# Defect Formation Through Boson Condensation

In Quantum Field Theory

{opening}
## 1 Introduction

The study of many body physics as well as the study of elementary particle physics has convinced us that at a very basic level Nature is ruled by quantum dynamical laws. On the other hand, we also know and observe several systems, such as superconductors, superfluids, crystals, ferromagnets, which behave as macroscopic quantum systems.

The question then arises of how the quantum dynamics may generate the observed macroscopic properties. In other words, how it happens that the macroscopic scale characterizing those systems is dynamically generated out of the microscopic scale of the quantum elementary components[1].

Moreover, we also observe a varieties of phenomena where quantum objects coexist and interact with extended macroscopic objects which show classical behavior, e.g. vortices in superconductors and superfluids, magnetic domains in ferromagnets, dislocations and other defects in crystals.

Thus, we are faced also with the question of the quantum origin of the macroscopically behaving extended objects and of their interaction with quanta[2]. Even for structures at cosmological scale, the question of their dynamical origin from elementary components asks for an answer consistent with quantum dynamical laws[3].

Macroscopic quantum systems are quantum systems not, of course, in the rather trivial sense that they are made by quantum components, but in the sense that, although they behave classically, nevertheless some of their macroscopic features cannot be understood without recurse to quantum theory. Quantum theory thus appears not confined to microscopic phenomena.

In this respect it is remarkable that these ”classical” systems present observable ordered patterns, e.g. crystal ordering, phase coherence, ferromagnetic ordering, etc.. Moreover, most extended objects present some topological singularity, and interesting enough, these topologically non-trivial defects only appear in systems presenting an ordered state.

The formation of defects in the course of phase transitions provides a further source of questions which are attracting much attention since it appears that defect formation during phase transitions may reveal unifying understanding of phenomena belonging to a wide range of energy scale[4].

The task of this paper is to review some of the main aspects in the Quantum Field Theory (QFT) description of topological defect formation, which also illustrate how to get the macroscopic scale out of the quantum dynamics. I will further mention some recent developments dealing with temperature effects on defect formation[5].

The paper is organized as follows. I will consider the problem of dynamical generation of order in quantum systems in Section 2. The key ingredients are the mechanism of spontaneous breakdown of symmetry (SBS) and the consequent appearance of Nambu-Golsdtone (NG) boson particles[6, 7] (such as phonons in crystals). In order to present a general, model independent discussion, I will use functional integration techniques. As we will see, NG modes manifest as long range correlations and thus they are responsible of the above mentioned change of scale, from microscopic to macroscopic. The coherent boson condensation of NG modes turns out to be the mechanism by which order is generated. From the point of view of the invariance properties of the theory, the mathematical structure of the contraction[8] of the symmetry group is the one controlling the SBS mechanism[9].

I will show how topologically non-trivial defects are generated in quantum systems by non-homogeneous boson condensation in Section 3. Here the so called boson transformation reveals to be the crucial tool. I will prove that topological defects only can be formed in the presence of NG modes, i.e. in the presence of ordering. This sheds some light on the mechanism by which defect formation occurs in phase transitions, i.e. in the presence of gradients of the order parameter. Interaction of defects with quanta is also very briefly considered in this Section. Explicit vortex solutions in terms of boson condensation are presented in Section 4.

Temperature effects and volume effects on SBS, on defects formation and on symmetry restoration are considered in Section 5. Contact with the problem of defect formation in phase transition processes is also made in this Section.

A preliminary remark to my subsequent discussion is the following.

The von Neumann theorem in Quantum Mechanics (QM) [10] states that for systems with a finite number of degrees of freedom all the representations of the canonical commutation relations are unitarily equivalent. This theorem actually states that in QM the physical system can only live in one single phase: unitary equivalence means indeed physical equivalence and thus there is no room ( no representations) to represent different physical phases. Fortunately, such a situation drastically changes in QFT where systems with infinitely many degrees of freedom are studied. In such a case the von Neumann theorem does not hold and infinitely many unitarily inequivalent representations of the canonical commutation relations do in fact exist[11]. It is such a richness of QFT which allows the description of different physical phases. The occurrence of spontaneous breakdown of symmetry and of the related NG boson condensation becomes thus possible in QFT.

Although one can set up many formal devices based on more or less sophisticate approximations, or even on semi-classical methods, which may nevertheless lead to phenomenologically successful results, it should be always understood that the proper theoretical framework where to operate dealing with phase transitions, defect formation and all that is the larger manifold of unitarily inequivalent representations provided by QFT.

