DMRG evaluation of the Kubo formula -- Conductance of strongly interacting quantum systems

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by  Dan Bohr

Europhysics Letters PREPRINT
DMRG evaluation of the Kubo formula –
6 Conductance of strongly interacting quantum systems
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2 D. Bohr1,2, P. Schmitteckert2 and P. Wo¨lfle2

n 1 MIC – Department of Micro and Nanotechnology
a Technical University of Denmark – Building 345E, DK-2800 Kgs. Lyngby, Denmark
J
2 TKM – Institut fu¨r Theorie der Kondensierten Materie
2
Universita¨t Karlsruhe, Wolfgang-Gaede-Str. 1, D-76128 Karlsruhe, Germany
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]
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e PACS.73.63.-b – Electronic transport in nanoscale materials and structures.
-
r
t
s
.
t
a Abstract. – In this paper we present a novel approach combining linear response theory
m
(Kubo) for the conductance and the Density Matrix Renormalization Group (DMRG). The
- systemconsideredisone-dimensionalandconsistsofnon-interactingtightbindingleadscoupled
d
to an interacting nanostructure via weak links. Electrons are treated as spinless fermions and
n
o two different correlation functions are used to evaluate theconductance.
c Exactdiagonalizationcalculationsinthenon-interactinglimitserveasabenchmarkforour
[ combinedKuboandDMRGapproachinthislimit. Includingbothweakandstronginteraction

we present DMRG results for an extended nanostructure consisting of seven sites. For the
2
stronglyinteractingstructureasimpleexplanationoftheposition oftheresonancesisgivenin
v
terms of hard-core particles moving freely on a lattice of reduced size.
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9
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1
1
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Introduction. – During the past decade improved experimental techniques have made
0
productionofandmeasurementsonone-dimensionalsystemspossible[1],andhenceledtoan
/
t increasing theoretical interest in these systems. Since its formulation in 1992 [2] the Density
a
m Matrix Renormalization Group method (DMRG) has been established as a very powerful,
quasi-exact method for numerical calculations of properties of (quasi) one-dimensional sys-
-
d tems.
n
Inthispaperwepresentanewapproachforcalculatinglinearresponseconductanceforone-
o
dimensional interacting nanostructures coupled to non-interacting tight binding leads. The
c
: method combines Kubo expressions for the conductance with numerical DMRG calculations
v
andisvalidforarbitraryinteractionstrength. Itfacilitatesaunifieddescriptionofstrongand
i
X weak interactions and provides conductance directly form a transport calculation, without
r relying on relations between equilibrium and transport properties.
a
We employ current-density and current-currentcorrelationfunctions to calculate the con-
ductance and in the non-interacting case compare to exact diagonalization calculations.
In the strongly interacting limit a simple interpretation of the position of the resonances
is given in terms of freely moving hard-core particles on a reduced size lattice [10], and
quantitative comparison with numerical DMRG results shows good agreement.
(cid:13)c EDPSciences
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EUROPHYSICSLETTERS
Left Lead Nanostructure Right Lead
M /2 sites M sites M /2 sites
L S L
1 2 n −1 n n −1 n M
1 1 2 2
Fig. 1 – One-dimensional interacting nanostructure with M sites, coupled to non-interacting tight
S
bindingleads. ThetotalsystemsizeisdenotedM,thenumberoflead sitesisM . Theinterdotand
L
interlead hoppingelements are tDot and t respectively,while thecontact between thenanostructure
and leads are via t and t .
L R
Model. – We are interested in studying the effect of correlations on transport within
a microscopic model of an interacting one-dimensional nanostructure coupled to two non-
interacting tight binding leads, as shown in Fig. 1. Electrons are treated as spinless and only
nearest neighbor interaction is considered. The corresponding Hamiltonian is
Hˆ0 = HˆNS + HˆL + HˆC, (1)
n2−1 n2−1
HˆNS = Ugc†jcj + −tDot(c†jcj−1+c†j−1cj)+Vc†jcjc†j−1cj−1 , (2)
jX=n1 j=Xn1+1(cid:0) (cid:1)
n1−1 M
HˆL = −t (c†ici−1+c†i−1ci)−t (c†ici−1+c†i−1ci), (3)
Xi=2 i=Xn2+1
HˆC = −tL(c†n1cn1−1+c†n1−1cn1)−tR(c†n2cn2−1+c†n2−1cn2)
+γ V(c† c c† c +c† c c† c ). (4)
V n1 n1 n1−1 n1−1 n2 n2 n2−1 n2−1
The parameterγ controlsthe smoothing ofthe interactionon the dot overthe contactlinks
V
asdiscussedin[3],andUg isagatevoltageonthe structure. Inthisworkwesett=tDot =1.
