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Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories Entanglement in Strongly Correlated Systems, Benasque 21st of February 2020 Patrick Emonts , E. Zohar, M. C. Banuls, I. Cirac MPI of Quantum Optics

Why do we need Lattice Gauge Theories? QED QCD L QED = iΨγ μ 𝜖 μ Ψ − eΨγ μ A μ Ψ − mΨΨ − 1 4 F μν F μν e − e − γ e + e + Small coupling α QED = e 2 1 4 π ≈ 137 Image adapted from Alexandre Deur, Stanley J. Brodsky, and Guy F. de Téramond, 2016, Progress in Particle and Nuclear Physics Slide 2 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

Path integral formalism in QFT pure QED S QED [ A μ ] = − 1 4 ∫ d x α F μν ( x α ) F μν ( x α ) = ∫ d x α 𝜖 μ A ν ( x α )𝜖 ν A μ ( x α ) vacuum expectation value Problems iSQED [ Aμ ]  Numerator oscillating ∫ D AO [ A μ ] e ⟨ Ω | O [ A μ ]| Ω ⟩ = iSQED [ Aμ ]  Integration measure ill-defined ∫ D Ae Slide 3 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

Wick rotation Shift to imaginary time t → − iτ Change of metric from Minkowski to Euclidean e iS M = e i ∫ d x α M L ( x α M ) ⟶ e − ∫ d x α E L ( x α E ) = e − S E Problems  Numerator converging  Integration measure ill-defined Slide 4 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

̃  ̃ Discretization: Lattice Gauge Theory x α a A μ → U μ = e iaA μ S E that agrees with S E in the continuum limit of vanishing a Find the lattice action S E [ U ] → S E [ A ]( a → 0 ) Kenneth G. Wilson, 1974, Physical Review D Slide 5 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

Vacuum expectation value in the action formalism Vacuum expectation value ∫ D UO [ U ] e − SE [ U ] ⟨ O [ U ]⟩ = with D U = ∏ x α d U μ ( x α ) ∫ D Ue − SE [ U ] Problems  Numerator converging  Integration with the Haar measure Slide 6 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

Lattice Systems Hilbert space H ⊂ H gauge fields ⊗ H fermions A general state | Ψ ⟩ = ∫ DG | G ⟩ ∣ Ψ F ( G )⟩ with DG = ∏ x , k dg ( x , k ) Erez Zohar and J. Ignacio Cirac, 2018, Physical Review D Patrick Emonts and Erez Zohar, 2020, SciPost Physics Lecture Notes Slide 7 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

Gauss law Gauss law ∑ k ( E k ( x ) − E k ( x − e i )) | phys ⟩ = 0 ∀ x E 2 ( x ) E 1 ( x − e 1 ) E 1 ( x ) Classical analogue in (cont.) E 2 ( x − e 2 ) electrodynamics ∇ ⋅ E = 0 Slide 8 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

Expectation value of an Observable Assume that O acts only on the gauge field and is diagonal in the group element basis: ⟨ O ⟩ = ⟨ Ψ | O | Ψ ⟩ ⟨ Ψ | Ψ ⟩ = ∫ DG ⟨ G | O | G ⟩ ⟨ Ψ F ( G )∣ Ψ F ( G )⟩ ∫ DG ′ ⟨ Ψ F ( G ′ )∣ Ψ F ( G ′ )⟩ = ∫ DGF O ( G ) p ( G ) ⟨ Ψ F ( G )∣ Ψ F ( G )⟩ ⟨ Ψ F ( G )∣ Ψ F ( G )⟩ with p ( G ) = Z ∫ DG ′ ⟨ Ψ F ( G ′ )∣ Ψ F ( G ′ )⟩ = Slide 9 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

The rest of this talk Expectation value ⟨ O ⟩ = ∫ DGF O ( G ) p ( G ) ⟨ Ψ F ( G )∣ Ψ F ( G )⟩ with p ( G ) = Z TODO List 1 How do we construct ∣ Ψ F ( G )⟩ ? 2 How do we efficiently calculate p ( G ) ? 3 Are those states useful? Slide 10 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

Creation of the fermionic state Desirable properties | Ψ ⟩ fulfills the Gauss law Definition of Ψ ∣ Ψ F ( G )⟩ allows efficient calculations of | Ψ ⟩ = ∫ DG | G ⟩ ∣ Ψ F ( G )⟩ the norm expectation values Choice for ∣ Ψ F ( G )⟩ We construct ∣ Ψ F ( G )⟩ with a tensor network. Patrick Emonts and Erez Zohar, 2020, SciPost Physics Lecture Notes Slide 11 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

Definition of Modes Gauss law in terms of our modes G 0 = E r − E l + E u − E d = r † + r + − r † − r − − l † + l + + l † − l − + u † + u + − u † − u − − d † + d + + d † − d − u − u + Definition of positive and negative l + r + modes Ψ r − l − a : { l + , r − , u − , d + } (neg. modes) b : { l − , r + , u + , d − } (pos. modes) d − d + Erez Zohar et al., 2015, Annals of Physics Slide 12 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

ω ( x , k ) ∏ A ( x ) x x , k T ij a † i ( x ) b † A ( x ) = exp ⎛ j ( x )⎞ ij ω ( x , k ) = ω k ( x ) Ω k ( x ) ω † k ( x ) ω 0 ( x ) = exp ( l † + ( x + e 1 ) r † − ( x )) exp ( l † − ( x + e 1 ) r † + ( x )) ∑ ⎜ ⎟ ⎠ ∏ ⎝ Creating a fermionic state The state ∣ ψ 0 ⟩ = ⟨ Ω V ∣ | Ω ⟩ Slide 13 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

