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arXiv:2310.10287 · Physical review. B./Physical review. B

Ferromagnetic Haldane state and dimer multiplet state of quantum ferromagnets

Shin Miyahara, Isao Maruyama

Published Jul 23, 2024• 3 citations
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Abstract

We present a theory of the realization of a ferromagnetic Haldane state in a spin-2 bilinear-biquadratic spin system on an orthogonal-dimer chain. The coexistence of a ferromagnetic state and a Haldane state is due to the rigorous correspondence between the eigenstates of a spin-2 model and a spin-$\frac{1}{2}$ Heisenberg model, i.e., ``eigensystem embedding.'' Numerical exact-diagonalization calculations indicate that the ground state in the model is a fractionally magnetized $M=3/4$ Haldane state. Moreover, a ferromagnetic-dimer multiplet state is an exact ground state on a lattice, where the direct product of dimer singlet states is the ground state in a spin-$\frac{1}{2}$ Heisenberg model that includes one-, two-, and three-dimensional orthogonal-dimer lattices. Eigensystem embedding demonstrates that a quantum ferromagnet can be obtained for an arbitrary spin $S\ensuremath{\geqq}2$ in any dimension and for any lattice in which anomalous ground states are realized in a spin-$\frac{1}{2}$ Heisenberg model.

DOI
10.1103/physrevb.110.014433

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We present a theory of the realization of a ferromagnetic Haldane state in a spin-2 bilinear-biquadratic spin system on an orthogonal-dimer chain. The coexistence of a ferromagnetic state and a Haldane state is due to the rigorous correspondence between the eigenstates of a spin-2 model and a spin-12\frac{1}{2}21​ Heisenberg model, i.e., ``eigensystem embedding.'' Numerical exact-diagonalization calculations indicate that the ground state in the model is a fractionally magnetized M=3/4M=3/4M=3/4 Haldane state. Moreover, a ferromagnetic-dimer multiplet state is an exact ground state on a lattice, where the direct product of dimer singlet states is the ground state in a spin-12\frac{1}{2}21​ Heisenberg model that includes one-, two-, and three-dimensional orthogonal-dimer lattices. Eigensystem embedding demonstrates that a quantum ferromagnet can be obtained for an arbitrary spin S\ensuremath≧2S\ensuremath{\geqq}2S\ensuremath≧2 in any dimension and for any lattice in which anomalous ground states are realized in a spin-12\frac{1}{2}21​ Heisenberg model.
We present a theory to realize fractionally magnetized quantum ferromagnet in aspin−S≧2\mathbf{spin}-S \geqq 2spin−S≧2bilinear-biquadratic spin systems. The origin of the fractionally magnetized state is spin-liquefaction of ferromagnetism which means partly emergent antiferromagnetism. The realization of the spin-liquefaction of ferromagnetism is guaranteed due to a rigorous eigenstate correspondence between theSpin−S\mathbf{Spin}-SSpin−Smodel andspin−12\mathbf{spin} - \frac{1}{2}spin−21​model, i. e., “eigensystem embedding”. This spin-liquefaction theory of ferromagnets can be generalized to any-dimensional lattices. As a typical case, we discuss the two-dimensional models on a square lattice with a numerical exact-diagonalization.
We present a theory to realize entangled quantum spin states with fractional magnetization. The origin of magnetization reduction is partly emergent antiferromagnetism, that is, the spin liquefaction of ferromagnetism. We study a ferromagnetic bilinear coupling region of the spin-SSS ($\ensuremath{\geqq}1$) bilinear-biquadratic spin chain based on (i) a rigorous eigenstate correspondence between the spin-SSS model and spin-12\frac{1}{2}21​ model and (ii) a numerical exact-diagonalization calculation up to S=3S=3S=3. As a result, we obtain a fractional magnetized $M=1\ensuremath{-}1/(2S)$ phase, where ground states have quantum entanglement-reflecting corresponding spin-12\frac{1}{2}21​ antiferromagnetic ground states in a ferromagnetic background. This spin-liquefaction theory of ferromagnets can be generalized to any-dimensional lattices even under a magnetic field. This fractional ferromagnetism opens another research field of quantum ferromagnets.
We systematically study the effects of higher-order interactions on the S=32,1S=\frac{3}{2},1S=23​,1 orthogonal dimer chains using exact diagonalization and density matrix renormalization group. Due to frustration and higher spin, there are rich quantum phases, including three Haldane phases, two gapless phases, and several magnetically ordered phases. To characterize these phases and their phase transitions, we study various physical quantities such as energy gap, energy-level crossing, fidelity susceptibility, spin correlation, entanglement spectrum, and central charge. According to our calculations, the biquadratic term can enhance the Haldane phase regions. In particular, we numerically identify that a Haldane phase in S=32S=\frac{3}{2}S=23​ case is adiabatically connected to the exact AKLT (Affleck, Kennedy, Lieb, and Tasaki) point when adding a bicubic term. Our study on the orthogonal dimer model, which is a 1D version of the Shastry-Sutherland model, provides insights into understanding the possible S=32,1S=\frac{3}{2},1S=23​,1 Haldane phases in quasi-1D and 2D frustrated magnetic materials.
The spin-12\frac{1}{2}21​ Heisenberg orthogonal-dimer chain is considered within the perturbative strong-coupling approach, which is developed from the exactly solved spin-12\frac{1}{2}21​ Ising-Heisenberg orthogonal-dimer chain with the Heisenberg intradimer and the Ising interdimer couplings. Although the spin-12\frac{1}{2}21​ Ising-Heisenberg orthogonal-dimer chain exhibits just intermediate plateaus at zero, one-quarter, and one-half of the saturation magnetization, the perturbative treatment up to second order stemming from this exactly solvable model additionally corroborates the fractional one-third plateau as well as the gapless Luttinger spin-liquid phase. It is evidenced that the approximate results obtained from the strong-coupling approach are in an excellent agreement with the state-of-the-art numerical data obtained for the spin-12\frac{1}{2}21​ Heisenberg orthogonal-dimer chain within the exact diagonalization and density-matrix renormalization group method. The nature of individual quantum ground states is comprehensively studied within the developed perturbation theory.
The quantum spin-12\frac{1}{2}21​ orthogonal-dimer chain with the Heisenberg intradimer and Ising interdimer interactions in a magnetic field is considered by a rigorous approach. The model conserves the zzz component of total spin on vertical Heisenberg bonds, and this property is used to calculate exactly the partition function using the transfer-matrix method. We have found the ground-state phase diagram of the given model in a magnetic field as well as the macroscopic degeneracy along field-induced transitions accompanied with the magnetization jumps. The model exhibits two intermediate fractional plateaux at one quarter and one half of the saturation magnetization. We have examined the effect of the exchange anisotropy in the XXZXXZXXZ Heisenberg intradimer interaction on the ground state. It is shown that the one-quarter and one-half plateaux may disappear from the magnetization curve for the ferromagnetic Heisenberg intradimer interaction. We have also studied rigorously the effect of frustrated interactions on the thermodynamic and magnetic properties of the model and show how the macroscopic degeneracy of the ground state is reflected in the low-temperature behavior of the magnetization, entropy, and specific heat. A possibility of observing enhanced magnetocaloric effect during the adiabatic demagnetization is discussed in detail.