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Physical review. B, Condensed matter

Implications of direct-product ground states in the one-dimensional quantumXYZandXYspin chains

Gerhard H. Muller, Robert E. Shrock

Published Nov 1, 1985• 92 citations
Publisher

Abstract

We state the conditions under which the general spin-s quantum XYZ ferromagnet (${H}_{\mathrm{\ensuremath{-}}}$) and antiferromagnet (${H}_{+}$) with an external magnetic field along one axis, specified by the Hamiltonian ${H}_{\ifmmode\pm\else\textpm\fi{}}$=\ifmmode\pm\else\textpm\fi{} ${\mathcal{J}}_{l=1}^{N}$ (${J}_{x}$${S}_{l}^{x}$${S}_{l+1}^{x}$+${J}_{y}$${S}_{l}^{y}$S $_{l+1}^{y}$+${J}_{z}$${S}_{l}^{z}$${S}_{l+1}^{z}$)-h ${\mathcal{J}}_{l=1}^{N}$${S}_{l}^{z}$ exhibits a fully ordered ground state described by a wave function which is a direct product of single-site wave functions. We present a detailed analysis of the implications for the zero-temperature dynamical properties of this model. In particular, we derive a rigorous relation between the three dynamic structure factors ${S}_{\ensuremath{\mu}\ensuremath{\mu}}$(q,\ensuremath{\omega}), \ensuremath{\mu}=x,y,z at T=0. For the special case of the s=(1/2) anisotropic XY model (${J}_{z}$=0), these relations are used to determine the dynamic structure factors ${S}_{\mathrm{xx}}$(q,\ensuremath{\omega}) and ${S}_{\mathrm{yy}}$(q,\ensuremath{\omega}) at T=0 and h=(${J}_{x}$${J}_{y}$${)}^{1/2}$ in terms of the known dynamic structure factor ${S}_{\mathrm{zz}}$(q,\ensuremath{\omega}).

