Basic Theoretical Physics: A Concise Overview P39 pot

At T = T0 a continuous phase change13 thus takes place, as for the case of Bose-Einstein condensation, in which the order parameter Ψ increases smoothly from zero for T ≥ T0 to finite val

Similar to Lagrangian formalism in classical mechanics, minimizing the

free energy with respect to Ψ ∗ provides the following Euler-Langrange

equa-tion for the variaequa-tion problem (53.5):

1

2meff(−i∇ − q e A)2Ψ ( r) + α · (T − T0 )Ψ + β · |Ψ|2Ψ + = 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 = μ0j s , since B = μ0H = curlA , and thus: curlH ≡ j s

Solving equation (53.6) forA ≡ 0 assuming spatially homogeneous states

and neglecting higher terms, one obtains for T ≥ T0 the trivial result Ψ ≡ 0, while for T < 0 the non-trivial expression

|Ψ| =



α2

2β (T0− T )

results In the first case the free energy is zero, while in the second case it is given by

F (T , V ) = − V · α2

4β · (T0− T )2.

On passing through T0the heat capacity

C := − ∂2F

∂T2

therefore changes discontinuously by an amount

ΔC = V α

2

2β .

At T = T0 a continuous phase change13 thus takes place, as for the case

of Bose-Einstein condensation, in which the order parameter Ψ increases smoothly from zero (for T ≥ T0) to finite values (for T < T0), 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.

13A discontinuous change of the specific heat is allowed by a continuous phase transition It is only necessary that the order parameter changes continuously

One obtains the coherence length ξ(T ) for the density of Cooper pairs

by assuming that Ψ = Ψ0+ δΨ ( r) (where, as above, A ≡ 0 and Ψ0 =



α

β · (T0− T )).

Using (53.6) we obtain:

− 2

2meff

d2(δΨ )

dx2 +(

α · (T − T0) + 3β |Ψ0|2)

· δΨ = 0 ,

which by assuming

δΨ ( r) ∝ e − x

ξ

leads to

ξ(T ) =



4meff

2α · (T0− T ) .

On the other hand one obtains the penetration depth λ(T ) of the magnetic

field assuming A = 0 and Ψ ≡ Ψ0 Thus

curl curlA(= grad divA − ∇2A) = μ0j s = − μ0· q2

e · |Ψ0|2A ,

so that with the assumption

A ∝ e y · e − x

the relation

λ(T ) =



β

αμ0q2· (T0− 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 c1 and Hc2 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 The super-conductivity does not disappear until the field Hc2 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 potentialA(r, t) corresponds to the standard fields

W ± and Z occurring in this field theory These massless particles “receive”

a mass M ± ≈ 90 GeV/c2via the so-called Higgs-Kibble mechanism, which

corresponds to the Meissner-Ochsenfeld effect in superconductivity This

cor-respondence rests on the possibility of translating the magnetic field

pene-tration depth λ into a mass Mλ, 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

eβ ·(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 , we also have μ ≡ 0.

Phonons are the quanta of the sound-wave field,

u(r, t) ∝ ei(k·r−ω k ·t) , magnons are the quanta of the spin-wave field (δ m ∝ ei(k·r−ω k ·t)), where the so-called dispersion relations ωk( ≡ 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 = cs · k + , where csis 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 = D · k2

+ ,

with so-called spin-wave stiffness D For the excitation energy εk and the

excitation frequency ωk 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

ekB T ω − 1 · g(ω)dω , where g(ω)dω = V · d3k

(2π)3 =V · k2

2π2



dk dω



dω , with

dk

dω =



dω dk

−1

From the dispersion relation ω( k) it follows that for ω → 0:

g(ω)dω =

⎧

⎪

⎪

V ·( ω2+ )dω 2π2c3 for phonons with sound velocitycs ,

V ·“ω1+ ”

dω 4π2D3 for magnons in f erromagnets , 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) ∝ ω2, and (ii) the

so-called sum rule, e.g.,

ωmax

0

dωg(ω) = 3N ,

are exactly known14, where ωmaxis the maximum eigenfrequency

14 The sum rule states that the total number of eigenmodes of a system of N coupled harmonic oscillators is 3N

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-inmatter-dependent 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 ω2-behavior from low

frequen-cies to the whole frequency range and simultaneously introduces a cut-off frequency ωDebye, 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 · ω2

2π2(ceff

i.e., for all frequencies

0≤ ω ≤ ωDebye, where the cut-off frequency ωDebye is chosen in such a way that the sum rule

is satisfied, i.e

V ω3 Debye

6π2(ceff

s )3

!

