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Journal of Magnetism and Magnetic Materials

Spin squeezing as a probe of symmetry-protected topological phases in an anisotropic four-spin cluster interaction

F. Valeh, S. Mahdavifar

Published Aug 3, 2026• 0 citations
Publisher
DOI
10.1016/j.jmmm.2026.174459

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Since the discovery of the quantum Hall effect, the lattice geometry's influence on charged particles in magnetic fields has been the subject of extensive research. Prototypical models such as the non-interacting Harper-Hofstadter model (HHm) exhibit fractionalization of the Bloch bands with non-trivial topology, manifesting in quantum (spin) Hall phases. Ultracold atomic gases with artificial magnetic fields enabled the experimental study of the non-interacting model, while the effect of strong interactions on the band properties remains an open problem. In this work we study the strongly-interacting bosonic Harper-Hofstadter-Mott model (HHMm) using a reciprocal cluster mean-field (RCMF) method. We obtain a rich groundstate phase diagram featuring striped superfluid, supersolid, and symmetry protected topological (SPT) phases. At filling $\nu=2$ (in contrast to fillings $\nu=1, 3$) the SPT phase has no fermionic counterpart. Incompressible metastable states at fractional filling are also observed. The SPT phases in the HHMm are promising candidates for realizing strongly-correlated topological phases using cold atoms.

•Sep 13, 2016•0 citations
ABSTRACT In this study, we investigate the emergence of non‐trivial topological phases in a 1D spin‐1/2 XX model induced by anisotropic four‐spin interactions. By employing the fermionization technique, we precisely diagonalize the system's Hamiltonian and analytically identify three critical lines. Subsequently, we calculate the energy gap, fidelity susceptibility, cluster local order parameters, winding number, and entanglement spectrum. Our findings identify three crucial boundaries that divide five gapped phases, each characterized by the coexistence of symmetry‐protected topological order and spontaneous symmetry breaking. Analyzing the winding number as a topological invariant allows us to distinguish these gapped phases as distinct non‐trivial topological phases, characterized by integer values of . Moreover, we observe topological phase transitions at quantum critical points, where the winding number changes are associated with the closing and reopening of the energy gap. Furthermore, we demonstrate that for a given winding number , the degeneracy of the lowest eigenvalue in the entanglement spectrum is generally . This work provides novel insights into the role of anisotropic four‐spin interactions in driving topological phenomena in low‐dimensional quantum systems, offering a deeper understanding of the interplay between topology and quantum fluctuations.
We study a three-leg spin-1/2 ladder with geometrically frustrated interleg interactions. We call this model an anisotropic triangular-strip (ATS) model. We numerically and field-theoretically show that its ground state belongs to a gapless symmetry-protected topological (SPT) phase. The numerical approach is based on density-matrix renormalization group analyses of the entanglement entropy and the entanglement spectrum. Whereas the entanglement entropy exhibits a critical behavior, the entanglement spectrum is nontrivially degenerate. These entanglement properties imply that the ground state is a gapless topological phase. We investigate the ATS model using a quantum field theory to support the numerical findings. When the frustrated interchain interaction is deemed a perturbation acting on the three spin chains, the frustrated interchain interaction almost isolates the second chain from the other two chains. However, at the same time, the second chain mediates a ferromagnetic interaction between the first and third chains. Therefore, the ground state of the ATS model is a gapless Tomonaga-Luttinger liquid weakly coupled to a spin-1 Haldane chain with irrelevant interactions. Last but not least, we show that the gapless SPT phase of the ATS model is a symmetry-protected critical phase. We point out that the symmetry protection of criticality is essential in characterization of the gapless SPT phase.
Since the discovery of the quantum Hall effect, the lattice geometry's influence on charged particles in magnetic fields has been the subject of extensive research. Prototypical models such as the non-interacting Harper-Hofstadter model (HHm) exhibit fractionalization of the Bloch bands with non-trivial topology, manifesting in quantum (spin) Hall phases. Ultracold atomic gases with artificial magnetic fields enabled the experimental study of the non-interacting model, while the effect of strong interactions on the band properties remains an open problem. In this work we study the strongly-interacting bosonic Harper-Hofstadter-Mott model (HHMm) using a reciprocal cluster mean-field (RCMF) method. We obtain a rich groundstate phase diagram featuring striped superfluid, supersolid, and symmetry protected topological (SPT) phases. At filling ν=2\nu=2ν=2 (in contrast to fillings ν=1,3\nu=1, 3ν=1,3) the SPT phase has no fermionic counterpart. Incompressible metastable states at fractional filling are also observed. The SPT phases in the HHMm are promising candidates for realizing strongly-correlated topological phases using cold atoms.