Statistical Mechanics by Konstantin K. Likharev - HTML preview
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~
t
~
~
t
,
for t
i.e.
,
0
T T ,
(4.26)
c
of the small deviation from the critical temperature – which is conveniently normalized as T T
~
c
t
t 1.
(4.27)
T c
Most other key variables follow a similar temperature behavior, with critical exponents frequently being the same for both signs of t~ – though typically with very different proportionality factors. In particular, the heat capacity at a fixed magnetic field behaves as24
~
c t
.
(4.28)
h
Similarly, the (normalized) low-field susceptibility 25
~
t
h h 0
.
(4.29)
Two other important critical exponents, and , describe the temperature behavior of the correlation function sksk’, whose dependence on the distance rkk’ between two spins may be well fitted by the following law,
1
r
s s
exp
(4.30)
k k'
kk' ,
d 2
r
kk
r c
'
with the correlation radius
~
r t
.
(4.31)
c
Finally, three more critical exponents, usually denoted , , and , describe the external field dependences of, respectively, c, , and r c at t~ 0. For example, may be defined as 1
h
~
.
(4.32)
t 0
(Other field exponents are used less frequently, and for their discussion, the interested reader is referred to the special literature that was cited above.)
24 The forms of this and other functions of are selected to make all critical exponents non-negative.
25 In most models of ferromagnetic phase transitions, this variable is proportional to the genuine low-field magnetic susceptibility m of the material – see, e.g., EM Eq. (5.111).
Chapter 4
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The leftmost column of Table 1 shows the ranges of experimental values of these critical exponents for various 3D physical systems featuring continuous phase transitions. One can see that their values vary from system to system, leaving no hope for a universal theory that would describe them all exactly. However, certain combinations of the exponents are much more reproducible – see the four bottom lines of the table.
Table 4.1. Major critical exponents of continuous phase transitions
Exponents and
Experimental
Landau’s
2D Ising
3D Ising
3D Heisenberg
combinations
range (3D)(a)
theory
model
model
Model(d)
0 – 0.14
0(b)
(c)
0.12
–0.14
0.32 – 0.39
1/2
1/8
0.31
0.3
1.3 – 1.4
1
7/4
1.25
1.4
4 – 5
3
15
5
?
0.6 – 0.7
1/2
1
0.64
0.7
0.05
0
1/4
0.05
0.04
( + 2 + )/2
1.00 0.005
1
1
1
1
– /
0.93 0.08
1
1
1
?
(2 – )/
1.02 0.05
1
1
1
1
(2 – )/ d
?
4/ d
1
1
1
(a)
Experimental data are from the monograph by A. Patashinskii and V. Pokrovskii, cited above.
(b) Discontinuity at T = T c – see below.
(c) Instead of following Eq. (28), in this case, ch diverges as ln t~ .
(d) With the order parameter defined as jB/B.
Historically the first (and perhaps the most fundamental) of these universal relations was derived in 1963 by J. Essam and M. Fisher:
2 2 .
(4.33)
It may be proved, for example, by finding the temperature dependence of the magnetic field value ht that changes the order parameter by the same amount as a finite temperature deviation t~ < 0 gives at h = 0.
Comparing Eqs. (26) and (29), we get
~
h t
t
.
(4.34)
By the physical sense of ht, we may expect that such a field has to affect the system’s free energy26 F by an amount comparable to the effect of a bare temperature change ~ t. Ensemble-averaging the last term 26 As was already discussed in Secs. 1.4 and 2.4, there is some dichotomy of terminology for free energies in literature. In models (21) and (23), the magnetic field effects are accounted for at the microscopic level, by the inclusion of the corresponding term into each particular value Em. From this point of view, the list of macroscopic Chapter 4
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of Eq. (23) and using the definition (25) of the order parameter , we see that the change of F (per particle) due to the field equals – ht and, according to Eq. (26), scales as ht t~ t~ (2 + ).
In order to estimate the thermal effect on F, let me first elaborate a bit more on the useful thermodynamic formula already mentioned in Sec. 1.3:
S
C T
,
(4.35)
X
T
X
where X means the variable(s) maintained constant at the temperature variation. In the standard “P-V”
thermodynamics, we may use Eqs. (1.35) for X = V, and Eqs. (1.39) for X = P, to write 2
2
S
F
S
G
C T
T
,
C T
T
.
(4.36)
V
2
P
2
T
T
T
T
V , N
P,
V ,
N
N
P, N
As was just discussed, in the ferromagnetic models of the type (21) or (23), at a constant field h, the role of G is played by F, so Eq. (35) yields
S
2
F
C T
T
.
(4.37)
h
T
T 2
h, N
h, N
The last form of this relation means that F may be found by double integration of (– Ch/ T) over temperature. In the limit t~ << 1, the factor T may be treated as a constant, so with Eq. (28) for ch
Ch, the free energy scales as the double integral of ch t~ – over t~ , i.e. as t~ (2 – ). Requiring this change to be proportional to the same power of t~ as the field-induced part of the energy, we finally get the Essam-Fisher relation (33).
Using similar reasoning, it is straightforward to derive a few other universal relations of critical exponents, including the Widom relation:
1,
(4.38)
very similar relations for other high-field exponents and (which I do not have time to discuss), and the Fisher relation:
2 .
(4.39)
A slightly more complex reasoning, involving the so-called scaling hypothesis, yields the following dimensionality-dependent Josephson relation
d 2 .
(4.40)
The second column of Table 1 shows that at least three of these relations are in very reasonable agreement with experiment, so their set may be used as a testbed for various theoretical approaches to continuous phase transitions.
variables in these systems does not include either P and V or their magnetic analogs, so we may take G F + PV
= F + const, and the equilibrium (at fixed h, T and N) corresponds to the minimum of the Helmholtz free energy F.
Chapter 4
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4.3. Landau’s mean-field theory
The highest-level (i.e. the most phenomenological) approach to continuous phase transitions, formally not based on any particular microscopic model (though implying either the Ising model (23) or one of its siblings), is the mean-field theory developed in 1937 by L. Landau, on the basis of prior ideas by P. Weiss – to be discussed in the next section. The main idea of this phenomenological approach is to represent the free energy’s change F at the phase transition as an explicit function of the order parameter (25). Since at T T c, the order parameter has to tend to zero, this change, F
F( T ) F( T ) ,
(4.41)
c
may be expanded into the Taylor series in , and only a few, most important first terms of that expansion retained. In order to keep the symmetry between two possible signs of the order parameter (i.e. between two possible spin directions in the Ising model) in the absence of the external field, at h = 0
this expansion should include only even powers of :
F
1
2
4
f
(
A T ) B( T ) ...,
at T T .
(4.42)
h0
h0
c
V
2
As Fig. 6 shows, at A( T) < 0, and B( T) > 0, these two terms are sufficient to describe the minimum of the free energy at 2 > 0, i.e. to calculate the stationary values of the order parameter; this is why Landau’s theory ignores higher terms of the Taylor expansion – which are much smaller at 0.
F
F
V
V
A 0
A 0
A / B
A 0
A 0
Fig. 4.6. The Landau free
energy (42) as a function of
0
(a) and (b) 2, for two signs
0
2
2
A
of the coefficient A( T), both
for B( T) > 0.
2 B
Now let us discuss the temperature dependence of the coefficients A and B. As Eq. (42) shows, first of all, the coefficient B( T) has to be positive for any sign of t~ ( T – T c), to ensure the equilibrium at a finite value of 2. Thus, it is reasonable to ignore the temperature dependence of B near the critical temperature altogether, i.e. use the approximation
B( T ) b .
0
(4.43)
On the other hand, as Fig. 6 shows, the coefficient A( T) has to change sign at T = T c, to be positive at T
> T c and negative at T < T c, to ensure the transition from = 0 at T > T c to a certain non-zero value of the order parameter at T < T c. Assuming that A is a smooth function of temperature, we may approximate it by the leading term of its Taylor expansion in the difference t~ T – T c < 0:
~
(
A T ) at , with a 0 ,
(4.44)
so Eq. (42) becomes
~
1
2
4
f
at b .
(4.45)
h0
2
Chapter 4
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In this rudimentary form, the Landau theory may look almost trivial, and its main strength is the possibility of its straightforward extension to the effects of the external field and of spatial variations of the order parameter. First, as the field terms in Eqs. (21) or (23) show, the applied field gives such systems, on average, the energy addition of – h per particle, i.e. – nh per unit volume, where n is the particle density. Second, since according to Eq. (31) (with > 0, see Table 1) the correlation radius diverges at t~ 0, in this limit, the spatial variations of the order parameter should be slow, 0.
