First order phase transitions - PhD students aren't always ...des/MACS_XMAS16.pdf · First order...
Transcript of 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
Mueller, Janke, Johnston Degeneracy/FSS 2/29
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.
Mueller, Janke, Johnston Degeneracy/FSS 3/29
Transitions - piccies
First order - melting Second order - Curie
Mueller, Janke, Johnston Degeneracy/FSS 4/29
Transitions - on/in a computer
Spins interact with nearest neighbours
Low-T, like to align - ordered phase
High-T, disordered phase
Mueller, Janke, Johnston Degeneracy/FSS 5/29
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)
Mueller, Janke, Johnston Degeneracy/FSS 6/29
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]
Mueller, Janke, Johnston Degeneracy/FSS 7/29
First and Second Order Transitions -Piccies
First order - discontinuities inmagnetization, energy (latentheat)
Second order - divergences inspecific heat, susceptibility
Mueller, Janke, Johnston Degeneracy/FSS 8/29
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
Mueller, Janke, Johnston Degeneracy/FSS 9/29
What does a first order system looklike (at PT) I?
Mueller, Janke, Johnston Degeneracy/FSS 10/29
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
Mueller, Janke, Johnston Degeneracy/FSS 11/29
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
Mueller, Janke, Johnston Degeneracy/FSS 12/29
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
Mueller, Janke, Johnston Degeneracy/FSS 13/29
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+ . . .
Mueller, Janke, Johnston Degeneracy/FSS 14/29
1st Order FSS: summary
Peaks grow as Ld
Transition point estimates shift as 1/Ld
Mueller, Janke, Johnston Degeneracy/FSS 15/29
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
Mueller, Janke, Johnston Degeneracy/FSS 16/29
Duality - geometric
Square lattice - self dualTriangle - hexagon dual
Mueller, Janke, Johnston Degeneracy/FSS 17/29
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 + . . .)
Mueller, Janke, Johnston Degeneracy/FSS 18/29
Duality - plaquette spin model
Plaquette Ising model
H = −∑�
σiσjσkσl
Dual to this
Hdual = −∑〈ij〉x
σiσj −∑〈ij〉y
τiτj −∑〈ij〉z
σiσjτiτj ,
Mueller, Janke, Johnston Degeneracy/FSS 19/29
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
Mueller, Janke, Johnston Degeneracy/FSS 20/29
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
Mueller, Janke, Johnston Degeneracy/FSS 21/29
(Non) Solutions
Blame the student (yours truly....)
Blame the student (yours truly....)
Think - What is special about plaquette model?
Mueller, Janke, Johnston Degeneracy/FSS 22/29
Groundstates: Plaquette
Degeneracy 23L
H = −∑�
σiσjσkσl
Mueller, Janke, Johnston Degeneracy/FSS 23/29
Groundstates: Dual++++
++
++
++
++
++
++(a)
(d)(c)
(b)
++ ++
++ ++
++ ++
++++
++
++
++
++
+
+ +
+
+ +
++
σ σττ
Degeneracy 23L
Hdual = −∑〈ij〉x
σiσj −∑〈ij〉y
τiτj −∑〈ij〉z
σiσjτiτj ,
Mueller, Janke, Johnston Degeneracy/FSS 24/29
Typical Ground state
Mueller, Janke, Johnston Degeneracy/FSS 25/29
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
+ . . .
Mueller, Janke, Johnston Degeneracy/FSS 26/29
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
Mueller, Janke, Johnston Degeneracy/FSS 27/29
Conclusions
Standard 1st order FSS: 1/L3 corrections in 3D
Exponential degeneracy: 1/L2 corrections in 3D
PhD students are not always wrong
Mueller, Janke, Johnston Degeneracy/FSS 28/29
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)
Mueller, Janke, Johnston Degeneracy/FSS 29/29