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arXiv:1207.0636 · Europhysics Letters (EPL)

Ground-state properties of linear-exchange quantum spin models

Bimla Danu, Brijesh Kumar, Ramesh V. Pai

Published Oct 1, 2012• 4 citations
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Abstract

We study a class of spin-1/2 quantum antiferromagnetic chains using DMRG technique. The exchange interaction in these models decreases linearly as a function of the separation between the spins, $J_{ij} = R-|i-j|$ for $|i-j| \le R$. For the separations beyond $R$, the interaction is zero. The range parameter $R$ takes positive integer values. The models corresponding to all the odd values of $R$ are known to have the same exact doubly degenerate dimer ground state as for the Majumdar-Ghosh (MG) model. In fact, R=3 is the MG model. For even $R$, the exact ground state is not known in general, except for R=2 (the Bethe ansatz solvable Heisenberg chain) and in the asymptotic limit of $R$ where the two MG dimer states again emerge as the exact ground state. In the present work, we numerically investigate the even-$R$ models whose ground state is not known analytically. In particular, for R=4, 6 and 8, we have computed a number of ground state properties. We find that, unlike R=2, the higher even-$R$ models are spin-gapped, and show strong dimer-dimer correlations of the MG type. Moreover, the spin-spin correlations decay very rapidly, albeit showing weak periodic revivals.

DOI
10.1209/0295-5075/100/27003

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We study a class of spin-1/2 quantum antiferromagnetic chains using DMRG technique. The exchange interaction in these models decreases linearly as a function of the separation between the spins, Jij=R−∣i−j∣J_{ij} = R-|i-j|Jij​=R−∣i−j∣ for ∣i−j∣≤R|i-j| \le R∣i−j∣≤R. For the separations beyond RRR, the interaction is zero. The range parameter RRR takes positive integer values. The models corresponding to all the odd values of RRR are known to have the same exact doubly degenerate dimer ground state as for the Majumdar-Ghosh (MG) model. In fact, R=3 is the MG model. For even RRR, the exact ground state is not known in general, except for R=2 (the Bethe ansatz solvable Heisenberg chain) and in the asymptotic limit of RRR where the two MG dimer states again emerge as the exact ground state. In the present work, we numerically investigate the even-RRR models whose ground state is not known analytically. In particular, for R=4, 6 and 8, we have computed a number of ground state properties. We find that, unlike R=2, the higher even-RRR models are spin-gapped, and show strong dimer-dimer correlations of the MG type. Moreover, the spin-spin correlations decay very rapidly, albeit showing weak periodic revivals.
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.
An antiferromagnetic spin S=1/2 model, the J1{\mathit{J}}_{1}J1​-J2{\mathit{J}}_{2}J2​-J3{\mathit{J}}_{3}J3​-J4{\mathit{J}}_{4}J4​-J5{\mathit{J}}_{5}J5​ model, is considered in two dimensions. J1{\mathit{J}}_{1}J1​,J2{\mathit{J}}_{2}J2​,J3{\mathit{J}}_{3}J3​,J4{\mathit{J}}_{4}J4​,J5{\mathit{J}}_{5}J5​ are the strengths of the nearest-neighbor, diagonal, next-nearest-neighbor, knight's-move-distance-away, and further-neighbor-diagonal interactions, respectively. It is known that when J1{\mathit{J}}_{1}J1​:J2{\mathit{J}}_{2}J2​:J3{\mathit{J}}_{3}J3​:J4{\mathit{J}}_{4}J4​:J5{\mathit{J}}_{5}J5​=1:1:1/2:1/2:1/4 the four columnar dimer states are the exact eigenstates. In a bosonic mean-field theory, we find that the columnar dimer state is the ground state. We compare the results obtained with those obtained in the same mean-field theory for the one-dimensional Majumdar-Ghosh (MG) model. For special values of the couplings, the dimer state is known to be the exact ground state of the MG model. The comparison of the mean-field results for both the models leads to the conjecture that the columnar dimer states are the exact ground states of the J1{\mathit{J}}_{1}J1​-J2{\mathit{J}}_{2}J2​-J3{\mathit{J}}_{3}J3​-J4{\mathit{J}}_{4}J4​-J5{\mathit{J}}_{5}J5​ model.
We present a new class of exactly solvable spin models. The solutions involve broken translational symmetry, in contrast to the Haldane gap, and constitute a new type of ground state for a spin-one chain. The models have ground states with precipitously short-range spin correlations and exhibit gaps to solitonic spin excitations. We construct a solvable model for each value of the total spin of the underlying atoms, with the solution corresponding to the natural generalization to the Majumdar-Ghosh state, of 'dimer' pairing, to larger spin magnitudes.
Interacting spin chains and ladders are known to support a plethora of quantum phases with complex ground-state phase diagrams. In this work, we study a large family of such models and determine precise, explicit conditions under which an exact dimer state is guaranteed to be the ground state. These general conditions are validated for various generalizations of the Majumdar-Ghosh model using exact diagonalization. Our results provide exact reference points in the phase diagrams of a wide class of spin chains and ladders, including those with anisotropic and arbitrary-range interactions.
We have obtained an analytic expression for the k-dependence of the excitation energy gap for an arbitrary double S = 1/2 spin chain by using the non-local unitary transformation and the variational method. We check that it explains the gap behaviour of various systems, which include the Haldane system and the dimer system in both extreme limits, and also the ladder model and the Majumdar - Ghosh model. The string order parameter, the dimer order parameter, and the local spin value are also calculated in the ground state. The ground-state energy exhibits a great stabilization by an antiferromagnetic bond dimerization, which might be realized in various new compounds. We also mention the relation of the convergence to the Haldane state with the spin-exchange symmetry of the model. The excited state has one domain wall of a local triplet type except in the vicinity of the Majumdar - Ghosh point, where a local triplet is decomposed into two S = 1/2 free spins moving among the singlet dimers.
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.