Hence, the effects of the gradient on F may be approximated by the first non-zero term of its expansion into the Taylor series in ()2. As a result, Eq. (45) may be generalized as
Landau
~
1
theory:
3
2
4
Δ F Δ fd r,
Δ
with f at
b
nh c 2
free energy
,
(4.46)
2
where c is a coefficient independent of . To avoid the unphysical effect of spontaneous formation of spatial variations of the order parameter, that factor has to be positive at all temperatures and hence may be taken for a constant in a small vicinity of T c – the only region where Eq. (46) may be expected to provide quantitatively correct results.
Let us find out what critical exponents are predicted by this phenomenological approach. First of all, we may find the equilibrium values of the order parameter from the condition of F having a minimum, F/ = 0. At h = 0, it is easier to use the equivalent equation F/(2) = 0, where F is given by Eq. (45) – see Fig. 6b. This immediately yields
~
at / b1/2
~
, for t ,
0
(4.47)
,0
t .
~
0
for
Comparing this result with Eq. (26), we see that in the Landau theory, = ½. Next, plugging the result (47) back into Eq. (45), for the equilibrium (minimal) value of the free energy, we get
2~2
~
a t / 2 b, for t ,
0
f
(4.48)
,
0
t .
~
0
for
From here and Eq. (37), the specific heat,
C
2
~
a / bT ,
for
t ,
0
h
c
~
(4.49)
V
,
0
0
for
t ,
has, at the critical point, a discontinuity rather than a singularity, so we need to prescribe zero value to the critical exponent .
In the presence of a uniform field, the equilibrium order parameter should be found from the condition f/ = 0 applied to Eq. (46) with = 0, giving
f
~
2 at 2 3
b nh 0 .
(4.50)
In the limit of a small order parameter, 0, the term with 3 is negligible, and Eq. (50) gives the solution (stable only for t~ 0):
nh
,
(4.51)
at~
2
Chapter 4
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so according to Eq. (29), = 1. On the other hand, at t~ 0 (or at sufficiently high fields at other temperatures), the cubic term in Eq. (50) is much larger than the linear one, and this equation yields 1/ 3
nh
,
(4.52)
2 b
so comparison with Eq. (32) yields = 3. Finally, according to Eq. (30), the last term in Eq. (46) scales as c2/ r 2
c . (If r c , the effects of the pre-exponential factor in Eq. (30) are negligible.) As a result, the gradient term’s contribution is comparable27 with the two leading terms in f (which, according to Eq.
(47), are of the same order), if
1/ 2
c
r
,
(4.53)
c
~
a t
so according to the definition (31) of the critical exponent , in the Landau theory it is equal to ½.
The third column in Table 1 summarizes the critical exponents and their combinations in Landau’s theory. It shows that these values are somewhat out of the experimental ranges, and while some of their “universal” relations are correct, some are not; for example, the Josephson relation would be only correct at d = 4 (not the most realistic spatial dimensionality :-) The main reason for this disappointing result is that describing the spin interaction with the field, the Landau mean-field theory neglects spin randomness, i.e. fluctuations. Though a quantitative theory of fluctuations will be discussed only in the next chapter, we can readily perform their crude estimate. Looking at Eq. (46), we see that its first term is a quadratic function of the effective “half-degree of freedom”, . Hence per the equipartition theorem (2.28), we may expect that the average square of its thermal fluctuations, within a d-dimensional volume with a linear size of the order of r c, should be of the order of T/2 (close to the critical temperature, T c/2 is a good enough approximation):
d
T
~ ~2
a t
r ~ c .
(4.54)
c
2
In order to be negligible, the variance has to be small in comparison with the average 2 ~ a t~ / b – see Eq. (47). Plugging in the T-dependences of the operands of this relation, and values of the critical exponents from the Landau theory, for > 0 we get the so-called Levanyuk- Ginzburg criterion of its validity:
d
~
/ 2
~
T
a t
a t
c
.
(4.55)
2 a c
b
We see that for any realistic dimensionality, d < 4, the order parameter’s fluctuations grow at t~ 0
faster than its average value, and hence the theory becomes invalid.
Thus the Landau mean-field theory is not a perfect approach to finding critical indices at continuous phase transitions in Ising-type systems with their next-neighbor interactions between the particles. However, any long-range interactions between particles increase the correlation radius r c, and 27 According to Eq. (30), the correlation radius may be interpreted as the distance at which the order parameter
relaxes to its equilibrium value if it is deflected from that value at some point. Since the law of such spatial change may be obtained by a variational differentiation of F, for the actual relaxation law, all major terms of (46) have to be comparable.
Chapter 4
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hence suppress the order parameter fluctuations. As one example, at laser self-excitation, the emerging coherent optical field couples essentially all photon-emitting particles in the electromagnetic cavity (resonator). As another example, in superconductors, the role of the correlation radius is played by the Cooper-pair size 0, which is typically of the order of 10-6 m, i.e. much larger than the average distance between the pairs (~10-8 m). As a result, the mean-field theory remains valid at all temperatures besides an extremely small temperature interval near T c – for bulk superconductors, of the order of 10–6 K.
The real strength of Landau’s mean-field theory is that despite its classical character, it may be readily generalized to a description of various Bose-Einstein condensates, i.e. quantum fluids. Of those generalizations, the most famous is the Ginzburg-Landau theory of superconductivity. It was developed in 1950, i.e. even before the microscopic-level explanation of this phenomenon by J. Bardeen, L.
Cooper, and R. Schrieffer in 1956-57. In this theory, the real order parameter is replaced with the modulus of a complex function , physically the wavefunction of the coherent Bose-Einstein condensate of Cooper pairs. Since each pair carries the electric charge q = –2 e and has zero spin, it interacts with the magnetic field in a way different from that described by the Heisenberg or Ising models. Namely, as was already discussed in Sec. 3.4, in the magnetic field, the del operator in Eq.
(46) has to be amended with the term – i( q/)A, where A is the vector potential of the total magnetic field B = A, including not only the external magnetic field H but also the field induced by the supercurrent itself. With the account for the well-known formula for the magnetic field energy, Eq. (46) is now replaced with
2
2
2
~
1
q
GL free
2
4
Δ f at b
i A
B
,
(4.56)
energy
2
2 m
20
where m is a phenomenological coefficient rather than the actual particle’s mass.
The variational minimization of the resulting Gibbs energy density g f – 0HM f –
HB + const28 over the variables and B (which is suggested for the reader’s exercise) yields two differential equations:
B
i
q *
q
i A
c.c.
,
(4.57a)
2 m
GL
0
equations
2
2
q
at
~ b 2
i A .
(4.57b)
2 m
The first of these Ginzburg-Landau equations (57a) should be no big surprise for the reader, because according to the Maxwell equations, in magnetostatics the left-hand side of Eq. (57a) has to be equal to the electric current density, while its right-hand side is the usual quantum-mechanical probability current density multiplied by q, i.e. the density j of the electric current of the Cooper pair condensate. (Indeed, after plugging = n 1/2exp{ i} into that expression, we come back to Eq. (3.84) which, as we already know, explains such macroscopic quantum phenomena as the magnetic flux quantization and the Meissner-Ochsenfeld effect.)
28 As an immediate elementary sanity check of this relation, resulting from the analogy of Eqs. (1.1) and (1.3), the minimization of g in the absence of superconductivity ( = 0) gives the correct result B = 0H. Note that this account of the difference between f and g is necessary here because (unlike Eqs. (21) and (23)), the Ginzburg-Landau free energy (56) does not take into account the effect of the field on each particle directly.
Chapter 4
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However, Eq. (57b) is new for us – at least for this course.29 Since the last term on its right-hand side is the standard wave-mechanical expression for the kinetic energy of a particle in the presence of a magnetic field,30 if this term dominates that side of the equation, Eq. (57b) is reduced to the stationary Schrödinger equation
E
ˆ
H , for the ground state of free Cooper pairs, with the energy E = – a t~ > 0.
However, in contrast to the usual (single-particle) Schrödinger equation, in which is determined by the normalization condition, the Cooper pair condensate density n = 2 is determined by the thermodynamic balance of the condensate with the ensemble of “normal” (unpaired) electrons, which plays the role of the uncondensed part of the particles in the usual Bose-Einstein condensate – see Sec.
3.4. In Eq. (57b), such balance is enforced by the first term b 2 on the right-hand side. As we have already seen, in the absence of magnetic field and spatial gradients, such term yields t~
1/2 ( T c
– T)1/2 – see Eq. (47).
As a parenthetic remark, from the mathematical standpoint, the term b 2, which is nonlinear in , makes Eq. (57b) a member of the family of the so-called nonlinear Schrödinger equations.
