Anomalous transport in random conformal field theory Per Moosavi - PowerPoint PPT Presentation

Anomalous transport in random conformal field theory Per Moosavi KTH Royal Institute of Technology Summer school on Quantum Transport and Universality Rome, September 16, 2019

Inhomogeneous CFT � L/ 2 � � H = d x v ( x ) T + ( x ) + T − ( x ) − L/ 2 − L/ 2 − L/ 2 − L/ 2 L/ 2 L/ 2 L/ 2 v ( x ) x a E.g.: Effective description of generalized quantum spin chain � � � � j +1 + S y j S y S x j S x j +1 − ∆ S z j S z h j S z H XXZ = − J j − j +1 j j j 1 / 20

Random CFT v v ( x ) = 1 − ξ ( x ) > 0 with Gaussian random function ξ ( x ) specified by E [ ξ ( x )] = 0 Γ( x − y ) = E [ ξ ( x ) ξ ( y )] Exact analytical results showing diffusion on top of ballistic motion: E ( x ) Spread of measured 1 t 0 arrival times t 1 ← − − → x x 0 x 1 P.M., PhD thesis (2018); Langmann, P.M., Phys. Rev. Lett. 122 (2019) Numerical demonstration of this diffusive effect in random integrable spin chains using generalized hydrodynamics Agrawal, Gopalakrishnan, Vasseur, Phys. Rev. B 99 (2019) 2 / 20

Outline ⋄ Introduction ⋄ Main tools ⋄ Applications ⋄ Random CFT

Outline ⋄ Introduction ⋄ Main tools ⋄ Applications ⋄ Random CFT

Minkowskian conformal field theory Spacetime: R + × S 1 with S 1 the circle of length L Conformal group ∼ = Diff + ( S 1 ) × Diff + ( S 1 ) with Diff + ( S 1 ) the group of orientation-preserving diffeomorphisms of the circle Right- and left-moving components of the energy-momentum tensor � � c = ∓ 2i δ ′ ( x − y ) T ± ( y ) ± i δ ( x − y ) T ′ 24 π i δ ′′′ ( x − y ) T ± ( x ) , T ± ( y ) ± ( y ) ± � � T ± ( x ) , T ∓ ( y ) = 0 in light-cone coordinates x ± = x ± vt Recall: T ± = T ± ( x ∓ ) with T + = T −− , T − = T ++ , and T + − = 0 = T − + E.g.: Schottenloher, A Mathematical Introduction to Conformal Field Theory (2008) 3 / 20

Observables and conformal transformations Primary fields Φ( x − , x + ) → f ′ ( x − ) ∆ + Φ f ′ ( x + ) ∆ − Φ Φ( f ( x − ) , f ( x + )) Energy-momentum tensor c T ± ( x ∓ ) → f ′ ( x ∓ ) 2 T ± ( f ( x ∓ )) − 24 π { f ( x ∓ ) , x ∓ } � f ′′ ( x ) � 2 f ∈ Diff + ( S 1 ) where { f ( x ) , x } = f ′′′ ( x ) f ′ ( x ) − 3 f ′ ( x ) 2 E.g.: Francesco, Mathieu, Sénéchal, Conformal Field Theory (1997) 4 / 20

Examples Non-interacting fermions � � T ± ( x ) = 1 π : ψ + ± ( x )( ∓ i ∂ x ) ψ − ± ( x ): + h.c. − 12 L 2 2 � � � � r ( x ) , ψ + ψ − ψ ± r ( x ) , ψ ± r ′ ( y ) = δ r,r ′ δ ( x − y ) r ′ ( y ) = 0 Local Luttinger model (renormalized) π ρ ± ( x ) 2 : − T ± ( x ) = π : � 12 L 2 ρ ± ( x ) = 1 + K ρ ± ( x ) + 1 − K ρ ± ( x ) = : ψ + ± ( x ) ψ − √ √ � ρ ∓ ( x ) ± ( x ): 2 K 2 K Voit, Rep. Prog. Phys. 58 (1995) Schulz, Cuniberti, Pieri, Fermi liquids and Luttinger liquids, p. 9 in Field Theories for Low-Dim. . . . (2000) 5 / 20

