Polarization-based indices in quantum many-body systems: validity and extension beyond one dimension

Yasuhiro Tada

arxiv: 2507.08294 · v3 · pith:ZCKGZX7Unew · submitted 2025-07-11 · ❄️ cond-mat.str-el

Polarization-based indices in quantum many-body systems: validity and extension beyond one dimension

Yasuhiro Tada This is my paper

Pith reviewed 2026-05-25 08:32 UTC · model grok-4.3

classification ❄️ cond-mat.str-el

keywords twist operatorpolarization indexgapped phasesgapless phasesquantum many-body systemssymmetry assumptionsmany-body indicesone dimension

0 comments

The pith

The twist operator expectation value sharply distinguishes gapped and gapless phases under symmetry assumptions, with a nontrivial extension beyond one dimension.

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

The paper formulates symmetry-based statements to clarify when the twist operator serves as a valid polarization index for phases in quantum many-body systems. It specifies the role of ground-state degeneracy in gapped systems and the additional assumptions needed to rule out gapless systems that mimic gapped behavior. The work then builds a controlled extension of this index to dimensions beyond one, showing that the extension cannot be obtained by simple generalization. This delineates the precise regime where such polarization-based quantities act as reliable many-body indices.

Core claim

Under explicitly stated symmetry-based assumptions, the expectation value of the twist operator sharply distinguishes gapped and gapless phases. Ground-state degeneracy has a specific role in statements for gapped systems, while further assumptions exclude gapless scenarios that could mimic gapped behavior in the thermodynamic limit. A meaningful and nontrivial extension of the index is constructed beyond one dimension.

What carries the argument

The twist operator, used as a polarization-based index whose expectation value is analyzed under symmetry assumptions to distinguish phases.

If this is right

The twist operator provides a sharp diagnostic for gapped versus gapless phases when assumptions hold.
Ground-state degeneracy must be accounted for in gapped phase statements.
Additional assumptions prevent gapless phases from mimicking gapped ones thermodynamically.
The index extends nontrivially to higher dimensions.
The regime of validity for polarization-based many-body indices is delineated.

Where Pith is reading between the lines

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

Numerical evaluation of the twist operator expectation could identify phases in models obeying the assumptions without separate gap measurements.
The symmetry-based framework could guide construction of similar indices for other operators or symmetries.
The higher-dimensional extension might link to polarization diagnostics in topological phases.

Load-bearing premise

Ground-state degeneracy and symmetry conditions are sufficient to exclude gapless systems that mimic gapped behavior in the thermodynamic limit.

What would settle it

A concrete counterexample would be a gapless quantum many-body system, satisfying the symmetry assumptions, where the twist operator expectation value indicates a gapped phase in the thermodynamic limit.

read the original abstract

The expectation value of the twist operator has been widely used as a polarization-based index for gapped and gapless phases in interacting quantum many-body systems. Although numerous studies support this usage in specific settings and rigorous results have established the validity of the criterion in important settings, the precise assumptions required for it to sharply distinguish gapped and gapless phases under more general conditions have not been fully clarified. In this work, we clarify the logical status of polarization-based indices by formulating symmetry-based statements under explicitly stated assumptions. We identify the role of ground-state degeneracy in the statements for gapped systems and clarify the distinct assumptions required to exclude gapless scenarios that could otherwise mimic gapped behavior in the thermodynamic limit. Building on this controlled framework, we construct a meaningful extension beyond one dimension, emphasizing that such an extension is nontrivial and cannot be obtained by a straightforward generalization of the one-dimensional twist operator. Our results delineate the regime in which polarization-based quantities are justified as sharply defined many-body indices.

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.

