lecture 28 phys 416 thursday december 2 fall 2021 1.7.7

Lecture 28 PHYS 416 Thursday December 2 Fall 2021

1. 7.7 Metals & the Fermi gas2. Ch 9: Order parameters, broken symmetry, and topology3. HW 9.5 Landau theory for the Ising model4. Quiz 10

Ch 7.7 Metals and the Fermi gas

History:• 1800’s Thermodynamics of ideal gas worked out

• But still no theory of materials – metals vs insulators

• 1896 Thomson discovers the electron

• 1900 Drude treats metals as a “gas” of electrons, applies thermodynamics – some success.

• 1927 Sommerfeld et al. applies QM, Fermi-Dirac statistics, many more successes…

• BUT, still lots of failures. Something is missing…

Drude modelled metals as materials where atoms give up outer electrons into a gas of charged particles that behave like an ideal gas. In particular, the equipartition theorem:

12mv0

2 =32kBT

Electrons move with constant kinetic energy, until they have a collision. This randomly re-directs their motion, and is characterized by a relaxation time t and a mean free path between collisions, lmfp. This can explain Ohm’s Law, when an electric field (or voltage) is applied. The average velocity, the current density, and the mean free path become:

vavg =−eEτm

j= ne2τm

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟E = σE lmfp = v0τ

(This discussion largely comes from the book “Solid State Physics” by Ashcroft & Mermin.)

Now we can see that a constant force (qE) leads to a constant speed instead of a constant acceleration. In effect, the scattering adds a frictional damping term to the force equation:

dp(t)dt=−

p(t)τ+ f (t)

A big success was explaining some properties of the thermal conductivity of metals. Thermal conductivity is defined by:

jQ ≡−κdT dxIn this Drude model, the thermal conductivity can be expressed as:

κ= 13 v

2τcV = 13 lmfpvcV

Applying the thermodynamics of an ideal gas, we write the specific heat as:

cV =32nkB

Take the ratio of thermal and electrical conductivities:

κσ=

13 cVmv

2

ne2=32kBe⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

2

T

This can explain (mostly) the Wiedemann-Franz Law:

κσT=

32kBe⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

2

=1.11 x 10−8 watt-ohm/K2

This differs from the data by a factor of 2 (not bad!). Drude, however, made a factor of 2 mistake that put him in agreement with the data, which really helped his theory! (It turned out that his theory also missed two factors of 100, that cancelled out.)

So, the Drude theory was a GREAT first start in developing a modern theory of materials. It still got many, many things wrong. Time to start talking about statistics.

Drude gets the average velocity from equipartition:12mv0

2 =32kBT

Maxwell-Boltzmann statistics are used for the electron velocity distribution:

fMB(v)= nm

2πkBT

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

3/2

e−mv2 /2kBT

By 1927, we had quantum mechanics, and especially Fermi-Dirac quantum statistics….

f (v)= (m / !)3

4π31

exp[( 12mv2−kBT0 ) / kBT ]+1

This Fermi-Dirac distribution changes everything. It will turn out that all transport processes depend only on the tiny slice of electrons within kT of the maximum energy.

n= dvf (v)∫

Let’s rederive the density of states and other basic properties. Again, consider a box of volume V, and determine the allowed states.

ψk (r)=1Veik⋅r ε(k)= !

2k 2

2mkx =

2πnxL...

p=!k, v= !km

, ε= 12mv

2 , λ=! k

wavefunction wave vector free particle energy

k-space density of states = V8π3

4πkF3

3

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟V8π3⎛

⎝⎜⎜⎜⎞

⎠⎟⎟⎟⎟=

kF3

6π2V

Consider the number of states within a sphere that could contain all of the particles. This is called the Fermi sphere, with a Fermi radius, corresponding to the Fermi energy….

k-space volume

Density of states

number of states

N = 2 ⋅kF3

6π2V =

kF3

3π2VAccount for the two possible spins:

vF =!mkF ∼108 cm/sSpeed of electrons at the Fermi level:

(more than 10 times bigger than Drude speed)

εclassical

= 32 kBT

EN= ε

FD= 2 dk

8π3! 2k 2

2m= 3

5 εF =35 kBTF

0

kF

∫

TF =εFkB∼104−105K

What is the energy of these electrons at the Fermi level? Classically, that is kT.Quantum mechanically, we get a Fermi temperature in excess of 10,000 K. That means even at room temperature, the electrons in metals are effectively at extremely low temperatures

We can now consider the specific heat, which experiments showed was about 100 times smaller than Drude theory predicts.

cV =∂u∂T⎛

⎝⎜⎜⎜⎞

⎠⎟⎟⎟⎟V

u=UV

U = 2 ε(k)k∑ fFD (ε(k)) fFD (ε)=

1e(ε−µ)/kBT +1

The density of states can be recast as: g(ε)= 32nεF

εεF

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

1/2

u = dε g(ε ) ε fFD (ε )∫Calculate the energy:

cV =π2

2kBTεF

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟nkBCalculate the specific heat:

This temperature ratio drops the specific heat by the necessary factor 0f ~100.

Summary:

1. Drude theory applied thermodynamics to the electron gas and explained Ohm’s Law, Wiedemann-Franz, and more. A great start….

2. Sommerfeld introduced QM and fermion statistics, and fixed many issues, especially specific heat.

3. BUT, this is just a start. This cannot explain:• Hall coefficients• Magnetoresistance• Thermoelectric effect• Low temperature s, k, cV• Why are some elements nonmetals?

What is the missing ingredient?

Symmetry! The 3D periodicity of crystals.

This can be illustrated by the shape of actual Fermi “spheres”:

Copyright Oxford University Press 2006 v2.0 --

Order parameters, brokensymmetry, and topology 9

9.1 Identify the broken symmetry246

9.2 Define the order parameter246

9.3 Examine the elementary exci-tations 250

9.4 Classify the topological de-fects 252

This chapter is slightly modified froma lecture given at the Santa Fe Insti-tute [153].

In elementary school, we were taught that there were three states ofmatter: solid, liquid, and gas. The ancients thought that there werefour: earth, water, air, and fire, which was considered sheer superstition.In junior high, the author remembers reading a book as a kid called TheSeven States of Matter [68]. At least one was ‘plasma’, which made upstars and thus most of the Universe,1 and which sounded rather like fire.

1They had not heard of dark matter.

Fig. 9.1 Quasicrystals. Crystals aresurely the oldest known of the broken-symmetry phases of matter. In thepast few decades, we have uncoveredan entirely new class of quasicrystals,here [95] with icosahedral symmetry.Note the five-fold structures, forbiddenin our old categories.

The original three, by now, have become multitudes. In importantand precise ways, magnets are a distinct form of matter. Metals aredifferent from insulators. Superconductors and superfluids are strikingnew states of matter. The liquid crystal in your wristwatch is one of ahuge family of different liquid crystalline states of matter [46] (nematic,cholesteric, blue phase I, II, and blue fog, smectic A, B, C, C∗, D, I,. . . ). There are over 200 qualitatively different types of crystals, notto mention the quasicrystals (Fig. 9.1). There are disordered states ofmatter like spin glasses, and states like the fractional quantum Hall effectwith excitations of charge e/3 like quarks. Particle physicists tell us thatthe vacuum we live within has in the past been in quite different states;in the last vacuum before this one, there were four different kinds oflight [42] (mediated by what is now the photon, the W+, the W−, andthe Z particle).When there were only three states of matter, we could learn about

each one and then turn back to learning long division. Now that thereare multitudes, though, we have had to develop a system. Our systemis constantly being extended and modified, because we keep finding newphases which do not fit into the old frameworks. It is amazing how the500th new state of matter somehow screws up a system which workedfine for the first 499. Quasicrystals, the fractional quantum hall effect,and spin glasses all really stretched our minds until (1) we understoodwhy they behaved the way they did, and (2) we understood how theyfit into the general framework.In this chapter, we are going to tell you the system. It consists of

four basic steps [117]. First, you must identify the broken symmetry(Section 9.1). Second, you must define an order parameter (Section 9.2).Third, you are told to examine the elementary excitations (Section 9.3).Fourth, you classify the topological defects (Section 9.4). Most of whatwe say in this chapter is taken from Mermin [117], Coleman [42], anddeGennes and Prost [46], which are heartily recommended.

The system for fitting newly discovered phases of matter into our existing framework:

1. Identify the broken symmetry2. Define an order parameter3. Examine the elementary excitations4. Classify the topological defects


8.3 What is a phase? Perturbation theory 219

∑

α

Pβαρλα =

∑

α

Pβα(τλα

√ρ∗α) =

∑

α

Qβα

√ρ∗βρ∗α

(τλα√ρ∗α)

=∑

α

(Qβατ

λα

)√ρ∗β = λ

(τλβ√ρ∗β

)= λρλβ . (8.17)

Now we turn to the main theorem underlying the algorithms for equi-librating lattice models in statistical mechanics.

