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arxiv: 2605.23769 · v1 · pith:BHZLUEZYnew · submitted 2026-05-22 · ❄️ cond-mat.str-el

Interaction-Split Edge Spectral Flow and Neutral Triplet Boundary Modes in a C = 2 Hubbard Pump

Yong-Feng Yang , Zhao-Rui Tian , Chen Cheng , Hong-Gang Luo This is my paper

Pith reviewed 2026-05-25 03:01 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords Hubbard modelThouless pumpmany-body Chern numberedge spectral flowneutral triplet modestopological Mott insulatorDMRG
0
0 comments X

The pith

A sliding modulation of the Hubbard interaction in a period-three chain at 2/3 filling yields a C=+2 many-body pump that splits boundary spectral flow and hosts neutral triplet edge modes.

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

The paper demonstrates that interactions reconstruct the boundary response in a many-body Thouless pump. In the period-three Hubbard model at filling two thirds, DMRG identifies a gapped phase with many-body Chern number plus two, pumping two charges per cycle. The spectrum separates the charge gap from the neutral spin gap. Under open boundaries the spectral flow splits into two distinct events, with a neutral spinful triplet excitation localized at the edge between them. This differs from a Hartree reference where both spin channels flow simultaneously.

Core claim

A period-three Hubbard chain at filling ρ=2/3 forms a correlated insulator with many-body Chern number C=+2 under a sliding spatial modulation of the on-site interaction. This state pumps two units of charge per cycle. The boundary spectral flow is split into two distinct edge events by correlations, between which the boundary charge is neutralized while a neutral, spinful triplet-like excitation is localized at the edge.

What carries the argument

The interaction-split edge spectral flow, which suppresses the boundary doublon channel and separates two edge events with an intervening neutral triplet mode.

If this is right

  • The ρ=2/3 regime remains in the C=+2 sector throughout the μ-U phase diagram.
  • The ρ=4/3 regime undergoes a spin-gap-closing transition from a C=-2 pump to a C=+1 topological Mott pump.
  • The charge gap separates from the lowest neutral spin gap in the excitation spectrum.
  • Open boundary conditions make the contrast with the spin-degenerate Hartree reference more pronounced.

Where Pith is reading between the lines

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

  • This splitting suggests that strong correlations can be tuned to create intermediate neutral boundary states in topological pumps.
  • Similar neutral triplet modes might appear in other interacting topological systems where charge and spin gaps differ.
  • Experimental detection could involve local spectroscopy to observe the triplet excitation during the pump cycle.

Load-bearing premise

Finite-size DMRG on open chains accurately determines the many-body Chern number of the infinite system and separates the charge and spin gaps without artifacts affecting the reported edge flow.

What would settle it

A calculation or measurement showing that the boundary spectral flow remains simultaneous in both spin channels without splitting, or that no neutral triplet excitation appears between the events.

Figures

Figures reproduced from arXiv: 2605.23769 by Chen Cheng, Hong-Gang Luo, Yong-Feng Yang, Zhao-Rui Tian.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of interaction-reconstructed boundary [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Interaction-split boundary spectral flow and neutral [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Ground-state phase diagram and filling-dependent [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

We show that a sliding spatial modulation of the on-site Hubbard interaction realizes a many-body Thouless pump whose boundary spectral flow is reconstructed by correlations. For a period-three Hubbard chain at filling $\rho=2/3$, density-matrix renormalization group (DMRG) calculations identify a correlated insulator with many-body Chern number $C=+2$, corresponding to two units of charge pumped per cycle. Its excitation spectrum separates the charge gap from the lowest neutral spin gap, revealing an interacting bulk response beyond a simple spin-degenerate band pump. Under open boundary conditions, this contrast becomes even more pronounced. A spin-degenerate Hartree/Aubry-Andr\'e-Harper reference pump exhibits simultaneous edge flow in the two spin channels, whereas the full modulated Hubbard model suppresses the boundary doublon channel and splits the spectral flow into two distinct edge events. Between these events, the boundary charge is neutralized while a neutral, spinful triplet-like excitation is localized at the edge. The global $\mu-U$ phase diagram reveals filling-dependent topology: the $\rho=2/3$ regime remains in the $C=+2$ sector, while $\rho=4/3$ regime undergoes a spin-gap-closing transition from a $C=-2$ pump to a distinct $C=+1$ topological Mott pump.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 0 minor

Summary. The manuscript studies a many-body Thouless pump realized via sliding spatial modulation of the on-site Hubbard interaction in a period-three chain at filling ρ=2/3. DMRG calculations identify a correlated insulator with many-body Chern number C=+2 (two units of charge pumped per cycle), with the excitation spectrum separating the charge gap from the lowest neutral spin gap. Under open boundaries the interaction suppresses the doublon channel relative to a Hartree reference, splitting the boundary spectral flow into two distinct edge events; between them a neutral spinful triplet-like mode is localized at the edge. The global μ-U phase diagram shows filling-dependent topology, with the ρ=2/3 regime remaining in the C=+2 sector while ρ=4/3 undergoes a spin-gap-closing transition from C=-2 to a distinct C=+1 topological Mott pump.

Significance. If the DMRG results hold in the thermodynamic limit, the work provides a concrete demonstration that electron correlations can reconstruct the boundary spectral flow of a many-body pump, producing split edge events and neutral triplet boundary modes absent from the non-interacting reference. The filling-dependent change in topology between ρ=2/3 and ρ=4/3 further illustrates how interactions can stabilize distinct topological Mott pumps.

major comments (3)
  1. [Abstract] Abstract: the claim of a C=+2 correlated insulator with split edge spectral flow and a neutral triplet mode rests on DMRG, yet no system sizes, bond dimensions, convergence checks, or error estimates are reported, leaving the central numerical claims without the validation details required to assess finite-size or truncation artifacts.
  2. [DMRG Calculations and Phase Diagram] DMRG Calculations and Phase Diagram sections: the extraction of the many-body Chern number C=+2 and the clean separation of charge versus neutral spin gaps on finite open chains lacks any finite-size scaling analysis with L or bond-dimension χ convergence, which is load-bearing for confirming that the reported split flow and localized triplet survive in the L→∞, χ→∞ limit.
  3. [Boundary Spectral Flow Analysis] Boundary Spectral Flow Analysis: the precise protocol used to assign the integer many-body Chern number (flux insertion versus direct spectral flow) and to resolve the two distinct edge events is not specified, preventing assessment of whether boundary-induced level crossings could merge or split the observed flow.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below, providing clarifications and committing to revisions where additional details are needed.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of a C=+2 correlated insulator with split edge spectral flow and a neutral triplet mode rests on DMRG, yet no system sizes, bond dimensions, convergence checks, or error estimates are reported, leaving the central numerical claims without the validation details required to assess finite-size or truncation artifacts.

    Authors: We agree that explicit reporting of DMRG parameters is essential. In the revised manuscript we will add a dedicated methods paragraph specifying the system sizes (L=24, 36, 48), bond dimensions (χ up to 2400), energy convergence thresholds (10^{-8}), and estimated truncation errors for the reported quantities. revision: yes

  2. Referee: [DMRG Calculations and Phase Diagram] DMRG Calculations and Phase Diagram sections: the extraction of the many-body Chern number C=+2 and the clean separation of charge versus neutral spin gaps on finite open chains lacks any finite-size scaling analysis with L or bond-dimension χ convergence, which is load-bearing for confirming that the reported split flow and localized triplet survive in the L→∞, χ→∞ limit.

    Authors: We acknowledge the value of explicit scaling analysis. While the present results already show consistent behavior across the accessed sizes, the revision will include additional panels demonstrating L- and χ-dependence of the Chern number, charge gap, and spin gap to strengthen the claim that the split flow and triplet mode persist in the thermodynamic limit. revision: yes

  3. Referee: [Boundary Spectral Flow Analysis] Boundary Spectral Flow Analysis: the precise protocol used to assign the integer many-body Chern number (flux insertion versus direct spectral flow) and to resolve the two distinct edge events is not specified, preventing assessment of whether boundary-induced level crossings could merge or split the observed flow.

    Authors: The many-body Chern number was obtained from the spectral flow under adiabatic flux insertion on periodic chains; the split edge events were identified by tracking the evolution of the lowest open-boundary eigenstates while monitoring their charge and spin densities to confirm edge localization. The revision will expand the relevant section with a step-by-step description of both protocols and the criteria used to distinguish edge from bulk crossings. revision: yes

Circularity Check

0 steps flagged

No circularity; results from direct DMRG numerics

full rationale

The paper's central claims (C=+2 insulator, split edge spectral flow, neutral triplet boundary mode) are obtained via density-matrix renormalization group computations on the modulated Hubbard model at ρ=2/3. The many-body Chern number is extracted numerically from the spectrum under interaction modulation, with no analytical derivation chain, fitted parameter renamed as prediction, or load-bearing self-citation that reduces the result to its own inputs by construction. The phase diagram and boundary analysis rest on the reported DMRG data rather than any self-referential loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the reliability of DMRG for extracting many-body topology and gaps in the modulated Hubbard model; no free parameters are fitted to produce the reported Chern number or edge events.

axioms (1)
  • domain assumption DMRG on finite chains yields the correct infinite-system many-body Chern number and gap structure
    The identification of C=+2 and the split edge flow depends on this numerical method being accurate for the model.

pith-pipeline@v0.9.0 · 5779 in / 1137 out tokens · 28263 ms · 2026-05-25T03:01:33.809036+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    For a period-three Hubbard chain at filling ρ=2/3, density-matrix renormalization group (DMRG) calculations identify a correlated insulator with many-body Chern number C=+2... splits the spectral flow into two distinct edge events... neutral, spinful triplet-like excitation

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matches
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The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
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uses
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The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

57 extracted references · 57 canonical work pages

  1. [1]

    M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys.82, 3045 (2010)

    work page 2010

  2. [2]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Topological insulators and su- perconductors, Rev. Mod. Phys.83, 1057 (2011)

    work page 2011

  3. [3]

    Rachel, Interacting topological insulators: a review, Reports on Progress in Physics81, 116501 (2018)

    S. Rachel, Interacting topological insulators: a review, Reports on Progress in Physics81, 116501 (2018)

    work page 2018

  4. [4]

    J. M. Luttinger, An exactly soluble model of a many- fermion system, Journal of Mathematical Physics4, 1154 (1963)

    work page 1963

  5. [5]

    F. D. M. Haldane, ’luttinger liquid theory’ of one- dimensional quantum fluids. i. properties of the luttinger model and their extension to the general 1d interact- ing spinless fermi gas, Journal of Physics C: Solid State Physics14, 2585 (1981)

    work page 1981

  6. [6]

    D. J. Thouless, Quantization of particle transport, Phys. Rev. B27, 6083 (1983)

    work page 1983

  7. [7]

    Q. Niu, D. J. Thouless, and Y.-S. Wu, Quantized hall conductance as a topological invariant, Physical Review B31, 3372 (1985)

    work page 1985

  8. [8]

    Resta, Quantum-mechanical position operator in ex- tended systems, Physical Review Letters80, 1800 (1998)

    R. Resta, Quantum-mechanical position operator in ex- tended systems, Physical Review Letters80, 1800 (1998)

    work page 1998

  9. [9]

    Nakajima, T

    S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, Topological thou- less pumping of ultracold fermions, Nature Physics12, 296 (2016)

    work page 2016

  10. [10]

    Lohse, C

    M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, A thouless quantum pump with ultracold bosonic atoms in an optical superlattice, Nature Physics 12, 350 (2016)

    work page 2016

  11. [11]

    Citro and M

    R. Citro and M. Aidelsburger, Thouless pumping and topology, Nature Reviews Physics5, 87 (2023)

    work page 2023

  12. [12]

    E. Berg, M. Levin, and E. Altman, Quantized pumping and topology of the phase diagram for a system of inter- acting bosons, Phys. Rev. Lett.106, 110405 (2011)

    work page 2011

  13. [13]

    T.-S. Zeng, W. Zhu, and D. N. Sheng, Fractional charge pumping of interacting bosons in one-dimensional super- lattice, Phys. Rev. B94, 235139 (2016)

    work page 2016

  14. [14]

