**7. Defects**

Although there is criticism on the validaties of the Landau Fermi-liquid concept (v. Löhneysen et al., 2006), a new insight for dealing macroscopic physical observables, *e.g.*, even domain (grain) boundaries and dislocations, combined with the concept of symmetry breaking (Anderson, 1997). Since the semi-phenomenological construction of the Ladau Fermi-liquid theory, it is possible to switch our intersts from the quasiparticle space to the order parameter space. This transformation to the order parameter space is familiar with that of Landau theory of phase transition. Toulouse & Kléman (1976) suggested an innovative idea that the defects in ordered media can be classified in terms of the *topological defects* of the corresponding order parameter. This topological concept was immediately applied to the vortex textures in superfluid 3He for showing that the textures with vorticity without vortex core can be constructed (Anderson & Toulouse, 1977). One of the most striking application of the topological defects theory may be the explanation of the entanglement free dislocation crossings (Poénaru & Toulouse, 1977); dislocations can be entangled if and only if they are members of separate class of *π*<sup>1</sup> (*R*). We will see the meaning of *π*<sup>1</sup> (*R*) later. A pedagogical review on the topological defects was given by Mermin (1979). Recenlty, Kleman & Friedel (2008) reviewed the practical interpretations of the homotopy group theory of the topological disclination, dislocations, and continuous defects.

### **7.1 Topological defects**

In a periodic solid, the elastic variable is the vector displacement field **u** (**x**). Changes of **u** by a Bravais lattice vector **R** leave the lattice unchanged. Thus the order parameter space for the solid is the three-dimensional space of displacements **u** with lattice points. A dislocation with a core along **l** is characterized by<sup>10</sup>

$$\oint \mathbf{du} = \oint\_{\mathcal{C}} \frac{\mathbf{du}}{\mathbf{ds}} \mathbf{ds} = \mathbf{R} \equiv \mathbf{b},\tag{206}$$

where *C* is a curve enclosing the core and **b** is called the *Burgers vector*. The construction of the Burgers vector is fully topological, but there are two nontopological descriptions; screw dislocations if **b** � **l** and edge dislocations if **b** ⊥ **l**. Since the Bravais lattice vector, which leaves after a single winding curve *C* passed to enclose the core, is discrete, the dislocation is, from homotopy group theory, *line*.

Let us look at the characters of dislocations with little rigor (Mermin, 1979). Suppose that *R* is an order-parameter space. There is a map *f* (*z*) of interval 0 ≤ *z* ≤ 1 into *R* defining the real-space loop or circles into order-parameter space to give closed curves in order-parameter space itself. A continuous directed curve described by maps *f* passing through a base point *x* in *R*. The close curve is decribed by *f* (0) = *f* (1) = *x*. Two loops *f* and *g* are *homotopic at x*, if there is a continous family of loops, all passing through *x*. The classes of homotopic loops at *x* form a group, called *π*<sup>1</sup> (*R*, *x*), and is known as the *fundamental group* of *R* at *x*. There are other groups associated with the mappings of the surface of an *n* sphere *Sn* in Euclidean *n* + 1 space

<sup>10</sup> Here, we follow the definition by Chaikin & Lubensky (1995).

38 Will-be-set-by-IN-TECH

magnons. All those quasiparticles should interact each other, in contrast to the typical considerations that quasiparticles hardly interacts each other. Lee et al. (2006) confirmed that there are interactions between phonon and magnetism, and Bodraykov (2007) in his phenomenological model reproduced the invar effect of the temperature dependent thermal expansion coeffient by considering the results of the interacting phonon-magnon (Kim, 1982).

