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Integrable discrete-time dynamics, generalized exclusion processes - - PowerPoint PPT Presentation

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrable discrete-time dynamics, generalized exclusion processes and "fused" matrix ansatz Matthieu VANICAT University of

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes

Integrable discrete-time dynamics, generalized exclusion processes and "fused" matrix ansatz

Matthieu VANICAT

University of Ljubljana

Annecy, September 2018

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes

1

Physical motivations and framework Markov dynamics and non-equilibrium stationary state Exclusion processes

2

Integrable discrete-time dynamics Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

3

Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

Physical motivations and framework

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

C1 C2 C3 Ci−1 Ci CN . . . . . . phase space The system can be in several different configurations. 4

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

C1 C2 C3 Ci−1 Ci CN . . . . . .

m(C1 → C2) m(C1 → C3) m(C1 → Ci−1) m(C1 → Ci)

phase space The system can be in several different configurations. During a time step, the system can jump from configuration C to another configuration C′ with probability m(C → C′). 4

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

C1 C2 C3 Ci−1 Ci CN . . . . . .

m(C1 → C2) m(C1 → C3) m(C1 → Ci−1) m(C1 → Ci)

phase space

Pt(C1) Pt(C2) Pt(C3) Pt(Ci−1) Pt(Ci) Pt(CN)

We denote by Pt(C) the probability for the system to be in configuration C at time t. 4

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

C1 C2 C3 Ci−1 Ci CN . . . . . .

m(C1 → C2) m(C1 → C3) m(C1 → Ci−1) m(C1 → Ci)

phase space

Pt(C1) Pt(C2) Pt(C3) Pt(Ci−1) Pt(Ci) Pt(CN)

We denote by Pt(C) the probability for the system to be in configuration C at time t. It obeys the master equation Pt+1(C) =

  • C′

m(C′ → C)Pt(C′) . 4

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

C1 C2 C3 Ci−1 Ci CN . . . . . .

m(C1 → C2) m(C1 → C3) m(C1 → Ci−1) m(C1 → Ci)

phase space

Pt(C1) Pt(C2) Pt(C3) Pt(Ci−1) Pt(Ci) Pt(CN)

We denote by Pt(C) the probability for the system to be in configuration C at time t. It obeys the master equation Pt+1(C) =

  • C′

m(C′ → C)Pt(C′) . 4

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

C1 C2 C3 Ci−1 Ci CN . . . . . .

m(C1 → C2) m(C1 → C3) m(C1 → Ci−1) m(C1 → Ci)

phase space

S(C1) S(C2) S(C3) S(Ci−1) S(Ci) S(CN)

We denote by S(C) the probability for the system to be in configuration C in the stationary state. 4

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

C1 C2 C3 Ci−1 Ci CN . . . . . .

m(C1 → C2) m(C1 → C3) m(C1 → Ci−1) m(C1 → Ci)

phase space

S(C1) S(C2) S(C3) S(Ci−1) S(Ci) S(CN)

We denote by S(C) the probability for the system to be in configuration C in the stationary state. We thus have S(C) =

  • C′

m(C′ → C)S(C′) . 4

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

C1 C2 C3 Ci−1 Ci CN . . . . . .

m(C1 → C2) m(C1 → C3) m(C1 → Ci−1) m(C1 → Ci)

phase space

S(C1) S(C2) S(C3) S(Ci−1) S(Ci) S(CN)

We denote by S(C) the probability for the system to be in configuration C in the stationary state. We thus have 0 =

  • C′

m(C′ → C)S(C′) − m(C → C′)S(C) .

4

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

C1 C2 C3 Ci−1 Ci CN . . . . . .

S(C1)m(C1 → C2) − S(C2)m(C2 → C1) = 0 S(C2)m(C2 → C3) − S(C3)m(C3 → C2) = 0 S(Ci−1)m(Ci−1 → Ci ) − S(Ci )m(Ci → Ci−1) = 0

phase space For a thermodynamic equilibrium, the detailed balance is fulfilled m(C′ → C)S(C′) = m(C → C′)S(C). 4

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

C1 C2 C3 Ci−1 Ci CN . . . . . . phase space

e

− E(C1) kB T

e

− E(C2)−E(C1) kB T

e

− E(C3)−E(C2) kB T

e

− E(Ci )−E(Ci−1) kB T

For a thermodynamic equilibrium, the detailed balance is fulfilled m(C′ → C)S(C′) = m(C → C′)S(C). From that we can easily compute the stationary distribution m(C → C′) m(C′ → C) = e

− E(C′)−E(C)

kBT

, S(C) = e

− E(C)

kBT

Z 4

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

C1 C2 C3 Ci−1 Ci CN . . . . . .

S(C1)m(C1 → C2) − S(C2)m(C2 → C1) = 0 S(C2)m(C2 → C3) − S(C3)m(C3 → C2) = 0 S(Ci−1)m(Ci−1 → Ci ) − S(Ci )m(Ci → Ci−1) = 0

phase space For a non-equilibrium stationary state, the detailed balance is broken m(C′ → C)S(C′) = m(C → C′)S(C). 4

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

C1 C2 C3 Ci−1 Ci CN . . . . . .

S(C1)m(C1 → C2) − S(C2)m(C2 → C1) = 0 S(C2)m(C2 → C3) − S(C3)m(C3 → C2) = 0 S(Ci−1)m(Ci−1 → Ci ) − S(Ci )m(Ci → Ci−1) = 0

phase space For a non-equilibrium stationary state, the detailed balance is broken m(C′ → C)S(C′) = m(C → C′)S(C). There are non vanishing probability currents flowing in the phase space in the stationary state. The system does not obey a Boltzmann statistics. S(C) = ? 4

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

The probabilities are encompassed in a vector |Pt =

C Pt(C)|C.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

The probabilities are encompassed in a vector |Pt =

C Pt(C)|C.

Master equation |Pt+1 = M|Pt, where MC′,C = m(C → C′) is a discrete-time Markov matrix

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

The probabilities are encompassed in a vector |Pt =

C Pt(C)|C.

Master equation |Pt+1 = M|Pt, where MC′,C = m(C → C′) is a discrete-time Markov matrix the entries are non-negative.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

The probabilities are encompassed in a vector |Pt =

C Pt(C)|C.

Master equation |Pt+1 = M|Pt, where MC′,C = m(C → C′) is a discrete-time Markov matrix the entries are non-negative. the sum over each column is one.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

The probabilities are encompassed in a vector |Pt =

C Pt(C)|C.

Master equation |Pt+1 = M|Pt, where MC′,C = m(C → C′) is a discrete-time Markov matrix the entries are non-negative. the sum over each column is one. The stationary state |S =

C S(C)|C satisfies

|S = M|S.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

The probabilities are encompassed in a vector |Pt =

C Pt(C)|C.

Master equation |Pt+1 = M|Pt, where MC′,C = m(C → C′) is a discrete-time Markov matrix the entries are non-negative. the sum over each column is one. The stationary state |S =

C S(C)|C satisfies

|S = M|S. In this talk: integrability of M means M = t(κ), with [t(x), t(y)] = 0, ∀x, y.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

(Generalized) exclusion processes

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

(Generalized) exclusion processes One-dimensional lattice with L sites

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

(Generalized) exclusion processes One-dimensional lattice with L sites Maximal number s of particles on each site

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

(Generalized) exclusion processes One-dimensional lattice with L sites Maximal number s of particles on each site The particles evolve randomly on the lattice

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

(Generalized) exclusion processes One-dimensional lattice with L sites Maximal number s of particles on each site The particles evolve randomly on the lattice The system is coupled to particle reservoirs at the boundaries

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

(Generalized) exclusion processes One-dimensional lattice with L sites Maximal number s of particles on each site The particles evolve randomly on the lattice The system is coupled to particle reservoirs at the boundaries A configuration is given by τ = (τ1, τ2, . . . , τL)

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

(Generalized) exclusion processes One-dimensional lattice with L sites Maximal number s of particles on each site The particles evolve randomly on the lattice The system is coupled to particle reservoirs at the boundaries A configuration is given by τ = (τ1, τ2, . . . , τL) τi is the number of particles on site i.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

Example: SSEP with sublattice parallel update β δ α γ The stochastic dynamics is defined in two steps:

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

Example: SSEP with sublattice parallel update β δ α γ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

Example: SSEP with sublattice parallel update p p β δ α γ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired

In the bulk, particles jump if possible to the left/right with proba p

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

Example: SSEP with sublattice parallel update p p β δ α γ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired

In the bulk, particles jump if possible to the left/right with proba p Exclusion principle

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

Example: SSEP with sublattice parallel update p p β δ α γ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired

In the bulk, particles jump if possible to the left/right with proba p Exclusion principle At the right boundary, particles leave with proba β or enter with δ

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

Example: SSEP with sublattice parallel update α γ β δ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired

In the bulk, particles jump if possible to the left/right with proba p Exclusion principle At the right boundary, particles leave with proba β or enter with δ

During the second step, neighboring even-odd sites are paired

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

Example: SSEP with sublattice parallel update p p p p p α γ β δ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired

In the bulk, particles jump if possible to the left/right with proba p Exclusion principle At the right boundary, particles leave with proba β or enter with δ

During the second step, neighboring even-odd sites are paired

In the bulk, particles jump if possible to the left/right with proba p

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

Example: SSEP with sublattice parallel update p p p p p α γ β δ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired

In the bulk, particles jump if possible to the left/right with proba p Exclusion principle At the right boundary, particles leave with proba β or enter with δ

During the second step, neighboring even-odd sites are paired

In the bulk, particles jump if possible to the left/right with proba p Exclusion principle

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

Example: SSEP with sublattice parallel update p p p p p α γ β δ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired

In the bulk, particles jump if possible to the left/right with proba p Exclusion principle At the right boundary, particles leave with proba β or enter with δ

During the second step, neighboring even-odd sites are paired

In the bulk, particles jump if possible to the left/right with proba p Exclusion principle At the left boundary, particles enter with proba α or leave with γ

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

p p β δ

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

p p β δ The associated discrete-time Markov matrix reads M = UeUo

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

p p β δ The associated discrete-time Markov matrix reads M = UeUo with operators Ue and Uo defined as Uo =

L−1 2

  • k=1

U2k−1,2k BL and Ue = B1

L−1 2

  • k=1

U2k,2k+1.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

p p β δ The associated discrete-time Markov matrix reads M = UeUo with operators Ue and Uo defined as Uo =

L−1 2

  • k=1

U2k−1,2k BL and Ue = B1

L−1 2

  • k=1

U2k,2k+1.

B =

  • 1 − α

γ α 1 − γ

  • U =

  

1 1 − p p p 1 − p 1

  

B =

  • 1 − δ

β δ 1 − β

  • |0

|1 |0 |1 |0 |1 |0 |1

|0 ⊗ |0 |0 ⊗ |1 |1 ⊗ |0 |1 ⊗ |1 |0 ⊗ |1 |1 ⊗ |0 |0 ⊗ |0 |1 ⊗ |1 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Markov dynamics and non-equilibrium stationary state Exclusion processes

1 2 3 4 5 B B U U U U U U B Ue Ue Uo

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Integrable discrete-time dynamics

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

How to construct an integrable Markov matrix M?

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

How to construct an integrable Markov matrix M? The building block is a matrix ˇ R(z) ∈ End(V ⊗ V)

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

How to construct an integrable Markov matrix M? The building block is a matrix ˇ R(z) ∈ End(V ⊗ V) Yang-Baxter equation

ˇ R23 (z1 − z2) ˇ R12 (z1 − z3) ˇ R23 (z2 − z3) = ˇ R12 (z2 − z3) ˇ R23 (z1 − z3) ˇ R12 (z1 − z2).

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

How to construct an integrable Markov matrix M? The building block is a matrix ˇ R(z) ∈ End(V ⊗ V) Yang-Baxter equation

ˇ R23 (z1 − z2) ˇ R12 (z1 − z3) ˇ R23 (z2 − z3) = ˇ R12 (z2 − z3) ˇ R23 (z1 − z3) ˇ R12 (z1 − z2).

We also require Unitarity: ˇ R(z)ˇ R(−z) = 1

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

How to construct an integrable Markov matrix M? The building block is a matrix ˇ R(z) ∈ End(V ⊗ V) Yang-Baxter equation

ˇ R23 (z1 − z2) ˇ R12 (z1 − z3) ˇ R23 (z2 − z3) = ˇ R12 (z2 − z3) ˇ R23 (z1 − z3) ˇ R12 (z1 − z2).

We also require Unitarity: ˇ R(z)ˇ R(−z) = 1 Markov property: 1| ⊗ 1|ˇ R(z) = 1| ⊗ 1|

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

How to construct an integrable Markov matrix M? The building block is a matrix ˇ R(z) ∈ End(V ⊗ V) Yang-Baxter equation

ˇ R23 (z1 − z2) ˇ R12 (z1 − z3) ˇ R23 (z2 − z3) = ˇ R12 (z2 − z3) ˇ R23 (z1 − z3) ˇ R12 (z1 − z2).

We also require Unitarity: ˇ R(z)ˇ R(−z) = 1 Markov property: 1| ⊗ 1|ˇ R(z) = 1| ⊗ 1| Example: SSEP (s = 1, V = C2) ˇ R(z) = 1 + zP 1 + z =

    

1

1 1+z z 1+z z 1+z 1 1+z

1

    

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

The integrable boundary conditions are given by K(z), K(z) ∈ End(V)

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

The integrable boundary conditions are given by K(z), K(z) ∈ End(V) Reflection equation ˇ R (z1 − z2)K1(z1)ˇ R(z1 + z2)K1(z2) = K1(z2)ˇ R(z1 + z2)K1(z1)ˇ R (z1 − z2). and a similar reflection equation for K(z).

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

The integrable boundary conditions are given by K(z), K(z) ∈ End(V) Reflection equation ˇ R (z1 − z2)K1(z1)ˇ R(z1 + z2)K1(z2) = K1(z2)ˇ R(z1 + z2)K1(z1)ˇ R (z1 − z2). and a similar reflection equation for K(z). We also require Unitarity: K(z)K(−z) = 1, K(z)K(−z) = 1

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

The integrable boundary conditions are given by K(z), K(z) ∈ End(V) Reflection equation ˇ R (z1 − z2)K1(z1)ˇ R(z1 + z2)K1(z2) = K1(z2)ˇ R(z1 + z2)K1(z1)ˇ R (z1 − z2). and a similar reflection equation for K(z). We also require Unitarity: K(z)K(−z) = 1, K(z)K(−z) = 1 Markov property: 1|K(z) = 1|, 1|K(z) = 1|

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

The integrable boundary conditions are given by K(z), K(z) ∈ End(V) Reflection equation ˇ R (z1 − z2)K1(z1)ˇ R(z1 + z2)K1(z2) = K1(z2)ˇ R(z1 + z2)K1(z1)ˇ R (z1 − z2). and a similar reflection equation for K(z). We also require Unitarity: K(z)K(−z) = 1, K(z)K(−z) = 1 Markov property: 1|K(z) = 1|, 1|K(z) = 1| Example: SSEP (s = 1, V = C2) K(z) =

 

(c−a)z+1 (a+c)z+1 2cz (a+c)z+1 2az (a+c)z+1 (a−c)z+1 (a+c)z+1

 

and K(z) =

 

(b−d)z−1 (b+d)z−1 2bz (b+d)z−1 2dz (b+d)z−1 (d−b)z−1 (b+d)z−1

 .

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Inhomogeneous transfer matrix

t(z|z) = tr0

  • K0(z)R0L (z − zL) . . . R01 (z − z1)K0(z)R10(z + z1) . . . RL0(z + zL)
  • Matthieu VANICAT

Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Inhomogeneous transfer matrix

t(z|z) = tr0

  • K0(z)R0L (z − zL) . . . R01 (z − z1)K0(z)R10(z + z1) . . . RL0(z + zL)
  • with R(z) = P.ˇ

R(z) and K(z) = tr0

  • K0(−z)R01(−2z)P01
  • .

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Inhomogeneous transfer matrix

t(z|z) = tr0

  • K0(z)R0L (z − zL) . . . R01 (z − z1)K0(z)R10(z + z1) . . . RL0(z + zL)
  • with R(z) = P.ˇ

R(z) and K(z) = tr0

  • K0(−z)R01(−2z)P01
  • .

Main property [t(x|z), t(y|z)] = 0

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Inhomogeneous transfer matrix

t(z|z) = tr0

  • K0(z)R0L (z − zL) . . . R01 (z − z1)K0(z)R10(z + z1) . . . RL0(z + zL)
  • with R(z) = P.ˇ

R(z) and K(z) = tr0

  • K0(−z)R01(−2z)P01
  • .

Main property [t(x|z), t(y|z)] = 0 Remarks: z = z1, z2, . . . , zL are the inhomogeneity parameters.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Inhomogeneous transfer matrix

t(z|z) = tr0

  • K0(z)R0L (z − zL) . . . R01 (z − z1)K0(z)R10(z + z1) . . . RL0(z + zL)
  • with R(z) = P.ˇ

R(z) and K(z) = tr0

  • K0(−z)R01(−2z)P01
  • .

Main property [t(x|z), t(y|z)] = 0 Remarks: z = z1, z2, . . . , zL are the inhomogeneity parameters. Usually one put z1 = z2 = · · · = zL = 0 to get

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Inhomogeneous transfer matrix

t(z|z) = tr0

  • K0(z)R0L (z − zL) . . . R01 (z − z1)K0(z)R10(z + z1) . . . RL0(z + zL)
  • with R(z) = P.ˇ

R(z) and K(z) = tr0

  • K0(−z)R01(−2z)P01
  • .

Main property [t(x|z), t(y|z)] = 0 Remarks: z = z1, z2, . . . , zL are the inhomogeneity parameters. Usually one put z1 = z2 = · · · = zL = 0 to get t′(0) = K ′

1(0) + 2 L−1

  • k=1

ˇ R′

k,k+1(0) + K ′ L(0)

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Inhomogeneous transfer matrix

t(z|z) = tr0

  • K0(z)R0L (z − zL) . . . R01 (z − z1)K0(z)R10(z + z1) . . . RL0(z + zL)
  • with R(z) = P.ˇ

R(z) and K(z) = tr0

  • K0(−z)R01(−2z)P01
  • .

Main property [t(x|z), t(y|z)] = 0 Remarks: z = z1, z2, . . . , zL are the inhomogeneity parameters. Usually one put z1 = z2 = · · · = zL = 0 to get t′(0) = K ′

1(0) + 2 L−1

  • k=1

ˇ R′

k,k+1(0) + K ′ L(0)

t′(0) is then interpreted as a continuous-time Markov matrix or as a quantum Hamiltonian.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

General method to get discrete-time process from transfer matrix

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

General method to get discrete-time process from transfer matrix We choose staggered inhomogeneity parameters z1 = z3 = z5 = · · · = zL = κ and z2 = z4 = z6 = · · · = zL−1 = −κ

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

General method to get discrete-time process from transfer matrix We choose staggered inhomogeneity parameters z1 = z3 = z5 = · · · = zL = κ and z2 = z4 = z6 = · · · = zL−1 = −κ A direct computation yields t(κ) = K1(κ)ˇ R23(2κ)ˇ R45(2κ) . . . ˇ RL−1,L(2κ) × ˇ R12(2κ)ˇ R34(2κ) . . . ˇ RL−2,L−1(2κ)K L(−κ),

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

General method to get discrete-time process from transfer matrix We choose staggered inhomogeneity parameters z1 = z3 = z5 = · · · = zL = κ and z2 = z4 = z6 = · · · = zL−1 = −κ A direct computation yields t(κ) = K1(κ)ˇ R23(2κ)ˇ R45(2κ) . . . ˇ RL−1,L(2κ) × ˇ R12(2κ)ˇ R34(2κ) . . . ˇ RL−2,L−1(2κ)K L(−κ), We can define a discrete-time Markov matrix as M = t(κ) = UeUo

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

General method to get discrete-time process from transfer matrix We choose staggered inhomogeneity parameters z1 = z3 = z5 = · · · = zL = κ and z2 = z4 = z6 = · · · = zL−1 = −κ A direct computation yields t(κ) = K1(κ)ˇ R23(2κ)ˇ R45(2κ) . . . ˇ RL−1,L(2κ) × ˇ R12(2κ)ˇ R34(2κ) . . . ˇ RL−2,L−1(2κ)K L(−κ), We can define a discrete-time Markov matrix as M = t(κ) = UeUo with operators Ue and Uo defined as Uo =

L−1 2

  • k=1

U2k−1,2k BL and Ue = B1

L−1 2

  • k=1

U2k,2k+1. where U = ˇ R(2κ), B = K(κ) and B = K(−κ).

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

1 2 3 4 5 B B U U U U U U B Ue Ue Uo

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

What is the associated stationary state?

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

What is the associated stationary state? We construct it in matrix product form.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

What is the associated stationary state? We construct it in matrix product form. The building block is a vector A(z) with non-commuting entries.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

What is the associated stationary state? We construct it in matrix product form. The building block is a vector A(z) with non-commuting entries. The commutation relations between the entries are given by Zamolodchikov-Faddeev (ZF) relation ˇ R(z1 − z2) A(z1) ⊗ A(z2) = A(z2) ⊗ A(z1)

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

What is the associated stationary state? We construct it in matrix product form. The building block is a vector A(z) with non-commuting entries. The commutation relations between the entries are given by Zamolodchikov-Faddeev (ZF) relation ˇ R(z1 − z2) A(z1) ⊗ A(z2) = A(z2) ⊗ A(z1) Example: SSEP (s = 1, V = C2) A(z) =

  • −z + E

z + D

  • Matthieu VANICAT

Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

What is the associated stationary state? We construct it in matrix product form. The building block is a vector A(z) with non-commuting entries. The commutation relations between the entries are given by Zamolodchikov-Faddeev (ZF) relation ˇ R(z1 − z2) A(z1) ⊗ A(z2) = A(z2) ⊗ A(z1) Example: SSEP (s = 1, V = C2) A(z) =

  • −z + E

z + D

  • The ZF relation implies DE − ED = D + E.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

We also need two boundary vectors W | and |V

  • Matthieu VANICAT

Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

We also need two boundary vectors W | and |V

  • The algebraic relations between the vectors and the entries of A(z)

are given by Ghoshal-Zamolodchikov (GZ) relations

  • W |A(z) =

W |K(z)A (−z) , A(z)|V = ¯ K(z)A (−z) |V .

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

We also need two boundary vectors W | and |V

  • The algebraic relations between the vectors and the entries of A(z)

are given by Ghoshal-Zamolodchikov (GZ) relations

  • W |A(z) =

W |K(z)A (−z) , A(z)|V = ¯ K(z)A (−z) |V . Example: SSEP (s = 1, V = C2) The GZ relations imply

  • W |
  • aE − cD − 1
  • = 0,

and

  • bD − dE − 1
  • |V

= 0.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Inhomogeneous ground-state |S(z1, z2, . . . , zL) = 1 ZL

  • W |A(z1) ⊗ A(z2) ⊗ · · · ⊗ A(zL)|V

,

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Inhomogeneous ground-state |S(z1, z2, . . . , zL) = 1 ZL

  • W |A(z1) ⊗ A(z2) ⊗ · · · ⊗ A(zL)|V

, Using ZF and GZ relations we can show t(zi|z)|S(z1, z2, . . . , zL) = |S(z1, z2, . . . , zL), for 1 ≤ i ≤ L,

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Inhomogeneous ground-state |S(z1, z2, . . . , zL) = 1 ZL

  • W |A(z1) ⊗ A(z2) ⊗ · · · ⊗ A(zL)|V

, Using ZF and GZ relations we can show t(zi|z)|S(z1, z2, . . . , zL) = |S(z1, z2, . . . , zL), for 1 ≤ i ≤ L, Hence the steady state of the discrete-time process is |S = 1 ZL

  • W |A(κ) ⊗ A(−κ) ⊗ A(κ) ⊗ · · · ⊗ A(κ)|V

.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Inhomogeneous ground-state |S(z1, z2, . . . , zL) = 1 ZL

  • W |A(z1) ⊗ A(z2) ⊗ · · · ⊗ A(zL)|V

, Using ZF and GZ relations we can show t(zi|z)|S(z1, z2, . . . , zL) = |S(z1, z2, . . . , zL), for 1 ≤ i ≤ L, Hence the steady state of the discrete-time process is |S = 1 ZL

  • W |A(κ) ⊗ A(−κ) ⊗ A(κ) ⊗ · · · ⊗ A(κ)|V

. We have indeed M|S = |S.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Example: SSEP with sublattice parallel update

|S = 1 ZL W |

  • −κ + E

κ + D

  • ⊗
  • κ + E

−κ + D

  • ⊗ · · · ⊗
  • −κ + E

κ + D

  • |V

.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Example: SSEP with sublattice parallel update

|S = 1 ZL W |

  • −κ + E

κ + D

  • ⊗
  • κ + E

−κ + D

  • ⊗ · · · ⊗
  • −κ + E

κ + D

  • |V

. S(1, 1, 0, 1, 0) = W |(κ + D)(−κ + D)(−κ + E)(−κ + D)(−κ + E)|V

  • Matthieu VANICAT

Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Example: SSEP with sublattice parallel update

|S = 1 ZL W |

  • −κ + E

κ + D

  • ⊗
  • κ + E

−κ + D

  • ⊗ · · · ⊗
  • −κ + E

κ + D

  • |V

. S(1, 1, 0, 1, 0) = W |(κ + D)(−κ + D)(−κ + E)(−κ + D)(−κ + E)|V

  • Normalization

ZL = W |(E + D)L|V = (a + c)L(b + d)L (ab − cd)L Γ L +

1 a+c + 1 b+d

  • Γ

1 a+c + 1 b+d

  • Matthieu VANICAT

Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Example: SSEP with sublattice parallel update

|S = 1 ZL W |

  • −κ + E

κ + D

  • ⊗
  • κ + E

−κ + D

  • ⊗ · · · ⊗
  • −κ + E

κ + D

  • |V

. S(1, 1, 0, 1, 0) = W |(κ + D)(−κ + D)(−κ + E)(−κ + D)(−κ + E)|V

  • Normalization

ZL = W |(E + D)L|V = (a + c)L(b + d)L (ab − cd)L Γ L +

1 a+c + 1 b+d

  • Γ

1 a+c + 1 b+d

  • Mean particle density

τi =

a a+c

  • L +

1 b+d − i

+

d b+d

  • i − 1 +

1 a+c

  • L +

1 a+c + 1 b+d − 1

,

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrability and transfer matrix Integrable discrete-time dynamics Stationary state in matrix product form

Example: SSEP with sublattice parallel update

|S = 1 ZL W |

  • −κ + E

κ + D

  • ⊗
  • κ + E

−κ + D

  • ⊗ · · · ⊗
  • −κ + E

κ + D

  • |V

. S(1, 1, 0, 1, 0) = W |(κ + D)(−κ + D)(−κ + E)(−κ + D)(−κ + E)|V

  • Normalization

ZL = W |(E + D)L|V = (a + c)L(b + d)L (ab − cd)L Γ L +

1 a+c + 1 b+d

  • Γ

1 a+c + 1 b+d

  • Mean particle density

τi =

a a+c

  • L +

1 b+d − i

+

d b+d

  • i − 1 +

1 a+c

  • L +

1 a+c + 1 b+d − 1

, Mean particle current J = 2κ

a a+c − d b+d

L +

1 a+c + 1 b+d − 1. Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Generalized exclusion processes

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Physical motivation: construct integrable processes where the exclusion constraint is relaxed: s > 1 instead of s = 1.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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SLIDE 88

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Physical motivation: construct integrable processes where the exclusion constraint is relaxed: s > 1 instead of s = 1. Procedure: use the integrable sublattice parallel update with R matrix acting on Cs+1 ⊗ Cs+1 instead of C2 ⊗ C2

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Physical motivation: construct integrable processes where the exclusion constraint is relaxed: s > 1 instead of s = 1. Procedure: use the integrable sublattice parallel update with R matrix acting on Cs+1 ⊗ Cs+1 instead of C2 ⊗ C2 Such a R matrix can be obtain through fusion procedure

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Physical motivation: construct integrable processes where the exclusion constraint is relaxed: s > 1 instead of s = 1. Procedure: use the integrable sublattice parallel update with R matrix acting on Cs+1 ⊗ Cs+1 instead of C2 ⊗ C2 Such a R matrix can be obtain through fusion procedure Fusion procedure in a nutshell R-matrix associated to the SSEP: spin 1/2 representation of the universal R-matrix of Y(ˆ sl2).

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Physical motivation: construct integrable processes where the exclusion constraint is relaxed: s > 1 instead of s = 1. Procedure: use the integrable sublattice parallel update with R matrix acting on Cs+1 ⊗ Cs+1 instead of C2 ⊗ C2 Such a R matrix can be obtain through fusion procedure Fusion procedure in a nutshell R-matrix associated to the SSEP: spin 1/2 representation of the universal R-matrix of Y(ˆ sl2). Generalized exclusion processes: higher spin representations of the universal R-matrix.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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SLIDE 92

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Physical motivation: construct integrable processes where the exclusion constraint is relaxed: s > 1 instead of s = 1. Procedure: use the integrable sublattice parallel update with R matrix acting on Cs+1 ⊗ Cs+1 instead of C2 ⊗ C2 Such a R matrix can be obtain through fusion procedure Fusion procedure in a nutshell R-matrix associated to the SSEP: spin 1/2 representation of the universal R-matrix of Y(ˆ sl2). Generalized exclusion processes: higher spin representations of the universal R-matrix. Constructed from spin 1/2 R-matrix: we take tensor products of spin 1/2 representation and project on the appropriate invariant subspace.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Example: Generalized SSEP

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Example: Generalized SSEP Consider the R-matrix R(z) of the SSEP and the projectors

Q(l) =

1

1 1 1

  • and

Q(r) =

  

1 1/2 1/2 1

  

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Example: Generalized SSEP Consider the R-matrix R(z) of the SSEP and the projectors

Q(l) =

1

1 1 1

  • and

Q(r) =

  

1 1/2 1/2 1

  

Fusion of the “second tensor space”

Ri,<jk>(z) = Q(l)

jk Rij

  • z − 1

2

  • Rik
  • z + 1

2

  • Q(r)

jk .

Ri,<jk>(z) acts on C2 ⊗ C3 and satisfies some Yang-Baxter equation.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Example: Generalized SSEP Consider the R-matrix R(z) of the SSEP and the projectors

Q(l) =

1

1 1 1

  • and

Q(r) =

  

1 1/2 1/2 1

  

Fusion of the “second tensor space”

Ri,<jk>(z) = Q(l)

jk Rij

  • z − 1

2

  • Rik
  • z + 1

2

  • Q(r)

jk .

Ri,<jk>(z) acts on C2 ⊗ C3 and satisfies some Yang-Baxter equation. Fusion of the “first tensor space”

R(z) = R<hi>,<jk>(z) = Q(l)

hi Rh,<jk>

  • z + 1

2

  • Ri,<jk>
  • z − 1

2

  • Q(r)

hi .

R(z) acts on C3 ⊗ C3 and satisfies the Yang-Baxter equation, Markov property and unitarity.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

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Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

We have explicitly

R(z) =                 1

z z+2 2 z+2 z(z−1) (z+1)(z+2) z (z+1)(z+2) 2 (z+1)(z+2) 2 z+2 z z+2 4z (z+1)(z+2) z2+z+2 (z+1)(z+2) 4z (z+1)(z+2) z z+2 2 z+2 2 (z+1)(z+2) z (z+1)(z+2) z(z−1) (z+1)(z+2) 2 z+2 z z+2

1                

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-98
SLIDE 98

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

A similar fusion procedure can also be applied to the K-matrices

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-99
SLIDE 99

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

A similar fusion procedure can also be applied to the K-matrices

K(z) = K<ij>(z) = Q(l)

ij Ki

  • z − 1

2

  • Rji(2z)Kj
  • z + 1

2

  • Q(r)

ij

K(z) = K <ij>(z) = Q(l)

ij K i

  • z − 1

2

  • Rji(2z)−1K j
  • z + 1

2

  • Q(l)

ij . Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-100
SLIDE 100

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

A similar fusion procedure can also be applied to the K-matrices

K(z) = K<ij>(z) = Q(l)

ij Ki

  • z − 1

2

  • Rji(2z)Kj
  • z + 1

2

  • Q(r)

ij

K(z) = K <ij>(z) = Q(l)

ij K i

  • z − 1

2

  • Rji(2z)−1K j
  • z + 1

2

  • Q(l)

ij . K(z) =

       

∗

4cz (2z−1)(c−a)+2

  • (2z−1)(a+c)+2

(2z+1)(a+c)+2 8c2z(2z−1)

  • (2z−1)(a+c)+2

(2z+1)(a+c)+2 8az (2z−1)(c−a)+2

  • (2z−1)(a+c)+2

(2z+1)(a+c)+2

∗

8cz (2z−1)(a−c)+2

  • (2z−1)(a+c)+2

(2z+1)(a+c)+2 8a2z(2z−1)

  • (2z−1)(a+c)+2

(2z+1)(a+c)+2 4az (2z−1)(a−c)+2

  • (2z−1)(a+c)+2

(2z+1)(a+c)+2

∗

       

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-101
SLIDE 101

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

A similar fusion procedure can also be applied to the K-matrices

K(z) = K<ij>(z) = Q(l)

ij Ki

  • z − 1

2

  • Rji(2z)Kj
  • z + 1

2

  • Q(r)

ij

K(z) = K <ij>(z) = Q(l)

ij K i

  • z − 1

2

  • Rji(2z)−1K j
  • z + 1

2

  • Q(l)

ij . K(z) =

       

∗

4cz (2z−1)(c−a)+2

  • (2z−1)(a+c)+2

(2z+1)(a+c)+2 8c2z(2z−1)

  • (2z−1)(a+c)+2

(2z+1)(a+c)+2 8az (2z−1)(c−a)+2

  • (2z−1)(a+c)+2

(2z+1)(a+c)+2

∗

8cz (2z−1)(a−c)+2

  • (2z−1)(a+c)+2

(2z+1)(a+c)+2 8a2z(2z−1)

  • (2z−1)(a+c)+2

(2z+1)(a+c)+2 4az (2z−1)(a−c)+2

  • (2z−1)(a+c)+2

(2z+1)(a+c)+2

∗

       

K(z) =

       

∗

4bz (2z+1)(b−d)−2

  • (2z−1)(b+d)−2

(2z+1)(b+d)−2 8b2z(2z+1)

  • (2z−1)(b+d)−2

(2z+1)(b+d)−2 8dz (2z+1)(b−d)−2

  • (2z−1)(b+d)−2

(2z+1)(b+d)−2

∗

8bz (2z+1)(d−b)−2

  • (2z−1)(b+d)−2

(2z+1)(b+d)−2 8d2z(2z+1)

  • (2z−1)(b+d)−2

(2z+1)(b+d)−2 4dz (2z+1)(d−b)−2

  • (2z−1)(b+d)−2

(2z+1)(b+d)−2

∗

       

. Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-102
SLIDE 102

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

The local update rules in the bulk are given by the operator U = ˇ R(2κ):

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-103
SLIDE 103

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

The local update rules in the bulk are given by the operator U = ˇ R(2κ):

01 − → 10 κ κ + 1 10 − → 01 κ κ + 1 12 − → 21 κ κ + 1 21 − → 12 κ κ + 1 02 − → 11 4κ (2κ + 1)(κ + 1); 02 − → 20 (2κ − 1)κ (2κ + 1)(κ + 1) 20 − → 11 4κ (2κ + 1)(κ + 1); 20 − → 02 (2κ − 1)κ (2κ + 1)(κ + 1) 11 − → 02 κ (2κ + 1)(κ + 1); 11 − → 20 κ (2κ + 1)(κ + 1);

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-104
SLIDE 104

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

On the left boundary the dynamics is encoded by B = K(κ)

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-105
SLIDE 105

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

On the left boundary the dynamics is encoded by B = K(κ)

0 − → 1 8aκ (2κ − 1)(c − a) + 2

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 ; 0 − → 2 8a2κ(2κ − 1)

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 1 − → 0 4cκ (2κ − 1)(c − a) + 2

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 ; 1 − → 2 4aκ (2κ − 1)(a − c) + 2

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 2 − → 0 8c2κ(2κ − 1)

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 ; 2 − → 1 8cκ (2κ − 1)(a − c) + 2

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-106
SLIDE 106

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

On the left boundary the dynamics is encoded by B = K(κ)

0 − → 1 8aκ (2κ − 1)(c − a) + 2

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 ; 0 − → 2 8a2κ(2κ − 1)

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 1 − → 0 4cκ (2κ − 1)(c − a) + 2

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 ; 1 − → 2 4aκ (2κ − 1)(a − c) + 2

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 2 − → 0 8c2κ(2κ − 1)

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 ; 2 − → 1 8cκ (2κ − 1)(a − c) + 2

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2

On the right boundary the dynamics is encoded by B = K(−κ)

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-107
SLIDE 107

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

On the left boundary the dynamics is encoded by B = K(κ)

0 − → 1 8aκ (2κ − 1)(c − a) + 2

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 ; 0 − → 2 8a2κ(2κ − 1)

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 1 − → 0 4cκ (2κ − 1)(c − a) + 2

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 ; 1 − → 2 4aκ (2κ − 1)(a − c) + 2

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 2 − → 0 8c2κ(2κ − 1)

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2 ; 2 − → 1 8cκ (2κ − 1)(a − c) + 2

  • (2κ − 1)(a + c) + 2

(2κ + 1)(a + c) + 2

On the right boundary the dynamics is encoded by B = K(−κ)

0 − → 1 8dκ (2κ − 1)(b − d) + 2

  • (2κ − 1)(b + d) + 2

(2κ + 1)(b + d) + 2 ; 0 − → 2 8d2κ(2κ − 1)

  • (2κ − 1)(b + d) + 2

(2κ + 1)(b + d) + 2 1 − → 0 4bκ (2κ − 1)(b − d) + 2

  • (2κ − 1)(b + d) + 2

(2κ + 1)(b + d) + 2 ; 1 − → 2 4dκ (2κ − 1)(d − b) + 2

  • (2κ − 1)(b + d) + 2

(2κ + 1)(b + d) + 2 2 − → 0 8b2κ(2κ − 1)

  • (2κ − 1)(b + d) + 2

(2κ + 1)(b + d) + 2 ; 2 − → 1 8bκ (2κ − 1)(d − b) + 2

  • (2κ − 1)(b + d) + 2

(2κ + 1)(b + d) + 2 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-108
SLIDE 108

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Remark: for physical applications, it might be useful to take κ = 1/2.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-109
SLIDE 109

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Remark: for physical applications, it might be useful to take κ = 1/2. In the bulk:

01 − → 10 1 3; 10 − → 01 1 3; 12 − → 21 1 3; 21 − → 12 1 3; 02 − → 11 2 3; 20 − → 11 2 3; 11 − → 02 1 6; 11 − → 20 1 6; 02 − → 20 0; 20 − → 02 0;

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-110
SLIDE 110

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Remark: for physical applications, it might be useful to take κ = 1/2. In the bulk:

01 − → 10 1 3; 10 − → 01 1 3; 12 − → 21 1 3; 21 − → 12 1 3; 02 − → 11 2 3; 20 − → 11 2 3; 11 − → 02 1 6; 11 − → 20 1 6; 02 − → 20 0; 20 − → 02 0;

At the left boundary:

0 − → 1 2a a + c + 1; 1 − → 2 a a + c + 1; 0 − → 2 0; 2 − → 1 2c a + c + 1; 1 − → 0 c a + c + 1; 2 − → 0 0;

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-111
SLIDE 111

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Remark: for physical applications, it might be useful to take κ = 1/2. In the bulk:

01 − → 10 1 3; 10 − → 01 1 3; 12 − → 21 1 3; 21 − → 12 1 3; 02 − → 11 2 3; 20 − → 11 2 3; 11 − → 02 1 6; 11 − → 20 1 6; 02 − → 20 0; 20 − → 02 0;

At the left boundary:

0 − → 1 2a a + c + 1; 1 − → 2 a a + c + 1; 0 − → 2 0; 2 − → 1 2c a + c + 1; 1 − → 0 c a + c + 1; 2 − → 0 0;

At the right boundary:

0 − → 1 2d b + d + 1; 1 − → 2 d b + d + 1; 0 − → 2 0; 2 − → 1 2b b + d + 1; 1 − → 0 b b + d + 1; 2 − → 0 0;

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-112
SLIDE 112

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

A(z) = Q(l)A

  • z − 1

2

  • ⊗ A
  • z + 1

2

  • where A(z) is the vector used for the SSEP.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-113
SLIDE 113

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

A(z) = Q(l)A

  • z − 1

2

  • ⊗ A
  • z + 1

2

  • where A(z) is the vector used for the SSEP.

A(z) =

    

  • z + 1

2

z − 1

2

  • − 2zE + E2

−2

  • z + 1

2

z − 1

2

  • + 2z(E − D) + ED + DE
  • z + 1

2

z − 1

2

  • + 2zD + D2

    

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-114
SLIDE 114

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

A(z) = Q(l)A

  • z − 1

2

  • ⊗ A
  • z + 1

2

  • where A(z) is the vector used for the SSEP.

A(z) =

    

  • z + 1

2

z − 1

2

  • − 2zE + E2

−2

  • z + 1

2

z − 1

2

  • + 2z(E − D) + ED + DE
  • z + 1

2

z − 1

2

  • + 2zD + D2

    

A direct computation shows that ˇ R(z1 − z2)A(z1) ⊗ A(z2) = A(z2) ⊗ A(z1).

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-115
SLIDE 115

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

A(z) = Q(l)A

  • z − 1

2

  • ⊗ A
  • z + 1

2

  • where A(z) is the vector used for the SSEP.

A(z) =

    

  • z + 1

2

z − 1

2

  • − 2zE + E2

−2

  • z + 1

2

z − 1

2

  • + 2z(E − D) + ED + DE
  • z + 1

2

z − 1

2

  • + 2zD + D2

    

A direct computation shows that ˇ R(z1 − z2)A(z1) ⊗ A(z2) = A(z2) ⊗ A(z1). and

  • W |K(z)A(−z) =

W |A(z) and K(z)A(−z)|V = A(z)|V .

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-116
SLIDE 116

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Steady state |S = 1 ZL

  • W |A(κ) ⊗ A(−κ) ⊗ A(κ) ⊗ · · · ⊗ A(κ)|V

.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-117
SLIDE 117

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Steady state |S = 1 ZL

  • W |A(κ) ⊗ A(−κ) ⊗ A(κ) ⊗ · · · ⊗ A(κ)|V

. It allows us to compute physical quantities Normalization ZL = (a + c)2L(b + d)2L (ab − cd)2L Γ

  • 2L +

1 a+c + 1 b+d

  • Γ
  • 1

a+c + 1 b+d

  • Matthieu VANICAT

Integrable discrete-time dynamics, generalized exclusion processes

slide-118
SLIDE 118

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Steady state |S = 1 ZL

  • W |A(κ) ⊗ A(−κ) ⊗ A(κ) ⊗ · · · ⊗ A(κ)|V

. It allows us to compute physical quantities Normalization ZL = (a + c)2L(b + d)2L (ab − cd)2L Γ

  • 2L +

1 a+c + 1 b+d

  • Γ
  • 1

a+c + 1 b+d

  • Mean particle current

J = 4κ

a a+c − d b+d

2L +

1 a+c + 1 b+d − 1.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-119
SLIDE 119

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Perspectives

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-120
SLIDE 120

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Perspectives Investigate the hydrodynamic limit of these generalized exclusion processes (phase transitions in the generalized ASEP, MFT in the generalized SSEP).

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-121
SLIDE 121

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Perspectives Investigate the hydrodynamic limit of these generalized exclusion processes (phase transitions in the generalized ASEP, MFT in the generalized SSEP). Study the higher “spin” representation with s particles allowed on the same site.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-122
SLIDE 122

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Perspectives Investigate the hydrodynamic limit of these generalized exclusion processes (phase transitions in the generalized ASEP, MFT in the generalized SSEP). Study the higher “spin” representation with s particles allowed on the same site. Compute other eigenvectors (excited states) in matrix product form.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-123
SLIDE 123

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Perspectives Investigate the hydrodynamic limit of these generalized exclusion processes (phase transitions in the generalized ASEP, MFT in the generalized SSEP). Study the higher “spin” representation with s particles allowed on the same site. Compute other eigenvectors (excited states) in matrix product form. Use time dependent matrix product states to describe the time evolution of the models.

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

slide-124
SLIDE 124

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Fusion procedure Definition of the process “Fused” matrix ansatz

Thank you!

Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes