Entanglement in stationary nonequilibrium states at high energies

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by  Marko Znidaric

Entanglement in stationary nonequilibrium states at high energies
Marko Zˇnidariˇc
Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia
(Dated: January 25, 2012)
In recent years it has been found that quantum systems can posses entanglement in equilibrium
thermal states provided temperature is low enough. In the present work we explore a possibility
of having entanglement in nonequilibrium stationary states. We show analytically that in a simple
one-dimensional spin chain there is entanglement even at the highest attainable energies, that is,
starting from an equilibrium state at infinite temperature, a sufficiently strong driving can induce
entanglement, even in the thermodynamic limit. We also show that dissipative dephasing, on the
other hand, destroys entanglement.
2 I. INTRODUCTION equilibrium
1 0.2
0 It has been realized that entanglement is rather ubiq-
2
uitous at sufficiently low temperatures. Ground states
0.1
n of interacting quantum systems typically posses entan-
a glement between different parts of a system. Similarly,
J when a system is at thermal equilibrium at sufficiently (cid:62) 0.0

4 lowtemperatureentanglementisoftenpresent. Thishas (cid:60)zz non(cid:45)eq. Μ
2
beenstudiedinanumberofworks, seee.g., Refs.[1]and (cid:45)0.1
references in [2], and is called thermal entanglement [3].
]
h Much less in known on the other hand about entan-
p glement in nonequilibrium stationary states. The main (cid:45)0.2
-
t difficulty is that solving a nonequilibrium system, either
n numerically or analytically, is much harder. Most stud- entangled
a (cid:45)0.3
ies of entanglement in a stationary nonequilibrium set-
u (cid:45)0.3 (cid:45)0.2 (cid:45)0.1 0.0 0.1 0.2 0.3
q ting have so far considered only small systems of 2 or 3
[ spin-1/2 particles [4]. It has been shown that currents (cid:60)z(cid:62)

can enhance entanglement [5]. In the present work we
2 present fully analytic results for entanglement in a sta- FIG. 1. (Color online) Expectation values of (cid:104)σjz(cid:105) ≡ (cid:104)z(cid:105) and
v (cid:104)σzσz (cid:105) ≡ (cid:104)zz(cid:105) in the bulk of a NESS for the XX model
tionarystateofanonequilibriumsystemofarbitrarysize. j j+1
5
without dephasing. The thick top curve is equilibrium states
1 The main questions we study are the dependence of en-
(allatinfinitetemperature),theshadedregionisNESSstates
4 tanglement in a nonequilibrium stationary state (NESS)
whichcanbereachedbynonzerodrivingµ. Asoneincreases
4 onthecouplingstrengthtothebath,thedrivingstrength
µ the NESS state moves downward, eventually reaching a
2. measuring how far we are from equilibrium, and the de- region of NESSs having entanglement (dark red region with
1 pendence on system size. The Hamiltonian part of the full circle). For details see text.
1 modelweconsiderisaone-dimensionalquantumspin1/2
1 chain with XX-type nearest-neighbor coupling,
:
v n−1 eventuallyreachingNESSinwhichonehasentanglement
Xi H = (cid:88)σjxσjx+1+σjyσjy+1. (1) between nearest neighbor spins, independent of system
size. By a sufficiently strong driving one can induce en-
r j=1
a tanglement even in an infinite temperature equilibrium
The XX model in its unitary setting is rather simple;
state. A nonequilibrium setting therefore enlarges the
it can be mapped to a system of noninteracting spinless
region of parameters for which entanglement is present.
fermions and can be easily diagonalized. A nonequilib-
In the presence of dephasing though, situation changes.
riumvariant,inwhichwecouplethesystemtoreservoirs
Entanglement is present only if dephasing is sufficiently
atchainends,canalsobesolvedanalytically,eveninthe
small, or if the system is small enough. In this diffusive
presence of dissipative dephasing. Nonequilibrium dy-
regime there is no entanglement in the thermodynamic
namics of the XX model with or without dephasing [6–
limit.
11] is rather rich, with transport varying from ballistic
to diffusive.
FortheXXmodelwithoutdephasing,whichshowsbal-
listic transport and is coherent in the bulk, our findings II. THERMAL ENTANGLEMENT
are summarized in Figure 1. As one turns on the driv-
ing µ one moves away from an equilibrium state at infi- Before going to nonequilibrium properties let us first
nitetemperature(indicatedbyanemptycircleinFig.1), remind ourselves of the entanglement in equilibrium
2
state. Results in this section are not new and will serve ΛPT
min
asareferencepointagainstwhichwewillcompareNESS
0.15
results. Foramoredetailedstudyofentanglementinthe
XY model see, for instance, Ref. [12]. As a separability 0.10
criterion we will always use the minimal eigenvalue of
0.05
thepartiallytransposedtwo-spinreduceddensitymatrix
ρPT , where transposition is done with respect to one T
j,j+1
spin. Itsnegativityisanecessaryandsufficientcondition 1 2 3 4 5 6
-0.05
for entanglement of two spin-1/2 particles [13] which is
the type of entanglement that we study throughout the -0.10
paper.
-0.15
We shall consider the grand-canonical state ρ at a
G
certain inverse temperature β = 1/T and chemical po-
tential φ,
FIG.2. MinimaleigenvalueofρPT (4)foragrand-canonical
j,j+1
state of XX model at φ=0 (3). A two-spin reduced density
exp(−βH −φM)
ρ = , Z =tr(e(−βH−φM)), (2) matrix ρ is entangled for T <1.94.
G Z j,j+1
where M = (cid:80)n σz is total magnetization. To calcu-
j=1 j
late the reduced density matrix of two nearest neighbor
spins in an infinite grand-canonical chain ρ we need all 0.0
G
two spin grand-canonical expectation values (cid:104)σασα(cid:48) (cid:105) ,
j j+1 G -0.2
where indices α,α(cid:48) label three Pauli matrices and an
identity, and (cid:104)A(cid:105) ≡ tr(ρ A). Due to the symmetry
G G -0.4
of the XX model the only nonzero terms are (cid:104)σz(cid:105) and
j G
(cid:104)σxσx (cid:105) = (cid:104)σyσy (cid:105) = (cid:104)H(cid:105) /(2n) and (cid:104)σzσz (cid:105) . > -0.6
j j+1 G j j+1 G G j j+1 G E
For an infinite chain the energy density and magneti- <
-0.8
zation can be found in [14], whereas correlations are in
Ref. [15]. In an infinite chain the relation (cid:104)σzσz (cid:105) =
j j+1 G -1.0
(cid:104)σz(cid:105)2 − (cid:104)σxσx (cid:105)2 holds [15]. In the thermodynamic
j G j j+1 G
limit n→∞ we have average magnetization and energy -1.2
density
-1.0 -0.5 0.0 0.5 1.0
1 (cid:90) π
(cid:104)σz(cid:105) = tanh(φ−2βcosk)dk, (3) <z>
j G π
0
(cid:104)σxσx (cid:105) = 1 (cid:90) πcos(k)tanh(φ−2βcosk)dk. FIG. 3. (Color online) Isocurves of constant β (solid curves)
j j+1 G π andφ(dashedcurves)inagrand-canonicalstateρ (2). Ex-
0 G
pectation values of energy density (cid:104)E(cid:105) ≡ 2(cid:104)σxσx (cid:105) and
Usingtheseonecaneasilywriteareduceddensitymatrix (cid:104)z(cid:105)≡(cid:104)σz(cid:105) (3)inatwo-spinnearestneighborredjucje+d1dGensity
of two nearest-neighbor spins, ρj,j+1 = 14[1+(cid:104)σjz(cid:105)G(σjz+ matrix ρjj,jG+1 uniquely determine β and φ. Iso-temperature
σz )+(cid:104)σxσx (cid:105) (σxσx +σyσy )+(cid:104)σzσz (cid:105) σzσz ], curves are at T = 1/β = 0,0.5,1,2,4,8,∞ (bottom to top),
j+1 j j+1 G j j+1 j j+1 j j+1 G j j+1
fromwhichonecanexpresstheminimaleigenvalueofthe whileiso-φcurvesareatφ=0,±0.25,±0.5,±1,±2(centerto
partially transposed reduced density matrix ρPT . The right/left). Thick (red) line is expectation values for NESS
j,j+1
result is states studied in the present work.
λPT = 1(cid:0)1−(cid:104)σxσx (cid:105)2 +(cid:104)σz(cid:105)2 −2√w(cid:1), (4)
min 4 j j+1 G j G
wherew =(cid:104)σxσx (cid:105)2 +(cid:104)σz(cid:105)2. Transitionfromanentan-
j j+1 G j G in a grand-canonical state. The relation between these
gled state to a separable state happens when λPT = 0, two expectation values and temperature T and chemical
√ min
which gives a condition ((cid:104)σxσx (cid:105) + 2)2 =1+(cid:104)σz(cid:105)2. potential φ can be inferred from Fig. 3. As we shall see,
j j+1 G j G
Numerically solving this equation for critical β at φ = all NESS states studied in the present work have energy
0 we get β ≈ 0.5162, or temperature T ≈ 1.94. densityequaltozero, andthereforelieonthethick(red)
c c
Thismeansthatbelowthistemperaturenearest-neighbor line in Fig. 3, that is, they have the same expectation
spins in a grand-canonical state of the XX model are en- valuesofenergyandmagnetizationasequilibriumgrand-
tangled, see Fig. 2. Critical β varies with φ only very canonical states at infinite temperature, β = 0. Note,
slightly, for instance, at φ=4 it is β ≈0.5146. however, that such NESS states are of course not grand-
c
Notethattheexpectationvaluesofmagnetizationand canonical as other expectation values, for instance the
energy density uniquely determine a two-spin nearest- current or (cid:104)σzσz (cid:105) do not have the same expectation
j j+1 G
neighbor reduced density matrix if the whole system is values as in the grand-canonical state.
3
III. NONEQUILIBRIUM XX MODEL B. Equilibrium, µ=0
A. Setting The equilibrium situation is obtained for µ=0 (other
parameters can be arbitrary). The NESS state is, in this
case, separable and equal to [9],
ThenonequilibriumdynamicsoftheXXmodelwillbe
described by the Lindblad master equation [16],
n
1 (cid:89)
ρ = (1+µ¯σz). (12)
d NESS 2n j
ρ=i[ρ,H]+Lbath(ρ)+Lbath(ρ)+Ldeph(ρ). (5) j=1
dt L R
This is exactly the grand-canonical state (2) with β =0
Each of the three dissipative terms is expressed in terms
and tanhφ = −µ¯. All equilibrium states, from which
of Lindblad operators L as
k NESS will be reached for nonzero µ, studied in the
presentworkthereforeareatinfinitetemperature. These
(cid:88)(cid:16) (cid:17)
Ldis(ρ)= [L ρ,L†]+[L ,ρL†] . (6) are indicated by a thick (red) line in Fig. 3.
k k k k
k
Dephasing is described by n Lindblad operators, each
C. Nonequilibrium and no dephasing, γ =0
acting only on the j-th spin, and being
(cid:114) Wefirstfocusonthecasewithγ =0becausethemain
γ
Ldeph = σz. (7) entanglementfeaturesarealreadypresentinthissimpler
j 2 j
XX model without dephasing. Because everything is in-
dependent of the system size n the results presented in
ThebathdissipatorLbath actsonlyonthe1stspinwhile this subsection are valid for any n.
L
LbRath acts on the last spin. Each involves two Lindblad Expectation values in the NESS that we need for a
operators, on the left end two-spin reduced density matrix are given in Ref. [8],
LL =(cid:114)Γ (1− µ +µ¯)σ+, LL =(cid:114)Γ (1+ µ −µ¯)σ−, −(cid:104)σxσy (cid:105)≡t=µ ΓLΓR , (13)
1 L 2 1 2 L 2 1 j j+1 (1+ΓLΓR)(ΓL+ΓR)
(8) µ (Γ −Γ )+Γ Γ (Γ +Γ )
while on the right end we have (cid:104)σ1z(cid:105)≡a1 =µ¯− 2 L(1+RΓ Γ )L(ΓR+LΓ ) R
L R L R
(cid:114) µ (cid:114) µ (cid:104)σz (cid:105)≡a=µ¯− µ (ΓL−ΓR)(1−ΓLΓR)
LR = Γ (1+ +µ¯)σ+, LR = Γ (1− −µ¯)σ−, 2,...,n−1 2 (1+Γ Γ )(Γ +Γ )
1 R 2 n 2 R 2 n L R L R
µ (Γ −Γ )−Γ Γ (Γ +Γ )
(9) (cid:104)σz(cid:105)≡a =µ¯− L R L R L R ,
where σj± = (σjx ± iσjy)/2. Relevant parameters are n n 2 (1+ΓLΓR)(ΓL+ΓR)
the two coupling strengths to the baths Γ , driving
L,R
strengthµthatdictatesthemagnetizationdifferencebe- aswellas(cid:104)σjzσkz(cid:105)=ajak−t2δj+1,k. Rangesthatthesepa-
tween chain ends, the average driving µ¯ and the dephas- rameters can take are t∈µ[0,1], then a ∈µ¯−µ[−1,1],
4 1 2
ing strength γ. Because driving coefficients have to be a∈µ¯− µ[−1,1], and a ∈µ¯− µ[−1,1].
2 n 2
real we must have Γ ≥ 0 as well as (for all four sign For equal coupling strengths at both ends, Γ = Γ ,
L,R L R
combinations) which we will use in all figures, the above expressions
simplify to t= µ Γ ,a =µ¯−Γt,a=µ¯,a =µ¯+Γt.
µ 21+Γ2 1 n
1± ±µ¯ ≥0. (10)
2
1. Two nearest-neighbor spins at the boundary
Inorderforthistoholdµandµ¯mustlieinsidearhombus
with corners (0,±1) and (±2,0) in a “µ−µ¯” plane. If
instead of µ and µ¯ we define the two parameters The reduced density matrix for two spins at the chain
end is
µ µ
2 +µ¯ ≡d, 2 −µ¯ ≡c, (11) x 0 0 0 
1 0 y 2it 0
ρ =  , (14)
i.e., µ = c+d,µ¯ = (d−c)/2, definition range is a sim- 12 4 0 −2it z 0 
ple square, c ∈ [−1,1] and d ∈ [−1,1]. The XX model 0 0 0 v
with or without dephasing has been studied in Refs. [6–
11]. Ouranalyticalcalculationsrelyheavilyontheexact with x=1+a+a +aa −t2, y =1+a−a −aa +t2,
1 1 1 1
NESS solution found in Refs. [7–9]. z = 1 − a + a − aa + t2, v = 1 − a − a + aa −
1 1 1 1
4
t2. Calculating ρPT we get only one eigenvalue that can
12
possibly be negative and which is
1 (cid:112)
λPT = (1+aa −t2− (a+a )2+4t2). (15)
min 4 1 1
In order to simplify expressions, from now on in this
3
subsection we take the same coupling at both ends,
Γ = Γ ≡ Γ. This has no qualitatively important con-
L R
sequences. In Fig. 4 we plot in black those regions of 2
(cid:71)
1.0
driving parameters for which ρ is entangled, that is,
12
λPT is negative. In addition to a “c−d” plane we also 1 0.5
min
0.0d
1.0 1.0 0
(cid:45)1.0 (cid:45)0.5
(cid:45)0.5
0.5 0.5
0.0
c 0.5 (cid:45)1.0
1.0
d 0.0 _Μ 0.0
(cid:45)0.5 (cid:45)0.5 FIG.5. (Coloronline)VerticalaxisisΓ(ΓL =ΓR),horizontal
axes are c and d. Parameters within the two symmetrically
(cid:45)1.0 (cid:45)1.0 placedshadedregions,existingforΓ>Γmin (18),correspond
(cid:45)1.0 (cid:45)0.5 0.0 0.5 1.0 (cid:45)0.5 0.0 0.5 toentangledboundarytwospinsintheNESS.Figure4shows
a cross-section of this plot at Γ=2.
c u
FIG. 4. Parameters for which ρ is entangled in the NESS
12
(black areas). Left: values of c and d (11) for which there Γ ≈ 1.08873. Regions of Γ and c and d for which ρ is
12
is entanglement. Right: values of u ≡ Γt = ((cid:104)σnz(cid:105)−(cid:104)σ1z(cid:105))/2 entangled can be seen in Fig. 5. It is instructive to plot
and µ¯ with an entangled ρ . Right plot is just a rescaled
12 how the NESS state differs from the equilibrium grand-
and rotated left figure. Dashed lines in the right plot denote
canonical state. As mentioned, the expectation value of
a region of allowed parameter values. Γ =Γ =2.
L R the energy is for our NESS states always zero [7, 8] and
onecanalwaysfindanequilibriumgrand-canonicalstate
showvaluesofu,definedasu≡Γt=((cid:104)σz(cid:105)−(cid:104)σz(cid:105))/2and
n 1 havingthesameexpectationvalueofenergyandmagne-
µ¯, which have physical interpretation as the difference of
tization,seeFig.3. Theexpectationvalueof(cid:104)σzσz (cid:105)in
magnetization at chain ends and the magnetization in- j j+1
theNESSontheotherhanddiffersfromtheoneinequi-
side the chain (at all spins apart from the 1st and the
librium. Therefore, we plot in Fig. 6 a region of allowed
last one). On the “c−d” plot the minimal c, i.e., the
expectation values of σz and σzσz in the bulk (away
beginning of the right tongue in the left frame of Fig. 4, j j j+1
from two boundary spins). Remember that in equilib-
is at
rium at infinite temperature (where (cid:104)σxσx (cid:105) =0) one
√ j j+1 G
3+8Γ2+4Γ4−2Γ(1+Γ2) 8+Γ2 has the relation (cid:104)σjzσjz+1(cid:105)G = (cid:104)σjz(cid:105)2G (top solid curve in
cmin = 1+2Γ2(1+Γ2) . (16) Fig.6). Onecanseethat forasufficientlystrong driving
µ, keeping µ¯ fixed, a NESS is reached in which ρ is
12
The functional form of the bottom boundary of this entangled (red region). What is even more interesting,
tongue, d(c,Γ), extending in c from c to c=1, is for sufficiently large (cid:104)z(cid:105), that is µ¯, an entangled state
min
is reached already for very small values of µ. That is,
√
d=1−(cid:0)1+w−c(1+3Γ2+Γ4)− v(cid:1), (17) for large average magnetization µ¯, one can reach entan-
glement from an infinite temperature equilibrium state
√
where w = Γ(1 + Γ2) 8+Γ2 and v = (1 + Γ2)[(9 + already with small driving. One can say that a small
5c2)Γ4+(1+c2)Γ6−4(−1+cw)−2Γ2(−6−2c2+cw)]. For stationary perturbation of an infinite temperature equi-
Γ > 1 the tongue extends all the way down to d = −1, librium state can induce entanglement. This is quite dif-
that is, d(c = 1,Γ) = −1. For Γ < 1 it is smaller, ferentfromthermalentanglement, whichispossibleonly
eventually disappearing for Γ<Γ ≈0.511, where at sufficiently low temperatures.
min
(cid:115) √ √
−3+(27−3 78)1/3+(27+3 78)1/3
Γ = . (18)
min 6
2. Nearest-neighbor spins in the bulk
In other words, ρ can be entangled only if the cou-
12
pling strength is in the range Γ < Γ < ∞. Entangle- We proceed to entanglement of two nearest-neighbor
min
ment tongues are the largest, i.e., c is the smallest, at spins in the bulk of the chain. Using again expectation
min
5
1.0 1.0 1.0
0.8 0.5 0.5
0.6
d 0.0 _Μ 0.0
0.4
(cid:45)0.5 (cid:45)0.5
(cid:62)
z
z 0.2
(cid:60) (cid:45)1.0 (cid:45)1.0
(cid:45)1.0 (cid:45)0.5 0.0 0.5 1.0 (cid:45)0.4(cid:45)0.2 0.0 0.2 0.4
0.0
c u
(cid:45)0.2
FIG. 7. Regions of entanglement in ρ for Γ = Γ = 1
j,j+1 L R
in “c−d” (left) and “u−µ¯” plane (right). There are two
(cid:45)0.4
symmetric pockets at large u, that is, at large magnetization
difference across the chain.
(cid:45)1.0 (cid:45)0.5 0.0 0.5 1.0
(cid:60)z(cid:62)
pocketgetssmaller,eventuallydisappearingwhenc =
min
FIG.6. (Coloronline)Expectationvaluesof(cid:104)zz(cid:105)≡(cid:104)σzσz (cid:105) 1. This bounds the range of possible couplings to
j j+1
and (cid:104)z(cid:105) ≡ (cid:104)σz(cid:105) in the bulk of NESS lie between top solid
j 1 √ (cid:113) √ 1 √ (cid:113) √
curveandbottomdashedcurve. Topsolidcurvecorresponds (1+ 2− −1+2 2)<Γ< (1+ 2− −1+2 2).
to equilibrium states. As one increases µ away from zero 2 2
(23)
(keeping all other parameters fixed) NESS moves vertically
in the plot. For sufficiently large µ an “entanglement smile” In the bulk we therefore have entanglement in NESS
region of entangled boundary two spins is reached, denoted only in this range of Γ. This can be seen in Fig. 8. In
by the dark (red) region. All is for Γ=1.088.
1.0
d 0.5
0.0
values (14) we get the reduced density matrix (cid:45)0.5
(cid:45)1.0
A 0 0 0 
+ 2.0
1 0 B 2it 0
ρ =  , (19)
j,j+1 4 0 −2it B 0 
0 0 0 A 1.5
−
where A± = (1±a)2 −t2 and B = 1−a2 +t2. The (cid:71)1.0
minimal eigenvalue of the partially transposed ρPT is
j,j+1
this time 0.5
(cid:45)1.0
1 (cid:112)
λPT = (1+a2−t2−2 a2+t2). (20) (cid:45)0.5
min 4
0.0
c
FromnowonweagainsetΓ =Γ =Γ. InFig.7wecan 0.5
L R
see regions of parameters for which ρ is entangled. 1.0
j,j+1
The main difference from the case of ρ (e.g., Fig. 4) is
12
that the entanglement is present in two pockets in the FIG.8. (Coloronline)VerticalaxisisΓ(Γ =Γ ),horizontal
L R
corners of Fig. 7. The lower edge of the right pocket are c and d. Two pockets at intermediate couplings 0.531 <
(near corner c=d=1) is Γ < 1.883 (23) are NESSs with entangled two-spin reduced
density matrix ρ .
√ √ i,i+1
c(1+3Γ2+Γ4)+2(1+Γ2)[−Γ 2+ A]
d= , (21)
1+Γ2+Γ4 Fig. 9 we also plot expectation values of magnetization
and (cid:104)σzσz (cid:105) in the bulk. One can see, that in contrast
√ j j+1
where A=1+Γ(3Γ+c2Γ+Γ3− 8c(1+Γ2)). It runs toentanglementintheboundarytwospins, hereonehas
from c to c=1, with entanglement only for sufficiently strong µ. Note how-
min
ever that all these states still have zero energy density.
√ √
4Γ( 2+Γ(−1+Γ 2)) Compare also the left plot in Fig. 9 with Fig. 1, where
c =3− . (22)
min 1+Γ2+Γ4 an enlarged central section of this plot (Γ=1) is shown.
Why can entanglement at the edge of the chain be
c is minimal, i.e., the pocket is the largest for Γ = reachedalreadyforsmallµwhilelargeµisneededinthe
min
1. When one increases or decreases Γ the size of this bulk? The answer lies in different magnetization values
6
1.0 1.0 valuesdependonthesystemsize. Undernonzerodriving
µ the spin current is proportional to ∼1/n, similarly as
long-range correlations. Because dephasing diminishes
0.5 0.5
(cid:62) (cid:62) off-diagonalmatrixelementsinthePaulibasis,itistobe
zz zz expectedthattheentanglementwillbesmallercompared
(cid:60) 0.0 (cid:60) 0.0
to a coherent XX model. This is indeed what happens.
For simplicity we will always use Γ = Γ = Γ and
L R
(cid:45)0.5 (cid:45)0.5 look at the central two spins described by the reduced
(cid:45)1.0 (cid:45)0.5 0.0 0.5 1.0 (cid:45)1.0 (cid:45)0.5 0.0 0.5 1.0 density matrix ρn/2,n/2+1. Using the exact solution for
NESS from Refs. [7, 9] one can easily obtain the reduced
(cid:60)z(cid:62) (cid:60)z(cid:62)
densitymatrixandtheminimaleigenvalueofρPT .
n/2,n/2+1
FIG. 9. (Color online) Expectation values in equilibrium, Expressions for λPT as well as boundaries of entangled
min
µ=0 (top full curve), and out of equilibrium (µ(cid:54)=0, region regionsaresimplebutlongandwedonotgivethemhere.
between dashed and full curve). Dark (red) region are NESS The main difference with the XX model is that the size
states with bulk nearest-neighbor entanglement. They are of the entanglement pockets depends on the dephasing
reached only for sufficiently strong driving µ. All is for Γ =
L strength γ. There are two symmetrically placed pockets,
Γ = Γ, on the left for Γ = 1, on the right for Γ = 1.5. An
R similarly as in the case of no dephasing. Regions of en-
enlarged central part of the left frame is shown in Fig. 1.
tangled NESS disappear if γ is larger than the critical
γ ,
c
√
attheedge. Becausethesystemisaballistictransporter
2+1−Γ−1/Γ
magnetization exhibits jump atthe boundary; it is µ¯−u γ = . (24)
c n−2
at the 1st spin and µ¯ at the 2nd, causing entanglement
for small µ. Also, regions of entangled NESS states are present only
Let us finally conclude the section on the XX model if the coupling strength Γ satisfies Γ <Γ<Γ , where
− +
with a discussion of non nearest-neighbor spins. Calcu-
latingthereduceddensitymatrixandtheminimaleigen- 1(cid:16) (cid:112) (cid:17)
Γ = w−γ(n−2)± (w−γ(n−2))2−4 , (25)
value of the partially transposed matri√x we see that the ± 2
conditionforentanglementis1−a2 <2 a2+t4,inother √
words,2|t|+µ¯2 >1. Thiscanneverbefulfilledandthere with w =1+ 2. These boundaries, which are a gener-
is no entanglement. alization of Eq.(23), can be seen in Fig. 10.
When one increases γ from zero the last entangled
NESS to survive at γ is the one with c = d = 1, i.e.,
c
µ = 2 and µ¯ = 0. At fixed γ and in the limit n → ∞
D. XX with nonzero dephasing
the critical dephasing strength goes to zero and entan-
glement disappears! With γ there is therefore a discon-
tinuous transition from entangled NESSs at γ = 0 (red
pocket in Fig. 1) to no entanglement for γ > 0. This is
illustrated in Fig. 11. One can see how the size of the
2.0
parameter region with entangled NESSs decreases when
(cid:71)
(cid:43) one increases γ or n. If nearest-neighbor spins ρ are
j,j+1
1.5 in the bulk but not in the middle of chain, j (cid:54)= n/2, the
situation is similar as for ρ . As one increases
n/2,n/2+1
(cid:71) dephasing γ or n there is a transition from entangled to
1.0 separable ρ at certain critical dephasing. The only
j,j+1
difference is that this critical dephasing weakly depends
2(cid:45)1 on the location j in the chain where the two spins are.
0.5
(cid:71) Critical γ weakly increases as one moves away from the
(cid:45) c
center of the chain.
For nearest-neighbor spins at the boundary, described
0.0
by the reduced density matrix ρ , entanglement is pos-
0.0 0.1 0.2 0.3 0.4 0.5 12
sible only if Γ is larger than some minimal value Γ .
m
Γ n(cid:45)2
This minimal value depends on γ and n in a compli-
cated way, however, it has a very simple thermodynamic
FIG. 10. Shaded region is a ra(cid:72)nge o(cid:76)f coupling strengths Γ limit. Namely, limn→∞Γm =1 irrespective of dephasing
(Eq.25)forwhichonecanhaveaNESSwithnearest-neighbor strength γ. For Γ > 1 dephasing therefore does not de-
entanglement in the middle of the XX chain with dephasing. stroyentanglementinρ . Theregionofentangledstates
12
stillhasthesameshapeasinzerodephasing,Fig.4,con-
For the XX model with dephasing NESS expectation tainingallstateswithc=1,onlyitswidthc changes.
min
7
1(cid:45)c 1.0 It shrinks to zero in the limit n → ∞ as well as for
min
0.5 0.8 γ → ∞. In the presence of dephasing there is no entan-
n
0.4 glement between non-nearest-neighbor spins.
0.6
0.3 d0.4
n(cid:61)6
0.2 10 0.2
0.1 16 0.0 IV. CONCLUSION
0.0 0.2 0.4 0.6 0.8 1.0
Γ
0.02 0.04 0.06 0.08 0.10 c
We have studied bipartite entanglement between two
spins in a nonequilibrium stationary state of the XX
FIG. 11. Left: Dependence of size of entangled region,
chain of any length. In the absence of dephasing we
1−c , on dephasing γ for three different chain sizes n =
min
show that the two boundary spins can become entan-
6,10,16. One can see that for γ >γ there are no entangled
c
regions (c gets larger than 1). Right: The extent of the gled for arbitrarily small driving away from an infinite
min
entangledregionforγ =0.04andn=6andn=10(twofull temperatureequilibriumstate. Inthebulk,drivingmust
curves). c for these two curves is marked with points in surpass a certain value in order to have nearest-neighbor
min
the left plot. For reference we also plot the entangled region entanglement. Coupling with the reservoirs also has to
without dephasing, γ =0 (dashed curve; the same data as in be sufficiently strong. Dephasing on the other hand de-
Fig. 7). The arrow shows the direction in which n increases; creases entanglement. Apart from the two boundary
only one quadrant of the “c−d” plane is shown. Both plots
spins, entanglement is present only if a product of de-
are for Γ=1 and ρ .
n/2,n/2+1 phasing strength and system size is sufficiently small.
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