phase transitions and renormalization group 182home.kias.re.kr/mkg/upload/kias-snu...

Phase Transitions and Renormalization Group ¶

Joonhyun Yeo

Konkuk University

KIAS-SNU Physics Wintercamp, December 17-23, 2016

Joonhyun Yeo (Konkuk) Phase Transitions and RG Wintercamp 2016 1 / 43

outline

1 Phase Transitions and Critical Phenomena

ä Phase transitions driven by Thermal fluctuationsä Basic concepts - Broken symmetry, Ordered parameters, etc.ä Critical phenomena and Universality - Scale Invariance

2 Theoretical tools - Equilibrium Statistical Mechanics

ä Models in Statistical Mechanicsä Mean Field Theoryä Upper and Lower Critical Dimensions

3 Idea of Renormalization Group

ä Block Spins

4 Implementation of Renormalization Group

ä Real Space RGä Migdal-Kadanoff approximation, Cumulant Expansions, etcä General Scaling Theory


phase diagrams

Typical phase diagram of classical

fluids

Continuous vs. Discontinuoustransitions (c.f. 2nd order vs. 1storder)

Path A: slowly compressing the gasmaintaining const. temp. (We haveto remove latent heat)

Path A: discontinuous changes inthermodynamic quantities (density,specific heat etc.) → first-order

Eq. of state (n(p,T )) has multiplesolutions on coexistence line:(nL > nG )

Path B: no discontinuity inthermodynamic quantity → 2nd orderor continuous


phase transitionsdynamical considerations

p vs. n = N/V diagram

Following path A (pressurizing gas),sample does not change into liquidinstantaneously and homogeneously

Droplets of liquid phase (present dueto fluctuations) will grow larger

A few (not the average) droplets willgrow to be very large: the process isgoverned by kinetics (nucleation)

As critical point is approached,density fluctuations continue to growto fill the container and live longer.


phase diagramsparamagnetic-ferromagnetic transition

T

H

Tc


ferromagnets and liquids

H ∼ p

T

H

Tc

Coexistence line

Critical point



M ∼ V



M vs. H p vs. V


critical phenomena and universality


basic conceptsbroken symmetry

High temperature states have fullrotational and translational symmetry→ uniform and istropic disorderd states

As temperature is lowered, phases withlower symmetry appear

Crystal: broken translational andorientational symmetry (only a discreteset of translations)

Liquid crystals: these symmetries arepartially broken

Magnetic systems:

ä Ising(Uniaxial): up-down (discrete)symmetry is broken

ä Heisenberg H = −∑

Jij~Si · ~Sj : 3d

rotational (continuous) symmetry →rotation around a fixed axis


basic conceptsorder parameter

On coexistence line, more than oneequilibrium phase exists.

Construct a thermodynamic functionthat is different in each phase →order parameter

Convenient to choose such that itbecomes zero in the disorderd phase

Shows non-analytic behavior nearphase transition


origin of universality


scale invariance


scale invariance

No charactersitic length scale

Correlation length ξ

scale invariance: ξ = 0 or ξ =∞ξ =∞: Critical point

Fluctuations at all length scales are important


power laws

Two physical laws: (x = [length])

f (x) = A exp(−x/a)

ä Constant “a” has a physical meaning.ä Charactersitic length of the problem

g(x) = B (x/c)−η

ä “c” is unimportant; can be absorbed into Bä Change of scale: x ≡ x/bä Corresponding change of scale for g : g ≡ bηgä The law takes the same form:

g(x) = B (x/c)−η

ä Scale invariant


renormalization group (RG)

Coarse-graining

RG Transformation: How does the system flow?

Useful tool when fluctuations at many length scales are important


primerEquilibrium statistical mechanics:

Z =∑

states

e−βH =∑

E

e−βE Ω(E )

=∑

E

exp[−β(E − kBT ln Ω(E ))]

=∑

E

exp[−β(E − TS(E ))]

∼ exp[−β(E − TS(E ))] = exp[−βF ]

Some (classical) spin models:

Heisenberg: H = −J∑〈i,j〉

~Si · ~Sj −∑

i~hi · ~Si , where ~S2

i =fixed.

