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Complexity of exact solutions of many body systems: nonequilibrium steady states - Tomaž Prosen
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
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. 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. 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. 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
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. 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. 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. 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. 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. 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. 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. ...
...
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. ...
...
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. ...
...
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
49. Some Monte-Carlo to warm up...
x
t
Tomaž Prosen Complexity of exact many-body solutions
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. 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. 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
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. 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. 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. 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. 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. 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. 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. Can we fully diagonalize U with a similar ansatz?
Tomaž Prosen Complexity of exact many-body solutions
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. 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. 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. 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. 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. 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. 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. 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. 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. 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