arXiv:1303.6662 · Physical Review B

Double zigzag spin chain in a strong magnetic field close to saturation

I. T. Shyiko, Ian P. McCulloch, J.V. Gumenjuk-Sichevska, Alexei K. Kolezhuk

Published Jul 8, 2013• 13 citations

Abstract

We study the ground state phase diagram of a frustrated spin tube in a strong external magnetic field. This model can be viewed as two coupled zigzag spin chains, or as a two-leg spin ladder with frustrating next-nearest-neighbor couplings along the legs, and its study is motivated by the physics of such materials as sulfolane-${\mathrm{Cu}}_{2}{\mathrm{Cl}}_{4}$ and ${\mathrm{BiCu}}_{2}{\mathrm{PO}}_{6}$. In magnetic fields right below the saturation, the system can be effectively represented as a dilute gas of two species of bosonic quasiparticles that correspond to magnons with inequivalent incommensurate momenta at two degenerate minima of the magnon dispersion. Using the method previously proposed and tested for frustrated spin chains, we calculate effective interactions in this two-component Bose gas. On this basis, we establish the phase diagram of nearly saturated frustrated spin tube, which is shown to include the two-component Luttinger liquid, two types of vector chiral phases, and phases whose physics is determined by the presence of bound magnons. We study the phase diagram of the model numerically by means of the density matrix renormalization group technique, and find a good agreement with our analytical predictions.

DOI: 10.1103/physrevb.88.014403

Related papers

Frustrated spin chains in strong magnetic field: Dilute two-component Bose gas regime

Alexei K. Kolezhuk, Fabian Heidrich‐Meisner, Sebastian Greschner +1 more

arXiv:1112.5720Physical Review BFeb 28, 201244 citations

Quantum phases of a frustrated four-leg spin tube

Marcelo Arlego, W. Brenig, Yousef Rahnavard +3 more

arXiv:1210.8199Physical Review BJan 9, 20139 citations

Ground-State Phase Diagram of an Anisotropic S = 1/2 Ladder with Different Leg Interactions

Takashi Tonegawa, Toshiya Hikihara, Kiyomi Okamoto +2 more

arXiv:1808.01090Journal of the Physical Society of JapanSep 12, 20188 citations

Bridging frustrated-spin-chain and spin-ladder physics: Quasi-one-dimensional magnetism of BiCu 2 PO 6

Alexander A. Tsirlin, Ioannis Rousochatzakis, Deepa Kasinathan +6 more

arXiv:1008.1771Physical Review BOct 18, 201062 citations

Magnetization plateaux and jumps in frustrated four-leg spin tubes in magnetic fields

H. D. Rosales, Marcelo Arlego, F. A. Gómez Albarracín

Journal of Physics Conference SeriesDec 8, 20141 citation

Magnetization plateaus and jumps in a frustrated four-leg spin tube under a magnetic field

F. A. Gómez Albarracín, Marcelo Arlego, H. D. Rosales

arXiv:1410.5502Physical Review BNov 4, 20149 citations

arXiv:1303.6662 · Physical Review B

Double zigzag spin chain in a strong magnetic field close to saturation

I. T. Shyiko, Ian P. McCulloch, J.V. Gumenjuk-Sichevska, Alexei K. Kolezhuk

Published Jul 8, 2013• 13 citations

arXiv HTML PDF

Abstract

{\mathrm{Cu}}_{2}{\mathrm{Cl}}_{4}

and

{\mathrm{BiCu}}_{2}{\mathrm{PO}}_{6}

. In magnetic fields right below the saturation, the system can be effectively represented as a dilute gas of two species of bosonic quasiparticles that correspond to magnons with inequivalent incommensurate momenta at two degenerate minima of the magnon dispersion. Using the method previously proposed and tested for frustrated spin chains, we calculate effective interactions in this two-component Bose gas. On this basis, we establish the phase diagram of nearly saturated frustrated spin tube, which is shown to include the two-component Luttinger liquid, two types of vector chiral phases, and phases whose physics is determined by the presence of bound magnons. We study the phase diagram of the model numerically by means of the density matrix renormalization group technique, and find a good agreement with our analytical predictions.