Ising: H = −J∑〈i,j〉 SiSj −

∑i hiSi , where Si = ±1.

Thermodynamic average:

〈Si 〉 =∂

∂(βhi )lnZ , β = 1/kBT


correlation function

G (i , j) ≡ 〈δSiδSj〉 = 〈(Si − 〈Si 〉)(Sj − 〈Sj〉)〉 = 〈SiSj〉 − 〈Si 〉〈Sj〉

=1

Z

∂2Z

∂(βhi )∂(βhj )−(

1

Z

∂Z

∂(βhi )

)(1

Z

∂Z

∂(βhj )

)=

∂2

∂(βhi )∂(βhj )lnZ =

∂〈Sj〉∂(βhi )

∼ exp

[−|ri − rj |

ξ

]ξ=Correlation length


white regions = fluctuations = nonzero δS


Ising model in 1 dimension

H = −J∑

i

σiσi+1 − h∑

i

σi ,

Z =∑

σi=±1

eK∑

i σiσi+1+B∑

i σi

with K ≡ βJ and B ≡ βh

Z =∑

σi=±1

Tσ1σ2Tσ2σ3 · · ·TσNσ1 = Tr TN = λN+ + λN

− −−−−→N→∞

λN+,

where λ± are eigenvalues of

Tσσ′ = eKσσ′+(B/2)(σ+σ′) =

(eK+B e−K

e−K eK−B

)λ± = eK coshB ±

√e2K sinh2 B + e−2K


No spontaneous magnetization in 1 dim:

m = 〈σi 〉 =1

N

∂

∂BlnZ

∣∣∣∣B=0

=eK sinhB

λ+

[1 +

eK coshB√e2K sinh2 B + e−2K

]B=0

= 0

Correlation function G (n) ≡ 〈σiσi+n〉:

G (n) =1

Z

∑σ1,σi ,σi+n

(Ti−1)

σ1σiσi (Tn)σiσi+n

σi+n

(TN−i−n+1

)σi+nσ1

=1

Z

∑σi ,σi+n

(TN−n

)σi+nσi

σi (Tn)σiσi+nσi+n

We can write for B = 0

Tσσ′ = cosh(σσ′K ) + sinh(σσ′K ) = coshK (1 + σσ′ tanhK )(T2)σσ′ = cosh2 K

∑σ′′

(1 + (σσ′′ + σ′′σ′) tanhK + σσ′ tanh2 K )

= 2 cosh2 K (1 + σσ′ tanh2 K )

(Tm)σσ′ = 2m−1 coshm K (1 + σσ′ tanhm K )


correlation function at B = 0

We have (Show this!)

G (n) =1

Z(2 coshK )N

(tanhn K + tanhN−n K

)with Z (B = 0) = (2 coshK )N (1 + tanhN K ). Therefore

G (n) −−−−→N→∞

tanhn K ≡ exp(−n/ξ),

where the correlation length ξ is

ξ = − 1

ln tanhK= − 1

ln tanh(J/kBT )

ξ −−−→T→0

1

2e2J/(kB T ) →∞

T = 0 is the critical point. System is at the lower critical dimension.


d=1

d=2

For d = 1, energy cost for a kink ∆E =2J and entropy gain ∆S = kB lnN.Free energy change ∆F = ∆E −T∆S =2J − kBT lnN.For any T > 0, F becomes lower whenthere is a kink.

· · ·

For d = 2, ∆E ∼ 2JP (P = perime-ter). Number of ways of closed walksof given P: ∼ (z − 1)P . Therefore,∆S ∼ kBP ln(z − 1).∆F ∼ (2J − kBT ln(z − 1))P. For T >Tc ∼ 2J/(kB ln(z − 1)), a droplet of re-versed spins proliferates. For T < Tc ,ordered state.


mean field theory

Hamiltonian

H = −1

2

∑i,j

Jijσiσj − h∑

i

σi ,

where Jij = J if (i , j) is a nearest neighbor pair and Jij = 0 otherwise.

