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arXiv:1802.00232 · Journal of the Physical Society of Japan

Magnetic Phase Diagrams and Magnetization Plateaus of the Spin-1/2 Antiferromagnetic Heisenberg Model on a Square-Kagome Lattice with Three Nonequivalent Exchange Interactions

Katsuhiro Morita, Takami Tohyama

Published Mar 14, 2018• 28 citations
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Abstract

Magnetization plateaus in quantum spin systems emerges in two-dimensional frustrated systems such as kagome lattice. The spin-1/2 antiferromagnetic Heisenberg model on square-kagome lattice is also appropriate for the study of magnetization plateau. Motivated by recent experimental finding of such a square-kagome lattice with nonequivalent three bonds, we investigate the phase diagrams and magnetization plateaus of the lattice by the exact diagonalization method. In addition to previously reported the 1/3 and 2/3 plateaus in the model with two equivalent bonds, we find a new 2/3 plateau whose magnetic structure is characterized by spontaneously broken four-fold rotational symmetry. The plateau appears only in the case of three nonequivalent bonds. We propose the possibility of finding plateaus including the new one.

DOI
10.7566/jpsj.87.043704

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Magnetization plateaus in quantum spin systems emerges in two-dimensional frustrated systems such as kagome lattice. The spin-1/2 antiferromagnetic Heisenberg model on square-kagome lattice is also appropriate for the study of magnetization plateau. Motivated by recent experimental finding of such a square-kagome lattice with nonequivalent three bonds, we investigate the phase diagrams and magnetization plateaus of the lattice by the exact diagonalization method. In addition to previously reported the 1/3 and 2/3 plateaus in the model with two equivalent bonds, we find a new 2/3 plateau whose magnetic structure is characterized by spontaneously broken four-fold rotational symmetry. The plateau appears only in the case of three nonequivalent bonds. We propose the possibility of finding plateaus including the new one.
The spin-12\frac{1}{2}21​ Heisenberg model on a kagome lattice is a typical frustrated quantum spin system. The basic structure of a kagome lattice is also present in the kagome-strip lattice in one dimension, where a similar type of frustration is expected. We thus study the magnetization plateaus of the spin-12\frac{1}{2}21​ Heisenberg model on a kagome-strip chain with three-independent antiferromagnetic exchange interactions using the density-matrix renormalization-group method. In a certain range of exchange parameters, we find twelve kinds of magnetization plateaus, nine of which have magnetic structures breaking translational and/or reflection symmetry spontaneously. The structures are classified by an array of five-site unit cells with specific bond-spin correlations. In a case with a nontrivial plateau, namely a 3/10 plateau, we find long-period magnetic structure with a period of four unit cells.
Plateaus can be observed in the zero-temperature magnetization curve of quantum spin systems at rational values of the magnetization. In one dimension, the appearance of a plateau is controlled by a quantization condition for the magnetization which involves the length of the local spin and the volume of a translational unit cell of the ground state. We discuss examples of geometrically frustrated quantum spin systems with large (in general unbounded) periodicities of spontaneous breaking of translational symmetry in the ground state. In two dimensions, we discuss the square, triangular and Kagomé lattices using exact diagonalization (ED) for up to N = 40 sites. For the spin-1/2 XXZ model on the triangular lattice we study the nature and stability region of a plateau at one third of the saturation magnetization. The Kagomé lattice gives rise to particularly rich behaviour with several plateaus in the magnetization curve and a jump due to local magnon excitations just below saturation.