Although there is criticism on the validaties of the Landau Fermi-liquid concept (v. Löhneysen et al., 2006), a new insight for dealing macroscopic physical observables, *e.g.*, even domain (grain) boundaries and dislocations, combined with the concept of symmetry breaking (Anderson, 1997). Since the semi-phenomenological construction of the Ladau Fermi-liquid theory, it is possible to switch our intersts from the quasiparticle space to the order parameter space. This transformation to the order parameter space is familiar with that of Landau theory of phase transition. Toulouse & Kléman (1976) suggested an innovative idea that the defects in ordered media can be classified in terms of the *topological defects* of the corresponding order parameter. This topological concept was immediately applied to the vortex textures in superfluid 3He for showing that the textures with vorticity without vortex core can be constructed (Anderson & Toulouse, 1977). One of the most striking application of the topological defects theory may be the explanation of the entanglement free dislocation crossings (Poénaru & Toulouse, 1977); dislocations can be entangled if and only if they are members of separate class of *π*<sup>1</sup> (*R*). We will see the meaning of *π*<sup>1</sup> (*R*) later. A pedagogical review on the topological defects was given by Mermin (1979). Recenlty, Kleman & Friedel (2008) reviewed the practical interpretations of the homotopy group theory of the topological

In a periodic solid, the elastic variable is the vector displacement field **u** (**x**). Changes of **u** by a Bravais lattice vector **R** leave the lattice unchanged. Thus the order parameter space for the solid is the three-dimensional space of displacements **u** with lattice points. A dislocation with

where *C* is a curve enclosing the core and **b** is called the *Burgers vector*. The construction of the Burgers vector is fully topological, but there are two nontopological descriptions; screw dislocations if **b** � **l** and edge dislocations if **b** ⊥ **l**. Since the Bravais lattice vector, which leaves after a single winding curve *C* passed to enclose the core, is discrete, the dislocation is,

Let us look at the characters of dislocations with little rigor (Mermin, 1979). Suppose that *R* is an order-parameter space. There is a map *f* (*z*) of interval 0 ≤ *z* ≤ 1 into *R* defining the real-space loop or circles into order-parameter space to give closed curves in order-parameter space itself. A continuous directed curve described by maps *f* passing through a base point *x* in *R*. The close curve is decribed by *f* (0) = *f* (1) = *x*. Two loops *f* and *g* are *homotopic at x*, if there is a continous family of loops, all passing through *x*. The classes of homotopic loops at *x* form a group, called *π*<sup>1</sup> (*R*, *x*), and is known as the *fundamental group* of *R* at *x*. There are other groups associated with the mappings of the surface of an *n* sphere *Sn* in Euclidean *n* + 1 space

d*s* = **R** ≡ **b**, (206)

**7. Defects**

disclination, dislocations, and continuous defects.

<sup>10</sup> Here, we follow the definition by Chaikin & Lubensky (1995).

d**u** = *C* d**u** d*s*

**7.1 Topological defects**

a core along **l** is characterized by<sup>10</sup>

from homotopy group theory, *line*.

called *πn* (*R*, *x*). The fundamental group is called the *first homotopy group* and *πn* is then called the *n*th homotopy group. For example, *π*<sup>2</sup> (*S*2, *x*) is called when the maps *f* at the base point *x* is on the sphere *S*, *i.e.*, say, *u* and *v* into *R*, of three dimensional order-parameter space. For an integer *Z*, if *π*<sup>1</sup> (*S*1) = *Z*, the group structure does not depend on the choice of base point *x* and the map is topologically the same as the circle. Any mapping of a loop into a sphere can be shrunk to a point, so that the fundamental group of the surface of a sphere consists only of the identity, *π*<sup>1</sup> (*S*1) = 0. Trivial point defects are associated with mappings that can be deformed to the constant map.

Let us say the group *G* containing translations as well as proper rotations, *i.e.* the proper part of the full Euclidean group. Let *H* be the subgroup of *G* containing those rigid body rotations that leave the reference system invariant. Naively the order-parameter space *R* is the coset space *G*/*H*. If we ignore of the rotational symmetries of crystals, the full proper Euclidean group can be replaced by the subgroup *T* (3) of translations. On the other hand the isotropy subgroup *H* becomes the subgroup of *T* (3) consisting of translations through Bravais lattice vectors. Since *T* (3) is parameterized by all of Euclidean three-space, it is connected and simply connected. Since *H* is discrete, we identifies *π*<sup>1</sup> (*G*/*H*) with *H* itself. Thus the line defects, in our case dislocations, are characterized by Bravais lattice vectors. The vector is usually known as the Burgers vectors. Because *H* is discrete *π*<sup>2</sup> (*G*/*H*) = 0. In this case, the homotopy theory says notopologically stable *point* defects. However, we know thermodynamically stable point defects, *e.g.*, vacancies or voids. The physical means of removing a void are reminiscent of the nonlocal means available for the elimination of topologically stable defects.

