The Rule 54: Completely Solvable Statistical Mechanics Model ofDeterministic Interacting DynamicsTomaž Prosen
Faculty of mathematics and physics, University of Ljubljana, Slovenia
WIS 2019, Osaka City University, 10 September 2019Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Questions and Motivation

Can matrix product ansatz be useful for encoding (time-dependent, or
steady) states of deterministic reversible interacting systems?
Find minimal interacting deterministic (1 + 1)d model about which we can‘know everything’ (without approximations and assumptions)
Check if the model has generic physical (say transport) properties!

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Questions and Motivation

Can matrix product ansatz be useful for encoding (time-dependent, or
steady) states of deterministic reversible interacting systems?
Find minimal interacting deterministic (1 + 1)d model about which we can‘know everything’ (without approximations and assumptions)
Check if the model has generic physical (say transport) properties!

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Questions and Motivation

Can matrix product ansatz be useful for encoding (time-dependent, or
steady) states of deterministic reversible interacting systems?
Find minimal interacting deterministic (1 + 1)d model about which we can‘know everything’ (without approximations and assumptions)
Check if the model has generic physical (say transport) properties!

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Questions and Motivation

Can matrix product ansatz be useful for encoding (time-dependent, or
steady) states of deterministic reversible interacting systems?
Find minimal interacting deterministic (1 + 1)d model about which we can‘know everything’ (without approximations and assumptions)
Check if the model has generic physical (say transport) properties!

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Outline

1

The model: Integrable reversible interacting cellular automaton (Rule 54)
A.Bobenko, M.Bordemann, C.Gunn, U.Pinkall, CMP 158,127(1993)

2

Rule 54 chain between stochastic soliton baths – steady state problem
TP and C. Mejia-Monasterio, J. Phys. A 49, 185003 (2016)see also: A. Inoue, S. Takesue, arXiv:1806.07099

3

Matrix product form of eigenvectors and diagionalization of Liouvillian
TP and B. Buča, J. Phys. A 50, 395002 (2017)4

Explicit matrix product form of time-dependent observables andanalytical evaluation of dynamic structure factor,
as well as the solution of inhomogeneous quench problem
K.Klobas, M.Medenjak, TP, M.Vanicat, Commun.Math.Phys.(2019)5

Exact large deviations for space-time extensive obervables in terms of an
inhomogeneous martrix product ansatz
B. Buča, J. P. Garrahan, TP, M. Vanicat, arXiv:1901.00845

6

Extra: New(?) integrable SO(3) symmetric classical-spin dynamics on
space-time discrete lattice from baxterized set-theretic solution of YB andRLL relations. Ž. Krajnik, TP, arXiv:1909.03799

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Outline

1

The model: Integrable reversible interacting cellular automaton (Rule 54)
A.Bobenko, M.Bordemann, C.Gunn, U.Pinkall, CMP 158,127(1993)

2

Rule 54 chain between stochastic soliton baths – steady state problem
TP and C. Mejia-Monasterio, J. Phys. A 49, 185003 (2016)see also: A. Inoue, S. Takesue, arXiv:1806.07099

3

Matrix product form of eigenvectors and diagionalization of Liouvillian
TP and B. Buča, J. Phys. A 50, 395002 (2017)4

Explicit matrix product form of time-dependent observables andanalytical evaluation of dynamic structure factor,
as well as the solution of inhomogeneous quench problem
K.Klobas, M.Medenjak, TP, M.Vanicat, Commun.Math.Phys.(2019)5

Exact large deviations for space-time extensive obervables in terms of an
inhomogeneous martrix product ansatz
B. Buča, J. P. Garrahan, TP, M. Vanicat, arXiv:1901.00845

6

Extra: New(?) integrable SO(3) symmetric classical-spin dynamics on
space-time discrete lattice from baxterized set-theretic solution of YB andRLL relations. Ž. Krajnik, TP, arXiv:1909.03799

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Outline

1

The model: Integrable reversible interacting cellular automaton (Rule 54)
A.Bobenko, M.Bordemann, C.Gunn, U.Pinkall, CMP 158,127(1993)

2

Rule 54 chain between stochastic soliton baths – steady state problem
TP and C. Mejia-Monasterio, J. Phys. A 49, 185003 (2016)see also: A. Inoue, S. Takesue, arXiv:1806.07099

3

Matrix product form of eigenvectors and diagionalization of Liouvillian
TP and B. Buča, J. Phys. A 50, 395002 (2017)4

Explicit matrix product form of time-dependent observables andanalytical evaluation of dynamic structure factor,
as well as the solution of inhomogeneous quench problem
K.Klobas, M.Medenjak, TP, M.Vanicat, Commun.Math.Phys.(2019)5

Exact large deviations for space-time extensive obervables in terms of an
inhomogeneous martrix product ansatz
B. Buča, J. P. Garrahan, TP, M. Vanicat, arXiv:1901.00845

6

Extra: New(?) integrable SO(3) symmetric classical-spin dynamics on
space-time discrete lattice from baxterized set-theretic solution of YB andRLL relations. Ž. Krajnik, TP, arXiv:1909.03799

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Outline

1

The model: Integrable reversible interacting cellular automaton (Rule 54)
A.Bobenko, M.Bordemann, C.Gunn, U.Pinkall, CMP 158,127(1993)

2

Rule 54 chain between stochastic soliton baths – steady state problem
TP and C. Mejia-Monasterio, J. Phys. A 49, 185003 (2016)see also: A. Inoue, S. Takesue, arXiv:1806.07099

3

Matrix product form of eigenvectors and diagionalization of Liouvillian
TP and B. Buča, J. Phys. A 50, 395002 (2017)4

Explicit matrix product form of time-dependent observables andanalytical evaluation of dynamic structure factor,
as well as the solution of inhomogeneous quench problem
K.Klobas, M.Medenjak, TP, M.Vanicat, Commun.Math.Phys.(2019)5

Exact large deviations for space-time extensive obervables in terms of an
inhomogeneous martrix product ansatz
B. Buča, J. P. Garrahan, TP, M. Vanicat, arXiv:1901.00845

6

Extra: New(?) integrable SO(3) symmetric classical-spin dynamics on
space-time discrete lattice from baxterized set-theretic solution of YB andRLL relations. Ž. Krajnik, TP, arXiv:1909.03799