For m = 〈σi 〉, write

σiσj = (σi + σj )m + (σi −m)(σj −m)−m2

and neglect the fluctuation O((σ −m)2) term.

Mean-field Hamiltonian

H ' −m

2

∑i,j

Jij (σi + σj ) +m2

2

∑i,j

Jij − h∑

i

σi ,

= −(Jmz + h︸︷︷︸heff

)∑

i

σi +1

2Jm2Nz ≡ HMF,

where z is the number of nearest neighbors.


Partition function:

ZMF =∑σi=±

e−βHMF

= e−βNzJm2/2∑σi=±

eβJmz+βh∑

i

σi

= e−βNzJm2/2 (2 cosh(βJmz + βh))N

Self-consistent equation:

m =1

N

∂

∂(βh)lnZMF,

m = tanh(βJzm + βh)

Mean-field free energy FMF = − lnZMF: The above can be obtained byminimizing

f [m] ≡ βFMF

N=

1

2βJzm2 − ln[2 cosh(βJzm + βh)]

with respect to m.


Tc = JzNear T ∼ Tc , |m| ∼ |φ| 1, so one canexpand

f [φ] ∼ 1

2(T − Tc

Tc)φ2 +

1

12φ4−βhφ+ · · ·


Changing h from h < 0 to h > 0for fixed T < Tc


critical exponents of MFTWhen h = 0, near T ∼ Tc ,

f [φ] ∼ t

2φ2 +

1

12φ4, t ≡ (T − Tc )/Tc .

Minimize f to get tφ+ (1/3)φ3 = 0;

φ = (Tc

T)m ' m =

0, if T > Tc

±√−3t, if T < Tc

â Order Parameter:

m ∼ (−t)β ; βMF =1

2, c.f. β3d ' 0.33, β2d = 1/8


â Susceptibility:

χ ≡ ∂m

∂h∼ ∂φ

∂h

In the presence of h, we have tφ+ 4uφ3 = h. From (t + 12uφ2)χ = 1,

χ =

1/t, if T > Tc

1/(−2t), if T < Tc

χ ∼ |t|−γ , γMF = 1 , c.f. γ3d ' 1.24, γ2d = 7/4

â m(h) at T = Tc : t = 0 → m3 ∼ h.

m ∼ h1/δ , δMF = 3 , c.f. δ3d ' 4.8


critical exponentsCorrelation length exponent

Consider the situation where the local field hi is applied, then m→ mi . We canwrite

HMF = −∑

i

∑j

Jijmj + hi

σi +1

2

∑i,j

Jijmimj .

From

mi =∂

∂(βhi )lnZ ,

self-consistent equation becomes

mi = tanh

(β∑

k

Jikmk + βhi

)

Correlation Function:

G (xi − xj ) ≡ 〈σiσj〉 − 〈σi 〉〈σj〉 =∂2

∂(βhi )∂(βhj )lnZ =

∂mi

∂(βhj )


G (xi − xj ) =1

cosh2(· · · )

[β∑

xk

J(xi − xk )G (xk − xj ) + δ(xi − xj )

]Now take hi to 0, then mi → m.Note that 1/ cosh2(· · · ) = 1−m2. Taking the Fourier transform, we have

G (q) = (1−m2)(βJ(q)G (q) + 1),

or

G (q) =1

11−m2 − βJ(q)

Since we are interested in large |xi − xj | behavior, we look at small q behaviors

J(q) =∑

x

J(x)e iq·x

=∑

x

J(x)[1 + iq · x− 1

2(q · x)2 + . . .] ' J(0)− cq2,

where we have used J(x) = J(−x) and J(0) = zJ = Tc .


Using the fact that m = 0 for t = (T/Tc )− 1 > 0 and m2 = −3t for t < 0 thus1/(1−m2) ' 1 + m2 ' 1 + (−3t),

G (q) =1

11−m2 − Tc

T + βcq2'

1

t+βcq2 , T > Tc1

(−2t)+βcq2 , T < Tc

Ornstein-Zernike form:

G (q) ∼ 1

q2 + ξ−2

We have

ξ ∼ |t|−ν , νMF =1

2, ν3d ' 0.63, ν2d = 1

We have diverging correlation length ξ at T = Tc .


