We apply a set of machine-learning (ML) techniques for the global exploration of the phase diagrams of two frustrated 2D Ising models with competing interactions. Based on raw Monte Carlo spin configurations generated for random system parameters, we apply principal-component analysis (PCA) and auto-encoders to achieve dimensionality reduction, followed by clustering using the DBSCAN method and a support-vector machine classifier to construct the transition lines between the distinct phases in both models. The results are in very good agreement with available exact solutions, with the auto-encoders leading to quantitatively superior estimates, even for a data set containing only 1400 spin configurations. In addition, the results suggest the existence of a relationship between the structure of the optimized auto-encoder latent space and physical characteristics of both systems. This indicates that the employed approach can be useful in perceiving fundamental properties of physical systems in situations where \emph{a priori} theoretical insight is unavailable.
We report on the development of two dual worm constructions that lead to cluster algorithms for efficient and ergodic Monte Carlo simulations of frustrated Ising models with arbitrary two-spin interactions that extend up to third-neighbours on the triangular lattice. One of these algorithms generalizes readily to other frustrated systems, such as Ising antiferromagnets on the Kagome lattice with further neighbour couplings. We characterize the performance of both these algorithms in a challenging regime with power-law correlations at finite wavevector.
We introduce a Generalized Loop Move (GLM) update for Monte Carlo simulations of frustrated Ising models on two-dimensional lattices with bond-sharing plaquettes. The GLM updates are designed to enhance Monte Carlo sampling efficiency when the system's low-energy states consist of an extensive number of degenerate or near-degenerate spin configurations, separated by large energy barriers to single spin flips. Through implementation on several frustrated Ising models, we demonstrate the effectiveness of the GLM updates in cases where both degenerate and near-degenerate sets of configurations ar…
We consider by means of Monte Carlo simulations the relaxation in the paramagnetic phase of the anti-ferromagnetic Ising model on a triangular lattice and of a fully-frustrated Ising model on a square lattice. In contradistinction to previous studies of the second model, we show that spin-spin correlation functions do not decay with a stretched-exponential law at low temperature but that both models display an exponential decay with logarithmic corrections that are interpreted as the signature of topological defects.
We show in this paper the results on the phase transition of the so-called fully frustrated simple cubic lattice with the Ising spin model. We use here the Monte Carlo method with the flat energy-histogram Wang-Landau technique which is very powerful to detect weak first-order phase transition. We show that the phase transition is clearly of first order, providing a definite answer to a question raised 25 years ago.
We study the dynamics of the critical two-dimensional fully-frustrated Ising model by means of Monte Carlo simulations. The dynamical exponent is estimated at equilibrium and is shown to be compatible with the value $z_c=2$. In a second step, the system is prepared in the paramagnetic phase and then quenched at its critical temperature $T_c=0$. Numerical evidences for the existence of logarithmic corrections in the aging regime are presented. These corrections may be related to the topological defects observed in other fully-frustrated models. The autocorrelation exponent is estimated to be $λ…
Ground-state phase diagram of the mixed spin-1/2 and spin-1 Ising-Heisenberg model on doubly decorated planar lattices is examined using the generalized decoration-iteration transformation. The main attention is devoted to the comparison of the ground-state properties of the quantum Ising-Heisenberg model and its semi-classical Ising analogue.
The critical relaxation from the low-temperature ordered state of the three-dimensional fully frustrated Ising model on a simple cubic lattice has been studied using the short time dynamics method. Particles with the periodic boundary conditions containing N = 262144 spins have been studied. Calculations have been performed by the standard Metropolis Monte Carlo algorithm. The static critical exponents of the magnetization and correlation radius have been obtained. The dynamic critical exponent of the model under study has been calculated.
Using extensive Monte Carlo simulations, we clarify the critical behaviour of the 3 dimensional simple cubic Ising Fully Frustrated system. We find two transition temperatures and two long range ordered phases. Within the present numerical accuracy, the transition at higher temperature is found to be second order and we have extracted the standard critical exponent using finite size scaling method. On the other hand, the transition at lower temperature is found to be first order. It is argued that entropy plays a major role on determining the low temperature state.
