Conductance in strongly correlated 1D systems: Real-Time Dynamics in DMRG

0.32 MB·English

by  Guenter Schneider

Conductance in strongly correlated 1D systems: Real-Time Dynamics in DMRG
Guenter Schneider∗ and Peter Schmitteckert†
Institut fu¨r Theorie der Kondensierten Materie, Universit¨at Karlsruhe, 76128 Karlsruhe, Germany
(Dated: February 6, 2008)
A new method to perform linear and finite bias conductance calculations in one dimensional
systemsbased on thecalculation of realtime evolution within theDensityMatrix Renormalization
6 Group (DMRG) is presented. We consider a system of spinless fermions consisting of an extended
0 interacting nanostructure attached to non-interacting leads. Results for the linear and finite bias
0 conductance through a seven site structure with weak and strong nearest-neighbor interactions
2 are presented. Comparison with exact diagonalization results in the non-interacting limit serve as
n verification of the accuracy of our approach. Our results show that interaction effects lead to an
a energy dependentself energy in the differential conductance.
J
PACSnumbers: 73.63.-b,72.10.Bg,71.27.+a,73.63.Kv
7
1

] During the past decade improved experimental tech- sity matrix for the time evolved states, its applicability
l niques have made production of and measurements on is very limited. Luo, Xiang and Wang [14] improved the
e
- one-dimensional systems possible [1], and hence led to methodbyextendingthe density matrixwiththe contri-
r
t an increasing theoreticalinterest in these systems. How- butions of the wave function at intermediate time steps.
s
. ever, the description of non-equilibrium transport prop- Schmitteckert [17] showed that the calculations can be
t
a erties, like the finite bias conductance of an interacting considerablyimprovedbyreplacingtheintegrationofthe
m nanostructure attached to leads, is a challenging task. timedependentSchr¨odingerequationwiththeevaluation
- For non-interacting particles, the conductance can be of the time evolution operator using a Krylov subspace
d
extracted from the transmission of the single particle method for matrix exponentials and by using the full fi-
n
levels [2, 3, 4]. Since the screening of electrons is re- nite lattice algorithm.
o
c duced by reducing the size of structures under investiga- Analternativeapproachis basedonthe wavefunction
[ tion, electron-electron correlations can no longer be ne- prediction[19]. There one first calculates aninitial state

glected. Recently several methods to calculate the zero with a static DMRG. One iteratively evolves this state
1
v bias conductance of strongly interacting nanostructures by combining the wave function prediction with a time
9 have been developed. One class of approaches consists evolution scheme. In contrast to the above mentioned
8 in extracting the conductance from an easier to calcu- full t-DMRG, one keeps only the wave functions at two
3
late equilibrium quantity, e.g. the conductance can be time steps in each DMRG step. In current implementa-
1
extracted from a persistent current calculation [5], from tions the time evolution is calculated by approximative
0
6 phase shifts in NRG calculations [6], or from approxi- schemes, like the Trotter decomposition [15, 16], or the
0 mative schemes based on the tunneling density of states Runge-Kuttamethod[18]. Inourwork,wecombinedthe
/ [7]. Alternatively one can evaluate the Kubo formula idea of the adaptive DMRG method with direct evalua-
t
a within Monte-Carlo simulations [8], or from DMRG cal- tion of the time evolution operator via a matrix expo-
m
culations [9]. In contrast, there are no general methods nential as described in Ref. [17]. Therefore our method
- availabletogetrigorousresultsforthefinitebiasconduc- involvesnoTrotterapproximations,thetimeevolutionis
d
tance. While the problem has been formally solved by unitary by construction,andit can be applied to models
n
o Meir andWingreenusing KeldyshGreens functions [10], beyond nearest-neighbor hopping.
