Sector dependence of the exact Kramers–Wannier duality in exactly solvable SO($n$) spin chains: the Ramond sector and its Spin($n$)$_1$ duality-defect interpretation

In the exactly solvable SO($n$)-symmetric spin chains of Chulliparambil, Zhang, and Tu [Phys. Rev. B 108, 094411 (2023)] — $n$ species of itinerant Majorana fermions coupled to a static $\mathbb{Z}_2$ gauge field, with a Spin($n$)$_1$ conformal critical point — the exact Kramers–Wannier duality exchanging the couplings $(J,h)$ is implemented by a Temperley–Lieb unitary that preserves the Hamiltonian's form only in the $w=-1$ (Neveu–Schwarz) $\mathbb{Z}_2$-flux sector. The unresolved question is the duality's complete sector structure on the full Hilbert space: its action on the $w=+1$ (Ramond) sector, together with the matching Spin($n$)$_1$ continuum interpretation in terms of duality defects or categorical symmetries that the authors' outlook explicitly defers to future work. The $n=1$ Ising analog is largely settled in the literature, whereas the general $n\ge2$ problem is open.

Background

The transverse-field Ising chain (TFI) is the paradigmatic exactly solvable quantum spin chain. Its two gapped phases (ordered and disordered) are related by the Kramers–Wannier (KW) duality, an exact mapping of the model at couplings $(J,h)$ to the model at $(h,J)$ that exchanges order and disorder variables; the self-dual point $J=h$ is the quantum critical point described by the Ising conformal field theory (CFT), as first exploited by Kramers and Wannier in 1941. Unlike an ordinary symmetry, the KW duality does not act as a unitary operator commuting with one Hamiltonian on a fixed Hilbert space; its correct interpretation, on the lattice and in the continuum, has become a central topic in the theory of non-invertible symmetries and topological duality defects.

Chulliparambil, Zhang, and Tu introduced a family of SO($n$)-symmetric spin chains that generalize the TFI chain ($n=1$). Each site carries a $2^n$-dimensional Hilbert space on which $2n+1$ matrices $\Gamma^1,\dots,\Gamma^{2n+1}$ act, generating a Clifford algebra (they square to the identity and anticommute pairwise). The Hamiltonian is exactly solvable through a Kitaev-style mapping onto $n$ species of itinerant Majorana fermions (fermionic operators that are their own Hermitian conjugates) coupled to a static $\mathbb{Z}_2$ gauge field $u_{j,j+1}$, schematically $H=-i\sum_{j,\alpha}\bigl(J\,u_{j,j+1}\,b^{2\alpha-1}_j b^{2\alpha}_{j+1}+h\,b^{2\alpha}_j b^{2\alpha-1}_j\bigr)$. The phase diagram contains a trivial gapped phase, a symmetry-protected topological (SPT) phase (a gapped phase, distinct from a product state as long as the symmetry is kept, with symmetry-protected boundary modes), and a critical point at $J=h$ described by the Spin($n$)$_1$ CFT of $n$ free massless Majorana fields. Two further TFI structures carry over: an infinite-dimensional Onsager algebra of conserved charges (the algebraic structure first found in Onsager's exact solution of the two-dimensional Ising model, generated here by an infinite family of commuting operators), and an exact KW duality between the two gapped phases. On the lattice the latter is seen by writing $H=\sqrt{2}\sum_{j,\alpha}\bigl(h\,E^{(\alpha)}_{2j-1}+J\,E^{(\alpha)}_{2j}\bigr)$ (up to a constant) in terms of Temperley–Lieb (TL) generators — operators $E^{(\alpha)}_j$, with $E^{(\alpha)}_j/\sqrt2$ a projector, obeying $(E^{(\alpha)}_j)^2=\sqrt{2}\,E^{(\alpha)}_j$, $E^{(\alpha)}_jE^{(\alpha)}_{j\pm1\,(\mathrm{mod}\,2N)}E^{(\alpha)}_j=E^{(\alpha)}_j$, and $[E^{(\alpha)}_j,E^{(\alpha)}_{j'}]=0$ for $|j-j'\,(\mathrm{mod}\,2N)|\ge2$ — whereupon the unitary $U=\prod_{\alpha=1}^{n}\prod_{j=1}^{2N-1}\exp\bigl(i\pi E^{(\alpha)}_j/(2\sqrt2)\bigr)$ shifts $E^{(\alpha)}_j\mapsto E^{(\alpha)}_{j+1}$ for $1\le j\le 2N-2$, a half-step translation of the TL chain that exchanges $h\leftrightarrow J$; the self-dual point is the Spin($n$)$_1$ critical point.

