Gauge Fields in Condensed Matter

The following sections are included:

• PREFACE

• CONTENTS

Most physical systems follow complicated nonlinear equations. For a small number of degrees of freedom, numerical methods may lead to a satisfactory theoretical understanding. In macroscopic many-body systems, however, this number is immensely large and such an approach is hopeless. The best we can achieve is an approximate understanding of the statistical behavior, averaged either with respect to thermal or to quantum mechanical fluctuations…

The phenomena that will be studied take place in real space. The most economical way of describing such phenomena is based on attributing to each space point x one or more variables, collectively denoted by ϕ_α(x). The set of these variables is called a field. The field ϕ_α(x) undergoes thermal as well as quantum fluctuations and the development of this part of our text will be devoted to reviewing the general description of the statistical properties of such a fluctuating field. Since several good textbooks are already available on this subject we can afford to be somewhat sketchy. Wherever statements require elaborate proofs these will be omitted and we shall be content with simple illustrations and examples. Hopefully, our short treatment will be sufficient to serve as an introduction for those unfamiliar with the relevant basic field theoretical techniques. At the very least we hope to clarify the notation, conventions, and language to be used in the further development of the theory.

The following sections are included:

• GENERAL FORMALISM

• VACUUM CONTRIBUTION Z[0]

• TWO-POINT FUNCTION AND SELF-ENERGY GRAPHS

• FOUR-POINT AND VERTEX FUNCTIONS

• SPECIAL FEATURES OF LOCAL THEORIES

• MOMENTUM SPACE DIAGRAMS

• RELATION BETWEEN VACUUM DIAGRAMS AND CORRELATION FUNCTIONS

The following sections are included:

• FREE MAGNETIC FIELDS

• GAUGE-FIXING FACTORS

• CHARGED SCALAR FIELDS

• FEYNMAN DIAGRAMS

• NON-ABELIAN GAUGE THEORIES

• NON-ABELIAN FIELD THEORIES INVOLVING THE SPACE GROUP

• NOTES AND REFERENCES

The following sections are included:

• EXTREMAL FIELD CONFIGURATIONS

• SADDLE-POINT APPROXIMATION OF INTEGRALS

• SADDLE-POINT APPROXIMATION FOR FLUCTUATING FIELDS

• CONNECTED CORRELATION FUNCTIONS

• SYMMETRY PROPERTIES OF EXTREMAL FIELD CONFIGURATIONS: CURRENTS AND CONSERVATION LAWS

• NOTES AND REFERENCES

• APPENDIX 4A. VACUUM DIAGRAMS

The following sections are included:

• EXTREMAL PRINCIPLE

• TREE GRAPHS AND VERTEX FUNCTIONS

• GRAPHICAL CONTENT OF THE EFFECTIVE ENERGY

• THE ONE-PARTICLE IRREDUCIBILITY OF Γ[0]

• PROPER SELF-ENERGY

• APPENDIX 5A. GRAPHS CONTRIBUTING TO (1/T)W[j]
  • Terms linear in j
  • Terms quadratic in j

• NOTES AND REFERENCES

The following sections are included:

• GENERAL REMARKS

• SINGLE RANDOM WALK

• MATERIAL RANDOM CHAINS IN THERMAL EQUILIBRIUM

• GRAND CANONICAL ENSEMBLE OF CLOSED ORIENTED RANDOM CHAINS AND DISORDER FIELDS

• CRITICAL REGION

• CLOSED RANDOM CHAINS AND LOOP DIAGRAMS

• OPEN RANDOM CHAINS AND LINES IN FEYNMAN DIAGRAMS

• FIELD FLUCTUATIONS VERSUS LINE FLUCTUATIONS

• ENSEMBLES OF FLUCTUATING LINES IN THE CRITICAL REGION

• GENERAL PROPERTIES OF LATTICE GREEN FUNCTIONS

• CALCULATION OF LATTICE GREEN FUNCTIONS VIA THE HOPPING EXPANSION

• IMPROVING THE CONVERGENCE OF HOPPING EXPANSION

• EXACT ONE- AND TWO-DIMENSIONAL COULOMB GREEN FUNCTION ON A SQUARE LATTICE

• FLUCTUATION DETERMINANTS

• FINITE-SIZE EFFECTS ON THE FLUCTUATION DETERMINANT AND QUANTUM STATISTICS OF A FREE FIELD

• SHORT-RANGE STERIC INTERACTIONS BETWEEN CHAINS VERSUS FIELD INTERACTIONS

• INCORPORATING LONG-RANGE FORCES

• INTERACTING RANDOM CHAINS AND FEYNMAN DIAGRAMS

• APPENDIX 6A. GREEN FUNCTIONS AND FLUCTUATION DETERMINANTS FOR NON-CUBIC LATTICES

• NOTES AND REFERENCES

The simplest physical system for which a gauge theory of stresses and defects can be developed is superfluid ⁴He. It possesses precisely the properties described in the Introduction: there exists an ordered ground state at zero temperature. Small disturbances introduce soft long-range excitations which are observable as sound waves. Stronger disturbances, for instance, the rotation of the container of the superfluid, lead to the formation of vortex lines which may be viewed as line-like defects. Moreover there exists a critical temperature at which a phase transition takes place leading to the disappearance of the superfluid state…

The following sections are included:

• DISORDER FIELDS FOR VORTEX LINES

• THERMODYNAMIC QUANTITIES

• NAMBU-GOLDSTONE MODES OF THE DISORDER FIELD

• INCLUSION OF HYDRODYNAMIC FORCES

• NOTES AND REFERENCES

The following sections are included:

• LONDON EQUATIONS

• ORDER PARAMETER

• ORDER FIELD

• GINZBURG-LANDAU ENERGY

• GORKOV’S DERIVATION

• GINZBURG-LANDAU EQUATIONS

• CRITICAL MAGNETIC FIELD

• THE TWO LENGTH SCALES AND TYPE I VERSUS TYPE II SUPERCONDUCTIVITY

• SINGLE VORTEX LINE AND CRITICAL FIELD H_C1

• THE FIELD H_C2 WHEN SUPERCONDUCTIVITY IS DESTROYED

• FLUCTUATION CORRECTIONS

• PHASE TRANSITION IN DISORDER FIELD THEORY AND DISORDER VERSION OF THE MEISSNER EFFECT

• NOTES AND REFERENCES

The following sections are included:

• MOTIVATION

• THE XY MODEL

• HIGH TEMPERATURE EXPANSION

• PHYSICAL INTERPRETATION

• ESTIMATE OF CRITICAL PROPERTIES VIA THE RATIO TEST

• APPENDIX 4A: GRAPHS OF ORDER β¹⁰

• NOTES AND REFERENCES

While the high temperature expansion is quite accurate in describing almost the entire high temperature phase (except in the immediate vicinity of the critical point), it fails completely in the low temperature phase. Other methods are therefore needed in order to complete our understanding of the model. The most prominent among them is based on the mean field theory plus fluctuation corrections.

In the general discussion in Chapter 2 we argued that superfluid ⁴He should be describable as an ensemble of vortex lines together with their long-range hydrodynamic interactions. Let us now verify this expectation within the XY model.

The following sections are included:

• THE VILLAIN APPROXIMATION

• THE VILLAIN MODEL

• HIGH AND LOW TEMPERATURE EXPANSIONS OF THE VILLAIN MODEL

• MONTE CARLO SIMULATION OF THE VILLAIN MODEL

• COMPARISON OF THE XY MODEL WITH ITS VILLAIN APPROXIMATION

• THE XY MODEL VERSUS VILLAIN MODEL OF SUPERFLUIDITY

• PERTURBATION EXPANSIONS OF THE XY MODEL (1/β-EXPANSION)

• HARTREE-FOCK RESUMMATION

• COMPARISON WITH THE MEAN-FIELD-PLUS-LOOP APPROACH

• VORTEX CORRECTIONS

• APPENDIX 7A. THE LOW TEMPERATURE EXPANSION OF THE VILLAIN APPROXIMATION

• NOTES AND REFERENCES

Up to now, we have dealt only with one type of gauge field, namely, the gauge field which describes superflow via a curl relation,

This gauge field was introduced in order to ensure that the b_i, field lines are closed. They are the rings of superflow arising in the disordered phase due to fluctuations which carry the liquid toward the ordered state. In the vortex-line representation of the XY model, we encountered another set of closed lines, namely, those of vortex lines. Vortex lines give rise to a second important gauge structure which we are now going to discuss.

The following sections are included:

• PATH-INTEGRAL REPRESENTATION

• MONOPOLE GAUGE FIELD-TRIPLE GAUGE THEORY

• OTHER REPRESENTATIONS OF CORRELATION FUNCTIONS IN THE VILLAIN MODEL

• HIGH TEMPERATURE EXPANSION FOR THE TWO-POINT CORRELATION FUNCTION AND SUSCEPTIBILITY

• RATIO TEST FOR THE SUSCEPTIBILITY EXPANSION

• ISOTROPY OF HIGH TEMPERATURE CORRELATION FUNCTIONS AT LARGE DISTANCES

• LOW TEMPERATURE EXPANSION OF THE CORRELATION FUNCTION IN THE VILLAIN MODEL

• NOTES AND REFERENCES

In the last chapters we showed that the calculation of correlation functions of the order parameter e^iγ(x) amounts to calculating the behaviour of a fixed set of magnetic monopoles within an ensemble of vortex lines. It is worthwhile pointing out that the model can be enriched in a very simple manner to contain, from the outset, a whole ensemble of such monopoles. All we have to do is to discretize the angle γ(x) so that it no longer covers the interval [0, 2π) continuously but only the fractional angles…

The following sections are included:

• FILMS OF SUPERFLUID HELIUM

• HIGH TEMPERATURE EXPANSION OF D = 2 XY MODEL

• VORTEX REPRESENTATION IN TWO DIMENSIONS

• LOW TEMPERATURE EXPANSION OF THE VILLAIN MODEL

• DETERMINATION OF THE TRANSITION POINT

• VILLAIN APPROXIMATION FOR D = 2

• VORTEX CORRECTIONS IN D = 2 XY MODEL

• CORRELATION FUNCTIONS IN THE 2-DIMENSIONAL VILLAIN MODEL

• EXPLICIT CALCULATION OF THE TWO-POINT CORRELATION FUNCTION AT LOW TEMPERATURES

• THE RENORMALIZATION GROUP

• COHERENCE LENGTH

• UNIVERSALITY OF THE CRITICAL STIFFNESS

• EXACT FORMULA FOR THE RENORMALIZATION OF SUPERFLUID DENSITY BY VORTICES

• EQUIVALENCE OF THE VILLAIN MODEL TO THE LIMITING CASE OF THE SINE-GORDON MODEL

• ESTIMATE OF THE CRITICAL TEMPERATURE IN THE SINE-GORDON MODEL

• SELF-CONSISTENT APPROXIMATION TO THE COHERENCE LENGTH OF THE XY AND SINE-GORDON MODELS

• SURFACE ROUGHENING AND CRYSTAL GROWTH

• NOTES AND REFERENCES

The following sections are included:

• EQUIVALENCE OF THE XY MODEL TO A LATTICE SUPERCONDUCTOR

• FROM LATTICE SUPERCONDUCTOR TO GINZBURG-LANDAU THEORY

• COMPARISON WITH APPROXIMATE DISORDER THEORY

• NATURAL CORE ENERGIES

• TYPE-I OR TYPE-II DISORDER THEORY

• QUANTUM VORTEX DYNAMICS IN FILMS OF SUPERFLUID HELIUM

• NOTES AND REFERENCES

The following sections are included:

• GENERAL DISCUSSION

• VORTEX-LINE REPRESENTATION OF A GENERAL LATTICE SUPERCONDUCTOR

• DISORDER FIELD THEORY OF VORTEX LINES IN LATTICE SUPERCONDUCTORS

• LIMITING CASES

• PHASE TRANSITIONS IN LATTICE SUPERCONDUCTORS

• TRICRITICAL POINT IN A SUPERCONDUCTOR

• NOTES AND REFERENCES

The following sections are included:

• INDEX

The following sections are included:

• CONTENTS

• PREFACE

The description of superfluid ⁴He in terms of a disorder field theory developed in Part II can serve as a prototype for the treatment of many other physical systems. For this to be true, these systems have to possess the following fundamental properties…

The following sections are included:

• DISPLACEMENT AND STRAIN

• ELASTIC ENERGY

• STRESS

• EXTERNAL BODY FORCES

• ELASTIC GREEN FUNCTION

• TWO-DIMENSIONAL ELASTICITY

• APPENDIX 1A. THE SYMMETRY CLASSES OF THE ELASTIC MATRIX

• NOTES AND REFERENCES

The following sections are included:

• GENERAL REMARKS

• DISLOCATION LINES AND BURGERS VECTOR

• DISCLINATIONS AND THE FRANK VECTOR

• INTERDEPENDENCE OF DISLOCATIONS AND DISCLINATIONS

• DEFECT LINES WITH INFINITESIMAL DISCONTINUITIES IN CONTINUOUS MEDIA

• MULTIVALUEDNESS OF THE DISPLACEMENT FIELD

• SMOOTHNESS PROPERTIES OF THE DISPLACEMENT FIELD AND WEINGARTEN’S THOOREM

• INTEGRABILITY CONSIDERATIONS

• DISLOCATION AND DISCLINATION DENSITIES

• MNEMONIC PROCEDURE FOR CONSTRUCTING DEFECT DENSITIES

• BRANCHING DEFECT LINES

• DEFECT DENSITY AND INCOMPATIBILITY

• DEFECTS IN TWO DIMENSIONS

• NOTES AND REFERENCES

The following sections are included:

• STRAIN AND STRESS AROUND DISLOCATION LINES

• ELASTIC INTERACTION ENERGY BETWEEN TWO DISLOCATION LINES

• NOTES AND REFERENCES

The following sections are included:

• ELASTIC PARTITION FUNCTION

• HELICITY DECOMPOSITION OF A VECTOR FIELD

• HELICITY DECOMPOSITION OF A TENSOR FIELD

• HELICITY FORM OF THE MAGNETIC ENERGY

• HELICITY FORM OF THE STRESS ENERGY

• THE TWO-DIMENSIONAL CASE

• NOTES AND REFERENCES

The following sections are included:

• THE SYMMETRIC STRESS GAUGE FIELD

• ELASTIC PARTITION FUNCTION FOR A FIXED GENERAL DEFECT DISTRIBUTION

• TWO-DIMENSIONAL DEFECTS

• NOTES AND REFERENCES

So far we have studied only static defect lines. For a complete understanding of the plastic properties of materials caused by defect lines it is necessary to know also their kinematic properties. In this text we shall be mainly interested in equilibrium thermodynamics of defects and stresses, where these properties are, to a large extent, irrelevant. For completeness, however, it will be useful to recapitulate the way defects move in a crystal.