c the evaluation of these formulas for interacting systems The Hamiltonian for the nanostructure attached to
v: is generally based on approximative schemes. leads, Hˆ =HˆS + HˆL + HˆC is given by
i
X In this work we propose a new concept of calculating m−1 m−1
finite bias conductance of nanostructures based on real Hˆ = −t (c†c +H.c.)+ µ n
r S S j j−1 g j
a timesimulationswithintheframeworkoftheDMRG[11, j=Xn+1 jX=n
12,13,14,15,16,17,18]. Itprovidesaunifieddescription m−1
1 1
of strong and weak interactions and works in the linear + V n − n − , (1a)
j j−1
(cid:18) 2(cid:19)(cid:18) 2(cid:19)
and finite bias regime, as long as finite size effects are j=Xn+1
treated properly. Hˆ = −t(c†c +H.c.), (1b)
L j j−1
In a first approach of real time dynamics within 1<j<nX,m<j≤M
DMRG, Cazalilla and Marston integrated the time- Hˆ = −t (c†c +c† c +H.c.). (1c)
dependent Schr¨odingerequationin the Hilbertspace ob- C C n n−1 m m−1
tained in a finite lattice ground state DMRG calcula- Individual sites are labeled according to Fig. 1, M =
S
tion [13]. Since this approach does not include the den- m − n is the size of the interacting nanostructure, µ
g
2
Left Lead Nano System Right Lead 0.6 µ = 0.8 a)
ML/2sites MSsites ML/2sites 0.4 µSD = 0.6
1 2 n−1tCn m−1tCm M e/h ] 00..02 µSSDD = 0.4
J [ -0.2
-0.4
-0.6
FIG.1: Nanostructureattachedtoleadsandschematicden-
µ = 0 b)
sity profile of theinitial wavepacket at T =0. 0.40 g
h ] -0.1
|J| [ e/0.35 -0.2
0.30
denotes a local external potential, which can be applied
0.40 c)
to the nanostructure, and V is a nearest-neighbor in- h ]
teraction term inside the nanostructure. The hopping e/0.35
elements in the leads, the structure, and coupling of J [ M = 384
0.30
M = 192
the structure to the leads are t, t , and t respectively.
Similartotheapproachin[17]weaSddanexCternalsource- h ]0.40 MM == 9468 d)
e/
drain potential Hˆ1 = µSD/2 jn=−11 nj − Mj=mnj , J [ 0.38
(cid:16) (cid:17)
to the unperturbed HamiltonianPHˆ and takePthe ground 0 20 40 60 80 100
Time T [ h/t ]
state|Ψ(T =0)iofHˆ+Hˆ ,obtainedbyastandardfinite
1
lattice DMRG calculation, as initial state at time T =0
FIG.2: Currentthrough asingle impuritywith t =0.5t at
C
[17]. In addition we target for the ground state of Hˆ. In nominalhalffillingN/M =0.5obtainedfromexactnumerical
actual calculations the switched external potential was diagonalization. (a) For different bias voltages (µg = −0.2).
smeared out over three lattice sites. (b) For different gate voltages (µSD = 0.4). System size is
We then perform a time evolution as described above M =96. (c,d) For different system sizes: (c) µg =−0.2, and
by applying the time evolution operator U = eiHˆT on (d) µg =0. (Seetext for details.)
|Ψ(T =0)i [22] [23], which leads to flow of the extended
wavepacketthroughthewholesystemuntilitisreflected
at the hard wall boundaries as described in [17]. the inverse of the system size 1/M. The period of oscil-
Theexpectationvalueofthe currentateachbondand lation strongly depends on the applied potential [Fig. 2
every time step is given by (a)] but is independent of the gate potential and system
size [Fig. 2 (b), (c)], and is given by T = 2π~/|µ |).
2e osc SD
J (t)=− Re{ihΨ(t)|t c†c |Ψ(t)i}. (2) This periodic contribution to the current is reminiscent
j,j−1 ~ j j j−1
of the Josephson contribution in the tunneling Hamilto-
FollowingRefs.[13,20]wedefinethecurrentthroughthe nian, obtained by gauge transforming the voltage into a
nanostructure as an average over the current in the left timedependentcouplingt˜C(T)=tCeiµSDT. Itispresent
and right contacts to the nanostructure even for zero gate potential, but the currents in the left
andrightleadsoscillatewithoppositephaseandcancelin
J(T)=[J (T)+J (T)]/2. (3) the current average Eq.(3). After the wavepackets have
n,n−1 m,m−1
finished one roundtrip, the currentoscillations reappear
ForthecalculationoftheDC-conductancethroughthe because of the additional phase shift due to the differ-
nanostructure the time evolution has to be carried out ent lengths of the left and right leads [Fig. 2 (b)]. The
forsufficientlylongtimesuntilaquasi-stationarystateis stationary current is given by a straightforwardaverage,
reachedandthesteadystatecurrentJ canbecalculated. because the oscillation period T is known. In general,
osc
If the stationary state corresponds to a well-defined ap- the density in the leads, and therefore also the current,
pliedexternalpotentialµ ,thedifferentialconductance depends on the system size and a finite size analysis has
SD
is given by g(µ ) = e∂J(µ )/∂µ . In the limit of a to be carried out in order to extract quantitative results
SD SD SD
smallappliedpotential,µ →0,thelinearconductance [Fig. 2 (c), see also discussion of Fig. 6]. Only in special
SD
is given by g(µ )=eJ(µ )/µ . cases (symmetry, half filled leads, and zero gate poten-
g g SD
We first consider the transport through a single im- tial) is the stationary current independent of the system
purity. The current rises from zero and settles into size [Fig. 2 (d)].
an oscillating quasi-stationary state (Fig. 2). After the Our result for the conductance through a single im-
wavepackets have travelled to the boundaries of the sys- purity in Fig. 3 is in excellent quantitative agreement
temandbacktothenanostructure,thecurrentfallsback with exact diagonalization results already for moderate
to zero and changes sign. The amplitudes of the oscil- system sizes and DMRG cutoffs. Accurate calculations
lations depend on µ and µ , and are proportional to for extended systems with interactions are more diffi-
SD g
3
1.5
1.0 t = 0.50 1
C
2g [ e/ h ]0000....2468 tC = 0.35 01..50Current J [ e/h] 2g [ e/ h ]000...468 VV// ttSS == 31
0.0 0.0
0 0.5 1 1.5 2 2.5 3 0.2
µ [ t ]
SD
0
-0.4 -0.2 0 0.2 0.4
FIG. 3: Current and differential conductance as function of µ [ t ]
applied potential through a single impurity with µg =0 and SD
half filled leads: N/M = 0.5. Circles (squares) show results
FIG. 5: Differential conductance as a function of bias volt-
for t =0.5t (0.35t). Systemsize was M =48 (M =96) and
C age through a 7 site nanostructure with nearest neighbor
n = 200 (400) states were kept in the DMRG. Lines are
Cut interaction. Parameters are t = 0.5t, t = 0.8t, and
exact diagonalization results for M =512. C S
N/M=0.5. Squares(circles) denoteweak (strong)interaction
with V/t = 1 (3). Lines are fits to a Lorentzian with an
S
energy dependent self energy Σ=iη +iη µ2. Dashed lines:
0 1
η = 0. System size is M = 144 (M = 192) and 600 (800)
1
cult, mainly for two reasons: 1.) The numerical effort states were keptin theDMRG.
required for our approach depends crucially on the time
to reach a quasi-stationary state. For the single impu-
rity, the quasi-stationary state is reached on a timescale
proportional to the inverse of the width of the conduc- strongly on µSD. To avoid these problems one has to re-
tance resonance, 4t~/t2, in agreement with the result in sortto the full t-DMRG [17], which does not suffer from
C
Ref. [20]. In general, extended structures with interac- the runaway error. A detailed analysis of the numerics
tions will take longer to reach a quasi-stationary state, of our approach will be published elsewhere [24].
and the time evolution has to be carried out to corre- In Fig. 5 we show results for the first differential con-
spondingly longer times. 2.) In the adaptive t-DMRG, ductance peak of an interacting 7-site nanostructure.
the truncation errorgrows exponentially due to the con- Carefulanalysisofthedatashows,thatinordertorepro-
tinued application of the wave function projection, and duce the line shape accurately, one has to introduce an
causes the sudden onset of an exponentially growing er- energydependentselfenergy. Sincetheeffectissmall,we
rorinthecalculatedtimeevolutionaftersometime. This approximateitbyacorrectionquadraticinthebiasvolt-
’runaway’time isstronglydependent onthe DMRGcut- age difference µ = µSD−µpeak. It is important to note
off,andwasfirstobservedinanadaptivet-DMRGstudy thatforthestronglyinteractingnanostructure,V/tS =3,
of spin transport by Gobert et al.[21]. We observe the the conductance peaks are very well separated. There-
sudden onset of an exponentially growing error in our fore the line shape is not overlapped by the neighboring
calculations as well, Fig. 4, but in addition to the de- peaks, and the fit is very robust. Performing the same
pendence on n , the ’runaway’ time now also depends analysis for a non-interacting nanostructurewith a com-
Cut
parableresonancewidth,weobtainnegligiblecorrections
to η in the selfenergy,indicating thatthe changeofthe
1
line shape is due to correlationeffects.