Theorem 8.4. (main theorem) A discrete dynamical system with afinite number of states can be guaranteed to converge to an equilibriumdistribution ρ∗ if the computer algorithm

• is Markovian (has no memory),

• is ergodic (can reach everywhere and is acyclic), and

• satisfies detailed balance.

Proof Let P be the transition matrix for our algorithm. Since the al-gorithm satisfies detailed balance, P has a complete set of eigenvectorsρλ. Since our algorithm is ergodic there is only one right eigenvector ρ1

with eigenvalue one, which we can choose to be the stationary distribu-tion ρ∗; all the other eigenvalues λ have |λ| < 1. Decompose the initialcondition ρ(0) = a1ρ∗ +

∑|λ|<1 aλρ

λ. Then24

24The eigenvectors closest to one willbe the slowest to decay. You can getthe slowest characteristic time τ fora Markov chain by finding the largest|λmax| < 1 and setting λn = e−n/τ .

ρ(n) = P · ρ(n− 1) = Pn · ρ(0) = a1ρ∗ +

∑

|λ|<1

aλλnρλ. (8.18)

Since the (finite) sum in this equation decays to zero, the density con-verges to a1ρ∗. This implies both that a1 = 1 and that our systemconverges to ρ∗ as n → ∞. !

Thus, to develop a new equilibration algorithm (Exercises 8.6, 8.8),one need only ensure that it is Markov, ergodic, and satisfies detailedbalance.

8.3 What is a phase? Perturbation theory

0 2 4 6kBT / (h2ρ2/3/m)

0

3/2

C v / N

k B

Fermi gasBose gas

Fig. 8.6 Bose and Fermi specificheats. The specific heats for the idealBose and Fermi gases. Notice the cuspat the Bose condensation temperatureTc. Notice that the specific heat of theFermi gas shows no such transition.

What is a phase? We know some examples. Water is a liquid phase,which at atmospheric pressure exists between 0 ◦C and 100 ◦C; the equi-librium density of H2O jumps abruptly downward when the water freezesor vaporizes. The Ising model is ferromagnetic below Tc and paramag-netic above Tc. Figure 8.6 plots the specific heat of a non-interacting gasof fermions and of bosons. There are many differences between fermionsand bosons illustrated in this figure,25 but the fundamental difference

25The specific heat of the Fermi gasfalls as the temperature decreases; atlow temperatures, only those single-particle eigenstates within a few kBTof the Fermi energy can be excited.The specific heat of the Bose gas ini-tially grows as the temperature de-creases from infinity. Both the Fermiand Bose gases have Cv/N → 0 asT → 0, as is always true (otherwise theentropy,

∫ T0 Cv/T dT would diverge).

is that the Bose gas has two different phases. The specific heat has acusp at the Bose condensation temperature, which separates the normalphase and the condensed phase.How do we determine in general how far a phase extends? Inside

phases the properties do not shift in a singular way; one can smoothly

What are phases of matter?


Order parameters, brokensymmetry, and topology 9

9.1 Identify the broken symmetry246

9.2 Define the order parameter246

9.3 Examine the elementary exci-tations 250

9.4 Classify the topological de-fects 252

This chapter is slightly modified froma lecture given at the Santa Fe Insti-tute [153].

In elementary school, we were taught that there were three states ofmatter: solid, liquid, and gas. The ancients thought that there werefour: earth, water, air, and fire, which was considered sheer superstition.In junior high, the author remembers reading a book as a kid called TheSeven States of Matter [68]. At least one was ‘plasma’, which made upstars and thus most of the Universe,1 and which sounded rather like fire.

1They had not heard of dark matter.

Fig. 9.1 Quasicrystals. Crystals aresurely the oldest known of the broken-symmetry phases of matter. In thepast few decades, we have uncoveredan entirely new class of quasicrystals,here [95] with icosahedral symmetry.Note the five-fold structures, forbiddenin our old categories.

The original three, by now, have become multitudes. In importantand precise ways, magnets are a distinct form of matter. Metals aredifferent from insulators. Superconductors and superfluids are strikingnew states of matter. The liquid crystal in your wristwatch is one of ahuge family of different liquid crystalline states of matter [46] (nematic,cholesteric, blue phase I, II, and blue fog, smectic A, B, C, C∗, D, I,. . . ). There are over 200 qualitatively different types of crystals, notto mention the quasicrystals (Fig. 9.1). There are disordered states ofmatter like spin glasses, and states like the fractional quantum Hall effectwith excitations of charge e/3 like quarks. Particle physicists tell us thatthe vacuum we live within has in the past been in quite different states;in the last vacuum before this one, there were four different kinds oflight [42] (mediated by what is now the photon, the W+, the W−, andthe Z particle).When there were only three states of matter, we could learn about

each one and then turn back to learning long division. Now that thereare multitudes, though, we have had to develop a system. Our systemis constantly being extended and modified, because we keep finding newphases which do not fit into the old frameworks. It is amazing how the500th new state of matter somehow screws up a system which workedfine for the first 499. Quasicrystals, the fractional quantum hall effect,and spin glasses all really stretched our minds until (1) we understoodwhy they behaved the way they did, and (2) we understood how theyfit into the general framework.In this chapter, we are going to tell you the system. It consists of

four basic steps [117]. First, you must identify the broken symmetry(Section 9.1). Second, you must define an order parameter (Section 9.2).Third, you are told to examine the elementary excitations (Section 9.3).Fourth, you classify the topological defects (Section 9.4). Most of whatwe say in this chapter is taken from Mermin [117], Coleman [42], anddeGennes and Prost [46], which are heartily recommended.


246 Order parameters, broken symmetry, and topology

9.1 Identify the broken symmetry(b)(a)

Fig. 9.2 Which is more symmet-ric? Cube and sphere. (a) The cubehas many symmetries. It can be ro-tated by 90◦, 180◦, or 270◦ about anyof the three axes passing through thefaces. It can be rotated by 120◦ or 240◦

about the corners and by 180◦ aboutan axis passing from the center throughany of the 12 edges. (b) The sphere,though, can be rotated by any angle.The sphere respects rotational invari-ance: all directions are equal. The cubeis an object which breaks rotationalsymmetry: once the cube is there, somedirections are more equal than others.

(a) (b)Ice Water

Fig. 9.3 Which is more symmet-ric? Ice and water. At first glance,water seems to have much less symme-try than ice. (a) The picture of ‘two-dimensional’ ice clearly breaks the ro-tational invariance; it can be rotatedonly by 120◦ or 240◦. It also breaksthe translational invariance; the crys-tal can only be shifted by certain spe-cial distances (whole number of latticeunits). (b) The picture of water hasno symmetry at all; the atoms are jum-bled together with no long-range pat-tern at all. However, water as a phasehas a complete rotational and transla-tional symmetry; the pictures will lookthe same if the container is tipped orshoved.

What is it that distinguishes the hundreds of different states of matter?Why do we say that water and olive oil are in the same state (the liquidphase), while we say aluminum and (magnetized) iron are in differentstates? Through long experience, we have discovered that most phasesdiffer in their symmetry.2

2This is not to say that different phasesalways differ by symmetries! Liquidsand gases have the same symmetry, andsome fluctuating phases in low dimen-sions do not break a symmetry. It issafe to say, though, that if the two ma-terials have different symmetries, theyare different phases.

Consider Figs 9.2, showing a cube and a sphere. Which is more sym-metric? Clearly, the sphere has many more symmetries than the cube.One can rotate the cube by 90◦ in various directions and not changeits appearance, but one can rotate the sphere by any angle and keep itunchanged.In Fig. 9.3, we see a two-dimensional schematic representation of ice

and water. Which state is more symmetric here? Naively, the ice looksmuch more symmetric; regular arrangements of atoms forming a lat-tice structure. Ice has a discrete rotational symmetry: one can rotateFig. 9.3(a) by multiples of 60◦. It also has a discrete translational sym-metry: it is easy to tell if the picture is shifted sideways, unless one shiftsby a whole number of lattice units. The water looks irregular and disor-ganized. On the other hand, if one rotated Fig. 9.3(b) by an arbitraryangle, it would still look like water! Water is not a snapshot; it is betterto think of it as a combination (or ensemble) of all possible snapshots.While the snapshot of the water shown in the figure has no symmetries,water as a phase has complete rotational and translational symmetry.