So far we have studied specific defect configurations with their elastic long range stress interactions. These interactions were described by a gauge field coupled locally to the defect densities. In the previously investigated case of superfluid ⁴He we saw that given a system of random lines with such a coupling, it is relatively easy to develop a disorder field theory for a grand canonical ensemble of such lines. This field theory permitted the study of phase transitions in which vortex lines condensed. In ⁴He this transition carried the superfluid into the normal state…

The Lindemann criterion gave us some important information on the nature of the melting transition: the transition occurs at a temperature slightly below which the atoms still perform, to a very good approximation, harmonic fluctuations about their mean positions. The nonlinearities of the interatomic potential are not very important. They merely provide a shift in the mean position and a softening of the elastic constants. If these two effects are taken into account, the crystal, even right below the melting temperature, can be treated practically as an ideal crystal. Defects are quite rare, due to their high energy, and we can describe the interactions of defect lines by the methods developed in the previous chapters…

The following sections are included:

• SETTING UP THE MODEL

• DEFECT REPRESENTATION OF LATTICE MODEL

• AN XY TYPE MODEL OF DEFECT MELTING

• APPENDIX 9A. DERIVATION OF DEFECT ENERGY (9.109) FROM STRESS ENERGY (9.90)

• NOTES AND REFERENCES

In the last chapter we focused attention mostly upon the gauge structure of stresses, since it leads to a simple defect representation of the melting model. Initially, however, when constructing the model for linear elasticity plus jump numbers n_ij, it was the defect gauge fields which played a primary role. The partition function (9.14) was invariant under the defect gauge transformations (9.15)–(9.18) and required a gaugefixing functional Φ which removed the gauge degeneracy. We stated that it was always possible to choose a gauge in which n_ij is quasi-symmetric and in which the symmetrized jump numbers have three components , , vanishing identically with the non-zero components satisfying the boundary conditions (9.20), (9.21). With Φ denoting the Kronecker δ enforcing these conditions, the partition function reads [note the differences with (9.14)]…

In the following two chapters we are going to analyze the thermal properties of the lattice model of defect melting in a way similar to that employed in the Villain model of the superfluid phase transition. For this we shall first expand the partition function Z of (9.40) into a high temperature series.

Let us now study the properties of the defect model and see whether it properly describes the phase transition of melting. We shall first consider the leading approximations at high and low temperatures, thereby obtaining crude estimates for the location and type of the transition. Then we shall proceed and calculate the corrections due to stress and defect graphs (which, in three dimensions, will be found to produce amazingly small effects).

For comparison we shall now investigate the melting transition in the cosine form of our model as defined in Eq. (9.147). For simplicity, let us omit the β dependent prefactors and consider the slightly modified partition function,…

Let us compare the properties of our defect models of melting with previous predictions on the behavior of two-dimensional defect melting as advanced by Kosterlitz, Thouless, Halperin, Nelson and Young (KTHNY).

Now that we possess a lattice model which respects all stack-up properties of the defects in a crystal it is possible to derive a consistent disorder field theory of crystalline defects. This field theory turns out to be rather different from the tentative construction presented in Chapter 8. In contrast to that theory for which a good deal of effort was spent in finding possible mechanisms for making the transition first-order, the proper disorder field theory to be proposed in this chapter will have a natural way of undergoing a first-order transition right at the mean field level. We shall see that as was the case with the field theory involving order fields [see Section 13.1] the disorder field theory will contain a temperature dependent quartic term which naturally causes a first-order transition (recall Fig. 13.1). The disorder field theory contains D complex disorder fields, one for every lattice direction. It describes all possible configurations of the defect tensor η_ij(x) and consequently a grand canonical ensemble of dislocations as well as disclinations.

In Chapter 15 we constructed a model of defect melting which, after a duality transformation, yielded a partition function with a sum over symmetric discrete defect tensors satisfying the conservation law . The diagonal parts of were integer, the off-diagonal parts half-integer. Such a sum contained only three independent sets of integer numbers which were not able to distinguish the full variety of all possible defect lines. So the question arises as to how we have to modify the model so that all defect lines appear explicitly in the partition function. In order to work towards an answer to this question let us first try and reformulate the continuous decomposition of the defect tensor according to dislocations and disclinations in such a way that it can be used on a lattice.

