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The Ising dual-reflection interface: $\mathbb{Z}_4$ symmetry and Majorana strong zero modes

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Pith's one-line read An Ising interface whose Kramers-Wannier plus reflection symmetry forces exact Majorana zero modes.

desk verdict A clean, exactly solvable spin-chain model with a genuinely new Z4 self-duality and exact Majorana strong zero modes; the central derivations check out, and only minor manuscript issues remain. read the letter →

arxiv 2412.06377 v4 pith:3QXV7PRQ submitted 2024-12-09 cond-mat.str-el cond-mat.quant-gascond-mat.supr-conhep-th

Juliane Graf , Federica Maria Surace , Marcus Berg , Sergej Moroz This is my paper · ORCID
classification cond-mat.str-elcond-mat.quant-gascond-mat.supr-conhep-th
keywords Z4symmetryKramers-WannierdualityMajoranastrongzeromodetransversefieldIsingchainnon-invertiblesymmetry-protectedtopologicalorderJordan-Wignertransformationself-duality
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 constructs an interface between two Kramers-Wannier dual phases of the transverse-field Ising chain and shows that, in an open chain, the combined operation of Kramers-Wannier duality and spatial reflection generates an exact Z4 symmetry. In the fermionic language this symmetry acts as a parity-dependent reflection about a Majorana site, and it forces the existence of exact Majorana strong zero modes that strictly commute with the Hamiltonian and anticommute with fermion parity. As a result, every energy eigenstate of the open chain is exactly twofold degenerate, and this degeneracy is claimed to survive arbitrary local symmetry-preserving interactions. In one coupling regime a second pair of strong zero modes gives a fourfold degeneracy in the thermodynamic limit. The closed chain instead exhibits a non-invertible symmetry and a single pair of exponentially localized zero modes.

What carries the argument

The central object is the symmetry operator S = R U_KW, a self-duality of order four whose square is the fermion parity; in the Majorana basis it is a parity-dependent reflection about the Majorana site N+1, mapping a to 2N+2-a. The argument relies on rewriting the Majorana chain in terms of modes ξ±_a that transform as S ξ±_a $S^{{-1}}$ = ±i P ξ±_a, which decouples the quadratic Hamiltonian into two chains H+ and H-. Locality then forces the edge Majorana η1 = ξ+_1 to commute exactly with the Hamiltonian, because any local S-invariant term containing η1 would map under S to a nonlocal right-end operator. The remaining exact zero modes are constructed explicitly by an iterative Bogoliubov-de Gennes procedure, yielding operators exponentially localized at the interface or edges.

What would settle it

Find a local Z4-symmetric interaction on the open chain whose matrix element couples η1 to a nearby Majorana; if such a term exists, the claimed exact twofold degeneracy would be lifted at finite system size. Concretely, exact diagonalization of finite open chains with S-invariant quartic Majorana terms can test whether the degenerate doublet remains pinned to zero energy for all such perturbations.

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Extended reading notes

Core claim

The central claim is that the open-chain Hamiltonian commutes with S = R U_KW, where R is spatial reflection and U_KW is the Kramers-Wannier unitary, and that $S^{2}$ equals the Ising parity Q, so S generates an exact Z4 symmetry containing the ordinary Z2 parity. In the Majorana representation obtained by the Jordan-Wigner transformation, S acts as a parity-dependent reflection about a Majorana site rather than about a link. This symmetry, together with locality, forces a pair of exact Majorana strong zero modes that strictly commute with the Hamiltonian and anticommute with fermion parity, giving an exact twofold degeneracy of all energy eigenstates in the open chain. In the J < h regime the same mechanism yields four Majorana strong zero modes and a fourfold degeneracy in the thermodynamic limit. The paper further argues that these zero modes are stable under generic local symmetry-preserving perturbations, including interactions, while the closed geometry supports only one pair of strong zero modes with exponentially small energy splitting.

Load-bearing premise

The robustness claim rests on the assumption that every local symmetry-preserving perturbation of the open fermion chain must be built from Majorana products in a finite spatial region and that no such term can contain the edge Majorana η1, because the symmetry would map it to a nonlocal right-end operator; the paper gives a compelling locality argument but not a formal proof of this no-coupling statement.

