Random field induced order in two dimensions(joint work with N. Crawford and prelude with A. van Enter)

Wioletta M. Ruszel
Mathematical Institute - Utrecht University - The Netherlands
Workshop ”Random Point Processes in Statistical Physics” - 30th June 2022

Table of contents

1. Introduction
2. The model
3. Main result
4. Prelude: spin-flop transitions
5. Ideas of the proof

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Introduction

Classical XY model2

Introduction: classical XY model• classical XY model (or O(2) vector model) one of the simplest
models with continuous symmetry• various applications in studies related to superfluid helium,thin-films, superconductivity, liquid crystals, dielectric plasma• Mermin-Wagner effect: no continuous symmetry breaking in d = 2• XY model related to integer valued GFF3

Introduction: Berezinskii-Kosterlitz-Thouless transitions• BKT transition: between the phases with exponentially andpower-law decaying correlation functions• phases linked to delocalization of their dual integer(Fröhlich-Spencer (1981), Kharash-Peled (2017), vanEngelenburg-Lis (2021), Aizenman-Harel-Peled-Shapiro (2021))• statistical reconstruction of DGFF from F(DGFF) undergoes BKTtransition (Garban-Sepulveda (2020))

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Introduction: adding frozen-in disorder• frozen-in disorder can have the potential to drastically alter somephysically interesting properties of homogeneous media• random field Ising model in d ≥ 3: first-order phase transition
persists for weak disorder (Bricmont-Kupiainen (1987), Imry-Ma(1975))• rounding effect of disorder for d = 2 for the RFIM (Aizenman-Wehr(1990)), unique Gibbs state for almost all realizations of disorder• no infinite-volume gradient measure under weak disorder (vanEnter-Külske (2006), Bovier-Külske (1995), Külske (1998))

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Introduction: XY with random field• since ’80ties discussion about ordering or not• Dotsenko-Feigelman (’81+’82) gave approximative arguments for noorder in d = 1, 2
• mean-field arguments and simulations suggesting order(Wehr-Niederberger-Sanchez-Palencia-Lewenstein (2006))• Gibbs-non Gibbs transitions (van Enter-R. (2009))• order for• Kac potentials (Crawford (2011) [2])• order for nearest-neighbour potential in d = 3 (Crawford (2014) [1])• question: what happens for d = 2?

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The model

The model• spin configuration σ = (σx )x∈Z2 , σx ∈ S1
• random field α = (αx (ω))x∈Z2 , i.i.d. N(0, 1) random variables,ω∈Ω• (random) Hamiltonian for Λ ⊂ Z2 :−HΛω (σ|σ 0 ) = −12X(σx − σy )2 + x∼y ;x,y ∈ΛXαx (ω)e2 · σxx∈Λ

where σ 0 is the boundary condition,  > 0 and
e1 = (1, 0), e2 = (0, 1)• Gibbs measure
0µω,σ(A) = ZΛ−1ΛZ Y

ν(dσx ) exp −βHΛω (σ|σ 0 )A x∈Λ

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The model: some notation• ΛN ⊂ Z2 , |ΛN | = N 2 , L = 2k for some k ∈ N and define box (block)
QL (z), z ∈ LZ2 by
QL (z) = z + {0, 1, · · · , L − 1}2• block average magnetization
Mz =X
1
σx
|QL |x∈QL (z)• for ω ∈ Ω defineDω = {union of regions for which there are too many small or large
boxes with ”large randomness”}

8Main result

Main result

Theorem (Crawford, R. 2021 [3])Let ξ ∈ (0, 1) be sufficiently small. ∃ 0 (ξ) > 0 so that for  < 0and for almost all ω ∃ Dω , N0 (ω) and δ > 0 such that
δ|Dω ∩ ΛN | ≤ Ce −c| log()| |ΛN | for all N ≥ N0 (ω).
Moreover ∃β0 () > 0 so that if β > β0 then for each z ∈ ΛN withQL (z) ∩ Dω = ∅, we haveEµω,e1 (Mz ) − e1
ΛN

≤ ξ.
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Prelude: spin-flop transitions

Prelude: spin-flop transitions• van Enter and R. (2008) [4] studied Gibbs-non-Gibbs phasetransitions for XY model on Z2• (formal) Hamiltonian of the double-layer system w.r.t specificconfiguration:XX
spec−βHτ (θ) = βJcos(θx − θy ) +ht (−1)|x| sin(θx )x∼yx• competition: interaction term wants angles close, external field
angles close to ±e2• induced discrete symmetry and ordering perpendicular to fieldaround ±e1 (→ spin-flop transitions)10Prelude: spin-flop transitions