DOI: 10.1103/physrevb.88.014403

Related papers

Frustrated spin chains in strong magnetic field: Dilute two-component Bose gas regime

Alexei K. Kolezhuk, Fabian Heidrich‐Meisner, Sebastian Greschner +1 more

We study the ground state of frustrated spin- $S$ chains in a strong magnetic field in the immediate vicinity of saturation. In strongly frustrated chains, the magnon dispersion has two degenerate minima at inequivalent momenta $\ifmmode\pm\else\textpm\fi{}Q$, and just below the saturation field the system can be effectively represented as a dilute one-dimensional lattice gas of two species of bosons that correspond to magnons with momenta around $\ifmmode\pm\else\textpm\fi{}Q$. We present a theory of effective interactions in such a dilute magnon gas that allows us to make quantitative predictions for arbitrary values of the spin. With the help of this method, we are able to establish the magnetic phase diagram of frustrated chains close to saturation and study phase transitions between several nontrivial states, including a two-component Luttinger liquid, a vector chiral phase, and phases with bound magnons. We study those phase transitions numerically and find a good agreement with our analytical predictions.We study the ground state of frustrated spin-$S$ chains in a strong magnetic field in the immediate vicinity of saturation. In strongly frustrated chains, the magnon dispersion has two degenerate minima at inequivalent momenta $\ifmmode\pm\else\textpm\fi{}Q$, and just below the saturation field the system can be effectively represented as a dilute one-dimensional lattice gas of two species of bosons that correspond to magnons with momenta around $\ifmmode\pm\else\textpm\fi{}Q$. We present a theory of effective interactions in such a dilute magnon gas that allows us to make quantitative predictions for arbitrary values of the spin. With the help of this method, we are able to establish the magnetic phase diagram of frustrated chains close to saturation and study phase transitions between several nontrivial states, including a two-component Luttinger liquid, a vector chiral phase, and phases with bound magnons. We study those phase transitions numerically and find a good agreement with our analytical predictions.

arXiv:1112.5720Physical Review BFeb 28, 201244 citations

Quantum phases of a frustrated four-leg spin tube

Marcelo Arlego, W. Brenig, Yousef Rahnavard +3 more

We study the ground-state phase diagram of a frustrated spin-1/2 four-leg tube. Using a variety of complementary techniques, namely, density matrix renormalization group, exact diagonalization, Schwinger boson mean-field theory, quantum Monte Carlo, and series expansion, we explore the parameter space of this model in the regime of all-antiferromagnetic exchange. In contrast to unfrustrated four-leg tubes, we uncover a rich phase diagram. Apart from the Luttinger liquid fixed point in the limit of decoupled legs, this comprises several gapped ground states, namely, a plaquette, an incommensurate, and an antiferromagnetic quasispin-2 chain phase. The transitions between these phases are analyzed in terms of total energy and static structure factor calculations and are found to be of (weak) first order. Despite the absence of long-range order in the quantum case, remarkable similarities to the classical phase diagram are uncovered, with the exception of the icommensurate regime, which is strongly renormalized by quantum fluctuations. In the limit of large leg exchange, the tube exhibits a deconfinement crossover from gapped magnon-like excitations to spinons.We study the ground-state phase diagram of a frustrated spin-1/2 four-leg tube. Using a variety of complementary techniques, namely, density matrix renormalization group, exact diagonalization, Schwinger boson mean-field theory, quantum Monte Carlo, and series expansion, we explore the parameter space of this model in the regime of all-antiferromagnetic exchange. In contrast to unfrustrated four-leg tubes, we uncover a rich phase diagram. Apart from the Luttinger liquid fixed point in the limit of decoupled legs, this comprises several gapped ground states, namely, a plaquette, an incommensurate, and an antiferromagnetic quasispin-2 chain phase. The transitions between these phases are analyzed in terms of total energy and static structure factor calculations and are found to be of (weak) first order. Despite the absence of long-range order in the quantum case, remarkable similarities to the classical phase diagram are uncovered, with the exception of the icommensurate regime, which is strongly renormalized by quantum fluctuations. In the limit of large leg exchange, the tube exhibits a deconfinement crossover from gapped magnon-like excitations to spinons.

arXiv:1210.8199Physical Review BJan 9, 20139 citations

Ground-State Phase Diagram of an Anisotropic S = 1/2 Ladder with Different Leg Interactions

