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Physica Scripta

Domain walls and magnetization plateaus in the finite-size anisotropic spin- 1 2 Heisenberg X X Z model in a longitudinal magnetic field

Vrushali Ramesh Kapase, Pradeep Thakur, P. Durganandini

Published Jul 27, 2026• 0 citations
Publisher

Abstract

Abstract We focus in this work on a numerical DMRG study of the one dimensional X X Z Heisenberg spin chain. We discuss the dependence of the energy spectrum, level crossings, magnetization, static structure factor, etc as a function of the magnetic field for finite size chains with periodic boundary condition and open boundary condition(OBC). We show that finite size effects lead to the emergence of a series of magnetization steps related to crossing of energy levels with unit change in the total z component of the spin. We also show interesting relations between the emergence of the magnetization steps and the appearance of domain walls in the local magnetization profiles with a fixed number of nodes. We connect the domain walls in the OBC case with the solitons of the sine-Gordon model with OBC.

DOI
10.1088/1402-4896/ae913a

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Abstract We focus in this work on a numerical DMRG study of the one dimensional X X Z Heisenberg spin chain. We discuss the dependence of the energy spectrum, level crossings, magnetization, static structure factor, etc as a function of the magnetic field for finite size chains with periodic boundary condition and open boundary condition(OBC). We show that finite size effects lead to the emergence of a series of magnetization steps related to crossing of energy levels with unit change in the total z component of the spin. We also show interesting relations between the emergence of the magnetization steps and the appearance of domain walls in the local magnetization profiles with a fixed number of nodes. We connect the domain walls in the OBC case with the solitons of the sine-Gordon model with OBC.
In this thesis, we deal with 1-dimensional anisotropic spin models more specifically with the chain and ladder geometries. These models are fundamental to the study of condensed matter theory and quantum magnetism because of their simplicity. Research into the chain model dates back to 1930s with Hans Bethe. One method to study these systems is through numerical analysis including renormalization group techniques. These allow for the diagonalization of the Hamiltonian without sacrificing computation to a large virtual Hilbert space. In this thesis, we present results for both the spin chain and spin ladder geometries in open boundary conditions. This work uses the density matrix renormalization group technique to calculate system energy. Initially we will present a study into the spin energy gap of the spin-\sfrac{1}{2} anisotropic (XXZ) Heisenberg chain in open boundary conditions (OBCs). The energy gap is shown to be reduced to half in the ground sector when compared to periodic boundary conditions (PBCs) due to edge effects. Secondly, a full phase diagram for the spin-\sfrac{1}{2} anisotropic Heisenberg ladder is presented which shows the emergence of a rich schematic with a variety of phases. Lastly, weak coupling limit maps are compared to quantum field theory phase transition predictions. This research was motivated by the lack of comprehensive phase diagram results for ladders and by real materials being investigated at the University of Kent.
We study the exact solution of the anisotropic spin-1/2 Heisenberg chain with a boundary magnetic field in the region where the bulk excitations are gapless. It is shown that near the boundary a bound state is created which is underscreened when \ensuremath{\gamma}/3 and which at low temperatures behaves like a single spin weakly coupled to the bulk. The IR fixed point in this case belongs to the universality class of the underscreened anisotropic Kondo model. We argue that the same fixed point appears in the boundary sine-Gordon model when the scaling dimension of the boundary term 2/31.
We discuss several problems concerning domain walls in the spin-S Ising model at zero temeprature in a magnetic field, H/(2S), applied in the x direction. Some results are also given for the planar (y-z) model in a transverse field. We treat the quantum problem in one dimension by perturbation theory at small H and numerically over a large range of H. We obtain the spin-density profile by fixing the spins at opposite ends of the chain to have opposite signs of Sz{\mathit{S}}_{\mathit{z}}Sz​. One dimensional is special in that there the quantum width of the wall is proportional to the size L of the system. We also study the quantitative features of the ``particle'' band which extends up to energies of order H above the ground state. Except for the planar limit, this particle band is well separated from excitations having energy J/S involving creation of more walls. At large S this particle band develops energy gaps and the lowest subband has tunnel splittings of order H${2}^{1\mathrm{\ensuremath{-}}2\mathit{S}}$. This scale of of energy gives rise to anomalous scaling with respect to (a) finite size, (b) temperature, or (c) random potentials. The intrinsic width of the domain wall and the pinning energy are also defined and calculated in certain limiting cases. The general conclusion is that quantum effects prevent the wall from being sharp and in higher dimension would prevent sudden excursions in the configuration of the wall.
At large anisotropies, the spectrum of the Heisenberg XXZ spin chain separates into ``bands'' with energies largely determined by the number of domain walls. The band structure is richer with open-boundary conditions: there are more bands and the bands develop intricate fine structures. We characterize and explain these structures and substructures in the open-boundary chain. The fine structures are explained using degenerate perturbation theory. We also present some dynamical consequences of these subband structures, through explicit time evolution of the wave function from initial states motivated by the fine-structure analysis.
The finite-size energy spectrum of the anisotropic Heisenberg chain in an external magnetic field is calculated for free and twisted boundary conditions. As with periodic boundary conditions, it is found that the spectra exhibit non-analytical terms which do not fit into the form predicted on the basis of conformal invariance unless extra commensurability conditions between the size of the system and the external field are introduced. Taking these conditions into account some scaling dimensions for associated models are derived.
We investigate the S=1/2 antiferromagnetic Heisenberg model with a 3$\times$3 checkerboard lattice structure in a longitudinal magnetic field. By using the stochastic series expansion quantum Monte Carlo (SSE-QMC) method, we obtain the properties of the non-plateau XY phase, 1/9, 3/9, 5/9, 7/9 magnetization plateau phases, and fully polarized phase. Then, we determine the precise phase transition critical points belonging to the 3D XY universality class through finite-size scaling. Moreover, we study the longitudinal and transverse dynamic spin structure factors of this model in different phases. For the non-plateau XY phase, the energy spectra present a gap between the low-energy gapless branch and the high-energy part under the competition of magnetic field and interaction. The gapless branch can be described by the spin wave theory in the canted antiferromagnetic phase of the effective "block spin" model. In the magnetization plateau phase, we identify that the excitation arises from localized disturbances within the sublattice, which are capable of spreading in momentum space. This study offers theoretical insights and interpretations for the characteristics of the ground state and the inelastic neutron scattering spectrum in two-dimensional quantum magnetic materials. Specifically, it focuses on materials with a checkerboard unit cell structure and odd-spin configurations under the influence of a longitudinal magnetic field.