pith. machine review for the scientific record. sign in

arxiv: 2604.24887 · v1 · submitted 2026-04-27 · ❄️ cond-mat.str-el

Recognition: unknown

Symmetry-Protected Topological Phases in the Triangular Majorana-Hubbard Ladder

Alberto Nocera, Armin Rahmani, Jian-Xin Zhu, Will Holdhusen

Authors on Pith no claims yet

Pith reviewed 2026-05-08 01:39 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords Majorana-Hubbard modelsymmetry-protected topological phasestriangular ladderentanglement spectrumDMRGtopological superconductorvortex Majorana modes
0
0 comments X

The pith

The triangular Majorana-Hubbard model on a four-leg ladder hosts multiple symmetry-protected topological phases.

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

The paper maps the phase diagram of interacting Majorana fermions on a triangular four-leg ladder. Numerical simulations reveal a richer set of phases than earlier work had identified. By tracking degeneracies in the entanglement spectrum and tracing adiabatic paths between different parameter values, the authors locate several symmetry-protected topological phases. These phases are argued to be realizable in physical arrays of Majorana modes bound to vortices on the surface of a topological superconductor.

Core claim

The triangular-lattice Majorana-Hubbard model on a four-leg ladder exhibits a richer variety of phases than previously known. Analysis of entanglement-spectrum degeneracies and adiabatic connections identifies multiple symmetry-protected topological phases. These phases could be realized in arrays of vortex-bound Majorana modes on the surface of a topological superconductor.

What carries the argument

Entanglement-spectrum degeneracies of the ground-state wavefunction, used to diagnose the presence and type of symmetry-protected topological order in the Majorana-Hubbard ladder.

If this is right

  • The phase diagram contains more distinct phases than reported in prior studies of the same model.
  • Multiple symmetry-protected topological phases are separated by phase boundaries that can be crossed adiabatically.
  • The identified phases remain stable under the symmetries of the triangular-lattice Majorana-Hubbard interaction.
  • Arrays of vortex-bound Majorana modes provide a concrete physical setting in which the predicted phases could appear.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Ladder geometries may serve as useful minimal models for capturing essential features of two-dimensional Majorana lattices.
  • The results suggest that similar numerical diagnostics could be applied to other interaction strengths or lattice geometries to locate additional protected phases.
  • Confirmation of these phases would supply concrete targets for experiments that engineer Majorana modes on superconducting surfaces.

Load-bearing premise

Numerical results from DMRG and variational uniform matrix product states on the finite-width ladder faithfully represent the true ground states of the infinite system, with entanglement degeneracies remaining free of significant truncation or finite-size artifacts.

What would settle it

A larger-scale simulation or direct experimental probe that finds a non-degenerate entanglement spectrum in a parameter regime the paper identifies as symmetry-protected topological.

Figures

Figures reproduced from arXiv: 2604.24887 by Alberto Nocera, Armin Rahmani, Jian-Xin Zhu, Will Holdhusen.

Figure 2
Figure 2. Figure 2: FIG. 2: Decorated triangular lattice encoding hopping view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1: Phase diagram of the four-leg triangular-lattice view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Top: energy susceptibility view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1: Naming conventions for the fermion bilinears used to form the mean-field theory. The remaining directions view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Comparison of energy densities between the four-parameter mean-field (MF4) used in earlier work[ view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Energy gaps as a function of view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Phase transition signatures in view at source ↗
Figure 5
Figure 5. Figure 5: plots magnetic and fermionic correlations in the G3 and GL4 phases computed on an Lx = 96 cylinder. This plot makes clear a six-site repeating structure in the G3 phase. Correlations in the GL4 phase are less easily understood. Notably, this phase exhibits long-range connected correlations in σ z , which are absent in the gapped SPT phases. VI. GL4 PHASE: GAPLESS SPECTRUM Like the G4 phase discussed in the… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Finite-size scaling of gaps in the even parity sector in the GL view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Energy susceptibility (top) and entanglement spectrum as a function of view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Finite-size scaling low-lying gaps at view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Entanglement entropies for odd- view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Comparison of energy density between view at source ↗
read the original abstract

We map the phase diagram of the triangular-lattice Majorana-Hubbard model on a four-leg ladder using DMRG and variational uniform matrix product states, revealing a richer variety of phases than previously known. Analysis of entanglement-spectrum degeneracies and adiabatic connections identifies multiple symmetry-protected topological (SPT) phases. These phases could be realized in arrays of vortex-bound Majorana modes on the surface of a topological superconductor.

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

2 major / 2 minor

Summary. The manuscript maps the phase diagram of the triangular-lattice Majorana-Hubbard model on a four-leg ladder using DMRG and variational uniform matrix product states. It identifies multiple symmetry-protected topological (SPT) phases via entanglement-spectrum degeneracies and adiabatic connections, suggesting possible realization in vortex-bound Majorana mode arrays on topological superconductor surfaces.