Takashi Tonegawa, Toshiya Hikihara, Kiyomi Okamoto +2 more

We explore the ground-state phase diagram of the $S=1/2$ two-leg ladder with different leg interactions. The $xy$ and $z$ components of the leg interactions between nearest-neighbor spins in the $a$ ( $b$ ) leg are respectively denoted by $J_{{\rm l},a}$ and $Δ_{\rm l} J_{{\rm l},a}$ ( $J_{{\rm l},b}$ and $Δ_{\rm l} J_{{\rm l},b}$ ). On the other hand, the $xy$ and $z$ components of the uniform rung interactions are respectively denoted by $Γ_{\rm r} J_{\rm r}$ and $J_{\rm r}$ . In the above, $Δ_{\rm l}$ and $Γ_{\rm r}$ are the $XXZ$ -type anisotropy parameters for the leg and rung interactions, respectively. This system has a frustration when $J_{{\rm l},a} J_{{\rm l},b}<0$ irrespective of the sign of $J_{\rm r}$ . The phase diagram on the $Δ_{\rm l}$ ( $|Δ_{\rm l}| \leq 1.0$ ) versus $J_{{\rm l},b}$ ( $-2.0\leq J_{{\rm l},b}\leq 3.0$ ) plane in the case where $J_{{\rm l},a}=0.2$ , $J_{\rm r}=-1.0$ , and $Γ_{\rm r} = 0.5$ is determined numerically. We employ the physical consideration, and the level spectroscopy and phenomenological renormalization-group analyses of the numerical date obtained by the exact diagonalization method. The resultant phase diagram contains the ferromagnetic, Haldane, N{é}el, nematic Tomonaga-Luttinger liquid (TLL), partial ferrimagnetic, and $XY1$ phases. Interestingly enough, the nematic TLL phase appears in the strong-rung unfrustrated region as well as in the strong-rung frustrated one. We perform the first-order perturbational calculations from the strong rung coupling limit to elucidate the characteristic features of the phase diagram. Furthermore, we make the density-matrix renormalization-group calculations for some physical quantities such as the energy gaps, the local magnetization, and the spin correlation functions to supplement the reliability of the phase diagram.We explore the ground-state phase diagram of the $S=1/2$ two-leg ladder with different leg interactions. The $xy$ and $z$ components of the leg interactions between nearest-neighbor spins in the $a$ ($b$) leg are respectively denoted by $J_{{\rm l},a}$ and $Δ_{\rm l} J_{{\rm l},a}$ ($J_{{\rm l},b}$ and $Δ_{\rm l} J_{{\rm l},b}$). On the other hand, the $xy$ and $z$ components of the uniform rung interactions are respectively denoted by $Γ_{\rm r} J_{\rm r}$ and $J_{\rm r}$. In the above, $Δ_{\rm l}$ and $Γ_{\rm r}$ are the $XXZ$-type anisotropy parameters for the leg and rung interactions, respectively. This system has a frustration when $J_{{\rm l},a} J_{{\rm l},b}<0$ irrespective of the sign of $J_{\rm r}$. The phase diagram on the $Δ_{\rm l}$ ($|Δ_{\rm l}| \leq 1.0$) versus $J_{{\rm l},b}$ ($-2.0\leq J_{{\rm l},b}\leq 3.0$) plane in the case where $J_{{\rm l},a}=0.2$, $J_{\rm r}=-1.0$, and $Γ_{\rm r} = 0.5$ is determined numerically. We employ the physical consideration, and the level spectroscopy and phenomenological renormalization-group analyses of the numerical date obtained by the exact diagonalization method. The resultant phase diagram contains the ferromagnetic, Haldane, N{é}el, nematic Tomonaga-Luttinger liquid (TLL), partial ferrimagnetic, and $XY1$ phases. Interestingly enough, the nematic TLL phase appears in the strong-rung unfrustrated region as well as in the strong-rung frustrated one. We perform the first-order perturbational calculations from the strong rung coupling limit to elucidate the characteristic features of the phase diagram. Furthermore, we make the density-matrix renormalization-group calculations for some physical quantities such as the energy gaps, the local magnetization, and the spin correlation functions to supplement the reliability of the phase diagram.

arXiv:1808.01090Journal of the Physical Society of JapanSep 12, 20188 citations

Bridging frustrated-spin-chain and spin-ladder physics: Quasi-one-dimensional magnetism of BiCu 2 PO 6