Desk Editor's Note private letter to a colleague

The paper sharpens the symmetry assumptions for the twist operator as a gapped/gapless index and sketches a nontrivial higher-D version, but the actual derivations and checks remain to be verified.

read the letter

This paper's core move is to state the symmetry assumptions explicitly so the twist operator expectation value can be used as a sharp index, while flagging the role of ground-state degeneracy for gapped cases and the extra conditions needed to keep gapless mimics out in the thermodynamic limit. It also claims a higher-dimensional construction that does not follow from a direct lift of the 1D operator. That framing is the main thing a reader should take away: the work treats the index as something that requires controlled statements rather than an automatic diagnostic. The authors do a clean job of separating the logical pieces and emphasizing why the higher-D step is not trivial. That kind of precision is useful in a subfield where the same quantity gets applied across many models without always spelling out the fine print. The abstract suggests the derivations rest on symmetry arguments rather than fitting or circular definitions, which keeps the circularity burden low. Still, without seeing the concrete handling of degeneracy or the explicit form of the higher-D operator, it is hard to judge how robust the exclusions are or whether the extension yields usable numbers on simple lattices. If the proofs stay at the level of formal statements without model checks, the practical payoff stays modest. The paper is aimed at condensed-matter theorists who already use polarization indices in 1D chains and want to know the safe operating range or try the idea in 2D. It is narrow but addresses a real source of ambiguity in the literature. I would send it to peer review; the topic is established enough that a careful referee can test whether the stated assumptions actually close the gaps the authors identify.

Referee Report

0 major / 3 minor

Summary. The manuscript formulates symmetry-based statements under explicitly stated assumptions to clarify the validity of the twist-operator expectation value as a polarization-based index distinguishing gapped and gapless phases in quantum many-body systems. It identifies the specific role of ground-state degeneracy in the gapped case and the additional assumptions required to exclude gapless scenarios that could mimic gapped behavior in the thermodynamic limit. Building on this framework, the work constructs a non-trivial extension of the index to higher dimensions that cannot be obtained by direct generalization of the one-dimensional operator.

Significance. If the derivations and constructions hold under the stated assumptions, the paper provides a valuable controlled framework that strengthens the justification for using polarization-based quantities as sharply defined many-body indices. The explicit delineation of assumptions and the demonstration that the higher-D extension is non-trivial represent a useful contribution to the literature on many-body topological and polarization diagnostics in condensed-matter theory.

minor comments (3)

The abstract states that the extension 'cannot be obtained by a straightforward generalization,' but a brief indication in the introduction of why a naive lift fails (e.g., loss of a key symmetry property) would help readers immediately grasp the non-triviality without waiting for the technical construction.
Notation for the twist operator and its expectation value should be introduced with a single consistent symbol and definition early in the text (ideally in a dedicated subsection) to avoid any ambiguity when the same quantity appears in both 1D and higher-D statements.
A short table or bullet list summarizing the precise assumptions for the gapped versus gapless cases would improve readability and make the logical distinctions easier to reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, including the accurate summary of our contributions in clarifying assumptions for twist-operator polarization indices and constructing a non-trivial higher-dimensional extension. The recommendation for minor revision is noted. No major comments are listed in the report, so we have no specific points requiring rebuttal or detailed response. We will address any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No circularity; derivation self-contained under stated assumptions

full rationale

The paper's central program is to clarify the logical status of polarization-based indices by formulating symmetry-based statements under explicitly stated assumptions, identify the role of ground-state degeneracy for gapped systems, exclude gapless mimics in the thermodynamic limit, and construct a nontrivial higher-D extension. No equations, parameters, or results are shown to reduce by construction to fitted inputs, self-definitions, or self-citation chains. The abstract and program description contain no load-bearing self-citations, ansatzes smuggled via prior work, or renaming of known results as new derivations. The derivation chain is therefore independent of the target claims and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard quantum many-body symmetry principles and the thermodynamic limit; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Symmetry-based statements can be formulated under explicitly stated assumptions to distinguish phases
Central to the clarification of logical status for the twist operator index.
domain assumption Ground-state degeneracy affects the statements for gapped systems
Identified as playing a role in the gapped case.

pith-pipeline@v0.9.0 · 5697 in / 1275 out tokens · 20136 ms · 2026-05-25T08:32:23.546903+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

[1]

