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Physical review. B, Condensed matter

New class of frustrated quantum spin systems with an exactly known ground state

Benoît Douçot, Ido Kanter

Published Jun 1, 1989• 10 citations
Publisher

Abstract

We present a new class of quantum systems whose ground state can be found exactly. This construction involves an assembly of finite frustrated clusters connected by fixed spins. By contrast to solvable dimerized models, our systems exhibit long-range magnetic order, providing a striking example of ordering by quantum fluctuations in the presence of frustration.

DOI
10.1103/physrevb.39.12399

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We present a new class of quantum systems whose ground state can be found exactly. This construction involves an assembly of finite frustrated clusters connected by fixed spins. By contrast to solvable dimerized models, our systems exhibit long-range magnetic order, providing a striking example of ordering by quantum fluctuations in the presence of frustration.
We outline the recent results on the ground state for a class of one- and two-dimensional frustrated quantum spin models with competing ferro(F)- and antiferromagnetic (AF) interactions. Frustrated spin systems are known to have many interesting properties due to large quantum fluctuations. As a result of these fluctuations usual mean-field approach gives quite crude (if not false) description of these systems. Therefore, exactly solvable models are very instructive at investigations of such systems. The exact ground state wave function of proposed models has a structure of the valence-bond state (VBS) type. One of the 1D model describes the transition line between the F and AF phase. The exact singlet ground state on this line has a double-spiral ordering. Using different approximation methods we study the magnetization curve in the AF phase. The second considered set of the 1D and 2D models has an exact non-degenerate ground state with exponentially decaying spin correlations. We also proposed the 1D and 2D electronic models with exact ground state represented in terms of singlet bond functions which are the generalization of the RVB functions including ionic states.
The ground state of a spin-1/2 Heisenberg chain with both frustration and long-range interactions is studied using Lanczos exact diagonalization. The evolution of the well-known dimerization transition of the system with short-range frustrated interactions (the J1-J2 chain) is investigated in the presence of additional unfrustrated interactions decaying with distance as 1/r(alpha). It is shown that the continuous (infinite-order) dimerization transition develops into a first-order transition between a long-range ordered antiferromagnetic state and a state with coexisting dimerization and critical spin correlations at wave number k=pi/2. The relevance of the model to real systems is discussed.
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.
We present a class of exactly solvable quantum spin models which consist of two Heisenberg subsystems coupled via a long-range Lieb-Mattis interaction. The total system is exactly solvable whenever the individual subsystems are solvable and allows to study the effects of frustration. We consider i) the antiferromagnetic linear chain and ii) the Lieb-Mattis antiferromagnet for the subsystem Hamiltonians and present i) the complete ground-state phase diagram and ii) the full thermodynamic phase diagram. We find a novel phase which exhibits order-from-disorder phenomena.
The existence of definite orders in frustrated quantum systems is related rigorously to the occurrence of fully factorized ground states below a threshold value of the frustration. Ground-state separability thus provides a natural measure of frustration: strongly frustrated systems are those that cannot accommodate for classical-like solutions. The exact form of the factorized ground states and the critical frustration are determined for various classes of nonexactly solvable spin models with different spatial ranges of the interactions. For weak frustration, the existence of disentangling transitions determines the range of applicability of mean-field descriptions in biological and physical problems such as stochastic gene expression and the stability of long-period modulated structures.