Statistical Physics of Loops on Lattices John Chalker Physics - - PowerPoint PPT Presentation

Statistical Physics of Loops on Lattices

John Chalker Physics Department, Oxford University

Work with: Adam Nahum (Oxford) Miguel Ortu˜ no, Pablo Serna & Andres Somoza (Murcia, Spain)

PRL 107, 110601 (2011), PRE 85, 031141 (2012), PRB 88, 134411 (2013) PRL 111, 100601 (2013) & forthcoming

Loop models

Continuum picture Phase transition

p short loops extended loops coupling constant

Lattice formulation

Fully-packed loops with n colours on lattice of (directed) links

Overview

Loop models on lattices & 3D classical stat mech

Transitions between short-loop phases and extended phases Continuum description as CP n−1 sigma model Loop length distribution in extended phase

Loop models & SU(n) spin systems in (2+1) dimensions

Valence bond liquid to N´ eel transitions

Loop models and non-intersecting random curves

Random curves appear in many contexts

2D random curves – zero-lines of random scalar field Lattice version – percolation hulls

Loop models and non-intersecting random curves 3D random curves – zero-lines of random 2-cpt field Lattice version – tricolour percolation

• Cf. cosmic strings, optical vortices, liquid crystal disclinations . . .

Phase transitions in loop models Z =

configs pnp(1 − p)n1−pnnloops To define model: specify lattice, link directions and nodes

Phase transitions in loop models Z =

configs pnp(1 − p)n1−pnnloops To define model: specify lattice, link directions and nodes

2D model Sample configuration

Phase transitions in loop models Z =

configs pnp(1 − p)n1−pnnloops To define model: specify lattice, link directions and nodes

Configuration of 3D model

Lattice designed so that:

p = 0

• nly short loops

p = 1

all curves extended

• Alternative has symmetry

under

p ↔ [1 − p]

Phase diagrams

3D Manhattan lattice 3D L-lattice

Translating between loops and spins

Extended vs local degrees of freedom

Background Recall high-temperature expansion for Ising model

Z = Tr eβJ

ij σiσj

σi = ±1 ∝ Tr

• ij

(1 + σiσj tanh βJ) ≡

• loops

[tanh βJ]loop length

1 J β tanh

Local Description and Continuum Theory

Z =

configs pnp(1 − p)n1−pnnloops

• Introduce n component complex unit vector

zl on each link l

• Choose action S so that

Z = N

l

• d

zl e−S

reproduces loop model in ‘high T’ expansion

• Identify symmetries of S

Local Description and Continuum Theory

Z =

configs pnp(1 − p)n1−pnnloops

• Introduce n component complex unit vector

zl on each link l

• Choose action S so that

Z = N

l

• d

zl e−S

reproduces loop model in ‘high T’ expansion

• Identify symmetries of S

Require e−S =

nodes

• p(

z†

A ·

zB)( z†

C ·

zD) + (1 − p)( z†

A ·

zD)( z†

C ·

zB)

• C

A D B

Expand

nodes[. . .]

loops contribute factors

• α,β,...γ
• d

z1 . . .

• d

zL z∗α

1 zα 2 z∗β 2 . . . z∗γ L zγ 1

n per loop via

• d

zℓ zα

ℓ z∗β ℓ

∝ δαβ

Local Description and Continuum Theory

Z =

configs pnp(1 − p)n1−pnnloops

• Introduce n component complex unit vector

zl on each link l

• Choose action S so that

Z = N

l

• d

zl e−S

reproduces loop model in ‘high T’ expansion

• Identify symmetries of S

Find: (i) local gauge invariance

• zl → eiϕl

zl

(ii) global rotational invariance

• zl → U

zl U ∈ SU(n)

Continuum limit: CP(n-1) σ-model

S = 1

2g

• ddr tr(∇Q)2

with

Qαβ = zαz∗β − δαβ/n

Phase transitions in CP n−1 model

Gauge-invariant degrees of freedom: ‘spins’

Q ≡ zz† − 1/n

Statistical mechanics:

S = 1 2g

• ddr tr(∇Q)2

& . . . ∝

• DQ . . . e−S
• DQe−S

Expect two phases for d > 2: Small g – Long range order Q(r) = Q0 + δQ(r) fluctuations δQ(r) small and Gaussian Large g – Disorder: no long-range correlations in Q(r)

Phase transitions in CP n−1 model

Gauge-invariant degrees of freedom: ‘spins’

Q ≡ zz† − 1/n

Correlations

Q1,2(0)Q2,1(r) ∝ G(r) – prob. 0 & r on same loop

Phase transitions in CP n−1 model

Gauge-invariant degrees of freedom: ‘spins’

Q ≡ zz† − 1/n

Correlations

Q1,2(0)Q2,1(r) ∝ G(r) – prob. 0 & r on same loop

Justification: return to lattice model e−S =

nodes

• p(

z†

A ·

zB)( z†

C ·

zD) + (1 − p)( z†

A ·

zD)( z†

C ·

zB)

• C

A D B

Expand

nodes[. . .]

loops contribute factors

• α,β,...γ
• d

z1 . . .

