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Appendix A Second

The most remarkable fact about a system of is that no more than one can occupy a particle state (Pauli’s exclusion principle). For no such restriction applies. That is, any number of bosons can occupy the same state. We shall formulate the in which creation and an- nihilation operators associated with each are used. This formalism is extremely useful in treating of many- and/or many-fermion systems. It is indispensable for the development of the superconductivity and theories.

A.1 Boson Creation and Annihilation Operators

The quantum state for a system of bosons (or fermions) can most conveniently be {  }  represented by a set of occupation numbers na with na being the numbers of bosons (or fermions) occupying the quantum particle-states a. This representation is called the occupation number representation or simply the number representation.  For bosons, the possible values for na are zero, one, two, or any positive integers:

 = , , , ··· . na 0 1 2 (A.1)

The many-boson state can best be represented by the distribution of particles (circles) in the states (boxes) as shown in Fig. A.1. Let us introduce operators na (without prime) whose eigenvalues are given by 0, 1,2,....Weassumethat

[na, nb] ≡ nanb − nbna = 0. (A.2)

It is convenient to introduce complex dynamic variables η and η† instead of di- rectly dealing with the number operators n. We attach labels a, b,...tothedynamic variables η and η† associated with the states a, b, . . . and assume that η and η† satisfy the following Bose commutation rules:

227 228 Appendix A Second Quantization

Fig. A.1 A many-boson state is represented by a set of boson numbers {n j } occupying the state {p j }

p 2 p 1 p0 p1 p2

, n 2 0, n 1 0, n 0 2, n1 0, n 2 1,

η ,η† = δ , η ,η = η† ,η† = . [ a b] ab [ a b] [ a b] 0 (A.3)

Let us set

η† η ≡ = † , a a na ( na) (A.4)

† which is Hermitian. Clearly, na ≡ ηaηa satisfy Equation (A.2).  We shall show that na has as eigenvalues all non-negative integers. Let n be an eigenvalue of n (dropping the suffix a) and |n an eigenket belonging to it. From the definition

n| η†η |n =n n|n . (A.5)

Now n| η†η |n is the squared length of the ket η|n and hence

n| η†η |n ≥0. (A.6)

Also by definition n|n > 0; hence from Equations (A.5) and (A.6), we obtain

n ≥ 0, (A.7) with the case of equality occurring only if

η|n =0. (A.8)

Consider now [η, n] ≡ [η, η†η] . We may use the following identities: A.1 Boson Creation and Annihilation Operators 229

[A, BC] = B[A, C] + [A, B]C , (A.9) [AB, C] = A[B, C] + [A, C]B, and obtain

[η, η†η] = η†[η, η] + [η, η†]η = η, ηn − nη. (A.10)

Hence

nη |n =(ηn − η) |n =(n − 1)η |n . (A.11)

Now if η |n = 0, then η |n is, according to Equation (A.11), an eigenket of n belonging to the eigenvalue n − 1. Hence for non-zero n, n − 1 is another eigenvalue. We can repeat the argument and deduce that if n − 1 = 0, then n − 2is another eigenvalue of n. Continuing this argument, we obtain a series of eigenvalues n, n −1, n −2,...,whichcanterminateonly with the value 0 because of inequality Equation (A.7). By a similar process, we can show from the Hermitean conjugate of Equation (A.10): nη† − η†n = η† that the eigenvalue of n has no upper limit (Problem A.1.1). Therefore, the eigenvalues of n are non-negative integers: 0, 1, 2,....(q.e.d) Let |φa be a normalized eigenket of na belonging to the eigenvalue 0 so that

|φ =η† η |φ = . na a a a a 0 (A.12)

By multiplying all these kets |φa together, we construct a normalized eigenket:

|⌽0 ≡|φa |φb ··· (A.13) which is a simultaneous eigenket of all n belonging to the eigenvalues zero. This ket is called the vacuum ket. It has the following property:

ηa|⌽0 =0 for any a. (A.14)

Using the commutation rules (A.3) we obtain a relation (dropping suffix a)

   η(η†)n − (η†)n η = n(η†)n −1, (A.15) which may be proved by induction (Problem A.1.2). Multiply Equation (A.15) by † η from the left and operate the result on |⌽0 . Using Equation (A.14) we obtain

  η†η(η†)n |φ =n(η†)n |φ , (A.16)

 indicating that (η†)n |ψ is an eigenket belonging to the eigenvalue n. The square  length of (η†)n |φ is 230 Appendix A Second Quantization

    φ|ηn (η†)n |φ =n φ|ηn −1(η†)n −1|φ =···=n!. (A.17)

We see from Equation (A.11) that η|n is an eigenket of n belonging to the eigenvalue n −1. Similary, we can show from [n,η†] = η† that η†|n is an eigenket of n belonging to the eigenvalue n + 1. Thus η, acting on the number eigenket, annihilates a particle while operator η† creates a particle. Therefore, η and η† are called annihilation and creation operators, respectively. From Equations   (A.16) and (A.17) we infer that if n1, n2,...areanynon-negative integers,

  − / †  †    ··· 1 2 η n1 η n2 ···|⌽ ≡| , , ··· (n1! n2! ) ( 1) ( 2) 0 n1 n2 (A.18)

 is a normalized simultaneous eigneket of all the n belonging to the eigenvalues n1,   n2, .... Variousketsobtainedbytakingdifferent n form a complete set of kets all orthogonal to each other. Following Dirac [1], we postulate that the quantum states for N bosons can be represented by a symmetric ket |α(1) |α(2) ···|α(N) ≡|α α ···α , S a b g a b g S (A.19) where S is the symmetrizing operator:

1 S ≡ √ P (A.20) N! P and P are permutation operators for the particle-indices (1, 2, ..., N). The ket in Equation (A.19) is not normalized but

−1/2 (n1! n2! ···) |αaαb ···αg S ≡|{n} (A.21) is a normalized ket representing the same state. Comparing Equations (A.21) and (A.18), we obtain

|α α ···α = η† η† ···η† |⌽ . a b g S a b g 0 (A.22)

The unnormalized symmetric kets |αaαb ···αg S for the system can be con- η† η† ···η† |⌽ structed by applying N creation operators a b g to the vacuum ket 0 .So far we have tacitly assumed that the total number of bosons is fixed at N .Ifthis number is variable, we can easily extend the theory to this case. Let us introduce a Hermitean operator N defined by = η† η = , N a a na (A.23) a a A.2 Observables 231 the summation extending over the whole set of boson states. Clearly, the operator N haseigenvalues0,1,2,...,andtheket|αaαb ···αg S is an eigenket of N belonging to the eigenvalue N . We may arrange kets in the order of N , i.e., zero-particle state, one-particle states, two-particle states, ...:

|⌽ ,η† |⌽ ,η† η†|⌽ , ··· . 0 a 0 a b 0 (A.24)

These kets are all orthogonal to each other. Two kets referring to the same number of bosons are orthogonal as before, and two referring to different numbers of bosons are orthogonal because they have different eigenvalues N . By normalizing the kets, we get a set of kets like (A.21) with no restriction on {n}. These kets form the basic kets in a representation where {na} are diagonal. Problem A.1.1. (a) Show that nη†−η†n = η†, by taking the Hermitian-conjugation of Equation (A.10) and also by using Equations (A.9). (b) Use this relation and obtain a series of eigenvalues n, n + 1, n + 2, ..., where n is an eigenvalue of n. Problem A.1.2. Prove Equation (A.15) by mathematical induction. Hint: use Equa- tions (A.9).

