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Table of Contents


Appendix A
Magnetism and the Ising Spin-Glass Model

A.1 General

Materials in which a state of magnetism can be induced are called magnetic materials. When magnetized, such materials cause a magnetic field (force) in the surrounding space.

The molecules of a material contain electrons which orbit around their nuclei and spin about their axes. Electronic motion equivalently represents an electric current with an associated magnetomotive force. That is, a magnetic material consists of a set of atomic level magnets arranged on a regular lattice which represents the crystalline structure of the material. In most molecules each magnetomotive force is neutralized by an opposite one, rendering the material nonmagnetic. However, in certain materials (such as iron), there is a resultant unneutralized magnetomotive force which constitutes a magnetic dipole. These dipoles are characterized with dipole moments due to three angular momenta, namely, orbital angular momentum, electron-spin angular momentum, and nuclear-spin angular momentum. The atomic magnets represent the associated ±1/2 spin atoms in which the angular spin is restricted to two distinct directions and represented in the so-called Ising model by Si = ±1 at a lattice site i.

In a magnetic material, the dipoles line up parallel with an externally applied magnetization field hext and internal field produced by other spins. When the dipoles remain aligned despite of removing external magnetization, the material is said to have acquired permanent magnetism. The readiness of a magnetic material to accept magnetism is termed as its permeability (μM).

In the process of magnetization, the dipole moment experienced per unit volume quantifies the extent of magnetization (of the medium) and is denoted by MM. The manner in which magnetic dipoles are arranged classifies the materials as paramagnetic, ferromagnetic, antiferromagnetic, and ferrimagnetic, as illustrated in Figure A.1.

With reference to the arrangements of Figure A.1, the nature of dipole interactions can be described qualitatively as follows:

Paramagnetic Disordered interaction with zero resultant magnetization.
Ferromagnetic Ordered strong interaction with a large resultant magnetization.
Antiferromagnetic Ordered interaction, but with zero resultant magnetization.
Ferrimagnetic Ordered interaction, with relatively weak resultant magnetization.


Figure A.1  Magnetic spin arrangements

A.2 Paramagnetism

Paramagnetism is a noncooperative magnetism arising from spontaneous moments, the orientation of which is randomized by the thermal energy kBT. The magnetic order is thus the net alignment achieved by the applied field in the face of the thermal disarray existing at the ambient temperature. The extent of magnetization is the time-average alignment of the moments, inasmuch as the orientations fluctuate constantly and are in no sense fixed. Ideal paramagnetism is characterized by a susceptibility which varies inversely proportional to temperature (Curie law) and the constant of proportionality is called the Curie constant.

At any given lattice location, the internal field pertinent to an atomic magnet due to the other (neighboring) spins is proportional to its own spin. Thus, the total magnetic field at the site i is given by:

where the summation refers to the contribution from all the neighboring atoms and the coefficient Jij measures the strengths of influence of spin Sj on the field at Si and is known as exchange interaction strengths.

(Another noncooperative magnetism known as diamagnetism is characterized by a negative temperature-dependent magnetic susceptibility.)

A.3 Ferromagnetism

As indicated earlier, ferromagnetism results from an ordered interaction. it is a cooperative magnetism with a long-range collinear alignment of all the moments, manifesting as the prevalence of magnetization even with no external magnetic field (spontaneous magnetization). This cooperative magnetism becomes prominent and dominates below a discrete temperature called Curie temperature (TC), and in spin-glass theory it is known as spin-glass temperature (TG).

A.4 Antiferromagnetism

Like ferromagnetism, this is also a cooperative process with a long-range order and spontaneous moments. The exchange coupling dominates the constraints on the moments in the cooperative process, and the exchange parameter for an antiferromagnetism is negative. That is, for a given direction of the moment of an ion, there is a neighboring ion with its moment pointing exactly in the opposite direction. Hence, there is no overall spontaneous magnetization. The transition temperature in this case is known as Néel temperature (TN).

A.5 Ferrimagnetism

This magnetism differs from the other types, as it involves two or more magnetic species that are chemically different. The species can have ferro- or anti ferromagnetic alignment; however, if the species with ferromagnetic moment dominate there would be a resultant spontaneous magnetization. The temperature dependence of ferrimagnetism is qualitatively similar to that of ferromagnetism.

A.6 Magnetization

As described earlier, a permanently magnetized sample say, of iron, typically contains a number of domains with the directions of magnetization being different in neighboring domains. The extent of magnetization MS that exists in the sample in the absence of any applied magnetic field is called spontaneous magnetization (more commonly known as retentivity or remnant magnetism in engineering). Its magnitude depends on the temperature in the manner sketched in Figure A.2.