Now let us recover the operations of the rigid body rotations. The fundamental group is identified with the double group. The three dimensional rotational parts of the group operations are lifted from SO(3) rotation to SU(2). Discrete operations in the space group with no translational part correspond to line defects, in which the local crystal structure rotates through an angle associated with a point group operation as the line is encircled. Such defects are known as disclinations. Although the *π*<sup>3</sup> (*R*) has the classification problem (Mermin, 1979), the domain walls (grain boundaries) are thought as the string bounded topological defects by the broken discrete symmetry (Vilenkin, 1985), hence so that textures evolve.

#### **7.2 Renormalization group and gauge theory**

Once the order parameter class is identified, the partition functions are divided by the corresponding topologically invariant number. Indeed, Burakovski et al. (2000) rederived the free energy density as a function of dislocation density depending on temperature, starting from the partition function computation for the indistinguishable dislocation loops. The resultant free energy density function is very similar to that out of the constitutional approach (Cotterill, 1980). The free-energy is in the form of the two-dimensional *xy* model, which should undergoes a two-dimensional Kosterlitz-Thouless phase transtion (Kosterlitz & Thouless, 1973). The theory break down point, *i.e.*, the melting temperature is also derived. They also showed that the dislocation density *ρ* cannot be a continuous function of temperature, but it is a gap function

$$\rho\left(T\right) = \begin{cases} 0, & T < T\_{m\nu} \\ \rho\left(T\_m\right) \sim b^{-2}, & T = T\_{m\nu} \end{cases} \tag{207}$$

where *Tm* is the melting temperature. The dislocation-mediated melting mechanism was observed in a bulk colloidal crystals (Alsayed et al., 2005). A good review on two-dimensional melting phenomena is given by Strandburg (1986; 1989).

renormalization group, while those for elementary excitations are conventional perturbative renormalization technique with the finite temperature Green's function. The effects of the identified topological defects on the elementary excitations can be dealt with the gauge field theory. The phase transition mechanism of each object can be understood by utilizing the fluctuation-dissipation theorem. Note that all the methods are based on the computation of

Towards the Authentic *Ab Intio* Thermodynamics 583

A reversed approach, seems more practical, is also possible. One can firstly identify the ideal structure and compute the equilibrium thermodynamics using the Green's function technique. The fundamental irreversible processes are also available from this technique. When one is faced a problem related with (topological) defects, the renormalization group

They usually raise a question on the practicality of the quantum field theoretic approach. Theories introduced here are nothing new, but tried to describe them in a unified way. Those theories have been developed and published from individual schools. This fact implies that the methods can be applied to materials engineering with a suitable software package and computers. There are the other approach for materials design. For example, Olson (2000) viewed a material as a system rather than a set of matters. Once an engineering target is given, one has to identifies the required properties in terms of materials properties and the corresponding structure with a suitable set of processes. The basic assumption is that the necessary theoretical and experimental tools are already prepared in a database and identified for the purpose. The length and time scales are covered from electronic scale to devices. A well organized spin-off company indeed does this business. The practicality awareness can be solved if we have a unified theoretical tool. This approach is an *ad hoc* combination of the current methods. The quantum field theoretic approach proposed in this article removes such

In view point of computer resources, the required computer performance of Olson's multiscase approach may be of order from peta-floating point operations per second (peta-FLOPS) to exa-FLOPS. The individual sectors of the theories introduced here utilize from a desktop personal computer to one of order tera-FLOPS. If there is a computer software package in a unified theoretical framework, the required computer performance may sit on the performance of order peta-FLOPS. Hence, the practicality will not be a major problem. In order to utilize a theoretical framework in engineering, a unified software package development is necessary. Unfortunately, such unified theoretical scheme has not been established yet. The reason can be thought in two-folds. Firstly, the theoretical maturity may not been achieved satisfactory by means of both theory itself and its implementation into a computer software package. When the author (hereafter referred as I) experienced first density functional calculation in 1994, I could remember the names of most researchers working in the density functional world. Now the number of citation of Kohn & Sham (1965) paper is approaching to 20,000 at the time of writing this manuscript. Most of them are using the software package as a black-box, due to the maturity of the software packages and the developments of computer technology. This example demonstrates the power of the matured

Secondly, the quantum field theory itself is too difficult to understand for beginners. Cardy

. . . and the the student, if he or she is lucky, has just about time to learn how to calculate the critical exponents of the Ising model in 4 − dimensions before the course comes to

The Ising model is considered as one of the simplest solvable "toy" model. In addition, many researchers in quantum field theory are working with their own "toys." The maturity of such

partition functions by quantum field theoretic manner.

and the defect gauge theory should be considered.

theory and the corresponding computer softwares.

