c

2009-05-25

Contents

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{

eview: Mean field approximation.

½„ O a arg mol l and Orronding N ar O N ighborO

½„ ola an av rag for all h OinO n ighboring or arg Oin

½„ Ôix all OinO o h av rag val

½„ find h n rgy of h arg Oin

½„ alla h ariion fnion for h arg Oin

½„ alla av rag n rgy for arg Oin

½„ alla av rag val of Oin

a

ations as developed in Lectre 8

ߪǡ ܰܰ ൌ ʹ݀

arg mol l 2

ڄdim nOionO N ar O N ighborO

8.6

ݏ ൌ

ଵ

ேே

σ

ߪ

௜

ߪ

௝

ேே

m an val Oin of (2d) N ar O N ighborO NN

8.7

݁ ൌ σ ߪ

௜

ߪ

௝

՜ ߪʹ݀ݏ

av rag n rgy of arg mol l

8.8

ܲ෠

ఙ

ൌ ݁

ିఉఙଶௗ௦

Bolman Probabiliy

8.9

ݖ ൌ ʹ …‘•Š ʹߚݏ݀

Pariion Ônion

8.11

ۦ݁ۧ ൌ ʹ݀ݏ –ƒŠሺʹߚ݀ݏሻ

Ex d n rgy a arg mol l

8.12

ۦݏۧ ൌ

ۦ௘ۧ

௘

ൌ –ƒŠሺʹߚ݀ݏሻ

Ex d Oin

No O on allaing h Pariion Ônion. Th Pariion Ônion iO h Om of Probabilii O
whih in or aO ı ± 1.

ݖ ൌ σ

ܲ෠

ఙ

ାଵ

ିଵ

ൌ ݁

ାଶఉ௦ௗ

൅ ݁

ିଶఉ௦ௗ

ൌ ʹ …‘•Š ʹߚݏ݀

No O on allaing h Ex d En rgy

ۦ݁ۧ ൌ െ

డ ୪୭୥ ௓ሺఉሻ

డఉ

ൌ െ

డ

డఉ

ሺŽ‘‰ ʹ ൅ Ž‘‰ …‘•Š ʹߚݏ݀ሻ

ൌ

ୱ୧୬୦ ଶఉௗ௦

ୡ୭ୱ୦ ଶఉௗ௦

ʹ݀ݏ ൌ ʹ݀ݏ –ƒŠሺʹߚ݀ݏሻ

Consistency {e irement

c

olionO o h x d Oin aion giv O O h r ir m nO wh r h M an Ôi ld

aroximaion iO onOiO n

8.12

ۦݏۧ ൌ

ۦ௘ۧ

௘

ൌ –ƒŠሺʹߚ݀ݏሻ

Ex d Oin

Criial T m rar for TranOiion Poin orO a O0 wh n Olo alO 45°

9.1

డۦ௦ۧ

డ௦

ൌ

డ

డ௦

ሺ–ƒŠሺʹߚ݀ݏሻሻ ൌ ʹߚ݀ •‡…Š

ଶ

ʹߚ݀ݏ Olo

డۦ௦ۧ

డ௦

ሃ

௦ୀ଴

ൌ ͳ ൌ ʹߚ݀ ൌ

ଶௗ

்

a O0 Olo 1 (

ۦۧ;<

9.2

Ó

ൌ ʹ݀ ൈ ܷ

riial m rar

wh r

U

C

iO an n rgy onv rOion faor (Oin iO ± 1 Ohold hav ± niO of n rgy)

