Diagrammatic Monte Carlo approach to angular momentum in quantum many-body systems Giacomo Bighin Institute of Science and Technology Austria Workshop on “Polarons in the 21st century” , ESI, Vienna, December 10th, 2019
Rotations in a many-body environment Rotations in a many-body environment and rotating impurities: Molecular physics/chemistry : molecules embedded into helium nanodroplets. J. P. Toennies and A. F. Vilesov, Angew. Chem. Int. Ed. 43 , 2622 (2004). Condensed matter : rotating molecules inside a ‘cage’ in perovskites. C. Eames et al, Nat. Comm. 6 , 7497 (2015). Ultracold matter : molecules and ions in a BEC. B. Midya, M. Tomza, R. Schmidt, and M. Lemeshko, Phys. Rev. A 94 , 041601(R) (2016). 2/13
Rotations in a many-body environment Rotations in a many-body environment and rotating impurities: Molecular physics/chemistry : molecules embedded into helium nanodroplets. J. P. Toennies and A. F. Vilesov, Angew. Chem. Int. Ed. 43 , 2622 (2004). Condensed matter : rotating molecules inside a ‘cage’ in perovskites. C. Eames et al, Nat. Comm. 6 , 7497 (2015). Ultracold matter : molecules and ions in a BEC. B. Midya, M. Tomza, R. Schmidt, and M. Lemeshko, Phys. Rev. A 94 , 041601(R) (2016). 2/13
Rotations in a many-body environment Rotations in a many-body environment and rotating impurities: • How to sample these diagrams at all diagrams? • How to describe rotations in a many-body Questions: 2/13 B. Midya, M. Tomza, R. Schmidt, and M. Lemeshko, Phys. Rev. A 94 , 041601(R) (2016). and ions in a BEC. Ultracold matter : molecules C. Eames et al, Nat. Comm. 6 , 7497 (2015). perovskites. molecules inside a ‘cage’ in Condensed matter : rotating J. P. Toennies and A. F. Vilesov, Angew. Chem. Int. Ed. 43 , 2622 (2004). helium nanodroplets. molecules embedded into Molecular physics/chemistry : environment in terms of Feynman orders using Diagrammatic Monte Carlo?
Feynman diagrams molecule Feynman diagrams and perturbation theory: molecule-phonon interaction The angulon Hamiltonian: phonons 3/13 J 2 � � � � ω k ˆ k λµ ˆ λµ (ˆ θ, ˆ ϕ )ˆ k λµ + Y λµ (ˆ θ, ˆ ϕ )ˆ ˆ B ˆ b † Y ∗ b † H = + + U λ ( k ) b k λµ b k λµ ���� k λµ k λµ � �� � � �� �
Feynman diagrams phonons How does angular momentum enter this picture? Feynman diagrams and perturbation theory: molecule-phonon interaction The angulon Hamiltonian: 3/13 molecule J 2 � � � � ω k ˆ k λµ ˆ λµ (ˆ θ, ˆ ϕ )ˆ k λµ + Y λµ (ˆ θ, ˆ ϕ )ˆ ˆ B ˆ b † Y ∗ b † H = + + U λ ( k ) b k λµ b k λµ ���� k λµ k λµ � �� � � �� � = + + + + . . .
Feynman diagrams phonons Fröhlich polaron Feynman diagrams and perturbation theory: molecule-phonon interaction The angulon Hamiltonian: 3/13 molecule J 2 � � � � ω k ˆ k λµ ˆ λµ (ˆ θ, ˆ ϕ )ˆ k λµ + Y λµ (ˆ θ, ˆ ϕ )ˆ ˆ B ˆ b † Y ∗ b † H = + + U λ ( k ) b k λµ b k λµ ���� k λµ k λµ � �� � � �� �
Feynman diagrams phonons Angulon Feynman diagrams and perturbation theory: molecule-phonon interaction The angulon Hamiltonian: 3/13 molecule J 2 � � � � ω k ˆ k λµ ˆ λµ (ˆ θ, ˆ ϕ )ˆ k λµ + Y λµ (ˆ θ, ˆ ϕ )ˆ ˆ B ˆ b † Y ∗ b † H = + + U λ ( k ) b k λµ b k λµ ���� k λµ k λµ � �� � � �� �
Feynman diagrams phonons here? momentum enter How does angular Angulon Feynman diagrams and perturbation theory: molecule-phonon interaction The angulon Hamiltonian: 3/13 molecule J 2 � � � � ˆ B ˆ ω k ˆ b † k λµ ˆ λµ (ˆ θ, ˆ ϕ )ˆ b † k λµ + Y λµ (ˆ θ, ˆ ϕ )ˆ Y ∗ H = + + U λ ( k ) b k λµ b k λµ ���� k λµ k λµ � �� � � �� �
Feynman rules Each vertex Not the same for angular enforced by an appropriate labeling. Usually momentum conservation is GB and M. Lemeshko, Phys. Rev. B 96 , 419 (2017). Each free propagator 4/13 Each phonon propagator � λ i µ i ( − 1 ) µ i G 0 ,λ i λ i µ i � λ i µ i ( − 1 ) µ i D λ i λ i µ i � � λ i λ j λ k ( − 1 ) λ i ⟨ λ i | | Y ( λ j ) | | λ k ⟩ µ i µ j µ k momentum, j and λ couple to | j − λ | , . . . , j + λ . � j ′ m ′