## 2 Spontaneous breakdown of symmetry and group contraction

In QFT the dynamics is described by a set of field equations for the interacting operator fields, say , also called the Heisenberg fields. These are the basic fields of the theory satisfying equal-time canonical commutation relations and the Heisenberg field equations

(1) |

where . is a functional of the field describing the interaction.

Observable phenomena are on the other hand described by observable (physical) operator fields (such as phonons), say . They also obey equal-time canonical commutation relations and satisfy free field equations, which, with convenient care in the renormalization procedure, may be written as

(2) |

The Hilbert space, say , for the physical states is the Fock space for the fields . Solving the dynamical problem thus means to compute by means of eq. (1) the matrix elements of in the space : this will relate the basic dynamics to the observable properties of the physical states. In this way we obtain the dynamical map between interacting fields and physical fields[1, 2]:

(3) |

Eq.(3) is also called the Haag expansion in the LSZ formalism[12]. I have to stress that the equality in (3) is a ”weak” equality: it must be understood as an equality among matrix elements computed in .

This a crucial point and a couple of remarks need to be made. First, I observe that the set of fields must be an irreducible set; however, it may happen that not all the elements of the set are known since the beginning. For example there might be composite (bound states) fields or even elementary quanta whose existence is ignored in a first recognition. Then the computation of the matrix elements in physical states will lead to the detection of unexpected poles in the Green’s functions, which signal the existence of the ignored quanta. One thus introduces the fields corresponding to these quanta and repeats the computation. This way of proceeding is called the self-consistent method[2]. In this connection, I note that it is not necessary to have a one-to-one correspondence between the sets and . This happens in fact when the set includes composite particles.

Another remark is that, as already mentioned, in QFT the Fock space for the physical states is not unique: one may have indeed several physical phases, e.g. for a metal the normal phase and the superconducting phase, and so on. Fock spaces describing different phases are unitarily inequivalent spaces and correspondingly we have different expectation values for certain observables and even different irreducible sets of physical quanta; for example, in ferromagnets this set includes magnon fields which do not exist in the non-magnetic phase, etc.. Thus, finding the dynamical map involves the ”choice” of the Fock space where the dynamics has to be realized: in other words the same dynamics (i.e. same Heisenberg fields and same Heisenberg field equations) may generate different physical phases.

Suppose now that the dynamics is invariant under some group of transformations of :

(4) |

with . Invariance of the dynamics means that the Heisenberg equations (or the Lagrangian from which they may be derived) are invariant (in form) under the transformations of .

One says that symmetry is spontaneously broken when the vacuum state in the Fock space is not invariant under the group but only under one of its subgroups[1, 2, 12].

Eq. (3) implies that when is transformed as in (4), then

(5) |

such that

(6) |

with belonging to some group of transformations .

Now it happens, as we will see below, that when symmetry is spontaneously broken, with the group contraction of [9]; when symmetry is not broken .

Since is the invariance group of the dynamics, eq. (3) requires that is the group under which free fields equations are invariant, i.e. also is a solution of (2). Since eq. (3) is a weak equality, depends on the choice of the Fock space (among the physically realizable unitarily inequivalent state spaces). Thus we see that the original (same) invariance of the dynamics may manifest itself in different symmetry groups for the fields according to different choices of the physical state space. Since this process is constrained by the dynamical equations (1), it is called the dynamical rearrangement of symmetry[1, 2, 13].

To be specific, let me consider, in the path-integral formalism, a complex scalar field interacting with a gauge field (Anderson-Higgs-Kibble type model)[14, 15, 16]. The lagrangian density is invariant under the global and the local gauge transformations:

(7) |

(8) |

respectively, where for and/or . The Lorentz gauge is used. I put .

Spontaneous breakdown of symmetry is introduced through the condition , with constant and I put . Here denotes the Heisenberg field. is the c-number field entering the functional integral. The generating functional, including the gauge constraint, is[17]

(9) |

with a convenient normalization. is an auxiliary field which guarantees the gauge condition. The rôle of the term is to specify the condition of symmetry breakdown under which we want to compute the path-integral[18, 19]. It may be given the physical meaning of the small external field triggering the symmetry breakdown. The limit must be made at the end of the computations.

As customary, I will use the notation to denote functional average and . Note that because of the invariance under .