cc
Th only Oabl OolionO ar h O

ۦ•ۧ ± 1 oinO for (2ȕd • 1) . Th Oolion a Oۦ•ۧൌͲ‹•ƒ

—•–ƒ„Ž‡•‘Ž—–‹‘ƒ•ƒ”‡–Š‡‡–ƒǦ•–ƒ„Ž‡•’‘–ƒ‡‘—••›‡–”›„”‡ƒ‹‰•‘Ž—–‹‘•
ሺʹȾ†ൌͳሻǤ

Chemical Potential: Total anergy is fixed (26:00

Ô

irO d riv h OaiOial aionO for mol l O ha ar in O ara iOola d ³ n rgy Oa

box O

¶

ଵ

ܧ

ଵ

¶

ଶ

ܧ

ଶ

¶

௜

ܧ

௜

wh r

ni

nmb r of box O ha oy h (i

h

) n rgy Oa

Ei

n rgy of h (i

h

) n rgy Oa

Maximi h nroy Obj o a onOrain h oal n rgy iO fix d
9.3

σ ¶

௜

ܧ

௜

௜

ൌ ܧ

்

oal n rgy E

T

iO fix d

9.4

σ ¶

௜

௜

ൌ ܰ

N iO oal nmb r of box O

9.5

௡

೔

ே

ൌ ܲሺ݅ሻ

robabily of h (i

h

) Oa

ۦ•ۧൌ•

2ȕd < 1 Omall ȕ highT

Plo Ohowing how diff r n m rar O (T1/ȕ) aff h ranOiion oin Oin ±0
Th in rO ion of h anh rv wih h

ۦOۧ O lin OhowO wh r h onOiO ny

r ir m nO ar m for h m an val of Oin.
Criial m rar iO wh n Olo of anh Olo

ۦOۧ O lin 45°

2ȕd 1

2ȕd > 1 larg ȕ lowT

ĸ N gaiv Oin.

PoOiiv Oin. ĺ

9.6

݁ ൌ σ

௡

೔

ா

೔

ே

௜

ൌ σ ܲሺ݅ሻܧ

௜

௜

ൌ

ா

೅

ே

av rag n rgy in a giv n box

9.7

ܵ ൌ െ σ ܲሺ݅ሻ Ž‘‰ ܲሺ݅ሻ

௜

c

nroy

½„ Probabiliy iO a fnion of (n

i

) n rgy iO a fnion of (n

i

)(Ei)

½„ Dnrod a Lagrang mlili r (L r 2)

½„ Maximi nroy Obj o ET b ing onOan

ܵ ൅ ߚ σ ¶

௜

ܧ

௜

௜

d fin a Lagrang mlili r

ଵ

௭

݁

ିఉா

d fin normali d n rgy fnion

9.8

ܵ ൅ ߚܧ ൌ െ σ ܲሺ݅ሻ Ž‘‰ ܲሺ݅ሻ

௜

൅ ߚ σ ܲሺ݅ሻܧ

௜

௜

9.9

ܲሺ݅ሻ ൌ

ଵ

௭

݁

ିఉா

೔

maximi (c ȕE) for a giv n i

no ha () (ariion fnion) inOr O h oal robabiliy alO on .

Chemical Potential: Total anergy and Total Nmber of Particles is fixed

c

ond d riv h OaiOial aionO for mol l O ha ar no in iOola d ³ n rgy Oa

box O

¶

ଵ

ܧ

ଵ

՞ ¶

ଶ

ܧ

ଶ

՞ ¶

௜

ܧ

௜

b Oid O n rgy aril O an alOo b xhang d b w n

box O

Now w do no know for rain h #of aril O in a box (r vioOly box O w r iOola d #of

aril O w r fix d)

9.10

σ ¶

௜

௜

ൌ ܰ

Toal #of aril O fix d #of aril O r box vari O

( aionO no d rivaion)

Probabiliy iO now a fnion of (i

h

) Oa and #of aril O

9.11

ܲሺ݅ǡ ܰሻ ൌ

ଵ

௭

݁

ିఉா

೔

ିఉఓே

Probabiliy iO now h rob. of b ing in h box of h i

h

Oa and ha box having ³n aril O

9.12

ݖ ൌ σ

݁

ିఉா

೔

ିఉఓே

௜ǡே

ariion fnion (oal robabiliy alO on )

wh r

ȕȝN iO h ff of h Lagrang mlili r of variabl nmb r of aril O r box

C

, C Dztning devicesdz to change magnetization and density

ȕ iO a ³ning d vi o hang n rgy in a box

½„ ȕ 1/T inr aO /d r aO m rar O addO/ObraO n rgy

ȝ iO a ³ning d vi o hang #aril O in a box

½„ hanging #aril O in a box hang O h d nOiy.