• d

zL z∗α

1 zα 2 z∗β 2 . . . z∗γ L zγ 1

n per loop via

• d

zℓ zα

ℓ z∗β ℓ

∝ δαβ Insert z1

0z∗2 0 and z2 rz∗1 r

into averages and use

• d

zℓ zα

ℓ z∗1 ℓ z2 ℓz∗β ℓ

∝ δα1δ2β

2 r 1

Phase transitions in CP n−1 model

Gauge-invariant degrees of freedom: ‘spins’

Q ≡ zz† − 1/n

Correlations

Qαβ(0)Qβα(r) ∝ G(r) – prob. 0 & r on same loop

Paramagnetic phase — only finite loops Critical point — fractal loops Ordered phase — Brownian loops escape to infinity

G(r) ∼ 1

re−r/ξ

G(r) ∼ r−(1+η) df = 5−η

2

G(r) ∼ r−2

Order parameter – prob. link lies on extended trajectory

Testing CP n−1 description: n = 2

Mapping to Heisenberg model for n = 2 via Sα = z†σαz Winding number vs coupling const Scaling collapse with Heisenberg exponents Fitted exponents

ν = 0.708(6) γ = 1.39(1)

Consistent with best estimates for classical Heisenberg model

ν = 0.7112(5) γ = 1.3960(9)

Putting the CP n−1 description to use

Computing the loop length distribution

Express moments as averages Example:

• Prob. for loop to pass through A, B & C

∝ Q1,2(A)Q2,3(B)Q3,1(C)

A B C

Hence

• d3r1 . . . d3rmQ1,2(r1) . . . Qm,1(rm) =

Am n(m − 1)!

• loops k

ℓm

k

(need ‘replica trick’ for m > n)

Evaluating correlators in the ordered phase

Want – for example

Q1,2(r1) . . . Qm,1(rm)

Long range order ⇒ Q(r) = Q0 + δQ(r) fluctuations δQ(r) small and correlations decay with separation

• rder parameter Qαβ

= B(zαz∗β − δαβ) independent of position

Features of loop length distribution

Consider system of size L ≡ Ld in extended phase Two components to loop population:

• loops of finite length ℓ
• ccupy fraction 1 − f of links
• loops of length ℓ ∼ L
• ccupy fraction f of links

Features of loop length distribution

Consider system of size L ≡ Ld in extended phase Two components to loop population:

• loops of finite length ℓ
• ccupy fraction 1 − f of links
• loops of length ℓ ∼ L
• ccupy fraction f of links

Correspondence in σ-model

Q(r) = Q0 + δQ(r)

• Length distribution of finite loops from fluctuations of δQ(r)
• Length distn of extended loops from avge on orientations of Q0

fraction f ≡ size of order parameter Q0

Results for moments

Putting everything together

using

Qαβ(r) = B(zαz∗β − δαβ) + δQαβ(r)

• loops k

ℓm

k

= n(m − 1)! 1 A m d3r1 . . . d3rmQ1,2(r1) . . . Qm,1(rm) ≈ n(m − 1)! BL A m |z1|2|z2|2 . . . |zm|2 = BL A m nΓ(n)Γ(m) Γ(n + m) ≡ (fL)mnΓ(n)Γ(m) Γ(n + m)

Loop length distribution

Consider system of size L ≡ Ld in extended phase Select link at random Length distribution Plink(ℓ) ≡ L−1ℓ

k δ(ℓ − ℓk)

• f loop passing through this link?

Find

Plink(ℓ) =    Cℓ−d/2 1 ≪ ℓ ≪ L2

n L

• 1 −

ℓ fL

n−1 L2 ≪ ℓ ≤ fL

Testing CP n−1 description: extended phase

Length distribution of long loops Plink(ℓ) = n L

• 1 − ℓ

fL n−1

Derived from average

• ver orientations of

CP n−1 order parameter Results from simulations

Poisson-Dirichlet distribution

For M random variables yi ≥ 0 with constraint M

i=1 yi = 1

Dirichlet distribution

Γ(Mα) [Γ(α)]M (y1, y2 . . . yM)α−1dy1 . . . dyM−1

Poisson-Dirichlet: limit M → ∞, α → 0 with θ ≡ Mα fixed. Order loop lengths ℓ1 ≥ ℓ2 ≥ . . . and normalise xi = ℓi/(fL) From calculation of moments find xi’s are PD with parameter θ = n for directed loops, and θ = n/2 for undirected loops

Why so simple and universal?

Stability of distribution under split-merge processes — if disconnection/reconnection determined only by loop length Long loops cross sample many times ⇒ mean field regime

Summary

Loop models are discretisation of

• Classical CP n−1 sigma models
• Quantum SU(n) antiferromagnets

Features

• Phase transitions between short-loop and extended phases
• Simple route to calculate length distribution of long loops