Transport in anisotropic model systems analyzed by a correlated projection superoperator technique

0.22 MB·English

by  Hendrik Weimer

Transport in anisotropic model systems analyzed by a correlated projection
superoperator technique
Hendrik Weimer,1 Mathias Michel,2 Jochen Gemmer,3 and Gu¨nter Mahler1
1Institute of Theoretical Physics I, University of Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany∗
2Advanced Technology Institute, Faculty of Engineering and Physical Sciences,
University of Surrey, Guildford GU2 7XH, United Kingdom
3Physics Department, University of Osnabru¨ck, Barbarastr. 7, 49069 Osnabru¨ck, Germany
(Dated: February 2, 2008)
By using a correlated projection operator, the time-convolutionless (TCL) method to derive a
quantummasterequationcanbeutilized toinvestigatethetransportbehaviorofquantumsystems
as well. Here, we analyze a three-dimensional anisotropic quantum model system according to
8
this technique. The system consists of Heisenberg coupled two-level systems in one direction and
0
weak random interactions in all other ones. Dependingon the partition chosen, we obtain ballistic
0
behavior along the chains and normal transport in the perpendicular direction. These results are
2
perfectly confirmed by thenumerical solution of thefull time-dependentSchr¨odinger equation.

n
a PACSnumbers: 05.60.Gg, 44.10.+i,73.23.Ad
J

7
1 I. INTRODUCTION lation, which has ad hoc been transferred to heat trans-
port simply by replacing the electrical current by a heat

]
current [9]. However, the justification of this replace-
h
The transport of different extensive quantities like en-
p ment remains unclear since there is no way of express-
ergy, heat, entropy, mass, charge, magnetization, etc.,
- ing a temperature gradientin terms of an addend to the
t through and within solid state systems is an inten-
n Hamiltonian of the system as before [10].
sively studied topic of nonequilibrium statistical dynam-
a Otherapproachestoheatconductivityinquantumsys-
u ics. Nevertheless,therearenumerousopenquestionscon-
tems are based on diagonalization of the Schr¨odinger
q cerning the type of transportespecially in small systems
equation[11], analyzing the level statistics of the Hamil-
[ far from the thermodynamic limit and in particular in
tonian [12, 13] or by an explicit coupling to some envi-
quantummechanics. Attheheartofmanyinvestigations
1 ronments of different temperature [14, 15]. In the latter
v is the classification into two main categories: normal or case, environments are described by a quantum master
9 diffusive transfer of the extensive quantity and ballistic
equation [16] in Liouville space. Here the temperature
6 transport featuring a divergence of the conductivity.
differences can, indeed, be described by a perturbation
6
Diffusivetransportoccurswheneverthesystemisgov- operator so that one may treat a thermal perturbation
2
. erned by a diffusion equation. In particular, this means in this extended state space similar as an electrical one
1
that excitations decay exponentially fast and the spa- in the Hilbert space [17].
0
tial variance of an initial excitation grows linear in time. TheHilbertspaceAverageMethod[18]allowsforadi-
8
0 Ballistic transport, however, is rather described by the rect investigation of the heat transport in quantum sys-
: equations of motion of a free particle. For the spatial tems from Schr¨odingerdynamics. By deriving a reduced
v
variance ofan excitationthis implies a quadratic growth dynamical equation for a class of design quantum sys-
i
X in time. tems,normalheattransportaswellasFourier’sLawhas
r Inthepresentpaper,wewillconcentrateonthetrans- beenconfirmed[19,20]. Recently,ithasbeenshownthat
a port of energy and heat in quantum systems. There are fordiffusivesystemsthe Hilbertspaceaveragemethodis
severaldifferentapproachesdiscussedintheliteratureto equivalenttoaprojectionoperatortechniquewithanex-
investigate the transport of those quantities in quantum tended projection superoperator [21, 22]. However, bal-
mechanics. One very famous ansatz is the investigation listicbehaviorcannotbeanalyzedwiththeHilbertspace
of heat transport in terms of the Green-Kubo formula Average Method in a straightforward manner since it is
[1–6]. A main advantage of this approach is certainly not obvious how to obtain time-dependent rates.
