53.4 Ginzburg-Landau Theory of Superconductivity 397 Similar to Lagrangian formalism in classical mechanics, minimizing the free energy with respect to Ψ ∗ provides the following Euler-Langrange equa- tion for the variation problem (53.5): 1 2m eff (−i∇−q e A) 2 Ψ(r)+α ·(T − T 0 )Ψ + β ·|Ψ| 2 Ψ + =0. (53.6) (Minimizing with respect to Ψ does not give anything new, only the complex conjugate result.) Minimizing F with respect to A on the other hand leads to the Maxwell equation curl curlA = μ 0 j s , since B = μ 0 H =curlA , and thus: curlH ≡ j s . Solving equation (53.6) for A ≡ 0 assuming spatially homogeneous states and neglecting higher terms, one obtains for T ≥ T 0 the trivial result Ψ ≡ 0, while for T<0 the non-trivial expression |Ψ| = α 2 2β (T 0 − T ) results. In the first case the free energy is zero, while in the second case it is given by F (T,V )=− V · α 2 4β ·(T 0 − T ) 2 . On passing through T 0 the heat capacity C := − ∂ 2 F ∂T 2 therefore changes discontinuously by an amount ΔC = V α 2 2β . At T = T 0 a continuous phase change 13 thus takes place, as for the case of Bose-Einstein condensation, in which the order parameter Ψ increases smoothly from zero (for T ≥ T 0 ) to finite values (for T<T 0 ), whereas the heat capacity increases discontinuously, as mentioned. Two characteristic lengths result from the Ginzburg-Landau theory of superconductivity. These are: a) the so-called coherence length ξ(T ) of the order parameter, and b) the so-called penetration depth λ(T ) of the magnetic induction. 13 A discontinuous change of the specific heat is allowed by a continuous phase transition. It is only necessary that the order parameter changes continuously. 398 53 Applications I: Fermions, Bosons, Condensation Phenomena One obtains the coherence length ξ(T ) for the density of Cooper pairs by assuming that Ψ = Ψ 0 + δΨ(r) (where, as above, A ≡ 0andΨ 0 = α β · (T 0 − T )). Using (53.6) we obtain: − 2 2m eff d 2 (δΨ) dx 2 + α · (T − T 0 )+3β|Ψ 0 | 2 · δΨ =0, which by assuming δΨ(r) ∝ e − x ξ leads to ξ(T )= 4m eff 2 α ·(T 0 − T ) . On the other hand one obtains the penetration depth λ(T ) of the magnetic field assuming A =0andΨ ≡ Ψ 0 .Thus curl curlA(= grad divA −∇ 2 A)=μ 0 j s = − μ 0 · q 2 e ·|Ψ 0 | 2 A , so that with the assumption A ∝ e y · e − x λ the relation λ(T )= β αμ 0 q 2 e · (T 0 − T ) results. Thus the magnetic induction inside a superconductor is compensated completely to zero by surface currents, which are only non-zero in a thin layer of width λ(T )(Meissner-Ochsenfeld effect, 1933 ). This is valid however only for sufficiently weak magnetic fields. In order to handle stronger fields, according to Abrikosov we must dis- tinguish between type I and type II superconductors, depending on whether ξ>λ √ 2 is valid or not. The difference therefore does not depend on the tem- perature. For type II superconductors between two critical magnetic fields H c 1 and H c 2 it is energetically favorable for the magnetic induction to pen- etrate inside the superconductor in the form of so-called flux tubes,whose diameter is given by 2λ, whereas the region in the center of these flux tubes where the superconductivity vanishes has a diameter of only 2ξ.Thesuper- conductivity does not disappear until the field H c 2 is exceeded. The function Ψ(r,t) in the Ginzburg-Landau functional (53.5) corre- sponds to the Higgs boson in the field theory of the electro-weak interac- tion, whereas the vector potential A(r,t) corresponds to the standard fields W ± and Z occurring in this field theory. These massless particles “receive” amassM W ± ,Z ≈ 90 GeV/c 2 via the so-called Higgs-Kibble mechanism,which 53.5 Debye Theory of the Heat Capacity of Solids 399 corresponds to the Meissner-Ochsenfeld effect in superconductivity. This cor- respondence rests on the possibility of translating the magnetic field pene- tration depth λ intoamassM λ , in which one interprets λ as the Compton wavelength of the mass, λ =: M λ c . It is certainly worth taking note of such relationships between low temperature and high energy physics. 