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arXiv:math-ph/9908019 · Physical Review Letters

Ground State Properties of a Fully Frustrated Quantum Spin System

Élliott H. Lieb, Peter Schupp

Published Dec 20, 1999• 53 citations
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Abstract

We find that ground states of the quantum Heisenberg antiferromagnet on the geometrically frustrated pyrochlore checkerboard lattice are singlets and can be expressed in terms of positive matrices. The magnetization at zero external field vanishes for each frustrated tetrahedral unit separately, and there is an upper bound of 1/8 in natural units on the susceptibility both for the ground state and at finite temperature. These results are the first exact ones in this field and generalize to some other lattices. The approach is also of interest for other spin systems.

DOI
10.1103/physrevlett.83.5362

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We find that ground states of the quantum Heisenberg antiferromagnet on the geometrically frustrated pyrochlore checkerboard lattice are singlets and can be expressed in terms of positive matrices. The magnetization at zero external field vanishes for each frustrated tetrahedral unit separately, and there is an upper bound of 1/8 in natural units on the susceptibility both for the ground state and at finite temperature. These results are the first exact ones in this field and generalize to some other lattices. The approach is also of interest for other spin systems.
We apply the contractor renormalization (CORE) method to the spin half Heisenberg antiferromagnet on the frustrated checkerboard and pyrochlore lattices. Their ground states are spin-gapped singlets which break lattice symmetry. Their effective Hamiltonians describe fluctuations of orthogonal singlet pairs on tetrahedral blocks, at an emergent low energy scale. We discuss low temperature thermodynamics and new interpretations of finite size numerical data. We argue that our results are common to many models of quantum frustration.
We present the results of an exact diagonalization study of the spin-1/21/21/2 Heisenberg antiferromagnet on a two-dimensional version of the pyrochlore lattice, also known as the square lattice with crossings or the checkerboard lattice. Examining the low-energy spectra for systems of up to 24 spins, we find that all clusters studied have nondegenerate ground states with total spin zero, and big energy gaps to states with higher total spin. We also find a large number of nonmagnetic excitations at energies within this spin gap. Spin-spin and spin-Peierls correlation functions appear to be short ranged, and we suggest that the ground state is magnetically disordered.
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.
We study the zero-temperature ground-state (GS) properties of the spin-1/2 anisotropic planar pyrochlore, using the coupled cluster method (CCM) implemented to high orders of approximation. The system comprises a J 1-J 2 model on the checkerboard lattice, with isotropic Heisenberg interactions of strength J 1 between all nearest-neighbour pairs of spins on the square lattice, and of strength J 2 between half of the next-nearest-neighbour pairs (in the checkerboard pattern). We calculate results for the GS energy and average local GS on-site magnetization, using various antiferromagnetic classical ground states as CCM model states. We also give results for the susceptibility of one of these states against the formation of crossed-dimer valence-bond crystalline (CDVBC) ordering. The complete GS phase diagram is presented for arbitrary values of the frustration parameter k≡J2/J1, and when each of the exchange couplings can take either sign.
The spin-1/2 Heisenberg antiferromagnet on the frustrated diamond-decorated square lattice is known to feature various zero-field ground-state phases, consisting of extended monomer-dimer and dimer-tetramer ground states as well as a ferrimagnetic regime. Using a combination of analytical arguments, density matrix renormalization group (DMRG), exact diagonalization as well as sign-problem-free quantum Monte Carlo (QMC) calculations, we investigate the properties of this system and the related Lieb lattice in the presence of a finite magnetic field, addressing both the ground-state phase diagram as well as several thermodynamic properties. In addition to the zero-field ground states, we find at high magnetic field a spin-canted phase with a continuously rising magnetization for increasing magnetic field strength as well as the fully polarized paramagnetic phase. At intermediate field strength, we identify a first-order quantum phase transition line between the ferrimagnetic and the monomer-dimer regime. This first-order line extends to finite temperatures, terminating in a line of critical points that belong to the universality class of the two-dimensional Ising model.
We study a spin- 1/2 Heisenberg model H=J 1 Sigma i=2 N S 1 S i +J 2 Sigma i=2 N S i S i+1 ; J 1 , J 2 >or=0 (Heisenberg star) which may be considered either as an essential structure element of a lattice with frustration or, alternatively, as an antiferromagnetic linear chain with a perturbation. We discuss general relations for the energy eigenvalues, the eigenstates as well as the spin correlation of the system in dependence on J 2 /J 1 . The correlation function (S 1 S i ) scales with 1/Z, where Z =N-1 is the number of nearest neighbours i of the central spin 1. In the mean-field limit (Z to infinity ) (S 1 S i ) goes to -0.25 for small J 2 /J 1 , but vanishes for strong frustration J 2 /J 1 . Adding exact numerical results for N=5, 7, ..., 23 we discuss the ground-state phase diagram, in particular, the ground-state spin correlations versus J 2 /J 1 . Analysing the spin correlation of frustrated and non-frustrated stars we suggest an upper bound -0.25 for the ground-state correlator (S i S j )0 of two antiferromagnetically interacting spins i and j in a non-frustrated Heisenberg spin- 1/2 antiferromagnet. We argue that any measured nearest-neighbour spin correlation (S i S j )0 larger than this bound indicates of frustration in quantum spin Heisenberg antiferromagnets.