Quasi-one-dimensional Ising models with defects of the "random local field" type: Imry-Ma phase in spaces with dimension higher than the lower critical one
The phase diagram in coordinates "temperature - concentration of defects" of quasi-one-dimensional Ising models with defects of the "random local field" type is investigated. The confrontation of the tendency to the emergence of the long-range order due to a weak interaction between one-dimensional spin chains and the tendency to the formation of the Imry-Ma phase in which the order parameter follows the fluctuations of the random field created by defects is studied. The possibility of the appearance of the Imry-Ma phase in a situation where the space dimension exceeds the lower critical dimension is shown. The question of the existence of the long-range order in the Ising model with random fields in space with the critical dimension dl = 2 is considered.
It is shown that the Imry-Ma theorem stating that in space dimensions d<4 the introduction of an arbitrarily small concentration of defects of the "random local field" type in a system with continuous symmetry of the n-component vector order parameter (O(n)model) leads to the long-range order collapse and to the occurrence of a disordered state, is not true if the anisotropic distribution of the defect-induced random local field directions in the n-dimensional space of the order parameter leads to the defect-induced effective anisotropy of the "easy axis" type. For a weakly anisotropic field d…
For the system with the n-component order parameter (O(n)-model), conditions for initiation of the Imry-Ma disordered state resulting from the influence of defects of the "random local anisotropy" type were discovered. The initiation of such a state was shown to be possible if the distribution of local anisotropy axes directions in the order parameter space is nearly isotropic, and the limiting degree of the distribution anisotropy was found. For a higher anisotropy in the distribution of local axes directions, the long-range order in the system holds true even in the presence of defects of th…
The Ising model in a random field and with power-law decaying ferromagnetic bonds is studied at zero temperature. Comparing the scaling of the energy contributions of the ferromagnetic domain wall flip and of the random field a la Imry-Ma we obtain a threshold value for the power $ρ$ of the long-range interaction, beyond which no critical behavior occurs. The critical threshold value is $ρ_c=3/2$, at a difference with the zero field model in which $ρ_c=2$. This prediction is confirmed by numerical computation of the ground states below, at, and above this threshold value. Some possible implica…
We demonstrate that in the system with anisotropic distribution of the defect-induced random local field directions in the n-dimensional space of vector order parameter with the O(n) symmetry, the defect-induced effective anisotropy arises for the space dimensionality 2<d<4. If the anisotropy constant exceeds the threshold value, then an inhomogeneous state predicted by Imry and Ma becomes energetically unfavorable, and the system goes back to the state with the long-range order.
The influence of defects of the "random local field" type with an anisotropic distribution of random fields on two-dimensional models with continuous symmetry of the vector order parameter is considered. In the case of weak anisotropy of random fields, with decreasing temperature there takes place a smooth transition from the paramagnetic phase with dynamic fluctuations of the order parameter to the Imry-Ma phase with static fluctuations caused by fluctuations of the random field of defects. In the case of strong anisotropy of random fields, defects lead to an effective decrease in the number …
The expression for the cubic-type-anisotropy constant created by defects of "random local anisotropy" type is derived. It is shown that the Imry-Ma theorem stating that in space dimensions d<4 the introduction of an arbitrarily small concentration of defects of the "random local anisotropy" type in a system with continuous symmetry of the n-component vector order parameter (O(n) model) leads to the long-range order collapse and to occurrence of a disordered state, is not true if an anisotropic distribution of the defect-induced random easy axes directions in the order parameter space creates a…
Systems with continuous symmetry of the vector order parameter containing defects of the "random local field" or "random local anisotropy" types are investigated. It is shown that the disordered Imry-Ma phase, in which the order parameter follows the spatial fluctuations of the random field or random anisotropy direction, can also occur in a coordinate space of dimension d higher than the lower critical dimension dl = 4. For this, the hyperplanes of dimension m < dl, must exist in the coordinate space in which there is a strong exchange interaction between spins, and the interaction between sp…
Random fields disorder Ising ferromagnets by aligning single spins in the direction of the random field in three space dimensions, or by flipping large ferromagnetic domains at dimensions two and below. While the former requires random fields of typical magnitude similar to the interaction strength, the latter Imry-Ma mechanism only requires infinitesimal random fields. Recently, it has been shown that for dilute anisotropic dipolar systems a third mechanism exists, where the ferromagnetic phase is disordered by finite-size glassy domains at a random field of finite magnitude that is considera…
We discuss with the aid of random walk arguments and exact numerical computations the magnetization properties of one-dimensional random field chains. The ground state structure is explained in terms of absorbing and non-absorbing random walk excursions. At low temperatures, the magnetization profiles follow those of the ground states except at regions where a local random field fluctuation makes thermal excitations feasible. This follows also from the non-absorbing random walks, and implies that the magnetization length scale is a product of these two scales. It is not simply given by the Imr…
We consider two-dimensional Ising models with randomly distributed ferromagnetic bonds and study the local critical behavior at defect lines by extensive Monte Carlo simulations. Both for ladder and chain type defects, non-universal critical behavior is observed: the critical exponent of the defect magnetization is found to be a continuous function of the strength of the defect coupling. Analyzing corresponding stability conditions, we obtain new evidence that the critical exponent $ν$ of the bulk correlation length of the random Ising model does not depend on dilution, i.e. $ν=1$.
