Rule 54 reversible cellular automaton An exactly solvable - PowerPoint PPT Presentation

Rule 54 reversible cellular automaton An exactly solvable microscopic model of interacting dynamics Katja Klobas University of Ljubljana 12th June, 2020 KK, M. Medenjak, T. Prosen, M. Vanicat, Commun. Math. Phys. 371 , 651–688 (2019) KK, M. Vanicat, J. P. Garrahan, T. Prosen, arXiv:1912.09742 (2019) KK, T. Prosen, arXiv:2004.01671 (2020)

Introduction Solitons Circuit formulation Conclusion Motivation ◮ A model in which “everything” can be done exactly and very explicitly. ◮ A classical cellular automaton, introduced in 1993 A. Bobenko, M. Bordemann, C. Gunn, U. Pinkall, Commun. Math. Phys. 158 , 127–134 (1993) ◮ Became popular in recent years: ◮ Classical: T. Prosen, C. Mejía-Monasterio, J. Phys. A: Math. Theor. 49 , 185003 (2016) T. Prosen, B. Buča, J. Phys. A: Math. Theor. 50 , 395002 (2017) A. Inoue, S. Takesue, J. Phys. A: Math. Theor. 51 , 425001 (2018) B. Buča, J. P. Garrahan, T. Prosen, M. Vanicat, Phys. Rev. E 100 , 020103 (2019) KK, M. Medenjak, T. Prosen, M. Vanicat, Commun. Math. Phys. 371 , 651–688 (2019) KK, M. Vanicat, J. P. Garrahan, T. Prosen, arXiv:1912.09742 (2019) KK, T. Prosen, arXiv:2004.01671 (2020) ◮ Quantum: S. Gopalakrishnan, Phys. Rev. B 98 , 060302 (2018) S. Gopalakrishnan, D. A. Huse, V. Khemani, R. Vasseur, Phys. Rev. B 98 , 220303 (2018) A. J. Friedman, S. Gopalakrishnan, R. Vasseur, Phys. Rev. Lett. 123 , 170603 (2019) V. Alba, J. Dubail, M. Medenjak, Phys. Rev. Lett. 122 , 250603 (2019) V. Alba, arXiv:2006.02788 (2020) Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 1 / 16

Introduction Solitons Circuit formulation Conclusion Definition of dynamics 1 -dim lattice of binary variables, with staggered time evolution: s t +1 s t +1 s t +1 x − 3 x − 1 x +1 s t s t s t s t s t s t x − 2 x +2 x − 2 x +2 x x s t − 1 s t − 1 s t − 1 s t − 1 s t − 1 s t − 1 − → − → x − 3 x − 1 x +1 x − 3 x − 1 x +1 s t − 2 s t − 2 s t − 2 x − 2 x x +2 s t − 1 s t s t +1 Local time evolution maps: s ′ 2 = χ ( s 1 , s 2 , s 3 ) = s 1 + s 2 + s 3 + s 1 s 3 (mod 2) s ′ 2 s 1 s 3 s 2 Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 2 / 16

Introduction Solitons Circuit formulation Conclusion Solitons move with fixed velocities ± 1 and obtain a delay while scattering. t x Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 3 / 16

Introduction Solitons Circuit formulation Conclusion MPS for time evolution of local observables KK, M. Medenjak, T. Prosen, M. Vanicat, Commun. Math. Phys. 371 , 651–688 (2019) Time evolution of local observables is mapped onto the problem of counting solitons in a section of the lattice of length 2 t + 1 . t 0 The computational complexity of the procedure grows as t 2 . Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 4 / 16

Introduction Solitons Circuit formulation Conclusion Example: Spatio-temporal density-density correlation function t −| x |− 2 2 � � t − 2 m − 3 � � t − 2 m − 2 �� C ( x, t ) = 2 − t − 1 � 4 m 2 − m m m =0 0 . 004 0 . 004 t = 100 0 . 0035 0 . 0035 t = 200 t = 300 0 . 003 0 . 003 0 . 0025 0 . 0025 C ( x, t ) 0 . 002 0 . 002 0 . 0015 0 . 0015 0 . 001 0 . 001 0 . 0005 0 . 0005 0 0 − 200 − 200 − 150 − 150 − 100 − 100 − 50 − 50 0 0 50 50 100 100 150 150 200 200 x Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 5 / 16