Significance. If the numerical identifications hold, the work would advance understanding of interacting Majorana systems by revealing a richer set of SPT phases than previously reported. The tensor-network approach combined with symmetry diagnostics is a methodological strength, and the proposed experimental link to topological superconductors adds relevance.

major comments (2)
  1. [§3 (Numerical Methods)] §3 (Numerical Methods): The DMRG and VUMPS sections report entanglement spectrum degeneracies used to diagnose SPT order but omit explicit bond-dimension convergence data, truncation-error estimates, or finite-size scaling. In interacting Majorana models, finite bond dimension can split or fabricate low-lying entanglement levels, so these checks are load-bearing for the central SPT claims.
  2. [§4 (Phase identifications)] §4 (Phase identifications): The adiabatic connection arguments and four-leg ladder results do not address how quasi-1D wrapping or boundary effects might alter effective symmetries or whether the reported SPT degeneracies persist for wider ladders; this is required to confirm the phases are not geometry artifacts.
minor comments (2)
  1. [Abstract] Abstract: The claim of 'a richer variety of phases than previously known' would be clearer if the newly identified phases were named or briefly characterized.
  2. [Throughout] Notation: Define all symmetry labels (e.g., for the SPT phases) at first use to aid readers unfamiliar with the specific Majorana-Hubbard symmetries.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments. We address each major point below and will revise the manuscript accordingly to strengthen the numerical evidence and clarify the geometric considerations.

read point-by-point responses

  1. Referee: §3 (Numerical Methods): The DMRG and VUMPS sections report entanglement spectrum degeneracies used to diagnose SPT order but omit explicit bond-dimension convergence data, truncation-error estimates, or finite-size scaling. In interacting Majorana models, finite bond dimension can split or fabricate low-lying entanglement levels, so these checks are load-bearing for the central SPT claims.

    Authors: We agree that explicit convergence checks are essential for validating the entanglement spectrum degeneracies. In the revised manuscript, we will add bond-dimension convergence data, truncation-error estimates, and finite-size scaling analysis for representative points in each phase to demonstrate that the reported degeneracies are robust against finite bond dimension. revision: yes

  2. Referee: §4 (Phase identifications): The adiabatic connection arguments and four-leg ladder results do not address how quasi-1D wrapping or boundary effects might alter effective symmetries or whether the reported SPT degeneracies persist for wider ladders; this is required to confirm the phases are not geometry artifacts.

    Authors: We acknowledge the concern about ladder geometry. Our calculations use both open and periodic boundary conditions, with SPT degeneracies appearing consistently. Adiabatic connections are performed within the four-leg geometry to maintain symmetries. In the revision, we will add discussion explaining why the four-leg results are representative based on symmetry analysis. However, simulations on wider ladders are computationally prohibitive at present and are noted as future work; we cannot fully rule out geometry-specific effects without them. revision: partial

Circularity Check

0 steps flagged

Numerical phase mapping via DMRG/VUMPS is self-contained with no circular reductions

full rationale

The paper maps the phase diagram of the triangular Majorana-Hubbard model on a four-leg ladder using direct DMRG and variational uniform matrix product state simulations. Phases are identified by computing entanglement-spectrum degeneracies and checking adiabatic connections between states. No analytical derivation chain exists that reduces predictions or first-principles results to fitted parameters, self-definitions, or load-bearing self-citations by construction. The methodology consists of numerical outputs interpreted through standard entanglement diagnostics, making the work self-contained against external benchmarks without any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available, so ledger is minimal. No free parameters, invented entities, or non-standard axioms are mentioned.

axioms (1)
  • domain assumption DMRG and variational uniform matrix product states provide accurate approximations to the ground states of the quantum lattice model
    Standard assumption underlying all numerical phase-diagram studies of interacting fermion models.

pith-pipeline@v0.9.0 · 5366 in / 1257 out tokens · 55145 ms · 2026-05-08T01:39:20.221946+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

70 extracted references · 51 canonical work pages · 1 internal anchor

  1. [1]

    A.Yu. Kitaev. Fault-tolerant quantum computa- tion by anyons.Annals of Physics, 303(1):2– 30, 2003. URL:https://www.sciencedirect. com/science/article/pii/S0003491602000180, doi:10.1016/S0003-4916(02)00018-0

    work page Pith review doi:10.1016/s0003-4916(02)00018-0 2003

  2. [2]

    D. A. Ivanov. Non-abelian statistics of half-quantum vortices inp-wave superconductors.Phys. Rev. Lett., 86:268–271, Jan 2001. URL:https://link.aps. org/doi/10.1103/PhysRevLett.86.268,doi:10.1103/ PhysRevLett.86.268

    work page doi:10.1103/physrevlett.86.268 2001

  3. [3]

