Magnetization plateaus in spin chains : Haldane gap for half-integer spins

We discuss zero-temperature quantum spin chains in a uniform magnetic field, with axial symmetry. For integer or half-integer spin, $S$ , the magnetization curve can have plateaus and we argue that the magnetization per site $m$ is topologically quantized as $n(S\ensuremath{-}m)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{integer}$ at the plateaus, where $n$ is the period of the ground state. We also discuss conditions for the presence of the plateau at those quantized values. For $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3/2$ and $m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ , we study several models and find two distinct types of massive phases at the plateau. One of them is argued to be a ``Haldane gap phase'' for half-integer $S$ .We discuss zero-temperature quantum spin chains in a uniform magnetic field, with axial symmetry. For integer or half-integer spin, $S$, the magnetization curve can have plateaus and we argue that the magnetization per site $m$ is topologically quantized as $n(S\ensuremath{-}m)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{integer}$ at the plateaus, where $n$ is the period of the ground state. We also discuss conditions for the presence of the plateau at those quantized values. For $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3/2$ and $m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$, we study several models and find two distinct types of massive phases at the plateau. One of them is argued to be a ``Haldane gap phase'' for half-integer $S$.

arXiv:cond-mat/9610168Physical Review LettersMar 10, 1997710 citations

Magnetization plateaus, Haldane-like gap, string order, and hidden symmetry in a spin- 12 tetrameric Heisenberg antiferromagnetic chain

Shou-Shu Gong, Gang Su

The ground-state properties of a spin $S=1/2$ tetrameric Heisenberg antiferromagnetic chain with alternating couplings ${\text{AF}}_{1}{\text{-AF}}_{2}{\text{-AF}}_{1}\text{-F}$ (AF and F denote antiferromagnetic and ferromagnetic couplings, respectively) are studied by means of the density-matrix renormalization-group method. Two plateaus of magnetization $m$ are found at $m=0$ and 1/4. It is shown that in such a spin-1/2 AF system, there is a gap from the singlet ground state to the triplet excited states in the absence of a magnetic field. The spin-spin correlation function decays exponentially, and the gapped states have a nonvanishing string order, which measures a hidden symmetry in the system. By a dual transformation, the string order is transformed into a ferromagnetic order and the hidden symmetry is unveiled to be a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ discrete symmetry, which is fully broken in the gapped states. This half-integer spin Heisenberg AF chain is in a Haldane-like phase suggesting that the present findings extend the substance of Haldane's scenario. A valence-bond-solid state picture is also proposed for the gapped states.The ground-state properties of a spin $S=1/2$ tetrameric Heisenberg antiferromagnetic chain with alternating couplings ${\text{AF}}_{1}{\text{-AF}}_{2}{\text{-AF}}_{1}\text{-F}$ (AF and F denote antiferromagnetic and ferromagnetic couplings, respectively) are studied by means of the density-matrix renormalization-group method. Two plateaus of magnetization $m$ are found at $m=0$ and 1/4. It is shown that in such a spin-1/2 AF system, there is a gap from the singlet ground state to the triplet excited states in the absence of a magnetic field. The spin-spin correlation function decays exponentially, and the gapped states have a nonvanishing string order, which measures a hidden symmetry in the system. By a dual transformation, the string order is transformed into a ferromagnetic order and the hidden symmetry is unveiled to be a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ discrete symmetry, which is fully broken in the gapped states. This half-integer spin Heisenberg AF chain is in a Haldane-like phase suggesting that the present findings extend the substance of Haldane's scenario. A valence-bond-solid state picture is also proposed for the gapped states.

arXiv:0809.0093Physical Review BSep 18, 200822 citations

Haldane Gap Spin Chains: Behaviour at Finite Temperatures and Fields

Paul Fendley, Robert Konik

APSMar 1, 20010 citations

A note on Haldane’s conjecture

Li-Hua Pan, Chang-De Gong

According to Haldane’s conjecture, there is a spin gap of the excitation spectrum in integer Heisenberg spin chains whereas half-integer spin chains have a gapless excitation spectrum. In an earlier publication (Hung and Gong 2005 Phys. Rev. B 71 054413), the Haldane gap and other properties of the spin-1 system are explained using a spin-1/2 alternating bond chain model. The ferromagnetic–ferromagnetic–antiferromagnetic Heisenberg spin-1/2 chain is expected to behave similarly to the uniform spin-3/2 Heisenberg chain for strong ferromagnetic coupling. And the general nonuniform spin-1/2 chain may be used to explain the properties of the integer and half-integer high spin- S chains, consistently. In this paper, we provide a detailed numerical verification for the above problems.According to Haldane’s conjecture, there is a spin gap of the excitation spectrum in integer Heisenberg spin chains whereas half-integer spin chains have a gapless excitation spectrum. In an earlier publication (Hung and Gong 2005 Phys. Rev. B 71 054413), the Haldane gap and other properties of the spin-1 system are explained using a spin-1/2 alternating bond chain model. The ferromagnetic–ferromagnetic–antiferromagnetic Heisenberg spin-1/2 chain is expected to behave similarly to the uniform spin-3/2 Heisenberg chain for strong ferromagnetic coupling. And the general nonuniform spin-1/2 chain may be used to explain the properties of the integer and half-integer high spin- S chains, consistently. In this paper, we provide a detailed numerical verification for the above problems.

Journal of Physics Condensed MatterApr 25, 20083 citations

Haldane gap in the spin- ½ double chain Heisenberg antiferromagnet

Kazuo Hida

Journal of Magnetism and Magnetic MaterialsFeb 1, 19922 citations

We discuss zero-temperature quantum spin chains in a uniform magnetic field, with axial symmetry. For integer or half-integer spin,

S

, the magnetization curve can have plateaus and we argue that the magnetization per site

m

is topologically quantized as $n(S\ensuremath{-}m)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{integer}$ at the plateaus, where

n

is the period of the ground state. We also discuss conditions for the presence of the plateau at those quantized values. For

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3/2

and

m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2

, we study several models and find two distinct types of massive phases at the plateau. One of them is argued to be a ``Haldane gap phase'' for half-integer

S

The ground-state properties of a spin

S=1/2

tetrameric Heisenberg antiferromagnetic chain with alternating couplings

{\text{AF}}_{1}{\text{-AF}}_{2}{\text{-AF}}_{1}\text{-F}

(AF and F denote antiferromagnetic and ferromagnetic couplings, respectively) are studied by means of the density-matrix renormalization-group method. Two plateaus of magnetization

m

are found at

m=0

and 1/4. It is shown that in such a spin-1/2 AF system, there is a gap from the singlet ground state to the triplet excited states in the absence of a magnetic field. The spin-spin correlation function decays exponentially, and the gapped states have a nonvanishing string order, which measures a hidden symmetry in the system. By a dual transformation, the string order is transformed into a ferromagnetic order and the hidden symmetry is unveiled to be a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ discrete symmetry, which is fully broken in the gapped states. This half-integer spin Heisenberg AF chain is in a Haldane-like phase suggesting that the present findings extend the substance of Haldane's scenario. A valence-bond-solid state picture is also proposed for the gapped states.