Abstract
We study a novel class of Renormalization Group flows which connect multicritical versions of the two-dimensional Yang-Lee edge singularity described by the conformal minimal models \( \mathcal{M} \)(2, 2n + 3). The absence in these models of an order parameter implies that the flows towards and between Yang-Lee edge singularities are all related to the spontaneous breaking of \( \mathcal{PT} \) symmetry and comprise a pattern of flows in the space of \( \mathcal{PT} \) symmetric theories consistent with the c-theorem and the counting of relevant directions. Additionally, we find that while in a part of the phase diagram the domains of unbroken and broken \( \mathcal{PT} \) symmetry are separated by critical manifolds of class \( \mathcal{M} \)(2, 2n + 3), other parts of the boundary between the two domains are not critical.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K.G. Wilson and J.B. Kogut, The Renormalization group and the epsilon expansion, Phys. Rep. 12 (1974) 75 [INSPIRE].
C. Domb and M.S. Green, Phase Transitions and Critical Phenomena. Volume 6, Academic Press (1976) [INSPIRE].
J. Polchinski, Renormalization and Effective Lagrangians, Nucl. Phys. B 231 (1984) 269 [INSPIRE].
D.J. Amit, Field Theory, the Renormalization Group, and Critical Phenomena, World Scientific (1984).
J.J. Binney, N.J. Dowrick, A.J. Fisher and M.E.J. Newman, The Theory of critical phenomena: An Introduction to the renormalization group, Clarendon Press (1992) [INSPIRE].
J.L. Cardy, Scaling and renormalization in statistical physics, Cambridge University Press (1996) [INSPIRE].
L.P. Kadanoff, Statistical physics: Statics, dynamics and renormalization, World Scientific (2000) [INSPIRE].
J. Berges, N. Tetradis and C. Wetterich, Nonperturbative renormalization flow in quantum field theory and statistical physics, Phys. Rep. 363 (2002) 223 [hep-ph/0005122] [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
A. Cappelli, C. Itzykson and J.B. Zuber, Modular Invariant Partition Functions in Two-Dimensions, Nucl. Phys. B 280 (1987) 445 [INSPIRE].
A.B. Zamolodchikov, Conformal Symmetry and Multicritical Points in Two-Dimensional Quantum Field Theory (in Russian), Sov. J. Nucl. Phys. 44 (1986) 529 [INSPIRE].
A.B. Zamolodchikov, Renormalization Group and Perturbation Theory Near Fixed Points in Two-Dimensional Field Theory, Sov. J. Nucl. Phys. 46 (1987) 1090 [INSPIRE].
A.W.W. Ludwig and J.L. Cardy, Perturbative Evaluation of the Conformal Anomaly at New Critical Points with Applications to Random Systems, Nucl. Phys. B 285 (1987) 687 [INSPIRE].
A.B. Zamolodchikov, From tricritical Ising to critical Ising by thermodynamic Bethe ansatz, Nucl. Phys. B 358 (1991) 524 [INSPIRE].
P. Dorey, C. Dunning and R. Tateo, New families of flows between two-dimensional conformal field theories, Nucl. Phys. B 578 (2000) 699 [hep-th/0001185] [INSPIRE].
C. Ahn and A. LeClair, On the classification of UV completions of integrable \( T\overline{T} \) deformations of CFT, JHEP 08 (2022) 179 [arXiv:2205.10905] [INSPIRE].
C.-N. Yang and T.D. Lee, Statistical theory of equations of state and phase transitions. Part 1. Theory of condensation, Phys. Rev. 87 (1952) 404 [INSPIRE].
T.D. Lee and C.-N. Yang, Statistical theory of equations of state and phase transitions. Part 2. Lattice gas and Ising model, Phys. Rev. 87 (1952) 410 [INSPIRE].
M.E. Fisher, Yang-Lee Edge Singularity and ϕ3 Field Theory, Phys. Rev. Lett. 40 (1978) 1610 [INSPIRE].
J.L. Cardy, Conformal Invariance and the Yang-Lee Edge Singularity in Two-dimensions, Phys. Rev. Lett. 54 (1985) 1354 [INSPIRE].