(2000) explained, at the preface of his book, this situation as

*ad hoc* awareness.

an end.

Since the inclusion of the symmetry classfication, the *renormalization group* approach is appropriate to order parameters, or phase fields. The renormalization group theory has been served as the most appropriate tool for investigating phase transitions and critical phenomena (Anderson, 1997; Fradkin, 1991; Goldenfeld, 1992; Zinn-Justin, 1997). The idea of the renormalization group approach is a sucessive scaling transformation of the system interaction in real space, known as coarse-graining, or cutoff the high momentum fluctuations larger than arbitrary value Λ to eliminate the ultraviolet catastrophe of the perturbation expansion (Wilson, 1975; 1983). During the successive scaling, the interaction Hamiltonian of the phase field should keep its symmetry. The resultant Hamiltonian is arrived at a *fixed point* independent on the scale of the system, so it gives the *universal* behavior of the critical phenomena. The singlularity in the order parameter equation gives the dissipation of the order parameter. The good examples of the RG procedure with the topological defects were given by Fradkin (1991).

Elegance is in the renormalization group theory, because it does not depend on the system scale but on the symmetry. For example, the dislocation-mediated melting transition is categorized as the problems of the vortex formation in superfluid 3He (Anderson & Toulouse, 1977). In this sense, transition phenomena in the world can be categorized by *universality class* defined by the critical exponents.

However, is it enough?

Teichler (1981) criticised the above approach, because it is unsolved how we see the deviation of the equations of motion of elementary excitations in an ordinary medium with topological defects to the corresponding equations in the ideal configuration. He also showed that the transformed single electron Hamiltonian for describing the distorted crystal by dislocation (Brown, 1845) is described in terms of *gauge fields* in addition to the well-known deformation potentials. Gauge means the redundant degrees of freedom in Hamiltonian or Lagrangian, but its transformation behavior characterizes the interaction. The quantization of gauge field is gauge boson; *e.g.*, photons are a kind of gauge bosons out of the gauge degree of freedom of electromagnetic fields.

The quantum motions of electrons in a gauge field due to the topological dislocation defects have been investigated (Bausch, Schmitz & Truski, 1999; Bausch, Schmitz & Turski, 1999; Turski & Mi ´nkowski, 2009). The main conclusions of the gauge field theory (Bausch et al., 1998) are that the motion of a quantum particle is described in a curved-twisted space, as in gravitational field, called Rieman-Cartan manifold and that the Hamiltonian is separated in two parts that the covariant part and noncovariant parts (Larzar, 2010). The gauge field method has been applied to the elastic properties of solids (Dietel & Kleinert, 2006; Kleinert & Jiang, 2003; Turski et al., 2007) by modelling the elastic energy of the lattices by harmonic displacement fields. The computed melting temperatures of selected bcc and fcc elemental crystals agrees acceptably with the experimental values (Kleinert & Jiang, 2003). The most striking result of the defect gauge theory is the Aharonov & Bohm (1949) type interferences near the dislocation. This fact indicates that the dislocations are not be classical objects anymore, but are qunatum mechanical ones.

## **8. Outlook and proposal**

We have arrived at the stage to consider the applications of the theoretical framework in materials science.