Alexander A. Tsirlin, Ioannis Rousochatzakis, Deepa Kasinathan +6 more

We derive and investigate the microscopic model of the quantum magnet ${\text{BiCu}}_{2}{\text{PO}}_{6}$ using band-structure calculations, magnetic susceptibility and high-field magnetization measurements, as well as exact diagonalization (ED) and density-matrix renormalization group (DMRG) techniques. The resulting quasi-one-dimensional spin model is a two-leg antiferromagnetic ladder with frustrating next-nearest-neighbor couplings along the legs. The individual couplings are estimated from band-structure calculations and by fitting the magnetic susceptibility with theoretical predictions, obtained using full diagonalizations. The nearest-neighbor leg coupling ${J}_{1}$ , the rung coupling ${J}_{4}$ , and one of the next-nearest-neighbor couplings ${J}_{2}$ amount to 120--150 K while the second next-nearest-neighbor coupling is ${J}_{2}^{\ensuremath{'}}\ensuremath{\simeq}{J}_{2}/2$. The spin ladders do not match the structural chains, and although the next-nearest-neighbor interactions ${J}_{2}$ and ${J}_{2}^{\ensuremath{'}}$ have very similar superexchange pathways, they differ substantially in magnitude due to a tiny difference in the O-O distances and in the arrangement of nonmagnetic ${\text{PO}}_{4}$ tetrahedra. An extensive ED study of the proposed model provides the low-energy excitation spectrum and shows that the system is in the strong rung coupling regime. The strong frustration by the next-nearest-neighbor couplings leads to a triplon branch with an incommensurate minimum. This is further corroborated by a strong-coupling expansion up to second order in the inter-rung coupling. Based on high-field magnetization measurements, we estimate the spin gap of $\ensuremath{\Delta}\ensuremath{\simeq}32\text{ }\text{K}$ and suggest the likely presence of antisymmetric Dzyaloshinskii-Moriya anisotropy and interladder coupling ${J}_{3}$ . We also provide a tentative description of the physics of ${\text{BiCu}}_{2}{\text{PO}}_{6}$ in magnetic field, in the light of the low-energy excitation spectra and numerical calculations based on ED and DMRG. In particular, we raise the possibility for a rich interplay between one- and two-component Luttinger liquid phases and a magnetization plateau at 1/2 of the saturation value.We derive and investigate the microscopic model of the quantum magnet ${\text{BiCu}}_{2}{\text{PO}}_{6}$ using band-structure calculations, magnetic susceptibility and high-field magnetization measurements, as well as exact diagonalization (ED) and density-matrix renormalization group (DMRG) techniques. The resulting quasi-one-dimensional spin model is a two-leg antiferromagnetic ladder with frustrating next-nearest-neighbor couplings along the legs. The individual couplings are estimated from band-structure calculations and by fitting the magnetic susceptibility with theoretical predictions, obtained using full diagonalizations. The nearest-neighbor leg coupling ${J}_{1}$, the rung coupling ${J}_{4}$, and one of the next-nearest-neighbor couplings ${J}_{2}$ amount to 120--150 K while the second next-nearest-neighbor coupling is ${J}_{2}^{\ensuremath{'}}\ensuremath{\simeq}{J}_{2}/2$. The spin ladders do not match the structural chains, and although the next-nearest-neighbor interactions ${J}_{2}$ and ${J}_{2}^{\ensuremath{'}}$ have very similar superexchange pathways, they differ substantially in magnitude due to a tiny difference in the O-O distances and in the arrangement of nonmagnetic ${\text{PO}}_{4}$ tetrahedra. An extensive ED study of the proposed model provides the low-energy excitation spectrum and shows that the system is in the strong rung coupling regime. The strong frustration by the next-nearest-neighbor couplings leads to a triplon branch with an incommensurate minimum. This is further corroborated by a strong-coupling expansion up to second order in the inter-rung coupling. Based on high-field magnetization measurements, we estimate the spin gap of $\ensuremath{\Delta}\ensuremath{\simeq}32\text{ }\text{K}$ and suggest the likely presence of antisymmetric Dzyaloshinskii-Moriya anisotropy and interladder coupling ${J}_{3}$. We also provide a tentative description of the physics of ${\text{BiCu}}_{2}{\text{PO}}_{6}$ in magnetic field, in the light of the low-energy excitation spectra and numerical calculations based on ED and DMRG. In particular, we raise the possibility for a rich interplay between one- and two-component Luttinger liquid phases and a magnetization plateau at 1/2 of the saturation value.