Another important member of this family is the Gross-Pitaevskii equation, 2
Gross-
at
~ b 2
2
U (r
) ,
(4.58) Pitaevskii
2 m
equation
which gives a reasonable (albeit approximate) description of gradient and field effects on Bose-Einstein condensates of electrically neutral atoms at T T c. The differences between Eqs. (58) and (57) reflect, first, the zero electric charge q of the atoms (so Eq. (57a) becomes trivial) and, second, the fact that the atoms forming the condensates may be readily placed in external potentials U(r) const (including the time-averaged potentials of optical traps – see EM Chapter 7), while in superconductors such potential profiles are much harder to create due to the screening of external electric and optical fields by conductors – see, e.g., EM Sec. 2.1.
Returning to the discussion of Eq. (57b), it is easy to see that its last term increases as either the external magnetic field or the density of current passed through a superconductor are increased, increasing the vector potential. In the Ginzburg-Landau equation, this increase is matched by a corresponding decrease of 2, i.e. of the condensate density n, until it is completely suppressed. This balance describes the well-documented effect of superconductivity suppression by an external magnetic field and/or the supercurrent passed through the sample. Moreover, together with Eq. (57a), naturally describing the flux quantization (see Sec. 3.4), Eq. (57b) explains the existence of the so-called Abrikosov vortices – thin magnetic-field tubes, each carrying one quantum 0 of magnetic flux – see Eq.
(3.86). At the core of the vortex, 2 is suppressed (down to zero at its central line) by the persistent, dissipation-free current of the superconducting condensate that circulates around the core and screens the rest of the superconductor from the magnetic field carried by the vortex.31 The penetration of such vortices into the so-called type-II superconductors enables them to sustain zero dc resistance up to very high magnetic fields of the order of 20 T, and as a result, to be used in very compact magnets –
including those used for beam bending in particle accelerators.
29 It is also discussed in EM Sec. 6.5.
30 See, e.g., QM Sec. 3.1.
31 See, e.g., EM Sec. 6.5.
Chapter 4
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Moreover, generalizing Eqs. (57) to the time-dependent case, just as it is done with the usual Schrödinger equation, one can describe other fascinating quantum macroscopic phenomena such as the
Josephson effects, including the generation of oscillations with frequency J = ( q/)V by weak links between two superconductors, biased by dc voltage V. Unfortunately, time/space restrictions do not allow me to discuss these effects in any detail in this course, and I have to refer the reader to special literature.32 Let me only note that in the limit T T c, and for not extremely pure superconductor crystals (in which the so-called non-local transport phenomena may be important), the Ginzburg-Landau equations are exact, and may be derived (and their parameters T c, a, b, q, and m determined) from the standard “microscopic” theory of superconductivity, based on the initial work by Bardeen, Cooper, and Schrieffer.33 Most importantly, such derivation proves that q = –2 e – the electric charge of a single Cooper pair.
4.4. Ising model: Weiss’ molecular-field approximation
The Landau mean-field theory is phenomenological in the sense that even within the range of its validity, it tells us nothing about the value of the critical temperature T c and other parameters (in Eq.
(46), the coefficients a, b, and c), so they have to be found from a particular “microscopic” model of the system under analysis. In this course, we would have time to discuss only the Ising model (23) for various dimensionalities d.
The most simplistic mean-field approach to this model is to assume that all spins are exactly equal, sk = , with an additional condition 2 1, ignoring for a minute the fact that in the genuine Ising model, sk may equal only +1 or –1. Plugging this relation into Eq. (23), we get34
F NJd 2 Nh .
(4.59)
This energy is plotted in Fig. 7a as a function of , for several values of h.
(a)
(b)
1
1
0.5
F
0
Fig. 4.7. Field dependences
0
h
h
c
c
h
of (a) the free energy profile
0
and (b) the order parameter
0.5
h
0.5
(i.e. magnetization), in the
2 J
1.0
1.5
crudest mean-field approach
1 1
0
1
1
to the Ising model.
32 See, e.g., M. Tinkham, Introduction to Superconductivity, 2nd ed., McGraw-Hill, 1996. A short discussion of the Josephson effects and Abrikosov vortices may be found in QM Sec. 1.6 and EM Sec. 6.5 of this series.
33 See, e.g., Sec. 45 in E. Lifshitz and L. Pitaevskii, Statistical Physics, Part 2, Pergamon, 1980.
34 In this naïve approach, we neglect the fluctuations of spin, i.e. their disorder. This full ordering assumption implies S = 0, so F E – TS = E, and we may use either notation for the system’s energy.
Chapter 4
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The plots show that at h = 0, the system may be in either of two stable states, with = 1, corresponding to two different spin directions (i.e. two different directions of magnetization), with equal energy.35 (Formally, the state with = 0 is also stationary, because at this point F/ = 0, but it is unstable because for the ferromagnetic interaction, J > 0, the second derivative 2 F/2 is always negative.) As the external field is increased, it tilts the potential profile, and finally at the critical field, h h 2 Jd ,
(4.60)
c
the state with = –1 becomes unstable, leading to the system’s jump into the only remaining state with opposite magnetization, = +1 – see the arrow in Fig. 7a. Application of the similar external field of the opposite polarity leads to the similar switching, back to = –1, at the field h = – h c, so the full field dependence of follows the hysteretic pattern shown in Fig. 7b.36
Such a pattern is the most visible experimental feature of actual ferromagnetic materials, with the coercive magnetic field Hc of the order of 103 A/m, and the saturated (or “remnant”) magnetization corresponding to fields B of the order of a few teslas. The most important property of these materials, also called permanent magnets, is their stability, i.e. the ability to retain the history-determined direction of magnetization in the absence of an external field, for a very long time. In particular, this property is the basis of all magnetic systems for data recording, including the ubiquitous hard disk drives with their incredible information density approaching 1 Terabit per square inch.37
So, already the simplest mean-field approximation (59) does give a (crude) description of the ferromagnetic ordering. However, it grossly overestimates the stability of these states with respect to thermal fluctuations. Indeed, in this theory, there is no thermally induced randomness at all, until T
becomes comparable with the height of the energy barrier separating two stable states, F
F( )
0 F( )
1 NJd ,
(4.61)
which is proportional to the number of particles. At N , this value diverges, and in this sense, the critical temperature is infinite, while numerical experiments and more refined theories of the Ising model show that actually its ferromagnetic phase is suppressed at T > T c ~ Jd – see below.
The accuracy of this theory may be dramatically improved by even an approximate account for thermally induced randomness. In this approach (suggested in 1907 by Pierre-Ernest Weiss), called the molecular-field approximation,38 random deviations of individual spin values from the lattice average, 35 The fact that the stable states always correspond to = 1, partly justifies the treatment, in this crude approximation, of the order parameter as a continuous variable.
36 Since these magnetization jumps are accompanied by (negative) jumps of the free energy F, they are sometimes called first-order phase transitions. Note, however, that in this simple theory, these transitions are between two physically similar fully-ordered phases.
37 For me, it was always shocking how little my graduate physics students knew about this fascinating (and very important) field of modern engineering, which involves so much interesting physics and fantastic electromechanical technology. For getting acquainted with it, I may recommend, for example, the monograph by C. Mee and E. Daniel, Magnetic Recording Technology, 2nd ed., McGraw-Hill, 1996.
38 In some texts, this approximation is called the “mean-field theory”. This terminology may lead to confusion, because the molecular-field theory belongs to a different, deeper level of the theoretical hierarchy than, say, the (more phenomenological) Landau-style mean-field theories. For example, for a given microscopic model, the Chapter 4
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s~ s ,
with s ,
(4.62)
k
k
k
are allowed, but considered small: s~ . This assumption allows us, after plugging the resulting k
expression s s~
into the first term on the right-hand side of Eq. (23),
k
k
E J
s~ s~
h
s
J
2
s~
s~
s~ s~
h
s ,
(4.63)
m
k k'
k
k
k '
k k '
k
k, k '
k
k, k '
k
ignore the last term in the square brackets. Making the replacement (62) in the terms proportional to s~ , k
we may rewrite the result as
E E ' NJd 2
h
s ,
(4.64)
m
m
ef k
k
where h ef is defined as the sum
h h
.
(4.65)
ef
2 Jd
This sum may be interpreted as an effective external field, which (besides the genuine external field h) takes into account the effect that would be exerted on spin sk by its 2 d next neighbors if they all had non-fluctuating (but possibly continuous) spin values sk’ = . Such addition to the external field, Weiss
molecular
h
h h
,
(4.66)
mol
ef
2 Jd
field
is called the molecular field – giving its name to Weiss’ approach.