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Outline

1

The model: Integrable reversible interacting cellular automaton (Rule 54)
A.Bobenko, M.Bordemann, C.Gunn, U.Pinkall, CMP 158,127(1993)

2

Rule 54 chain between stochastic soliton baths – steady state problem
TP and C. Mejia-Monasterio, J. Phys. A 49, 185003 (2016)see also: A. Inoue, S. Takesue, arXiv:1806.07099

3

Matrix product form of eigenvectors and diagionalization of Liouvillian
TP and B. Buča, J. Phys. A 50, 395002 (2017)4

Explicit matrix product form of time-dependent observables andanalytical evaluation of dynamic structure factor,
as well as the solution of inhomogeneous quench problem
K.Klobas, M.Medenjak, TP, M.Vanicat, Commun.Math.Phys.(2019)5

Exact large deviations for space-time extensive obervables in terms of an
inhomogeneous martrix product ansatz
B. Buča, J. P. Garrahan, TP, M. Vanicat, arXiv:1901.00845

6

Extra: New(?) integrable SO(3) symmetric classical-spin dynamics on
space-time discrete lattice from baxterized set-theretic solution of YB andRLL relations. Ž. Krajnik, TP, arXiv:1909.03799

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Outline

1

The model: Integrable reversible interacting cellular automaton (Rule 54)
A.Bobenko, M.Bordemann, C.Gunn, U.Pinkall, CMP 158,127(1993)

2

Rule 54 chain between stochastic soliton baths – steady state problem
TP and C. Mejia-Monasterio, J. Phys. A 49, 185003 (2016)see also: A. Inoue, S. Takesue, arXiv:1806.07099

3

Matrix product form of eigenvectors and diagionalization of Liouvillian
TP and B. Buča, J. Phys. A 50, 395002 (2017)4

Explicit matrix product form of time-dependent observables andanalytical evaluation of dynamic structure factor,
as well as the solution of inhomogeneous quench problem
K.Klobas, M.Medenjak, TP, M.Vanicat, Commun.Math.Phys.(2019)5

Exact large deviations for space-time extensive obervables in terms of an
inhomogeneous martrix product ansatz
B. Buča, J. P. Garrahan, TP, M. Vanicat, arXiv:1901.00845

6

Extra: New(?) integrable SO(3) symmetric classical-spin dynamics on
space-time discrete lattice from baxterized set-theretic solution of YB andRLL relations. Ž. Krajnik, TP, arXiv:1909.03799

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

ss Rule 54
Integrable reversible cellular automaton:
x,t+1

sx,t+2x+2,t+1

sx

1,t+2

sx+2,t+2

sx+1,t+2

Figure 1. Schematic representation of the time evolution of a section of the lattice.
In the first time-step, only the sites x and x + 2 are updated, while the states on the0sitess2x = 1χ(sand1 ,xs+
remaintime step,) = s1unchanged.
+ s2 + s3In+thes1seconds3 (mod2) the sites x ± 1 are
2 ,1s3
updated, and x, x + 2 do not change.

s1

s2
s02

s3

Figure 2. In the figure, the RCA 54 is presented diagrammatically. The updated
value s02 , i.e. the square with a red-border, depends on the values s1 , s2 , s3 (the top
2
3
6
three
s3 )2as +defined
0 × 20 + 1 ×
21 squares)+ 1 × 2and+is 0given
× 2by+the1 local
× 24map+ 1s02×=25(s+0 × 2in7 (2.3).
= 54
1 , s02 ,×Bobenko et al., Commun. Math. Phys. 158, 127 (1993)
where Me : s 7! s0 and Mo : s 7! s00 are maps from {0, 1}Z to {0, 1}Z , defined by the
local three-site updates(((sx 1 , sx , sx+1 ); x ⌘ 0 (mod 2),sx ;x ⌘ 0 (mod 2),
0
00
sx =
sx =
(2.2)
sx
x ⌘ 1 (mod 2),(sx 1 , sx , sx+1 ); x ⌘ 1 (mod 2).

The schematic representation of the time evolution is presented in Figure 1. The RCA
54 is described by the binary function ,(s1 , s2 , s3 ) = s1Tomaž
+ s2 +
s3 + s1 sReversible(modCellular2). Automata and Statistical Mechanics(2.3)
Prosen3

Integrable reversible cellular automaton: Rule 54
Figure 1. Schematic representation of the time evolution of a section of the lattice.
In the first time-step, only the sites x and x + 2 are updated, while the states on the0sitess2x = 1χ(sand1 ,xs+
remaintime step,) = s1unchanged.
+ s2 + s3In+thes1seconds3 (mod2) the sites x ± 1 are
2 ,1s3
updated, and x, x + 2 do not change.

s1

s2
s02

s3

Figure 2. In the figure, the RCA 54 is presented diagrammatically. The updated
value s02 , i.e. the square with a red-border, depends on the values s1 , s2 , s3 (the top
2
3
6
three
s3 )2as +defined
0 × 20 + 1 ×
21 squares)+ 1 × 2and+is 0given
× 2by+the1 local
× 24map+ 1s02×=25(s+0 × 2in7 (2.3).
= 54
1 , s02 ,×Time-dependent
matrix productansatzPhys.
for interacting
dynamics
4
Bobenko
et al., Commun.
Math.
158, reversible
127 (1993)
where Me : s 7! s0 and Mo : s 7! s00 are maps from {0, 1}Z to {0, 1}Z , defined by the
local three-site updates(((sx 1 , sx , sx+1 ); x ⌘ 0 (mod 2),sx ;x ⌘ 0 (mod 2),
0
00
sx =
sx =
(2.2)
sx
x ⌘ 1 (mod 2),(sx 1 , sx , sx+1 ); x ⌘ 1 (mod 2).

The schematic representation of the time evolution is presented in Figure 1. The RCA
54 is described by the binary function ,(s1 , s2 , s3 ) = s1 + s2 + s3 + s1 s3(mod 2).

(2.3)

The rules (2.3) are diagrammatically expressed in Figure 2. The complete time evolution
is obtained by alternately
applying the maps Me and Mo . Note that Me and Mo
Figure 3. Time evolution of a random initial configuration. Particles move withTomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Two color version (low density):Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Two color version (medium density):Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Two color version (high density):Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Constructing a Markov matrix: deterministic bulk + stochastic boundaries

TP and C.Mejia-Monasterio, JPA 49, 185003 (2016)

Describe an evolution of probability state vector for n−cell automatonp(t) = U t p(0)p = (p0 , p1 , . . . , p2n −1 ) ≡ (ps1 ,s2 ,...,sn ; sj ∈ {0, 1})

s2

s1

s4

s3

sn-2

s5

Tomaž Prosen

sn

sn-1

Reversible Cellular Automata and Statistical Mechanics

2
1

4

P

PL

n

3

...

P

P

Ue

n-1

...

P

P

P

P

R

Uo

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

2
1

4

P

PL

n

3

Ue

n-1

...

P

P

...

P

P

P

R

P

Uo

U = Uo Ue ,R
Ue = P123 P345 · · · Pn−3,n−2,n−1 Pn−1,n,L
Uo = Pn−2,n−1,n · · · P456 P234 P12.

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

2
1

4

n

3

P

Ue

n-1

...

P

P

P

R

1

PL

P

...

P

P

Uo





P=






1
1
1
1
1








1


1

U = Uo Ue ,R
Ue = P123 P345 · · · Pn−3,n−2,n−1 Pn−1,n,L
Uo = Pn−2,n−1,n · · · P456 P234 P12.

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

2
1

4
3

P

P

L

n
n-1

...

P

P

...

Ue

PR

P

1




P=






1
1
1
1
1








1


1

P

P

Uo
α
0

L
P =
1−α
0

γ
 1−γ
R
P =
0
0


U = Uo Ue ,R
Ue = P123 P345 · · · Pn−3,n−2,n−1 Pn−1,n,L
Uo = Pn−2,n−1,n · · · P456 P234 P12.

Tomaž Prosen

0
β
0
1−β
γ
1−γ
0
0

α
0
1−α
0
0
0
δ
1−δ


0
β


0
1−β

0
0


δ
1−δ

Reversible Cellular Automata and Statistical Mechanics

Some Monte-Carlo to warm up...

txTomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Holographic ergodicity

Theorem
The 2n ×2n matrix U is irreducible and aperiodic
for generic values of driving parameters, more
precisely, for an open set 0 < α, β, γ, δ < 1.

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Holographic ergodicity

Theorem
The 2n ×2n matrix U is irreducible and aperiodic
for generic values of driving parameters, more
precisely, for an open set 0 < α, β, γ, δ < 1.

Consequence (via Perron-Frobenius theorem):
Nonequilibrium steady state (NESS), i.e. fixed
point of U
Up = pis unique, and any initial probability state vector
is asymptotically (in t) relaxing to p.

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Holographic ergodicity
Theorems

The 2n ×2n matrix U is irreducible and aperiodic
for generic values of driving parameters, more
precisely, for an open set 0 < α, β, γ, δ < 1.

Consequence (via Perron-Frobenius theorem):
Nonequilibrium steady state (NESS), i.e. fixed
point of U
Up = ptgap

is unique, and any initial probability state vector
is asymptotically (in t) relaxing to p.

Idea of the proof:
Show that for any pair of configurations s, s0 ,such t0 exists that
(U t )s,s0 > 0,∀t ≥ t0 .
Tomaž Prosens'
Reversible Cellular Automata and Statistical Mechanics

Unified matrix ansatz for NESS and decay modes: some magic at work

[TP and B. Buča, JPA 50, 395002 (2017)]

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Unified matrix ansatz for NESS and decay modes: some magic at work

[TP and B. Buča, JPA 50, 395002 (2017)]

Consider a pair of matrices:
1 1 0 0
 0 0 0 0 
W0 =  ξ ξ 0 0 ,
0 0 0 00
 0
W1 = 
 0
0

0
0
0
0

0
ξ
0
1


0
1 
,
0 
ω


W=

W0
W1



and Ws0 (ξ, ω) := Ws (ω, ξ).

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Unified matrix ansatz for NESS and decay modes: some magic at work
[TP and B. Buča, JPA 50, 395002 (2017)]

Consider a pair of matrices:
1 1 0 0
 0 0 0 0 
W0 =  ξ ξ 0 0 ,
0 0 0 00
 0
W1 = 
 0
0

0
0
0
0

0
ξ
0
1


0
1 
,
0 
ω


W=

W0
W1



and Ws0 (ξ, ω) := Ws (ω, ξ). These satisfy a remarkable bulk relation:
P123 W1 SW2 W30 = W1 W20 W3 S
or component-wise
Ws SWχ(ss 0 s 00 ) Ws000 = Ws Ws00 Ws 00 S.
where S is a “delimiter” matrix


0
 1

S =
0
0

Tomaž Prosen

1
0
0
0

0
0
0
1


0
0 
.
1 
0

Reversible Cellular Automata and Statistical Mechanics

Suppose there exists pairs and quadruples of vectors hls |, hlss0 0 |, |rss 0 i, |rs0 i, and ascalar parameter λ, satisfying the following boundary equations
P123 hl1 |W2 W30 = hl012 |W3 S,
R
P12|r12 i = W10 S|r20 i,

P123 W10 W2 |r30 i = λW10 S|r23 i,
L 0
P12
hl12 | = λ−1 hl1 |W2 S.

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Suppose there exists pairs and quadruples of vectors hls |, hlss0 0 |, |rss 0 i, |rs0 i, and ascalar parameter λ, satisfying the following boundary equations
P123 hl1 |W2 W30 = hl012 |W3 S,
R
P12|r12 i = W10 S|r20 i,

P123 W10 W2 |r30 i = λW10 S|r23 i,
L 0
P12
hl12 | = λ−1 hl1 |W2 S.

Then, the following probability vectors
p
0

p

≡

≡

0
p12...n = hl1 |W2 W30 W4 · · · Wn−3
Wn−2 |rn−1,n i,0p012...n = hl012 |W3 W40 · · · Wn−3 Wn−2
Wn−1 |rn0 i,

satisfy the NESS fixed point condition
Ue p = p0 ,Tomaž Prosen

Uo p0 = p.

