Matrix product solutions of boundary driven quantum chains

Transcription

1 Matrix product solutions of boundary driven quantum chains Faculty of mathematics and physics, University of Ljubljana, Slovenia Edinburgh, 11 September 2015

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 Open problems Topical Review, J. Phys. A: Math. Theor. 48, (2015) Collaborators involved in this work: Enej Ilievski (now at University of Amsterdam), Slava Popkov (Cologne/Florence), Marko Medenjak (PhD student, Ljubljana)

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 ( ) 2L µρl µ {L µl µ, ρ}.

4 Nonequilibrium quantum transport problem in one-dimension Canonical markovian master equation for the many-body density matrix: The Lindblad (L-GKS) equation: X dρ = L ρ := i[h, ρ] + 2Lµ ρl µ {L µ Lµ, ρ}. dt µ Pn 1 Bulk: Fully coherent, local interactions,e.g. H = x=1 hx,x+1. Boundaries: Fully incoherent, ultra-local dissipation, jump operators Lµ supported near boundaries x = 1 or x = n.

5 Prime example of exactly solvable NESS: boundary driven XXZ chain Steady state Lindblad equation ˆLρ = 0: i[h, ρ ] = ( ) 2L µρ L µ {L µl µ, ρ } µ The XXZ Hamiltonian: n 1 H = (2σ x + σ x+1 + 2σx σ + x+1 + σxσ z x+1) z x=1 and symmetric boundary (ultra local) Lindblad jump operators: L L 1 1 = 2 (1 µ)ε σ+ 1, L R 1 1 = 2 (1 + µ)ε σ+ n, L L 1 2 = 2 (1 + µ)ε σ 1, L R 1 2 = 2 (1 µ)ε σ n. Two key boundary parameters: ε System-bath coupling strength µ Non-equilibrium driving strength (bias)

6 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 = ( ) n 0 A s1 A s2 A sn 0 σ s 1 σ s2 σ sn A0 A + = 0 0 A A 0 (s 1,...,s n) {+,,0} n

7 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 = ( ) n 0 A s1 A s2 A sn 0 σ s 1 σ s2 σ sn A0 A + = 0 0 A A 0 (s 1,...,s n) {+,,0} n A 0 = A + = A = ak 0 k k, k=0 a + k k +1, k k=0 a k +1 r, k k=

8 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 = ( ) n 0 A s1 A s2 A sn 0 σ s 1 σ s2 σ sn A0 A + = 0 0 A A 0 (s 1,...,s n) {+,,0} n A 0 = A + = A = ak 0 k k, k=0 a + k k +1, k k=0 a k +1 r, k k= ak 0 = cos((s k)η) cos η :=, a + k = sin((k + 1)η) ε tan(ηs) := 2i sin η a k = cos((2s k)η) s is a q deformed complex spin q = e iη

9 cf. Asymmetric simple exclusion process (ASEP) R1 R2 Markovian model on a 2 L dimensional probability state vector p(t): The asymmetric exclusion model with open boundaries α d dt p = Mp q 1 β RESERVOIR 1 L RESERVOIR γ [from talk of K. Mallick] δ K. Mallick Exact Solutions in Nonequilibrium Statistical Mechanics

10 cf. Asymmetric simple exclusion process (ASEP) Markovian model on a 2 L dimensional probability state vector p(t): The asymmetric exclusion model d with open boundaries dt p = Mp α q 1 β RESERVOIR 1 L RESERVOIR γ [from talk of K. Mallick] δ Nonequilibrium steady state (NESS): a fixed point probability state vector p K. Mallick Exact Solutions in Nonequilibrium Statistical Mechanics Mp = 0

11 Matrix Product Ansatz (MPA) Derrida, Evans, Hakim & Pasquier (1993): Let A 0, A 1 be a pair of matrices, and L, R a pair of left and right vacua. MPA : p s1,s 2,...,s L = L A s1 A s2 A sl R, s j {0, 1}

12 Matrix Product Ansatz (MPA) Derrida, Evans, Hakim & Pasquier (1993): Let A 0, A 1 be a pair of matrices, and L, R a pair of left and right vacua. MPA : p s1,s 2,...,s L = L A s1 A s2 A sl R, s j {0, 1} Asking such MPA p to solve the Markov fixed point condition Mp = 0 results in a single algebraic relation in the bulk with two boundary conditions A 1A 0 qa 0A 1 = (1 q)(a 0 + A 1) L (αa 0 γa 1) = L, (βa 1 δa 0) R = R

13 Observables in NESS: From insulating to ballistic transport For < 1, J n 0 (ballistic) For > 1, J exp( constn) (insulating) For = 1, J n 2 (anomalous) a 0.1 b Σ j z j J n

14 Two-point spin-spin correlation function in NESS ( x C n, y ) = σxσ z y z σx σ z y z n for isotropic case = 1 (XXX ) C(ξ 1, ξ 2) = π2 ξ1(1 ξ2) sin(πξ1) sin(πξ2), for ξ1 < ξ2 2n

15 Yang-Baxter formulation of non-equilibrium integrability TP, Ilievski and Popkov, NJP(2013) Cholesky-factor (amplitude operator) Ω = Ω n(ε) satisfies the square-root Lindblad equation: [H, Ω n(ε)] = iεσ z Ω n 1(ε) + iεω n 1(ε) σ z.

16 Yang-Baxter formulation of non-equilibrium integrability TP, Ilievski and Popkov, NJP(2013) Cholesky-factor (amplitude operator) Ω = Ω n(ε) satisfies the square-root Lindblad equation: [H, Ω n(ε)] = iεσ z Ω n 1(ε) + iεω n 1(ε) σ z. This follows after considering the Lax operator L End(C 2 H a) ( ) sin(ϕ + ηs z L(ϕ, s) = s ) (sin η)s s (sin η)s + s sin(ϕ ηs z s) where S s ±,z is the highest-weight complex-spin irep of U q(sl 2) over H a: S z s = (s k) k k, and writing S + s = S s = k=0 k=0 k=0 sin(k + 1)η k k + 1, sin η sin(2s k)η k + 1 k. sin η Ω n(ε) = 0 a L 1,aL 2,a L n,a 0 a, with ϕ = π 2, tan(ηs) := ε 2i sin η.

17 Steady state Lindblad eq. in fact follows from telescoping series using the local operator divergence condition (or Sutherland equation) [h x,x+1, L x,al x+1,a] = ~L x,a L x+1,a L x,a ~L x+1,a, ~L x,a(ϕ, s) ϕl x,a(ϕ, s).

18 Steady state Lindblad eq. in fact follows from telescoping series using the local operator divergence condition (or Sutherland equation) [h x,x+1, L x,al x+1,a] = ~L x,a L x+1,a L x,a ~L x+1,a, ~L x,a(ϕ, s) ϕl x,a(ϕ, s). This, in turn, is equivalent to YBE (or so-called RLL relation) Ř 1,2(ϕ 2 ϕ 1)L 1,a(ϕ 1, s)l 2,a(ϕ 2, s) = L 1,a(ϕ 2, s)l 2,a(ϕ 1, s)ř 1,2(ϕ 2 ϕ 1) where R 1,2 = Ř 1,2P 1,2 is the 6-vertex R-matrix yielding the XXZ hamiltonian as h 1,2 = 2 ϕ Ř 1,2 ϕ=0.

19 However, using the concept of universal R-matrix there is an infinitely dimensional matrix interwining the representation parameters Ř a1,a 2 L x,a1 (ϕ 1, s 1)L x,a2 (ϕ 2, s 2) = L x,a1 (ϕ 2, s 2)L x,a2 (ϕ 1, s 1)Ř a1,a 2

20 However, using the concept of universal R-matrix there is an infinitely dimensional matrix interwining the representation parameters Ř a1,a 2 L x,a1 (ϕ 1, s 1)L x,a2 (ϕ 2, s 2) = L x,a1 (ϕ 2, s 2)L x,a2 (ϕ 1, s 1)Ř a1,a 2 Remarkably, together with the property 0 a1 0 a2 Ř a1,a 2 = 0 a1 0 a2, Ř a1,a 2 0 a1 0 a2 = 0 a1 0 a2 this immediately implies the commutatvity [W n(ϕ 1, s 1), W n(ϕ 2, s 2)] = 0 of the generalised highest-weight transfer matrix W n(ϕ, s) = 0 a L 1,a(ϕ, s) L n,a(ϕ, s) 0 a, Ω n(ε) = W n(π/2, s(ε)).