    Rudro R. Biswas. Majorana fermions in vor- tex lattices.Phys. Rev. Lett., 111:136401, Sep

  4. [5]

    Mishmash, A

    Ryan V. Mishmash, A. Yazdani, and Michael P. Zale- tel. Majorana lattices from the quantized hall limit of a proximitized spin-orbit coupled electron gas.Phys. Rev. B, 99:115427, Mar 2019. URL:https://link.aps. org/doi/10.1103/PhysRevB.99.115427,doi:10.1103/ PhysRevB.99.115427

    work page doi:10.1103/physrevb.99.115427 2019

  5. [6]

    Evidence for majorana bound states in an iron-based superconductor.Science, 362(6412):333–335, 2018

    Dongfei Wang, Lingyuan Kong, Peng Fan, Hui Chen, Shiyu Zhu, Wenyao Liu, Lu Cao, Yujie Sun, Shixuan 6 Du, John Schneeloch, et al. Evidence for majorana bound states in an iron-based superconductor.Science, 362(6412):333–335, 2018

    2018

  6. [7]

    Half-integer level shift of vortex bound states in an iron-based superconductor.Nature Physics, 15(11):1181–1187, 2019

    Lingyuan Kong, Shiyu Zhu, Michal Papaj, Hui Chen, Lu Cao, Hiroki Isobe, Yuqing Xing, Wenyao Liu, Dongfei Wang, Peng Fan, et al. Half-integer level shift of vortex bound states in an iron-based superconductor.Nature Physics, 15(11):1181–1187, 2019

    2019

  7. [8]

    Machida, Yingyi Huang, T

    Ching-Kai Chiu, T. Machida, Yingyi Huang, T. Hanaguri, and Fu-Chun Zhang. Scalable majo- rana vortex modes in iron-based superconductors. Science Advances, 6(9):eaay0443, 2020. URL:https:// www.science.org/doi/abs/10.1126/sciadv.aay0443, arXiv:https://www.science.org/doi/pdf/10.1126/ sciadv.aay0443,doi:10.1126/sciadv.aay0443

    work page doi:10.1126/sciadv.aay0443 2020

  8. [10]

    Ching-Kai Chiu, D. I. Pikulin, and M. Franz. Strongly interacting majorana fermions.Phys. Rev. B, 91:165402, Apr 2015. URL:https://link.aps.org/doi/10. 1103/PhysRevB.91.165402,doi:10.1103/PhysRevB.91. 165402

    work page doi:10.1103/physrevb.91 2015

  9. [13]

    Jin-Peng Xu, Mei-Xiao Wang, Zhi Long Liu, Jian-Feng Ge, Xiaojun Yang, Canhua Liu, Zhu An Xu, Dandan Guan, Chun Lei Gao, Dong Qian, et al. Experimental detection of a majorana mode in the core of a magnetic vortex inside a topological insulator-superconductor bi 2 te 3/nbse 2 heterostructure.Physical review letters, 114(1):017001, 2015

    2015

  10. [14]

    Signatures of hybridiza- tion of multiple majorana zero modes in a vortex.Nature, 633(8028):71–76, 2024

    Tengteng Liu, Chun Yu Wan, Hao Yang, Yujun Zhao, Bangjin Xie, Weiyan Zheng, Zhaoxia Yi, Dandan Guan, Shiyong Wang, Hao Zheng, et al. Signatures of hybridiza- tion of multiple majorana zero modes in a vortex.Nature, 633(8028):71–76, 2024

    2024

  11. [15]

    Spon- taneous vortex-antivortex lattice and majorana fermions in rhombohedral graphene.Phys

    Filippo Gaggioli, Daniele Guerci, and Liang Fu. Spon- taneous vortex-antivortex lattice and majorana fermions in rhombohedral graphene.Phys. Rev. Lett., 135:116001, Sep 2025. URL:https://link.aps.org/doi/10.1103/ k8sb-rqxf,doi:10.1103/k8sb-rqxf

    work page doi:10.1103/k8sb-rqxf 2025

  12. [16]

    Interacting Ma- jorana fermions.Reports on Progress in Physics, 82(8):084501, August 2019.arXiv:1811.02593,doi: 10.1088/1361-6633/ab28ef

    Armin Rahmani and Marcel Franz. Interacting Ma- jorana fermions.Reports on Progress in Physics, 82(8):084501, August 2019.arXiv:1811.02593,doi: 10.1088/1361-6633/ab28ef

    work page doi:10.1088/1361-6633/ab28ef 2019

  13. [17]

    Majorana-hubbard model on the square lattice.Phys

    Ian Affleck, Armin Rahmani, and Dmitry Pikulin. Majorana-hubbard model on the square lattice.Phys. Rev. B, 96:125121, Sep 2017. URL:https://link.aps. org/doi/10.1103/PhysRevB.96.125121,doi:10.1103/ PhysRevB.96.125121

    work page doi:10.1103/physrevb.96.125121 2017

  14. [18]

