Topological phase space disconnection has been recently found to be a general phenomenon in many-body spin system with anisotropic interaction. We show that the power law divergence of magnetic reversal time at the energy signaling such disconnection is generic for long-range interacting systems with an exponent proportional to the number of particles. We also study the modifications induced putting the system in contact with a thermal bath. Using the canonical formalism we analyze the magnetic reversal times at any temperature. Moreover, due to the divergence of reversal time at the energy disconnection threshold we can recover, using saddle point approximation, a simple exponential dependence on the inverse temperature showing the explicit relevance of the energy disconnection threshold for finite many-body interacting systems at finite temperature. This sets a general framework to understand the emergence of ferromagnetism in finite magnetic systems starting from microscopic models without phenomenological on-site barriers.
Topological phase space disconnection has been recently found to be a general phenomenon in many-body spin system with anisotropic interaction. We show that the power law divergence of magnetic reversal time at the energy signaling such disconnection is generic for long-range interacting systems with an exponent proportional to the number of particles. We also study the modifications induced putting the system in contact with a thermal bath. Using the canonical formalism we analyze the magnetic reversal times at any temperature. Moreover, due to the divergence of reversal time at the energy disconnection threshold we can recover, using saddle point approximation, a simple exponential dependence on the inverse temperature showing the explicit relevance of the energy disconnection threshold for finite many-body interacting systems at finite temperature. This sets a general framework to understand the emergence of ferromagnetism in finite magnetic systems starting from microscopic models without phenomenological on-site barriers.
We demonstrate the existence of a topological disconnection threshold, recently found by Borgonovi [J. Stat. Phys. 116, 1435 (2004)], for generic 1-d anisotropic Heisenberg models interacting with an interparticle potential R(-alpha) when 0<alpha<1(here R is the distance among spins). We also show that alpha if is greater than the embedding dimension then the ratio between the disconnected energy region and the total energy region goes to zero when the number of spins becomes very large. On the other hand, numerical simulations in d=2,3 for the long-range case support the conclusion that such a ratio remains finite for large N values. The disconnection threshold can thus be thought of as a distinctive property of anisotropic long-range interacting systems.
When interacting spin systems possess non-zero magnetization or topological entanglement entropy below the transition temperature, they often serve as classical or quantum self-correcting memory. In particular, their memory time grows exponentially in the system size due to polynomially growing energy barrier, as in the 2D Ising model and 4D Toric code. Here, we argue that this is not always the case. We demonstrate that memory time of classical clock model (a generalization of ferromagnet to q-state spins) may be polynomially long even when the system possesses finite magnetization. This weak violation of the Arrhenius law occurs above the percolation temperature, but below the transition temperature, a regime where excitation droplets percolate the entire lattice, yet the system retains a finite magnetization. We present numerical evidences for polynomial scaling as well as analytical argument showing that energy barrier is effectively suppressed and is only logarithmically divergent. We also suggest an intriguing possibility of experimentally observing the precession of magnetization vectors at experimentally relevant time scale.
Quantum thermalization of closed systems -- i.e. the relaxation of local properties to their thermodynamic value along the Hamiltonian evolution -- is generically expected in non-integrable systems at finite energy density. In this work we show that quantum spin systems with at least one spin component coupled through long-range ferromagnetic interactions, and a second spin component coupled to a field, cannot in general thermalize to a paramagnetic state whenever they have a finite, negative energy per spin. The long-range interactions we consider decay as a power law of the distance, ∼r−α with α<d, where d is the number of spatial dimensions of the system. The lack of thermalization is generic to an infinite family of Hamiltonians, and it is not associated to any form of integrability. It can instead be understood by the anomalous features of the thermodynamics of the system in the paramagnetic phase, which cannot be reflected in the properties of the eigenstates. This leads to a breakdown of eigenstate typicality and of the eigenstate thermalization hypothesis. Interestingly, the non-thermal states emerging at long times in the dynamics display long-range multipartite entanglement, elevating the breakdown of thermalization to a resource.