arXiv:1008.1771Physical Review BOct 18, 201062 citations

Magnetization plateaux and jumps in frustrated four-leg spin tubes in magnetic fields

H. D. Rosales, Marcelo Arlego, F. A. Gómez Albarracín

We study the ground state phase diagram of a frustrated spin-1/2 four-leg tube in an external magnetic field. We explore the parameter space of this model in the regime of all-antiferromagnetic exchange couplings by means of three different approaches: density matrix renormalization group (DMRG), a low-energy effective Hamiltonian (LEH) and a Hartree variational approach (HVA). We find that in the limit of weakly interacting plaquettes, singlet and triplet states play an important role in the formation of magnetization plateaux. We study the transition regions numerically and analytically, and find that they are described, at first order in a strong- coupling expansion, by an XXZ spin-1/2 chain in a magnetic field. These results are consistent with the DMRG and HVA calculations.We study the ground state phase diagram of a frustrated spin-1/2 four-leg tube in an external magnetic field. We explore the parameter space of this model in the regime of all-antiferromagnetic exchange couplings by means of three different approaches: density matrix renormalization group (DMRG), a low-energy effective Hamiltonian (LEH) and a Hartree variational approach (HVA). We find that in the limit of weakly interacting plaquettes, singlet and triplet states play an important role in the formation of magnetization plateaux. We study the transition regions numerically and analytically, and find that they are described, at first order in a strong- coupling expansion, by an XXZ spin-1/2 chain in a magnetic field. These results are consistent with the DMRG and HVA calculations.

Journal of Physics Conference SeriesDec 8, 20141 citation

Magnetization plateaus and jumps in a frustrated four-leg spin tube under a magnetic field

F. A. Gómez Albarracín, Marcelo Arlego, H. D. Rosales

We study the ground state phase diagram of a frustrated spin-1/2 four-leg spin tube in an external magnetic field. We explore the parameter space of this model in the regime of all-antiferromagnetic exchange couplings by means of three different approaches: analysis of low-energy effective Hamiltonian, a Hartree variational approach, and density matrix renormalization group for finite clusters. We find that in the limit of weakly interacting plaquettes, low-energy singlet, triplet, and quintuplet states play an important role in the formation of fractional magnetization plateaus. We study the transition regions numerically and analytically, and find that they are described, at first order in a strong-coupling expansion, by an $XXZ$ spin-1/2 chain in a magnetic field; the second-order terms give corrections to the $XXZ$ model. All techniques provide consistent results which allow us to predict the existence of fractional plateaus in an important region in the space of parameters of the model.We study the ground state phase diagram of a frustrated spin-1/2 four-leg spin tube in an external magnetic field. We explore the parameter space of this model in the regime of all-antiferromagnetic exchange couplings by means of three different approaches: analysis of low-energy effective Hamiltonian, a Hartree variational approach, and density matrix renormalization group for finite clusters. We find that in the limit of weakly interacting plaquettes, low-energy singlet, triplet, and quintuplet states play an important role in the formation of fractional magnetization plateaus. We study the transition regions numerically and analytically, and find that they are described, at first order in a strong-coupling expansion, by an $XXZ$ spin-1/2 chain in a magnetic field; the second-order terms give corrections to the $XXZ$ model. All techniques provide consistent results which allow us to predict the existence of fractional plateaus in an important region in the space of parameters of the model.

arXiv:1410.5502Physical Review BNov 4, 20149 citations

{\mathrm{Cu}}_{2}{\mathrm{Cl}}_{4}

and

{\mathrm{BiCu}}_{2}{\mathrm{PO}}_{6}

We study the ground state of frustrated spin-

S

chains in a strong magnetic field in the immediate vicinity of saturation. In strongly frustrated chains, the magnon dispersion has two degenerate minima at inequivalent momenta $\ifmmode\pm\else\textpm\fi{}Q$, and just below the saturation field the system can be effectively represented as a dilute one-dimensional lattice gas of two species of bosons that correspond to magnons with momenta around $\ifmmode\pm\else\textpm\fi{}Q$. We present a theory of effective interactions in such a dilute magnon gas that allows us to make quantitative predictions for arbitrary values of the spin. With the help of this method, we are able to establish the magnetic phase diagram of frustrated chains close to saturation and study phase transitions between several nontrivial states, including a two-component Luttinger liquid, a vector chiral phase, and phases with bound magnons. We study those phase transitions numerically and find a good agreement with our analytical predictions.