0 0 0 1

κ q−1   , U=   0 1 0. . .0 0 0 1. . .0 ... ... ... ... ... 0 0. . .0 1 1 0. . . . . .0   .(5) It turns out that this representation is essentially valid also in the gapped ground state sector of the Hamiltonian Hwith small correctionsO(1/L) which vanish in the thermodynamic limit. We consider insulators and metals respec- tiv...

work page
[2]

Xiao-Gang Wen,Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons(Oxford University Press, Oxford, 2004). 5

work page 2004
[3]

Bei Zeng, Xie Chen, Duan-Lu Zhou, and Xiao-Gang Wen,Quantum Information Meets Quantum Matter: From Quantum Entanglement to Topological Phases of Many-Body Systems, Quantum Science and Technology (Springer, New York, 2019)

work page 2019
[4]

Theory of the insulating state,

Walter Kohn, “Theory of the insulating state,” Phys. Rev.133, A171–A181 (1964)

work page 1964
[5]

Quantum-mechanical position operator in extended systems,

Raffaele Resta, “Quantum-mechanical position operator in extended systems,” Phys. Rev. Lett.80, 1800–1803 (1998)

work page 1998
[6]

Electron localization in the insulating state,

Raffaele Resta and Sandro Sorella, “Electron localization in the insulating state,” Phys. Rev. Lett.82, 370–373 (1999)

work page 1999
[7]

What makes an insulator different from a metal?

Raffaele Resta, “What makes an insulator different from a metal?” AIP Conference Proceedings535, 67–78 (2000)

work page 2000
[8]

Why are insulators insulating and met- als conducting?

Raffaele Resta, “Why are insulators insulating and met- als conducting?” Journal of Physics: Condensed Matter 14, R625 (2002)

work page 2002
[9]

Quantum mechanical po- sition operator and localization in extended systems,

A. A. Aligia and G. Ortiz, “Quantum mechanical po- sition operator and localization in extended systems,” Phys. Rev. Lett.82, 2560–2563 (1999)

work page 1999
[10]

How localized is an extended quantum system?

G. Ortiz and A.A. Aligia, “How localized is an extended quantum system?” physica status solidi (b)220, 737–743 (2000)

work page 2000
[11]

Polar- ization and localization in insulators: Generating func- tion approach,

Ivo Souza, Tim Wilkens, and Richard M. Martin, “Polar- ization and localization in insulators: Generating func- tion approach,” Phys. Rev. B62, 1666–1683 (2000)

work page 2000
[12]

Variational description of mott insulators,

Manuela Capello, Federico Becca, Michele Fabrizio, San- dro Sorella, and Erio Tosatti, “Variational description of mott insulators,” Phys. Rev. Lett.94, 026406 (2005)

work page 2005
[13]

Reconstruction of the polar- ization distribution of the rice-mele model,

M. Yahyavi and B. Het´ enyi, “Reconstruction of the polar- ization distribution of the rice-mele model,” Phys. Rev. A95, 062104 (2017)

work page 2017
[14]

Ground-state properties of the hydrogen chain: Dimerization, insulator-to-metal transition, and magnetic phases,

Mario Motta, Claudio Genovese, Fengjie Ma, Zhi-Hao Cui, Randy Sawaya, Garnet Kin-Lic Chan, Natalia Chep- iga, Phillip Helms, Carlos Jim´ enez-Hoyos, Andrew J. Mil- lis, Ushnish Ray, Enrico Ronca, Hao Shi, Sandro Sorella, Edwin M. Stoudenmire, Steven R. White, and Shi- wei Zhang (Simons Collaboration on the Many-Electron Problem), “Ground-state properties ...

work page 2020
[15]

Order parameter to characterize valence-bond-solid states in quantum spin chains,

Masaaki Nakamura and Synge Todo, “Order parameter to characterize valence-bond-solid states in quantum spin chains,” Phys. Rev. Lett.89, 077204 (2002)

work page 2002
[16]

Scaling of the po- larization amplitude in quantum many-body systems in one dimension,

