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TRANSCRIPT
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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
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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
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Critical Phenomena
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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
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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
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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,
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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.
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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).
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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
.
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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
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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 !
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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 .
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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
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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.
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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.
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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+ .
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Perturbation expansion
Nonlinear interaction term:
Hint[S] = u
4! |qi|
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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|
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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
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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
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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
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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
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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
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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 .
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Field Theory Approach to Critical Phenomena
Perturbation expansion
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Perturbation expansion
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 .