PhD research

My PhD work focuses on time-reversal-symmetry-breaking superconductors: materials where the superconducting state does more than simply conduct without resistance. It also breaks an important symmetry of the underlying equations, opening a path to rich microscopic structure and experimentally observable signatures.

What the work is about

The core task is to build microscopic theories that are physically motivated enough to say something real about experiment, rather than remaining abstract formal exercises.

That means:

• constructing models of unconventional superconducting order
• understanding how symmetry breaking appears in those models
• relating the theory back to measurable behaviour

Why it mattered beyond the thesis

Working at that level made a separate problem impossible to ignore. Scientific thinking is often split across too many tools, too many formats, and too much friction between writing, calculation, and collaboration.

That is a large part of why my research life expanded into software and systems-building. QuantaLumin, TutorLumin, and LuminOS all came from the same pressure: to make serious intellectual work more coherent.

• Research overview
• Software
• About

The background chapters establish the conventional superconducting baseline, then narrow toward the thesis mechanism: TRSB from internal winding in multicomponent superconductors, the BdG and topological language used to analyse microscopic representatives, and the low-energy phase dynamics of coupled superconducting phases.

This thesis now supports a clear final boundary. Experimentally, both LaNiC${}_2$ and LaNiGa${}_2$ show superconducting time-reversal-symmetry breaking. The materials-faithful programme developed here therefore asks whether the minimal microscopic TRSB mechanisms advanced in the thesis can recover that fact on shared Wannier bases rather than only in reduced toy models.

The answer is scientifically useful even where it is negative. On the dense Fermi-aligned Wannier bases, the present reduced closure set does not recover TRSB as the thermodynamic ground state. For LaNiC${}_2$, the seeded mixed-parity branch collapses onto the singlet control. For LaNiGa${}_2$, interorbital unitary and nonunitary TRSB branches remain as self-consistent solutions, but both lie above the singlet control in free energy. This is not a demonstration that the materials are experimentally singlet superconductors. It is a demonstration that the current minimal materials-faithful closures do not yet explain the observed TRSB.

That distinction also clarifies the status of the two microscopic pictures developed across the thesis. The nonunitary multiorbital triplet mechanism has now been tested in a minimal materials-faithful form, and on the present reduced basis it is not selected as the preferred state. The singlet frustration or loop-supercurrent mechanism is not yet falsified by the material calculation, because the chapter-15 comparison used a conventional singlet control rather than the full frustrated multicomponent winding closure on the imported Wannier basis. The unified-theory chapter therefore remains a conceptual synthesis of the two languages, while a direct material-faithful realization of that equivalence remains open.

That is an appropriate place for a PhD thesis to stop. The thesis has established a reusable BdG and qttree framework, a real QE to Wannier to materials-study pipeline, a set of topological and reduced-model consequence studies, and a falsifiable materials comparison that sharply narrows what still needs to be explained. The remaining work is a coherent postdoctoral programme: richer closure families on the imported bases, explicit frustrated singlet loop closures in the materials setting, broader low-energy orbital reductions, and tighter quantitative comparison to experiment. Those are no longer missing foundations. They are the next research programme.

This tree holds thesis-facing result chapters only: cleaned narrative text, selected figures, and stable data artifacts. Shared reusable code lives in the common qttree package, while broader exploratory or supporting computational material is kept outside the thesis narrative.

This section is the thesis-facing result narrative. It is intentionally limited to three chapters. The computational notebooks and reusable implementations live in QuLab; this section keeps the argument, the selected evidence, and the conclusions.

1. Majorana Resonance Study in a 2D SSH Extension with Soft Walls develops the topology-led boundary problem. A Weyl–SSH descendant model is used as a controlled laboratory for how Majorana-like near-zero spectral weight changes when a clean hard edge is replaced by a tunable impurity wall.
2. Internally Antisymmetric Non-Unitary Spin-Triplet Pairing Theory of LaNiX2 {X=C,Ga} tests the proposed internally antisymmetric nonunitary triplet mechanism for LaNiC2 and LaNiGa2. The chapter separates robust BdG diagnostics from the harder question of whether local interactions or materials-faithful Hamiltonians actually select the state.
3. Microscopic Theories of Time-Reversal Symmetry Breaking in Superconductors through Loop Supercurrent Ordering collects the loop-supercurrent mechanism, the branch language, the Josephson-island implementation route, the observable estimates, and the positive and negative microscopic-selection lessons into one coherent chapter.

Together these chapters cover the current qulab.research results as a thesis story rather than as a research-log index.

This section keeps thesis-level technical material without interrupting the main argument. Appendices should be narrow, cited from the chapter that uses them, and limited to derivations, reference tables, or supplementary constructions needed by the defended thesis.

This thesis took shape over a long period of reading, modelling, coding, and revision. I am grateful to the supervisors, collaborators, colleagues, and friends who helped sharpen the argument, challenged the weaker ideas, and kept the work moving when it would have been easier to leave it unfinished.

I am grateful to the University of Kent, and to the School of Engineering, Mathematics and Physics in particular, for the institutional setting in which this work was carried out. I also owe thanks to the wider condensed-matter and superconductivity community whose papers, preprints, seminars, and conversations provided both the materials context and the conceptual pressure that pushed this thesis toward its final form.

Finally, I thank my family and those close to me for their patience throughout the long and uneven process of finishing this work.

Adaptive Phase-Space Expansion Plan (archive note)

The live adaptive-scan workflow for this result family is now maintained in the canonical Notebook bundle:

~/Projects/Research/Notebook/content/unconventional-superconductivity/frustration-mediated-loop-supercurrent

That Notebook bundle retains code_40_adaptive_phase_space_expansion.py, the publishable and quick-run parameter presets, and the operational notes for targeted hybrid-global checks. The PhD tree now keeps only thesis-facing chapters, figures, and stable artifacts.

Internal Maintainer Notes (Not Thesis Content)

This directory is a thesis content tree; keep workflow notes here for maintainers/agents only.

To attach a missing PDF to an existing Papis entry (library papers), use:

1papis -l papers addto --doc-folder /home/henry/Resources/Papers/<papis_id> -f /path/to/file.pdf --file-name '{doc[ref]}.pdf'

Replace <papis_id> with the entry folder id and /path/to/file.pdf with the downloaded file path.

AGENTS.md

Guidelines for automated agents (and humans) modifying manuscripts intended for the Physical Review family of journals, with Physical Review Letters (PRL)–specific rules called out explicitly.

Primary references:

• Physical Review Style and Notation Guide (styleguide-pr.pdf)
• PRL Information for Authors (journals.aps.org/prl/authors)

This repository’s manuscript sources should follow the conventions below.

1) General principles

• Prefer APS/Physical Review conventions over local preferences.
• Keep style choices consistent within a manuscript (notation, capitalization, abbreviations, units).
• Avoid introducing new terminology or abbreviations unless required; define unavoidable acronyms on first use.
• Treat mathematical expressions as part of sentences (punctuate and capitalize accordingly).

2) Manuscript parts and ordering

Maintain the standard Physical Review ordering unless the target journal requires otherwise:

• Title
• Author list + affiliations + byline footnotes
• Abstract (see PRL rule below)
• Main text
• Acknowledgments
• Optional: author contributions / disclaimers / conflict-of-interest / ethics / data availability (see sections below)
• References
• Appendixes (PRL differs—see §8)

3) Titles

• Titles must be self-contained, simple, and concise.
• Avoid nonstandard abbreviations, acronyms, and terminology.
• Do not start titles with unnecessary leading words (e.g., “A”, “An”, “The”, “On”).

PRL title capitalization

• PRL uses Title Case: capitalize the first letter of each word except conjunctions, prepositions, and articles (unless preceded by a colon or em dash).

(Non-PRL Physical Review journals commonly use sentence case; keep the repository’s target-journal setting consistent.)

4) Author names, affiliations, and byline footnotes

• Use a consistent author name form across publications (full first names recommended where possible).
• Collaboration/group names may appear in parentheses between the author list and the institution list.
• Byline footnotes are for contact/locator information (e.g., “Present address: …”, “Also at …”, “Deceased.”).
• End byline footnotes with a period except for email addresses and URLs.

5) Abstracts

• Abstracts should be concise and proportional to article length.
• PRL: abstract length is limited to ≤ 600 characters.
• PRL Comments/Replies: do not require abstracts. Errata do not require abstracts.

6) Headings and section structure

6.1 Physical Review (general) heading levels

Use the journal’s standard hierarchy and format headings consistently.

6.2 PRL heading style (important)

PRL generally uses run-in headings, not freestanding headings:

• Run-in heading: paragraph indent, italic heading, first word capitalized, followed by an em dash, then text.
  • Example: Introduction—Text follows here

If a further level is used:

• Paragraph indent, roman heading, colon, em space, then text.
  • Example: Global fit: Text…

Theorems/lemmas/proofs:

• The leading single word (e.g., “Theorem 1”, “Proof”) may be italic on first appearance, but do not italicize long multi-sentence blocks.

7) Citations, references, and numbering

7.1 Consecutive numbering

• In the body of the paper, cite references, figures, and tables consecutively in numerical order.

7.2 Reference callouts (Physical Review conventions)

• For PRL (and PRA/PRC/PRD/PRE and other “Letters” styles), references typically appear as on-line numerals in square brackets:
  • Example: … as shown in Ref. [1].
• Place bracketed reference numbers inside punctuation where appropriate and ensure spacing from the preceding word/symbol.

7.3 Figures and tables

• Figures: Arabic numerals (1, 2, 3, …). Multi-part figures labeled (a), (b), … and cited as Fig. 1(a).
• Tables: Roman numerals (I, II, III, …).

8) Equations and mathematical material

• Display equations that are important, long, complex, or referenced later; keep only the simplest expressions inline.
• Number displayed equations consecutively with Arabic numerals in parentheses: (1), (2), (3), ….
• Place equation numbers flush to the extreme right of the equation line.
• Ensure punctuation of equations matches the surrounding sentence (commas/periods, etc.).

9) Units, abbreviations, hyphenation, and symbols

9.1 Abbreviations

• Single-word abbreviations: lowercase, usually unpunctuated (e.g., obs, av).
• Acronyms: initial letters of a phrase, roman letters (e.g., DWBA, bcc).
• Define acronyms at first occurrence unless truly standard for the target audience.

9.2 Hyphenation

• Avoid hyphens that serve no useful purpose (e.g., prefer cutoff, not cut-off; output, not out-put).
• Prefixes/suffixes are usually closed up (e.g., nonradioactive), but use hyphens where closing creates ambiguity or awkward doubling, or when attaching to proper nouns (e.g., non-Fermi).

9.3 Symbols in text and math

• Use Greek letters as symbols rather than spelling them out when they function as variables/symbols.
• Use upright (roman) type for English words, standard abbreviations, chemical symbols, and many multi-letter abbreviations.

10) Figures and tables quality

• Ensure all text in figures is readable at final journal size.
• Use consistent axis labeling and significant figures; include a leading zero (e.g., 0.2, not .2).
• Prefer accessible color palettes for online figures.

11) Data availability statements

APS journals strongly encourage sharing data/code/software that support results.

PRL placement and styling

• Place the data availability statement after the Acknowledgments (and any contribution/disclaimer paragraphs) and before References/Appendixes.
• Use a run-in heading:
  • Data availability—Text follows…
• Add a reference to the dataset/software in the reference list and cite it in the statement (include creators, year, repository, and persistent identifier such as a DOI).

12) Appendixes (PRL differs)

• Most Physical Review journals: appendixes appear after Acknowledgments and before References.

PRL

• Appendixes appear after the References in an End Matter section.
• Heading style examples:
  • Appendix—Text… (single appendix)
  • Appendix A—Text… (multiple appendixes)
  • Appendix: Survey of results—Text… (single appendix with subtitle)

13) What automated agents must do (repo hygiene)

When changing manuscript sources:

1. Preserve journal-specific formatting (especially PRL run-in headings, abstract character limit, appendix placement).
2. Keep all numbering consistent:
   • reference order, figure order, table order, equation numbers.
3. Do not introduce style drift:
   • capitalization rules, hyphenation patterns, units, and notation must remain consistent.
4. Keep changes minimal:
   • avoid rewriting for voice unless explicitly requested; focus on correctness and APS style.
5. If you change references:
   • ensure every reference is cited in the text and appears in the correct numerical order.
6. If you modify figures/tables:
   • ensure captions and callouts remain consistent (Fig. 1(a), Table I, etc.).

14) Quick PRL checklist

• Title is in Title Case (PRL rule).
• Abstract ≤ 600 characters (PRL rule).
• Headings are run-in with em dash (Introduction—…) (PRL rule).
• Data availability uses run-in heading and is placed before references (PRL rule).
• Appendixes are after References with PRL appendix heading style (PRL rule).
• References/figures/tables cited consecutively and formatted correctly.

AGENTS.md

Guidelines for automated agents (and humans) modifying manuscripts intended for the Physical Review family of journals, with Physical Review Letters (PRL)–specific rules called out explicitly.

Primary references:

• Physical Review Style and Notation Guide (styleguide-pr.pdf)
• PRL Information for Authors (journals.aps.org/prl/authors)

This repository’s manuscript sources should follow the conventions below.

1) General principles

• Prefer APS/Physical Review conventions over local preferences.
• Keep style choices consistent within a manuscript (notation, capitalization, abbreviations, units).
• Avoid introducing new terminology or abbreviations unless required; define unavoidable acronyms on first use.
• Treat mathematical expressions as part of sentences (punctuate and capitalize accordingly).

2) Manuscript parts and ordering

Maintain the standard Physical Review ordering unless the target journal requires otherwise:

• Title
• Author list + affiliations + byline footnotes
• Abstract (see PRL rule below)
• Main text
• Acknowledgments
• Optional: author contributions / disclaimers / conflict-of-interest / ethics / data availability (see sections below)
• References
• Appendixes (PRL differs—see §8)

3) Titles

• Titles must be self-contained, simple, and concise.
• Avoid nonstandard abbreviations, acronyms, and terminology.
• Do not start titles with unnecessary leading words (e.g., “A”, “An”, “The”, “On”).

PRL title capitalization

• PRL uses Title Case: capitalize the first letter of each word except conjunctions, prepositions, and articles (unless preceded by a colon or em dash).

(Non-PRL Physical Review journals commonly use sentence case; keep the repository’s target-journal setting consistent.)

4) Author names, affiliations, and byline footnotes

• Use a consistent author name form across publications (full first names recommended where possible).
• Collaboration/group names may appear in parentheses between the author list and the institution list.
• Byline footnotes are for contact/locator information (e.g., “Present address: …”, “Also at …”, “Deceased.”).
• End byline footnotes with a period except for email addresses and URLs.

5) Abstracts

• Abstracts should be concise and proportional to article length.
• PRL: abstract length is limited to ≤ 600 characters.
• PRL Comments/Replies: do not require abstracts. Errata do not require abstracts.

6) Headings and section structure

6.1 Physical Review (general) heading levels

Use the journal’s standard hierarchy and format headings consistently.

6.2 PRL heading style (important)

PRL generally uses run-in headings, not freestanding headings:

• Run-in heading: paragraph indent, italic heading, first word capitalized, followed by an em dash, then text.
  • Example: Introduction—Text follows here

If a further level is used:

• Paragraph indent, roman heading, colon, em space, then text.
  • Example: Global fit: Text…

Theorems/lemmas/proofs:

• The leading single word (e.g., “Theorem 1”, “Proof”) may be italic on first appearance, but do not italicize long multi-sentence blocks.

7) Citations, references, and numbering

7.1 Consecutive numbering

• In the body of the paper, cite references, figures, and tables consecutively in numerical order.

7.2 Reference callouts (Physical Review conventions)

• For PRL (and PRA/PRC/PRD/PRE and other “Letters” styles), references typically appear as on-line numerals in square brackets:
  • Example: … as shown in Ref. [1].
• Place bracketed reference numbers inside punctuation where appropriate and ensure spacing from the preceding word/symbol.

7.3 Figures and tables

• Figures: Arabic numerals (1, 2, 3, …). Multi-part figures labeled (a), (b), … and cited as Fig. 1(a).
• Tables: Roman numerals (I, II, III, …).

8) Equations and mathematical material

• Display equations that are important, long, complex, or referenced later; keep only the simplest expressions inline.
• Number displayed equations consecutively with Arabic numerals in parentheses: (1), (2), (3), ….
• Place equation numbers flush to the extreme right of the equation line.
• Ensure punctuation of equations matches the surrounding sentence (commas/periods, etc.).

9) Units, abbreviations, hyphenation, and symbols

9.1 Abbreviations

• Single-word abbreviations: lowercase, usually unpunctuated (e.g., obs, av).
• Acronyms: initial letters of a phrase, roman letters (e.g., DWBA, bcc).
• Define acronyms at first occurrence unless truly standard for the target audience.

9.2 Hyphenation

• Avoid hyphens that serve no useful purpose (e.g., prefer cutoff, not cut-off; output, not out-put).
• Prefixes/suffixes are usually closed up (e.g., nonradioactive), but use hyphens where closing creates ambiguity or awkward doubling, or when attaching to proper nouns (e.g., non-Fermi).

9.3 Symbols in text and math

• Use Greek letters as symbols rather than spelling them out when they function as variables/symbols.
• Use upright (roman) type for English words, standard abbreviations, chemical symbols, and many multi-letter abbreviations.

10) Figures and tables quality

• Ensure all text in figures is readable at final journal size.
• Use consistent axis labeling and significant figures; include a leading zero (e.g., 0.2, not .2).
• Prefer accessible color palettes for online figures.

11) Data availability statements

APS journals strongly encourage sharing data/code/software that support results.

PRL placement and styling

• Place the data availability statement after the Acknowledgments (and any contribution/disclaimer paragraphs) and before References/Appendixes.
• Use a run-in heading:
  • Data availability—Text follows…
• Add a reference to the dataset/software in the reference list and cite it in the statement (include creators, year, repository, and persistent identifier such as a DOI).

12) Appendixes (PRL differs)

• Most Physical Review journals: appendixes appear after Acknowledgments and before References.