Introduction Solitons Circuit formulation Conclusion Multi-time correlation functions at the same position KK, M. Vanicat, J. P. Garrahan, T. Prosen, arXiv:1912.09742 (2019) Can we efficiently encode the probabilities of finding a given configuration in time? t b k . . . q b 1 b 0 x equilibrium state p Generically the complexity grows exponentially. Our case: q is an MPS with bond dimension 3 (the same as p ). Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 6 / 16

Introduction Solitons Circuit formulation Conclusion Time-space duality KK, T. Prosen, arXiv:2004.01671 (2020) What happens when the roles of space and time are reversed? Motivation: dual unitary circuits B. Bertini, P. Kos, T. Prosen, Phys. Rev. Lett. 123 , 210601 (2019) Solitons speed up while scattering t (instead of slowing down) x Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 7 / 16

Introduction Solitons Circuit formulation Conclusion Circuit representation of dynamics s ′ s ′ s ′ 1 2 3 U s ′ 1 s ′ 2 s ′ s 1 s 2 s 3 = δ s ′ 3 1 ,s 1 δ s ′ 2 ,χ ( s 1 ,s 2 ,s 3 ) δ s ′ U 3 ,s 3 s 1 s 2 s 3 Time evolution is then: . . . U o U e t U o U e x Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 8 / 16

Introduction Solitons Circuit formulation Conclusion We define dual gates ˆ U as s ′ s 3 3 U s ′ 1 s ′ 2 s ′ 3 ,s 3 U s 2 s 1 s ′ ˆ s ′ s 1 s 2 s 3 = δ s ′ 3 1 ,s 1 δ s ′ 2 s 2 ˆ U s 2 s 3 s ′ 2 2 s ′ s 1 1 and obtain the rotated picture: U e ˆ U o ˆ U e ˆ ˆ U o · · · Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 9 / 16

Introduction Solitons Circuit formulation Conclusion Problem: ˆ U non-deterministic 1   0 1     0     1 1 ˆ = ˆ UP = P ˆ   U = U   0 1     1     1 1   0 P projector on the allowed subspace s ′ s 1 P s ′ 1 s ′ 2 s ′ 1 s ′ s 2 s 1 s 2 s 3 = δ s ′ 3 1 ,s 1 δ s ′ 2 ,s 2 δ s ′ 3 ,s 3 (1 − δ s 1 ,s 3 δ s 2 , 1 ) 2 P s ′ s 3 3 “Time-configurations” (0 , 1 , 0) and (1 , 1 , 1) are forbidden! Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 10 / 16

Introduction Solitons Circuit formulation Conclusion Evolution in space is local and deterministic on the reduced subspace of allowed configurations. ˜ U e U o ˆ ˆ U e ˆ U o − → U e and ˜ U o can be expressed in terms of deterministic gates with support 7 . ˜ Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 11 / 16

Introduction Solitons Circuit formulation Conclusion Revisiting multi-time correlation functions ◮ Multi-time correlation function of 2 m observables in maximum entropy state: 2 n C (2 n ) a 1 ,a 2 ,a 3 ,...,a 2 m ( p ∞ ) = 2 − 2 n ◮ Gray dots: one-site vectors ω = � � 1 1 Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 12 / 16

Introduction Solitons Circuit formulation Conclusion ◮ U is deterministic, ≡ , ≡ ◮ Light-cone structure 2 − 2 m ≡ 2 − 2 m ◮ This is general Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 13 / 16

Introduction Solitons Circuit formulation Conclusion ◮ RCA54 dual operator ˆ U is not deterministic, but it has some nontrivial structure: , ≡ ≡ Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 14 / 16

Introduction Solitons Circuit formulation Conclusion Layer after layer of dual gates can therefore be removed: C a 1 ,a 2 ,a 3 ,...,a 2 m ( p ∞ ) = 2 − 2 m = 2 − 2 m . . . . . . . . . This is equivalent to the MPS for correlation functions corresponding to the maximum entropy state. It can be generalized to a class of equilibrum states. Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 15 / 16

Introduction Solitons Circuit formulation Conclusion Summary and outlook ◮ A more algebraic formulation that does not explicitly depend on the quasiparticle interpretation of dynamics ◮ Open questions: ◮ How far can we push this approach? ◮ Can this be done for a class of models? Other cellular automata? J. W. P. Wilkinson, KK, T. Prosen, J. P. Garrahan, arXiv:2006.06556 (2020) Stochastic generalizations? Quantum generalizations? A. J. Friedman, S. Gopalakrishnan, R. Vasseur, Phys. Rev. Lett. 123 , 170603 (2019) Katja Klobas (UL) RCA54: a solvable interacting model ICTP, June 2020 16 / 16