boring (?) first-order phase transitions - macs.hw.ac.ukdes/edinburgh_june14_talk.pdf · boring (?)...

Boring (?) first-orderphase transitions

Des JohnstonEdinburgh, June 2014

Johnston First Order 1/34

Plan of talk

First and Second Order Transitions

Finite size scaling (FSS) at first order transitions

Non-standard FSS


First and Second Order Transitions(Ehrenfest)

First-order phase transitions exhibit a discontinuity in thefirst derivative of the free energy with respect to somethermodynamic variable.Second-order transitions are continuous in the firstderivative but exhibit discontinuity in a second derivative ofthe free energy.


First and Second Order Transitions(“modern”)

First-order phase transitions are those that involve a latentheat.Second-order transitions are also called continuous phasetransitions. They are characterized by a divergentsusceptibility, an infinite correlation length, and apower-law decay of correlations near criticality.


First Order Transitions - Piccies

First order - melting First order - field driven


Second Order Transition

H = −∑〈ij〉

σiσj ; Z(β) =∑{σ}

exp(−βH)


First and Second Order Transitions -Piccies

First order - discontinuities inmagnetization, energy (latentheat)

Second order - divergences inspecific heat, susceptibility


Looking at transitions

Cook up a (lattice) model

Identify order parameter, measure/calculate things

Look for transition

Continuous - extract critical exponents

Finite Size Scaling (FSS)


Define a model: q-state Potts

Hamiltonian

Hq = −∑〈ij〉

δσi,σj

Ising : H = −∑〈ij〉

σiσj

Evaluate a partition function

Z(β) =∑{σ}

exp(−βHq)

Derivatives of free energy give observables (energy,magnetization..)

F (β) = lnZ(β)


Measure 1001 Different ObservablesOrder parameter

M = (qmax{ni} −N)/(q − 1)

Per-site quantities denoted by e = E/N and m = M/N

u(β) = 〈E〉/N,C(β) = β2N [〈e2〉 − 〈e〉2],

B(β) = [1− 〈e4〉3〈e2〉2

].

m(β) = 〈|m|〉,χ(β) = β N [〈m2〉 − 〈|m|〉2],

U2(β) = [1− 〈m2〉3〈|m|〉2

].


Continuous Transitions - Criticalexponents

(Continuous) Phase transitions characterized by criticalexponents

Define t = |T − Tc|/Tc

Then in general, ξ ∼ t−ν , M ∼ tβ, C ∼ t−α, χ ∼ t−γ

Can be rephrased in terms of the linear size of a system Lor N1/d

ξ ∼ N1/d, M ∼ N−β/νd, C ∼ Nα/νd, χ ∼ Nγ/νd


Scaling

Another exponent....

〈ψ(0)ψ(r)〉 ∼ r−d+2−η

Scaling relations

α = 2− νd ; α+ 2β + γ = 2

Two independent exponents


Now, first order....

Formally νd = 1, α = 1

i.e. Volume scaling

C ∼ N ; ξ ∼ N

Squeezing a delta function onto a lattice


What does a first order system looklike (at PT) I?


What does a first order system looklike (at PT) II?

Hysteresis

-1.5

-1.4

-1.3

-1.2

-1.1

-1

-0.9

-0.8

1.24 1.26 1.28 1.3 1.32 1.34 1.36 1.38

E

β

COLD HOT


What does a first order system looklike (at PT) III?

Phase coexistence

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

-1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7

P(E

)

E


Heuristic two-phase model

A fraction Wo in q ordered phase(s), energy eo

A fraction Wd = 1−Wo in disordered phase, energy ed

The hat =⇒ quantities evaluated at β∞

Neglect fluctuations within the phases


Energy moments

Energy moments become

〈en〉 = Woeno + (1−Wo)e

nd

And the specific heat then reads:

CV (β, L) = Ldβ2(⟨e2⟩− 〈e〉2

)= Ldβ2Wo(1−Wo)∆e

2

Max of CmaxV = Ld (β∞∆e/2)2 at Wo = Wd = 0.5

Volume scaling


FSS: Specific HeatProbability of being in any of the states

po ∝ e−βLdfo and pd ∝ e−βL

dfd

Time spent in the ordered states ∝ qpo

Wo/Wd ' qe−Ldβfo/e−βL

dfd

Expand around β∞

0 = ln q + Ld∆e(β − β∞) + . . .

