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arXiv:1711.07041 · Physical review. B./Physical review. B

SU(N) spin-wave theory: Application to spin-orbital Mott insulators

Zhao-Yang Dong, Wei Wang, Jian‐Xin Li

Published May 8, 2018• 15 citations
arXivHTMLPDF

Abstract

We present the application of the $\mathrm{SU}(N)$ spin-wave theory to spin-orbital Mott insulators whose ground states exhibit magnetic orders. When taking both spin and orbital degrees of freedom into account rather than projecting Hilbert space onto the Kramers doublet, which is the lowest spin-orbital locked energy levels, the $\mathrm{SU}(N)$ spin-wave theory should take the place of the $\mathrm{SU}(2)$ one due to the inevitable spin-orbital multipole exchange interactions. To implement the application, we introduce an efficient general local mean-field method, which involves all local fluctuations, and develop the $\mathrm{SU}(N)$ linear spin-wave theory. Our approach is tested firstly by calculating the multipolar spin-wave spectra of the $\mathrm{SU}(4)$ antiferromagnetic model. Then, we apply it to spin-orbital Mott insulators. It is revealed that the Hund's coupling would influence the effectiveness of the isospin-$1/2$ picture when the spin-orbital coupling is not large enough. We further carry out the $\mathrm{SU}(N)$ spin-wave calculations of two materials, $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ and ${\mathrm{Sr}}_{2}{\mathrm{IrO}}_{4}$, and find that the magnonic and spin-orbital excitations are consistent with experiments.