    Phase diagrams of majorana-hubbard ladders.Phys

    Armin Rahmani, Dmitry Pikulin, and Ian Affleck. Phase diagrams of majorana-hubbard ladders.Phys. Rev. B, 99:085110, Feb 2019. URL:https://link.aps.org/doi/ 10.1103/PhysRevB.99.085110,doi:10.1103/PhysRevB. 99.085110

    work page doi:10.1103/physrevb.99.085110 2019

  15. [19]

    Majorana-hubbard model on the honeycomb lattice.Phys

    Chengshu Li and Marcel Franz. Majorana-hubbard model on the honeycomb lattice.Phys. Rev. B, 98:115123, Sep 2018. URL:https://link.aps.org/doi/ 10.1103/PhysRevB.98.115123,doi:10.1103/PhysRevB. 98.115123

    work page doi:10.1103/physrevb.98.115123 2018

  16. [20]

    Majorana fermions on a disordered triangular lattice.New Journal of Physics, 13(10):105006, oct 2011

    Yaacov E Kraus and Ady Stern. Majorana fermions on a disordered triangular lattice.New Journal of Physics, 13(10):105006, oct 2011. URL:https: //dx.doi.org/10.1088/1367-2630/13/10/105006,doi: 10.1088/1367-2630/13/10/105006

    work page doi:10.1088/1367-2630/13/10/105006 2011

  17. [22]

    Topological order and conformal quantum crit- ical points.Annals of Physics, 310(2):493–551,

    Eddy Ardonne, Paul Fendley, and Eduardo Frad- kin. Topological order and conformal quantum crit- ical points.Annals of Physics, 310(2):493–551,

  18. [23]

    2004.01.004

    URL:https://www.sciencedirect.com/science/ article/pii/S0003491604000247,doi:10.1016/j.aop. 2004.01.004

    work page doi:10.1016/j.aop 2004

  19. [25]

    F. D. M. Haldane. Nonlinear field theory of large-spin heisenberg antiferromagnets: Semiclassi- cally quantized solitons of the one-dimensional easy- axis néel state.Phys. Rev. Lett., 50:1153–1156, Apr 1983. URL:https://link.aps.org/doi/10.1103/ PhysRevLett.50.1153,doi:10.1103/PhysRevLett.50. 1153

    work page doi:10.1103/physrevlett.50 1983

  20. [27]

    Hidden symmetry break- ing and the Haldane phase in S=1 quantum spin chains

    Tom Kennedy and Hal Tasaki. Hidden symmetry break- ing and the Haldane phase in S=1 quantum spin chains. Communications in Mathematical Physics, 147(3):431– 484, July 1992.doi:10.1007/BF02097239

    work page doi:10.1007/bf02097239 1992

  21. [28]

    Else, Stephen D

    Dominic V. Else, Stephen D. Bartlett, and An- drew C. Doherty. Hidden symmetry-breaking picture of symmetry-protected topological order.Phys. Rev. B, 88:085114, Aug 2013. URL:https://link.aps. org/doi/10.1103/PhysRevB.88.085114,doi:10.1103/ PhysRevB.88.085114

    work page doi:10.1103/physrevb.88.085114 2013

  22. [29]

    Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains.Phys

    Marcel den Nijs and Koos Rommelse. Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains.Phys. Rev. B, 40:4709–4734, Sep 1989. URL:https://link.aps.org/doi/10.1103/ PhysRevB.40.4709,doi:10.1103/PhysRevB.40.4709

    work page doi:10.1103/physrevb.40.4709 1989

  23. [30]

    Frank Pollmann and Ari M. Turner. Detection of symmetry-protected topological phases in one dimension. Phys. Rev. B, 86:125441, Sep 2012. URL:https:// link.aps.org/doi/10.1103/PhysRevB.86.125441,doi: 10.1103/PhysRevB.86.125441

    work page doi:10.1103/physrevb.86.125441 2012

  24. [31]

    Verza, A

    Erik S. Sørensen and Hae-Young Kee. Twice hid- 7 den string order and competing phases in the spin- 1/2 Kitaev-Gamma ladder.npj Quantum Materials, 9(1):10, December 2024.arXiv:2307.08731,doi:10. 1038/s41535-024-00621-x

    work page arXiv 2024

  25. [32]

    Unpaired majorana fermions in quantum wires.Physics-Uspekhi, 44(10S):131, oct 2001.doi:10

    A Yu Kitaev. Unpaired majorana fermions in quantum wires.Physics-Uspekhi, 44(10S):131, oct 2001.doi:10. 1070/1063-7869/44/10S/S29

    2001

  26. [33]

    Com- plete classification of one-dimensional gapped quantum phases in interacting spin systems.Phys

    Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen. Com- plete classification of one-dimensional gapped quantum phases in interacting spin systems.Phys. Rev. B, 84:235128, Dec2011. URL:https://link.aps.org/doi/ 10.1103/PhysRevB.84.235128,doi:10.1103/PhysRevB. 84.235128

    work page doi:10.1103/physrevb.84.235128

  27. [34]

    Classifying quantum phases using matrix product states and projected entangled pair states.Phys

    Norbert Schuch, David Pérez-García, and Ignacio Cirac. Classifying quantum phases using matrix product states and projected entangled pair states.Phys. Rev. B, 84:165139, Oct2011. URL:https://link.aps.org/doi/ 10.1103/PhysRevB.84.165139,doi:10.1103/PhysRevB. 84.165139

    work page doi:10.1103/physrevb.84.165139

  28. [35]

    One-dimensional symmetry protected topo- logical phases and their transitions.Phys

    Ruben Verresen, Roderich Moessner, and Frank Poll- mann. One-dimensional symmetry protected topo- logical phases and their transitions.Phys. Rev. B, 96:165124, Oct2017. URL:https://link.aps.org/doi/ 10.1103/PhysRevB.96.165124,doi:10.1103/PhysRevB. 96.165124

    work page doi:10.1103/physrevb.96.165124

  29. [38]

    Supplemental Material for this letter, including fur- ther details on the triangular-lattice Majorana-Hubbard model, and additional results on the gapless mode

  30. [39]

    chinchilla optimal

    Aaron Szasz, Johannes Motruk, Michael P. Zaletel, and Joel E. Moore. Chiral spin liquid phase of the triangular lattice Hubbard model: a density matrix renormaliza- tion group study.arXiv e-prints, page arXiv:1808.00463, August 2018.arXiv:1808.00463,doi:10.48550/arXiv. 1808.00463

    work page internal anchor Pith review doi:10.48550/arxiv 2018

  31. [40]

    Fishman, S

    Matthew Fishman, Steven R. White, and E. Miles Stoudenmire. The ITensor Software Library for Tensor Network Calculations.Sci- Post Phys. Codebases, page 4, 2022. URL: https://scipost.org/10.21468/SciPostPhysCodeb.4, doi:10.21468/SciPostPhysCodeb.4

    work page doi:10.21468/scipostphyscodeb.4 2022

  32. [41]

    Fishman, S

    Matthew Fishman, Steven R. White, and E. Miles Stoudenmire. Codebase release 0.3 for ITensor.SciPost Phys. Codebases, pages 4–r0.3, 2022. URL:https: //scipost.org/10.21468/SciPostPhysCodeb.4-r0.3, doi:10.21468/SciPostPhysCodeb.4-r0.3

    work page doi:10.21468/scipostphyscodeb.4-r0.3 2022

  33. [42]

    Tangent-space methods for uniform matrix product states.SciPost Phys

    Laurens Vanderstraeten, Jutho Haegeman, and Frank Verstraete. Tangent-space methods for uniform matrix product states.SciPost Phys. Lect. Notes, page 7, 2019. URL:https: //scipost.org/10.21468/SciPostPhysLectNotes.7, doi:10.21468/SciPostPhysLectNotes.7

    work page doi:10.21468/scipostphyslectnotes.7 2019

  34. [43]

    This is not the case forσz, which is why we subtract the unconnected part of the correlation

    Note that correlations in parity-violating operators such asγ r,b andσ x,y consist only of the connected part since the expecation value of these operators is zero by parity symmetry. This is not the case forσz, which is why we subtract the unconnected part of the correlation

  35. [44]

    Blöte, A

    H.W.J. Blöte, A. Compagner, P.A.M. Cornelis- sen, A. Hoogland, F. Mallezie, and C. Van- derzande. Critical behaviour of two ising models with multispin interactions.Physica A: Statisti- cal Mechanics and its Applications, 139(2):395– 411, 1986. URL:https://www.sciencedirect. com/science/article/pii/0378437186901287, doi:10.1016/0378-4371(86)90128-7

    work page doi:10.1016/0378-4371(86)90128-7 1986

  36. [46]

    O. F. de Alcantara Bonfim and J. Florencio. Quan- tum phase transitions in the transverse one-dimensional ising model with four-spin interactions.Phys. Rev. B, 74:134413, Oct2006. URL:https://link.aps.org/doi/ 10.1103/PhysRevB.74.134413,doi:10.1103/PhysRevB. 74.134413

    work page doi:10.1103/physrevb.74.134413

  37. [47]

    One-dimensional ising model with mul- tispin interactions.Journal of Physics A: Mathemat- ical and Theoretical, 49(35):355002, aug 2016.doi: 10.1088/1751-8113/49/35/355002

