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IEEE International Magnetic Conference - Short Papers (INTERMAG Short Papers)

Two-Dimensional Fractionally Magnetized Quantum Ferromagnet

Shin Miyahara, Isao Maruyama

Published May 5, 2024• 0 citations
Publisher

Abstract

We present a theory to realize fractionally magnetized quantum ferromagnet in a$\mathbf{spin}-S \geqq 2$bilinear-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 the$\mathbf{Spin}-S$model and$\mathbf{spin} - \frac{1}{2}$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.

DOI
10.1109/intermagshortpapers61879.2024.10576809

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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 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 $S\ensuremath{\geqq}2$ 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 compare quantum Hall systems at filling factors $\ensuremath{\nu}=2$ to $\ensuremath{\nu}=\frac{2}{3}$ and 25\frac{2}{5}52​, corresponding to the exact filling of two lowest electron or composite fermion (CF) Landau levels. The two fractional states are examples of CF liquids with spin dynamics. There is a close analogy between the ferromagnetic (spin polarization P=1P=1P=1) and paramagnetic (P=0)(P=0)(P=0) incompressible ground states that occur in all three systems in the limits of large and small Zeeman spin splitting. However, the excitation spectra are different. At $\ensuremath{\nu}=2$, we find spin domains at half-polarization (P=12)(P=\frac{1}{2})(P=21​), while antiferromagnetic order seems most favorable in the CF systems. The transition between P=0P=0P=0 and 1, as seen when, e.g., the magnetic field is tilted, is also studied by exact diagonalization in toroidal and spherical geometries. The essential role of an effective CF-CF interaction is discussed, and the experimentally observed incompressible half-polarized state is found in some models.
The spin phase diagram of interacting electrons in a quantum wire under a high magnetic field is investigated by finite-system exact diagonalization. Various phases emerge as a function of interaction strength and aspect ratio. Stable spin-polarized states correspond to two-dimensional fractional quantum Hall states of filling fractions 1/q,1/q,1/q, with qqq odd. Spin-singlet ground states for six electrons with filling fractions around 25−−12\frac{2}{5}--\frac{1}{2}52​−−21​ are inspected in detail, and a real-space singlet pairing state peculiar to wire systems is found for a larger aspect ratio.
We give a rigorous proof of the existence of long-range antiferromagnetic order in the ground state of several two-dimensional spin-1/2 Heisenberg systems.Three types of systems are considered.The first type consists of an even number, N , of coupled square lattices with antiferromagnetic nearest-neighbor Heisenberg interactions.In this case, we prove the existence of long-range order (LRO) in the ground state for an interplane to inplane coupling ratio r in the range 0.16 ≤ r ≤ 2.1 for N ≥ 4. Further, we prove that the antiferromagnetic bilayer with ferromagnetic next-nearest-neighbor (nnn) interplane couplings possesses LRO in the ground state for 0.21 ≤ r < 1/4, where r is the absolute value of the ratio of the ferromagnetic and antiferromagnetic couplings.The final example consists of two antiferromagnetic spin-1/2 square lattices that are coupled via an antiferromagnetic nnn interplane coupling r.For r = 1, this system is effectively a spin-1 square lattice and has an ordered ground state.We show that LRO exists in the region 0.85 ≤ r ≤ 1. * ) Note that E r q in the notation of KLS corresponds to 2 + r -E q in our notation.
The noninteracting electronic structures of tight-binding models on bipartite lattices with unequal numbers of sites in the two sublattices have a number of unique features, including the presence of spatially localized eigenstates and flat bands. When a uniform on-site Hubbard interaction UUU is turned on, Lieb proved rigorously that at half-filling ($\ensuremath{\rho}=1$) the ground state has a nonzero spin. In this paper we consider a ``CuO2{\mathrm{CuO}}_{2}CuO2​ lattice'' (also known as ``Lieb lattice,'' or as a decorated square lattice), in which ``ddd orbitals'' occupy the vertices of the squares, while ``ppp orbitals'' lie halfway between two ddd orbitals; both ddd and ppp orbitals can accommodate only up to two electrons. We use exact determinant quantum Monte Carlo (DQMC) simulations to quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of UUU and temperature; we have also calculated the projected density of states, and the compressibility. We study both the homogeneous (H) case, Ud=Up{U}_{d}={U}_{p}Ud​=Up​, originally considered by Lieb, and the inhomogeneous (IH) case, ${U}_{d}\ensuremath{\ne}{U}_{p}$. For the H case at half-filling, we found that the global magnetization rises sharply at weak coupling, and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all UUU. For the IH system at half-filling, we argue that the case ${U}_{p}\ensuremath{\ne}{U}_{d}$ falls under Lieb's theorem, provided they are positive definite, so we used DQMC to probe the cases Up=0,Ud=U{U}_{p}=0,{U}_{d}=UUp​=0,Ud​=U and Up=U,Ud=0{U}_{p}=U,{U}_{d}=0Up​=U,Ud​=0. We found that the different environments of ddd and ppp sites lead to a ferromagnetic insulator when Ud=0{U}_{d}=0Ud​=0; by contrast, Up=0{U}_{p}=0Up​=0 leads to to a metal without any magnetic ordering. In addition, we have also established that at density $\ensuremath{\rho}=1/3$, strong antiferromagnetic correlations set in, caused by the presence of one fermion on each ddd site.