Abstract. We study three-leg antiferromagnetic Heisenberg model with the periodic boundary conditions in the rung direction. Since the rungs form regular triangles, spin frustration is induced. We use the density-matrix renormalization group method to investigate the ground state. We find that the spin excitations are always gapped to remove the spin frustration as long as the rung coupling is nonzero. We also visibly confirm spin-Peierls dimerization order in the leg direction. Both the spin gap and the dimerization order are basically enhanced as the rung coupling increases. 1.
We study three-leg antiferromagnetic Heisenberg model with the periodic boundary conditions in the rung direction. Since the rungs form regular triangles, spin frustration is induced. We use the density-matrix renormalization group method to investigate the ground state. We find that the spin excitations are always gapped to remove the spin frustration as long as the rung coupling is nonzero. We also visibly confirm spin-Peierls dimerization order in the leg direction. Both the spin gap and the dimerization order are basically enhanced as the rung coupling increases.
We present a density matrix renormalization group study of the ground-state properties of spin-1/2 frustrated J1-J3 Heisenberg n(l)-leg ladders (with n(l) up to 8). For strong frustration (J(3)/J(1) approximately 0.5), both even-leg and odd-leg ladders display a finite gap to spin excitations, which we argue remains finite in the two-dimensional limit. In this regime, on odd-leg ladders the ground state is spontaneously dimerized, in agreement with the Lieb-Schultz-Mattis prediction, while on even-leg ladders the dimer correlations decay exponentially. The magnitude of the dimer order parameter decreases as the number of legs increases, consistent with a two-dimensional spin-liquid ground state.
The frustrated ladder with alternate ferromagnetic exchange $\ensuremath{-}{J}_{F}$ and antiferromagnetic exchange
JA 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-
21 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-
21 string correlation functions
g1(N) and
g2(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
g2(N) is related to the range of ground-state spin correlations. Convergence to ${g}_{2}(\ensuremath{\infty})$ is from below, and
g1(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
JF,JL, and
JA 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-
21 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
JA. 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
J2/J1=1/2.
The frustrated ladder with alternate ferromagnetic(F) exchange
−JF and AF exchange
JA to first neighbors and F exchange
−JL 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
εT is finite for the exchanges considered. Spin-1/2 string correlation functions
g1(N) and
g2(N) are defined for an even number
N of consecutive spins in systems with two spins per unit cell; the ladder has string order
g2(∞)>0 and
g1(∞)=0. The minimum
N∗ of
g2(N) is related to the range of ground-state spin correlations. Convergence to
g2(∞) is from below, and
g1(N) decreases exponentially for
N≥N∗. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of
JF,
JL and
JA reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the
J1−J2 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
JL=JF/2≤JA, that is reversed on interchanging
−JF and
JA. Inversion symmetry is spontaneously broken in the dimer phase of the
J1−J2 model where the Kekulé diagrams are the doubly degenerate ground states at
J2/J1=1/2.
We perform a systematic investigation on an asymmetric zigzag spin ladder with interleg exchange
J1 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
J2=J1/2, 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.