On the long-range order of Lieb-Mattis model of quantum antiferromagnet

The spontaneous magnetization m and the root-mean-square order parameter ${\mathit{m}}_{0}$ of the Lieb-Mattis model for arbitrary temperatures and spin values s are obtained. For the ratio r(T,s)=m/ ${\mathit{m}}_{0}$ , the value r(T,s)= \ensuremath{\surd}3 is found.The spontaneous magnetization m and the root-mean-square order parameter ${\mathit{m}}_{0}$ of the Lieb-Mattis model for arbitrary temperatures and spin values s are obtained. For the ratio r(T,s)=m/${\mathit{m}}_{0}$, the value r(T,s)= \ensuremath{\surd}3 is found.

Physical review. B, Condensed matterJan 1, 19928 citations

Ground-State Long-Range Order of the Two-Dimensional Quantum Antiferromagnet

Yukiyasu Ozeki, Hidetoshi Nishimori, Yasuhiro Tomita

Springer series in synergeticsJan 1, 19890 citations

Long Range Order in Antiferromagnetic Quantum Spin Systems

之康尾関, Yukiyasu Ozeki

Jan 1, 19890 citations

Spontaneous symmetry breaking in the Lieb-Mattis model of antiferromagnetism

T. A. Kaplan, Wolfgang von der Linden, Peter Horsch

We calculate the order parameter m in the infinite-range isotropic-interaction quantum model of antiferromagnetism due to Lieb and Mattis (LM). The Hamiltonian ${\mathit{H}}_{1}$ in this model contains pure Heisenberg interactions. m is the mean value of the z component ${\mathit{m}}_{\mathit{s}}^{\mathit{z}}$ of the staggered spin per particle ${\mathbf{m}}_{\mathit{s}}$ in the ground state of ${\mathit{H}}_{1}$ plus a symmetry-breaking term ${\mathit{H}}_{\mathit{B}}$ due to an infinitesimal staggered field B in the thermodynamic limit (TL). We find a value of \ensuremath{\surd}3 for the ratio r=m/ ${\mathit{m}}_{0}$ , ${\mathit{m}}_{0}$ being the root-mean-square value of ${\mathit{m}}_{\mathit{s}}^{\mathit{z}}$ for B==0 in the TL. m and ${\mathit{m}}_{0}$ measure, respectively, the spontaneous symmetry breaking and the amount of long-range order in the symmetry-unbroken state. This value is expected on the basis of our recent argument that r= \ensuremath{\surd}3 if ${\mathbf{m}}_{\mathit{s}}$ behaves in the TL as a classical vector with no magnitude fluctuations---behavior that we show holds for the Lieb-Mattis model. The addition to ${\mathit{H}}_{1}$ of a small easy-axis anisotropic term is also discussed briefly. These studies allow calculation of the size dependence of quantities that we have recently shown to become lower bounds on m in the TL (bounds that are valid for short-range-interaction models as well). In the LM model, low-lying energy eigenstates with excitation energies of order ${\mathit{N}}^{\mathrm{\ensuremath{-}}1}$, where N is the number of spins, form a continuum (with no gap) in the TL. These are closely related to similar states shown to exist for the nearest-neighbor isotropic Heisenberg antiferromagnet. We show that this continuum has a density of states that is O( ${\mathit{N}}^{0}$ ); i.e., it is not extensive.We calculate the order parameter m in the infinite-range isotropic-interaction quantum model of antiferromagnetism due to Lieb and Mattis (LM). The Hamiltonian ${\mathit{H}}_{1}$ in this model contains pure Heisenberg interactions. m is the mean value of the z component ${\mathit{m}}_{\mathit{s}}^{\mathit{z}}$ of the staggered spin per particle ${\mathbf{m}}_{\mathit{s}}$ in the ground state of ${\mathit{H}}_{1}$ plus a symmetry-breaking term ${\mathit{H}}_{\mathit{B}}$ due to an infinitesimal staggered field B in the thermodynamic limit (TL). We find a value of \ensuremath{\surd}3 for the ratio r=m/${\mathit{m}}_{0}$, ${\mathit{m}}_{0}$ being the root-mean-square value of ${\mathit{m}}_{\mathit{s}}^{\mathit{z}}$ for B==0 in the TL. m and ${\mathit{m}}_{0}$ measure, respectively, the spontaneous symmetry breaking and the amount of long-range order in the symmetry-unbroken state. This value is expected on the basis of our recent argument that r= \ensuremath{\surd}3 if ${\mathbf{m}}_{\mathit{s}}$ behaves in the TL as a classical vector with no magnitude fluctuations---behavior that we show holds for the Lieb-Mattis model. The addition to ${\mathit{H}}_{1}$ of a small easy-axis anisotropic term is also discussed briefly. These studies allow calculation of the size dependence of quantities that we have recently shown to become lower bounds on m in the TL (bounds that are valid for short-range-interaction models as well). In the LM model, low-lying energy eigenstates with excitation energies of order ${\mathit{N}}^{\mathrm{\ensuremath{-}}1}$, where N is the number of spins, form a continuum (with no gap) in the TL. These are closely related to similar states shown to exist for the nearest-neighbor isotropic Heisenberg antiferromagnet. We show that this continuum has a density of states that is O(${\mathit{N}}^{0}$); i.e., it is not extensive.

