schladming11a

8/10/2019 schladming11a

1/37

Renormalization Group: Applications in

Statistical Physics

Uwe C. Tauber

Department of Physics, Virginia Tech

Blacksburg, VA 24061-0435, USAemail: [email protected]

http://www.phys.vt.edu/~tauber/utaeuber.html

Schladming International Winter School

Physics at All Scales: The Renormalization Group

26 February 5 March, 2011


2/37

Lecture plan

Critical Phenomena

Continuous phase transitionsScaling theoryLandauGinzburgWilson HamiltonianGaussian approximationWilsons momentum shell renormalization groupDimensional expansion and critical exponentsLiterature

Field Theory Approach to Critical Phenomena

Perturbation expansion and Feynman diagramsUltraviolet and infrared divergences, renormalizationRenormalization group equation and critical exponentsLiterature


3/37

Critical Phenomena


4/37

Landau expansion; mean-field theory

Expand free energy (density) in terms of order parameter (scalarfield) near a continuous(second-order) phase transitionat Tc:

f() = r

22 +

u

4!4 +. . . h ,

r=a(T Tc), u>0; conjugate fieldh breaks Z(2) symmetry

.

f() = 0 equation of state:

h(T, ) =r(T) +u

63 ;

stability: f() =r+ u22 >0.

Critical isotherm at T =Tc:h(Tc, ) =

u6

3.

Spontaneous order parameterfor

r

+-

0Tc

h < 0

h > 0

T


5/37

Thermodynamic singularities at critical point

Isothermal order parameter susceptibility:

V1T = hT =r+u

22 T

V = 1/r1 r>0

1/2|r|1 r


6/37

Scaling hypothesis for free energy

Postulate: (sing.) free energy generalized homogeneousfunction:

fsing(, h) = ||2f h/|| , = T TcTc ;two-parameter scaling, with scaling functions f, f(0) = const.

Landau theory: critical exponents= 0, = 3/2.

Specific heat:

Ch=0 = VTT2c

2fsing

2

h=0

=C || .

Equation of state:

(, h) = fsing

h

= ||2 f

h/||

.

Coexistence line h= 0,


7/37

Scaling relations

Critical isotherm: -dependence in f must cancel prefactor,

f(x)

x(2)/ as x

; hence

(0, h) h(2)/ =h1/ , = / . Isothermal susceptibility:

V = h, h=0 = || , =+ 2( 1) .

Eliminate scaling relations: = , +(1 +) = 2 =+ 2+ , =(

1) ;

only two independent(static) critical exponents.

Mean-field: = 0, = 12 , = 1, = 3, = 32 (dim. analysis).

Experimental exponent values different, but still universal:depend only on symmetry, dimension . . ., notmicroscopic details.


8/37

Thermodynamic self-similarity in the vicinity ofTc

Temperature dependence of the specific heat near the normal- tosuperfluid transition of He 4, shown in successively reduced scales.From: M.J. Buckingham and W.M. Fairbank, in: Progress in low temperature

physics, Vol. III, ed. C.J. Gorter, 80112, North-Holland (Amsterdam, 1961).


9/37

LandauGinzburgWilson Hamiltonian

Coarse-grained Hamiltonian, order parameter field S(x):

H[S] = ddx r2S(x)2 +12[S(x)]2 + u4!S(x)4 h(x) S(x) ,where r=a(T T0c), u>0, h(x) local external field;gradient term [S(x)]2 suppresses spatial inhomogeneities.Probability density for configuration S(x): Boltzmann factor

Ps[S] = exp(H[S]/kBT)/Z[h] ,canonical partition function and momentsfunctional integrals:Z[h] =

D[S] eH[S]/kBT , = S(x) =

D[S] S(x) Ps[S] .

Integral measure: discretize x xi D[S] = idS(xi); or employ Fourier transform: S(x) =

ddq(2)d

S(q) eiqx,

D[S] = q

dS(q)

V

= q,q1>0dRe S(q) dIm S(q)

V

.


10/37

LandauGinzburg approximation

Most likely configuration GinzburgLandau equation:

0 = H[S]S(x)

= r 2 + u6

S(x)2 S(x) h(x) .Linearize S(x) =+S(x): h(x) r 2 + u22 S(x).Fourier transform

OrnsteinZernickesusceptibility:

0(q) = 1

r+ u22 +q2

= 1

2 +q2, =

1/r1/2 r>0

1/|2r|1/2 r


11/37

Scaling hypothesis for correlation function

Scaling ansatz, defines Fisher exponent and correlation length :

C(, q) = |q|2+

C(q) , = ||

.