We study the ground-state phase diagram of the spin-1/21/21/2 antiferromagnetic Heisenberg model on the square-kagome lattice using infinite projected entangled-pair states (iPEPS). By systematically varying the ratio of exchange couplings on triangular and square plaquettes, we establish a complete quantum phase diagram in the thermodynamic limit. In the intermediate-coupling regime, we identify four distinct nonmagnetic phases that are unambiguously characterized as valence-bond crystals (VBCs) by their symmetry-inequivalent patterns of strong and weak spin-spin correlations. These include a plaquette crossed-dimer VBC, a loop-six VBC stabilized around the isotropic point, a generalized pinwheel VBC with reduced rotational symmetry, and a decorated loop-six VBC proximate to ferrimagnetic order. We determine the phase boundaries using a combination of bond-resolved correlation functions, entanglement entropy, and magnetization. For transitions not accompanied by sharp entanglement signatures, we extract the spin gap from finite-field simulations, allowing us to distinguish gapped and gapless VBC phases. At larger coupling ratios, the system undergoes transitions into imperfect and perfect ferrimagnetic states. Our results resolve long-standing ambiguities in the square-kagome Heisenberg model and provide a quantitatively reliable reference for future theoretical and experimental studies of frustrated quantum magnets.
We study the ground state of the S=12S=\frac{1}{2}S=21​ Heisenberg model on the checkerboard lattice in a magnetic field by the density matrix renormalization group method with the sine-square deformation. We obtain magnetization plateaus at M/Msat=0,14,38,12M/{M}_{\mathrm{sat}}=0,\frac{1}{4},\frac{3}{8},\frac{1}{2}M/Msat​=0,41​,83​,21​, and 34\frac{3}{4}43​, where Msat{M}_{\mathrm{sat}}Msat​ is the saturated magnetization. The obtained 34\frac{3}{4}43​ plateau state is consistent with the exact result, and the 12\frac{1}{2}21​ plateau is found to have a four-spin resonating loop structure similar to the six-spin loop structure of the 13\frac{1}{3}31​ plateau of the kagome lattice. Different four-spin loop structures are obtained in the 14\frac{1}{4}41​ and 38\frac{3}{8}83​ plateaus but no corresponding states exist in the kagome lattice. The 38\frac{3}{8}83​ plateau has a unique magnetic structure of three types of four-spin local quantum states in a $4\sqrt{2}\ifmmode\times\else\texttimes\fi{}2\sqrt{2}$ magnetic unit cell with a 16-fold degeneracy.
Frustrated spin systems on kagome lattices have long been considered to be a promising candidate for realizing exotic spin-liquid phases. Recently, there has been a lot of renewed interest in these systems with the discovery of materials such as volborthite and herbertsmithite that have kagomelike structures. In the presence of an external magnetic field, these frustrated systems can give rise to magnetization plateaus of which the plateau at m=13m=\frac{1}{3}m=31​ is considered to be the most prominent. Here, we study the problem of the antiferromagnetic spin-12\frac{1}{2}21​ quantum XXZ Heisenberg model on a kagome lattice by using a Jordan-Wigner transformation that maps the spins onto a problem of fermions coupled to a Chern-Simons gauge field. This mapping relies on being able to define a consistent Chern-Simons term on the lattice. Such a lattice Chern-Simons term had previously only been written for the square lattice and was used to successfully study the unfrustrated Heisenberg antiferromagnet on the square lattice. At a mean-field level, these ideas have also been applied to frustrated systems by ignoring the details of the Chern-Simons term. However, fluctuations are generally strong in these models and are expected to affect the mean-field physics. Using a recently developed method to rigorously extend the Chern-Simons term to the frustrated kagome lattice, we can now formalize the Jordan-Wigner transformation on the kagome lattice. We then discuss the possible phases that can arise at the mean-field level from this mapping and focus specifically on the case of 13\frac{1}{3}31​ filling (m=13m=\frac{1}{3}m=31​ plateau) and analyze the effects of fluctuations in our theory. We show that in the regime of XY\text{XY}XY anisotropy, the ground state at the 13\frac{1}{3}31​ plateau is equivalent to a bosonic fractional quantum Hall Laughlin state with filling fraction 12\frac{1}{2}21​ and that at the 59\frac{5}{9}95​ plateau it is equivalent to the first bosonic Jain daughter state at filling fraction 23\frac{2}{3}32​.
The magnetization process of the spin-1/21/21/2 antiferromagnetic Heisenberg model on two-dimensional square-kagome lattice is studied theoretically. The metamagnetic jumps exist in the magnetization process at the higher edge of the 1/31/31/3 and 2/32/32/3 plateaus. The parameter-dependencies of the critical field and the magnitude of the magnetization jump at the higher edge of the 1/31/31/3 plateau are obtained by using the approximated state in the unit cell and compared with the numerical results of the exact diagonalization of 42 sites.