arXiv:cond-mat/9809086

Gap's in the antiferromagnetic Heisenberg model

Andreas Fledderjohann, Michael Karbach, K.-H. Mütter

Published Sep 4, 1998• 0 citations

Abstract

We study the one-dimensional spin-1/2 antiferromagnetic Heisenberg model exposed to an external field, which is a superposition of a homogeneous field $h_{3}$ and a small periodic field of strength $h_{1}$. For the case of a transverse staggered field a gap opens, which scales with $h_{1}^{ε_{1}}$, where $ε_{1}=ε_{1}(h_{3})$ is given by the critical exponent $η_{1}(M(h_{3}))$ defined through the transverse structure factor of the model at $h_{1}=0$. For the case of a longitudinal periodic field with wave vector $q=π/2$ and strength $h_{q}$ a plateau is found in the magnetization curve at $M=1/4$. The difference of the upper- and lower magnetic field scales with $h_{3}^{u}-h_{3}^{l}\sim h_{q}^{ε_{3}}$, where $ε_{3}=ε_{3}(h_{3})$ is given by the critical exponent $η_{3}(M(h_{3}))$ defined through the longitudinal structure factor of the model at $h_{q}=0$.

DOI: 10.48550/arxiv.cond-mat/9809086

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Soft modes, gaps, and magnetization plateaus in one-dimensional spin- 12 antiferromagnetic Heisenberg models

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Scaling behavior of the energy gap of the XXZ model in a transverse magnetic field

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arXiv:cond-mat/9809086

Gap's in the antiferromagnetic Heisenberg model

Andreas Fledderjohann, Michael Karbach, K.-H. Mütter

Published Sep 4, 1998• 0 citations

arXiv HTML PDF

Abstract

We study the one-dimensional spin-1/2 antiferromagnetic Heisenberg model exposed to an external field, which is a superposition of a homogeneous field

h_{3}

and a small periodic field of strength

h_{1}

. For the case of a transverse staggered field a gap opens, which scales with

h_{1}^{ε_{1}}

, where

ε_{1}=ε_{1}(h_{3})

is given by the critical exponent

η_{1}(M(h_{3}))

defined through the transverse structure factor of the model at

h_{1}=0

. For the case of a longitudinal periodic field with wave vector

q=π/2

and strength

h_{q}

a plateau is found in the magnetization curve at

M=1/4

. The difference of the upper- and lower magnetic field scales with

h_{3}^{u}-h_{3}^{l}\sim h_{q}^{ε_{3}}

, where

ε_{3}=ε_{3}(h_{3})

is given by the critical exponent

η_{3}(M(h_{3}))

defined through the longitudinal structure factor of the model at

h_{q}=0

DOI: 10.48550/arxiv.cond-mat/9809086

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Andreas Fledderjohann, C. Gerhardt, Michael Karbach +2 more

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We have studied the ground state phase diagram of a spin ladder with alternating rung exchange $J^{n}_{\perp} = J_{\perp}[1 + (-1)^{n} δ]$ in a magnetic filed, in the limit where the rung coupling is dominant. In this limit the model is mapped onto an $XXZ$ Heisenberg chain in a uniform and staggered longitudinal magnetic fields, where the amplitude of the staggered field is $\sim δ$ . We have shown that the magnetization curve of the system exhibits a plateau at magnetization equal to the half of the saturation value. The width of a plateau scales as $δ^ν$ , where $ν=4/5$ in the case of ladder with isotropic antiferromagnetic legs and $ν=2$ in the case of ladder with isotropic ferromagnetic legs. We have calculated four critical fields ( $H^{\pm}_{c1}$ and $H^{\pm}_{c2}$ ) corresponding to transitions between different magnetic phases of the system. We have shown that these transitions belong to the universality class of the commensurate-incommensurate transition.We have studied the ground state phase diagram of a spin ladder with alternating rung exchange $J^{n}_{\perp} = J_{\perp}[1 + (-1)^{n} δ]$ in a magnetic filed, in the limit where the rung coupling is dominant. In this limit the model is mapped onto an $XXZ$ Heisenberg chain in a uniform and staggered longitudinal magnetic fields, where the amplitude of the staggered field is $\sim δ$. We have shown that the magnetization curve of the system exhibits a plateau at magnetization equal to the half of the saturation value. The width of a plateau scales as $δ^ν$, where $ν=4/5$ in the case of ladder with isotropic antiferromagnetic legs and $ν=2$ in the case of ladder with isotropic ferromagnetic legs. We have calculated four critical fields ($H^{\pm}_{c1}$ and $H^{\pm}_{c2}$) corresponding to transitions between different magnetic phases of the system. We have shown that these transitions belong to the universality class of the commensurate-incommensurate transition.

