We examine n component spin systems with arbitrary two spin interactions (of unspecified range) within a general framework to highlight some new subtleties present in incommensurate systems. We determine the ground states of all translationally invariant O(n>1) systems and prove that barring commensurability effects they are always spiral- no other ground states are possible. We study the effect of thermal fluctuations on the ground states to discover a novel odd-even n effect. Soft spin analysis suggests that algebraic long range order is possible in certain frustrated incommensurate even n systems while their odd n counterparts exhibit an exponential decay of correlations. We illustrate that many frustrated incommensurate continuous spin systems display smectic like thermodynamics. We report on a generalized Mermin-Wagner-Coleman theorem for all two dimensional systems (of arbitrary range) with analytic kernels in momentum space. A new relation between generalization Mermin-Wagner-Coleman bounds and dynamics is further reported. We suggest a link between a generalized Mermin-Wagner-Coleman theorem to divergent decoherence (or bandwidth) time scale in the quantum context. A generalization of the Peierls bound for commensurate systems with long range interactions is also discussed. We conclude with a discussion of O(n) spin dynamics in the general case.
We investigate the existence and the properties of fully separable (fully factorized) ground states in quantum spin systems. Exploiting techniques of quantum information and entanglement theory, we extend a recently introduced method and construct a general, self-contained theory of ground state factorization in frustration-free quantum spin models defined on lattices in any spatial dimension and for interactions of arbitrary range. We show that, quite generally, nonexactly solvable translationally invariant models in presence of an external uniform magnetic field can admit exact, fully factorized ground state solutions. Unentangled ground states occur at finite values of the Hamiltonian parameters satisfying well-defined balancing conditions between the applied field and the interaction strengths. These conditions are analytically determined together with the type of magnetic orderings compatible with factorization and the corresponding values of the fundamental observables such as energy and magnetization. The method is applied to a series of examples of increasing complexity, including translationally invariant models with short, long, and infinite ranges of interaction, as well as systems with spatial anisotropies, in lower and higher dimensions. We also illustrate how the general method, besides yielding a large series of exact results for complex models in any dimension, recovers, as particular cases, the results previously achieved on simple models in low dimensions exploiting direct methods based on factorized mean-field ansatz.
The question is posed of how the ground state of the Heisenberg Hamiltonian $H=\ensuremath{-}\ensuremath{\Sigma}{F}_{\mathrm{ij}}{\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{j}$ depends on the magnitude
s of
N interacting spins, particularly in the case of long-ranged oscillatory interactions
Fij. It is discussed whether fixing the geometry and the bond strengths
Fij suffices to determine the nature of the spin correlations in the ground state, and a review is given of known instances when this is the case; those are special situations in ferromagnetism and antiferromagnetism when qualitative ground-state properties such as "lack of nodes" can be proved to be independent of
s. These are valuable examples for the application of semiclassical methods, which are strictly valid only for $s\ensuremath{\rightarrow}\ensuremath{\infty}$ and depend on the convergence of a series in powers of ${s}^{\ensuremath{-}1}$. But these examples are, after all, only special cases, and it is argued that, in general, the nature of the ground state can depend sensitively on
s. The following situation is considered in some detail: An oscillatory interaction which leads to a ferromagnetic ground state in the correspondence limit $s\ensuremath{\gg}1$, but for which the ferromagnetic state of saturation magnetization may be unstable for small quantum mechanical spins, e.g.,
s=21or1. Two distinct types of interaction are considered which lead to this result, and it is seen that the ferromagnetic instability is a consequence, not so much of the long range of the interaction as of the presence of some relatively strong antiferromagnetic (negative) bonds. However, the variational approach which is used casts no light on the nature of the true ground state or of the thermal properties, problems which are increasingly interesting in these instances when semiclassical procedures are seen to fail.
We introduce a general analytic approach to the study of factorization points and factorized ground states in quantum cooperative systems. The method allows us to determine rigorously the existence, location, and exact form of separable ground states in a large variety of, generally nonexactly solvable, spin models belonging to different universality classes. The theory applies to translationally invariant systems, irrespective of spatial dimensionality, and for spin-spin interactions of arbitrary range.
Some measurements have shown that the second-order exchange interaction is non-negligible in ferromagnetic compounds whose microscopic interactions are described by means of half-odd integer quantum spins. In these spin systems the ground state is either ferromagnetic or antiferromagnetic when the bilinear exchange interaction is dominant. Instead, in ferromagnetic systems characterized by bilinear and biquadratic exchange interactions of comparable magnitude, the energy minimum occurs when spins are in a canting ground-state. To this aim, a one-dimensional (1D) quantum spin chain and a two-dimensional (2D) lattice of quantum spins subjected to periodic boundary conditions are modeled via the generalized quantum Heisenberg Hamiltonian containing, in addition to the isotropic and short-range bilinear exchange interaction of the Heisenberg type, a second-order interaction, the isotropic and short-range biquadratic exchange interaction between nearest-neighbors quantum spins. For these 1D and 2D quantum systems a generalization of the Mermin–Wagner–Hohenberg theorem (also known as Mermin–Wagner–Berezinksii or Coleman theorem) is given. It is demonstrated, by means of quantum statistical arguments, based on Bogoliubov’s inequality, that, at any finite temperature, (1) there is absence of long-range order and that (2) the law governing the vanishing of the order parameter is the same as in the bilinear case for both 1D and 2D quantum ferromagnetic systems. The physical implications of the absence of a spontaneous spin symmetry breaking in 1D spin chains and 2D spin lattices modeled via a generalized quantum Heisenberg Hamiltonian are discussed.
We establish dimerization in
O(n)-invariant quantum spin chains with large enough
n, in a much larger region of the phase diagram than previous results. This entails identifying two distinct infinite volume ground states which are gapped, are translations by one unit of each other, and which both have exponentially decaying correlations. We show these states are the limits of finite-volume, finite temperature states as the graph tends to
Z and temperature tends to zero in any order. Our method relies on a probabilistic representation of the quantum system in terms of random loops, and an adaptation of a method developed for loop
O(n) models on the hexagonal lattice by Duminil-Copin, Peled, Samotij and Spinka.