Riemann surfaces for KPZ fluctuations in finite volume Sylvain - PowerPoint PPT Presentation

Riemann surfaces for KPZ fluctuations in finite volume Sylvain Prolhac Laboratoire de physique th´ eorique Universit´ e Paul Sabatier, Toulouse 3 September 2020 RAQIS20, Annecy

I ASEP and KPZ fluctuations II Bethe ansatz and Riemann surfaces III Height fluctuations of TASEP IV g → ∞ ⇒ KPZ fluctuations

KPZ fluctuations in 1 + 1 dimension universal statistics random field h ( x, t ) time evolution of probabilities integrable Interface growth Directed polymer Random unitary dynamics in random medium height h ( x, t ) entanglement entropy free energy (Nahum-Ruhman-Vijay-Haah 2017) i j � E = ε i,j (Takeuchi-Sano 2010) ( i,j ) ∈ path 1D classical / quantum fluids Localization Driven particles with few conservation laws conductance g current normal modes hydrodynamics (Prior-Somoza-Ortu˜ no 2005) (Van Beijeren 2012, Spohn 2014) log g ≃ − 2 L/ℓ + α ( L/ℓ ) 1 / 3 h ∂ t ρ = ∂ 2 x ρ + ∂ x J ( ρ ) + ∂ x ξ

Kardar-Parisi-Zhang (KPZ) equation Stable thermodynamic phase growing inside metastable phase √ x h ( x, t ) − λ ( ∂ x h ( x, t )) 2 + KPZ equation ∂ t h ( x, t ) = ν ∂ 2 D ξ ( x, t ) Gaussian white noise ξ � ξ ( x, t ) � = 0 h ( x, t ) � ξ ( x, t ) ξ ( x ′ , t ′ ) � = δ ( x − x ′ ) δ ( t − t ′ ) δh 2 λδt Boundary conditions for system of size ℓ δx 2 λδt � δh = • periodic h ( x + ℓ, t ) = h ( x, t ) 1 − � δh � 2 • open ∂ x h (0 , t ) = ρ − ∂ x h ( ℓ, t ) = ρ + x δx Singular non-linear stochastic PDE (Hairer, Kupiainen, Gubinelli-Perkowski) Only one parameter λ after rescaling space, time, height in finite volume Large scale behaviour: two fixed points under renormalization group flow • Edwards-Wilkinson λ → 0 (repulsive) z = 2 interface at equilibrium • KPZ fixed point λ → ∞ (attractive) z = 3 / 2 irreversible evolution Universality (at fixed points, but also RG flow EW → KPZ)

Asymmetric simple exclusion process (ASEP) Continuous time Markov process q L sites, N particles, exclusion 1 Total time-integrated current Q ( t ) = � L i =1 ( H i ( t ) − H i (0)) � e γQ ( t ) � � q 1 �C| e tM ( γ ) | P 0 � = C∈ Ω ∆ = ( q 1 / 2 + q − 1 / 2 ) / 2 ≥ 1 M ( γ ) ∼ H XXZ twisted and non-Hermitian √ KPZ equation at large scales for typical height fluctuations when 1 − q ∼ λ/ L ∆ − 1 ∼ λ 2 /L • Edwards-Wilkinson fixed point λ → 0: SSEP q = 1 • KPZ fixed point λ → ∞ : TASEP q = 0 (∆ → ∞ ) sufficient Conditioning on small / large height for ASEP beyond KPZ regime ⇒ crossover phase separation / conformal invariance (Karevski-Sch¨ utz 2017)

KPZ fluctuations in finite volume: several approaches KPZ fixed point in finite volume: random field h ( x, t ) x ≡ x + 1  flat h 0 ( x ) = 0  Initial condition h ( x, 0) = h 0 ( x ): sharp wedge h 0 ( x ) = −| x | / 0  stationary h 0 ( x ) = b ( x ) Brownian bridge General n -point statistics P ( h ( x 1 , t 1 ) > u 1 , . . . , h ( x n , t n ) > u n ) TASEP: expansion over Bethe eigenstates • Euler-Maclaurin asymptotics: singularities, very tedious (P. 2016) • Riemann surfaces: analytic continuation eigenstates ⇒ simpler expressions same kind of structures TASEP and KPZ (P. 2020) TASEP: integral formula for propagator ⇒ rigorous approach (Baik-Liu 2018) Replica method: continuum ⇒ attractive δ -Bose gas (Brunet-Derrida 2000)

I ASEP and KPZ fluctuations II Bethe ansatz and Riemann surfaces 3 2 1 C (1 − y ) L = ( − 1) N − 1 y N ∼ � C 0 � 1 � 2 � 3 � 1 0 1 2 3 4 III Height fluctuations of TASEP IV g → ∞ ⇒ KPZ fluctuations

Bethe ansatz for TASEP (∆ → ∞ ) ASEP 0 < q < 1 (∆ > 1) TASEP q = 0 � 1 − y j � L N � y j − qy k Bethe C (1 − y j ) L = ( − 1) N − 1 y N e Lγ = − j equations 1 − qy j qy j − y k k =1 � � � N � N y j 1 − q 1 − q Eigenvalue E 1 − y j − j =1 j =1 1 − qy j 1 − y j � � x σ ( j ) � � N � � e γ 1 − y j y − k (1 − y j ) x k e γx k Eigenvector ψ ( � x ) A σ ( � y ) det j 1 − qy j j,k σ ∈ S N j =1 � �� 1 − y j �� N � L � N y j Gaudin det. � qy j − y k det ∂ y i log 1 − qy j y j − qy k � ψ ( � x ) | ψ ( � x ) � N + ( L − N ) y j k =1 i,j j =1 “Mean field” Bethe equations for TASEP: parameter C = e Lγ � N k =1 y k ⇒ compact Riemann surface R N Symmetric functions of N Bethe roots y j ⇒ meromorphic functions on R N

Bethe root functions y j ( C ) Domains y j ( C \ R − ) L = 12 N = 4 3 C (1 − y ) L + ( − 1) N y N = 0 | C | = 0 . 1 | C ∗ | | C | = 0 . 5 | C ∗ | | C | = | C ∗ | | C | = 2 | C ∗ | Generalized Cassini ovals | C | | 1 − y | L = | y | N 2 y 6 y 7 | C | = 100 | C ∗ | | C | < | C ∗ | | C | = | C ∗ | | C | > | C ∗ | y 5 1 y 8 y 4 y 3 0 y 2 y 1 y 9 � 1 y 12 L solutions y j ( C ) analytic in C \ R − Generators of analytic continuations y 10 y 11 � 2 A ∞ A 0 A 0 , A ∞ : y j → y k C ∗ 0 Group G = S L A − 1 A − 1 � 3 ∞ 0 iff L, N co-prime � 1 0 1 2 3 4

Riemann surface R 1 ∼ � C for a single Bethe root L sheets glued together along cuts ( −∞ , C ∗ ), Riemann surface R 1 ⇒ ( C ∗ , 0) according to analytic continuations of y j [ C, j ], C ∈ C , j ∈ [ [1 , L ] ] Identifications Meromorphic function y on R 1 y − 1 (0) = [0 , 1] = . . . = [0 , N ] y ([ C, j ]) = y j ( C ) y − 1 ( ∞ ) = [0 , N + 1] = . . . = [0 , L ] y − 1 (1) = [ ∞ , 1] = . . . = [ ∞ , L ] ρ y − 1 ( − Covering map π 1 : R 1 → � 1 − ρ ) = [ C ∗ − i ǫ, 1] = [ C ∗ + i ǫ, N ] C π 1 ([ C, j ]) = C = [ C ∗ − i ǫ, N + 1] = [ C ∗ + i ǫ, L ] Riemann-Hurwitz formula: genus Covering map π : M → N degree d Ramification index e p , p ∈ M : � g M = d ( g N − 1) + 1 + 1 p ∈M ( e p − 1) winding number π (circle around p ) 2 Ramification points p ∈ M : e p ≥ 2 Euler charac. χ = 2 − 2 g = V − E + F ⇒ branch points π ( p ) ∈ N for graph on M linking ramif. points R 1 : genus g = 0 ⇔ R 1 ∼ � Riemann sphere C

Riemann surface R N for sym. functions N Bethe roots Eigenstate: choice N Bethe roots among L Riemann surface R N : [ C, J ] ⇒ Symmetric functions s ( y j 1 ( C ) , . . . , y j N ( C )) C ∈ C , J ⊂ [ [1 , L ] ], | J | = N R N → � � Covering map π N : C TASEP height fluctuations: tr π N = [ C, J ] �→ C J Several connected components Genus if L and N not co-prime L \ N L \ N 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 2 1 · · · · · · · · 2 0 · · · · · · · · 3 1 1 · · · · · · · 3 0 0 · · · · · · · 4 1 2 1 · · · · · · 4 0 0 0 · · · · · · 5 1 1 1 1 · · · · · 5 0 0 0 0 · · · · · 6 1 2 3 2 1 · · · · 6 0 0 0 0 0 · · · · · · · · · · 7 1 1 1 1 1 1 7 0 0 1 1 0 0 · · · · 8 1 2 1 6 1 2 1 8 0 0 2 1 2 0 0 · · 9 1 1 4 1 1 4 1 1 9 0 0 1 7 7 1 0 0 10 1 2 1 3 11 3 1 2 1 10 0 0 4 8 7 8 4 0 0

I ASEP and KPZ fluctuations II Bethe ansatz and Riemann surfaces III Height fluctuations of TASEP � � � n � � d C 1 . . . d C n P ( H i ℓ ( t ℓ ) ≥ H ℓ , ℓ ∈ [ [1 , n ] ]) = (2i π ) n C n | C n | <...< | C 1 | ℓ =1 J ℓ ⊂ [ [1 ,L ] ] , | J ℓ | = N � � � � �� � [ Cℓ,Jℓ ] L µ ([ C, · ]) η ([ C, · ]) − N µ ([ C, · ]) d C NL L − N µ ([ C, · ]) 2 + � n H ℓ − H ℓ − 1 +( t ℓ − t ℓ − 1 ) O C L − N L L ℓ =1 e � � � NL d B � n − 1 B µ ([ C ℓ B, · ]) µ ([ C ℓ +1 B, · ]) ( C ℓ − C ℓ +1 ) e L − N γ ℓ =1 IV g → ∞ ⇒ KPZ fluctuations

Generating function of TASEP height Height H i ( t ) at site i and time t Height increments H i ( t ) − H i (0) ∈ N Initial height H i (0) = � i j =1 ( N L − n i ) One-point generating function average height e γ � L � i =1 ( H i ( t ) − H i (0)) � � �C| e tM ( γ ) | P 0 � M ( γ ) ∼ H XXZ = C∈ Ω Markov property (memoryless) ⇒ generating function at times 0 < t 1 < . . . < t n � n � n � � 1 (e − γ ℓ S iℓ e ( t ℓ − t ℓ − 1 ) M ( � n � γ ℓ ( H iℓ ( t ℓ ) − H iℓ (0)) � γ ℓ S iℓ | P 0 � � � m = ℓ γ m /L ) ) e = �C| e ℓ =1 ℓ =1 C∈ Ω ℓ = n � 1 � � N S i |C� = j =1 [ x j ] i |C� with [ x ] i positions counted from site i L