PRL

• Appendixes appear after the References in an End Matter section.
• Heading style examples:
  • Appendix—Text… (single appendix)
  • Appendix A—Text… (multiple appendixes)
  • Appendix: Survey of results—Text… (single appendix with subtitle)

13) What automated agents must do (repo hygiene)

When changing manuscript sources:

1. Preserve journal-specific formatting (especially PRL run-in headings, abstract character limit, appendix placement).
2. Keep all numbering consistent:
   • reference order, figure order, table order, equation numbers.
3. Do not introduce style drift:
   • capitalization rules, hyphenation patterns, units, and notation must remain consistent.
4. Keep changes minimal:
   • avoid rewriting for voice unless explicitly requested; focus on correctness and APS style.
5. If you change references:
   • ensure every reference is cited in the text and appears in the correct numerical order.
6. If you modify figures/tables:
   • ensure captions and callouts remain consistent (Fig. 1(a), Table I, etc.).

14) Quick PRL checklist

• Title is in Title Case (PRL rule).
• Abstract ≤ 600 characters (PRL rule).
• Headings are run-in with em dash (Introduction—…) (PRL rule).
• Data availability uses run-in heading and is placed before references (PRL rule).
• Appendixes are after References with PRL appendix heading style (PRL rule).
• References/figures/tables cited consecutively and formatted correctly.

INT manuscript revision changelog

Changed

• Expanded the Hubbard-Kanamori definitions, INT tensor notation, nonunitarity diagnostic, projection argument, Kanamori feedback description, numerical settings, and reproducibility note in main.tex.
• Added publication figures:
  • figures/self_consistent_convergence_example.png
  • figures/wannier_band_validation.png
• Updated publication index metadata to name the LaNiX2 (X = Ga, C)-inspired scope explicitly.

Data and scripts used

• Figure generation code: henry/qulab/qulab/research/int/scripts/generate_figures.py.
• Convergence data source: qulab.core.scmft.benchmarks.self_consistent.repaired_int_kanamori.repaired_int_seed_comparison_benchmark.
• Wannier validation data source: henry/qulab/qulab/research/int/data/lanix2_wannier/LaNiGa2_full_soc_icarus/qe_bands_validation_aligned.json.
• Material Hamiltonians:
  • LaNiC2_zhang2018_soc_icarus/wannier90_hr.dat
  • LaNiGa2_full_soc_icarus/wannier90_hr.dat

TODO / missing data

• Full high-symmetry DFT-vs-Wannier validation for the active LaNiGa2 SOC basis remains missing.
• The convergence history stores channel weights, residuals, and free energy; it does not store literal per-iteration Delta_upup and Delta_downdown amplitudes.
• Full-basis unrestricted Kanamori Hartree/Fock feedback remains a production follow-up rather than a completed figure in this manuscript.

Assumptions

• The code default U'=U-2J_H and J_P=J_H is the intended Kanamori convention unless explicitly overridden.
• The LaNiGa2 Gamma-Z validation overlay is a survey-level smoke check, not publication-grade full-path validation.

This chapter presents a general framework for building the most general quadratic Hamiltonian consistent with the physical structure of the problem. That structure includes lattice translations, boundaries, inhomogeneous geometries, and defects; optional particle-hole (Nambu) doubling; internal structure such as sublattices, orbitals, and intra-cell positions; spin; and whatever set of symmetry generators is imposed for the model under consideration.

The same formalism is intended to cover both single-order and multi-order settings, including spatial textures, flux- or current-like phases, and regimes in which different orders either compete or coexist. In that sense, the aim of the chapter is not only to write down particular Hamiltonians, but to establish a general construction that can be applied across the later microscopic models in the thesis.

We also explain how this microscopic framework connects to, and generalises, the more macroscopic symmetry-based approach associated with Ginzburg and Landau.

Unified structure: lattice shifts ⊗ Nambu ⊗ internal ⊗ spin

Hilbert-space factorization and index order

We fix the tensor-product order

All operators are written to respect this order. When Nambu space is absent, the factor ${\mathcal{H}}_{\text{Nambu}}$ is simply omitted and the formulas are interpreted in the normal, non-doubled space.

Shift operators as the lattice backbone (real space)

Definition of the shift symbol and the “/𝕄” convention

Fix a finite lattice with shape $\mathbf{N}$ and boundary-condition label $\mathrm{bc}$. Let 𝕄 be a defect/mask operator acting on ${\mathcal{H}}_{\text{lat}}$.

Start from the “pure translation” shift 𝕊${}_{\mathbf{\delta }}^{(\mathbf{N},\mathrm{bc})}$ defined by its action on site kets $|\mathbf{R}⟩$:

where $\mathrm{bc}$ specifies how $\mathbf{R}+\mathbf{\delta }$ is interpreted at the boundary.

To model inhomogeneous geometry, impurities, vacancies, or any other spatial inhomogeneity, let 𝕄 be a fixed site mask: a diagonal operator in the site basis with eigenvalues $m_{\mathbf{R}}\in \{0,1\}$, selecting the active lattice degrees of freedom.

This equation defines the slash notation “$/\mathbb{M}$”: the shift only connects active sites. Equivalently, one may work directly in the restricted active-site subspace. The same masking can be written compactly as:

Embedding into the full Hilbert space

Embed the masked shift into the full Hilbert space using the fixed order:

Accordingly, any quadratic term may be written as a sum of objects of the form (shift on ${\mathcal{H}}_{\text{lat}}$) ⊗ (matrix on ${\mathcal{H}}_{\text{Nambu}}\otimes {\mathcal{H}}_{\text{int}}\otimes {\mathcal{H}}_{\text{spin}}$).

When translation is a good quantum number: 𝕊 becomes the unitary phase $e^{i\mathbf{k}\cdot \mathbf{\delta }}$

If the lattice is translation-invariant (typically periodic $\mathrm{bc}$ and no defects so $\mathbb{M}={\mathrm{𝟙}}_{\text{lat}}$), define Bloch plane-wave kets via the discrete Fourier transform 𝓕:

Then each shift is diagonal in the $\mathbf{k}$ basis:

So any translation-invariant lattice Hamiltonian written in the shift notation,

where $\mathcal{D}$ is a chosen (typically finite) set of lattice displacement vectors $\mathbf{\delta }$ connecting sites/unit cells (e.g. n.n., n.n.n., etc.). For normal (number-conserving) hopping terms, ${\mathbb{T}}_{\mathbf{\delta }}$ is understood to act trivially on Nambu space, i.e. ${\mathbb{T}}_{\mathbf{\delta }}={\mathrm{𝟙}}_{\text{Nambu}}\otimes t_{\mathbf{\delta }}$ with $t_{\mathbf{\delta }}$ acting on ${\mathcal{H}}_{\text{int}}\otimes {\mathcal{H}}_{\text{spin}}$; spin dependence can encode spin-orbit coupling, whereas spin-independent hopping corresponds to $t_{\mathbf{\delta }}\propto {\mathrm{𝟙}}_{\text{spin}}$. All coefficients are independent of $\mathbf{R}$ for a translation-invariant model, becomes block-diagonal in $\mathbf{k}$:

so the lattice part has reduced to the unitary phase factors $e^{i\mathbf{k}\cdot \mathbf{\delta }}$ (the lattice representation of translations, i.e. Bloch’s theorem) [1, 2].

In 1D with lattice spacing $a$ and nearest-neighbour shifts $\delta =\pm a$, the basic unitaries are $e^{\pm ika}$. Restricting $\mathcal{D}$ to nearest neighbours (n.n.), next-nearest neighbours (n.n.n.), etc. amounts to retaining a finite set of displacement vectors $\mathbf{\delta }$, and therefore produces a trigonometric polynomial in momentum. For example, a single-band 1D tight-binding model with hopping amplitudes $t_n$ to the $n$th neighbour has

On a square lattice with spacing $a$, keeping only n.n. hopping $t$ gives the familiar cosine dispersion

while adding an n.n.n. hopping $t^{\prime }$ contributes additional harmonics, e.g. $-4t^{\prime }\cos (k_{x}a)\cos (k_{y}a)$.

In multi-orbital models the same structure holds, but $t_n$ and $t,t^{\prime }$ are replaced by matrices ${\mathbb{T}}_{\mathbf{\delta }}$ acting on ${\mathcal{H}}_{\text{int}}$ (and possibly spin/Nambu). The resulting Bloch Hamiltonian is a matrix-valued trigonometric polynomial; in particular,

so cosine terms typically appear in Hermitian (even) hopping channels, whereas sine terms commonly appear in antisymmetric or purely imaginary (odd) inter-orbital couplings (e.g. hybridization terms). For example, an inter-orbital hopping channel along $\hat{x}$ with ${\mathbb{T}}_{\pm a\hat{x}}\propto \pm i|1⟩⟨2|$ produces an off-diagonal Bloch matrix element proportional to $\sin (k_{x}a)$.

Intra-cell position phases and a clean Bloch convention

The discussion above accounts for the Bravais-lattice (unit-cell translation) part of Bloch’s theorem, where shifts contribute factors $e^{i\mathbf{k}\cdot \mathbf{\delta }}$. For multi-orbital/unit-cell models there is an additional, purely conventional choice: whether intra-cell orbital positions ${\mathbf{r}}_a$ are included explicitly in the Bloch basis. The following gauge transformation implements that convention, converting a “naive” $\mathbf{h}(\mathbf{k})$ built only from $\mathbf{\delta }$ into the corresponding cell-periodic (orbital-position-aware) Bloch Hamiltonian.

