. A 66, 032110 (2002)

    2002

  7. [7]

    Holzhey, F

    C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424, 443 (1994)

    1994

  8. [8]

    Vidal, J

    G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entan- glement in Quantum Critical Phenomena, Phys. Rev. Lett. 90, 227902 (2003)

    2003

Show all 104 references
  1. [9]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 2004, P06002 (2004)

    2004

  2. [10]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42, 504005 (2009)

    2009

  3. [11]

    Kitaev and J

    A. Kitaev and J. Preskill, Topological Entanglement En- tropy, Phys. Rev. Lett. 96, 110404 (2006)

    2006

  4. [12]

    Levin and X.-G

    M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96, 110405 (2006)

    2006

  5. [13]

    Ryu and Y

    S. Ryu and Y. Hatsugai, Entanglement entropy and the Berry phase in the solid state, Phys. Rev. B 73, 245115 (2006)

    2006

  6. [14]

    Li and F

    H. Li and F. D. M. Haldane, Entanglement Spectrum as a Generalization of Entanglement Entropy: Iden- tification of Topological Order in Non-Abelian Frac- tional Quantum Hall Effect States, Phys. Rev. Lett. 101, 010504 (2008)

    2008

  7. [15]

    Pollmann, A

    F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Entanglement spectrum of a topological phase in one dimension, Phys. Rev. B 81, 064439 (2010)

    2010

  8. [16]

    A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Unitary-projective entanglement dynamics, Phys. Rev. B 99, 224307 (2019)

    2019

  9. [17]

    Skinner, J

    B. Skinner, J. Ruhman, and A. Nahum, Measurement- Induced Phase Transitions in the Dynamics of Entan- glement, Phys. Rev. X 9, 031009 (2019)

    2019

  10. [18]

    Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno ef- fect and the many-body entanglement transition, Phys. Rev. B 98, 205136 (2018); Measurement-driven entan- glement transition in hybrid quantum circuits, Phys. Rev. B 100, 134306 (2019). 9

    2018

  11. [19]

    S. Choi, Y. Bao, X.-L. Qi, and E. Altman, Quan- tum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition, Phys. Rev. Lett. 125, 030505 (2020)

    2020

  12. [20]

    M. J. Gullans and D. A. Huse, Dynamical Purification Phase Transition Induced by Quantum Measurements, Phys. Rev. X 10, 041020 (2020)

    2020

  13. [21]

    Jian, Y.-Z

    C.-M. Jian, Y.-Z. You, R. Vasseur, and A. W. W. Lud- wig, Measurement-induced criticality in random quan- tum circuits, Phys. Rev. B 101, 104302 (2020)

    2020

  14. [22]

    M. P. A. Fisher, V. Khemani, A. Nahum, and S. Vi- jay, Random Quantum Circuits, Annu. Rev. Condens. Matter Phys. 14, 335 (2023)

    2023

  15. [23]

    R. B. Potts, Some generalized order-disorder transfor- mations, Math. Proc. Camb. Philos. Soc.48, 106 (1952)

    1952

  16. [24]

    F. Y. Wu, The Potts model, Rev. Mod. Phys. 54, 235 (1982)

    1982

  17. [25]

    R. J. Baxter, Potts model at the critical temperature, J. Phys. C 6, L445 (1973)

    1973

  18. [26]

    Nienhuis, A

    B. Nienhuis, A. N. Berker, E. K. Riedel, and M. Schick, First- and Second-Order Phase Transitions in Potts Models: Renormalization-Group Solution, Phys. Rev. Lett. 43, 737 (1979)

    1979

  19. [27]

    Nienhuis, E

    B. Nienhuis, E. K. Riedel, and M. Schick, Variational renormalisation-group approach to the q-state Potts model in two dimensions, J. Phys. A 13, L31 (1980)

    1980

  20. [28]

    Nauenberg and D

    M. Nauenberg and D. J. Scalapino, Singularities and Scaling Functions at the Potts-Model Multicritical Point, Phys. Rev. Lett. 44, 837 (1980)

    1980

  21. [29]

    Binder, Static and dynamic critical phenomena of the two-dimensional q-state Potts model, J

    K. Binder, Static and dynamic critical phenomena of the two-dimensional q-state Potts model, J. Stat. Phys. 24, 69 (1981)

    1981

  22. [30]

