Ground state properties of antiferromagnetic spin-1/2 Heisenberg systems in external fields

Andreas Fledderjohann

Published May 23, 2001• 0 citations

Abstract

The present work investigates the influence of external perturbations on the ground states of spin-1/2 antiferromagnetic Heisenberg chains and two-dimensional spin lattices. The lifting of the well-known zero-energy excitations (soft modes) of the unperturbed Heisenberg chain due to suitably chosen periodicities of applied perturbations is presented. The evolution of excitation energies and transition matrix elements is described by a complete set of differential equations and scaling solutions for small perturbations are discussed. A variety of topic quasi-one-dimensional spin systems (spin-Peierls chain, spin ladders in different geometries) are then embedded in the descriptive scheme of periodically perturbed Heisenberg chains. In particular, the energy gaps and, most noticeable, the formation of magnetization plateaus occurring in these systems are discussed in a single prediction scheme related to the soft mode positions of the previously unperturbed chain. Plateaus due to various kinds of perturbations are furthermore shown to be observable in static correlation functions of the perturbed chain. Particular emphasis is laid on the consideration of processes of spontaneous breaking of symmetry. The latter aspect is discussed for translationally invariant perturbations that do not provide an explicit change of the periodicity of the spin system. The presented investigations finally turn to two-dimensional spin systems first addressing the formation of a non-zero staggered magnetization (long range order) for increased two-dimensionality in terms of a translationally invariant perturbed (helical) spin system. Finally, first stages of a present study on the formation of magnetization plateaus in square spin lattices with frustrating diagonal couplings are given.

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Exact magnetization plateaus and phase transitions in spin- S Heisenberg antiferromagnets in arbitrary dimensions

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Study of the excited states of the quantum antiferromagnets

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Niederenergetische Anregungen in eindimensionalen Quanten-Antiferromagneten

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Jan 1, 19990 citations

Magnetization properties of some quantum spin ladders

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Ground state properties of antiferromagnetic spin-1/2 Heisenberg systems in external fields

Andreas Fledderjohann

Published May 23, 2001• 0 citations

OpenAlex

Abstract

The present work investigates the influence of external perturbations on the ground
states of spin-1/2 antiferromagnetic Heisenberg chains and two-dimensional spin
lattices. The lifting of the well-known zero-energy excitations (soft modes) of
the unperturbed Heisenberg chain due to suitably chosen periodicities of applied
perturbations is presented. The evolution of excitation energies and transition
matrix elements is described by a complete set of differential equations and scaling
solutions for small perturbations are discussed. A variety of topic
quasi-one-dimensional spin systems (spin-Peierls chain, spin ladders in different
geometries) are then embedded in the descriptive scheme of periodically perturbed
Heisenberg chains. In particular, the energy gaps and, most noticeable, the formation
of magnetization plateaus occurring in these systems are discussed in a single
prediction scheme related to the soft mode positions of the previously unperturbed
chain. Plateaus due to various kinds of perturbations are furthermore shown to be
observable in static correlation functions of the perturbed chain. Particular emphasis
is laid on the consideration of processes of spontaneous breaking of symmetry.
The latter aspect is discussed for translationally invariant perturbations that
do not provide an explicit change of the periodicity of the spin system. The
presented investigations finally turn to two-dimensional spin systems first addressing
the formation of a non-zero staggered magnetization (long range order) for increased
two-dimensionality in terms of a translationally invariant perturbed (helical) spin
system. Finally, first stages of a present study on the formation of magnetization
plateaus in square spin lattices with frustrating diagonal couplings are given.