When orbitals sit at different intra-cell positions ${\mathbf{r}}_a$, define the diagonal phase operator on internal space

Concretely, let $|\mathbf{R},a⟩:=|\mathbf{R}⟩\otimes |a⟩$ denote the real-space orbital basis (and, if present, include spin as an additional tensor factor $|\sigma ⟩$). A standard orbital Bloch basis at fixed $\mathbf{k}$ is

with $N_c$ the number of unit cells. The “position-phase” (cell-periodic) convention instead attaches the intra-cell phase,

In this notation, $\mathbb{D}(\mathbf{k})$ is precisely the internal-space unitary that maps between the two orbital Bloch conventions. At the level of operators, let $c_{\mathbf{R},a}$ annihilate an electron in orbital $a$ of unit cell $\mathbf{R}$. The corresponding orbital Bloch operators are

If ${\mathbf{h}}^{\text{(naive)}}(\mathbf{k})$ is built ignoring intra-cell positions, the “cell-periodic gauge” convention is

This is a change of basis within the orbital/sublattice Bloch basis (a $\mathbf{k}$-dependent gauge choice), not a change of momentum. The band basis is obtained separately by diagonalizing $\mathbf{h}(\mathbf{k})$ at each $\mathbf{k}$, i.e. by a unitary $U(\mathbf{k})$ such that

and defining band states $|\mathbf{k},n⟩:={\sum }_a|\mathbf{k},a{⟩}_{\text{cp}}{U}_{an}(\mathbf{k})$ (with the same construction for the corresponding field operators). Equivalently, the band annihilation operators are ${\gamma }_{\mathbf{k},n}:={\sum }_a{U}_{an}^{\ast }(\mathbf{k})c_{\mathbf{k},a}^{\text{(cp)}}$, i.e. ${\mathbf{\gamma }}_{\mathbf{k}}=U^{†}(\mathbf{k}){\mathbf{c}}_{\mathbf{k}}^{\text{(cp)}}$.

In particular, each band eigenstate at fixed $\mathbf{k}$ is a superposition of the orbital basis states in the same $\mathbf{k}$ sector, with coefficients given by the eigenvectors of $\mathbf{h}(\mathbf{k})$. For example, if the internal space consists of four sites arranged in a ring within the unit cell and only intra-cell hopping is present, then $U(\mathbf{k})$ can be chosen $\mathbf{k}$-independent and reduces to a discrete Fourier transform on the ring, so the band index $n$ may be identified with a discrete internal (cluster) momentum $q=2\pi \ell /4$ with $\ell =0,1,2,3$ (i.e. $q\in \{0,\pi /2,\pi ,3\pi /2\}$).

More generally, whenever the orbitals within a unit cell carry an internal discrete translation symmetry (e.g. a cyclic ordering of $M$ orbitals), one can define an internal translation operator $\mathbb{U}$ acting on ${\mathcal{H}}_{\text{int}}$ by its action on the orbital basis,

Its eigenstates are internal Bloch modes $|q⟩\propto {\sum }_{a=0}^{M-1}{e}^{iqa}|a⟩$ with eigenvalues $e^{iq}$, where $q=2\pi \ell /M$ and $\ell =0,\dots ,M-1$. If the internal couplings respect this symmetry (i.e. $[\mathbf{h}(\mathbf{k}),\mathbb{U}]=0$ for each $\mathbf{k}$), then $\mathbf{h}(\mathbf{k})$ can be block-diagonalized in $q$, and the band label can be taken as a pair $(q,\nu )$ (internal momentum plus residual band index within each $q$ sector).

This “position-phase” correction underlies consistent symmetry actions for multi-sublattice/orbital Bloch bases [3, 4].

Quadratic Hamiltonians in normal and Nambu form

General normal-state quadratic form

In the absence of Nambu doubling,

with ${\Psi }_{\mathbf{k}}$ valued in ${\mathcal{H}}_{\text{int}}\otimes {\mathcal{H}}_{\text{spin}}$.

Multiple normal orders, such as density waves, orbital order, and loop-current-like hopping patterns, enter additively:

Nambu-doubled quadratic form

If you include pairing, introduce a Nambu spinor ${\Phi }_{\mathbf{k}}$ valued in ${\mathcal{H}}_{\text{Nambu}}\otimes {\mathcal{H}}_{\text{int}}\otimes {\mathcal{H}}_{\text{spin}}$ and write

A standard block structure is

with $\mathbf{h}$ and $𝚫$ acting on ${\mathcal{H}}_{\text{int}}\otimes {\mathcal{H}}_{\text{spin}}$.

Multiple pairing orders are additive:

Particle–hole symmetry in BdG systems is intrinsic and constrains $\mathcal{H}$ accordingly [5].

Encoding textures, currents, and frustration

Loop/current/flux phenomena are encoded as phases attached to internal and/or link-resolved structures.

Site/orbital phase texture operator

Define a diagonal phase field on $(\mathbf{R},a)$ (acts on ${\mathcal{H}}_{\text{lat}}\otimes {\mathcal{H}}_{\text{int}}$):

Normal bilinears are dressed by conjugation:

Pairing-type bilinears are dressed by the transpose on the right:

Link-based phases

For bond-resolved terms

Loop currents correspond to nontrivial gauge-invariant loop products of these phases around cycles.

Competition vs coexistence from symmetry-allowed invariants

Given multiple order components $\{O_{\alpha }\}$ (normal or pairing), symmetry decides which invariants can appear in

and whether phase-sensitive terms that lock relative phases are allowed [6, 7].

Accordingly, the couplings are not introduced independently, but are derived from the generator constraints discussed below.

Generator-based construction of the most general symmetry-allowed quadratic Hamiltonian

The construction may be written in a form that will be used throughout the later models in the thesis.

Basis expansion

Let $\{{\tau }_{\ell }\}$ be a Pauli basis on ${\mathcal{H}}_{\text{Nambu}}$, $\{{\lambda }_i\}$ a Hermitian basis on ${\mathcal{H}}_{\text{int}}$, and $\{{\sigma }_j\}$ a Pauli basis on ${\mathcal{H}}_{\text{spin}}$.

In Nambu-doubled form

In the normal (non-Nambu) case

The scalar form factors $f(\mathbf{k})$ are lattice harmonics selected by symmetry.

Generator representations

For each generator $g$, construct $U_g(\mathbf{k})$ acting on ${\mathcal{H}}_{\text{Nambu}}\otimes {\mathcal{H}}_{\text{int}}\otimes {\mathcal{H}}_{\text{spin}}$ as

When Nambu space is absent, the factor $U_g^{(\text{Nambu})}$ and the $\tau$ expansion are omitted.

Constraints

For unitary spatial symmetries

or $\mathcal{H}\to \mathbf{h}$ in the normal case.

Time reversal (antiunitary), if imposed

and similarly for $\mathbf{h}$.

In BdG form, the intrinsic particle–hole constraint is

which is the basis for the standard symmetry classification of gapped free-fermion phases [5, 8, 9].

General form of the quadratic Hamiltonian

Quadratic model

or, when Nambu doubling is included,

Example: Friedel oscillations

A natural benchmark of the normal-state implementation is the response of a conductor to a single local impurity. In a clean metal the density is uniform, but a defect mixes states across the Fermi surface and produces oscillations with characteristic wavevector $2k_F$. It provides a natural introductory benchmark because it tests several parts of the framework at once: real-space masking, boundary conditions, impurity insertion, diagonalisation, and local observables such as the local density of states.

Friedel oscillations therefore serve as an early benchmark of Quantum Tensor Tree before we turn to self-consistent mean-field calculations. The calculation below is performed for a 2D square lattice with nearest-neighbour hopping, an open circular mask, and a single impurity at the origin. The oscillatory rings in the LDOS are the Friedel oscillations themselves, while the radial line cut shows that the observed period agrees with the expected value ${\lambda }_F/2=\pi /k_F$. This allows the analytical derivation to be compared directly with the numerical results. The same figure also exposes the weak lattice anisotropy that survives beyond the isotropic continuum approximation.

The canonical chapter-local Python benchmark driver for this example is friedel_qpi_native_figures.py. It is a thin wrapper over qttree benchmark construction and plotting, replacing the earlier Julia draft while still emitting the simple LDOS and QPI figures reused later in the methodology. The older tightbinding_lattice.py name is kept only as a compatibility alias while the thesis scripts are being consolidated.

The microscopic Hamiltonian used in the simulation is the normal-state tight-binding model

where the sum runs over nearest-neighbour pairs of active sites retained by the circular mask $\mathbb{M}$, so that the boundary is open, and the impurity is represented by a local onsite potential at the origin. The LDOS shown below is then

evaluated at $E=\mu$. In the present low-filling benchmark, this energy is identified with the Fermi level used in the analytical estimate. For the numerical example shown here, we take $L=41$, radius $R=20$, $t=1$, $V_{\mathrm{imp}}=3t$, $\mu =-3.81t$, and $\eta =0.01t$.