We study a 3-dimensional Ising model in which the tendency to order due to short-range ferromagnetic interactions is frustrated by competing long-range (Coulombic) interactions. Complete ferromagnetic ordering is impossible for any nonzero value of the frustration parameter, but the system displays a variety of phases characterized by periodically modulated structures. We have performed extensive Monte-Carlo simulations which provide strong evidence that the microphase separation transition between paramagnetic and modulated phases is a fluctuation-induced first-order transition. Additional tr…
The spin-1/2 Ising-Heisenberg model on the triangulated Kagome (triangles-in-triangles) lattice is exactly solved by establishing a precise mapping correspondence to the simple spin-1/2 Ising model on Kagome lattice. It is shown that the disordered spin liquid state, which otherwise occurs in the ground state of this frustrated spin system on assumption that there is a sufficiently strong antiferromagnetic intra-trimer interaction, is eliminated from the ground state by arbitrary but non-zero Dzyaloshinskii-Moriya anisotropy.
We investigate a frustrated Ising spin system on the garnet lattice composed of a specific network of corner-sharing triangles. By means of Monte Carlo simulations with the heat bath algorithm, we discuss the magnetic properties at finite temperatures. It is shown that the garnet spin system with the nearest-neighbor couplings does not exhibit any magnetic transitions, yielding the large residual entropy at zero temperature. We also investigate the effect of the long-range dipolar interaction systematically to determine the phase diagram of the Ising model on the garnet lattice. We find that t…
A frustrated Ising model on a diamond hierarchical lattice is studied. We obtain the exact partition function of this model and calculate the transition temperature, specific heat, entropy, magnetization, and ferromagnetic correlation function. Depending on the magnitude of a parameter giving the frustration, there exist three types of ground states: ferromagnetic, classical spin-liquid with highly developed short-range order, and paramagnetic. The dependence of the zero-temperature entropy on the frustration parameter has an infinite number of steps. The temperature dependence of the specific…
The geometrically frustrated spin-1/2 Ising-Heisenberg model on triangulated Husimi lattices is exactly solved by combining the generalized star-triangle transformation with the method of exact recursion relations. The ground-state and finite-temperature phase diagrams are rigorously calculated along with both sublattice magnetizations of the Ising and Heisenberg spins. It is evidenced that the Ising-Heisenberg model on triangulated Husimi lattices with two or three inter-connected triangles-in-triangles units displays in a highly frustrated region a quantum disorder irrespective of temperatur…
Here we consider the Ising-Heisenberg model in the expanded Kagomé lattice, also known as triangle-dodecagon (3-12) or star lattice. This model can still be understood as a decorated honeycomb lattice. Assuming that the Heisenberg spins are at the vertices of the triangle while other spins are of the Ising type. Thus, this model is equivalent to an effective Kagomé Ising lattice, through the decoration transformation technique. Thus this means that the model is exactly solvable so we can study the most relevant properties of this model. Like the phase diagram at zero temperature, exhibiting a …
Through the direct decoration transformation approach, we obtain a general solution for the pentagonal Ising model, showing its equivalence to the isotropic free-fermion eight-vertex model. We study the ground-state phase diagram, in which one ferromagnetic (FM) state, one ferrimagnetic (FIM) state, and one frustrated state are found. Using the exact solution of the pentagonal Ising model, we discuss the finite-temperature phase diagrams and find a phase transition between the FIM state and the disordered state as well as a phase transition between the disordered state and the FM state. We als…
The phase transitions occurring in the frustrated Ising square antiferromagnet with first- ($J_1 < 0$) and second- ($J_2 < 0$) nearest-neighbor interactions are studied within the framework of the effective-field theory with correlations based on the different cluster sizes and for a wide range of $R = J_2/|J_1|$. Despite the simplicity of the model, it has proved difficult to precisely determine the order of the phase transitions. In contrast to the previous effective-field study, we have found a first-order transition line in the region close to $R = -0.5$ not only between the superantiferro…
Critical and in the highly frustrated regime also dynamical properties of the $J_1-J_2$ Ising model with competing nearest-neighbor $J_1$ and second-nearest-neighbor $J_2$ interactions on a honeycomb lattice are investigated by standard Monte Carlo and parallel tempering simulations. The phase boundary is determined as a function of the coupling ratio for the phase transition between the paramagnetic and ferromagnetic states within $R \equiv J_2/|J_1| \in [-1/4,0]$. It is confirmed that at least for $R \geq -0.2$ the transition remains second-order and complies with the standard Ising universa…
