Paulo Bedaque2016

CLUSTER EXPANSION.

• Method To approximately

compute The Thermodynamic .properties of gases

• Results in an expansion in powers of The density .Thus

, it works well at low density . It's a wellcontrolled approximation in the sense That it becomes

better and better as the density becomes smaller .

• It's an example ( maybe your first ) of a diagrammaticexpansion . Many features generalize To many other

diagrammatic expansions .

the goal is To compute :

# of particles

Bari;÷

F

- PEE,p÷n+€ Vari .ir ;DZ

= 5¥ , dirham NN ! h potential between/ particlesTo account for % it makes Zindistiguishability ; dimensionless ; justifiedjustified by quantum by quantum calculationcalculation

short for Var ; - lrj )

12= (2¥59'An,

.us#ixrie-PiEviF

- -¥ I ;iP%=¥, city

's

fij = e-Ptfij vanishes at large distances

( where Vijeo ) and it is well behaved

even if V :j → 00 ( as itr→o

frequently does )= In then

JIE,d3 . if ( " fi ;)

÷ ztfntfz } + . . . tfnfz } + . . .

+ . . .

= of 's one f two f 'sEach one of these Term has a graphical representation :

fiz = § ,

f , } = § , fizfsyi §§ , fzf , }=@% , . . .

If we omit the particle index inside the circles wemean The sum of all graphs with That same Topology :

flztfhtfzqt . . . = :flzfxitfnfsb + fsyfsot . . . = § §

flz £3 + fczfz } + . . . = §\o

.

z= ¥ .io#sII.&riCi+fytfEtitEentIfEt" ]

T.to g% a. .

=

¥µµ÷nµ ["

+N2wtu"'

fdtridmfz

tsxnxn

unsigned v"→g⇒+ 3NlN;DlIsV"

' 3 Sdtndhrrdhr

, fish

+.

. . ]

fdln d3n fz = fdhndlrz

(EPVCK

"

!, )

A fd3Rxr(e-PVCM

. , )

lR=m# = SIR §dr4tr2(EPV " - i )±- ri - in T )

assuming

Var ) = Vcr )

fdtndtrdrsfizf }q=fd3md3vz (EPVWI " ) . , ) fd }r,d}n , (e-

PVC 'T "l, )

d ' "

=§d}nd}n (EPVMMHD

2htwo connected

components

=v2(§irc#r(e- ever.DZ 88

fdkid 'm

dlr

, fnf , ]

= fdtndmdhr } ( EPVCMN . , ) ( e-PVCM " )

- , )=

SIR Skeske . . .

o

,IR - lrnlrztir } one connected- = V ftp.fd#.. . component¢ , = RI . us f §\o¢ - ri - Nz because of Translationinvariance , the integrand isindependent of The center - of - masscoordinate and There is alwaysone factor of fd3R=V for

every connected

component-

we are ultimately interested in F , not Z :

F= - Kot hit = - KIN hµt , th [ it ft eflotf of

+ . - ' ]

-

97 7ideal gas =fo

connected disconnectedgraphs graph= to + 8 +

flotf It . . ÷( 8+86+58 . , . . .5+ . .

= to + of + fb + EE - { 82 + . . .

But 88 = q÷, [ v soodrhtriceevatht

( 8j=##2vCdr← ceevaty'

so The contribution of The disconnected diagrams vanish , atleast if we include graphs only up to two ling

We 'll proof it using The " replica trice ? '

lnZ = lim ZNR . 1

%→o Trnumber of

"

replicas"

ZNR can be computed as The partition function of a"

replica" system consisting of NR copies ( replicas ) of The

original system , i.e. , a gas with NR different

species ofparticles with The same mass and interactions . Particles

of different species don't interact with each other :

index of Thereplica

-

z"

- SII,d3m :.sn#II.diie.d3mi...sII.d3rYed3mYr

* e-P[{ , EI + E. varied t.E.lt#n+.E.varinrB+-

replica I replica 2.

- partition function of a system made of NRreplicas of the original system .

We can now perform The cluster expansion on The replicasystem . We get The same graphs as in the original theory

e¥pt the bubbles representing the particles may come fromdifferent replica :

@

6 Eta §

or f on °replica indices ,

not particle indices

The

only Thing That cannot happen is a line between

different replicas as there is no potential between replicas( Vcvi

'. it) =o )

% ordinates

of particlesbelonging to

different replicas

The consequence of this is that any graph in The

replica Theory equals The graph in The original onemultiplied by a factor of NF , where a is Thenumber of )

connected components of the graph :

§ ¥3 " R § ) f- other NR f- /

ff# miff , . . .

