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At finite N and for sufficiently small boson density, the problem is strictly equivalent to gapped spinons (the Schwinger bosons) interacting with a fluctuating bond field. It has been understood that the low-energy physics of such a system will be that of a lattice gauge theory coupled to charged matter fields. But the nature (group) of the gauge field crucially depends on the lattice geometry and in the present case where the lattice is not bipartite, it should generically be of Z2 type and has two types of phases (Refs.). If the effective Z2 gauge theory is in a deconfined phase (which should be the case for large enough N), the ground state is qualitatively close to the mean-field one, with gapped and unconfined spinons. If the phase is instead confined, the gauge fluctuations are strong and cannot be neglected, the mean-field picture is not valid any more.
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At finite N and for sufficiently small boson density, the problem is strictly equivalent to gapped spinons (the Schwinger bosons) interacting with a fluctuating bond field. It has been understood that the low-energy physics of such a system will be that of a lattice gauge theory coupled to charged matter fields. But the nature (group) of the gauge field crucially depends on the lattice geometry and in the present case where the lattice is not bipartite, it should generically be of Z 2 type and has two types of phases (Refs.). If the effective Z 2 gauge theory is in a deconfined phase (which should be the case for large enough N), the ground state is qualitatively close to the mean-field one, with gapped and unconfined spinons. If the phase is instead confined, the gauge fluctuations are strong and cannot be neglected, the mean-field picture is not valid any more.
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For the sake of simplicity we develop only the formulas corresponding to a unique soft mode. This is the case of the Néel order in presence of a Dzyaloshinskii-Moriya interaction: due to the U(1) symmetry of the original Hamiltonian, the Goldstone mode is unique. The general case is straightforward but more cumbersome in writing.
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m AF 2 is delicate to normalize because of the fluctuations of the spin length on each site. Fluctuations also increase with the strength of the Dzyaloshinskii-Moriya coupling, making the normalization partly arbitrary.
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m A F 2 is delicate to normalize because of the fluctuations of the spin length on each site. Fluctuations also increase with the strength of the Dzyaloshinskii-Moriya coupling, making the normalization partly arbitrary.
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If q0 0, the spinon dispersion is folded in the magnetic Brillouin zone.
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If q 0 0, the spinon dispersion is folded in the magnetic Brillouin zone.
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For S>1/2, we note that single-ion anisotropies are also present and may affect the form of magnetic order.
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For S > 1 / 2, we note that single-ion anisotropies are also present and may affect the form of magnetic order.
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