    Taddia, E

    L. Taddia, E. Cornfeld, D. Rossini, L. Mazza, E. Sela, and R. Fazio, Topological fractional pumping with alkaline- earth-like atoms in synthetic lattices, Phys. Rev. Lett. 118, 230402 (2017)

    work page 2017

  15. [15]

    Y. Kuno, K. Shimizu, and I. Ichinose, Various topologi- cal mott insulators and topological bulk charge pumping in strongly-interacting boson system in one-dimensional superlattice, New Journal of Physics19, 123025 (2017)

    work page 2017

  16. [16]

    Requist and E

    R. Requist and E. K. U. Gross, Accurate formula for the macroscopic polarization of strongly correlated materi- 6 als, The Journal of Physical Chemistry Letters9, 7045 (2018)

    work page 2018

  17. [17]

    Nakagawa, T

    M. Nakagawa, T. Yoshida, R. Peters, and N. Kawakami, Breakdown of topological thouless pumping in the strongly interacting regime, Phys. Rev. B98, 115147 (2018)

    work page 2018

  18. [18]

    Greschner, S

    S. Greschner, S. Mondal, and T. Mishra, Topological charge pumping of bound bosonic pairs, Phys. Rev. A 101, 053630 (2020)

    work page 2020

  19. [19]

    J¨ urgensen, S

    M. J¨ urgensen, S. Mukherjee, C. J¨ org, and M. C. Rechts- man, Quantized fractional thouless pumping of solitons, Nature Physics19, 420 (2023)

    work page 2023

  20. [20]

    J¨ urgensen, J

    M. J¨ urgensen, J. Steiner, G. Refael, and M. C. Rechts- man, Multiband fractional thouless pumps, Phys. Rev. Lett.135, 166601 (2025)

    work page 2025

  21. [21]

    Bertok, F

    E. Bertok, F. Heidrich-Meisner, and A. A. Aligia, Split- ting of topological charge pumping in an interacting two- component fermionic rice-mele hubbard model, Phys. Rev. B106, 045141 (2022)

    work page 2022

  22. [22]

    Walter, Z

    A.-S. Walter, Z. Zhu, M. G¨ achter, J. Minguzzi, S. Roschinski, K. Sandholzer, K. Viebahn, and T. Esslinger, Quantization and its breakdown in a hubbard–thouless pump, Nature Physics19, 1471 (2023)

    work page 2023

  23. [23]

    Viebahn, A.-S

    K. Viebahn, A.-S. Walter, E. Bertok, Z. Zhu, M. G¨ achter, A. A. Aligia, F. Heidrich-Meisner, and T. Esslinger, Inter- actions enable thouless pumping in a nonsliding lattice, Phys. Rev. X14, 021049 (2024)

    work page 2024

  24. [24]

    Juli` a-Farr´ e, J

    S. Juli` a-Farr´ e, J. Arg¨ uello-Luengo, L. Henriet, and A. Dauphin, Quantized thouless pumps protected by in- teractions in dimerized rydberg tweezer arrays, Phys. Rev. A110, 023328 (2024)

    work page 2024

  25. [25]

    Parida, D

    R. Parida, D. Sen, and T. Mishra, Topological phase transition through tunable nearest-neighbor interactions in a one-dimensional lattice, Phys. Rev. B112, 085124 (2025)

    work page 2025

  26. [26]

    Huang, Y

    B. Huang, Y. Ke, W. Liu, and C. Lee, Topological pump- ing induced by spatiotemporal modulation of interaction, Physica Scripta99, 065997 (2024)

    work page 2024

  27. [27]

    Ravets, N

    S. Ravets, N. Pernet, N. Mostaan, N. Goldman, and J. Bloch, Thouless pumping in a driven-dissipative kerr resonator array, Phys. Rev. Lett.134, 093801 (2025)

    work page 2025

  28. [28]

    Bai and Z

    C. Bai and Z. Liang, Fractional thouless pumping of soli- tons: A unique manifestation of bulk-edge correspon- dence of nonlinear eigenvalue problems, Phys. Rev. A 112, 052207 (2025)

    work page 2025

  29. [29]

    A. P. Chaudhari, M. J¨ urgensen, and M. C. Rechtsman, Quantized pumping in disordered nonlinear thouless pumps (2025), arXiv:2512.11394 [cond-mat.mes-hall]

    work page arXiv 2025

  30. [30]