Reversible Cellular Automata and Statistical Mechanics

Suppose there exists pairs and quadruples of vectors hls |, hlss0 0 |, |rss 0 i, |rs0 i, and ascalar parameter λ, satisfying the following boundary equations
P123 hl1 |W2 W30 = hl012 |W3 S,
R
P12|r12 i = W10 S|r20 i,

P123 W10 W2 |r30 i = λW10 S|r23 i,
L 0
P12
hl12 | = λ−1 hl1 |W2 S.

Then, the following probability vectors
p
p0

≡

≡

0
p12...n = hl1 |W2 W30 W4 · · · Wn−3
Wn−2 |rn−1,n i,0p012...n = hl012 |W3 W40 · · · Wn−3 Wn−2
Wn−1 |rn0 i,

satisfy the NESS fixed point condition
Ue p = p0 ,Uo p0 = p.

Proof: Observe the bulk relations
P123 W1 SW2 W30=

W1 W20 W3 S,

P123 W10 W2 W30 S=

W10 SW20 W3

to move the delimiter S around, when it ‘hits’ the boundary observe one of theboundary equations. After the full cycle, you obtain Uo Ue p = λλ−1 p.
Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

This yields a consistent system of equations which uniquely determine theunknown parameters, namely for the left boundary:ξ=(α + β − 1) − λ−1 β,
λ−2 (β − 1)ω=

λ−1 (α − λ−1 ),β−1and for the right boundary:ξ=λ(γ − λ),δ−1

ω=γ + δ − 1 − λδ,
λ2 (δ − 1)yielding
ξ

=

ω

=

(γ(α + β − 1) − β)(β(γ + δ − 1) − γ),(α − δ(α + β − 1))2(δ(α + β − 1) − α)(α(γ + δ − 1) − δ),
(γ − β(γ + δ − 1))2

and explicit expressions for the boundary vectors..

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Can we diagonalize U with a similar ansatz?

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Can we diagonalize U with a similar ansatz?
Yes, a good deal of decay modes can be written as a compact MPA with
explicitly positionally dependent matrices
W(x) ,

W0(x)depending on x ∈ {2, 3, . . . , n − 1} via multiplicative momentum variable z,
containing linear combinations of{1, z x , z −x }Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Can we diagonalize U with a similar ansatz?
Yes, a good deal of decay modes can be written as a compact MPA with
explicitly positionally dependent matrices
W(x) ,

W0(x)depending on x ∈ {2, 3, . . . , n − 1} via multiplicative momentum variable z,
containing linear combinations of{1, z x , z −x }For example:
W(x) = (e11 ⊗ W(ξz, ω/z) + e22 ⊗ W(ξ/z, ωz))


0
 0

F+ = 
0
0

0
0
0
0

0
0
ξω−1
ωz 2

0


0
z 
,
0 
ξz 2

0
 0
F− =  00Tomaž Prosen





0
0
0
0

c z x F+ + c− z −x F−ξω − 118 + e12 ⊗ +
0
0
ξω−1
ξz 2

0

0



1
ξ2 z 3.
0
ω + ξ11−1z2

Reversible Cellular Automata and Statistical Mechanics

The Bethe-like equations for the Markov spectrum

Ue p(z) = ΛL p0 (z),Uo p0 (z) = ΛR p(z).

ΛR (γz − ΛR )z(α + β − 1) − βΛL=,(β − 1)Λ2L(δ − 1)zΛL (αz − ΛL )z(γ + δ − 1) − δΛR=,(δ − 1)Λ2R(β − 1)zz 2n−6−4p =(α + β − 1)p (γ + δ − 1)p
4p
Λ4p
L ΛR

Im Λ

0.5

0.0

.
-0.5

-0.5

0.0

0.5
Re Λ

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

1.0

The time-dependent problem

Consider a (commutative C ∗ ) algebra of observables on infinite lattice x ∈ Z.

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

The time-dependent problem

Consider a (commutative C ∗ ) algebra of observables on infinite lattice x ∈ Z.
Ultralocal basis {[0]x , [1]x }:[α]x (s) = δα,sx ,([α]x [β]y )(s) = [α]x (s) [β]y (s),Tomaž Prosen

α, β ∈ {0, 1}, s ∈ {0, 1}Z

Reversible Cellular Automata and Statistical Mechanics

The time-dependent problem

Consider a (commutative C ∗ ) algebra of observables on infinite lattice x ∈ Z.
Ultralocal basis {[0]x , [1]x }:[α]x (s) = δα,sx ,([α]x [β]y )(s) = [α]x (s) [β]y (s),α, β ∈ {0, 1}, s ∈ {0, 1}Zr-local basis centred on site x:[α1 α2 . . . αr ]x ≡ [α1 ]x−b r c [α2 ]x−b r c+1 · · · [αr ]x+b r −1 c .
2Tomaž Prosen

22

Reversible Cellular Automata and Statistical Mechanics

The time-dependent problem

Consider a (commutative C ∗ ) algebra of observables on infinite lattice x ∈ Z.
Ultralocal basis {[0]x , [1]x }:[α]x (s) = δα,sx ,([α]x [β]y )(s) = [α]x (s) [β]y (s),α, β ∈ {0, 1}, s ∈ {0, 1}Zr-local basis centred on site x:[α1 α2 . . . αr ]x ≡ [α1 ]x−b r c [α2 ]x−b r c+1 · · · [αr ]x+b r −1 c .
2

2

2

Using unit element 1 = [0]x + [1]x , we can extend the support of each r-local
basis element as[α1 α2 . . . αr ]x ≡ 1x−b r +2 c · [α1 α2 . . . αr ]x · 1x+b r +1 c ≡2

2≡ [0α1 α2 . . . αr 0]x + [0α1 α2 . . . αr 1]x + [1α1 α2 . . . αr 0]x + [1α1 α2 . . . αr 1]x .

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Separable (strongly clustering) states p defined by expectation values p(x) of
ultralocal observablesh[α1 α2 . . . αr ]x ip = px−b 2r c (α1 ) · px−b 2r c+1 (α2 ) · · · px+b r −1 c (αr ).
2Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Separable (strongly clustering) states p defined by expectation values p(x) of
ultralocal observablesh[α1 α2 . . . αr ]x ip = px−b 2r c (α1 ) · px−b 2r c+1 (α2 ) · · · px+b r −1 c (αr ).
2

Two examples of separable states that we consider:1

A maximum entropy statepx (0) = px (1) = 1/2,

2

An inhomogeneous initial state(px (0) = px (1) = 1/2,
px (0) = 1, px (1) = 0.