DOI
10.1103/physrevb.97.205106

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We present the application of the SU(N)\mathrm{SU}(N)SU(N) spin-wave theory to spin-orbital Mott insulators whose ground states exhibit magnetic orders. When taking both spin and orbital degrees of freedom into account rather than projecting Hilbert space onto the Kramers doublet, which is the lowest spin-orbital locked energy levels, the SU(N)\mathrm{SU}(N)SU(N) spin-wave theory should take the place of the SU(2)\mathrm{SU}(2)SU(2) one due to the inevitable spin-orbital multipole exchange interactions. To implement the application, we introduce an efficient general local mean-field method, which involves all local fluctuations, and develop the SU(N)\mathrm{SU}(N)SU(N) linear spin-wave theory. Our approach is tested firstly by calculating the multipolar spin-wave spectra of the SU(4)\mathrm{SU}(4)SU(4) antiferromagnetic model. Then, we apply it to spin-orbital Mott insulators. It is revealed that the Hund's coupling would influence the effectiveness of the isospin-1/21/21/2 picture when the spin-orbital coupling is not large enough. We further carry out the SU(N)\mathrm{SU}(N)SU(N) spin-wave calculations of two materials, \ensuremathα\ensuremath-RuCl3\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}\ensuremathα\ensuremath-RuCl3​ and Sr2IrO4{\mathrm{Sr}}_{2}{\mathrm{IrO}}_{4}Sr2​IrO4​, and find that the magnonic and spin-orbital excitations are consistent with experiments.
Motivated by recent progress in the realization of artificial gauge fields and SU(NNN) Mott insulators using alkaline-earth-like atoms in optical lattices, we develop an unbiased SU(NNN) real-space dynamical mean-field theory approach to study the effect of spin-orbit coupling and on-site Hubbard interaction UUU on SU(3) fermionic systems. We investigate the behavior of the local magnetization, double occupancies, and the triple occupancy versus the Hubbard interaction across the metal to Mott insulator transition. We map out the magnetic phase diagram in the large-UUU limit and show that the spin-orbit coupling can stabilize long-range orders such as ferromagnets, spirals, and stripes with different orientations in SU(3) Mott insulators.
The SU(4)-symmetric spin-orbital model on the honeycomb lattice was recently studied in connection to correlated insulators such as the eg{e}_{g}eg​ Mott insulator Ba3CuSb2O9{\mathrm{Ba}}_{3}{\mathrm{CuSb}}_{2}{\mathrm{O}}_{9}Ba3​CuSb2​O9​ and the insulating phase of magic-angle twisted bilayer graphene at quarter filling. Here we provide a unified discussion of these systems by investigating an extended model that includes the effects of Hund's coupling and anisotropic, orbital-dependent exchange interactions. Using a combination of mean-field theory, linear flavor-wave theory, and variational Monte Carlo, we show that this model harbors a quantum spin-orbital liquid over a wide parameter regime around the SU(4)-symmetric point. For large Hund's coupling, a ferromagnetic antiferro-orbital ordered state appears, while a valence-bond crystal combined with a vortex orbital state is stabilized by dominant orbital-dependent exchange interactions.
Motivated by the structure of TDAE-\c60, we derive a multicomponent superexchange Hamiltonian for the spins and orbital (``isospin'') degrees of freedom in the Mott-insulating phase. We explore its phase diagram and identify points of special interest: an SU(4) antiferromagnet (solved by Sutherland) and two ferromagnet⊗\otimes⊗antiferromagnet points where the ground state is known. For the ferromagnetic regime, we apply interchain mean field theory and derive an expression for the Curie temperature where spin ordering occurs and a lower Ne{é}l temperature for a conjectured isospin ordering.
The enhancement of the spin-space symmetry from the usual SU(2)\mathrm{SU}(2)SU(2) to SU(N)\mathrm{SU}(N)SU(N) with N>2N>2N>2 is promising for finding nontrivial quantum spin liquids, but the realization of SU(N)\mathrm{SU}(N)SU(N) spin systems in real materials is still challenging. Although there is a proposal in cold atomic systems, in magnetic materials with a spin-orbital degree of freedom it is difficult to achieve the SU(N)\mathrm{SU}(N)SU(N) symmetry by fine tuning. Here we propose a new mechanism by which the SU(4)\mathrm{SU}(4)SU(4) symmetry emerges in the strong spin-orbit coupling limit. In d1d^1d1 transition metal compounds with edge-sharing anion octahedra, the spin-orbit coupling gives rise to strongly bond-dependent and apparently SU(4)\mathrm{SU}(4)SU(4)-breaking hopping between the Jeff=3/2J_\textrm{eff}=3/2Jeff​=3/2 quartets. However, in the honeycomb structure, a gauge transformation maps the system to an SU(4)\mathrm{SU}(4)SU(4)-symmetric Hubbard model, which means that the system has a hidden symmetry in spite of its large spin-orbit coupling. In the strong repulsion limit at quarter filling, as expected in ααα-ZrCl3,_3,3​, the low-energy effective model is the SU(4)\mathrm{SU}(4)SU(4) Heisenberg model on the honeycomb lattice, which cannot have a trivial gapped ground state and is expected to host a gapless spin-orbital liquid. In such quantum spin-orbital liquids, both the spin and orbital degrees of freedom become fractionalized and correlated together at low temperature due to the strong frustrated interactions between them. Similarly to spinons in pure quantum spin liquids, quantum spin-orbital liquids can host not only spinon excitations, but also fermionic "orbitalon" excitations at low temperature, which we have named here in distinction from orbitons in the symmetry-broken Jahn-Teller phases.
We present a comprehensive study of the spin excitations---as measured by the dynamical spin structure factor $S(q,\ensuremath{\omega})$---of the so-called block-magnetic state of low-dimensional orbital-selective Mott insulators. We realize this state via both a multi-orbital Hubbard model and a generalized Kondo-Heisenberg Hamiltonian. Due to various competing energy scales present in the models, the system develops periodic ferromagnetic islands of various shapes and sizes, which are antiferromagnetically coupled. The $2\ifmmode\times\else\texttimes\fi{}2$ particular case was already found experimentally in the ladder material BaFe2Se3\mathrm{Ba}{\mathrm{Fe}}_{2}{\mathrm{Se}}_{3}BaFe2​Se3​ that becomes superconducting under pressure. Here we discuss the electronic density as well as Hubbard and Hund coupling dependence of $S(q,\ensuremath{\omega})$ using density matrix renormalization group method. Several interesting features were identified: (1) An acoustic (dispersive spin-wave) mode develops. (2) The spin-wave bandwidth establishes a new energy scale that is strongly dependent on the size of the magnetic island and becomes abnormally small for large clusters. (3) Optical (dispersionless spin excitation) modes are present for all block states studied here. In addition, a variety of phenomenological spin Hamiltonians have been investigated but none matches entirely our results that were obtained primarily at intermediate Hubbard UUU strengths. Our comprehensive analysis provides theoretical guidance and motivation to crystal growers to search for appropriate candidate materials to realize the block states, and to neutron scattering experimentalists to confirm the exotic dynamical magnetic properties unveiled here, with a rich mixture of acoustic and optical features.
Mott insulators, in which strong Coulomb interactions fully localize electrons on single atomic sites, play host to an incredibly rich and exciting array of strongly correlated physics. One can naturally extend this concept to cluster Mott insulators, wherein electrons localize not on single atoms but across clusters of atoms, forming "molecules in solids". The resulting localized degrees of freedom incorporate the full spectrum of electronic degrees of freedom, spin, orbital, and charge. These serve as the building blocks for cluster Mott insulators, and understanding them is an important first step toward understanding the many-body physics that emerges in candidate cluster Mott insulators. Here, we focus on elementary building blocks, neglecting some of the complexity present in real materials which can often obfuscate the underlying principles at play. Through an extensive set of exact theoretical calculations on clusters of varying geometry, number of orbitals, and number of electrons, we uncover some of the basic organizing principles of cluster Mott phases, particularly when interactions dominate and negate a simple single-particle picture. These include, for example, the identification of an additional "cluster Hund's rule", of cluster ground states that are best understood from a purely interacting perspective, and of several localized degrees of freedom which are protected by an unusual combination of discrete spatial or orbital symmetries. Finally, we discuss the impact of adding additional terms, relevant to material candidates, on the phase diagrams presented throughout, as well as the potential next steps in the journey to building a more complete picture of cluster Mott insulators.