Solve for specific heat peak

βCmaxV (L) = β∞ − ln q

Ld∆e+ . . .


FSS: Binder CumulantEnergetic Binder cumulant

B(β, L) = 1− 〈e4〉3〈e2〉2

Use (again)

〈en〉 = Woeno + (1−Wo)e

nd

to get location of min: βBmin

(L)

βBmin

(L) = β∞ −ln(qe2o/e

2d)

Ld∆e+ . . .

1

LdFSS


An Aside

You can do this more carefully

Pirogov-Sinai Theory (Borgs/Kotecký)

Z(β) =[e−βL

dfd + qe−βLdfo] [

1 +O(Lde−L/L0)]

Z(β) ' 2√qe−βL

d(fd+f0)/2 cosh

(βLd(fd − f0)

2+

1

2ln q

)

x =βLd(fd − f0)

2+

1

2ln q ∼ Ld∆e(β − β∞)

2+

1

2ln q + . . .


A Bit Dull....

Peaks grow like Ld = V = N

Critical temperatures shift like 1/Ld

No other numbers (?)


Some other numbers - Fixed BCFixed boundary conditions

Z(β) =[e−β(L

dfd+2dLd−1fd) + qe−β(Ldfo+2dLd−1fo)

][1 + . . .]

x =Ld(fd − f0)

2+ dLd−1(fd − f0) +

1

2ln q

∼ aLd(β − β∞) + bLd−1 + . . .

(sincefd = a1 + ed(β − β∞) + . . . , fo = a2 + eo(β − β∞) + . . .)

∆β ∼ 1

L

It is clear that there can now be O(1/L) corrections


Fitting the data - 8-state Potts

Estimated from peak in χ:

βc(L) = β∞ +a

L+ . . .


Some other numbers - degeneracy

A 3D plaquette Ising model

Its dual

A big critical temperature discrepancy (30σ)


A 3D Plaquette Ising action

3D cubic, spins on vertices

H = −1

2

∑[i,j,k,l]

σiσjσkσl

NOT

H = −∑

[i,j,k,l]

UijUjkUklUli , Uij = ±1 Z2 Lattice Gauge


And the dual

An anisotropically coupled Ashkin-Teller model

Hdual = −1

2

∑〈ij〉x

σiσj −1

2

∑〈ij〉y

τiτj −1

2

∑〈ij〉z

σiσjτiτj ,


The Problem

Original model: L = 8, 9, ..., 26, 27, periodic bc, 1/Ld fits

β∞ = 0.549994(30)

Dual model: L = 8, 10, ..., 22, 24, periodic bc, 1/Ld fits

β∞dual = 1.31029(19)

β∞ = 0.55317(11)

Estimates are about 30 error bars apart.


A Solution...

Degeneracy

Modified FSS


Groundstates: Plaquette

Persists into low temperature phase: degeneracy 23L


Ground state


1st Order FSS with ExponentialDegeneracy

Normally q is constant

Suppose instead q ∝ eL

βCmaxV (L) = β∞ − ln q

Ld∆e+ . . .

βBmin

(L) = β∞ −ln(qe2o/e

2d)

Ld∆e+ . . .

becomeβC

maxV (L) = β∞ − 1

Ld−1∆e+ . . .

βBmin

(L) = β∞ −ln(e2o/e

2d)

Ld−1∆e+ . . .


Conclusions

Standard 1st order FSS: 1/L3 corrections in 3D

Fixed BC: 1/L (surface tension)

Exponential degeneracy: 1/L2 in 3D

Further applications may be higher-dimensional variants ofthe gonihedric model, ANNNI models, spin icesystems,“orbital” compass models, . . .


References

K. Binder, Rep. Prog. Phys. 50, 783 (1987)

C. Borgs and R. Kotecký, Phys. Rev. Lett. 68, 1734 (1992)

W. Janke, Phys. Rev. B 47, 14757 (1993)

M. Mueller, W. Janke and D. A. Johnston, Phys. Rev. Lett.112, 200601 (2014)


boring (?) first-order phase transitions - macs.hw.ac.ukdes/edinburgh_june14_talk.pdf · boring (?)...

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