    P. G. Harper, The general motion of conduction electrons in a uniform magnetic field, with application to the dia- magnetism of metals, Proceedings of the Physical Society. Section A68, 879 (1955)

    work page 1955

  31. [31]

    Aubry and G

    S. Aubry and G. Andr´ e, Analyticity breaking and ander- son localization in incommensurate lattices, Ann. Israel Phys. Soc3, 18 (1980)

    work page 1980

  32. [32]

    See Supplemental Material appended to this manuscript for detailed DMRG numerical parameters, exact defini- tions of the thermodynamic gaps and boundary observ- ables, the Fukui many-body Chern number calculation, and finite-size scaling analysis

    work page

  33. [33]

    S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett.69, 2863 (1992)

    work page 1992

  34. [34]

    S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B48, 10345 (1993)

    work page 1993

  35. [35]

    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)

    work page 2022

  36. [36]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys.82, 1959 (2010)

    work page 1959

  37. [37]

    Fukui, Y

    T. Fukui, Y. Hatsugai, and H. Suzuki, Chern numbers in discretized Brillouin zone: Efficient method of computing (spin) hall conductances, Journal of the Physical Society of Japan74, 1674 (2005)

    work page 2005

  38. [38]

    C. N. Varney, K. Sun, M. Rigol, and V. Galitski, Topolog- ical phase transitions for interacting finite systems, Phys. Rev. B84, 241105(R) (2011)

    work page 2011

  39. [39]

    L.-J. Lang, X. Cai, and S. Chen, Edge states and topo- logical phases in one-dimensional optical superlattices, Phys. Rev. Lett.108, 220401 (2012)

    work page 2012

  40. [40]

    Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zil- berberg, Topological states and adiabatic pumping in quasicrystals, Phys. Rev. Lett.109, 106402 (2012)

    work page 2012

  41. [41]

    Y. E. Kraus and O. Zilberberg, Topological equivalence between the fibonacci quasicrystal and the harper model, Phys. Rev. Lett.109, 116404 (2012)

    work page 2012

  42. [42]

    Paiva and R

    T. Paiva and R. R. dos Santos, Metal-insulator transition in one-dimensional hubbard superlattices, Phys. Rev. B 58, 9607 (1998)

    work page 1998

  43. [43]

    L. W. Clark, L.-C. Ha, C.-Y. Xu, and C. Chin, Quantum dynamics with spatiotemporal control of interactions in a stable bose-einstein condensate, Phys. Rev. Lett.115, 155301 (2015)

    work page 2015

  44. [44]

    Yamazaki, S

    R. Yamazaki, S. Taie, S. Sugawa, and Y. Takahashi, Submicron spatial modulation of an interatomic interac- tion in a bose-einstein condensate, Phys. Rev. Lett.105, 050405 (2010)

    work page 2010

  45. [45]

    Landig, L

    R. Landig, L. Hruby, N. Dogra, M. Landini, R. Mottl, T. Donner, and T. Esslinger, Quantum phases from com- peting short- and long-range interactions in an optical lattice, Nature532, 476 (2016)

    work page 2016

  46. [46]

    V. D. Vaidya, Y. Guo, R. M. Kroeze, K. E. Ballantine, A. J. Koll´ ar, J. Keeling, and B. L. Lev, Tunable-range, photon-mediated atomic interactions in multimode cav- ity qed, Phys. Rev. X8, 011002 (2018)

    work page 2018

  47. [47]

    P. Wang, G. Yu, Y. H. Kwan, Y. Jia, S. Lei, S. Klemenz, F. A. Cevallos, R. Singha, T. Devakul, K. Watanabe, T. Taniguchi, S. L. Sondhi, R. J. Cava, L. M. Schoop, S. A. Parameswaran, and S. Wu, One-dimensional lut- tinger liquids in a two-dimensional moir´ e lattice, Nature 605, 57 (2022)

    work page 2022

  48. [48]

    X. Yang, Y. Zhang, L. Chen, K. Aso, W. Yamamori, R. Moriya, K. Watanabe, T. Taniguchi, T. Sasagawa, N. Nakatsuji, M. Koshino, Y. Yamada-Takamura, Y. Os- hima, and T. Machida, Intrinsic one-dimensional moir´ e superlattice in large-angle twisted bilayer wte(2), ACS Nano19, 13007 (2025)

    work page 2025

  49. [49]

