Correlations and field theory inside the arctic circle [or Arctic - PowerPoint PPT Presentation

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Correlations and field theory inside the arctic circle [or Arctic quenches] ephan 1 Jean-Marie St´ 1 Max-Planck-Institut f¨ ur Physik Complexer Systeme (Dresden) Firenze 2015 in collaboration with N. Allegra, J. Dubail, M. Haque and J. Viti.

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Outline Arctic circle in Statistical mechanics 1 Dimers Six vertex A simple toy model 2 Model and Motivations Correlations in the bulk Dirac action in curved space Consequences and generalizations 3 Dimers and vertex models Back to real time

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Classical Dimers in 2d dimers with hardcore constraint. Exactly solvable: free fermions. Z = det ( . . . ) [Kasteleyn, Fisher] Critical system Long distance limit: Dirac field or free gaussian compact field S = g � dxdy ( ∇ ϕ ) 2 , ϕ = ϕ + 2 π 4 π � − 1 /g � − g C dd ( r , r ′ ) = � r − r ′ � C mm ( r , r ′ ) = � r − r ′ � � � ,

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Aztec diamond, and the arctic circle [Jokusch, Propp and Shor, 1995]

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Aztec diamond, and the arctic circle [Jokusch, Propp and Shor, 1995] Image at http://tuvalu.santafe.edu/ ∼ moore/gallery.html

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Lots of variations on this Theory for the shape [Kenyon, Okounkov, Sheffield]

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Can be induced by boundary conditions � T L y � � ψ 0 � � � Z = ψ 0

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Density profile for dimers � � ρ ( x, y ) = 1 2 + 1 2 y − 1 √ π arctan 1 − 2 x 2 − 2 y 2 [Cohn, Elkies and Propp, Duke. Math. Journ 1996]

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Six vertex with domain wall boundary conditions [Korepin, Izergin, Zinn-Justin, Colomo, Pronko,. . . ] Conjecture for the arctic curve [Colomo & Pronko, J. Stat. Phys 2010]

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Field theory inside the circle?

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations A simple toy model � c † | ψ 0 � = x | 0 � x< 0 H = − 1 � � � c † x c x +1 + h.c 2 x Imaginary time evolution � e − 2 RH � � ψ 0 � � � Z = ψ 0

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Motivations (1/3) TASEP in continuous time [Rost, Zeit. Wahrs. 1981] . Arctic circle phenomenology. 2 R Quantum mechanics from a domain wall intial state. [Antal R´ acz R´ akos Sch¨ utz 1999 ; Antal Krapivsky R´ akos 2008 ; . . . ]

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Motivation (2/3): Filling fraction quenches � � � c † j c j +1 + c † � ε k c † H = − j +1 c j = k c k j k Fermi level π k l k r Fermions, with certain Fermi levels ( k l , k r ) | ψ 0 � = | k l � ⊗ | k r � and let evolve with H ( k l + k r ) at time t > 0 2

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Motivation (2/3): Filling fraction quenches Fermi level π k l k r Limiting cases: k l = π , k r = 0 is the domain wall quench. k l = k r [Eisler, Karevski, Platini & Peschel, 2008] [Calabrese & Cardy, 2008] [JMS & Dubail, 2011]

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Motivation (3/3) low energy local quenches k l = k r � 2 � � � ψ 0 | e − τH tot | ψ 0 � � L ( τ ) = Keep in mind τ → it , but only at the end. F ( τ ) = − ln L ( τ ) is a free energy! [JMS & Dubail, 2011] L/ 2 L/ 2 τ

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Motivation (3/3) low energy local quenches k l = k r � 2 � � � ψ 0 | e − τH tot | ψ 0 � � L ( τ ) = Keep in mind τ → it , but only at the end. F ( τ ) = − ln L ( τ ) is a free energy! [JMS & Dubail, 2011] Loschmidt echo L/ 2 L/ 2 F ( τ ) = c � L � πτ �� � � 4 ln π sinh � � L � � τ

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Motivation (3/3) low energy local quenches k l = k r � 2 � � � ψ 0 | e − τH tot | ψ 0 � � L ( τ ) = Keep in mind τ → it , but only at the end. F ( τ ) = − ln L ( τ ) is a free energy! [JMS & Dubail, 2011] Loschmidt echo L/ 2 L/ 2 F ( τ ) = c � L � πτ �� � � 4 ln π sinh � � L � � Back to real time τ � � πt �� F ( t ) = c L � � 4 ln π sin � � L � �

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Symmetric case L A = L B = L/ 2 The Loschmidt echo is periodic � �� F ( t ) = c L � πt � � 4 ln π sin � � L � � 1.4 L = 128 CFT 1.2 F ( τ ) = − log L ( τ ) 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 ( v F /L ) t = τ

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Non symmetric case L A = L/ 3 Loschmidt echo a 3 ( a + 1) 6 ( a + 2)(2 a + 1) F ( a ) = c 4 ln L + c � � � � 24 ln � � ( a − 1) 7 � � a is one of the solutions of it = 2 L � 1 � b − 1 � + 2 � a − b �� 3 ln 3 ln π b + 1 a + b b 2 = a a + 2 2 a + 1

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Non symmetric case L A = L/ 3 2.1 L=6144 2 CFT F ( τ ) = − log L ( τ ) 1.9 1.8 1.7 1.6 0 2 / 3 4 / 3 2 ( v F /L ) t = τ

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Non symmetric case L A = L/ 3 2.1 L=6144 2 CFT F ( τ ) = − log L ( τ ) 1.9 1.8 1.7 1.6 0 2 / 3 4 / 3 2 ( v F /L ) t = τ 2 L A 2 L B 2 L t

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Motivation (end) Here it is not that simple, because (need both!) 1 Conservation of the number of particles 2 Inhomogeneous initial state Naive low energy-field theory does not work.

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Correlations inside the “circle” Want to compute = � ψ 0 | e − ( R + y ) H c † x e − ( y − y ′ ) H c x ′ e − ( R − y ′ ) H | ψ 0 � � � c † x ( y ) c x ′ ( y ′ ) � ψ 0 | e − 2 RH | ψ 0 � � c † ( k ) = e − ikx c † [ H, c † ( k )] = ε ( k ) c † ( k ) , x x ∈ Z

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Appearance of the Hilbert transform � e − ( R − y ) H c † ( k ) e − ( y ′ − y ) H c ( k ′ ) e − ( R + y ) H � � � � ψ 0 � ψ 0 � � � � c † ( k, y ) c ( k ′ , y ′ ) = � ψ 0 | e − 2 RH | ψ 0 � One can show that = e − iR [˜ ε ( k ) − ˜ ε ( k ′ )] e yε ( k ) − y ′ ε ( k ′ ) � � c † ( k, y ) c ( k ′ , y ′ ) � k − k ′ + i 0 + � 2 i sin 2 � π dq 2 πε ( q ) cot k − q ε ( k ) = Hilbert transform of ε ( k ) = pv ˜ . 2 − π

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Bosonisation trick c † ( k ) → : e iϕ ( k ) : c ( k ) → : e − iϕ ( k ) : c † ( k ) c ( k ) → ∂ϕ ( k ) Fusion of two vertex operators: : e αϕ ( k ) : : e βϕ ( k ′ ) := e αβ � ϕ ( k ) ϕ ( k ′ ) � : e αϕ ( k )+ βϕ ( k ′ ) : Need to define Normal order : c † x c x : � ϕ ( k ) ϕ ( k ′ ) � = log sin k − k ′ 2

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Comments The result is exact. This completely solves the problem in principle. Real space correlations: inverse Fourier transform+stationary phase approximation. Stationary points x + iydε ( k ) + Rd ˜ ε ( k ) = 0 k dk

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Density profile x ( y ) c x ( y ) � = 1 x � c † π arccos R 2 − y 2 � Z = e − R 2 / 2

Arctic circle in Statistical mechanics A simple toy model Consequences and generalizations Dirac in curved space ψ † ( x, y ) + e i ( π/ 4+ θ ∗ ( x,y )) x = e − i ( π/ 4+ θ ( x,y )) † ( x, y ) c † √ √ ψ 2 π 2 π where e − 1 2 [ σ ( x,y )+ σ ( x ′ ,y ′ )] � � ψ † ( x, y ) ψ ( x ′ , y ′ ) = � � z ( x,y ) − z ( x ′ ,y ′ ) sin 2 R 2 − y 2 + i arcth y x z ( x, y ) = arcsin � R R 2 − x 2 − y 2 � θ ( x, y ) = − z ( x, y ) x − � R 2 − x 2 − y 2 σ ( x, y ) = log Dirac theory with a metric ds 2 = e 2 σ ( dx 2 + dy 2 )