We study the thermal phase transitions of the four-fold degenerate phases (the plaquette and single stripe states) in two-dimensional frustrated Ising model on the Shastry-Sutherland lattice using Monte Carlo simulations. The critical Ashkin-Teller-like behavior is identified both in the parameter regions with the plaquette and single stripe phases, respectively. The four-state Potts-critical end points differentiating the continuous transitions from the first-order ones are estimated based on finite-size scaling analyses. Furthermore, similar behavior of the transition to the four-fold single…
Phase transitions of the mixed spin-1/2 and spin-1 Ising-Heisenberg model on several decorated planar lattices consisting of interconnected diamonds are investigated within the framework of the generalized decoration-iteration transformation. The main attention is paid to the systematic study of the finite-temperature phase diagrams in dependence on the lattice topology. The critical behaviour of the hybrid quantum-classical Ising-Heisenberg model is compared with the relevant behaviour of its semi-classical Ising analogue. It is shown that both models on diamond-like decorated planar lattices…
The entanglement between spin-1/2 interstitial Heisenberg dimers in the mixed spin-(1,1/2) Ising-XXZ double sawtooth ladder is investigated at low temperature. Here, we consider a cyclic four-spin exchange interaction in square plaquette of each block, and investigate the effects of this amazing interaction on the bipartite entanglement between spin-1/2 interstitial dimers. Interestingly, we observe a remarkable difference in concurrence behavior with respect to the cyclic four-spin exchange interaction and magnetic field. Also, the critical points at which the concurrence vanishes are changed…
We present a mean field analysis of a fully frustrated Ising spin model on an Ising lattice gas. This is equivalent to a degenerate Blume-Emery-Griffiths model with frustration, which we analyze for different values of the quadrupolar interaction. This model might be useful in the study of structural glasses and related systems with disorder.
We use the effective-field theory with correlations based on different cluster sizes to investigate phase diagrams of the frustrated Ising antiferromagnet on the honeycomb lattice with isotropic interactions of the strength $J_1 < 0$ between nearest-neighbour pairs and $J_2 < 0$ between next-nearest neighbour pairs of spins. We present results for the ground-state energy as a function of the frustration parameter $R = J_2 /|J_1|$. We find that the cluster-size has a considerable effect on the existence and location of a tricritical point in the phase diagram at which the phase transition chang…
Magnetic frustrations and dimensionality play an important role in determining the nature of the magnetic long-range order and how it melts at temperatures above the ordering transition $T_N$. In this work, we use large-scale Monte Carlo simulations to study these phenomena in a class of frustrated Ising spin models in two spatial dimensions. We find that the melting of the magnetic long-range order into an isotropic gas-like paramagnet proceeds via an intermediate stage where the classical spins remain anisotropically correlated. This correlated paramagnet exists in a temperature range $T_N <…
The study of frustrated spin systems often requires time-consuming numerical simulations. As the simplest approach, the classical Ising model is often used to investigate the thermodynamic behavior of such systems. Exploiting the small correlation lengths in frustrated Ising systems, we develop a method for obtaining a first approximation to the energetic properties of frustrated two-dimensional Ising systems using small networks of less than 30 spins. These small networks allow much faster numerical simulations, and more importantly, analytical calculation of the properties from the partition…
We have investigated by Monte-Carlo simulation the phase diagram of a three-dimensional Ising model with nearest-neighbor ferromagnetic interactions and small, but long-range (Coulombic) antiferromagnetic interactions. We have developed an efficient cluster algorithm and used different lattice sizes and geometries, which allows us to obtain the main characteristics of the temperature-frustration phase diagram. Our finite-size scaling analysis confirms that the melting of the lamellar phases into the paramgnetic phase is driven first-order by the fluctuations. Transitions between ordered phases…
