arXiv:cond-mat/0201004 · Physical review. B, Condensed matter

Ground state and excitation of an asymmetric spin ladder model

Shu Chen, H. Büttner, Johannes Voit

Published Feb 10, 2003• 33 citations

Abstract

We perform a systematic investigation on an asymmetric zigzag spin ladder with interleg exchange ${J}_{1}$ and different exchange integrals ${J}_{2}\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}$ on both legs. In the weak frustration limit, the spin model can be mapped to a revised double frequency sine-Gordon model by using bosonization. Renormalization-group analysis shows that the Heisenberg critical point flows to an intermediate-coupling fixed point with gapless excitations and a vanishing spin velocity. When the frustration is large, a spin gap opens and a dimer ground state is realized. Fixing ${J}_{2}{=J}_{1}/2,$ we find, as a function of $\ensuremath{\delta},$ a continuous manifold of Hamiltonians with dimer product ground states, interpolating between the Majumdar-Ghosh and sawtooth spin-chain model. While the ground state is independent of the alternating next-nearest-neighbor exchange $\ensuremath{\delta},$ the gap size of excitations is found to decrease with increasing $\ensuremath{\delta}.$ We also extend our study to a two-dimensional double layer model with an exactly known ground state.

DOI: 10.1103/physrevb.67.054412

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arXiv:cond-mat/0201004 · Physical review. B, Condensed matter

Ground state and excitation of an asymmetric spin ladder model

Shu Chen, H. Büttner, Johannes Voit

Published Feb 10, 2003• 33 citations

arXiv HTML PDF

Abstract

We perform a systematic investigation on an asymmetric zigzag spin ladder with interleg exchange

{J}_{1}

and different exchange integrals

{J}_{2}\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}

on both legs. In the weak frustration limit, the spin model can be mapped to a revised double frequency sine-Gordon model by using bosonization. Renormalization-group analysis shows that the Heisenberg critical point flows to an intermediate-coupling fixed point with gapless excitations and a vanishing spin velocity. When the frustration is large, a spin gap opens and a dimer ground state is realized. Fixing

{J}_{2}{=J}_{1}/2,

we find, as a function of

\ensuremath{\delta},

a continuous manifold of Hamiltonians with dimer product ground states, interpolating between the Majumdar-Ghosh and sawtooth spin-chain model. While the ground state is independent of the alternating next-nearest-neighbor exchange

\ensuremath{\delta},

the gap size of excitations is found to decrease with increasing

\ensuremath{\delta}.

We also extend our study to a two-dimensional double layer model with an exactly known ground state.

DOI: 10.1103/physrevb.67.054412

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Phase Diagram of an Asymmetric Spin Ladder

Shu Chen, H. Büttner, Johannes Voit

We investigate an asymmetric zigzag spin ladder with different exchange integrals on both legs using bosonization and renormalization group approaches. When the leg exchange integrals and frustration both are sufficiently small, renormalization group analysis shows that the Heisenberg critical point flows to an intermediate-coupling fixed point with gapless excitations and a vanishing spin velocity. When they are large, a spin gap opens and a dimer liquid is realized. Here, we find a continuous manifold of Hamiltonians with dimer product ground states, interpolating between the Majumdar-Ghosh and sawtooth spin-chain model.We investigate an asymmetric zigzag spin ladder with different exchange integrals on both legs using bosonization and renormalization group approaches. When the leg exchange integrals and frustration both are sufficiently small, renormalization group analysis shows that the Heisenberg critical point flows to an intermediate-coupling fixed point with gapless excitations and a vanishing spin velocity. When they are large, a spin gap opens and a dimer liquid is realized. Here, we find a continuous manifold of Hamiltonians with dimer product ground states, interpolating between the Majumdar-Ghosh and sawtooth spin-chain model.

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Exact columnar dimer ground state and quantum phase transitions in a frustrated coupled spin ladder model