9.2 Define the order parameter

Particle physics and condensed-matter physics have quite different philo-sophies. Particle physicists are constantly looking for the building blocks.Once pions and protons were discovered to be made of quarks, the fo-cus was on quarks. Now quarks and electrons and photons seem to bemade of strings, and strings are hard to study experimentally (so far).Condensed-matter physicists, on the other hand, try to understand whymessy combinations of zillions of electrons and nuclei do such interest-ing simple things. To them, the fundamental question is not discoveringthe underlying quantum mechanical laws, but in understanding and ex-plaining the new laws that emerge when many particles interact.3

As one might guess, we do not always keep track of all the electronsand protons. We are always looking for the important variables, theimportant degrees of freedom. In a crystal, the important variables arethe motions of the atoms away from their lattice positions. In a magnet,the important variable is the local direction of the magnetization (anarrow pointing to the ‘north’ end of the local magnet). The local mag-netization comes from complicated interactions between the electrons,

3The particle physicists use order parameter fields too; their quantum fields also hide lots of details about what their quarksand gluons are composed of. The main difference is that they do not know what their fields are composed of. It ought to bereassuring to them that we do not always find our greater knowledge very helpful.



9.1 Identify the broken symmetry(b)(a)

Fig. 9.2 Which is more symmet-ric? Cube and sphere. (a) The cubehas many symmetries. It can be ro-tated by 90◦, 180◦, or 270◦ about anyof the three axes passing through thefaces. It can be rotated by 120◦ or 240◦

about the corners and by 180◦ aboutan axis passing from the center throughany of the 12 edges. (b) The sphere,though, can be rotated by any angle.The sphere respects rotational invari-ance: all directions are equal. The cubeis an object which breaks rotationalsymmetry: once the cube is there, somedirections are more equal than others.

(a) (b)Ice Water

Fig. 9.3 Which is more symmet-ric? Ice and water. At first glance,water seems to have much less symme-try than ice. (a) The picture of ‘two-dimensional’ ice clearly breaks the ro-tational invariance; it can be rotatedonly by 120◦ or 240◦. It also breaksthe translational invariance; the crys-tal can only be shifted by certain spe-cial distances (whole number of latticeunits). (b) The picture of water hasno symmetry at all; the atoms are jum-bled together with no long-range pat-tern at all. However, water as a phasehas a complete rotational and transla-tional symmetry; the pictures will lookthe same if the container is tipped orshoved.

What is it that distinguishes the hundreds of different states of matter?Why do we say that water and olive oil are in the same state (the liquidphase), while we say aluminum and (magnetized) iron are in differentstates? Through long experience, we have discovered that most phasesdiffer in their symmetry.2

2This is not to say that different phasesalways differ by symmetries! Liquidsand gases have the same symmetry, andsome fluctuating phases in low dimen-sions do not break a symmetry. It issafe to say, though, that if the two ma-terials have different symmetries, theyare different phases.

Consider Figs 9.2, showing a cube and a sphere. Which is more sym-metric? Clearly, the sphere has many more symmetries than the cube.One can rotate the cube by 90◦ in various directions and not changeits appearance, but one can rotate the sphere by any angle and keep itunchanged.In Fig. 9.3, we see a two-dimensional schematic representation of ice

and water. Which state is more symmetric here? Naively, the ice looksmuch more symmetric; regular arrangements of atoms forming a lat-tice structure. Ice has a discrete rotational symmetry: one can rotateFig. 9.3(a) by multiples of 60◦. It also has a discrete translational sym-metry: it is easy to tell if the picture is shifted sideways, unless one shiftsby a whole number of lattice units. The water looks irregular and disor-ganized. On the other hand, if one rotated Fig. 9.3(b) by an arbitraryangle, it would still look like water! Water is not a snapshot; it is betterto think of it as a combination (or ensemble) of all possible snapshots.While the snapshot of the water shown in the figure has no symmetries,water as a phase has complete rotational and translational symmetry.

9.2 Define the order parameter

Particle physics and condensed-matter physics have quite different philo-sophies. Particle physicists are constantly looking for the building blocks.Once pions and protons were discovered to be made of quarks, the fo-cus was on quarks. Now quarks and electrons and photons seem to bemade of strings, and strings are hard to study experimentally (so far).Condensed-matter physicists, on the other hand, try to understand whymessy combinations of zillions of electrons and nuclei do such interest-ing simple things. To them, the fundamental question is not discoveringthe underlying quantum mechanical laws, but in understanding and ex-plaining the new laws that emerge when many particles interact.3

As one might guess, we do not always keep track of all the electronsand protons. We are always looking for the important variables, theimportant degrees of freedom. In a crystal, the important variables arethe motions of the atoms away from their lattice positions. In a magnet,the important variable is the local direction of the magnetization (anarrow pointing to the ‘north’ end of the local magnet). The local mag-netization comes from complicated interactions between the electrons,

3The particle physicists use order parameter fields too; their quantum fields also hide lots of details about what their quarksand gluons are composed of. The main difference is that they do not know what their fields are composed of. It ought to bereassuring to them that we do not always find our greater knowledge very helpful.


9.2 Define the order parameter 247

space

M

x

Physical space Order parameter

Fig. 9.4 Magnetic order parame-ter. For a magnetic material at a giventemperature, the local magnetization|M| = M0 will be pretty well fixed,but the energy is often nearly indepen-dent of the direction M = M/M0 ofthe magnetization. Often, the magne-tization changes directions in differentparts of the material. (That is why notall pieces of iron are magnetic!) Wetake the magnetization as the order pa-rameter for a magnet; you can think ofit as an arrow pointing to the north endof each atomic magnet. The currentstate of the material is described by anorder parameter field M(x). It can beviewed either as an arrow at each pointin space. or as a function taking pointsin space x into points on the sphere.This sphere S2 is the order parameterspace for the magnet.

and is partly due to the little magnets attached to each electron andpartly due to the way the electrons dance around in the material; thesedetails are for many purposes unimportant.The important variables are combined into an ‘order parameter field’.

In Fig. 9.4, we see the order parameter field for a magnet.4 At each 4Most magnets are crystals, which al-ready have broken the rotational sym-metry. For some ‘Heisenberg’ magnets,the effects of the crystal on the mag-netism is small. Magnets are really dis-tinguished by the fact that they breaktime-reversal symmetry: if you reversethe arrow of time, the magnetizationchanges sign.

position x = (x, y, z) we have a direction for the local magnetizationM(x). The length of M is pretty much fixed by the material, but thedirection of the magnetization is undetermined. By becoming a magnet,this material has broken the rotational symmetry. The order parameterM labels which of the various broken symmetry directions the materialhas chosen.The order parameter is a field; at each point in our magnet, M(x)

tells the local direction of the field near x. Why would the magne-tization point in different directions in different parts of the magnet?Usually, the material has lowest energy when the order parameter fieldis uniform, when the symmetry is broken in the same way throughoutspace. In practice, though, the material often does not break symmetryuniformly. Most pieces of iron do not appear magnetic, simply becausethe local magnetization points in different directions at different places.The magnetization is already there at the atomic level; to make a mag-net, you pound the different domains until they line up. We will see inthis chapter that much of the interesting behavior we can study involvesthe way the order parameter varies in space.The order parameter field M(x) can be usefully visualized in two

different ways. On the one hand, one can think of a little vector attachedto each point in space. On the other hand, we can think of it as amapping from real space into order parameter space. That is, M is afunction which takes different points in the magnet onto the surface ofa sphere (Fig. 9.4). As we mentioned earlier, mathematicians call the


9.3 Examine the elementary excitations 251

Long after phonons were understood, Jeffrey Goldstone started tothink about broken symmetries and order parameters in the abstract.He found a rather general argument that, whenever a continuous sym-metry (rotations, translations, SU(3), . . . ) is broken, long-wavelengthmodulations in the symmetry direction should have low frequencies (seeExercise 9.14). The fact that the lowest energy state has a broken sym-metry means that the system is stiff; modulating the order parameterwill cost an energy rather like that in eqn 9.2. In crystals, the bro-ken translational order introduces a rigidity to shear deformations, andlow-frequency phonons (Fig. 9.8). In magnets, the broken rotationalsymmetry leads to a magnetic stiffness and spin waves (Fig. 9.9). Innematic liquid crystals, the broken rotational symmetry introduces anorientational elastic stiffness (they pour, but resist bending!) and rota-tional waves (Fig. 9.10).

Fig. 9.9 Magnets: spin waves.Magnets break the rotational invari-ance of space. Because they resisttwisting the magnetization locally, butdo not resist a uniform twist, they havelow-energy spin wave excitations.