Ryohei Kobayashi, Yuya O. Nakagawa, Yoshiki Fukusumi, and Masaki Oshikawa, “Scaling of the po- larization amplitude in quantum many-body systems in one dimension,” Phys. Rev. B97, 165133 (2018)

work page 2018
[17]

Polariza- tion amplitude near quantum critical points,

Shunsuke C. Furuya and Masaaki Nakamura, “Polariza- tion amplitude near quantum critical points,” Phys. Rev. B99, 144426 (2019)

work page 2019
[18]

Inequivalent berry phases for the bulk polarization,

Haruki Watanabe and Masaki Oshikawa, “Inequivalent berry phases for the bulk polarization,” Phys. Rev. X8, 021065 (2018)

work page 2018
[19]

Bosonic in- teger quantum hall effect as topological pumping,

Masaya Nakagawa and Shunsuke Furukawa, “Bosonic in- teger quantum hall effect as topological pumping,” Phys. Rev. B95, 165116 (2017)

work page 2017
[20]

Correlated zak in- sulator in organic antiferromagnets,

Takahiro Misawa and Makoto Naka, “Correlated zak in- sulator in organic antiferromagnets,” Phys. Rev. B108, L081120 (2023)

work page 2023
[21]

Rigorous index theory for one-dimensional interacting topological insulators,

Hal Tasaki, “Rigorous index theory for one-dimensional interacting topological insulators,” Journal of Mathemat- ical Physics64, 041903 (2023)

work page 2023
[22]

Quantized polarization in a generalized rice-mele model at arbitrary filling,

Yasuhiro Tada, “Quantized polarization in a generalized rice-mele model at arbitrary filling,” Phys. Rev. B110, 045113 (2024)

work page 2024
[23]

Topological phase transition and𭟋 2 index fors= 1 quantum spin chains,

Hal Tasaki, “Topological phase transition and𭟋 2 index fors= 1 quantum spin chains,” Phys. Rev. Lett.121, 140604 (2018)

work page 2018
[24]

Identifica- tion of gapless phases by squaring a twist operator,

Hang Su, Yuan Yao, and Akira Furusaki, “Identifica- tion of gapless phases by squaring a twist operator,” ArXiv:2506.02496

work page arXiv
[25]

Two soluble models of an antiferromagnetic chain,

Elliott Lieb, Theodore Schultz, and Daniel Mattis, “Two soluble models of an antiferromagnetic chain,” Annals of Physics16, 407–466 (1961)

work page 1961
[26]

A proof of part of hal- dane’s conjecture on spin chains,

Ian Affleck and Elliott Lieb, “A proof of part of hal- dane’s conjecture on spin chains,” Letters in Mathemat- ical Physics12, 57–69 (1986)

work page 1986
[27]

See supplemental materials

work page
[28]

Density matrix formulation for quan- tum renormalization groups,

Steven R. White, “Density matrix formulation for quan- tum renormalization groups,” Phys. Rev. Lett.69, 2863– 2866 (1992)

work page 1992
[29]

The ITensor Software Library for Ten- sor Network Calculations,

Matthew Fishman, Steven R. White, and E. Miles Stoudenmire, “The ITensor Software Library for Ten- sor Network Calculations,” SciPost Phys. Codebases , 4 (2022)

work page 2022
[30]

Efficient nu- merical simulations with Tensor Networks: Tensor Net- work Python (TeNPy),

Johannes Hauschild and Frank Pollmann, “Efficient nu- merical simulations with Tensor Networks: Tensor Net- work Python (TeNPy),” SciPost Phys. Lect. Notes , 5 (2018)

work page 2018
[31]

mvmc–open-source software for many-variable varia- tional monte carlo method,

Takahiro Misawa, Satoshi Morita, Kazuyoshi Yoshimi, Mitsuaki Kawamura, Yuichi Motoyama, Kota Ido, Takahiro Ohgoe, Masatoshi Imada, and Takeo Kato, “mvmc–open-source software for many-variable varia- tional monte carlo method,” Computer Physics Commu- nications235, 447–462 (2019)

work page 2019
[32]