    J. L. Cardy, M. Nauenberg, and D. J. Scalapino, Scal- ing theory of the Potts-model multicritical point, Phys. Rev. B 22, 2560 (1980)

    1980

  23. [31]

    Nienhuis, E

    B. Nienhuis, E. K. Riedel, and M. Schick, q-state Potts model in general dimension, Phys. Rev. B 23, 6055 (1981)

    1981

  24. [32]

    V. S. Dotsenko, Critical behaviour and associated con- formal algebra of the Z 3 Potts model, Nucl. Phys. B 235, 54 (1984)

    1984

  25. [33]

    V. S. Dotsenko and V. A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nucl. Phys. B 240, 312 (1984)

    1984

  26. [34]

    A. B. Zamolodchikov and V. A. Fateev, Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points inZN - symmetric statistical systems, J. Exp. Theor. Phys. 62, 215 (1985)

    1985

  27. [35]

    parafermion currents

    A. B. Zamolodchikov and V. A. Fateev, Representations of the algebra of “parafermion currents” of spin 4 /3 in two-dimensional conformal field theory. Minimal models and the tricritical potts Z 3 model, Theor. Math. Phys. 71, 451 (1987)

    1987

  28. [36]

    Kl¨ umper, A

    A. Kl¨ umper, A. Schadschneider, and J. Zittartz, Inver- sion relations, phase transitions and transfer matrix ex- citations for special spin models in two dimensions, Z. Physik B 76, 247 (1989)

    1989

  29. [37]

    Buddenoir and S

    E. Buddenoir and S. Wallon, The correlation length of the Potts model at the first-order transition point, J. Phys. A 26, 3045 (1993)

    1993

  30. [38]

    Nishino and K

    T. Nishino and K. Okunishi, Numerical Latent Heat Observation of the q = 5 Potts Model, J. Phys. Soc. Jpn. 67, 1492 (1998)

    1998

  31. [39]

    Delfino and J

    G. Delfino and J. Cardy, The field theory of the q → 4+ Potts model, Phys. Lett. B 483, 303 (2000)

    2000

  32. [40]

    Ozeki, K

    Y. Ozeki, K. Kasono, N. Ito, and S. Miyashita, Nonequi- librium relaxation analysis for first-order phase transi- tions, Physica A 321, 271 (2003)

    2003

  33. [41]

    W. Shun, X. Zhi-Yuan, C. Jing, N. Bruce, and X. Tao, Phase Transitions of Ferromagnetic Potts Models on the Simple Cubic Lattice, Chin. Phys. Lett. 31, 070503 (2014)

    2014

  34. [42]

    Liang and F

    N. Liang and F. Zhong, Renormalization-group theory for cooling first-order phase transitions in Potts models, Phys. Rev. E 95, 032124 (2017)

    2017

  35. [43]

    S. Iino, S. Morita, N. Kawashima, and A. W. Sandvik, Detecting Signals of Weakly First-order Phase Transi- tions in Two-dimensional Potts Models, J. Phys. Soc. Jpn. 88, 034006 (2019)

    2019

  36. [44]

    Morita and N

    S. Morita and N. Kawashima, Calculation of higher- order moments by higher-order tensor renormalization group, Comput. Phys. Commun. 236, 65 (2019)

    2019

  37. [45]

    D’Emidio, A

    J. D’Emidio, A. A. Eberharter, and A. M. L¨ auchli, Diag- nosing weakly first-order phase transitions by coupling to order parameters, SciPost Phys. 15, 061 (2023)

    2023

  38. [46]

    D. B. Kaplan, J.-W. Lee, D. T. Son, and M. A. Stephanov, Conformality lost, Phys. Rev. D 80, 125005 (2009)

    2009

  39. [47]

    C. Wang, A. Nahum, M. A. Metlitski, C. Xu, and T. Senthil, Deconfined Quantum Critical Points: Sym- metries and Dualities, Phys. Rev. X 7, 031051 (2017)

    2017

  40. [48]

    Gorbenko, S

    V. Gorbenko, S. Rychkov, and B. Zan, Walking, weak first-order transitions, and complex CFTs, J. High En- erg. Phys. 2018 (10), 108; Walking, Weak first-order transitions, and Complex CFTs II. Two-dimensional Potts model at Q >4, SciPost Phys. 5, 050 (2018)