½„ ȕȝN iO all d h Ch mial o nial

9.13

ۦ¶

௜

ۧ ൌ െ

డ ୪୭୥ ௭

డሺఉఓሻ

x d #aril O in a box

9.14

¶

ൌ

ଵ

ே

ܲሺ݅ǡ ܰሻ

av rag #aril O in a box

How do yo hooO ȝ ?

½„ alla av rag #aril O r box

½„ h n O ȝ o wha yo hink h av rag iO

Consistency {e irement: Phase Transition (38:45

d nOiy in h lai flid iO h (#of OinO) / (volm )

aial N b om O h (#of OinO) (down Oi iO my Oin r r O nO a mol l )

Yo modify ³h (v rial axiO) hang (#of OinO)

wh n h iO down f w r OinO

h-axiO iO loO ly r la d o h h nial o nial

c

olid bl lin

haO ranOiion lin a jm in magn iaion or d nOiy in a gaO-flid

V ry low m rar rOiOO o Oom fini m rar . Th Oonan oO Oymm ry
br aking r gion.

R d lin

oninoO ranOiion lin . A high r m rar O h ± ı Oin magn iOm or

gaO-flid ranOiion orO Omooohly no jmO.

Gr n lin

m a-Oabl Oa . Liid in gaO Oa or gaO in liid Oa . No hang in h ±

ı Oin ori naion. An nOabl Oa ha d rioa O wih im .

Õ

ƒ––‹…‡‘†‡Žǣ‡’‡”ƒ–—”‡Ȃ•’‹’Ž‘–

ۦıۧƒ˜‡”ƒ‰‡‘”‡š’‡…–ƒ–‹‘˜ƒŽ—‡‘ˆ•’‹
T

m rar axiO

h

O

in axiO on of ± Oa O

h

T

ۦıۧ> 0
Liid

ĥ
ۦıۧ< 0

c

ystem Behavior approaching Critical Point (46:00

c

ide Note: Correlation Fnction

ܵ݌݅¶

ሺ݉ǡ ܦሻ ൌ ۦߪሺܽሻߪሺܾሻۧ

orr laion fnion

wh r

m

faor d nding on h m rar

D

diOan in lai oinO O araing (a) (b)

½„ av rag val of h rod of h m aOr m n a wo diff r n oinO (ı(a) and ı(b)

½„ m aOr O h d gr

of orr laion b w n Oin a (b) and Oin a (a)

ÿ

igh critical T: Correlation Fnction is an axponential crve

D

niially a a v ry high m rar

½„ h Oin orr laion iO ~0 wh n (a)(b) ar n ighborO a a v ry high m rar

½„ if (a) iO alla robabiliy (b) iO

½„ orif (a) iO alla av rag of (b)

c

ond Oar o low r h m rar

½„ orr laion b om O  0

½„ mor lik ly > 0 aO if (a) /down h n (b) mor lik ly o b /down

½„ ha iO (b) mor lik ly o b in Oam Oa aO (a) b aO of l OO n rgy r .

Wih gaO-liid m anO wh h r a mol l xiOO a lai oin

½„ a high T Probabiliy iO 50-50 wh h r mol l xiOO

Th orr laion b w n O ond n ighborO iO aroxima ly h Oar of orr laion b w n
firO n ighborO.

½„ Do O no d nd on how lai iO drawn.

AO h diOan Or adO yo loO h Oam fraion of orr laion making h orr laion fnion

xon nial in diOan

N ሺ

ܦሻ ൌ ͳ ൅

ଵ

௫

ܥ

ଵ

ܦ

ଵ

൅

ଵ

௫

మ

ܥ

ଶ

ܦ

ଶ

൅

ଵ

௫

య

ܥ

ଷ

ܦ

ଷ

൅ ڮ

9.16

ۦߪሺܽሻߪሺܾሻۧ ൌ ݁

ି௠ሺ௧ሻ஽

orr laion fnion a high (o low) m rar O

T

Ô

irO mov from high m rar owardO T

h

h mial o nial

T

m rar

Th Oolid lin aally a rO from T§0 o T

C

(riial

m rar )

h

T

C

wh r
D

iO lai oinO diOan b w n (a) and (b) (aial D

m

d ndO on h m rar .
r r O nO h diOan yo hav o mov for h orr laion o dro by a faor of ( )