We consider a one-dimensional lattice of Ising-type variables where the ferromagnetic exchange interaction J between neighboring sites is frustrated by a long-ranged antiferromagnetic interaction of strength g between the sites i and j, decaying as ${|i\ensuremath{-}j|}^{\ensuremath{-}\ensuremath{\alpha}}$, with $\ensuremath{\alpha}>1$. For $\ensuremath{\alpha}$ smaller than a certain threshold ${\ensuremath{\alpha}}_{0}$, which is larger than 2 and depends on the ratio J/g, the ground state consists of an ordered sequence of segments with equal length and alternating magnetization. The width of the segments depends on both $\ensuremath{\alpha}$ and the ratio J/g. Our Monte Carlo study shows that the on-site magnetization vanishes at finite temperatures and finds no indication of any phase transition. Yet, the modulation present in the ground state is recovered at finite temperatures in the two-point correlation function, which oscillates in space with a characteristic spatial period. The latter depends on $\ensuremath{\alpha}$ and J/g and decreases smoothly from the ground-state value as the temperature is increased. Such an oscillation of the correlation function is exponentially damped over a characteristic spatial scale, the correlation length, which asymptotically diverges roughly as the inverse of the temperature as T=0 is approached. This suggests that the long-range interaction causes the Ising chain to fall into a universality class consistent with an underlying continuous symmetry. The ${e}^{\ensuremath{\Delta}/T}$ temperature dependence of the correlation length and the uniform ferromagnetic ground state, characteristic of the g=0 discrete Ising symmetry, are recovered for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{0}$.
A quantum Heisenberg model with anisotropic coupling and all-to-all interaction has been analyzed using the Bose-Einstein statistics. In F. Borgonovi, G. L. Celardo, M. Maianti, and E. Pedersoli, J. Stat. Phys. 116, 1435 (2004), the existence of a disconnection in the classical phase space has been proved analytically. Such topological disconnection occurs when the energy exceeds a suitable threshold value, called the topological nonconnectivity threshold. We address here the problem of finding quantum signatures of such a threshold. An independent definition of the quantum nonconnectivity threshold, motivated by considerations strictly valid in the quantum regime, is given. We also discuss the dynamical relevance of the quantum border with respect to the quantum magnetic reversal. Contrary to the classical case the magnetization can flip even below the classical threshold through macroscopic quantum tunneling. We evaluate the time scale for magnetic reversal from statistical and spectral properties for a small number of particles and in the semiclassical limit.
The d=1 Ising ferromagnet and spin glass with long-range power-law interactions J r^-a is studied for all interaction range exponents a by a renormalization-group transformation that simultaneously projects local ferromagnetism and antiferromagnetism. In the ferromagnetic case, J>0, a finite-temperature ferromagnetic phase occurs for interaction range 0.742, the phase transition temperature discontinuously drops to zero and for a>2 there is no ordered phase above zero temperature, also as predicted by rigorous results. At the other end, on approaching a=0.74 from above, namely increasing the range of the interaction, the phase transition temperature diverges to infinity, meaning that, at all non-infinite temperatures, the system is ferromagnetically ordered. Thus, the equivalent-neighbor interactions regime is entered before (a > 0) the neighbors become equivalent, namely before the interactions become equal for all separations. The critical exponents alpha,beta, gamma,delta,eta,nu are calculated, from a large recursion matrix, varying as function of a. For antiferromagnetic J<0, all triplets of spins at all ranges have competing interactions and this highly frustrated system does not have an ordered phase. In the spin-glass system, where all couplings for all separations are randomly ferromagnetic or antiferromagnetic (with probability p), a finite-temperatures spin-glass phase is obtained in the absence of antiferromagnetic phase. In the spin-glass phase, the signature chaotic behavior under scale change occurs in a richer version than previously: In the long-range interaction of this system, the interactions at every separation become chaotic, yielding a piecewise chaotic interaction function.