A.2 Observables

We wish to express observable physical quantities (observables) for the system of identical bosons in terms of η and η† These observables are by postulate symmetric functions of the boson variables. An observable may be written in the form: y( j) + z(ij) +···≡Y + Z +··· , (A.25) j i j where y( j) is a function of the dynamic variables of the jth boson, z(ij) that of the dynamic variables of theith and jth bosons, and so on. ≡ ( j) ( j) |α( j) First, consider Y j y . Because y acts only on the ket of the jth boson, we obtain y( j) |α(1) |α(2) ···|α( j) ··· x1 x2 x j = |α(1) |α(2) ···|α( j) ··· α( j)|y( j)|α( j) . x1 x2 a a (x j ) a (A.26)

α( j)| ( j) |α( j) ≡ α | |α The matrix element a y x j a y x j does not depend on the parti- cle index j since we consider identical bosons. Summing Equation (A.26) over all 232 Appendix A Second Quantization j’s and applying operator S to the result, we obtain SY |α(1) |α(2) ··· x1 x2 = S |α(1) |α(2) ···|α(1) |α( j) ··· α | y |α . x1 x2 x1 a a x j j a (A.27)

Because Y is symmetric, we can replace SY by YSfor the lhs. After straightforward calculations, we obtain from Equation (A.27) Y η† η† ···|⌽ = η† η† ···η† η† η† ···|⌽ α | y |α x1 x2 0 x1 x2 x j−1 a x j+1 0 a x j j a = η† η† η† ···η† η† ···|⌽ δ α | y |α . a x1 x2 x j−1 x j+1 0 bx j a b a b j (A.28)

Using the commutation rules and the property (A.14) we can show that η η† η† ···|⌽ = η† η† ···η† η† ···|⌽ δ (A.29) b x1 x2 0 x1 x2 x j−1 x j+1 0 bx j j

(Problem A.2.1). Using this relation, we obtain from Equation (A.28) Y η† η† ···|⌽ = η† η α | y |α η† η† ···|⌽ . (A.30) x1 x2 0 a b a b x1 x2 0 a b

η† η† ···|⌽ Since the kets x1 x2 0 form a complete set, we obtain = η† η α | |α . Y a b a y b (A.31) a b

In a similar manner Z in Equation (A.25) can be expressed by (Problem A.2.2) = η† η†η η α α | |α α , Z a b d c a b y c d (A.32) a b c d α α | |α α ≡ α(1)| α(2)| (12)|α(2) |α(1) . a b y c d a b z d c (A.33)

Problem A.2.1. Prove Equation (A.29). Hint: Start with cases of one- and two- particle-state kets. Problem A.2.2. Prove Equation (A.32) by following those steps similar to (A.27)- (A.31). A.3 Fermion Creation and Annihilation Operators 233

A.3 Fermion Creation and Annihilation Operators

In this section we treat a system of identical fermions in a parallel manner. The quantum states for fermions, by postulate, are represented by antisymmetric kets: |α α ···α ≡ |α(1) |α(2) ···|α(N) , a b g A A a b g (A.34) where

1 A ≡ √ δP P, (A.35) N! P is the antisymmetrizing operator, with δP being +1or−1 according to whether P is even or odd. Each antisymmetric ket in Equation (A.34) is characterized such that it changes its sign if an odd permutation of particle indices is applied to it, and the fermion states a, b,...,g are all different. Similar to a boson system, we can intro- duce observables n1, n2, . . . each with eigenvalues 0 or 1, representing the number of fermions in the states α1, α2,...respectively. The many-fermion occupation-number state can be represented as shown in Figure A.2. † † We can also introduce a set of linear operators (η, η ), one pair (ηa,ηa) for each state αa, satisfying the Fermi anticommutation rules:

{η ,η†}≡η η† + η†η = δ , {η ,η }={η† ,η†}= . a b a b b a ab a b a b 0 (A.36)

The number of fermions in the state αa is again represented by

= η† η = † . na a a na (A.37)

Fig. A.2 A many-fermion state is represented by the set of the numbers {n j } occupying the states p 2 p 1 p0 p1 p2 {p ≡ 2πL−1 j}. Each n is j j , n 1, n 0, n 1, n 1, n 0 , restricted to 0 or 1 2 1 0 1 2 234 Appendix A Second Quantization

Using Equations (A.36), we obtain

2 = η† η η† η = η† − η† η η = η† η = 2 − = . na a a a a a(1 a a) a a a na or na na 0 (A.38)  |  If an eigenket of na belonging to the eigenvalue na is denoted by na , then Equa- tion (A.38) yields

2 − |  =  2 −  |  = . (na na) na (na na) na 0 (A.39) |  =   − =  Since na 0, we obtain na(na 1) 0, meaning that the eigenvalues na are either 0 or 1 as required:

 = . na 0or1 (A.40)

Similarly to bosons, we can show that

|α α ···α = η† η† ···η† |⌽ , a b g A a b g 0 (A.41) which is normalized to unity. Observables describing the system of fermions can be expressed in terms of op- erators η and η†, and the results have the same forms as for the case of bosons, Equations (A.31) and (A.32). In summary both states and observables for a system of (fermions, bosons) can be expressed in terms of creation and annihilation opera- tors. This formalism, called the second quantization formalism, has some notable advantages over the usual Schrodinger¨ formalism. First, the permutation- property of the quantum particles is represented simply in the form of Bose commu- tation (or Fermi anticommutation) rules. Second, observables in second quantization are defined for an arbitrary number of particles so that the formalism may apply to systems in which the number of particles is not fixed but variable. Third, and most importantly, all relevant quantities (states and observables) can be defined referring only to the single-particle states. This property allows us to describe the motion of the many-body system in the 3D space. We shall explain this in the following section. This is a natural description. In fact, relativistic quantum field theory can be developed only in second quantization.