Low critical T: Correlation Fnction is an Constant crve

A m rar v ry loO o ro

½„ Th biaO of on Oin h Omall O rrbaionO aO O a hang

½„ will biaO all h OinO o b in on dir ion or h oh r (av rag Oin)

D

f yo fr (a) h n (b) will on av rag hav Oam dir ion aO (a). onin O o all (b) (o

infiniy)

9.17

ۦߪሺܽሻߪሺܾሻۧ ՜ N¶ݏݐܽ¶ݐ

will aroah Oom onOan aO yo (D) inr aO .

onOan alOo d ndO on m rar . a v ry low T Orr b om O v ry rigid

Plot of the Correlation Fnction at different Temperatres

Correlation Crves at high T Critical Temperatre

½„ lo of

-m() D

Th rv iO xon nial.

½„ A v ry high m rar O (T§ ’) h orr laionO dro raidly. Th bondO b w

n llO

o w ak o ov rom h high m rar ( n rgy)

½„ AO h m rar fallO Th orr laion d lin b om O mor gradal aO diOan O

inr aO

½„ a Oom oin fallO aO h ow r law r ah O a fini limi for larg D

½„ Th blak arrow ( ՞ ) Ohow (m) inv rO orr laion l ngh.DiOan b w

n rv and

axiO.

½„ AO inv rO orr laion l ngh inr aO O m b om O v ry Omall and (

-m() D

) v ry

gradally go O o ro

D

Corr laion fnion vO D (lai diOan )

Th

bl rv O

ar T’ ĺT

C

m rar O

Th

r d rv O

ar T§0 ĺT

C

m rar O

Th riial m rar (T

C

) iO wh r h wo O O of

rv O aroah ah oh r.

A D0 all orr laionO ar on

ۦɐሺƒሻɐሺƒሻۧൌͳ

C

Crv hikn OO O d o r r O n varioO m rar O
(hik bl T§ ’ )

(m dim bl high T)

(hin bl T§ T

C

)

(hik r d T§ 0 )

(m dim r d low T)

(hin r d T§ T

C

- )

Correlation Crves at low T Critical Temperatre

½„ iniially (T § 0 ) r d lin iO Oraigh

½„ aO T inr aO O r d lin OhfO Olighly b Oill a Oraigh lin

½„ fallO gradally o infiniy no fini limi a h nd (larg D)

Correlations near the Critical Temperatre

wha divid O h wo aroah O iO h riial b havior (wh n h rv O m )

n ih r onOan o infiniy

nor xon nially falling

b w n h wo rv O

fallO off aO a ow r law (roof n d d)

orr laionO ar long rang ¾ 1/D (o Oom ow r)

Microscopic Behavior near the Critical Temperatre (62:00

AO yo aroah T look miroOoially a h magn O h orr laion law llO yo h
magn iO fill d wih ³blobO of Oin mor or l OO in h Oam dir ion.

AO yo aroah h riial oin h blobO g bigg r blobO wihin blobO and h y nd o fad .

½„ r Oal h lai oinO Oo ha h blobO ar roghly h Oam Oi (Oom wha lik

d r aOing miroOo Oal )

½„ r m mb r ha h Oal of h lai do O no aff h orr laionO

½„ h onraO b w

n h () (down) blobO will g w ak r and w ak r

A h riial oin Th bigg O blobO grow o infiniyar infini ly fad d

A h riial oin h blobO b om a O lf-Oimilar fraal Orr .

½„ a v ry Oal yo O

h Oam hing

½„ Th blobO nd o look mor and mor alik

½„ b on Omall r fraal Oal O h d nOiy onraO b om O Orong r and Orong r

Yo an ³n h Oal and m rar o adjO h Oi of h blobO.
D

f yo alOo adjO h d nOiy (ȕȝ) o rn down h onraO h n h fraal Orr will look

h Oam on v ry Oal .

