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Physical Review Letters

Signature of Néel order in exact spectra of quantum antiferromagnets on finite lattices

B. Bernu, C. Lhuillier, Laurent Pierre

Published Oct 26, 1992• 384 citations
Publisher

Abstract

We show how the broken symmetries of the N\'eel state are embodied in the exact spectrum of the triangular Heisenberg antiferromagnet on finite lattices as small as N=21 (spectra up to N=36 have been computer). We present the first numerical evidence of an extensive set of low-lying levels that are below the softest magnons and collapse to the ground state in the thermodynamic limit. This set of quantum states represents the quantum counterpart of the classical N\'eel ground state. We develop an approach relying on the symmetry analysis and finite-size scaling and we provide new arguments in favor of an ordered ground state for the S=1/2 triangular Heisenberg model.

DOI
10.1103/physrevlett.69.2590

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We show how the broken symmetries of the N'eel state are embodied in the exact spectrum of the triangular Heisenberg antiferromagnet on finite lattices as small as N=21 (spectra up to N=36 have been computer). We present the first numerical evidence of an extensive set of low-lying levels that are below the softest magnons and collapse to the ground state in the thermodynamic limit. This set of quantum states represents the quantum counterpart of the classical N'eel ground state. We develop an approach relying on the symmetry analysis and finite-size scaling and we provide new arguments in favor of an ordered ground state for the S=1/2 triangular Heisenberg model.
We have realized a thorough quantum mechanical analysis of an extensive number of low lying levels of the Heisenberg Hamiltonian on small periodic samples. Striking qualitative differences appear between the spectra of the triangular lattice and those of the Kagomé lattice. Symmetries and finite size scaling of the low lying levels of the triangular antiferromagnetic point to an à la Néel symmetry breaking, and the dynamical structure factors corroborate this conclusion. The Kagomé antiferromagnet spectra are qualitatively different: no signature of a symmetry breaking has been exhibited and the dynamical structure factors of the low lying levels display broad bands. The Heisenberg antiferromagnet on a Kagomé lattice is a serious candidate for a spin liquid.
We have studied the Heisenberg model on the triangular lattice using quantum Monte Carlo techniques (up to 144 sites) and exact diagonalization (up to 36 sites). By studying the spin gap as a function of the system size we have obtained robust evidence for a gapless spectrum, confirming the existence of long-range N\'eel order. Our best estimate is that in the thermodynamic limit the order parameter ${m}^{\ifmmode\dagger\else\textdagger\fi{}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.41\ifmmode\pm\else\textpm\fi{}0.02$ is reduced by about 5959%59 from its classical value and the ground state energy per site is ${e}_{0}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}0.5458\ifmmode\pm\else\textpm\fi{}0.0001$ in units of the exchange coupling. We have identified the ground state correlations that are important at short distances.
In magnetically ordered systems, the breaking of SU(2) symmetry in the thermodynamic limit is associated with the appearance of a special type of low-lying excitations in finite-size energy spectra, the so-called tower of states (TOS). In the present work, we numerically demonstrate that there is a correspondence between the SU(2) tower of states and the lower part of the ground-state entanglement spectrum (ES). Using state-of-the-art density matrix renormalization group (DMRG) calculations, we examine the ES of the 2D antiferromagnetic J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg model on both the triangular and kagome lattice. At large ferromagnetic J2{J}_{2}J2​, the model exhibits a magnetically ordered ground state. Correspondingly, its ES contains a family of low-lying levels that are reminiscent of the energy tower of states. Their behavior (level counting, finite-size scaling in the thermodynamic limit) sharply reflects TOS features, and is characterized in terms of an effective entanglement Hamiltonian that we provide. At large system sizes, TOS levels are divided from the rest by an entanglement gap. Our analysis suggests that (TOS) entanglement spectroscopy provides an alternative tool for detecting and characterizing SU(2)-broken phases using DMRG.
Based on the triangular lattice and its depletions there are three simple frustrated antiferromagnetic Heisenberg models in two dimensions.The first two, the triangular and kagοmé lattices, have been examined in the recent past.The triangular lattice seems to have a łong range order whereas the kagomé does not show the long range order.But these results are still con-' troversial.This work is concentrated on a third type of this lattice family in order to improve the understanding of the connection between the long range order and coordination number in low dimensional systems.Bets has described the geometric properties of this lattice.It has a coordination number 5, which lies precisely between coordination numbers 6 and 4 of the other two lattices.The low-lying spectra and the correlation functions of finite lattices have been examined to discuss the possibility of a long range ordered ground state in the I/7-depleted triangular lattice.The low-lying spectrum is generated by an exact diagonalization, and the tower of states behavior points to a long range ordered ground state.
The spin-1/2 antiferromagnetic Heisenberg and X Y models are investigated on the triangular lattice by numerically diagonalizing finite size systems (up to 21 spins). The ground state energies of both models are found to be close to other theoretical predictions except those of Marland and Betts who performed similar analysis to ours. We have numerically constructed a resonating valence bond state of Anderson. It has a relatively large projection on the ground state of the finite Heisenberg model, although the absolute value of the projection decreases linearly with the system size. The trial function proposed by Miyashita for the X Y model does not have an appreciable projection on the numerically exact state. We present evidence for the absence of a sublattice long range order in the Heisenberg model and the existence of the same quantity in the X Y model.