    2018

  41. [49]

    Ma and Y.-C

    H. Ma and Y.-C. He, Shadow of complex fixed point: Approximate conformality of Q >4 Potts model, Phys. Rev. B 99, 195130 (2019)

    2019

  42. [50]

    Benini, C

    F. Benini, C. Iossa, and M. Serone, Conformality Loss, Walking, and 4D Complex Conformal Field Theories at Weak Coupling, Phys. Rev. Lett. 124, 051602 (2020)

    2020

  43. [51]

    A. F. Faedo, C. Hoyos, D. Mateos, and J. G. Subils, Holographic Complex Conformal Field Theories, Phys. Rev. Lett. 124, 161601 (2020)

    2020

  44. [52]

    Giombi, R

    S. Giombi, R. Huang, I. R. Klebanov, S. S. Pufu, and G. Tarnopolsky, O(N ) model in 4 < d <6: Instantons and complex CFTs, Phys. Rev. D 101, 045013 (2020)

    2020

  45. [53]

    Nahum, Fixed point annihilation for a spin in a fluc- tuating field, Phys

    A. Nahum, Fixed point annihilation for a spin in a fluc- tuating field, Phys. Rev. B 106, L081109 (2022)

    2022

  46. [54]

    S. Han, D. J. Schultz, and Y. B. Kim, Complex fixed points of the non-Hermitian Kondo model in a Luttinger liquid, Phys. Rev. B 107, 235153 (2023)

    2023

  47. [55]

    Haldar, O

    A. Haldar, O. Tavakol, H. Ma, and T. Scaffidi, Hid- den Critical Points in the Two-Dimensional O( n > 2) Model: Exact Numerical Study of a Complex Confor- mal Field Theory, Phys. Rev. Lett. 131, 131601 (2023)

    2023

  48. [56]

    J. L. Jacobsen and K. J. Wiese, Lattice Realization of Complex Conformal Field Theories: Two-Dimensional Potts Model with Q >4 States, Phys. Rev. Lett. 133, 077101 (2024)

    2024

  49. [57]

    Y. Tang, H. Ma, Q. Tang, Y.-C. He, and W. Zhu, Reclaiming the Lost Conformality in a Non-Hermitian Quantum 5-State Potts Model, Phys. Rev. Lett. 133, 076504 (2024)

    2024

  50. [58]

    C. N. Yang and T. D. Lee, Statistical Theory of Equa- tions of State and Phase Transitions. I. Theory of Con- 10 densation, Phys. Rev. 87, 404 (1952); T. D. Lee and C. N. Yang, Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model, Phy...

    1952

  51. [59]

    M. E. Fisher, Yang-Lee Edge Singularity and ϕ3 Field Theory, Phys. Rev. Lett. 40, 1610 (1978)

    1978

  52. [60]

    J. L. Cardy, Conformal Invariance and the Yang-Lee Edge Singularity in Two Dimensions, Phys. Rev. Lett. 54, 1354 (1985)

    1985

  53. [61]

    Couvreur, J

    R. Couvreur, J. L. Jacobsen, and H. Saleur, Entan- glement in Nonunitary Quantum Critical Spin Chains, Phys. Rev. Lett. 119, 040601 (2017)

    2017

  54. [62]

    Chang, J.-S

    P.-Y. Chang, J.-S. You, X. Wen, and S. Ryu, En- tanglement spectrum and entropy in topological non- Hermitian systems and nonunitary conformal field the- ory, Phys. Rev. Research 2, 033069 (2020)

    2020

  55. [63]

    C. H. Lee, Exceptional Bound States and Negative Entanglement Entropy, Phys. Rev. Lett. 128, 010402 (2022)

    2022

  56. [64]

    Guo, Y.-C

    Y.-B. Guo, Y.-C. Yu, R.-Z. Huang, L.-P. Yang, R.-Z. Chi, H.-J. Liao, and T. Xiang, Entanglement entropy of non-Hermitian free fermions, J. Phys.: Condens. Matter 33, 475502 (2021)

    2021

  57. [65]

    Tu, Y.-C

    Y.-T. Tu, Y.-C. Tzeng, and P.-Y. Chang, R´ enyi en- tropies and negative central charges in non-Hermitian quantum systems, SciPost Phys. 12, 194 (2022)

    2022

  58. [66]

