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arXiv:1704.04024 · Physical review. B./Physical review. B

Exact diagonalization and cluster mean-field study of triangular-lattice XXZ antiferromagnets near saturation

Daisuke Yamamoto, Hiroshi Ueda, Ippei Danshita, Giacomo Marmorini, Tsutomu Momoi, Tokuro Shimokawa

Published Jul 25, 2017• 31 citations
arXivHTMLPDF

Abstract

Quantum magnetic phases near the magnetic saturation of triangular-lattice antiferromagnets with XXZ anisotropy have been attracting renewed interest since it has been suggested that a nontrivial coplanar phase, called the $\ensuremath{\pi}$-coplanar or $\mathrm{\ensuremath{\Psi}}$ phase, could be stabilized by quantum effects in a certain range of anisotropy parameter $J/{J}_{z}$ besides the well-known 0-coplanar (known also as $V)$ and umbrella phases. Recently, Sellmann et al. [Phys. Rev. B 91, 081104(R) (2015)] claimed that the $\ensuremath{\pi}$-coplanar phase is absent for $S=1/2$ from an exact-diagonalization analysis in the sector of the Hilbert space with only three down-spins (three magnons). We first reconsider and improve this analysis by taking into account several low-lying eigenvalues and the associated eigenstates as a function of $J/{J}_{z}$ and by sensibly increasing the system sizes (up to 1296 spins). A careful identification analysis shows that the lowest eigenstate is a chirally antisymmetric combination of finite-size umbrella states for $J/{J}_{z}\ensuremath{\gtrsim}2.218$ while it corresponds to a coplanar phase for $J/{J}_{z}\ensuremath{\lesssim}2.218$. However, we demonstrate that the distinction between 0-coplanar and $\ensuremath{\pi}$-coplanar phases in the latter region is fundamentally impossible from the symmetry-preserving finite-size calculations with fixed magnon number. Therefore, we also perform a cluster mean-field plus scaling analysis for small spins $S\ensuremath{\le}3/2$. The obtained results, together with the previous large-$S$ analysis, indicate that the $\ensuremath{\pi}$-coplanar phase exists for any $S$ except for the classical limit $(S\ensuremath{\rightarrow}\ensuremath{\infty})$ and the existence range in $J/{J}_{z}$ is largest in the most quantum case of $S=1/2$.

DOI
10.1103/physrevb.96.014431

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Quantum magnetic phases near the magnetic saturation of triangular-lattice antiferromagnets with XXZ anisotropy have been attracting renewed interest since it has been suggested that a nontrivial coplanar phase, called the \ensuremathπ\ensuremath{\pi}\ensuremathπ-coplanar or \ensuremathΨ\mathrm{\ensuremath{\Psi}}\ensuremathΨ phase, could be stabilized by quantum effects in a certain range of anisotropy parameter J/JzJ/{J}_{z}J/Jz​ besides the well-known 0-coplanar (known also as V)V)V) and umbrella phases. Recently, Sellmann et al. [Phys. Rev. B 91, 081104(R) (2015)] claimed that the \ensuremathπ\ensuremath{\pi}\ensuremathπ-coplanar phase is absent for S=1/2S=1/2S=1/2 from an exact-diagonalization analysis in the sector of the Hilbert space with only three down-spins (three magnons). We first reconsider and improve this analysis by taking into account several low-lying eigenvalues and the associated eigenstates as a function of J/JzJ/{J}_{z}J/Jz​ and by sensibly increasing the system sizes (up to 1296 spins). A careful identification analysis shows that the lowest eigenstate is a chirally antisymmetric combination of finite-size umbrella states for J/Jz\ensuremath≳2.218J/{J}_{z}\ensuremath{\gtrsim}2.218J/Jz​\ensuremath≳2.218 while it corresponds to a coplanar phase for J/Jz\ensuremath≲2.218J/{J}_{z}\ensuremath{\lesssim}2.218J/Jz​\ensuremath≲2.218. However, we demonstrate that the distinction between 0-coplanar and \ensuremathπ\ensuremath{\pi}\ensuremathπ-coplanar phases in the latter region is fundamentally impossible from the symmetry-preserving finite-size calculations with fixed magnon number. Therefore, we also perform a cluster mean-field plus scaling analysis for small spins S\ensuremath≤3/2S\ensuremath{\le}3/2S\ensuremath≤3/2. The obtained results, together with the previous large-SSS analysis, indicate that the \ensuremathπ\ensuremath{\pi}\ensuremathπ-coplanar phase exists for any SSS except for the classical limit (S\ensuremath→\ensuremath∞)(S\ensuremath{\rightarrow}\ensuremath{\infty})(S\ensuremath→\ensuremath∞) and the existence range in J/JzJ/{J}_{z}J/Jz​ is largest in the most quantum case of S=1/2S=1/2S=1/2.
The triangular lattice of S=1/2 spins with XXZ anisotropy is a ubiquitous model for various frustrated systems in different contexts. We determine the quantum phase diagram of the model in the plane of the anisotropy parameter and the magnetic field by means of a large-size cluster mean-field method with a scaling scheme. We find that quantum fluctuations break up the nontrivial continuous degeneracy into two first-order phase transitions. In between the two transition boundaries, the degeneracy-lifting results in the emergence of a new coplanar phase not predicted in the classical counterpart of the model. We suggest that the quantum phase transition to the nonclassical coplanar state can be observed in triangular-lattice antiferromagnets with large easy-plane anisotropy or in the corresponding optical-lattice systems.
Coplanar model states for applications of the coupled cluster method (CCM) to problems in quantum magnetism are those in which all spins lie in a plane, whereas three-dimensional (3D) model states are, by contrast, non-coplanar ones in which all the spins do not lie in any single plane. A crucial first step in applying the CCM to any such lattice quantum spin system is to perform a passive rotation of the local spin axes so that all spins in the model state appear mathematically to point in the same (say, downwards z-)direction. Whereas this process leads to terms with only real coefficients in the rotated Hamiltonian for coplanar model states, an additional complication arises for 3D model states where the corresponding coefficients can become complex-valued. We show here for the first time how high-order implementations of the CCM can be performed for such Hamiltonians. We explain in detail why the extension of the computational implementation of the CCM when going from coplanar to 3D model states is a non-trivial task that has not hitherto been undertaken. To illustrate these new developments, we present results for three cases: (a) the spin-half one-dimensional Ising ferromagnet in an applied transverse magnetic field (as an exactly solvable test model to use as a yardstick for the viability and accuracy of our new methodology); (b) the spin-half triangular-lattice Heisenberg antiferromagnet in the presence of an external magnetic field; and (c) the spin-S triangular-lattice XXZ antiferromagnet in the presence of an external magnetic field, for the cases 12≤S≤5\frac{1}{2} \le S \le 5 21​≤S≤5 . For 3D model states the sets of algebraic CCM equations for the ket- and bra-state correlation coefficients become complex-valued, but ground-state expectation values of all physical observables are manifestly real numbers, as required, and as we explicitly demonstrate in all three applications. Indeed, excellent correspondence is seen with the results of other methods, where they exist, for these systems. In particular, our CCM results demonstrate explicitly that coplanar ordering is favoured over non-coplanar ordering for the triangular-lattice spin-half Heisenberg antiferromagnet at all values of the applied external magnetic field, whereas for the anisotropic XXZ model non-coplanar ordering can be favoured in some regions of the parameter space. Specifically, we present a precise determination of the boundary (i.e., the critical value of the XXZ anisotropy parameter Δ\Delta Δ ) between a 3D ground state and a coplanar ground state for the XXZ model for values for the external magnetic field near to saturation, for values of the spin quantum number S≤5S \le 5S≤5 . Although the CCM calculations are computationally intensive for this frustrated model, especially for high spin quantum numbers, our accurate new results certainly improve our understanding of it.
Coplanar model states for applications of the coupled cluster method (CCM) to problems in quantum magnetism are those in which all spins lie in a plane, whereas three-dimensional (3D) model states are, by contrast, non-coplanar ones in which all the spins do not lie in any single plane. A crucial first step in applying the CCM to any such lattice quantum spin system is to perform a passive rotation of the local spin axes so that all spins in the model state appear mathematically to point in the same (say, downwards z-)direction. Whereas this process leads to terms with only real coefficients in the rotated Hamiltonian for coplanar model states, an additional complication arises for 3D model states where the corresponding coefficients can become complex-valued. We show here for the first time how high-order implementations of the CCM can be performed for such Hamiltonians. We explain in detail why the extension of the computational implementation of the CCM when going from coplanar to 3D model states is a non-trivial task that has not hitherto been undertaken. To illustrate these new developments, we present results for three cases: (a) the spin-half one-dimensional Ising ferromagnet in an applied transverse magnetic field (as an exactly solvable test model to use as a yardstick for the viability and accuracy of our new methodology); (b) the spin-half triangular-lattice Heisenberg antiferromagnet in the presence of an external magnetic field; and (c) the spin-S triangular-lattice XXZ antiferromagnet in the presence of an external magnetic field, for the cases <= S <= 5. For 3D model states the sets of algebraic CCM equations for the ket- and bra-state correlation coefficients become complex-valued, but ground-state expectation values of all physical observables are manifestly real numbers, as required, and as we explicitly demonstrate in all three applications. Indeed, excellent correspondence is seen with the results of other methods, where they exist, for these systems. In particular, our CCM results demonstrate explicitly that coplanar ordering is favoured over non-coplanar ordering for the triangular-lattice spin-half Heisenberg antiferromagnet at all values of the applied external magnetic field, whereas for the anisotropic XXZ model non-coplanar ordering can be favoured in some regions of the parameter space. Specifically, we present a precise determination of the boundary (i.e., the critical value of the XXZ anisotropy parameter Delta) between a 3D ground state and a coplanar ground state for the XXZ model for values for the external magnetic field near to saturation, for values of the spin quantum number S <= 5. Although the CCM calculations are computationally intensive for this frustrated model, especially for high spin quantum numbers, our accurate new results certainly improve our understanding of it.
The magnetization process of the S=1/2 Kagome-lattice quantum antiferromagnet is investigated using the numerical exact diagonalization up to 39-site clusters, comparing with the triangular-lattice antiferromagnet. Our previous finite-size scaling analysis with the rhombic clusters indicated the “magnetization ramp” as a novel field-induced quantum phase transition at 1/3 of the saturation magnetization. As another possible exotic behavior, we focus on the feature of the magnetization curve at the 2/3 of the saturation. The same critical exponent analysis indicates that a different singular behavior occurs at the 2/3 magnetization of the Kagome-lattice antiferromagnet, while no anomaly in the triangular-lattice one.
The quantum triangular XXZ model has recently enjoyed a wealth of new theoretical results, especially in relation to the modeling of the Ba3CoSb2O9{\mathrm{Ba}}_{3}{\mathrm{CoSb}}_{2}{\mathrm{O}}_{9}Ba3​CoSb2​O9​ compound. In particular, it has been understood that in a longitudinal magnetic field the umbrella (cone) phase, classically stable in all the easy-plane region of the ground-state phase diagram, is considerably reduced by the effect of quantum fluctuations. We provide more quantitative information for this phenomenon at arbitrary value of the site spin SSS, by employing the dilute Bose gas expansion, valid in the high-field regime; our results improve the available estimates of the 1/S1/S1/S expansion. We quantify the extent to which a higher spin suppresses the effect of quantum fluctuations. Besides, we show how in three-dimensional layered systems a relatively small antiferromagnetic interlayer coupling has a similar consequence of bringing back the umbrella phase in some part of the phase diagram.
We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone Γ(q=0)\Gamma(\mathbf q=0)Γ(q=0) and at the magnetic ordering wavevectors ±[Q=(4π/3,0)]\pm[\mathbf Q= (4\pi/3,0)]±[Q=(4π/3,0)], the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.