Functional Renormalization - from quantum gravity and dark energy to ultracold atoms and condensed matter March 07-10, 2017 IWH Heidelberg, Germany Fate of Kosterlitz-Thouless Physics in Driven Open Quantum Systems Sebastian Diehl Institute for Theoretical Physics, University of Cologne collaboration: L. Sieberer, E. Altman (Berkeley) : G. Wachtel (Toronto) L. He (Cologne)
Universality in low dimensions: 2D • continuous phase rotations: low temperature high temperature • correlations h φ ( r ) φ ∗ (0) i ⇠ r − α ∼ e − r/ ξ • superfluidity ρ s 6 = 0 ρ s = 0 • KT transition: unbinding of vortex-antivortex pairs … also for out-of-equilibrium systems? … new universal phenomena tied to non-equilibrium?
Experimental Platform: Exciton-Polariton Systems Imamoglu et al., PRA 1996 Kasprzak et al., Nature 2006 photons E relaxation excitons pump lower polaritons k loss • phenomenological description: stochastic driven-dissipative Gross-Pitaevskii-Eq � r 2 � 2 m � µ + i ( γ p � γ l ) + ( λ � i κ ) | φ | 2 i ∂ t φ = φ + ζ u h ζ ⇤ ( t, x ) ζ ( t 0 , x 0 ) i = γδ ( t � t 0 ) δ ( x � x 0 ) pump & loss rates two-body loss propagation elastic collisions microscopic derivation and linear fluctuation analysis: Szymanska, Keeling, Littlewood PRL (04, 06); PRB (07)); Wouters, Carusotto PRL (07,10)
Experimental Platform: Exciton-Polariton Systems • Bose condensation seen despite non-equilibrium conditions stationary state! Kasprzak et al., Nature 2006 • stochastic driven-dissipative Gross-Pitaevskii-Eq � r 2 � Szymanska, Keeling, Littlewood PRL (04, 06); 2 m � µ + i ( γ p � γ l ) + ( λ � i κ ) | φ | 2 i ∂ t φ = φ + ζ u PRB (07)); Wouters, Carusotto PRL (07,10) • mean field • neglect noise • homogeneous solution φ ( x , t ) = φ 0 • naively, just as Bose condensation in equilibrium! • Q: What is “non-equilibrium” about it?
“What is non-equilibrium about it?” • rewrite stochastic Gross-Pitaevski equation i ∂ t φ c = δ H c − i δ H d + ξ δφ ∗ δφ ∗ c c Z d d x [ r α | φ c | 2 + K α | r φ c | 2 + λ α | φ ∗ c φ c | 4 ] , H α = α = c, d u • couplings located in the complex plane: Im incoherent/ irrev. dynamics ⇔ H d example: two-body processes Re λ elastic two-body collisions u Im λ u inelastic two-body losses Re coherent/ reversible dynamics ⇔ H c
“What is non-equilibrium about it?”: Field theory • Representation of stochastic Langevin dynamics as MSRJD functional integral i ∂ t φ c = δ H c − i δ H d Z ⇔ q ] e iS [ φ c , φ ∗ c , φ q , φ ∗ q ] + ξ Z = D [ φ c , φ ∗ c , φ q , φ ∗ δφ ∗ δφ ∗ c c δ ¯ Z ⇢ S [ φ c ] � Z ¯ { φ ∗ c i ∂ t φ c − H c + i H d } S = S = φ ∗ + c.c. + i 2 γφ ∗ q φ q q δφ ∗ t, x t, x c • Equilibrium conditions signalled by presence of symmetry under: H. K. Janssen (1976); C. Aron et al, J Stat. Mech (2011) generalisation to quantum systems T β φ c ( t, x ) = φ ∗ c ( − t, x ) , (Keldysh functional integral) q ( − t, x ) + i L. Sieberer, A. Chiochetta, U. Tauber, T β φ q ( t, x ) = φ ∗ c ( − t, x ) 2 T ∂ t φ ∗ A. Gambassi, SD , PRB (2015) • Implication 1 [equivalence]: (classical) fluctuation-dissipation c ( ω , q ) i = 2 T h φ c ( ω , q ) φ ∗ ω [ h φ c ( ω , q ) φ ∗ q ( ω , q ) � h φ c ( ω , q ) φ ∗ q ( ω , q ) i correlations responses (imaginary part) ➡ equilibrium conditions as a symmetry
“What is non-equilibrium about it?”: Geometric interpretation • Implication 2: geometric constraint equilibrium dynamics non-equilibrium dynamics no symmetry symmetry Im Im protected Re Re • • coherent and dissipative dynamics may coherent and driven-dissipative dynamics do occur simultaneously occur simultaneously • • but they are not independent they result from different dynamical resources ➡ what are the physical consequences of the spread in the complex plane? Review: L. Sieberer, M. Buchhold, SD, Keldysh Field Theory for Driven Open Quantum Systems , Reports on Progress in Physics (2016)
Outline • mapping of the driven-dissipative GPE to KPZ-type equation • fundamental difference to conventional context: KPZ variable: condensate phase, compact ➡ weak non-equilibrium drive: two competing scales • smooth non-equilibrium fluctuations -> emergent KPZ length scale L ∗ • non-equilibrium vortex physics -> emergent length scale L v • result: different sequence in 2D and 1D ➡ strong non-equilibrium drive: new first order phase transition (one dimension)