its computability after having diagonalized the system’s Usingacorrelatedsuperprojectionoperatorwithinthe
Hamiltonian. Derivedonthebasisoflinearresponsethe- derivation of the time-convolutionless (TCL) quantum
ory the Kubo formula has originallybeen formulated for master equation leads to a reduced dynamical descrip-
electrical transport [7, 8], where an external potential tion of the investigated system. The main advantage of
can be written as an addend to the Hamiltonian of the thecorrelatedTCLmethodreferstoitsperturbationthe-
system. Basically one finds a current-current autocorre- oretical character. Thus it is a systematic expansion in
some perturbational parameter.
To use this alternative method for an investigation of
the transport behavior of a quantum system, it is nec-
∗Electronicaddress: [email protected] essary to partition the microscopic system described by
2
the Hamiltonian Hˆ into mesoscopic subunits. While the
µ=1
complete dynamics is governedby the Schr¨odingerequa-
tion of the full system according to its density operator
i
ρˆ˙ = [Hˆ,ρˆ] (t)ρˆ, (1) µ=2
−~ ≡L
we aim at deriving a closed reduced dynamical equation
for the subunits chosen. Formally, this partitioning is µ=3
done by introducing a projection superoperator that
P
projects onto the relevant part of the full density matrix
(a) (b)
ρˆ [16], here the spatial energy distribution within the
system. However, by a straightforward application of
FIG.1: Partitionschemesforinvestigatingthetransportper-
the projection superoperator on the above equation, the
pendicular (a) or parallel (b) to the spin chains. Each spin
dynamicsofthe reducedsystemisnolongerunitary,but
is represented by a dot, solid lines indicate Heisenberg in-
described by teractions along the chains, dashed lines represent random
interactions. The diagonal couplings within each plane have
ρˆ˙ = (t)ρˆ. (2) been left out for clarity [except thelower left corner of (a)].
P PL
These effective equations of motion for the relevant
part ρˆ can either be written as an integro-differential
transportanisotropiesemergingfromananisotropic(but
P
equation (Nakajima-Zwanzig equation [23, 24]) or as a
coherent) interaction.
time-convolutionless (TCL) master equation [16], which
The model we are going to investigate is a three-
is an ordinary linear differential equation of first order.
dimensionalmodeloftwo-levelsystemsdepictedinFig.1.
Bothmethodsallowforasystematicperturbativeexpan-
In terms of Pauli operators the local Hamiltonian of the
sion. In the TCL expansion series the first-order term
network is given by
typically vanishes and thus the leading order is given by
(cf. [21, 22]) ∆E
Hˆ = σˆ(i), (4)
loc 2 z
t Xi
ρˆ˙ = dt (t) (t ) ρˆ. (3)
1 1
P −Z PL L P with the localenergy splitting ∆E defining the basic en-
0 ergy unit within our model.
In x direction, the two-levelsystems are coupled via a
However, in order to obtain a converging perturbation
series expansion should not be chosen arbitrarily: A Heisenberg interaction
P
“wrong” projection superoperator may lead to a break-
down of the expansion [21]. Hˆ = σˆ(i) σˆ(i+1), (5)
H
⊗
Xi
II. DESCRIPTION OF THE MODEL with the Pauli spin vectors σˆ(i) =(σˆ(i),σˆ(i),σˆ(i)) at site
x y z
i.
