Thermodynamics of some layered Ising models

Thermodynamic properties of some inhomogeneous Ising models with layered distribution of couplings are studied. In particular, the specific heat is investigated in detail, both analytically and numerically. It is shown that different ordering mechanisms, namely ordering of finite regions and global ordering of infinite range, can occur in different temperature ranges. This results in remarkable structures of the specific heat curves. In addition we investigate the case where the random distribution of couplings extends over an infinite distance in one space direction. The ordinary Ising singularity then changes to one of infinite order provided the transition temperature remains finite.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köln     

Ising models with two-and three-spin interactions

Three Ising models with two- and three-spin interactions are shown to be exactly solvable. Their critical behaviour: non-universal dependence of the critical exponents on the interaction parameters in one case, absence of finite critical temperature in the second case and logarithmic singularity of the specific heat in the third one, is different from the recent conjecture that such models belong to the Baxter-Wu universality class.     

Slow relaxation in microcanonical warming of a Ising lattice

We study the warming process of a semi-infinite cylindrical Ising lattice initially ordered and coupled at the boundary to a heat reservoir. The adoption of a proper microcanonical dynamics allows a detailed study of the time evolution of the system. As expected, thermal propagation displays a diffusive character and the spatial correlations decay exponentially in the direction orthogonal to the heat flow. However, we show that the approach to equilibrium presents an unexpected slow behavior. In particular, when the thermostat is at infinite temperature, correlations decay to their asymptotic values by a power law. This can be rephrased in terms of a correlation length vanishing logarithmically with time. At finite temperature, the approach to equilibrium is also a power law, but the exponents depend on the temperature in a non-trivial way. This complex behavior could be explained in terms of two dynamical regimes characterizing finite and infinite temperatures, respectively. When finite sizes are considered, we evidence the emergence of a much more rapid equilibration, and this confirms that the microcanonical dynamics can be successfully applied on finite structures. Indeed, the slowness exhibited by correlations in approaching the asymptotic values are expected to be related to the presence of an unsteady heat flow in an infinite system.     

Quenched bond-dilute Ising ferromagnet in square lattice: Thermodynamical properties

Within an effective field framework which substantially improves the Molecular Field Approximation, we calculate the phase diagram, magnetization, specific heat and susceptibility associated with the quenched bond-dilute Ising ferromagnet in square lattice. The results are qualitatively (and within certain extent quantitatively) satisfactory; in particular the effects, on the specific heat and susceptibility, of the (eventually) coexisting finite and infinite clusters are exhibited.     

Scaling theory and finite systems

A renormalization group transformation is introduced with the help of which critical properties of infinite systems can be related to finite systems. As a numerical example the method is applied to the two-dimensional Ising model. The critical point and critical point exponent are computed in addition to the amplitude of the logarithmic singularity in the specific heat.     

Ising Model on an Infinite Ladder Lattice

In this paper we propose an Ising model on an infinite ladder lattice, which is made of two infinite Ising spin chains with interactions. It is essentially a quasi-one-dimessional Ising model because the length of the ladder lattice is infinite, while its width is finite. We investigate the phase transition and dynamic behavior of Ising model on this quasi-one-dimessional system. We use the generalized transfer matrix method to investigate the phase transition of the system. It is found that there is no nonzero temperature phase transition in this system. At the same time, we are interested in Glauber dynamics. Based on that, we obtain the time evolution of the local spin magnetization by exactly solving a set of master equations.     

Ising Model on an Infinite Ladder Lattice

In this paper we propose an Ising model on an infinite ladder lattice, which is made of two infinite Ising spin chains with interactions. It is essentially a quasi-one-dimessional Ising model because the length of the ladder lattice is infinite, while its width is finite. We investigate the phase transition and dynamic behavior of Ising model on this quasi-one-dimessional system. We use the generalized transfer matrix method to investigate the phase transition of the system. It is found that there is no nonzero temperature phase transition in this system. At the same time, we are interested in Glauber dynamics. Based on that, we obtain the time evolution of the local spin magnetization by exactly solving a set of master equations.     

