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arXiv:1205.1955 · Physical Review B

Quantum disorder in the spatially completely anisotropic triangular lattice

Philipp Hauke

Published Jan 11, 2013• 35 citations
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Abstract

Spin liquids are important for a fundamental understanding of quantum magnetism and may find applications in quantum computing, but it is still not clear under which general circumstances these exotic, quantum-disordered phases occur. To gather deeper insights, we analyze a generalization of the spatially anisotropic triangular lattice (SATL), the spatially completely anisotropic triangular lattice (SCATL), using Takahashi's modified spin-wave theory, complemented by exact diagonalizations. For Heisenberg interactions, calculations on the SATL predict quantum disorder for several materials in which experiments have found magnetic long-range order. In the SCATL, the corresponding parameter values yield ordered ground states and this discrepancy disappears. We also study the model with $XY$ interactions, which can be implemented with current technology in ultracold atoms in optical lattices. This allows to estimate the stability of quantum-disordered phases towards imperfect implementations of the coupling parameters. For both kinds of interactions, we find indications for extended quantum-disordered phases. In part of the parameter space of the $XY$ model, exact diagonalization hints at a gapped state with chiral long-range order. In both models, our results suggest that two gapped nonmagnetic regions, identified as distinct in the SATL, could actually be continuously connected via the additional anisotropy of the SCATL. Further, we find that different kinds of order are always separated by disordered phases. We propose that this is a quite universal feature of two-dimensional frustrated antiferromagnets with continuous symmetry in the couplings. The SCATL may therefore---besides being relevant for experiments---yield fundamental insight into quantum-disordered phases.