The exact solution splits the Hilbert space into flux sectors labeled by the gauge-invariant $\mathbb{Z}_2$ flux $w=\prod_j u_{j,j+1}$: $w=-1$ is the Neveu–Schwarz (NS) sector, where the itinerant Majoranas obey antiperiodic boundary conditions, and $w=+1$ is the Ramond (R) sector, where they are periodic and $n$ fermionic zero modes appear. At the self-dual point the exact finite-$N$ partition function splits as $Z=Z_{\rm NS}+Z_{\rm R}$, the fermion-parity-projected contributions of the two sectors, and in the continuum limit — after dropping the divergent vacuum energy common to both sectors — these two pieces reproduce the modular-invariant torus partition function $\widetilde Z=\sum_{\mathfrak a}|\chi_{\mathfrak a}|^2$ of the Spin($n$)$_1$ CFT: the identity and vector primaries reside in the NS sector and the spinor primary(ies) in the R sector, whose zero-mode degeneracy matches the spinor-representation dimension. The sectors are physically essential — depending on $n$ and the phase, the ground state(s) occupy one or both of them.

The open subtlety concerns the duality's action on this sector structure. Under $U$ the bulk generators shift as $E^{(\alpha)}_j\mapsto E^{(\alpha)}_{j+1}$ for $1\le j\le 2N-2$, but the images of the two boundary generators are $U E^{(\alpha)}_{2N-1}U^\dagger=\frac{1}{\sqrt2}\bigl(1-w\,\Gamma^{2\alpha-1,2n+1}_N\Gamma^{2\alpha}_1\bigr)$ and $U E^{(\alpha)}_{2N}U^\dagger=\frac{1}{\sqrt2}\bigl(1+w\,\Gamma^{2\alpha,2\alpha-1}_1\bigr)$, while the generators themselves are $w$-independent, $E^{(\alpha)}_{2N}=\frac{1}{\sqrt2}\bigl(1+\Gamma^{2\alpha-1,2n+1}_N\Gamma^{2\alpha}_1\bigr)$ and $E^{(\alpha)}_{1}=\frac{1}{\sqrt2}\bigl(1-\Gamma^{2\alpha,2\alpha-1}_1\bigr)$. The dual images therefore coincide with $E^{(\alpha)}_{2N}$ and $E^{(\alpha)}_{1}$ only for $w=-1$: the TL shift closes cyclically — and hence $U$ maps the Hamiltonian onto the dual Hamiltonian — only in the NS sector. The authors state this, remarking that the KW duality is by no means a conventional symmetry, but do not analyze what the duality does in the R sector; their outlook defers the topological-defect and categorical-symmetry understanding of the duality to future work.

For $n=1$ (the plain Ising chain) the analogous questions are substantially clarified by modern work: Seiberg and Shao traced how the Majorana chain's spin-structure sectors (R/NS and their GSO projections, i.e. sums over the fermionic spin structures) map under Jordan–Wigner to Ising Hilbert spaces with and without a KW duality defect, and related constructions give non-invertible KW symmetry operators on tensor-product Hilbert spaces and their boundary-condition dependence (with the duality-twisted case still contested as of 2026, as Zhou and Shi showed). For $n\ge2$ Majorana species coupled to a static $\mathbb{Z}_2$ gauge field — where the continuum categorical symmetry is the Spin($n$)$_1$ modular tensor category (the category of topological superselection sectors of the Spin($n$)$_1$ CFT, with their fusion and braiding rules) rather than a $\mathbb{Z}_n$ Tambara–Yamagami category (the fusion-category structure underlying $\mathbb{Z}_n$ KW duality) — no corresponding analysis exists.

Problem Statement

In the exactly solvable SO($n$) spin chains of Chulliparambil, Zhang, and Tu, the exact Kramers–Wannier duality $h\leftrightarrow J$ is implemented by the Temperley–Lieb unitary $U=\prod_{\alpha=1}^{n}\prod_{j=1}^{2N-1}\exp\bigl(i\pi E^{(\alpha)}_j/(2\sqrt2)\bigr)$, which preserves the Hamiltonian's form only in the $w=-1$ (Neveu–Schwarz) $\mathbb{Z}_2$-flux sector. Determine the complete sector structure of this duality on the full Hilbert space, specifically:

1. Ramond sector: what does $U$ — or a suitably completed or modified duality operator — do to the Hamiltonian and the states of the $w=+1$ (Ramond) sector, in which the dual images of the two boundary Temperley–Lieb generators carry the flux-dependent factors $\frac{1}{\sqrt2}\bigl(1-w\,\Gamma^{2\alpha-1,2n+1}_N\Gamma^{2\alpha}_1\bigr)$ and $\frac{1}{\sqrt2}\bigl(1+w\,\Gamma^{2\alpha,2\alpha-1}_1\bigr)$ and hence fail to reproduce $E^{(\alpha)}_{2N}$ and $E^{(\alpha)}_{1}$? In particular, is the Ramond-sector Hamiltonian mapped to itself, to a duality-twisted (defect) Hamiltonian, or to a distinct sector, and how does the TL shift close in that case?