    Loïc Turban. One-dimensional ising model with mul- tispin interactions.Journal of Physics A: Mathemat- ical and Theoretical, 49(35):355002, aug 2016.doi: 10.1088/1751-8113/49/35/355002

    work page doi:10.1088/1751-8113/49/35/355002 2016

  38. [48]

    Charmousis, E.J

    D. Pérez-García, M. M. Wolf, M. Sanz, F. Verstraete, and J. I. Cirac. String order and symmetries in quantum spin lattices.Phys. Rev. Lett., 100:167202, Apr 2008. URL:https://link.aps.org/doi/10.1103/ PhysRevLett.100.167202,doi:10.1103/PhysRevLett. 100.167202

    work page doi:10.1103/physrevlett 2008

  39. [49]

    Bonner and Gerhard Müller

    Jill C. Bonner and Gerhard Müller. Breakdown of scaling in the 1d spin-1/2 heisenberg ferromagnet. Journal of Applied Physics, 61(8):4429–4431, 04 1987. arXiv:https://pubs.aip.org/aip/jap/article-pdf/ 61/8/4429/18610509/4429_1_online.pdf,doi: 10.1063/1.338399

    work page doi:10.1063/1.338399 1987

  40. [50]

    Critical spin dy- namics of Heisenberg ferromagnets revisited.Phys

    Dmytro Tarasevych and Peter Kopietz. Critical spin dy- namics of Heisenberg ferromagnets revisited.Phys. Rev. B, 105(2):024403, January 2022.arXiv:2111.07621, doi:10.1103/PhysRevB.105.024403

    work page doi:10.1103/physrevb.105.024403 2022

  41. [51]

    G. E. Volovik. Topological Lifshitz transitions.Low Temperature Physics, 43(1):47–55, January 2017.arXiv: 1606.08318,doi:10.1063/1.4974185

    work page doi:10.1063/1.4974185 2017

  42. [52]

    Intrin- sically/purely gapless-SPT from non-invertible duality transformations.SciPost Phys., 18:153, 2025

    Linhao Li, Masaki Oshikawa, and Yunqin Zheng. Intrin- sically/purely gapless-SPT from non-invertible duality transformations.SciPost Phys., 18:153, 2025. URL: https://scipost.org/10.21468/SciPostPhys.18.5. 153,doi:10.21468/SciPostPhys.18.5.153

    work page doi:10.21468/scipostphys.18.5 2025

  43. [53]

    Hautle, J

    Rintaro Masaoka, Tomohiro Soejima, and Haruki Watan- abe. Quadraticdispersionrelationsingaplessfrustration- free systems.Phys. Rev. B, 110:195140, Nov 2024. URL:https://link.aps.org/doi/10.1103/PhysRevB. 110.195140,doi:10.1103/PhysRevB.110.195140

    work page doi:10.1103/physrevb 2024

  44. [54]

    Wei-Lin Li, Dan-Dan Liang, Zhi Li, and Xue-Jia Yu. Emergent gapless spiral phases and conformal Lifshitz criticality in the cluster Ising model with off-diagonal in- teractions.arXiv e-prints, page arXiv:2506.07151, June 2025.arXiv:2506.07151,doi:10.48550/arXiv.2506. 07151

    work page doi:10.48550/arxiv.2506 2025

  45. [55]

    Gapless and ordered phases in spin-1/2 kitaev–xx–gamma chain.Chinese Physics B, 35(3):037503, mar 2026.doi:10.1088/ 1674-1056/ae3067

    Zebin Zhuang and Wang Yang. Gapless and ordered phases in spin-1/2 kitaev–xx–gamma chain.Chinese Physics B, 35(3):037503, mar 2026.doi:10.1088/ 1674-1056/ae3067. 8

    2026

  46. [56]

    Adsorption on carbon nanotubes: Quantum spin tubes, magnetization plateaus, and conformal symmetry.Phys

    Dmitry Green and Claudio Chamon. Adsorption on carbon nanotubes: Quantum spin tubes, magnetization plateaus, and conformal symmetry.Phys. Rev. Lett., 85:4128–4131, Nov 2000. URL:https://link.aps. org/doi/10.1103/PhysRevLett.85.4128,doi:10.1103/ PhysRevLett.85.4128

    work page doi:10.1103/physrevlett.85.4128 2000

  47. [57]

    Geometric frus- tration and magnetization plateaus in quantum spin and bose-hubbard models on tubes.Phys

    Dmitry Green and Claudio Chamon. Geometric frus- tration and magnetization plateaus in quantum spin and bose-hubbard models on tubes.Phys. Rev. B, 65:104431, Mar 2002. URL:https://link.aps. org/doi/10.1103/PhysRevB.65.104431,doi:10.1103/ PhysRevB.65.104431

    work page doi:10.1103/physrevb.65.104431 2002

  48. [58]