The direction in which the spontaneous magnetization is rendered usually lies along one of several easy axes defined by the crystal structure of the material. Upon heating to the critical or Curie temperature TC, the spontaneous magnetism vanishes continuously. The immediate neighborhood of TC is called the critical region and where βC varies rather little from one ferromagnetic material to another and is typically about 1/3.


Figure A.2  Spontaneous magnetization versus temperature TC: Curie point

The property of the material in the neighborhood of the critical point is known as critical phenomenon which is substantially independent of the detailed microscopic constitution of the system considered.

The forces which tend to align the spins in a ferromagnet have electrostatic and quantum mechanical origins. The phase transition which occurs at the Curie temperature in zero magnetic field can be described as an order-disorder transition. In the high temperature (paramagnetic) phase, one in which the fluctuations in the orientation of each spin variable are entirely random so that the configurational average is zero, the state is a disordered one. In the ferromagnetic phase, on the other hand, inasmuch as the spins point preferably in one particular direction, the state is said to be ordered.

The interaction of the spins in a magnetic system dictates implicitly the minimum energy considerations which can be mathematically specified by a Hamiltonian. That is, considering the potential energy corresponding to Equation (A.1), it can be specified by a Hamiltonian HM given by:

A.7 Spin-Glass Model [116]

The phenomenon of ferromagnetism was modeled by Ising in 1925 [35] as due to the interaction between the spins of certain electrons in the atom making up a crystal. As mentioned earlier, each particle of the crystal is associated with a spin coordinate S. Assuming the particles are rigidly fixed in the lattice and either neglecting the vibrations of the cyrstal or assuming that they act independently of the spin configuration (so that they can be treated separately), the spin is modeled as a scalar quantity (instead of a vector) which can assume two dichotomous levels, namely, S = ±1 (+1 refers to the up spin and -1 is the down spin). The interaction between two particles located at the ith and jth lattice points is postulated as Eij = -JijSiSj if i and j are nearest-neighbors; otherwise Eij = 0. This postulation assumes only nearest-neighbor interaction. That is, the energy is -J if the nearest neighbors have the same spin and +J if they have unlike spins; and the zero energy corresponds to the average of these two. Thus the parameter J is a measure of the strength of i to j coupling or the exchange parameter determined by the physical properties of the system. It is positive for a ferromagnetic system and negative for an antiferromagnetic system.

In the Ising model, it is further postulated that the particle can interact with an external magnetic field as well. Denoting the magnetic moment mM assigned to a lattice point for an external magnetic field hM applied, the associated energy of interaction of the ith particle can be written as EM = -mMhMSi. The corresponding thermodynamics of the system with M lattice points can be specified by a partition function * as:


*The definition and details on partition function are presented in Section.A.13.

where aij = 1 if i and j are nearest neighbors, otherwise aij = 0, and further kB is the Boltzmann constant and T is the temperature.

In terms of this partition function, the internal energy Eint per particle and the magnetization MM per particle can be written as:

and

where

The Ising partition function indicated above quantifies the scalar spin interaction which does not depend on the spin orientation with the lattice distance being fixed.

Although the Ising model is not considered as a very realistic model for ferromagnetism, it is considered as a good emulation of a binary substitutional alloy as well as an interesting model of a gas or liquid.

A.8 The Hamiltonian

The simplest model for a d-dimensional magnetic system is the Heisenberg Hamiltonian (in the absence of external field) given by:

where Si, Sj, etc. are the m-component spin vectors. When m = 1, the corresponding model is the Ising model. These vectors at the positions ri and rj, etc. (in the d-dimensional space) interact with each other, the strength of such interaction(s) being Jij as mentioned before. In a general case, Jij depends on ri, rj, etc. The exchange interaction Jij on the right-hand side of Equation (A.7) is called short-ranged or long-ranged interaction depending on whether .

Commensurate with spin theory, the spontaneous magnetization is a measure of the long-range orientational (spin) order in space. It is an order parameter that distinguishes the ferromagnetic phase from the paramagnetic phase or the so-called phase transition.

A.9 Bonds and Sites

In a magnetic system controlled by the spins, the “disorder” discussed earlier can be introduced either at the “bonds” or at the “sites”. The bond-random model assumes the exchange bond strengths as independent random variables with a specific probability density function (pdf). For example, when the exchange interaction is such that each spin is assumed to interact with every other spin in the system (specified as the lattice coordination number tending to infinity), the corresponding pdf of Jij is shown to be gaussian.

