Microscopic topological order parameter and thermodynamic-limit phase boundary of the gapped phase of the $S=1/2$ three-leg spin nanotube

The $S=1/2$ three-leg antiferromagnetic spin tube — three coupled spin chains with frustrated triangular cross-sections — hosts a narrow gapped phase around the rung-symmetric line, realized as a spontaneously dimerized state with doubled periodicity. Otsuka and Sakai (2012) showed on $L=4$ clusters that this phase has a two-fold degenerate singlet ground state and that a local SU(2)-twist Berry phase is quantized inside it, and suggested that this quantized Berry phase can be formulated as a topological order parameter, but they did not construct such an order parameter in the thermodynamic limit, and their phase-boundary estimate retains a finite-size discrepancy with the earlier singlet–triplet boundary at larger leg coupling. Later work (Fuji, Nishimoto, Nakada, and Oshikawa 2014) settled part of the question in the strong rung-coupling limit: the gapped phase is a dimer phase, and the dimer–Tomonaga–Luttinger-liquid transition is an SU(2) Berezinskii–Kosterlitz–Thouless transition whose critical point is fixed by level spectroscopy. What remains open is the construction of a thermodynamic-limit microscopic topological order parameter and its relation to the doubled-unit-cell singlet pattern, and the quantitative gapped–gapless phase boundary at intermediate and weak rung coupling, where the source's singlet–singlet and the earlier singlet–triplet boundary estimates disagree.