Thus,

hrZ = lim Zl±µR→0 NRreplica

= lim 1+gvaph=

NR→° NR.

turnout,["rcoatlohnfiatttnryfwothsonwphatdt]connected graphs( = connected graphs .

This is called The

" linked cluster Theorem"

Versions of the linked cluster theorem appear in several diagrammatic

expansions and can be proved by similar means . They all havethe form :

E connected graphsE all graphs

=

e.

EXAMPLE : hand sphere gas to 2¥ order ( first beyond ideal

gas ) in The cluster expansion .

molecules have

00 , r < a

analog:÷kae: " r ' :{ •

, na

F = . uat bµyp - " at I. v at Fdr r2 (Et

- at §dr r'

= - Ija= - volume excluded

by one molecule

I - of

= - Rest In ¥3 + KBT zN÷,÷ up N÷ ice

fractionof volumeexcluded

equation of state : P = - 0£ f ,µ = KBT Nj + guy ro KBT

= eat if [ 1 + qtr \ first Term ofan expansion in powers

of Nw

since f- [ +÷ro+ ... ]± 6. +0K¥' ) ,

The l± correction corresponds To a reduction of the volume , as

expected on physical grounds for a repulsive hand coreinteraction .

LOW TEMPERATURE EXPANSION.

OF LATTICE SYSTEMS.

Let us illustrate The low T expansion using the ZD Isingmode :

o O

•

H = - J E

0

0

•

0

g

links & %

•

•

•

0 T son over links ;•

o

o

• J > 0favor aligned only nearest neighbors% in each site of nearest neighbor interacta ZD square lattice spinsThere is a jitl or . I

variable

.(called " spins

" )

- PH PJE oisj

Z = E e = { e lines

[ Gi ) { si }5

Sum over all

spin configurations

Eis" of " £±i

"

The state with the smallest energy is The one with all

sit +1 ( or all 5i= - l ) . The state with The secondlower energy has all spins 5 ;

=+l except for onesingle one with ( oiit ) . One could expect That , atlow enough Temperature , The state wl the smallest energy

will dominate the sum in Z . After that , The secondlargest contribution would come from the states with

The second lowest energy and so on . Let us Then break up

The sum in Z in That order :

z = E LJ€ixF"

flipping all

His spins giveconfig . WI sameenergy

=E2*T€| + v( iesj

'

+zvE2eJ6+¥CEPftovC*F

. ]d

0

mu seesspain's up * ¥

bragging

a

* :*:* .to At Y $* § *• • • • •

• • •

• • • •

• • • • • • • •

•

•

•

• a •

0 • co

o o • •

0 a •

•

0 0 •• • • •

• • • • • • • a •

•

•

•

•

o

• •

• • • a

•

•

•

•

• • • •

• • •

0

•

•

•

•

• • • •

•

0 0 0• • • •

• • o •

It's better To draw only the " broken " links ( The ones between

spins up and down ) !

z=eeIi[into .ie?i*EEt:*i#tikidkkw..tit

Each graph corresponds to an expression according to the rules :

. . . - ( each ) €3 , EZPJbroken link

combinatorial factor i⇒op # of ways The graph can be embeddedin the lattice

we can use the low T expansion To compote other quantitiesbesides Z

. For instance

, The Itkin i

( Jo Jotr )

= 1-

T F Z £ , Bar e-

PA

all

80 or Jar

spin at TWO sprung flippedpoints separatedby R ± (etssjz [ ¥

+ z ¥85 + . . . ]÷ eTJz[ 1 tv I 8ft + . . . ]± 1 + OC E8PJ ) .

± and 5o+R

Correlated even at

large R

Of course , the low T expansion can be generalized To manyother models

.

HIGHTEMPERATUREEXPANSION OF LATTICE Syste

-

Again , we use The ZD Ising model . First we write The Boltzmann

←re¥aa

= ashes ( vanishes )=⇐F;

%?[,

'

Then

z - ,§ ,

EP " . gq

, e@Exs0i0s-ggyI.soshpJCitsi8tshpJI-CcoshPJJggint.s

( It oiostshes )it

for every linkThere's a Term wl " I

"

and another w/ " gisstghpj"

.