From the point of view of statistical physics, at fixed parameters of the system (including the order parameter ), the first term on the right-hand side of Eq. (64) is merely a constant energy offset, and h ef is just another constant, so
h , for s ,
1
E '
const
h s
(4.67)
m
,
with
k
k
k
ef
k
ef
h
s
k
, for
1.
ef
k
Such separability of the energy means that in the molecular-field approximation the fluctuations of different spins are independent of each other, and their statistics may be examined individually, using the energy spectrum k. But this is exactly the two-level system that was the subject of Problems 2.2-2.4. Actually, its statistics is so simple that it is easier to redo it from scratch, rather than to use the results of those exercises (which would require changing notation). Indeed, according to the Gibbs distribution (2.58)-(2.59), the equilibrium probabilities of the states sk = 1 may be found as 1
h / T
h
h
h
W
e ef
,
with Z
ef
ef
ef
exp
exp
2cosh
.
(4.68)
Z
T
T
T
From here, we may readily calculate F = – T ln Z and all other thermodynamic variables, but let us immediately use Eq. (68) to calculate the statistical average of sj, i.e. the order parameter:
e h / T
/
ef
e h T
ef
h
s ( )
1 W ( )
1 W
ef
tanh
.
(4.69)
j
2cosh h / T
ef
T
molecular-field approach may be used for the (approximate) calculation of the parameters a, b, and T c participating in Eq. (46) – the starting point of the Landau theory.
Chapter 4
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Now comes the punch line of Weiss’ approach: plugging this result back into Eq. (65), we may write the condition of self-consistency of the molecular-field approach:
h
Self-
h h 2 Jd
ef
tanh
.
(4.70) consistency
ef
T
equation
This is a transcendental equation, which evades an explicit analytical solution, but whose properties may be readily analyzed by plotting both its sides as functions of the same argument, so the stationary state(s) of the system corresponds to the intersection point(s) of these plots.
First of all, let us explore the field-free case ( h = 0), when h ef = h mol 2 dJ, so Eq. (70) is reduced to
2 Jd
tanh
,
(4.71)
T
giving one of the patterns sketched in Fig. 8, depending on the dimensionless parameter 2 Jd/ T.
LHS
1
RHS
Fig. 4.8. The ferromagnetic phase transition
0
in Weiss’ molecular-field theory: two sides
0
0
of Eq. (71) sketched as functions of for
three different temperatures: above T c (red),
1
below T c (blue), and equal to T c (green).
If this parameter is small, the right-hand side of Eq. (71) grows slowly with (see the red line in Fig. 8), and there is only one intersection point with the left-hand side plot, at = 0. This means that the spin system has no spontaneous magnetization; this is the so-called paramagnetic phase. However, if the parameter 2 Jd/ T exceeds 1, i.e. if T is decreased below the following critical value, T 2 Jd ,
(4.72) Critical
c
(“Curie”)
temperature
the right-hand side of Eq. (71) grows, at small , faster than its left-hand side, so their plots intersect it at three points: = 0 and = 0 – see the blue line in Fig. 8. It is almost evident that the former stationary point is unstable, while the two latter points are stable. (This fact may be readily verified by using Eq. (68) to calculate F. Now the condition F/ h=0 = 0 returns us to Eq. (71), while calculating the second derivative, for T < T c we get 2 F/2 > 0 at = 0, and 2 F/2 < 0 at = 0). Thus, below T c the system is in the ferromagnetic phase, with one of two possible directions of the average spontaneous magnetization, so the critical ( Curie 39) temperature given by Eq. (72) marks the transition between the paramagnetic and ferromagnetic phases. (Since the stable minimum value of the free energy F is a continuous function of temperature at T = T c, this phase transition is continuous.) Now let us repeat this graphics analysis to examine how each of these phases responds to an external magnetic field h 0. According to Eq. (70), the effect of h is just a horizontal shift of the straight-line plot of its left-hand side – see Fig. 9. (Note a different, here more convenient, normalization of both axes.)
39 Named after Pierre Curie, rather than his (more famous) wife Marie Skłodowska-Curie.
Chapter 4
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(a)
(b)
2 dJ
2 dJ
h h
c
h 0
h 0
h h c
h h c
0
h
0
h
ef
ef
Fig. 4.9 External field effects on
h h c
(a) a paramagnet ( T > T c), and
2 dJ
2 dJ
(b) a ferromagnet ( T < T c).
In the paramagnetic case (Fig. 9a) the resulting dependence h ef( h) is evidently continuous, but the coupling effect ( J > 0) makes it steeper than it would be without spin interaction. This effect may be quantified by the calculation of the low-field susceptibility defined by Eq. (29). To calculate it, let us notice that for small h, and hence small h ef, the function tanh is approximately equal to its argument so Eq. (70) is reduced to
2 Jd
2 Jd
h h
h ,
for
h 1.
(4.73)
ef
ef
ef
T
T
Solving this equation for h ef, and then using Eq. (72), we get
h
h
h
.
(4.74)
ef
1 2 Jd / T
1 T / T
c
Recalling Eq. (66), we can rewrite this result for the order parameter:
h h
h
ef
,
(4.75)
T
T T
c
c
so the low-field susceptibility
Curie-
1
Weiss
,
for T T .
(4.76)
c
0
law
h h
T T c
This is the famous Curie-Weiss law, which shows that the susceptibility diverges at the approach to the Curie temperature T c.
In the ferromagnetic case, the graphical solution (Fig. 9b) of Eq. (70) gives a qualitatively different result. A field increase leads, depending on the spontaneous magnetization, either to the further saturation of h mol (with the order parameter gradually approaching 1), or, if the initial was negative, to a jump to positive at some critical (coercive) field h c. In contrast with the crude approximation (59), at T > 0 the coercive field is smaller than that given by Eq. (60), and the magnetization saturation is gradual, in a good (semi-quantitative) accordance with experiment.
To summarize, the Weiss molecular-field approach gives an approximate but realistic description of the ferromagnetic and paramagnetic phases in the Ising model, and a very simple prediction (72) of the temperature of the transition between them, for an arbitrary dimensionality d of the cubic lattice. It also enables the calculation of other parameters of Landau’s mean-field theory for this model – an exercise left for the reader. Moreover, the molecular-field approximation allows one to obtain similarly reasonable analytical results for some other models of phase transitions – see, e.g., the exercise problems at the end of Sec. 6.
Chapter 4
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4.5. Ising model: Exact and numerical results
In order to evaluate the main prediction (72) of the Weiss theory, let us now discuss the exact (analytical) and quasi-exact (numerical) results obtained for the Ising model, going from the lowest value of dimensionality, d = 0, to its higher values. The zero dimensionality means that the spin has no nearest neighbors at all, so the first term of Eq. (23) vanishes. Hence Eq. (64) is exact, with h ef = h, and so is its solution (69). Now we can simply use Eq. (76), with J = 0, i.e. T c = 0, reducing this result to the so-called Curie law:
1
.
(4.77) Curie
T
law
It shows that the system is paramagnetic at any temperature. One may say that for d = 0 the Weiss molecular-field theory is exact – or even trivial. (However, in some sense it is more general than the Ising model because as we know from Chapter 2, it gives the exact result for a fully quantum-mechanical treatment of any two-level system, including spin-½.) Experimentally, the Curie law is approximately valid for many so-called paramagnetic materials, i.e. 3D systems of particles with spontaneous spins and sufficiently weak interaction between them.
The case d = 1 is more complex but has an exact analytical solution. A simple way to obtain it for a uniform chain is to use the so-called transfer matrix approach.40 For this, first of all, we may argue that most properties of a 1D system of N >> 1 spins (say, put at equal distances on a straight line) should not change too much if we bend that line gently into a closed ring (Fig. 10), assuming that spins s 1 and sN interact exactly as all other next-neighbor pairs. Then the energy (23) becomes E Js s Js s ... Js s hs hs ... hs
.
(4.78)
m
1 2
2 3
N 1
1
2
N
...
sN 1 sN
s
1
s
2
Fig. 4.10. The closed-ring
version of the 1D Ising system.
s
3
...
Let us regroup the terms of this sum in the following way:
h
h
h
h
h
h
E
,
(4.79)
m
s Js s s
s
Js s
s
...
s
Js s
s
1
1 2
2
2
2 3
3
N
N 1
1
2
2
2
2
2
2
so the group inside each pair of parentheses depends only on the state of two adjacent spins. The corresponding statistical sum,
40 This approach was developed in 1941 by H. Kramers and G. Wannier. Note that for the field-free case h = 0, an even simpler solution of the problem, valid for chains with an arbitrary number N of “spins” and arbitrary coefficients Jk, is possible. For that, one needs to calculate the explicit relation between the statistical sums Z for systems with N and ( N + 1) spins first, and then apply it sequentially to systems starting from Z = 2. I am leaving this calculation for the reader’s exercise.