Fig. 9.10 Nematic liquid crystals:rotational waves. Nematic liquidcrystals also have low-frequency rota-tional waves.

In superfluids, an exotic broken gauge symmetry10 leads to a stiffness

10See Exercise 9.8.

which results in the superfluidity. Superfluidity and superconductivityreally are not any more amazing than the rigidity of solids. Is it notamazing that chairs are rigid? Push on a few atoms on one side and,109 atoms away, atoms will move in lock-step. In the same way, de-creasing the flow in a superfluid must involve a cooperative change ina macroscopic number of atoms, and thus never happens spontaneouslyany more than two parts of the chair ever drift apart.The low-frequency Goldstone modes in superfluids are heat waves!

(Do not be jealous; liquid helium has rather cold heat waves.) This isoften called second sound, but is really a periodic temperature modula-tion, passing through the material like sound does through a crystal.Just to round things out, what about superconductors? They have

also got a broken gauge symmetry, and have a stiffness that leads tosuperconducting currents. What is the low-energy excitation? It doesnot have one. But what about Goldstone’s theorem?Goldstone of course had conditions on his theorem which excluded

superconductors. (Actually, Goldstone was studying superconductorswhen he came up with his theorem.) It is just that everybody forgotthe extra conditions, and just remembered that you always got a low-frequency mode when you broke a continuous symmetry. We condensed-matter physicists already knew why there is no Goldstone mode for su-perconductors; P. W. Anderson had shown that it was related to thelong-range Coulomb interaction, and its absence is related to the Meiss-ner effect. We call the loophole in Goldstone’s theorem the Anderson–Higgs mechanism.11

11In condensed-matter language, theGoldstone mode produces a charge-density wave, whose electric fields areindependent of wavelength. This givesit a finite frequency (the plasma fre-quency) even at long wavelength. Inhigh-energy language the photon eatsthe Goldstone boson, and gains a mass.The Meissner effect is related to the gapin the order parameter fluctuations (!times the plasma frequency), which thehigh-energy physicists call the mass ofthe Higgs boson.

We end this section by bringing up another exception to Goldstone’stheorem; one we have known about even longer, but which we do nothave a nice explanation for. What about the orientational order incrystals? Crystals break both the continuous translational order and thecontinuous orientational order. The phonons are the Goldstone modes

Excitations:



for the translations, but there are no orientational Goldstone modesin crystals.12 Rotational waves analogous to those in liquid crystals(Fig. 9.10) are basically not allowed in crystals; at long distances theytear up the lattice. We understand this microscopically in a clunky way,but do not have an elegant, macroscopic explanation for this basic factabout solids.

9.4 Classify the topological defects

When the author was in graduate school, the big fashion was topologicaldefects. Everybody was studying homotopy groups, and finding exoticsystems to write papers about. It was, in the end, a reasonable thing todo.13 It is true that in a typical application you will be able to figure out13The next fashion, catastrophe theory,

never became particularly important. what the defects are without homotopy theory. You will spend foreverdrawing pictures to convince anyone else, though. Most importantly,homotopy theory helps you to think about defects.

Fig. 9.11 Dislocation in a crystal.Here is a topological defect in a crys-tal. We can see that one of the rows ofatoms on the right disappears half-waythrough our sample. The place where itdisappears is a defect, because it doesnot locally look like a piece of the per-fect crystal. It is a topological defectbecause it cannot be fixed by any localrearrangement. No reshuffling of atomsin the middle of the sample can changethe fact that five rows enter from theright, and only four leave from the left!The Burger’s vector of a disloca-tion is the net number of extra rowsand columns, combined into a vector(columns, rows).

A defect is a tear in the order parameter field. A topological defectis a tear that cannot be patched. Consider the piece of two-dimensionalcrystal shown in Fig. 9.11. Starting in the middle of the region shown,there is an extra row of atoms. (This is called a dislocation.) Away fromthe middle, the crystal locally looks fine; it is a little distorted, but thereis no problem seeing the square grid and defining an order parameter.Can we rearrange the atoms in a small region around the start of theextra row, and patch the defect?No. The problem is that we can tell there is an extra row without

ever coming near to the center. The traditional way of doing this is totraverse a large loop surrounding the defect, and count the net numberof rows crossed by the loop. For the loop shown in Fig. 9.12, there aretwo rows going up and three going down; no matter how far we stayfrom the center, there will always be an extra row on the right.How can we generalize this basic idea to other systems with broken

symmetries? Remember that the order parameter space for the two-dimensional square crystal is a torus (see Fig. 9.7), and that the orderparameter at a point is that translation which aligns a perfect square gridto the deformed grid at that point. Now, what is the order parameter farto the left of the defect (a), compared to the value far to the right (d)?The lattice to the right is shifted vertically by half a lattice constant;the order parameter has been shifted half-way around the torus. As

12In two dimensions, crystals provide another loophole in a well-known result, known as the Mermin–Wagner theorem. Hohen-berg, Mermin, and Wagner, in a series of papers, proved in the 1960s that two-dimensional systems with a continuous symmetrycannot have a broken symmetry at finite temperature. At least, that is the English phrase everyone quotes when they discussthe theorem; they actually prove it for several particular systems, including superfluids, superconductors, magnets, and trans-lational order in crystals. Indeed, crystals in two dimensions do not break the translational symmetry; at finite temperatures,the atoms wiggle enough so that the atoms do not sit in lock-step over infinite distances (their translational correlations decayslowly with distance). But the crystals do have a broken orientational symmetry: the crystal axes point in the same directionsthroughout space. (Mermin discusses this point in his paper on crystals.) The residual translational correlations (the localalignment into rows and columns of atoms) introduce long-range forces which force the crystalline axes to align, breaking thecontinuous rotational symmetry.


9.4 Classify the topological defects 253

dcb

d

agfe

a

cd

gf

e

ga b

b

c

f ed

abdc

defg

Fig. 9.12 Loop around the dislo-cation mapped onto order parameterspace. Consider a closed loop aroundthe defect. The order parameter fieldu changes as we move around the loop.The positions of the atoms around theloop with respect to their local ‘ideal’lattice drift upward continuously as wetraverse the loop. This precisely cor-responds to a path around the orderparameter space; the path passes oncearound the hole of the torus. A paththrough the hole corresponds to an ex-tra column of atoms.Moving the atoms slightly will deformthe path, but will not change the num-ber of times the path winds throughor around the hole. Two paths whichtraverse the torus the same number oftimes through and around are equiva-lent.

shown in Fig. 9.12, as you progress along the top half of a clockwise loopthe order parameter (position of the atom within the unit cell) movesupward, and along the bottom half again moves upward. All in all,the order parameter circles once around the torus. The winding numberaround the torus is the net number of times the torus is circumnavigatedwhen the defect is orbited once.

Fig. 9.13 Hedgehog defect. Mag-nets have no line defects (you cannotlasso a basketball), but do have pointdefects. Here is shown the hedgehogdefect, M(x) = M0 x. You cannot sur-round a point defect in three dimen-sions with a loop, but you can encloseit in a sphere. The order parameterspace, remember, is also a sphere. Theorder parameter field takes the enclos-ing sphere and maps it onto the or-der parameter space, wrapping it ex-actly once. The point defects in mag-nets are categorized by this wrappingnumber; the second homotopy group ofthe sphere is Z, the integers.

Why do we call dislocations topological defects? Topology is the studyof curves and surfaces where bending and twisting is ignored. An orderparameter field, no matter how contorted, which does not wind aroundthe torus can always be smoothly bent and twisted back into a uni-form state. If along any loop, though, the order parameter winds eitheraround the hole or through it a net number of times, then enclosed inthat loop is a defect which cannot be bent or twisted flat; the windingnumber (an integer) cannot change in a smooth and continuous fashion.

How do we categorize the defects for two-dimensional square crystals?Well, there are two integers: the number of times we go around the cen-tral hole, and the number of times we pass through it. In the traditionaldescription, this corresponds precisely to the number of extra rows andcolumns of atoms we pass by. This was named the Burger’s vector inthe old days, and nobody needed to learn about tori to understand it.We now call it the first homotopy group of the torus:

Π1(T2) = Z× Z, (9.6)

where Z represents the integers. That is, a defect is labeled by two



for the translations, but there are no orientational Goldstone modesin crystals.12 Rotational waves analogous to those in liquid crystals(Fig. 9.10) are basically not allowed in crystals; at long distances theytear up the lattice. We understand this microscopically in a clunky way,but do not have an elegant, macroscopic explanation for this basic factabout solids.