DOI
10.1103/physrevb.32.5845

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We state the conditions under which the general spin-s quantum XYZ ferromagnet (H\ensuremath−{H}_{\mathrm{\ensuremath{-}}}H\ensuremath−​) and antiferromagnet (H+{H}_{+}H+​) with an external magnetic field along one axis, specified by the Hamiltonian H\ifmmode±\else\textpm\fi{H}_{\ifmmode\pm\else\textpm\fi{}}H\ifmmode±\else\textpm\fi​=\ifmmode\pm\else\textpm\fi{} Jl=1N{\mathcal{J}}_{l=1}^{N}Jl=1N​ ({J}_{x}$${S}_{l}^{x}$${S}_{l+1}^{x}+{J}_{y}$${S}_{l}^{y}S l+1y_{l+1}^{y}l+1y​+{J}_{z}$${S}_{l}^{z}$${S}_{l+1}^{z})-h {\mathcal{J}}_{l=1}^{N}$${S}_{l}^{z} exhibits a fully ordered ground state described by a wave function which is a direct product of single-site wave functions. We present a detailed analysis of the implications for the zero-temperature dynamical properties of this model. In particular, we derive a rigorous relation between the three dynamic structure factors S\ensuremathμ\ensuremathμ{S}_{\ensuremath{\mu}\ensuremath{\mu}}S\ensuremathμ\ensuremathμ​(q,\ensuremath{\omega}), \ensuremath{\mu}=x,y,z at T=0. For the special case of the s=(1/2) anisotropic XY model (Jz{J}_{z}Jz​=0), these relations are used to determine the dynamic structure factors Sxx{S}_{\mathrm{xx}}Sxx​(q,\ensuremath{\omega}) and Syy{S}_{\mathrm{yy}}Syy​(q,\ensuremath{\omega}) at T=0 and h=({J}_{x}$${J}_{y}$${)}^{1/2} in terms of the known dynamic structure factor Szz{S}_{\mathrm{zz}}Szz​(q,\ensuremath{\omega}).
We have found the exact (factorized) ground state of a general class of ferrimagnets in the presence of a magnetic field which includes the anisotropic and long-range interactions for arbitrary dimensional space. In particular cases our model represents many spin chains with bond alternation of antiferromagnetic-ferromagnetic coupling, ferrimagnetic spin ladders, and also homogeneous spin-sss models. The factorized ground state is a product of single-particle kets on a bipartite lattice composed of two different spins $(\ensuremath{\rho},\ensuremath{\sigma})$ which is characterized by two angles, a biangle state. The spin-wave analysis around the exact ground state shows two branch of excitations which are the origin of two dynamics of the model. The signature of these dynamics is addressed as a peak and a broaden bump in the specific heat.
We report the exact dimer phase, in which the ground states are described by product of singlet dimer, in the extended XYZ model by generalizing the isotropic Majumdar-Ghosh model to the fully anisotropic region. We demonstrate that this phase can be realized even in models when antiferromagnetic interaction along one of the three directions. This model also supports three different ferromagnetic (FM) phases, denoted as x-FM, y-FM and z-FM, polarized along the three directions. The boundaries between the exact dimer phase and FM phases are infinite-fold degenerate. The breaking of this infinite-fold degeneracy by either translational symmetry breaking or [Formula: see text] symmetry breaking leads to exact dimer phase and FM phases, respectively. Moreover, the boundaries between the three FM phases are critical with central charge [Formula: see text] for free fermions. We characterize the properties of these boundaries using entanglement entropy, excitation gap, and long-range spin-spin correlation functions. These results are relevant to a large number of one dimensional magnets, in which anisotropy is necessary to isolate a single chain out from the bulk material. We discuss the possible experimental signatures in realistic materials with magnetic field along different directions and show that the anisotropy may resolve the disagreement between theory and experiments based on isotropic spin-spin interactions.
The spin fluctuations parallel to the external magnetic field in the ground state of the one-dimensional (1D) s=12s=\frac{1}{2}s=21​ Heisenberg antiferromagnet are dominated by a two-parameter set of collective excitations. In a cyclic chain of N sites and magnetization 0<Mz<N/2,0<{M}_{z}<N/2,0<Mz​<N/2, the ground state, which contains 2Mz{2M}_{z}2Mz​ spinons, is reconfigured as the physical vacuum for a different species of quasiparticles, identifiable in the framework of the coordinate Bethe ansatz by characteristic configurations of Bethe quantum numbers. The dynamically dominant excitations are found to be scattering states of two such quasiparticles. For $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty},$ these collective excitations form a continuum in $(q,\ensuremath{\omega})$ space with an incommensurate soft mode. Their matrix elements in the dynamic spin structure factor ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ are calculated directly from the Bethe wave functions for finite N. The resulting line- shape predictions for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ complement the exact results previously derived via algebraic analysis for the exact two-spinon part of ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ in the zero-field limit. They are relevant for neutron-scattering experiments on quasi-1D antiferromagnetic compounds in a strong magnetic field.
We report the exact dimer phase, in which the ground states are described by product of singlet dimer, in the extended XYZ model by direct generalizing the isotropic Majumdar-Ghosh model to the fully anisotropic regime. This model also supports three different ferromagnetic (FM) phases, denoted as xxx-FM, yyy-FM and zzz-FM. The boundaries between the dimer phase and FM phases are infinite-fold degenerate. The breaking of this infinite-fold degeneracy by either translational symmetry or Z2\mathbb{Z}_2Z2​ symmetry lead to dimer phase and FM phases, respectively. Moreover, the boundaries between the three FM phases are critical with central charge c=1c=1c=1. We characterize the properties of these boundaries using entanglement entropy, excitation gap and long-range spin-spin correlations. These results are relevant to a large number of quasi-one-dimensional magnets, in which anisotropy is necessary to isolate a single chain out from the bulk material. We discuss the possible experimental signatures in true materials with magnetic field along different directions, and show that the anisotropy may resolve the long-standing disagreements between theory and experiments.
One-dimensional quantum spin-1/2 XY models admit the rigorous analysis not only of their static properties (i.e. the thermodynamic quantities and the equal-time spin correlation functions) but also of their dynamic properties (i.e. the different-time spin correlation functions, the dynamic susceptibilities, the dynamic structure factors).This becomes possible after exploiting the Jordan-Wigner transformation which reduces the spin model to a model of spinless noninteracting fermions.A number of dynamic quantities (e.g.related to transverse spin operator or dimer operator fluctuations) are entirely determined by two-fermion excitations and can be examined in much detail.We consider the spin-1/2 XY chain in a transverse ( z) magnetic field with the Hamiltonianand calculate the dynamic structure factorsis the dimer operator.The results for the dynamic transverse structure factor S zz (κ, ω) and for the dynamic dimer structure factor S DD (κ, ω) are known, whereas the analysis of the dynamic structure factor S zD (κ, ω) = (S Dz (κ, ω)) has not been reported so far.We compare different two-fermion dynamic quantities contrasting their generic and specific features.