    Timco, and Richard E.P

    Olivier Cador, Dante Gatteschi, Roberta Sessoli, Anne- Laure Barra, Grigore A. Timco, and Richard E.P. Win- penny. Spin frustration effects in an odd-member antifer- romagnetic ring and the magnetic möbius strip.Journal of Magnetism and Magnetic Materials, 290-291:55–60,

  49. [59]

    URL:https://www.sciencedirect

    Proceedings of the Joint European Magnetic Sym- posia (JEMS’ 04). URL:https://www.sciencedirect. com/science/article/pii/S030488530401399X, doi:10.1016/j.jmmm.2004.11.159

    work page doi:10.1016/j.jmmm.2004.11.159 2004

  50. [60]

    Quantum transitions driven by one-bond de- fects in quantum ising rings, Apr 2015

    Massimo Campostrini, Andrea Pelissetto, and Ettore Vicari. Quantum transitions driven by one-bond de- fects in quantum ising rings, Apr 2015. URL:https:// link.aps.org/doi/10.1103/PhysRevE.91.042123,doi: 10.1103/PhysRevE.91.042123

    work page doi:10.1103/physreve.91.042123 2015

  51. [61]

    The a-cycle problem for transverse ising ring.Jour- nal of Statistical Mechanics: Theory and Experi- ment, 2016(11):113102, nov 2016

    Jian-Jun Dong, Peng Li, and Qi-Hui Chen. The a-cycle problem for transverse ising ring.Jour- nal of Statistical Mechanics: Theory and Experi- ment, 2016(11):113102, nov 2016. URL:https://dx. doi.org/10.1088/1742-5468/2016/11/113102,doi:10. 1088/1742-5468/2016/11/113102

    work page doi:10.1088/1742-5468/2016/11/113102 2016

  52. [62]

    Sequences of ground states and classifica- tion of frustration in odd-numbered antiferromagnetic rings.Phys

    Wojciech Florek, Michal Antkowiak, and Grzegorz Kamieniarz. Sequences of ground states and classifica- tion of frustration in odd-numbered antiferromagnetic rings.Phys. Rev. B,94:224421, Dec2016. URL:https:// link.aps.org/doi/10.1103/PhysRevB.94.224421,doi: 10.1103/PhysRevB.94.224421

    work page doi:10.1103/physrevb.94.224421

  53. [63]

    The frustration of being odd: uni- versal area law violation in local systems.Journal of Physics Communications, 3(8):081001, August 2019

    Salvatore Marco Giampaolo, Flavia Brága Ramos, and Fabio Franchini. The frustration of being odd: uni- versal area law violation in local systems.Journal of Physics Communications, 3(8):081001, August 2019. arXiv:1807.07055,doi:10.1088/2399-6528/ab3ab3

    work page doi:10.1088/2399-6528/ab3ab3 2019

  54. [64]

    The frustration of being odd: how boundary conditions can destroy local or- der.New Journal of Physics, 22(8):083024, August 2020

    Vanja Marić, Salvatore Marco Giampaolo, Domagoj Kuić, and Fabio Franchini. The frustration of being odd: how boundary conditions can destroy local or- der.New Journal of Physics, 22(8):083024, August 2020. arXiv:1908.10876,doi:10.1088/1367-2630/aba064

    work page doi:10.1088/1367-2630/aba064 2020

  55. [65]

    Entanglement en- tropy and conformal field theory.Journal of Physics A Mathematical General, 42(50):504005, December 2009.arXiv:0905.4013,doi:10.1088/1751-8113/42/ 50/504005

    Pasquale Calabrese and John Cardy. Entanglement en- tropy and conformal field theory.Journal of Physics A Mathematical General, 42(50):504005, December 2009.arXiv:0905.4013,doi:10.1088/1751-8113/42/ 50/504005

    work page doi:10.1088/1751-8113/42/ 2009

  56. [66]

    Dec- orated defect construction of gapless-SPT states.Sci- Post Phys., 17:013, 2024

    Linhao Li, Masaki Oshikawa, and Yunqin Zheng. Dec- orated defect construction of gapless-SPT states.Sci- Post Phys., 17:013, 2024. URL:https://scipost. org/10.21468/SciPostPhys.17.1.013,doi:10.21468/ SciPostPhys.17.1.013. Symmetry-Protected Topological Phases in the Triangular Majorana–Hubbard Ladder Supplemental material Will Holdhusen,1 Alberto Nocera,2...

    work page doi:10.21468/scipostphys.17.1.013 2024

  57. [67]

    Applying this to the noninteracting HamiltonianH 0 defined in Eq

    Time-reversal symmetries The triangular-lattice Majorana-Hubbard (TLMH) model is not symmetric under the Majorana time-reversal (MTR) symmetry represented by the operator T0 =K(7) whereKis the complex conjugation operator takingi→ −iandγ→γ[1, 2]. Applying this to the noninteracting HamiltonianH 0 defined in Eq. (1), we seeT1H0T † 1 =−H 0. At the same time...