D

f yo ar abl o g idown o h fin O lai Oal (anm Oi )

D

³BlobO of Oin n ar T

C

dark for moOly Oin gr y for moOly down Oin

Corr laion d ndO on inv rO orr laion l ngh (

՞ arrow)

DiOOaa O aO lai diOan (D) inr aO O. A r Ol of h

orr laion fnion ĺ 0 aO h diOan inr aO O (ow r law)

C

½„ h OinO ar abOol ly blak

½„ h down OinO ar abOol ly whi

!"#$
D

f (a)(b) ar no adja n h orr laion

ۦı(a) ı(b)ۧ d ndO on h blob Oi b w n (a)(b)

D

sotropic strctres near T

C

(72:30

Aroahing h T

C

h aal lai Orr iO no v ry Oymm ri. Th Oymm ry iO no

irlar b r anglar.

½„ Th ro ri O of h magn ar no iOoroi

½„ Th orr laion fnion will diff r if m aOr d from h horional or 45° angl .

AO yo g loO r o h riial oin T

C

h OyO m loO O m mory of h baOi nd rlying lai

h ory and b om O irlar iOoroi

A h riial oin

½„ oml roaional Oymm ry

½„ irlar iOoroi

½„ anno diOngiOh any lai dir ion

Coplings other than Nearest Neighbor (74:00

Rah r han h Oiml n ar O v rial or horional n ighbor w old hav d Oign d or
Corr laion fnion o inld diagonal n ighborO or n ighborO wi r mov d.

A high r m rar O h bondO b om w ak r and h xra orr laionO do no aff h
fnion. A low m rar O h OyO m loO O rak of h lai Orr O d. Th only aff

old b an inr aO of h riial oin.

Critical Light Wave Opalescence

d finiionO for oal O n h viOal ro ry of Oom hing having a milky brighn OO and a

lay of olorO from h Orfa

ak a ranOar n liid and n h m rar and d nOiy owardO h riial oin.

½„ blobO of v ry l ngh Oal form (aO abov )

½„ h ff iO ligh wav O of v ry wav l ngh will b Oa r d

away from h riial oin h blobO will hav a O ifi Oi and wav l nghO gr a r han ha
O

i will no b Oa r d. AO yo aroah h riial oin blob Oi will inr aO and long r

wav l nghO will b Oa r d

Vari O aO h riial oinO of varioO liidO Liid arond 306° Ô H llim mh low r

‰

antm Field Theory, {enormalization

ThiO b havior iO h fondaion of QÔT.

AO QÔT OyO mO hav Oymm ry (roaional iOoroi «) i wold b onv ni n o d Orib a
lai Orr foor h varioO ro ri O. Dn hiO mann r all h individal d gr O of fr dom
wold b r r O n d aO O ara lai Oi O.

Th robl m (d Orib d abov ) iO ha h lai Orr will in rf r wih h roaional
invarian h r will b orn rO o aff iOoroy.

To r Oor h Omooh roaional and ranOlaional invarian yo n h OyO m o h riial

oin or wh r h orr laion oin iO mh long r han h lai l ngh. A ha oin h

³lai arifaO diOa ar.

Th ³ning o riial oin haO o do wih r normaliaion groO.

‰

estions.

D

sotropy of CMB, Cosmology

Th iOoroy of h CMB iO Oimilar o blob Orr n ar h riial oin.
Th main aO in oOmology ha ali O o riial d nOiy iO h d nOiy flaionO ha

xiO d loO o h Orfa of laO Oa ring.

First, cecond order Phase Transitions

firO ord r haO ranOiion

½„ h aroah iO from T§0 ĺ T

C

½„ a firO ord r abrb raOiion in magn iaion will or

½„ orO aO Oonan oO Oymm ry br aking.

O

ond ord r haO ranOiion

½„ h aroah iO from T§high ĺ T

C

½„ h magn iaion hang will b Omooh aO h m rar aoroahO T

C