Thermodynamic susceptibility:

(, q= 0)

2

|

|(2) =

|

| , =(2

) .

Spatial correlationsforx :

C(, x) = |x|(d2+)C(x/) (d2+) ||(d2+) .S(x)S(0) S

2

=2

()2

hyperscaling relations:=

2(d 2 +) , 2 = d .

Mean-field values: = 12 , = 0 (OrnsteinZernicke).

Diverging spatial correlations induce thermodynamic singularities !


12/37

Gaussian approximation

High-temperature phase, T >Tc: neglect nonlinear contributions:

H0[S] = ddq(2)d

12r+q2 |S(q)|2 h(q)S(q) .

Linear transformation

S(q) =S(q) h(q)

r+q2,

q

. . .=

ddq

(2)d . . .:

Z0[h] =D[S] exp(H0[S]/kBT) =

= exp

1

2kBT

q

|h(q)|2r+q2

D[

S] exp

q

r+q2

2kBT |

S(q)|2

S(q)S(q)0

=(kBT)

2

Z0[h](2)2d 2Z0[h]

h(q) h(q)

h=0

=C0(q) (2)d(q+q) , C0(q) =

kBT

r+q2 .


13/37

Gaussian model: free energy and specific heat

F0[h] = kBTln Z0[h] = 12

q

|h(q)|2r+q2

+kBT V ln2 kBT

r+q2

.

Leading singularity in specific heat:

Ch=0 = T

2F0T2

h=0

VkB(aT0c)

2

2

q

1

(r+q2)2 .

d>4: integral UV-divergent; regularized by cutoff (Brillouin zone boundary) = 0 as in mean-field theory;

d=dc= 4: integral diverges logarithmically: 0

k3

(1 +k2)2 dk ln() ;

d


14/37

Renormalization group program in statistical physics

Goal: critical(IR) singularities; perturbatively inaccessible. Exploit fundamental new symmetry:

divergent correlation length induces scale invariance. Analyze theory in ultraviolet regime: integrate out

short-wavelength modes / renormalize UV divergences. Rescale onto original Hamiltonian, obtain recursion relations

for effective, now scale-dependent running couplings. Under such RG transformations:

Relevantparameters grow: set to 0: critical surface. Certain couplings approach IR-stable fixed point:

scale-invariant behavior.

Irrelevant couplings vanish: origin ofuniversality. Scale invariance at critical fixed point infer correct IR

scaling behavior from (approximative) analysis of UV regime derivation of scaling laws.

Dimensional expansion: = dc

dsmall parameter, permits

perturbational treatment computationof critical exponents.


15/37

Wilsons momentum shell renormalization group

RG transformation steps:

(1) Carry out the partition integral over allFourier components S(q) with wavevectors /b |q| , where b>1:eliminates short-wavelength modes.

(2) Scale transformation with the same scaleparameterb>1:x x =x/b , q q =b q .

/b

Accordingly, we also need to rescale the fields:

S(x) S

(x

) =b

S(x) , S(q) S

(q

) =bd

S(q) .

Proper choice of rescaled Hamiltonian assumes original form scale-dependent effective couplings, analyze dependence on b.Notice semi-groupcharacter: RG transformation has no inverse.


16/37

Momentum shell RG: Gaussian model

H0[S] = q

r+q2

2 |S(q)|2 h(q) S(q)

,

where 0. Other couplings irrelevant: vi vi =byivi, yi >0.

After single RG transformation:

fsing(, h,

{ui

},

{vi

}) =bdfsingb

1/, bdh,ui+

ui

bxi,

vi

byi.

After sufficiently many 1 RG transformations:fsing(, h, {ui}, {vi}) = bdfsing

b/, b(d+2)/2h, {ui }, {0}

.

Choose matching condition b

|

| = 1

scaling form:

fsing(, h) = ||df

h/||(d+2)/2 .Correlation function scaling law: use b =/

C(, x,{

ui}

,{

vi}

) =b2Cb/, x

b,{

u

i },{

0}

C(x/)

|x|d2+ .


18/37

Perturbation expansion

Nonlinear interaction term:

Hint[S] = u

4! |qi|


19/37

First-order correction to two-point function

ConsiderS(q)S(q) =C(q) (2)d(q+q) for h= 0; to O(u):

S(q)S(q)1 u

4! |qi|


20/37

Wilson RG procedure: first-order recursion relations

Split field variables in outer (S>) / inner (S.