The following sections are included:

• TORQUE STRESSES

• GENERAL FORM OF THE ELASTIC ENERGY

• CANONICAL FORMALISM FOR HIGHER GRADIENT THEORIES

• SECOND-GRADIENT ELASTICITY

• CANONICAL FORMALISM FOR SECOND-GRADIENT ELASTICITY

• NOTES AND REFERENCES

Our goal is to investigate how our model of defect melting must be modified if we are to account for the higher gradient terms in linear elasticity. For this it is first necessary to understand the elastic properties of a continuous material which has undergone a given set of plastic deformations.

The following sections are included:

• THE PARTITION FUNCTION OF GENERAL DEFECT LINES IN THREE DIMENSIONS

• COSINE FORM OF THE PARTITION FUNCTION

• DISORDER FIELDS FOR DISLOCATIONS AND DISCLINATIONS

• TOWARDS A QUANTUM DEFECT DYNAMICS OF MOVING DEFECTS IN TWO DIMENSIONS

• NOTES AND REFERENCES

The defects in different physical systems have the common property that, in the continuum limit, certain closed contour integrals over field variables do not vanish due to singularities. If the field variables are spatial distortions, as in the case of crystals, the defects in the continuum may be efficiently described by means of differential geometry. A crystal filled with dislocations and disclinations turns out to have the same geometric properties as an affine space with torsion and curvature, respectively. Now, according to Einstein and later researchers, such a space forms the basis for a coordinate independent description of gravity. Mass points generate curvature, spinning matter gives rise to curvature and torsion. We may therefore expect many features of gravitational matter to coincide with those of crystalline defects…

The following sections are included:

• GRAVITY AND GEOMETRY

• MINKOWSKI GEOMETRY FORMULATED IN GENERAL COORDINATES
  • Local basis tetrads
  • Vectors and tensors
  • Connections and covariant derivatives

• TORSION TENSOR

• CURVATURE TENSOR AS A COVARIANT CURL OF THE CONNECTION

• TORSION AND CURVATURE FROM DEFECTS

• DIFFERENTIAL GEOMETRIC PROPERTIES OF METRIC-AFFINE SPACES WITH CURVATURE AND TORSION

• CIRCUIT INTEGRALS IN METRIC-AFFINE SPACES WITH CURVATURE AND TORSION

• SOME EXAMPLES OF COORDINATE SYSTEMS WITH DEFECTS

• IDENTITIES FOR CURVATURE AND TORSION TENSORS

• CURVATURE FROM EMBEDDING

• GEODESIC COORDINATES IN CURVED SPACE

The following sections are included:

• INVARIANT ACTION

• ENERGY-MOMENTUM TENSOR AND SPIN DENSITY

• SYMMETRIC ENERGY-MOMENTUM TENSOR OF THE GRAVITATIONAL FIELD AND DEFECT DENSITY

The following sections are included:

• LOCAL LORENTZ INVARIANCE AND NON-HOLONOMIC COORDINATES

• FIELD EQUATIONS WITH GRAVITATIONAL SPINNING MATTER

According Noether’s theorem, the invariance of the action under general coordinate transformations and local Lorentz transformations must be associated with certains conservation laws. For the following considerations, it will be convenient to consider and as independent variables and rename as . Then, from the derivation in (4.71) and (4.72), it follows that varying the action in at fixed gives the canonical energy-momentum tensor…

The following sections are included:

• LOCAL LORENTZ TRANSFORMATIONS

• LOCAL TRANSLATIONS

• LOCAL FOUR-FERMION INTERACTION DUE TO TORSION

• NOTES AND REFERENCES

After the preparations in the previous chapters it is now easy to construct a geometric theory of stresses and defects. With the defects described by the geometry of an affine space, we have to find an appropriate way of incorporating the correct long-range elastic interactions between the defects into the theory.

The following sections are included:

• SUMMARY AND OUTLOOK

• INDEX