J.L. Cardy and G. Mussardo, S Matrix of the Yang-Lee Edge Singularity in Two-Dimensions, Phys. Lett. B 225 (1989) 275 [INSPIRE].
C.M. Bender, V. Branchina and E. Messina, Critical behavior of the \( \mathcal{PT} \)-symmetric iϕ3 quantum field theory, Phys. Rev. D 87 (2013) 085029 [arXiv:1301.6207] [INSPIRE].
G. Mussardo, R. Bonsignori and A. Trombettoni, Yang-Lee zeros of the Yang-Lee model, J. Phys. A 50 (2017) 484003 [arXiv:1708.06444] [INSPIRE].
H.-L. Xu and A.B. Zamolodchikov, 2D Ising Field Theory in a magnetic field: the Yang-Lee singularity, JHEP 08 (2022) 057 [arXiv:2203.11262] [INSPIRE].
B.-B. Wei and R.-B. Liu, Lee-Yang Zeros and Critical Times in Decoherence of a Probe Spin Coupled to a Bath, Phys. Rev. Lett. 109 (2012) 185701 [arXiv:1206.2077].
N. Matsumoto, M. Nakagawa and M. Ueda, Embedding the Yang-Lee quantum criticality in open quantum systems, Phys. Rev. Res. 4 (2022) 033250 [arXiv:2012.13144] [INSPIRE].
C. Li and F. Yang, Lee-Yang zeros in the Rydberg atoms, Front. Phys. 18 (2023) 22301 [arXiv:2203.16128] [INSPIRE].
R. Shen, T. Chen, F. Qin, Y. Zhong and C.H. Lee, Proposal for Observing Yang-Lee Criticality in Rydberg Atomic Arrays, Phys. Rev. Lett. 131 (2023) 080403 [arXiv:2302.06662] [INSPIRE].
M. Lencsés, A. Miscioscia, G. Mussardo and G. Takács, Multicriticality in Yang-Lee edge singularity, JHEP 02 (2023) 046 [arXiv:2211.01123] [INSPIRE].
G. von Gehlen, Non-Hermitian tricriticality in the Blume-Capel model with imaginary field, Int. J. Mod. Phys. B 08 (1994) 3507 [hep-th/9402143].
C.M. Bender and S. Boettcher, Real spectra in nonHermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998) 5243 [physics/9712001] [INSPIRE].
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 (2018) 11 [INSPIRE].
Y. Ashida, Z. Gong and M. Ueda, Non-Hermitian physics, Adv. Phys. 69 (2020) 249 [arXiv:2006.01837] [INSPIRE].
C.M. Bender, Introduction to PT-Symmetric Quantum Theory, Contemp. Phys. 46 (2005) 277 [quant-ph/0501052] [INSPIRE].
C.M. Bender, D.C. Brody and H.F. Jones, Extension of PT symmetric quantum mechanics to quantum field theory with cubic interaction, Phys. Rev. D 70 (2004) 025001 [Erratum ibid. 71 (2005) 049901] [hep-th/0402183] [INSPIRE].
C.M. Bender, N. Hassanpour, S.P. Klevansky and S. Sarkar, PT-symmetric quantum field theory in D dimensions, Phys. Rev. D 98 (2018) 125003 [arXiv:1810.12479] [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
O.A. Castro-Alvaredo, B. Doyon and F. Ravanini, Irreversibility of the renormalization group flow in non-unitary quantum field theory, J. Phys. A 50 (2017) 424002 [arXiv:1706.01871] [INSPIRE].
I.R. Klebanov, V. Narovlansky, Z. Sun and G. Tarnopolsky, Ginzburg-Landau description and emergent supersymmetry of the (3, 8) minimal model, JHEP 02 (2023) 066 [arXiv:2211.07029] [INSPIRE].
A.B. Zamolodchikov, Thermodynamic Bethe ansatz for RSOS scattering theories, Nucl. Phys. B 358 (1991) 497 [INSPIRE].
V.P. Yurov and A.B. Zamolodchikov, Truncated conformal space approach to scaling Lee-Yang model, Int. J. Mod. Phys. A 5 (1990) 3221 [INSPIRE].