Background

A quantum spin tube, also called a spin nanotube, is a quasi-one-dimensional frustrated magnet built from three spin chains (the legs) coupled by exchange interactions (the rungs) that close around the chain direction, so that every cross-section of the system is a triangle. With antiferromagnetic rung couplings the triangular cross-section frustrates the rung spins, and the tube combines strong one-dimensional quantum fluctuations with geometric frustration. The minimal model is the $S=1/2$ three-leg antiferromagnetic Heisenberg tube with leg coupling $J_1$ and rung couplings $J_r$, studied since the late 1990s in part because of molecular and solid-state realizations such as the copper molecular magnet $[(CuCl_2tachH)_3Cl]Cl_2$. The founding surprise was that the symmetric tube, in which all three rung couplings are equal and every cross-section is an equilateral triangle, develops a finite spin gap — an energy gap between the singlet ground state and the lowest magnetic excitation — when the rung coupling is sufficiently strong, although an odd number of coupled half-integer-spin chains would be gapless without frustration (Kawano and Takahashi 1997). Strong-coupling and field-theoretical work converged on a dimer picture of the gapped phase: the spins form singlet pairs arranged in a pattern with doubled real-space periodicity, a resonating doubled-unit-cell singlet pattern, so that translation by one rung is spontaneously broken, the ground state is two-fold degenerate in the total-spin $S=0$ sector, and the lowest singlet excitation at momentum $k=\pi$ becomes degenerate with the ground state in the thermodynamic limit (Lüscher et al. 2004; Okunishi et al. 2005; Sakai et al. 2005; reviewed in Sakai et al. 2010). In the strong-rung limit each triangle retains, besides its total spin, a two-valued scalar chirality — the sense of the 120-degree spin arrangement around the triangle — which acts as an additional pseudo-spin degree of freedom (Fouet et al. 2006). Modulating one rung coupling $J'_r$ away from the other two (asymmetry $\alpha=J'_r/J_r$, isosceles cross-sections) connects the tube to unfrustrated limits: for $\alpha\to 0$ it becomes a three-leg ladder and for $\alpha\to\infty$ a single chain plus a two-leg ladder, both gapless Tomonaga–Luttinger liquids, that is, critical one-dimensional phases described by a conformal field theory of central charge $c=1$ with algebraically decaying correlations. The gapped phase must therefore end in phase transitions on both sides of the symmetric line. Density-matrix renormalization group (DMRG) calculations by Nishimoto and Arikawa (2008) found the gap to collapse at rung asymmetry of order one percent, while Sakai, Sato, Okunishi, Otsuka, Okamoto, and Itoi (2008), combining bosonized effective theories with DMRG and exact diagonalization, argued that a narrow gapped region survives around $\alpha=1$ and that the gapped–gapless transitions are of the Berezinskii–Kosterlitz–Thouless (BKT) type, the vortex-unbinding transition of one-dimensional systems in which the gap on the critical side closes exponentially slowly rather than algebraically. The source of the present problem, Otsuka and Sakai (2012), approached the tube from the topological side. The Haldane phase of the spin-1 Heisenberg chain is the paradigm of hidden topological order in one dimension: it is detected by a nonlocal string order parameter (the expectation value of spin operators separated by a product of spin-flip operators along the chain) or, in the local-gauge formulation of Hatsugai (2006), by a quantized Berry phase: twisting the exchange on one bond by an SU(2) phase $\theta$ and evaluating $\gamma=\mathrm{Arg}\prod_C\langle\psi_{\theta_j}|\psi_{\theta_{j+1}}\rangle$ along a discretized twist path gives a value that is quantized in a gapped phase and can change only when the gap closes. For spin tubes, string order parameters had identified Haldane phases in integer-spin tubes (Charrier, Capponi, Oshikawa, and Pujol 2010). Using exact diagonalization of small clusters (length $L=4$), Otsuka and Sakai found that the $k=\pi$ singlet excitation of the asymmetric $S=1/2$ tube becomes degenerate with the ground state under a $1/L^2$ extrapolation (confirming the doubled-period dimer pattern), that the local twist Berry phase is quantized inside the spin-gap region and vanishes outside, and that a phenomenological renormalization analysis of the singlet–singlet gap gives a phase boundary $\alpha_c(J_1)$ broadly consistent with earlier singlet–triplet estimates but with visible finite-size discrepancies at larger leg coupling. They suggested that the quantized Berry phase characterizing the spin-gap phase can be formulated as a microscopic topological order parameter, but did not construct one in the thermodynamic limit, and they acknowledged the finite-size discrepancy between their singlet–singlet boundary and the earlier singlet–triplet estimate at larger $J_1$. Subsequently Fuji, Nishimoto, Nakada, and Oshikawa (2014) mapped the strong-rung asymmetric tube onto an effective spin-orbital chain — one real spin-$1/2$ (the rung total spin) plus one pseudo-spin-$1/2$ (the rung chirality) per rung, with the rung asymmetry acting as a transverse field on the chirality — and proved by Abelian bosonization that the symmetric tube is in the dimer phase for every half-odd-integer spin and that the dimer–TLL transition driven by rung modulation is an SU(2)-symmetric BKT transition of the sine-Gordon theory, with level spectroscopy fixing $|J'_r-J_r|/J_1\approx 0.283$ in the strong-rung limit and DMRG confirming that the dimer phase survives a finite modulation. Their analysis is restricted to strong rung coupling; they state that the intermediate and weak rung-coupling regimes remain untreated. The Lieb–Schultz–Mattis (LSM) theorem — the statement that a one-dimensional spin system with a half-odd-integer spin per translation unit cell cannot have a unique, symmetric, gapped ground state — implies that the gapped phase of the tube must show a two-fold ground-state degeneracy due to translation-symmetry breaking, as in the dimer picture, and the source invokes it in exactly this role.