Analytical Derivation in $d$ dimensions

Take an isotropic normal state with quadratic dispersion

and a point impurity

The retarded Green’s function of the clean system is

For an isotropic continuum band this has the Hankel-function form

and therefore, for $k_{\omega }r\gg 1$,

To first order in the impurity strength,

so the correction to the LDOS is

At fixed energy, then, the impurity produces oscillations with wavelength $\pi /k_{\omega }$ and envelope $r^{-(d-1)}$.

If instead one integrates over occupied states to obtain the density modulation,

the extra oscillatory integral contributes one further power of $1/r$, so asymptotically

This is the general $d$-dimensional Friedel law for the integrated density. The often-quoted $1/r^d$ decay therefore refers to the density integrated to the Fermi level, whereas the LDOS measured at fixed energy decays one power more slowly.

In particular,

Specialisation to the low-filling square lattice

Near the bottom of the square-lattice band,

so the lattice model reduces to the continuum form above with effective mass $m^{\ast }=1/(2t{a}^2)$. In two dimensions one may therefore write

which at large distance reduces to

For the benchmark shown above, $\mu /t=-3.81$ places the Fermi level close to the band bottom, so the continuum estimate is already accurate:

The observed ring spacing agrees with this prediction, so the example provides a compact validation of the real-space geometry, impurity implementation, and LDOS evaluation that are used throughout the later numerical work.

Thesis Role

This chapter is the topology-led boundary result of the thesis. It asks whether a local impurity wall inside a periodic superconducting SSH system can be tuned into an effective internal boundary carrying the arc physics of the clean open model.

The answer is deliberately finite-device and model-based. The clean parent is a two-dimensional Weyl-SSH construction in which fixed (k_y) slices behave as SSH chains [10, 11, 12]. The wall calculation does not claim a materials-faithful model of LaNiX2, and it does not identify a thermodynamic quantum critical point. It shows how a finite wall transfers Weyl/Majorana-arc spectral weight from the bulk continuum to a near-zero internal-boundary branch, and how self-consistency turns the same wall into a suppression of the local anomalous field.

A journal-style version of this work is archived with this chapter: soft impurity walls PRB manuscript. An early version was presented at ExoSup 2022, the Cargese Summer School on Exotic Superconductivity.

Clean Weyl-SSH Parent

The normal-state model has two sublattices, intra- and intercell SSH hoppings (v) and (w), and a diagonal interchain hopping (t_d). For the translation-invariant parent, each (k_y) slice is an SSH chain along (x) with effective hoppings

For (\delta\epsilon=0), the slice winding is

so (\nu(k_y)=1) when (|w_1(k_y)|>|v_1(k_y)|) and zero otherwise. The stacked SSH model therefore supplies a (k_y)-dependent one-dimensional winding number. Inter-sublattice spinless pairing becomes an effective (p)-wave gap after projection onto the winding Weyl bands; forced zeros of that gap intersect the Fermi pockets to create Bogoliubov-Weyl/Majorana nodes; and the change of the fixed-(k_y) one-dimensional BdG invariant across those nodes produces Majorana arcs on an edge.

Internal Wall Geometry

The imposed-pairing BdG problem uses an inter-sublattice cell pairing (\Delta_{AB}=\Delta_0), usually (\Delta_0=0.3), in the Nambu basis ((c_A,c_B,c_A^\dagger,c_B^\dagger)^T). The wall is a scalar onsite potential

where (W_\ell) is a one-cell-thick support of length (\ell). The wall is inside a periodic torus, not an imposed open boundary. The real-space plots use centered unit-cell coordinates, so the plotted wall is at (x=0). Unless stated otherwise, real-space densities are cell-resolved sums over the (A) and (B) sites in each unit cell.

Fixed-(\Delta_0) Arc Transfer

The imposed-(\Delta_0) calculation isolates the quasiparticle boundary problem. For representative wall strengths, the wall-projected spectral function (A(k_y,E)) shows the boundary branch detach from the bulk response and approach the near-zero hard-wall arc.

To quantify the transfer, the calculation labels the branch by hard-wall ancestry rather than selecting a new local minimum at each parameter point. For each sampled (k_y), the tracker seeds the wall-weighted positive-energy mode at the largest simulated wall strength and continues that state downward in (V) by eigenvector overlap. When nearby eigenvalues have comparable single-vector overlaps, a small nearby-state subspace is used; ambiguous steps are logged and can be marked in the viewer. This makes the tracked branch a reproducible object, distinct from a purely visual ridge fit to (A(k_y,E)).

The (k_y=0) point of the tracked branch provides a finite-size order parameter for the wall-driven transfer. Let (\epsilon_{\mathrm{arc}}(0;V)) be the wall-projected ADOS branch energy at (k_y=0), i.e. the distance from (E=0) to the selected positive-energy peak. The normalized order parameter is

In practice the hard-wall reference is represented by the largest simulated wall strength. The (41\times41) full-wall data show a sharp finite-device onset between (V=2.5) and (V=3). This is physically useful: it marks the wall strength where the tracked branch changes from a bulk-scale spectral feature into a hard-wall descendant. It is not reported as a critical point. Window scans of the Landau-style form

do not give a stable (V_c) or (\beta): fits that include the jump pin (V_c) to the first fitted point, while fits beginning after the jump drive (V_c) to an unphysical lower bound. A thermodynamic quantum-critical interpretation would require a size-scaling collapse of (\Psi_{\mathrm{arc}}(V,L)) and a consistent gap-closing diagnostic, neither of which is obtained from the present data.

Localization And Majorana Character

The same transfer is visible in real space. The LDOS profiles below show wall-normal localization of selected spectral contributions at fixed (k_y) and energy. The component-resolved wavefunction shows the corresponding signed BdG amplitudes for the strong-wall near-zero mode. Together these diagnostics check that the tracked spectral branch is not merely a relabelled bulk eigenvalue.

In the BdG basis (\Psi=(c_A,c_B,c_A^\dagger,c_B^\dagger)^T), a wall eigenstate has spinor (\psi=(u_A,u_B,v_A,v_B)^T). Particle-hole symmetry pairs the state at energy (E) with a partner at (-E). At an exact Majorana zero mode the spinor can be gauge-fixed so that (v_\alpha=e^{i\phi}u_\alpha^\ast). The strong-wall state is therefore read as Majorana-like when the tracked branch approaches this particle-hole self-conjugate limit while remaining on the arc connecting the projected Weyl-node endpoints.

Self-Consistent Wall Feedback

The imposed calculation treats the wall as a scatterer in a fixed BdG background. The self-consistent calculation asks a stronger question: how does the anomalous field respond to the wall?

The interaction is decoupled through the local Gor’kov field

with the local mean-field pairing field

Here (\kappa_{AB}(r)) and (\Delta_{AB}(r)) are local inter-sublattice cell fields. The plots show (|\Delta_{AB}(r)|), or averages of this magnitude over wall and off-wall cell sets, so the displayed quantity is insensitive to the global sign convention for the real pairing gauge used in the calculation. This self-consistent field is distinct from the imposed constant (\Delta_0) used in the fixed-pairing BdG problem.

The self-consistent result is the main physical correction to the fixed-pairing picture. The wall is not only a scattering potential for quasiparticles. It also creates a local depression of the anomalous field: (|\Delta_{AB}(r)|) is strongly suppressed on the wall support while the off-wall condensate remains finite. Attempts to fit the wall mean to the same finite-(V_c) ansatz as the fixed-(\Delta_0) arc are unstable. The safer interpretation is wall-local depletion with an empirical algebraic tail over the simulated interval, with an exponent of order unity rather than a claimed critical exponent.

| (V) | wall mean (|\Delta_{AB}|) | off-wall mean (|\Delta_{AB}|) | | —: | —: | —: | | 1.0 | 0.318 | 0.331 | | 1.1 | 0.234 | 0.329 | | 1.2 | 0.139 | 0.326 | | 1.3 | 0.0846 | 0.324 | | 1.5 | 0.0480 | 0.321 | | 2.0 | 0.0271 | 0.327 | | 3.0 | 0.0150 | 0.337 |

Local Marker And Symmetry Controls

The clean winding invariant is exact only in the chiral slices of the translation-invariant model. The scalar onsite wall is a local chiral-symmetry-breaking perturbation because it contributes a same-sublattice onsite term on the wall cells. This does not invalidate the wall-transfer calculation; it clarifies what is being tested. The bulk away from the wall remains governed by the chiral SSH structure, while the wall locally breaks that symmetry and acts as a boundary-forming perturbation.

The local chiral marker is therefore used as a diagnostic, not as a final quantized invariant:

where (\Gamma=+1) on (A), (\Gamma=-1) on (B), and (Q=1-2P_-) is the flattened occupied-state projector. The comparison below shows that a scalar onsite wall and a chiral hopping cut are distinct boundary mechanisms.

Finite-Size Diagnostics

The current data support a finite-device arc-transfer crossover rather than a thermodynamic quantum critical point. The diagnostics below are kept in this chapter because they document the negative result: the onset can be sharp on a single device, but the fitted exponent and finite-size onset do not stabilize.