D.X. Horváth, K. Hódśagi and G. Takács, Chirally factorised truncated conformal space approach, Comput. Phys. Commun. 277 (2022) 108376 [arXiv:2201.06509] [INSPIRE].
A.B. Zamolodchikov, Integrable field theory from conformal field theory, Adv. Stud. Pure Math. 19 (1989) 641 [INSPIRE].
V.A. Fateev and A.B. Zamolodchikov, Conformal field theory and purely elastic S matrices, Int. J. Mod. Phys. A 5 (1990) 1025 [INSPIRE].
A. Koubek and G. Mussardo, φ1,2 deformation of the M2,2n+1 conformal minimal models, Phys. Lett. B 266 (1991) 363 [INSPIRE].
P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 639 [hep-th/9607167] [INSPIRE].
F.A. Smirnov, Exact S-matrices for ϕ1,2 perturbated minimal models of conformal field theory, Int. J. Mod. Phys. A 6 (1991) 1407 [INSPIRE].
V.A. Fateev, The Exact relations between the coupling constants and the masses of particles for the integrable perturbed conformal field theories, Phys. Lett. B 324 (1994) 45 [INSPIRE].
P.G.O. Freund, T.R. Klassen and E. Melzer, S Matrices for Perturbations of Certain Conformal Field Theories, Phys. Lett. B 229 (1989) 243 [INSPIRE].
A.B. Zamolodchikov, Mass scale in the sine-Gordon model and its reductions, Int. J. Mod. Phys. A 10 (1995) 1125 [INSPIRE].
P. Fonseca and A.B. Zamolodchikov, Ising field theory in a magnetic field: Analytic properties of the free energy, hep-th/0112167 [INSPIRE].
P. Cejnar, P. Stránský, M. Macek and M. Kloc, Excited-state quantum phase transitions, J. Phys. A 54 (2021) 133001 [arXiv:2011.01662] [INSPIRE].
H. Kausch, G. Takács and G. Watts, On the relation between Φ(1,2) and Φ(1,5) perturbed minimal models, Nucl. Phys. B 489 (1997) 557 [hep-th/9605104] [INSPIRE].
X. Peng, H. Zhou, B.-B. Wei, J. Cui, J. Du and R.-B. Liu, Experimental Observation of Lee-Yang Zeros, Phys. Rev. Lett. 114 (2015) 010601 [arXiv:1403.5383].
Y. Nakayama, Is there supersymmetric Lee-Yang fixed point in three dimensions?, Int. J. Mod. Phys. A 36 (2021) 2150176 [arXiv:2104.13570] [INSPIRE].
Acknowledgments
GM acknowledges the grants Prin 2017-FISI and PRO3 Quantum Pathfinder. The work of ML was supported by the National Research Development and Innovation Office of Hungary under the postdoctoral grant PD-19 No. 132118 and partially by the OTKA Grant K 134946. GT was partially supported by the Ministry of Culture and Innovation and the National Research, Development and Innovation Office (NKFIH) through the OTKA Grant K 138606, and also by the Ministry of Culture and Innovation and the National Research, Development and Innovation Office under Grant Nr. TKP2021-NVA-02. AM is partially supported from the German Research Foundation DFG under Germany’s Excellence Strategy — EXC 2121 Quantum Universe — 390833306 and from the European Union’s Horizon 2020 research. ML, GM and GT are also grateful for the hospitality of KITP Santa Barbara during the program “Integrability in String, Field, and Condensed Matter Theory”, where a part of this work was completed. This collaboration was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958, and by the CNR/MTA Italy-Hungary 2023–2025 Joint Project “Effects of strong correlations in interacting many-body systems and quantum circuits”. AM is grateful to the Galileo Galilei Institute for Theoretical Physics in Firenze for hosting him during LACES 2022.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2304.08522
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Lencsés, M., Miscioscia, A., Mussardo, G. et al. \( \mathcal{PT} \) breaking and RG flows between multicritical Yang-Lee fixed points. J. High Energ. Phys. 2023, 52 (2023). https://doi.org/10.1007/JHEP09(2023)052
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2023)052