ω ( x , k ) x , k ω ( x , k ) = ω k ( x ) Ω k ( x ) ω † k ( x ) ω 0 ( x ) = exp ( l † + ( x + e 1 ) r † − ( x )) exp ( l † − ( x + e 1 ) r † + ( x )) ⎟ ⎠ ⎝ ∑ ∏ ∏ ⎜ Creating a fermionic state The state ∣ ψ 0 ⟩ = ⟨ Ω V ∣ A ( x ) | Ω ⟩ x T ij a † i ( x ) b † A ( x ) = exp ⎛ j ( x )⎞ x 01 x 11 x 21 ij x 00 x 10 x 20 Slide 13 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

⎟ ⎠ ∑ ⎝ ⎜ Creating a fermionic state The state ∣ ψ 0 ⟩ = ⟨ Ω V ∣ ∏ ω ( x , k ) ∏ A ( x ) | Ω ⟩ x x , k ω 01 , h ω 11 , h T ij a † i ( x ) b † A ( x ) = exp ⎛ j ( x )⎞ x 01 x 11 x 21 ij ω 00 , v ω 10 , v ω 20 , v ω ( x , k ) = ω k ( x ) Ω k ( x ) ω † k ( x ) ω 00 , h ω 10 , h ω 0 ( x ) = exp ( l † + ( x + e 1 ) r † − ( x )) x 00 x 10 x 20 exp ( l † − ( x + e 1 ) r † + ( x )) Slide 13 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

Moving towards local symmetry Lattice Gauge theory We demand a local symmetry ∑ x G ( x ) | Ψ ⟩ = 0 → G ( x ) | Ψ ⟩ = 0 x 01 x 11 x 21 x 00 x 10 x 20 Erez Zohar et al., 2015, Annals of Physics Erez Zohar and Michele Burrello, 2016, New Journal of Physics Slide 14 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

Local symmetry – The state Substitution ± ( x ) → e ± iθ ( x ) r † r † ± ( x ) ± ( x ) → e ± iθ ( x ) u † u † ± ( x ) x 01 x 11 x 21 x 00 x 10 x 20 Slide 15 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

Fermionic state Fermionic state ∣ ψ ( G )⟩ = ⟨ Ω v ∣ ∏ ω ( x ) ∏ A ( x ) | Ω ⟩ U Φ ( x ) ∏ x x x  Gauge invariance of | Ψ ⟩ by constructing Ψ ( G )  Obeys all demanded symmetries ? Efficient to calculate with Slide 16 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

⎟ ⎜ ⎠ ∑ ⎝ Is ∣ Ψ F ( G )⟩ special? The fermionic state ∣ Ψ F ( G )⟩ ∣ Ψ F ( G )⟩ = ⟨ Ω v ∣ ∏ ω ( x ) ∏ A ( x ) | Ω ⟩ U Φ ( x ) ∏ x x x T ij a † i ( x ) b † A ( x ) = exp ⎛ j ( x )⎞ ij ω ( x ) = ω 0 ( x ) ω 1 ( x ) Ω ( x ) ω † 1 ( x ) ω † 0 ( x ) ω 0 ( x ) = exp ( l † + ( x + e 1 ) r † − ( x )) exp ( l † − ( x + e 1 ) r † + ( x )) ω 1 ( x ) = exp ( d † + ( x + e 2 ) u † − ( x )) exp ( d † − ( x + e 2 ) u † + ( x )) Slide 17 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

Gaussian States Definition Fermionic Gaussian states are represented by density operators that are exponentials of a quadratic form in Majorana operators. ρ = K exp (− i 4 γ T Gγ ) Covariance matrix Covariance matrix for a state Φ : ⟨ Φ |[ γ a , γ b ]| Φ ⟩ Γ ab = i 2 ⟨[ γ a , γ b ]⟩ = i 2 ⟨ Φ | Φ ⟩ Sergey Bravyi, 2005, Quantum Inf. and Comp. Slide 18 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

⏟ ∏ ) ⏟ ⏟ ⏟ ⏟⏟ ⏟ ⏟⏟⏟ ⏟ ⏟⏟ ⏟ Calculating the Norm and the Observables ∣ ψ ( G )⟩ = ⟨ Ω v ∣ ∏ ω ( x ) ∏ A ( x ) | Ω ⟩ U Φ ( x ) x x x ❀ Γ in ( G ) ❀ Γ M A Physical-Physical correlations i , j = ( A B Γ M − B T B Physical-Virtual correlations D ) C Virtual-Virtual correlations Norm ⟨ ψ ( G )∣ ψ ( G )⟩ = √ det ( 1 − Γ in ( G ) M D 2 Slide 19 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC

The whole framework Build the state ∣ Ψ ( G ′ )⟩ Draw new gauge field configuration ∣ G ′ ⟩ Calculate the acceptance probability Accept or decline [Measure observables] by computing ⟨ Ψ ( G ′ )∣ Ψ ( G ′ )⟩ the new configuration G ′ Slide 20 Combining Tensor Networks and Monte Carlo for Lattice Gauge Theories | 21st of February 2020 | PE, EZ, MCB, IC