Note on a theorem of bloch concerning possi- ble causes of superconductivity,

D. Bohm, “Note on a theorem of bloch concerning possi- ble causes of superconductivity,” Phys. Rev.75, 502–504 (1949)

work page 1949
[33]

Two no-go theo- rems on superconductivity,

Yasuhiro Tada and Tohru Koma, “Two no-go theo- rems on superconductivity,” J. Stat. Phys.165, 455–470 (2016)

work page 2016
[34]

Topological approach to luttinger’s theorem and the fermi surface of a kondo lattice,

Masaki Oshikawa, “Topological approach to luttinger’s theorem and the fermi surface of a kondo lattice,” Phys. Rev. Lett.84, 3370–3373 (2000)

work page 2000
[35]

Lieb-schultz-mattis theorem in higher dimensions from approximate magnetic translation sym- metry,

Yasuhiro Tada, “Lieb-schultz-mattis theorem in higher dimensions from approximate magnetic translation sym- metry,” Phys. Rev. Lett.127, 237204 (2021)

work page 2021
[36]

Projectification of point group symmetries with a background flux and lieb-schultz-mattis theorem,

Yasuhiro Tada and Masaki Oshikawa, “Projectification of point group symmetries with a background flux and lieb-schultz-mattis theorem,” ArXiv:2505.00927

work page arXiv
[37]

Solitons in polyacetylene,

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett.42, 1698–1701 (1979). 6 SUPPLEMENT AL MA TERIALS A. One-dimensional fermions We consider interacting fermions in one-dimension, H= L−1X i=0 tc† i ci+1 + (h.c.) + X i,k Vknini+k,(S1) whereLis assumed to beL∈3Z. We suppose that the filling isν=N/L= 1/3 and the interac...

work page 1979

[1] [1]

0 0 0 1

κ q−1   , U=   0 1 0. . .0 0 0 1. . .0 ... ... ... ... ... 0 0. . .0 1 1 0. . . . . .0   .(5) It turns out that this representation is essentially valid also in the gapped ground state sector of the Hamiltonian Hwith small correctionsO(1/L) which vanish in the thermodynamic limit. We consider insulators and metals respec- tiv...

work page

[2] [2]

Xiao-Gang Wen,Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons(Oxford University Press, Oxford, 2004). 5

work page 2004

[3] [3]

Bei Zeng, Xie Chen, Duan-Lu Zhou, and Xiao-Gang Wen,Quantum Information Meets Quantum Matter: From Quantum Entanglement to Topological Phases of Many-Body Systems, Quantum Science and Technology (Springer, New York, 2019)

work page 2019

[4] [4]

Theory of the insulating state,

Walter Kohn, “Theory of the insulating state,” Phys. Rev.133, A171–A181 (1964)

work page 1964

[5] [5]

Quantum-mechanical position operator in extended systems,

Raffaele Resta, “Quantum-mechanical position operator in extended systems,” Phys. Rev. Lett.80, 1800–1803 (1998)

work page 1998

[6] [6]

Electron localization in the insulating state,

Raffaele Resta and Sandro Sorella, “Electron localization in the insulating state,” Phys. Rev. Lett.82, 370–373 (1999)

work page 1999

[7] [7]

What makes an insulator different from a metal?

Raffaele Resta, “What makes an insulator different from a metal?” AIP Conference Proceedings535, 67–78 (2000)

work page 2000

[8] [8]

Why are insulators insulating and met- als conducting?

Raffaele Resta, “Why are insulators insulating and met- als conducting?” Journal of Physics: Condensed Matter 14, R625 (2002)

work page 2002

[9] [9]

Quantum mechanical po- sition operator and localization in extended systems,

A. A. Aligia and G. Ortiz, “Quantum mechanical po- sition operator and localization in extended systems,” Phys. Rev. Lett.82, 2560–2563 (1999)

work page 1999

[10] [10]

How localized is an extended quantum system?