Kubo Expressions. – Using linear response in applied source-drain voltage, V (t), the
SD
current is given by
Hˆ = Hˆ0+δHˆ, (5)
t
J˜n(t) = J¯−iZ dt′ ψ0 [J˜n(t),δH˜(t′)] ψ0 , (6)
(cid:10) (cid:11) −∞ (cid:10) (cid:12) (cid:12) (cid:11)
(cid:12) (cid:12)
J˜ (t) = −it c˜†(t)c˜ (t)−c˜† (t)c˜ (t) , (7)
n n n n−1 n−1 n
(cid:2) (cid:3)
where Hˆ0 is the Hamiltonian in eq. 1, the applied voltage perturbation is δHˆ(t)=VSD(t)Nˆ,
A˜(t) = eiHˆ0tAˆe−iHˆ0t denotes the interaction picture time evolution of the operator Aˆ, and
ψ0 denotes the ground state. Note that in this approach A˜(t) contains all correlations of
(cid:12)the(cid:11)unbiasedstructure,since we apply it to the quasi-exactgroundstate givenby the DMRG
(cid:12)
procedure. The number operator is taken as a symmetric combination of the left and right
lead operators,Nˆ = 1(Nˆ −Nˆ ), and J¯is the equilibrium currentincluded for completeness
2 L R
and henceforth neglected in all numerical calculations.
The Kuboconductanceinthe DC limit, g ≡ e2 J˜ /V ,canbe expressedintermsoftwo
h SD
(cid:10) (cid:11)
D. Bohr, P. Schmitteckertand P. Wo¨lfle: DMRG evaluation of Kubo formula 3
different correlators,
e2 4πiη
gJjN = −h(cid:10)ψ0(cid:12)Jˆnj(Hˆ0−E0)2+η2Nˆ(cid:12)ψ0(cid:11), (8)
(cid:12) (cid:12)
gJJ = eh2(cid:10)ψ0(cid:12)Jˆn1 (Hˆ80πη−(HEˆ00)−2+E0η)2 2Jˆn2(cid:12)ψ0(cid:11), (9)
(cid:12) (cid:12)
(cid:2) (cid:3)
where the positions n1 and n2 are defined in Fig. 1. Analogous Kubo expressions were used
by Louis and Gros in [4], where Quantum Monte Carlo calculations for the density-density
correlator were performed.
Damped Boundary Conditions. – To improve the finite size scaling and to facilitate the
use of sufficiently short leads we use exponentially damped boundary conditions, decreasing
the hopping elements towards the end of the leads exponentially as shown for the right lead
in eq. 10,(1)
[−t,···,−t,−t,−t,···,−t] → [−t,···,−t,−td,−td2,···,−tdMD−1,−tdMD], (10)
MD MD
| {z } | {z }
whered<1. The improvementofthe finite sizescalingreliesontwopropertiesofthe DBC’s:
(1) They allow for use of a smaller η and (2) serve as a particle bath for the nanostructure.
The first property is caused by the introduction of exponentially small energyscales in the
system thus reducing the finite size level splitting at the Fermi energy at half filling. The
secondpropertycanbeunderstoodfromthefactthattheenergycostofaddingorremovinga
particlefromthedampedregionisoftheorderoftheexponentiallysmallhoppingelement.(2)
The DBC’s introduce two more parameters in the model, the number of damped bonds
M and the damping factor d, and these must take values such that physical quantities do
D
not depend sensitively on the particular choice.
Numerical Calculations. – Before actual numerical calculations can be performed the
parameters of the model, M , d, and η, must be determined. This is done using exact
D
diagonalization calculations for the non-interacting systems, specifically the resonant value
at U = 0.(3) For fixed M we do indeed find a range of d values that produce essentially
g D
identical physical results, indicating the range of validity of the DBC’s. Additionally we find
that the actual value of M is not significant (for reasonably large values) as long as the
D
corresponding value of d is tuned such that the damping at the edge reaches values of the
same order of magnitude. The leads used are sufficiently long to keep the damped region
separated from the nanostructure, thus allowing Friedel oscillations at the structure edge to
decay before reaching the damped region.
The magnitude of the parameter η is bounded by physical arguments; from below by the
fact that it should be larger than the finite size level splitting to allow transport, and from
(1)ModifiedBC’sinconnection withDMRGwereintroducedbyVekicandWhitein[5]usingsoftboundary
conditionstoreducefinitesizeeffects. NotethatexponentialdampingcorrespondstothehoppingHamiltonian
intheNumericalRenormalizationGroup,whichmodelsthelogarithmicdiscretization.