The mixed spin-(1/2, 1) Ising model on two fully frustrated triangles-in-triangles lattices is exactly solved with the help of the generalized star-triangle transformation, which establishes a rigorous mapping correspondence with the equivalent spin-1/2 Ising model on a triangular lattice. It is shown that the mutual interplay between the spin frustration and single-ion anisotropy gives rise to various spontaneously ordered and disordered ground states, which differ mainly in an occurrence probability of the non-magnetic spin state of the integer-valued decorating spins. We have convincingly e…
Ground state and thermodynamics of geometrically frustrated spin-1/2 Ising-Heisenberg model on two different but topologically related triangles-in-triangles lattices is investigated in particular. A rigorous mapping based on generalized star-triangle transformations enables one to derive exact analytical results for the free and internal energy, spontaneous magnetization, entropy and specific heat from the known exact results for the corresponding spin-1/2 Ising model on a triangular lattice. It is demonstrated that several unconventional quantum ordered and disordered phases may become the g…
We consider two fully frustrated Ising models: the antiferromagnetic triangular model in a field of strength, $h=H T k_B$, as well as the Villain model on the square lattice. After a quench from a disordered initial state to T=0 we study the nonequilibrium dynamics of both models by Monte Carlo simulations. In a finite system of linear size, $L$, we define and measure sample dependent "first passage time", $t_r$, which is the number of Monte Carlo steps until the energy is relaxed to the ground-state value. The distribution of $t_r$, in particular its mean value, $< t_r(L) >$, is shown to obey…
Thermodynamic and magnetocaloric properties of geometrically frustrated Ising spin clusters of selected shapes and sizes are studied by exact enumeration. In the ground state the magnetization and the entropy show step-wise variations with an applied magnetic field. The number of steps, their widths and heights depend on the cluster shape and size. While the character of the magnetization plateau heights is always increasing, the entropy is not necessarily decreasing function of the field, as one would expect. For selected clusters showing some interesting ground-state properties, the calculat…
Magnetic and magnetocaloric properties of geometrically frustrated antiferromagnetic Ising (IA) and ferromagnetic spin ice (SI) models on a nanocluster with a `Star of David' topology, including next-nearest-neighbor (NNN) interactions, are studied by an exact enumeration. In an external field applied in characteristic directions of the respective models, depending on the NNN interaction sign and magnitude, the ground state magnetization of the IA model is found to display up to three intermediate plateaus at fractional values of the saturation magnetization, while the SI model shows only one …
We discuss three exactly solvable spin models of geometric frustration. First, we discuss a 1-parameter subfamily of the 16 vertex model, which can be mapped to a planar Ising model and solved via Fisher-Dubedát decorations. We then consider a 1-parameter family generalization of the Villain's fully frustrated model, which interpolates between Onsager's 2D Ising model and the Villain one. We then discuss spin ice models on a tree, which can be solved exactly using recursions \textit{a lá Bethe}.
We investigate the role of a transverse field on the Ising square antiferromagnet with first-($J_1$) and second-($J_2$) neighbor interactions. Using a cluster mean-field approach, we provide a telltale characterization of the frustration effects on the phase boundaries and entropy accumulation process emerging from the interplay between quantum and thermal fluctuations. We found that the paramagnetic (PM) and antiferromagnetic phases are separated by continuous phase transitions. On the other hand, continuous and discontinuous phase transitions, as well as tricriticality, are observed in the p…
We study the nature of a low-temperature phase in the frustrated honeycomb-lattice Ising model with first- and second-neighbor antiferromagnetic (AF) interactions, $J_1$ and $J_2$, respectively, for $R = J_2/J_1 > 1/4$. It is known that for $R < 1/4$ there is a phase transition at low temperatures to the AF phase. Nevertheless, little is known about the critical behavior of the model for $R > 1/4$, except for recent effective field results which detected no phase transition down to zero temperature. Our Monte Carlo results suggest that for $R > 1/4$ there is at least one peculiar phase transit…
To understand the origin of the dynamical transition, between high temperature exponential relaxation and low temperature nonexponential relaxation, that occurs well above the static transition in glassy systems, a frustrated spin model, with and without disorder, is considered. The model has two phase transitions, the lower being a standard spin glass transition (in presence of disorder) or fully frustrated Ising (in absence of disorder), and the higher being a Potts transition. Monte Carlo results clarify that in the model with (or without) disorder the precursor phenomena are related to the…
