Boundary-driven Lindblad dynamics of random quantum spin chains : strong disorder approach for the relaxation, the steady state and the current

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by  Cecile Monthus

Boundary-driven Lindblad dynamics of random quantum spin chains :
strong disorder approach for the relaxation, the steady state and the current
C´ecile Monthus
Institut de Physique Th´eorique, Universit´e Paris Saclay, CNRS, CEA, 91191 Gif-sur-Yvette, France
The Lindblad dynamics of the XX quantum chain with large random fields hj (the couplings
Jj can be either uniform or random) is considered for boundary-magnetization-drivings acting on
the two end-spins. Since each boundary-reservoir tends to impose its own magnetization, we first
7
study the relaxation spectrum in the presence of a single reservoir as a function of the system size
1
via some boundary-strong-disorder renormalization approach. The non-equilibrium-steady-state in
0
the presence of two reservoirs can be then analyzed from the effective renormalized Linbladians
2
associated to the two reservoirs. The magnetization is found to follow a step profile, as found

r previously in otherlocalized chains. Thestrong disorder approach allows to computeexplicitly the
a
location of the step of the magnetization profile and the corresponding magnetization-current for
M
each disordered sample in terms of therandom fields and couplings.

5
1
I. INTRODUCTION

]
n
Inthefieldofrandomquantumspinchains,theinterplayofdisorderanddissipationhasattractedalotofattention
n
recently. As a first example, the coupling to a dissipative bath of harmonic oscillatorswith some spectral function as
-
s in the spin-boson model [1] has been analyzed via Strong Disorder Renormalization [2–9]. As a second example, the
i
d Lindbladdynamics with boundary-drivingand/ordephasing hasbeen studiedfor Many-Body-Localizationmodels in
. various regimes [10–15].
t
a Among the various descriptions of open quantum systems [16], one of the most effective is indeed the Lindblad
m
equation for the density matrix ρ(t)
-
d ∂ρ(t)
n = [ρ(t)]= [ρ(t)]+ [ρ(t)] (1)
∂t L U D
o
c
where the Lindblad operator contains the unitary evolution as if the system of Hamiltonian H were isolated
[ L

2 [ρ(t)] i[H,ρ(t)] (2)
U ≡−
v
2 and the dissipative contribution defined in terms of some set of operators L that describe the interaction with the
α
0 reservoirs (see example in section II)
1
2
1 1
0 [ρ(t)]= γ L ρ(t)L† L†L ρ(t) ρ(t)L†L (3)
. D α α α− 2 α α − 2 α α
1 Xα (cid:18) (cid:19)
0
so that the trace of the density matrix is conserved by the dynamics
7
1
∂
: Tr(ρ(t))=0 (4)
v ∂t
i
X
The first advantage of this formulation of the dynamics as a Quantum Markovian Master Equation is that the
r relaxation properties can be studied from the spectrum of the Lindblad operator [17–19] with possible metastability
a
phenomena [20]. This spectralanalysisalsoallowsto make some link with the RandomMatrix Theoryofeigenvalues
statistics[21]. Thesecondadvantageisthatthisframeworkisveryconvenienttostudythenon-equilibriumtransport
properties [23–31] with many exact solutions [32–37]. In addition, many important ideas that have been developed
in the context of classical non-equilibrium systems (see the review [38] and references therein) have been adapted to
the Lindblad description of non-equilibrium dissipative quantum systems, in particular the large deviation formalism
to access the full-counting statistics [39–46], the additivity principle [47] and the fluctuation relations [48].
In the present paper, we consider the XX chain of N spins with random fields h and couplings J (that can be
j j
either uniform or random)
N N
H = h σz +J (σxσx +σyσy ) = h σz +2J (σ+σ− +σ−σ+ ) (5)
j j j j j+1 j j+1 j j j j j+1 j j+1
j=1 j=1
X(cid:2) (cid:3) X(cid:2) (cid:3)
2
and analyze the Lindblad dynamics in the presence of two boundary-magnetization-drivings acting on the two end-
spins. We focus on the strong disorder regime where the scale of the random fields (h ) is much bigger than the
j
couplings (J ). The paper is organized as follows. In section II, we introduce the notations for the boundary-
j
magnetization-drivingsandwe recallthe spectralanalysisofthe Lindbladianinthe ladderformulation,aswellasthe
notion of tilted-Lindbladian to access the full-counting statistics of the exchanges with one reservoir. In section III,
weshowhowthisformalismworksinpracticeonthesimplestcaseofN =2spins. Wethenturntothecaseofachain
of arbitrary length N : in section IV, we analyze the relaxation properties of a long chain in contact with a single
reservoir,whileinsectionV,weanalyzethe non-equilibrium-steady-stateforthe chaincoupledtotworeservoirs: the
magnetizationprofileandthe magnetizationcurrentarecomputedinthe strongdisorderregime. Ourconclusionsare
summarized in section VI.
II. LINDBLAD DYNAMICS WITH BOUNDARY-MAGNETIZATION-DRIVING
A. Boundary-magnetization-driving on the end-spins σ and σ
1 N
The standard boundary-magnetization-driving on the first spin σ is based on the dissipative operator of Eq. 3
1
with the two operators α=1,2
L =σ+
1 1
L =σ− (6)
2 1
and the corresponding amplitudes
1+µ
γ =Γ
1
2
1 µ
γ =Γ − (7)
2
2
leading to
1+µ 1 1
spin1[ρ] =Γ σ+ρσ− σ−σ+ρ ρσ−σ+
D 2 1 1 − 2 1 1 − 2 1 1
(cid:18) (cid:19)
1 µ 1 1
+Γ − σ−ρσ+ σ+σ−ρ ρσ+σ− (8)
2 1 1 − 2 1 1 − 2 1 1
(cid:18) (cid:19)
Using the identities
1 σz
σ−σ+ = − 1
1 1 2
1+σz
σ+σ− = 1 (9)
1 1 2
Eq 8 becomes
1+µ 1 µ Γ Γµ
spin1[ρ] =Γ σ+ρσ−+ − σ−ρσ+ ρ+ (σzρ+ρσz) (10)
D 2 1 1 2 1 1 − 2 4 1 1
(cid:18) (cid:19)
The physical meaning of this dissipative operator is that it tends to impose the magnetization (+µ) on the spin 1
with a characteristic relaxation rate of order Γ.
A simple way to generate a non-equilibrium steady-state is to consider a similar boundary-magnetization-driving
on the last spin σ that tend to impose another magnetization µ′ =µ with some rate Γ′, so that the corresponding
N
6
dissipative operator reads
1+µ′ 1 µ′ Γ′ Γ′µ′
spinN[ρ] =Γ′ σ+ρσ− + − σ−ρσ+ ρ+ (σz ρ+ρσz ) (11)
D 2 N N 2 N N − 2 4 N N
(cid:18) (cid:19)
3
B. Ladder Formulation of the Lindbladian
Since the Lindblad operator acts on the density matrix ρ(t) of the chain of N spins that can be expanded in the
σz basis
ρ(t)= ... ... ρ (t)S ,...,S ><T ,...,T (12)
S1,..,SN;T1,...,TN | 1 N 1 N|
S1X=±1 SNX=±1T1X=±1 TNX=±1
in terms of the 4N coefficients
ρ (t)=<S ,...,S ρ(t)T ,...,T > (13)
S1,..,SN;T1,...,TN 1 N| | 1 N
it can be technically convenient to ’vectorize’ the density matrix of the spin chain [19, 47, 49–51], i.e. to consider
that these 4N coefficients are the components of a ket describing the state of a spin ladder
ρ(t)>Ladder= ... ... ρ (t)S ,...,S > T ,...,T > (14)
| S1,..,SN;T1,...,TN | 1 N ⊗| 1 N
S1X=±1 SNX=±1T1X=±1 TNX=±1
To translatethe LindbladoperatorofEq. 1in this ladderformulation,one needsto considerthe product(Aρ(t)B)
where A and B are two arbitrary matrices
Aρ(t)B = ... ... ρ (t)AS ,...,S ><T ,...,T B
S1,..,SN;T1,...,TN | 1 N 1 N|
S1X=±1 SNX=±1T1X=±1 TNX=±1
= ... ... S′,...,S′ ><T′,...,T′
| 1 N 1 N|
S′=±1 S′ =±1T′=±1 T′ =±1
1X NX 1X NX
... ... <S′,...,S′ AS ,...,S >ρ (t)<T ,...,T B T′,...,T′ > (15)
1 N| | 1 N S1,..,SN;T1,...,TN 1 N| | 1 N
S1X=±1 SNX=±1T1X=±1 TNX=±1
and to write the corresponding ket
Aρ(t)B >Ladder= ... ... S′,...,S′ > T′,...,T′ >
| | 1 N ⊗| 1 N