With Sd=Kd/(2)d = 1/2d1d/2(d/2) and = 0 to O(u):

r =b2r+ u2

A(r)= b2r+ u2

Sd /b

pd1

r+p2dp,

u =b4du

1 3u2

B(r)

=b4du

1 3u

2 Sd

/b

pd1 dp

(r+p2)2

.

r 1: fluctuation contributions disappear, Gaussian theory; r 1: expand

A(r) =Sdd2 1 b2d

d 2 r Sdd4 1 b4d

d 4 +O(r2) ,

B(

r) =

Sd

d4 1

b4d

d 4 +O

(r

) .

Diff i l RG fl fi d i di i l i


21/37

Differential RG flow, fixed point, dimensional expansion

Differential RG flow: set b=e with 0:dr()

d = 2r() +u()

2 Sdd2

r()u()

2 Sdd4 +O(ur2, u2),

du()

d = (4 d)u() 3

2u()2Sd

d4 +O(ur, u2) .

Renormalization group fixed points: dr()/d= 0 =du()/d.

Gauss: u0 = 0 Ising: uI Sd= 23(4 d)4d, d4, uI stable for d


22/37

Critical exponents

Deviation from true Tc: =r rI T Tc.Recursion relation for this (relevant) running coupling:

d()

d = ()

2 u()

2 Sd

d4 .Solve near Ising fixed point: () = (0) exp

2 3

.

Compare with() =(0) e 1 = 2

3

.Consistently to order = 4 d:

=1

2+

12+O(2) , = 0 +O(2) .

Note at d=dc= 4: u() = u(0)/[1 + 3 u(0) /162]

logarithmic correctionsto mean-field exponents.Renormalization group procedure:

Derive scaling laws. Two relevant couplings

independent critical exponents.

Compute scaling exponents via power series in =dc d.

S l t d lit t


23/37

Selected literature: J.J. Binney, N.J. Dowrick, A.J. Fisher, and M.E.J. Newman, The theory

of critical phenomena, Oxford University Press (Oxford, 1993).

J. Cardy, Scaling and renormalization in statistical physics, CambridgeUniversity Press (Cambridge, 1996).

M.E. Fisher, The renormalization group in the theory of critical behavior,Rev. Mod. Phys. 46, 597616 (1974).

N. Goldenfeld, Lectures on phase transitions and the renormalizationgroup, AddisonWesley (Reading, 1992).

S.-k. Ma, Modern theory of critical phenomena, BenjaminCummings(Reading, 1976).

G.F. Mazenko, Fluctuations, order, and defects, WileyInterscience(Hoboken, 2003).

A.Z. Patashinskii and V.L. Pokrovskii, Fluctuation theory of phase

transitions, Pergamon Press (New York, 1979). U.C. Tauber, Critical Dynamics A Field Theory Approach to

Equilibrium and Non-equilibrium Scaling Behavior, Cambridge UniversityPress (Cambridge, 201?), Chap. 1; seehttp://www.phys.vt.edu/~tauber/utaeuber.html.

K.G. Wilson and J. Kogut, The renormalization group and the

expansion, Phys. Rep. 12 C, 75200 (1974).

S i


24/37

Some exercises1. Landau theory for the6 model.

Consider the effective free energy

f() =r

22 +

u

4!4 +

v

6!6 h .

Here, r = a(T T0), v > 0, and h denotes an external field.(a) Show that for u> 0, there is a second-order phase transition at h = 0 and T = T0 with the usual mean-fieldcritical exponents , , , and . Why can vbe neglected near the critical point ?(b) Compute t, t, t, and t at the tricritical point u= 0.

(c) Now assume u= |u| < 0 and h = 0. Show that there is a first-order transition at rd = 5u2/8v, and

calculate the jump in the order parameter and the associated free-energy barrier.(d) For non-zero external field h = 0 and u< 0, find parametric equations rc(|u|, v) and hc(|u|, v) for two

additionalsecond-ordertransition lines, with all three continuous phase boundaries merging at the tricritical pointu= 0, h = 0.