    Ryu and J

    S. Ryu and J. Yoon, Unitarity of Symplectic Fermions in α Vacua with Negative Central Charge, Phys. Rev. Lett. 130, 241602 (2023)

    2023

  59. [67]

    Hsieh and P.-Y

    C.-T. Hsieh and P.-Y. Chang, Relating non-Hermitian and Hermitian quantum systems at criticality, SciPost Phys. Core 6, 062 (2023)

    2023

  60. [68]

    Fossati, F

    M. Fossati, F. Ares, and P. Calabrese, Symmetry- resolved entanglement in critical non-Hermitian sys- tems, Phys. Rev. B 107, 205153 (2023)

    2023

  61. [69]

    Rottoli, M

    F. Rottoli, M. Fossati, and P. Calabrese, Entanglement Hamiltonian in the non-Hermitian SSH model, J. Stat. Mech. 2024, 063102 (2024)

    2024

  62. [70]

    Xue and C.-H

    W.-T. Xue and C.-H. Lee, Topologically protected neg- ative entanglement, arXiv:2403.03259

  63. [71]

    C. M. Bender and S. Boettcher, Real Spectra in Non- Hermitian Hamiltonians Having PT Symmetry, Phys. Rev. Lett. 80, 5243 (1998)

    1998

  64. [72]

    C. M. Bender, Making sense of non-Hermitian Hamilto- nians, Rep. Prog. Phys. 70, 947 (2007)

    2007

  65. [73]

    V. V. Konotop, J. Yang, and D. A. Zezyulin, Nonlinear waves in PT -symmetric systems, Rev. Mod. Phys. 88, 035002 (2016)

    2016

  66. [74]

    El-Ganainy, K

    R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non- Hermitian physics and PT symmetry, Nat. Phys. 14, 11 (2018)

    2018

  67. [75]

    Cipolloni and J

    G. Cipolloni and J. Kudler-Flam, Entanglement En- tropy of Non-Hermitian Eigenstates and the Ginibre Ensemble, Phys. Rev. Lett. 130, 010401 (2023)

    2023

  68. [76]

    Nakata, T

    Y. Nakata, T. Takayanagi, Y. Taki, K. Tamaoka, and Z. Wei, New holographic generalization of entanglement entropy, Phys. Rev. D 103, 026005 (2021)

    2021

  69. [77]

    Mollabashi, N

    A. Mollabashi, N. Shiba, T. Takayanagi, K. Tamaoka, and Z. Wei, Pseudo-Entropy in Free Quantum Field Theories, Phys. Rev. Lett. 126, 081601 (2021); Aspects of pseudoentropy in field theories, Phys. Rev. Research 3, 033254 (2021)

    2021

  70. [78]

    Murciano, P

    S. Murciano, P. Calabrese, and R. M. Konik, General- ized entanglement entropies in two-dimensional confor- mal field theory, J. High Energ. Phys. 2022 (5), 152

    2022

  71. [79]

    W.-z. Guo, S. He, and Y.-X. Zhang, Constructible real- ity condition of pseudo entropy via pseudo-Hermiticity, J. High Energ. Phys. 2023 (5), 21

    2023

  72. [80]

    K. Doi, J. Harper, A. Mollabashi, T. Takayanagi, and Y. Taki, Pseudoentropy in dS /CFT and Timelike Entanglement Entropy, Phys. Rev. Lett. 130, 031601 (2023); Timelike entanglement entropy, J. High Energ. Phys. 2023 (5), 52

    2023

  73. [81]

    Narayan, de Sitter space, extremal surfaces, and time entanglement, Phys

    K. Narayan, de Sitter space, extremal surfaces, and time entanglement, Phys. Rev. D 107, 126004 (2023)

    2023

  74. [82]

    Kanda, M

    H. Kanda, M. Sato, Y.-k. Suzuki, T. Takayanagi, and Z. Wei, AdS/BCFT with brane-localized scalar field, J. High Energ. Phys. 2023 (3), 105

    2023

  75. [83]

    Kanda, T

    H. Kanda, T. Kawamoto, Y.-k. Suzuki, T. Takayanagi, K. Tasuki, and Z. Wei, Entanglement phase transition in holographic pseudo entropy, J. High Energ. Phys. 2024 (3), 60

    2024

  76. [84]