Editorial extensions

If this is right

  • For an open chain in the J > h regime, every energy eigenstate is exactly twofold degenerate at finite system size, not merely in the thermodynamic limit, because two exact Majorana strong zero modes commute strictly with the Hamiltonian.
  • For J < h, four Majorana strong zero modes lead to a fourfold degeneracy of energy eigenstates in the thermodynamic limit, with the finite-size splitting between the two doublets exponentially small.
  • The exact zero modes are stable under generic local perturbations that preserve the Z4 symmetry, including quartic Majorana interactions, because no such perturbation can couple the protected edge Majorana to the rest of the chain without becoming nonlocal.
  • On a closed chain the Z4 symmetry is replaced by a non-invertible symmetry projecting onto the even parity sector, and the model retains a pair of interface-localized strong zero modes whose degeneracy is exponentially accurate.
  • The distinct ground-state degeneracies of the two open-chain regimes suggest a Z4-protected distinction between the phases, although the closed-chain double degeneracy and the absence of a conventional string order parameter complicate a standard SPT classification.

Reading between the lines

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

  • Our inference: if the exactness claim survives non-perturbative checks, this is a rare example of a symmetry-enforced exact (rather than asymptotic) Majorana strong zero mode, and the same construction of composing a duality with reflection could be explored for other self-dualities to produce Z_{2m} symmetries and additional zero modes.
  • Our inference: the provided quantum circuit realizations make a direct experimental test feasible on current digital quantum hardware; measuring the energy splitting between S-charge sectors of a finite open chain would distinguish exact two-fold degeneracy from exponentially small splitting.
  • Our inference: the failure to construct a string order parameter may indicate that the two phases are distinguished by a quantized invariant carried by the Z4 charge of the zero modes rather than by a conventional entanglement string; a tensor-network study of the open-chain ground states could look for such an invariant.
  • Our inference: the non-invertible symmetry of the closed chain, valid away from criticality, is an unusual feature that may carry an anomaly; studying its fusion rules and anomalies could connect this lattice model to known results on non-invertible symmetries in one dimension.
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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

0 major / 4 minor

Summary. The paper introduces a transverse-field Ising chain with an interface between Kramers-Wannier dual phases, and shows that the combined Kramers-Wannier and spatial reflection operator S generates a Z4 symmetry in the open chain (S^2 = Ising parity Q). In the closed chain, the symmetry becomes non-invertible. After a Jordan-Wigner transformation, the model becomes a quadratic Majorana chain in which S acts as a parity-dependent reflection about a Majorana site; a change of basis to ξ± modes decouples the chain into two commuting subsystems. For the open chain, the authors construct exact Majorana strong zero modes in the J>h and J<h regimes and argue that they are robust under local S-preserving interactions; for the closed chain, they construct approximate strong zero modes localized at the two interfaces. The paper closes with a discussion of possible SPT order and a list of future directions.

Significance. The paper establishes a new exactly solvable model exhibiting a Z4 self-duality symmetry (self-quadrality) that is not a conventional unitary symmetry, and it provides explicit Majorana strong zero modes whose associated degeneracy is argued to be robust against local symmetry-preserving interactions. The derivations are transparent: the identity S^2=Q is proved step-by-step in Eq. (8), the ξ± decomposition is verified in Sec. 3.2, and all zero-mode operators are given with their commutators and normalization factors in Secs. 4 and E. The robustness argument in Sec. 4.1.3 is essentially rigorous and can be made fully explicit with a short support-based proof. These features make the model a valuable testbed for generalized symmetries, self-dualities, and strong zero modes. The authors are also honest about the limitations of the SPT interpretation, explicitly stating that the two regimes are not conventional SPT phases.

minor comments (4)
  1. [Abstract and Sec. 6] The abstract promises that the authors 'develop quantum circuit realizations of our model', but the main text and appendices contain no explicit gate decomposition or circuit construction for either the interface Hamiltonian or the strong zero modes; the only related element is the sequential-circuit representation of U_KW in Eq. (17). Please either add a concrete circuit implementation or revise the abstract to match the actual content of the paper.
  2. [Sec. 3.2, Eq. (20)] The notation for the reflected index is difficult to follow: the text defines both \hat{a}=2N+1-a and \tilde{a}=2N+2-a, but Eq. (20) uses an ambiguous placement of the hat/tilde (e.g., 'η^2j−1'). Please adopt a consistent notation such as \eta_{\hat{a}} and \eta_{\tilde{a}}, and double-check the transformation of the special mode η_{N+1} against the stated definitions.
  3. [Sec. 4.1.3] The robustness argument for η1 is stated in words only. To make it rigorous for arbitrary local interaction monomials, add a short lemma: if a Majorana monomial A has connected support I containing the index 1, then S A S^{-1} contains η1 and has support {1} ∪ (2N+2 - I), which is disconnected for |I| < N/2; hence no finite-range S-invariant term can contain η1. This would also cover multi-Majorana terms beyond the 'adjacent' cases explicitly mentioned.
  4. [Sec. 5] Since the section concludes that the two phases 'cannot be SPTs in the conventional sense' and the attempted string-order construction is unsuccessful, the presence of 'SPT phases' in the manuscript title may overstate the contribution. Consider rewording the title or the outlook to emphasize that the SPT interpretation remains an open question.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed Z4 symmetry is verified by direct computation, and the strong zero modes are constructed from the BdG equations rather than imported from a self-citation.

