arXiv:1303.5537 · Condensed Matter Physics

Ground state of a spin-1/2 Heisenberg-Ising two-leg ladder with XYZ intra-rung coupling

Taras Verkholyak, Jozef Strečka

Published Mar 1, 2013• 7 citations

Abstract

The quantum spin-1/2 two-leg ladder with an anisotropic X Y Z Heisenberg intra-rung interaction and Ising inter-rung interactions is treated by means of a rigorous approach based on the unitary transformation.The particular case of the considered model with X -X intra-rung interaction resembles a quantum compass ladder with additional frustrating diagonal Ising interactions.Using an appropriately chosen unitary transformation, the model under investigation may be reduced to a transverse Ising chain with composite spins, and one may subsequently find the ground state quite rigorously.We obtain a ground-state phase diagram and analyze the interplay of the competition between several factors: the X Y Z anisotropy in the Heisenberg intra-rung coupling, the Ising interaction along the legs, and the frustrating diagonal Ising interaction.The investigated model shows extraordinarily diverse ground-state phase diagrams including several unusual quantum ordered phases, two different disordered quantum paramagnetic phases, as well as discontinuous or continuous quantum phase transitions between those phases.

DOI: 10.5488/cmp.16.13601

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arXiv:1303.5537 · Condensed Matter Physics

Ground state of a spin-1/2 Heisenberg-Ising two-leg ladder with XYZ intra-rung coupling

Taras Verkholyak, Jozef Strečka

Published Mar 1, 2013• 7 citations

arXiv HTML PDF

Abstract

DOI: 10.5488/cmp.16.13601

Related papers

Quantum phase transitions in the exactly solved spin-1/2 Heisenberg–Ising ladder

Taras Verkholyak, Jozef Strečka

The ground-state behaviour of the frustrated quantum spin-1/2 two-leg ladder with the Heisenberg intra-rung and Ising inter-rung interactions is examined in detail. The investigated model is transformed into the quantum Ising chain with composite spins in an effective transverse and longitudinal field by employing either the bond-state representation or the unitary transformation. It is shown that the ground state of the Heisenberg–Ising ladder can be descended from three exactly solvable models: the quantum Ising chain in a transverse field, the ‘classical’ Ising chain in a longitudinal field or the spin-chain model in a staggered longitudinal–transverse field. The last model serves in evidence of the staggered bond phase with alternating singlet and triplet bonds on the rungs of a two-leg ladder, which appears at moderate values of the external magnetic field and consequently leads to a fractional plateau at a half of the saturation magnetization. The ground-state phase diagram totally consists of five ordered and one quantum paramagnetic phase, which are separated from each other either by the lines of discontinuous or continuous quantum phase transitions. The order parameters are exactly calculated for all five ordered phases and the quantum paramagnetic phase is characterized through different short-range spin–spin correlations.The ground-state behaviour of the frustrated quantum spin-1/2 two-leg ladder with the Heisenberg intra-rung and Ising inter-rung interactions is examined in detail. The investigated model is transformed into the quantum Ising chain with composite spins in an effective transverse and longitudinal field by employing either the bond-state representation or the unitary transformation. It is shown that the ground state of the Heisenberg–Ising ladder can be descended from three exactly solvable models: the quantum Ising chain in a transverse field, the ‘classical’ Ising chain in a longitudinal field or the spin-chain model in a staggered longitudinal–transverse field. The last model serves in evidence of the staggered bond phase with alternating singlet and triplet bonds on the rungs of a two-leg ladder, which appears at moderate values of the external magnetic field and consequently leads to a fractional plateau at a half of the saturation magnetization. The ground-state phase diagram totally consists of five ordered and one quantum paramagnetic phase, which are separated from each other either by the lines of discontinuous or continuous quantum phase transitions. The order parameters are exactly calculated for all five ordered phases and the quantum paramagnetic phase is characterized through different short-range spin–spin correlations.

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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

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Takashi Tonegawa, Kiyomi Okamoto, Toshiya Hikihara +1 more

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arXiv:1510.04928Journal of Physics Conference SeriesFeb 5, 20162 citations

S

σ

-type spins) are aligned in the same direction. Whereas, in anisotropic antiferromagnetic phase, both

S

and

σ

-type of spins are anti-ferromagnetically aligned with each other, two nearest

S

spins along the rung form an anisotropic singlet bond whereas two nearest

σ

spins form an Ising bond. In large Heisenberg rung exchange interaction limit, spins on each leg are ferromagnetically aligned, but spins on different legs are anti-ferromagnetically aligned. The thermodynamic quantities like

Cv(T)

χ(T)

and

S(T)

are also calculated using the transfer matrix method for different phase. The magnetic gap in the SRFM and the SLFM can be notice from

χ(T)

and

Cv(T)

curves.

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.