Using large-scale density-matrix renormalzation group calculations and minimally augmented spin-wave theory, we demonstrate that the phase diagram of the quantum
S=21J1--
J3 ferro-antiferromagnetic model on the honeycomb lattice differs dramatically from the classical one. It hosts the double-zigzag and Ising-
z phases as unexpected intermediaries between ferromagnetic and zigzag states that are also extended beyond their classical regions of stability. In broad agreement with quantum order-by-disorder arguments, these collinear phases replace the classical spiral state.
We study the ground-state (GS) properties of the frustrated spin-
21 J1--
J2--
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) and next-next-nearest-neighbor (
J3>0) exchanges, for the case
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=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
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). The magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter
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) 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), 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, 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
101<J2<51.
Strongly correlated systems with geometric frustrations can host the emergent phases of matter with unconventional properties. Here, we study the spin
S=1 Heisenberg model on the honeycomb lattice with the antiferromagnetic first- (
J1) and second-neighbor (
J2) interactions ($0.0\ensuremath{\le}{J}_{2}/{J}_{1}\ensuremath{\le}0.5$) by means of density-matrix renormalization group. In the parameter regime ${J}_{2}/{J}_{1}\ensuremath{\lesssim}0.27$, the system sustains a N\'eel antiferromagnetic phase. At the large
J2 side ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.32$, a stripe antiferromagnetic phase is found. Between the two magnetic ordered phases $0.27\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.32$, we find a nonmagnetic intermediate region with a plaquette valence-bond order. Although our calculations are limited within a six unit-cell width on cylinder, we present evidence that this plaquette state could be a strong candidate for this nonmagnetic region in the thermodynamic limit. We also briefly discuss the nature of the quantum phase transitions in the system. We gain further insight into the nonmagnetic phases in the spin-1 system by comparing its phase diagram with the spin-
1/2 system.
We use the coupled-cluster method in high orders of approximation to make a comprehensive study of the ground-state (GS) phase diagram of the spin-1/2
J1-
J2-
J3 model on a two-dimensional honeycomb lattice with antiferromagnetic (AFM) interactions up to third-nearest neighbors. Results are presented for the GS energy and the average local onsite magnetization. With the nearest-neighbor coupling strength ${J}_{1}\ensuremath{\equiv}1$, we find four magnetically ordered phases in the parameter window ${J}_{2},{J}_{3}\ensuremath{\in}[0,1]$, namely, the N\'eel, striped, and N\'eel-II collinear AFM phases, plus a spiral phase. The N\'eel-II phase appears as a stable GS phase in the classical version of the model only for values
J3<0. Each of these four ordered phases shares a boundary with a disordered quantum paramagnetic (QP) phase, and at several widely separated points on the phase boundaries the QP phase has an infinite susceptibility to plaquette valence-bond crystalline order. We identify all of the phase boundaries with good precision in the parameter window studied, and we find three tricritical quantum critical points therein at (a) $({J}_{2}^{{c}_{1}},{J}_{3}^{{c}_{1}})=(0.51\ifmmode\pm\else\textpm\fi{}0.01,0.69\ifmmode\pm\else\textpm\fi{}0.01)$ between the N\'eel, striped, and QP phases; (b) $({J}_{2}^{{c}_{2}},{J}_{3}^{{c}_{2}})=(0.65\ifmmode\pm\else\textpm\fi{}0.02,0.55\ifmmode\pm\else\textpm\fi{}0.01)$ between the striped, spiral, and QP phases; and (c) $({J}_{2}^{{c}_{3}},{J}_{3}^{{c}_{3}})=(0.69\ifmmode\pm\else\textpm\fi{}0.01,0.12\ifmmode\pm\else\textpm\fi{}0.01)$ between the spiral, N\'eel-II, and QP phases.
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.
Searching for spin-liquid states has long been attracting both experimentalists and theorists. In this paper, we review recent density matrix renormalization group studies of the spin-½ XY model on the honeycomb lattice, with first-neighbor (J1 = 1) and frustrating second-neighbor (J2 > 0) interactions. For the intermediate frustration regime 0.22 ≲ J2 ≲ 0.36, there exists a surprising antiferromagnetic Ising phase, with ordered moments pointing along the z-axis, despite the absence of any SzSz interactions in the Hamiltonian. Surrounding this phase as a function of J2 are antiferromagnetic phases with the moments pointing in the xy-plane for small J2 and a close competition between an xy-plane magnetic collinear phase and a dimer phase for large values of J2. No spin-liquid phases was found in the XY model even with the third-neighbor (J3 > 0) interactions.