In the site-disorder models, the randomness is due to a finite (nonzero) fraction of sites being occupied (randomly) by spins and the remaining sites being occupied by nonmagnetic atoms/molecules. The state of the disorder arises due to the finite temperature. That is, thermal fluctuations tend to flip the spins from down to up or from up to down, and thus upset the tendency of each spin to align with its field. At the so-called absolute zero temperature (-273° C or 0° K), such thermal fluctuations vanish. Conventionally, the effect of thermal fluctuations in an Ising model is depicted by Glauber dynamics with a stochastic rule:

where the pdf depends on the temperature (T) with difference functional choices. Usually the following sigmoidal (S-shaped) function is considered for p(hi):

where β = 1/kBT with kB = 1.38 × 10-16 erg/K and is known as the Boltzmann constant. The dynamic rule of symmetry for the state Si can be written as:

The factor β (or the temperature T) controls the steepness of the sigmoid near h = 0 as shown in Figure A.3.


Figure A.3  Glauber’s rule of symmetry for the state Si

A.10 The Hard Spin

The Ising spin is referred to as hard, as the spins can have only fixed (finite) values, namely +1 and -1. In other words, the spins are characterized by a weighting function specified by:

A.11 Quenching and Annealing

The disorder in a magnetic spin system can be classified as either quenched or annealed. Such a classification arises from the spatial ergodic property, namely, the physical properties of a macroscopically large system are identical with the same property averaged over all possible spatial configurations. Suppose a large sample is divided into a large number of smaller subunits (each of which is statistically large) in such a way that each of these subunits can be regarded as an independent number of ensemble systems characterized by a random distribution; and, as each of the subunits is statistically large, surface interaction can be assumed as negligible.

If the original system is annealed, it refers to each of the subunits acquiring all possible configurations in them because positions of the impurities are not frozen. Suppose the original system is quenched. Neglecting surface interactions, there is a frozen rigidity meaning that the positions of the impurities are clamped down. The result is that the characteristics of the quenched systems could only be the superposition of the corresponding characteristics of the subunits. The spin-glass systems have the common features of the quenched disorder.

A.12 Frustration

The interactions between spins are normally in conflict with each other leading to frustration. For example, the spin arrangement in a unit of square lattice (called a plaquette) can be of two possibilities as illustrated in Figure A.4.


Figure A.4  Plaquettes of a spin system
A: Unfrustrated interaction; B: Frustrated interaction; +, -: Two opposing spins

The individual bound energies are minimized if the two spins connected by an arbitrary bond <ij> are parallel to each other for + sign and antiparallel to each other for - sign. In plaquette A, all the bond energies can be minimized simultaneously whereas the same is not possible in plaquette B. Hence, plaquette B is called frustrated. In other words, those plaquettes where topological constraints prevent the neighboring spins from adopting a configuration with bond energy minimized are called frustrated. The extent of frustration is quantified by a frustration function dictated by the product of bond strengths taken over a closed contour of connected bonds.

A.13 Partition Function

In a physical system with a set of ν states, each of which has an energy level φν, at a finite temperature (T > 0) the fluctuation of states attains a thermal equilibrium around a constant value. Under this condition each of the possible states v occurs with a probability Pν = (1/Z)exp(-φν/kBT) where Z is a normalization factor equal to Σνexp(-φν/kBT). That is associated with discrete states φν(ν = 1,2,...) (each of which occurs with a probability under thermal equilibrium), a controlling function which determines the average energy known as the sum of the states or the partition function is defined by:

In the presence of an applied magnetic field (mMhM), the partition function corresponding to the Ising model can be specified via Equations (A.9) and (A.11) as:

where Σi is taken over all spins and Σ<ik> is taken over all pairs of direct neighbors. Further, is over the 2M combinations of the M spins. The associated energy E and magnetic moment mM can be specified in terms of Z* as:

The relations specified by Equations (A.12), (A.13), and (A.14) are explicitly stated in the thermodynamics viewpoint with β = kBT in Equations (A.3), (A.4), and (A.5), respectively.

A.14 Ising Model: A Summary

Let a string of M identical units, numbered as 1, 2, 3, ..., (M-1), M each identified with a state variable x1, x2, x3, ..., xM represent a one-dimensional interacting system. A postulation of nearest-neighbor interaction is assumed which specifies that each unit interacts with its two direct neighbors only (see Figure. A.5).