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s
s s
s
1
1 2
2
s
s s
s
2
2 3
3
s
s s
s
N
N 1
1
Z exp h
J
h
exp h
J
h
...exp h
J
h
, (4.80)
s for
,
1
2
2
2
2
2
2
k
T
T
T
T
T
T
T
T
T
k ,
1 2,... N
still has 2 N terms, each corresponding to a certain combination of signs of N spins. However, each operand of the product under the sum may take only four values, corresponding to four different combinations of its two arguments:
exp
J h/ T, for s s ,1
k
k 1
s
s s
s
k
k k 1
k 1
exp h
J
h
exp
J h/ T, for s s ,1
(4.81)
2 T
T
2 T
k
k 1
exp J / T,
for
s s
.
1
k
k 1
These values do not depend on the site number k,41 and may be represented as the elements Mj,j’ (with j, j’ = 1, 2) of the so-called transfer matrix
exp
J h /T
exp J/T
M
,
(4.82)
exp J/T
exp
J h /T
so the whole statistical sum (80) may be recast as a product:
Z M
M
...
.
(4.83)
j j
j j
M j j M j j
j
1 2
2 3
N
,
1 2
1 N
N 1
k
According to the basic rule of matrix multiplication, this sum is just
Tr N
Z
M .
(4.84)
Linear algebra tells us that this trace may be represented just as
N
N
Z ,
(4.85)
where are the eigenvalues of the transfer matrix M, i.e. the roots of its characteristic equation, exp
J h /T
exp J/T
.
(4.86)
exp J/T
J h /T
0
exp
A straightforward solution of this equation yields two roots:
1/ 2
J
h
h
4 J
2
exp
cosh
sinh
exp
.
(4.87)
T
T
T
T
The last simplification comes from the condition N >> 1 – which we need anyway, to make the ring model sufficiently close to the infinite linear 1D system. In this limit, even a small difference of the exponents, + > -, makes the second term in Eq. (85) negligible, so we finally get N
NJ
h
4
2 h
J
1/ 2
N
Z exp
cosh
sinh
exp
.
(4.88)
T
T
T
T
41 This is a result of the “translational” (or rather rotational) symmetry of the system, i.e. its invariance to the index replacement k k + 1 in all terms of Eq. (78).
Chapter 4
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From here, we can find the free energy per particle:
1/ 2
F
T
1
h
h
4 J
2
ln
J T lncosh sinh
exp
,
(4.89)
N
N
Z
T
T
T
and then use thermodynamics to calculate such variables as entropy – see the first of Eqs. (1.35).
However, we are mostly interested in the order parameter defined by Eq. (25): sj. The conceptually simplest approach to the calculation of this statistical average would be to use the sum (2.7), with the Gibbs probabilities Wm = Z-1exp{- Em/ T}. However, the number of terms in this sum is 2 N, so for N >> 1 this approach is completely impracticable. Here the analogy between the canonical pair {–
P, V} and other generalized force-coordinate pairs {F, q}, in particular {0H(r k), m k} for the magnetic field, discussed in Secs. 1.1 and 1.4, becomes invaluable – see in particular Eq. (1.3b). (In our normalization (22), and for a uniform field, the pair {0H(r k), m k} becomes { h, sk}.) Indeed, in this analogy the last term of Eq. (23), i.e. the sum of N products (– hsk) for all spins, with the statistical average (– Nh), is similar to the product PV, i.e. the difference between the thermodynamic potentials F
and G F + PV in the usual “P-V thermodynamics”. Hence, the free energy F given by Eq. (89) may be understood as the Gibbs energy of the Ising system in the external field, and the equilibrium value of the order parameter may be found from the last of Eqs. (1.39) with the replacements – P h, V N:
F
F N
N
/
,
i.e.
.
(4.90)
h
h
T
T
Note that this formula is valid for any model of ferromagnetism, of any dimensionality, if it has the same form of interaction with the external field as the Ising model.
For the 1D Ising ring with N >> 1, Eqs. (89) and (90) yield
h
2 h
4 J 1/2
1
2 J
sinh
sinh
exp
,
giving
h
exp
0
.
(4.91)
T
T
T
h
T
T
This result means that the 1D Ising model does not exhibit a phase transition, i.e., in this model T c = 0.
However, its susceptibility grows, at T 0, much faster than the Curie law (77). This gives us a hint that at low temperatures the system is “virtually ferromagnetic”, i.e. has the ferromagnetic order with some rare random violations. (Such violations are commonly called low-temperature excitations.) This interpretation may be confirmed by the following approximate calculation. It is almost evident that the lowest-energy excitation of the ferromagnetic state of an open-end 1D Ising chain at h = 0 is the reversal of signs of all spins in one of its parts – see Fig. 11.
Fig. 4.11. A Bloch wall in an open-end
+
+
+
+
-
-
-
-
1D Ising chain.
Indeed, such an excitation (called either the “magnetic domain wall” or just the Bloch wall 42) involves the change of sign of just one product sksk’, so according to Eq. (23), its energy EW (defined as 42 Named after Felix Bloch who was the first one to discuss such excitations. More complex excitations such as skyrmions (see, e.g., A. Fert et al., Nature Review Materials 2, 17031 (2017)) have higher energies.
Chapter 4
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the difference between the values of Em with and without the excitation) equals 2 J, regardless of the wall position.43 Since in the ferromagnetic Ising model, the parameter J is positive, EW > 0. If the system
“tried” to minimize its internal energy, having any wall in the system would be energy-disadvantageous.
However, thermodynamics tells us that at T 0, the system’s thermal equilibrium corresponds to the minimum of the free energy F E – TS, rather than just energy E.44 Hence, we have to calculate the Bloch wall’s contribution FW to the free energy. Since in an open-end linear chain of N >> 1 spins, the wall can take ( N – 1) N positions with the same energy EW, we may claim that the entropy SW
associated with this excitation is ln N, so
F E TS 2 J T ln N .
(4.92)
W
W
W
This result tells us that in the limit N , and at T 0, the Bloch walls are always free-energy-beneficial, thus explaining the absence of the perfect ferromagnetic order in the 1D Ising system. Note, however, that since the logarithmic function changes extremely slowly at large values of its argument, one may argue that a large but finite 1D system should still feature a quasi-critical temperature 2 J
" T "
,
(4.93)
c
ln N
below which it would be in a virtually complete ferromagnetic order. (The exponentially large susceptibility (91) is another manifestation of this fact.)
Now let us apply a similar approach to estimate T c of a 2D Ising model, with open borders. Here the Bloch wall is a line of a certain total length L – see Fig. 12. (For the example presented in that figure, counting from the left to the right, L = 2 + 1 + 4 + 2 + 3 = 12 lattice periods.)
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + - - -
+ + + + + + - - -
+ + - - - - - - -
Fig. 4.12. A Bloch wall in a 2D Ising system.
- - - - - - - - -
Evidently, the additional energy associated with such a wall is E W = 2 JL, while the wall’s entropy SW may be estimated using the following reasoning. A continuous Bloch wall may be thought about as the path of a “Manhattan pedestrian” crossing the system between its nodes. At each junction of straight segments of the path, the pedestrian may select 3 choices of 4 possible directions (except the one that leads backward), so for a path without self-crossings, there are 3( L-1) 3 L options for a walk starting from a certain point. Now taking into account that the open borders of a square-shaped lattice with N spins have a length of the order of N 1/2, and the Bloch wall may start from any of them, there are approximately M ~ N 1/23 L different walks between two borders. Again estimating SW as ln M, we get 43 For the closed-ring model (Fig. 10) such analysis gives an almost similar prediction, with the difference that in that system, the Bloch walls may appear only in pairs, so EW = 4 J, and S W = ln[ N( N – 1)] 2ln N.
44 This is a very vivid application of one of the core results of thermodynamics. If the reader is still uncomfortable with it, they are strongly encouraged to revisit Eq. (1.42) and its discussion.
Chapter 4
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F E TS 2 JL T ln
1/ 2 3 (2 ln )
3
/ 2 ln
.
(4.94)
W
W
N L
W
L J T
T N
(Actually, since L scales as N 1/2 or higher, at N the last term in Eq. (94) is negligible.) We see that the sign of the derivative FW / L depends on whether the temperature is higher or lower than the following critical value:
2 J
T
82
.
1
J .
(4.95)
c
ln 3
At T < T c, the free energy’s minimum corresponds to L 0, i.e. the Bloch walls are free-energy-detrimental, and the system is in the purely ferromagnetic phase.
So, for d = 2 the simple estimate predicts a non-zero critical temperature of the same order as the Weiss theory (according to Eq. (72), in this case T c = 4 J). The major approximation implied in the calculation leading to Eq. (95) is disregarding possible self-crossings of the “Manhattan walk”. The accurate counting of such self-crossings is rather difficult. It had been carried out in 1944 by L.