9.4 Classify the topological defects

When the author was in graduate school, the big fashion was topologicaldefects. Everybody was studying homotopy groups, and finding exoticsystems to write papers about. It was, in the end, a reasonable thing todo.13 It is true that in a typical application you will be able to figure out13The next fashion, catastrophe theory,

never became particularly important. what the defects are without homotopy theory. You will spend foreverdrawing pictures to convince anyone else, though. Most importantly,homotopy theory helps you to think about defects.

Fig. 9.11 Dislocation in a crystal.Here is a topological defect in a crys-tal. We can see that one of the rows ofatoms on the right disappears half-waythrough our sample. The place where itdisappears is a defect, because it doesnot locally look like a piece of the per-fect crystal. It is a topological defectbecause it cannot be fixed by any localrearrangement. No reshuffling of atomsin the middle of the sample can changethe fact that five rows enter from theright, and only four leave from the left!The Burger’s vector of a disloca-tion is the net number of extra rowsand columns, combined into a vector(columns, rows).

A defect is a tear in the order parameter field. A topological defectis a tear that cannot be patched. Consider the piece of two-dimensionalcrystal shown in Fig. 9.11. Starting in the middle of the region shown,there is an extra row of atoms. (This is called a dislocation.) Away fromthe middle, the crystal locally looks fine; it is a little distorted, but thereis no problem seeing the square grid and defining an order parameter.Can we rearrange the atoms in a small region around the start of theextra row, and patch the defect?No. The problem is that we can tell there is an extra row without

ever coming near to the center. The traditional way of doing this is totraverse a large loop surrounding the defect, and count the net numberof rows crossed by the loop. For the loop shown in Fig. 9.12, there aretwo rows going up and three going down; no matter how far we stayfrom the center, there will always be an extra row on the right.How can we generalize this basic idea to other systems with broken

symmetries? Remember that the order parameter space for the two-dimensional square crystal is a torus (see Fig. 9.7), and that the orderparameter at a point is that translation which aligns a perfect square gridto the deformed grid at that point. Now, what is the order parameter farto the left of the defect (a), compared to the value far to the right (d)?The lattice to the right is shifted vertically by half a lattice constant;the order parameter has been shifted half-way around the torus. As

12In two dimensions, crystals provide another loophole in a well-known result, known as the Mermin–Wagner theorem. Hohen-berg, Mermin, and Wagner, in a series of papers, proved in the 1960s that two-dimensional systems with a continuous symmetrycannot have a broken symmetry at finite temperature. At least, that is the English phrase everyone quotes when they discussthe theorem; they actually prove it for several particular systems, including superfluids, superconductors, magnets, and trans-lational order in crystals. Indeed, crystals in two dimensions do not break the translational symmetry; at finite temperatures,the atoms wiggle enough so that the atoms do not sit in lock-step over infinite distances (their translational correlations decayslowly with distance). But the crystals do have a broken orientational symmetry: the crystal axes point in the same directionsthroughout space. (Mermin discusses this point in his paper on crystals.) The residual translational correlations (the localalignment into rows and columns of atoms) introduce long-range forces which force the crystalline axes to align, breaking thecontinuous rotational symmetry.


212 Calculation and computation

nary alloys, and the liquid–gas transition. The Ising model is the mostextensively studied lattice model in physics. Like the ideal gas in theprevious chapters, the Ising model will provide a tangible applicationfor many topics to come: Monte Carlo (this section), low- and high-temperature expansions (Section 8.3, Exercise 8.1), relations betweenfluctuations, susceptibility, and dissipation (Exercises 8.2 and 10.6), nu-cleation of abrupt transitions (Exercise 11.4), coarsening and phase sep-aration (Section 11.4.1, Exercise 11.6), and self-similarity at continuousphase transitions (Exercise 12.1).

Fig. 8.1 The 2D square-latticeIsing model. It is traditional to de-note the values si = ±1 as up anddown, or as two different colors.

The Ising model has a lattice of N sites i with a single, two-statedegree of freedom si on each site that may take values ±1. We will beprimarily interested in the Ising model on square and cubic lattices (in2D and 3D, Fig. 8.1). The Hamiltonian for the Ising model is

H = −∑

〈ij〉

Jsisj −H∑

i

si. (8.1)

Here the sum 〈ij〉 is over all pairs of nearest-neighbor sites,5 and J is5In simulations of finite systems, wewill avoid special cases at the edgesof the system by implementing peri-odic boundary conditions, where cor-responding sites on opposite edges arealso neighbors.

the coupling between these neighboring sites. (For example, there arefour neighbors per site on the square lattice.)

0 2 4 6 8kBT/J

0

1

m(T)

Ferromagnetic Paramagnetic

Fig. 8.2 Ising magnetization. Themagnetization m(T ) per spin for the3D cubic lattice Ising model. At lowtemperatures there is a net magneti-zation, which vanishes at temperaturesT > Tc ≈ 4.5.

8.1.1 Magnetism

The Ising model was originally used to describe magnets. Hence thedegree of freedom si on each site is normally called a spin, H is calledthe external field, and the sum M =

∑i si is termed the magnetization.

The energy of two neighboring spins −Jsisj is −J if the spins areparallel, and +J if they are antiparallel. Thus if J > 0 (the usual case)the model favors parallel spins; we say that the interaction is ferromag-netic.6 At low temperatures, the spins will organize themselves to either

6‘Ferromagnetic’ is named after iron(Fe), the most common material whichhas a spontaneous magnetization.

mostly point up or mostly point down, forming a ferromagnetic phase.If J < 0 we call the interaction antiferromagnetic; the spins will tend toalign (for our square lattice) in a checkerboard antiferromagnetic phaseat low temperatures. At high temperatures, independent of the sign ofJ , we expect entropy to dominate; the spins will fluctuate wildly in aparamagnetic phase and the magnetization per spin m(T ) = M(T )/N iszero (see Fig. 8.2).7

7The Ising model parameters are rescaled from the microscopic ones. The Ising spin si = ±1 represents twice the z-componentof a spin-1/2 atom in a crystal, σzi = si/2. The Ising interactions between spins, Jsisj = 4Jσzi σ

zj , is thus shifted by a factor of

four from the z–z coupling between spins. The coupling of the spin to the external magnetic field is microscopically gµBH ·σzi ,where g is the gyromagnetic ratio for the spin (close to two for the electron) and µB = e!/2me is the Bohr magneton. Hencethe Ising external field is rescaled from the physical one by gµB/2. Finally, the interaction between spins in most materialsis not so anisotropic as to only involve the z-component of the spin; it is usually better approximated by the dot productσi · σj = σxi σ

xj + σyi σ

yj + σzi σ

zj , used in the more realistic Heisenberg model. (Unlike the Ising model, where σzi commutes with

H and the spin configurations are the energy eigenstates, the quantum and classical Heisenberg models differ.) Some materialshave anisotropic crystal structures which make the Ising model at least approximately valid.






H = −∑

〈ij〉

Jsisj −H∑

i

si. (8.1)



0 2 4 6 8kBT/J

0

1

m(T)



8.1.1 Magnetism









xj + σyi σ

yj + σzi σ




8.1 The Ising model 213

8.1.2 Binary alloys

B

A A

A

B

B B B

B A B

A BB

B

AFig. 8.3 The Ising model as a bi-nary alloy. Atoms in crystals natu-rally sit on a lattice. The atoms in al-loys are made up of different elements(here, types A and B) which can ar-range in many configurations on thelattice.

The Ising model is quite a convincing model for binary alloys. Imaginea square lattice of atoms, which can be either of type A or B (Fig. 8.3).(A realistic alloy might mix roughly half copper and half zinc to makeβ-brass. At low temperatures, the copper and zinc atoms each sit ona cubic lattice, with the zinc sites in the middle of the copper cubes,together forming an ‘antiferromagnetic’ phase on the body-centered cu-bic (bcc) lattice. At high temperatures, the zincs and coppers freelyinterchange, analogous to the Ising paramagnetic phase.) The transi-tion temperature is about 733 ◦C [195, section 3.11]. We set the spinvalues A = +1 and B = −1. Let the number of the two kinds of atomsbe NA and NB, with NA +NB = N , let the interaction energies (bondstrengths) between two neighboring atoms be EAA, EBB, and EAB ,and let the total number of nearest-neighbor bonds of the three possibletypes be NAA, NBB and NAB. Then the Hamiltonian for our binaryalloy is

Hbinary = −EAANAA − EBBNBB − EABNAB. (8.2)

Since each site interacts only with its nearest neighbors, this must be theIsing model in disguise. Indeed, one finds8 J = 1/4(EAA +EBB − 2EAB)and H = EAA − EBB .To make this a quantitative model, one must include atomic relax-

ation effects. (Surely if one kind of atom is larger than the other, itwill push neighboring atoms off their sites. We simply include this re-laxation into the energies in our Hamiltonian 8.2.) We must also incor-porate thermal position fluctuations into the Hamiltonian, making it afree energy.9 More elaborate Ising models (with three-site and longer-range interactions, for example) are commonly used to compute realisticphase diagrams for alloys [197]. Sometimes, though, the interactions in-troduced by relaxations and thermal fluctuations have important long-range pieces, which can lead to qualitative changes in the behavior—forexample, they can change the transition from continuous to abrupt.