    D. C. Brody, Biorthogonal quantum mechanics, J. Phys. A 47, 035305 (2014)

    2014

  77. [85]

    Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1966)

    T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1966)

    1966

  78. [86]

    Berry, Physics of Nonhermitian Degeneracies, Czech

    M. Berry, Physics of Nonhermitian Degeneracies, Czech. J. Phys. 54, 1039 (2004)

    2004

  79. [87]

    W. D. Heiss, The physics of exceptional points, J. Phys. A 45, 444016 (2012)

    2012

  80. [88]

    Carlon, M

    E. Carlon, M. Henkel, and U. Schollw¨ ock, Density ma- trix renormalization group and reaction-diffusion pro- cesses, Eur. Phys. J. B 12, 99 (1999)

    1999

  81. [89]

    Yamamoto, M

    K. Yamamoto, M. Nakagawa, M. Tezuka, M. Ueda, and N. Kawakami, Universal properties of dissipative Tomonaga-Luttinger liquids: Case study of a non- Hermitian XXZ spin chain, Phys. Rev. B 105, 205125 (2022)

    2022

  82. [90]

    Fishman, S

    M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor Software Library for Tensor Network Calcula- tions, SciPost Phys. Codebases , 4 (2022)

    2022

  83. [91]

    Fishman, S

    M. Fishman, S. R. White, and E. M. Stoudenmire, Codebase release 0.3 for ITensor, SciPost Phys. Code- bases , 4 (2022)

    2022

  84. [92]

    T. E. Lee, Anomalous Edge State in a Non-Hermitian Lattice, Phys. Rev. Lett. 116, 133903 (2016)

    2016

  85. [93]

    Yao and Z

    S. Yao and Z. Wang, Edge States and Topological In- variants of Non-Hermitian Systems, Phys. Rev. Lett. 121, 086803 (2018)

    2018

  86. [94]

    F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Biorthogonal Bulk-Boundary Corre- spondence in Non-Hermitian Systems, Phys. Rev. Lett. 121, 026808 (2018)

    2018

  87. [95]

    Yokomizo and S

    K. Yokomizo and S. Murakami, Non-Bloch Band The- ory of Non-Hermitian Systems, Phys. Rev. Lett. 123, 066404 (2019)

    2019

  88. [96]

    Zhang, Z

    K. Zhang, Z. Yang, and C. Fang, Correspondence be- tween Winding Numbers and Skin Modes in Non- Hermitian Systems, Phys. Rev. Lett. 125, 126402 (2020)

    2020

  89. [97]

    Okuma, K

    N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topological Origin of Non-Hermitian Skin Effects, Phys. Rev. Lett. 124, 086801 (2020)

    2020

  90. [98]

    A. J. Parzygnat, T. Takayanagi, Y. Taki, and Z. Wei, SVD entanglement entropy, J. High Energ. Phys. 2023 (12), 123

    2023

  91. [99]

    Petermann, Calculated spontaneous emission factor 11 for double-heterostructure injection lasers with gain- induced waveguiding, IEEE J

    K. Petermann, Calculated spontaneous emission factor 11 for double-heterostructure injection lasers with gain- induced waveguiding, IEEE J. Quantum Electron. 15, 566 (1979)

    1979

  92. [100]

    Wiersig, Petermann factors and phase rigidities near exceptional points, Phys

    J. Wiersig, Petermann factors and phase rigidities near exceptional points, Phys. Rev. Research 5, 033042 (2023)

    2023

  93. [101]

    A. M. L¨ auchli, Operator content of real-space en- tanglement spectra at conformal critical points, arXiv:1303.0741

    arXiv

  94. [102]

    G. Y. Cho, A. W. W. Ludwig, and S. Ryu, Universal entanglement spectra of gapped one-dimensional field theories, Phys. Rev. B 95, 115122 (2017)

    2017

  95. [103]

    W. Tang, F. Verstraete, and J. Haegeman, Matrix prod- uct state fixed points of non-Hermitian transfer matri- ces, Phys. Rev. B 111, 035107 (2025)

    2025

  96. [104]

    Nakagawa, N

    M. Nakagawa, N. Kawakami, and M. Ueda, Non- Hermitian Kondo Effect in Ultracold Alkaline-Earth Atoms, Phys. Rev. Lett. 121, 203001 (2018)

    2018

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