full rationale

The paper's central claim is that the Hamiltonian (1) commutes with S=R U_KW and that the open chain hosts exact Majorana strong zero modes. This is not circular: the Hamiltonian is defined independently on spin sites with specific couplings, S is defined separately as a composition of a Kramers-Wannier unitary and a spatial reflection, and the commutator is checked explicitly (Eqs. (5)-(8) and Eq. (27)). The model is admittedly designed so that S is a symmetry (Sec. 2.1, 'we demand that a composition ... constitute a symmetry'), but a designed model with a verified symmetry is not a circular derivation. The strong zero modes are not assumed or fitted: eta1=xi+_1 is exact by the vanishing boundary field h1=0, and the interface/edge modes eta, eta-prime, and eta-double-prime are obtained by explicit iterative solutions of the Bogoliubov-de Gennes equations in Appendix E, with commutators evaluated in Eqs. (33), (39), and (42). No parameter is fitted to data used later as a prediction. Citations to earlier work (e.g., [16] for the Majorana form of U_KW) are background or technical and are not used to justify the new claims; there are no load-bearing self-citations. The robustness argument in Sec. 4.1.3 rests on a locality/support statement about S-invariant perturbations, which is an ordinary physics argument rather than an import of an unproved uniqueness claim. Finally, the abstract promises quantum circuit realizations that the text does not provide, but this is an omission, not circularity. No step in the derivation chain reduces by construction to its own input.

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

The central claims rest on the Jordan-Wigner and Kramers-Wannier transformations, the standard definition of strong zero modes, and a locality assumption used to prove robustness of the edge zero mode under S-invariant interactions. The coupling constants J, h, J0 and J'0 are model inputs, not fitted parameters; results hold for all non-critical values. No invented entities are introduced; the xi+-xi- Majoranas are a change of basis.

assumptions (5)
  • standard math The Jordan-Wigner mapping (12)-(14) exactly maps the spin Hamiltonian (1) to the quadratic Majorana Hamiltonian (15), with Ising parity Q mapped to fermion parity P.
    Invoked in Sec. 3.1; standard but load-bearing because all zero-mode results are derived in the Majorana representation.
  • standard math The Kramers-Wannier unitary (2) has the transformation laws (3)-(4), in particular U_KW Z_1 U_KW^{-1} = Q X_1 X_N.
    Derived in Appendix A and used in Sec. 2.2 to prove S^2=Q; the exact convention choice affects where the interface field sits.
  • domain assumption The closed-chain spin model (9) corresponds to the fermion Hamiltonian (28) with antiperiodic Majorana boundary conditions only in the Q=+1 sector; the odd-parity sector is treated as a separate fermion model.
    Stated in Sec. 3.2; the paper explicitly notes that the fermion S symmetry in the P=-1 sector does not correspond to a spin symmetry.
  • domain assumption Robustness under perturbations assumes local S-invariant operators are products of Majoranas in a finite spatial region and that no such operator can contain the edge Majorana eta1.
    The locality argument in Sec. 4.1.3 is plausible but not formalised as a theorem; if violated, the exact degeneracy could be lifted.
  • standard math The Bogoliubov-de Gennes equations provide a complete single-particle description of the quadratic fermion model, and zero-energy solutions correspond to strong zero modes.
    Used throughout Sec. 4 and Appendix E; this is a standard framework for quadratic Majorana systems.