In the present paper we consider a three-dimensional In the y and z directions, we use a random interac-
(3D)modelcomposedoftwo-levelsystems. Thecoupling tion matrix HˆR to couple both adjacent sites and next
between the atoms is anisotropic, i.e., in one direction neighbor sites lying diagonally opposite [see lower left
dominated by a Heisenberg interaction whereas the cou- corner of Fig. 1(a)]. The nonzero matrix elements are
pling in all other directions is random. The choice of taken from a Gaussian ensemble with zero mean and a
random couplings ensures that the interaction is unbi- variance s2. While each matrix element is taken from
ased as it does not have any special symmetry. Two- the same ensemble, the geometry of the system requires
level atoms or spin-1/2 systems [25] allow to study a that we do not have translational invariance within the
large variety of quantum effects from quantum infor- random interaction.
mation processing to solid state theory, described by a To investigate the transport in x or z direction, re-
rather simple interaction, making them interesting both spectively (cf. Fig. 1), we perform a partition of the
from an experimental and theoretical point of view. Of model into N subunits. A layer of n two-level systems
particular interest are the transport properties of sys- is grouped together into a new local subsystem, coupled
tems containing 1D and 2D spin structures, e.g., the in- to adjacent layers by the connections between pairs of
vestigations of heat transport in cuprates, in which a two level systems. Because of the anisotropy within the
dramatic heat transport anisotropy has been reported model we can study the transport perpendicular to the
[26, 27]. While the anisotropy is mainly attributed to Heisenbergchainsinthez direction[Fig.1(a)]andalong
anisotropic phonon scattering processes, we investigate the chains in x direction [Fig. 1(b)].
3
partitioningintosubunitsyieldsthefollowingmesoscopic
δǫ n Hamiltonianconsistingofalocalandaninteractionpart
⊗ ⊗ ··· ⊗
∆E Hˆ =Hˆ +Hˆ
L I
N N−1
µ=1 µ=2 µ=N = Hˆ (µ)+ Hˆ (µ,µ+1). (8)
L I
µX=1 µX=1
FIG. 2: N subunits with ground state and first excitation
bandof width δε containing nenergy levels each. Black dots
Here Hˆ (µ) of subunit µ consists of the constant local
specify the initial states used. L
energy splitting, the Heisenberg interaction, and the in-
ternal random couplings of each subunit [cf. gray planes
in Fig 1(a)]. Since λ λ the effect of the internal
The coupling strength of an arbitrary interaction ma- R H
≪
trix Vˆ is defined as random couplings on the spectrum of HˆL(µ) may be ne-
glected. Therefore the bandwidth δε is determined by
η = 1 Tr Vˆ†Vˆ , (6) the Heisenberg interaction given by
dq { }
δε=8λ . (9)
H
with d being the dimension of the matrix (see [18]). For
therandominteractionwechoosethevariances2 insuch The last term in Eq. (8), Hˆ (µ,µ+1), denotes the in-
a way that η = 1 for all interaction matrices coupling I
teraction between the subunits which is purely random
adjacent subunits.
here,i.e.,containspartsoftherandominteractionHamil-
The complete Hamiltonian of the full model system is tonian Hˆ only.
thus described by R
Hˆ =Hˆ +λ Hˆ +λ Hˆ . (7)
loc H H R R
B. Derivation of the TCL master equation
Because of the normalization of the interaction matrices
the numbers λH andλR define the coupling strengthbe- The correlatedprojection superoperator introduced
P
tween different sites. The coupling strengths λH for the in Sec. I is of the type as suggested by Breuer [22] and
Heisenberg interaction and λR for the random interac- reads
tionarechosensothatλ λ ∆E,whichisknown
R H
as the weak coupling limit.≪ ≪ ρˆ= Tr Πˆ ρˆ 1Πˆ P 1Πˆ , (10)
µ µ µ µ
Regardlessofthepartitionschemechosen(inthexorz P { }n ≡ n
Xµ Xµ
direction)eachsubunitcanbeseenasamoleculeconsist-
ingofseveralenergybands. However,thesolutionforthe with Πˆ being the standard projection operators
complete system is computationally unfeasible for more µ
than a few sites. If we restrict ourselves to initial states
Πˆ = n n , (11)
whereonly onesite is excited(orsuperpositionsthereof) µ µ µ
| ih |
the Heisenberg interaction does not allow to leave this Xnµ
subspace of the total Hilbert space. By choosing also
the random interaction to conserve this subspace we re- and nµ the eigenstate of HˆL(µ) in the one-particle ex-
| i
strict all further investigations to the single excitation citation subspace, i.e. the states in the band of subunit
subspace. Figure 2 gives a graphical representation of µ (cf. Fig. 2). Consequently, the number Pµ is just the
our system, with δε being the width of the first energy excitationprobabilityofsubunitµ. Thischoiceof thus
P
band (all higher excitation bands are neglected here). implementsthepartitioningschemerequiredforstudying
Interpreting our model system in terms of a magnetic transport behavior.
system, i.e., the two-level atoms representing coupled Switchingtotheinteractionpicture,pluggingboththe
spins in a magnetic field for example, the considered en- Hamiltonian (8) and the projection (10) into Eq. (3) we
ergytransportisequivalenttospintransportinagapless get
system (i.e. ∆E =0).
λ2 t
P˙ = R dt Tr Πˆ [Hˆ (t),[Hˆ (t ),Πˆ ]] P
µ −n~2 Z 1 { µ R R 1 ν } ν
III. TRANSPORT IN THE z DIRECTION Xν 0 (12)
forthesecondorderTCLexpansion. Thetimedependen-
A. Partitioning scheme ciesofthe couplingoperatorsrefertothe transformation
into the interaction picture and are defined as
Let us consider the transport perpendicular to the
Heisenberg chains (in the z direction) first. Then, the Hˆ (t)=eiHˆLtHˆ e−iHˆLt. (13)
R R
4
By exploiting that Πˆ projects onto eigenstates of Unfortunately, this function is not square integrable due
µ
Hˆ (µ)wecanevaluatethetracebyusingtheblockstruc- to singularities at the boundaries of the spectrum. How-
L
tureoftheinteractionHˆ (µ,µ+1)betweenadjacentsub- ever, due to symmetry we have
I
units (see [19, 20]), resulting in
δε δε/2
g2(E)dE =2 g2(E)dE. (21)
dPµ = γ 2P P P (14) Z0 Z0
µ µ µ+1 µ−1
dt − − −
(cid:0) (cid:1) In order to avoid the singularity at E = 0 we renormal-
with the decay rate ize the number of states in the band. We introduce the
regularized integral
2λ2 n sin(ω t)
γµ = n~R2 Xk,l |hkµ|HˆR|lµ+1i|2 ωklkl . (15) FΛ(n)=2 δε/2α2g2(E)dE
Z
Λ
The frequency ω refers to the transition between the δε/2 α2n2
kl
=2 dE, (22)
eigenstatesk,l. Equation(14)isbasicallyarateequation Z π2E(δε E)
for the probabilities to find an excitation in subunit µ. Λ −
with α being the factor that renormalizes the number
of states. We assume that for a band consisting of only
C. Decay rate a few levels n˜ (but still enough to define a density of
states), the density of states is approximately constant.
For a constant density of states g˜(E) we simply have
Since the interaction between two adjacent subunits
is a random matrix with the above described proper-
δε/2 n˜2
ties, all matrix elements are approximately of the same 2 g˜2(E)dE = , (23)
size. That means that the rate does not depend on Z δε
0
the subunit µ (γ = γ). Furthermore, we can assume
µ
therefore our renormalization prescription is given by
k Hˆ l 2 1, finding
µ R µ+1
|h | | i| ≈
n˜2
2λ2 sinω t F (n˜)= . (24)
γ = R kl . (16) Λ δε
n~2 ω
Xk,l kl
Using this result to solve Eq. (22) for α at constant n˜
yields
In the following the double sum is treated analogous to
the derivation of Fermi’s Golden Rule. π
Since the sine cardinal(sinc) ofEq. (16)is a represen- α= . (25)
2ln(δε/Λ 1)
tation of the Dirac δ-distribution −
p
Thisallowsustocalculatethephysicallimitoftherenor-
sin(ω t)
kl
πδt(ωkl)= lim , (17) malization procedure, i.e.,
t→∞ ωkl
n2
we may approximate the rate for not too small t by lim F (n)= (26)
Λ
Λ→0 δε
2πλ2
γ R δ(E E ). (18) whichisthesamevalueasforaconstantdensityofstates.
≈ n~ k− l
Xk,l This finally leads to the relaxation rate
Replacing the double sum over integrals in the energy 2πλ2n
γ = R . (27)
space we arrive at ~δε
2πλ2 δε The approximation introduced by Fermi’s Golden Rule
γ R g2(E)dE (19) is only valid in the linear regime (see [18]), i.e.,
≈ n~ Z
0
4π2nλ2
with the state density g(E), i.e., the integral over the R 1. (28)
δε2 ≪
square of the density of states. Since we have neglected
the internal random interaction completely, the state
density of the first excitation subspace is just given by D. Solution of the TCL master equation
the state density of a Heisenberg spin chain
2n 1 Figure 3 shows both the numerical results for the so-
g(E)= . (20) lution of the full Schr¨odinger equation and the solution
πδε 1 2E 1 2 ofthe rateequation(14), accordingto the abovederived
q − δε −
(cid:0) (cid:1)
5
1 the interaction transforms into
0.8 P1(t)
bability 00..46 P2(t) HˆR(t)==eeiiH(ˆHˆLHt+HˆHˆRLe)−tHiˆHˆRLte,−i(HˆH+HˆL)t (32)
o
r
P 0.2 where Eq. (31) has been used. Note that this is not the
P3(t) standardinteractionpicture as usedabove,but a special
0
oneallowingustotreatthetransportinthexdirectionin
0 500 1000 1500 2000
t[unitsof ~/∆E] a similarmanner as in the z direction. According to this
transformation the derivation of the second order TCL
master equation in Sec. IIIB, especially Eqs. (12), (14)
FIG. 3: Perpendicular transport: probability to find the ex-
citation in subunit µ = 1,2,3. Comparison of the numerical and (15), remain unchanged.
solution of the Schr¨odinger equation (crosses) and second-
order TCL (lines). (N = 3, n = 600, λR = 5·10−4∆E,
λH =6.25·10−2∆E) B. Decay rate
However, the computation of the rate (16) is different
approximation for the rate γ [cf. Eq. (27)]. Both are in here. For calculating the local band structure we con-
reasonably good agreement. siderjustarandommatrixofdimensionn,drawnfroma
Equation(14)isadiscreteversionofthediffusionequa- Gaussianunitaryensemble. Fromrandommatrixtheory
tion, which does notchange when regardingthe thermo- [28] it is known that the density of levels ζ(x) for such
dynamiclimit(n,N ,nλ2 =const). Foraδ-shaped a random Hermitian matrix consisting of elements with
→∞ R
excitationatt=0itssolutionisaGaussianfunction,the zeromeanandunitvarianceforbothrealandimaginary
variance of which grows linear in time. Therefore, it is parts is given by
evident that the heat transport is normal perpendicular
1
to the chains.
ζ(x)= 2n x2. (33)
π −
p
Mapping this to a density of energy levels leads to
IV. TRANSPORT IN THE x DIRECTION
8n δε2
g(E)= E2. (34)
A. Partitioning scheme πδεr 4 −
We rescale the variance to the interaction strength λ ,
R
In the following let us concentrate on the transport
which gives for the bandwidth
in the x direction, i.e., parallel to the chains. Thus, we
have a slightly different partition of the total Hamilto- δε=4√nλ . (35)
R
nian. Besides the local energy splitting, the local part
HˆL of the mesoscopic Hamiltonian (8) contains random In order to check whether our local Hamiltonian HˆL
interactions only: canbe approximatedby such a randommatrix,we com-
pare the eigenvalues E(x) of both matrices. Using
N
Hˆ = Hˆ (µ)+λ Hˆ (µ) . (29) dE 1
L loc R R
= (36)
µX=1(cid:2) (cid:3) dx g[E(x)]
Incontrast,theinteractionbetweenthesubunitsconsists and separating variables yields
of a Heisenberg and a random part,
8n δε2
E2dE =dx, (37)
N−1 πδεr 4 −
Hˆ = λ Hˆ (µ,µ+1)+λ Hˆ (µ,µ+1) . (30)
I H H R R
µX=1(cid:2) (cid:3) with the state density (34). This expression cannot be
solved analytically for E, so we compare the numerical
In the one-particle excitation subspace the commuta- solution for discrete values of x with the eigenvalues of
tor relations Hˆ . As Fig. 4 shows, Hˆ may indeed be approximated
L L
by a random matrix drawn from a Gaussian unitary en-
[Hˆ ,Hˆ ]=[Hˆ ,Hˆ ]=0 (31) semble. However,by plugging Eq.(35) into Eq.(28) one
H L H R
getsaconstantvalueofπ2/4whichisdefinitelynotsmall
are satisfied. If the dynamics induced by the local part compared to one. Thus, the requirement for the linear
Hˆ and the Heisenberg Hˆ is absorbed in the transfor- regimeisviolatedandthederivationoftherateaccording
L H
mation into the interaction picture, the random part of to Fermi’s Golden Rule can no longer be applied.
6
0.03 1
]
E
∆ 0.02 0.8
of 0.01 (t) 0.6
units −0.001 P1 0.4
)[
(x −0.02 0.2
E
−0.03
0 200 400 600 0 100 200 300
x t[unitsof ~/∆E]
FIG. 4: Comparison of the eigenvalues of HˆL and a random FIG. 5: Parallel transport: probability to find the excitation
matrix drawn from a Gaussian unitary ensemble. (n = 600, in the first subunit (µ = 1). Comparison of the numerical
λR =5·10−4∆E) solution of the Schr¨odinger equation (crosses) and second-
order TCL (solid line). (Same parameters as for Fig. 3)
The approximation used in Sec. IIIB is analogous to
Fermi’sGoldenRule. AlltransitionsinEq.(16)fromlto
where the diagonal elements ρˆs are the occupation
kareweightedbytherespectivevalueofthesincfunction P µµ
whichchangesits shape forincreasingtimes to approach probabilities Pµs. The off-diagonal elements of Pρˆint can
adeltapeakfort . Inthesituationdescribedabove becomputedbyreplacingtheprojector(11)withanother
the decay takes p→lac∞e within the linear regime, i.e., at one projecting out off-diagonal elements as well. The
an intermediate time scale. That means that all possi- dynamics of the diagonal and the off-diagonal elements
bletransitionfrequenciesaredistributedbelowthepeak. decouple so that diagonal initial states remain diagonal
Thus the sum in Eq. (16) can be approximated by the for all time.
area under the peak (see [18]). Thus using Eq. (39) for the inverse transformation we
This is not the case here. The decay happens on a getthetime-dependentsolutionoftheprobabilitiesinthe
much shorter time scale, when the peak is extremely Schr¨odinger picture. In Fig. 5 the numerical solution of
broad. Therefore almost any transition frequency be- theSchr¨odingerequationiscomparedwiththeTCLpre-
longs to the maximum of the peak. Thus the sinc in diction. Again, there is a very good agreement between
Eq. (16) should better be approximated by the maxi- the exact solution and our second order approximation.
mum value of the peak, instead of the area under the
peak. The maximum value grows with time according
to t. Thus the double sum over sinc functions could be
approximatedby n2t. This means that we get the relax-
ation rate D. Spatial variance
2nλ2
γ = R t. (38) To classify the transport behavior in the x direction
~2
a very large system has to be considered, so that the
initial excitation does not reach the boundaries of the
system during the relaxation time. Since the solution of
C. Solution of the TCL master equation
the time-dependent Schr¨odinger equation becomes un-
feasible the second-order TCL prediction has been used
The solution of Eq. (14) with the diffusion coefficient
forsubsequentnumericalintegration. Thevarianceofan
(38) defines the occupation probabilities in the interac-
excitation initially at µ=µ ,
0
tion picture Pint. Note that in the other direction the
µ
occupation probabilities of the interaction picture have
been equivalent to the occupation probabilities in the N
σ2(t)= P(s)(t)(µ µ )2, (40)
Schr¨odinger picture. This is not the case for the present µ − 0
situation because of the special choice of the interaction µX=1
picture. Remember that we have used not only the local
Hamiltonian for the transformation into the interaction shown in Fig. 6 grows quadratic in time, i.e., the trans-
picture,butalsoapartoftheinter-subsysteminteraction port is ballistic. Here we have considered a system with
[cf. Eq. (32)]. N = 300 subunits and an initial excitation at µ = 150
0
Since we are interestedin the occupationprobabilities solving the TCL master equation. This is also valid in
in the Schr¨odinger picture Pµs we need to calculate the the thermodynamic limit as γ(t) does not change. Nu-
inverse transformation of the density operator merical investigations show that the transport behavior
is largely independent of γ(t). Ballistic transport is ob-
ρˆs =e−iHˆHt ρˆinteiHˆHt, (39) served as long as λ t γ(t) on all relevant time-scales.
H
P P ≫
7
1000 methodareinverygoodaccordancewiththerealdynam-
icalbehaviorofthesystem. Havingestablishedamethod
750
which efficiently describes the dynamical properties of a
2σ 500 complex quantum model the transport behavior can be
classified by either an analytic analysis of the solution
250 of the reduced dynamical equations or by a numerical
investigation. Here, the simplicity of the reduced equa-
0 tions in comparison to the exact system of differential
0 100 200 300
equations allows us to investigate the dynamical proper-
t[unitsof ~/∆E]
ties of a much larger system which is not accessible to a
direct investigation.
FIG. 6: Variance of an excitation initially at subunit µ0 =
The analysis shows that the model features two very
150. Second-order TCL prediction (crosses) and quadratic
fit (solid line). (N = 300, n = 600, λR = 5 · 10−4, different types of transport behavior in the x and z di-
λH = 6.25 · 10−2) rections, perpendicular or parallel to the chains, respec-
tively. In the z direction we have found a standard sta-
tistical decay behavior following a diffusion equation on
V. CONCLUSIONS the basis of the mesoscopic subunits. In this way dif-
fusive behavior has been derived from first principles
on a mesoscopic scale whereas the dynamics on the mi-
In the present paper we have demonstrated how the
abstract method of correlated projection superoperators croscopic scale (i.e., of a single spin) is obviously non-
diffusive. This indicates that the transport behavior is
for the TCL master equation [21, 22] can be used to an-
alyzethe transportbehaviorofa three-dimensionalsolid not only a property of a system per se, but also depends
on the way we are looking at it. In contrast the model
state model: a system of coupled two-level atoms with
an anisotropic interaction. The analysis is based on the shows ballistic behavior parallel to the chains which is
demonstrated by the features of the reduced dynamical
following preconditions:
equations. Note that this behavior is similar to investi-
1. a partitioning scheme in position space to consider gationsoflargeanisotropieswithintheheatconductivity
the transport in one direction of the model, thus of cuprates [26, 27].
introducing a projection superoperator In conclusion this method of a correlated projection
superoperators within TCL allows to investigate the dy-
2. the convergence of the TCL expansion in second
namical behavior of 3D model systems on a mesoscopic
order (a wrong projection superoperator leads to
scale. It is useful both in the case of a statistical decay
a diverging expansion, or large higher than second
according to a diffusion equation and the ballistic case,
orders)
where time dependent rates are important.
3. an approximation scheme for computing the decay
rate to avoid numerical integration
Acknowledgments
According to those central points a reduced dynamical
description of the complex quantum model is derived
which canbe analyzed,e.g., to classify the transportbe- We thank H.-P. Breuer, M. Henrich, F. Rempp, G.
havior of the system. Reuther, H. Schmidt, H. Schr¨oder, J. Teifel and P. Vi-
ByacomparisonoftheTCLpredictionwiththe exact dal for fruitful discussions. Financial support by the
numerical solution of the complete Schr¨odinger equation Deutsche Forschungsgemeinschaft is gratefully acknowl-
ofourmodelsystemwehaveshownthattheresultsofthe edged.
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