Physical Review

One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System

Chongming Yang, Chuan‐Zhang Yang

Published Oct 7, 1966• 733 citations

Abstract

The ground-state energy $2f$ per lattice site for an infinite system is studied as a function of $\ensuremath{\Delta}$ and of the magnetization $y$. Analyticity properties of $f(\ensuremath{\Delta}, y)$ are proved. The behavior of $f(\ensuremath{\Delta}, y)$ at and near $y=0$ and $y=1$ are investigated.

DOI: 10.1103/physrev.150.327

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One-Dimensional Chain of Anisotropic Spin-Spin Interactions. III. Applications

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Physical Review

One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System

Chongming Yang, Chuan‐Zhang Yang

Published Oct 7, 1966• 733 citations

Publisher

Abstract

The ground-state energy

2f

per lattice site for an infinite system is studied as a function of

\ensuremath{\Delta}

and of the magnetization

y

. Analyticity properties of

f(\ensuremath{\Delta}, y)

are proved. The behavior of

f(\ensuremath{\Delta}, y)

at and near

y=0

and

y=1

are investigated.

DOI: 10.1103/physrev.150.327

Related papers

One-Dimensional Chain of Anisotropic Spin-Spin Interactions. III. Applications

Chongming Yang, Chuan‐Zhang Yang

The magnetization---versus---magnetic-field curve of the one-dimensional chain at $T=0$ is discussed. Also, the corresponding $p\ensuremath{-}V$ diagram of the quantum lattice gas (q.l.g.) is discussed. The ground-state wave function for the case $\ensuremath{\Delta}>1$, in a q.l.g. of a finite number of particles on an infinite lattice is analyzed in detail, giving the "surface energy" near the ends. In the Appendix some general properties of the q.l.g. in one, two, or three dimensions are given.The magnetization---versus---magnetic-field curve of the one-dimensional chain at $T=0$ is discussed. Also, the corresponding $p\ensuremath{-}V$ diagram of the quantum lattice gas (q.l.g.) is discussed. The ground-state wave function for the case $\ensuremath{\Delta}>1$, in a q.l.g. of a finite number of particles on an infinite lattice is analyzed in detail, giving the "surface energy" near the ends. In the Appendix some general properties of the q.l.g. in one, two, or three dimensions are given.

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Satoshi Fujiki, Donald D. Betts

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Zero-temperature properties of quantum spin models on the triangular lattice II: the antiferromagnet

Satoshi Fujiki, Donald D. Betts

The calculation of two- and four-spin correlations of the [Formula: see text] antiferromagnet has been extended to an N = 21 site triangular lattice. By fitting a quadratic in 1/N to the nearest neighbour transverse spin correlations, we have estimated the ground-state energy per bond on the infinite lattice to be E0/3NJ = −0.2716 ± 0.005. The nearest neighbour longitudinal correlation is estimated to be [Formula: see text]. A short-range order parameter, the chirality, is defined and estimated for the infinite lattice. From the N dependence of the three sublattice or helical magnetization, the spin correlation is conjectured to decay algebraically as [Formula: see text].The calculation of two- and four-spin correlations of the [Formula: see text] antiferromagnet has been extended to an N = 21 site triangular lattice. By fitting a quadratic in 1/N to the nearest neighbour transverse spin correlations, we have estimated the ground-state energy per bond on the infinite lattice to be E0/3NJ = −0.2716 ± 0.005. The nearest neighbour longitudinal correlation is estimated to be [Formula: see text]. A short-range order parameter, the chirality, is defined and estimated for the infinite lattice. From the N dependence of the three sublattice or helical magnetization, the spin correlation is conjectured to decay algebraically as [Formula: see text].