A.4 Heisenberg Equation of Motion

In the Schrodinger¨ Picture (SP), the energy eigenvalue equation is

H |E =E |E , (A.42) where H is the Hamiltonian and E the eigenvalue. In the position representation this equation is written as A.4 Heisenberg Equation of Motion 235

H(x1, −iѨ/Ѩx1, x2, −iѨ/Ѩx2, ···)⌿(x1, x2, ···) = E⌿, (A.43) where ⌿ is the for the system. We consider a one-dimensional motion for conceptional and notational simplicity. [For a three-dimensional motion, (x, p) should be replaced by (x, y, z, px , py, pz) = (r, p).] If the number of electrons N is large, the wave function ⌿ contains many electron variables (x1, x2, ···). This complexity in dealing with many electron variables can be avoided if we use the second quantization formulation and the Heisenberg Picture (HP), which will be shown in this section. If the Hamiltonian H is the sum of single-particle Hamiltonians: H = h( j), (A.44) j this Hamiltonian H can be represented by = α | |α η† η ≡ η† η , H a h b a b hab a b (A.45) a b a b

† where ηa (ηa) are annihilation (creation) operators associated with particle-state a and satisfying the Fermi commutation rules. In the HP a variable ξ(t) changes in time, following the Heisenberg equation of motion:

dξ(t) − i = [H,ξ] ≡ Hξ − ξ H. (A.46) dt

† Setting ξ = ηa, we obtain

† dη − i a = [H,η† ], (A.47) dt a whose Hermitian conjugate is given by

dη i a = ([H,η† ])† =−[H,η]. (A.48) dt a a

† † By postulate the physical observable ξ is Hermitian: ξ = ξ. Variables ηa and ηa are not Hermitian, but both obey the same Heisenberg equation of motion. We introduce Equation (A.45) into Equation (A.47), and calculate the commu- † tator [H,ηa]. In such a calculation, identities Equations (A.9) and the following identities: 236 Appendix A Second Quantization

[A, BC] ={A, B}C − B{A, C} , (A.49) [AB, C] = A{B, C}−{A, C}B , are very useful. Note: The negative signs on the right-hand terms in Equations (A.49) occur when the cyclic order is destroyed. We obtain from Equations (A.47) and (A.48)

† dη −i a = h [η†η ,η† ] = h η†{η ,η† }= h η† (A.50) dt cb c b a cb c b a ca c c b c b c dη i a = h η . (A.51) dt ac c c

† Equation (A.50) means that the change of the one-body operator ηa is determined by the one-body Hamiltonian h. This is the main advantage of working in the HP. Equations (A.50) and (A.51) are valid for any single-particle state {a}. In the field operator language Equation (A.51) reads

Ѩψ(r, t) i = h(r, −iѨ/Ѩr)ψ(r, t), (A.52) Ѩt which is formally identical to the Schrodinger¨ equation of motion for a particle. The ψ represents a field operator, however. If the system Hamiltonian H contains an interparticle interaction

1 V = d3r d3r  v(r − r)ψ†(r, t)ψ†(r, t)ψ(r, t)ψ(r, t), (A.53) 2 the evolution equation for ψ(r, t) is nonlinear (Problem A.4.2):

Ѩψ(r, t) i = h(r, −iѨ/Ѩr)ψ(r, t) Ѩt + d3r v(r − r)ψ†(r, t)ψ(r, t)ψ(r, t). (A.54)

In quantum field theory the basic dynamical variables are particle-field operators. The quantum statistics of the particles are given by the Bose commutation or the A.4 Heisenberg Equation of Motion 237

Fermi commutation rules satisfied by the field operators. The evolution equations of the field operators are intrinsically nonlinear when the interparticle interaction is present.

Problem A.4.1. Verify that the equation of motion Equation (A.50) holds for bosons. Problem A.4.2. Use Equation (A.46) to verify Equation (A.54) Bibliography

Superconductivity

Introductory and Elementary Books Lynton, E. A.: Superconductivity, (Methuen, London, 1962). Vidali, G.: Superconductivity, (Cambridge University Press, Cambridge, England, 1993).

General Textbooks at About the Same Level as the Present Text Feynman, R. P., Leighton R. B. and Sands, M.: Feynman Lectures on , Vol. 3, (Addison- Wesley, Reading, MA, 1965), pp. 1–19. Feynman, R. P.: Statistical Mechanics, (Addison-Wesley, Reading, MA, 1972), pp. 265–311. Rose-Innes, A. C. and Rhoderick, E. H.: Introduction to Superconductivity, 2nd ed., (Pergamon, Oxford, England, 1978).

Advanced Texts and Monographs Abrikosov, A. A.: Fundamentals of The Theory of Metals, A. Beknazarov, trans., (North Holland- Elsevier, Amsterdam, 1988). Fujita, S. and Godoy, S.: Quantum Statistical Theory of Superconductivity, (Plenum-Springer, New York, 1996). Gennes, P.: Superconductivity of Metals and Alloys, (Benjamin, Menlo Park, CA, 1966). Rickayzen, G.: Theory of Superconductivity, (Interscience, New York, 1965). Saint-James, D., Thomas, E. J. and Sarma, G.: Type II Superconductivity, (Pergamon, Oxford, England, 1969). Schafroth, M. R.: Solid State Physics, Vol. 10, F. Seitz and D. Turnbull eds., (Academic, New York, 1960), p. 488. Schrieffer, J. R.: Theory of Superconductivity, (Benjamin, New York, 1964). Tilley, D.R. and Tilley, J.: Superfluidity and Superconductivity, 3rd ed., (Adam Hilger, Bristol, England, 1990). Tinkham, M.: Introduction to Superconductivity, (McGraw-Hill, New York, 1975).

High-Temperature Superconductivity

Anderson, P. W.: Theory of Superconductivity in High-Tc Cuprates, (Princeton University Press, Princeton, NJ, 1997). Burns, G.: High-Temperature Superconductivity, An Introduction, (Academic, New York, 1992).

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Fujita, S. and Godoy, S.: Theory of High Temperature Superconductivity, (Kluwer, Dordrecht, Netherland, 2001). Ginsberg D. M., ed.: Physical Properties of High-Temperature Superconductors, (World Scientific, Singapore, 1989)-(series). Halley, J. W., ed.: Theory of High-Temperature Superconductivity, (Addison-Wesley, Redwood City, CA, 1988). Kresin,V.Z.andWolfS.A.:Fundamentals of Superconductivity, (Plenum, New York, 1990). Kresin,W.Z.:Novel Superconductivity, (Plenum, New York, 1989). Lindquist, S., et al., eds.: Towards the Theoretical Understanding of High-Tc Superconductivity, Vol. 14, (World Scientific, Singapore, 1988). Lynn, J. W., ed.: High Temperature Superconductivity, (Springer-Verlag, New York, 1990). Owens, F. J. and Poole, C. P.: New Superconductors, (Plenum, New York, 1996). Phillips, J. C.: Physics of High-Tc Superconductors, (Academic Press, San Diego, CA, 1989). Poole, C. P., Farach, H. A. and Creswick, R. J.: Superconductivity, (Academic, New York, 1995). Sheahen, T. P.: Introduction to High-Temperature Superconductivity, (Plenum, New York, 1994). Waldram J. R.: Superconductivity of Metals and Cuprates, (Institute of Physics Publishing, Bristol, 1996).

