We study a 1D-Quantum Ising Model in transverse field driven out of equilibrium by performing a composite quantum quench to deduce the asymptotic properties of the transverse magnetization stationary state via the analysis of the spectral function. What emerges is that, in correspondence of the dynamical phase transition transition predicted for this model, the spectral function vanishes giving a hint of the criticality. This result suggests also that linear response experiment can be used to detect this kind of phase transition.
We suggest the new definition of the magnetization. For the two - dimensional Ising model with the free boundary conditions we calculate this magnetization.
We suggest the new definition of the magnetization. For the two - dimensional Ising model with the free boundary conditions we calculate any derivative of this magnetization for zero magnetic field.
We give a rigorous proof of the existence of spontaneous magnetization at finite temperature for the Ising spin model defined on the Sierpinski carpet fractal. The theorem is inspired by the classical Peierls argument for the two dimensional lattice. Therefore, this exact result proves the existence of spontaneous magnetization for the Ising model in low dimensional structures, i.e. structures with dimension smaller than 2.
Properties of the two dimensional Ising model with fixed magnetization are deduced from known exact results on the two dimensional Ising model. The existence of a continuous phase transition is shown for arbitrary values of the fixed magnetization when crossing the boundary of the coexistence region. Modifications of this result for systems of spatial dimension greater than two are discussed.
The relevant approximate spontaneous magnetization relations for the simple-cubic and body-centered-cubic Ising lattices have recently been obtained analytically by a novel approach that conflates the Callen--Suzuki identity with a heuristic odd-spin correlation magnetization relation. By exploiting this approach, we study an approximate analytic spontaneous magnetization expression for the face-centered-cubic Ising lattice. We report that the results of the analytic relation obtained in this work are nearly consistent with those derived from the Monte Carlo simulation.
Magnetization process and adiabatic demagnetization of the antiferromagnetic Ising spin clusters with the shape of regular polyhedra (Platonic solids) are exactly examined within the framework of a simple graph-theoretical approach. While the Ising cube as the only unfrustrated (bipartite) spin cluster shows just one trivial plateau at zero magnetization, the other regular Ising polyhedra (tetrahedron, octahedron, icosahedron and dodecahedron) additionally display either one or two intermediate plateaux at fractional values of the saturation magnetization. The nature of highly degenerate groun…
We present a theoretical framework which is generally applicable to the study of time scales of activated processes in systems with Brownian type dynamics. This framework is applied to a prototype system: magnetization reversal times in the 2D Ising model. Direct simulation results for the magnetization reversal times, spanning more than five orders of magnitude, are compared with theoretical predictions; the two agree in most cases within 20%.
We consider the Ising spin system, which stems out from the corresponding Multiple-spin exchange (MSE) Hamiltonian, on the special one--dimensional lattice, diamond-plaquette chain. Using the technique e of transfer-matrix we obtain the exact expression for system free energy with the aid of which we obtain the magnetization function. Analyzing magnetization curves for varies values of temperature and couplin constants we found the magnetization plateaux at 1/3 and 2/3 of the full moment. The corresponding microscopic spin configurations are unknown by virtue of high frustration.