8Check this yourself. Adding an overall shift −CN to the Ising Hamiltonian, one can see that

HIsing = −J∑

〈ij〉

sisj −H∑

i

si − CN = −J (NAA +NBB −NAB)−H (NA −NB) − CN, (8.3)

since NA−NB corresponds to the net magnetization, NAA+NBB is the number of parallel neighbors, and NAB is the numberof antiparallel neighbors. Now, use the facts that on a square lattice there are twice as many bonds as spins (NAA + NBB +NAB = 2N), and that for every A atom there must be four bonds ending in an A (4NA = 2NAA + NAB , and similarly4NB = 2NBB +NAB). Solve for and remove N , NA, and NB from the Hamiltonian, and rearrange into the binary alloy form(eqn 8.2); you should find the values for J and H above and C = 1/2(EAA +EBB + 2EAB).9To incorporate thermal fluctuations, we must do a partial trace, integrating out the vibrations of the atoms around theirequilibrium positions (as in Section 6.6). This leads to an effective free energy for each pattern of lattice occupancy {si}:

F{si} = −kBT log

(∫dP∫

atom ri of type si near site i

dQe−H(P,Q)/kBT

h3N

)

= H{si}− TS{si}. (8.4)

The entropy S{si} due to these vibrations will depend upon the particular atomic configuration si, and can often be calculatedexplicitly (Exercise 6.11(b)). F{si} can now be used as a lattice Hamiltonian, except with temperature-dependent coefficients;those atomic configurations with more freedom to vibrate will have larger entropy and will be increasingly favored at highertemperature.



8.1.3 Liquids, gases, and the critical point

P

Liquidpoint

Triplepoint Gas

Solid

Critical

Temperature T

Pres

sure

Fig. 8.4 P–T phase diagram fora typical material. The solid–liquidphase boundary corresponds to achange in symmetry, and cannot end.The liquid–gas phase boundary typi-cally does end; one can go continuouslyfrom the liquid phase to the gas phaseby increasing the pressure above Pc, in-creasing the temperature above Tc, andthen lowering the pressure again.

The Ising model is also used as a model for the liquid–gas transition. Inthis lattice gas interpretation, up-spins (si = +1) count as atoms anddown-spins count as a site without an atom. The gas is the phase withmostly down-spins (negative ‘magnetization’), with only a few up-spinatoms in the vapor. The liquid phase is mostly atoms (up-spins), witha few vacancies.The Ising model description of the gas phase seems fairly realistic.

The liquid, however, seems much more like a crystal, with atoms sittingon a regular lattice. Why do we suggest that this model is a good wayof studying transitions between the liquid and gas phase?Unlike the binary alloy problem, the Ising model is not a good way

to get quantitative phase diagrams for fluids. What it is good for is tounderstand the properties near the critical point. As shown in Fig. 8.4,one can go continuously between the liquid and gas phases; the phaseboundary separating them ends at a critical point Tc, Pc, above whichthe two phases blur together seamlessly, with no jump in the densityseparating them.

cDown

Up

T

H

CriticalpointTemperature T

Exte

rnal

fiel

d

Fig. 8.5 H–T phase diagram forthe Ising model. Below the criticaltemperature Tc, there is an an up-spinand a down-spin ‘phase’ separated bya jump in magnetization at H = 0.Above Tc the behavior is smooth as afunction of H.

The Ising model, interpreted as a lattice gas, also has a line H = 0along which the density (magnetization) jumps, and a temperature Tc

above which the properties are smooth as a function of H (the param-agnetic phase). The phase diagram in Fig. 8.5 looks only topologicallylike the real liquid–gas coexistence line in Fig. 8.4, but the behavior nearthe critical point in the two systems is remarkably similar. Indeed, wewill find in Chapter 12 that in many ways the behavior at the liquid–gascritical point is described exactly by the three-dimensional Ising model.

8.1.4 How to solve the Ising model

How do we solve for the properties of the Ising model?

(1) Solve the one-dimensional Ising model, as Ising did.10

10This is a typical homework exercisein a textbook like ours; with a few hints,you can do it too.

(2) Have an enormous brain. Onsager solved the two-dimensional Isingmodel in a bewilderingly complicated way. Since Onsager, manygreat minds have found simpler, elegant solutions, but all wouldtake at least a chapter of rather technical and unilluminating ma-nipulations to duplicate. Nobody has solved the three-dimensionalIsing model.

(3) Perform the Monte Carlo method on the computer.1111Or do high-temperature expansions,low-temperature expansions, transfer-matrix methods, exact diagonalizationof small systems, 1/N expansions in thenumber of states per site, 4− ε expan-sions in the dimension of space, . . .

The Monte Carlo12 method involves doing a kind of random walk

12Monte Carlo is a gambling center inMonaco. Lots of random numbers aregenerated there.

through the space of lattice configurations. We will study these methodsin great generality in Section 8.2. For now, let us just outline the heat-bath Monte Carlo method.

Heat-bath Monte Carlo for the Ising model

• Pick a site i = (x, y) at random.



8.1.3 Liquids, gases, and the critical point

P

Liquidpoint

Triplepoint Gas

Solid

Critical

Temperature T

Pres

sure

Fig. 8.4 P–T phase diagram fora typical material. The solid–liquidphase boundary corresponds to achange in symmetry, and cannot end.The liquid–gas phase boundary typi-cally does end; one can go continuouslyfrom the liquid phase to the gas phaseby increasing the pressure above Pc, in-creasing the temperature above Tc, andthen lowering the pressure again.

The Ising model is also used as a model for the liquid–gas transition. Inthis lattice gas interpretation, up-spins (si = +1) count as atoms anddown-spins count as a site without an atom. The gas is the phase withmostly down-spins (negative ‘magnetization’), with only a few up-spinatoms in the vapor. The liquid phase is mostly atoms (up-spins), witha few vacancies.The Ising model description of the gas phase seems fairly realistic.

The liquid, however, seems much more like a crystal, with atoms sittingon a regular lattice. Why do we suggest that this model is a good wayof studying transitions between the liquid and gas phase?Unlike the binary alloy problem, the Ising model is not a good way

to get quantitative phase diagrams for fluids. What it is good for is tounderstand the properties near the critical point. As shown in Fig. 8.4,one can go continuously between the liquid and gas phases; the phaseboundary separating them ends at a critical point Tc, Pc, above whichthe two phases blur together seamlessly, with no jump in the densityseparating them.

cDown

Up

T

H

CriticalpointTemperature T

Exte

rnal

fiel

d

Fig. 8.5 H–T phase diagram forthe Ising model. Below the criticaltemperature Tc, there is an an up-spinand a down-spin ‘phase’ separated bya jump in magnetization at H = 0.Above Tc the behavior is smooth as afunction of H.

The Ising model, interpreted as a lattice gas, also has a line H = 0along which the density (magnetization) jumps, and a temperature Tc

above which the properties are smooth as a function of H (the param-agnetic phase). The phase diagram in Fig. 8.5 looks only topologicallylike the real liquid–gas coexistence line in Fig. 8.4, but the behavior nearthe critical point in the two systems is remarkably similar. Indeed, wewill find in Chapter 12 that in many ways the behavior at the liquid–gascritical point is described exactly by the three-dimensional Ising model.

8.1.4 How to solve the Ising model

How do we solve for the properties of the Ising model?

(1) Solve the one-dimensional Ising model, as Ising did.10

10This is a typical homework exercisein a textbook like ours; with a few hints,you can do it too.

(2) Have an enormous brain. Onsager solved the two-dimensional Isingmodel in a bewilderingly complicated way. Since Onsager, manygreat minds have found simpler, elegant solutions, but all wouldtake at least a chapter of rather technical and unilluminating ma-nipulations to duplicate. Nobody has solved the three-dimensionalIsing model.

(3) Perform the Monte Carlo method on the computer.1111Or do high-temperature expansions,low-temperature expansions, transfer-matrix methods, exact diagonalizationof small systems, 1/N expansions in thenumber of states per site, 4− ε expan-sions in the dimension of space, . . .