Background

Solid State Physics Ashcroft, N. W. and Mermin, N. D.: Solid State Physics, (Saunders, Philadelphia, PA, 1976). Harrison, W. A.: Solid State Theory, (Dover, New York, 1979). Haug, A.: Theoretical Solid State Physics, I, (Pergamon, Oxford, England, 1972). Isihara, A.: , (Oxford University Press, New York, 1991). Kittel, C.: Introduction to Solid State Physics, 6th ed., (Wiley, New York, 1986). Seitz, F.: Modern Theory of Solids, (McGraw-Hill, New York, 1940). Sommerfeld, A. and Bethe, H.: Handbuch der Physik 24, Part 2, (Springer, Berlin, 1933). Wannier,G.H.:Elements of Solid State Theory, (Cambridge University Press, London, 1960). Wilson, A. H.: Theory of Metals, 2nd ed., (Cambridge University Press, London, 1953).

Mechanics Goldstein, H.: , (Addison Wesley, Reading, MA, 1950). Kibble, T. W. B.: Classical Mechanics, (McGraw-Hill, London, 1966). Marion, J. B.: Classical Dynamics, (Academic, New York, 1965). Symon, K. R.: Mechanics, 3rd ed., (Addison-Wesley, Reading, MA, 1971).

Quantum Mechanics Alonso, M. and Finn, E. J.: Fundamental University Physics, III Quantum and Statistical Physics, (Addison-Wesley, Reading, MA, 1989). Dirac, P. A. M.: Principles of , 4th ed., (Oxford University Press, London, 1958). Gasiorowitz, S.: Quantum Physics, (Wiley, New York, 1974). Liboff, R. L.: Introduction to Quantum Mechanics, (Addison-Wesley, Reading, MA, 1992). McGervey,J.D.:Modern Physics, (Academic Press, New York, 1971). Pauling, L. and Wilson, E. B.: Introduction to Quantum Mechanics (McGraw-Hill, New York, 1935). Powell, J. L. and Crasemann, B.: Quantum Mechanics, (Addison-Wesley, Reading, MA, 1961). Bibliography 241

Electricity and Magnetism Griffiths, D. J.: Introduction to Electrodynamics, 2nd ed., (Prentice-Hall, Englewood Cliffs, NJ, 1989). Lorrain, P. and Corson, D. R.: , (Freeman, San Fransisco, CA, 1978). Wangsness, R. K.: Electromagnetic Fields, (Wiley, New York, 1979).

Thermodynamics Andrews, F. C.: Thermodynamics: Principles and Applications, (Wiley, New York, 1971). Bauman,R.P.:Modern Thermodynamics with Statistical Mechanics, (Macmillan, New York, 1992). Callen, H. B.: Thermodynamics, (Wiley, New York, 1960). Fermi, E.: Thermodynamics, (Dover, New York, 1957). Pauli, W.: Thermodynamics and the Kinetic Theory of Gases, (Dover, New York, 2000). Pippard, A. B.: Thermodynamics: Applications, (Cambridge University Press, Cambridge, Eng- land, 1957).

Statistical Physics (undergraduate) Baierlein, R.: Thermal Physics, (Cambridge U.P., Cambridge, UK, 1999). Carter, A. H.: Classical and Statistical Thermodynamics, (Prentice-Hall, Upper Saddle River, NJ, 2001). Fujita, S.: Statistical and Thermal Physics, I and II, (Krieger, Malabar, FL, 1986). Kittel, C. and Kroemer, H.: Thermal Physics, (Freeman, San Francisco, CA, 1980). Mandl, F.: Statistical Physics, (Wiley, London, 1971). Morse,P.M.:Thermal Physics, 2nd ed., (Benjamin, New York, 1969). Reif, F.: Fundamentals of Statistical and Thermal Physics, (McGraw-Hill, New York, 1965). Rosser, W. G. V.: Introduction to Statistical Physics, (Horwood, Chichester, England, 1982). Terletskii, Ya. P.: Statistical Physics, N. Froman, trans., (North-Holland, Amsterdam, 1971). Zemansky,M.W.:Heat and Thermodynamics, 5th ed., (McGraw-Hill, New York, 1957).

Statistical Physics (graduate) Davidson, N.: Statistical Mechanics, (McGraw-Hill, New York, 1969). Feynman, R. P.: Statistical Mechanics, (Benjamin, New York, 1972). Finkelstein, R. J.: Thermodynamics and Statistical Physics, (Freeman, San Fransisco, CA, 1969). Goodstein, D. L.: States of Matter, (Prentice-Hall, Englewood Cliffs, NJ, 1975). Heer, C. V.: Statistical Mechanics, Kinetic Theory, and Stochastic Processes, (Academic Press, New York, 1972). Huang, K.: Statistical Mechanics, 2nd ed., (Wiley, New York, 1972). Isihara, A.: Statistical Physics, (Academic, New York, 1971). Kestin,J.andDorfman,J.R.:Course in Statistical Thermodynamics, (Academic, New York, 1971). Landau, L. D. and Lifshitz, E. M.: Statistical Physics, 3rd ed., Part 1, (Pergamon, Oxford, England, 1980). Lifshitz,E.M.andPitaevskii,L.P.:Statistical Physics, Part 2, (Pergamon, Oxford, England, 1980). McQuarrie, D. A.: Statistical Mechanics, (Harper and Row, New York, 1976). Pathria, R. K.: Statistical Mechanics, (Pergamon, Oxford, England, 1972). Robertson, H. S.: Statistical Thermodynamics, (Prentice Hall, Englewood Cliffs, NJ, 1993). Wannier,G.H.:Statistical Physics, (Wiley, New York, 1966). Index

