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:

st−1

x−3

st−2

x−2

st−1

x−1

st−2

x

st−1

x+1

st−2

x+2

st−1 − →

st−1

x−3

st−1

x−1

st−1

x+1

st

x−2

st

x

st

x+2

st − →

st+1

x−3

st

x−2

st+1

x−1

st

x

st+1

x+1

st

x+2

st+1

Local time evolution maps: s′

2 = χ(s1, s2, s3) = s1 + s2 + s3 + s1s3

(mod 2)

s1 s2 s3 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. x t

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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 2t + 1. t The computational complexity of the procedure grows as t2.

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 C(x, t) = 2−t−1

t−|x|−2 2

• m=0

4m

• 2

t − 2m − 3 m

• −

t − 2m − 2 m

• 0.0005

0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 −200 −150 −100 −50 50 100 150 200 C(x, t) x t = 100 t = 200 t = 300 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 −200 −150 −100 −50 50 100 150 200

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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?

x t equilibrium state p b0 b1 . . . bk q

Generically the complexity grows exponentially. Our case: q is an MPS with bond dimension 3 (the same as p).

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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)

t x Solitons speed up while scattering (instead of slowing down)

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Introduction Solitons Circuit formulation Conclusion

Circuit representation of dynamics

s′

1

s′

2

s′

3

s1 s2 s3 U

U s′

1s′ 2s′ 3

s1s2s3 = δs′

1,s1δs′ 2,χ(s1,s2,s3)δs′ 3,s3

Time evolution is then: U e U o U e U o . . . t x

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Introduction Solitons Circuit formulation Conclusion

We define dual gates ˆ U as

s′

3

s′

2

s′

1

s3 s2 s1 ˆ U

ˆ U s′

1s′ 2s′ 3

s1s2s3 = δs′

1,s1δs′ 3,s3U s2s1s′ 2

s2s3s′

2

and obtain the rotated picture: ˆ U e ˆ U o ˆ U e ˆ U o· · ·

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Introduction Solitons Circuit formulation Conclusion

Problem: ˆ U non-deterministic ˆ U =             1 1 1 1 1 1 1 1             = ˆ UP = P ˆ U P projector on the allowed subspace

s′

1

s′

2

s′

3

s1 s2 s3 P

P s′

1s′ 2s′ 3

s1s2s3 = δs′

1,s1δs′ 2,s2δs′ 3,s3(1 − δs1,s3δs2,1)

“Time-configurations” (0, 1, 0) and (1, 1, 1) are forbidden!

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Introduction Solitons Circuit formulation Conclusion

Evolution in space is local and deterministic on the reduced subspace of allowed configurations.

ˆ U oˆ U eˆ U o

− →

˜ Ue

˜ U e and ˜ U o can be expressed in terms of deterministic gates with support 7.

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Introduction Solitons Circuit formulation Conclusion

Revisiting multi-time correlation functions

◮ Multi-time correlation function of 2m observables in maximum entropy state: C(2n)

a1,a2,a3,...,a2m(p∞) = 2−2n

2n ◮ 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 2−2m ≡ 2−2m ◮ This is general

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Introduction Solitons Circuit formulation Conclusion

◮ RCA54 dual operator ˆ U is not deterministic, but it has some nontrivial structure: ≡ , ≡

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Introduction Solitons Circuit formulation Conclusion

Layer after layer of dual gates can therefore be removed: Ca1,a2,a3,...,a2m(p∞) = 2−2m . . . . . . = 2−2m . . . 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.

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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)

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