In this paper we show quantum fluctuation effect of fully frustrated Ising spin systems. Quantum annealing has been expected to be an efficient method to find ground state of optimization problems. However it is not clear when to use the quantum annealing. In order to clarify when the quantum annealing works well, we have to study microscopic properties of quantum annealing. In fully frustrated Ising spin systems, there are macroscopically degenerated ground states. When we apply quantum annealing to fully frustrated systems, we cannot obtain each ground state with the same probability. This n…
We consider itinerant spinless fermions as moving defects in a dilute two-dimensional frustrated Ising system where they occupy site vacancies. Fermions interact via local spin fluctuations and we analyze coupled self-consistent mean-field equations of the Green functions after expressing the spin and fermion operators in terms of Grassmann variables. The specific heat and effective mass are analyzed with the solutions satisfying the symmetry imposed by the coupling layout. At low temperature, we find that these solutions induce stripes along the lines of couplings with the same sign, and that…
Some strongly frustrated magnets such as the "spin-ice" compounds fail to produce any magnetic order at finite temperatures even in the presence of magnetic field. Still they have very unusual low-temperature thermodynamic properties related to the field-induced ground state transitions. Here we show that general qualitative picture of such peculiar thermodynamics can be obtained in the antiferromagnetic Ising model on the triangular Husimi lattice. The analytical results for this model show magnetic plateaus, entropy spikes, crossing points and peculiarities in magnetic susceptibility and spe…
We study the spin-1/2 Ising-XXZ model on a decorated honeycomb lattice composed of five spins per unit cell, one Ising spin, and four Heisenberg spins. This model involving the Heisenberg exchange interaction is one of the few models that can be exactly solvable through the generalized star-triangle transformation. The significance of this model is its close relationship to the fully decorated quantum Heisenberg honeycomb lattice since 4/5 of the particles are Heisenberg spins. We investigate the phase diagram at zero temperature and identify a relevant quantum spin frustrated phase resulting …
Spin-spin correlations are calculated in frustrated hierarchical Ising models that exhibit chaotic renormalization-group behavior. The spin-spin correlations, as a function of distance, behave chaotically. The far correlations, but not the near correlations, are sensitive to small changes in temperature or frustration, with temperature changes having a larger effect. On the other hand, the calculated free energy, internal energy, and entropy are smooth functions of temperature. The recursion-matrix calculation of thermodynamic densities in a chaotic band is demonstrated. The leading Lyapunov e…
Low-temperature magnetization curves and thermodynamics of a frustrated spin-1/2 Heisenberg triangular bilayer with the antiferromagnetic intradimer interaction and either ferromagnetic or antiferromagnetic interdimer interaction are investigated in a highly frustrated parameter region, where localized many-magnon eigenstates provide the most dominant contribution to magnetothermodynamics. Low-energy states of the highly frustrated spin-1/2 Heisenberg triangular bilayer can be accordingly found from a mapping correspondence with an effective triangular-lattice spin-1/2 Ising model in a field. …
Many-body systems subject to a high-frequency drive can show intriguing thermalization behavior. Prior to heating to a featureless infinite-temperature state, these systems can spend an exponentially long time in prethermal phases characterized by various kinds of order. Here, we uncover the rich non-equilibrium phase diagram of a driven frustrated two-dimensional Ising magnet with competing short-range ferromagnetic and long-range dipolar interactions. We show that the ordered stripe and nematic phases, which appear in equilibrium as a function of temperature, underpin subsequent prethermal p…
We study ordering mechanisms which are induced by the quantum fluctuation in fully frustrated Ising spin systems. Since there are many degenerated states in frustrated systems, "order by thermal disorder" often takes place due to a kind of entropy effect. To consider "order by quantum disorder" in fully frustrated Ising spin systems, we apply transverse field as quantum fluctuation. There exists a ferromagnetic correlation in each sublattice. The sublattice correlation at zero temperature is enlarged due to transverse field. The quantum fluctuation enhances the solid order at zero temperatures…
Motivated by the recent success of tensor networks to calculate the residual entropy of spin ice and kagome Ising models, we develop a general framework to study frustrated Ising models in terms of infinite tensor networks %, i.e. tensor networks that can be contracted using standard algorithms for infinite systems. This is achieved by reformulating the problem as local rules for configurations on overlapping clusters chosen in such a way that they relieve the frustration, i.e. that the energy can be minimized independently on each cluster. We show that optimizing the choice of clusters, inclu…