Onsager; since then his calculations have been redone in several easier ways, but even they are rather cumbersome, and I will not have time to discuss them.45 The final result, however, is surprisingly simple:
2 J
T
J
(4.96) Onsager’s
c
ln1 2
2.269 ,
exact result
i.e. showing that the simple estimate (95) is off the mark by only ~20%.
The Onsager solution, as well as all alternative solutions of the problem that were found later, are so “artificial” (2D-specific) that they do not give a clear way towards their generalization to other (higher) dimensions. As a result, the 3D Ising problem is still unsolved analytically. Nevertheless, we do know T c for it with extremely high precision – at least to the 6th decimal place. This has been achieved by numerical methods; they deserve a discussion because of their importance for the solution of other similar problems as well. Conceptually, the task is rather simple: just compute, to the desired precision, the statistical sum of the system (23):
J
h
Z exp s s
s
.
(4.97)
k k '
k
s for
,
1
T { , }'
T
k
k k
k
k ,
1 2,..., N
As soon as this has been done for a sufficient number of values of the dimensionless parameters J/ T and h/ T, everything is easy; in particular, we can compute the dimensionless function F / T ln Z ,
(4.98)
and then find the ratio J/ T c as the smallest value of the parameter J/ T at which the ratio F/ T (as a function of h/ T) has a minimum at zero field. However, for any system of a reasonable size N, the
“exact” computation of the statistical sum (97) is impossible, because it contains too many terms for any supercomputer to handle. For example, let us take a relatively small 3D lattice with N = 101010 = 103
spins, which still features substantial boundary artifacts even using the periodic boundary conditions, so its phase transition is smeared about T c by ~ 3%. Still, even for such a crude model, Z would include 45 For that, the interested reader may be referred to either Sec. 151 in the textbook by Landau and Lifshitz, or Chapter 15 in the text by Huang.
Chapter 4
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21,000 (210)100 (103)100 10300 terms. Let us suppose we are using a modern exaflops-scale supercomputer performing 1018 floating-point operations per second, i.e. ~1026 such operations per year.
With those resources, the computation of just one statistical sum would require ~10(300-26) = 10274 years.
To call such a number “astronomic” would be a strong understatement. (As a reminder, the age of our Universe is close to 1.41010 years – a very humble number in comparison.)
This situation may be improved dramatically by noticing that any statistical sum,
Em
Z exp
,
(4.99)
m
T
is dominated by terms with lower values of Em. To find those lowest-energy states, we may use the following powerful approach (belonging to a broad class of numerical Monte-Carlo techniques), which essentially mimics one (randomly selected) path of the system’s evolution in time. One could argue that for that we would need to know the exact laws of evolution of statistical systems,46 that may differ from one system to another, even if their energy spectra Em are the same. This is true, but since the genuine value of Z should be independent of these details, it may be evaluated using any reasonable kinetic model that satisfies certain general rules. In order to reveal these rules, let us start from a system with just two states, with energies Em and Em’ Em + – see Fig. 13.
Wm'
E
E
m'
m
Fig. 4.13. Deriving the detailed
E
balance relation.
m
W
m
In the absence of quantum coherence between the states (see Sec. 2.1), the equations for the time evolution of the corresponding probabilities Wm and Wm’ should depend only on the probabilities (plus certain constant coefficients). Moreover, since the equations of quantum mechanics are linear, these master equations should be also linear. Hence, it is natural to expect them to have the following form, Master
dW
dW
equations
m W Γ W Γ ,
m' W Γ W Γ ,
(4.100)
m'
m
m
m'
dt
dt
where the coefficients and have the physical sense of the rates of the corresponding transitions (see Fig. 13); for example, dt is the probability of the system’s transition into the state m’ during an infinitesimal time interval dt, provided that at the beginning of that interval it was in the state m with full certainty: Wm = 1, Wm’ = 0.47 Since for the system with just two energy levels, the time derivatives of the probabilities have to be equal and opposite, Eqs. (100) describe a redistribution of the probabilities between the energy levels, while keeping their sum W = Wm + Wm’ constant. According to Eqs. (100), at t , the probabilities settle to their stationary values related as W
m'
.
(4.101)
W
m
46 Discussion of such laws in the task of physical kinetics, which will be briefly reviewed in Chapter 6.
47 The calculation of these rates for several particular cases is described in QM Secs. 6.6, 6.7, and 7.6 – see, e.g., QM Eq. (7.196), which is valid for a very general model of a quantum system.
Chapter 4
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Now let us require these stationary values to obey the Gibbs distribution (2.58); from it W
E E
m'
m
m'
exp
exp 1.
(4.102)
Wm
T
T
Comparing these two expressions, we see that the rates have to satisfy the following detailed balance relation:
exp .
(4.103) Detailed
T
balance
Now comes the final step: since the rates of transition between two particular states should not depend on other states and their occupation, Eq. (103) has to be valid for each pair of states of any multi-state system. (By the way, this relation may serve as an important sanity check: the rates calculated using any reasonable model of a quantum system have to satisfy it.)
The detailed balance yields only one equation for two rates and ; if our only goal is the calculation of Z, the choice of the other equation is not too critical. A very simple choice is
,
1
if
,
0
(4.104)
exp / T otherwise,
,
where is the energy change resulting from the transition. This model, which evidently satisfies the detailed balance relation (103), is very popular (despite the unphysical cusp this function has at = 0), because it enables the following simple Metropolis algorithm (Fig. 14).
set up an initial state
- flip a random spin
- calculate
- calculate ()
generate random
(0 1)
Fig. 4.14. A crude scheme of
<
>
reject
compare
accept
the Metropolis algorithm for
spin flip
spin flip
the Ising model simulation.
The calculation starts by setting a certain initial state of the system. At relatively high temperatures, the state may be generated randomly; for example, in the Ising system, the initial state of each spin sk may be selected independently, with a 50% probability. At low temperatures, starting the calculations from the lowest-energy state (in particular, for the Ising model, from the ferromagnetic state sk = sgn( h) = const) may give the fastest convergence. Now one spin is flipped at random, the Chapter 4
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SM: Statistical Mechanics
corresponding change of the energy is calculated,48 and plugged into Eq. (104) to calculate (). Next, a pseudo-random number generator is used to generate a random number , with the probability density being constant on the segment [0, 1]. (Such functions are available in virtually any numerical library.) If the resulting is less than (), the transition is accepted, while if > (), it is rejected. Physically, this means that any transition down the energy spectrum ( < 0) is always accepted, while those up the energy profile ( > 0) are accepted with the probability proportional to exp{–/ T}.49 After sufficiently many such steps, the statistical sum (99) may be calculated approximately as a partial sum over the states passed by the system. (It may be better to discard the contributions from a few first steps, to avoid the effects of the initial state choice.)
This algorithm is extremely efficient. Even with the very modest computers available in the 1980s, it has allowed simulating a 3D Ising system of (128)3 spins to get the following result: J/ Tc
0.221650 0.000005. For all practical purposes, this result is exact – so perhaps the largest benefit of the possible future analytical solution of the infinite 3D Ising problem will be a virtually certain Nobel Prize for its author. Table 2 summarizes the values of T c for the Ising model. Very visible is the fast improvement of the prediction accuracy of the molecular-field approximation, because it is asymptotically correct at d .
Table 4.2. The critical temperature T c (in the units of J) of the Ising model of a ferromagnet ( J > 0), for several values of dimensionality d d
Molecular-field approxima
tion – Eq. (72) Exact value
Exact value’s source
0
0
0
Gibbs distribution
1
2
0
Transfer matrix theory
2
4
2.269…
Onsager’s solution
3
6
4.513…
Numerical simulation
Finally, I need to mention the renormalization-group (“RG”) approach,50 despite its low efficiency for Ising-type problems. The basic idea of this approach stems from the scaling law (30)-(31): at T = T c the correlation radius r c diverges. Hence, the critical temperature may be found from the requirement for the system to be spatially self-similar. Namely, let us form larger and larger groups (“blocks”) of adjacent spins, and require that all properties of the resulting system of the blocks approach those of the initial system, as T approaches T c.
Let us see how this idea works for the simplest nontrivial (1D) case described by the statistical sum (80). Assuming N to be even (which does not matter at N ), and adding an inconsequential constant C to each exponent (for the purpose that will be clear soon), we may rewrite this expression as 48 Note that a flip of a single spin changes the signs of only (2 d + 1) terms in the sum (23), i.e. does not require the re-calculation of all (2 d +1) N terms of the sum, so the computation of takes just a few multiply-and-accumulate operations even at N >> 1.
49 The latter step is necessary to avoid the system’s trapping in local minima of its multidimensional energy profile Em( s 1, s 2,…, sN).
