Download pdf - How random is topological disorder? Phase transitions on ...web.mst.edu/~vojtat/files/SantaBarbaraTalk.pdfPhase transitions, disorder, and universality • eﬀects of disorder on

How random is topological disorder?Phase transitions on random lattices

Thomas VojtaDepartment of Physics, Missouri University of Science and Technology

Santa Barbara, April 12, 2019

• Phase transitions, disorder, and universality

• Random Voronoi-Delaunay lattices

• Disorder fluctuations

• Generalization to broad class of lattices

• Phase transitions on random lattices

• Anderson localization on random lattices

Phys. Rev. Lett., 113, 120602 (2014)Eur. Phys. J. B 88, 314 (2015)

Hatem Barghathi Martin Puschmann, Philipp Cain, and

Michael Schreiber

Phase transitions, disorder, and universality

• effects of disorder on thermal and quantum phase transitions governed byuniversal criteria based on symmetry and dimensionality

Harris criterion governs stability of clean critical points: clean critical point stableagainst weak disorder if correlation length exponent fulfills dν > 2

Imry-Ma criterion controls phase coexistence in the presence of weak disorder:macroscopic phase coexistence stable against domain formation in dimensions d > 2(discrete symmetry) or d > 4 (continuous symmetry)⇒ 1st-order phase transitions rounded in dimensions d ≤ 2.

Anderson localization governs spatial character of quantum states for weak disorder:all wave functions are localized in dimensions d ≤ 2 (orthogonal case)

Harris criterion

Harris’ insight:variation of average local Tc(x) in correlation volume must be smaller than distancefrom global Tc

variation of average Tc in volume ξd

∆〈Tc(x)〉 ∼ ξ−d/2

distance from global critical pointT − Tc ∼ ξ−1/ν

∆〈Tc(x)〉 < T −Tc ⇒ dν > 2

ξ

+TC(1),

+TC(4),

+TC(2),

+TC(3),

• if clean critical point fulfills Harris criterion dν > 2 ⇒ stable against disorderexample: 3D classical Heisenberg magnet: ν = 0.711

• if dν > 2 is violated ⇒ disorder relevant, character of transition changesexample: 3D classical Ising magnet: ν = 0.628

Imry-Ma criterion

Imry and Ma:compare energy gain from forming domain thattakes advantage of disorder with energy cost ofdomain wall

disorder energy gain:

Edis ∼ Ld/2

domain wall energy:

EDW ∼

{

Ld−1 (discrete)Ld−2 (continuous)

L

discrete

continuous

• if Edis > ERF for L → ∞: domain formation favorable

• phase coexistence destroyed by domain formation in d ≤ 2 (discrete symmetry) ord ≤ 4 (continuous symmetry)

• 1st-order phase transitions rounded in dimensions d ≤ 2

Violations of universal criteria on random lattices

• criteria fulfilled for vast majority of systems withrandom interactions or impurities

• many apparent violations for topological disorder(random connectivity)

3D Ising ferromagnet• clean critical behavior even though Harris criterionviolated, dν < 2

2D 8-state Potts model• phase transition remains 1st order despite Imry-Macriterion

2D Contact process• clean critical behavior even though Harris criterionviolated, no trace of exotic infinite-randomnessphysics found in diluted system

2D Tight-binding Hamiltonian• energy-level statistics of metallic phase

Random Voronoi lattice

What causes the failure of the Harris, Imry-Ma, and localization criteriaon random (Voronoi) lattices??







Voronoi-Delaunay construction

• construct cell structure from set of random latticesites

Voronoi cell of site:

• contains all points in the plane (in space) closer togiven site than to any other

• sites whose Voronoi cells share an edge (a face)considered neighbors

Delaunay triangulation (tetrahedrization):

• graph consisting of all bonds connecting pairs ofneighbors

• dual lattice to Voronoi lattice

Algorithms

• generating Voronoi lattice or Delaunay triangulation is prototypical problem incomputational geometry

• many different algorithms exist

• efficient algorithm inspired by Tanemura et al., uses empty circumcircle propertyup to 50002 sites in 2d and 4003 sites in 3d

• computer time scales roughly linearly with number of sites

• 106 sites in 2d: about 30 seconds on PC106 sites in 3d: about 3 min

Properties of random Voronoi lattices

• lattice sites at independent random positions

• local coordination number qi fluctuates2d: 〈q〉 = 6, σq ≈ 1.333d: 〈q〉 = 2 + (48/35)π2 ≈ 15.54, σq ≈ 3.36

• random connectivity (topology) generates disorder in physical system

4 6 8 10 12 14 16 18q

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

P

Moderate disorder!

