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Complexity of exact solutions of many body systems: nonequilibrium steady states - Tomaž Prosen

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Let’s face complexity September 4-8, 2017

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Complexity of exact solutions of many body systems: nonequilibrium steady states - Tomaž Prosen

  1. 1. Complexity of exact solutions of many body systems: nonequilibrium steady states Tomaž Prosen Faculty of mathematics and physics, University of Ljubljana, Slovenia Let’s face complexity, Como, 4-8 September 2017 Tomaž Prosen Complexity of exact many-body solutions
  2. 2. Program of the talk Boundary driven interacting quantum chain paradigm and exact solutions via Matrix Product Ansatz New conservation laws and exact bounds on transport coefficients Boundary driven classical chains: reversible cellular automata Topical Review article, J. Phys. A: Math. Theor. 48, 373001 (2015) Collaborators involved in this work: E. Ilievski (Amsterdam), V. Popkov (Bonn), I. Pižorn (now in finance), C. Mejia-Monasterio (Madrid), B. Buča (Split) Tomaž Prosen Complexity of exact many-body solutions
  3. 3. Nonequilibrium quantum transport problem in one-dimension Canonical markovian master equation for the many-body density matrix: The Lindblad (L-GKS) equation: dρ dt = ˆLρ := −i[H, ρ] + µ 2LµρL† µ − {L† µLµ, ρ} . Tomaž Prosen Complexity of exact many-body solutions
  4. 4. Nonequilibrium quantum transport problem in one-dimension Canonical markovian master equation for the many-body density matrix: The Lindblad (L-GKS) equation: dρ dt = ˆLρ := −i[H, ρ] + µ 2LµρL† µ − {L† µLµ, ρ} . Bulk: Fully coherent, local interactions,e.g. H = n−1 x=1 hx,x+1. Boundaries: Fully incoherent, ultra-local dissipation, jump operators Lµ supported near boundaries x = 1 or x = n. Tomaž Prosen Complexity of exact many-body solutions
  5. 5. A warmup and a teaser: noneq. phase transition in a driven XY chain TP & I. Pižorn, PRL 101, 105701 (2008) H = n j 1+γ 2 σx j σx j+1 + 1−γ 2 σy j σy j+1 + hσz j L1 = 1 2 ΓL 1 σ− 1 L3 = 1 2 ΓR 1 σ− n L2 = 1 2 ΓL 2 σ+ 1 L4 = 1 2 ΓR 2 σ+ n C(j, k) = σz j σz k − σz j σz k hc = |1 − γ2 | 0 Γ 1 0 h 1 10 9 10 8 10 7 10 6 10 5 10 4 Cres 00.0010.002 0 0.5 1. 1.5 Re Β ImΒ h 0.9 hc Π 2 1 0 1 2 Π Φ 0 0.001 0.002 Re Β h 0.3 hc Tomaž Prosen Complexity of exact many-body solutions
  6. 6. Fluctuation of spin-spin correlation in NESS and "wave resonators" Near neQPT: Scaling variable z = (hc − h)n2 Tomaž Prosen Complexity of exact many-body solutions
  7. 7. Fluctuation of spin-spin correlation in NESS and "wave resonators" Near neQPT: Scaling variable z = (hc − h)n2 Scaling ansatz: C2j+α,2k+β = Ψα,β (x = j/n, y = k/n, z) Tomaž Prosen Complexity of exact many-body solutions
  8. 8. Fluctuation of spin-spin correlation in NESS and "wave resonators" Near neQPT: Scaling variable z = (hc − h)n2 Scaling ansatz: C2j+α,2k+β = Ψα,β (x = j/n, y = k/n, z) Certain combination Ψ(x, y) = (∂/∂x + ∂/∂y )(Ψ0,0 (x, y) + Ψ1,1 (x, y)) obeys the Helmholtz equation!!! ∂2 ∂x2 + ∂2 ∂y2 + 4z Ψ = ”octopole antenna sources” Tomaž Prosen Complexity of exact many-body solutions
  9. 9. Paradigm of exactly solvable NESS: boundary driven XXZ chain Steady state Lindblad equation ˆLρ∞ = 0: i[H, ρ∞] = µ 2Lµρ∞L† µ − {L† µLµ, ρ∞} The XXZ Hamiltonian: H = n−1 x=1 (2σ+ x σ− x+1 + 2σ− x σ+ x+1 + ∆σz x σz x+1) and symmetric boundary (ultra local) Lindblad jump operators: LL 1 = 1 2 (1 − µ)ε σ+ 1 , LR 1 = 1 2 (1 + µ)ε σ+ n , LL 2 = 1 2 (1 + µ)ε σ− 1 , LR 2 = 1 2 (1 − µ)ε σ− n . Two key boundary parameters: ε System-bath coupling strength µ Non-equilibrium driving strength (bias) Tomaž Prosen Complexity of exact many-body solutions
