Physical review. B, Condensed matter

Long-range order of the Lieb-Mattis model of a quantum antiferromagnet

I. G. Gochev, N. S. Tonchev

Published Jan 1, 1992• 8 citations

Abstract

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.

DOI: 10.1103/physrevb.45.480

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Physical review. B, Condensed matter

Long-range order of the Lieb-Mattis model of a quantum antiferromagnet

I. G. Gochev, N. S. Tonchev

Published Jan 1, 1992• 8 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevb.45.480

Related papers

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.

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The existence of long-range order is proved under certain conditions for the antiferromagnetic XYZ model on the simple cubic or the square lattice. In particular, the spin-1/2 XXZ model on the square lattice is shown to have ground-state long-range order if the exchange anisotropy Δ satisfies 0≦ Δ <0.20 or Δ >1.72, which improves the result of Kubo and Kishi. The existence of long-range order of the z -component of the spin operator is proved for the XXZ model with XY -like anisotropy (0≦ Δ ≦1) under certain conditions. A similar result is shown to hold for the long-range order in the x -direction for the Ising-like model ( Δ ≧1). The XXZ model on the two-dimensional hexagonal lattice is proved to have finite ground-state long-range order for any value of Δ (≧0) if S ≧1 and for Δ >2.55 when S =1/2.The existence of long-range order is proved under certain conditions for the antiferromagnetic XYZ model on the simple cubic or the square lattice. In particular, the spin-1/2 XXZ model on the square lattice is shown to have ground-state long-range order if the exchange anisotropy Δ satisfies 0≦ Δ <0.20 or Δ >1.72, which improves the result of Kubo and Kishi. The existence of long-range order of the z -component of the spin operator is proved for the XXZ model with XY -like anisotropy (0≦ Δ ≦1) under certain conditions. A similar result is shown to hold for the long-range order in the x -direction for the Ising-like model ( Δ ≧1). The XXZ model on the two-dimensional hexagonal lattice is proved to have finite ground-state long-range order for any value of Δ (≧0) if S ≧1 and for Δ >2.55 when S =1/2.

Journal of the Physical Society of JapanJan 15, 198927 citations

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Tigran Hakobyan

The Lieb—Mattis theorem about ordering of energy levels is generalized to frustrated antiferromagnetic spin-1/2 ladder model with diagonal and four-spin interaction.

Physics of Atomic NucleiFeb 1, 20102 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.

We prove the bound

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

(\ensuremath{\beta})\ensuremath{\ge} \ensuremath{\surd}3 \ensuremath{\sigma}(\ensuremath{\beta}) between the spontaneous staggered magnetization

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

(\ensuremath{\beta}) and the long range order parameter \ensuremath{\sigma}(\ensuremath{\beta}) in the quantum Heisenberg antiferromagnets. The same bound has been known for the Heisenberg ferromagnets from the works of Griffiths, and of Dyson, Lieb, and Simon, but extensions to the antiferromagnets had been lacking for more than two decades. When combined with the results on nonvanishing \ensuremath{\sigma}(\ensuremath{\beta}) from the Dyson-Lieb-Simon method, our bound proves the existence of symmetry breaking in the physically natural equilibrium states and ground states obtained by applying an infinitesimal symmetry breaking field.