Properties of random Voronoi lattices

• lattice sites at independent random positions

• local coordination number qi fluctuates2d: 〈q〉 = 6, σq ≈ 1.333d: 〈q〉 = 2 + (48/35)π2 ≈ 15.54, σq ≈ 3.36

• random connectivity (topology) generates disorder in physical system

4 6 8 10 12 14 16 18q

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

P

Moderate disorder!







Coordination number fluctuations in 2d

Voronoi diluted

• divide large system intoblocks of size Lb

• Calculate block-averagecoordination number

Qµ =1

Nb,µ

∑

i∈µ

qi

• fluctuations in Voronoilattice suppressed

Coordination number fluctuations in 2d – II

3 10 30 100 300 1000Lb , dl

10-5

10-4

10-3

10-2

10-1σ Q

VoronoiDilutedVoronoi (link)

1 10 100dl

6

6.1

6.2

6.3

[Q] µ

10-3

10-2

10-1

[Q] µ

− 6

• standard deviationσ2Q(Lb) =

[

(Qµ − q̄)2]

µ

• Voronoi lattice: σQ ∼ L−3/2b

• diluted lattice: σQ ∼ L−1b ∼ N

−1/2b

• also study link-distance clusters

• σQ ∼ L−3/2b as for the real-space

clusters

Topological constraint

• What is the reason for the suppressed disorder fluctuations in the Voronoi lattice??

Euler equation for Delaunay triangulation:(graph of N lattice sites, E edges, F facets, i.e., triangles)

N − E + F = χ

χ: Euler characteristic, topological invariant of the underlying surfacetorus topology (periodic boundary conditions): χ = 0

• each triangle has three edges, and each edge is shared by two triangles, 3F = 2E⇒ average coordination number precisely q̄ = 6 for any disorder configuration

• also follows from fixed angle sum of 180◦ in a triangle

Topological constraint introduces anticorrelations between disorder fluctuations

• fluctuations stem from surface: σQ(Lb) ∼ L(d−1)/2b /Ld

b = L−(d+1)/2b = L

−3/2b

Correlation function

0 1 2 3 4 5 6 7 8r

-0.5

0.0

0.5

1.0

1.5

2.0C

(r),

D(r

)

0 1 2 3 4 5r10-5

10-4

10-3

10-2

10-1

100

|C(r

)|, |D

(r)|

C(r)D(r)

C(r) =1

N

∑

ij

(qi − q̄)(qj − q̄)δ(r− rij)

σ2Q,bulk(r) = D(r) =

2π

Nr

∫ r

0

dr′ r′C(r′)

⇒ bulk contribution to fluctuations negligible beyond 5 or 6 n.n. distances

Rare regions

rare region:

• large block with average coordinationnumber Qµ > QRR where QRR isconstant larger than global average q̄.

• rare region probability

PRR ∼ exp(−aL3b)

⇒ rare regions more strongly suppressedthan in generic random systems forwhich PRR ∼ exp(−aL2

b)0 2500 5000 7500

Lb3

10-6

10-4

10-2

100

PR

R

QRR=6.05QRR=6.10QRR=6.20







How general is the suppression of fluctuations?

Two dimensions:

• topological constraint: Euler eq. N − E + F = χ and triangle relation 3F = 2E

• holds for all random triangulations (with short-range bonds)

• also holds for all tilings with arbitrary quadrilaterals (using 4F = 2E)

⇒ broad class of random lattices with fixed total coordination, σQ ∼ L−3/2b

101

102

103

Lb

10-5

10-3

10-1

σQ

Examples:

• random Voronoi lattices

• lattices with random bond-exchangedefects

• quasiperiodic Penrose andAmmann-Beenker tilings

Three dimensions

Three-dimensional random Voronoi lattice

• Euler equation N −E +F −C = χ contains extra degree of freedom, number C of3d cells (tetrahedra)

• total coordination number not fixed (solid angle sum in tetrahedron not constant)

