Exact asymptotics of the current in boundary dissipated quantum chains in large external fields

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by  Zala Lenarčič

Exactasymptoticsofthecurrentinboundarydissipatedquantumchainsinlargeexternalfields
Zala Lenarcˇicˇ1 and Tomazˇ Prosen2
1J. Stefan Institute, SI-1000 Ljubljana, Slovenia and
2Faculty of Mathematics and Physics, University ofLjubljana, SI-1000Ljubljana, Slovenia
Boundarydrivenquantummasterequationforageneralinhomogeneous(non-integrable)anisotropicHeisen-
bergspin1/2chain,oranequivalentnearestneighborinteractingspinlessfermionchain,isconsideredinthe
presenceofastrongexternalfieldf. Wepresentanexactclosedformexpressionforlargef asymptoticsof
thecurrentinthepresenceofpureincoherentsourceandsinkdissipationattheboundaries. Inapplicationwe
demonstratearbitrarylargecurrentrectificationinthepresenceofinteraction.
PACSnumbers:75.10.Pq,05.60.Gg,03.65.Yz
5
1
0 Introduction.– The study of quantum dynamics far from systemisgovernedbytheHamiltonianH =H +V,
xxz
2
equilibrium is currently at the forefront of theoretical [1]
n−1
n (but also experimental [2, 3]) condensed matter physics re- (cid:88)
H =H +H , H = 2α (σ+σ− +σ−σ+ ),
a search. Although most of non-equilibrium phenomena stud- xxz xx z xx j j j+1 j j+1
J j=1
ied are directly time dependent, such as e.g. in the so-called
0 n−1 n
quantum quenches [4], there is a growing interest in study- (cid:88) (cid:88)
1 H = ∆ σzσz , V = f σz. (1)
ingsteady-statephenomena[5,6]especiallyunderlargether- z j j j+1 j j

j=1 j=1
] mal/chemicalbiasand/orexternalfields[7–10]. Aprominent
l
e examples of this sort are dielectric breakdown in the Mott whereparametersα ,∆ maybesitedependent. Fortheho-
j j
- insulator [11–18] or rectification effects in nanoscale heat mogeneouscaseα = 1,∆ = ∆willbeused. f determine
r j j j
t or particle transport [19–21]. Particularly fruitful direction the profile of magnetic field. Using Wigner-Jordan transfor-
s
. whichhasbeendevelopedrecently[22]isahybridapproach mationtheproblemcanbemappedtospinlessfermionchain
t
a of open quantum systems formalism [23], where the bulk of whereα maptofreehoppingamplitudes,∆ tointeractions
j j
m
afinite(possiblylarge)interactingquantumsystemistreated and f to electric potential. Density matrix ρ of the system
j
- fullycoherentlywithaunitarygenerator,whereastheenviron- may be described using the Liouville master equation in the
d
mentisconsideredtoactlocallyonthesystem’sboundariesin Lindbladform[22,23]
n
termsofincoherentmarkovianquantumnoise.Remarkably,it
o
dρ(t)
c hasbeendemonstratedthatsuchanapproachallowsforinter- =Lˆρ:=−i[H,ρ(t)]+Dˆρ, (2)
[ esting exactly solvable, integrable instances [24–27] and can dt

1 be phenomenologically justified even for non-small system- whereDˆρ = (cid:80) 2L ρ(t)L† −{L†L ,ρ(t)}describescou-
v environment couplings in terms of the so-called repeated in- k k k k k
plingtothebathattheedges. Here,onlytheextremecaseof
5 teractionsprotocol[28]. √
pure source/sink on the left/right end, L = εσ+,L =
9 √ 1 1 2
3 An exact treatment of nonequilibrium transport has been εσ−, will be considered. Nonequilibrium steady state
n
2 sofarpossibleonlyforpseudo-forcedrivingoriginatedfrom (NESS) ρ = lim ρ(t) is found as the fixed point
∞ t→∞
0 the bias in markovian processes at the system’s boundaries. Lˆρ = 0. Theeffectofdrivingwillbecharacterizedbythe
∞
.