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Pith. "Pith review of The Ising dual-reflection interface: $\mathbb{Z}_4$ symmetry and Majorana strong zero modes." pith.science (2026). https://pith.science/paper/3QXV7PRQ

@misc{pith2026241206377,
  author       = {Pith},
  title        = {Pith review of: The Ising dual-reflection interface: $\mathbbZ_4$ symmetry and Majorana strong zero modes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3QXV7PRQ}},
  note         = {Machine review of arXiv:2412.06377}
}
abstract

We investigate an interface in the transverse field quantum Ising chain connecting an ordered ferromagnetic phase and a disordered paramagnetic phase that are Kramers-Wannier duals of each other. Unlike prior studies focused on non-invertible defects, this interface exhibits a symmetry that combines Kramers-Wannier transformation with spatial reflection. We demonstrate that, under open boundary conditions, this setup gives rise to a discrete $\mathbb{Z}_4$ symmetry, encompassing the conventional $\mathbb{Z}_2$ Ising parity as a subgroup, while in a closed geometry a non-invertible symmetry emerges. Using the Jordan-Wigner transformation, we map the spin chain onto a solvable quadratic Majorana fermion system. In this formulation, the $\mathbb{Z}_4$ symmetry is realized manifestly as a parity-dependent reflection with respect to a Majorana site, in contrast to the conventional reflection which mirrors with respect to the central link of the Majorana chain. Additionally, we construct Majorana strong zero modes that retain the $\mathbb{Z}_4$ symmetry, ensure degeneracies of all energy eigenstates, and are robust under generic local symmetry-preserving perturbations of the fermion model, including interactions. Finally, we develop quantum circuit realizations of our model paving the way towards the creation of exact Majorana strong zero modes with digital quantum hardware.

Figures

Figures reproduced from arXiv: 2412.06377 by the authors.

Figure 1
Figure 1. An interface between ferromagnetic and paramagnetic Ising regions that are Kramers [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Schematic action of the Kramers-Wannier transformation combined with spatial reflec [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Schematic action of the Kramers-Wannier transformation combined with spatial reflec [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Decomposition of the open quadratic Majorana chain with [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: The two decoupled Majorana chains corresponding to the closed chain Hamiltonian ( [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Quantum phase diagram of the fermion model: In an open geometry the two regions [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

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Works this paper leans on

37 extracted references · 7 canonical work pages

  1. [18]

    S. D. Pace, G. Delfino, H. T. Lam and O. M. Aksoy, Gauging modulated symmetries: Kramers-wannier dualities and non-invertible reflections, arXiv preprint arXiv:2406.12962 (2024), doi:10.48550/arXiv.2406.12962

    arXiv doi:10.48550/arxiv.2406.12962 2024

  2. [1]

    R. M. Nandkishore and M. Hermele, Fractons, Annual Review of Condensed Matter Physics 10(1), 295 (2019), doi:10.1146/annurev-conmatphys-031218-013604

    doi:10.1146/annurev-conmatphys-031218-013604 2019

  3. [2]

    Pretko, X

    M. Pretko, X. Chen and Y . You, Fracton phases of matter, International Journal of Modern Physics A 35(06), 2030003 (2020), doi:10.1142/S0217751X20300033

    doi:10.1142/s0217751x20300033 2020

  4. [3]

    McGreevy, Generalized symmetries in condensed matter , Annual Review of Condensed Matter Physics 14(1), 57 (2023), doi:10.1146/annurev-conmatphys-040721-021029

    J. McGreevy, Generalized symmetries in condensed matter , Annual Review of Condensed Matter Physics 14(1), 57 (2023), doi:10.1146/annurev-conmatphys-040721-021029

    doi:10.1146/annurev-conmatphys-040721-021029 2023

  5. [4]

    Shao, What’s done cannot be undone: Tasi lectures on non-invertible symmetry , arXiv:2308.00747 (2023), doi:10.48550/arXiv.2308.00747

    S.-H. Shao, What’s done cannot be undone: Tasi lectures on non-invertible symmetry , arXiv:2308.00747 (2023), doi:10.48550/arXiv.2308.00747

    arXiv doi:10.48550/arxiv.2308.00747 2023

  6. [5]

    P. R. Gomes, An introduction to higher-form symmetries, SciPost Physics Lecture Notes p. 074 (2023), doi:10.21468/SciPostPhysLectNotes.74

    doi:10.21468/scipostphyslectnotes.74 2023

  7. [6]

    Bhardwaj, L

    L. Bhardwaj, L. E. Bottini, L. Fraser-Taliente, L. Gladden, D. S. Gould, A. Platschorre and H. Tillim, Lectures on generalized symmetries , Physics Reports 1051, 1 (2024), doi:10.1016/j.physrep.2023.11.002

    doi:10.1016/j.physrep.2023.11.002 2024

  8. [7]