The Monte Carlo12 method involves doing a kind of random walk

12Monte Carlo is a gambling center inMonaco. Lots of random numbers aregenerated there.

through the space of lattice configurations. We will study these methodsin great generality in Section 8.2. For now, let us just outline the heat-bath Monte Carlo method.

Heat-bath Monte Carlo for the Ising model

• Pick a site i = (x, y) at random.



Fig. 8.7 Perturbation theory.(a) Low-temperature expansions forthe cubic Ising model magnetization(Fig. 8.2) with successively largernumbers of terms. (b) The high- andlow-temperature expansions for theIsing and other lattice models are sumsover (Feynman diagram) clusters. Atlow T , Ising configurations are smallclusters of up-spins in a backgroundof down-spins (or vice versa). Thiscluster of four sites on the cubic latticecontributes to the term of order x20 ineqn 8.19, because flipping the clusterbreaks 20 bonds.

0 1 2 3 4 5Temperature kBT/J

0

1

Mag

netiz

atio

n m

(T)

m(T)Series to x10

Series to x20

Series to x30

Series to x40

Series to x54

(b)(a)

extrapolate the behavior inside a liquid or magnetic phase under smallchanges in external conditions. Perturbation theory works inside phases.More precisely, inside a phase the properties are analytic (have conver-gent Taylor expansions) as functions of the external conditions.Much of statistical mechanics (and indeed of theoretical physics) is

devoted to calculating high-order perturbation theories around specialsolvable limits. (We will discuss linear perturbation theory in space andtime in Chapter 10.) Lattice theories at high and low temperaturesT have perturbative expansions in powers of 1/T and T , with Feyn-man diagrams involving all ways of drawing clusters of lattice points(Fig. 8.7(b)). Gases at high temperatures and low densities have virialexpansions. Metals at low temperatures have Fermi liquid theory, wherethe electron–electron interactions are perturbatively incorporated bydressing the electrons into quasiparticles. Properties of systems nearcontinuous phase transitions can be explored by perturbing in the di-mension of space, giving the ε-expansion. Some of these perturbationseries have zero radius of convergence; they are asymptotic series (seeExercise 1.5).For example the low-temperature expansion [48, 135] (Exercises 8.19

and 8.18) of the magnetization per spin of the cubic-lattice three-dimensionalIsing model (Section 8.1) starts out [22]

m =1− 2x6 − 12x10 + 14x12 − 90x14 + 192x16 − 792x18 + 2148x20

− 7716x22 + 23262x24 − 79512x26 + 252054x28

− 846628x30 + 2753520x32 − 9205800x34

+ 30371124x36 − 101585544x38 + 338095596x40

− 1133491188x42 + 3794908752x44 − 12758932158x46

+ 42903505303x48 − 144655483440x50

+ 488092130664x52 − 1650000819068x54+ . . . , (8.19)

where x = e−2J/kBT is the probability to break a bond (parallel energy−J to antiparallel energy +J).26 This series was generated by carefully

26This heroic calculation (27 terms)was not done to get really accurate low-temperature magnetizations. Variousclever methods can use these expan-sions to extrapolate to understand thesubtle phase transition at Tc (Chap-ter 12). Indeed, the m(T ) curve shownin both Figs 8.2 and 8.7(a) was notmeasured directly, but was generatedusing a 9, 10 Pade approximant [47]. considering the probabilities of low-energy spin configurations, formed



Fig. 8.7 Perturbation theory.(a) Low-temperature expansions forthe cubic Ising model magnetization(Fig. 8.2) with successively largernumbers of terms. (b) The high- andlow-temperature expansions for theIsing and other lattice models are sumsover (Feynman diagram) clusters. Atlow T , Ising configurations are smallclusters of up-spins in a backgroundof down-spins (or vice versa). Thiscluster of four sites on the cubic latticecontributes to the term of order x20 ineqn 8.19, because flipping the clusterbreaks 20 bonds.

0 1 2 3 4 5Temperature kBT/J

0

1

Mag

netiz

atio

n m

(T)

m(T)Series to x10

Series to x20

Series to x30

Series to x40

Series to x54

(b)(a)

extrapolate the behavior inside a liquid or magnetic phase under smallchanges in external conditions. Perturbation theory works inside phases.More precisely, inside a phase the properties are analytic (have conver-gent Taylor expansions) as functions of the external conditions.Much of statistical mechanics (and indeed of theoretical physics) is

devoted to calculating high-order perturbation theories around specialsolvable limits. (We will discuss linear perturbation theory in space andtime in Chapter 10.) Lattice theories at high and low temperaturesT have perturbative expansions in powers of 1/T and T , with Feyn-man diagrams involving all ways of drawing clusters of lattice points(Fig. 8.7(b)). Gases at high temperatures and low densities have virialexpansions. Metals at low temperatures have Fermi liquid theory, wherethe electron–electron interactions are perturbatively incorporated bydressing the electrons into quasiparticles. Properties of systems nearcontinuous phase transitions can be explored by perturbing in the di-mension of space, giving the ε-expansion. Some of these perturbationseries have zero radius of convergence; they are asymptotic series (seeExercise 1.5).For example the low-temperature expansion [48, 135] (Exercises 8.19

and 8.18) of the magnetization per spin of the cubic-lattice three-dimensionalIsing model (Section 8.1) starts out [22]

m =1− 2x6 − 12x10 + 14x12 − 90x14 + 192x16 − 792x18 + 2148x20

− 7716x22 + 23262x24 − 79512x26 + 252054x28

− 846628x30 + 2753520x32 − 9205800x34

+ 30371124x36 − 101585544x38 + 338095596x40

− 1133491188x42 + 3794908752x44 − 12758932158x46

+ 42903505303x48 − 144655483440x50

+ 488092130664x52 − 1650000819068x54+ . . . , (8.19)

where x = e−2J/kBT is the probability to break a bond (parallel energy−J to antiparallel energy +J).26 This series was generated by carefully

26This heroic calculation (27 terms)was not done to get really accurate low-temperature magnetizations. Variousclever methods can use these expan-sions to extrapolate to understand thesubtle phase transition at Tc (Chap-ter 12). Indeed, the m(T ) curve shownin both Figs 8.2 and 8.7(a) was notmeasured directly, but was generatedusing a 9, 10 Pade approximant [47]. considering the probabilities of low-energy spin configurations, formed

Phase transitions cannot be described using perturbation theory.

Exercises 276

This allows us to integrate terms in the free energy by parts; by subtracting a total divergence r(uv) fromthe free energy we can exchange a term urv for a term �vru. For example, we can subtract a term�r ·

�mb(m2, T )rm

�from the free energy 9.8:

F Ising{m,T}= A(m2, T ) +mb(m2, T )r2m

+ C(m2, T )(rm)2

�r�mb(m2, T ) ·rm

�

= A(m2, T ) + C(m2, T )(rm)2

�r�mb(m2, T )

�·rm

= A(m2, T ) + C(m2, T )(rm)2

��b(m2, T ) + 2m2b0(m2, T )

�(rm)2

= A(m2, T )

+�C(m2, T )� b(m2, T )

� 2m2b0(m2, T )�(rm)2, (9.9)

replacing�mb(m2, T )

�(r2m) with the equivalent term �(rm)(r(mb(m2, T )rm) · rm). Thus we may

absorb the b term proportional to r2m into an altered c = C(m2, T )� b(m2, T )� 2m2b0(m2, T ) term times(rm)2:

F Ising{m,T} = A(m2, T ) + c(m2, T )(rm)2. (9.10)

[5] (Perhaps) assume the order parameter is small.13

If we assume m is small, we may Taylor expand A and c in powers of m2, yielding A(m2, T ) = f0 +(µ(T )/2)m2 + (g/4!)m4 and c(m2, T ) = 1/2K, leading to the traditional Landau free energy for the Isingmodel:

F Ising = 1/2K(rm)2 + f0 + (µ(T )/2)m2 + (g/4!)m4, (9.11)

where f0, g, and K can also depend upon T . (The factors of 1/2 and 1/4! are traditional.)The free energy density of eqn 9.11 is one of the most extensively studied models in physics. The field theoristsuse � instead of m for the order parameter, and call it the �4 model. Ken Wilson added fluctuations to thismodel in developing the renormalization group (Chapter 12).

