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arXiv:cond-mat/0304106 · Physical review. B, Condensed matter

Scalar chiral ground states of spin ladders with four-spin exchanges

Tsutomu Momoi, Toshiya Hikihara, Masaaki Nakamura, Xiao Hu

Published May 16, 2003• 34 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.67.174410

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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 Z2\ifmmode×\else\texttimes\fiZ2\ifmmode×\else\texttimes\fiZ2{Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}Z2​\ifmmode×\else\texttimes\fiZ2​\ifmmode×\else\texttimes\fiZ2​ 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.
The zero-temperature phases of a generalized two-leg spin ladder with four-spin exchanges are discussed by means of a low-energy field theory approach starting from an SU(4) quantum critical point. The latter fixed point is shown to be a rich multicritical point that unifies different competing dimerized orders and a scalar chirality phase that breaks spontaneously the time-reversal symmetry. The quantum phase transition between these phases is dominated by spin-singlet fluctuations and belongs to the Luttinger universality class due to the existence of a U(1)\mathrm{U}(1)U(1) self-duality symmetry.
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 study the ground-state phase diagram of a spin-12\frac{1}{2}21​ XXZ model with a chirality-chirality interaction (CCI) on a two-leg ladder. This model offers a minimal setup to study an interplay between spin and chirality degrees of freedom. The spin-chirality duality transformation allows us to relate the regimes of weak and strong CCIs. By applying the Abelian bosonization and the duality, we obtain a rich phase diagram that contains distinct gapped featureless and ordered phases. In particular, N\'eel and vector chiral orders appear for easy-axis anisotropy, while two distinct symmetry-protected topological (SPT) phases appear for easy-plane anisotropy. The two SPT phases can be viewed as twisted variants of the Haldane phase. We also present an effective description in terms of (spinor) hard-core bosons, which reveals critical behavior on the self-dual line in the easy-axis and easy-plane regimes. We perform numerical simulations to confirm the predicted phase structure and critical properties. We further demonstrate that the two SPT phases and a trivial phase are distinguished by topological indices in the presence of certain symmetries. A similar phase structure is expected in a spin-12\frac{1}{2}21​ XXZ ladder with four-spin ring exchange.
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 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 study two-leg S=1/2 ladders with general isotropic exchange interactions between spins on neighboring rungs, whose ground state can be found exactly in a form of finitely correlated (matrix product) wave function. Two families of models admitting an exact solution are found: one yields translationally invariant ground states and the other describes spontaneously dimerized models with twofold degenerate ground state. Several known models with exact ground states (Majumdar–Ghosh and Shastry–Sutherland spin-1/2 chains, Affleck–Kennedy–Lieb–Tasaki spin-1 chain, Δ-chain, Bose–Gayen ladder model) can be obtained as particular cases from the general solution of the first family, which includes also a set of models with only bilinear interactions. Those two families of models have nonzero intersection, which enables us to determine exactly the phase boundary of the second-order transition into the dimerized phase and to study the properties of this transition. The structure of elementary excitations in the dimerized phase is discussed on the basis of a variational ansatz. For a particular class of models, we present exact wave functions of the elementary excitations becoming gapless at second-order transition lines. We also propose a generalization of the Bose–Gayen model which has a rich phase diagram with all phase boundaries being exact.