    Kawakami, H

    T. Kawakami, H. Tateish, D. Yoshida, X. Yang, N. Nakatsuji, L. Chen, K. Aso, Y. Yamada-Takamura, Y. Oshima, Y. Zhang, T. Machida, K. Kato, and M. Koshino, One-dimensional electronic states in a moir´ e superlattice of twisted bilayer wte2 (2026), arXiv:2601.21228 [cond-mat.mes-hall]

    work page arXiv 2026

  50. [50]

    Jolie, C

    W. Jolie, C. Murray, P. S. Weiß, J. Hall, F. Portner, N. Atodiresei, A. V. Krasheninnikov, C. Busse, H.-P. Komsa, A. Rosch, and T. Michely, Tomonaga-luttinger liquid in a box: Electrons confined within mos 2 mirror- twin boundaries, Phys. Rev. X9, 011055 (2019)

    work page 2019

  51. [51]

    B. Deng, H. Ahn, J. Wang, G. Moon, C. Han, N. Dongre, 7 C. Lei, G. Scuri, J. Sung, E. Brutschea, K. Watanabe, T. Taniguchi, F. Zhang, M.-H. Jo, and H. Park, Epitaxi- ally defined luttinger liquids on mos2 bicrystals, Physical Review Letters134, 046301 (2025)

    work page 2025

  52. [52]

    Q. Yao, H. Jung, K. Kong, C. De, J. Kim, J. D. Den- linger, and H. W. Yeom, Robust luttinger liquid state of 1d dirac fermions in a van der waals system Nb 9Si4Te18, Nano Letters23, 7961 (2023)

    work page 2023

  53. [53]

    ¨Ostlund and S

    S. ¨Ostlund and S. Rommer, Thermodynamic limit of den- sity matrix renormalization, Phys. Rev. Lett.75, 3537 (1995)

    work page 1995

  54. [54]

    Dukelsky, M

    J. Dukelsky, M. A. Mart´ ın-Delgado, T. Nishino, and G. Sierra, Equivalence of the variational matrix product method and the density matrix renormalization group applied to spin chains, Europhysics Letters43, 457 (1998). S1 Supplemental Material for Supplemental Material for: Interaction-Split Edge Spectral Flow and Neutral Triplet Boundary Modes in aC= 2...

    work page 1998

  55. [55]

    Substituting this into the interaction term generates the non-interacting Hartree control Hamiltonian: HH =−t X i,σ ˆc† i,σˆci+1,σ + H.c

    yields the standard Hartree mean-field decoupling: ˆni,↑ˆni,↓ ≈ ⟨ˆni,↑⟩ˆni,↓ +⟨ˆni,↓⟩ˆni,↑ − ⟨ˆni,↑⟩⟨ˆni,↓⟩.(S4) By enforcing the global SU(2) spin symmetry of our un- magnetized ground state alongside the commensurate fill- ing constraint, the average density per spin channel is uniform:⟨ˆni,↑⟩=⟨ˆni,↓⟩=ρ/2. Substituting this into the interaction term gen...

    work page

  56. [56]

    In a mean-field pump, the spin-up and spin-down edge modes cross the Fermi level simultane- ously

    Spin-Degenerate Edge Transport: BecauseH ↑ and H↓ are strictly identical and uncoupled in the Hartree limit, their corresponding topological edge states are per- fectly degenerate. In a mean-field pump, the spin-up and spin-down edge modes cross the Fermi level simultane- ously. The emergence of split charge transport, the ex- tended topological plateau, ...

    work page

  57. [57]

    Consequently, creating a charge excitation or a spin-flip excitation requires promoting an identical electron across the same single-particle bandgapE g

    The Absence of Bulk Spin-Charge Separation: In any time-reversal symmetric band insulator governed by HH, the fundamental quasiparticle remains the electron. Consequently, creating a charge excitation or a spin-flip excitation requires promoting an identical electron across the same single-particle bandgapE g. A Hartree descrip- tion strictly dictates tha...

    work page