(2)In principle properties (1) and (2) of the DBC’s could be obtained by using longer non-damped leads.
However theseleadswouldhavetobeexponentially long makingsuchadirectapproachimpossible.
(3)Consideringstructures consistingofanoddnumber ofsites has theadvantage that the central resonance
(by symmetry) remains at Ug = 0 for half filled leads. Due to the bath property of the DBC’s it is safe to
assumethathalffillingismaintainedinthepartsoftheleadsthatareclosetothenanostructure. Incontrast
the strongly damped regions act like particle baths and therefore cannot maintain half filling for non-zero
external potential.
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EUROPHYSICSLETTERS
above by the broadening of physical results by any finite η, and should thus be much smaller
than the width of the resonances we wish to resolve. It is important to note, that η is an
inherent property of any transport calculation and can only be avoided if one finds a way to
obtain transport properties from equilibrium properties.
The conductances in eqs. 8 and 9 are given in terms of ground state correlators and
hence DMRG is directly applicable. To evaluate the correlators we use the correction vector
DMRG [6–8] in the zero frequency limit. Calculating, e.g., the correlator in eq. 9 is done by
formulating the linear problems,
1
Hˆ0−E0+iηJˆnj(cid:12)ψ0(cid:11) = (cid:12)φj(cid:11) ⇒ Jˆnj(cid:12)ψ0(cid:11) = (cid:2)Hˆ0−E0+iη(cid:3)(cid:12)φj(cid:11), (11)
(cid:12) (cid:12) (cid:12) (cid:12)
which can be solved for φ by a linear solver. Having solved for the correction vector φ
j j
the conductance is found(cid:12)as(cid:11)the vector overlap, (cid:12) (cid:11)
(cid:12) (cid:12)
φ = φR +i φI , (12)
j j j
(cid:12)(cid:12) (cid:11) (cid:12)(cid:12) 8π(cid:11)e2 (cid:12)(cid:12) (cid:11)
g = − φI φR . (13)
JJ h 1 2
(cid:10) (cid:12) (cid:11)
(cid:12)
Inour DMRG calculationswe targetapartfrom the groundstate alsothe realandimaginary
parts of the two correction vectors, φ1 and φ2 , as well as the states Nˆ ψ0 and Jˆn1,2 ψ0
to ensure that the DMRG basis is su(cid:12)ita(cid:11)ble for(cid:12)de(cid:11)scribing the conductance(cid:12)acc(cid:11)urately [7,8(cid:12)]. (cid:11)
(cid:12) (cid:12) (cid:12) (cid:12)
Itshouldbementionedthatthedampedboundaryconditionsmaketheconvergenceratein
numerical calculations much slower. In addition any finite external gate voltage, U , changes
g
the particle number in the structure and the excess particles come from the bath property
of the DBC’s. We therefore face the problem that the damping should be sufficiently strong
to provide a reasonable particle bath but at the same time a strong damping decreases the
couplingofthehighlydampedregiontotherestofthesystem. Toremedytheslowconvergence
inthe DMRG calculationswe turn onthe damping in steps andperformseveralfinite system
DMRG sweeps for each such damping step. In other words, we perform a complete finite
lattice calculation employing typically 11 sweeps and then initiate the scaling sweeps. This
allowsDMRGtograduallyoptimizethebasistoincludethedampingintheleadsandprovides
amoregradualdecoupling ofthe dampedregionsfromthe restofthe system,thus improving
the convergence rate at the cost of more DMRG iterations.
Nevertheless the resolvent equations, eq. 11, are still ill-conditioned and standard solvers
liketheConjugateGradientMethoddonotconverge. WeuseinsteadapreconditionedDavid-
sontypesolversimilartoRamasesha[9]modifiedwithaGauss-Seidelenhancedblockdiagonal
preconditioner. The DMRG calculations presented in Fig. 2 were done using up to m=1200
states. In our DMRG implementation we do not fix the number of states per block to be
m but rather fix the dimension of the target space to be at least m2. In the calculations
presented this corresponds to an increase of block states of typically 15%−30%.
Results. – Here we present DMRG and (in the non-interacting limit) exact diagonal-
ization calculations for a single resonant level, Fig. 2(a), and a nanostructure consisting of
seven sites coupled symmetrically to two non-interacting leads. For the extended structure
wepresentresultsinthe noninteractinglimit, Fig.2(b),andforweakandstronginteraction,
Fig. 2(c) and 2(d).