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We study the one-dimensional spin-1/2 antiferromagnetic Heisenberg model exposed to an external field, which is a superposition of a homogeneous field

h_{3}

and a small periodic field of strength

h_{1}

. For the case of a transverse staggered field a gap opens, which scales with

h_{1}^{ε_{1}}

, where

ε_{1}=ε_{1}(h_{3})

is given by the critical exponent

η_{1}(M(h_{3}))

defined through the transverse structure factor of the model at

h_{1}=0

. For the case of a longitudinal periodic field with wave vector

q=π/2

and strength

h_{q}

a plateau is found in the magnetization curve at

M=1/4

. The difference of the upper- and lower magnetic field scales with

h_{3}^{u}-h_{3}^{l}\sim h_{q}^{ε_{3}}

, where

ε_{3}=ε_{3}(h_{3})

is given by the critical exponent

η_{3}(M(h_{3}))

defined through the longitudinal structure factor of the model at

h_{q}=0

We study the one-dimensional spin-1/2 model with nearest and next-to-nearest-neighbor couplings exposed to a homogeneous magnetic field

{h}_{3}

and a dimer field with period q and strength $\ensuremath{\delta}$. The latter generates a magnetization plateau at $M=(1\ensuremath{-}q/\ensuremath{\pi})/2$, which evolves with strength $\ensuremath{\delta}$ of the perturbation as ${\ensuremath{\delta}}^{\ensuremath{\epsilon}}$, where $\ensuremath{\epsilon}=\ensuremath{\epsilon}{(h}_{3},\ensuremath{\alpha})$ is related to the $\ensuremath{\eta}$ exponent which describes the critical behavior of the dimer structure factor, if the perturbation is switched off $(\ensuremath{\delta}=0)$. We also discuss the appearance of magnetization plateaus in ladder systems with l legs.

{\mathit{H}}_{\mathit{c}1}

(=\ensuremath{\Delta}), where \ensuremath{\Delta} is the Haldane gap. It is also found that the transverse spin correlation has the asymptotic form 〈

{\mathit{S}}_{0}^{\mathit{x}}

{\mathit{S}}_{\mathit{r}}^{\mathit{x}}

〉\ensuremath{\sim}(-1

{)}^{\mathit{r}}

${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$, and the transverse staggered susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{st}}^{\mathit{x}\mathit{x}}$ diverges between

{\mathit{H}}_{\mathit{c}1}

and

{\mathit{H}}_{\mathit{c}2}

(=4). The exponent \ensuremath{\eta} has a minimum (\ensuremath{\eta}\ensuremath{\simeq}0.3) at magnetization m\ensuremath{\simeq}1/3 and \ensuremath{\eta}\ensuremath{\simeq}0.5 at

{\mathit{H}}_{\mathit{c}1}

and

{\mathit{H}}_{\mathit{c}2}

. If the system is quasi-one-dimensional, even small interchain couplings can create a canted N\'eel order, within a mean-field approximation for interchain interactions. This is consistent with a recent NMR measurement for Ni(

{\mathrm{C}}_{2}

{\mathrm{H}}_{8}

{\mathrm{N}}_{2}

{)}_{2}

{\mathrm{NO}}_{2}

(

{\mathrm{ClO}}_{4}

) (NENP) at low temperature.

The infinite Heisenberg XXZ spin-

\frac{1}{2}

chain in the gapped antiferromagnetic regime has two degenerate vacua and kink topological excitations (which are also called spinons) interpolating between these vacua as elementary excitations. Application of an arbitrary weak staggered longitudinal magnetic field

h

induces a long-range attractive potential between two adjacent spinons leading to their confinement into ``meson'' bound states. Exploiting the integrability of the XXZ model in the deconfined phase

h=0

, we perform perturbative calculations of the energy spectra of the two-spinon bound states in the weak confinement regime at $h\ensuremath{\rightarrow}+0$, using the strength of the staggered magnetic field

h

as a small parameter. Both transverse and longitudinal dynamical structure factors of the local spin operators are calculated as well in the two-spinon approximation in the weak confinement regime to the leading order in

h

We study the low-energy excitations of the spin-

\frac{1}{2}

antiferromagnetic Heisenberg chain and

N

-leg

(N=2,3,4)

ladders in a staggered magnetic field

{h}_{s}

. We show that

{h}_{s}

induces gap and midgap states in all the cases and we will examine their field scaling behavior. A modified boundary scheme is devised to extract accurate bulk excitation behavior. The gap values converge rapidly as

N

increases, leading to a field scaling exponent $\ensuremath{\gamma}=\frac{1}{2}$ for both the longitudinal and transverse gaps of the square lattice $(N\ensuremath{\rightarrow}\ensuremath{\infty})$. The midgap states induced by the boundary edge effects share the bulk gap scaling exponents but their overall scaling behavior in the large-

N

limit needs further investigation.

We have studied the ground state phase diagram of a spin ladder with alternating rung exchange

J^{n}_{\perp} = J_{\perp}[1 + (-1)^{n} δ]

in a magnetic filed, in the limit where the rung coupling is dominant. In this limit the model is mapped onto an

XXZ

Heisenberg chain in a uniform and staggered longitudinal magnetic fields, where the amplitude of the staggered field is

\sim δ

. We have shown that the magnetization curve of the system exhibits a plateau at magnetization equal to the half of the saturation value. The width of a plateau scales as

δ^ν

, where

ν=4/5

in the case of ladder with isotropic antiferromagnetic legs and

ν=2

in the case of ladder with isotropic ferromagnetic legs. We have calculated four critical fields (

H^{\pm}_{c1}

and

H^{\pm}_{c2}

) corresponding to transitions between different magnetic phases of the system. We have shown that these transitions belong to the universality class of the commensurate-incommensurate transition.