21 However, using the concept of universal R-matrix there is an infinitely dimensional matrix interwining the representation parameters Ř a1,a 2 L x,a1 (ϕ 1, s 1)L x,a2 (ϕ 2, s 2) = L x,a1 (ϕ 2, s 2)L x,a2 (ϕ 1, s 1)Ř a1,a 2 Remarkably, together with the property 0 a1 0 a2 Ř a1,a 2 = 0 a1 0 a2, Ř a1,a 2 0 a1 0 a2 = 0 a1 0 a2 this immediately implies the commutatvity [W n(ϕ 1, s 1), W n(ϕ 2, s 2)] = 0 of the generalised highest-weight transfer matrix W n(ϕ, s) = 0 a L 1,a(ϕ, s) L n,a(ϕ, s) 0 a, Ω n(ε) = W n(π/2, s(ε)). Namely, n W n(ϕ 1, s 1)W n(ϕ 2, s 2) = 0 a1 0 a2 Ř a1,a 2 L x,a1 (ϕ 1, s 1)L x,a2 (ϕ 2, s 2) 0 a1 0 a2 x=1 n = 0 a1 0 a2 L x,a1 (ϕ 2, s 2)L x,a2 (ϕ 1, s 1)Ř a1,a 2 0 a1 0 a2 x=1 = W n(ϕ 2, s 2)W n(ϕ 1, s 1)

22 Quasi-local conservation laws Local conserved operators [H, Q m] = [Q m, Q l ] = 0, typically derived via Algebraic Bethe Ansatz machinery Q m = m ϕ log tr al x n (ϕ, 1 2 ) ϕ= η 2 = x q(m) x, Q 1 H satisfy extensivity property ( = tr ( )/ tr 1): Q 2 m n.

23 Quasi-local conservation laws Local conserved operators [H, Q m] = [Q m, Q l ] = 0, typically derived via Algebraic Bethe Ansatz machinery Q m = m ϕ log tr al x n (ϕ, 1 2 ) ϕ= η 2 = x q(m) x, Q 1 H satisfy extensivity property ( = tr ( )/ tr 1): Definition (quasi-locality): Q 2 m n. Nonlocal operator A End((C 2 ) n ), with n independent (a k 1 n k )A for any locally supported fixed a k and with extensivity property is called quasi-local. A A n

24 Observations: (TP, PRL106(2011); TP and Ilievski, PRL111(2013); TP, NPB886(2014); Pereira et al., JSTAT2014) 1 All local Q m are spin-flip invariant since PW n(ϕ, 1 2 )P 1 = W n(ϕ, 1 2 ) 2 However, for generic s C PQ m = Q mp, P = (σ x ) n. PW n(ϕ, s)p 1 = W n(π ϕ, s) T W n(ϕ, s). 3 Derivative w.r.t. s at the scalar point s = 0 is quasi-local if η = πl/m Z n(ϕ) = c n sw n(ϕ, s) s=0, Z n (ϕ)z n(ϕ) n for Reϕ π 2 < π 2m and almost conserved, e.g., for ϕ = π 2 [H, Z n] = σ z 1 σ z n. 4 Q = i(z Z ) has been the first known quasi-local CL of odd parity PQ = QP