G. Ortiz and A.A. Aligia, “How localized is an extended quantum system?” physica status solidi (b)220, 737–743 (2000)

work page 2000

[11] [11]

Polar- ization and localization in insulators: Generating func- tion approach,

Ivo Souza, Tim Wilkens, and Richard M. Martin, “Polar- ization and localization in insulators: Generating func- tion approach,” Phys. Rev. B62, 1666–1683 (2000)

work page 2000

[12] [12]

Variational description of mott insulators,

Manuela Capello, Federico Becca, Michele Fabrizio, San- dro Sorella, and Erio Tosatti, “Variational description of mott insulators,” Phys. Rev. Lett.94, 026406 (2005)

work page 2005

[13] [13]

Reconstruction of the polar- ization distribution of the rice-mele model,

M. Yahyavi and B. Het´ enyi, “Reconstruction of the polar- ization distribution of the rice-mele model,” Phys. Rev. A95, 062104 (2017)

work page 2017

[14] [14]

Ground-state properties of the hydrogen chain: Dimerization, insulator-to-metal transition, and magnetic phases,

Mario Motta, Claudio Genovese, Fengjie Ma, Zhi-Hao Cui, Randy Sawaya, Garnet Kin-Lic Chan, Natalia Chep- iga, Phillip Helms, Carlos Jim´ enez-Hoyos, Andrew J. Mil- lis, Ushnish Ray, Enrico Ronca, Hao Shi, Sandro Sorella, Edwin M. Stoudenmire, Steven R. White, and Shi- wei Zhang (Simons Collaboration on the Many-Electron Problem), “Ground-state properties ...

work page 2020

[15] [15]

Order parameter to characterize valence-bond-solid states in quantum spin chains,

Masaaki Nakamura and Synge Todo, “Order parameter to characterize valence-bond-solid states in quantum spin chains,” Phys. Rev. Lett.89, 077204 (2002)

work page 2002

[16] [16]

Scaling of the po- larization amplitude in quantum many-body systems in one dimension,

Ryohei Kobayashi, Yuya O. Nakagawa, Yoshiki Fukusumi, and Masaki Oshikawa, “Scaling of the po- larization amplitude in quantum many-body systems in one dimension,” Phys. Rev. B97, 165133 (2018)

work page 2018

[17] [17]

Polariza- tion amplitude near quantum critical points,

Shunsuke C. Furuya and Masaaki Nakamura, “Polariza- tion amplitude near quantum critical points,” Phys. Rev. B99, 144426 (2019)

work page 2019

[18] [18]

Inequivalent berry phases for the bulk polarization,

Haruki Watanabe and Masaki Oshikawa, “Inequivalent berry phases for the bulk polarization,” Phys. Rev. X8, 021065 (2018)

work page 2018

[19] [19]

Bosonic in- teger quantum hall effect as topological pumping,

Masaya Nakagawa and Shunsuke Furukawa, “Bosonic in- teger quantum hall effect as topological pumping,” Phys. Rev. B95, 165116 (2017)

work page 2017

[20] [20]

Correlated zak in- sulator in organic antiferromagnets,

Takahiro Misawa and Makoto Naka, “Correlated zak in- sulator in organic antiferromagnets,” Phys. Rev. B108, L081120 (2023)

work page 2023

[21] [21]

Rigorous index theory for one-dimensional interacting topological insulators,

Hal Tasaki, “Rigorous index theory for one-dimensional interacting topological insulators,” Journal of Mathemat- ical Physics64, 041903 (2023)

work page 2023

[22] [22]

Quantized polarization in a generalized rice-mele model at arbitrary filling,

Yasuhiro Tada, “Quantized polarization in a generalized rice-mele model at arbitrary filling,” Phys. Rev. B110, 045113 (2024)

work page 2024

[23] [23]

Topological phase transition and𭟋 2 index fors= 1 quantum spin chains,

Hal Tasaki, “Topological phase transition and𭟋 2 index fors= 1 quantum spin chains,” Phys. Rev. Lett.121, 140604 (2018)

work page 2018

[24] [24]

