arXiv:cond-mat/0508620 · Physical Review B

Phase diagram of S=12 two-leg XXZ spin-ladder systems

Keigo Hijii, Atsuhiro Kitazawa, K. Nomura

Published Jul 20, 2005• 46 citations

Abstract

We investigate the ground-state phase diagram of the $\mathrm{S}=\frac{1}{2}$ two-leg $XXZ$ spin-ladder system with an isotropic interchain coupling. In this model, there is the Berezinskii-Kosterlitz-Thouless transition which occurs at the $XY$-Haldane and $XY$-rung singlet phase boundaries. It was difficult to determine the transition line using traditional methods. We overcome this difficulty using the level spectroscopy method combined with the twisted boundary condition method, and we check the consistency. We find out that the phase boundary between $XY$ phase and Haldane phase lies on $\ensuremath{\Delta}=0$ line. And we show that there exist two different $XY$ phases, which we can distinguish investigating a $XX$ correlation function.

DOI: 10.1103/physrevb.72.014449

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Phase diagram of the S=12 two-leg spin ladder with staggered bond alternation

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

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arXiv:cond-mat/0508620 · Physical Review B

Phase diagram of S=12 two-leg XXZ spin-ladder systems

Keigo Hijii, Atsuhiro Kitazawa, K. Nomura

Published Jul 20, 2005• 46 citations

arXiv HTML PDF

Abstract

We investigate the ground-state phase diagram of the

\mathrm{S}=\frac{1}{2}

two-leg

XXZ

spin-ladder system with an isotropic interchain coupling. In this model, there is the Berezinskii-Kosterlitz-Thouless transition which occurs at the

XY

-Haldane and

XY

-rung singlet phase boundaries. It was difficult to determine the transition line using traditional methods. We overcome this difficulty using the level spectroscopy method combined with the twisted boundary condition method, and we check the consistency. We find out that the phase boundary between

XY

phase and Haldane phase lies on

\ensuremath{\Delta}=0

line. And we show that there exist two different

XY

phases, which we can distinguish investigating a

XX

correlation function.

DOI: 10.1103/physrevb.72.014449

Related papers

Ground-state phase diagram of a spin- 1 2 frustrated XXZ ladder

Takuhiro Ogino, Ryui Kaneko, Satoshi Morita +1 more

We study the ground-state phase diagram of a spin- $\frac{1}{2}$ frustrated XXZ ladder, in which two antiferromagnetic chains are coupled by competing rung and diagonal interactions, ${J}_{\ensuremath{\perp}}$ and ${J}_{\ifmmode\times\else\texttimes\fi{}}$. Previous studies on the isotropic model have revealed that a fluctuation-induced effective dimer attraction between the legs stabilizes the columnar dimer (CD) phase in the highly frustrated regime ${J}_{\ensuremath{\perp}}\ensuremath{\approx}2{J}_{\ifmmode\times\else\texttimes\fi{}}$, especially for ferromagnetic ${J}_{\ensuremath{\perp},\ifmmode\times\else\texttimes\fi{}}<0$. By means of effective field theory and numerical analyses, we extend this analysis to the XXZ model, and we obtain a rich phase diagram. The diagram includes four gapped featureless phases with no symmetry breaking: the rung singlet (RS) and Haldane phases as well as their twisted variants, the RS* and Haldane* phases, which are all distinct in the presence of certain symmetries. Significantly, the Haldane-CD transition point in the isotropic model turns out to be a crossing point of two transition lines in the XXZ model, and the stripe N\'eel and RS* phases appear between these lines. This indicates a nontrivial interplay between the effective dimer attraction and the exchange anisotropy. In the easy-plane regime, the four featureless phases and two critical phases are found to compete in a complex manner depending on the signs of ${J}_{\ensuremath{\perp},\ifmmode\times\else\texttimes\fi{}}$.We study the ground-state phase diagram of a spin-$\frac{1}{2}$ frustrated XXZ ladder, in which two antiferromagnetic chains are coupled by competing rung and diagonal interactions, ${J}_{\ensuremath{\perp}}$ and ${J}_{\ifmmode\times\else\texttimes\fi{}}$. Previous studies on the isotropic model have revealed that a fluctuation-induced effective dimer attraction between the legs stabilizes the columnar dimer (CD) phase in the highly frustrated regime ${J}_{\ensuremath{\perp}}\ensuremath{\approx}2{J}_{\ifmmode\times\else\texttimes\fi{}}$, especially for ferromagnetic ${J}_{\ensuremath{\perp},\ifmmode\times\else\texttimes\fi{}}<0$. By means of effective field theory and numerical analyses, we extend this analysis to the XXZ model, and we obtain a rich phase diagram. The diagram includes four gapped featureless phases with no symmetry breaking: the rung singlet (RS) and Haldane phases as well as their twisted variants, the RS* and Haldane* phases, which are all distinct in the presence of certain symmetries. Significantly, the Haldane-CD transition point in the isotropic model turns out to be a crossing point of two transition lines in the XXZ model, and the stripe N\'eel and RS* phases appear between these lines. This indicates a nontrivial interplay between the effective dimer attraction and the exchange anisotropy. In the easy-plane regime, the four featureless phases and two critical phases are found to compete in a complex manner depending on the signs of ${J}_{\ensuremath{\perp},\ifmmode\times\else\texttimes\fi{}}$.

arXiv:2206.13048Physical review. B./Physical review. BOct 3, 202213 citations

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Kiyomi Okamoto

The ground state of the $S=1/2$ two-leg ladder with the staggered bond alternations is investigated. We have determined the precise phase diagram on the $\ensuremath{\delta}\ensuremath{-}{J}^{\ensuremath{'}}$ plane $(\ensuremath{\delta}$ and ${J}^{\ensuremath{'}}$ are the magnitudes of the bond alternation and the rung interaction, respectively), employing the level spectroscopic analysis of the numerical diagonalization data obtained by the Lanczos algorithm with the twisted boundary condition. The phase boundary between the leg-dimer state and the rung-dimer state near $(\ensuremath{\delta}{,J}^{\ensuremath{'}})=(0,0)$ is of the form ${J}^{\ensuremath{'}}\ensuremath{\propto}{\ensuremath{\delta}}^{0.69},$ which is consistent with the prediction ${J}^{\ensuremath{'}}\ensuremath{\propto}{\ensuremath{\delta}}^{2/3}$ by Mart\'{\i}n-Delgado, Duleksky, and Sierra, and also by Wang and Nersesyan, but not with ${J}^{\ensuremath{'}}\ensuremath{\propto}{\ensuremath{\delta}}^{2}$ by Cabra and Grynberg.The ground state of the $S=1/2$ two-leg ladder with the staggered bond alternations is investigated. We have determined the precise phase diagram on the $\ensuremath{\delta}\ensuremath{-}{J}^{\ensuremath{'}}$ plane $(\ensuremath{\delta}$ and ${J}^{\ensuremath{'}}$ are the magnitudes of the bond alternation and the rung interaction, respectively), employing the level spectroscopic analysis of the numerical diagonalization data obtained by the Lanczos algorithm with the twisted boundary condition. The phase boundary between the leg-dimer state and the rung-dimer state near $(\ensuremath{\delta}{,J}^{\ensuremath{'}})=(0,0)$ is of the form ${J}^{\ensuremath{'}}\ensuremath{\propto}{\ensuremath{\delta}}^{0.69},$ which is consistent with the prediction ${J}^{\ensuremath{'}}\ensuremath{\propto}{\ensuremath{\delta}}^{2/3}$ by Mart\'{\i}n-Delgado, Duleksky, and Sierra, and also by Wang and Nersesyan, but not with ${J}^{\ensuremath{'}}\ensuremath{\propto}{\ensuremath{\delta}}^{2}$ by Cabra and Grynberg.

Physical review. B, Condensed matterJun 24, 200317 citations

Ground state phase diagram of an S = 1 2 two-leg Heisenberg spin ladder system with negative four-spin interaction

Keigo Hijii, Tôru Sakai

We investigate the ground-state phase diagram and critical properties of the $S=1/2$ two-leg Heisenberg spin ladder system with a negative four-spin interaction using a numerical exact diagonalization method. Using the perturbation theory in the strong negative rung-coupling limit, we derive an $S=1$ bilinear-biquadratic chain as an effective model. We discuss the ground-state phase diagram in this limit. Next we numerically determine a phase boundary between the rung singlet phase and the columnar dimer (CD) phase by the phenomenological renormalization group method, and one between the CD phase and the Haldane phase by the twisted boundary condition method. We confirm that the phase transition between the CD phase and the Haldane phase is of second order and this universality class is described by the $k=2$ $SU(2)$ Wess-Zumino-Novikov-Witten nonlinear $\ensuremath{\sigma}$ model, calculating the central charge and scaling dimensions.We investigate the ground-state phase diagram and critical properties of the $S=1/2$ two-leg Heisenberg spin ladder system with a negative four-spin interaction using a numerical exact diagonalization method. Using the perturbation theory in the strong negative rung-coupling limit, we derive an $S=1$ bilinear-biquadratic chain as an effective model. We discuss the ground-state phase diagram in this limit. Next we numerically determine a phase boundary between the rung singlet phase and the columnar dimer (CD) phase by the phenomenological renormalization group method, and one between the CD phase and the Haldane phase by the twisted boundary condition method. We confirm that the phase transition between the CD phase and the Haldane phase is of second order and this universality class is described by the $k=2$ $SU(2)$ Wess-Zumino-Novikov-Witten nonlinear $\ensuremath{\sigma}$ model, calculating the central charge and scaling dimensions.

Physical Review BSep 4, 20138 citations

FrustratedS= 1/2 Two-Leg Ladder with Different Leg Interactions

Takashi Tonegawa, Kiyomi Okamoto, Toshiya Hikihara +1 more

We explore the ground-state phase diagram of the S = 1/2 two-leg ladder. The isotropic leg interactions J 1, a and J 1, b between nearest neighbor spins in the legs a and b , respectively, are different from each other. The xy and z components of the uniform rung interactions are denoted by J r and Δ J r , respectively, where Δ is the XXZ anisotropy parameter. This system has a frustration when J 1, a J 1, b < 0 irrespective of the sign of J r . The phase diagrams on the Δ (0≤Δ<1) versus J 1, b plane in the cases of J 1, a = − 0.2 and J 1, a = 0.2 with J r = −1 are determined numerically. We employ the physical consideration, the level spectroscopy analysis of the results obtained by the exact diagonalization method and also the density-matrix renormalization-group method. It is found that the non-collinear ferrimagnetic (NCFR) state appears as the ground state in the frustrated region of the parameters. Furthermore, the direct-product triplet-dimer (TD) state in which all rungs form the TD pair is the exact ground state , when J 1, a + J 1, b = 0 and 0≤ Δ ≲ 0.83. The obtained phase diagrams consist of the TD, XY and Haldane phases as well as the NCFR phase.We explore the ground-state phase diagram of the S = 1/2 two-leg ladder. The isotropic leg interactions J 1, a and J 1, b between nearest neighbor spins in the legs a and b , respectively, are different from each other. The xy and z components of the uniform rung interactions are denoted by J r and Δ J r , respectively, where Δ is the XXZ anisotropy parameter. This system has a frustration when J 1, a J 1, b < 0 irrespective of the sign of J r . The phase diagrams on the Δ (0≤Δ<1) versus J 1, b plane in the cases of J 1, a = − 0.2 and J 1, a = 0.2 with J r = −1 are determined numerically. We employ the physical consideration, the level spectroscopy analysis of the results obtained by the exact diagonalization method and also the density-matrix renormalization-group method. It is found that the non-collinear ferrimagnetic (NCFR) state appears as the ground state in the frustrated region of the parameters. Furthermore, the direct-product triplet-dimer (TD) state in which all rungs form the TD pair is the exact ground state , when J 1, a + J 1, b = 0 and 0≤ Δ ≲ 0.83. The obtained phase diagrams consist of the TD, XY and Haldane phases as well as the NCFR phase.

arXiv:1608.02064Journal of Physics Conference SeriesApr 20, 20175 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

Opening of the Haldane gap in anisotropic two- and four-leg spin ladders

Eugene H. Kim, Jenő Sólyom

We study the opening of the Haldane gap in two-leg and four-leg anisotropic spin ladders using bosonization and renormalization group methods, and we determine the phase diagram as a function of the interchain coupling and relative anisotropy ${J}^{z}{/J}^{\mathrm{xy}}.$ It is found that the opening of the Haldane gap is qualitatively different for the two cases considered. For the two-leg ladder the Haldane gap opens for arbitrarily small interchain coupling, independent of ${J}^{z}{/J}^{\mathrm{xy}},$ and the Haldane phase exists in a large region of parameter space. For the four-leg ladder the opening of the Haldane gap is strongly dependent on both the interchain coupling as well as ${J}^{z}{/J}^{\mathrm{xy}},$ and the Haldane phase exists only in a narrow region about the isotropic antiferromagnet.We study the opening of the Haldane gap in two-leg and four-leg anisotropic spin ladders using bosonization and renormalization group methods, and we determine the phase diagram as a function of the interchain coupling and relative anisotropy ${J}^{z}{/J}^{\mathrm{xy}}.$ It is found that the opening of the Haldane gap is qualitatively different for the two cases considered. For the two-leg ladder the Haldane gap opens for arbitrarily small interchain coupling, independent of ${J}^{z}{/J}^{\mathrm{xy}},$ and the Haldane phase exists in a large region of parameter space. For the four-leg ladder the opening of the Haldane gap is strongly dependent on both the interchain coupling as well as ${J}^{z}{/J}^{\mathrm{xy}},$ and the Haldane phase exists only in a narrow region about the isotropic antiferromagnet.

arXiv:cond-mat/9903241Physical review. B, Condensed matterDec 1, 199917 citations

We investigate the ground-state phase diagram of the

\mathrm{S}=\frac{1}{2}

two-leg

XXZ

spin-ladder system with an isotropic interchain coupling. In this model, there is the Berezinskii-Kosterlitz-Thouless transition which occurs at the

XY

-Haldane and

XY

XY

phase and Haldane phase lies on

\ensuremath{\Delta}=0

line. And we show that there exist two different

XY

phases, which we can distinguish investigating a

XX

correlation function.