25 Observations: (TP, PRL106(2011); TP and Ilievski, PRL111(2013); TP, NPB886(2014); Pereira et al., JSTAT2014) 1 All local Q m are spin-flip invariant since PW n(ϕ, 1 2 )P 1 = W n(ϕ, 1 2 ) 2 However, for generic s C PQ m = Q mp, P = (σ x ) n. PW n(ϕ, s)p 1 = W n(π ϕ, s) T W n(ϕ, s). 3 Derivative w.r.t. s at the scalar point s = 0 is quasi-local if η = πl/m Z n(ϕ) = c n sw n(ϕ, s) s=0, Z n (ϕ)z n(ϕ) n for Reϕ π 2 < π 2m and almost conserved, e.g., for ϕ = π 2 [H, Z n] = σ z 1 σ z n. 4 Q = i(z Z ) has been the first known quasi-local CL of odd parity PQ = QP

26 Observations: (TP, PRL106(2011); TP and Ilievski, PRL111(2013); TP, NPB886(2014); Pereira et al., JSTAT2014) 1 All local Q m are spin-flip invariant since PW n(ϕ, 1 2 )P 1 = W n(ϕ, 1 2 ) 2 However, for generic s C PQ m = Q mp, P = (σ x ) n. PW n(ϕ, s)p 1 = W n(π ϕ, s) T W n(ϕ, s). 3 Derivative w.r.t. s at the scalar point s = 0 is quasi-local if η = πl/m Z n(ϕ) = c n sw n(ϕ, s) s=0, Z n (ϕ)z n(ϕ) n for Reϕ π 2 < π 2m and almost conserved, e.g., for ϕ = π 2 [H, Z n] = σ z 1 σ z n. 4 Q = i(z Z ) has been the first known quasi-local CL of odd parity PQ = QP

27 Observations: (TP, PRL106(2011); TP and Ilievski, PRL111(2013); TP, NPB886(2014); Pereira et al., JSTAT2014) 1 All local Q m are spin-flip invariant since PW n(ϕ, 1 2 )P 1 = W n(ϕ, 1 2 ) 2 However, for generic s C PQ m = Q mp, P = (σ x ) n. PW n(ϕ, s)p 1 = W n(π ϕ, s) T W n(ϕ, s). 3 Derivative w.r.t. s at the scalar point s = 0 is quasi-local if η = πl/m Z n(ϕ) = c n sw n(ϕ, s) s=0, Z n (ϕ)z n(ϕ) n for Reϕ π 2 < π 2m and almost conserved, e.g., for ϕ = π 2 [H, Z n] = σ z 1 σ z n. 4 Q = i(z Z ) has been the first known quasi-local CL of odd parity PQ = QP

28 Observations: (TP, PRL106(2011); TP and Ilievski, PRL111(2013); TP, NPB886(2014); Pereira et al., JSTAT2014) 1 All local Q m are spin-flip invariant since PW n(ϕ, 1 2 )P 1 = W n(ϕ, 1 2 ) 2 However, for generic s C PQ m = Q mp, P = (σ x ) n. PW n(ϕ, s)p 1 = W n(π ϕ, s) T W n(ϕ, s). 3 Derivative w.r.t. s at the scalar point s = 0 is quasi-local if η = πl/m Z n(ϕ) = c n sw n(ϕ, s) s=0, Z n (ϕ)z n(ϕ) n for Reϕ π 2 < π 2m and almost conserved, e.g., for ϕ = π 2 [H, Z n] = σ z 1 σ z n. 4 Q = i(z Z ) has been the first known quasi-local CL of odd parity PQ = QP

29 Implication for linear response spin transport Green-Kubo formulae express the conductivities in terms of current a.c.f. κ(ω) = lim lim t n β n t 0 dt e iωt J(t )J(0) β When d.c. conductivity diverges, one defines a Drude weight D which in linear response expresses as D = lim κ(ω) = 2πDδ(ω) + κ reg(ω) lim t n β 2tn t 0 dt J(t )J(0) β.

30 Implication for linear response spin transport Green-Kubo formulae express the conductivities in terms of current a.c.f. κ(ω) = lim lim t n β n t 0 dt e iωt J(t )J(0) β When d.c. conductivity diverges, one defines a Drude weight D which in linear response expresses as D = lim κ(ω) = 2πDδ(ω) + κ reg(ω) lim t 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, estimating Drude weight in terms of local conserved operators Q j, [H, Q j ] = 0: β JQ m 2 β D lim n 2n Qm 2 m β where operators Q m are chosen mutually orthogonal Q mq k β = 0 for m k.