Physical review. B, Condensed matterSep 1, 199036 citations

Coexistence of the ferromagnetic and antiferromagnetic long-range orders in the generalized antiferromagnetic Heisenberg model on a bipartite lattice

Guang-Shan Tian

The concept of ferrimagnetism was first proposed by Neel to explain why some materials have a macroscopic magnetization but no ferromagnetic long-range order, when the temperature T is lower than a phase transition temperature T c . In this article, based on a theorem of Lieb and Mattis, we show in a mathematically rigorous way that the global ground states of the generalized antiferromagnetic Heisenberg model on a bipartite lattice with unequal sublattice points have both ferromagnetic and antiferromagnetic long-range orders with the latter being predominant. Our rigorous results conform to Neel's theory.The concept of ferrimagnetism was first proposed by Neel to explain why some materials have a macroscopic magnetization but no ferromagnetic long-range order, when the temperature T is lower than a phase transition temperature T c . In this article, based on a theorem of Lieb and Mattis, we show in a mathematically rigorous way that the global ground states of the generalized antiferromagnetic Heisenberg model on a bipartite lattice with unequal sublattice points have both ferromagnetic and antiferromagnetic long-range orders with the latter being predominant. Our rigorous results conform to Neel's theory.

Journal of Physics A Mathematical and GeneralApr 7, 199422 citations

The spontaneous magnetization m and the root-mean-square order parameter

{\mathit{m}}_{0}

of the Lieb-Mattis model for arbitrary temperatures and spin values s are obtained. For the ratio r(T,s)=m/

{\mathit{m}}_{0}

, the value r(T,s)= \ensuremath{\surd}3 is found.

We calculate the order parameter m in the infinite-range isotropic-interaction quantum model of antiferromagnetism due to Lieb and Mattis (LM). The Hamiltonian

{\mathit{H}}_{1}

in this model contains pure Heisenberg interactions. m is the mean value of the z component

{\mathit{m}}_{\mathit{s}}^{\mathit{z}}

of the staggered spin per particle

{\mathbf{m}}_{\mathit{s}}

in the ground state of

{\mathit{H}}_{1}

plus a symmetry-breaking term

{\mathit{H}}_{\mathit{B}}

due to an infinitesimal staggered field B in the thermodynamic limit (TL). We find a value of \ensuremath{\surd}3 for the ratio r=m/

{\mathit{m}}_{0}

{\mathit{m}}_{0}

being the root-mean-square value of

{\mathit{m}}_{\mathit{s}}^{\mathit{z}}

for B==0 in the TL. m and

{\mathit{m}}_{0}

measure, respectively, the spontaneous symmetry breaking and the amount of long-range order in the symmetry-unbroken state. This value is expected on the basis of our recent argument that r= \ensuremath{\surd}3 if

{\mathbf{m}}_{\mathit{s}}

behaves in the TL as a classical vector with no magnitude fluctuations---behavior that we show holds for the Lieb-Mattis model. The addition to

{\mathit{H}}_{1}

of a small easy-axis anisotropic term is also discussed briefly. These studies allow calculation of the size dependence of quantities that we have recently shown to become lower bounds on m in the TL (bounds that are valid for short-range-interaction models as well). In the LM model, low-lying energy eigenstates with excitation energies of order ${\mathit{N}}^{\mathrm{\ensuremath{-}}1}$, where N is the number of spins, form a continuum (with no gap) in the TL. These are closely related to similar states shown to exist for the nearest-neighbor isotropic Heisenberg antiferromagnet. We show that this continuum has a density of states that is O(

{\mathit{N}}^{0}

); i.e., it is not extensive.