-2 -1 0 1 2

Magnetization m per spin

µ > 0µ = 0µ < 0

13Notice that this approximation is not valid for abrupt phase transitions, where the order parameter is large until the transitionand zero afterward. Landau theories are often used anyhow for abrupt transitions (see Fig. 11.2(a)), and are illuminating ifnot controlled.We shall see in Chapter 12 that Landau theory also fails when the order parameter is small near a continuous phase transition.For example, just below Tc in the Ising model, the average M is small but simulations show large, scale-invariant fluctuationsin the magnetization (Exercise 8.1). These lead to power law singularities F Ising{m,T} = M1+�(a + bM/(Tc � T )� + . . . )where in 3D � = 4.7898 . . . and � = 0.326 . . . .

Exercises 277

Fig. 9.4 Landau free energy density for the Ising model 9.11, at positive, zero, and negative values of the quadraticterm µ.

Notice that the Landau free energy density has a qualitative change at µ = 0. For positive µ it has a singleminimum at m = 0; for negative µ it has two minima at m = ±

p�6µ/g. Is this related to the transition in

the Ising model from the paramagnetic phase (m = 0) to the ferromagnetic phase at Tc?The free energy density already incorporates (by our assumptions) fluctuations in m on length scales smallcompared to the coarse-graining length W . If we ignored fluctuations on scales larger than W then thefree energy of the whole system14 would be given by the volume times the free energy density, and themagnetization at a temperature T would be given by minimizing the free energy density. The quadratic termµ(T ) would vanish at Tc, and if we expand µ(T ) ⇠ a(T � Tc) + . . . we find m = ±

p6a/g

pTc � T near the

critical temperature.

Fig. 9.5 Fluctuations on all scales. A snapshot of the Ising model at Tc. Notice that there are fluctuations on alllength scales.

This is qualitatively correct, but quantitatively wrong. The magnetization does vanish at Tc with a powerlaw m ⇠ (Tc � T )� , but the exponent � is not generally 1/2; in two dimensions it is �2D = 1/8 and inthree dimensions it is �3D ⇡ 0.32641 . . . . These exponents (particularly the presumably irrational one in 3D)cannot be fixed by keeping more or di↵erent terms in the analytic Landau expansion.(d) Show that the power-law �Landau = 1/2 is not changed in the limit T ! Tc even when one adds anotherterm (h/6!)m6 into eqn 9.11. (That is, show that m(T )/(T � Tc)

� goes to a constant as T ! Tc.) (Hint:You should get a quadratic equation for m2. Keep the root that vanishes at T = Tc, and expand in powersof h.) Explore also the alternative phase transition where g ⌘ 0 but h > 0; what is � for that transition?

Suppose we keep more terms in the expansion for A(m2, T ). Ignoring all fluctuations, rm ! 0 so

F ising = f0 +µ(T )2

m2 +g4!m4 +

h6!m6 + . . . .

14The total free energy is convex (Fig. 11.2(a)). The free energy density F in Fig. 9.4 can have a barrier if a boundary betweenthe phases is thicker than the coarse-graining length. The total free energy also has singularities at phase transitions. F canbe analytic because it is the free energy of a finite region; thermal phase transitions do not occur in finite systems.

Exercises 276

This allows us to integrate terms in the free energy by parts; by subtracting a total divergence r(uv) fromthe free energy we can exchange a term urv for a term �vru. For example, we can subtract a term�r ·

�mb(m2, T )rm

�from the free energy 9.8:

F Ising{m,T}= A(m2, T ) +mb(m2, T )r2m

+ C(m2, T )(rm)2

�r�mb(m2, T ) ·rm

�

= A(m2, T ) + C(m2, T )(rm)2

�r�mb(m2, T )

�·rm

= A(m2, T ) + C(m2, T )(rm)2

��b(m2, T ) + 2m2b0(m2, T )

�(rm)2

= A(m2, T )

+�C(m2, T )� b(m2, T )

� 2m2b0(m2, T )�(rm)2, (9.9)

replacing�mb(m2, T )

�(r2m) with the equivalent term �(rm)(r(mb(m2, T )rm) · rm). Thus we may

absorb the b term proportional to r2m into an altered c = C(m2, T )� b(m2, T )� 2m2b0(m2, T ) term times(rm)2:

F Ising{m,T} = A(m2, T ) + c(m2, T )(rm)2. (9.10)

[5] (Perhaps) assume the order parameter is small.13

If we assume m is small, we may Taylor expand A and c in powers of m2, yielding A(m2, T ) = f0 +(µ(T )/2)m2 + (g/4!)m4 and c(m2, T ) = 1/2K, leading to the traditional Landau free energy for the Isingmodel:

F Ising = 1/2K(rm)2 + f0 + (µ(T )/2)m2 + (g/4!)m4, (9.11)

where f0, g, and K can also depend upon T . (The factors of 1/2 and 1/4! are traditional.)The free energy density of eqn 9.11 is one of the most extensively studied models in physics. The field theoristsuse � instead of m for the order parameter, and call it the �4 model. Ken Wilson added fluctuations to thismodel in developing the renormalization group (Chapter 12).

-2 -1 0 1 2


µ > 0µ = 0µ < 0

13Notice that this approximation is not valid for abrupt phase transitions, where the order parameter is large until the transitionand zero afterward. Landau theories are often used anyhow for abrupt transitions (see Fig. 11.2(a)), and are illuminating ifnot controlled.We shall see in Chapter 12 that Landau theory also fails when the order parameter is small near a continuous phase transition.For example, just below Tc in the Ising model, the average M is small but simulations show large, scale-invariant fluctuationsin the magnetization (Exercise 8.1). These lead to power law singularities F Ising{m,T} = M1+�(a + bM/(Tc � T )� + . . . )where in 3D � = 4.7898 . . . and � = 0.326 . . . .

9.5 Landau theory for the Ising model


Exercises 263

-2 -1 0 1 2


µ > 0µ = 0µ < 0

Fig. 9.22 Landau free energy density for the Isingmodel 9.19, at positive, zero, and negative values ofthe quadratic term µ.

Notice that the Landau free energy density has aqualitative change at µ = 0. For positive µ it hasa single minimum at m = 0; for negative µ it hastwo minima at m = ±

√−6µ/g. Is this related to

the transition in the Ising model from the param-agnetic phase (m = 0) to the ferromagnetic phaseat Tc?The free energy density already incorporates (byour assumptions) fluctuations in m on lengthscales small compared to the coarse-graininglength W . If we ignored fluctuations on scaleslarger than W then the free energy of the wholesystem33 would be given by the volume times thefree energy density, and the magnetization at atemperature T would be given by minimizing thefree energy density. The quadratic term µ(T )would vanish at Tc, and if we expand µ(T ) ∼a(T − Tc) + . . . we find m = ±

√6a/g

√Tc − T

near the critical temperature.

Fig. 9.23 Fluctuations on all scales. A snapshotof the Ising model at Tc. Notice that there are fluctu-ations on all length scales.

This is qualitatively correct, but quantitativelywrong. The magnetization does vanish at Tc

with a power law m ∼ (Tc − T )β, but the ex-ponent β is not generally 1/2; in two dimensionsit is β2D = 1/8 and in three dimensions it isβ3D ≈ 0.32641 . . . . These exponents (particularlythe presumably irrational one in 3D) cannot befixed by keeping more or different terms in theanalytic Landau expansion.(d) Show that the power-law βLandau = 1/2 is notchanged in the limit T → Tc even when one addsanother term (h/6!)m6 into eqn 9.19. (That is,show that m(T )/(T − Tc)

β goes to a constant asT → Tc.) (Hint: You should get a quadraticequation for m2. Keep the root that vanishes atT = Tc, and expand in powers of h.) Explore alsothe alternative phase transition where g ≡ 0 buth > 0; what is β for that transition?As we see in Fig. 9.23 there is no length W abovewhich the Ising model near Tc looks smooth anduniform. The Landau free energy density gets cor-rections on all length scales; for the infinite systemthe free energy has a singularity at Tc (makingour power-series expansion for F Ising inadequate).The Landau free energy density is only a starting-point for studying continuous phase transitions;34

33The total free energy is convex (Fig. 11.2(a)). The free energy density F in Fig. 9.22 can have a barrier if a boundary betweenthe phases is thicker than the coarse-graining length. The total free energy also has singularities at phase transitions. F canbe analytic because it is the free energy of a finite region; thermal phase transitions do not occur in finite systems.34An important exception to this is superconductivity, where the Cooper pairs are large compared to their separation. Becausethey overlap so many neighbors, the fluctuations in the order parameter field are suppressed, and Landau theory is valid evenvery close to the phase transition.






H = −∑

〈ij〉

Jsisj −H∑

i

si. (8.1)



0 2 4 6 8kBT/J

0

1

m(T)



8.1.1 Magnetism









xj + σyi σ

yj + σzi σ



lecture 28 phys 416 thursday december 2 fall 2021 1.7.7

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