M. Mahapatra, Rakesh Kumar

We study a spin-half frustrated coupled ladder system, in which ladders with leg, rung, and diagonal interactions are linked via nearest-neighbor coupling. By introducing a leg-symmetric inter-ladder interaction that connects the left-to-left and right-to-right legs of adjacent ladders, the model is found to possess an exact dimer ground state, characterized by a product of two-spin singlets forming a columnar dimer phase. We analyze this model using bond-operator mean-field theory (BOMFT) and the density matrix renormalization group (DMRG) to probe the phase transitions that occur as one traverses the coupling space. The BOMFT reveals three distinct phases: a double-stripe ordered phase, a Néel ordered phase, and a quantum disordered dimerized phase. The critical points for the transitions are at $ J_1 = -0.81 $ (double-stripe to dimerized) and at $ J_1 = 2.81 $ (dimerized to Néel phase). Further, the DMRG results corroborate the exact ground state and refine the critical points to $ J_1 = -0.79 $ and $ J_1 = 2.29 $ for the respective transitions. Additionally, another transition is identified as the Néel order vanishes for $ J_1 \ge 4.5 $. The model can alternatively be represented as a network of orthogonal zigzag and fully frustrated spin ladders, offering a structural framework conducive to quantum materials engineering.We study a spin-half frustrated coupled ladder system, in which ladders with leg, rung, and diagonal interactions are linked via nearest-neighbor coupling. By introducing a leg-symmetric inter-ladder interaction that connects the left-to-left and right-to-right legs of adjacent ladders, the model is found to possess an exact dimer ground state, characterized by a product of two-spin singlets forming a columnar dimer phase. We analyze this model using bond-operator mean-field theory (BOMFT) and the density matrix renormalization group (DMRG) to probe the phase transitions that occur as one traverses the coupling space. The BOMFT reveals three distinct phases: a double-stripe ordered phase, a Néel ordered phase, and a quantum disordered dimerized phase. The critical points for the transitions are at $ J_1 = -0.81 $ (double-stripe to dimerized) and at $ J_1 = 2.81 $ (dimerized to Néel phase). Further, the DMRG results corroborate the exact ground state and refine the critical points to $ J_1 = -0.79 $ and $ J_1 = 2.29 $ for the respective transitions. Additionally, another transition is identified as the Néel order vanishes for $ J_1 \ge 4.5 $. The model can alternatively be represented as a network of orthogonal zigzag and fully frustrated spin ladders, offering a structural framework conducive to quantum materials engineering.

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Spin-1/2 string correlations and singlet-triplet gaps of frustrated ladders with ferromagnetic (F) legs and alternate F and AF rungs

Chatterjee, Monalisa, Manoranjan Kumar, Z. G. Soos

The frustrated ladder with alternate ferromagnetic(F) exchange $-J_F$ and AF exchange $J_A$ to first neighbors and F exchange $-J_L$ to second neighbors is studied by exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations in systems of $2N$ spins-1/2 with periodic boundary conditions. The ground state is a singlet $(S = 0)$ and the singlet-triplet gap $\varepsilon_T$ is finite for the exchanges considered. Spin-1/2 string correlation functions $g_1(N)$ and $g_2(N)$ are defined for an even number $N$ of consecutive spins in systems with two spins per unit cell; the ladder has string order $g_2(\infty)> 0$ and $g_1(\infty) = 0$ . The minimum $N^*$ of $g_2(N)$ is related to the range of ground-state spin correlations. Convergence to $g_2(\infty)$ is from below, and $g_1(N)$ decreases exponentially for $N \geq N^*$ . Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of $J_F$ , $J_L$ and $J_A$ reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the $J_1-J_2$ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-1/2 systems. The nondegenerate singlet ground state of ladder is a bond-order wave, a Kekulé VB diagrams at $J_L = J_F/2 \leq J_A$ , that is reversed on interchanging $-J_F$ and $J_A$ . Inversion symmetry is spontaneously broken in the dimer phase of the $J_1-J_2$ model where the Kekulé diagrams are the doubly degenerate ground states at $J_2/J_1 = 1/2$ .The frustrated ladder with alternate ferromagnetic(F) exchange $-J_F$ and AF exchange $J_A$ to first neighbors and F exchange $-J_L$ to second neighbors is studied by exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations in systems of $2N$ spins-1/2 with periodic boundary conditions. The ground state is a singlet $(S = 0)$ and the singlet-triplet gap $\varepsilon_T$ is finite for the exchanges considered. Spin-1/2 string correlation functions $g_1(N)$ and $g_2(N)$ are defined for an even number $N$ of consecutive spins in systems with two spins per unit cell; the ladder has string order $g_2(\infty)> 0$ and $g_1(\infty) = 0$. The minimum $N^*$ of $g_2(N)$ is related to the range of ground-state spin correlations. Convergence to $g_2(\infty)$ is from below, and $g_1(N)$ decreases exponentially for $N \geq N^*$. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of $J_F$, $J_L$ and $J_A$ reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the $J_1-J_2$ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-1/2 systems. The nondegenerate singlet ground state of ladder is a bond-order wave, a Kekulé VB diagrams at $J_L = J_F/2 \leq J_A$, that is reversed on interchanging $-J_F$ and $J_A$. Inversion symmetry is spontaneously broken in the dimer phase of the $J_1-J_2$ model where the Kekulé diagrams are the doubly degenerate ground states at $J_2/J_1 = 1/2$.