1 In this Letter we combine this approach with a real force f, spincurrentfromsitej toj+1
0 resultingfromanexternalfieldgradient,andformulateasys-
5 tematicperturbationtheoryofthesteadystatecurrentin1/f. Jj =iαj(σj+σj−+1−σj−σj++1), (3)
1
We show that under a plausible assumption about the zeroth
: whichisinthelimitt → ∞,i.e. forNESS,siteindependent,
v order of the steady state density matrix an explicit asymp-
Xi totic f → ∞ expression for the current can be derived for J = tr(Jjρ∞)/trρ∞.
Largefield,largeinteractionasymptotics.–WesplittheLi-
ageneralinhomogeneous(andnonintegrable)XXZspin1/2
r ouvillianintodiagonalandoff-diagonalpartsLˆ=fLˆ +Lˆ ,
a chain. For long chains one can demonstrate arbitrarily large 0 1
relative current rectification under reversal f → −f. Our Lˆ ρ=−i[V˜ +H˜ ,ρ], Lˆ ρ=−i[H ,ρ]+Dˆρ, (4)
0 z 1 xx
formula is clearly corroborated by numerical computations
onsmallchains,wherefurtherresonanceeffectsofunusually whereH˜ = (cid:80)n−1∆˜ σzσz ,V˜ = (cid:80)n f˜σz with∆˜ ≡
z j=1 j j j+1 j=1 j j j
largecurrentsatparticularvaluesoff arediscussedandpartly ∆ /f,f˜ ≡f /f. Weconsider1/f asaformalsmallparam-
j j j
explained. Theresultappliesaswellforlargeanisotropy(in-
eter,assumingf (cid:29)α ,ε. Notethatbothdiagonalgenerators,
j
teraction)asymptotics. V orH ,neednotbelarge. Wemayeitherfixf˜ −f˜ = 1,
z n 1
Quantumtransportinboundary&fielddrivenspinchain.– inwhichcasef representsthetotalfield/potentialdropacross
We consider an open inhomogeneous XXZ spin 1/2 chain thesystemwhile∆j =O(1),orconsiderthelargeanisotropy
exposed to an additional magnetic field, so that bulk of the (interaction) asymptotics ∆˜ = O(1) while f˜ = O(f−1).
j j
2
Expanding NESS in powers of f−1, ρ = (cid:80)∞ f−pρ(p), Steady state spin current.– We would like to identify the
∞ p=0
differentordersareconnectedwiththerecurrencerelation: leading order terms in the current J = tr(J ρ )/trρ . In
j ∞ ∞
ourdiagrammaticnotation trisasumofcoefficientsatcon-
Lˆ0ρ(p+1)+Lˆ1ρ(p) =0, Lˆ0ρ(0) =0. (5) figurationswiththesameupperandlowerindexateachsite.
Therefore trρ = a + O(f−2). Similarly, trJ ρ is a
∞ 0 j ∞
Each order of ρ∞ can be expanded in terms of a complete sumoverconfigurationsinρ withσ±σ∓ and1 ±σzelse-
Paulibasisρ(p) = (cid:80) M(p) (cid:81) σsj,wheres ∈ ∞ j j+1 l l
{s1,...,sn} s1,...,sn j j j where. SupposewecalculatecurrentfromtheoperatorJn/2.
{0,z,+,−},andσ0 =1. Conditionforρ(0) inEq.(5)issat- We will show that then the leading contribution comes from
isfied for exponentiallymany products (cid:81) σsj for which the the components with 1 +σz for l < n/2 and 1 −σz for
j j l l l l
sum (cid:80)n f ξ +(cid:80)n−1∆ (cid:0)(ξ +ξ ) (mod 2)(cid:1) van- l>n/2+1.