Invariance of the path-integral under the change of variables (7) (and/or (8) ) leads to

(10) |

This is one of the Ward-Takahashi identities. Such identities carry the symmetry content of the theory. In momentum space the propagator for the Heisenberg field has the general form

(11) |

and are renormalization constants. The integration in eq.(10) picks up the pole contribution at , and leads to

(12) |

The Goldstone theorem[6] is thus proved[18, 19]: if the symmetry is spontaneously broken (), a massless mode exists, whose interpolating Heisenberg field is . It is the NG boson mode. Since it is massless it manifests as a long range correlation mode. Notice that the NG mode is an elementary field. In other models it may appear as a bound state, e.g. the magnon in ferromagnets[20]. Note that[18, 19] is independent of , although the phase of determines the one of : as in ferromagnets, once an external magnetic field is switched on, the system is magnetized independently of the strength of the external field.

The analysis of the two-point functions of the theory shows[17] that the model contains a massless negative norm state (ghost), besides the NG massless mode , and a massive vector field . The dynamical maps are:

(13) |

(14) |

(15) |

where the functionals and are to be determined within a particular model. These relations are weak equalities and are equivalent to the familiar LSZ reduction formula[12]. It will be also used . In eqs. (13)-(15) denotes the NG mode, the ghost mode, the massive vector field and the massive matter field. Their field equations are

(16) |

(17) |

with . We also have

(18) |

The field equations for and are

(19) |

with . One may then require that the current is the only source of the gauge field in any observable process. This amounts to impose the condition: , i.e.

(20) |

where and denotes two generic physical states. Eq.(20) are the classical Maxwell equations. The condition leads to the Gupta-Bleuler-like condition

(21) |

where and are the positive-frequency parts of the corresponding fields. Thus we see that and do not participate to any observable reaction. Note in fact that they are present in the matrix in the combination . It is to be remarked that the NG boson do not disappear from the theory: we will see that in situations in which the vacuum is not translationally invariant, the NG fields can have observable effects.

The study of the dynamical rearrangement of the symmetry shows that local gauge transformations of the Heisenberg fields

(22) |

are induced by the in-field transformations

(23) |

(24) |

The global transformation is induced by

(25) |

(26) |

Note that under the above in-field transformations the in-field equations and the matrix are invariant and that is changed by an irrelevant c-number. We thus see that the original invariance cannot be lost even at the level of the physical fields, although it can manifest there in a different symmetry group structure.

Eq. (25) shows that the physical field translates by a constant when the Heisenberg field undergoes the global phase transformation (and vice versa): The global invariance group is dynamically rearranged into the one-parameter constant translation group. This last one is the group contraction of global .

Notice that translation by a constant is an invariant transformation for the field equation if and only if is a massless field. Thus the rearrangement into the contraction of global group has the same content as the Goldstone theorem[9]. It is also interesting to note that while is a compact group its contraction is not compact. Since the number operator of field changes under the translation (25) we say that we have coherent boson condensation.

It must be stressed that the translation by a constant (25) must be actually understood as the limit for of the transformation

(27) |

with a normalizable solution of the field equation: . Eq. (25) induces infrared singularities in Feynman diagrams with many momentumless and energyless lines. These are smeared out by use of (27). However, notice that matrix elements are well defined even when (25) is used. Also, the function appearing in the generator of (27) makes it well defined[19, 20, 21].

Since different physical phases (unitarily inequivalent representations) are associated to different NG boson condensation densities, we see that boson translations, by inducing variation of NG boson condensation, represent transitions through physically different phases. In particular, I will discuss non-homogeneous boson condensation induced by transformations such as (27).

The dynamical rearrangement of symmetry has been studied in many models of physical interest. In the case of global invariance group, for example in ferromagnets[20], or in systems with isospin vector fields[21], the NG boson condensation is controlled by the group, which is the group contraction of [9]. Unfortunately, for lack of space I cannot report about these and some other interesting cases.

## 3 Observable effects of non-homogeneous boson transformations

Translations of bosonic physical fields (not necessarily massless) by space-time dependent functions, say , satisfying the same field equation of the translated physical field, are called boson transformations[1, 2, 19]. Eq. (27) is thus an example of boson transformation.

Let denote the Heisenberg field obtained through the dynamical mapping when the physical field undergoes the boson transformation. The boson transformation theorem can be then proved, which states that is also a solution of the Heisenberg field equation[1, 2, 19].

The proof of the theorem consists in showing that the boson-transformed fields, say , differ from only by an dependent factor and therefore are solutions of the same field equations.

In the absence of a gauge field, under boson transformation the order parameter gets space-time dependence given by

(28) |

where the expansion of around is used ( when ). Note that the modulus of changes and in the limit only its phase is changed.

In the case a gauge field is present, it can be shown that any space-time dependence of the term can be eliminated by a gauge transformation when is a regular (i.e. Fourier transformable) function and the only effect is the appearance of a phase factor in the order parameter: , with a constant.