We perform large scale simulations of the frustrated Ising lattice gas, a three-dimensional lattice model of a structural glass, using the parallel tempering technique. We evaluate the spin and density overlap distributions, and the corresponding non-linear susceptibilities, as a function of the chemical potential. We then evaluate the relaxation functions of the spin and density self-overlap, and study the behavior of the relaxation times. The results suggest that the spin variables undergo a transition very similar to the one of the Ising spin glass, while the density variables do not show a…
We study in this paper the resistivity encountered by Ising itinerant spins traveling in the so-called $J_1-J_2$ frustrated simple cubic Ising lattice. For the lattice, we take into account the interactions between nearest-neighbors and next-nearest-neighbors, $J_1$ and $J_2$ respectively. Itinerant spins interact with lattice spins via a distance-dependent interaction. We also take into account an interaction between itinerant spins. The lattice is frustrated in a range of $J_2$ in which we show that it undergoes a very strong first-order transition. Using Monte Carlo simulation, we calculate…
We describe a novel type of magnetic domain wall which, in contrast to Bloch or Neel walls, is non-localized and, in a certain temperature range, non-monotonic. The wall appears as the mean-field solution of the two-dimensional ferromagnetic Ising model frustrated by the long-ranged dipolar interaction. We provide experimental evidence of this wall delocalization in the stripe-domain phase of perpendicularly magnetized ultrathin magnetic films. In agreement with experimental results, we find that the stripe width decreases with increasing temperature and approaches a finite value at the Curie-…
Charge correlations in dense ionic fluids give rise to novel effects such as long-range screening and colloidal stabilization which are not predicted by the classic Debye-Huckel theory. We show that a Coulomb or charge-frustrated Ising model, which accounts for both long-range Coulomb and short-range molecular interactions, simply describes some of these ionic correlations. In particular, we obtain at mean field level and in simulations, a non-monotonic dependence of the screening length on the temperature. Using a combination of simulations and mean field theories, we study how the correlatio…
We study critical and magnetic properties of a bilayer Ising system consisting of two triangular planes A and B, with the antiferromagnetic (AF) coupling $J_{\rm A}$ and the ferromagnetic (FM) one $J_{\rm B}$ for the respective layers, which are coupled by the interlayer interaction $J_{\rm AB}$ by using Monte Carlo simulations. When $J_{\rm A}$ and $J_{\rm B}$ are of the same order, the unfrustrated FM plane orders first at a high temperature $T_{c1} \sim J_{\rm B}$. The spontaneous FM order then exerts influence on the other frustrated AF plane as an effective magnetic field, which subsequen…
We study the challenging thermal phase transition to stripe order in the frustrated square-lattice Ising model with couplings J1<0 (nearest-neighbor, ferromagnetic) and J2>0 (second-neighbor, antiferromagnetic) for g=J2/|J1|>1/2. Using Monte Carlo simulations and known analytical results, we demonstrate Ashkin-Teller criticality for g>= g*, i.e., the critical exponents vary continuously between those of the 4-state Potts model at g=g* and the Ising model for g -> infinity. Thus, stripe transitions offer a route to realizing a related class of conformal field theories with conformal charge c=1 …
We present a theoretical study aimed to elucidate the origin of the inverse symmetry breaking transition observed in ultrathin magnetic films with perpendicular anisotropy. We study the behavior of the dipolar frustrated Ising model in a mean field approximation as well as two other models with simple domain walls. By a numerical analysis we show that the internal degrees of freedom of the domain walls are decisive for the presence of the inverse symmetry breaking transition. In particular, we show that in a sharp domain wall model the inverse transition is absent. At high temperatures the add…
Ising models with frustrated next-nearest-neighbor interactions present a rich morphology of modulated phases. These phases, however, assemble and relax slowly, which hinders their computational study. In two dimensions, strong fluctuations further hamper determining their equilibrium phase behavior from theoretical approximations. The exact numerical transfer matrix (TM) method, which bypasses these difficulties, can serve as a benchmark method once its own numerical challenges are surmounted. Building on our recent study [Hu and Charbonneau, Phys. Rev. B 103, 094441 (2021)], in which we eval…