Identifica- tion of gapless phases by squaring a twist operator,

Hang Su, Yuan Yao, and Akira Furusaki, “Identifica- tion of gapless phases by squaring a twist operator,” ArXiv:2506.02496

work page arXiv

[25] [25]

Two soluble models of an antiferromagnetic chain,

Elliott Lieb, Theodore Schultz, and Daniel Mattis, “Two soluble models of an antiferromagnetic chain,” Annals of Physics16, 407–466 (1961)

work page 1961

[26] [26]

A proof of part of hal- dane’s conjecture on spin chains,

Ian Affleck and Elliott Lieb, “A proof of part of hal- dane’s conjecture on spin chains,” Letters in Mathemat- ical Physics12, 57–69 (1986)

work page 1986

[27] [27]

See supplemental materials

work page

[28] [28]

Density matrix formulation for quan- tum renormalization groups,

Steven R. White, “Density matrix formulation for quan- tum renormalization groups,” Phys. Rev. Lett.69, 2863– 2866 (1992)

work page 1992

[29] [29]

The ITensor Software Library for Ten- sor Network Calculations,

Matthew Fishman, Steven R. White, and E. Miles Stoudenmire, “The ITensor Software Library for Ten- sor Network Calculations,” SciPost Phys. Codebases , 4 (2022)

work page 2022

[30] [30]

Efficient nu- merical simulations with Tensor Networks: Tensor Net- work Python (TeNPy),

Johannes Hauschild and Frank Pollmann, “Efficient nu- merical simulations with Tensor Networks: Tensor Net- work Python (TeNPy),” SciPost Phys. Lect. Notes , 5 (2018)

work page 2018

[31] [31]

mvmc–open-source software for many-variable varia- tional monte carlo method,

Takahiro Misawa, Satoshi Morita, Kazuyoshi Yoshimi, Mitsuaki Kawamura, Yuichi Motoyama, Kota Ido, Takahiro Ohgoe, Masatoshi Imada, and Takeo Kato, “mvmc–open-source software for many-variable varia- tional monte carlo method,” Computer Physics Commu- nications235, 447–462 (2019)

work page 2019

[32] [32]

Note on a theorem of bloch concerning possi- ble causes of superconductivity,

D. Bohm, “Note on a theorem of bloch concerning possi- ble causes of superconductivity,” Phys. Rev.75, 502–504 (1949)

work page 1949

[33] [33]

Two no-go theo- rems on superconductivity,

Yasuhiro Tada and Tohru Koma, “Two no-go theo- rems on superconductivity,” J. Stat. Phys.165, 455–470 (2016)

work page 2016

[34] [34]

Topological approach to luttinger’s theorem and the fermi surface of a kondo lattice,

Masaki Oshikawa, “Topological approach to luttinger’s theorem and the fermi surface of a kondo lattice,” Phys. Rev. Lett.84, 3370–3373 (2000)

work page 2000

[35] [35]

Lieb-schultz-mattis theorem in higher dimensions from approximate magnetic translation sym- metry,

Yasuhiro Tada, “Lieb-schultz-mattis theorem in higher dimensions from approximate magnetic translation sym- metry,” Phys. Rev. Lett.127, 237204 (2021)

work page 2021

[36] [36]

Projectification of point group symmetries with a background flux and lieb-schultz-mattis theorem,

Yasuhiro Tada and Masaki Oshikawa, “Projectification of point group symmetries with a background flux and lieb-schultz-mattis theorem,” ArXiv:2505.00927

work page arXiv

[37] [37]

Solitons in polyacetylene,

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett.42, 1698–1701 (1979). 6 SUPPLEMENT AL MA TERIALS A. One-dimensional fermions We consider interacting fermions in one-dimension, H= L−1X i=0 tc† i ci+1 + (h.c.) + X i,k Vknini+k,(S1) whereLis assumed to beL∈3Z. We suppose that the filling isν=N/L= 1/3 and the interac...

work page 1979