50 Initially developed in the quantum field theory in the 1950s, it was adapted to statistics by L. Kadanoff in 1966, with a spectacular solution of the so-called Kubo problem by K. Wilson in 1972, later awarded with a Nobel Prize.
Chapter 4
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SM: Statistical Mechanics
h
J
h
Z exp s s s
s
C .
(4.105)
k
k k 1
k 1
s 1
T
T
T
k
N
2
2
,
1 2,...
k
Let us group each pair of adjacent exponents to recast this expression as a product over only even numbers k,
h
J
h
h
Z exp s s
s
s
s
2 C ,
(4.106)
k 1
k
k 1
k 1
k 1
s 1
T
T
T
T
k
N
2
2
2,4,...
k
and carry out the summation over two possible states of the internal spin sk explicitly:
h
J
h
h
exp
s
s
s
s
C
k 1
k 1 k 1
2
k
2 T
T
T
2
1
T
Z
s
k
N
h
J
h
h
k
1
2,4,... exp s s s
s
C
k 1
k 1 k 1
2
k
(4.107)
2 T
T
T
2
1
T
J
h
h
2cosh s s
s
s
C
k
k
1
1
exp
k 1 k 1 2 .
s 1
T
T
T
k 2,4,...
k
N
2
Now let us require this statistical sum (and hence all statistical properties of the system of two-spin blocks) to be identical to that of the Ising system of N/2 spins, numbered by odd k:
J'
h'
Z' exp s s s C' ,
(4.108)
k 1 k 1
k 1
s 1
T
T
k 2,4,...,
k
N
with some different parameters h’, J’, and C’, for all four possible values of sk-1 = 1 and sk+1 = 1.
Since the right-hand side of Eq. (107) depends only on the sum ( sk-1 + sk+1), this requirement yields only three (rather than four) independent equations for finding h’, J’, and C’. Of them, the equations for h’
and J’ depend only on h and J (but not on C),51 and may be represented in an especially simple form, RG
x 1
( y)2
y ( x y)
x'
, y'
,
(4.109) equations
( x y 1
)( xy)
1 xy
for 1D Ising
model
if the following notation is used:
J
h
x exp 4 , y exp 2 .
(4.110)
T
T
Now the grouping procedure may be repeated, with the same result (109)-(110). Hence these equations may be considered as recurrence relations describing repeated doubling of the spin block size.
Figure 15 shows (schematically) the trajectories of this dynamic system on the phase plane [ x, y]. (Each trajectory is defined by the following property: for each of its points { x, y}, the point { x’, y’} defined by the “mapping” Eq. (109) is also on the same trajectory.) For the ferromagnetic coupling ( J > 0) and h > 0, we may limit the analysis to the unit square 0 x, y 1. If this flow diagram had a stable fixed point with x’ = x = x 0 (i.e. T/ J < ) and y’ = y = 1 (i.e. h = 0), then the first of Eqs. (110) would immediately give us the critical temperature of the phase transition in the field-free system: 51 This might be expected because physically C is just a certain constant addition to the system’s energy.
However, the introduction of that constant was mathematically necessary, because Eqs. (107) and (108) may be reconciled only if C’ C.
Chapter 4
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SM: Statistical Mechanics
4 J
T
.
(4.111)
c
1
ln( / x )
However, Fig. 15 shows that the only fixed point of the 1D system is x = y = 0, which (at a finite coupling J) should be interpreted as T c = 0. This is of course in agreement with the exact result of the transfer-matrix analysis but does not provide much additional information.
y
exp 2 h / T
h 0
1
T 0
T
Fig. 4.15. The RG flow
diagram of the 1D Ising
system (schematically).
0
h
1 x exp{4 J / T}
Unfortunately, for higher dimensionalities, the renormalization-group approach rapidly becomes rather cumbersome and requires certain approximations whose accuracy cannot be easily controlled. For the 2D Ising system, such approximations lead to the prediction T c 2.55 J, i.e. to a substantial difference from the exact result (96).
4.6. Exercise problems
4.1. Calculate the entropy, the internal energy, and the specific heat cV of the van der Waals gas, and discuss the results. For the gas with temperature-independent cV, find the relation between V and T
during an adiabatic process.
4.2. Use two different approaches to calculate the coefficient ( E/ V) T for the van der Waals gas and the change of temperature of such a gas, with a temperature-independent CV, at its very fast expansion.
4.3. For real gases, the Joule-Thomson coefficient ( T/ P) H (and hence the gas temperature change at its throttling, see Problem 1.11) inverts its sign at crossing the so-called inversion curve T inv( P). Calculate this curve for the van der Waals gas.
4.4. Calculate the difference CP – CV for the van der Waals gas, and compare the result with that for an ideal classical gas.
4.5. For the van der Waals model, calculate the temperature dependence of the phase-equilibrium pressure P 0( T) and the latent heat ( T), in the low-temperature limit T << T c.
4.6. Perform the same tasks as in the previous problem, in the opposite limit: in close vicinity of the critical point T c.
Chapter 4
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SM: Statistical Mechanics
4.7. Calculate CV and CP for the stable gas-liquid system described by the van der Waals equation, for V = V c and 0 < T c – T << T c.
4.8. Calculate the critical values P c, V c, and T c for the so-called Redlich-Kwong model of the real gas, with the following equation of state:52
a
NT
P
,
V V Nb T 1/ 2
V Nb
with constant parameters a and b.
Hint: Be prepared to solve a cubic equation with particular (numerical) coefficients.
4.9. Calculate the critical values P c, V c, and T c for the phenomenological Dieterici model, with the following equation of state:53
NT
a
P
exp
,
V b
NTV
with constant parameters a and b. Compare the value of the dimensionless factor P c V c/ NT c with those given by the van der Waals and Redlich-Kwong models.
4.10. In the crude sketch shown in Fig. 3b, the derivatives dP/ dT of the phase transitions liquid-gas (“vaporization”) and solid-gas (“sublimation”), at the triple point, are different, with
dP
dP
v
s
.
dT T
dT
t
T
T t
T
Is this occasional? What relation between these derivatives can be obtained from thermodynamics?
4.11. Use the Clapeyron-Clausius formula (17) to calculate the latent heat of the Bose-Einstein condensation, and compare the result with that obtained in the solution of Problem 3.21.
4.12. As was discussed in Sec. 4.1 of the lecture notes, properties of systems with first-order phase transitions (such as the van der Waals gas) change qualitatively at the critical temperature: at T < T c, the system may include two different phases of the same substance. However, since the difference in the density of these phases, in equilibrium, is a continuous function of the difference T c – T, this change itself is sometimes considered a continuous phase transition between the purely gaseous phase and the mixed gas-liquid “phase”. From this viewpoint, what are the most reasonable analogs of the critical exponents , , and , which were defined in Sec. 4.2, for such a continuous transition? Evaluate these exponents for the van der Waals model.
52 This equation of state, suggested in 1948, describes most real gases better than not only the original van der Waals model, but also other two-parameter alternatives, such as the Berthelot, modified-Berthelot, and Dieterici models, though some approximations with more fitting parameters (such as the Soave-Redlich-Kwong model) work even better.
53 For practical applications, this approximation is currently less popular than the Redlich-Kwong model (also with two fitting parameters), whose analysis was the task of the previous problem.
Chapter 4
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SM: Statistical Mechanics
4.13.
(i) Compose such an effective single-particle Hamiltonian that the corresponding stationary Schrödinger equation coincides with the Gross-Pitaevski equation (58).
(ii) Use this Gross-Pitaevskii Hamiltonian, with the particular confining potential U(r) =
m2 r 2/2, to calculate the ground-state energy E of N >> 1 confined particles, by assuming the natural trial solution exp{– r 2/2 r 2
0 }.54
(iii) Explore the function E( r 0) for positive and negative values of the constant b, and interpret the results.
(iv) For b < 0 with small b , estimate the largest number N of particles that may form a metastable Bose-Einstein condensate.
4.14. Superconductivity may be suppressed by a sufficiently strong magnetic field. In the simplest case of a bulk, long cylindrical sample of a type-I superconductor, placed into an external magnetic field Hext parallel to its axis, this suppression takes a simple form of a simultaneous transition of the whole sample from the superconducting state to the “normal” (non-superconducting) state at a certain value Hc( T) of the field’s magnitude. This critical field gradually decreases with temperature from its maximum value Hc(0) at T 0 to zero at the critical temperature T c. Assuming that the function Hc( T) is known, calculate the latent heat of this phase transition as a function of temperature, and spell out its values at T 0 and T = T c.
Hint: In this particular context, “bulk sample” means a sample much larger than the intrinsic length scales of the superconductor (such as the London penetration depth L and the coherence length
).55 For such bulk superconductors, magnetic properties of the superconducting phase may be well described as perfect diamagnetism, with B = 0 inside it.
