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arXiv:cond-mat/0106584 · Journal of Physics Condensed Matter

The field-induced gap due to four-spin exchange in a spin ladder

敦志 中須, Keisuke Totsuka, Yasumasa Hasegawa, Kiyomi Okamoto, Tôru Sakai

Published Aug 2, 2001• 17 citations
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Abstract

The effect of the four-spin cyclic exchange interaction at each plaquette in the $S=1/2$ two-leg spin ladder is investigated at T=0, especially focusing on the field-induced gap. The strong rung coupling approximation suggests that it yields a plateau at half of the saturation moment ($m=1/2$) in the magnetization curve, which corresponds to a field-induced spin gap with a spontaneous breaking of the translational symmetry. A precise phase diagram at $m=1/2$ is also presented based on the level spectroscopy analysis of the numerical data obtained by Lanczos method. The boundary between the gapless and plateau phases is confirmed to be of the Kosterlitz-Thouless (KT) universality class.

DOI
10.1088/0953-8984/13/33/321

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•Dec 20, 2022•0 citations
The effect of the four-spin cyclic exchange interaction at each plaquette in the S=1/2S=1/2S=1/2 two-leg spin ladder is investigated at T=0, especially focusing on the field-induced gap. The strong rung coupling approximation suggests that it yields a plateau at half of the saturation moment (m=1/2m=1/2m=1/2) in the magnetization curve, which corresponds to a field-induced spin gap with a spontaneous breaking of the translational symmetry. A precise phase diagram at m=1/2m=1/2m=1/2 is also presented based on the level spectroscopy analysis of the numerical data obtained by Lanczos method. The boundary between the gapless and plateau phases is confirmed to be of the Kosterlitz-Thouless (KT) universality class.
We have studied the ground state phase diagram of a spin ladder with alternating rung exchange J⊥n=J⊥[1+(−1)nδ]J^{n}_{\perp} = J_{\perp}[1 + (-1)^{n} δ]J⊥n​=J⊥​[1+(−1)nδ] in a magnetic filed, in the limit where the rung coupling is dominant. In this limit the model is mapped onto an XXZXXZXXZ Heisenberg chain in a uniform and staggered longitudinal magnetic fields, where the amplitude of the staggered field is ∼δ\sim δ∼δ. We have shown that the magnetization curve of the system exhibits a plateau at magnetization equal to the half of the saturation value. The width of a plateau scales as δνδ^νδν, where ν=4/5ν=4/5ν=4/5 in the case of ladder with isotropic antiferromagnetic legs and ν=2ν=2ν=2 in the case of ladder with isotropic ferromagnetic legs. We have calculated four critical fields (Hc1±H^{\pm}_{c1}Hc1±​ and Hc2±H^{\pm}_{c2}Hc2±​) corresponding to transitions between different magnetic phases of the system. We have shown that these transitions belong to the universality class of the commensurate-incommensurate transition.
The magnetization process of the S=1/2S=1/2S=1/2 antiferromagnetic spin ladder with the four-spin cyclic exchange interaction at T=0T=0T=0 is studied by the exact diagonalization of finite clusters and size scaling analyses. It is found that a magnetization plateau appears at half the saturation value if the ratio of the four- and two-spin exchange coupling constants J4{J}_{4}J4​ is larger than the critical value ${J}_{4c}=0.05\ifmmode\pm\else\textpm\fi{}0.04.$ The phase transition with respect to J4{J}_{4}J4​ at J4c{J}_{4c}J4c​ is revealed to be the Kosterlitz-Thouless type.
We investigate, at T =0, the magnetization process of the S =1 frustrated two-leg spin ladder having the leg interaction ( J ), the rung interaction ( J 0 ) and the diagonal interaction ( J d ), especially focusing on the appearance and the disappearance of the magnetization plateaux. We use the degenerate perturbation theory, the level spectroscopy analysis of the numerical diagonalization data and the DMRG. When J d =0, the M s /2 plateau ( M s is the saturation magnetization) appears for J 0 / J > 1.463. Then there is no M s /2 plateau for the so-called isotropic case J = J 0 . In case of J d > 0, J d helps the M s /2 plateau. And also, the M s /4 and (3/4) M s plateaux appear when J d is sufficiently large. These two plateaux are associated with the spontaneous breaking of the translation symmetry along the leg direction. We show the plateau phase diagram on the J d versus J plane, and also the magnetic phase diagram on the H versus J d plane for fixed J .
The magnetization process of the S=1/2 antiferromagnetic spin ladder at T=0 is studied by the exact diagonalization of finite clusters and size scaling analyses. It is found that a magnetization plateau appears at half the saturation value (m=1/2) in the presence of a sufficiently large next-nearest-neighbor exchange interaction to yield the frustration, when the rung coupling is larger than the leg one. The phase diagram at m=1/2 is given by the analysis based on the conformal invariance. The magnetization curves are also presented in several cases.
Translational Symmetry Broken Magnetization Plateau of the S=12S={{1}\over{2}}S=21​ Anisotropic Spin Ladder with Ferromagnetic Rung Interaction
The magnetization process of the S=1/2S=1/2S=1/2 anisotropic spin ladder with the ferromagnetic rung interaction is investigated using the numerical diagonalization of finite-size clusters. It is found that the translational symmetry broken magnetization plateau would appear at half the saturation magnetization, when the competing anisotropies are sufficiently large. The phase diagram with respect to the anisotropies and several magnetization curves are presented.
At zero temperature, thermal fluctuation is eliminated and phase transitions will occur due to quantum fluctuations that arise from the Heisenberg uncertainty principle. Magnetic insulators, described by the Heisenberg model Hamiltonian, are a known class of physical systems that can undergo quantum phase transitions when submitted to a magnetic field. The magnetic field induces the transition by closing energy gaps through the Zeeman effect. Examples of systems that undergo these transitions are the antiferromagnetic spin-1 chain, the antiferromagnetic spin- 1/2 ladder, the ferrimagnetic mixed spin-1 and spin- 1/2 chains and ladders. The presence of a gap in the energy spectrum with zero magnetic field leads to a magnetization plateau in the magnetization curve. We use the density matrix renormalization group to investigate the magnetization curves of the mixed spin-1 and spin- 1/2 ladder, for antiferromagnetic and ferro- magnetic couplings between the ladder legs J⊥. For J⊥ > 0, the ground-state is ferrimagnetic with the total spin equal to 1/3 of the saturation value, in accord with the Lieb-Mattis theorem. The magnetization curve presents a plateau at total magnetization 1/3 of the saturation value, the 1/3-plateau since the ground state has a gap to excitations that increase the total spin by 1 unit. Decreasing J⊥ below zero, the ground state becomes a singlet, but the 1/3-plateau survives down to a critical value J⊥ = Jc. Given that the gap closes with the magnetization fixed, it is a Kosterlitz-Thouless transition type. To determine Jc, we have made a finite-size scale analysis of the plateau width.