31 Implication for linear response spin transport Green-Kubo formulae express the conductivities in terms of current a.c.f. κ(ω) = lim lim t n β n t 0 dt e iωt J(t )J(0) β When d.c. conductivity diverges, one defines a Drude weight D which in linear response expresses as D = lim κ(ω) = 2πDδ(ω) + κ reg(ω) lim t 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, estimating Drude weight in terms of local conserved operators Q j, [H, Q j ] = 0: β JQ m 2 β D lim n 2n Qm 2 m β where operators Q m are chosen mutually orthogonal Q mq k β = 0 for m k. Considering the spin current J = i x (σ+ x σ x+1 σ x σ + x+1), being of odd parity PJ = JP, one has JQ j 0, so Mazur bound is trivial.

32 Mazur bound with the novel quasi-local CL However, almost conserved odd quasi-local operator Q, or correspondingly extended holomorphic family K = {Q(ϕ)}, can be used to bound the spin Drude weight: Fractal Drude weight bound D β D Z := sin2 (πl/m) sin 2 (π/m) 1.0 ( 1 m ( )) 2π 2π sin, = cos m ( ) πl m 0.8 D Z, D K TP, PRL 106 (2011); Ilievski and TP, CMP 318 (2013); TP and Ilievski, PRL 111 (2013)

33 Another type of quasilocal conserved operators... exists even in the isotropic XXX spin 1/2 chain!

34 Another type of quasilocal conserved operators... exists even in the isotropic XXX spin 1/2 chain! Theorem [Ilievski, Medenjak, TP, arxiv: ] Traceless operators X s(t), s 1 Z, t R, defined as 2 X s(t) = [τ s(t)] n { T s( it)t s ( it)}, τ s(t) = t 2 ( s ) 2, where T s(λ) = tr al(λ, s) x n, T s (λ) λ T s(λ), are quasilocal for all s, t and linearly independent from {Q m; m 1} for s > 1 2

35 Another type of quasilocal conserved operators... exists even in the isotropic XXX spin 1/2 chain! Theorem [Ilievski, Medenjak, TP, arxiv: ] Traceless operators X s(t), s 1 Z, t R, defined as 2 X s(t) = [τ s(t)] n { T s( it)t s ( it)}, τ s(t) = t 2 ( s ) 2, where T s(λ) = tr al(λ, s) x n, T s (λ) λ T s(λ), are quasilocal for all s, t and linearly independent from {Q m; m 1} for s > and it resolves (!) the 2014 controversy with GGE (Wouters et al. PRL 2014, Pozsgay et al. PRL 2014) see: (Ilievski, De Nardis, Wouters, Caux, Essler, TP, arxiv: )

36 Analytical solution of NESS for Hubbard chain TP, PRL 112 (2014) Extremely boundary driven open fermi Hubbard chain n 1 H n = j=1 (σ + j σ j+1 + τ + j τ j+1 + H.c.) + u 4 n j=1 σ z j τ z j L 1 = εσ + 1, L 2 = ετ + 1, L 3 = εσ n, L 4 = ετ n.

37 Analytical solution of NESS for Hubbard chain TP, PRL 112 (2014) Extremely boundary driven open fermi Hubbard chain n 1 H n = j=1 (σ + j σ j+1 + τ + j τ j+1 + H.c.) + u 4 n j=1 σ z j τ z j L 1 = εσ + 1, L 2 = ετ + 1, L 3 = εσ n, L 4 = ετ n. Key ansatz for NESS density matrix ˆLρ = 0, again a decomposition a-la Cholesky: ρ = Ω nω n.