4.15. In some textbooks, the discussion of thermodynamics of superconductivity is started by displaying, as self-evident, the following formula:
2
F
H
,
n T
F s T
0
c T V
2
where F s and F n are the free energy values in the superconducting and non-superconducting (“normal”) phases, and Hc( T) is the critical value of the magnetic external field. Is this formula correct, and if not, what qualification is necessary to make it valid? Assume that all the conditions of a simultaneous field-induced phase transition in the whole sample, spelled out in the previous problem, are satisfied.
4.16. Consider a system of N = 3 Ising “spins” ( sk = 1) with similar ferromagnetic coupling J of each pair of them, in thermal equilibrium.
(i) Calculate the order parameter and the low-field susceptibility / h h=0.
(ii) Use the low-temperature limit of the result for to predict it for a similar “ring” with an arbitrary N, and verify your prediction by a direct calculation (in this limit).
(iii) Discuss the relation between the last result, in the limit N , and Eq. (91).
54 This task is essentially the first step of the variational method of quantum mechanics – see, e.g., QM Sec. 2.9.
55A discussion of these parameters, as well as of the difference between the type-I and type-II superconductivity, may be found in EM Secs. 6.4-6.5. However, those details are not needed for the solution of this problem.
Chapter 4
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SM: Statistical Mechanics
4.17. Calculate the average energy, entropy, and heat capacity of a uniform three-site ring of Ising-type “spins” ( sk = 1), with antiferromagnetic coupling (of magnitude J) between the sites, in thermal equilibrium at temperature T, with no external magnetic field. Find the asymptotic behavior of its heat capacity for low and high temperatures, and give an interpretation of the results.
4.18. Using the results discussed in Sec. 5 for the infinite 1D Ising model, calculate the average energy, free energy, entropy, and heat capacity (all per one “spin”) as functions of temperature T and external field h. Sketch the temperature dependence of the heat capacity for various values of the h/ J
ratio, and give a physical interpretation of the result.
4.19. Calculate the specific heat (per “spin”) for the d-dimensional Ising problem in the absence of the external field, in the molecular-field approximation. Sketch the temperature dependence of C and compare it with the corresponding plot in the previous problem’s solution.
4.20. Prove that in the limit T T c, the molecular-field approximation, applied to the Ising model with a spatially constant order parameter, gives results similar to those of Landau’s mean-field theory with certain coefficients a and b. Calculate these coefficients and list the critical exponents defined by Eqs. (26), (28), (29), and (32), given by this approximation.
4.21. Assuming that the statistical sum ZN of a field-free, open-ended 1D Ising system of N
“spins” with arbitrary coefficients Jk is known, calculate ZN+1. Then use the result to obtain an explicit expression for ZN and compare it with Eq. (88).
4.22. Use the molecular-field approximation to calculate the critical temperature and the low-field susceptibility of a d-dimensional cubic lattice of spins, described by the so-called classical Heisenberg model:56
E J
s s
h s .
m
k
k'
k
k, ,k'
k
Here, in contrast to the (otherwise, very similar) Ising model (23), the spin of each site is described as a classical 3D vector s k = { sxk, syk, szk} of unit length: 2
s = 1.
k
4.23. Use the molecular-field approximation to calculate the coefficient a in Landau’s expansion (46) for a 3D cubic lattice of spins described by the classical Heisenberg model (whose analysis was the subject of the previous problem).
4.24. Use the molecular-field approximation to calculate the critical temperature of the ferromagnetic transition for the d-dimensional cubic Heisenberg lattice of arbitrary (either integer or half-integer) quantum spins s.
Hint: This model is described by Eq. (4.21) of the lecture notes, with σˆ now meaning the vector operator of spin s, in units of Planck’s constant .
56 This classical model is formally similar to the generalization of the genuine (quantum) Heisenberg model (21) to arbitrary spin s, and serves as its infinite-spin limit.
Chapter 4
Page 36 of 36
SM: Statistical Mechanics
Chapter 5. Fluctuations
This chapter discusses fluctuations of macroscopic variables, mostly in thermodynamic equilibrium. In particular, it describes the intimate connection between fluctuations and dissipation ( damping) in dynamic systems coupled to multi-particle environments. This connection culminates in the Einstein relation between the diffusion coefficient and mobility, the Nyquist formula, and its quantum-mechanical generalization – the fluctuation-dissipation theorem. An alternative approach to the same problem, based on the Smoluchowski and Fokker-Planck equations, is also discussed in brief.
5.1. Characterization of fluctuations
At the beginning of Chapter 2, we discussed the notion of averaging a variable f over a statistical ensemble – see Eqs. (2.7) and (2.10). Now, the fluctuation of the variable is defined simply as its deviation from the average f :
~
f f f ;
(5.1)
Fluctuation
this deviation is, generally, also a random variable. The most important property of any fluctuation is that its average over the same statistical ensemble equals zero; indeed:
~
f f f
f f f f .
0
(5.2)
As a result, such an average cannot be used to characterize fluctuations’ intensity, and the simplest meaningful characteristic of the intensity is the variance (sometimes called “dispersion”):
~
Variance:
f
f f 2
2
.
(5.3)
definition
The following simple property of the variance is frequently convenient for its calculation:
~ f f f 2
2
f 2
2
2
2
f f f
f
2
2
2
f
f ,
(5.4a)
so, finally:
Variance
~
via
2
2
2
f
f
f .
(5.4b)
averages
As the simplest example, consider a variable that takes only two values, 1, with equal probabilities Wj = ½. For such a variable, the basic Eq. (2.7) yields 1
f W f
;
(5.5a)
j
j
1
1
1 0
j
2
2
however,
2
2
2
2
~
f
W f ½( )1 ½( )1 ,1
that
so
2
2
2
f
f
f
1 0.
(5.5b)
j
j
j
The square root of the variance,
1/ 2
r.m.s.
~
2
f
f
,
(5.6)
fluctuation
© K. Likharev
SM: Statistical Mechanics
is called the root-mean-square ( r.m.s. ) fluctuation. An advantage of this measure is that it has the same dimensionality as the variable itself, so the ratio f/ f is dimensionless, and is used to characterize the relative intensity of fluctuations.
As has been mentioned in Chapter 1, all results of thermodynamics are valid only if the fluctuations of thermodynamic variables (the internal energy E, entropy S, etc.) are relatively small.1 Let us make a simple estimate of the relative intensity of fluctuations, taking as the simplest example a system of N independent, similar parts (e.g., particles), and an extensive variable N
F f .
(5.7)
k
k 1
where all single-particle functions fk are similar, besides that each of them depends on the state of only
“its own” ( k th) part. The statistical average of such F is evidently N
F f N f ,
(5.8)
k 1
while the variance of its fluctuations is
N
N
N
N
~
~ ~
~
~
~ ~
~ ~
2
F
F F f
f
f f
f f
.
(5.9)
k
k '
k
k '
k k'
k 1
k' 1
k , k '1
k , k '1
Now we may use the fact that for two independent variables:
~ ~
~
~
f f
f
f
,
0
k'
for k ;
(5.10)
k
k '
k
k '
indeed, the first of these equalities may be used as the mathematical definition of their independence.
Hence, only the terms with k’ = k make nonzero contributions to the right-hand side of Eq. (9):
~
N
2
~2
~
2
F
f N f .
(5.11)
k
k , k '
k , k ' 1
Comparing Eqs. (8) and (11), we see that the relative intensity of fluctuations of the variable F,
F
1
f
Relative
,
(5.12) fluctuation
1/ 2
F
N
f
estimate
tends to zero as the system size grows ( N ). It is this fact that justifies the thermodynamic approach to typical physical systems, with the number N of particles of the order of the Avogadro number N A ~
1024. Nevertheless, in many situations even small fluctuations of variables are important, and in this chapter, we will calculate their basic properties, starting with the variance.
It should be comforting for the reader to notice that for one very important case, such a calculation has already been done in our course. Indeed, for any generalized coordinate q and generalized momentum p that give quadratic contributions of the type (2.46) to the system’s 1 Let me remind the reader that up to this point, the averaging signs … were dropped in most formulas, for the sake of notation simplicity. In this chapter, I have to restore these signs to avoid confusion. The only exception will be temperature – whose average, following (probably, bad :-) tradition, will be still called just T everywhere, besides the last part of Sec. 3, where temperature fluctuations are discussed explicitly.
Chapter 5
Page 2 of 44
SM: Statistical Mechanics
Hamiltonian (as in a harmonic oscillator), we have derived the equipartition theorem (2.48), valid in the classical limit. Since the average values of these variables, in the thermodynamic equilibrium, equal zero, Eq. (6) immediately yields their r.m.s. fluctuations: