We prove the bound ${\mathit{m}}_{\mathit{s}}$(\ensuremath{\beta})\ensuremath{\ge} \ensuremath{\surd}3 \ensuremath{\sigma}(\ensuremath{\beta}) between the spontaneous staggered magnetization ${\mathit{m}}_{\mathit{s}}$(\ensuremath{\beta}) and the long range order parameter \ensuremath{\sigma}(\ensuremath{\beta}) in the quantum Heisenberg antiferromagnets. The same bound has been known for the Heisenberg ferromagnets from the works of Griffiths, and of Dyson, Lieb, and Simon, but extensions to the antiferromagnets had been lacking for more than two decades. When combined with the results on nonvanishing \ensuremath{\sigma}(\ensuremath{\beta}) from the Dyson-Lieb-Simon method, our bound proves the existence of symmetry breaking in the physically natural equilibrium states and ground states obtained by applying an infinitesimal symmetry breaking field.
We prove the bound ms(\ensuremath{\beta})\ensuremath{\ge} \ensuremath{\surd}3 \ensuremath{\sigma}(\ensuremath{\beta}) between the spontaneous staggered magnetization ms(\ensuremath{\beta}) and the long range order parameter \ensuremath{\sigma}(\ensuremath{\beta}) in the quantum Heisenberg antiferromagnets. The same bound has been known for the Heisenberg ferromagnets from the works of Griffiths, and of Dyson, Lieb, and Simon, but extensions to the antiferromagnets had been lacking for more than two decades. When combined with the results on nonvanishing \ensuremath{\sigma}(\ensuremath{\beta}) from the Dyson-Lieb-Simon method, our bound proves the existence of symmetry breaking in the physically natural equilibrium states and ground states obtained by applying an infinitesimal symmetry breaking field.
We calculate the order parameter m in the infinite-range isotropic-interaction quantum model of antiferromagnetism due to Lieb and Mattis (LM). The Hamiltonian H1 in this model contains pure Heisenberg interactions. m is the mean value of the z component msz of the staggered spin per particle ms in the ground state of H1 plus a symmetry-breaking term HB due to an infinitesimal staggered field B in the thermodynamic limit (TL). We find a value of \ensuremath{\surd}3 for the ratio r=m/m0, m0 being the root-mean-square value of msz for B==0 in the TL. m and m0 measure, respectively, the spontaneous symmetry breaking and the amount of long-range order in the symmetry-unbroken state. This value is expected on the basis of our recent argument that r= \ensuremath{\surd}3 if ms behaves in the TL as a classical vector with no magnitude fluctuations---behavior that we show holds for the Lieb-Mattis model. The addition to H1 of a small easy-axis anisotropic term is also discussed briefly. These studies allow calculation of the size dependence of quantities that we have recently shown to become lower bounds on m in the TL (bounds that are valid for short-range-interaction models as well). In the LM model, low-lying energy eigenstates with excitation energies of order ${\mathit{N}}^{\mathrm{\ensuremath{-}}1}$, where N is the number of spins, form a continuum (with no gap) in the TL. These are closely related to similar states shown to exist for the nearest-neighbor isotropic Heisenberg antiferromagnet. We show that this continuum has a density of states that is O(N0); i.e., it is not extensive.
We discuss the nature of symmetry breaking in quantum antiferromagnets. We show that published arguments concerning the order parameter for the 2d s =1/2 Heisenberg antiferromagnet have to be reconsidered. These had led to 3 and 1 for the ratio r of the mean z -component m of the staggered spin per particle m s z in an infinitesimal staggered field B , to the r.m.s. value m 0 of m s z with B =0, in the thermodynamic limit. We give arguments which put doubt on both values. Also we prove r ≧1, given m 0 ≠0.
We investigate the existence of N\'eel long-range order in a spatially anisotropic antiferromagnetic Heisenberg model. As the anisotropical coupling ${J}_{\ensuremath{\perp}}$ increases from 0 to 1, the model interpolates between an infinite number of decoupled one-dimensional gapped spin chains and a two-dimensional gapless isotropic Heisenberg antiferromagnet. By applying a method based on reflection positivity [F. J. Dyson, E. H. Lieb, and B. Simon, J. Stat. Phys. 18, 335 (1978)], we determine the upper bound of the critical values of ${J}_{\ensuremath{\perp}}^{c}$ such that when ${J}_{\ensuremath{\perp}}>~{J}_{\ensuremath{\perp}}^{c},$ the ground state of this model has an antiferromagnetic long-range order. For the case of spin S=1, our result confirms and strengthens previous results derived by numerical calculations on finite-size samples, while for other values of S, our results are new.
We look at the general SU(2) invariant spin-1 Heisenberg model. This family includes the well-known Heisenberg ferromagnet and antiferromagnet as well as the interesting nematic (biquadratic) and the largely mysterious staggered-nematic interaction. Long range order is proved using the method of reflection positivity and infrared bounds on a purely nematic interaction. This is achieved through the use of a type of matrix representation of the interaction making clear several identities that would not otherwise be noticed. Using the reflection positivity of the antiferromagnetic interaction one can then show that the result is maintained if we also include an antiferromagnetic interaction that is sufficiently small.
The ground-state properties of the quasi-one-dimensional spin-1/2 antiferromagnetic Heisenberg model is investigated by using a variational method. Spins on chains along the x direction are antiferromagnetically coupled with exchange J>0, while spins between chains in the y direction are coupled either ferromagnetically (${J}^{\ensuremath{'}}<0$) or antiferromagnetically (${J}^{\ensuremath{'}}>0$). The staggered and the colinear antiferromagnetic magnetizations are computed and their dependence on the anisotropy parameter $\ensuremath{\lambda}=|{J}^{\ensuremath{'}}|/J$ is analyzed. It is found that an infinitesimal interchain coupling parameter is sufficient to stabilize a long-range order with either a staggered magnetization ms (${J}^{\ensuremath{'}}>0$) or a colinear antiferromagnetic magnetization mcaf (${J}^{\ensuremath{'}}<0$), both behaving as $\ensuremath{\simeq}{\ensuremath{\lambda}}^{1/2}$ for $\ensuremath{\lambda}\ensuremath{\rightarrow}0$.
We study quantum phase transitions in Heisenberg antiferromagnetic chains with a staggered power-law decaying long-range interactions. Employing the density-matrix renormalization group (DMRG) algorithm and the fidelity susceptibility as the criticality measure, we establish more accurate values of quantum critical points than the results obtained from the spin-wave approximation, quantum Monte Carlo, and DMRG in literatures. The deviation is especially evident for strong long-range interactions. We extend isotropic long-range interactions to the anisotropic cases and find that kaleidoscope of quantum phases emerge from the interplay of anisotropy of the long-range exchange interaction and symmetry breaking. We demonstrate nonfrustrating long-range interactions induce the true long-range order in Heisenberg antiferromagnetic chains with a continuous symmetry breaking, lifting the restrictions imposed by the Mermin-Wagner theorem.