Boring (?) first-order phase transitions

Boring (?) first-order

phase transitions

Des Johnston

Plan of talk

First and Second Order Transitions

Finite size scaling (FSS) at first order transitions

First and Second Order Transitions

(Ehrenfest)

First-order phase transitions exhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable.

Second-order transitions are continuous in the first

derivative but exhibit discontinuity in a second derivative of the free energy.

First and Second Order Transitions

(“modern”)

First-order phase transitions are those that involve a latent heat.

Second-order transitions are also called continuous phase transitions. They are characterized by a divergent

susceptibility, an infinite correlation length, and a power-law decay of correlations near criticality.

First Order Transitions - Piccies

Second Order Transition

H=−X

hiji

σiσj ; Z(β) =X

{σ}

First and Second Order Transitions

-Piccies

First order - discontinuities in magnetization, energy (latent heat)

Second order - divergences in specific heat, susceptibility

Looking at transitions

Cook up a (lattice) model

Identify order parameter, measure/calculate things

Look for transition

Continuous - extractcritical exponents

Define a model:

q

-state Potts

Hamiltonian

Hq =−

X

hiji

δσi,σj



Ising: H=−

X

hiji

σiσj





Evaluate a partition function

Z(β) =X

{σ}

exp(−βH_q)

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

Measure 1001 Different Observables

Order parameter

M = (qmax{ni} −N)/(q−1)

Per-site quantities denoted bye=E/N andm=M/N

u(β) = hEi/N,

C(β) = β2N[he2i − hei2],

B(β) = [1− he

4_i 3he2_i2].

m(β) = h|m|i,

χ(β) = β N[hm2i − h|m|i2],

U2(β) = [1−

hm2_i 3h|m|i2].

Continuous Transitions - Critical

exponents

(Continuous) Phase transitions characterized by critical exponents

Definet=|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 systemL

orN1/d

Scaling

Another exponent....

hψ(0)ψ(r)i ∼r−d+2−η

Scaling relations

α= 2−νd ; α+ 2β+γ = 2

Now, first order....

Formallyνd= 1, α= 1

i.e. Volume scaling

C ∼N ; ξ∼N

What does a first order system look

like (at PT) I?

What does a first order system look

like (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

What does a first order system look

like (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)

Heuristic two-phase model

A fractionWo inq ordered phase(s), energyˆeo

A fractionWd = 1−Wo in disordered phase, energyeˆd

The hat =⇒ quantities evaluated atβ∞

Energy moments

Energy moments become

heni=Woeˆno + (1−Wo)ˆend

And the specific heat then reads:

CV(β, L) =Ldβ2

e2− hei2=Ldβ2Wo(1−Wo)∆ˆe2

Max ofC_Vmax=Ld(β∞∆ˆe/2)2 atWo =Wd= 0.5

FSS: Specific Heat

Probability of being in any of the states

po ∝e−βL

d_foˆ

andpd ∝e−βL d_fdˆ

Time spent in the ordered states∝qpo

Wo/Wd'qe−L

d_βfoˆ

/e−βLdfˆd

Expand aroundβ∞

0 = lnq+Ld∆ˆe(β−β∞) +. . .

Solve for specific heat peak

βCmaxV (L) =β∞− lnq

FSS: Binder Cumulant

Energetic Binder cumulant

B(β, L) = 1− he

4_i 3he2_i2

Use (again)

heni=Woeˆno + (1−Wo)ˆend

to get location of min: βBmin(L)

βBmin(L) =β∞−ln(qˆe

2 o/ˆe2d)

Ld_∆ˆe +. . .

1

An Aside

Youcando this more carefully

Pirogov-Sinai Theory (Borgs/Kotecký)

Z(β) =

h

e−βLdfd_+qe−βLdfo

i h

1 +O(Lde−L/L0) i

Z(β)'2√qe−βLd(fd+f0)/2_cosh

βLd(fd−f0)

2 +

1 2lnq

x= βL

d_(f

d−f0)

2 +

1 2lnq∼

Ld∆ˆe(β−β∞)

2 +

1

A Bit Dull....

Peaks grow likeLd=V =N

Critical temperatures shift like1/Ld

Some other numbers - Fixed BC

Fixedboundary conditions

Z(β) =he−β(Ldfd+2dLd−1fd)˜ _+qe−β(Ldfo+2dLd−1fo˜)i_{[1 +. . .]}

x = L

d_(f

d−f0)

2 +dL

d−1_{( ˜f}

d−f˜0) +

1 2lnq

∼ aLd(β−β∞) +bLd−1+. . .

(since

˜

fd=a1+ ˜ed(β−β∞) +. . . , f˜o =a2+ ˜eo(β−β∞) +. . .)

∆β ∼ 1

L

Fitting the data - 8-state Potts

Estimated from peak inχ:

βc(L) =β∞+

a

Some other numbers - degeneracy

A 3D plaquette Ising model

Its dual

A 3D Plaquette Ising action

3Dcubic, spins onvertices

H = −1

2 X

[i,j,k,l]

σiσjσkσl

NOT

H =− X

[i,j,k,l]

And the dual

An anisotropically coupled Ashkin-Teller model

Hdual =−

1 2

X

hiji_x

σiσj−1

2 X

hiji_y

τiτj−1

2 X

hiji_z

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/Ldfits

β_dual∞ = 1.31029(19)

β∞= 0.55317(11) Estimates are about 30 error bars apart.

A Solution...

Degeneracy

Groundstates: Plaquette

1st Order FSS with Exponential

Degeneracy

Normallyqis constant

Suppose instead q ∝eL

βCmaxV _{(L) =β}∞− lnq

Ld_∆ˆe+. . .

βBmin(L) =β∞−ln(qˆe

2 o/ˆe2d)

Ld_∆ˆe +. . .

become

βCmaxV (L) =β∞− 1

Ld−1_∆ˆe+. . .

βBmin(L) =β∞−ln(ˆe

2 o/ˆe2d)

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 of the gonihedric model, ANNNI models, spin ice

References

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

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

W. Janke, Phys. Rev. B47, 14757 (1993)

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