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arXiv:cond-mat/0206102 · Physical Review Letters

Spin-Chirality Duality in a Spin Ladder with Four-Spin Cyclic Exchange

Toshiya Hikihara, Tsutomu Momoi, Xiao Hu

Published Feb 27, 2003• 54 citations
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Abstract

Effect of four-spin cyclic exchange on magnetism is studied in the two-leg S=1/2 ladder. We develop an exact spin-chirality duality transformation, under which the system is self-dual when the four-spin exchange J4 is half of the two-spin exchange. Using the density-matrix renormalization-group method and the duality relation, we find that the four-spin exchange makes the vector-chirality correlation dominant. A "chirality short-range resonating-valence-bond" phase is identified for the first time at large J4.

DOI
10.1103/physrevlett.90.087204

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Effect of four-spin cyclic exchange on magnetism is studied in the two-leg S=1/2 ladder. We develop an exact spin-chirality duality transformation, under which the system is self-dual when the four-spin exchange J4 is half of the two-spin exchange. Using the density-matrix renormalization-group method and the duality relation, we find that the four-spin exchange makes the vector-chirality correlation dominant. A "chirality short-range resonating-valence-bond" phase is identified for the first time at large J4.
We study the spin-1/2 two-leg Heisenberg ladder with four-spin ring exchanges under a magnetic field. We introduce an exact duality transformation which is an extension of the spin-chirality duality developed previously and yields a new self-dual surface in the parameter space. We then determine the magnetic phase diagram using the numerical approaches of the density-matrix renormalization-group and exact diagonalization methods. We demonstrate the appearance of a magnetization plateau and the Tomonaga-Luttinger liquid with dominant vector-chirality quasi-long-range order for a wide parameter regime of strong ring exchange. A "nematic" phase, in which magnons form bound pairs and the magnon-pairing correlation functions dominate, is also identified.
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.
We propose a mechanism of a vector chiral long-range order in two-leg spin-12\frac{1}{2}21​ and spin-1 antiferromagnetic ladders with four-spin exchanges and a Zeeman term. It is known that for one-dimensional quantum systems, spontaneous breakdown of continuous symmetries is generally forbidden. Any vector chiral order hence does not appear in spin-rotationally [SU(2)]-symmetric spin ladders. However, if a magnetic field is added along the Sz{S}^{z}Sz axis of ladders and the SU(2) symmetry is reduced to the U(1) one, the zzz component of a vector chiral order can emerge with the remaining U(1) symmetry unbroken. Making use of Abelian bosonization techniques, we actually show that a certain type of four-spin exchange can yield a vector chiral long-range order in spin-12\frac{1}{2}21​ and spin-1 ladders under a magnetic field. In the chiral-ordered phase, the Z2{Z}_{2}Z2​ interchain-parity (i.e., chain-exchange) symmetry is spontaneously broken. We also consider effects of perturbations breaking the parity symmetry.
We show that scalar chiral order can be induced by four-spin exchanges in the two-leg spin ladder, using the spin-chirality duality transformation and matrix-product ansatz. Scalar-chiral-ordered states are found to be exact ground states in a family of spin ladder models. In this scalar chiral phase, there is a finite energy gap above the doubly degenerate ground states and a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry is fully broken. It is also shown that the SU(4)-symmetric model, which is self-dual under the duality transformation, is on a multicritical point surrounded by the staggered dimer phase, the staggered scalar chiral phase, and the gapless phase.
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 $\ensuremath{\omega}=0$.
The magnetization process of the S=1/2S=1/2S=1/2 antiferromagnetic spin ladder with the four-spin cyclic exchange interaction at T=0T=0T=0 is studied by the exact diagonalization of finite clusters and size scaling analyses. It is found that a magnetization plateau appears at half the saturation value if the ratio of the four- and two-spin exchange coupling constants J4{J}_{4}J4​ is larger than the critical value ${J}_{4c}=0.05\ifmmode\pm\else\textpm\fi{}0.04.$ The phase transition with respect to J4{J}_{4}J4​ at J4c{J}_{4c}J4c​ is revealed to be the Kosterlitz-Thouless type.