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 Phase transition Continuum picture short loops extended loops p 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 configs p n p (1 − p ) n 1 − p n n loops Z = � To define model: specify lattice, link directions and nodes

Phase transitions in loop models configs p n p (1 − p ) n 1 − p n n loops Z = � To define model: specify lattice, link directions and nodes 2D model Sample configuration

Phase transitions in loop models configs p n p (1 − p ) n 1 − p n n loops Z = � To define model: specify lattice, link directions and nodes Configuration of 3D model Lattice designed so that: p = 0 only short loops p = 1 all curves extended � Alternative has symmetry � p ↔ [1 − p ] under

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 e βJ � � ij � σ i σ j σ i = ± 1 � ∝ Tr (1 + σ i σ j tanh βJ ) � ij � � [tanh βJ ] loop length ≡ J β tanh 1 loops

Local Description and Continuum Theory configs p n p (1 − p ) n 1 − p n n loops Z = � • Introduce n component complex unit vector � z l on each link l z l e −S � • Choose action S so that Z = N � d � l reproduces loop model in ‘high T’ expansion • Identify symmetries of S

Local Description and Continuum Theory configs p n p (1 − p ) n 1 − p n n loops Z = � • Introduce n component complex unit vector � z l on each link l z l e −S � • Choose action S so that Z = N � d � l reproduces loop model in ‘high T’ expansion • Identify symmetries of S Require A D � � e −S = � z † z † z † z † p ( � A · � z B )( � C · � z D ) + (1 − p )( � A · � z D )( � C · � z B ) nodes C B Expand � nodes [ . . . ] n per loop via loops contribute factors ℓ z ∗ β 2 z ∗ β 2 . . . z ∗ γ L z γ � z ℓ z α ∝ δ αβ � � z L z ∗ α 1 z α � d � d � z 1 . . . d � α,β,...γ 1 ℓ

Local Description and Continuum Theory configs p n p (1 − p ) n 1 − p n n loops Z = � • Introduce n component complex unit vector � z l on each link l z l e −S � Z = N � • Choose action S so that d � l reproduces loop model in ‘high T’ expansion • Identify symmetries of S z l → e i ϕ l � � z l Find: (i) local gauge invariance z l → U � U ∈ SU ( n ) � z l (ii) global rotational invariance CP(n-1) σ -model Continuum limit: Q αβ = z α z ∗ β − δ αβ /n S = 1 d d r tr( ∇ Q ) 2 � with 2 g

Phase transitions in CP n − 1 model Q ≡ zz † − 1 /n Gauge-invariant degrees of freedom: ‘spins’ Statistical mechanics: D Q . . . e −S � S = 1 � d d r tr( ∇ Q ) 2 � . . . � ∝ & � D Qe −S 2 g Expect two phases for d > 2 : Small g – Long range order Q ( r ) = Q 0 + 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 Q ≡ zz † − 1 /n Gauge-invariant degrees of freedom: ‘spins’ � Q 1 , 2 ( 0 ) Q 2 , 1 ( r ) � ∝ G ( r ) – prob. 0 & r on same loop Correlations

Phase transitions in CP n − 1 model Q ≡ zz † − 1 /n Gauge-invariant degrees of freedom: ‘spins’ � Q 1 , 2 ( 0 ) Q 2 , 1 ( r ) � ∝ G ( r ) – prob. 0 & r on same loop Correlations Justification: return to lattice model A D � � e −S = � z † z † z † z † A · � C · � z D ) + (1 − p )( � A · � C · � p ( � z B )( � z D )( � z B ) nodes C B Expand � nodes [ . . . ] n per loop via loops contribute factors 2 z ∗ β 2 . . . z ∗ γ L z γ ℓ z ∗ β � � z L z ∗ α 1 z α � z ℓ z α ∝ δ αβ � d � z 1 . . . d � d � 1 ℓ α,β,...γ 1 0 z ∗ 2 r z ∗ 1 Insert z 1 0 and z 2 r into averages r ℓ z ∗ β z ℓ z α ℓ z ∗ 1 ℓ z 2 ∝ δ α 1 δ 2 β � d � and use ℓ 0 2

Phase transitions in CP n − 1 model Q ≡ zz † − 1 /n Gauge-invariant degrees of freedom: ‘spins’ � Q αβ ( 0 ) Q βα ( r ) � ∝ G ( r ) – prob. 0 & r on same loop Correlations Paramagnetic phase G ( r ) ∼ 1 r e − r/ξ — only finite loops Critical point d f = 5 − η G ( r ) ∼ r − (1+ η ) — fractal loops 2 Ordered phase G ( r ) ∼ r − 2 — Brownian loops escape to infinity Order parameter – prob. link lies on extended trajectory

Testing CP n − 1 description: n = 2 Mapping to Heisenberg model for n = 2 via S α = z † σ α z Winding number Scaling collapse with vs coupling const Heisenberg exponents ν = 0 . 708(6) γ = 1 . 39(1) Fitted exponents Consistent with best estimates for ν = 0 . 7112(5) γ = 1 . 3960(9) classical Heisenberg model

Putting the CP n − 1 description to use Computing the loop length distribution Express moments as averages B Example: Prob . for loop to pass through A, B & C A ∝ � Q 1 , 2 ( A ) Q 2 , 3 ( B ) Q 3 , 1 ( C ) � C Hence A m � � d 3 r 1 . . . d 3 r m � Q 1 , 2 ( r 1 ) . . . Q m, 1 ( r m ) � = � ℓ m k � n ( m − 1)! loops k (need ‘replica trick’ for m > n )

Evaluating correlators in the ordered phase � Q 1 , 2 ( r 1 ) . . . Q m, 1 ( r m ) � Want – for example Long range order ⇒ Q ( r ) = Q 0 + δQ ( r ) fluctuations δQ ( r ) small and correlations decay with separation = B ( z α z ∗ β − δ αβ ) independent of position order parameter Q αβ 0

Features of loop length distribution Consider system of size L ≡ L d in extended phase Two components to loop population: • loops of finite length ℓ occupy fraction 1 − f of links • loops of length ℓ ∼ L occupy fraction f of links

Features of loop length distribution Consider system of size L ≡ L d in extended phase Two components to loop population: • loops of finite length ℓ occupy fraction 1 − f of links • loops of length ℓ ∼ L occupy fraction f of links Correspondence in σ -model Q ( r ) = Q 0 + δQ ( r ) • Length distribution of finite loops from fluctuations of δQ ( r ) • Length distn of extended loops from avge on orientations of Q 0 fraction f ≡ size of order parameter Q 0

Results for moments Putting everything together Q αβ ( r ) = B ( z α z ∗ β − δ αβ ) + δQ αβ ( r ) using � 1 � m � � � ℓ m d 3 r 1 . . . d 3 r m � Q 1 , 2 ( r 1 ) . . . Q m, 1 ( r m ) � k � n ( m − 1)! = A loops k � m � B L �| z 1 | 2 | z 2 | 2 . . . | z m | 2 � ≈ n ( m − 1)! A � m n Γ( n )Γ( m ) � B L = A Γ( n + m ) ( f L ) m n Γ( n )Γ( m ) ≡ Γ( n + m )

Loop length distribution Consider system of size L ≡ L d in extended phase Select link at random Length distribution P link ( ℓ ) ≡ L − 1 ℓ � � k δ ( ℓ − ℓ k ) � of loop passing through this link? Find  Cℓ − d/ 2 1 ≪ ℓ ≪ L 2  P link ( ℓ ) = � n − 1 � L 2 ≪ ℓ ≤ f L n ℓ 1 −  L f L

Testing CP n − 1 description: extended phase Length distribution of long loops Results from simulations � n − 1 � P link ( ℓ ) = n 1 − ℓ L f L Derived from average over orientations of CP n − 1 order parameter

Poisson-Dirichlet distribution For M random variables y i ≥ 0 with constraint � M i =1 y i = 1 Dirichlet distribution Γ( Mα ) [Γ( α )] M ( y 1 , y 2 . . . y M ) α − 1 d y 1 . . . d y M − 1 Poisson-Dirichlet: limit M → ∞ , α → 0 with θ ≡ Mα fixed. Order loop lengths ℓ 1 ≥ ℓ 2 ≥ . . . and normalise x i = ℓ i / ( f L ) From calculation of moments find x i ’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