arXiv:hep-th/9809066 · Physical review. B, Condensed matter

Antiferromagnetic S=12 Heisenberg chain and the two-flavor massless Schwinger model

Yutaka Hosotani

Published Sep 1, 1999• 4 citations

Abstract

An antiferromagnetic $S=\frac{1}{2}$ Heisenberg chain is mapped to the two-flavor massless Schwinger model at $\ensuremath{\theta}=\ensuremath{\pi}.$ The electromagnetic coupling constant and velocity of light in the Schwinger model are determined in terms of the Heisenberg coupling and lattice spacing in the spin chain system.

DOI: 10.1103/physrevb.60.6198

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arXiv:hep-th/9809066 · Physical review. B, Condensed matter

Antiferromagnetic S=12 Heisenberg chain and the two-flavor massless Schwinger model

Yutaka Hosotani

Published Sep 1, 1999• 4 citations

arXiv HTML PDF

Abstract

An antiferromagnetic

S=\frac{1}{2}

Heisenberg chain is mapped to the two-flavor massless Schwinger model at

\ensuremath{\theta}=\ensuremath{\pi}.

The electromagnetic coupling constant and velocity of light in the Schwinger model are determined in terms of the Heisenberg coupling and lattice spacing in the spin chain system.

DOI: 10.1103/physrevb.60.6198

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日本物理学会講演概要集. 年会Mar 15, 19960 citations

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We review the correspondence between strongly coupled lattice multiflavor Schwinger models and SU(N) antiferromagnetic chains. We show that finding the low lying states of the gauge models is equivalent to solving an SU(N) Heisenberg antiferromagnetic chain. For the two-flavor lattice Schwinger model the massless excitations correspond to gapless states of the Heisenberg chain, while the massive states are created by fermion transport in the ground state of the spin chain. Our analysis shows explicitly how spinons may arise in lattice gauge theories.We review the correspondence between strongly coupled lattice multiflavor Schwinger models and SU(N) antiferromagnetic chains. We show that finding the low lying states of the gauge models is equivalent to solving an SU(N) Heisenberg antiferromagnetic chain. For the two-flavor lattice Schwinger model the massless excitations correspond to gapless states of the Heisenberg chain, while the massive states are created by fermion transport in the ground state of the spin chain. Our analysis shows explicitly how spinons may arise in lattice gauge theories.

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The phenomenological expression $\ensuremath{\chi}T∕(N{g}^{2}{\ensuremath{\mu}}_{B}^{2}∕k)={C}_{1n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{1n}∕T)+{C}_{2n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{2n}∕T)$ describes very accurately the temperature dependence of the magnetic susceptibility computed for antiferromagnetic rings of Heisenberg spins $S=1$ , whose size $n$ is even and ranges from 6 to 20. This expression has been obtained through a strategy justified by scaling considerations together with finite size numerical calculations. For $n$ large, the coefficients of the expression converge towards ${C}_{1}=0.125$ , ${W}_{1}=0.451J$ , ${C}_{2}=0.564$ , ${W}_{2}=1.793J$ ( $J$ is the exchange constant), which are appropriate for describing the susceptibility of the spin-1 Haldane chain. The Curie constant, the paramagnetic Curie--Weiss temperature, the correlation length at $T=0$ and the Haldane gap are found to be closely related to these coefficients. With this expression, a very good description of the magnetic behavior of ${\mathrm{Y}}_{2}\mathrm{Ba}\mathrm{Ni}{\mathrm{O}}_{5}$ and of $\mathrm{Ni}{({\mathrm{C}}_{2}{\mathrm{H}}_{8}{\mathrm{N}}_{2})}_{2}\mathrm{N}{\mathrm{O}}_{2}\mathrm{Cl}{\mathrm{O}}_{4}$ (NENP), the archetype of the Haldane gap systems, is achieved over the whole temperature range.The phenomenological expression $\ensuremath{\chi}T∕(N{g}^{2}{\ensuremath{\mu}}_{B}^{2}∕k)={C}_{1n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{1n}∕T)+{C}_{2n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{2n}∕T)$ describes very accurately the temperature dependence of the magnetic susceptibility computed for antiferromagnetic rings of Heisenberg spins $S=1$, whose size $n$ is even and ranges from 6 to 20. This expression has been obtained through a strategy justified by scaling considerations together with finite size numerical calculations. For $n$ large, the coefficients of the expression converge towards ${C}_{1}=0.125$, ${W}_{1}=0.451J$, ${C}_{2}=0.564$, ${W}_{2}=1.793J$ ($J$ is the exchange constant), which are appropriate for describing the susceptibility of the spin-1 Haldane chain. The Curie constant, the paramagnetic Curie--Weiss temperature, the correlation length at $T=0$ and the Haldane gap are found to be closely related to these coefficients. With this expression, a very good description of the magnetic behavior of ${\mathrm{Y}}_{2}\mathrm{Ba}\mathrm{Ni}{\mathrm{O}}_{5}$ and of $\mathrm{Ni}{({\mathrm{C}}_{2}{\mathrm{H}}_{8}{\mathrm{N}}_{2})}_{2}\mathrm{N}{\mathrm{O}}_{2}\mathrm{Cl}{\mathrm{O}}_{4}$ (NENP), the archetype of the Haldane gap systems, is achieved over the whole temperature range.