Magnetization process of the mixed spin-1/2 and spin-3/2 Ising-Heisenberg diamond chain is examined by combining three exact analytical techniques: Kambe projection method, decoration-iteration transformation and transfer-matrix method. Multiple frustration-induced plateaus in a magnetization process of this geometrically frustrated system are found provided that a relative ratio between the antiferromagnetic Heisenberg- and Ising-type interactions exceeds some particular value. By contrast, there is just a single magnetization plateau if the frustrating Heisenberg interaction is sufficiently …
Magnetizations of ferro- and antiferromagnetic Ising models with frustration on diamond hierarchical lattices are exactly obtained at zero temperature. For the zero-field classical spin-liquid phase found in [Kobayashi {\it et al}, J. Phys. Soc. Jpn. 78, 074004 (2009) ], for which frustrating interactions play an important role, an infinitely small applied magnetic field can induce an infinitely small magnetization, despite classical Ising models that have discrete energy levels. In antiferromagnetic systems, the magnetization cannot saturate under finite magnetic fields owing to the competiti…
Recently we have performed large-scale average magnetization studies of the Ising model for the square, honeycomb, triangular, and simple cubic lattice. We want to complement those studies with the structurally more complicated body-centered cubic lattice Ising model in this paper. We have relevantly calculated the order parameter or average magnetization expression for the body-centered cubic Ising lattice based on the recently developed interrelation, given by $\langleσ_{0, i}\rangle= \langle\tanh[K(σ_{1,i}+σ_{2,i}+\dots +σ_{z,i})+H]\rangle$. We will use the same conjectures for the odd spin…
The spin-1/2 Ising-Heisenberg model on diamond-like decorated Bethe lattices is exactly solved in the presence of the longitudinal magnetic field by combining the decoration-iteration mapping transformation with the method of exact recursion relations. In particular, the ground state and low-temperature magnetization process of the ferrimagnetic version of the considered model is investigated in detail. Three different magnetization scenarios with up to two consecutive fractional magnetization plateaus were found, whereas the intermediate magnetization plateau may either correspond to the clas…
The partition functions for two-dimensional nearest neighbour Ising model in a non-zero magnetic field have been derived for finite square lattices with the help of graph theoretical procedures, show-bit algorithm, enumeration of configurations and transfer matrix methods. A new recurrence relation valid for all lattice sizes is proposed. A novel method of computing the critical temperature has been demonstrated. The magnetization and susceptibility for infinite lattices in the presence of magnetic field is estimated. The critical exponent δ pertaining to the magnetization has also been comput…
In 2011 I reviewed the calculation by Onsager and Kaufman of the spontaneous magnetization of the square-lattice Ising model, which Onsager announced in 1949 but never published. I have recently been alerted to further original papers that bear on the subject. It is quite clear that the draft paper on which I relied was indeed written by Onsager, who was working on the problem with Kaufman, and that they had two derivations of the result.
A method for the study of the ground states of lattice-gas models or equivalent spin models with extended-range interactions is developed. It is shown that effect of longer-range interactions can be studied in terms of the solution of the ground-state problem for a model with short-range interactions. The method is applied to explain the emergence of fractional magnetization plateaus in TmB4 that is regarded as a strong Ising magnet on the Shastry-Sutherland lattice.
The frustrated spin-1/2 Ising-Heisenberg ladder with Heisenberg intra-rung and Ising inter-rung interactions is exactly solved in a longitudinal magnetic field by taking advantage of the local conservation of the total spin on each rung and the transfer-matrix method. We have rigorously calculated the ground-state phase diagram, magnetization process, magnetocaloric effect and basic thermodynamic quantities for the model, which can be alternatively viewed as an Ising-Heisenberg tetrahedral chain. It is demonstrated that a stepwise magnetization curve with an intermediate plateau at a half of t…
The spin-1 Ising-Heisenberg diamond chain in a magnetic field is exactly solved by a rigorous treatment based on the transfer-matrix method. An exact ground-state phase diagram includes in total three unconventional quantum ground states with a quantum entanglement of the decorating spin-1 Heisenberg dimers apart from two ground states with a classical spin arrangement. It is evidenced that all three values of the magnetization allowed for the spin-1 diamond chain without translationally broken symmetry by the Oshikawa-Yamanaka-Affleck criterion can become evident in an outstanding stepwise ma…
We consider the zero-temperature random-field Ising model in the presence of an external field, on ladders and in one dimension with finite range interactions, for unbounded continuous distributions of random fields, and show that there is no jump discontinuity in the magnetizations for any quasi-one dimensional model. We show that the evolution of the system at an external field can be described by a stochastic matrix and the magnetization can be obtained using the eigenvector of the matrix corresponding to the eigenvalue one, which is continuous and differentiable function of the external fi…
Finite-size scaling functions are investigated both for the mean-square magnetization fluctuations and for the probability distribution of the magnetization in the one-dimensional Ising model. The scaling functions are evaluated in the limit of the temperature going to zero (T -> 0), the size of the system going to infinity (N -> oo) while N[1-tanh(J/k_BT)] is kept finite (J being the nearest neighbor coupling). Exact calculations using various boundary conditions (periodic, antiperiodic, free, block) demonstrate explicitly how the scaling functions depend on the boundary conditions. We also s…
The axial next-nearest-neighbor Ising model is studied in two dimensions at finite temperature using the density matrix renormalization group. The model exhibits phase transition of the second-order between the antiphase in low temperature and the modulated phase in high temperature. Observing the domain wall free energy, we confirm that the modulation period in high-temperature side is well explained by the free-fermion picture.