Problem Statement

For the $S=1/2$ three-leg antiferromagnetic Heisenberg spin tube with leg coupling $J_1$ and rung couplings $J_r$ and $J'_r$ (asymmetry $\alpha=J'_r/J_r$), establish a complete identification of the gapped phase and of its boundaries in the thermodynamic limit: (1) construct a microscopic topological order parameter for the gapped phase — extending the $L=4$ local SU(2)-twist Berry-phase computation to a well-defined thermodynamic-limit diagnostic, or an equivalent nonlocal (string-type) operator — and establish its relation to the resonating doubled-unit-cell singlet pattern and to the two-fold degenerate $S=0$ ground state found in that phase; and (2) complete the thermodynamic-limit phase diagram in the $(\alpha,J_1)$ plane beyond the strong-rung limit in which the BKT nature of the transitions and the transition point $|J'_r-J_r|/J_1\approx 0.283$ are already established, fixing the gapped–gapless boundaries at intermediate and weak rung coupling and resolving the discrepancy noted by the source at larger leg coupling between its singlet–singlet and the earlier singlet–triplet boundary estimates.

References

1. Yuichi Otsuka and Toru Sakai, Topological aspects of the quantum spin nanotube, J. Phys.: Conf. Ser. 344, 012009 (2012). DOI: 10.1088/1742-6596/344/1/012009. https://doi.org/10.1088/1742-6596/344/1/012009
2. Kenro Kawano and Minoru Takahashi, Three-leg antiferromagnetic Heisenberg ladder with frustrated boundary condition; ground state properties, J. Phys. Soc. Jpn. 66, 4001 (1997). DOI: 10.1143/JPSJ.66.4001. https://doi.org/10.1143/JPSJ.66.4001
3. A. Lüscher, R. M. Noack, G. Misguich, V. N. Kotov, and F. Mila, Soliton binding and low-lying singlets in frustrated odd-legged S=1/2 spin tubes, Phys. Rev. B 70, 060405(R) (2004). DOI: 10.1103/PhysRevB.70.060405. https://doi.org/10.1103/PhysRevB.70.060405
4. Jürgen Schnack, Hiroyuki Nojiri, Paul Kögerler, Geoffrey J. T. Cooper, and Leroy Cronin, Magnetic characterization of the frustrated three-leg ladder compound [(CuCl2tachH)3Cl]Cl2, Phys. Rev. B 70, 174420 (2004). DOI: 10.1103/PhysRevB.70.174420. https://doi.org/10.1103/PhysRevB.70.174420
5. Kouichi Okunishi, Shin-ichiro Yoshikawa, Tôru Sakai, and Seiji Miyashita, Low-energy excitations of the S=1/2 quantum spin tube with the triangular lattice structure, Prog. Theor. Phys. Suppl. 159, 297 (2005). DOI: 10.1143/PTPS.159.297. https://doi.org/10.1143/PTPS.159.297
6. Tôru Sakai, Munehisa Matsumoto, Kouichi Okunishi, Kiyomi Okamoto, and Masahiro Sato, Energy gap of spin nanotube, Physica E 29, 633 (2005). DOI: 10.1016/j.physe.2005.06.044. https://doi.org/10.1016/j.physe.2005.06.044
7. J.-B. Fouet, A. Läuchli, S. Pilgram, R. M. Noack, and F. Mila, Frustrated three-leg spin tubes: from spin 1/2 with chirality to spin 3/2, Phys. Rev. B 73, 014409 (2006). DOI: 10.1103/PhysRevB.73.014409. https://doi.org/10.1103/PhysRevB.73.014409
8. Yasuhiro Hatsugai, Quantized Berry phases as a local order parameter of a quantum liquid, J. Phys. Soc. Jpn. 75, 123601 (2006). DOI: 10.1143/JPSJ.75.123601. https://doi.org/10.1143/JPSJ.75.123601