    Luo, Q.-R

    R. Luo, Q.-R. Wang and Y .-N. Wang,Lecture notes on generalized symmetries and applica- tions, Physics Reports 1065, 1 (2024), doi:10.1016/j.physrep.2024.02.002

    doi:10.1016/j.physrep.2024.02.002 2024

Show all 37 references
  1. [8]

    Gromov and L

    A. Gromov and L. Radzihovsky, Colloquium: Fracton matter, Reviews of Modern Physics 96(1), 011001 (2024), doi:10.1103/RevModPhys.96.011001

    2024 doi

  2. [9]

    Sch ¨afer-Nameki, Ictp lectures on (non-) invertible generalized symmetries , Physics Re- ports 1063, 1 (2024), doi:10.1016/j.physrep.2024.01.007

    S. Sch ¨afer-Nameki, Ictp lectures on (non-) invertible generalized symmetries , Physics Re- ports 1063, 1 (2024), doi:10.1016/j.physrep.2024.01.007

    2024 doi

  3. [10]

    Fr ¨ohlich, J

    J. Fr ¨ohlich, J. Fuchs, I. Runkel and C. Schweigert,Kramers-Wannier duality from conformal defects, Phys. Rev. Lett. 93, 070601 (2004), doi:10.1103/PhysRevLett.93.070601

    2004 doi

  4. [11]

    Grimm, Spectrum of a duality-twisted Ising quantum chain, Journal of Physics A: Math- ematical and General 35(3), L25 (2002), doi:10.1088/0305-4470/35/3/101

    U. Grimm, Spectrum of a duality-twisted Ising quantum chain, Journal of Physics A: Math- ematical and General 35(3), L25 (2002), doi:10.1088/0305-4470/35/3/101

    2002 doi

  5. [12]

    Fr ¨ohlich, J

    J. Fr ¨ohlich, J. Fuchs, I. Runkel and C. Schweigert, Duality and defects in rational conformal field theory, Nuclear Physics B 763(3), 354 (2007), doi:10.1016/j.nuclphysb.2006.11.017. 31

    2007 doi

  6. [13]

    Aasen, R

    D. Aasen, R. S. Mong and P. Fendley, Topological defects on the lattice: I. The Ising model, Journal of Physics A: Mathematical and Theoretical 49(35), 354001 (2016), doi:10.1088/1751-8113/49/35/354001

    2016 doi

  7. [14]

    Hauru, G

    M. Hauru, G. Evenbly, W. W. Ho, D. Gaiotto and G. Vidal, Topological conformal defects with tensor networks, Phys. Rev. B 94, 115125 (2016), doi:10.1103/PhysRevB.94.115125

    2016 doi

  8. [15]

    Seiberg and S.-H

    N. Seiberg and S.-H. Shao, Majorana chain and Ising model–(non-invertible) trans- lations, anomalies, and emanant symmetries , SciPost Physics 16(3), 064 (2024), doi:10.21468/SciPostPhys.16.3.064

    2024 doi

  9. [16]

    Seiberg, S

    N. Seiberg, S. Seifnashri and S.-H. Shao, Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space , SciPost Physics 16(6), 154 (2024), doi:10.21468/SciPostPhys.16.6.154

    2024 doi

  10. [17]

    Carqueville, M

    N. Carqueville, M. Del Zotto and I. Runkel, Topological defects, arXiv:2311.02449 (2023), doi:10.1016/B978-0-323-95703-8.00098-7

    2023 arXiv

  11. [19]

    Alicea and P

    J. Alicea and P. Fendley, Topological phases with parafermions: theory and blueprints , Annual Review of Condensed Matter Physics 7(1), 119 (2016), doi:10.1146/annurev- conmatphys-031115-011336

    2016 doi

  12. [20]

    P. Fendley, Strong zero modes and eigenstate phase transitions in the XYZ/interacting Ma- jorana chain, Journal of Physics A: Mathematical and Theoretical 49(30), 30LT01 (2016), doi:10.1088/1751-8113/49/30/30LT01

    2016 doi

  13. [21]

    J. Kemp, N. Y . Yao, C. R. Laumann and P. Fendley, Long coherence times for edge spins, Journal of Statistical Mechanics: Theory and Experiment 2017(6), 063105 (2017), doi:10.1088/1742-5468/aa73f0

    2017 doi

  14. [22]

    A. Y . Kitaev,Unpaired Majorana fermions in quantum wires, Physics-uspekhi 44(10S), 131 (2001), doi:10.1070/1063-7869/44/10S/S29