2. Gaussian approximation for the Heisenberg model.Isotropic magnets with continuous rotational spin symmetry are described by the Heisenberg model. Thecorresponding effective LandauGinzburgWilson Hamiltonian reads

H[S] = ddxn

=1

r2

[S

(x)]2

+1

2[S

(x)]

2+

u

4!

n

=1

[S

(x)]2

[S

(x)]2 h

(x) S

(x) ,

where S(x) is an n-component order parameter vector field.(a) Determine the two-point correlation functions in the high- and low-temperature phases in harmonic (Gaussian)approximation.Notice: For T < Tc, it is useful to expand about the spontaneous magnetization: e.g., S

(x) = (x) for = 1, . . . , n 1, and Sn (x) = + (x); then = 0 = . The components along and perpendicular to must be carefully distinguished.

(b) For d < dc= 4, compute the specific heat in Gaussian approximation on both sides of the phase transition,and show that Ch=0 = C||

(4d)/2. Compute the universal amplitude ratio C+/C = 2d/2n.

More exercises


25/37

More exercises

3. First-order recursion relations for the Heisenberg model.For the n-component Heisenberg model above, derive the renormalization group recursion relations

r

= b2

r+n+ 2

6u A(r)

, u

= b

4du

1

n+ 8

6u B(r)

.

Determine the associated RG fixed points and discuss their stability. Compute the critical exponent to first orderin = 4 d.

4. RG flow equations for the n-vector model with cubic anisotropy.The O(n) rotational invariance of the Hamiltonian in the previous problems is broken by additional quartic termswith cubic symmetry,

H[S] =

d

dx

n=1

v

4![S

(x)]4

.

(a) Derive the differential RG flow equations for the running couplings r(), u(), and v().(b) Discuss the ensuing RG fixed points and their stability as function of the number n of order parametercomponents, and compute the associated correlation length critical exponents .


26/37

Field Theory Approach to Critical Phenomena



27/37


O(n)-symmetric Hamiltonian (set kBT= 1):

H[S] =ddxn

=1r

2S

(x)

2

+

1

2[S

(x)]

2

+

u

4!

n

=1

S

(x)

2

S

(x)

2.Construct perturbation expansion for

ijS

i Sj

:

ijS

i Sj eHint[S]

0eHint[S]0 = ijS

i Sj

l=0

(Hint[S])l

l!

0l=0 (Hint[S])ll! 0 .Diagrammatic representation:

Propagator C0(q) = (r+q2)1;

Vertexu6 .

= (q)C

0

q

=

u

6

Generating functionalfor correlation functions (cumulants):

Z[h] =

exp

ddx

hS

,

iSi

(c)=

i(ln)Z[h]

hi h=0

.

Vertex functions


28/37

Vertex functions connectedFeynman diagrams:

u u

+ +

uu

u

+

u u

Dyson equation:= + + + ...

+=

propagator self-energy: C(q)1 =C0(q)1 (q).Generating functional for vertex functions, =ln Z[h]/h:

[] =

ln

Z[h] +

ddx

h ,

(N){i}

=N

i[]

i h=0;

(2)(q) =C(q)1 , 4

i=1

S(qi)

c=

4i=1

C(qi) (4)({qi})

one-particle irreducibleFeynman graphs.

Perturbation series in nonlinear coupling u loop expansion.

Explicit results


29/37

Explicit results

Two-point

vertex

function totwo-loop

order:

+

uu

u

+

u u

(2)

(q) =r+q2

+

n+ 2

6 uk1

r+k2

n+ 2

6 u

2 k

1

r+k2

k

1

(r+k2)2

n+ 2

18 u2 k

1

r+k2 k1

r+k2

1

r+ (q k k)2 ;four-point vertex function to one-loop order:

(4)(

{qi = 0

}) =u

n+ 8

6

u2 k1

(r+k2

)2

. u u

Ultraviolet and infrared divergences


30/37

Ultraviolet and infrared divergencesFluctuation correction to four-point vertex function:

d4

as ,ultravioletdivergences for d>dc= 4: upper critical dimension.Power counting in terms of arbitrary momentum scale :

[x] =1, [q] =, [S(x)] =1+d/2;

[r] =2 relevant, [u] =4d marginalat dc= 4; only divergent vertex functions: (2)(q), (4)({qi = 0});

field dimensionless at lower critical dimensiondlc= 2.

Dimension regimes and dimensional regularization


31/37

Dimension regimes and dimensional regularization

dimension perturbation O(n)-symmetric criticalinterval series 4 field theory behavior

d dlc= 2 IR-singular ill-defined no long-rangeUV-convergent u relevant order (n 2)

2 < d< 4 IR-singular super-renormalizable non-classicalUV-convergent u relevant exponents

d=dc= 4 logarithmic IR-/ renormalizable logarithmicUV-divergence umarginal corrections

d> 4 IR-regular non-renormalizable mean-fieldUV-divergent u irrelevant exponents

Integrals in dimensional regularization: even for non-integer d, :

ddk(2)d

k2

(+k2)s =(+d/2)(s d/2)

2d d/2 (d/2)(s) s+d/2 ;

discard divergent surface integrals;

UV singularities dimensional polesin Euler functions.