2. Continuum interpretation: what is the interpretation of that duality action in the Spin($n$)$_1$ continuum limit — as a topological duality defect or, more generally, a categorical symmetry of these chains — consistent with the Spin($n$)$_1$ primary content (identity and vector in the NS sector, the spinor primary $s$ for odd $n$ or the two spinor primaries $s_\pm$ for even $n$ in the R sector)?

Item 1 is the open point the source itself flags; item 2 is the topological-defect/categorical-symmetry understanding of the same duality that the source's outlook defers to future work, and is sought as the continuum interpretation of the lattice result of item 1 rather than as an independent classification task. Together the two items constitute one objective: completing the Kramers–Wannier duality structure of these chains on the full Hilbert space (both flux sectors). The $n=1$ (Ising-chain) corner is substantially understood in the non-invertible-symmetry literature and serves as the anchor and consistency check; the target is the general-$n$ answer, in particular $n\ge2$, where $n$ Majorana species couple to the static $\mathbb{Z}_2$ gauge field and the continuum categorical symmetry is the Spin($n$)$_1$ modular tensor category.

References

1. Sreejith Chulliparambil, Hua-Chen Zhang, Hong-Hao Tu, "Symmetry-protected topological phases, conformal criticalities, and duality in exactly solvable SO($n$) spin chains", Phys. Rev. B 108, 094411 (2023). DOI: 10.1103/PhysRevB.108.094411; arXiv:2305.03398. https://arxiv.org/abs/2305.03398
2. H. A. Kramers, G. H. Wannier, "Statistics of the Two-Dimensional Ferromagnet. Part I", Phys. Rev. 60, 252 (1941). DOI: 10.1103/PhysRev.60.252. https://doi.org/10.1103/PhysRev.60.252
3. Nathan Seiberg, Shu-Heng Shao, "Majorana chain and Ising model – (non-invertible) translations, anomalies, and emanant symmetries", SciPost Phys. 16, 064 (2024). DOI: 10.21468/SciPostPhys.16.3.064; arXiv:2307.02534. https://arxiv.org/abs/2307.02534
4. Hua-Chen Zhang, Germán Sierra, "Kramers-Wannier self-duality and non-invertible translation symmetry in quantum chains: a wave-function perspective", J. High Energ. Phys. 2025, 157 (2025). DOI: 10.1007/JHEP05(2025)157; arXiv:2410.06727. https://arxiv.org/abs/2410.06727
5. Nathan Seiberg, Sahand Seifnashri, Shu-Heng Shao, "Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space", SciPost Phys. 16, 154 (2024). DOI: 10.21468/SciPostPhys.16.6.154; arXiv:2401.12281. https://arxiv.org/abs/2401.12281
6. Huan-Qiang Zhou, Qian-Qian Shi, "Non-invertible symmetries and boundary conditions for the transverse-field Ising model", arXiv:2603.21290 (2026). https://arxiv.org/abs/2603.21290
7. Lei Su, "$\mathbb{Z}_2$ lattice gauge theories: fermionic gauging, transmutation, and Kramers-Wannier dualities", arXiv:2510.20893 (2025). https://arxiv.org/abs/2510.20893
8. Yash Chugh, Kusum Dhochak, Uma Divakaran, Prithvi Narayan, Amit Kumar Pal, "Exactly solvable one-dimensional quantum models with gamma matrices", Phys. Rev. E 106, 024114 (2022). DOI: 10.1103/PhysRevE.106.024114; arXiv:2201.06588. https://arxiv.org/abs/2201.06588
9. Mateo Fontaine, Koudai Sugimoto, Shunsuke Furukawa, "Symmetry, topology, duality, chirality, and criticality in a spin-1/2 XXZ ladder with a four-spin interaction", Phys. Rev. B 109, 134413 (2024). DOI: 10.1103/PhysRevB.109.134413; arXiv:2311.07053. https://arxiv.org/abs/2311.07053
10. Kazuhiko Minami, "Conserved charges of series of solvable lattice models", Nucl. Phys. B 1012, 116844 (2025). DOI: 10.1016/j.nuclphysb.2025.116844; arXiv:2410.20798. https://arxiv.org/abs/2410.20798
11. Rui Xian Siew, Shailesh Chandrasekharan, Ribhu K. Kaul, "Transverse field $\gamma$-matrix spin chains", Phys. Rev. D 110, 094510 (2024). DOI: 10.1103/PhysRevD.110.094510; arXiv:2406.04120. https://arxiv.org/abs/2406.04120
12. Akhil Pravin Furtado, Kusum Dhochak, "Topological phases and Edge states in an exactly solvable Gamma matrix model", arXiv:2507.10509 (2025). https://arxiv.org/abs/2507.10509
13. Xue-Jia Yu, Limei Xu, Hai-Qing Lin, "Topological physics in quantum critical systems", Phys. Rep. 1160, 1–56 (2026). DOI: 10.1016/j.physrep.2025.10.005; arXiv:2601.00184. https://arxiv.org/abs/2601.00184