Given a problem, we have to identify the order paramers governed by a certain set of symmetries. Based on the symmetry we need to classify the possible topological defects. The problem is separated into two parts: the scale independent universal properties and the elementary excitations. The tools for the universal properties are the homotopy theory and the 40 Will-be-set-by-IN-TECH

Since the inclusion of the symmetry classfication, the *renormalization group* approach is appropriate to order parameters, or phase fields. The renormalization group theory has been served as the most appropriate tool for investigating phase transitions and critical phenomena (Anderson, 1997; Fradkin, 1991; Goldenfeld, 1992; Zinn-Justin, 1997). The idea of the renormalization group approach is a sucessive scaling transformation of the system interaction in real space, known as coarse-graining, or cutoff the high momentum fluctuations larger than arbitrary value Λ to eliminate the ultraviolet catastrophe of the perturbation expansion (Wilson, 1975; 1983). During the successive scaling, the interaction Hamiltonian of the phase field should keep its symmetry. The resultant Hamiltonian is arrived at a *fixed point* independent on the scale of the system, so it gives the *universal* behavior of the critical phenomena. The singlularity in the order parameter equation gives the dissipation of the order parameter. The good examples of the RG procedure with the topological defects were

Elegance is in the renormalization group theory, because it does not depend on the system scale but on the symmetry. For example, the dislocation-mediated melting transition is categorized as the problems of the vortex formation in superfluid 3He (Anderson & Toulouse, 1977). In this sense, transition phenomena in the world can be categorized by *universality class*

Teichler (1981) criticised the above approach, because it is unsolved how we see the deviation of the equations of motion of elementary excitations in an ordinary medium with topological defects to the corresponding equations in the ideal configuration. He also showed that the transformed single electron Hamiltonian for describing the distorted crystal by dislocation (Brown, 1845) is described in terms of *gauge fields* in addition to the well-known deformation potentials. Gauge means the redundant degrees of freedom in Hamiltonian or Lagrangian, but its transformation behavior characterizes the interaction. The quantization of gauge field is gauge boson; *e.g.*, photons are a kind of gauge bosons out of the gauge degree of freedom

The quantum motions of electrons in a gauge field due to the topological dislocation defects have been investigated (Bausch, Schmitz & Truski, 1999; Bausch, Schmitz & Turski, 1999; Turski & Mi ´nkowski, 2009). The main conclusions of the gauge field theory (Bausch et al., 1998) are that the motion of a quantum particle is described in a curved-twisted space, as in gravitational field, called Rieman-Cartan manifold and that the Hamiltonian is separated in two parts that the covariant part and noncovariant parts (Larzar, 2010). The gauge field method has been applied to the elastic properties of solids (Dietel & Kleinert, 2006; Kleinert & Jiang, 2003; Turski et al., 2007) by modelling the elastic energy of the lattices by harmonic displacement fields. The computed melting temperatures of selected bcc and fcc elemental crystals agrees acceptably with the experimental values (Kleinert & Jiang, 2003). The most striking result of the defect gauge theory is the Aharonov & Bohm (1949) type interferences near the dislocation. This fact indicates that the dislocations are not be classical

We have arrived at the stage to consider the applications of the theoretical framework in

Given a problem, we have to identify the order paramers governed by a certain set of symmetries. Based on the symmetry we need to classify the possible topological defects. The problem is separated into two parts: the scale independent universal properties and the elementary excitations. The tools for the universal properties are the homotopy theory and the

given by Fradkin (1991).

However, is it enough?

of electromagnetic fields.

**8. Outlook and proposal**

materials science.

objects anymore, but are qunatum mechanical ones.

defined by the critical exponents.

renormalization group, while those for elementary excitations are conventional perturbative renormalization technique with the finite temperature Green's function. The effects of the identified topological defects on the elementary excitations can be dealt with the gauge field theory. The phase transition mechanism of each object can be understood by utilizing the fluctuation-dissipation theorem. Note that all the methods are based on the computation of partition functions by quantum field theoretic manner.

A reversed approach, seems more practical, is also possible. One can firstly identify the ideal structure and compute the equilibrium thermodynamics using the Green's function technique. The fundamental irreversible processes are also available from this technique. When one is faced a problem related with (topological) defects, the renormalization group and the defect gauge theory should be considered.