9. S. Nishimoto and M. Arikawa, Low-lying excitations of the three-leg spin tube: a density-matrix renormalization group study, Phys. Rev. B 78, 054421 (2008). DOI: 10.1103/PhysRevB.78.054421. https://doi.org/10.1103/PhysRevB.78.054421
10. Tôru Sakai, Masahiro Sato, Kouichi Okunishi, Yuichi Otsuka, Kiyomi Okamoto, and Chigak Itoi, Quantum phase transitions of the asymmetric three-leg spin tube, Phys. Rev. B 78, 184415 (2008). DOI: 10.1103/PhysRevB.78.184415. https://doi.org/10.1103/PhysRevB.78.184415 (arXiv:0807.4769)
11. Tôru Sakai, Masahiro Sato, Kiyomi Okamoto, Kouichi Okunishi, and Chigak Itoi, Quantum spin nanotubes — frustration, competing orders and criticalities, J. Phys.: Condens. Matter 22, 403201 (2010). DOI: 10.1088/0953-8984/22/40/403201. https://doi.org/10.1088/0953-8984/22/40/403201 (arXiv:1007.5102)
12. D. Charrier, S. Capponi, M. Oshikawa, and P. Pujol, Quantum phase transitions in three-leg spin tubes, Phys. Rev. B 82, 075108 (2010). DOI: 10.1103/PhysRevB.82.075108. https://doi.org/10.1103/PhysRevB.82.075108 (arXiv:1005.0711)
13. Yang Zhao, Shou-Shu Gong, Yong-Jun Wang, and Gang Su, Low-energy effective theory and two distinct critical phases in a spin-1/2 frustrated three-leg spin tube, Phys. Rev. B 86, 224406 (2012). DOI: 10.1103/PhysRevB.86.224406. https://doi.org/10.1103/PhysRevB.86.224406 (arXiv:1212.1767)
14. Yohei Fuji, Satoshi Nishimoto, Hitoshi Nakada, and Masaki Oshikawa, Quantum criticality in an asymmetric three-leg spin tube: a strong rung-coupling perspective, Phys. Rev. B 89, 054425 (2014). DOI: 10.1103/PhysRevB.89.054425. https://doi.org/10.1103/PhysRevB.89.054425 (arXiv:1311.3041)
15. X. Plat, Y. Fuji, S. Capponi, and P. Pujol, Selection of factorizable ground state in a frustrated spin tube: order by disorder and hidden ferromagnetism, Phys. Rev. B 91, 064411 (2015). DOI: 10.1103/PhysRevB.91.064411. https://doi.org/10.1103/PhysRevB.91.064411
16. K. Yonaga and N. Shibata, Ground state phase diagram of twisted three-leg spin tube in magnetic field, J. Phys. Soc. Jpn. 84, 094706 (2015). DOI: 10.7566/JPSJ.84.094706. https://doi.org/10.7566/JPSJ.84.094706
17. Takaichi Isoyama and Kiyohide Nomura, Discrete symmetries and the Lieb–Schultz–Mattis theorem, Prog. Theor. Exp. Phys. (2017). DOI: 10.1093/ptep/ptx139. https://doi.org/10.1093/ptep/ptx139 (arXiv:1605.03385)
18. Yuta Tachibana, Yuki Ueno, Tomosuke Zenda, Kiyomi Okamoto, and Tôru Sakai, Quantum phase transition of the twisted spin tube, J. Phys. Soc. Jpn. Conf. Proc. 30, 011082 (2020). DOI: 10.7566/JPSCP.30.011082. https://doi.org/10.7566/JPSCP.30.011082 (arXiv:1910.02587)
19. Takanori Sugimoto and Takami Tohyama, Quasi-fractionalization of edge spin in chirality-assisted cluster-based Haldane state on triangular spin tube, Commun. Phys. 6, 291 (2023). DOI: 10.1038/s42005-023-01399-3. https://doi.org/10.1038/s42005-023-01399-3 (arXiv:2108.00681)
20. João C. Getelina, Zekun Zhuang, Premala Chandra, Piers Coleman, Peter P. Orth, and S. L. Sondhi, Quantum order by disorder in frustrated spin nanotubes, Phys. Rev. B 109, 075137 (2024). DOI: 10.1103/PhysRevB.109.075137. https://doi.org/10.1103/PhysRevB.109.075137 (arXiv:2303.04690)