We explore the ground-state phase diagram of the

S=1/2

two-leg ladder with different leg interactions. The

xy

and

z

components of the leg interactions between nearest-neighbor spins in the

a

(

b

) leg are respectively denoted by

J_{{\rm l},a}

and

Δ_{\rm l} J_{{\rm l},a}

(

J_{{\rm l},b}

and

Δ_{\rm l} J_{{\rm l},b}

). On the other hand, the

xy

and

z

components of the uniform rung interactions are respectively denoted by

Γ_{\rm r} J_{\rm r}

and

J_{\rm r}

. In the above,

Δ_{\rm l}

and

Γ_{\rm r}

are the

XXZ

-type anisotropy parameters for the leg and rung interactions, respectively. This system has a frustration when

J_{{\rm l},a} J_{{\rm l},b}<0

irrespective of the sign of

J_{\rm r}

. The phase diagram on the

Δ_{\rm l}

(

|Δ_{\rm l}| \leq 1.0

) versus

J_{{\rm l},b}

(

-2.0\leq J_{{\rm l},b}\leq 3.0

) plane in the case where

J_{{\rm l},a}=0.2

J_{\rm r}=-1.0

, and

Γ_{\rm r} = 0.5

is determined numerically. We employ the physical consideration, and the level spectroscopy and phenomenological renormalization-group analyses of the numerical date obtained by the exact diagonalization method. The resultant phase diagram contains the ferromagnetic, Haldane, N{é}el, nematic Tomonaga-Luttinger liquid (TLL), partial ferrimagnetic, and

XY1

phases. Interestingly enough, the nematic TLL phase appears in the strong-rung unfrustrated region as well as in the strong-rung frustrated one. We perform the first-order perturbational calculations from the strong rung coupling limit to elucidate the characteristic features of the phase diagram. Furthermore, we make the density-matrix renormalization-group calculations for some physical quantities such as the energy gaps, the local magnetization, and the spin correlation functions to supplement the reliability of the phase diagram.

We derive and investigate the microscopic model of the quantum magnet

{\text{BiCu}}_{2}{\text{PO}}_{6}

using band-structure calculations, magnetic susceptibility and high-field magnetization measurements, as well as exact diagonalization (ED) and density-matrix renormalization group (DMRG) techniques. The resulting quasi-one-dimensional spin model is a two-leg antiferromagnetic ladder with frustrating next-nearest-neighbor couplings along the legs. The individual couplings are estimated from band-structure calculations and by fitting the magnetic susceptibility with theoretical predictions, obtained using full diagonalizations. The nearest-neighbor leg coupling

{J}_{1}

, the rung coupling

{J}_{4}

, and one of the next-nearest-neighbor couplings

{J}_{2}

amount to 120--150 K while the second next-nearest-neighbor coupling is ${J}_{2}^{\ensuremath{'}}\ensuremath{\simeq}{J}_{2}/2$. The spin ladders do not match the structural chains, and although the next-nearest-neighbor interactions

{J}_{2}

and ${J}_{2}^{\ensuremath{'}}$ have very similar superexchange pathways, they differ substantially in magnitude due to a tiny difference in the O-O distances and in the arrangement of nonmagnetic

{\text{PO}}_{4}

tetrahedra. An extensive ED study of the proposed model provides the low-energy excitation spectrum and shows that the system is in the strong rung coupling regime. The strong frustration by the next-nearest-neighbor couplings leads to a triplon branch with an incommensurate minimum. This is further corroborated by a strong-coupling expansion up to second order in the inter-rung coupling. Based on high-field magnetization measurements, we estimate the spin gap of $\ensuremath{\Delta}\ensuremath{\simeq}32\text{ }\text{K}$ and suggest the likely presence of antisymmetric Dzyaloshinskii-Moriya anisotropy and interladder coupling

{J}_{3}

. We also provide a tentative description of the physics of

{\text{BiCu}}_{2}{\text{PO}}_{6}

in magnetic field, in the light of the low-energy excitation spectra and numerical calculations based on ED and DMRG. In particular, we raise the possibility for a rich interplay between one- and two-component Luttinger liquid phases and a magnetization plateau at 1/2 of the saturation value.

XXZ

spin-1/2 chain in a magnetic field; the second-order terms give corrections to the

XXZ

model. All techniques provide consistent results which allow us to predict the existence of fractional plateaus in an important region in the space of parameters of the model.