A Blurred Fermi surface, 43 Abrikosov, 103, 104, 105, 108, 132, 138, 209 Bohr magneton, 184 Abrikosov structure, 105, 108 Boltzmann distribution, 188, 201 Ac Josephson effect, 143, 150–153, 154 Boltzmann factor, 97 Acoustic , 42, 103, 111, 161, 178, 194, Bose commutation rule, 21, 61, 65, 79, 227 195, 200, 221 Bose-condensed state, 55 Affinity between electrons and , 19 Bose distribution function, 80, 188 Anderson, 143, 144, 214, 215 Bose–Einstein Condensation (BEC), 73, Anderson and Rowell, 143, 144, 214 79–83, 84–86, 88, 157, 178, 187, Angle-Resolved PhotoEmission Spectroscopy 213–214, 215 (ARPES), 88, 196 Boson enhancement effect, 147 Angular frequency, 17, 18, 116, 125, 150 Bound electron-pair, 27 Anisotropic optical-phonon-exchange, 193 Brillouin zone, 18, 43, 161, 175, 184, 195 Antiferromagnetic, 173, 174, 182, 205, 215 Bulk limit, 17, 37, 38, 54, 57, 80, 84, 165, 211 Antiferromagnetic insulator, 173, 182, 205, 215 C Antisymmetric state, 65, 69 Cassimir, 207 Cause of superconductivity, 10, 18, 19 B Center-of-mass (CM), 52, 61, 62, 63, 64, 65, Bardeen, Cooper and Schrieffer (BCS), 10, 11, 68, 69, 70, 73, 94, 113, 178 12, 18, 29–43, 45, 46, 50, 53, 55, 58, 59, Charge state, 97, 159 60, 70, 79, 86, 88, 95, 105, 111, 114, 115, Chu and Wu, 215 132, 135, 136, 137, 155, 162, 163, 164, Circumference, 114, 118, 127, 134, 220 167, 168, 173, 194, 210–213, 215, 220, 222 Classical (Boltzmann) distribution func- BCS coherence length, 105, 167 tion, 188 BCS energy gap equation, 38, 137 CM motion, 62, 65, 69, 70, 178 BCS formula, 38, 46, 58, 59, 87 Coherence length, 105, 108, 132, 136, 138, BCS Hamiltonian, 12, 18, 29, 30, 32, 39, 53, 157, 161, 167, 168, 204, 209, 213, 223 59, 111, 114, 155, 162, 163, 210, 211, Coherent, 116 212, 222 Complex long-range order, 6 BCS-like Hamiltonian, 162 Complex order parameter, 131, 208 BCS picture, 115, 220 Concentration, 105, 168, 173, 175, 181, 183, Bednorz, 9, 157, 197, 215 189, 197, 215 Bednorz and Muller,¨ 9, 157, 215 Condensate density, 95, 136 Bessel function, 152 Condensed pairon, 45, 56, 59, 60, 86, 91, 95, Binding energy, 37, 39–40, 75, 92, 96, 140, 96, 103, 106, 120, 126, 127, 131, 132, 133, 169, 205, 211, 221 134, 135, 136, 137, 140, 152, 153, 154, Bloch system, 31, 34, 37, 211 207, 208, 209, 220, 221, 224 Bloch theorem, 159 Constriction, 126, 144, 145

243 244 Index

Continuous with no energy gap, 79 Density of states, 11, 17, 75, 77, 79, 87, 94, Cooper, 4, 10–12, 15, 27, 30, 39, 41–42, 136, 178, 181, 184, 185, 186, 188, 189, 51–54, 59, 60, 61, 70, 75–77, 79, 108, 135, 190, 204, 211 139, 140, 157, 162, 166, 173, 175, 181, Density-wave modes, 19 187, 193–196, 210, 213, 219, 221, 223 Discontinuous drop, 85 Cooper Hamiltnian, 53 Dispersion relation, 73, 75, 80, 87, 88, 110, Cooper pair, 4, 10, 12, 15, 27, 30, 41–42, 51, 153, 155, 157, 161, 166, 178, 187, 189, 59, 61, 75–77, 108, 135, 140, 157, 162, 194, 195, 200, 213, 214, 219, 220, 221 173, 175, 181, 187, 193–196, 210, 219, Displacement, 15, 19, 20, 21, 108, 110, 121 221, 223 Distinguishable particle, 65 Cooper’s equation, 54, 76, 77, 210 Dome-shaped curve, 175 Copper plane, 157, 159, 160, 161, 163, 166, Doping, 87, 157, 173–179, 183, 184, 189, 190, 173, 174, 175, 176, 177, 178, 182, 183, 195, 197, 198, 199, 202, 204, 205, 215 189, 190, 195, 202, 205 Down-spin, 51, 125, 185, 186 Cosine law, 159 d-wave, 157, 162, 173, 175, 178, 193–196 Coulomb interaction, 24, 27, 30, 31, 193, 212 d-wave Cooper pair, 157, 162, 175, 193–196 Coulomb repulsion, 10, 11, 29, 30, 163, d-wave type, 193, 195 168, 222 Creation operator, 47, 67, 70, 73, 134, 230, 235 E Critical current, 116, 127, 221, 223–224, 225 Effective Hamiltonian, 24, 25, 121 Critical field, 8, 9, 39–40, 103, 104, 107, 115, Effective interaction, 27, 193 127, 136, 138, 167, 224 Effective mass, 29, 30, 32, 78, 114, 159, 160, Critical temperature, 1, 7, 8, 9, 11, 12, 29, 39, 163, 184, 198, 204 45, 49, 57, 58, 59, 73, 81, 84, 86, 87, 103, Ehrenfest–Oppenheimer–Bethe (EOB) rule, 104, 107, 131, 136, 157, 166–168, 173, 61–62, 68, 70 176, 177, 183, 204, 209, 211, 213, 215, Eigenvalue equation, 46, 48, 49, 60, 76, 77, 234 219, 221, 222, 223 Electric current density, 114, 126 Cu-Fermi sheet, 162, 175, 190, 195 Electron, 10, 11, 12, 15–27, 29, 30, 31, 32, 36, Cu-Fermi surface, 175, 177, 178, 205 37, 40, 42, 43, 45, 46, 47, 50, 51, 56, 59, Cuprate superconductor, 9, 100, 157, 166, 169, 60, 62, 64, 65, 69, 70, 73, 75, 78, 79, 89, 170, 215, 222 90, 91, 97, 99, 101, 103, 111, 114, 115, Current density, 94, 99, 114, 122, 126, 132, 124, 137, 147, 159, 161, 162, 163, 168, 199, 209, 220 173, 175, 176, 177, 178, 184, 185, 190, Curvature inversion, 177, 178, 181, 184, 189, 193, 194, 195, 197, 198, 204, 205, 207, 190, 195, 196, 204, 205 208, 209–210, 211, 212, 213, 215, 216, Curvature sign change, 176, 184 219, 220, 221, 222, 223, 235 Cusp, 101 Electron band, 12, 13, 30, 42, 89, 95, 173, 222 Cyclotron frequency, 187, 188, 189 Electron gas system, 23, 26 Cyclotron resonance, 159 Electronic heat capacity, 6, 7, 167, 168 Electron-pair operator, 79 Electron–phonon system, 23, 25 D “Electron” wave packet, 162, 195 dc Josephson effect, 4 Elementary fermion, 61, 68, 70, 71 Deaver, 4, 115 Energy-eigenvalue equation, 48, 60, 76 Debye’s continuum model, 18 Energy gap, 6, 7, 11, 12, 29, 36, 37, 38, 39, Deformation potential approximation, 20 40–41, 45–60, 79, 88, 89, 90, 91, 92, 95, Deformation potential model, 194 96, 97, 99, 100, 101, 131, 133, 136, 137, de Haas–van Alphen (dHvA) oscillation, 159 138, 140, 157, 165, 168–170, 209, 211, Delta function, 184, 187, 189 212, 216, 219, 221, 222 Delta-function singularity, 189 Energy gap equation, 36, 38, 40–41, 45–60, Density operator, 25, 46, 47, 48, 50, 68, 137, 165, 211 113, 123, 124, 125, 131, 134, 135, 140, Energy momentum relation, 75, 79, 111, 120, 153–154, 209 137, 139 Index 245