A spin-1/2 Ising-Heisenberg branched chain constituted by regularly alternating Ising spins and Heisenberg dimers involving an additional side branching is exactly solved in a magnetic field by the transfer-matrix method. The spin-1/2 Ising-Heisenberg branched chain involves two different Ising and one Heisenberg coupling constants. The overall ground-state phase diagram is formed by three different ground states emergent depending on a mutual interplay between the magnetic field and three considered coupling constants: the modulated quantum antiferromagnetic phase, the quantum ferrimagnetic p…
We employ an effective-field theory with correlations in order to study the phase diagram and ground-state magnetizations of a selectively diluted Ising antiferromagnet on triangular and honeycomb lattices. Dilution of different sublattices with generally unequal probabilities results in a rather intricate phase diagram in the sublattice dilution parameters space. In the case of the frustrated triangular lattice antiferromagnet the selective dilution affects the degree of frustration which can lead to some peculiar phenomena, such as reentrant behavior of long-range order or unsaturated sublat…
The ground state and zero-temperature magnetization process of the spin-1/2 Ising-Heisenberg model on two-dimensional triangles-in-triangles lattices is exactly calculated using eigenstates of the smallest commuting spin clusters. Our ground-state analysis of the investigated classical--quantum spin model reveals three unconventional dimerized or trimerized quantum ground states besides two classical ground states. It is demonstrated that the spin frustration is responsible for a variety of magnetization scenarios with up to three or four intermediate magnetization plateaus of either quantum o…
A complete solution of the ground-state problem for an Ising model on the Shastry-Sutherland lattice with an additional interaction along the diagonals of "empty" squares in an applied magnetic field is presented. A rigorous proof is given that this interaction gives rise to a plateau at one-half of the saturation magnetization. Such a fractional plateau has been observed in some rare-earth-metal tetraborides, in particular, in strong Ising magnets ErB4 (where it is the only one) and TmB4 (where it is the broadest one), but its origin has remained unclear. Our study sheds new light on the solu…
We study the dynamics caused by transport of transverse magnetization in one dimensional transverse Ising chain at zero temperature. We observe that a class of initial states having product structure in fermionic momentum-space and satisfying certain criteria, produce spatial variation in transverse magnetization. Starting from such a state, we obtain the transverse magnetization analytically and then observe its dynamics in presence of a homogeneous constant field $Γ$. In contradiction with general expectation, whatever be the strength of the field, the magnetization of the system does not be…
Using a combinatorial method, the partition functions for two-dimensional nearest neighbour Ising models have been derived for a square lattice of 16 sites in the presence of the magnetic field. A novel hierarchical method of enumeration of all the configurations for any arrangement of sites has been proposed. This enumeration has been executed by a systematic analysis of the appropriate diagrams without employing any algorithmic approach or computational tools. The resulting algebraic eqn in terms of the magnetic field and nearest neighbour interaction energies may then provide a methodology …
The spin-1/2 Ising-Heisenberg branched chain composed of regularly alternating Ising spins and Heisenberg dimers involving an additional side branching is rigorously solved in a magnetic field by the transfer-matrix approach. The ground-state phase diagram, the magnetization process and the concurrence measuring a degree of bipartite entanglement within the Heisenberg dimers are examined in detail. Three different ground states were found depending on a mutual interplay between the magnetic field and two different coupling constants: the modulated quantum antiferromagnetic phase, the quantum f…
A new approximation of the cluster variational method is introduced for the three-dimensional Ising model on the simple cubic lattice. The maximal cluster is, as far as we know, the largest ever used in this method. A message-passing algorithm, generalized belief propagation, is used to minimize the variational free energy. Convergence properties and performance of the algorithm are investigated. The approximation is used to compute the spontaneous magnetization, which is then compared to previous results. Using the present results as the last step in a sequence of three cluster variational ap…
Starting from an exact formal identity for the two-state transverse Ising model and using correlation inequalities rigorous upper bounds for the critical temperature and the critical transverse field are obtained which improve effective results.
The ground state and magnetization process of the mixed spin-(1,1/2) Ising diamond chain is exactly solved by employing the generalized decoration-iteration mapping transformation and the transfer-matrix method. The decoration-iteration transformation is first used in order to establish a rigorous mapping equivalence with the corresponding spin-1 Blume-Emery-Griffiths chain in a non-zero magnetic field, which is subsequently exactly treated within the framework of the transfer-matrix technique. It is shown that the ground-state phase diagram includes just four different ground states and the l…
We provide a general scheme to predict and derive the contribution to the noise spectrum of a stochastic sequence of pulses from the distribution of pulse parameters. An example is the magnetization noise spectra of a 2D Ising system near its phase transition. At $T\le T_c$, the low frequency spectra is dominated by magnetization flips of nearly the entire system. We find that both the predicted and the analytically derived spectra fit those produced from simulations. Subtracting this contribution leaves the high frequency spectra which follow a power law set by the critical exponents.
In order to provide an experimental access to the statistical theory of Lee and Yang (1952) the density function g(theta) of zeros on the Lee-Yang circle has been determined for the first time by analyzing isothermal magnetization data, m(H), of the Ising ferromagnet FeCl2 in axial magnetic fields, H, at temperatures 34 < T < 99K. The validity of our approach is demonstrated by the perfect agreement of magnetic specific heat data as calculated from g(theta) and m(H) via Maxwells relation, respectively. Moreover, the correct in-plane exchange constant of FeCl2 emerges from the series expansion …
The magnetization probability density in d=2 and 3 dimensional Ising models in slab geometry of volume $L_{\parallel}^{d-1} \times L_{\perp}$ is computed through Monte-Carlo simulation at the critical temperature and zero magnetic field. The finite-size scaling of this distribution and its dependence on the system aspect-ratio $ρ=\frac{L_{\perp}}{L_{\parallel}}$ and boundary conditions is discussed. In the limiting case $ρ\to 0$ of a macroscopically large slab ($L_{\parallel} \gg L_{\perp}$) the distribution is found to scale as a Gaussian function for all tested system sizes and boundary cond…
Magnetization plateaus and phase diagrams of the extended Ising model on the Shastry-Sutherland lattice with the first $(J_1)$, second $(J_2)$, third $(J_3)$ fourth $(J_4)$ and fifth $(J_5)$ nearest-neighbour spin couplings are studied by the classical Monte Carlo method. It is shown that switching on $J_4$ and $J_5$ interactions (in addition to usually considered $J_1, J_2$ and $J_3$ interactions) changes significantly the picture of magnetization processes found for $J_4=J_5=0$ and leads to stabilization of new macroscopic magnetic phases (plateaus) with fractional magnetization. In particul…
The Ising chain consisting of the antiferromagneticaly coupled ferromagnetic trimer is considered in the external magnetic field. In the framework of the transfer-matrix formalism the thermodynamics of the system is described. The magnetization per site (m) is obtained as the explicit function of the external magnetic field (H). The corresponding plots of m(H) are drawn. Two qualitatively different regions of the values of coupling constants are established: weak antiferromagnetic coupling (J_A<3J_F) and the strong antiferromagnetic coupling $(J_A\geq3J_F)$. For the latter case the magnetizati…
A three-dimensional layered Ising-Antiferromagnet with a ferromagnetic intra-layer coupling to z neighbors, zJ > 0, and an antiferromagnetic interlayer coupling to z' neighbors, z'J' < 0, is investigated by Monte Carlo simulations on a hexagonal lattice. The physical nature of the anomalous temperature bahavior of the sublattice magnetizations, which is found for certain values of r=zJ/z'J' and z' in magnetic fields is explained in terms of successive phase transitions. They take place on the ferromagnetic 2-dimensional spin-down sublattice at T = T_c^{2d}, smeared by a finite stabilizing mole…
The slow dynamics (10^-6 s - 10^4 s) of the magnetization in the paramagnetic phase, predicted by Glauber for 1d Ising ferromagnets, has been observed with ac susceptibility and SQUID magnetometry measurements in a molecular chain comprising alternating Co{2+} spins and organic radical spins strongly antiferromagnetically coupled. An Arrhenius behavior with activation energy Delta=152 K has been observed for ten decades of relaxation time and found to be consistent with the Glauber model. We have extended this model to take into account the ferrimagnetic nature of the chain as well as its heli…
We investigate the Ising model in one, two, and three dimensions using a cumulant method that allows us to determine the Lee-Yang zeros from the magnetization fluctuations in small lattices. By doing so with increasing system size, we are able to determine the convergence point of the Lee-Yang zeros in the thermodynamic limit and thereby predict the occurrence of a phase transition. The cumulant method is attractive from an experimental point of view since it uses fluctuations of measurable quantities, such as the magnetization in a spin lattice, and it can be applied to a variety of equilibri…
The spontaneous magnetization relations for the 2D triangular and the 3D cubic lattices of the Ising model are derived by a new tractable easily calculable mathematical method. The result obtained for the triangular lattice is compared with the already available result to test and investigate the relevance the new mathematical method. From this comparison, it is seen that the agreement of our result is almost the same or almost equivalent to the previously obtained exact result. The new approach is, then, applied to the long-standing 3D cubic lattice, and the corresponding expression for the s…
Lars Onsager announced in 1949 that he and Bruria Kaufman had proved a simple formula for the spontaneous magnetization of the square-lattice Ising model, but did not publish their derivation. It was three years later when C. N. Yang published a derivation in Physical Review. In 1971 Onsager gave some clues to his and Kaufman's method, and there are copies of their correspondence in 1950 now available on the Web and elsewhere. Here we review how the calculation appears to have developed, and add a copy of a draft paper, almost certainly by Onsager and Kaufman, that obtains the result.
Energy eigenvalues and order parameters are calculated by exact diagonalization for the transverse Ising model on square lattices of up to 6x6 sites. Finite-size scaling is used to estimate the critical parameters of the model, confirming universality with the three-dimensional classical Ising model. Critical amplitudes are also estimated for both the energy gap and the ground-state energy.
We consider the diamond chain with S=1/2 XYZ vertical dimers which interact with the intermediate sites via the interaction of the Ising type. We also suppose all four spins form the diamond-shaped plaquette to have different g-factors. The non-uniform g-factors within the quantum spin dimer as well as the XY-anisotropy of the exchange interaction lead to the non-conserving magnetization for the chain. We analyze the effects of non-conserving magnetization as well as the effects of the appearance of negative g-factors among the spins from the unit cell. A number of unusual frustrated states fo…
Spin-1/2 orthogonal-dimer chain composed of regularly alternating Ising and Heisenberg dimers is exactly solved in a presence of the magnetic field by the transfer-matrix method. It is shown that the ground-state phase diagram involves in total six different phases. Besides the ferromagnetic phase with fully polarized spins one encounters the singlet antiferromagnetic and modulated antiferromagnetic phases manifested in zero-temperature magnetization curves as zero magnetization plateau, the frustrated ferrimagnetic and singlet ferrimagnetic phases causing existence of an intermediate one-half…
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 investigate roughening transitions in the context of transverse-field Ising models. As a modification of the transverse Ising model with short range interactions, which has been shown to exhibit domain wall roughening, we have looked into the possibility of a roughening transition for the case of long-range interactions, since such a system is physically realized in the insulator LiHoF4. The combination of strong Ising anisotropy and long-range forces lead naturally to the formation of domain walls but we find that the long-range forces destroy the roughening transition.