53.5 Debye Theory of the Heat Capacity of Solids In the following consider the contributions of phonons, magnons and similar bosonic quasiparticles to the heat capacity of a solid. Being bosonic quasi- particles they have the particle-number expectation value n(ε) T,μ = 1 e β·(ε−μ) − 1 . But since for all these quasiparticles the rest mass vanishes such that they can be generated in arbitrary number without requiring work μdN,wealso have μ ≡ 0. Phonons are the quanta of the sound-wave field, u(r,t) ∝ e i(k·r− ω k ·t) , magnons are the quanta of the spin-wave field (δm ∝ e i(k·r− ω k ·t) ), where the so-called dispersion relations ω k (≡ ω(k)) for the respective wave fields are different, i.e., as follows. For wavelengths λ := 2π k , which are much larger than the distance between nearest neighbors in the system considered, we have for phonons: ω k = c s ·k + , where c s is the longitudinal or transverse sound velocity and terms of higher order in k are neglected. Magnons in antiferromagnetic crystals also have a linear dispersion relation ω ∝ k, whereas magnons in ferromagnetic systems have a quadratic dispersion, ω k = D · k 2 + , 400 53 Applications I: Fermions, Bosons, Condensation Phenomena with so-called spin-wave stiffness D. For the excitation energy ε k and the excitation frequency ω k one always has of course the relation ε k ≡ ω k . For the internal energy U of the system one thus obtains (apart from an arbitrary additive constant): U(T,V,N)= ∞ 0 ω 1 e ω k B T − 1 ·g(ω)dω, where g(ω)dω = V · d 3 k (2π) 3 = V ·k 2 2π 2 dk dω dω, with dk dω = dω dk −1 . From the dispersion relation ω(k)itfollowsthatforω → 0: g(ω)dω = ⎧ ⎪ ⎨ ⎪ ⎩ V · ( ω 2 + ) dω 2π 2 c 3 s for phonons with sound velocityc s , V · “ ω 1 2 + ” dω 4π 2 D 3 2 for magnons in ferromagnets , where the terms + indicate that the above expressions refer to the asymp- totes for ω → 0. For magnons in antiferromagnetic systems a similar formula to that for phonons applies; the difference is only that the magnon contribution can be quenched by a strong magnetic field. In any case, since for fixed k there are two linearly independent transverse sound waves with the same sound velocity c (⊥) s plus a longitudinal sound wave with higher velocity c (|) s ,one uses the effective sonic velocity given by 1 c (eff) s 3 := 2 c (⊥) s 3 + 1 c (|) s 3 . However, for accurate calculation of the contribution of phonons and magnons to the internal energy U (T,V,N) one needs the complete behav- ior of the density of excitations g(ω), of which, however, e.g., in the case of phonons, only (i) the behavior at low frequencies, i.e. g(ω) ∝ ω 2 , and (ii) the so-called sum rule, e.g., ω max 0 dωg(ω)=3N, are exactly known 14 ,whereω max is the maximum eigenfrequency. 14 The sum rule states that the total number of eigenmodes of a system of N coupled harmonic oscillators is 3N. 53.5 Debye Theory of the Heat Capacity of Solids 401 In the second and third decade of the twentieth century the Dutch physi- cist Peter Debye had the brilliant idea of replacing the exact, but matter- dependent function g(ω) by a matter-independent approximation, the so- called Debye approximation (see below), which interpolates the essential properties, (i) and (ii), in a simple way, such that a) not only the low-temperature behavior of the relevant thermodynamic quantities, e.g., of the phonon contribution to U (T,V,N), b) but also the high-temperature behavior can be calculated exactly and analytically, c) and in-between a