⇒ 3d random Voronoi lattices not in class of lattices with suppressed fluctuations(preliminary numerics: fluctuations still decay faster than central limit theorem)

101

102

103

Lb

10-5

10-3

10-1

σQ

3D lattices with fixed totalcoordination:

• random lattices built fromrhombohedra (solid angle sum ≡ 4π)

• 3d versions of bond-exchange lattices

• coordination number fluctuationsσQ ∼ L−2

b







Physical systems on Voronoi lattices

Hamiltonians defined on random Voronoi lattice:

• various classical and quantum spin systems

• Josephson junction arrays and lattice bosons,

• nonequilibrium reaction-diffusion processes

• tight-binding Hamiltonian

How does the random connectivity change the character of the transition?

Disorder produced by coordination fluctuations

Example: classical magnet

• Hamiltonian H = −J∑

〈ij〉SiSj

(interactions between nearest-neighbor sites of random lattice)

• local (mean) field hi = −J∑′

j〈Sj〉 = −Jqi〈S〉

⇒ local coordination number determines local critical temperature

Coordination number fluctuations of random lattice determine (bare) disorderfluctuations of many-particle system.

• applies to all systems in which the total coupling strengths is a superposition ofnearest neighbor contributions

Harris-Luck and Imry-Ma criteria

Stability of clean critical point:

• critical point stable if ∆〈Tc(x)〉 < T − Tc

⇒ ξ−(d+1)/2 < ξ−1/ν ⇒ (d+ 1)ν > 2 (rather than dν > 2)

• equivalent to Luck’s generalization of Harris criterion, dν >1

1− ωwith wandering exponent ω = (d− 1)/(2d)

Phase coexistence:

• macroscopic phase coexistence possible if disorder energy gain less than domainwall energy, Edis ∼ L(d−1)/2 and EDW ∼ Ld−1

⇒ phase coexistence possible, first-order phase transitions survive

Disorder anticorrelations of Voronoi lattice explain all apparent violations ofthe Harris and Imry-Ma criteria in 2d mentioned in the introduction

Disorder renormalizations

• coordination number determines bare disorder

What about disorder renormalizations?

• in some systems anticorrelations are protected,e.g., random transverse-field Ising chain (Hoyoset al., EPL, 93 (2011) 30004)

• in general, anticorrelations are fragile, weakuncorrelated disorder will be generated under RG

How strong is this uncorrelated disorder?

• fluctuations of local Tc of an Ising model on a

random Voronoi lattice ⇒ σ(Tc) ∼ L−3/2b

• uncorrelated disorder, if any, invisible for L . 100 20 30 50 100Lb

10-5

10-4

10-3

σ2 (Tc)

⇒ uncorrelated disorder created by RG likely unobservable in experiment and numerics







2D Anderson model on Voronoi-Delaunay lattice

H =∑

〈i,j〉

|i〉〈j|+∑

i

υi |i〉〈i|

vi ∈ [−W/2,W/2] random potentials

• multifractal analysis

• finite-size scaling

-3.0 -2.5 -2.00

5

10

15

20

E

αq=0

6.0 6.2 6.4

E

60

120

180

240

300

500

1000

2000

L =

All states are localized, even in the absence of potential disorder, W = 0

Localization on 3D Voronoi-Delaunay lattice

• two Anderson transitions close to theband edges, even for W = 0

• finite size scaling:correlation length exponent ν ≈ 1.6

• all results agree with usual orthogonaluniversality class

Phase diagram for W 6= 0

Conclusions

• Euler equation imposes topological constraint on coordination numbers in 2d

⇒ broad class of random lattices with strong disorder anticorrelations

• in higher dimensions, lattices in this class exist, but they are not generic

• modified Harris-Luck criterion (d+ 1)ν > 2

• modified Imry-Ma criterion allows first-order transitions to survive

This broad class of topological disorder is less random than generic disorder

Anderson localization:

• tight-binding model on random VD lattice is in usual orthogonal universality class

• universal properties of localization not influenced by anticorrelations ⇒ quantuminterference is crucial

H. Barghathi + T.V., Phys. Rev. Lett., 113, 120602 (2014);

M. Puschmann, P. Cain, M. Schreiber + T.V., Eur. Phys. J. B 88, 314 (2015)