Figure A.5  A string of M cooperative interacting units

Let the interaction between two neighbors be φ(x, y). The corresponding probability for a given state of the system is proportional to the Boltzmann potential, namely, exp[-β{φ(x1, x2) + φ(x2, x3) + ... + φ(xM, x1)}] so that the corresponding partition function can be written as a summing (or integration) function as defined in Equation (A.11). That is:

On the basis of the reasoning from probability calculus, the following eigenvalue problem can be associated with the summing function:

where λ has a number of different eigenvalues λv, to each of which there belongs one eigenvector av. Also, an orthogonality relation for the a’s prevails, given by:

Hence, the following integral equation can be specified:

Using the above relations, Z reduces to:

and

where λ1 is the largest eigenvalue.

In the so-called one-dimensional Ising model as stated earlier, the variable x is the spin S which is restricted to just the two dichotomous values ±1. The interaction φ(x, y) is therefore a matrix. In the case of a linear array of spins forming a closed loop (Figure A.5), the interaction φ(x, y) simplifies to:

where J refers to the exchange coupling coefficient.

The Ising Hamiltonian with hM = 0 has a symmetry; that is, if the sign of every spin is reversed, it remains unchanged. Therefore, for each configuration in which a given spin Si has the value +1, there is another one obtained by reversing all the spins, in which case it has the value -1; and both configurations have, however, the same statistical weight. The magnetization per spin is therefore zero, which is appparently valid for any temperature and for any finite system.

Thus, within the theoretical framework of the Ising model, the only way to obtain a nonzero, spontaneous magnetization is to consider an infinite system, that is, to take the thermodynamic limit (see Figure A.6).

Considering single spins which can be flipped back and forth randomly between the dichotomous values ±1 in a fixed external magnetic field h = hext, the average magnetization refers to average of S given by <S> = prob(+1).(+1) + prob(-1).(-) which reduces to tanh(Λh) via Equation (A.9a).

Further considering many-spins, the fluctuating values of hi at different sites can be represented by a mean value <hi> = Σj Jij <Si> + hext. That is, the overall scenario of fluctuating many-spins is focused into a single average background field. This mean field approximation becomes exact in the limit of infinite range interactions, where each spin interacts with all the others so that the principle of central limit theorem comes into vogue.

A.15 Total Energy

Considering the Ising model, the state of the system is defined by a configuration of + (up) and - (down) spins at the vertices of a square lattice in the plane.

Each edge of the lattice in the Ising model is considered as an interaction and contributes an energy φ(S, S’) to the total energy where S, S’ are the spins at the ends of the edge. Thus in the Ising model, the total energy of a state σ is:

The corresponding partition function is then defined by (as indicated earlier):

A.16 Sigmoidal Function

The spin-glass magnetization (Figure A.6) exhibits a distinct S-shaped transition to a state of higher magnetization. This S-shaped function as indicated earlier is referred to as the sigmoidal function. In the thermodynamic limiting case for an infinite system at T < TC, the sigmoidal function tends to be a step-function as illustrated. In the limit, the value of MM at h = 0 is not well defined, but the limit of MM as h → 0 from above or below the value zero is ±MS(T).


Figure A.6  Magnefization of the one-dimensional Ising chain as a function of the magnetic field (a) Finite system (T > Tc); (b) Infinite system (T < Tc)

(The method of calculat.ing the actual values of magnetization under thermodynamic considerations is dimension-dependent. For the one-dimensional case, there is no ferromagnetic state. For the two-dimensional case, the solutions at zero magnetic field due to Onsager and for a nonzero field by Yang are available. For three dimensions, no exact solution has been found; however, there are indications of a possible ferromagnetic state.)

A.17 Free Energy

Pertinent to N atoms, the partition function is summed over all possible states, all the 2N combinations of the spins, Si = ±1. That is:

The multiple summation over all states is known as the trace TΣ. Hence the average activation <Si> of limit i is given by:

For Z = (hi):

By defining a free-energy term as FE = -(kBT)log(Z), Equadon (A.26) reduces to:

and the correlation function <SiSj> becomes:

This is useful in deriving the Boltzmann machine algorithm. Free energy is like an exponentially weighted sum of energies. That is exp(-FE/kBT) = Z = Σν exp(-βφν) and exp(-FE/kBT)/Z = Σν exp(-φνT)/Z = Σν pν = 1 depicts the sum of the probability of the states (which is just 1).

A.18 Entropy

The difference between average energy <φν> and the free energy FE is given by:

The above expression, except for the kBT term depicts a quantity called entropy . That is:

and in terms of , the free-energy can be written as

The entropy refers to:

  Width of the probability distribution pν.
  Larger corresponding to more states ν that have appreciable probability.
  Average amount of additional information required to specify one of the states; or larger entropy, to the more uncertain of the actual state ν.


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