j=1 j sj j=1 j sj sj+1
ishes, where ξsj = ±1 for sj = ± and ξsj = 0 other- Weclaimthatinordertoobtainthecurrentfrom trJn/2ρ∞
wise.However,exactnumericaldiagonalizationofLˆonsmall uptoO(f−(n+2))onlythetermsρ˜= ρ(0) +ρ1,R +ρ1,L +
systems of even size n ≤ 8 shows that for strong field or ρ2,R+ρ2,Linthedensitymatrixarerelevant. Expression
anisotropyρ(0)isadiagonalprojector
ρ =
1,R
ρ(0) =a0(cid:89)n2 12(1j +σjz) (cid:89)n 21(1j −σjz) (6) 2af1(cid:0)↑↑ ...... ↑↑ ↓↑(cid:12)(cid:12)↑↓ ↓↓ ↓↓ ...... ↓↓(cid:1)+ ifann (cid:0)↑↑ ...... ↑↑ ↓↑(cid:12)(cid:12)↓↑ ↓↓ ↓↓ ...... ↓↓(cid:1)
j=1 j=n2+1 2: ⇓ n:⇑
with step-like magnetization profile. In case of large +a2(cid:0)↑ ... ↑ ↓(cid:12)(cid:12)↓ ↑ ↓ ... ↓(cid:1)+ ian−1(cid:0)↑ ... ↑ ↓(cid:12)(cid:12)↓ ↑ ↓ ... ↓(cid:1)
anisotropy this has already been observed [29] and given f2 ↑ ... ↑ ↑ ↓ ↓ ↓ ... ↓ fn−1 ↑ ... ↑ ↑ ↓ ↓ ↓ ... ↓
physical motivation suitable for our limit as well. Thus, we 3: ⇓ n−1:⇑
conjecture that ρ(0) has the form of Eq. (6) for a system of
. .
arbitrary even length n. If n is odd ρ(0) contains two terms .. ..
of the form (6) with a jump in magnetization between sites n n
(n−1,n+1) or (n+1,n+3). Further on we assume even n, 2 : ⇓ 2 +2:⇑
2 2 2 2
tgieonne.ralizing the solution to arbitrary n at the end of deriva- +fann22 (cid:0)↑↑ ...... ↑↑ ↓↑(cid:12)(cid:12)↓↓ ↓↓ ...... ↓↓ ↑↓(cid:1)+ ifan2n2++11(cid:0)↑↑ ...... ↑↑ ↓↑(cid:12)(cid:12)↓↓ ↓↓ ...... ↓↓ ↑↓(cid:1)
Weintroducethefollowingnotationforlocaloperators
Dˆ :⇒
1(1+σz)≡(cid:0)↑(cid:1), σ− ≡(cid:0)↓(cid:1),
2 ↑ ↑ withap ∈Rintroducesasequenceoftermsconnectedbythe
1(1−σz)≡(cid:0)↓(cid:1), σ+ ≡(cid:0)↑(cid:1), (7) recurrence relation (5), which takes upper ↑n/2+1 in ρ(1) (in
2 ↓ ↓ redcolor)totherightboundaryandbacktoitsinitialposition.
Aftereachstepcoefficientinvolvedisforaf−1 higherinex-
i.e.,(cid:0)µν(cid:1) ≡ |ν(cid:105)(cid:104)µ|,ν,µ ∈ {↑,↓}. E.g.,ρ(0) isrepresentedas pansion,beingrealuntiltheactionofDˆ andimaginaryafter.
asinglecomponent ρ is constructed symmetrically, taking upper ↓ to the
1,L n/2
leftboundaryandwillbeexpandedintermsofcoefficientsb
ρ(0) =a0(cid:0)↑↑ ...... ↑↑(cid:12)(cid:12)↓↓ ...... ↓↓(cid:1). (8) withb1 =a1.Otherρ2,R(L)originatefromthesecondtermopf
ρ(1)inEq.(9)andcorrespondtoshiftinglower↑ (↓ )
Vertical bar marks the center of the system. It is instructive n/2+1 n/2
to think of the effect that different parts of Lˆhave on a cho- totheright(left)boundary.
sentermM(p) (cid:81) σsj insuchnotation. TheactionofLˆ Fromthisconstructionitisclearthattheleadingcontribu-
s1,...,sn j j 0 tiontothecurrentisoforderf−n,namely
is diagonal, i.e. it does not change the operator product to
whichitisapplied,whileunitarypartofLˆ causesexchange
ofneighboringpairsifdifferent,(↑ ↓ )o1r(↓ ↑ ). Dissi- J =−αn2 2(an+bn) +O(f−(n+2)) (10)
j j+1 j j+1 fna
patorDˆistheonlyprocesstorelateimaginaryandrealpartof 0
coefficientsinvolvedintherecurrencerelationandistrivialif since real coefficients a ,b get subtracted, while nth order
1 1
bothµ,ν arealignedwiththeboundarydriving. coefficient in ρ and ρ (equivalently for ρ and ρ )
1,R 2,R 1,L 2,L
Using the recurrence relation (5) higher order ρ(p) can be are complex conjugated and sum up yielding a factor 2 in
generatedfromρ(0). E.g. thefirstorderρ(1)thenconsistsof Eq.(10).Anyotheroperatorproducttermhavingσ± σ∓
n/2 n/2+1
ρ(1) =a1(cid:104)(cid:0)↑↑ ...... ↑↑ ↓↑(cid:12)(cid:12)↑↓ ↓↓ ...... ↓↓(cid:1)+(cid:0)↑↑ ...... ↑↑ ↑↓(cid:12)(cid:12)↓↑ ↓↓ ...... ↓↓(cid:1)(cid:105). ainndf−pr1odjeucetotorso1biv+iouσsizafdodristoiomnealis>tepns/n2e+ed1edistoofchreigatheeirtofrrodmer
(9) ρ(0). To calculate the spin currentwe have to express a ,b
n n
However, wewillnotstrivetobuiltthewholeNESSdensity with a . In the procedure each step yields an equation that
0
matrix, instead we will relate perturbatively only its compo- relatescoefficientsfromtheconsecutiveordersviarecurrence
nentrelevantfortheNESSspincurrent. (5). Explicitrelationsobtainedfromρ are:
1,R
3
Step1: Stepn/2+1(actionofdissipatorDˆ):
(cid:16) (cid:17)
−2iαn2aa0+=2i ∆˜n2−1+a∆˜0αn2+n21+f˜n2 −f˜n2+1 a1 =(011) an2+1 =−2(cid:2)∆˜n−1+∆˜n−a1n2−ε∆˜n −f˜n+f˜n(cid:3) (14)
1 ∆˜n−1+∆˜n+1−f˜n+1+f˜n 2 2 2
2 2 2 2
Stepn/2+1+k,1≤k <n/2:
Stepk, 1<k <n/2:
ak = ∆˜n2+k−1+∆˜n2+kak+−∆1˜αn2n−2+1k−−1∆˜n2 −f˜n2+k+f˜n2 an2+1+k = ∆˜n−k−1+∆˜n−ka+n2∆˜+kn−α1n−−k∆˜n −f˜n−k+f˜n
(12) 2 2 2
(15)
Stepn/2:
Combining Eqs. (11 - 15) we finally obtain a as a function
an−1αn−1 n
an2 = ∆˜n−1+∆˜n−21−∆˜n −f˜n+f˜n (13) ogfouas0lya,ndparametersofHamiltonian. bn isexpressedanalo-
2 2 2
n−1
an = − εαn−1 2(cid:81) αn2+k−1 αn2+k , (16)
a0 2(cid:16)∆˜n−1+∆˜ n−1−∆˜ n−f˜n+f˜n(cid:17)2 k=1 (cid:16)∆˜ n+k+∆˜ n+k−1+∆˜ n−1−∆˜ n−f˜n+k+f˜n(cid:17)2
2 2 2 2 2 2 2 2 2
n−1
bn = − εα1 2(cid:81) αn2−k+1 αn2−k . (17)
a0 2(cid:16)∆˜1+∆˜ n+1−∆˜ n−f˜n+1+f˜1(cid:17)2 k=1 (cid:16)∆˜ n−k+∆˜ n−k+1+∆˜ n+1−∆˜ n−f˜n+1+f˜n−k+1(cid:17)2
2 2 2 2 2 2 2 2 2
Forahomogeneouschainandfieldwhichisoddwithrespect thetwocomponentsinρ(0)havedifferentweights,causingan
tothecenterf =−f expressionforthecurrentsimpli- additionalprefactorinthelatterequation.
j n+1−j
fies In Fig. 1 we compare numerical and analytical results
n−1 for the spin current J at linear field profile and homoge-
J = f2εn (cid:16)∆˜ −f˜n+f˜n2(cid:17)−2 2(cid:89) (cid:16)2∆˜ −f˜n2+k+f˜n2(cid:17)−2. n4,eo6u,8s.HaAmtiltnoni=an w4itwhe∆sho=w 5alsino saynsteemxasmopflesiwzeithn in=-
k=1
(18) homogeneous Hamiltonian with parameters {α1,α2,α3} =
{0.1,0.3,0.5},{∆ ,∆ ,∆ } = {−2.5,−5,−1}. Negative
1 2 3
One should note that a shift of f˜ for a constant does not ∆ are chosen for clarity since J(−∆ ,−f ) = J(∆ ,f ).
j j j j j j
change the result. Obtained expression for the current is ap- Evidently, expression(19)perfectlycapturesthedependence
propriateforageneralmonotonicmagneticfield,however,in J(g)atlargeenoughgradientsgandindicatesthepositionof
thecaseoflinearfieldprofilef −f ≡githasespecially someoftheresonancesinthenumericalresult. Fromthecon-
j j−1
simpleformintermsofagradientg struction of the perturbative solution it is clear that poles in
Eqs. (16,17) appear when at some level of perturbative hier-
8ε (cid:18)2∆ (cid:19)−2 n2(cid:89)−1(cid:18)2∆ (cid:19)−2 archytheactionofdiagonalLˆ0 onconsideredoperatorprod-
J = −n −k . (19)
gn g g uctistrivial. Consequently,thehierarchyisdisconnectedand
k=1 ourexpressionnotvalidanymore. Additionalresonancesob-
For systems of odd size n one should consider J (or servedinnumericalresultsoriginatefromotheroperatorprod-
(n−1)/2
J ). The leading order contributions originate from ucts which are annulled by Lˆ that are not contained in ρ˜.
(n+1)/2 0
the part of ρ(0) with jump in magnetization between sites In the resonances the hierarchy can be altered in a way that
(n−1,n+1) (or (n+1,n+3)). The imaginary coefficient rel- yieldsacontributiontothecurrentinlowerorderoff−1[30].
2 2 2 2
evant for the current is then obtained as in system of even The range of validity of asymptotic expression (19), which
sizen−1,havingcontributiononlyfromtheboundarythatis is g ∈ (−∞,0)∪(gc,∞) for ∆ > 0, is set by the furthest
closertothemagnetizationjump. Thecurrentforoddn,ho- resonance,yieldingacriticalgradientgc =2∆.
mogeneousXXZ chainandfieldofoddsymmetryw.r.t. the Fluxrectification.–Asymmetryofthedrivingisrevealedby
centerofthesystemisthensimplyJ(n) = J(n−1)/4. For changingthedirectionofthedrivingforce, f → −f, result-
an inhomogeneous chain or non-symmetric monotonic field ing in J(f) (cid:54)= J(−f). Rectifying behavior is conveniently
4
interaction prefers aligned spins, magnetic field with g > 0
100 num,(cid:68)(cid:61)5 n(cid:61)4 dictates the same orientation of spins as the Lindblad driv-
ana
10(cid:45)2 naunma,(cid:68)inh ing,resultinginanenhancedinsulatingbehaviorofNESS.If
g < 0 their effects are contrary, manifested in the enlarged
J
10(cid:45)4 spincurrent. RelationJ(−∆ ,−f )=J(∆ ,f )generalizes
j j j j
10(cid:45)6 theargumentto∆j >0.
For gradients above the critical value the rectification co-
10(cid:45)2 n(cid:61)6 efficientR canbecalculatedforanysystemsizenusingthe
currentasymptotics(16-19). InFig.2weplotthedependence
10(cid:45)4 of R on the system size n (even only) for a homogeneous
J 10(cid:45)6 chainandlinearfieldprofileatthreedifferentgradients,g =
10(cid:45)8 3∆,5∆,10∆. Circles correspond to exact results for small
systems. Ourresultsclearlyshowthatinthethermodynamic
limitRreachestheextremevaluelim R=sgn(∆),cor-
n→∞
n(cid:61)8 responding to insulating behavior for gradients of opposite
10(cid:45)4 signthananisotropy∆.
J 10(cid:45)8 Discussion.– We have developed a systematic asymptotic
expansionofthespin/particlecurrentinquantumspin/fermion
10(cid:45)12 chains with nearest neighbor interaction driven with inho-
mogeneous external fields and dissipated with incoherent
(cid:45)20 (cid:45)10 0 10 20
source/sink baths at the chain ends. Our results have been
g
worked out in detail on an example of an inhomogeneous
XXZ spin 1/2 chain, driven by a transverse magnetic field
Figure1. (Coloronline)SpincurrentJ asafunctionofgradientg
gradient. We can point out two mechanisms for enhanc-
forlinearfieldprofileandanisotropy∆ = 5andε = 1insystems
of size n = 4,6,8 obtained numerically (num) and from Eq. (19) ing current rectification: (i) rectification coefficient R → 1
(ana). Curveatn=4labeledby∆ showssamecomparisonfor in thermodynamic limit n → ∞, at fixed large gradient
inh
aninhomogeneouschainwithparametersgiveninthetext. f/(n − 1), for any nonzero anisotropy/interaction ∆ (cid:54)= 0,
as a result of our asymptotic current formula; (ii) for reso-
nant— field (gradient) values where our asymptotic analysis
1.0 (cid:230) (cid:230) (cid:230)(cid:224) (cid:230)(cid:224) (cid:230)(cid:224) (cid:230)(cid:224) (cid:230)(cid:224) (cid:230)(cid:224) (cid:230)(cid:224) (cid:230)(cid:224) (cid:230)(cid:224) (cid:230)(cid:224) (cid:230)(cid:224) (cid:230)(cid:224) fails,thecurrentisofhigherorderin1/f foronedirectionof
0.8 (cid:224) (cid:224) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) tchoeeffifieclidenftRtha→n in1tehveenotahtefir.xeTdhifisncitleeasrilzyeynie.lNdsotreectthiafitcathtieorne
R (cid:236) (cid:236) (cid:230)(cid:230) g(cid:61)3(cid:68) is no rectification effect in noninteracting chains ∆ = 0, in
(cid:236)
0.6 (cid:236) (cid:224)(cid:224) g(cid:61)5(cid:68) agreementwith[31]. Ouranalysisalsoappliestoreveallarge
anisotropyasymptoticsatvanishingexternalfieldsbysetting
(cid:236)(cid:236) g(cid:61)10(cid:68)
(cid:236) ∆ = f∆˜ ,f = 0inEqs.(16,17),yieldingforthehomoge-
10 20 30 j j j
neousXXZ chainJ =ε2−γ∆−n, γ =n2/4−n/2−1for
n
evennandJ =ε2−δ∆−(n−1), δ =(n−1)2/4−(n−1)/2+1
foroddn,asnotednumericallyin[29].
Figure2. (Coloronline)DependenceoffluxrectificationRonsys-
tem size n for ∆ > 0,ε = 1 and gradients g = 3∆,5∆,10∆ Discussions with M. Zˇnidaricˇ and E. Ilievski, as well as
obtained from Eqs. (20,19). Circles correspond to exact numerics supportbythegrantsP1-0044,J1-5349oftheSlovenianRe-
forsmallsystems. searchAgencyareacknowledged.
quantifiedbyrectificationcoefficient[31]
J(f)−J(−f) [1] A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalattore,
R= , (20)
J(f)+J(−f) Rev.Mod.Phys.83,863(2011).
[2] T.Langen,R.GeigerandJ.Schmiedmayer, AnnualReviewof
whichiszeroincase ofnorectificationandreachesextreme CondensedMatterPhysics6,null(2015),[arXiv:1408.6377].
values±1whenthesystemforfieldsinonedirectionbehaves [3] I.Bloch,J.DalibardandW.Zwerger, Rev.Mod.Phys.80,885
(2008).
asaperfectinsulator.
[4] P.CalabreseandJ.Cardy, Phys.Rev.Lett.96,136801(2006).
For one direction of the dissipative Lindblad driving the
[5] M.Fagotti,M.Collura,F.H.L.EsslerandP.Calabrese, Phys.
anisotropy ∆ and the field f have a cooperative effect and
j j Rev.B89,125101(2014).
competingonefortheotherdirection[32]. Fixingdissipative [6] B.Woutersetal., Phys.Rev.Lett.113,117202(2014).
drivingtoourinitialchoiceandassuming∆j < 0,whenthe [7] D.BernardandB.Doyon, JournalofPhysicsA:Mathematical
5
andTheoretical45,362001(2012). systems(OxfordUniv.Press,2002).
[8] V. Jaksˇic´ and C.-A. Pillet, Communications in Mathematical [24] T.Prosen, NewJ.Phys.10,040326(2008).
Physics226,131(2002). [25] T.Prosen, Phys.Rev.Lett.106,217206(2011).
[9] A.Amaricci,C.Weber,M.CaponeandG.Kotliar, Phys.Rev. [26] E.IlievskiandT.Prosen, NuclearPhysicsB882,485(2014).
B86,085110(2012). [27] T.Prosen, Phys.Rev.Lett.112,030603(2014).
[10] M.EcksteinandP.Werner, Phys.Rev.B88,075135(2013). [28] D.KarevskiandT.Platini,Phys.Rev.Lett.102,207207(2009).
[11] Y.Taguchi,T.MatsumotoandY.Tokura,Phys.Rev.B62,7015 [29] G.Benenti,G.Casati,T.Prosen,D.RossiniandM.Zˇnidaricˇ,
(2000). Phys.Rev.B80,035110(2009).
[12] T. Oka, R. Arita and H. Aoki, Phys. Rev. Lett. 91, 066406 [30] Reordering within ρ can be motivated by degeneracy of
∞
(2003). eigenstatesofHamiltonianH.Itturnsoutthataresonancein
[13] M. Eckstein, T. Oka and P. Werner, Phys. Rev. Lett. 105, thecurrentappearswhenever|0(cid:105)=|↑1 ···↑n↓n+1 ···↓n(cid:105),
2 2
146404(2010). whichisnearlyaneigenstateforlargeanisotropyorlargefield,
[14] F.Heidrich-Meisneretal., Phys.Rev.B82,205110(2010). isdegeneratewithanothereigenstate|i(cid:105)(withlessofinsulating
[15] Z. Lenarcˇicˇ and P. Prelovsˇek, Phys. Rev. Lett. 108, 196401 nature)withboundaryspinsatleastpartiallycompatiblewith
(2012). theboundarydriving.Suchadegeneracyreducestheinsulating
[16] T.Oka, Phys.Rev.B86,075148(2012). effectof|0(cid:105)(cid:104)0|andpromotes|0(cid:105)(cid:104)i|,|i(cid:105)(cid:104)0|,|i(cid:105)(cid:104)i|inρ .
∞
[17] C.Aron, Phys.Rev.B86,085127(2012). [31] G.T.Landi,E.Novais,M.J.deOliveiraandD.Karevski,Phys.
[18] G. Mazza, A. Amaricci, M. Capone and M. Fabrizio, Rev.E90,042142(2014).
arXiv:1412.6415(2014). [32] If the field and the boundary driving try to align spins in the
[19] M. Terraneo, M. Peyrard and G. Casati, Phys. Rev. Lett. 88, samemannerfor∆<0theinsulatingstatewithstep-likemag-
094302(2002). netization(compatiblewithsource/sink)willnotbedegenerate
[20] B. Li, L. Wang and G. Casati, Phys. Rev. Lett. 93, 184301 withanyothereigenstateofH atarbitraryfieldstrength[30].
(2004). In this sense the field and the interaction cooperate and sepa-
[21] C.W.Chang, D.Okawa, A.MajumdarandA.Zettl, Science rateitsenergyfromothereigenstates.Byreversingthefieldor
314,1121(2006). theboundarydrivingdegeneraciesbecomepossible.Thesame
[22] H. Wichterich, M. J. Henrich, H.-P. Breuer, J. Gemmer and effectat∆>0isobtainedwiththefieldofoppositesign.
M.Michel, Phys.Rev.E76,031115(2007).
[23] F.PetruccioneandH.-P.Breuer, Thetheoryofopenquantum