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Prelude: low-energy clusters percolatea12Prelude: low-energy clusters percolate
Ideas of the proof:• identify each point a in the dual lattice (Z2 )∗ with surrounding cubein Z2Pspec• write: βHτ (θ) = a∈(Z2 )∗ Φβ (a, θ)• for fixed δ > 0 and configuration θ, look at the graph with vertex setVδ = {a ∈ (Z2 )∗ : Φβ (a, θ) ≤ (inf Φβ ) + δ}
• let Cδ be the ”low-energy percolation cluster” on the sites of Gδ• with probability going to 1, ∃!{|Cδ | = ∞} for β > βc for at least
one t.i. Gibbs measure• discrete symmetry breaking via splitting {∃!|Cδ | = ∞} into disjointevents
13Ideas of the proof

Ideas of the proof: What happens in small boxes?

Let us first look at small boxes Q` :• Hamiltonian with free boundary condition:−HQ` (σ) = −=XX
1(σx − σy )2 + α̂x (ω)e2 · σx2x∼y ;x,y ∈Q`
x∈Q`XX(cos(θx − θy ) − 1) + α̂x (ω) sin(θx )x∼y ;x,y ∈Q`with α̂x = αx − |Q1` |x∈Q`

Pz∈Q` αz

• optimize for θx = Ψ + θ̂x :−HQ` (σ) ≈ −12Xx∼y ;x,y ∈Q`(θ̂x −θ̂y )2 + cos(Ψ)

Xx∈Q`α̂x (ω)θ̂x +O(|Xαx |)

x∈Q`

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Ideas of the proof: ground states• the optimal deviation isθ̂x = cos(Ψ)(−∆−1 )α̂x = cos(Ψ)gxN• this leads to!
XX22−1−HQ` (θ) =supcos (Ψ)α̂x (−∆ )α̂x +O |αx |2(θx )x∈Q` ≈(Ψ)
x∈Q`

x∈Q`

and Ψ ∈ {0, π}• the first term is of order 2 `2 log(`) and the error term typically `• energetic costly ifEQ` (σ) =12X(σx − σy )2 ≥ C 2 log(`)`2x∼y ;x,y ∈Q`

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Ideas of the proof: construct contours• artificial microscopic surface tension by considering two scales ` andL, such that `  ”interaction length”  L,(Presutti, Zahradnik)• the larger L, the worse GPS bounds• fundamental length scale: −1 | log()|−1/2 (width of 0 ↔ π)
• choose ` ∼ −1 | log()|−1/2−1/64 and L ∼ −1 | log()|−1/2+1/64• contours will be defined relative to L
• roughly a box Q` will behave bad if either the Dirichlet energy is toolarge or the average configuration is far from ±e1• a contour Γ is a maximally connected union of boxes QL so thatwithin 2L ∃ Q` which is bad

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Ideas of the proof: constructing contours

ψz0 = 1Ql (z 0 )z0QL (z) z

ψz0 = 0ψz0 = 1 if in an enlarged neighbourhood the Dirichlet energy is small
EQ` (σ) ≤ C 2 | log()|1+χ |Q` |17

Ideas of the proof: constructing contours

ψz1 = 1ψz1 = −1
Ql (z 0 )z0QL (z) z
ψz1 = 0ψz1 = ±1 if in an enlarged neighbourhood the average orientation is close
to ±e118

Ideas of the proof: constructing contours
LLΨz = ψz0 ψz1 = 1

Ψz = ψz0 ψz1 = −1

Ψz = ψz0 ψz1 = 0contour Γ, union of squares on scale L separates Ψ = 1 from Ψ = −1
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Ideas of the proof: construct new configuration σ from σ• regular regions: if the thickened region δ(R) has not too many boxes
with randomness which behaves atypical (dirty boxes QL0 )• show thatP(Γ is not regular)≤ C exp −| log()|δ ]|boxes on scale L0 which cover Γ|• on regular contours Γ, construct σ from σ:• on ΛN \ δ(Γ) set σ = σ or the reflection of σ on e2• on δ(Γ) do surgery and interpolate smoothly to σ

• regular contours can contain dirty boxes but energy contribution
small• Peierls argument: large contours are unlikely and surgery provides
energy gain
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Ideas of the proof: bulk of a contour
δL (Γ)