S′=±1 S′ =±1T′=±1 T′ =±1
1X NX 1X NX
... ... <S′,...,S′ AS ,...,S >ρ (t)<T′,...,T′ BT T ,...,T >
1 N| | 1 N S1,..,SN;T1,...,TN 1 N| | 1 N
S1X=±1 SNX=±1T1X=±1 TNX=±1
=A BT ρ(t)>Ladder (16)
⊗ |
where BT denotes the transpose of the matrix B. As a consequence, the Lindblad operator governing the evolution
of the ket ρ(t)>Ladder
|
∂ ρ(t)>Ladder
| = Ladder ρ(t)>Ladder (17)
∂t L |
can be translated from Eqs 2 and 3 and reads
1 1
Ladder = i(H I I HT)+ γ L (L†)T L†L I I (L†L )T
L − ⊗ − ⊗ α α⊗ α − 2 α α⊗ − 2 ⊗ α α
α (cid:18) (cid:19)
X
1 1
= i(H I I H)+ γ L L∗ L†L I I LTL∗ (18)
− ⊗ − ⊗ α α⊗ α− 2 α α⊗ − 2 ⊗ α α
α (cid:18) (cid:19)
X
For the chain of Eq. 5, the unitary part reads in terms of the Pauli matrices of the spin ladder
N
Ladder = i h σz +2J (σ+σ− +σ−σ+ )
U − j j j j j+1 j j+1
j=1
X(cid:2) (cid:3)
N
+i h τz +2J (τ+τ− +τ−τ+ ) (19)
j j j j j+1 j j+1
j=1
X(cid:2) (cid:3)
4
while the dissipative operators of Eqs 8 and 11 become
1+µ 1 µ Γ Γµ
Ladder =Γ σ+τ++ − σ−τ− + (σz +τz) (20)
DSpin1 2 1 1 2 1 1 − 2 4 1 1
(cid:18) (cid:19)
and
1+µ′ 1 µ′ Γ′ Γ′µ′
Ladder =Γ′ σ+τ++ − σ−τ− + (σz +τz) (21)
DSpinN 2 N N 2 N N − 2 4 N N
(cid:18) (cid:19)
C. Spectral Decomposition of the Ladder Lindbladian
The ladder formulation of the Lindbladian described above is especially useful to use the very convenient bra-ket
notations to denote the Right and Left eigenvectors associated to the 4N eigenvalues λ
n
Ladder ψR > =λ ψR >
L | λn n| λn
<ψL Ladder =λ <ψL (22)
λn|L n λn|
with the orthonormalization
<ψL ψR >=δ (23)
λn| λm nm
and the identity decomposition
4N−1
1= ψR ><ψL (24)
| λn λn|
n=0
X
The spectral decomposition of the Lindbladian
4N−1
Ladder = λ ψR ><ψL (25)
L n| λn λn|
n=0
X
then allows to write the solution for the dynamics in terms of the initial condition at t=0 as
4N−1
ρLadder(t)>= eλnt ψR ><ψL ρLadder(t=0)> (26)
| | λn λn|
n=0
X
The trace of the density matrix ρ(t) corresponds in the Ladder Formulation to
Tr(ρ(t))= ... ρ (t)= ... <S ,..,S <S ,...,S ρ(t)>Ladder (27)
S1,..,SN;S1,...,SN 1 N|⊗ 1 N|
S1X=±1 SNX=±1 S1X=±1 SNX=±1
Its conservation by the dynamics (Eq 4) means that the eigenvalue
λ =0 (28)
0
is associated to the Left eigenvector
<ψL = ... <S ,..,S <S ,...,S (29)
λ0=0| 1 N|⊗ 1 N|
S1X=±1 SNX=±1
while the corresponding Right Eigenvector corresponds to the steady state towards which any initial condition will
converges
ρLadder(t + )>= ψR > (30)
| → ∞ | λ0=0
The other (4N 1) eigenvalues λ with negative real parts describe the relaxation towards this steady state.
n6=0
−
5
D. Tilted-Lindbladian L(s) to measure the exchanges with the boundary-reservoir on spin 1
As mentioned in the Introduction, the method of ’tilting’ the master equation to access the full-counting statistics
developed for classical non-equilibrium models (see the review [38] and references therein) has been adapted to the
Lindblad framework[39–47]as follows. To keepthe informationonthe globalnumber N of’magnetizationparticles’
t
that have been exchanged with the reservoir acting on the spin 1 since the initial condition at t=0, it is convenient
to decompose the Lindbladian into three terms
Ladder = Ladder+ Ladder+ Ladder (31)
L L0 L+ L−
where
1+µ
Ladder =Γ σ+τ+
L+ 2 1 1
1 µ
Ladder =Γ − σ−τ− (32)
L− 2 1 1
describe respectively the processes corresponding to an increase (N N +1) and a decrease (N N 1) by
t t t t
an elementary ’magnetization particle’, while Ladder contains all the→other terms of the Lindbladia→n that−do not
L0
correspond to an exchange with the reservoir acting on spin 1 (N N ). As a consequence, the eigenvalue λ (s)
t t 0
→
with the largest real-part of the tilted-Lindbladian by the parameter s
Ladder(s)= Ladder+es Ladder+e−s Ladder (33)
L L0 L+ L−
allows to obtain the statistics of the number N in the large-time regime via
t
ln<esNt >
λ (s)= lim (34)
0 t→+∞ t
In particular, the expansion up to second order in s
s2
λ (s)=sI + F +O(s3) (35)
0 av
2
gives the averagedcurrent entering from the reservoir acting on the spin 1
<N >
t
I = lim (36)
av
t→+∞ t
and the fluctuation
(<N2 > <N >2)
F = lim t − t (37)
t→+∞ t
More generally, the whole large-deviationproperties of the probability distribution Pt(I) of the current I = Ntt
P (I) e−tΦ(I) (38)
t
t→≃+∞
can be obtained as the Legendre transform of the tilted eigenvalue of Eq. 34
Φ(I)=max(sI λ (s)) (39)
0
s −
E. Notation
In the remaining of this paper, the ladder formulation of the Lindblad operator described above will be always
used, so that the explicit mention ’Ladder’ will be dropped from now on in order to simplify the notations.
6
III. STRONG-DISORDER APPROACH FOR N =2 SPINS
Toseehowtheformalismrecalledinthe previoussectionworksinpractice,itisusefultofocusfirstonthesimplest
example of N = 2 spins. In addition, to motivate the Strong-Disorder approach for long chains N 1 that will be
≫
described in the following sections, we will consider that the only term of the Linbladian that couples the two spins
per(1,2) = i2J (τ+τ−+τ−τ+ σ+σ− σ−σ+) (40)
L 1 1 2 1 2 − 1 2 − 1 2
is a perturbation with respect to all the other terms that do not couple the two spins
unper = spin1(s)+ spin2 (41)
L L L
A. Spectral decomposition of Lspin1(s)
The tilted Lindbladian of Eq. 31 for the spin 1
Γ Γµ 1+µ 1 µ
spin1(s)=ih (τz σz) + (σz +τz)+esΓ σ+τ++e−sΓ − σ−τ−
L 1 1 − 1 − 2 4 1 1 2 1 1 2 1 1
has the following four eigenvalues that do not depend on the tilting parameter s in contrast to some corresponding
eigenvectors written in the basis (σz,τz):
1 1
(0) The eigenvalue λspin1(s)=0 is associated to
n=0
<λspin1(L)(s) =e−s <++ +<
n=0 | | −−|
1+µ 1 µ
λspin1(R)(s)> =es ++>+ − > (42)
| n=0 2 | 2 |−−
(1) The eigenvalue λspin1(s)= Γ is associated to
n=1 −
1 µ 1+µ
<λspin1(L)(s) =e−s − <++ <
n=1 | 2 |− 2 −−|
λspin1(R)(s)> =es ++> > (43)
| n=1 | −|−−
(2) The eigenvalue λspin1(s)= Γ +i2h is associated to
n=2 −2 1
<λspin1(L)(s) =< +
n=2 | − |
λspin1(R)(s)> = +> (44)
| n=2 |−
(4) The eigenvalue λspin1(s)= Γ i2h is associated to
n=3 −2 − 1
<λspin1(L)(s) =<+
n=3 | −|
λspin1(R)(s)> = + > (45)
| n=3 | −
B. Spectral decomposition of Lspin2
The non-tilted Lindbladian for the spin N =2
Γ′ Γ′µ′ 1+µ′ 1 µ′
spin2 =ih (τz σz) + (σz +τz)+Γ′ σ+τ++Γ′ − σ−τ−
L 2 2 − 2 − 2 4 2 2 2 2 2 2 2 2
has the following four eigenvalues and eigenvectors in the basis (σz,τz) :
2 2
(0) The eigenvalue λspin2 =0 is associated to
m=0
<λspin2(L) =<++ +<
m=0 | | −−|
1+µ′ 1 µ′
λspin2 > = ++>+ − > (46)
| m=0(R) 2 | 2 |−−
7
(1) The eigenvalue λspin2 = Γ′ is associated to
m=1 −
1 µ′ 1+µ′
<λspin2(L) = − <++ <
m=1 | 2 |− 2 −−|
λspin2(R) > = ++> > (47)
| m=1 | −|−−
(2) The eigenvalue λspin2 = Γ′ +i2h is associated to
m=2 −2 2
<λspin2(L) =< +
m=2 | − |
λspin2(R) > = +> (48)
| m=2 |−
(4) The eigenvalue λspin2 = Γ′ i2h is associated to
n=3 −2 − 2
<λspin2(L) =<+
m=3 | −|
λspin2(R) > = + > (49)
| m=3 | −
C. Second-Order perturbation theory in the coupling Lper(1,2)
The unperturbed Lindbladian of Eq. 41 is the sum of the two independent Lindbladians discussed above, so its 16
eigenvalues are simply given by the sum of eigenvalues for n=0,1,2,3 and m=0,1,2,3
λunper =λspin1+λspin2 (50)
n,m n m
while the left and right eigenvectors are given by the corresponding tensor-products
<λunper(L) =<λspin1(L) <λspin2(L)
n,m | n |⊗ m |
λunper(R) > = λspin1(R) > λspin2(R) > (51)
| n,m | n ⊗| m
Here we are interested into the eigenvalue λ (s) with the largest real part of the tilted Lindbladian (Eq. 33). The
0
corresponding unperturbed eigenvalue vanishes
λunper (s) =0 (52)
n=0,m=0
but it will become non-zero and depend on the parameter s when the coupling between the two spins is taken into
account by the second-order perturbation theory
<λunper(L) per(1,2) λunper(R) ><λunper(L) per(1,2) λunper(R) >
λ (s)= 0,0 |L | n,m n,m |L | 0,0 (53)
0 λunper λunper
(n,mX)6=(0,0) 0,0 − n,m
The application of the perturbation per(1,2) to the left unperturbed eigenvector
L
<λunper(L) per(1,2) =i2J (e−s 1)(<λunper(L) <λunper(L) ) (54)
0,0 |L 1 − 3,2 |− 2,3 |
and to the right unperturbed eigenvector
es(1+µ)(1 µ′) (1 µ)(1+µ′)
per(1,2) λunper(R) > =i2J − − − (λunper(L) > λunper(L) >) (55)
L | n,m 1 4 | 3,2 −| 2,3
shows that the formula of Eq. 53 only involves the two intermediate states (n = 3,m = 2) and (n = 2,m = 3) with
the unperturbed complex-conjugated eigenvalues
Γ+Γ′
λunper = +i2(h h )
n=3,m=2 − 2 2− 1
Γ+Γ′
λunper = i2(h h ) (56)
n=2,m=3 − 2 − 2− 1
8
and becomes
<λunper(L) per(1,2) λunper(R) ><λunper(L) per(1,2) λunper(R) >
λ (s) = 0,0 |L | 3,2 3,2 |L | 0,0
0 0 λunper
− 3,2
<λunper(L) per(1,2) λunper(R) ><λunper(L) per(1,2) λunper(R) >
+ 0,0 |L | 2,3 2,3 |L | 0,0
0 λunper
− 2,3
Γ+Γ′
=J2(1 e−s)[es(1+µ)(1 µ′) (1 µ)(1+µ′)]
1 − − − − Γ+Γ′ 2+4(h h )2
2 2− 1
D
= (es 1)(1+µ)(1 µ′)+(e−s 1)(1 µ)(1+(cid:0) µ′) (cid:1) (57)
2 − − − −
(cid:2) (cid:3)
where we have introduced the notation
Γ+Γ′
D 2J2 (58)
≡ 1 Γ+Γ′ 2+4(h h )2
2 2− 1
(cid:0) (cid:1)
D. Averaged current and fluctuations
The expansion of the eigenvalue of Eq. 57 up to second order in s (Eq. 35)
D
λ (s) = s2(µ µ′)+s2(1 µµ′) +O(s3) (59)
0
2 − −
(cid:2) (cid:3)
yields the averagedcurrent (Eq 36)
<N >
I = lim t =D(µ µ′) (60)
av
t→+∞ t −
and the fluctuation (Eq 37)
(<N2 > <N >2)
F = lim t − t =D(1 µµ′) (61)
t→+∞ t −
E. Large deviations
To compute the function Φ(I) that governs the large-deviation form of the probability distribution P (I) of the
t
current I = Nt (Eq. 38), we need the Legendre transform of Eq. 39
t
Φ(I)=max(Is λ (s))=Is λ (s ) (62)
0 I 0 I
s − −
where s is the location of the maximum determined by the solution of the equation
I
D
I =λ′(s) = es(1+µ)(1 µ′) e−s(1 µ)(1+µ′) (63)
0 2 − − −
Since this is a second-order equation in the vari(cid:2)able es (cid:3)
2I
0=(1+µ)(1 µ′)e2s es (1 µ)(1+µ′) (64)
− − D − −
with the discriminant
2I 2
∆= +4(1 µ2)(1 (µ′)2) (65)
D − −
(cid:18) (cid:19)
one obtains that the positive roots reads
I 2+(1 µ2)(1 (µ′)2)+ I (1 µ)(1+µ′)
esI = D − − D = − (66)
q(cid:0) (cid:1) (1+µ)(1−µ′) I 2+(1 µ2)(1 (µ′)2) I
D − − − D
q
(cid:0) (cid:1)
9
so that the large deviation function of Eq. 62 finally reads
D
Φ(I) =IsI (esI 1)(1+µ)(1 µ′)+(e−sI 1)(1 µ)(1+µ′)
− 2 − − − −
(cid:2) I2+D2(cid:3)(1 µ2)(1 (µ′)2)+I
=D(1 µµ′) I2+D2(1 µ2)(1 (µ′)2)+Iln − − (67)
− − − − "p D(1+µ)(1−µ′) #
p
It vanishes Φ(I )=0 at I =D(µ µ′) of Eq. 60 as it should.
av av
−
F. Discussion
Insummary,besidesthemagnetizationsµandµ′ oftheboundarydrivings,theimportantparameterintheaveraged
current I , in the fluctuation F and more generally in the whole large-deviation function Φ(I) is the parameter D
av
introduced in Eq. 58 that contains the difference of the two random fields (h h ) in the denominator. In the
2 1
−
remaining of the paper, we focus on the ’Strong-Disorder regime’ where the scale of the random fields h is much
j
bigger than the scale of the couplings J that can be either uniform or random
j
(h h )2 J2 (68)
j+1− j ≫ j
so that it is valid to use perturbationtheory in the hoppings to evaluate various observables,as shownin this section
on the example of N =2 spins.
IV. RENORMALIZATION APPROACH FOR THE RELAXATION WITH A SINGLE RESERVOIR
WhenthequantumchainofN spinsissubjecttothe singleboundary-magnetization-driving(Eq20)ofparameters
(Γ,µ) on the spin 1 (while there is no driving on the last spin N), the stationary state is the trivial tensor-product
with the magnetization µ for all spins
1+µ 1 µ
λR >= N S =1,T =1>+ − S = 1,T = 1> (69)
| n=0 ⊗j=1 2 | j j 2 | j − j −
(cid:18) (cid:19)
but it is neverthelessinteresting to analyze the behavior ofthe relaxationrate Γ as a function of the systemsize N.
N
A. Boundary Strong Disorder Renormalization for the relaxation rate ΓN
The idea is that in the Strong Disorder regime for the random fields (Eq. 68), there exists a strong hierarchy
between the relaxation rates
Γ Γ ... Γ =Γ (70)
N+1 N 1
≪ ≪ ≪
i.e. the first spin σ in contact with the reservoir is the first to equilibrate with rate Γ = Γ, then the second spin
1 1
σ will equilibrate with some slower rate Γ , and so on. The aim is thus to introduce a Boundary Strong Disorder
2 2
Renormalization procedure in order to compute iteratively the relaxation rates Γ .
N
So we decompose the Lindbladian for the chain of (N +1) spins into
= unper + per
LN+1 LN+1 LN+1
unper = +ih (τz σz )
LN+1 LN N+1 N+1− N+1
per =i2J (τ+τ− +τ−τ+ σ+σ− σ−σ+ ) (71)
LN+1 N N N+1 N N+1− N N+1− N N+1
in order to take into account the coupling term per by perturbation theory in the hopping J .
LN+1 N
10
B. Structure of the four lowest modes of L
N
When the strong hierarchy of Eq. 70 exists, one may restrict the Lindbladian to its four lowest modes
N
L
3
lowest = λ(N) ψR ><ψL (72)
LN i | λ(N) λ(N)|
n=0 i i
X
that havethe following structure for the last spin N (while all the previousspins j =1,..,N 1 havealreadyrelaxed
−
towards equilibrium) :
(0) The vanishing eigenvalue λ(N) =0 representing the equilibrium is associated to the left an right eigenvectors
0
<λ(N)L =<S =+,T =++<S = ,T =
0 | N N | N − N −|
1+µ 1 µ
λ(N)R > = S =+,T =+>+ − S = ,T = > (73)
| 0 2 | N N 2 | N − N −
(1) The real eigenvalue λ(N) = Γ is associated to
1 − N
1 µ 1+µ
<λ(N)L = − <S =+,T =+ <S = ,T =
1 | 2 N N |− 2 N − N −|
λ(N)R > = S =+,T =+> S = ,T = > (74)
| 1 | N N −| N − N −
(2-3) The complex eigenvalue λ(2N) =−Γ2N +i(2hN +ωN) is associated to
<λ(N)L =<S = ,T =+
2 | N − N |
λ(N)R > = S = ,T =+> (75)
| 2 | N − N
while the complex-conjugate eigenvalue λ(3N) =−Γ2N −i(2hN +ωN) is associated to
<λ(N)L =<S =+,T =
3 | N N −|
λ(N)R > = S =+,T = > (76)
| 3 | N N −
C. Properties of the unperturbed Lindbladian Lun
N+1
The lowest modes sector of the decoupled unperturbed Lindbladian reads
unper = lowest+ih (τz σz )
LN+1 LN N+1 N+1− N+1
3
= λ(N) λ(N)R ><λ(N)L +ih (τz σz ) (77)
n | n n | N+1 N+1− N+1
n=0
X
so that its eigenstates are simply tensor-products of eigenstates of each term
unper λ(N)R > S ,T > =λ(unper) λ(N)R > S ,T >
LN+1 | n ⊗| N+1 N+1 n,SN+1,TN+1| n ⊗| N+1 N+1
<λ(N)L <S ,T unper =λ(unper) <λ(N)L <S ,T (78)
n |⊗ N+1 N+1|LN+1 n,SN+1,TN+1 n |⊗ N+1 N+1|
and the corresponding eigenvalues are simply the sums
λ(unper) =λ(N)+ih (T S ) (79)
n,SN+1,TN+1 n N+1 N+1− N+1
In particular, the four eigenvalues corresponding to n=0 (with S = 1 and T = 1) have no real part as
N+1 N+1
± ±
a consequence of λ(N) = 0. After taking into account the perturbation per of Eq. 71, these four eigenvalues will
n=0 LN+1
correspond to the four slowest modes of , with the structure analog to Eq. 72.
N+1
L
Since the perturbation has no diagonal contribution, we need to consider the second-order perturbation theory for
the eigenvalues. Let us first consider the two complex-conjugate non-degenerate eigenvalues
λ(unper) = i2h
0,+,− − N+1
λ(unper) =+i2h (80)
0,−,+ N+1
before we turn to the two-dimensional degenerate subspace
λ(unper) =λ(unper) =0 (81)
n=0,++ n=0,−−