It is known that, after a quench to zero temperature (T=0), two-dimensional (d=2) Ising ferromagnets with short-range interactions do not always relax to the ordered state. They can also fall in infinitely long-lived striped metastable states with a finite probability. In this paper, we study how the abundance of striped states is affected by long-range interactions. We investigate the relaxation of d=2 Ising ferromagnets with power-law interactions by means of Monte Carlo simulations at both T=0 and T≠0. For T=0 and the finite system size, the striped metastable states are suppressed by long-range interactions. In the thermodynamic limit, their occurrence probabilities are consistent with the short-range case. For T≠0, the final state is always ordered. Further, the equilibration occurs at earlier times with an increase in the strength of the interactions.
We study the relaxation of the local ferromagnetic order in the transverse field quantum Ising chain with power-law decaying interactions \sim 1/r^\alpha ∼ 1 / r α . We prepare the system in the GHZ state and study the time evolution of the probability distribution function (PDF) of the order parameter within a block of \ell ℓ when quenching the transverse field. The model is known to support long range order at finite temperature for \alpha\leq 2.0 α ≤ 2.0 . In this regime, quasi-localized topological magnetic defects are expected to strongly affect the equilibration of the full probability distribution. We highlight different dynamical regimes where gaussification mechanism may be slowed down by confinement and eventually breaks. We further study the PDF dynamics induced by changing the effective dimensionality of the system; we mimic this by quenching the range of the interactions. As a matter of fact, the behavior of the system crucially depends on the value of \alpha α governing the unitary evolution.
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 $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{0}$.
We study the ground state of a
d-dimensional Ising model with both long-range (dipole-like) and nearest-neighbor ferromagnetic (FM) interactions. The long-range interaction is equal to ${r}^{\ensuremath{-}p}$,
p>d, while the FM interaction has strength
J. If
p>d+1 and
J is large enough the ground state is FM, while if $d<p\ensuremath{\leqslant}d+1$ the FM state is not the ground state for any choice of
J. In
d=1 we show that for any
p>1 the ground state has a series of transitions from an antiferromagnetic state of period 2 to
2h-periodic states of blocks of sizes
h with alternating sign, the size
h growing when the FM interaction strength
J is increased (a generalization of this result to the case $0<p\ensuremath{\leqslant}1$ is also discussed). In $d\ensuremath{\geqslant}2$ we prove, for $d<p\ensuremath{\leqslant}d+1$, that the dominant asymptotic behavior of the ground-state energy agrees for large
J with that obtained from a periodic striped state conjectured to be the true ground state. The geometry of contours in the ground state is discussed.
The ferromagnetic Ising model with antiferromagnetic dipole interactions is investigated by means of Monte Carlo simulations, focusing on the characterization of the phase transitions between the tetragonal liquid and stripe of width h phases. The dynamic evolution of the physical observables is analyzed within the short-time regime for 0.5≤δ≤1.3, where δ is the ratio between the short-range exchange and the long-range dipole interaction constants. The obtained results for the interval 0.5≤δ≤1.2 indicate that the phase transition line between the h=1 stripe and tetragonal liquid phases is continuous. This finding contributes to clarifying the controversy about the order of this transition. This controversy arises from the difficulties introduced in the simulations due to the presence of long-range dipole interactions, such as an important increase in the simulation times that limits the system size used, strong finite size effects, as well as to the existence of multiple metastable states at low temperatures. The study of the short-time dynamics of the model allows us to avoid these hindrances. Moreover, due to the fact that the finite-size effects do not significantly affect the power-law behavior exhibited in the observables within the short-time regime, the results could be attributed to those corresponding to the thermodynamic limit. As a consequence of this, a careful characterization of the critical behavior for the whole transition line is performed by giving the complete set of critical exponents.
Extensive Monte Carlo simulations are employed in order to study the dynamic critical behaviour of the one-dimensional Ising magnet, with algebraically decaying long-range interactions of the form 1/rd + σ, with σ = 0.75. The critical temperature, as well as the critical exponents, are evaluated from the power-law behaviour of suitable physical observables when the system is quenched from uncorrelated states, corresponding to infinite temperature, to the critical point. These results are compared with those obtained from the dynamic evolution of the system when it is annealed at the critical point from the ordered state. Also, the critical temperature in the infinite interaction limit is obtained by means of a finite-range scaling analysis of data measured with different truncated interaction ranges. All the estimated static critical exponents (γ/ν, β/ν, and 1/ν) are in good agreement with renormalization group (RG) results and previously reported numerical data obtained under equilibrium conditions. On the other hand, the dynamic exponent of the initial increase of the magnetization (θ) was close to RG predictions. However, the dynamic exponent z of the time correlation length is slightly different to the RG results probably due to the fact that it may depend on the specific dynamics used or because the two-loop expansion used in the RG analysis may be insufficient.
We study the zero-temperature phase diagram of Ising spin systems in two dimensions in the presence of competing interactions: long-range antiferromagnetic and nearest-neighbor ferromagnetic of strength
J. We first introduce the notion of a ``corner energy,'' which shows, when the antiferromagnetic interaction decays faster than the fourth power of the distance, that a striped state is favored with respect to a checkerboard state when
J is close to
Jc, the transition to the ferromagnetic state, i.e., when the length scales of the uniformly magnetized domains become large. Next, we perform detailed analytic computations on the energies of the striped and checkerboard states in the cases of antiferromagnetic interactions with exponential decay and with power-law decay ${r}^{\ensuremath{-}p}$,
p>2, which depend on the Manhattan distance instead of the Euclidean distance. We prove that the striped phase is always favored compared to the checkerboard phase when the scale of the ground-state structure is very large. This happens for $J\ensuremath{\lesssim}{J}_{c}$ if
p>3, and for
J sufficiently large if $2<p\ensuremath{\leqslant}3$. Many of our considerations involving rigorous bounds carry over to dimensions greater than two and to more general short-range ferromagnetic interactions.