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arXiv:cond-mat/9701089 · Physical review. B, Condensed matter

Models with exact ground states connecting smoothly the S= 12 dimer and S=1 Haldane phases of one-dimensional spin chains

Alexei K. Kolezhuk, H.-J. Mikeska

Published Nov 1, 1997• 38 citations
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Abstract

We investigate the isotropic two-leg $S=$$\frac{1}{2}$ ladder with general bilinear and biquadratic exchange interactions between spins on neighboring rungs, and determine the Hamiltonians that have a matrix-product wave function as an exact ground state. We demonstrate that a smooth change of parameters leads one from the $S=$$\frac{1}{2}$ dimer and Majumdar-Ghosh chains to the $S=1$ chain with biquadratic exchange, which proves that these model systems are in the same phase. The family of Hamiltonians also includes a set of models of isotropic $S=$$\frac{1}{2}$ chains with only bilinear nearest- and next-nearest-neighbor interactions. The ground state for these models becomes unstable at a line of first-order phase transitions to the ferromagnetic state.

DOI
10.1103/physrevb.56.r11380

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We investigate the isotropic two-leg S=$$\frac{1}{2} ladder with general bilinear and biquadratic exchange interactions between spins on neighboring rungs, and determine the Hamiltonians that have a matrix-product wave function as an exact ground state. We demonstrate that a smooth change of parameters leads one from the S=$$\frac{1}{2} dimer and Majumdar-Ghosh chains to the S=1S=1S=1 chain with biquadratic exchange, which proves that these model systems are in the same phase. The family of Hamiltonians also includes a set of models of isotropic S=$$\frac{1}{2} chains with only bilinear nearest- and next-nearest-neighbor interactions. The ground state for these models becomes unstable at a line of first-order phase transitions to the ferromagnetic state.
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.
Using the matrix product formalism, we define a multiparameter family of spin models on one-dimensional chains, with nearest- and next-nearest-neighbor interactions for which exact analytical expressions can be found for its doubly degenerate ground states. The family of Hamiltonians depends on five continuous parameters and the Majumdar-Ghosh model is a particular point in this parameter space. The model can also be represented as a zigzag spin ladder in which there are extra three-particle couplings between spins on vertices of triangles. The doubly degenerate ground states of these models models have a very simple structure, they are the product of entangled (dimer) states on adjacent sites (or consecutive rungs of the ladder in the zigzag ladder picture). At the Majumdar-Ghosh point, these entangled states become the spin-singlets pertaining to this model. We will also calculate, in closed form, the two point correlation functions, both for finite sizes of the chain and in the thermodynamic limit.
Inspired by the exact solution of the Majumdar-Ghosh model, a family of one-dimensional, translationally invariant spin Hamiltonians is constructed. The exchange coupling in these models is antiferromagnetic, and decreases linearly with the separation between the spins. The coupling becomes identically zero beyond a certain distance. It is rigorously proved that the dimer configuration is an exact, superstable ground-state configuration of all the members of the family on a periodic chain. The ground state is twofold degenerate, and there exists an energy gap above the ground state. The Majumdar-Ghosh Hamiltonian with a twofold degenerate dimer ground state is just the first member of the family. The scheme of construction is generalized to two and three dimensions, and illustrated with the help of some concrete examples. The first member in two dimensions is the Shastry-Sutherland model. Many of these models have exponentially degenerate, exact dimer ground states.
We investigate the ground-state phase diagram and critical properties of the S=1/2S=1/2S=1/2 two-leg Heisenberg spin ladder system with a negative four-spin interaction using a numerical exact diagonalization method. Using the perturbation theory in the strong negative rung-coupling limit, we derive an S=1S=1S=1 bilinear-biquadratic chain as an effective model. We discuss the ground-state phase diagram in this limit. Next we numerically determine a phase boundary between the rung singlet phase and the columnar dimer (CD) phase by the phenomenological renormalization group method, and one between the CD phase and the Haldane phase by the twisted boundary condition method. We confirm that the phase transition between the CD phase and the Haldane phase is of second order and this universality class is described by the k=2k=2k=2 SU(2)SU(2)SU(2) Wess-Zumino-Novikov-Witten nonlinear $\ensuremath{\sigma}$ model, calculating the central charge and scaling dimensions.
We study the ground-state phase diagram of a spin-12\frac{1}{2}21​ frustrated XXZ ladder, in which two antiferromagnetic chains are coupled by competing rung and diagonal interactions, ${J}_{\ensuremath{\perp}}$ and ${J}_{\ifmmode\times\else\texttimes\fi{}}$. Previous studies on the isotropic model have revealed that a fluctuation-induced effective dimer attraction between the legs stabilizes the columnar dimer (CD) phase in the highly frustrated regime ${J}_{\ensuremath{\perp}}\ensuremath{\approx}2{J}_{\ifmmode\times\else\texttimes\fi{}}$, especially for ferromagnetic ${J}_{\ensuremath{\perp},\ifmmode\times\else\texttimes\fi{}}<0$. By means of effective field theory and numerical analyses, we extend this analysis to the XXZ model, and we obtain a rich phase diagram. The diagram includes four gapped featureless phases with no symmetry breaking: the rung singlet (RS) and Haldane phases as well as their twisted variants, the RS* and Haldane* phases, which are all distinct in the presence of certain symmetries. Significantly, the Haldane-CD transition point in the isotropic model turns out to be a crossing point of two transition lines in the XXZ model, and the stripe N\'eel and RS* phases appear between these lines. This indicates a nontrivial interplay between the effective dimer attraction and the exchange anisotropy. In the easy-plane regime, the four featureless phases and two critical phases are found to compete in a complex manner depending on the signs of ${J}_{\ensuremath{\perp},\ifmmode\times\else\texttimes\fi{}}$.
We discuss spin-SSS antiferromagnetic Heisenberg chains with three-spin interactions, next-nearest-neighbor interactions, and bond alternation. First, we prove rigorously that there exist parameter regions of the exact dimerized ground state in this system. This is a generalization of the Majumdar-Ghosh model to arbitrary SSS. Next, we discuss the ground-state phase diagram of the models by introducing several effective field theories and the universality classes of the transitions are described by the level-2S2S2S SU(2)\mathrm{SU}(2)SU(2) Wess-Zumino-Witten model and the Gaussian model. Finally, we determine the phase diagrams of S=1S=1S=1 and S=3/2S=3/2S=3/2 systems by using exact diagonalization and level spectroscopy.