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We study a model of three $S=1/2$ antiferromagnetic Heisenberg spin chains weakly coupled by on-rung and plaquette-diagonal interchain interactions. It is shown that the model exhibits a critical phase with central charge $C=2$ and belongs to the class of ``chirally stabilized'' liquids recently introduced by Andrei, Douglas, and Jerez. By allowing anisotropic interactions in spin space, we find an exact solution at a Toulouse point which captures all universal properties of the model, including the SU(2) symmetric case. At this critical point the massless degrees of freedom are described in terms of an effective $S=1/2$ Heisenberg spin chain and two critical Ising models. We discuss the spectral properties of the model, compute spin-spin correlation functions, and estimate the NMR relaxation rate.We study a model of three $S=1/2$ antiferromagnetic Heisenberg spin chains weakly coupled by on-rung and plaquette-diagonal interchain interactions. It is shown that the model exhibits a critical phase with central charge $C=2$ and belongs to the class of ``chirally stabilized'' liquids recently introduced by Andrei, Douglas, and Jerez. By allowing anisotropic interactions in spin space, we find an exact solution at a Toulouse point which captures all universal properties of the model, including the SU(2) symmetric case. At this critical point the massless degrees of freedom are described in terms of an effective $S=1/2$ Heisenberg spin chain and two critical Ising models. We discuss the spectral properties of the model, compute spin-spin correlation functions, and estimate the NMR relaxation rate.

arXiv:cond-mat/9806369Physical review. B, Condensed matterOct 1, 199825 citations

An antiferromagnetic

S=\frac{1}{2}

Heisenberg chain is mapped to the two-flavor massless Schwinger model at

\ensuremath{\theta}=\ensuremath{\pi}.

The electromagnetic coupling constant and velocity of light in the Schwinger model are determined in terms of the Heisenberg coupling and lattice spacing in the spin chain system.

S=1

, whose size

n

is even and ranges from 6 to 20. This expression has been obtained through a strategy justified by scaling considerations together with finite size numerical calculations. For

n

large, the coefficients of the expression converge towards

{C}_{1}=0.125

{W}_{1}=0.451J

{C}_{2}=0.564

{W}_{2}=1.793J

(

J

is the exchange constant), which are appropriate for describing the susceptibility of the spin-1 Haldane chain. The Curie constant, the paramagnetic Curie--Weiss temperature, the correlation length at

T=0

and the Haldane gap are found to be closely related to these coefficients. With this expression, a very good description of the magnetic behavior of

{\mathrm{Y}}_{2}\mathrm{Ba}\mathrm{Ni}{\mathrm{O}}_{5}

and of

\mathrm{Ni}{({\mathrm{C}}_{2}{\mathrm{H}}_{8}{\mathrm{N}}_{2})}_{2}\mathrm{N}{\mathrm{O}}_{2}\mathrm{Cl}{\mathrm{O}}_{4}

(NENP), the archetype of the Haldane gap systems, is achieved over the whole temperature range.

We study a model of three

S=1/2

antiferromagnetic Heisenberg spin chains weakly coupled by on-rung and plaquette-diagonal interchain interactions. It is shown that the model exhibits a critical phase with central charge

C=2

and belongs to the class of ``chirally stabilized'' liquids recently introduced by Andrei, Douglas, and Jerez. By allowing anisotropic interactions in spin space, we find an exact solution at a Toulouse point which captures all universal properties of the model, including the SU(2) symmetric case. At this critical point the massless degrees of freedom are described in terms of an effective

S=1/2

Heisenberg spin chain and two critical Ising models. We discuss the spectral properties of the model, compute spin-spin correlation functions, and estimate the NMR relaxation rate.