38 Walking graph state representation of the solution Ω n = e W n(0,0) a e1 a e2 a en n j=1 σ bx (e j ) j τ by (e j ) j.

39 Lax form of NESS for the Hubbard chain Popkov and TP, PRL 114, (2015) Amplitude Lindblad equation can again be reduced to LOD [h x,x+1, L x,al x+1,a] = ~L x,a L x+1,a L x,a ~L x+1,a plus appropriate boundary conditions for the dissipators, solved with: Ω n(ε) = 0 a L 1,aL 2,a L n,a 0 a, again forming a commuting family [Ω n(ε), Ω n(ε )] = 0.

40 Lax operator admits very appealing factorization L x,a(ε) = S α a T β a X a(u, ε)σx α τx β α,β {+,,0,z} S S S 0 T X T T S z T z

41 Lax operator admits very appealing factorization L x,a(ε) = S α a T β a X a(u, ε)σx α τx β α,β {+,,0,z} S S S 0 T X T T S z T z S α a and T β a are particular commuting ([S α a, T β a ] = 0) representations of an extended CAR: {S +, S } = 2(S 0 S z ), [S α, S 0 ] = [S α, S z ] = 0.

42 Bonus: Quadratically extensive conserved quantities of the Hubbard chain Let the lattice of n + 1 sites be Λ n [ n/2, n/2]. Then: where Z n = i(d/dε)ω n ε=0 Z n = 1 2 n/2 1 x= n/2 (σ + x σ x+1+τ + x τ x+1)+ u 2 Q Λn = i(z n Z n ) x<y x,y Λ n ( 1) x y σ + x P (σ) x+1,y 1 σ y τ + x P (τ) x+1,y 1 τ y, P (σ) x,y := σ z xσ z x+1 σ z y, P (τ) x,y := τ z x τ z x+1 τ z y, and P (σ,τ) x,y 1 if x > y, satisfying the almost conservation condition [H Λn, Q Λn ] = 1 2 (σz n/2 + τ z n/2 σ z n/2 τ z n/2).

43 Bonus: Quadratically extensive conserved quantities of the Hubbard chain Let the lattice of n + 1 sites be Λ n [ n/2, n/2]. Then: where Z n = i(d/dε)ω n ε=0 Z n = 1 2 n/2 1 x= n/2 (σ + x σ x+1+τ + x τ x+1)+ u 2 Q Λn = i(z n Z n ) x<y x,y Λ n ( 1) x y σ + x P (σ) x+1,y 1 σ y τ + x P (τ) x+1,y 1 τ y, P (σ) x,y := σ z xσ z x+1 σ z y, P (τ) x,y := τ z x τ z x+1 τ z y, and P (σ,τ) x,y 1 if x > y, satisfying the almost conservation condition [H Λn, Q Λn ] = 1 2 (σz n/2 + τ z n/2 σ z n/2 τ z n/2). Q Λn is quadratically extensive, as Q 2 Λ n qn 2

44 Strict lower bounds on Green-Kubo diffusion constants in terms of quadratically extensive almost conserved quantities PRE 89,012142(2014) Theorem: Take an arbitrary self-adjoint local current density operator j, satisfying j = 0, where is an infinite temperature (tracial) state, and define a spatiotemporal correlation function of the infinite lattice dynamics as C(x, t) = lim j(0, 0)j(x, t). (1) n Assuming that C(t) := x= C(x, t) exists for any t, that D := dt C(t) and D := dt t C(t) exist as well, and that Q Λ n has a well defined component along j, Q j := lim n jq Λn, the following inequality holds where v is the Lieb-Robinson group velocity. D Qj 2 8vq. (2)

45 Strict lower bounds on Green-Kubo diffusion constants in terms of quadratically extensive almost conserved quantities PRE 89,012142(2014) Theorem: Take an arbitrary self-adjoint local current density operator j, satisfying j = 0, where is an infinite temperature (tracial) state, and define a spatiotemporal correlation function of the infinite lattice dynamics as C(x, t) = lim j(0, 0)j(x, t). (1) n Assuming that C(t) := x= C(x, t) exists for any t, that D := dt C(t) and D := dt t C(t) exist as well, and that Q Λ n has a well defined component along j, Q j := lim n jq Λn, the following inequality holds where v is the Lieb-Robinson group velocity. D Qj 2 8vq. (2) For example, for the Hubbard chain, our bound evaluates to D c,s 2 3u. 2 for spin and charge currents j c,s = 2i [ σ + σ σ σ + ± (τ + τ τ τ + ) ] satisfying continuity equations i[h Λn, σ z x ± τ z x ] = j c,s x j c,s x 1. (Agrees with DMRG numerics!)