Conclusion

Soft impurity walls provide a controlled finite-device route from clean momentum-space topology to a real-space internal boundary in a superconducting SSH lattice. In the fixed-(\Delta_0) problem, the (k_y=0) branch begins as a bulk-scale spectral feature and moves toward the near-zero hard-wall branch as (V) grows. In the self-consistent problem, the same wall suppresses (\kappa_{AB}(r)) and (\Delta_{AB}(r)) on the wall support while preserving a finite off-wall condensate.

The conservative conclusion is a linked chain of evidence: clean slice winding, slab Majorana arcs, wall-projected spectral evolution, eigenvector-continuous arc tracking, real-space mode localization, and self-consistent suppression of the wall pairing field. The current evidence does not justify a reported critical exponent. A stronger quantum-critical claim would require stable finite-size scaling of the order parameter and the relevant quasiparticle gap, performed on the same selected states. Until then, the result is a sharp and useful finite-device onset of Weyl-arc transfer, not an identified thermodynamic phase transition.

Numerical data and figures were generated with the qulab.research.ssh_2d module in QuLab [13].

Introduction

Paper 1 established a microscopic loop-supercurrent route to time-reversal symmetry breaking (TRSB) by branch ranking in a current-channel BdG closure, with strict winding/circulation filters and a triplet-penalty decomposition. The objective here is a narrower continuation question: starting from a loop-favored winding texture, what transition structure is implied when the normal-state sector is extended by Rashba spin–orbit coupling and Zeeman splitting?

Within the full thesis hierarchy, this chapter should be read as an effective topological sidecar rather than as a literal materials model. Its role is to ask what kinds of class-D topological structure can be induced once a TRSB superconducting texture has already been selected microscopically. The later LaNiX${}_2$ materials chapter is where the normal-state Hilbert space is earned from crystallography, DFT, and Wannierization. The present chapter instead probes a reduced orbital block that is best interpreted as a topology-facing descendant of that programme, especially on the LaNiGa${}_2$ side where nonsymmorphic low-energy structure and boundary physics are part of the materials motivation.

This manuscript is intentionally scoped as an analytic-first topological probe (Tier 1). Its central deliverable is an explicit mass-inversion transition fan at the time-reversal invariant momenta (TRIM) and a corresponding piecewise-constant proxy index. Numerical results are deliberately restricted to targeted confirmations (representative bulk band cuts and strip spectra) across a matched proxy transition. We do not present full two-parameter Chern/gap heatmaps here.

Additional diagnostic (real-space topology in open geometry). In open-boundary settings, topology can be monitored using local real-space markers built from projectors (local Chern marker) and their associated redistribution/flow (“marker currents”). We do not study non-equilibrium marker-current dynamics here; we include only a static local-marker visualization as a companion to the strip spectrum.

Scope and Model Hierarchy

To keep claims clean, we separate three layers:

1. Established from paper 1: microscopic loop-branch selection in the current-channel model and the decomposition crossing controlled by diagonal triplet penalty.
2. New in this follow-up: fixed-pattern topological probe (same orbital structure, added SOC+Zeeman, fixed winding pairing texture) + an optional real-space local-marker view in open geometry.
3. Not claimed here: fully self-consistent re-optimization of all paper-1/current-channel mean fields after adding SOC+Zeeman.

Baseline from Paper 1 (context only)

The paper-1 Hamiltonian is

\begin{equation} H = H_0 + H_s + H_t, \end{equation} with onsite singlet pairing field

\begin{equation} \Delta_i = U_s\langle \hat c_{i\downarrow}\hat c_{i\uparrow}\rangle, \end{equation} and current-channel HS saddle

\begin{equation} a_{ij}=U_t\langle \hat J_{ij}\rangle,\qquad \hat J_{ij}=it\sum_\sigma(\hat c^\dagger_{i\sigma}\hat c_{j\sigma}-\hat c^\dagger_{j\sigma}\hat c_{i\sigma}). \end{equation} Branch ranking in paper 1 used

\begin{equation} F_{\rm eff}=F_{\rm BdG}+\Chi_{\rm trip}^{\rm edge}+\Chi_{\rm trip}^{\rm diag}, \end{equation} with strict loop acceptance criteria (e.g., (\Delta F_{\rm eff}<0), winding index (m=\pm1), circulation coherence).

[Insert paper-1 anchor figures with explicit provenance.]

Fixed-Texture SOC+Zeeman Extension

We add spinful SOC+Zeeman terms to the normal block while keeping a winding pairing texture motivated by the paper-1 loop branch.

A. Spinful normal block

\begin{equation} h_{\rm topo}(\mathbf{k}) = h_{\rm orb}(\mathbf{k})\otimes\sigma_0 + \lambda(\sin k_y,\sigma_x-\sin k_x,\sigma_y) + V_z\sigma_z, \end{equation}

with (\sigma_i) acting in spin space and (h_{\rm orb}(\mathbf k)) the orbital tight-binding block (parameters stated in figure captions and run logs).

Optional flux decoration (kept OFF in this paper). A natural multi-orbital extension is to allow Peierls phases on selected hoppings to represent engineered flux patterns in the orbital block. We define the bookkeeping for this extension (to keep codepaths consistent with Paper B), but set the flux parameter (\Phi=0) throughout Paper A.

B. BdG Hamiltonian

\begin{equation} \mathcal H_{\mathrm{BdG}}(\mathbf k)= \begin{pmatrix} h_{\rm topo}(\mathbf k) & \Delta \ \Delta^\dagger & -h_{\rm topo}^T(-\mathbf k) \end{pmatrix}. \end{equation}

The particle sector has dimension (4\times2=8) (orbital (\times) spin) and the full BdG matrix is (16\times16).

C. Fixed intracell winding pairing texture

Pairing is onsite singlet and orbital-diagonal with a fixed intracell winding pattern

\begin{equation} (\Delta_A,\Delta_B,\Delta_C,\Delta_D) = \Delta_0(1,e^{i\pi/2},e^{i\pi},e^{i3\pi/2}). \end{equation}

This texture is imposed (seeded from the loop program) and not re-optimized under SOC+Zeeman.

D. Conventions and sign governance (Hewitt integration)

We fix a single set of conventions for all reported spectra, proxy indices, and (optional) Chern/marker checks.

Real-space unit cell and orbital order. Each Bravais cell contains four orbitals in the order

\begin{equation} (A,B,C,D)\equiv (0,0),(1,0),(0,1),(1,1), \end{equation} and the Bloch orbital spinor is

\begin{equation} \hat c_{\mathbf k}=\big(\hat c_{\mathbf k,A},\hat c_{\mathbf k,B},\hat c_{\mathbf k,C},\hat c_{\mathbf k,D}\big)^T. \end{equation}

Brillouin-zone orientation and integration convention. We use ((k_x,k_y)\in[-\pi,\pi)\times[-\pi,\pi)) with the standard right-handed orientation. Whenever a Chern sign is referenced (e.g. for optional selected-point validation), it is computed with the oriented area element (dk_x\wedge dk_y). TRIM are ordered as (\Gamma=(0,0), X=(\pi,0), Y=(0,\pi), M=(\pi,\pi)).

High-symmetry path direction. Band plots use the directed path (\Gamma\to X\to M\to \Gamma) (unless explicitly stated otherwise in the caption/run log).

Spin and Nambu (BdG) basis ordering. The spinful particle basis is ordered as

\begin{equation} \hat c_{\mathbf k}= \big(\hat c_{\mathbf k,A\uparrow},\hat c_{\mathbf k,A\downarrow}, \hat c_{\mathbf k,B\uparrow},\hat c_{\mathbf k,B\downarrow}, \hat c_{\mathbf k,C\uparrow},\hat c_{\mathbf k,C\downarrow}, \hat c_{\mathbf k,D\uparrow},\hat c_{\mathbf k,D\downarrow}\big)^T. \end{equation} The Nambu spinor is

\begin{equation} \Psi_{\mathbf k}=\big(\hat c_{\mathbf k},,\hat c^\dagger_{-\mathbf k}\big)^T, \end{equation} so (\mathcal H_{\rm BdG}(\mathbf k)) is (16\times16).

Sign-sensitive labels. Winding-like labels and some sign conventions can flip under equivalent relabelings (orbital permutations, unit-cell embedding changes, or altered symmetry-operator phases). We therefore treat signs as declared conventions and keep them fixed across all scripts and figures.

TRIM Structure and Analytic Mass-Inversion Proxy

The time-reversal invariant momenta (TRIM) are the four points (K) satisfying (K=-K) modulo reciprocal lattice vectors. For a square lattice:

\begin{equation} \Gamma=(0,0),\quad X=(\pi,0),\quad Y=(0,\pi),\quad M=(\pi,\pi). \end{equation}

Gap closings at these points control a clean analytic transition structure.