1 µ = 1.6
µSD = 1.4 The linear conductance as a function of applied gate
urrent J [ e/h ]00..68 µµSSDD = = 1 1.2.0 SD pcthioeetnetsnlaytmiasemlcanalonlnab-pienptcleiaerldaccuetlxianttegerdn7ai-nslipttehoetnesanantmioaseltimrsuaucntsuendree.r,Waisfeabssetufuoffidrye-
C0.4 µSD = 0.6 and use a bias voltage of µSD = 2·10−4. For half filled
leads, the result for the linear conductance calculated
0.2 with a fixed number of fermions, N/M =0.5, is qualita-
0 10 20 30 40
Time T [ h/t ] tively correct, but the conductance peaks are shifted to
higherenergiesrelativetotheexpectedpeakpositionsat
FIG.4: Exponentiallygrowingerrorintheadaptivet-DMRG
the energy levels of the non-interacting system (Fig. 6).
for large bias voltages and times: Current through a non-
interacting 7 site nanostructure with tC = 0.5t, tS = 0.8t, Varyingthegatepotentialµg increasesthechargeonthe
and µg = 0. System size is M = 144 and N/M = 0.5. The nanostructure by unity whenever an energy level of the
numberofstateskeptintheDMRGweren =600(circles) nanostructuremovesthroughtheFermilevel[Fig.6(b)].
Cut
and 1000 (squares). Lines are exact diagonalization results. The density in the leads varies accordingly [Fig. 6 (c)].
Since the number of fermions in the system is restricted
4
a) N = 48 within 1% of the conductance for a single channel.
1.0 N = 47
N = 46 Wehaveintroducedanewconceptofextractingthefi-
N = 45
2g [ e/ h ]0.5 Nn == 04.45 ncaitlecubliaatsioannsd. lVineerayraccocnudruatcetaqnucaenftriotmatirveealretsimuletsevaoreluptioosn-
sible, as long as finite size effects are taken into account.
Our results for the linear conductance compare favor-
0.0
3 b) ably both in accuracyand computationaleffort with the
NDot2 DMRG evaluationofthe Kubo formula[9]. Calculations
1 of strongly interacting systems show correlationinduced
0.51 c) corrections to the resonance line shape.
ρLeads0.50 We profited from many discussions with Peter W¨olfle.
0.49 The authors acknowledge the support from the DFG
0 0.5 1 1.5 through project B2.10 of the Center for Functional
µ [ t ]
g
Nanostructures, and from the Landesstiftung Baden-
FIG. 6: Transport through a non-interacting 7-site nanos- Wu¨rttemberg under project B710.
tructure with t = 0.5t and t = 0.8t. The energy levels of
C S
the nanostructure are indicated by dashed vertical lines. (a)
LinearconductancefordifferentN. Theresultafterapplying
finite size corrections is shown as straight line (see text for
details). (b)Numberoffermionsonthe7-sitenanostructure. ∗ Electronic address: [email protected]
(c) Density ρ=(N −NDot)/(M −Ms) in the leads. System † Electronic address: [email protected]
size is M =96 and the number of states kept in the DMRG [1] L. L. Sohn, L. P. Kouwenhoven, and G. Sch¨on, eds.,
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g
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actingsystemrequiresroughlythe samenumericaleffort (1992).
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densitywaveregime,thetimetoreachaquasi-stationary
256403 (2002).
state is longer, and a correspondingly larger system size
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(2004).
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S [17] P. Schmitteckert,Phys. Rev.B 70, R121302 (2004).
V/t = 3
h] S [18] A. E. Feiguin and S. R. White, Phys. Rev. B 72,
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µg [t] [21] D. Gobert et al., Phys.Rev.E 71, 036102 (2005).
[22] Itisequallypossibletopreparethesystemintheground-
FIG. 7: Linear conductance through an interacting 7 site state of the Hamiltonian Hˆ, and to perform the time
system with tC = 0.5t and tS = 0.8t for weak (squares) and evolution via U =ei(Hˆ+Hˆ1)T.
strong(circles)interaction. SystemsizeisM =96(M =192) [23] Weuseatimestep of∆T =~/2t.Ineach adaptivetime
and 400 (600) states were kept in the DMRG. Finite size steponeintermediatetimestepistargetedand2sweeps
corrections havebeen included. Lines are guides to theeye. are performed.
[24] G. Schneiderand P. Schmitteckert,tobe published.