AT hight, sir ; TGHPJ < < I

= @shetJf+vHps5' +zv@he⇒6+6v#pt8tu÷tshpFt . . . )

µ] Di

Mi #:

D Dlinks where

D t.fi#the oisjtghpjwas chosen

fief €t

Notice that graphs where an odd number of lines meet

-

, L ,

17,

.. .

always vanish since

# = gF± , £± , 902 TSHPJ

=o

k↳ = ¥± , of ± , § , ⇒ , 54+545 XBBHPJ = o¥f? ' %{,a

, a 4424434,40452 ESHPJJ =o

a oz

Jq , jell

At high T ( KAT D J ), PJ ( < I , TGHPJ KI and

the larger The number of lines in a graph , the smallest itscontributions .

IT turns out that only connected graph contribute Tobut ( or F ) . This version of The linked cluster Theoremcan be proved , again , using the replica trick .

Let us use the high T expansion To compute the

spin - spin correlation :

( 50 60+12 )

p q = ¥ EQ , Is ( ' +99

tshps ) roar .

spins atTwo sites separatedby R

Since now there's an extra 5 and ftp. The non - vanishing Theterms coming from sir ; tghfj should now have an o± numberof 5 's at The sites O and OTR . For instance ,

¥.la?.*=g;5j.yyIrYX4HlsK44a%4eEshe

comes from ( ltsisj Tghfj )

= e@heD5

The

disconnected grphs like

•r.

Di

cancel in The ratio ( Aso , , ) ÷ lzgg , ago , , IP#=

# t.I.t.ro " "DtD+ . . .( . - tart . . . ) ( It

Dt Edt . . . )= ftp.T

= - .

+ •-h• + . . .

The dominant contribution is given by the graph connecting 0 and OTRwith The smallest number of links ( consequently , the smallest power

of TSHPJ c < l ) : as

( a . for ) = • .=o

~ (tghps)R~(psJ= e

' RhettTo Gotti /

R links

exponentialdecay

w| distance

This form of the correlation indicates Thet far away spinsare decorated

, as opposed To The low T limit where theymove Together .

duality As you may have notice , The combinator .es of the graphs-

in the low and high Temperature expansion is similar .

In fact , The high T graphs on the deft the one formedby the plaquelts ) are in one - to - one correspondence with the low T

graphs in the original lattice

:

*

x x

x

×

•=•÷• ×

×

× ×

× . --0-9

. x×

×

* ×

×

61.-6 ××

•

.no X

x

× ×. - # -* x ×. . -

Q. -

.

"

×

'' toad × ×

Kol x××←gx

+

× x × × ××

Ti

fxtxtxgx 1

k¥ceoriginal originallattice dual latticelatethe duel lattice of a

ZD square lattice is

another ZD square lattice

This relation between diagrams means That there is a relation

between the partition function at some value of p and the

partition function at another value of p :

zip )

=

( EHYT [ 1 + v(deT4

.

Gshpyav

%;Dq÷←

+ " . ]

=

4

zcf ) =

[ it v Hp's) + . . . ]

so we find That :

zcp )e2M . ¥1

e-2€

= TSHFJ.

(coshpjyzv with

If p is large ( lowt ) , F is small ( high F) and vice . versa .This relation

,

called Kvamevs - Wauuier duality , allows us tofind The low T behavior of The system if the high T behavior

is known ( and vice - versa ) . In general , the dual of amodel is a different model . The ZD Ising model is dual To

itself or self.de# . The dual of the 3D Ising model on acubic late

, for instance , is dual To the Zz - gauge modelalso in a cubic lattice

. This is perhaps the simplest example

of a duality in stat neck . ( field Theory .As we will see later , There is a value of Ptpc at which

ZCP ) is not analytic . That means That Z ( § ) should also not beanalytic at The corresponding f=§c . But The power series of ZCP )

in EZPJ is identical to The expansion of ZCF ) in tghfjso Their radius o§ convergence is The sane and Pc=Fc .Using duality we can find pc=§c :

e-295

= tghfcs =tshfJa←

e- ZRJ = EFJ - e- FJ÷- + @PcJ

si

#fc .-

Pc = ¥ hc HK ) .This Temperature is The critical

Temperature below which The spinsare ordered .

b }y+4 ,)

tbs by' i )