Tomaž Prosen∀x ∈ Z.

for x ≤ 0
for x > 0

Reversible Cellular Automata and Statistical Mechanics

Dynamics: Time automorphism of algebra of observablesat (s0 ) = a(st )Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Dynamics: Time automorphism of algebra of observablesat (s0 ) = a(st )For 3-site observables, dynamical automorphism is defined as([α χ(α, β, γ) γ]y ; x = y ,Ux [α β γ]y =[α β γ]y ;|x − y | ≥ 2,
while for any r-local observable it is defined asa t-staggered linear homomorphism
at+1 = U(t)at(Q
Ux ;U(t) = Qx∈2Z
x∈2Z+1 Ux ;Tomaž Prosen

t≡0
t≡1

(mod 2),(mod 2).

Reversible Cellular Automata and Statistical Mechanics

Time-dependent matrix product ansatz
Theorem (Klobas et al. 18): Time evolution of a local observable [1]x reads
X[1]tx =
cs−t ,...,st (t)[s−t s−t+1 · · · st ]x ,s−t ,...,st ∈{0,1}where the amplitudes cs−t ,...,st (t) ∈ {0, 1} can be represented as MPA
cs−t ,...st (t) = hl(t)|Vs−t Ws−t+1 Vs−t+2 · · · Wst−1 Vst |r i++ hl 0 |Vs0−t Ws0−t+1 Vs0−t+2 · · · Ws0t−1 Vs0t |r 0 (t)i.

Vs , Ws , Vs0 , Ws0 ∈ End(V), s ∈ {0, 1}, are linear operators over auxiliary
Hilbert space V = lsp{|c, w , n, ai; c, w ∈ N0 , n ∈ {0, 1, 2}, a ∈ {0, 1}}, andcan be explicitly expressed in terms of ladder operators and projectors
X
T
c+ =
|c + 1, w , n, aihc, w , n, a|,c− = c+ ,c,w ,n,a

+

w =

X

|c, w + 1, n, aihc, w , n, a|,w− = w+T,c,w ,n,aec2 w2 n2 a2 ,c1 w1 n1 a1 = |c2 , w2 , n2 , a2 ihc1 , w1 , n1 , a1 |,X
en2 a2 ,n1 a1 =|c, w , n2 , a2 ihc, w , n1 , a1 |,c,wTomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

V0 = e00,00 + e10,00 + e20,00 + c+ e10,01 + e01,01 + c+ w+ e11,01 + e21,01 ++ e0001,0001 + e0011,0001 + e0021,0001 ,V1 = e00,10 + e10,20 + e20,20 + e00,11 + e10,21 + e20,21 + e01,11 ++ w+ e11,21 + w+ e21,21 + e0001,0011 + e0011,0021 + e0021,0021 ,W0 = c− w+ (e00,00 + e10,00 + e20,00 ) + w+ e10,01 + w+ e01,01 +2+ c+ w+ e11,01 + w+ e21,01 + e1111,0001 + e0001,0001 + e0011,0001 + e0021,0001 ,W1 = c− w+ (e00,10 + e10,20 + e20,20 ) + w+ e01,11 + c+ w+ e11,21 ++ c+ w+ e21,21 + e0001,0011 + e0011,0021 + e0021,0021 ,V00 = V0T − (e0001,1111 + e0101,1211 + e0101,1110 ) ,V10 = V1T ,W00 = W0T − (e0001,1111 + e0000,1211 ) ,W10 = W1T − (e0021,1111 + e0021,1121 + e0121,1211 + e0121,1221 ) .

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

V0 = e00,00 + e10,00 + e20,00 + c+ e10,01 + e01,01 + c+ w+ e11,01 + e21,01 ++ e0001,0001 + e0011,0001 + e0021,0001 ,V1 = e00,10 + e10,20 + e20,20 + e00,11 + e10,21 + e20,21 + e01,11 ++ w+ e11,21 + w+ e21,21 + e0001,0011 + e0011,0021 + e0021,0021 ,W0 = c− w+ (e00,00 + e10,00 + e20,00 ) + w+ e10,01 + w+ e01,01 +2+ c+ w+ e11,01 + w+ e21,01 + e1111,0001 + e0001,0001 + e0011,0001 + e0021,0001 ,W1 = c− w+ (e00,10 + e10,20 + e20,20 ) + w+ e01,11 + c+ w+ e11,21 ++ c+ w+ e21,21 + e0001,0011 + e0011,0021 + e0021,0021 ,V00 = V0T − (e0001,1111 + e0101,1211 + e0101,1110 ) ,V10 = V1T ,W00 = W0T − (e0001,1111 + e0000,1211 ) ,W10 = W1T − (e0021,1111 + e0021,1121 + e0121,1211 + e0121,1221 ) .

The time-dependent auxiliary space boundary vectors take the following form:hl(t)| = h0, t, 0, 0|,|r i = |0, 0, 0, 0i + |0, 0, 0, 1i + |0, 0, 0, 2i,

hl 0 | = h0, 0, 0, 1| + h0, 0, 1, 1| + h0, 0, 2, 1| + h0, 1, 0, 1| + h0, 1, 2, 1|,(1)|r 0 (t)i = |0, t + 1, 0, 0i.

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Proof: ‘Real space,time ansatz
inverseforscatteringtransform’
Time-dependent
matrixreal
product
interacting
reversible dynamics

Figure 4.

8

An example of an allowed configuration (the bottom-most saw). The

The weight ofleft MPA hl(t)|Vs−t W
s−t+1 Vs−t+2 · · · Wst−1 Vst |r i is 1 (or 0) if
red, blue and green-coloredsites
denote three distinct solitons. The blue soliton goesthe configuration
(s
,
s
,
.
.
.
,
s
)
be obtained
a light-cone
−t
−t+1
t
through the site (0, 0), while thecan
red (cannot)
and green solitons
originatein
outsideof the lightwith the left-moverat the origin!
cone. Alternatively,
we can think of this as solitons that start at the bottom andpropagate in the negative time. The blue soliton goes through the top site, while thered and green one escape the light-cone and cannot reach the origin.
Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Exact dynamical structure factor

C (x, t) = h[1]x [1]t0 ip − h[1]x ip h[1]t0 ip = h[1]x [1]t0 ip −Tomaž Prosen

1
4

Reversible Cellular Automata and Statistical Mechanics

Exact dynamical structure factor

C (x, t) = h[1]x [1]t0 ip − h[1]x ip h[1]t0 ip = h[1]x [1]t0 ip −1
4

Using time-dependent MPA:
 1
x+t
x+t
1 
t− x+t0 x+t
2|r i + hl 0 |T 2 V10 T 0 t− 2 |r 0 (t)i −C (x, t) = 2t+1 hl(t)|T 2 V1 T
2
4
with
T = (V0 + V1 )(W0 + W1 ),T0= (W00 + W10 )(V00 + V10 ),Tomaž Prosen

T = (W0 + W1 )(V0 + V1 ),0

T = (V00 + V10 )(W00 + W10 ).