The spinless single resonant level is generically non-interacting and serves as a testing
ground for the approach. The exact result for the conductance in the symmetrically coupled
case canbe shownto be a Lorentzianof full width 4t′2 athalf maximum, where t′ =t =t .
L R
D. Bohr, P. Schmitteckertand P. Wo¨lfle: DMRG evaluation of Kubo formula 5
11111.....00000 gggggJJJJJJJJJJ 111111......000000 1.0 fJJ
ggggJJJJ1111NNNN 000000......888888 fJ1N
/h]/h]/h]/h]/h] 0000000000..........8888686666 LfffffJJJJJ11JJJNN gggggg000000000000............666666444444−−−−−−111111 000000 111111 /h] 00..68
22222eeeee 00000.....44444 UUUUUUgggggg 2e 0.4
g[g[g[g[g[ g[
00000.....22222 0.2
00000.....00000 0.0
−−−−−44444 −−−−−22222 00000 22222 44444 0 1 2
UUUUUggggg Ug
(a) Single resonant level, MS = 1 and M = (b)Sevensitenanostructure,MS =7andM =
102. f’sdenoteexactdiagonalizationresults,g’s 150,inthenoninteractinglimit,V =0.0. Exact
denoteDMRGresults,andLdenotestheexact diagonalizationcalculation.
Lorentzian in the infinite lead limit. The inset
showsanenlargementoftheresonancepeak.
1.0 gJJ 3.5 1.0 gJJ 3.5
h] 0.8 NgJD1N 23..50 h] 0.8 NgJD1N 23..50
/ 0.6 2.0 D / 0.6 2.0 D
2e 0.4 1.5 N 2e 0.4 1.5 N
g[ 1.0 g[ 1.0
0.2 0.5 0.2 0.5
0.0 0.0 0.0 0.0
0 1 2 3 4 0 1 2 3 4 5 6 7 8
U U
g g
(c)Sevensitenanostructure, MS =7andM = (d)Sevensitenanostructure,MS =7andM =
150,intheLuttinger Liquidregime,V =1.0. 150,inthechargedensitywaveregime,V =5.0.
Fig. 2 – Conductance, g, and number of particles on the dot, N , versus external potential U for
D g
a single resonant level and for an extended nanostructure consisting of seven sites. g denotes the
JJ
current-current correlator, and g denotes the current-density correlator. The left/right contact
JjN
hopping elements are t = 0.5 and the parameters of the DBC’s are M = 30 and d = 0.8. For
L/R D
the calculations above we use η = 1/M. For the interacting spectra notice the offset of resonance
positionsoftheorderV ascomparedtothenoninteractingcase. Theinteractiononthenanostructure
is smoothed overthe contacts with γ =0.5.
V
In Fig. 2(a) we show exact diagonalization and DMRG calculations for the single resonant
level and the two sets are virtually indistinguishable. This verifies that the truncation error
introduced by the DMRG is negligible. Furthermore we have plotted the exact Lorentzian
result, and the agreementbetween the three curvesis verygood, demonstrating the accuracy
of our combined Kubo and DMRG approach.
There is a systematic difference between the current-current and the current-density cor-
relators, specifically close to resonances the current-density correlator generally gives better
results. This is due to the additional energy dependent broadening given by Hˆ0 − E0 in
the current-current correlator. The opposite is true in the tails where the current-current
correlator is more reliable since it is less sensitive to changes of the particle number.
In numerical calculations the parameter η is always finite making the expected form of
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EUROPHYSICSLETTERS
theconductancepeaksthatofanareanormalizedLorentzian(L )ofhalf-widthη convoluted
A
with the “bare” physicalresult. Assuming as a first approximationthat the latter is a height
normalized Lorentzian (L ) of width Γ, the expression for the expected numerical results is
H
of the general form
Γ η+Γ/2
(L ∗L )(x)= . (14)
A H 2 (x−x0)2+(η+Γ/2)2
To leading order in η the conductance at the resonantlevel is then given by gres ≈1−2η/Γ,
which demonstrates that one needs smallη to reachthe unitary limit, gres =1. Howeverη is
knownfrom the input and Γ can be extracted from the results. Thus a conductance value on
resonanceof1−2η/Γis explainedentirelybythe broadeningbythe finite leadsandtherefore
suggeststhatinfiniteleadsinthiscasewouldyieldtheunitarylimit. Ourcalculationsindicate,
that the peak width is only slightly affected by the interaction on the nanostructure, as long
as the nanostructure remains in the Luttinger liquid regime. However, once the structure is
driveninto thechargedensitywaveregimethe peakwidthdecreasesrapidly. Amoredetailed
study of the resonance shapes is considered future work.