reasonable interpolation is given. The Debye approximation extrapolates the ω 2 -behavior from low frequen- cies to the whole frequency range and simultaneously introduces a cut-off frequency ω Debye , i.e., in such a way that the above-mentioned sum rule is satisfied. Thus we have for phonons: g(ω) ≈ −→ g Debye (ω)= V · ω 2 2π 2 (c eff s ) 3 , (53.7) i.e., for all frequencies 0 ≤ ω ≤ ω Debye , where the cut-off frequency ω Debye is chosen in such a way that the sum rule is satisfied, i.e. Vω 3 Debye 6π 2 (c eff s ) 3 ! =3N. Furthermore, the integral U(T,V,N)= ω Debye 0 dωg Debye (ω) · ω e ω k B T − 1 can be evaluated for both low and high temperatures, i.e., for k B T ω Debye and ω Debye , viz in the first case after neglecting exponentially small terms if the upper limit of the integration interval, ω = ω Debye , is replaced by ∞.Inthisway one finds the low-temperature behavior U(T,V,N)= 9Nπ 4 15 · ω Debye · k B T ω Debye 4 . The low-temperature contribution of phonons to the heat capacity C V = ∂U ∂T is thus ∝ T 3 . 402 53 Applications I: Fermions, Bosons, Condensation Phenomena Similar behavior, U ∝ V ·T 4 (the Stefan-Boltzmann law )isobservedfor a photon gas, i.e., in the context of black-body radiation; however this is valid at all temperatures, essentially since for photons (in contrast to phonons) the value of N is not defined. Generally we can state: a) The low-temperature contribution of phonons, i.e., of sound-wave quanta, thus corresponds essentially to that of light-wave quanta, photons;the velocity of light is replaced by an effective sound-wave velocity, considering the fact that light-waves are always transverse, whereas in addition to the two transverse sound-wave modes there is also a longitudinal sound-wave mode. b) In contrast, the high-temperature phonon contribution yields Dulong and Petits’s law; i.e., for k B T ω Debye one obtains the exact result: U(T,V,N)=3Nk B T. This result is independent of the material properties of the system con- sidered: once more essentially universal behavior, as is common in ther- modynamics. In the same way one can show that magnons in ferromagnets yield a low- temperature contribution to the internal energy ∝ V · T 5 2 which corresponds to a low-temperature contribution to the heat capacity ∝ T 3 2 . This results from the quadratic dispersion relation, ω(k) ∝ k 2 , for magnons in ferromagnets. In contrast, as already mentioned, magnons in antiferromag- nets have a linear dispersion relation, ω(k) ∝ k, similar to phonons. Thus in antiferromagnets the low-temperature magnon contribution to the specific heat is ∝ T 3 as for phonons. But by application of a strong magnetic field the magnon contribution can be suppressed. – In an earlier section, 53.1, we saw that electrons in a metal produce a con- tribution to the heat capacity C which is proportional to the temperature T . For sufficiently low T this contribution always dominates over all other contributions. However, a linear contribution, C ∝ T , is not character- istic for metals but it also occurs in glasses below ∼ 1 K. However in glasses this linear term is not due to the electrons but to so-called two- level “tunneling states” of local atomic aggregates. More details cannot be given here. 53.6 Landau’s Theory of 2nd-order Phase Transitions 403 53.6 Landau’s Theory of 2nd-order Phase Transitions The Ginzburg-Landau theory of superconductivity, which was described in an earlier subsection, is closely related to Landau’s theory of second-order phase transitions [48]. The Landau theory is described in the following. One begins with a real or complex scalar (or vectorial or tensorial) order parameter η(r), which marks the onset of order at the critical temperature, e.g., the onset of superconductivity. In addition, the