Outline ⋄ Introduction ⋄ Main tools ⋄ Applications ⋄ Random CFT

Projective unitary representations of diffeomorphisms Proj. unitary reps. U ± ( f ) of f ∈ � Diff + ( S 1 ) given by � L/ 2 U ± ( f ) = I ∓ i ε d x ζ ( x ) T ± ( x ) + o ( ε ) − L/ 2 for infinitesimal f ( x ) = x + εζ ( x ) with ζ ( x + L ) = ζ ( x ) Meaning of projective: U ± ( f 1 ) U ± ( f 2 ) = e ± i cB ( f 1 ,f 2 ) / 24 π U ± ( f 1 ◦ f 2 ) E.g.: Khesin, Wendt, The Geometry of Infinite-Dimensional Groups (2009) Gawędzki, Langmann, P.M., J. Stat. Phys. 172 (2018) 6 / 20

Virasoro-Bott group and Virasoro algebra Bott cocycle � L/ 2 B ( f 1 , f 2 ) = 1 2 ( x )] ′ log[ f ′ d x [log f ′ 1 ( f 2 ( x ))] 2 − L/ 2 Virasoro-Bott group: Central extension of Diff + ( S 1 ) given by B ( f 1 , f 2 ) Corresponding Lie algebra: The Virasoro algebra � � n + m + c 12( n 3 − n ) δ n + m, 0 = ( n − m ) L ± L ± n , L ± m � � L ± n , L ∓ = 0 m and � � � ∞ T ± ( x ) = 2 π n − c e ± 2 π i nx L ± 24 δ n, 0 L L 2 n = −∞ E.g.: Khesin, Wendt, The Geometry of Infinite-Dimensional Groups (2009) Gawędzki, Langmann, P.M., J. Stat. Phys. 172 (2018) 7 / 20

Adjoint action Using the Bott cocycle: c U ± ( f ) T ± ( x ) U ± ( f ) − 1 = f ′ ( x ) 2 T ± ( f ( x )) − 24 π { f ( x ) , x } U ± ( f ) T ∓ ( x ) U ± ( f ) − 1 = T ∓ ( x ) Given a smooth L -periodic function v ( x ) > 0 , define � x � L/ 2 v 0 1 = 1 1 d x ′ d x ′ f ( x ) = v ( x ′ ) v 0 L v ( x ′ ) 0 − L/ 2 Then f ∈ � Diff + ( S 1 ) and U ( f ) = U + ( f ) U − ( f ) gives � L/ 2 � � U ( f ) HU ( f ) − 1 = d x v 0 T + ( x ) + T − ( x ) + c-number − L/ 2 8 / 20

Outline ⋄ Introduction ⋄ Main tools ⋄ Applications ⋄ Random CFT

Non-equilibrium dynamics Focus on heat transport in inhomogeneous CFT: Only need reps. of Diff + ( S 1 ) Can also do both heat and charge transport: Need reps. of Map( S 1 , G ) ⋊ Diff + ( S 1 ) Simplest example: G = U (1) as for the local Luttinger model 9 / 20

Time evolution from smooth-profile states Non-equilibrium initial states defined by � L/ 2 G = d x β ( x ) v ( x )[ T + ( x ) + T − ( x )] − L/ 2 with smooth inverse-temperature profile β ( x ) Recipe to compute � � e − G O 1 ( x 1 ; t 1 ) . . . O n ( x n ; t n ) �O 1 ( x 1 ; t 1 ) . . . O n ( x n ; t n ) � neq = Tr � e − G � Tr for O j ( x ; t ) = e i Ht O j ( x )e − i Ht 10 / 20

Energy density and heat current The energy density operator � � E ( x ) = v ( x ) T + ( x ) + T − ( x ) and the heat current operator J ( x ) = v ( x ) 2 � � T + ( x ) − T − ( x ) satisfy ∂ t E ( x ) + ∂ x J ( x ) = 0 � � ∂ t J ( x ) + v ( x ) ∂ x v ( x ) E ( x ) + S ( x ) = 0 with � � S ( x ) = − c v ( x ) v ′′ ( x ) − 1 2 v ′ ( x ) 2 12 π 11 / 20