They usually raise a question on the practicality of the quantum field theoretic approach. Theories introduced here are nothing new, but tried to describe them in a unified way. Those theories have been developed and published from individual schools. This fact implies that the methods can be applied to materials engineering with a suitable software package and computers. There are the other approach for materials design. For example, Olson (2000) viewed a material as a system rather than a set of matters. Once an engineering target is given, one has to identifies the required properties in terms of materials properties and the corresponding structure with a suitable set of processes. The basic assumption is that the necessary theoretical and experimental tools are already prepared in a database and identified for the purpose. The length and time scales are covered from electronic scale to devices. A well organized spin-off company indeed does this business. The practicality awareness can be solved if we have a unified theoretical tool. This approach is an *ad hoc* combination of the current methods. The quantum field theoretic approach proposed in this article removes such *ad hoc* awareness.

In view point of computer resources, the required computer performance of Olson's multiscase approach may be of order from peta-floating point operations per second (peta-FLOPS) to exa-FLOPS. The individual sectors of the theories introduced here utilize from a desktop personal computer to one of order tera-FLOPS. If there is a computer software package in a unified theoretical framework, the required computer performance may sit on the performance of order peta-FLOPS. Hence, the practicality will not be a major problem.

In order to utilize a theoretical framework in engineering, a unified software package development is necessary. Unfortunately, such unified theoretical scheme has not been established yet. The reason can be thought in two-folds. Firstly, the theoretical maturity may not been achieved satisfactory by means of both theory itself and its implementation into a computer software package. When the author (hereafter referred as I) experienced first density functional calculation in 1994, I could remember the names of most researchers working in the density functional world. Now the number of citation of Kohn & Sham (1965) paper is approaching to 20,000 at the time of writing this manuscript. Most of them are using the software package as a black-box, due to the maturity of the software packages and the developments of computer technology. This example demonstrates the power of the matured theory and the corresponding computer softwares.

Secondly, the quantum field theory itself is too difficult to understand for beginners. Cardy (2000) explained, at the preface of his book, this situation as

. . . and the the student, if he or she is lucky, has just about time to learn how to calculate the critical exponents of the Ising model in 4 − dimensions before the course comes to an end.

The Ising model is considered as one of the simplest solvable "toy" model. In addition, many researchers in quantum field theory are working with their own "toys." The maturity of such

• **H**: external magnetic field

• *K*: propagtor, Green's function

• *M*ˆ *<sup>μ</sup>*: magnetization in *μ*-direction • *N*: number of particles of the system • *Ni*: number of particles of *i*th component

• ℘: Cauchy principal value • **P**ˆ: momentum operator

• **R**: Bravais lattice vector

• �: real part extraction operator

• **S***i*: localized atomic spin at the site *i*

• *T* (3): Euler translational subgroup • *U*ˆ : unary external potential operator

• *T*0: Bose-Einstein condensation temperature

• L: Landau function • **M**: total magnetization

• *P*: pressure

• *S*: entropy

• *T*: temperature

• *T*C: Curie temperature • *TL*: liquifying temperature • *Tm*: melting temperature

• *V*: volume of the system

• *b*: norm of Burgers vector

• *a*: lattice constant

• **b**: Burgers vector

• �: imaginary part extraction operator

• <sup>K</sup><sup>ˆ</sup> : grand canonical Hamiltonian operator

• **P***m*: eigenvalue of the momentum operator

• *R*: interaction range; order-parameter space (depending on context)

Towards the Authentic *Ab Intio* Thermodynamics 585

• *S* (*q*): mean density fluctuation as function of wavevector *q*

• *SO* (3): special orthogonal rotation group at three dimension • *SU* (2): special unitary rotation group at two dimension

• *T*<sup>0</sup> [*n*]: kinetic energy functional of the noninteracting system of density *n*

• *Z*: partition function; renormalization factor; an integer (depending on context)

• *an* (*H*, *T*): expansion coefficient of the Landau function as function of *H* and *T*

• *bn*, *cn*, *dn*: expansion coefficients of the Landau function

• *Jij*: interaction of spins between *i*th and *j*th sites

toys seems enough to evolve to as tools. Covering many topics in the renormalization group and quantum field theory is not so practical to learn in depth for typical material scientists. Such study will be done by the experts. The pedagogy for understanding the quantum field theory is also equally important.

To date there is no further fundamental theoretical tool in describing our physical world rather than the quantum field theory. It is hardly necessary to go beyond the quantum field theory for materials scientists. I would like to emphasize that we are standing at the starting point for a long journey to utilize the unified method with the reinterpreted terminology, *ab initio* thermodynamics authentically.