46 The pictorial proof of the theorem.. A n,t := 1 t t 0 dt ( τ t (J Λ k n ) α n Q Λ n ) Since A 2 n,t 0, we have A 2 n,t 0 for any t = t n, α R, n Z + : tn tn dt dt 1 tn t τ 2 t (J Λ )τ k t (J n Λ ) dt 2α k n 0 nt τ t (J Λ )Q k Λ n + α2 n n 2 Q2 Λ n Τ t b n 2 t 1 k n 2 v Μ 1 k n t n c 2 v Μ j x Τ t b n 2 n k n n n 2 x

47 Lai-Sutherland SU(3) model with degenerate dissipative driving Ilievski and TP, NPB 882 (2014) n 1 H = x=1 h x,x+1

48 Lai-Sutherland SU(3) model with degenerate dissipative driving Ilievski and TP, NPB 882 (2014) n 1 H = x=1 h x,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).

49 Lai-Sutherland SU(3) model with degenerate dissipative driving Ilievski and TP, NPB 882 (2014) n 1 H = x=1 h x,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).

50 Closing with a list of open problems Why only pure particle-source on one-end and pure particle-sink on the other-end boundary conditions seem to be generally exactly solvable? More general boundary conditions need to be discussed. Perhaps generalising the notion of Sklyanin Reflection Algebra. The interpretation in terms of quantum symmetries are missing for some solutions, e.g. of open Hubbard chain. No apparent link to integralility structures (R& L-matrix) proposed by Shastry. No exact solutions of Liouvillian decay modes of boundary driven integrable chains known so far.

51 Closing with a list of open problems Why only pure particle-source on one-end and pure particle-sink on the other-end boundary conditions seem to be generally exactly solvable? More general boundary conditions need to be discussed. Perhaps generalising the notion of Sklyanin Reflection Algebra. The interpretation in terms of quantum symmetries are missing for some solutions, e.g. of open Hubbard chain. No apparent link to integralility structures (R& L-matrix) proposed by Shastry. No exact solutions of Liouvillian decay modes of boundary driven integrable chains known so far.

52 Closing with a list of open problems Why only pure particle-source on one-end and pure particle-sink on the other-end boundary conditions seem to be generally exactly solvable? More general boundary conditions need to be discussed. Perhaps generalising the notion of Sklyanin Reflection Algebra. The interpretation in terms of quantum symmetries are missing for some solutions, e.g. of open Hubbard chain. No apparent link to integralility structures (R& L-matrix) proposed by Shastry. No exact solutions of Liouvillian decay modes of boundary driven integrable chains known so far.

53 Closing with a list of open problems Why only pure particle-source on one-end and pure particle-sink on the other-end boundary conditions seem to be generally exactly solvable? More general boundary conditions need to be discussed. Perhaps generalising the notion of Sklyanin Reflection Algebra. The interpretation in terms of quantum symmetries are missing for some solutions, e.g. of open Hubbard chain. No apparent link to integralility structures (R& L-matrix) proposed by Shastry. No exact solutions of Liouvillian decay modes of boundary driven integrable chains known so far.

54 Closing with a list of open problems Why only pure particle-source on one-end and pure particle-sink on the other-end boundary conditions seem to be generally exactly solvable? More general boundary conditions need to be discussed. Perhaps generalising the notion of Sklyanin Reflection Algebra. The interpretation in terms of quantum symmetries are missing for some solutions, e.g. of open Hubbard chain. No apparent link to integralility structures (R& L-matrix) proposed by Shastry. No exact solutions of Liouvillian decay modes of boundary driven integrable chains known so far.