Γdirty small box

clean small box• on δL (Γ) \ Γ the cubes Q` are good and Ψ of constant sign
• for dirty boxes set σ x = e1 and for clean boxes
λ,Dλ,Dσ x = (cos(gx,Q), sin(gx,Q)) (replace g N by g λ,D )``0• energy gain: −HδL (Γ) (σ) + HδL (Γ) (σ) ≥ C 2 | log()|1−δ |Γ|21

Ideas of the proof: surgery on the boundary of a contour
Collar of Γ C +δL (Γ)

ΓM+M−

C−

middle strip inthe collar

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Ideas of the proof: surgery on the boundary of a contour

Modification 1:• there exists a thickened region around M± with constant signsign(e1 · σx ) and such that σ is close to ±e1 on the outer boundary• flip configuration on M± if it has sign ∓• energy gain: −HΛN (σ (1) ) ≥ −HΛN (σ)

23Ideas of the proof: surgery on the boundary of a contour
Modification 2:
δL (Γ)

Γdirty small box

clean small box

(2)

(1)

• σx = σx outside thickened region around dirty boxes(2)(1)• σx = gradient descent in blue region to σx(2)(1)• σx = σx otherwise00• energy gain: | − HΛN (σ (2) |e1 ) + HΛN (σ (1) |e1 )| ≤ 2 | log()|δ |Γ|24

Ideas of the proof: approximate Hamiltonian• replace gxN by the massive field
λ,D−1gx,Q= (−∆DQ + λ) αxλ,D• perform the c.o.v. φx = θx − cos(θx )gx,Qand obtain−KR (φ|τ ) =Xcos(φx − φy ) − 1 +x∼yand mx =1Xmx cos2 (φx )4 xλ,D 22y ∼x (∇e gQ ) ∼  | log()|P• then we have−HR (φ|τ ) ≈ −KR (φ|τ )• error small if ER ≤ C 2 | log()||R| and αx is typical inside R• need energy cost of average angle to be of same order as before,choose λ = 2 | log()|1+η25

Ideas of the proof: ground states for auxiliary Hamiltonian• think of maximizers of −K that satisfy
X−Cxy (φx − φy ) + Vx φx = 0y ∼xwith Cxy =sin(φx −φy )cos(φx )mx
and Vx = − sin(φx )2φφx −φyx

• we can write the above equation as(−LC + V )φ = 0
and interpret it as discrete elliptic PDE with random mass
• prove that maximizer is unique and uniformly close to ±e1 for points
inside boundary layer O(L)26

Ideas of the proof: surgery on the boundary of a contour

Modification 3:• construct σ (3) on M± \ D ± by replacing σ (2) with optimizer of −K• angle in bulk configuration is inversion of c.o.v. of minimizers
00• energy gain: | − HΛN (σ (3) |e1 ) + HΛN (σ (2) |e1 )| ≤ C 2 | log()|δ |Γ|Modification 4:
match σ (3) with σ inside the ”middle strip”, by forcing σ (3) towards ±e1
Glue together: flip interior components of Γ27

Ideas of the proof: Peierls argument• after all modifications we obtain−HΛN (S(σ)|e1 ) + HΛN (σ|e1 ) ≥ C 2 | log()|1−δ• then1µω,eΛN (Ψx (σ) 6= 1)
X≤
1(Γregular)

Γ

≤CX

1µω,e
ΛN (Ψx (σ) 6= 1, Γ is largest surr. QL )(sp(Γ),Ψ(Γ))∞
XrL2 /`2 −c()βC1r C2e1< e − 2 c()β for β  1

r =1where C1r is the upper bound on ] connected sets Γ surrounding QL|Γ|/`2with r boxes of size L2 and C2is ] of Ψ(Γ) for given δ(Γ)

28Discussion• skeleton of contour argument robust• implementation of modifications differs from d = 3 (patching ofalmost ground states)• random field behaves worse in d = 2, new bounds were needed• possible extensions are for example considering general boundary
conditions or O(N) spins• interpretation in terms of connections to IVGFF?

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References[1] N. Crawford. On random field induced ordering in the classical xy
model. J. Stat. Phys., 142:11–42, 2011.
[2] N. Crawford. Random field induced order in low dimension i. Comm.
Math. Phys., 328:203–249, 2014.
[3] N. Crawford and W.M.Ruszel. Random field induced order in two
dimensions. ArXiv:, 2111.00241v1, 2021.
[4] A.C.D. van Enter and W.M. Ruszel. Gibbsianness versus
non-gibbisanness of time-evolved planar rotor models. Stoch. Proc.
Appl., 119:1866–1888, 2009.

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Thank you for your attention!

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