Renormalization


32/37

RenormalizationSusceptibility 1 =C(q= 0)1 = (2)(q= 0) ==r rc

rc=

n+ 2

6 uk

1

rc+k2+O(u2) =

n+ 2

6

u Kd

(2)d

d2

d 2 ,

(non-universal) Tc-shift: additive renormalization.

(q)1 =q2 +

1 n+ 26

u

k

1

k2(+k2)

.

Multiplicative renormalization:absorb UV poles at = 0 into renormalizedfields and parameters:

SR =Z1/2S S

(N)R =ZN/2S (N) ;R=Z

2 , uR=Zuu Add4 , Ad =

(3 d/2)2

d1

d/2

.

Normalization pointoutside IR regime, R= 1 or q=:

O(uR) : Z = 1 n+ 26

uR

, Zu= 1 n+ 8

6

uR

;

O(u2R

) : ZS

= 1 +n+ 2

144

u2R

.

Renormalization group equation


33/37

Renormalization group equation

Unrenormalizedquantities cannotdepend on arbitrary scale :

0 =

d

d

(N)

(, u) =

d

d ZN/2S (N)R (, R, uR) renormalization groupequation:

+

N

2 S+R

R+u

uR

(N)R (, R, uR) = 0.

with Wilsons flowand RG beta functions:

S=

0ln ZS= n+ 2

72 u2R+O(u

3R) ,

=

0

lnR

= 2 + n+ 26

uR+O(u2R) ,

u=

0

uR=uR

d 4 +

0

ln Zu

=uR+ n+ 86 uR+O(u2R).0

Ru

u

Method of characteristics


34/37

Method of characteristicsSusceptibility (q) = (2)(q)1:

R(, R, uR, q)1 =2 R(R, uR, q/)

1 .

solve RG equation: method of characteristics

() = ,

R

()1 =R

(1)1 2 exp

1

S()

d

,u(l)

u(1)

(l)(1)

with running couplings, initial values (1) =R, u(1) =uR:

d()

d

= () () , du()

d

=u() .

Near infrared-stable RG fixed point: u(u) = 0, u(u

)> 0

() R , R(R, q)1 2 2+S R(R, u, q/ )1,

matching= |q|/ scaling form with = S , = 1/.

Critical exponents


35/37

p

Systematic = 4 d expansion:u=uR

+n+ 8

6

uR+O(u2R) u0 = 0 , uH= 6 n+ 8 +O(2) ;

IR stability: u(u)> 0

0Ru

u

d>4: Gaussianfixed point u0 = 0,= 12 (mean-field); d


36/37

D.J. Amit, Field theory, the renormalization group, andcritical phenomena, World Scientific (Singapore, 1984).

M. Le Bellac, Quantum and statistical field theory, OxfordUniversity Press (Oxford, 1991).

C. Itzykson and J.M. Drouffe, Statistical field theory, Vol. I,

Cambridge University Press (Cambridge, 1989). G. Parisi, Statistical field theory, AddisonWesley (Redwood

City, 1988).

P. Ramond, Field theory A modern primer,

BenjaminCummings (Reading, 1981). J. Zinn-Justin, Quantum field theory and critical phenomena,

Clarendon Press (Oxford, 1993).

Some exercises


37/37

1. Relationship between cumulants and vertex functions.By means of appropriate derivatives of the generating functional for the vertex functions, establish therelations

(2)

(q) = C(q)1

, 4

i=1S(qi)

c =

4i=1

C(qi) (4)

({qi})

between the two- and four-point vertex functions and cumulants.

2. Explicit two-loop p erturbation theory for the vertex functions.

Confirm the explicit two-loop result for (2)(q) and the one-loop expression for (4)({qi = 0}).

3. Singular contribution to the two-loop propagator self-energy.Employ Feynman parametrization

1

Ar Bs =

(r+s)

(r) (s)

10

xr1 (1 x)s1

[x A+ (1 x) B]r+s dx

to extract the UV-singular part of the two-loop integral

D(q) =

k

1

+k2

k

1

+k2

1

+ (q kk)2 ,

D(q)

q2

sing.q=0

= A2d

8 .

schladming11a

Documents