Energy density and heat current – Results Given smooth L -periodic functions v ( x ) and β ( x ) defining the time evolution and the initial state as above, then � � 1 1 x + ) �E ( x ; t ) � ∞ x − ) + F (˚ neq = F (˚ − v ( x ) S ( x ) 2 v ( x ) � � neq = 1 �J ( x ; t ) � ∞ x − ) − F (˚ x + ) F (˚ 2 in the thermodynamic limit L → ∞ with x ± = f − 1 ( f ( x ) ± v 0 t ) ˚ and � � � β ′ ( x ) � 2 6 β ( x ) 2 + cv ( x ) 2 β ′′ ( x ) + v ′ ( x ) β ′ ( x ) πc β ( x ) − 1 F ( x ) = 12 π 2 β ( x ) v ( x ) β ( x ) 12 / 20

Thermal conductivity Dynamically: �� �� � � ∂ � κ th ( ω ) = β 2 R + d t e i ωt d x ∂ t �J ( x ; t ) � ∞ � neq ∂ ( δβ ) R δβ =0 for a kink-like initial profile β ( x ) = β + δβW ( x ) with height δβ or equivalently Green-Kubo formula: � β � � � � c, ∞ R 2 d x d x ′ ∂ x ′ [ − W ( x ′ )] R + d t e i ωt J ( x ; t ) J ( x ′ ; i τ ) κ th ( ω ) = β d τ β 0 � � with �· · · � β = �· · · � neq β ( x )= β P.M., PhD thesis (2018) 13 / 20

Thermal conductivity – Results On general grounds Re κ th ( ω ) = D th πδ ( ω ) + Re κ reg th ( ω ) Given a smooth v ( x ) , then � � ωβ � 2 � D th = πvc th ( ω ) = πc Re κ reg 1 + I ( ω ) 3 β 6 β 2 π with � � � � � � x ∂ x ′ � � v d � x R 2 d x d x ′ − W ( x ′ ) I ( ω ) = 1 − cos ω v ( x ) v ( � x ) x ′ where v is arbitrary in the thermodynamic limit 14 / 20

Full counting statistics “Full counting statistics of energy transfers in inhomogeneous nonequilibrium states of (1+1)D CFT” Gawędzki, Kozłowski, arXiv:1906.04276 (2019) 15 / 20

Alternative approach Standard Euclidean CFT in curved spacetime with the metric h = d x 2 + v ( x ) 2 d τ 2 (imaginary time τ = it ) Dubail, Stéphan, Viti, Calabrese, SciPost Phys. 2 (2017) Dubail, Stéphan, Calabrese, SciPost Phys. 3 (2017) 16 / 20

Outline ⋄ Introduction ⋄ Main tools ⋄ Applications ⋄ Random CFT

Ballistic and anomalous/normal diffusive contributions Recall: Random CFT with v ( x ) = v/ [1 − ξ ( x )] and Gaussian random function ξ ( x ) specified by E [ ξ ( x )] = 0 and Γ( x − y ) = E [ ξ ( x ) ξ ( y )] After averaging: D th = πvc 3 β � 2 �� � ωβ � � ωx � th ( ω ) = πc 2 ( ω/v ) 2 Λ( x ) cos Re κ reg d x e − 1 1 + 6 β 2 π v R th ( ω ) = πc ω → 0 Re κ reg L th = lim 6 β Γ 0 � x � x � with Λ( x ) = 0 d x 1 0 d x 2 Γ( x 1 − x 2 ) and Γ 0 = R d x Γ( x ) 17 / 20

Wave propagation in random media Solving random PDEs for the expectations E ( x ; t ) and J ( x ; t ) of E ( x ; t ) and J ( x ; t ) in an arbitrary state with E ( x ; 0) = e 0 ( x ) and J ( x ; 0) = 0 gives: � � � G E + ( x − y ; t ) + G E E [ E ( x ; t )] = d y − ( x − y ; t ) e 0 ( y ) R � � � G J + ( x − y ; t ) + G J E [ J ( x ; t )] = d y − ( x − y ; t ) e 0 ( y ) R ± ( x ; t ) and G J with G E ± ( x ; t ) expressed in terms of G ± ( x ; t ) = θ ( ± x )e − ( x ∓ vt ) 2 / 2Λ( x ) � 2 π Λ( x ) Propagation-diffusion equation � � v − 1 ∂ t ± ∂ x − γ ( x ) ∂ 2 G ± ( x ; t ) = 0 ( ± x > 0 , t > 0 ) t with temporal diffusion coeff. γ ( x ) Boon, Grosfils, Lutsko, Euro. Phys. Lett. 63 (2003) Langmann, P.M., Phys. Rev. Lett. 122 (2019) 18 / 20