arXiv:2403.05238Mar 8, 20240 citations

Variety of order-by-disorder phases in the asymmetric J1−J2 zigzag ladder: From the delta chain to the J1−J2 chain

Tomoki Yamaguchi, Stefan‐Ludwig Drechsler, Y. Ohta +1 more

We study an asymmetric ${J}_{1}\ensuremath{-}{J}_{2}$ zigzag ladder consisting of two different spin- $\frac{1}{2}$ antiferromagnetic (AFM; ${J}_{2}, \ensuremath{\gamma}{J}_{2}>0$) Heisenberg legs coupled by zigzag-shaped ferromagnetic (FM; ${J}_{1}<0$ ) interleg interaction. On the basis of density-matrix renormalization group based calculations, the ground-state phase diagram is obtained as functions of $\ensuremath{\gamma}$ and ${J}_{2}/|{J}_{1}|$ . It contains four kinds of frustration-induced ordered phases except a trivial FM phase. Two of the ordered phases are valence bond solid (VBS) with spin-singlet dimerization, which is a rather conventional order by disorder. Still, it is interesting to note that the VBS states possesses an Affleck-Kennedy-Lieb-Tasaki--type topological hidden order. The remaining two phases are ferrimagnetic orders, each of which is distinguished by commensurate or incommensurate spin-spin correlation. It is striking that the ferrimagnetic orders are not associated with geometrical symmetry breaking; instead, the global spin-rotation symmetry is broken. In other words, the system lowers its energy via the FM interleg interaction by polarizing both of the AFM Heisenberg legs. This is a rare type of order by disorder. Besides, the incommensurate ferrimagnetic state appears as a consequence of the competition between a polarization and a critical Tomonaga-Luttinger--liquid behavior in the AFM Heisenberg legs.We study an asymmetric ${J}_{1}\ensuremath{-}{J}_{2}$ zigzag ladder consisting of two different spin-$\frac{1}{2}$ antiferromagnetic (AFM; ${J}_{2}, \ensuremath{\gamma}{J}_{2}>0$) Heisenberg legs coupled by zigzag-shaped ferromagnetic (FM; ${J}_{1}<0$) interleg interaction. On the basis of density-matrix renormalization group based calculations, the ground-state phase diagram is obtained as functions of $\ensuremath{\gamma}$ and ${J}_{2}/|{J}_{1}|$. It contains four kinds of frustration-induced ordered phases except a trivial FM phase. Two of the ordered phases are valence bond solid (VBS) with spin-singlet dimerization, which is a rather conventional order by disorder. Still, it is interesting to note that the VBS states possesses an Affleck-Kennedy-Lieb-Tasaki--type topological hidden order. The remaining two phases are ferrimagnetic orders, each of which is distinguished by commensurate or incommensurate spin-spin correlation. It is striking that the ferrimagnetic orders are not associated with geometrical symmetry breaking; instead, the global spin-rotation symmetry is broken. In other words, the system lowers its energy via the FM interleg interaction by polarizing both of the AFM Heisenberg legs. This is a rare type of order by disorder. Besides, the incommensurate ferrimagnetic state appears as a consequence of the competition between a polarization and a critical Tomonaga-Luttinger--liquid behavior in the AFM Heisenberg legs.

arXiv:1911.05270Physical review. B./Physical review. BMar 9, 202021 citations

Spin- 12 string correlations and singlet-triplet gaps of frustrated ladders with ferromagnetic legs and alternate ferromagnetic and antiferromagnetic rungs