The conclusion is that when a gauge field is present, the boson transformation with regular is equivalent to a gauge transformation. On the contrary, in a theory with global invariance only, non-singular boson transformations of the NG fields can produce non-trivial physical effects (like linear flow in superfluidity).

I want to stress that, in the case of global phase transformations as well as in the case of local gauge transformations, the proof of the boson transformation theorem relies on the fact that is a regular function. If one wants to consider functions with some singularities (divergence and topological singularities) one has to carefully exclude the singularity regions when integrating on space and/or time. For example, if is singular on the axis of a cylinder (at ) one must exclude the singular line by a cylindrical surface of infinitesimal radius. The phase of the order parameter will be singular on that line. This means that SBS does not occur in that region (the core): there we have the ”normal” state rather than the ordered one. Provided one uses such care, the boson transformation can be safely (and advantageously) used also in the case of singular .

The boson theorem has relevant physical meaning since it shows that the same dynamics may describe homogeneous and non-homogeneous phenomena. When a theory allows SBS, there always exist solutions of the field equations with space and/or time-dependent vacuum. These solutions are obtained from the translationally invariant ones by the boson transformation of the NG field: they results from a local Bose condensation of the particles. This directly leads us to the mechanism of formation of extended objects (defect formation).

Notice that in local gauge theories the boson transformation must be compatible with Heisenberg field equations but also with the physical state condition (21). Under the boson transformation with , changes as and

(29) |

Eq. (20) is thus violated when the Gupta-Bleuler-like condition is imposed. In order to restore it, the shift in must be compensated by means of the transformation on :

(30) |

with a convenient c-number function . The dynamical maps of the various Heisenberg operators are not affected by (30) since they contain and in a combination such that the changes of and of compensate each other provided

(31) |

Eq. (31) thus obtained is the Maxwell equation for the massive potential vector [17, 22]. The classical ground state current turns out to be

(32) |

The term is the Meissner current, while is the boson current.

The macroscopic field and current are thus given in terms of the boson condensation function.

The (classical) boson current is given by , i.e. by variations in the non-homogeneous boson condensate: boson condensation functions must play a rôle in phase transitions where boson condensate indeed changes.

Let me now show that boson transformation functions carrying topological singularities are only allowed for massless bosons[2, 23, 24].

Suppose the function for the boson transformation of the field carries a topological singularity and is thus path-dependent:

(33) |

On the other hand, , which is related with observables, is single-valued, i.e. . Recall that is solution of the equation and suppose there is a non-zero mass term:

(34) |

From the definition of and the regularity of it follows that

(35) |

which leads to which implies .

Thus (33) is only compatible with massless equation for .

The quantity is given by (35) with . From this equation, can be determined. The topological charge is defined as

(36) |

Here is a contour enclosing the singularity and a surface with as boundary. does not depend on the path provided this does not cross the singularity. The dual tensor is

(37) |

and satisfies the continuity equation

(38) |

This equation completely characterizes the topological singularity [2, 24].

Let me now observe that all the macroscopic ground state effects do not occur for regular (). In fact, from (31) we obtain for regular which implies zero classical current () and zero classical field (), since the Meissner and the boson current cancel each other.

In conclusion, the vacuum current appears only when has topological singularities and these can be created only by condensation of massless bosons, e.g. by condensation of NG bosons when SBS occurs. This explains why topological defects appear in the process of phase transitions, where NG modes are present and gradients in their condensate densities are nonzero.

On the other hand, the appearance of space-time order parameter is no guarantee that persistent ground state currents (and fields) will exist: if is a regular function, the space-time dependence of can be gauged away by an appropriate gauge transformation.

Since the boson transformation with regular does not affect observable quantities, the matrix (13) is actually given by

(39) |

This is in fact independent of the boson transformation with regular :

(40) |

since for regular . However, for singular : includes the interaction of the quanta and with the defect classical field and current.

Thus we see how quantum fluctuations may interact and have effects on classically behaving macroscopic defects: our picture includes interaction of quanta with macroscopic objects. Much more can be said on the interaction of extended objects with quanta; however, for shortness I will not discuss more on that.

## 4 The vortex solution

The meaning of Eq. (35) with is of course[2, 22, 24, 25]

(41) |

(42) |

with the Green’s function satisfying . Eq.(41) gives upon path integration

(43) |

which is indeed solution of . The classical vector potential is

(44) |

(45) |

The electromagnetic tensor and the vacuum current are[2, 22, 24]

(46) |

(47) |

respectively, and satisfy .