Low frequency phase dynamics • driven-dissipative stochastic GPE � r 2 � 2 m � µ + i ( γ p � γ l ) + ( λ � i κ ) | φ | 2 i ∂ t φ = φ + ζ u • integrate out fast amplitude fluctuations: φ ( x , t ) = ( M 0 + χ ( x , t )) e i θ ( x ,t ) see also: G. Grinstein et al., PRL 1993 ∂ t θ = D r 2 θ + λ ( r θ ) 2 + ξ particles deposited h ( x , t ) λ at rate phase diffusion phase nonlinearity Markov noise Kardar, Parisi, Zhang, form of the KPZ equation gravitational PRL (1986) field • spin wave becomes non-linear x surface roughening, fire spreading, • nonlinearity: single-parameter measure of non-equilibrium strength bacterial colony growth.. (ruled out in equilibrium by symmetry) Im Im λ = 0 λ 6 = 0 Re Re non-equilibrium equilibrium
2 Dimensions L v E. Altman, L. Sieberer, L. Chen, SD, J. Toner, PRX (2015) G. Wachtel, L. Sieberer, SD, E. Altman, PRB (2016) L. Sieberer, G. Wachtel, E. Altman, SD, PRB (2016)
Im Physical implication I: Smooth KPZ fluctuations λ 6 = 0 • RG flow of the effective dimensionless KPZ coupling parameter Re non-equilibrium g 2 = λ 2 ∆ D 3 FRG analysis: Canet, Chate, Delamotte, Wschebor, PRL (2010), PRE (2012) g strong coupling: disordered / rough non-equilibrium phase weak coupling: equilibrium phase g ( L ∗ ) = 1 1 1 2 3 d • general trend: non-equilibrium effects in systems with soft mode are • enhanced in d = 1,2 • softened in d = 3 (below a threshold)
Im Physical implication I: Smooth KPZ fluctuations λ 6 = 0 • RG flow of the effective dimensionless KPZ coupling parameter Re non-equilibrium g 2 = λ 2 ∆ D 3 FRG analysis: Canet, Chate, Delamotte, Wschebor, PRL (2010), PRE (2012) g strong coupling: disordered / rough non-equilibrium phase weak coupling: equilibrium phase g ( L ∗ ) = 1 1 1 2 3 d • 2D: implication: a length scale is generated • exponentially large for 16 π L ∗ = a 0 e g 2 • weak nonequilibrium λ • small noise level ∆ microscopic (healing) length
Physical implications I: Absence of quasi-LRO • long-range behavior of two-point/ spatial coherence function: h φ ⇤ ( r ) φ (0) i ⇡ n 0 e �h [ θ ( x ) � θ (0)] 2 i leading order cumulant expansion 16 π • L ∗ = a 0 e g 2 generated length scale distinguishes two regimes: h φ ∗ ( r ) φ (0) i sub-exponential non- equilibrium disordered (rough) phase ∼ r − α e − r 2 χ , χ ≈ 0 . 37 ( d = 2) algebraic quasi-long range order (Kosterlitz-Thouless phase) r L ∗ ➡ algebraic order absent in any two-dimensional driven open system at the largest distances ➡ but crossover scale exponentially large for small deviations from equilibrium
Physical implications II: Non-equilibrium Kosterlitz-Thouless • KPZ equation for phase variable ∂ t θ = D r 2 θ + λ ( r θ ) 2 + ξ anti-vortex • compact nature of phase allows for vortex defects in 2D! vortex • in 2D equilibrium: perfect analogy between vortices and electric charges • log(r) interactions, forces 1 / ( ✏ r ) Z ✏ − 1 • dielectric constant = superfluid stiffness d 2 r r P ( r ) P = ( ε − 1) E ext = T>T KT $ T<T KT $ Normal$=$plasma$ superfluid$=$dipole$gas$$ superfluid = dipole gas normal fluid = plasma metallic$screening$ (“vortex$insulator”)$ metallic screening ✏ − 1 → 0 ✏ − 1 > 0 ➡ how is this scenario modified in the driven system?
Duality approach • KPZ equation for phase variable ∂ t θ = D r 2 θ + λ ( r θ ) 2 + ξ ψ t, x = √ ρ t, x e i θ t, x • phase compactness = local discrete gauge invariance of θ t, x 7! θ t, x + 2 π n t, x θ t, x ∈ [0 , 2 π ) , n t, x ∈ Z ➡ needs to be taught to the KPZ equation: • deterministic part: lattice regularization � D sin( θ x − θ x + a ) + λ X ∂ t θ x = − 2 (cos( θ x − θ x + a ) − 1) + η x a unit lattice direction =: L [ θ ] t, x noise deterministic
Duality approach • KPZ equation for phase variable ∂ t θ = D r 2 θ + λ ( r θ ) 2 + ξ ψ t, x = √ ρ t, x e i θ t, x • phase compactness = local discrete gauge invariance of θ t, x 7! θ t, x + 2 π n t, x θ t, x ∈ [0 , 2 π ) , n t, x ∈ Z ➡ needs to be taught to the KPZ equation: • temporal part: stochastic update θ t, x ∈ [0 , ✓ t + ✏ , x = ✓ t, x + ✏ ( L [ ✓ ] t, x + ⌘ t, x ) + 2 ⇡ n t, x • NB: phase can jump, continuum limit eps -> 0 ill defined, derivatives discrete θ t, x ∈ [0 , 2 π ) θ t, x ∈ [0 , 2 π )
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