Related papers

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

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

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.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.

arXiv:cond-mat/9810272Physical review. B, Condensed matterJan 1, 199912 citations

Exact magnetization plateaus and phase transitions in spin- S Heisenberg antiferromagnets in arbitrary dimensions

V. Ravi Chandra, Naveen Surendran

We generalize a class of Heisenberg antiferromagnets in one, two, and three dimensions, which have been shown to exhibit magnetization plateaus for spin- $\frac{1}{2}$ . In a certain parameter range of the general model, which is formally defined in $D$ dimensions, we obtain the exact ground state(s) in the presence of an external magnetic field for arbitrary values of spin. In this range, the magnetization remains a constant as a function of the external field, except at some special values of the field where there is a jump from one plateau to the next. The plateaus are formed at certain specific fractions of the full magnetization which are determined by the spin and the lattice. Our general spin- $S$ result reproduces the known cases for spin- $\frac{1}{2}$ in various lattices. Furthermore, we argue that outside the exact regime, the mechanism for the plateau formation is different. This results in first order phase transitions along some of the plateaus as the coupling constant is varied. We rigorously show the existence of such transitions for some particular cases. Finally, we numerically analyze a spin- $1$ model in one dimension using exact diagonalization to obtain its complete phase diagram. It agrees with our analytic results.We generalize a class of Heisenberg antiferromagnets in one, two, and three dimensions, which have been shown to exhibit magnetization plateaus for spin-$\frac{1}{2}$. In a certain parameter range of the general model, which is formally defined in $D$ dimensions, we obtain the exact ground state(s) in the presence of an external magnetic field for arbitrary values of spin. In this range, the magnetization remains a constant as a function of the external field, except at some special values of the field where there is a jump from one plateau to the next. The plateaus are formed at certain specific fractions of the full magnetization which are determined by the spin and the lattice. Our general spin-$S$ result reproduces the known cases for spin-$\frac{1}{2}$ in various lattices. Furthermore, we argue that outside the exact regime, the mechanism for the plateau formation is different. This results in first order phase transitions along some of the plateaus as the coupling constant is varied. We rigorously show the existence of such transitions for some particular cases. Finally, we numerically analyze a spin-$1$ model in one dimension using exact diagonalization to obtain its complete phase diagram. It agrees with our analytic results.

arXiv:cond-mat/0604258Physical Review BJul 21, 200613 citations

Magnetization plateaus in frustrated antiferromagnetic quantum spin models

A. Honecker, J. Schulenburg, Johannes Richter

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arXiv:cond-mat/0309425Journal of Physics Condensed MatterMar 4, 2004185 citations

Study of the excited states of the quantum antiferromagnets

Mohammad G. Merdan

We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone $\Gamma(\mathbf q=0)$ and at the magnetic ordering wavevectors $\pm[\mathbf Q= (4\pi/3,0)]$ , the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone $\Gamma(\mathbf q=0)$ and at the magnetic ordering wavevectors $\pm[\mathbf Q= (4\pi/3,0)]$, the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.