The position of the resonances can be described by the addition spectrum,
UND−1→ND = END−1−END, (15)
g 0 0
whereEND isthe energyofthe isolatednanostructureoccupiedbyN particles. In the large
0 D
interaction limit the kinetic energy of the particles can be approximated by freely moving
fermions on an effective lattice of size M∗ =M −N . In this approximation one describes
S S D
the interacting fermions by effective hard-core particles of the size of the interaction range,
compare [10,11]. Thus eq. 15 can be expressed as
ND πn ND−1 πn
UND−1→ND = V +2t cos( )− cos( ) , (16)
g (cid:16)nX=1 MS −ND nX=1 MS −(ND−1) (cid:17)
where N should be small enough that the nanostructure is still in a delocalized state.
D
In an effective charging model the additional splitting of the levels due to the interaction
is linear in the charging interaction V. By contrast, in our microscopic model the interaction
leads to an overall offset for the non-central peaks, while their mutual splitting is governed
by the kinetic energy, ∼t.
In tab. I we show a comparisonof resonance positions as predicted by the reduced lattice
(RL)modelineq.16,aspredictedbyexactdiagonalization(ED)oftheisolatednanostructure,
and resonancesfound in our DMRG calculations for interactionstrengths V =5, 20, 30. The
position of the outermost resonance from 0 → 1 particle fits fairly well for both predictions,
while the next ones deviate somewhat. The RL prediction for the transition 2 → 3 is not
expected to be accurate since N =3 is a localized charge density wave like state. All exact
D
diagonalizationpredictions are correct to lowest order in t/V as expected.
Conclusion. – In this work we have presented a new approach for linear conductance
calculationsofinteractingone-dimensionalnanostructures,combininglinearresponseforcon-
ductanceandDMRG.We havebenchmarkedthisnewapproachagainstexactdiagonalization
calculations in the non-interactingcase and found excellent agreement,which servesas a real
testfortherealspaceDMRG.Fortheresonantlevelwealsocomparedourresultstotheexact
Lorentzian result, and found excellent agreement.
For the interacting case we have presented conductance curves for a seven site nanostruc-
ture in both the Luttinger Liquid (V = 1) and the charge density wave (V = 5) regimes,
D. Bohr, P. Schmitteckertand P. Wo¨lfle: DMRG evaluation of Kubo formula 7
TableI–Tableofpeak positions forthe M =7 sitestructure withinteraction V =5,20,30, as pre-
S
dicted by the reduced lattice (RL) model, by exact diagonalization (ED) of the isolated nanostructure,
and as found from the conductance peaks in our DMRG calculations. The RL prediction for N =3
D
is not expected to be accurate since the nanostructure is in a localized charge density wave like state.
Except for the RL prediction for V =5, N =3, all predictions are correct to linear order in t/V.
D
V 5 20 30
N 1 2 3 1 2 3 1 2 3
D
UND−1→ND RL 6.73 5.50 2.76 21.73 20.50 17.76 31.73 30.50 27.76
g
UND−1→ND ED 6.77 5.88 3.85 21.75 20.63 18.03 31.74 30.59 27.94
g
UND−1→ND DMRG 6.76 5.79 3.66 21.74 20.59 17.97 31.74 30.60 27.95
g
thusdemonstratingtheversatilityofourapproach. Wefindthelargestconductancewhenthe
particle number in the structure fluctuates, in agreement with physical intuition.
Inthe largeinteractionlimit we haveshownthat a simple picture basedoneffective hard-
core particles moving freely on a reduced size lattice describes the position of the resonances
quite well. However, the peak width is strongly decreased by strong interaction.
We expect that further finetuning of the method and numerical parameters will lead to
significantly moreprecise resultsfacilitating calculationsfor morecomplicatedstructuresand
allow to quantitatively describe resonance peaks for strongly interacting and extended struc-
tures.
∗∗∗
This work was performed at TKM, Universit¨at Karlsruhe and we profited from many
discussions with colleagues. In particular we would like to thank Ferdinand Evers, Gert-
Ludwig Ingold, Gu¨nter Schneider, and Ralph Werner for their help in clarifying concepts.
D. B. is grateful for the hospitality of TKM during this work. The authors acknowledge the
support from the DFG through project B2.10 of the Center for Functional Nanostructures,
and from the Landesstiftung Baden-Wu¨rttemberg under project B710.
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