  10. 10. Cholesky decomposition of NESS and Matrix Product Ansatz (for µ = 1) TP, PRL106(2011); PRL107(2011); Karevski, Popkov, Schütz, PRL111(2013) ρ∞ = ( tr R)−1 R, R = ΩΩ† Ω = (s1,...,sn)∈{+,−,0}n 0|As1 As2 · · · Asn |0 σs1 ⊗ σs2 · · · ⊗ σsn = 0|L(ϕ, s)⊗x n |0 Tomaž Prosen Complexity of exact many-body solutions
  11. 11. Cholesky decomposition of NESS and Matrix Product Ansatz (for µ = 1) TP, PRL106(2011); PRL107(2011); Karevski, Popkov, Schütz, PRL111(2013) ρ∞ = ( tr R)−1 R, R = ΩΩ† Ω = (s1,...,sn)∈{+,−,0}n 0|As1 As2 · · · Asn |0 σs1 ⊗ σs2 · · · ⊗ σsn = 0|L(ϕ, s)⊗x n |0 A0 = ∞ k=0 a0 k |k k|, A+ = ∞ k=0 a+ k |k k+1|, A− = ∞ k=0 a− k |k+1 r|, 0 1 2 3 4 Tomaž Prosen Complexity of exact many-body solutions
  12. 12. Cholesky decomposition of NESS and Matrix Product Ansatz (for µ = 1) TP, PRL106(2011); PRL107(2011); Karevski, Popkov, Schütz, PRL111(2013) ρ∞ = ( tr R)−1 R, R = ΩΩ† Ω = (s1,...,sn)∈{+,−,0}n 0|As1 As2 · · · Asn |0 σs1 ⊗ σs2 · · · ⊗ σsn = 0|L(ϕ, s)⊗x n |0 A0 = ∞ k=0 a0 k |k k|, A+ = ∞ k=0 a+ k |k k+1|, A− = ∞ k=0 a− k |k+1 r|, 0 1 2 3 4 where (for symmetric driving): ϕ = π 2 tan(ηs) := ε 2i sin η Tomaž Prosen Complexity of exact many-body solutions
  13. 13. Cholesky decomposition of NESS and Matrix Product Ansatz (for µ = 1) TP, PRL106(2011); PRL107(2011); Karevski, Popkov, Schütz, PRL111(2013) ρ∞ = ( tr R)−1 R, R = ΩΩ† Ω = (s1,...,sn)∈{+,−,0}n 0|As1 As2 · · · Asn |0 σs1 ⊗ σs2 · · · ⊗ σsn = 0|L(ϕ, s)⊗x n |0 A0 = ∞ k=0 a0 k |k k|, A+ = ∞ k=0 a+ k |k k+1|, A− = ∞ k=0 a− k |k+1 r|, 0 1 2 3 4 where (for symmetric driving): ϕ = π 2 tan(ηs) := ε 2i sin η Ω(ϕ, s) = 0|L(ϕ, s)⊗x n |0 plays a role of a highest-weight transfer matrix cor- responding to non-unitary representation of the Lax operator [Ω(ϕ, s), Ω(ϕ , s )] = 0, ∀s, s , ϕ, ϕ Tomaž Prosen Complexity of exact many-body solutions
  14. 14. Uq(sl2) Lax operator L(ϕ, s) = sin(ϕ + ηSz s ) (sin η)S− s (sin η)S+ s sin(ϕ − ηSz s ) where S±,z s is the highest-weight complex-spin irrep: Sz s = ∞ k=0 (s − k)|k k|, S+ s = ∞ k=0 sin(k + 1)η sin η |k k + 1|, S− s = ∞ k=0 sin(2s − k)η sin η |k + 1 k|. Tomaž Prosen Complexity of exact many-body solutions
  15. 15. Uq(sl2) Lax operator L(ϕ, s) = sin(ϕ + ηSz s ) (sin η)S− s (sin η)S+ s sin(ϕ − ηSz s ) where S±,z s is the highest-weight complex-spin irrep: Sz s = ∞ k=0 (s − k)|k k|, S+ s = ∞ k=0 sin(k + 1)η sin η |k k + 1|, S− s = ∞ k=0 sin(2s − k)η sin η |k + 1 k|. Example: Fundamental rep. of auxiliary space s = 1/2, generates local charges: Q( ) = ∂ −1 ϕ log traL⊗x n (ϕ, 1 2 )|ϕ= η 2 = x q ( ) x Tomaž Prosen Complexity of exact many-body solutions
  16. 16. Uq(sl2) Lax operator L(ϕ, s) = sin(ϕ + ηSz s ) (sin η)S− s (sin η)S+ s sin(ϕ − ηSz s ) where S±,z s is the highest-weight complex-spin irrep: Sz s = ∞ k=0 (s − k)|k k|, S+ s = ∞ k=0 sin(k + 1)η sin η |k k + 1|, S− s = ∞ k=0 sin(2s − k)η sin η |k + 1 k|. Example: Fundamental rep. of auxiliary space s = 1/2, generates local charges: Q( ) = ∂ −1 ϕ log traL⊗x n (ϕ, 1 2 )|ϕ= η 2 = x q ( ) x Q(2) = H = x (2σ+ x σ− x+1 + 2σ− x σ+ x+1 + ∆σz x σz x+1) [Q( ) , Q( ) ] = 0. Tomaž Prosen Complexity of exact many-body solutions
  17. 17. Observables in NESS: From insulating to ballistic transport For |∆| < 1, J ∼ n0 (ballistic) For |∆| > 1, J ∼ exp(−constn) (insulating) For |∆| = 1, J ∼ n−2 (anomalous) 0 20 40 60 80 100 1.0 0.5 0.0 0.5 1.0 j Σj z a 5 10 50 100 10 4 0.001 0.01 0.1 n J b 10 20 30 40 10 17 10 14 10 11 10 8 10 5 0.01 Tomaž Prosen Complexity of exact many-body solutions
  18. 18. Two-point spin-spin correlation function in NESS C x n , y n = σz x σz y − σz x σz y for isotropic case ∆ = 1 (XXX) C(ξ1, ξ2) = − π2 2n ξ1(1 − ξ2) sin(πξ1) sin(πξ2), for ξ1 < ξ2 Tomaž Prosen Complexity of exact many-body solutions
  19. 19. Application: linear response spin transport Green-Kubo formulae express the conductivities in terms of current a.c.f. κ(ω) = lim t→∞ lim n→∞ β n t 0 dt eiωt J(t )J(0) β When d.c. conductivity diverges, one defines a Drude weight D κ(ω) = 2πDδ(ω) + κreg(ω) which in linear response expresses as D = lim t→∞ lim n→∞ β 2tn t 0 dt J(t )J(0) β. Tomaž Prosen Complexity of exact many-body solutions
