First order phase transitions - PhD students aren't always ...des/MACS_XMAS16.pdf · First order...

First order phasetransitions - PhD students

aren’t always wrong

Marco Mueller (good guy), Wolfhard Janke (good guy),Des Johnston (bad guy)MACS Xmas, Dec 2016

Mueller, Janke, Johnston Degeneracy/FSS 1/29

Plan of talk

Phase transitions, first and second order

Phase transitions on a computer (lattice models)

A problem (with simulations of first order transitions)

A solution


First and Second Order Transitions

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.


Transitions - piccies

First order - melting Second order - Curie


Transitions - on/in a computer

Spins interact with nearest neighbours

Low-T, like to align - ordered phase

High-T, disordered phase


Transitions - on/in a computer

Hamiltonian q-state Potts, σ = 1 . . . q

Hq = −∑〈ij〉

δσi ,σj

Evaluate a partition function, β = 1/kbT

Z (β) =∑{σ}

exp(−βHq)

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

F (β) = ln Z (β)


Measure 1001 Different Observables

Order parameter

M = (q max{ni} − N)/(q − 1)

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

u(β) = 〈E 〉/N ,C(β) = β2 N [〈e2〉 − 〈e〉2].

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


First and Second Order Transitions -Piccies

First order - discontinuities inmagnetization, energy (latentheat)

Second order - divergences inspecific heat, susceptibility


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 L

ξ ∼ L, M ∼ L−β/ν , C ∼ Lα/ν , χ ∼ Lγ/ν


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


1st Order FSS: Heuristic two-phasemodel

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

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

Ignore transits


1st Order FSS: Energy moments

Energy moments become

〈en〉 = Woeno + (1−Wo)en

d

And the specific heat then reads:

CV (β,L) = Ldβ2(⟨

e2⟩− 〈e〉2

)= Ldβ2Wo(1−Wo)∆e2

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

Volume scaling


1st Order FSS: Specific Heat peakshift

Probability of being in any of the states

Wo ∼ q exp(−βLd fo) , Wd ∼ exp(−βLd fd )

Take logs, expand around β∞

ln(Wo/Wd ) = ln q + βLd (fd − fo)

= ln q + Ld ∆e(β − β∞)

Solve for specific heat peak Wo = Wd , ln(Wo/Wd ) = 0

βCmaxV (L) = β∞ − ln q

Ld ∆e+ . . .


1st Order FSS: summary

Peaks grow as Ld

Transition point estimates shift as 1/Ld


Strong First Order Transition

Plaquette Ising model

H = −∑�

σiσjσkσl

Only inaccurate (yours truly...) determination of transitionpoint

Same for dual model - only inaccurate (yours truly ...)determination of transition point


Duality - geometric

Square lattice - self dualTriangle - hexagon dual


Duality - spin models

Spins on orginal lattice↔ spins on dual lattice

High-T↔ Low-Ttanhβ = e−2β∗

Ising, square lattice

Z (β) ' (1 + N tanh(β)4 + . . .)

Z (β∗) ' (1 + N exp(−2β∗)4 + . . .)


Duality - plaquette spin model

Plaquette Ising model

H = −∑�

σiσjσkσl

Dual to this

Hdual = −∑〈ij〉x

σiσj −∑〈ij〉y

τiτj −∑〈ij〉z

σiσjτiτj ,


Exercise for a starting PhD student(Marco Mueller)

Simulate 3d plaquette model and dual

Do a better job than, ahem, before at determining transitionpoint of both

Obtain consistent estimates of transition point


A Problem

Determine critical point(s) L = 8 . . . 27, periodic bc, 1/L3

fits

Original model:β∞ = 0.549994(30)

Dual model:β∞dual = 1.31029(19)

β∞ = 0.55317(11)

Estimates are about 30 error bars apart


(Non) Solutions

Blame the student (yours truly....)

Blame the student (yours truly....)

Think - What is special about plaquette model?


Groundstates: Plaquette

Degeneracy 23L

H = −∑�

σiσjσkσl


Groundstates: Dual++++

++

++

++

++

++

++(a)

(d)(c)

(b)

++ ++

++ ++

++ ++

++++

++

++

++

++

+

+ +

+

+ +

++

σ σττ

Degeneracy 23L

Hdual = −∑〈ij〉x

σiσj −∑〈ij〉y

τiτj −∑〈ij〉z

σiσjτiτj ,


Typical Ground state


1st Order FSS with ExponentialDegeneracy

Normally q is constant

Suppose instead q ∝ eL ( q = e(3 ln 2)L )

βCmaxV (L) = β∞ − ln q

Ld ∆e+ . . .

becomesβCmax

V (L) = β∞ − 3 ln 2Ld−1∆e

+ . . .


Quality of fits

6

26

21

16

11L

max

β∞ +a/L3

βCmaxV

6

26

21

16

11

Lm

ax

β∞ +a/L3

βBmin

4 9 14 19 24Lmin

6

26

21

16

11

Lm

ax

β∞ +a/L2

4 9 14 19 24Lmin

6

26

21

16

11

Lm

ax

β∞ +a/L2

0.00.10.20.30.40.50.60.70.80.91.0

Q

Forcing a fit to 1/L3 gives much poorer quality


Conclusions

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

Exponential degeneracy: 1/L2 corrections in 3D

PhD students are not always wrong


References

G.K. Savvidy and F.J. Wegner, Nucl. Phys. B 413, 605(1994).

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

M. Mueller, D. A. Johnston and W. Janke, Nucl. Phys. B 888(2014) 214; Nucl. Phys. B 894 (2015) 1.

M. Mueller, D. A. Johnston and W. Janke, MPLB 29 (2015)1550109; Physics Procedia 57, 68 (2014)


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