    2001 doi

  15. [23]

    C. T. Olund, N. Y . Yao and J. Kemp,Boundary strong zero modes, arXiv:2305.16382 (2023), doi:10.48550/arXiv.2305.16382

    2023 doi arXiv

  16. [24]

    F. Yan, R. Konik and A. Mitra, Duality defect in a deformed transverse-field Ising model , arXiv:2410.17317 (2024), doi:10.48550/arXiv.2410.17317

    2024 doi

  17. [25]

    E. Lieb, T. Schultz and D. Mattis, Two soluble models of an antiferromagnetic chain, Annals of Physics 16(3), 407 (1961), doi:10.1016/0003-4916(61)90115-4

    1961 doi

  18. [26]

    X. Chen, A. Dua, M. Hermele, D. T. Stephen, N. Tantivasadakarn, R. Vanhove and J.-Y . Zhao, Sequential quantum circuits as maps between gapped phases , Phys. Rev. B 109, 075116 (2024), doi:10.1103/PhysRevB.109.075116

    2024 doi

  19. [27]

    Shankar, Quantum field theory and condensed matter: an introduction , Cambridge University Press (2017)

    R. Shankar, Quantum field theory and condensed matter: an introduction , Cambridge University Press (2017). 32

    2017

  20. [28]

    G. B. Mbeng, A. Russomanno and G. E. Santoro, The quantum Ising chain for beginners , SciPost Physics Lecture Notes p. 082 (2024), doi:10.21468/SciPostPhysLectNotes.82

    2024 doi

  21. [29]

    Shapourian, K

    H. Shapourian, K. Shiozaki and S. Ryu, Many-body topological invariants for fermionic symmetry-protected topological phases , Physical review letters 118(21), 216402 (2017), doi:10.1103/PhysRevLett.118.216402

    2017 doi

  22. [30]

    Shiozaki, H

    K. Shiozaki, H. Shapourian and S. Ryu, Many-body topological invariants in fermionic symmetry-protected topological phases: Cases of point group symmetries, Physical Review B 95(20), 205139 (2017), doi:10.1103/PhysRevB.95.205139

    2017 doi

  23. [31]

    Calabrese and J

    P. Calabrese and J. Cardy, Quantum quenches in 1+ 1 dimensional conformal field the- ories, Journal of Statistical Mechanics: Theory and Experiment 2016(6), 064003 (2016), doi:10.1088/1742-5468/2016/06/064003

    2016 doi

  24. [32]

    O’Brien and P

    E. O’Brien and P. Fendley, Lattice supersymmetry and order-disorder coexistence in the tricritical ising model , Physical review letters 120(20), 206403 (2018), doi:10.1103/PhysRevLett.120.206403

    2018 doi

  25. [33]

    Aasen, P

    D. Aasen, P. Fendley and R. S. Mong, Topological defects on the lattice: dualities and degeneracies, arXiv:2008.08598 (2020), doi:10.48550/arXiv.2008.08598

    2020 doi arXiv

  26. [34]

    X. Mi, M. Sonner, M. Y . Niu, K. W. Lee, B. Foxen, R. Acharya, I. Aleiner, T. I. Andersen, F. Arute, K. Arya, A. Asfaw, J. Atalaya et al., Noise-resilient edge modes on a chain of superconducting qubits, Science 378(6621), 785 (2022), doi:10.1126/science.abq5769

    2022 doi

  27. [35]

    Monroe, W

    C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V . Gorshkov, P. W. Hess, R. Is- lam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko et al., Programmable quan- tum simulations of spin systems with trapped ions , Rev. Mod. Phys. 93, 025001 (2021), doi:10.1103/RevMo...

    2021 doi

  28. [36]

    A. De, A. Lerose, D. Luo, F. M. Surace, A. Schuckert, E. R. Bennewitz, B. Ware, W. Morong, K. S. Collins, Z. Davoudi et al., Observation of string-breaking dynamics in a quantum simulator, arXiv:2410.13815 (2024), doi:10.48550/arXiv.2410.13815

    2024 doi arXiv

  29. [37]

    Steinert, P

    L.-M. Steinert, P. Osterholz, R. Eberhard, L. Festa, N. Lorenz, Z. Chen, A. Trautmann and C. Gross, Spatially tunable spin interactions in neutral atom arrays , Phys. Rev. Lett. 130, 243001 (2023), doi:10.1103/PhysRevLett.130.243001. 33

    2023 doi

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