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arXiv:0811.2456 · Physical Review B

Dynamics in a two-leg spin ladder with a four-spin cyclic interaction

Satoshi Nishimoto, Mitsuhiro Arikawa

Published Mar 27, 2009• 9 citations
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Abstract

We study the two-leg Heisenberg ladder with four-spin cyclic interaction using the (dynamical) density-matrix renormalization-group method. We demonstrate the dependence of the low-lying excitations in the spin wave, staggered dimer order, and scalar-chirality order structure factors on the four-spin cyclic interaction. We find that the cyclic interaction enhances spin-spin correlations with wave vector around momentum $({q}_{x},{q}_{y})=(\frac{\ensuremath{\pi}}{2},0)$. Also, the presence of long-range order in the staggered dimer and scalar-chirality phases is confirmed by a $\ensuremath{\delta}$-function peak contribution of the structure factors at energy $\ensuremath{\omega}=0$.

DOI
10.1103/physrevb.79.113106

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We study the two-leg Heisenberg ladder with four-spin cyclic interaction using the (dynamical) density-matrix renormalization-group method. We demonstrate the dependence of the low-lying excitations in the spin wave, staggered dimer order, and scalar-chirality order structure factors on the four-spin cyclic interaction. We find that the cyclic interaction enhances spin-spin correlations with wave vector around momentum (qx,qy)=(\ensuremathπ2,0)({q}_{x},{q}_{y})=(\frac{\ensuremath{\pi}}{2},0)(qx​,qy​)=(2\ensuremathπ​,0). Also, the presence of long-range order in the staggered dimer and scalar-chirality phases is confirmed by a \ensuremathδ\ensuremath{\delta}\ensuremathδ-function peak contribution of the structure factors at energy \ensuremathω=0\ensuremath{\omega}=0\ensuremathω=0.
We present the phase diagram of the S=1/2S=1/2S=1/2 Heisenberg model on the two leg ladder with cyclic four-spin exchange, determined by a combination of exact diagonalization and density-matrix renormalization group techniques. We find six different phases and regimes: the rung singlet phase, a ferromagnetic phase, two symmetry broken phases with staggered dimers and staggered scalar chiralities, and a gapped region with dominant vector chirality, or collinear spin correlations. We localize the phase transitions and investigate their nature.
Four-spin exchange interaction has been raising intriguing questions regarding the exotic phase transitions it induces in two-dimensional quantum spin systems. In this context, we investigate the effects of a cyclic four-spin exchange in the quasi-1D limit by considering a general NNN-leg spin ladder. We show by means of a low-energy approach that depending on its sign, this ring exchange interaction can engender either a staggered or a uniform dimerization from the conventional phases of spin ladders. The resulting quantum phase transition is found to be described by the SU(2)N{}_{N}N​ conformal field theory. This result, as well as the fractional value of the central charge at the transition, is further confirmed by a large-scale numerical study performed by means of exact diagonalization and density matrix renormalization group approaches for $N\ensuremath{\le}4$.
Using the (dynamical) density-matrix renormalization group method, we study the low-energy physics of three-leg antiferromagnetic Heisenberg model where the periodic boundary conditions are applied in the rung direction. We confirm that the spin excitations are always gapped as long as the intraring couplings form a regular triangle. From precise finite-size-scaling analyses of the spin gap and dimerization order parameter, we also find that the spin gap is collapsed by very small asymmetric modulation of the intraring couplings. Moreover, the dynamical spin structure factors on the intraleg and interleg correlations are calculated. It is demonstrated that the low-lying structure of the interleg spectra is particularly affected by the asymmetric modulation.
We study the ground state and low-lying excited states of the Kitaev-Heisenberg model on a two-leg ladder geometry using the density-matrix renormalization-group and Lanczos exact diagonalization methods. The Kitaev and Heisenberg interactions are parametrized as $K=sin\ensuremath{\phi}$ and $J=cos\ensuremath{\phi}$ with an angle parameter $\ensuremath{\phi}$. Based on the results for several types of order parameters, excitation gaps, string order parameter, and entanglement spectra, the $\ensuremath{\phi}$-dependent ground state phase diagram is determined. Remarkably, the phase diagram is quite similar to that of the Kitaev-Heisenberg model on a honeycomb lattice, exhibiting the same long-range-ordered states, namely rung singlet (analog to N\'eel in three dimensions), zigzag, ferromagnetic, and stripy, and the presence of gapped spin liquids around the exactly solvable Kitaev points $\ensuremath{\phi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$. We also calculate the expectation value of a plaquette operator corresponding to a $\ensuremath{\pi}$-flux state in order to establish how the gapped Kitaev spin liquid extends away from the $\ensuremath{\phi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$. Furthermore, we determine the dynamical spin structure factor and discuss the effect of the Kitaev interaction on the spin-triplet dispersion.
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We investigate an S=1/2S=1/2S=1/2 two-leg spin ladder with a cyclic four-spin exchange interaction whose interaction constant is denoted by J4J_4J4​, by using the density matrix renormalization group method. The interchain and the intrachain interaction constant are denoted by JrungJ_{rung}Jrung​ and JlegJ_{\rm leg}Jleg​, respectively and assumed to be antiferromagnetic. It turns out that a spin gap between the singlet (Stotz=0S_{tot}^z=0Stotz​=0) and the triplet (Stotz=1S_{tot}^z=1Stotz​=1) states vanishes at J4/Jleg≃0.3J_4/J_{leg} \simeq 0.3J4​/Jleg​≃0.3 for Jrung=JlegJ_{rung}=J_{leg}Jrung​=Jleg​. This result is in contrast with the fact that the S=1/2S=1/2S=1/2 antiferromagnetic Heisenberg ladder, that is the case of Jrung≠Jleg,J4=0J_{rung} \neq J_{leg}, J_4=0Jrung​=Jleg​,J4​=0, has a spin gap for all nonzero value of interchain interaction Jrung>0J_{\rm rung}>0Jrung​>0. We find a larger value of the correlation length for the spin-pair correlation function than a linear size LLL of the system at J4/Jleg=0.3J_4/J_{\rm leg}=0.3J4​/Jleg​=0.3 and Jrung=JlegJ_{rung}=J_{leg}Jrung​=Jleg​: the correlation length ξξξ is about 204 times of the lattice constant for L=84 for these values of interactions. We also find that the string correlation function decays rather algebraically than exponentially at J4/Jleg=0.3J_4/J_{leg}=0.3J4​/Jleg​=0.3 and Jrung=JlegJ_{rung}=J_{leg}Jrung​=Jleg​. These results suggest that there is a quantum phase transition at J4/Jleg≃0.3J_4/J_{leg} \simeq 0.3J4​/Jleg​≃0.3 for Jrung=JlegJ_{rung}=J_{leg}Jrung​=Jleg​. We estimate a phase boundary where the spin gap vanishes in a J4/Jleg−Jrung/JlegJ_4/J_{leg}-J_{rung}/J_{leg}J4​/Jleg​−Jrung​/Jleg​ plane and obtain a consistent result with that by a perturbation theory for Jrung/Jleg>1J_{\rm rung}/J_{leg}>1Jrung​/Jleg​>1.