A. TRIM masses

Let (\epsilon_n(K,\mu)) be eigenvalues of (h_{\rm orb}(K)). Define the TRIM mass

\begin{equation} m_{n,K}(\mu,V_z) = V_z^2-\left(\epsilon_n(K,\mu)^2+\Delta_0^2\right). \end{equation} Candidate transition lines (“fan”) are

\begin{equation} V_z=\sqrt{\epsilon_n(K,\mu)^2+\Delta_0^2}. \end{equation}

B. Proxy index

Assign chirality weights

\begin{equation} \eta_\Gamma=\eta_M=+1,\qquad \eta_X=\eta_Y=-1, \end{equation} and define

\begin{equation} C_{\rm proxy}(\mu,V_z)=\frac12\sum_{n,K}\eta_K\left[\mathrm{sgn}(m_{n,K})-\mathrm{sgn}(m_{n,K}|_{V_z=0})\right]. \end{equation} This index is a proxy: it is intended to organize candidate transitions associated with TRIM mass inversions. It does not replace a full 2D Chern evaluation away from TRIM.

In class D, a gapped phase with integer (C) supports (|C|) chiral Majorana edge modes (with sign fixed by orientation convention), so the proxy is used only to pre-organize where this edge content can change through a bulk gap closing. Conceptually, this continuation keeps a common circulation thread: real-space loop circulation in paper 1, order-parameter winding in the fixed texture, and momentum-space Berry-curvature circulation in the topological sector.

Results (Tier 1)

A. Analytic proxy map and transition fan

[Insert fig_code_03_analytic_chern_proxy_map_01.png and cite the generated CSV analytic_chern_proxy_map.csv.] Captions must state grid sizes and all parameter values used.

B. Representative bulk band cut across a proxy transition

We select a representative chemical potential and SOC strength (stated in the figure caption/run log) and plot bulk BdG dispersions along a standard high-symmetry path (e.g., (\Gamma!!-!X!!-!M!!-!\Gamma)) at two Zeeman fields straddling the first proxy line.

[Insert fig_code_04_bulk_bands_analytic_transition_cut_01.png.] In-panel text reports a mesh-estimated minimum gap (\min_{\mathbf k} E_{\min}(\mathbf k)).

C. Strip bulk—boundary diagnostic across matched points

We compute strip spectra (open (x), periodic (k_y)) at two Zeeman fields bracketing the same proxy transition and color eigenvalues by an explicit edge-localization weight.

[Insert fig_code_05_bulk_boundary_correspondence_01.png and cite bulk_boundary_correspondence_summary.txt.]

D. Optional companion: local topology marker in open geometry (static)

At the same strip parameters, we optionally compute a static local marker built from the occupied-state projector in the open geometry, yielding a spatial profile that distinguishes bulk-like regions from boundary reconstructions. This is included solely as a companion visualization alongside the edge-weight strip spectrum.

[Optional insert fig_code_05b_local_chern_marker_strip_01.png and cite local_marker_strip_summary.txt.]

Limitations and Interpretation

1. Fixed texture. The winding pairing texture is imposed and not re-optimized under SOC+Zeeman; no claim of a self-consistent SOC+Zeeman extension is made.
2. Proxy nature. (C_{\rm proxy}) organizes transitions associated with TRIM mass inversions; it is not a proof away from TRIM.
3. Targeted numerics. Numerical evidence here is restricted to representative band cuts and strip spectra across a matched transition; full 2D Chern/gap maps are deferred to a separate 2D completion.
4. Local marker role. The open-geometry marker is a static visualization and does not replace bulk Chern evaluation; no dynamical marker-current claims are made.

Conclusion

We introduced an analytic TRIM mass-inversion construction for a fixed-texture SOC+Zeeman extension seeded from a loop/TRSB winding pattern, yielding an explicit transition fan and a piecewise-constant proxy index. Representative bulk and strip diagnostics support spectral reorganization and bulk–boundary contrast across a matched proxy transition, with an optional static local-marker companion in open geometry. This provides an analytic-first organizing framework and a concrete continuation target for a subsequent referee-proof 2D topological characterization.

References

1. Paper 1 baseline (loop-supercurrent/TRSB program): ../microscopic-loop-supercurrent-trsb/index.md.
2. T. Fukui, Y. Hatsugai, and H. Suzuki, “Chern Numbers in Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Conductances,” J. Phys. Soc. Jpn. 74, 1674 (2005).
3. A. Altland and M. R. Zirnbauer, “Nonstandard Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures,” Phys. Rev. B 55, 1142 (1997).
4. X.-L. Qi, T. L. Hughes, and S.-C. Zhang, “Chiral topological superconductor from the quantum Hall state,” Phys. Rev. B 82, 184516 (2010).
5. A. Kitaev, “Periodic table for topological insulators and superconductors,” AIP Conf. Proc. 1134, 22 (2009).
6. T. Hewitt, Topological Insulators and Superconductors in One Dimension: Chiral Ladder Models and Symmetry Constraints (PhD thesis, University of Kent, 2023).
7. M. D. Caio, M. Möller, M. A. Cazalilla, and M. Lewenstein, “Topological marker currents in Chern insulators,” Nat. Phys. 15, 257–261 (2019).
8. G. Möller and N. R. Cooper, “Synthetic gauge fields for lattices with multi-orbital unit cells: routes towards a (\pi)-flux dice lattice,” New J. Phys. 20, 073025 (2018).

This chapter extends the same symmetry and tensor-product structure used for quadratic, free-fermion, and BdG models to quartic, two-body lattice Hamiltonians. The aim is to write the most general interacting Hamiltonian consistent with the physical structure of the problem: lattice translations, boundaries, inhomogeneous geometries, and defects implemented through masks; Nambu doubling when it is useful for pairing-channel bookkeeping and later mean-field decouplings; internal structure such as sublattices, orbitals, and intra-cell positions; spin; and the chosen set of symmetry generators.

The organising principle is that quartic Hamiltonians may be constructed systematically as symmetry-constrained combinations of products of bilinears, where each bilinear is the second-quantized lift of a single-particle operator written in the same tensor-product order as in the quadratic chapter. In this way, the interacting theory remains compatible with later mean-field reductions: once an interaction has been specified, the admissible quadratic orders are precisely the symmetry-allowed bilinears that can appear as decoupling channels.

From the condensed-matter side, this is the natural language in which on-site Hubbard interactions, extended density-density couplings, and multiorbital local interactions are usually formulated before one chooses a particular approximation scheme or pairing channel. [14, 15, 16, 17]

Unified structure: lattice ⊗ Nambu ⊗ internal ⊗ spin

We keep the same single-particle tensor-product order

When Nambu space is absent, that factor is omitted.

Define a composite internal index

and write fermion operators as

with the same intra-cell position-phase convention as in the quadratic chapter if desired.

From single-particle operators to many-body operators

Second-quantized lift of a single-particle operator

Given any single-particle operator (X) acting on (\mathcal H) (in the fixed tensor order), define its many-body (second-quantized) lift

where (m,n) range over the full one-particle basis (site ⊗ internal ⊗ spin, and Nambu if used).

This construction provides the bridge between the quadratic and quartic theories. Quadratic Hamiltonians are sums of operators of the form (\widehat X), whereas quartic Hamiltonians are built from products of such operators, typically in normal-ordered form.

Normal ordering and a canonical quartic container

A symmetry-compatible and nonredundant container for many interactions is

with Hermitian (X_\mu) and a real symmetric coupling matrix (g_{\mu\nu}=g_{\nu\mu}) (after choosing a Hermitian operator basis). Normal ordering removes the quadratic “Hartree” pieces from the algebraic definition, so that quadratic terms are handled in the quadratic chapter and quartic terms remain genuinely interacting.

Accordingly, the operators (X_\mu) should already respect the lattice structure, including shifts, masks, and boundaries, while symmetry acts on the remaining tensor factors.

Real-space parametrization using shifts and masks

Masked shifts and local projectors

Retain the mask ( \mathbb M ) on (\mathcal H_{\text{lat}}) and the masked shift

For interactions it is often convenient to also use site projectors on lattice space:

These give a clean “operator density at (\mathbf R)” construction.

Local bilinears as building blocks

Let (\Gamma) be any Hermitian matrix acting on (\mathcal H_{\text{(Nambu)}}\otimes\mathcal H_{\text{int}}\otimes\mathcal H_{\text{spin}}). Define the local bilinear (operator density) at (\mathbf R)

Equivalently, (\widehat O_\Gamma(\mathbf R)=\widehat X) with

Common choices of (\Gamma) include the charge-density channel (\Gamma=\mathbb 1), the spin-density channels (\Gamma=\sigma_j), the orbital-density channels (\Gamma=\lambda_i), and, when Nambu space is retained, combined channels of the form (\Gamma=\tau_\ell\otimes\lambda_i\otimes\sigma_j).

Two-site (finite-range) quartic terms via displacements

A large class of lattice interactions can be written as sums over displacements (\boldsymbol\delta\in\mathcal D):

This is the interacting analogue of restricting a quadratic model to a finite displacement set (\mathcal D).

Masking is implemented by restricting (\mathbf R) to active sites (or inserting (m_{\mathbf R}m_{\mathbf R+\boldsymbol\delta}) in the sum).

Special cases include the on-site Hubbard interaction, obtained with (\boldsymbol\delta=\mathbf 0) and (\Gamma) chosen to resolve the spin densities, or directly as (Un_{\uparrow}n_{\downarrow}); nearest-neighbour density interactions, for which (\Gamma=\mathbb 1) and (|\boldsymbol\delta|=a); spin-exchange terms with (\Gamma=\sigma_j) and (|\boldsymbol\delta|=a); and orbital-exchange or Kugel-Khomskii-type structures involving (\Gamma=\lambda_i) and (\Gamma=\lambda_i\otimes\sigma_j).