Bose–Einstein condensation in the three-sphere and in the infinite slab: Analytical results

We study the finite size effects on Bose–Einstein condensation (BEC) of an ideal non-relativistic Bose gas in the three-sphere (spatial section of the Einstein universe) and in a partially finite box which is infinite in two of the spatial directions (infinite slab). Using the framework of grand-canonical statistics, we consider the number of particles, the condensate fraction and the specific heat. After obtaining asymptotic expansions for large system size, which are valid throughout the BEC regime, we describe analytically how the thermodynamic limit behaviour is approached. In particular, in the critical region of the BEC transition, we express the chemical potential and the specific heat as simple explicit functions of the temperature, highlighting the effects of finite size. These effects are seen to be different for the two different geometries. We also consider the Bose gas in a one-dimensional box, a system which does not possess BEC in the sense of a phase transition even in the infinite volume limit.     

Inhomogeneous Ising models with diagonal structure on a square lattice

We study inhomogeneous Ising models on a square lattice. The nearest neighbour couplings are allowed to be of arbitrary strength and sign such that the coupling distribution is translationally invariant in diagonal direction. We calculate the partition function and free energy for a random coupling distribution of finite period. The phase transition is universally of Ising type. The transition temperature is independent of specific details of the coupling distribution. In particular, unexpected results for the absence of a phase transition are derived. Special examples are considered in detail, phase diagrams and critical temperature are determined. We calculate ground state energy and ground state degeneracy or, equivalently, rest entropy for “pure” frustration models, i.e. models with couplings of fixed strength but arbitrary sign, which never show a phase transition at a finite temperature.     

Spin-spin correlations in ferromagnetic nanosystems

Using exact diagonalization, Monte-Carlo, and mean-field techniques, characteristic temperature scales for ferromagnetic order are discussed for the Ising and the classical anisotropic Heisenberg model on finite lattices in one and two dimensions. The interplay between nearest-neighbor exchange, anisotropy and the presence of surfaces leads, as a function of temperature, to a complex behavior of the distance-dependent spin-spin correlation function, which is very different from what is commonly expected. A finite experimental observation time is considered in addition, which is simulated within the Monte-Carlo approach by an incomplete statistical average. We find strong surface effects for small nanoparticles, which cannot be explained within a simple Landau or mean-field concept and which give rise to characteristic trends of the spin-correlation function in different temperature regimes. Unambiguous definitions of crossover temperatures for finite systems and an effective method to estimate the critical temperature of corresponding infinite systems are given.     

Multicanonical sampling of the space of states of ℋ(2, <Emphasis Type="Italic">n</Emphasis>)-vector models

Problems of temperature behavior of specific heat are solved by the entropy simulation method for Ising models on a simple square lattice and a square spin ice (SSI) lattice with nearest neighbor interaction, models of hexagonal lattices with short-range (SR) dipole interaction, as well as with long-range (LR) dipole interaction and free boundary conditions, and models of spin quasilattices with finite interaction radius. It is established that systems of a finite number of Ising spins with LR dipole interaction can have unusual thermodynamic properties characterized by several specific-heat peaks in the absence of an external magnetic field. For a parallel multicanonical sampling method, optimal schemes are found empirically for partitioning the space of states into energy bands for Ising and SSI models, methods of concatenation and renormalization of histograms are discussed, and a flatness criterion of histograms is proposed. It is established that there is no phase transition in a model with nearest neighbor interaction on a hexagonal lattice, while the temperature behavior of specific heat exhibits singularity in the same model, in case of LR interaction. A spin quasilattice is found that exhibits a nonzero value of residual entropy.     

Competition between Ising spins and classical n-vector spins on the honeycomb and the diced lattice

The competition between ordering and disordering is investigated for mixed spin models of Ising spins and classical n-vector spins on the honeycomb and the diced lattice. The critical indices of the specific heat and the spontaneous magnetization for the mixed spin models turn out to be the same as those for the two-dimensional Ising model.     

Point defects in the honeycomb Ising lattice

A plane isotropic honeycomb Ising lattice is considered with randomly distributed defects, namely missing lattice spins (including the three adjacent bonds). The impurities are in thermodynamic equilibrium through a chemical potential. We find a rescaled temperature and a finite cusp-like specific heat at the critical point.     