The anisotropic superfluid 3He-A in aerogel provides an interesting example of a system with continuous symmetry in the presence of random anisotropy disorder. Recent NMR experiments allow us to discuss two regimes of the orientational disorder, which have different NMR properties. One of them, the (s)-state, is identified as the pure Larkin-Imry-Ma state. The structure of another state, the (f)-state, is not very clear: probably it is the Larkin-Imry-Ma state contaminated by the network of the topological defects pinned by aerogel.
We show that phase transitions in Ising systems with planar defects, i.e., disorder perfectly correlated in two dimensions are destroyed by smearing. This is caused by effects similar to but stronger than the Griffiths phenomena: Exponentially rare spatial regions can develop true static long-range order even when the bulk system is still in its disordered phase. Close to the smeared transition, the order parameter is very inhomogeneous in space, with the thermodynamic (average) order parameter depending exponentially on temperature. We determine the behavior using extremal statistics, and we …
We consider whether it is possible to find ground states of frustrated spin systems by solving them locally. Using spin glass physics and Imry-Ma arguments in addition to numerical benchmarks we quantify the power of such local solution methods and show that for the average low-dimensional spin glass problem outside the spin- glass phase the exact ground state can be found in polynomial time. In the second part we present a heuristic, general-purpose hierarchical approach which for spin glasses on chimera graphs and lattices in two and three dimensions outperforms, to our knowledge, any other …
We present results of large-scale Monte Carlo simulations for a three-dimensional Ising model with short range interactions and planar defects, i.e., disorder perfectly correlated in two dimensions. We show that the phase transition in this system is smeared, i.e., there is no single critical temperature, but different parts of the system order at different temperatures. This is caused by effects similar to but stronger than Griffiths phenomena. In an infinite-size sample there is an exponentially small but finite probability to find an arbitrary large region devoid of impurities. Such a rare …
We study in Ising spin glasses the finite-size effects near the spin-glass transition in zero field and at the de Almeida-Thouless transition in a field by Monte Carlo methods and by analytical approximations. In zero field, the finite-size scaling function associated with the spin-glass susceptibility of the Sherrington-Kirkpatrick mean-field spin-glass model is of the same form as that of one-dimensional spin-glass models with power-law long-range interactions in the regime where they can be a proxy for the Edwards-Anderson short-range spin-glass model above the upper critical dimension. We …
We have studied a two dimensional lattice model of Coulomb glass for a wide range of disorders at $T\sim 0$. The system was first annealed using Monte Carlo simulation. Further minimization of the total energy of the system was done using Baranovskii et al algorithm followed by cluster flipping to obtain the pseudo ground states. We have shown that the energy required to create a domain of linear size L in d dimensions is proportional to $L^{d-1}$. Using Imry-Ma arguments given for random field Ising model, one gets critical dimension $d_{c}\geq 2$ for Coulomb glass. The investigations of doma…
The one-dimensional long-range Ising spin glass provides useful insights into the properties of finite dimensional spin glasses with short-range interactions. The defect energy renormalization group equations derived for it by Kotliar, Anderson and Stein have been re-examined and a new fixed point has been found. The fixed points which have previously been studied are found to be inappropriate. It is shown that the renormalization group equations themselves directly imply that the spin glass phase is replica symmetric, in agreement with droplet model expectations, when the range exponent $σ$ l…
We propose a microscopic mechanism of the long-range order in two-dimensional space induced by random local fields of crystal defects. The anisotropic distribution of defect-induced random local field directions in the n-dimensional space of vector order parameter with the O(n) symmetry is shown to give rise to the defect-induced effective anisotropy in the system. The expression for the effective anisotropy constant is found. A weak anisotropy of the "easy axis" type transforms the X-Y model and the Heisenberg model to the class of Ising models, and brings into existence the long-range order …
Machine Learning Topological Order from Defect Partition Functions
Shahrood University of Technology（沙赫鲁德理工大学）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京国际数学研究中心）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
We calculate the naive defect energy $ΔE$ of Ising spin glass(SG) models in two dimensions using conjugate boundary conditions. We predict that, in the $\pm J$ model, the averaged value $\bar{ΔE}$ converges to some non-zero value in the thermodynamic limit in contrast with $\bar{ΔE} = 0$ in the Gaussian model. This prediction is incompatible with previous ones but supports a recent Monte Carlo prediction of the presence of the SG phase at finite temperatures in the $\pm J$ Ising model. We also calculate the interface free energy to confirm it.
Ferromagnetic Ising systems with competing interactions are considered in the presence of a random field. We find that in three space dimensions the ferromagnetic phase is disordered by a random field which is considerably smaller than the typical interaction strength between the spins. This is the result of a novel disordering mechanism triggered by an underlying spin-glass phase. Calculations for the specific case of the long-range dipolar LiHo_xY_{1-x}F_4 compound suggest that the above mechanism is responsible for the peculiar dependence of the critical temperature on the strength of the r…
Laboratoire de Physique de l’École Normale Supérieure, Université PSL, CNRS, Sorbonne Université, Université Paris Cité（巴黎高等师范学院物理实验室，巴黎文理研究大学，法国国家科学研究中心，索邦大学，巴黎西岱大学）；Department of Physics, Indian Institute of Technology - Kanpur（坎普尔印度理工学院物理系）
We analyze the effect of adding quenched disorder along a defect line in the 2D conformal minimal models using replicas. The disorder is realized by a random applied magnetic field in the Ising model, by fluctuations in the ferromagnetic bond coupling in the Tricritical Ising model and Tricritical Three-state Potts model (the $ϕ_{12}$ operator), etc.. We find that for the Ising model, the defect renormalizes to two decoupled half-planes without disorder, but that for all other models, the defect renormalizes to a disorder-dominated fixed point. Its critical properties are studied with an expan…
A quantum phase transition from paramagnetic to ferromagnetic phase is driven by a time-dependent external magnetic field. For any rate of the transition the evolution is non-adiabatic and finite density of defects is excited in the ferromagnetic state. The density of excitations has only logarithmic dependence on the transition rate. This is much weaker than any usual power law scaling predicted for pure systems by the Kibble-Zurek mechanism.
We present results from Monte Carlo simulations of the one-dimensional Ising spin glass with power-law interactions at low temperature, using the parallel tempering Monte Carlo method. For a set of parameters where the long-range part of the interaction is relevant, we find evidence for large-scale droplet-like excitations with an energy that is independent of system size, consistent with replica symmetry breaking. We also perform zero-temperature defect energy calculations for a range of parameters and find a stiffness exponent for domain walls in reasonable, but by no means perfect agreement…
The effect of long-range correlated quenched structural defects on the critical ultrasound attenuation and sound velocity dispersion is studied for three-dimensional Ising-like systems. A field-theoretical description of the dynamic critical effects of ultrasound propagation in solids is performed with allowance for both fluctuation and relaxation attenuation mechanisms. The temperature and frequency dependences of the dynamical scaling functions of the ultrasound critical characteristics are calculated in a two-loop approximation for different values of the correlation parameter $a$ of the We…
Dipolar Bose-Einstein condensates in an array of double-well potentials realize an effective transverse Ising model with peculiar inter-layer interactions, that may result under proper conditions in an anomalous first-order ferromagnetic-antiferromagnetic phase transition, and nontrivial phases due to frustration. The considered setup as well allows the study of Kibble-Zurek defect formation, whose kink statistics follows that expected from the universality class of the mean-field transverse Ising model in 1D. Furthermore, random occupation of each layer of the stack leads to random effective …