Monalisa Chatterjee, Manoranjan Kumar, Z. G. Soos

The frustrated ladder with alternate ferromagnetic exchange $\ensuremath{-}{J}_{F}$ and antiferromagnetic exchange ${J}_{A}$ to first neighbors and ferromagnetic exchange $\ensuremath{-}{J}_{L}$ to second neighbors is studied by exact diagonalization and density matrix renormalization group calculations in systems of $2N$ spins- $\frac{1}{2}$ with periodic boundary conditions. The ground state is a singlet $(S=0)$ and the singlet-triplet gap ${\ensuremath{\varepsilon}}_{T}$ is finite for the exchanges considered. Spin- $\frac{1}{2}$ string correlation functions ${g}_{1}(N)$ and ${g}_{2}(N)$ are defined for an even number $N$ of consecutive spins in systems with two spins per unit cell; the ladder has string order ${g}_{2}(\ensuremath{\infty})>0$ and ${g}_{1}(\ensuremath{\infty})=0$. The minimum ${N}^{*}$ of ${g}_{2}(N)$ is related to the range of ground-state spin correlations. Convergence to ${g}_{2}(\ensuremath{\infty})$ is from below, and ${g}_{1}(N)$ decreases exponentially for $N\ensuremath{\ge}{N}^{*}$. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of ${J}_{F}, {J}_{L}$ , and ${J}_{A}$ reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin- $\frac{1}{2}$ systems. The nondegenerate singlet ground state of the ladder is a bond-order wave, a Kekul\'e VB diagram at ${J}_{L}={J}_{F}/2\ensuremath{\le}{J}_{A}$, that is reversed on interchanging $\ensuremath{-}{J}_{F}$ and ${J}_{A}$ . Inversion symmetry is spontaneously broken in the dimer phase of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model where the Kekul\'e diagrams are the doubly degenerate ground states at ${J}_{2}/{J}_{1}=1/2$ .The frustrated ladder with alternate ferromagnetic exchange $\ensuremath{-}{J}_{F}$ and antiferromagnetic exchange ${J}_{A}$ to first neighbors and ferromagnetic exchange $\ensuremath{-}{J}_{L}$ to second neighbors is studied by exact diagonalization and density matrix renormalization group calculations in systems of $2N$ spins-$\frac{1}{2}$ with periodic boundary conditions. The ground state is a singlet $(S=0)$ and the singlet-triplet gap ${\ensuremath{\varepsilon}}_{T}$ is finite for the exchanges considered. Spin-$\frac{1}{2}$ string correlation functions ${g}_{1}(N)$ and ${g}_{2}(N)$ are defined for an even number $N$ of consecutive spins in systems with two spins per unit cell; the ladder has string order ${g}_{2}(\ensuremath{\infty})>0$ and ${g}_{1}(\ensuremath{\infty})=0$. The minimum ${N}^{*}$ of ${g}_{2}(N)$ is related to the range of ground-state spin correlations. Convergence to ${g}_{2}(\ensuremath{\infty})$ is from below, and ${g}_{1}(N)$ decreases exponentially for $N\ensuremath{\ge}{N}^{*}$. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of ${J}_{F}, {J}_{L}$, and ${J}_{A}$ reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-$\frac{1}{2}$ systems. The nondegenerate singlet ground state of the ladder is a bond-order wave, a Kekul\'e VB diagram at ${J}_{L}={J}_{F}/2\ensuremath{\le}{J}_{A}$, that is reversed on interchanging $\ensuremath{-}{J}_{F}$ and ${J}_{A}$. Inversion symmetry is spontaneously broken in the dimer phase of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model where the Kekul\'e diagrams are the doubly degenerate ground states at ${J}_{2}/{J}_{1}=1/2$.

Physical review. B./Physical review. BMar 28, 20241 citation

We perform a systematic investigation on an asymmetric zigzag spin ladder with interleg exchange

{J}_{1}

and different exchange integrals

{J}_{2}\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}

{J}_{2}{=J}_{1}/2,

we find, as a function of

\ensuremath{\delta},

\ensuremath{\delta},

the gap size of excitations is found to decrease with increasing

\ensuremath{\delta}.

We also extend our study to a two-dimensional double layer model with an exactly known ground state.

J_{\perp}>0

). The estimated gap for the strong coupling limit (

J_{\perp}/J_1 \to \infty

) is 0.28

J_1

. Although the interaction is homogeneous and only nearest-neighbor, the ground states of the system are dimerized and break the translational symmetry in the thermodynamic limit. Introducing the next-nearest neighbor coupling (

J_2

), we can see that the system is solved exactly. The ground state wave function is completely dimer-ordered. Using density matrix renomalization group algorithm, we show numerically that the original model (

J_2=0

) has the same nature with the exactly solvable model. The ground state properties of the ladder with a higher odd number of legs are also discussed.

The frustrated ladder with alternate ferromagnetic(F) exchange

-J_F

and AF exchange

J_A

to first neighbors and F exchange

-J_L

to second neighbors is studied by exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations in systems of

2N

spins-1/2 with periodic boundary conditions. The ground state is a singlet

(S = 0)

and the singlet-triplet gap

\varepsilon_T

is finite for the exchanges considered. Spin-1/2 string correlation functions

g_1(N)

and

g_2(N)

are defined for an even number

N

of consecutive spins in systems with two spins per unit cell; the ladder has string order

g_2(\infty)> 0

and

g_1(\infty) = 0

. The minimum

N^*

g_2(N)

is related to the range of ground-state spin correlations. Convergence to

g_2(\infty)

is from below, and

g_1(N)

decreases exponentially for

N \geq N^*

. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of

J_F

J_L

and

J_A

reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the

J_1-J_2

model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-1/2 systems. The nondegenerate singlet ground state of ladder is a bond-order wave, a Kekulé VB diagrams at

J_L = J_F/2 \leq J_A

, that is reversed on interchanging

-J_F

and

J_A

. Inversion symmetry is spontaneously broken in the dimer phase of the

J_1-J_2

model where the Kekulé diagrams are the doubly degenerate ground states at

J_2/J_1 = 1/2

We study an asymmetric ${J}_{1}\ensuremath{-}{J}_{2}$ zigzag ladder consisting of two different spin-

\frac{1}{2}

antiferromagnetic (AFM; ${J}_{2}, \ensuremath{\gamma}{J}_{2}>0$) Heisenberg legs coupled by zigzag-shaped ferromagnetic (FM;

{J}_{1}<0

) interleg interaction. On the basis of density-matrix renormalization group based calculations, the ground-state phase diagram is obtained as functions of $\ensuremath{\gamma}$ and

{J}_{2}/|{J}_{1}|

. It contains four kinds of frustration-induced ordered phases except a trivial FM phase. Two of the ordered phases are valence bond solid (VBS) with spin-singlet dimerization, which is a rather conventional order by disorder. Still, it is interesting to note that the VBS states possesses an Affleck-Kennedy-Lieb-Tasaki--type topological hidden order. The remaining two phases are ferrimagnetic orders, each of which is distinguished by commensurate or incommensurate spin-spin correlation. It is striking that the ferrimagnetic orders are not associated with geometrical symmetry breaking; instead, the global spin-rotation symmetry is broken. In other words, the system lowers its energy via the FM interleg interaction by polarizing both of the AFM Heisenberg legs. This is a rare type of order by disorder. Besides, the incommensurate ferrimagnetic state appears as a consequence of the competition between a polarization and a critical Tomonaga-Luttinger--liquid behavior in the AFM Heisenberg legs.

The frustrated ladder with alternate ferromagnetic exchange $\ensuremath{-}{J}_{F}$ and antiferromagnetic exchange

{J}_{A}

to first neighbors and ferromagnetic exchange $\ensuremath{-}{J}_{L}$ to second neighbors is studied by exact diagonalization and density matrix renormalization group calculations in systems of

2N

spins-

\frac{1}{2}

with periodic boundary conditions. The ground state is a singlet

(S=0)

and the singlet-triplet gap ${\ensuremath{\varepsilon}}_{T}$ is finite for the exchanges considered. Spin-

\frac{1}{2}

string correlation functions

{g}_{1}(N)

and

{g}_{2}(N)

are defined for an even number

N

of consecutive spins in systems with two spins per unit cell; the ladder has string order ${g}_{2}(\ensuremath{\infty})>0$ and ${g}_{1}(\ensuremath{\infty})=0$. The minimum

{N}^{*}

{g}_{2}(N)

is related to the range of ground-state spin correlations. Convergence to ${g}_{2}(\ensuremath{\infty})$ is from below, and

{g}_{1}(N)

decreases exponentially for $N\ensuremath{\ge}{N}^{*}$. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of

{J}_{F}, {J}_{L}

, and

{J}_{A}

reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-

\frac{1}{2}

systems. The nondegenerate singlet ground state of the ladder is a bond-order wave, a Kekul\'e VB diagram at ${J}_{L}={J}_{F}/2\ensuremath{\le}{J}_{A}$, that is reversed on interchanging $\ensuremath{-}{J}_{F}$ and

{J}_{A}

. Inversion symmetry is spontaneously broken in the dimer phase of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model where the Kekul\'e diagrams are the doubly degenerate ground states at

{J}_{2}/{J}_{1}=1/2