Nov 10, 20130 citations

Niederenergetische Anregungen in eindimensionalen Quanten-Antiferromagneten

S. Brehmer

In the present work low energy excitations in various antiferromagnetic systems are investigated.The first part is devoted to the antiferromagnetic Heisenberg chain with two different alternating magnetic ions having spins S A and S B .The system can be characterised as a ferrimagnet with a long range ordered ground state.Linear spin wave theory (LSWT) leads to a qualitatively correct description of two excitation branches.The low energy spectrum contains ferromagnetic magnons which are discussed in the introductory chapter.The LSWT results are compared with the numerical data from exact diagonalisation for the case S A =1 and S B =1/2.In the second part of the thesis, the quantum phase transitions from the Néel phase to the Heisenberg phase and from the Néel phase to the Haldane phase for half-odd-integer spins S =1 /2, 3/2 and integer spins S =1 , 2, respectively, are studied in context of the Haldane conjecture.The models are analysed by means of high-order series expansions for the ground state properties and excitation energies.The expansion parameter is the anisotropy λ where the unperturbed case λ =0corresponds to the Ising model.For the integer-S systems the following scenario is obtained: The phase transitions from the Néel phase to the Haldane phase are determined by the condensation of the so-called spin zero defects (SZD) at finite anisotropy.These domain wall type excitations can be characterised as S z =0states embedded in the antiferromagnetic order.The critical exponents β of the order parameter and η of the correlation function are those of the two-dimensional Ising model.In contrast to the above scenario, the half-odd-integer S =1/2 chain undergoes the Kosterlitz-Thouless phase transition at the isotropic point.In the case of the S =3/2 chain the critical behaviour of the static structure factor S (zz) (π) is consistent with the exponentially divergent correlation length at the isotropic point.Finally, kink excitations of the S =1 /2 Heisenberg antiferromagnet chain with nearest and next nearest neighbour interactions (the zigzag chain) are investigated.Here a kink is defined as a free spin S =1 /2 mediating two dimerised ground states.The phase diagram of the zigzag chain contains the Majumdar-Ghosh model as well as the S =1/2 spin ladder and the S =1Heisenberg antiferromagnet.Starting from the Majumdar-Ghosh model, a two-kink variational wave function with S tot =1is constructed.The variational parameter is the characteristic distance between the two kinks.The variational wave function describes the low energy triplet excitation of the gapped zigzag chain in a qualitatively correct way.Within this approach, the elementary excitation of the spin ladder as well as the Haldane triplet of the S =1chain appears as a bound state of two kinks.The phase diagram of the zigzag chain contains the incommensurate phase where the wave vector of the lowest mode k 0 runs from k 0 =0to k 0 = π.At the qualitative level, this behaviour is correctly described by the variational approach.In the present work low energy excitations in various antiferromagnetic systems are investigated.The first part is devoted to the antiferromagnetic Heisenberg chain with two different alternating magnetic ions having spins S A and S B .The system can be characterised as a ferrimagnet with a long range ordered ground state.Linear spin wave theory (LSWT) leads to a qualitatively correct description of two excitation branches.The low energy spectrum contains ferromagnetic magnons which are discussed in the introductory chapter.The LSWT results are compared with the numerical data from exact diagonalisation for the case S A =1 and S B =1/2.In the second part of the thesis, the quantum phase transitions from the Néel phase to the Heisenberg phase and from the Néel phase to the Haldane phase for half-odd-integer spins S =1 /2, 3/2 and integer spins S =1 , 2, respectively, are studied in context of the Haldane conjecture.The models are analysed by means of high-order series expansions for the ground state properties and excitation energies.The expansion parameter is the anisotropy λ where the unperturbed case λ =0corresponds to the Ising model.For the integer-S systems the following scenario is obtained: The phase transitions from the Néel phase to the Haldane phase are determined by the condensation of the so-called spin zero defects (SZD) at finite anisotropy.These domain wall type excitations can be characterised as S z =0states embedded in the antiferromagnetic order.The critical exponents β of the order parameter and η of the correlation function are those of the two-dimensional Ising model.In contrast to the above scenario, the half-odd-integer S =1/2 chain undergoes the Kosterlitz-Thouless phase transition at the isotropic point.In the case of the S =3/2 chain the critical behaviour of the static structure factor S (zz) (π) is consistent with the exponentially divergent correlation length at the isotropic point.Finally, kink excitations of the S =1 /2 Heisenberg antiferromagnet chain with nearest and next nearest neighbour interactions (the zigzag chain) are investigated.Here a kink is defined as a free spin S =1 /2 mediating two dimerised ground states.The phase diagram of the zigzag chain contains the Majumdar-Ghosh model as well as the S =1/2 spin ladder and the S =1Heisenberg antiferromagnet.Starting from the Majumdar-Ghosh model, a two-kink variational wave function with S tot =1is constructed.The variational parameter is the characteristic distance between the two kinks.The variational wave function describes the low energy triplet excitation of the gapped zigzag chain in a qualitatively correct way.Within this approach, the elementary excitation of the spin ladder as well as the Haldane triplet of the S =1chain appears as a bound state of two kinks.The phase diagram of the zigzag chain contains the incommensurate phase where the wave vector of the lowest mode k 0 runs from k 0 =0to k 0 = π.At the qualitative level, this behaviour is correctly described by the variational approach.