Canadian Journal of PhysicsJan 1, 198725 citations

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H. Q. Lin, Victor J. Emery

Physical review. B, Condensed matterAug 1, 198911 citations

Magnetic Properties of Two-Dimensional Dipolar Squares: Boundary Geometry Dependence

Ryoko Sugano, Katsuyoshi Matsushita, Akiyoshi Kuroda +2 more

By means of the molecular dynamics simulation on gradual cooling processes, we investigate magnetic properties of classical spin systems only with the magnetic dipole-dipole interaction, which we call dipolar systems. Focusing on their finite-size effect, particularly their boundary geometry dependence, we study two finite dipolar squares cut out from a square lattice with $Φ=0$ and $π/4$ , where $Φ$ is an angle between the direction of the lattice axis and that of the square boundary. Distinctly different results are obtained in the two dipolar squares. In the $Φ=0$ square, the ``from-edge-to-interior freezing'' of spins is observed. Its ground state has a multi-domain structure whose domains consist of the two among infinitely (continuously) degenerated Luttinger-Tisza (LT) ground-state orders on a bulk square lattice, i.e., the two antiferromagnetically aligned ferromagnetic chains (af-FMC) orders directed in parallel to the two lattice axes. In the $Φ=π/4$ square, on the other hand, the freezing starts from the interior of the square, and its ground state is nearly in a single domain with one of the two af-FMC orders. These geometry effects are argued to originate from the anisotropic nature of the dipole-dipole interaction which depends on the relative direction of sites in a real space of the interacting spins.By means of the molecular dynamics simulation on gradual cooling processes, we investigate magnetic properties of classical spin systems only with the magnetic dipole-dipole interaction, which we call dipolar systems. Focusing on their finite-size effect, particularly their boundary geometry dependence, we study two finite dipolar squares cut out from a square lattice with $Φ=0$ and $π/4$, where $Φ$ is an angle between the direction of the lattice axis and that of the square boundary. Distinctly different results are obtained in the two dipolar squares. In the $Φ=0$ square, the ``from-edge-to-interior freezing'' of spins is observed. Its ground state has a multi-domain structure whose domains consist of the two among infinitely (continuously) degenerated Luttinger-Tisza (LT) ground-state orders on a bulk square lattice, i.e., the two antiferromagnetically aligned ferromagnetic chains (af-FMC) orders directed in parallel to the two lattice axes. In the $Φ=π/4$ square, on the other hand, the freezing starts from the interior of the square, and its ground state is nearly in a single domain with one of the two af-FMC orders. These geometry effects are argued to originate from the anisotropic nature of the dipole-dipole interaction which depends on the relative direction of sites in a real space of the interacting spins.

arXiv:cond-mat/0701467Journal of the Physical Society of JapanMar 26, 20075 citations

The ground-state energy

2f

per lattice site for an infinite system is studied as a function of

\ensuremath{\Delta}

and of the magnetization

y

. Analyticity properties of

f(\ensuremath{\Delta}, y)

are proved. The behavior of

f(\ensuremath{\Delta}, y)

at and near

y=0

and

y=1

are investigated.

The magnetization---versus---magnetic-field curve of the one-dimensional chain at

T=0

is discussed. Also, the corresponding $p\ensuremath{-}V$ diagram of the quantum lattice gas (q.l.g.) is discussed. The ground-state wave function for the case $\ensuremath{\Delta}>1$, in a q.l.g. of a finite number of particles on an infinite lattice is analyzed in detail, giving the "surface energy" near the ends. In the Appendix some general properties of the q.l.g. in one, two, or three dimensions are given.

Φ=0

and

π/4

, where

Φ

is an angle between the direction of the lattice axis and that of the square boundary. Distinctly different results are obtained in the two dipolar squares. In the

Φ=0

square, the ``from-edge-to-interior freezing'' of spins is observed. Its ground state has a multi-domain structure whose domains consist of the two among infinitely (continuously) degenerated Luttinger-Tisza (LT) ground-state orders on a bulk square lattice, i.e., the two antiferromagnetically aligned ferromagnetic chains (af-FMC) orders directed in parallel to the two lattice axes. In the

Φ=π/4

square, on the other hand, the freezing starts from the interior of the square, and its ground state is nearly in a single domain with one of the two af-FMC orders. These geometry effects are argued to originate from the anisotropic nature of the dipole-dipole interaction which depends on the relative direction of sites in a real space of the interacting spins.