We study the ground-state phase diagram of a spin-

\frac{1}{2}

frustrated XXZ ladder, in which two antiferromagnetic chains are coupled by competing rung and diagonal interactions, ${J}_{\ensuremath{\perp}}$ and ${J}_{\ifmmode\times\else\texttimes\fi{}}$. Previous studies on the isotropic model have revealed that a fluctuation-induced effective dimer attraction between the legs stabilizes the columnar dimer (CD) phase in the highly frustrated regime ${J}_{\ensuremath{\perp}}\ensuremath{\approx}2{J}_{\ifmmode\times\else\texttimes\fi{}}$, especially for ferromagnetic ${J}_{\ensuremath{\perp},\ifmmode\times\else\texttimes\fi{}}<0$. By means of effective field theory and numerical analyses, we extend this analysis to the XXZ model, and we obtain a rich phase diagram. The diagram includes four gapped featureless phases with no symmetry breaking: the rung singlet (RS) and Haldane phases as well as their twisted variants, the RS* and Haldane* phases, which are all distinct in the presence of certain symmetries. Significantly, the Haldane-CD transition point in the isotropic model turns out to be a crossing point of two transition lines in the XXZ model, and the stripe N\'eel and RS* phases appear between these lines. This indicates a nontrivial interplay between the effective dimer attraction and the exchange anisotropy. In the easy-plane regime, the four featureless phases and two critical phases are found to compete in a complex manner depending on the signs of ${J}_{\ensuremath{\perp},\ifmmode\times\else\texttimes\fi{}}$.

The ground state of the

S=1/2

two-leg ladder with the staggered bond alternations is investigated. We have determined the precise phase diagram on the $\ensuremath{\delta}\ensuremath{-}{J}^{\ensuremath{'}}$ plane $(\ensuremath{\delta}$ and ${J}^{\ensuremath{'}}$ are the magnitudes of the bond alternation and the rung interaction, respectively), employing the level spectroscopic analysis of the numerical diagonalization data obtained by the Lanczos algorithm with the twisted boundary condition. The phase boundary between the leg-dimer state and the rung-dimer state near $(\ensuremath{\delta}{,J}^{\ensuremath{'}})=(0,0)$ is of the form ${J}^{\ensuremath{'}}\ensuremath{\propto}{\ensuremath{\delta}}^{0.69},$ which is consistent with the prediction ${J}^{\ensuremath{'}}\ensuremath{\propto}{\ensuremath{\delta}}^{2/3}$ by Mart\'{\i}n-Delgado, Duleksky, and Sierra, and also by Wang and Nersesyan, but not with ${J}^{\ensuremath{'}}\ensuremath{\propto}{\ensuremath{\delta}}^{2}$ by Cabra and Grynberg.

We investigate the ground-state phase diagram and critical properties of the

S=1/2

two-leg Heisenberg spin ladder system with a negative four-spin interaction using a numerical exact diagonalization method. Using the perturbation theory in the strong negative rung-coupling limit, we derive an

S=1

bilinear-biquadratic chain as an effective model. We discuss the ground-state phase diagram in this limit. Next we numerically determine a phase boundary between the rung singlet phase and the columnar dimer (CD) phase by the phenomenological renormalization group method, and one between the CD phase and the Haldane phase by the twisted boundary condition method. We confirm that the phase transition between the CD phase and the Haldane phase is of second order and this universality class is described by the

k=2

SU(2)

Wess-Zumino-Novikov-Witten nonlinear $\ensuremath{\sigma}$ model, calculating the central charge and scaling dimensions.

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.

{J}^{z}{/J}^{\mathrm{xy}}.

It is found that the opening of the Haldane gap is qualitatively different for the two cases considered. For the two-leg ladder the Haldane gap opens for arbitrarily small interchain coupling, independent of

{J}^{z}{/J}^{\mathrm{xy}},

and the Haldane phase exists in a large region of parameter space. For the four-leg ladder the opening of the Haldane gap is strongly dependent on both the interchain coupling as well as

{J}^{z}{/J}^{\mathrm{xy}},

and the Haldane phase exists only in a narrow region about the isotropic antiferromagnet.