The line singularity for the vortex solution can be parameterized by a single line parameter and by the time parameter . The static vortex solution is then obtained by setting and , with denoting the line coordinate. is non-zero only on the line at (we can consider more lines but here I limit myself to one line, for simplicity). Thus,

(48) |

(49) |

Eq.(46) shows that these vortices are purely magnetic. Eq.(41) gives

(50) |

and , i.e., by using the identity ,

(51) |

Note that is satisfied.

Straight infinitely long vortex is specified by with . The only non vanishing component of are . Eq.(51) gives[2, 22, 24]

(52) |

(53) |

which give

(54) |

Use of these results gives the vector potential, and the vacuum current. The only non-zero components of these fields are , , , and .

Notice that the condition (38) can be shown to be violated if the line singularity has isolated end points inside the system. Thus consistency with the continuity equation (38) implies that either the string is infinite, or that it form a closed loop. Also, if there are more than one string, the end points of different strings can be connected in a vertex, eq.(38) resulting then in a condition for the relative string tensions , with denoting different strings.

A circular loop: , . , , .

A straight line along the third axis moving in the direction with velocity is given by , , from which and .

## 5 Finite temperature and finite volume effects

Consider the invariant model at finite temperature. The breakdown of symmetry condition in the homogeneous condensation case is[26]

(55) |

where . denotes the temperature dependent vacuum state in Thermo Field Dynamics [2]. Note that the statistical average of any operator is given by .

I omit to consider here the presence of other fields (such as the ghost fields) for brevity. The fields , and may undergo translation transformations by c-number functions, say , and , respectively, controlling the respective condensate structures. I write . Usual gauge transformations are induced by using , and .

The homogeneous boson condensation of the Higgs field alone (, and ) leads to

(56) |

(57) |

where and denote the Higgs field mass and self-coupling, respectively, at and is assumed to be non-zero, is the gauge field mass and is the (electric charge) coupling between and . and denote the physical fields.

Eqs.(56) are actually self-consistent equations since also depends on . In the discontinuous phase transition case the free energy should be examined[26]. The proper phase transition point is defined by the equality between the ordered and the disordered free energy phase.

As eqs. (56) show that thus recovering the original zero temperature symmetry breaking. We have phase transition to the (disordered) phase at the critical temperature such that

(58) |

Above the phase transition point , and , we have[26]

(59) |

Full symmetry restoration (i.e. ) occurs at such that thermal contributions in (58) compensate each other, and then also . The gauge field mass goes to zero not at , but at such that

(60) |

The vortex solution arises in the non-homogeneous condensation case obtained by introducing space dependence in the condensate functions. Introducing the cylindrical coordinates, the asymptotic gauge field configuration is imposed by considering the angle function as gauge function at infinity

(61) |

Here is the winding number and we see that, although, as already observed, the NG bosons do not enter the physical spectrum, nevertheless their condensation is directly related to the topological charge. For we assume (the vortex ansatz)[27]

(62) |

where is the Higgs field shift for the homogeneous condensation and

(63) |

The vortex ansatz leads to the temperature dependent vortex equations

(64) |

As these equations reduce to the usual vortex equations. One can show[26] that in the vortex case the masses are given by

(65) |

These masses act as potential terms in the field equations and only at spatial infinity () ordinary mass interpretation is recovered. We have in fact the asymptotic behavior

(66) |

gives the size of the gauge field core and the Higgs field core. As the Higgs field core increases and the gauge field core becomes smaller. At one obtains the pure gauge field core. Above symmetry is restored. The discussion on temperature dependence of is similar to the one for the homogeneous case.

The ’t Hooft-Polyakov monopole and the sphaleron solutions at finite temperature are discussed in [26].

Let me now discuss the effects of the finite size of the system on the boson condensate and the relation between finite size and temperature effects. This will help to understand how temperature variations near control the defect size (and thus the defect number)[5].

In the case of large but finite volume we expect that the condition of symmetry breakdown is still satisfied “inside the bulk” far from the boundaries. However, “near” the boundaries, one might expect “distortions” in the order parameter: (or even ). “Near” the system boundaries we may have non-homogeneous order parameter. Non-homogeneities in the boson condensation will “smooth out” in the limit. Suppose the integration in eq. (10) is over the finite (but large) volume and use

(67) |

which, as well known, approaches as : . Now

(68) |

then, using for small , it is

(69) |

(70) |

with . Thus, only if , otherwise . Eq. (12) (the Goldstone theorem) is thus recovered in the infinite volume limit ().

On the other hand, assume that NG modes are there, i.e.