fluctuations of the vector potential A(r) of the magnetic induction B(r) are important. In contrast, most phase transitions are first-order, e.g., the liquid-vapor type or magnetic phase transition below the critical point, and of course the transition from the liquid into the solid state, since at the phase transition a discontinuous change in (i) density Δ, (ii) magnetization ΔM, and/or (iii) entropy ΔS occurs, which is related to a heat of transition Δl = T ·ΔS . These discontinuities always appear in first-order derivatives of the rele- vant thermodynamic potential. For example, we have S = − ∂G(T,p,N) ∂T or M = − ∂F g (T,H) ∂H , and so it is natural to define a first-order phase transition as a transition for which at least one of the derivatives of the relevant thermodynamic potentials is discontinuous. In contrast, for a second-order phase transition, i.e., at the critical point of a liquid-vapor system, or at the Curie temperature of a ferromagnet, the N´eel temperature of an antiferromagnet, or at the onset of superconductivity, all first-order derivatives of the thermodynamic potential are continuous. At these critical points considered by Landau’s theory, which we are going to describe, there is thus neither a heat of transformation nor a discontinu- ity in density, magnetization or similar quantity. In contrast discontinuities and/or divergencies only occur for second-order (or higher) derivatives of the thermodynamic potential, e.g., for the heat capacity − T · ∂ 2 F g (T,V,H,N, ) ∂T 2 and/or the magnetic susceptibility χ = − ∂ 2 F g (T,V,H,N, ) ∂H 2 . Thus we have the following definition due to Ehrenfest: For n-th order phase transitions at least one n-th order derivative, ∂ n F g ∂X i 1 ∂X i n , 404 53 Applications I: Fermions, Bosons, Condensation Phenomena of the relevant thermodynamic potential, e.g., of F g (T,V,N,H, ),where the X i k are one of the variables of this potential, is discontinuous and/or divergent, whereas all derivatives of lower order are continuous. Ehrenfest’s definition is mainly mathematical. Landau recognized that the above examples, for which the ordered state is reached by falling below a critical temperature T c , are second-order phase transitions in this sense and that here the symmetry in the ordered state always forms a subgroup of the symmetry group of the high-temperature disordered state (e.g., in the ferromagnetic state one only has a rotational symmetry restricted to rotations around an axis parallel to the magnetization, whereas in the disordered phase there is full rotational symmetry. Further central notions introduced by Landau into the theory are a) as mentioned, the order parameter η, i.e., a real or complex scalar or vecto- rial (or tensorial) quantity 15 which vanishes everywhere in the disordered state, i.e., at T>T c , increasing continuously nonetheless at T<T c to finite values; and b) the conjugate field, h, associated with the order parameter, e.g., a mag- netic field in the case of a ferromagnetic system, i.e., for η ≡ M. On the basis of phenomenological arguments Landau then assumed that for the Helmholtz free energy F(T,V,h) of the considered systems with pos- itive coefficients A, α and b (see below) in the vicinity of T c , in the sense of a Taylor expansion, the following expression should apply (where the second term on the r.h.s., with the change of sign at T c ,isanimportantpointof Landau’s ansatz, formulated for a real order parameter): F (T,V,h)= min η V d 3 r 1 2 A ·(∇η) 2 + α ·(T − T c ) ·η 2 + b 2 η 4 − h ·η . (53.8) By minimization w.r.t. η,for∇η ! = 0 plus h ! = 0, one then obtains similar results as for the above Ginzburg-Landau theory of superconductivity, e.g., η(T ) ≡ η 0 (T ):=0 at T>T c , but |η(T )|≡η 0 (T ):= α ·(T c − T ) b at T<T c , and for the susceptibility χ := ∂η ∂h |h→0 = 1 α ·(T − T c ) at T>T c and χ = 1 2α · (T c − T ) at T<T c . 