Equal numbers of ±pairons, 37, 41, 56, 59, Ginzburg–Landau Theory, 6, 131–141, 113, 114, 126, 165, 208, 215, 219 208–209 Equation-of-motion method, 41, 45, 50, 59, GL wavefunction, 131, 132, 134, 135, 139, 60, 165 140, 152, 153–154, 208, 209 Gorter, 207 F Gorter–Cassimir’s Model, 207 Factorization, 54, 135 Grand-canonical-ensemble average, 12, 53 Fairbank, 4, 115 Ground pairon, 11, 33, 34, 42, 60, 92, 124, 125 Fermi anticommutation rule, 10, 22, 47, 52, 62, Ground state, 6, 11, 29–43, 45, 49, 50, 55, 56, 65, 79, 233, 234 59, 70, 73, 75, 76, 89, 95, 113, 135, 136, Fermi distribution function, 54, 99, 186 137, 139, 163–165, 169, 210, 211, 212, Fermi sea, 75 215, 216, 219 Fermi surface, 10, 12, 27, 29, 30, 42, 43, 75, Ground state energy, 6, 11, 29, 35, 37, 38, 39, 77, 78, 89, 111, 114, 125, 159, 160, 161, 40, 45, 59, 75, 76, 89, 135, 136, 137, 139, 162, 163, 166, 167, 173–178, 181, 183, 164, 165, 169, 211, 212, 215, 219 184, 185, 189, 190, 193, 194, 195, 196, 204, 205, 210, 212, 219, 222 H Fermi temperature, 166, 183 Hall angle, 88, 197, 200, 202 Fermi velocity, 19, 75, 87, 88, 114, 166, 167, Hall coefficient, 88, 173, 174, 176, 181, 183, 219, 222 184, 197, 199, 201–202, 204 Feynman, 22, 23, 24, 32, 34, 61, 70, 146, 147, Hall effect, 200, 215–216, 227 149, 154, 214 Harmonic approximation, 16, 17 , 22, 23, 24, 32, 34 Harmonic equation of motion, 20 Floating magnet, 1, 2 Harmonic oscillator, 20, 21 Fluxoid, 123 Heat capacity, 6, 7, 15, 16, 17, 43, 81, 82, 83, Flux quantization, 1, 3–4, 6, 12, 13, 41, 103, 85, 107, 167, 168, 190, 207, 221 106, 108, 113–129, 146, 157, 211, 219, 220 Heavy fermion, 10, 18, 56, 61, 62, 65, 66, 68, Flux quantum, 4, 104, 106, 147, 220 69, 70, 71, 79, 80, 99, 101, 133, 134, 135, Frohlich,¨ 15, 19, 22, 23, 209, 210 146, 147, 153, 198, 214, 216, 227, 233–234 Frohlich¨ Hamiltonian, 15, 19, 22, 210 Heisenberg equation of motion, 234–237 Frohlich¨ interaction Hamiltonian, 23 Heisenberg picture, 235 Fugacity, 81, 82 Hermitian conjugate, 22, 235 Full Hamiltonian, 73–75 High critical temperature, 87, 157, 166–168 High-Tc cuprate superconductor, 100, 157, 215 G High temperature superconductor, 9, 157–171, Gap, 6, 7, 11, 12, 36, 37, 38, 39, 40–41, 45–60, 215, 222 79, 89, 90, 91, 92, 95, 96, 97, 100, 101, Hohenberg’s theorem, 83, 214 126, 131, 133, 136, 137, 140, 165, 170, “Hole”, 11, 12, 29, 30, 31, 32, 33, 34, 36, 37, 209, 211, 212, 213, 216, 221 39, 42, 43, 51, 53, 73, 74, 88, 89, 93, 96, Gap equation, 36, 37, 38, 40–41, 45–60, 137, 111, 114, 117, 125, 131, 161, 162, 163, 165, 211 174, 175, 176, 177, 183, 184, 185, 189, Gauge-choice, 120 190, 195, 197, 198, 200, 201, 202, 204, Generalized BCS Hamiltonian, 18, 29, 30, 59, 205, 210, 211, 212, 219, 222, 224, 231 111, 155, 162, 163, 212, 222 “Hole” wave packet, 162, 195 Generalized BCS model, 173, 215 Hyperboloidal Fermi surface, 42, 43, 222 Generalized energy-gap equation, 57 Generalized London penetration depth, 121 I Generalized London’s equation, 121 Impurity, 198, 203 Generalized Ohm’s law, 92, 100 Inflection point, 181, 184, 189, 190, 195 Giaever and Megerle, 7, 90, 91, 98 Instantaneous, 24 Ginzburg–Landau equation, 132 Interaction strength, 30, 37, 135, 140, 163, Ginzburg and Landau (GL), 105, 131, 132, 168, 169, 194, 209, 210 133, 134, 135, 136, 138, 139, 140, 152, Interference, 5, 12, 143–147, 153, 155, 153, 154, 208, 209 214, 220 246 Index

Intermediate state, 223, 224 Longitudinal acoustic phonon, 161, 194 Internal energy density, 81, 84 Longitudinal optical phonon, 88, 157, 178 Interpairon distance, 86, 87, 166, 167, 213, 222 Longitudinal wave, 19, 110 Ionicity, 173, 175, 182, 190 Long-range order, 6, 214 I–V curve, 89, 90, 91, 92, 96, 97, 98, 100, 101, Lorentz-magnetic force, 115, 127, 202, 220 143, 144, 169–170, 221, 222 Lower critical field, 8, 103, 104, 224