Asymptotic behavior of fluctuations for the 1D Ising model in zero-temperature limit

Fluctuation of the average spin for one-dimensional Ising spins with nearest neighbor interactions are studied. The distribution function for the average spin is calculated for a finite volume, finite temperature, and finite magnetic field. As the volume increases and the temperature diminishes at zero magnetic field, there are two limits in which the probability distribution shows quite different behaviors: in the thermodynamic limit as the volume goes to infinity for finite temperature, small deviations of the fluctuations are described by a Gaussian distribution, and in the limit as the temperature vanishes for a finite volume, the ground states are realized with probability one. The crossover between these limits is analyzed via a ratio of the correlation length to the volume. The helix-coil transition in a polypeptide is discussed as an application.     

Microcanonical Phase Diagram of the BEG and Ising Models

The density of states of long-range Blume-Emery-Griffiths (BEG) and short-range Ising models are obtained by using Wang-Landau sampling with adaptive windows in energy and magnetization space. With accurate density of states, we are able to calculate the microcanonical specific heat of fixed magnetization introduced by Kastner et al. in the regions of positive and negative temperature. The microcanonical phase diagram of the Ising model shows a continuous phase transition at a negative temperature in energy and magnetization plane. However the phase diagram of the long-range model constructed by peaks of the microcanonical specific heat looks obviously different from the Ising chart.     

Microcanonical Phase Diagram of the BEG and Ising Models

The density of states of long-range Blume-Emery-Griffiths(BEG) and short-range Ising models are obtained by using Wang-Landau sampling with adaptive windows in energy and magnetization space.With accurate density of states,we are able to calculate the microcanonical specific heat of fixed magnetization introduced by Kastner et al.in the regions of positive and negative temperature.The microcanonical phase diagram of the Ising model shows a continuous phase transition at a negative temperature in energy and magnetization plane.However the phase diagram of the long-range model constructed by peaks of the microcanonical specific heat looks obviously different from the Ising chart.     

A new comprehensive study of the 3D random-field Ising model via sampling the density of states in dominant energy subspaces

The three-dimensional bimodal random-field Ising model is studied via a new finite temperature numerical approach. The methods of Wang-Landau sampling and broad histogram are implemented in a unified algorithm by using the N-fold version of the Wang-Landau algorithm. The simulations are performed in dominant energy subspaces, determined by the recently developed critical minimum energy subspace technique. The random-fields are obtained from a bimodal distribution, that is we consider the discrete (±Δ) case and the model is studied on cubic lattices with sizes 4≤L ≤20. In order to extract information for the relevant probability distributions of the specific heat and susceptibility peaks, large samples of random-field realizations are generated. The general aspects of the model's scaling behavior are discussed and the process of averaging finite-size anomalies in random systems is re-examined under the prism of the lack of self-averaging of the specific heat and susceptibility of the model.     

Finite-size behaviour of the two-dimensional ANNNI model

The two-dimensional axial next-nearest neighbour Ising (ANNNI) model of finite size with periodic boundary conditions is studied by the Monte Carlo method. The model shows an interesting finite size dependence in connection with its oscillatory correlations pretending for finite systems a Lifshitz point in one part of the phase diagram, while the infinite system appears to display one in another part of the phase diagram.     

An Ising ferromagnet with discontinuous long-range order

An infinite one-dimensional Ising ferromagnetM with long-range interactions is constructed and proved to have the following properties. (1)M has an order-disorder phase transition at a finite temperature. (2) Any Ising ferromagnet of the same structure asM, but with interactions tending to zero with distance more rapidly than those ofM, cannot have a phase-transition. (3) The long-range-order parameter (thermal average of the spin-spin correlation at infinite distance) jumps discontinuously from zero in the disordered phase to a finite value in the ordered phase. All three properties have been conjectured by Anderson and Thouless to hold for a particular Ising ferromagnet which is relevant to the theory of the Kondo effect. AlthoughM is not identical to Anderson's model, the results proved forM support the validity of the physical arguments of Anderson and Thouless.     

单壁碳纳米管声子谱及比热计算

利用卷曲法计算了有限长单壁碳纳米管的声子色散关系．讨论了单壁碳纳米管的比热随管径、温度的变化趋势．结果表明碳管的比热随温度、管径的增大而增大,并逐渐趋于一恒定数值．根据色散关系的计算结果,给出了有限长（5,5）型单壁碳纳米管的振动模式以及部分振动模式的频率随长度的变化关系． 关键词： 碳纳米管 声子 比热