In this formulation, the lattice geometry is encoded in (\mathcal D) and (\mathbb M), while the (\Gamma)-structure is treated as internal, spin, and, when present, Nambu algebra subject to symmetry.

Examples relevant to later chapters

The interaction classes most relevant to the present thesis are those that lead naturally to superconducting, bond-resolved, and multiorbital mean-field channels. The general quartic framework becomes concrete in the following examples.

On-site attractive Hubbard interaction and singlet pairing

The simplest superconducting example is the on-site attractive Hubbard interaction [14, 15]

In the present formalism this is an on-site quartic term, corresponding to (\boldsymbol\delta=\mathbf 0), and it is the natural starting point for on-site spin-singlet pairing. Decoupling in the anomalous channel gives

so that the resulting quadratic theory contains terms of the form

together with the corresponding Hartree shifts. This is the minimal interaction underlying the lattice BdG constructions used later in the thesis and the standard microscopic bridge to the BCS/Gor’kov mean-field description. [18, 19]

Nearest-neighbour density interactions and bond-resolved channels

A second important class consists of finite-range density interactions such as

This is the simplest example with a nontrivial displacement set (\mathcal D), and therefore makes explicit contact with the harmonic structure discussed above. In the superconductivity literature this is the standard extension beyond the on-site Hubbard term when one wants nonlocal charge, bond, or pairing channels. [15, 20, 21] On the square lattice one obtains

so the interaction already carries the lattice harmonics that later distinguish different ordering patterns. Decoupling in the particle-hole sector can favour charge order, whereas decoupling in the bond-singlet pairing sector gives

The symmetry of the bond pattern then distinguishes extended (s)-wave from (d_{x^2-y^2})-type pairing. This example is therefore the direct interacting analogue of the finite-displacement quadratic models discussed in the preceding chapter.

Bond-current interactions and loop-supercurrent channels

The loop-supercurrent chapters are naturally connected to interactions written in terms of bond-current operators. For an oriented bond (b=(\mathbf R,\mathbf R’)), define

A quartic current-channel interaction may then be written as

In the bilinear-product language, this is simply a coupling between bond bilinears rather than on-site densities. A mean-field decoupling introduces bond fields

which enter the quadratic Hamiltonian as directed bond terms, or equivalently as self-consistent imaginary hopping amplitudes. It is in this sense that current-channel interactions provide a microscopic route to time-reversal-breaking loop-current or loop-supercurrent states, including the loop-supercurrent constructions discussed later in the thesis. [22]

Multiorbital local interactions

When several orbitals or sublattices are retained inside the same unit cell, local interactions acquire a richer internal structure. The standard local multiorbital parametrization goes back to Kanamori, while the corresponding spin-orbital exchange descendants are often summarized as Kugel-Khomskii-type interactions. [16, 17] A standard multiorbital form is

written here for a single site or unit cell with orbital labels (a,b). Such terms are naturally expanded in the (\lambda_i\otimes\sigma_j) basis introduced above. They can generate orbital polarisation, spin exchange, interorbital singlet pairing, or more specialised multicomponent pairing penalties and couplings. This is precisely the class of interaction structure needed once internal degrees of freedom inside a unit cell become central to the later microscopic superconducting models.

Canonical interaction-vertex form and fermionic constraints

Vertex tensor form (most general quartic interaction)

In a general basis label (p=(\mathbf R,\alpha)) (or ((\mathbf k,\alpha)) in momentum space),

Fermion statistics and Hermiticity impose antisymmetry in the incoming legs,

antisymmetry in the outgoing legs,

and Hermiticity,

The bilinear-product container ( \sum g_{\mu\nu}:\widehat X_\mu\widehat X_\nu: ) is a structured way to parameterize such (V) while keeping symmetry constraints tractable.

Translation-invariant case: momentum conservation and lattice harmonics

Assume periodic boundaries and (\mathbb M=\mathbb 1). Translation invariance implies momentum conservation (up to a reciprocal lattice vector (\mathbf G)):

A common reduced parametrization uses transfer momentum (\mathbf q):

Finite-range interactions become trigonometric polynomials

If in real space you kept a finite displacement set (\mathcal D), then the (\mathbf q)-dependence is a finite harmonic expansion:

where each (V_{\boldsymbol\delta}) is a matrix in the internal/spin (and possibly Nambu-channel) indices.

This is the interaction analogue of the quadratic “Bloch polynomial” in (\mathbf k).

Generator-based symmetry constraints for quartic Hamiltonians

Symmetry action on fermion fields

Let a unitary spatial symmetry (g) act on the one-particle Hilbert space by

with (U_g(\mathbf k)) constructed exactly as in the quadratic chapter:

When Nambu space is absent, the factor (U_g^{(\text{Nambu})}) is omitted.

Time reversal (\mathsf T) (antiunitary) acts as

i.e. complex conjugation in coefficients plus the unitary matrix (U_{\mathsf T}) on internal/spin (and possibly Nambu) indices.

Constraint on the interaction vertex

In momentum space, invariance under a unitary symmetry (g) imposes

together with momentum conservation.

For time reversal (\mathsf T),

These are the direct interacting analogues of the quadratic constraints (U_g,\mathcal H(\mathbf k),U_g^\dagger=\mathcal H(g\mathbf k)) and (U_{\mathsf T},\mathcal H(\mathbf k)^*,U_{\mathsf T}^\dagger=\mathcal H(-\mathbf k)), but now acting on a rank-4 vertex.

Constraint in the bilinear-product container

If the interaction is written as

and the symmetry maps the basis by

then invariance is the matrix condition

For antiunitary symmetries, include complex conjugation of coefficients; with a Hermitian basis one typically works with real (g) after enforcing constraints.

In practice, one computes (R_g) by acting with (U_g) on the single-particle operators (X_\mu), and then enforces (g=R_g g R_g^T) as a system of linear constraints.

Basis expansions for interacting channels

Operator basis on internal, spin, and Nambu space

As in the quadratic chapter, one chooses Hermitian bases ({\tau_\ell}) for Nambu space when it is present, ({\lambda_i}) for the internal sector, and ({\sigma_j}) for spin.

Define channel matrices

Then local bilinears are

and finite-range interactions can be expanded as

Symmetry constraints act only on the index structure ((\ell,i,j)) and on the displacement classes (\boldsymbol\delta) (or their orbits under the point group), exactly mirroring the quadratic form-factor selection.

Mean-field bridge: recovering symmetry-allowed quadratic orders from quartic interactions

A quartic term written as a product of bilinears provides an immediate mean-field/Hubbard–Stratonovich entry point:

with an analogous construction for pairing-type decouplings when Nambu space is retained.

Two consequences follow. First, the allowed order parameters are precisely the symmetry-allowed bilinears: the generator constraints of the quadratic chapter determine which (\widehat O_\mu) may acquire expectation values without explicitly breaking the imposed symmetries. Second, the question of competition or coexistence among candidate orders is inherited from the symmetry-allowed invariants. Once a set of channels ({\widehat O_\mu}) has been selected, the Landau-type couplings among the associated mean fields are constrained by the same generator logic, with coefficients determined in principle by the microscopic couplings (g_{\mu\nu}).

Mean-field terms should therefore not be introduced independently, but obtained by decoupling a symmetry-allowed quartic Hamiltonian in symmetry-identified channels. This is the same logic that underlies standard superconducting mean-field theory, microscopic derivations of Ginzburg-Landau theory, and symmetry-based Landau expansions of unconventional order parameters. [23, 19, 24, 25]

Symmetry-first construction of quartic Hamiltonians

The construction proceeds in a natural sequence. One first fixes the lattice structure by specifying the lattice shape (\mathbf N), the boundary conditions, the mask (\mathbb M), and a finite displacement set (\mathcal D). One then chooses a bilinear operator basis of the form

with (\Gamma_\mu) drawn from (\tau\otimes\lambda\otimes\sigma), or from (\lambda\otimes\sigma) when Nambu space is absent. The quartic Hamiltonian is then written in the container

or, equivalently, in a displacement-resolved form with couplings (g^{(\boldsymbol\delta)}).

At that stage one imposes the intrinsic fermionic constraints, namely antisymmetry and Hermiticity, either directly on the vertex (V) or implicitly through the use of Hermitian bilinear bases and symmetric coupling matrices (g). The next step is to construct the generator representations (U_g(\mathbf k)) exactly as in the quadratic chapter, including the intra-cell phase conventions, and from these obtain the induced action (R_g) on the basis (X_\mu). Solving the resulting linear constraints, (g=R_g g R_g^T) together with the antiunitary variants, yields the most general symmetry-allowed coupling space.

If desired, this space may then be organised further by projection into irreducible representations of the point group or spin-rotation group, and may subsequently be reduced by controlled approximations such as mean-field decoupling, random-phase approximation, or functional-renormalization-group truncations.

General form of the quartic Hamiltonian

A symmetry-compatible quartic model can be written as