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Unconventional mechanisms of magnetic ordering in frustrated models of quantum magnetism

Tommaso Coletta

Published Jan 1, 2013• 0 citations
Publisher

Abstract

In this thesis we investigate in the semiclassical approximation how unconventional magnetic ordering, that is magnetic ordering which is not predicted at the classical level, emerges in a variety of frustrated models of quantum magnetism. The first part of this thesis presents our semiclassical investigation of the magnetization processes of the anisotropic triangular lattice Heisenberg antiferromanget and of the J1 − J2 square lattice Heisenberg antiferromanget. Existing analytical, numerical and experimen- tal evidences report that these systems exhibit a magnetization plateau respectively at 1/3 and 1/2 of the saturation value and whose structure can be understood in simple terms as a collinear spin structure. We develop a strategy to estimate the semiclassical energy of collinear spin structures away from their classical stability point. In this framework we are able to predict the stabilization of a collinear up-up-down structure with a magnetization equal to one third of the saturation value in a wide anisotropy range for the triangular lattice Heisen- berg antiferromanget. Similarly we find that the four sublattice up-up-up-down structure with magnetization equal to 1/2 of the saturation value is stabilized deep into the classically striped ordered region of the J1 − J2 square lattice Heisenberg antiferromanget. In the case of the anisotropic triangular lattice Heisenberg antiferromagnet a brief digression is made to the zero field case. The semiclassical approach confirms that quantum fluctuations favor collinear structures over the classical spiral states in either the weakly and strongly coupled chain regimes. The second part of the thesis is devoted to the study of fully frustrated transverse field Ising models. These models have emerged as an alternative tool to access the physics of the quan- tum dimer models (QDM) of Rokhsar and Kivelson at the purely kinetic point. We studied the classical phase diagram of this model for the square and honeycomb lattice geometries. In either case the effect of quantum fluctuations and the relevance to the physics of the QDM is discussed. In the case of the honeycomb lattice geometry we find that the classical phase diagram is rather involved with columnar and plaquette states separated by an infinite series of states of mixed nature. From the study of quantum fluctuations we conjecture that the sqrt(12) × sqrt(12) plaquette phase dominates over the columnar structure down to zero transverse field in the limit S = 1/2. In the case of the square lattice geometry we find that at the classical level the plaquette state is stabilized below the polarized state down to zero transverse field. In agreement with the most recent quantum Monte Carlo results, we show that for S = 1/2 quantum fluctuations favor columnar oder in the whole field range below saturation even though this structure is never classically stable in this parameter range.

