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

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

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)

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} .

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.

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

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)

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”

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)

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

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

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

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

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|.

L(ϕ, s) =
sin(ϕ + ηSz
s ) (sin η)S−
s
(sin η)S+
s sin(ϕ − ηSz
s )
where S±,z
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.

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

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

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.

κ(ω) = lim
t→∞
lim
n→∞
β
n
t
0
dt eiωt
J(t )J(0) β
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.

κ(ω) = lim
t→∞
lim
n→∞
β
n
t
0
dt eiωt
J(t )J(0) β
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 → ∞.

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

Spin ﬂip:
P = (σx
)⊗n
PQ( )
= Q( )
P
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!

Spin ﬂip:
P = (σx
)⊗n
PQ( )
= Q( )
P
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!

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

ρ∞ = 1 + iεQ−
+ O(ε2
) Q−
= i(Z − Z†
)
where
|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)]

ρ∞ = 1 + iεQ−
+ O(ε2
) Q−
= i(Z − Z†
)
where
|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

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)]

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)]

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

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↑

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 .

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.

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 .

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.

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

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.

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

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).

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).

What about strongly correlated steady states in conservative deterministic
classical interacting systems with stochastic boundary driving?

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)

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

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

...
...
P P P PR
PL
P P P
1
2
3
4
n-1
n
Ue
Uo

...
...
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 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











...
...
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




Some Monte-Carlo to warm up...
x
t

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.

The 2n
×2n
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.

The 2n
×2n
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

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

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).

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



 .

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.

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..

Can we fully diagonalize U with a similar ansatz?

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

More magic
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 .

More magic
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.

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.

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
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.

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

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

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

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