  20. 20. Application: linear response spin transport Green-Kubo formulae express the conductivities in terms of current a.c.f. κ(ω) = lim t→∞ lim n→∞ β n t 0 dt eiωt J(t )J(0) β When d.c. conductivity diverges, one defines a Drude weight D κ(ω) = 2πDδ(ω) + κreg(ω) which in linear response expresses as D = lim t→∞ lim n→∞ β 2tn t 0 dt J(t )J(0) β. For integrable quantum systems, Zotos, Naef and Prelovšek (1997) suggested to use Mazur/Suzuki (1969/1971) bound: D ≥ lim n→∞ β 2n m JQ(m) 2 β [Q(m)]2 β with conserved Q(m) chosen mutually orthogonal Q(m) Q(k) β = 0 for m = k. Tomaž Prosen Complexity of exact many-body solutions
  21. 21. Application: linear response spin transport Green-Kubo formulae express the conductivities in terms of current a.c.f. κ(ω) = lim t→∞ lim n→∞ β n t 0 dt eiωt J(t )J(0) β When d.c. conductivity diverges, one defines a Drude weight D κ(ω) = 2πDδ(ω) + κreg(ω) which in linear response expresses as D = lim t→∞ lim n→∞ β 2tn t 0 dt J(t )J(0) β. For integrable quantum systems, Zotos, Naef and Prelovšek (1997) suggested to use Mazur/Suzuki (1969/1971) bound: D ≥ lim n→∞ β 2n m JQ(m) 2 β [Q(m)]2 β with conserved Q(m) chosen mutually orthogonal Q(m) Q(k) β = 0 for m = k. Note that conserved operators Q(m) need not be strictly local, but may be only pseudo-local for the lower bound to exist in thermodynamic limit n → ∞. Tomaž Prosen Complexity of exact many-body solutions
  22. 22. Spin reversal symmetry and the XXZ spin transport controversy Spin flip: P = (σx )⊗n Hamiltonian and all the local charges are even under P PQ( ) = Q( ) P while the spin current is odd J = i x (σ+ x σ− x+1 − σ− x σ+ x+1) PJ = −JP Tomaž Prosen Complexity of exact many-body solutions
  23. 23. Spin reversal symmetry and the XXZ spin transport controversy Spin flip: P = (σx )⊗n Hamiltonian and all the local charges are even under P PQ( ) = Q( ) P while the spin current is odd J = i x (σ+ x σ− x+1 − σ− x σ+ x+1) PJ = −JP Consequence: JQ( ) β = 0 and Mazur bound (using only local charges) vanishes! Tomaž Prosen Complexity of exact many-body solutions
  24. 24. Spin reversal symmetry and the XXZ spin transport controversy Spin flip: P = (σx )⊗n Hamiltonian and all the local charges are even under P PQ( ) = Q( ) P while the spin current is odd J = i x (σ+ x σ− x+1 − σ− x σ+ x+1) PJ = −JP Consequence: JQ( ) β = 0 and Mazur bound (using only local charges) vanishes! However, various numerical and heuristic theoretical results accumulated until 2011 indicated that finite temperature Drude weight in the massless (critical) regime |∆| < 1 is strictly positive! Tomaž Prosen Complexity of exact many-body solutions
  25. 25. Perturbative NESS: Quasilocal charge odd under spin reversal! ρ∞ = 1 + iεQ− + O(ε2 ) Q− = i(Z − Z† ) where Z = cn∂s 0|L(ϕ, s)⊗x n |0 |s=0 Tomaž Prosen Complexity of exact many-body solutions
  26. 26. Perturbative NESS: Quasilocal charge odd under spin reversal! ρ∞ = 1 + iεQ− + O(ε2 ) Q− = i(Z − Z† ) where Z = cn∂s 0|L(ϕ, s)⊗x n |0 |s=0 Note that 1 PZ = −Z† P, hence PQ− = −Q− P 2 Q− is pseudo-local in the easy-plane (gapless) regime |∆| < 1, [Q− ]2 = O(n), jx Q− = O(1) 3 Q− is almost conserved (satisfying conservation law property) [H, Q− ] = 2σz 1 − 2σz n [TP, PRL106 (2011)] Tomaž Prosen Complexity of exact many-body solutions
  27. 27. Perturbative NESS: Quasilocal charge odd under spin reversal! ρ∞ = 1 + iεQ− + O(ε2 ) Q− = i(Z − Z† ) where Z = cn∂s 0|L(ϕ, s)⊗x n |0 |s=0 Note that 1 PZ = −Z† P, hence PQ− = −Q− P 2 Q− is pseudo-local in the easy-plane (gapless) regime |∆| < 1, [Q− ]2 = O(n), jx Q− = O(1) 3 Q− is almost conserved (satisfying conservation law property) [H, Q− ] = 2σz 1 − 2σz n [TP, PRL106 (2011)] Moreover, replacing 0| • |0 −→ tr a(•), for root-of-unity anisotropies ∆ = cos(πl/m), l, m ∈ Z, Q− becomes exactly conserved for periodic boundary conditions [Hpbc, Q− ] = 0 TP, NPB886(2014); Pereira et al., JSTAT2014 Tomaž Prosen Complexity of exact many-body solutions