The physical origin of the backbendings in the equations of state of finite but not necessarily small systems is studied in the Ising model with fixed magnetization (IMFM) by means of the topological properties of the observable distributions and the analysis of the largest cluster with increasing lattice size. Looking at the convexity anomalies of the IMFM thermodynamic potential, it is shown that the order of the transition at the thermodynamic limit can be recognized in finite systems independently of the lattice size. General statistical mechanics arguments and analytical calculations sugg…
In this paper we show how an infinite system of coupled Toda-type nonlinear differential equations derived by one of us can be used efficiently to calculate the time-dependent pair-correlations in the Ising chain in a transverse field. The results are seen to match extremely well long large-time asymptotic expansions newly derived here. For our initial conditions we use new long asymptotic expansions for the equal-time pair correlation functions of the transverse Ising chain, extending an old result of T.T. Wu for the 2d Ising model. Using this one can also study the equal-time wavevector-depe…
The ground state and the thermodynamics of a spin-1/2 asymmetric diamond Ising--Heisenberg chain are considered. For the $XYZ$ anisotropic Heisenberg interaction, the exact calculations of the free energy, entropy, heat capacity, magnetization, and magnetic susceptibility are performed using the method of decoration-iteration transformation. In the case of antiferromagnetic interactions (Ising and $XXZ$ anisotropic Heisenberg ones), the ground state, magnetization process, temperature dependence of the magnetization, magnetic susceptibility, and heat capacity are investigated. The influence of…
We investigate the influence of aperiodic perturbations on the critical behaviour at a second order phase transition. The bond and site problems are compared for layered systems and aperiodic sequences generated through substitution. In the bond problem, the interactions between the layers are distributed according to an aperiodic sequence whereas in the site problem, the layers themselves follow the sequence. A relevance-irrelevance criterion introduced by Luck for the bond problem is extended to discuss the site problem. It involves a wandering exponent for pairs, which can be larger than th…
We consider the transverse Ising model in one dimension with nearest-neighbour interaction and calculate exactly the longitudinal spin-spin correlation for a class of excited states. These states are known to play an important role in the perturbative treatment of one-dimensional transverse Ising model with frustrated second-neighbour interaction. To calculate the correlation, we follow the earlier procedure of Wu, use Szego's theorem and also use Fisher-Hartwig conjecture. The result is that the correlation decays algebraically with distance ($n$) as $1/\surd n$ and is oscillatory or non-osci…
The magnetization process of the spin-1/2 antiferromagnetic XXZ model with Ising-like anisotropy in the ground state is investigated. We show numerically that the Ising-like XXZ models on square and cubic lattices show a first-order phase transition at some critical magnetic field. We estimate the value of the critical field and the magnetization jump on the basis of the Maxwell construction. The magnetization jump in the Ising-limit is investigated by means of perturbation theory. Based on our numerical results, we briefly discuss the phase diagram of the extended Bose-Hubbard model in the ha…
The standard Metropolis algorithm and the parallel tempering method are used to examine magnetization processes in the Ising model with the long-range RKKY interaction on the Shastry-Sutherland lattice. It is shown that the Ising model with RKKY interaction exhibits, depending on the value of the Fermi wave vector $k_F$, the reach spectrum of magnetic solutions, which is manifested in the appearance of new magnetization plateaus on the magnetization curve. In particular, we have found the following set of individual magnetization plateaus with fractional magnetization $m/m_s$=1/18, 1/9, 1/8, 1…
We study the zero temperature coarsening dynamics in an Ising chain in presence of a dynamically induced field that favors locally the `-' phase compared to the `+' phase. At late times, while the `+' domains still coarsen as $t^{1/2}$, the `-' domains coarsen slightly faster as $t^{1/2}\log (t)$. As a result, at late times, the magnetization decays slowly as, $m(t)=-1 +{\rm const.}/{\log (t)}$. We establish this behavior both analytically within an independent interval approximation (IIA) and numerically. In the zero volume fraction limit of the `+' phase, we argue that the IIA becomes asympt…
We study the surface magnetization of aperiodic Ising quantum chains. Using fermion techniques, exact results are obtained in the critical region for quasiperiodic sequences generated through an irrational number as well as for the automatic binary Thue-Morse sequence and its generalizations modulo p. The surface magnetization exponent keeps its Ising value, beta_s=1/2, for all the sequences studied. The critical amplitude of the surface magnetization depends on the strength of the modulation and also on the starting point of the chain along the aperiodic sequence.
In this paper we study the critical behavior of the two-dimensional antiferromagnetic Ising model in both uniform longitudinal ($H$) and transverse ($Ω$) magnetic fields. Using the effective-field theory (EFT) with correlation in single site cluster we calculate the phase diagrams in the $H-T$ and $Ω-T$ planes for the square ($z=4$) lattices. We have only found second order phase transitions for all values of fields and reentrant behavior was not observed.