DOI
10.1103/physrevb.87.014415

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Spin liquids are important for a fundamental understanding of quantum magnetism and may find applications in quantum computing, but it is still not clear under which general circumstances these exotic, quantum-disordered phases occur. To gather deeper insights, we analyze a generalization of the spatially anisotropic triangular lattice (SATL), the spatially completely anisotropic triangular lattice (SCATL), using Takahashi's modified spin-wave theory, complemented by exact diagonalizations. For Heisenberg interactions, calculations on the SATL predict quantum disorder for several materials in which experiments have found magnetic long-range order. In the SCATL, the corresponding parameter values yield ordered ground states and this discrepancy disappears. We also study the model with XYXYXY interactions, which can be implemented with current technology in ultracold atoms in optical lattices. This allows to estimate the stability of quantum-disordered phases towards imperfect implementations of the coupling parameters. For both kinds of interactions, we find indications for extended quantum-disordered phases. In part of the parameter space of the XYXYXY model, exact diagonalization hints at a gapped state with chiral long-range order. In both models, our results suggest that two gapped nonmagnetic regions, identified as distinct in the SATL, could actually be continuously connected via the additional anisotropy of the SCATL. Further, we find that different kinds of order are always separated by disordered phases. We propose that this is a quite universal feature of two-dimensional frustrated antiferromagnets with continuous symmetry in the couplings. The SCATL may therefore---besides being relevant for experiments---yield fundamental insight into quantum-disordered phases.
We study the ground-state phases of the S =1/2 Heisenberg quantum antiferromagnet on the spatially anisotropic triangular lattice (SATL) and on the square lattice with up to next-next-nearest-neighbor coupling (the J 1 J 2 J 3 model), making use of Takahashi's modified spin-wave (MSW) theory supplemented by ordering vector optimization. We compare the MSW results with exact diagonalization and projected-entangled-pair-states calculations, demonstrating their qualitative and quantitative reliability. We find that the MSW theory correctly accounts for strong quantum effects on the ordering vector of the magnetic phases of the models under investigation: in particular, collinear magnetic order is promoted at the expense of non-collinear (spiral) order, and several spiral states that are stable at the classical level disappear from the quantum phase diagram. Moreover, collinear states and non-collinear ones are never connected continuously, but they are separated by parameter regions in which the MSW theory breaks down, signaling the possible appearance of a non-magnetic ground state. In the case of the SATL, a large breakdown region appears also for weak couplings between the chains composing the lattice, suggesting the possible occurrence of a large non-magnetic region continuously connected with the spin-liquid state of the uncoupled chains. This shows that the MSW theory is—despite its apparent simplicity—a versatile tool for finding candidate regions in the case of spin-liquid phases, which are among prime targets for relevant quantum simulations.
We study quantum disordered ground states of the two-dimensional Heisenberg-Kitaev model on the triangular lattice using a Schwinger boson approach. Our aim is to identify and characterize potential gapped quantum spin liquid phases that are stabilized by anisotropic Kitaev interactions. For antiferromagnetic Heisenberg and Kitaev couplings and sufficiently small spin SSS, we find three different symmetric Z2{Z}_{2}Z2​ spin liquid phases, separated by two continuous quantum phase transitions. Interestingly, the gap of elementary excitations remains finite throughout the transitions. The first spin liquid phase corresponds to the well-known zero-flux state in the Heisenberg limit, which is stable with respect to small Kitaev couplings and develops ${120}^{\ensuremath{\circ}}$ order in the semiclassical limit at large SSS. In the opposite Kitaev limit, we find a different spin liquid ground state, which is a quantum disordered version of a magnetically ordered state with antiferromagnetic chains, in accordance with results in the classical limit. Finally, at intermediate couplings, we find a spin liquid state with unusual spin correlations. Upon spinon condensation, this state develops Bragg peaks at incommensurate momenta in close analogy to the magnetically ordered Z2 vortex crystal phase, which has been analyzed in recent theoretical works.
Disordered quantum antiferromagnets in two-dimensional compounds have been a focus of interest in the last years due to their exotic properties. However, with very few exceptions, the ground states of the corresponding Hamiltonians are notoriously difficult to simulate making their characterization and detection very elusive, both theoretically and experimentally. Here we propose a method to signal quantum disordered antiferromagnets by doing exact diagonalization in small lattices using random boundary conditions and averaging the observables of interest over the different disorder realizations. We apply our method to study the Heisenberg spin-1/2 model in an anisotropic triangular lattice. In this model, the competition between frustration and quantum fluctuations might lead to some spin-liquid phases as predicted from different methods ranging from spin-wave mean-field theory to 2D-DMRG or PEPS. Our method accurately reproduces the ordered phases expected of the model and signals quantum disordered phases by the presence of a large number of quasidegenerate ground states together with an undefined local order parameter. The method presents a weak dependence on finite-size effects.
We investigate a system of frustrated hardcore bosons, modeled by an XY antiferromagnet on the spatially anisotropic triangular lattice, using Takahashi's modified spin-wave (MSW) theory. In particular, we implement ordering vector optimization on the ordered reference state of MSW theory, which leads to significant improvement of the theory and accounts for quantum corrections to the classically ordered state. The MSW results at zero temperature compare favorably to exact diagonalization (ED) and projected entangled-pair state (PEPS) calculations. The resulting zero-temperature phase diagram includes a one-dimensional (1D) quasi-ordered phase, a 2D Néel ordered phase and a 2D spiraling ordered phase. Strong indications coming from the ED and PEPS calculations, as well as from the breakdown of MSW theory, suggest that the various ordered or quasi-ordered phases are separated by spin-liquid phases with short-range correlations, in analogy to what has been predicted for the Heisenberg model on the same lattice. Within MSW theory, we also explore the finite-temperature phase diagram. In agreement with the Berezinskii–Kosterlitz–Thouless (BKT) theory, we find that zero-temperature long-range-ordered phases turn into quasi-ordered phases (up to a BKT transition temperature), while zero-temperature quasi-ordered phases become short-range correlated at finite temperature. These results show that, despite its simplicity, MSW theory is very well suited to describing ordered and quasi-ordered phases of frustrated XY spins (or, equivalently, of frustrated lattice bosons) both at zero and finite temperatures. While MSW theory, just as other theoretical methods, cannot describe spin-liquid phases, its breakdown provides a fast and reliable method for singling out Hamiltonians that may feature these intriguing quantum phases. We thus suggest a tool for guiding our search for interesting systems whose properties are necessarily studied with a physical quantum simulator instead of theoretical methods.
Employing exact diagonalization, we systematically study the anisotropic Heisenberg model which is related to rare-earth triangular-lattice materials. We probe its full 3D phase diagram afresh and identify a large region of quantum spin liquid (QSL) phase which can extend to the QSL region of the J1–J2{J}_{1}\text{--}{J}_{2}J1​–J2​ triangular Heisenberg model. Furthermore, we explore the magnetization curves of different phases and reproduce the 1/3-magnetization plateau in the quantum spin liquid phase region. More importantly, to study the possible chemical disorders in real materials, we consider the randomness of exchange interactions and find no spin glass order. And there is a large region of random-singlet phase which contains strongly random spin networks, dominated by two-spin singlets, four-spin singlets and other singlet domains. Our comprehensive ED study can give detailed insightful understanding of the microscopic Hamiltonian related to the YbMgGaO4{\mathrm{YbMgGaO}}_{4}YbMgGaO4​ and some other related rare-earth triangular-lattice materials.
Based on exact diagonalization and density matrix renormalization group method, we show that an anisotropic triangular lattice Heisenberg spin model has three distinct quantum phases. In particular, a spin-liquid phase is present in the weak interchain coupling regime, which is characterized by an anisotropic spin structure factor with an exponential-decay spin correlator along the weaker coupling direction, consistent with the Cs2CuCl4{\mathrm{Cs}}_{2}\mathrm{Cu}{\mathrm{Cl}}_{4}Cs2​CuCl4​ compounds. In the obtained phase diagram, the spin-liquid phase is found to persist up to a relatively large critical anisotropic coupling ratio ${J}^{\ensuremath{'}}∕J=0.78$, which is stabilized by strong quantum fluctuations, with a parity symmetry distinct from two magnetic ordered states in the stronger coupling regime.