This thesis presents a theoretical study of the Heisenberg model on a two dimensional lattice with asymmetric couplings along x and y directions. By gradually varying the coupling along y-direction we traverse the 1D, quasi 1D and 2D domains of this model. Primary focus of the research revolves around exploring the merits and de-merits of the staggered-flux state, a prototypical disordered resonating valance bond state, as a candidate wavefunction within this domain. Novel features like the fractionalization of spin-1 `magnon' excitations into spin-1/2 `spinon' excitations, which qualitatively explain the experimentally observed
(π,0) quantum anomaly, demand further investigation into this novel wave function. Theoretical framework involving construction of a Gutzwiller projected staggered flux state, a wavefunction depending on variational parameters, is developed. A numerical method based on MonteCarlo simulation is used to optimize the variational wave function and calculate the relevant physical observables. Exploring the gradual evolution of these observables with the coupling ratio
γ=Jy/Jx provides us with a clear picture of the evolution of spin excitations on this RVB state, as the dimensionality of the model is changed from 1D to 2D.
Fractionalization remains one of the most fascinating manifestations of strong interactions in quantum many-body systems. In quantum magnetism, the existence of spinons—collective magnetic excitations that behave as quasiparticles with fractional quantum numbers—is proven in spin chains, but the criteria for their appearance in higher dimensions remain disputed. Motivated by experiments reporting the observation of spinons at high energies in the square-lattice Heisenberg antiferromagnet, we adopt the approach of extrapolating from where spinons are well defined. We study the dynamical properties of a Gutzwiller-projected wave function, the staggered-flux state, on a rectangular spin-1/2 Heisenberg lattice as a function of the spatial coupling ratio, γ = J y / J x . By studying the spectrum and the spinon separation distribution we show how, as the system evolves from one-dimensional (1D) towards 2D, the spinons become progressively more confined over most of reciprocal space, but remain deconfined at specific wave vectors.
We present a systematic study of the anisotropic spin-1/2 Heisenberg model in staggered magnetic fields in two dimensions. To mimic real materials, we consider a system of coupled, antiferromagnetic chains, whose interchain interaction can be either ferro- or antiferromagnetic. When the staggered field is commensurate with the magnetic interactions, an energy gap opens immediately and follows a power law as a function of the applied field, similar to the situation in one dimension. When the field competes with the interactions, a quantum phase transition (QPT) occurs from a gapless, magnetically ordered phase at low fields to a gapped, disordered regime. We use a continuous-time Monte Carlo method to compute the staggered moment of the ordered phases and the spin gap of the disordered phases. We deduce the phase diagrams as functions of the anisotropy ratio and the applied field, and calculate the scaling behavior of the models in both quantities. We show that in the competitive case, the staggered field acts to maintain a regime of quasi-1D behavior around the QPT, and we discuss as a consequence the nature of the crossover from one-dimensional (1D) to 2D physics.
The ground-state properties of the spin-1/2 antiferromagnetic Heisenberg model with spatially anisotropic couplings on a square lattice are investigated by a spin-rotation-invariant Green's-function approach and by Lanczos diagonalizations on lattices up to 36 sites supplemented by finite-size scaling. We focus on the anisotropy-driven transition from the N\'eel state to a paramagnetic state with antiferromagnetic short-range order and on the spatial dependence of spin correlation functions. Our principal result is that a rather sharp crossover in the magnetic behavior occurs at the coupling ratio ${R}_{0}\ensuremath{\simeq}0.2 {(R=J}_{y}{/J}_{x}).$
The ground-state properties of the spin-(1/2 Heisenberg antiferromagnet on a square lattice are studied by using a simple variational wave function that interpolates continuously between the N\'eel state and short-range resonating-valence-bond states. Exact calculations of the variational energy for small systems show that the state with the lowest energy has long-range antiferromagnetic order. The staggered magnetization in this state is approximately 70% of its maximum possible value. The variational estimate of the ground-state energy is substantially lower than the value obtained for the nearest-neighbor resonating-valence-bond wave function.
We study the spin-
21 antiferromagnetic Heisenberg model on an $\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}N$ square lattice for even
N's up to 14. Previously, the nonlinear sigma model perturbatively predicted that its spin-rotational symmetry breaks asymptotically with $N\ensuremath{\rightarrow}\ensuremath{\infty}$, i.e., when it becomes two dimensional (2D). However, we identify a critical width
Nc=10 for which this symmetry breaks spontaneously. It shows the signature of a dimensional transition from one dimensional (1D) including quasi-1D to 2D. The finite-size effect differs from that of the $N\ifmmode\times\else\texttimes\fi{}N$ lattice. The ground-state (GS) energy per site approaches the thermodynamic limit value, in agreement with the previously accepted value, by one order of
1/N faster than when using $N\ifmmode\times\else\texttimes\fi{}N$ lattices in the literature. Methodwise, we build and variationally solve a matrix product state (MPS) on a chain, converting the
N sites in each rung into an effective site. We show that the area law of entanglement entropy does not apply when
N increases in our method and the reduced density matrix of each effective site has a saturating number of dominant diagonal elements with increasing
N. These two characteristics make the MPS rank needed to obtain a desired energy accuracy quickly saturate when
N is large, making our algorithm efficient for large
N's. Furthermore, the latter enables space reduction in MPS. Within the framework of MPS, we prove a theorem that the spin-spin correlation at infinite separation is the square of staggered magnetization and demonstrate that the eigenvalue structure of a building MPS unit of $\ensuremath{\langle}g|g\ensuremath{\rangle},|g\ensuremath{\rangle}$ being the GS is responsible for order, disorder, and quasi-long-range order.