Jan 1, 19990 citations

Magnetization properties of some quantum spin ladders

Kunj Tandon, Siddhartha Lal, Swapan Kumar Pati +2 more

The experimental realization of various spin ladder systems has prompted their detailed theoretical investigations. Here we study the evolution of ground-state magnetization with an external magnetic field for two different antiferromagnetic systems: a three-legged spin- $1/2$ ladder, and a two-legged spin- $1/2$ ladder with an additional diagonal interaction. The finite system density-matrix renormalization-group method is employed for numerical studies of the three-chain system, and an effective low-energy Hamiltonian is used in the limit of strong interchain coupling to study the two- and three-chain systems. The three-chain system has a magnetization plateau at one-third of the saturation magnetization. The two-chain system has a plateau at zero magnetization due to a gap above the singlet ground state. It also has a plateau at half of the saturation magnetization for a certain range of values of the couplings. We study the regions of transitions between plateaus numerically and analytically, and find that they are described, at first order in a strong-coupling expansion, by an $\mathrm{XXZ}$ spin- $1/2$ chain in a magnetic field; the second-order terms give corrections to the $\mathrm{XXZ}$ model. We also study numerically some low-temperature properties of the three-chain system, such as the magnetization, magnetic susceptibility and specific heat.The experimental realization of various spin ladder systems has prompted their detailed theoretical investigations. Here we study the evolution of ground-state magnetization with an external magnetic field for two different antiferromagnetic systems: a three-legged spin-$1/2$ ladder, and a two-legged spin-$1/2$ ladder with an additional diagonal interaction. The finite system density-matrix renormalization-group method is employed for numerical studies of the three-chain system, and an effective low-energy Hamiltonian is used in the limit of strong interchain coupling to study the two- and three-chain systems. The three-chain system has a magnetization plateau at one-third of the saturation magnetization. The two-chain system has a plateau at zero magnetization due to a gap above the singlet ground state. It also has a plateau at half of the saturation magnetization for a certain range of values of the couplings. We study the regions of transitions between plateaus numerically and analytically, and find that they are described, at first order in a strong-coupling expansion, by an $\mathrm{XXZ}$ spin-$1/2$ chain in a magnetic field; the second-order terms give corrections to the $\mathrm{XXZ}$ model. We also study numerically some low-temperature properties of the three-chain system, such as the magnetization, magnetic susceptibility and specific heat.

arXiv:cond-mat/9806111Physical review. B, Condensed matterJan 1, 199963 citations

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.

We generalize a class of Heisenberg antiferromagnets in one, two, and three dimensions, which have been shown to exhibit magnetization plateaus for spin-

\frac{1}{2}

. In a certain parameter range of the general model, which is formally defined in

D

dimensions, we obtain the exact ground state(s) in the presence of an external magnetic field for arbitrary values of spin. In this range, the magnetization remains a constant as a function of the external field, except at some special values of the field where there is a jump from one plateau to the next. The plateaus are formed at certain specific fractions of the full magnetization which are determined by the spin and the lattice. Our general spin-

S

result reproduces the known cases for spin-

\frac{1}{2}

in various lattices. Furthermore, we argue that outside the exact regime, the mechanism for the plateau formation is different. This results in first order phase transitions along some of the plateaus as the coupling constant is varied. We rigorously show the existence of such transitions for some particular cases. Finally, we numerically analyze a spin-

1

model in one dimension using exact diagonalization to obtain its complete phase diagram. It agrees with our analytic results.

\Gamma(\mathbf q=0)

and at the magnetic ordering wavevectors

\pm[\mathbf Q= (4\pi/3,0)]

, the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.

1/2

ladder, and a two-legged spin-

1/2

ladder with an additional diagonal interaction. The finite system density-matrix renormalization-group method is employed for numerical studies of the three-chain system, and an effective low-energy Hamiltonian is used in the limit of strong interchain coupling to study the two- and three-chain systems. The three-chain system has a magnetization plateau at one-third of the saturation magnetization. The two-chain system has a plateau at zero magnetization due to a gap above the singlet ground state. It also has a plateau at half of the saturation magnetization for a certain range of values of the couplings. We study the regions of transitions between plateaus numerically and analytically, and find that they are described, at first order in a strong-coupling expansion, by an

\mathrm{XXZ}

spin-

1/2

chain in a magnetic field; the second-order terms give corrections to the

\mathrm{XXZ}

model. We also study numerically some low-temperature properties of the three-chain system, such as the magnetization, magnetic susceptibility and specific heat.