  28. 28. Mazur bound with the novel quasi-local CLs Almost conserved, odd quasi-local operators Q(ϕ) can be used to rigorously bound spin Drude weight: Fractal Drude weight bound D β ≥ DZ := sin2 (πl/m) sin2 (π/m) 1 − m 2π sin 2π m , ∆ = cos πl m 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 DZ,DK TP, PRL 106 (2011); TP and Ilievski, PRL 111 (2013) Agrees very well with numerical simulations on finite systems based on DMRG [Karrasch et al, PRL108 (2012); PRB87 (2013)] and wave-function sampling of exact dynamics [Steinigeweg et al. PRL112 (2014)] Tomaž Prosen Complexity of exact many-body solutions
  29. 29. Another model: Charge and spin diffusion in inf. temp. Hubbard model Hamiltonian is rewritten from fermionic to spin-ladder formulation: H = −t L−1 i=1 s∈{↑,↓} (c† i,s ci+1,s + h.c.) + U L i=1 ni↑ni↓, = − t 2 L−1 i=1 (σx i σx i+1 + σy i σy i+1 + τx i τx i+1 + τy i τy i+1) + U 4 L i=1 (σz i + 1)(τz i + 1). And the following simplest boundary driving channels are considered: L1,2 = ε(1 µ) σ± 1 , L3,4 = ε(1 ± µ) σ± L L5,6 = ε(1 µ) τ± 1 , L7,8 = ε(1 ± µ) τ± L [TP and M. Žnidarič, PRB 86, 125118 (2012)] Tomaž Prosen Complexity of exact many-body solutions
  30. 30. Diffusion in 1D Hubbard model: DMRG results, PRB 86, 125118 (2012). 0.4 0.45 0.5 0.55 0.6 0 20 40 60 80 100 ni,s i 1 10 10 100 j(c) /(L∇n) L-1 0.2 U=1.0 0.5 2.0 Tomaž Prosen Complexity of exact many-body solutions
  31. 31. 0 0.002 0.004 0.006 0.008 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 <ni↑ni↓>c (i-0.5)/L (b) L=16,32,64 0.4 0.45 0.5 0.55 0.6 ni,s (a) 10 100 0.01 0.1 ∆ n L n2↑-n1↑ n1↑-nL↑ Tomaž Prosen Complexity of exact many-body solutions
  32. 32. Analytical solution of NESS for Hubbard chain TP, PRL 112, 030603 (2014) Extremely boundary driven open fermi Hubbard chain Hn = n−1 j=1 (σ+ j σ− j+1 + τ+ j τ− j+1 + H.c.) + u 4 n j=1 σz j τz j L1 = √ εσ+ 1 , L2 = √ ετ+ 1 , L3 = √ εσ− n , L4 = √ ετ− n . Tomaž Prosen Complexity of exact many-body solutions
  33. 33. Analytical solution of NESS for Hubbard chain TP, PRL 112, 030603 (2014) Extremely boundary driven open fermi Hubbard chain Hn = n−1 j=1 (σ+ j σ− j+1 + τ+ j τ− j+1 + H.c.) + u 4 n j=1 σz j τz j L1 = √ εσ+ 1 , L2 = √ ετ+ 1 , L3 = √ εσ− n , L4 = √ ετ− n . Key ansatz for NESS density matrix ˆLρ∞ = 0, again a decomposition a-la Cholesky: ρ∞ = ΩnΩ† n. Tomaž Prosen Complexity of exact many-body solutions
  34. 34. Walking graph state representation of the solution 0 1 2 3 4 1 2 3 2 5 2 7 2 1 2 3 2 5 2 7 2 Ωn = e∈Wn(0,0) ae1 ae2 · · · aen n j=1 σ bx (ej ) j τ by (ej ) j . Tomaž Prosen Complexity of exact many-body solutions
  35. 35. Lax form of NESS for the Hubbard chain Popkov and TP, PRL 114, 127201 (2015) Amplitude Lindblad equation can again be reduced to Local Yang Baxter equation [hx,x+1, Lx,aLx+1,a] = ~Lx,a Lx+1,a − Lx,a ~Lx+1,a plus appropriate boundary conditions for the dissipators, solved with: Ωn(ε) = 0|aL1,aL2,a · · · Ln,a|0 a, again forming a commuting family [Ωn(ε), Ωn(ε )] = 0. Tomaž Prosen Complexity of exact many-body solutions
  36. 36. Lax operator admits very appealing factorization Lx,a(ε) = α,β∈{+,−,0,z} Sα a Tβ a Xa(u, ε)σα x τβ x 1 2 3 2 5 2 7 2 1 2 3 2 5 2 7 2 0 1 1 2 2 3 3 4 4 S 1 2 3 2 5 2 7 2 1 2 3 2 5 2 7 2 0 1 1 2 2 3 3 4 4 S 1 2 3 2 5 2 7 2 1 2 3 2 5 2 7 2 0 1 1 2 2 3 3 4 4 S0 T0 1 2 3 2 5 2 7 2 1 2 3 2 5 2 7 2 0 1 1 2 2 3 3 4 4 X 1 2 3 2 5 2 7 2 1 2 3 2 5 2 7 2 0 1 1 2 2 3 3 4 4 T 1 2 3 2 5 2 7 2 1 2 3 2 5 2 7 2 0 1 1 2 2 3 3 4 4 T 1 2 3 2 5 2 7 2 1 2 3 2 5 2 7 2 0 1 1 2 2 3 3 4 4 Sz Tz Tomaž Prosen Complexity of exact many-body solutions