J M Josephson, 1, 4–6, 12, 13, 70, 92, 103, 115, Macrowavefunction, 126, 208 143–155, 157, 193, 214–215, 219, 220 Magnetic flux, 1, 3, 4, 5, 8, 12, 104, 106, 122, Josephson current, 5, 146, 147–150, 153 123, 136, 138, 145, 146, 149, 220, 224 Josephson–Feynman equation, 149, 154 Magnetic impurity, 198 Josephson frequency, 151, 154, 214 Magnetic pressure, 1, 106, 127 Josephson interference, 5, 12, 214, 220 Magnetic property, 3, 103, 181 Josephson junction, 4, 5, 143, 144, 146, 147, Magnetization, 104, 105, 107, 131, 183, 185, 153, 154, 214 187, 188, 189, 208, 223 Josephson theory, 214–215 Main superconducting property, 103, 157 Josephson tunneling, 4, 92, 143–147, 153, Major properties of a superconductor, 219 193, 214 Many-boson state of motion, 114 Markoffian approximation, 26, 27 K Massless boson, 83, 84, 85, 146, 147, 168, Kinetic theory, 198–202 214, 222 k-q representation, 63, 74 Massless particle, 79, 128 k-vector, 18, 54, 111, 116, 159, 160, 162 Maximum heat capacity, 85, 168, 221 Meissner, 1–3, 8, 9, 13, 39, 103, 104, 106, 108, L 121, 127–128, 136, 157, 207, 219, 220 La2−xSrxCuO4, 174, 182, 183, 199 Meissner effect, 1–3, 8, 13, 39, 127–128, 136, Landau, 6, 10, 105, 131–141, 183, 184, 188, 157, 219, 220 208–209, 215 Meissner pressure, 106, 127 Laser, 5, 12, 115, 116, 143, 146, 147, 153, 155, Meissner state, 103, 104, 106, 121, 127 214, 220 Metallic compound, 8 Lattice-dependent anisotropy, 193 Mixed state, 8, 104, 224 Lattice dynamics, 15–19 Molar heat capacity, 81, 82, 83, 85 Layered lattice structure, 222 Momentum, 19, 22, 24, 26, 27, 31, 39, 42, 46, Layered structure, 157–160 47, 49, 50, 53, 54, 56, 63, 64, 65, 68, 69, Linear, 20, 43, 49, 73, 78, 79, 80, 88, 110, 114, 70, 74–79, 80, 81, 84, 87, 88, 99, 111, 113, 119, 120, 131, 132, 133, 135, 136, 138, 114, 115, 116, 117, 118, 119, 120, 122, 139, 140, 150, 152, 153, 155, 157, 161, 123, 124, 125, 126, 127, 128, 131, 134, 166, 178, 187, 189, 194, 195, 197, 200, 135, 137, 138, 139, 140, 146, 155, 162, 203, 209, 213, 214, 215, 216, 219, 220, 175, 176, 193, 194, 195, 198, 208, 209, 221, 222, 224, 233, 236, 237 210, 215, 220, 221 Linear dispersion relation, 73, 80, 88, 153, 155, Motional diamagnetism, 188 157, 161, 166, 178, 187, 189, 200, 213, Moving pairon, 12, 73, 75, 77–79, 89, 90, 92, 214, 219, 220, 221 95, 96, 136, 137, 138, 139, 167, 170, 178, Linearly, 79, 150, 224 212, 221, 222 Liquid nitrogen, 9 Muller,¨ 9, 157, 197, 215 London, F., 86, 119–123, 126, 128, 132, 139, 207–208, 209, 213 N London, H., 119–123, 126, 128, 132, 139, Nd2−xCexCuO4, 173, 174, 175, 177, 178, 183, 207–208, 209, 213 190, 197, 199, 205 London–London, 119, 207–208 Neck-like hyperboloidal Fermi surface, 42 London rigidity, 208 Neel´ temperature, 173, 174, 182, 205 London’s equation, 113, 119–123, 128, Negative energy, 77, 91 132, 209 Negative-energy state, 91 Index 247

Negative ion, 161, 175 Pairon energy gap, 89, 97, 101, 131, 133, 136, Negative resistance region, 97, 99, 101 137, 140, 169, 170, 209, 221 Net momentum, 24, 26, 53, 63, 64, 65, 75, 76, Pairon with zero net momentum, 75 77, 80, 135, 198 Pair wavefunction, 77 Never-decaying supercurrent, 1, 3 Parent material, 173, 175, 182, 205 Nonmagnetic impurity, 223 Pauli, 12, 42, 64, 68, 69, 70, 183, 184, 187, 227 Normal coordinate, 16 Pauli exclusion principle, 64 Normal current, 6, 12, 114, 121, 126, 146, Pauli paramagnetism, 184 150, 154 Penetration depth, 3, 105, 106, 111, 113, Normal metal, 43, 170 119–123, 128, 131, 133, 138–140, 208, Normal-mode frequency, 16, 17 220, 222, 223 n-type, 173 Perfect-diamagnetic, 3 Nuclear force, 161, 209 Permutation symmetry, 68, 69, 234 Number operator, 30, 52, 62, 63, 65, 69, 74, Perovskite structure, 182, 183, 190, 205 79, 124, 227 Phase change, 1, 6–7, 13, 73, 157, 205, Number of phonons, 17 219, 221 Phase diagram, 9, 107, 174, 183, 199, 205 O Phase difference, 116, 117, 119, 120, 122, 128, Occupation number, 56, 62, 64, 67, 227, 233 148, 149, 150, 214 Ochsenfeld, 1, 207 Phonon, 10, 11, 12, 15–27, 29, 30, 31, 32, 37, O-Fermi surface, 174, 176, 177, 195, 204, 205 41, 42, 50, 56, 71, 77, 88, 103, 108–110, Ohm’s law, 91, 92, 96, 100, 150, 154, 200, 201 114, 125, 137, 161, 162, 163, 168, 174, One-body density matrix, 48 176, 178, 188, 193–195, 198, 200, 201, One-body density operator, 48 205, 209–210, 212, 219, 221, 222, 223 One-body quantum Liouville equation, 125 Phonon exchange attraction, 10, 12, 23–27, 30, One-body trace, 46, 134 77, 137, 163, 174, 193–195, 205, 219 Onsager’s formula, 159 Phonon scattering, 15, 201 Onsager’s quantum flux, 188 , 68, 70, 71, 93, 96, 97, 146, 147, Onsager’s solution, 215 155, 219 Optical phonon, 88, 108–110, 111, 157, 161, Physical vacuum state, 31, 34, 37, 165 162, 173, 174, 175, 178, 193, 195, 196, pi (π) meson, 111 221, 222 Planck distribution, 17, 93, 95, 99 Optical phonon exchange, 157, 161, 173, 174, Planck distribution function, 17, 93, 95, 99 175, 193, 195, 196 Point contact, 144, 145 Order of phase transition, 83 Point-like repulsion, 131, 133, 135, 140 Positive ion, 175 P Principal-axis transformation, 16 Pair-annihilation operator, 37, 111, 125 p-type, 173 Pair creation, 33, 34, 37, 56, 67, 97, 111, 125, 162, 165, 175, 176, 195, 210 Q Pairing interaction Hamiltonian, 210 Quantum Hall effect (QHE), 215–216, 227 Pairon, 10, 11, 12, 29, 30, 31, 32, 33, 34, Quantum Liouville equation, 25, 47, 124, 125, 37, 39, 40, 41–42, 43, 45, 49, 53, 54, 55, 154, 193 56, 59–65, 69, 73, 75, 77–79, 86, 88, 89, Quantum Liouville operator, 25 91–101, 108, 111, 113, 114, 115, 116, 120, Quantum nonlinear equation, 136 121, 125, 127, 128, 131–140, 143, 146, Quantum statistical factor, 95, 99, 146 147, 148, 150, 152, 153–154, 155, 157, Quantum statistical postulate, 61 160, 162, 165, 167, 169, 170, 175, 177, Quantum statistical theory, 1, 12–13, 25, 178, 181, 187, 189, 194, 195–196, 198, 73–88, 225 201, 203, 208, 210, 211, 212, 213, 215, Quantum Theory of Conducting Matter, 216 219–224 Quantum transition rate, 94 Pairon energy, 55, 89, 92, 95, 97, 98, 101, 131, Quantum tunneling, 6, 39, 89–101, 133, 136, 133, 136, 137, 140, 169, 170, 209, 212, 221 143, 168–170, 212, 213, 221 248 Index