15 For tensor order parameters there are considerable complications. 53.7 Molecular Field Theories; Mean Field Approaches 405 A k-dependent susceptibility can also be defined. With h(r)=h k · e ik·r and the ansatz η(r):=η 0 (T )+η k ·e ik·r we obtain for T>T c : χ k (T ):= ∂η k ∂h k = 1 2Ak 2 + α ·(T − T c ) and for T<T c : χ k (T )= 1 2Ak 2 +2α ·(T c − T ) , respectively. Thus one can write χ k (T ) ∝ 1 k 2 + ξ −2 , with the so-called thermal coherence length 16 ξ(T )= 2A α ·(T − T c ) and = A α ·|T − T c | for T>T c and T<T c , respectively. But it does not make sense to pursue this ingeniously simple theory further, since it is as good or bad as all molecular field theories,as described next. 53.7 Molecular Field Theories; Mean Field Approaches In these theories complicated bilinearHamilton operators describing interact- ing systems, for example, in the so-called Heisenberg model, H = − l,m J l,m ˆ S l · ˆ S m , 16 The concrete meaning of the thermal coherence length ξ(T ) is based on the fact that in a snapshot of the momentary spin configuration spins at two different places, if their separation |r −r | is much smaller than ξ(T ), are almost always parallel, whereas they are uncorrelated if their separation is ξ. 406 53 Applications I: Fermions, Bosons, Condensation Phenomena are approximated by linear molecular field operators 17 e.g., H ≈ −→ H MF := − l 2 m J l,m ˆ S m T · ˆ S l + l,m J l,m ˆ S m T · ˆ S l T . (53.9) The last term in H MF , where the factor 2 is missing, in contrast to the first term in the summation, is actually a temperature-dependent constant, and only needed if energies or entropies of the system are calculated. Moreover, the sign of this term is opposite to that of the first term; in fact, the last term is some kind of double-counting correction to the first term. The approximations leading from Heisenberg’s model to the molecular field theory are detailed below. Furthermore, from molecular field theories one easily arrives at the Lan- dau theories by a Taylor series leading from discrete sets to continua, e.g., S(r l ) → S(r):=S(r l )+(r −r l ) ·∇S(r) |r=r l + 1 2! 3 i,k=1 (x i − (r l ) i ) ·(x k − (r l ) k ) · ∂ 2 S(r) ∂x i ∂x k |r=r l + . The name molecular field theory is actually quite appropriate. One can in fact write H MF ≡−gμ B l H MF l (T ) ˆ S l + . In this equation H MF l (T ) is the effective magnetic field given by 2 m J l,m ˆ S m T /(gμ B ); g is the Land´efactor, μ B = μ 0 e 2m e the Bohr magneton,ande and m e are the charge and mass of an electron respectively; μ 0 and = h 2π are, as usual, the vacuum permeability and the reduced Planck constant. Van der Waals’ theory is also a kind of molecular field theory, which in the vicinity of the critical point is only qualitatively correct, but quantitatively wrong. The approximation 18 neglects fluctuations (i.e., the last term in the 17 Here operators are marked by the hat-symbol, whereas the thermal expectation value ˆ S l T is a real vector. 18 One may also say incorrectness, since neglecting all fluctuations can be a very severe approximation. . the high-temperature behavior can be calculated exactly and analytically, c) and in-between a reasonable interpolation is given. The Debye approximation extrapolates the ω 2 -behavior from low. ) is based on the fact that in a snapshot of the momentary spin configuration spins at two different places, if their separation |r −r | is much smaller than ξ(T ), are almost always parallel,. high-temperature disordered state (e.g., in the ferromagnetic state one only has a rotational symmetry restricted to rotations around an axis parallel to the magnetization, whereas in the disordered phase there