G (x) ∼∫

dd q

(2π)d

e iq·x

ξ−2 + q2

=

∫ ∞0

dq qd−1∫

dΩ

(2π)de iqx cos θ 1

ξ−2 + q2

=1

xd

∫ ∞0

dy yd−1∫

dΩ

(2π)de iy cos θ x2

y2 + (x/ξ)2

≡ 1

|x|d−2Y (|x|/ξ)

For d = 3

Y (|x|/ξ) =1

4πexp[−|x|/ξ]

At T = Tc ,

G (x) ∼ 1

|x|d−2+η, ηMF = 0 , η3d ' 0.02, η2d = 1/4


Ising model in ∞ dimensions

Hamiltonian with infinite-range interaction

H = − J

2N

∑i 6=j

σiσj − h∑

i

σi ,

where 1/N factor is necessary to have H ∼ O(N)ä Partition function:

Z =∑

σi=±1

exp

[βJ

2N

(∑

i

σi )2 − N

+ βh

∑i

σi

]

=e−βJ/2∑

σi=±1

exp

[βJ

2N(∑

i

σi )2 + βh

∑i

σi

]

Use Hubbard-Stratonovich transformation

ea2/2 =1√2π

∫ ∞−∞

dx e−x2/2+ax


Z =1√2π

e−βJ/2∑

σi=±1

∫ ∞−∞

dx exp

[−x2

2+

√βJ

Nx + βh

∑i

σi

]

=

√NβJ

2πe−βJ/2

∑σi=±1

∫ ∞−∞

dy exp

[−NβJ

2y2 + βJy + βh

∑i

σi

]

=

√NβJ

2πe−βJ/2

∫ ∞−∞

dy exp

[−NβJ

2y2

]2 cosh(βJy + βh)N

In the limit N →∞, the integral is dominated by the saddle point given by

y = tanh(βJy + βh)

This is exactly the MF self-consistent equation.(Dimensionless) free energy per site

f =βF

N= − lnZ

N' βJ

2y2 − ln[2 cosh(βJy + βh)],

where y is the solution of the aove equtiion.MFT↔ z →∞ or d →∞ limit ↔ infinite-range interaction


upper critical dimension

â Fluctuation

δS ≡ 1

Vξ

∫Vξ

dd x δS(x), δS(x) ≡ S(x)− 〈S〉, Vξ ∼ ξd

â Ginzburg criterion: Self-consistency of MFT if

〈(δS)2〉 < 〈S〉2

Recall that 〈S〉 = m ∼√

(−t), where t = (T − Tc )/Tc .

〈(δS)2〉 =

1

V 2ξ

∫Vξ

dd x

∫Vξ

dd x′ 〈δS(x)δS(x′)〉

=1

V 2ξ

∫Vξ

dd x

∫Vξ

dd x′ G (x− x′)

=1

Vξ

∫Vξ

dd xG (x)


Recall that, within MFT,

G (q) ∼ 1

q2 + ξ−2, ξ ∼ t−1/2

G (x) ∼∫

dd q

(2π)d

e iq·x

q2 + ξ−2= ξ2−d

∫dd k

(2π)d

e ik·y

k2 + 1

= ξ2−d f (y) , k = ξq, y = x/ξ

Therefore

〈(δS)2〉 =

∫Vξ

dd x

VξG (x) = ξ2−d

∫V1

dd y

V1f (y) = (const.)ξ2−d


Since m2 ∼ ξ−2, the above inequality becomes

ξ2−d < (const.)ξ−2, (const.) < ξd−4 ∼ |T − Tc |(4−d)/2

If d > 4, the condition is always satisfied near T ∼ Tc

If d < 4, the condition is not satisfied and fluctuations become importantnear T ∼ Tc ; MFT breaks down.

duc = 4 is the upper critical dimension of the Ising model (universality class).


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