Feynman rules Each vertex from D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, “Quantum Theory of Angular Momentum” . theoretical atomic spectroscopy) Diagrammatic theory of angular momentum (developed in the context of GB and M. Lemeshko, Phys. Rev. B 96 , 419 (2017). Each free propagator 4/13 Each phonon propagator � λ i µ i ( − 1 ) µ i G 0 ,λ i λ i µ i � λ i µ i ( − 1 ) µ i D λ i λ i µ i � � λ i λ j λ k ( − 1 ) λ i ⟨ λ i | | Y ( λ j ) | | λ k ⟩ µ i µ j µ k
Angulon spectral function: first and second order Self-energy (first order) Dyson equation Self-energy (second order) GB and M. Lemeshko, Phys. Rev. B 96 , 419 (2017). 5/13 = = +
What about higher orders? At order n : n integrals, and higher angular momentum couplings (3 n -j symbols). 6/13 = + + + + . . . + + + . . . + + . . .
Diagrammatic Monte Carlo Numerical technique for summing all Feynman diagrams 1 . Usually: structureless particles (Fröhlich polaron, Holstein polaron), or particles Molecules 2 ? Connecting DiagMC and molecular simulations! 1 N. V. Prokof’ev and B. V. Svistunov, Phys. Rev. Lett. 81 , 2514 (1998). 2 GB, T.V. Tscherbul, M. Lemeshko, Phys. Rev. Lett. 121 , 165301 (2018). 7/13 = + + + + … + + + … with a very simple internal structure (e.g. spin 1 / 2 ).
Diagrammatic Monte Carlo DiagMC idea: set up a stochastic process sampling among all diagrams 1 . 1 N. V. Prokof’ev and B. V. Svistunov, Phys. Rev. Lett. 81 , 2514 (1998). etc... Number of variables varies with the topology! Configuration space: diagram topology, phonons internal variables, times, 8/13 H int Green’s function Hamiltonian for an impurity problem: ˆ H = ˆ H imp + ˆ H bath + ˆ G ( τ ) = + + + + . . . = all Feynman diagrams How: ergodicity, detailed balance w 1 p ( 1 → 2 ) = w 2 p ( 2 → 1 ) Result: each configuration is visited with probability ∝ its weight.
Diagrammatic Monte Carlo Configuration space: diagram topology, phonons internal variables, times, namic limit: no finite-size efgects or systematic Works in continuous time and in the thermody- quantization. A Monte Carlo technique that works in second 8/13 1 N. V. Prokof’ev and B. V. Svistunov, Phys. Rev. Lett. 81 , 2514 (1998). etc... Number of variables varies with the topology! DiagMC idea: set up a stochastic process sampling among all diagrams 1 . Green’s function H int errors. Hamiltonian for an impurity problem: ˆ H = ˆ H imp + ˆ H bath + ˆ G ( τ ) = + + + + . . . = all Feynman diagrams How: ergodicity, detailed balance w 1 p ( 1 → 2 ) = w 2 p ( 2 → 1 ) Result: each configuration is visited with probability ∝ its weight.
Result: the time the stochastic process spends with diagrams of length . One can fill a histogram afuer each update and get the Updates We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. will be proportional to G Green’s function. 9/13
Result: the time the stochastic process spends with diagrams of length . One can fill a histogram afuer each update and get the Updates We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. will be proportional to G Green’s function. 9/13
Result: the time the stochastic process spends with diagrams of length . One can fill a histogram afuer each update and get the Updates We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. will be proportional to G Green’s function. 9/13
Result: the time the stochastic process spends with diagrams of length . One can fill a histogram afuer each update and get the Updates We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. will be proportional to G Green’s function. 9/13
Result: the time the stochastic process spends with diagrams of length . One can fill a histogram afuer each update and get the Updates We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. will be proportional to G Green’s function. 9/13
Result: the time the stochastic process spends with diagrams of length . One can fill a histogram afuer each update and get the Updates We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. will be proportional to G Green’s function. 9/13
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