Quasi-electron, 11, 12, 29, 36, 40, 41, 45, 49, 103–111, 113, 114, 115, 116, 119, 121, 51, 55, 56, 57, 59, 60, 89, 90, 91, 96, 137, 127, 131, 132, 136, 138, 143, 144, 147, 168, 207, 211, 212, 213, 221, 222 148, 153, 154, 157–171, 181, 189, 197, Quasi-electron energy, 11, 45, 137, 168, 204, 207, 208, 209, 212, 213, 214, 215, 211, 212 219, 220, 221, 222, 223, 224 Quasi-electron energy gap, 11, 137, 168, Supercurrent, 1, 3–4, 5, 6, 9, 12, 13, 32, 41, 73, 211, 212 91, 92, 103, 104, 105, 106, 113–129, 132, Quasi-wavefunction, 47, 113, 116–118, 134, 138, 139, 143, 144, 145, 146, 147, 123–126, 143, 148, 152, 153–154 149, 150, 151, 152, 153, 154, 155, 160, 208, 209, 211, 214, 215, 220, 221, 222, R 223, 224 Reduced Hamiltonian, 30–32, 39, 40, 41, 49, Supercurrent interference, 146, 155 50, 54, 124, 125, 163 Supercurrent tunneling, 143, 144, 214 Relative momentum, 10, 36, 52, 63, 74, 75, 89, Superelectron, 119, 128, 132, 135, 139, 140, 90, 94, 95, 96, 105, 123, 159, 189, 190, 147, 209 202, 212 Superelectron model, 132, 135, 139, 140, 147 Resonating valence bond, 215 Super-to-normal transition, 181 Revised London equation, 128, 208 Susceptibility, 178, 181–191 Ring-shaped cylindrical superconductor, 3 Symmetric state, 65, 69 Ring supercurrent, 3–4, 73, 113–116, 117, 118, 134, 153, 154 T Thermodynamic critical field, 107, 115, S 167, 224 Schrodinger¨ equation, 126, 154, 236 Thin film, 8, 90, 159, 222 Second quantization formulation, 10, 209, 235 Third-order phase transition, 83, 85 Self-focusing power, 146, 147, 153, 214, 220 Threshold voltage, 90, 91, 92, 96, 99, 100, Sharp Fermi surface, 10, 30, 43 170, 221 Sharp phase change, 1, 6–7, 13, 219 T-linear behavior, 197 Shockley’s formula, 159 Torque magnetometry, 159 Short-ranged nuclear force, 161 Total pair number, 79 Shoulder, 168 T-quadratic behavior, 197, 203, 205 Silsbee’s rule, 223–224 Transverse acoustic, 110 Simple cubic lattice, 108 Transverse optical, 110 Single-particle state, 234, 236 Transverse wave, 19 Small pocket, 161, 162, 175, 176, 195 Tunneling junction, 144, 145 Spin-statistics theorem, 68, 69, 70 Two energy gaps, 36, 89, 97, 100, 157, Stationary, 41, 47, 49, 51, 54, 55, 75, 76, 168–170, 222 77, 89, 90, 106, 125, 126, 138, 140, 163, Two-fluid model, 207 212, 220 Two-particle composite, 62–68 Stephan-Boltzmann law, 85 Type II magnetic behavior, 8, 13, 103, 157 Supercondensate, 11, 12, 37, 41, 42–43, 45, Type II superconductor, 9, 103–108, 111, 132, 49, 50, 51, 54, 56, 57, 58, 59, 60, 90, 92, 209, 224 95, 96, 99, 101, 103, 106, 108, 111, 113, Type I superconductor, 8, 103, 107, 111, 136, 115, 116, 124, 125, 126, 128, 132, 134, 161, 166, 170, 224 136, 137, 147, 150, 151, 152, 153, 154, 160, 162, 165, 175, 207, 208, 215, 219, U 220, 221, 223 Unpaired electron, 11, 36, 40, 45, 51, 56, Superconducting quantum interference device 59, 137 (SQUID), 5, 116, 143, 149, 153, 214 Upper critical field, 8, 9, 103, 104, 138, 224 Superconducting state, 8, 9, 12, 39, 41, 87, 88, Up-spin, 51, 185, 186 89, 103, 104, 105, 106, 111, 127, 128, 157, 169, 174, 177, 205, 212, 220, 221, 222 V Superconductor, 1–13, 29, 30, 37, 39, 41, 42, Vacuum ket, 21, 229, 230 43, 45, 57, 87, 90, 91, 92, 94, 97, 99, 100, van Hove, 18 Index 249 van Hove singularity, 18 Weak-coupling approximation, 26 Voltage standard, 154 Weak link, 144, 145 Vortex, 103, 104, 105, 106, 108, 132, 138, 207, 209, 224 Y Vortex structure, 104, 105, 132, YBa2Cu3O7, 157, 158 138, 209 Yukawa, 111, 209

W Z Wave function, 35, 46, 47, 48, 59, 76, Zero-momentum boson, 80, 84 77, 235 Zero-momentum pairon, 31, 39, 79 Wavelength, 18, 19, 147, 154, 161, 195 Zero resistance, 1, 4, 6, 13, 103, 157, 219, 223