DOI
10.5075/epfl-thesis-5977

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In this thesis we investigate in the semiclassical approximation how unconventional magnetic ordering, that is magnetic ordering which is not predicted at the classical level, emerges in a variety of frustrated models of quantum magnetism. The first part of this thesis presents our semiclassical investigation of the magnetization processes of the anisotropic triangular lattice Heisenberg antiferromanget and of the J1 − J2 square lattice Heisenberg antiferromanget. Existing analytical, numerical and experimen- tal evidences report that these systems exhibit a magnetization plateau respectively at 1/3 and 1/2 of the saturation value and whose structure can be understood in simple terms as a collinear spin structure. We develop a strategy to estimate the semiclassical energy of collinear spin structures away from their classical stability point. In this framework we are able to predict the stabilization of a collinear up-up-down structure with a magnetization equal to one third of the saturation value in a wide anisotropy range for the triangular lattice Heisen- berg antiferromanget. Similarly we find that the four sublattice up-up-up-down structure with magnetization equal to 1/2 of the saturation value is stabilized deep into the classically striped ordered region of the J1 − J2 square lattice Heisenberg antiferromanget. In the case of the anisotropic triangular lattice Heisenberg antiferromagnet a brief digression is made to the zero field case. The semiclassical approach confirms that quantum fluctuations favor collinear structures over the classical spiral states in either the weakly and strongly coupled chain regimes. The second part of the thesis is devoted to the study of fully frustrated transverse field Ising models. These models have emerged as an alternative tool to access the physics of the quan- tum dimer models (QDM) of Rokhsar and Kivelson at the purely kinetic point. We studied the classical phase diagram of this model for the square and honeycomb lattice geometries. In either case the effect of quantum fluctuations and the relevance to the physics of the QDM is discussed. In the case of the honeycomb lattice geometry we find that the classical phase diagram is rather involved with columnar and plaquette states separated by an infinite series of states of mixed nature. From the study of quantum fluctuations we conjecture that the sqrt(12) × sqrt(12) plaquette phase dominates over the columnar structure down to zero transverse field in the limit S = 1/2. In the case of the square lattice geometry we find that at the classical level the plaquette state is stabilized below the polarized state down to zero transverse field. In agreement with the most recent quantum Monte Carlo results, we show that for S = 1/2 quantum fluctuations favor columnar oder in the whole field range below saturation even though this structure is never classically stable in this parameter range.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
We demonstrate the versatility, simplicity, and power of the minimally augmented spin-wave theory in studying phase diagrams of the quantum spin models in which unexpected magnetically ordered phases occur or the existing ones expand beyond their classical stability regions. We use this method to obtain approximate phase diagrams of the two paradigmatic spin-12\frac{1}{2}21​ models on the honeycomb lattice: the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ ferro-antiferromagnetic and ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic XXZXXZXXZ models. For the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ case, various combinations of the XXZXXZXXZ anisotropies are analyzed. In a dramatic deviation from their classical phase diagrams, which host significant regions of the noncollinear spiral phases, quantum fluctuations stabilize several unconventional collinear phases and significantly extend conventional ones to completely supersede spiral states. These results are in close agreement with the available density-matrix renormalization group calculations. The applicability of this approach to the other models and its potential extension to different types of orders are discussed.
We study the spin-1/2 and spin-1 [Formula: see text] Heisenberg antiferromagnets (HAFs) on an infinite, anisotropic, two-dimensional triangular lattice, using the coupled cluster method. With respect to an underlying square-lattice geometry the model contains antiferromagnetic (J1 > 0) bonds between nearest neighbours and competing [Formula: see text] bonds between next-nearest-neighbours across only one of the diagonals of each square plaquette, the same diagonal in each square. In a topologically equivalent triangular-lattice geometry the model has two sorts of nearest-neighbour bonds, with [Formula: see text] bonds along parallel chains and with J1 bonds providing an interchain coupling. The model thus interpolates between an isotropic HAF on the square lattice at one extreme (κ = 0) and a set of decoupled chains at the other (κ → ∞), with the isotropic HAF on the triangular lattice in between at κ = 1. For the spin-1/2 [Formula: see text] model, we find a weakly first-order (or possibly second-order) quantum phase transition from a Néel-ordered state to a helical state at a first critical point at κc1 = 0.80 ± 0.01, and a second critical point at κc2 = 1.8 ± 0.4 where a first-order transition occurs between the helical state and a collinear stripe-ordered state. For the corresponding spin-1 model we find an analogous transition of the second-order type at κc1 = 0.62 ± 0.01 between states with Néel and helical ordering, but we find no evidence of a further transition in this case to a stripe-ordered phase.
We report on a systematic study of two-dimensional, periodic, frustrated Ising models with quantum dynamics introduced via a transverse magnetic field. The systems studied are the triangular and kagom\'e lattice antiferromagnets, fully frustrated models on the square and hexagonal (honeycomb) lattices, a planar analog of the pyrochlore antiferromagnet, a pentagonal lattice antiferromagnet, as well as two quasi-one-dimensional lattices that have considerable pedagogical value. All of these exhibit a macroscopic degeneracy at T=0T=0T=0 in the absence of the transverse field, which enters as a singular perturbation. We analyze these systems with a combination of a variational method at weak fields, a perturbative Landau-Ginzburg-Wilson approach from large fields, as well as quantum Monte Carlo simulations utilizing a cluster algorithm. Our results include instances of quantum order arising from classical criticality (triangular lattice) or classical disorder (pentagonal and probably hexagonal) as well as notable instances of quantum disorder arising from classical disorder (kagom\'e). We also discuss the effect of finite temperature, as well as the interplay between longitudinal and transverse fields---in the kagom\'e problem the latter gives rise to a nontrivial phase diagram with bond-ordered and bond-critical phases in addition to the disordered phase. We also note connections to quantum-dimer models and thereby to the physics of Heisenberg antiferromagnets in short-ranged resonating valence-bond phases that have been invoked in discussions of high-temperature superconductivity.
We study the ground-state (GS) properties of the frustrated spin-12\frac{1}{2}21​ J1{J}_{1}J1​--J2{J}_{2}J2​--J3{J}_{3}J3​ Heisenberg model on the two-dimensional honeycomb lattice with ferromagnetic nearest-neighbor (${J}_{1}=\ensuremath{-}1$) exchange and frustrating antiferromagnetic next-nearest-neighbor (J2>0{J}_{2}>0J2​>0) and next-next-nearest-neighbor (J3>0{J}_{3}>0J3​>0) exchanges, for the case J3=J2{J}_{3}={J}_{2}J3​=J2​. We use the coupled-cluster method implemented to high orders of approximation, complemented by the Lanczos exact diagonalization of a large finite lattice with 32 sites, in order to calculate the GS energy, magnetic order parameter, and spin-spin correlation functions. In one scenario we find a quantum phase transition point between regions characterized by ferromagnetic order and a form of antiferromagnetic (``striped'') collinear order at ${J}_{2}^{c}\ensuremath{\approx}0.1095\ifmmode\pm\else\textpm\fi{}0.0005$, which is below the corresponding hypothetical transition point at J2cl=17{J}_{2}^{\mathrm{cl}}=\frac{1}{7}J2cl​=71​ ($\ensuremath{\approx}0.143$) for the classical version of the model, in which we momentarily ignore the intervening noncollinear spiral phase in the region 110<J2<15\frac{1}{10}<{J}_{2}<\frac{1}{5}101​<J2​<51​. Hence we see that quantum fluctuations appear to stabilize somewhat the collinear antiferromagnetic order in preference to the ferromagnetic order in this model. We compare results for the present ferromagnetic case (with ${J}_{1}=\ensuremath{-}1$) to previous results for the corresponding antiferromagnetic case (with J1=+1{J}_{1}=+1J1​=+1). The magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter J2{J}_{2}J2​. However, there are considerable differences in the behavior of the order parameters for the two models for ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}0.6$. For example, the quasiclassical collinear magnetic long-range order for the antiferromagnetic model (with J1=+1{J}_{1}=+1J1​=+1) breaks down at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.60$, whereas the ``equivalent'' point for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) occurs at ${J}_{2}^{c}\ensuremath{\approx}0.11$. Unlike in the antiferromagnetic model (with J1=+1{J}_{1}=+1J1​=+1), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical antiferromagnetic phases (with N\'eel and striped order) for J2c1<J2<J2c2{J}_{2}^{{c}_{1}}<{J}_{2}<{J}_{2}^{{c}_{2}}J2c1​​<J2​<J2c2​​, with ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.47$, we find no clear indications at all in the ferromagnetic model for an intermediate magnetically disordered phase between the corresponding phases exhibiting ferromagnetic and striped order. Instead the evidence for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) points to one of two scenarios: either there is a direct first-order transition between the two magnetically ordered phases, as mentioned above; or there exists an intervening phase between them in the very narrow range $0.10\ensuremath{\lesssim}{J}_{2}\ensuremath{\lesssim}0.12$, which is probably a remnant of the spiral phase that exists in the classical counterpart of the model over the larger range 110<J2<15\frac{1}{10}<{J}_{2}<\frac{1}{5}101​<J2​<51​.
In this thesis, we investigate strongly correlated phenomena in quantum spin systems. In the first part of this work, we study geometrically frustrated antiferromagnets (AFMs). Generalizing the SU(2) Heisenberg Hamiltonian to Sp(N) symmetry, we obtain, in the large-N limit, the mean-field phase diagrams for the planar pyrochlore and cubic AFMs. We then use gauge theories to consider fluctuation effects about their respective mean-field configurations. We find, in addition to conventional Neel states, a plethora of novel magnetically disordered phases: two kinds of spin liquids, Z2 in 2+1D and U(1)in 3+1D, and several valence bond solids such as two and three-dimensional plaquette and columnar singlet states. We use the same approach to study the diamond lattice AFM which possesses extended classical ground state degeneracy. We demonstrate that quantum and entropic fluctuations lift this degeneracy in different ways. In the second part of the thesis, we study ultracold spinor atoms confined in optical lattices. We first demonstrate the feasibility of experimental realization of rotor models using ultracold spin-one Bose atoms in a spin-dependent and disordered optical lattice. We show that the ground state of such disordered rotor models with quadrupolar interactions can exhibit biaxial nematic ordering in the disorder-averaged sense, and suggest an imaging experiment to detect the biaxial nematicity in such systems. Finally, using variational wavefunction methods, we study the Mott phases and superfluid-insulator transition of spin-three bosons in an optical lattice with an anisotropic two dimensional optical trap. We chart out the phase diagrams for Mott states with n = 1 and n = 2 atoms per lattice site. We show that the long-range dipolar interaction stabilizes a state characterized by antiferromagnetic chains made of ferromagnetically aligned spins. We also obtain the mean-field phase boundary for the superfluid-insulator transition, and show that inside the superfluid phase and near the superfluid-insulator phase boundary, the system undergoes a first order antiferromagnetic-ferromagnetic spin ordering transition.