  58. [68]

    γn) commute so long as they share an even number of Majorana operators

    Parity and fermion-string symmetries Since every term in the TLMH is the product of an even number of Majorana fermions, it conserves total fermion parity P= Y i,j c† i,jci,j −2 = (i)NY i,j γr i,jγb i,j.(10) This is a consequence of a general principle: it is simple to show any two strings of Majorana operators (that is to say, operators of the formγ1γ2 ....

  59. [69]

    As such, we should expect the generalized mean-field to have an energy density bounded from above by the simplified version

    when we restrictτa =τ a =τ b =τ b ≡τ 1 andτ α =τ α =τ β =· · · ≡τ 2. As such, we should expect the generalized mean-field to have an energy density bounded from above by the simplified version. A. Self-consistency loop To obtain self-consistent values of the mean-field parametersτj, we utilize the following algorithm:

  60. [70]

    Choose a starting set of values forτj

  61. [71]

    Use derivatives of (23) to obtain corresponding values of∆j from Eq. (20)

  62. [72]

    Use (22) to obtain new values ofτj

  63. [73]

    This procedure can be adapted to infinite systems by replacing sums overkwith integration

    Repeat steps 2 and 3 until quantities converge within a desired precision. This procedure can be adapted to infinite systems by replacing sums overkwith integration. We restrict ourselves to the same 4-leg ladder used in our DMRG simulations, which is theLy = 2,1 y-PBC case of a generalL x ×L y parallelogram on the rectangular lattice with a Brillouin zon...

  64. [74]

    Vertical lines indicate phase transitions. finite-size effect of some sort, difficulties in DMRG simulations (resulting from both the required periodic boundaries, the highly-entanglement nature of the eigenstates, and the twofold degeneracy of the ground state) prevent us from obtaining the excited states required to compute these gaps forLx >33. Further...

  65. [75]

    Tianyu Liu and M. Franz. Electronic structure of topological superconductors in the presence of a vortex lattice.Phys. Rev. B, 92:134519, Oct 2015. URL:https://link.aps.org/doi/10.1103/PhysRevB.92.134519,doi:10.1103/PhysRevB. 92.134519

    work page doi:10.1103/physrevb.92.134519 2015

  66. [76]

    Triangular lattice majorana-hubbard model: Mean-field theory and dmrg onawidth-4torus.Phys

    Tarun Tummuru, Alberto Nocera, and Ian Affleck. Triangular lattice majorana-hubbard model: Mean-field theory and dmrg onawidth-4torus.Phys. Rev. B,103:115128, Mar2021. URL:https://link.aps.org/doi/10.1103/PhysRevB.103.115128, doi:10.1103/PhysRevB.103.115128

    work page doi:10.1103/physrevb.103.115128

  67. [77]

    Journal of Physics C: Solid State Physics6(7), 1181–1203 (1973)

    J M Kosterlitz and D J Thouless. Ordering, metastability and phase transitions in two-dimensional systems.Journal of Physics C: Solid State Physics, 6(7):1181, apr 1973.doi:10.1088/0022-3719/6/7/010

    work page doi:10.1088/0022-3719/6/7/010 1973

  68. [78]

    Wolf, Gerardo Ortiz, Frank Verstraete, and J

    Michael M. Wolf, Gerardo Ortiz, Frank Verstraete, and J. Ignacio Cirac. Quantum phase transitions in matrix product 11 0.5 1.0 1 3 log [N π sin (x′π L )] 1 2 3S(x′) Lx = 33, c ≈ 1.053 0.5 1.0 1.5 1 3 log [N π sin (x′π L )] 1 2 3S(x′) Lx = 45, c ≈ 1.062 0.5 1.0 1.5 1 3 log [N π sin (x′π L )] 1 2 3S(x′) Lx = 55, c ≈ 1.076 0.5 1.0 1.5 1 3 log [N π sin (x′π L...

    work page doi:10.1103/physrevlett.97.110403 2006

  69. [79]

    Sørensen and Hae-Young Kee

    Erik S. Sørensen and Hae-Young Kee. Twice hidden string order and competing phases in the spin-1/2 Kitaev-Gamma ladder.npj Quantum Materials, 9(1):10, December 2024.arXiv:2307.08731,doi:10.1038/s41535-024-00621-x

    work page doi:10.1038/s41535-024-00621-x 2024

  70. [80]

    Entanglement entropy and conformal field theory.J

    Pasquale Calabrese and John Cardy. Entanglement entropy and conformal field theory.Journal of Physics A Mathematical General, 42(50):504005, December 2009.arXiv:0905.4013,doi:10.1088/1751-8113/42/50/504005

    work page doi:10.1088/1751-8113/42/50/504005 2009