= 3N

Furthermore, the integral

U (T , V, N ) =

ωDebye

0

dωgDebye(ω) · ω

ekB T ω − 1

can be evaluated for both low and high temperatures, i.e., for

k B T  ωDebye and Debye, viz in the first case after neglecting exponentially small terms if the upper limit of the integration interval, ω = ωDebye, is replaced by ∞ In this way

one finds the low-temperature behavior

U (T , V, N ) = 9N π

4

15 · ωDebye·



k B T

ωDebye

4

The low-temperature contribution of phonons to the heat capacity

C V = ∂U

∂T

is thus ∝ T3

Similar behavior, U ∝ V · T4 (the Stefan-Boltzmann law ) is observed for

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 Debye

one obtains the exact result:

U (T , V, N ) = 3N k 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 · T5

which corresponds to a low-temperature contribution to the heat capacity

∝ T3

This results from the quadratic dispersion relation, ω(k) ∝ k2, 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 ∝ T3 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

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 potentialA(r)

of the magnetic inductionB(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

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)

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 densdiscontinu-ity, magnetization or similar quantdiscontinu-ity 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 · ∂2F g(T , V, H, N, )

∂T2

and/or the magnetic susceptibility

χ = − ∂2F g(T , V, H, N, )

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 ∂X ,

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 Tc, 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 tensovecto-rial) quantity15which 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 ferromagmag-netic 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 Tc, is an important point of Landau’s ansatz, formulated for a real order parameter):

F (T , V, h) =

min

η



V

d3r

,

1 2

$

A · (∇η)2+ α · (T − T c) · η2+ b

2η

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 ) ≡ η0(T ) := 0 at T > T c , but

|η(T )| ≡ η0(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

2α · (T c − T ) at T < T c

15

For tensor order parameters there are considerable complications.

A k-dependent susceptibility can also be defined With

h( r) = h k · eik·r

and the ansatz

η( r) := η0 (T ) + ηk · eik·r

we obtain for T > Tc:

χ k(T ) := ∂η k

∂h k

2Ak2+ α · (T − T c) and for T < T c:

2Ak2+ 2α · (T c − T ) ,

respectively

Thus one can write

χ k (T ) ∝ 1

k2+ ξ −2 , with the so-called thermal coherence length16

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 bilinear Hamilton operators describing

interact-ing systems, for example, in the so-called Heisenberg model,

H = −

l,m

J l,m Sˆl · ˆ S m ,

16The 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.

are approximated by linear molecular field operators17 e.g.,

H −→ H ≈ MF:=−

l

*

2

m

J l,m

1

ˆ

S m

2

T

+

· ˆ S l+



l,m

J l,m

1

ˆ

S m

2

T ·1Sˆl

2

T

(53.9) The last term inHMF, 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

(xi − (r l)i)· (x k − (r l)k)· ∂2S( r)

∂x i ∂x k |r=r l

+

The name molecular field theory is actually quite appropriate One can in

fact write

HMF≡ −gμ B



l

H lMF(T ) ˆ S l +

In this equation HMF

l (T ) is the effective magnetic field given by

2

m

J l,m

1

ˆ

S m

2

T /(gμ B) ;

g is the Land´e factor,

μ B= μ0e

2m e the Bohr magneton, and e and m e are the charge and mass of an electron

respectively; μ0 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 approximation18 neglects fluctuations (i.e., the last term in the

17 Here operators are marked by the hat-symbol, whereas the thermal expectation value S l  T is a real vector

18 One may also say incorrectness, since neglecting all fluctuations can be a very

severe approximation.