Reversible Cellular Automata and Statistical Mechanics

We find normal hydrodynamic scaling!
For maximum entropy (‘infinite temperature’) state, tMPA yields
t−|x|−22!
!!
t − 2m − 3
t − 2m − 2
42−
C (x, t) = 2
m
m
m=0




t 2
4
1
√ exp −
|x| −
.
'
t
2
Time-dependent matrix
product
ansatzforinteracting
reversible dynamics
16 tπ
−t−1

X

m

0.004

t = 100
t = 200
t = 300

0.0035
0.003
C(x, t)170.00250.002
0.00150.001
0.0005
0

200

150

100

50

Tomaž Prosen

0
x

50

100

150

200

Reversible Cellular Automata and Statistical Mechanics

Exact solution of inhomogeneous quench problem
Time-dependent matrix product ansatz for interacting reversible dynamics
Time-dependent matrix product ansatz for interacting reversible dynamics

30

0.5

⇢ˆ(x, t)

0.4
0.3
0.2
0.1

t = 100
t = 200
t = 300

0200
Figure C1. One realization of the inhomogeneous quench up to t = 74. Between theempty space on the right and the area filled with left and right moving solitons, there
is a section, where all the particles move to the right and do not scatter, corresponding
to the free regime.
x tρ̂(x, t) = h[1] ip

inhom

100

0

x

100

200

300

−t
= the
h[1]
ip with the velocity 1 and its shape does not change. On the left sright
x movesFigure 5. Density profile at di↵erent times t after the quench. The ballistic f

inhom
profile has a shapeof an erf that moves with a velocity 1/2 and interpolatesbp

1/3 towards the center and 1/2 to the left. Its width scales as ⇠ t.
Exact solution exhibits the following simple asymptotic
behavior:
Quasi-free regime
diagonal m, ⇢(m, t),8!
0; b t−x+1>
t  x,
c


>
>
2
>
t
1
<⇢( t x1, t 1); x + t ⌘ 0 (mod 2) and
Figure C2. A path of a soliton (red bordered squares) that moves to the left witht < x < t,
2
ρ̂
t
≥
x
≥
−
+
1,
t
=
1
−
−
the e↵ective velocity 1/3. Due to propagation rules, the solitons can not scatter more33 ⇢ˆ(x, t) = >2 , t); x + t ⌘ 1 (mod 2) and t < x < t,⇢( t x+1
frequently, therefore this is the slowest possible e↵ective speed.
>2>>supersite labeled by n = k,

:1
2

;

x

t,

Thermalizing(diffusive)
regime which can be calculated efficiently, using the tMPA representation
t 8
t 6
t 4
t 2
t
 t
√  ⇢(m,1t) = 2 2m (L(m, t) + R(m, t)) , m
t
7
t
5
t
3
t 1
···
= hl(t)|((V0 + V1 )(W0 + W1 )) V0 (W0 V0 )t m |ri ,t =L(m, t) (5lim ρ̂ − + ζ t,(C.1)− erf(2ζ))t→∞
212t) = hl0| ((V00 + V10)(W00 + W10 ))m V00 (W00 V00)t m |r0(t)i .
R(m,
n=4
n=3
n=2
n=1

All the sites with xt are empty, while a site with x  t 1 is occupied if thereis
In orderto calculate the matrix elements L(x, t) and R(x, t), the following redTomaž Prosen
Reversible Cellular Automata and Statistical
Mechanics
T0T0a right moving soliton going through it, in which case the whole supersite has to beExact large deviations[B.Buča, J.P.Garrahan, T.Prosen, M.Vanicat, arXiv:1901.00845]
Large deviation theory for arbitrary observable of the form:
OT =T−1 N−1
X
Xh

t+1/2

t
fx (sxt , sx+1
) + gx (sxt+1/2 , sx+1 )i

t=0 x=1Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Exact large deviations[B.Buča, J.P.Garrahan, T.Prosen, M.Vanicat, arXiv:1901.00845]
Large deviation theory for arbitrary observable of the form:
OT =T−1 N−1
X
Xh

t+1/2

i

(2)

(3)

t
fx (sxt , sx+1
) + gx (sxt+1/2 , sx+1 )t=0 x=1

Tilted Markov generator:Ũ(s) = Uo G (s) Ue F (s).
(1)

(2)

(3)

(N−1)

F (s) = F12 F23 F34 . . . FN−1,N
where

 (x)
f0,0

 0
(x)
F =
 0
0

0
(x)

f0,1
0
0

and
0
0
(x)

f1,0
0

(1)

(N−1)

G (s) = G12 G23 G34 . . . GN−1,N

0

0 
,
0 

(x)

0

fs,s 0 ≡ e sfx (s,s )

(x)

f1,1

and similar for G (x) .

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Inhomogeneous matrix ansatz cancellation mechanism

There exist 3 × 3 matrices satisying bulk algebraic conditions:(j−1) (j)(j) (j+1)(j)(j+1)
fs 0 s 00 Ws(j−1) Ws 0 Xs 00 = Xs(j−1) Vχ(ss 0 s 00 ) Vs 00 ,

fss 0

(j−2)

gss 0

(j−1)

(j−1)

gs 0 s 00 Xs(j−2) Vs 0Tomaž Prosen

(j)

(j−1)

(j)

Vs 00 = Ws(j−2) Wχ(ss 0 s 00 ) Xs 00 ,Reversible Cellular Automata and Statistical Mechanics

Inhomogeneous matrix ansatz cancellation mechanism

There exist 3 × 3 matrices satisying bulk algebraic conditions:
(j)
(j+1)
(j) (j+1)(j−1) (j)fs 0 s 00 Ws(j−1) Ws 0 Xs 00 = Xs(j−1) Vχ(ss 0 s 00 ) Vs 00 ,

fss 0

(j−2)

gss 0

(j−1)

(j−1)

gs 0 s 00 Xs(j−2) Vs 0

(j−1)

(j)

(j)

Vs 00 = Ws(j−2) Wχ(ss 0 s 00 ) Xs 00 ,and boundary equations(2)(1) (2)

X

(3)

(3)

0
= hlsχ(ss0 s 00 ) |Vs 00 ,fss 0 fs 0 s 00 hls |Ws 0 Xs 00
0

(N−1)

0 f
Rssmm
mm0

m,m0 =0,1

X
m,m0 =0,10|rmm0 i0
Lmm
ss 0 gmm0 hlmm0 |(1)(N−2) (N−1)(N−1) 0
gss 0 gs 0 s 00 Xs(N−2) Vs 0
|rs 00 iTomaž Prosen(N−1)= λR Xs|rs00 i,(2)

= λL hls |Xs 0 ,
(N−2)

= Ws

|rχ(ss 0 s 00 )s 00 i,

Reversible Cellular Automata and Statistical Mechanics

Inhomogeneous matrix ansatz cancellation mechanism
There exist 3 × 3 matrices satisying bulk algebraic conditions:

(j)
(j+1)
(j) (j+1)(j−1) (j)fs 0 s 00 Ws(j−1) Ws 0 Xs 00 = Xs(j−1) Vχ(ss 0 s 00 ) Vs 00 ,

fss 0

(j−2)

gss 0

(j−1)

(j−1)

gs 0 s 00 Xs(j−2) Vs 0

(j−1)

(j)

(j)

Vs 00 = Ws(j−2) Wχ(ss 0 s 00 ) Xs 00 ,and boundary equations(2)(1) (2)

X

(3)

(3)

0
= hlsχ(ss0 s 00 ) |Vs 00 ,fss 0 fs 0 s 00 hls |Ws 0 Xs 00
0

(N−1)

0 f
Rssmm
mm0

m,m0 =0,1
0

Xm,m0 =0,1|rmm0 i(N−1)

= λR Xs0
Lmm
ss 0 gmm0 hlmm0 |

(2)

(1)

(N−2) (N−1) (N−2) (N−1) 0
gs 0 s 00 Xs
Vs 0|rs 00 i

gss 0|rs00 i,= λL hls |Xs 0 ,
(N−2)

= Ws

|rχ(ss 0 s 00 )s 00 i,

such that MPA:
ps1 ,...,sN=

ps01 ,...,sN=

hls1 |Ws(2)Ws(3)· · · Ws(N−3)
Ws(N−2)|rsN−1 sN i
2
3
N−3
N−2
hls01 s2 |Vs(3)Vs(4)· · · Vs(N−2)
Vs(N−1)|rs0N i,
3
4
N−2
N−1

solves the eigenalue equationŨ(s)p = Λ(s)pand Λ(s) = e

θ(s)is a root of third order polynomial.
Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

1.4

0.2

θ(s)
N

1.2

ϕ(x)
N

0.1

1.0

∝ 1/N

0.8

0.0

0.0

0.2

0.6

x

d θ(s)

0.4 −ds N

0.0
0.2

0.6

N =10
N =30
N =500∝N

0.2 d2 θ(s)

0.4

ds2 N

2

1

0

Tomaž Prosen

s

1

∝ 1/N

2

3

Reversible Cellular Automata and Statistical Mechanics

Conclusions and Outlook

Interacting integrable model about which we can compute everything:quenches, non-equilibrium steady states with baths, relaxation rates,
dynamical structure factor, large deviations etc.
Generalizations (stochastic/unitary branching)?
Link to Yang-Baxter integrability missing?
Testbed for computing diffusive corrections to generalized hydroduynamics.
See e.g.: S. Gopalakrishnan, D. Huse, V. Khemani, R. Vasseur,
PRB 98, 220303 (2018)
The work supported by Slovenian Research Agency (ARRS) andTomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Conclusions and Outlook

Interacting integrable model about which we can compute everything:quenches, non-equilibrium steady states with baths, relaxation rates,
dynamical structure factor, large deviations etc.
Generalizations (stochastic/unitary branching)?
Link to Yang-Baxter integrability missing?
Testbed for computing diffusive corrections to generalized hydroduynamics.
See e.g.: S. Gopalakrishnan, D. Huse, V. Khemani, R. Vasseur,
PRB 98, 220303 (2018)
The work supported by Slovenian Research Agency (ARRS) andTomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Conclusions and Outlook

Interacting integrable model about which we can compute everything:quenches, non-equilibrium steady states with baths, relaxation rates,
dynamical structure factor, large deviations etc.
Generalizations (stochastic/unitary branching)?
Link to Yang-Baxter integrability missing?
Testbed for computing diffusive corrections to generalized hydroduynamics.
See e.g.: S. Gopalakrishnan, D. Huse, V. Khemani, R. Vasseur,
PRB 98, 220303 (2018)
The work supported by Slovenian Research Agency (ARRS) andTomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Extra: Reversible-deterministic (symplectic) dynamical map over (S 2 )×N
Family of rational-symplectic maps over S 2 × S 2 (S1 · S1 = S2 · S2 = 1):1Φτ (S1 , S2 ) =σ 2 S1 + τ 2 S2 + τ S1 × S2 , σ 2 S2 + τ 2 S1 + τ S2 × S1 ,
2
2
σ +τ

1
2
σ
:=
1 + S1 · S2 ,2defining discrete space-time many-body symplectic dynamics(classical local Floquet circuit)
2t+1
2t+1
(S2x
, S2t+1Φτ (S2t, S2tsymmetric), dynamics(S2t+2, S2t+2) = Φτlattice(S2t+12x
2x+1 )in=integrable
2x−1
2x space-time
2x−1 , S52x ),KPZ physics
SO(3)2x+1
on discrete

..
.

x
⌧

S2t+3
2x
⌧

S2t+2
2x

⌧
⌧

S2t+1
2x
⌧

S2t
2x

⌧
⌧
⌧
⌧

2t
S2t
2x+1 S2x+2

⌧

⌧
⌧

2t
S2t
2x+3 S2x+4

t
···
S2t
2x+5

Fig. 1: Classical local symplectic circuit representation of discrete space-time dyTomažspins
ProsenAutomata
and Statisticalnamics of the model. Lattice(taking Reversible
values onCellular2-spheres)are denotedby Mechanics

Integrability: baxterized ‘set-theoretic’ R- and L- matrices

Defining I a identity map over S 2 we find (note that Φ∞ (S1 , S2 ) = (S2 , S1 )):

 
 
 
 

 
Φλ ⊗ I ◦ I ⊗ Φλ+µ ◦ Φµ ⊗ I = I ⊗ Φµ ◦ Φλ+µ ⊗ I ◦ I ⊗ Φλ .

