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Heisenberg antiferromagnet on triangulated trees

Monti, F., András Sütö

Published Jun 10, 1992• 2 citations
Publisher

Abstract

We study the spin-1/2 antiferromagnetic Heisenberg Hamiltonians with nearest-neighbour pair interaction on some graphs with an odd number of vertices, called triangulated trees. The model has valence bond ground states in which there is a localized spinon (1/2 spin) on some site. The space of ground states for a tree of L triangles is (L+1) dimensional. In the limit of infinite volume we find an infinity of pure ground states. To any (infinite) branch of the tree there belongs a spin-Peierls ground state without spinon. There is an infinite family of equivalent pure ground states differing only locally, in the position of the spinon. In any finite-volume ground state the pair correlations decay exponentially. For the chain of triangles we show the existence of a gap in the energy spectrum above the ground state

DOI
10.5169/seals-116504

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We study the spin-1/2 antiferromagnetic Heisenberg Hamiltonians with nearest-neighbour pair interaction on some graphs with an odd number of vertices, called triangulated trees. The model has valence bond ground states in which there is a localized spinon (1/2 spin) on some site. The space of ground states for a tree of L triangles is (L+1) dimensional. In the limit of infinite volume we find an infinity of pure ground states. To any (infinite) branch of the tree there belongs a spin-Peierls ground state without spinon. There is an infinite family of equivalent pure ground states differing only locally, in the position of the spinon. In any finite-volume ground state the pair correlations decay exponentially. For the chain of triangles we show the existence of a gap in the energy spectrum above the ground state
To understand the role of local sublattice imbalance in low-energy spectra of s=12s=\frac{1}{2}s=21​ quantum antiferromagnets, we study the s=12s=\frac{1}{2}s=21​ quantum nearest neighbor Heisenberg antiferromagnet on the coordination 3 Cayley tree. We perform many-body calculations using an implementation of the density matrix renormalization group (DMRG) technique for generic tree graphs. We discover that the bond-centered Cayley tree has a quasidegenerate set of a low-lying tower of states and an ``anomalous'' singlet-triplet finite-size gap scaling. For understanding the construction of the first excited state from the many-body ground state, we consider a wave function ansatz given by the single-mode approximation, which yields a high overlap with the DMRG wave function. Observing the ground-state entanglement spectrum leads us to a picture of the low-energy degrees of freedom being ``giant spins'' arising out of sublattice imbalance, which helps us analytically understand the scaling of the finite-size spin gap. The Schwinger-boson mean-field theory has been generalized to nonuniform lattices, and ground states have been found which are spatially inhomogeneous in the mean-field parameters.
A special class of Heisenberg spin Hamiltonians with short-range antiferromagnetically signed interactions is described. It is demonstrated via a simple group-theoretic argument that the ground states to many of these Hamiltonians are Kekule states, involving singlet spin-pairing between nearest-neighbour pairs of sites. A number of examples are given and previous related work is indicated. In addition to standard applications to exchange-coupled atomic ions, the relevance of this work for the simple valence-bond model of conjugated pi -electron networks is mentioned.
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.
We study a class of infinite-range spin-(1/2 quantum Heisenberg antiferromagnets. We show that, in addition to ground states with macroscopic net spin in one or several sublattices (``N\'eel states''), there are valence-bond (VB) states that are local spin singlets. We show that there is a class of perturbations that stabilizes the VB state. We characterize the excitation spectrum and find an order parameter for the VB states. Finally, we extend our method to find perturbations that stabilize microscopic as well as macroscopic chiral states.
We consider a resonating-valence-bond state as a trial wave function for the ground state of the antiferromagnetic Heisenberg model on the square lattice. The energy and the staggered spin-spin correlation functions are calculated on a small lattice and are compared with the results of the exact diagonalization of Oitmaa and Betts. The trial energy is only about 4.7% different from the exact ground-state energy on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. The importance of the singlet pairs between the next-nearest antiferromagnetic sites is inferred. An argument against the previous interpretation of the numerical data as being evidence of long-range order is presented.
We recall a simple class of translation invariant states for an infinite quantum spin chain, which was introduced by Accardi. Those states have exponential decay of correlation functions, and a subclass contains the groun states of a certain class of finite-range interactions. We consider, in particular, a family of states for integer spin chains, containing as its simplest member the ground state of a spin-1 Heisenberg antiferromagnet recently studied by Affleck, Kennedy, Lieb and Tasaki. For this family we compute explicitly the correlation functions and other properties.