  37. 37. Lax operator admits very appealing factorization Lx,a(ε) = α,β∈{+,−,0,z} Sα a Tβ a Xa(u, ε)σα x τβ x 1 2 3 2 5 2 7 2 1 2 3 2 5 2 7 2 0 1 1 2 2 3 3 4 4 S 1 2 3 2 5 2 7 2 1 2 3 2 5 2 7 2 0 1 1 2 2 3 3 4 4 S 1 2 3 2 5 2 7 2 1 2 3 2 5 2 7 2 0 1 1 2 2 3 3 4 4 S0 T0 1 2 3 2 5 2 7 2 1 2 3 2 5 2 7 2 0 1 1 2 2 3 3 4 4 X 1 2 3 2 5 2 7 2 1 2 3 2 5 2 7 2 0 1 1 2 2 3 3 4 4 T 1 2 3 2 5 2 7 2 1 2 3 2 5 2 7 2 0 1 1 2 2 3 3 4 4 T 1 2 3 2 5 2 7 2 1 2 3 2 5 2 7 2 0 1 1 2 2 3 3 4 4 Sz Tz Sα a and Tβ a are particular commuting ([Sα a , Tβ a ] = 0) representations of an extended CAR: {S+ , S− } = 2(S0 − Sz ), [Sα , S0 ] = [Sα , Sz ] = 0. Tomaž Prosen Complexity of exact many-body solutions
  38. 38. Lai-Sutherland SU(3) model with degenerate dissipative driving Ilievski and TP, NPB 882 (2014) H = n−1 x=1 hx,x+1 Tomaž Prosen Complexity of exact many-body solutions
  39. 39. Lai-Sutherland SU(3) model with degenerate dissipative driving Ilievski and TP, NPB 882 (2014) H = n−1 x=1 hx,x+1 Again, Cholesky factor of NESS has a Lax form, where the auxiliary space now needs 2 oscillator modes and a complex spin (basis labelled by inf. 3D lattice). Tomaž Prosen Complexity of exact many-body solutions
  40. 40. Lai-Sutherland SU(3) model with degenerate dissipative driving Ilievski and TP, NPB 882 (2014) H = n−1 x=1 hx,x+1 Again, Cholesky factor of NESS has a Lax form, where the auxiliary space now needs 2 oscillator modes and a complex spin (basis labelled by inf. 3D lattice). NESS manifold is infinitely degenerate (in TD limit). NESSes can be labeled by the fixed number of green particles (‘doping’/chemical potential in TDL). Tomaž Prosen Complexity of exact many-body solutions
  41. 41. What about strongly correlated steady states in conservative deterministic classical interacting systems with stochastic boundary driving? Tomaž Prosen Complexity of exact many-body solutions
  42. 42. Integrable reversible cellular automaton, the rule 54 sS = χ(sW, sN, sE) = sN + sW + sE + sWsE (mod 2) Expressed as 8-bit numbers, where the value of the ith bit is F(q), these eight rules are 204 (11001100), 51 (00110011), 150 (10010110), 105 (01101001), 108 (01101100), 147 (10010011), 54 (00110110), 201 (11001001). Warning: These binary numbers have to be read from the right to the left in order to get the values of CQ, Cj, . . . , CÁ. ÷÷÷ Fig. 3. Rule 54 0 × 20 + 1 × 21 + 1 × 22 + 0 × 23 + 1 × 24 + 1 × 25 + 0 × 26 + 0 × 27 = 54 Bobenko et al., Commun. Math. Phys. 158, 127 (1993) Tomaž Prosen Complexity of exact many-body solutions
  43. 43. Integrable reversible cellular automaton, the rule 54 sS = χ(sW, sN, sE) = sN + sW + sE + sWsE (mod 2) rules are 204 (11001100), 51 (00110011), 150 (10010110), 105 (01101001), 108 (01101100), 147 (10010011), 54 (00110110), 201 (11001001). Warning: These binary numbers have to be read from the right to the left in order to get the values of CQ, Cj, . . . , CÁ. ÷÷÷ Fig. 3. Rule 54 0 × 20 + 1 × 21 + 1 × 22 + 0 × 23 + 1 × 24 + 1 × 25 + 0 × 26 + 0 × 27 = 54 Bobenko et al., Commun. Math. Phys. 158, 127 (1993) h we choose in preference to F5 since it has a 0 vacuum state, i.e.,a state s preserved under the system.1 he system equivalent to F4 was considered in [1]. It does not possess particle tures against the vacuum: computation shows that the particle world-lines ence of √s or black squares) are approximately vertical, i.e.at rest. rom now on in all figures faces with ı = 1 are shaded or colored black. Vac (v = 0) will always be white. Figures 4 and 5 show typical evolutions o ule. Figure 4 has a region of random initial conditions within a backgroun Fig. 5 shows the collision of two oppositely directed particles. Notice tha particles pass through one another completely, but with a time delay (phase- acteristic of solitons. 4. Evolution of rule 54 from random initial configurationTomaž Prosen Complexity of exact many-body solutions
  44. 44. Constructing a Markov matrix: deterministic bulk + stochastic boundaries T.P., C.Mejia-Monasterio, J. Phys. A: Math. Theor. 49, 185003 (2016) Describe an evolution of probability state vector for n−cell automaton p(t) = Ut p(0) p = (p0, p1, . . . , p2n−1) ≡ (ps1,s2,...,sn ; sj {0, 1}) s1 s2 s3 s4 s5 sn-2 sn-1 sn Tomaž Prosen Complexity of exact many-body solutions
  45. 45. ... ... P P P PR PL P P P 1 2 3 4 n-1 n Ue Uo Tomaž Prosen Complexity of exact many-body solutions
  46. 46. ... ... P P P PR PL P P P 1 2 3 4 n-1 n Ue Uo U = UoUe, Ue = P123P345 · · · Pn−3,n−2,n−1PR n−1,n, Uo = Pn−2,n−1,n · · · P456P234PL 12. Tomaž Prosen Complexity of exact many-body solutions
  47. 47. ... ... P P P PR PL P P P 1 2 3 4 n-1 n Ue Uo U = UoUe, Ue = P123P345 · · · Pn−3,n−2,n−1PR n−1,n, Uo = Pn−2,n−1,n · · · P456P234PL 12. P =           1 1 1 1 1 1 1 1           Tomaž Prosen Complexity of exact many-body solutions
  48. 48. ... ... P P P PR PL P P P 1 2 3 4 n-1 n Ue Uo U = UoUe, Ue = P123P345 · · · Pn−3,n−2,n−1PR n−1,n, Uo = Pn−2,n−1,n · · · P456P234PL 12. P =           1 1 1 1 1 1 1 1           PL =    α 0 α 0 0 β 0 β 1 − α 0 1 − α 0 0 1 − β 0 1 − β    PR =    γ γ 0 0 1 − γ 1 − γ 0 0 0 0 δ δ 0 0 1 − δ 1 − δ    Tomaž Prosen Complexity of exact many-body solutions
  49. 49. Some Monte-Carlo to warm up... x t Tomaž Prosen Complexity of exact many-body solutions
  50. 50. Theorem: Holographic ergodicity 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 Complexity of exact many-body solutions
  51. 51. Theorem: Holographic ergodicity 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 = p is unique, and any initial probability state vector is asymptotically (in t) relaxing to p. Tomaž Prosen Complexity of exact many-body solutions
  52. 52. Theorem: Holographic ergodicity 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 = p 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, s , such t0 exists that (Ut )s,s > 0, ∀t ≥ t0. s s' tgap Tomaž Prosen Complexity of exact many-body solutions
  53. 53. Spectrum of the Markov matrix n=8 -0.5 0.0 0.5 1.0 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 ReΛ ImΛ n=10 -0.5 0.0 0.5 1.0 -0.5 0.0 0.5 ReΛ ImΛ n=12 -0.5 0.0 0.5 1.0 -0.5 0.0 0.5 ReΛ ImΛ Tomaž Prosen Complexity of exact many-body solutions
  54. 54. Unified matrix ansatz for NESS and decay modes: some magic at work [TP and Buča, arXiv:1705.06645] Tomaž Prosen Complexity of exact many-body solutions
  55. 55. Unified matrix ansatz for NESS and decay modes: some magic at work [TP and Buča, arXiv:1705.06645] Consider a pair of matrices: W0 =     1 1 0 0 0 0 0 0 ξ ξ 0 0 0 0 0 0     , W1 =     0 0 0 0 0 0 ξ 1 0 0 0 0 0 0 1 ω     and Ws (ξ, ω) := Ws (ω, ξ). Tomaž Prosen Complexity of exact many-body solutions
  56. 56. Unified matrix ansatz for NESS and decay modes: some magic at work [TP and Buča, arXiv:1705.06645] Consider a pair of matrices: W0 =     1 1 0 0 0 0 0 0 ξ ξ 0 0 0 0 0 0     , W1 =     0 0 0 0 0 0 ξ 1 0 0 0 0 0 0 1 ω     and Ws (ξ, ω) := Ws (ω, ξ). These satisfy a remarkable bulk relation: P123W1SW2W3 = W1W2W3S or component-wise Ws SWχ(ss s )Ws = Ws Ws Ws S. where S is a “delimiter” matrix S =     0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0     . Tomaž Prosen Complexity of exact many-body solutions
  57. 57. Suppose there exists pairs and quadruples of vectors ls |, lss |, |rss , |rs , and a scalar parameter λ, satisfying the following boundary equations P123 l1|W2W3 = l12|W3S, PR 12|r12 = W1S|r2 , P123W1W2|r3 = λW1S|r23 , PL 12 l12| = λ−1 l1|W2S. Tomaž Prosen Complexity of exact many-body solutions
  58. 58. Suppose there exists pairs and quadruples of vectors ls |, lss |, |rss , |rs , and a scalar parameter λ, satisfying the following boundary equations P123 l1|W2W3 = l12|W3S, PR 12|r12 = W1S|r2 , P123W1W2|r3 = λW1S|r23 , PL 12 l12| = λ−1 l1|W2S. Then, the following probability vectors p ≡ p12...n = l1|W2W3W4 · · · Wn−3Wn−2|rn−1,n , p ≡ p12...n = l12|W3W4 · · · Wn−3Wn−2Wn−1|rn , satisfy the NESS fixed point condition Uep = p , Uop = p. Tomaž Prosen Complexity of exact many-body solutions
  59. 59. Suppose there exists pairs and quadruples of vectors ls |, lss |, |rss , |rs , and a scalar parameter λ, satisfying the following boundary equations P123 l1|W2W3 = l12|W3S, PR 12|r12 = W1S|r2 , P123W1W2|r3 = λW1S|r23 , PL 12 l12| = λ−1 l1|W2S. Then, the following probability vectors p ≡ p12...n = l1|W2W3W4 · · · Wn−3Wn−2|rn−1,n , p ≡ p12...n = l12|W3W4 · · · Wn−3Wn−2Wn−1|rn , satisfy the NESS fixed point condition Uep = p , Uop = p. Proof: Observe the bulk relations P123W1SW2W3 = W1W2W3S, P123W1W2W3S = W1SW2W3 to move the delimiter S around, when it ‘hits’ the boundary observe one of the boundary equations. After the full cycle, you obtain UoUep = λλ−1 p. Tomaž Prosen Complexity of exact many-body solutions
  60. 60. This yields a consistent system of equations which uniquely determine the unknown parameters, namely for the left boundary: ξ = (α + β − 1) − λ−1 β λ−2(β − 1) , ω = λ−1 (α − λ−1 ) β − 1 , and for the right boundary: ξ = λ(γ − λ) δ − 1 , ω = γ + δ − 1 − λδ λ2(δ − 1) , yielding ξ = (γ(α + β − 1) − β)(β(γ + δ − 1) − γ) (α − δ(α + β − 1))2 , ω = (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 Complexity of exact many-body solutions
  61. 61. Can we fully diagonalize U with a similar ansatz? Tomaž Prosen Complexity of exact many-body solutions
  62. 62. More magic Define inhomogeneous 2-site 8 × 8 matrix operators Z (k) s s = e11 ⊗ Ws (ξz, ω z )Ws (ξz, ω z ) + e22 ⊗ Ws (ξ z , ωz)Ws ( ξ z , ωz) + e12 ⊗ A (k) s s , Z (k) ss = e11 ⊗ Ws (ξz, ω z )Ws (ξz, ω z ) + e22 ⊗ Ws ( ξ z , ωz)Ws ( ξ z , ωz) + e12 ⊗ A (k) ss , Y (k) ss = e11 ⊗ Ws (ξz, ω z )SWs (ξz, ω z ) + e22 ⊗ Ws ( ξ z , ωz)SWs ( ξ z , ωz) + e12 ⊗ B (k) ss , Y (k) ss = ze11 ⊗ Ws (ξz, ω z )Ws (ξz, ω z )S + 1 z e22 ⊗ Ws (ξ z , ωz)Ws ( ξ z , ωz)S + e12 ⊗ B (k) ss Tomaž Prosen Complexity of exact many-body solutions
  63. 63. More magic Define inhomogeneous 2-site 8 × 8 matrix operators Z (k) s s = e11 ⊗ Ws (ξz, ω z )Ws (ξz, ω z ) + e22 ⊗ Ws (ξ z , ωz)Ws ( ξ z , ωz) + e12 ⊗ A (k) s s , Z (k) ss = e11 ⊗ Ws (ξz, ω z )Ws (ξz, ω z ) + e22 ⊗ Ws ( ξ z , ωz)Ws ( ξ z , ωz) + e12 ⊗ A (k) ss , Y (k) ss = e11 ⊗ Ws (ξz, ω z )SWs (ξz, ω z ) + e22 ⊗ Ws ( ξ z , ωz)SWs ( ξ z , ωz) + e12 ⊗ B (k) ss , Y (k) ss = ze11 ⊗ Ws (ξz, ω z )Ws (ξz, ω z )S + 1 z e22 ⊗ Ws (ξ z , ωz)Ws ( ξ z , ωz)S + e12 ⊗ B (k) ss The most general form of their nilpotent parts A(k) , B(k) can be obtained by requireing the following inhomogeneous quadratic algebra to hold P123Y (k) 12 Z (k+1) 34 = Z (k) 12 Y (k+1) 34 , P234Z (k) 12 Y (k+1) 34 = Y (k) 12 Z (k+1) 34 . which has a solution of the form A (k) ss = c+A+ ss z2k + c−A− ss z−2k , B (k) ss = c+B+ ss z2k + c−B− ss z−2k with a simple, explicit form of 4 × 4 matrices A± ss , B± ss . Tomaž Prosen Complexity of exact many-body solutions
  64. 64. More magic Define inhomogeneous 2-site 8 × 8 matrix operators Z (k) s s = e11 ⊗ Ws (ξz, ω z )Ws (ξz, ω z ) + e22 ⊗ Ws (ξ z , ωz)Ws ( ξ z , ωz) + e12 ⊗ A (k) s s , Z (k) ss = e11 ⊗ Ws (ξz, ω z )Ws (ξz, ω z ) + e22 ⊗ Ws ( ξ z , ωz)Ws ( ξ z , ωz) + e12 ⊗ A (k) ss , Y (k) ss = e11 ⊗ Ws (ξz, ω z )SWs (ξz, ω z ) + e22 ⊗ Ws ( ξ z , ωz)SWs ( ξ z , ωz) + e12 ⊗ B (k) ss , Y (k) ss = ze11 ⊗ Ws (ξz, ω z )Ws (ξz, ω z )S + 1 z e22 ⊗ Ws (ξ z , ωz)Ws ( ξ z , ωz)S + e12 ⊗ B (k) ss The most general form of their nilpotent parts A(k) , B(k) can be obtained by requireing the following inhomogeneous quadratic algebra to hold P123Y (k) 12 Z (k+1) 34 = Z (k) 12 Y (k+1) 34 , P234Z (k) 12 Y (k+1) 34 = Y (k) 12 Z (k+1) 34 . which has a solution of the form A (k) ss = c+A+ ss z2k + c−A− ss z−2k , B (k) ss = c+B+ ss z2k + c−B− ss z−2k with a simple, explicit form of 4 × 4 matrices A± ss , B± ss . z can be understood as a rapidity, or quasi-momentum, and c−, c+ are two free parameters. Tomaž Prosen Complexity of exact many-body solutions
  65. 65. The inhomogeneous Matrix Product Ansatz p = l12|Z (1) 34 Z (2) 56 . . . Z (m) n−3,n−2|rn−1,n , p = l12|Z (1) 34 Z (2) 56 . . . Z (m) n−3,n−2|rn−1,n , where m = n 2 − 2. Tomaž Prosen Complexity of exact many-body solutions
  66. 66. The inhomogeneous Matrix Product Ansatz p = l12|Z (1) 34 Z (2) 56 . . . Z (m) n−3,n−2|rn−1,n , p = l12|Z (1) 34 Z (2) 56 . . . Z (m) n−3,n−2|rn−1,n , where m = n 2 − 2. If in addition to bulk relations P123Y (k) 12 Z (k+1) 34 = Z (k) 12 Y (k+1) 34 , P234Z (k) 12 Y (k+1) 34 = Y (k) 12 Z (k+1) 34 , the following boundary equations hold: P123 l12|Z (1) 34 = l12|Y (1) 34 , P123PR 34Y (m) 12 |r34 = Z (m) 12 |r34 , P234Z (m) 12 |r34 = ΛRY (m) 12 |r34 , PL 12P234 l12|Y (1) 34 = ΛL l12|Z (1) 34 , then Up = Λp, Λ = ΛLΛR. Tomaž Prosen Complexity of exact many-body solutions
  67. 67. The inhomogeneous Matrix Product Ansatz p = l12|Z (1) 34 Z (2) 56 . . . Z (m) n−3,n−2|rn−1,n , p = l12|Z (1) 34 Z (2) 56 . . . Z (m) n−3,n−2|rn−1,n , where m = n 2 − 2. If in addition to bulk relations P123Y (k) 12 Z (k+1) 34 = Z (k) 12 Y (k+1) 34 , P234Z (k) 12 Y (k+1) 34 = Y (k) 12 Z (k+1) 34 , the following boundary equations hold: P123 l12|Z (1) 34 = l12|Y (1) 34 , P123PR 34Y (m) 12 |r34 = Z (m) 12 |r34 , P234Z (m) 12 |r34 = ΛRY (m) 12 |r34 , PL 12P234 l12|Y (1) 34 = ΛL l12|Z (1) 34 , then Up = Λp, Λ = ΛLΛR. Proof: Sweeping Y operator left and right with the bulk relations while being absorbed or created at the boundary via boundary equations. Tomaž Prosen Complexity of exact many-body solutions
  68. 68. The Bethe-like equations for the Markov spectrum BEqs yield an overdetermined system of equations for ΛL,R, ξ, ω, z, c+, c−, and the boundary vectors, which form a closed set for the three main variables, the left and right part of the eigenvalue ΛL,R and the quasi-momentum z: z(α + β − 1) − βΛL (β − 1)Λ2 L = ΛR(γz − ΛR) (δ − 1)z , z(γ + δ − 1) − δΛR (δ − 1)Λ2 R = ΛL(αz − ΛL) (β − 1)z , z4m+2−4p = (α + β − 1)p(γ + δ − 1)p Λ4p L Λ4p R . -0.5 0.0 0.5 1.0 -0.5 0.0 0.5 Re Λ ImΛ Tomaž Prosen Complexity of exact many-body solutions
  69. 69. Conclusions Conservative systems driven by uneven dissipative boundaries are a fruitful paradigm of statistical physics, both quantum and classical Intimate connection between steady states and current-like conservation laws of bulk dynamics Inhomogeneous matrix product ansatz for Markov eigenvectors, perhaps ubiquitous in integrable boundary driven chains The work supported by Slovenian Research Agency (ARRS) and Tomaž Prosen Complexity of exact many-body solutions
  70. 70. Conclusions Conservative systems driven by uneven dissipative boundaries are a fruitful paradigm of statistical physics, both quantum and classical Intimate connection between steady states and current-like conservation laws of bulk dynamics Inhomogeneous matrix product ansatz for Markov eigenvectors, perhaps ubiquitous in integrable boundary driven chains The work supported by Slovenian Research Agency (ARRS) and Tomaž Prosen Complexity of exact many-body solutions
  71. 71. Conclusions Conservative systems driven by uneven dissipative boundaries are a fruitful paradigm of statistical physics, both quantum and classical Intimate connection between steady states and current-like conservation laws of bulk dynamics Inhomogeneous matrix product ansatz for Markov eigenvectors, perhaps ubiquitous in integrable boundary driven chains The work supported by Slovenian Research Agency (ARRS) and Tomaž Prosen Complexity of exact many-body solutions

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