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Integrability: baxterized ‘set-theoretic’ R- and L- matrices

Defining I a identity map over S 2 we find (note that Φ∞ (S1 , S2 ) = (S2 , S1 )):


 
 
 
 
 
Φλ ⊗ I ◦ I ⊗ Φλ+µ ◦ Φµ ⊗ I = I ⊗ Φµ ◦ Φλ+µ ⊗ I ◦ I ⊗ Φλ .
Furthermore, defining a matrix valued function L(λ) : S 2 → End(C2 )L(S; λ) = 1 +1
S · σ,
2iλwe find that RLL relation is obeyed:
L(S2 ; λ)L(S1 ; µ) = L(S02 ; µ)L(S01 ; λ),Tomaž Prosen(S01 , S02 ) := Φλ−µ (S1 , S2 ).

Reversible Cellular Automata and Statistical Mechanics

Integrability: baxterized ‘set-theoretic’ R- and L- matrices
Defining I a identity map over S 2 we find (note that Φ∞ (S1 , S2 ) = (S2 , S1 )):

 
 
 
 

 
Φλ ⊗ I ◦ I ⊗ Φλ+µ ◦ Φµ ⊗ I = I ⊗ Φµ ◦ Φλ+µ ⊗ I ◦ I ⊗ Φλ .
Furthermore, defining a matrix valued function L(λ) : S 2 → End(C2 )L(S; λ) = 1 +1
S · σ,
2iλwe find that RLL relation is obeyed:
L(S2 ; λ)L(S1 ; µ) = L(S02 ; µ)L(S01 ; λ),(S01 , S02 ) := Φλ−µ (S1 , S2 ).

Defining monodromy-matrix
←−−−
!
N/2−1
Y
T (S0 , S1 , . . . , SN−1 ; λ, µ) = tr
L(S2x+1 ; λ)L(S2x ; µ) ,x=0

subsequent application of RLL relations shows manifest conservation of its
trace (for any λ while µ = λ − τ )t+2
T (St0 . . . StN−1 ; λ, λ−τ ) = T (St+1. . . St+1
. . . St+20
N−1 ; λ−τ, λ) = T (S0N−1 ; λ, λ−τ ).

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics

Local integrals of motion

Qkeven=∂λk log |T (λ, λ − τ )|2 |λ=− i ,

Qkodd

=

∂λk log |T (λ, λ − τ )|2 |λ=τ − i ,22Tomaž Prosen

k = 0, 1, 2 . . .

Reversible Cellular Automata and Statistical Mechanics

Space-Time Self-Duality
10Žiga Krajnik, Tomaž Prosen

operation that would facilitate their automated computation, similarly as in thequantum case [21].

S01

S02

S01

˜⌧

⌧
S1

S02

S2

S1

S2

Fig. 2: Definition of the dual 2-spin symplectic mapping ˜⌧ by a simple relabelling0of the domain(out-of-box(S01 ,(in-to-box
S02 ) = Φarrows)Φ̃τ (S1argument, S01 ). pairs.
τ (S1 , Sand
2 ), image(S2 , S2 ) =arrows)

2.5 Space-time self-duality
The two-body propagator (1) specifies the temporal dynamics. Knowing the valuesof two adjacent spins at the same time allows us to compute their time evolution
at a latter time:⌧(S1 , S2 ) !
(S01 , S02 ).
(37)
Prosen
Reversible
Automata
andwould
Statistical
We are interested in the Tomaž
existenceof a dual
spatialCellular
propagator,that
al- Mechanics

Space-Time Self-Duality
10Žiga Krajnik, Tomaž Prosen

operation that would facilitate their automated computation, similarly as in thequantum case [21].

S01

S02

S01

˜⌧

⌧
S1

S02

S2

S1

S2

Fig. 2: Definition of the dual 2-spin symplectic mapping ˜⌧ by a simple relabelling0of the domain(out-of-box(S01 ,(in-to-box
S02 ) = Φarrows)Φ̃τ (S1argument, S01 ). pairs.
τ (S1 , Sand
2 ), image(S2 , S2 ) =arrows)0Ξ ◦ Φ̃τ = Φτ ◦ (−Ξ),

2.5 Space-time
Ξ(S,self-duality
S ) := (S, −S0 ),(−Ξ)(S, S0 ) := (−S, S0 ).

The two-body propagator (1) specifies the temporal dynamics. Knowing the valuesof two adjacent spins at the same time allows us to compute their time evolution
at a latter time:⌧(S1 , S2 ) !
(S01 , S02 ).
(37)
We are interested in the existence of a dual spatial propagator, that would allow us to uniquely evolve any pair of temporally consequent spins with the same
spatial coordinate to a consequent
pair at Reversible
one spatial
pointAutomata
to the right,
i.e. we MechanicsTomaž Prosen
Cellularand Statistical

KPZ physics: Spin-Spin dynamical correlation function

C (x, t) = hSxt S00 i − hSxt ihS00 i
=0

(b)

=0

4000

(a)
10 3

3000

2000

10 2

10 4

t

0

t:

10 3

10 5

600
1200
1800
2400
3000
Gauss
KPZ

10 4

10.0

7.5

5.0

2.5

0.0

|C(x, t)|2

t 3 C(x, t)

1000

1000

2000

10 6

3000

2.5

5.0

7.5

2

x/t 3

10.0

4000
2000

1500

1000

500

0

500

1000

1500

2000

x

Tomaž Prosen

Reversible Cellular Automata and Statistical Mechanics