Quantum Monte Carlo simulation for the spin-drag conductance of the Hubbard model

0.23 MB·English

by  K. Louis

Quantum Monte Carlo simulation for the spin-drag conductance of the Hubbard
model
Kim Louis and C. Gros
Fakult¨at 7, Theoretische Physik, University of the Saarland, 66041 Saarbru¨cken, Germany.
(Dated: February 2, 2008)
In the situation of two electro-statically coupled conductors a current in one conductor may
5
induce a current in the other one. We will study this phenomenon, called Coulomb drag, in the
0
Hubbard chain where the two “conductors” are given by fermions with different spin orientation.
0
WiththeaidofaMonteCarlo(MC)approachwhichwepresentedinarecentpaperwecalculatethe
2
TransconductanceindifferentvariantsoftheHubbardchain(with/withoutimpurityandadditional

n [long-ranged] interactions) for different fillings.
a
J PACSnumbers: 75.30.Gw,75.10.Jm,78.30.-j

1
1
I. INTRODUCTION fermions with opposite spin orientation.

] In the Section II we present the model and give some
l
e central definitions for the spin-drag problem. The Sec.
- The Coulomb-drag effect describes how two conduc- III contains the technical details on the subject of the
r
tors (only coupled by the Coulomb force) may influence
t MC simulations. The MC method of our choice9,10 was
s each other. Since the Coulomb repulsion is relatively
. a variant of the Stochastic Series Expansion (SSE) as
t small, a sizeable effect will only arise when the two con-
a introduced in Refs. 11,12,13,14. This method allows an
m ductors are very close to each other. This condition can investigationoftheone-dimensionalHubbardmodel.15,16
be met in mesoscopic systems—where with the advent
- In the following Sec. IVA we present our results for
d of new technologies (e.g., carbon nanotubes) the prob-
the standard Hubbard model and compare with analyt-
n lem ofCoulombdrag attractedmore andmore attention
ical predictions from bosonization theory. To obtain a
o (e.g.,Refs.1,2,3,4)—orinthespin-dragproblem. Forin-
spin-polarized current we add in Sec. IVB an impurity
c
steadofconsideringtwoconductorsonemaylookatdrag
[ to the system, which acts like a combination of a one-
effects between different fermion species, e.g., fermions
site chemicalpotentialandaone-site magneticfield. We
2 with different spin orientation. Since fermions with dif-
show that such a “magnetic” impurity can produce the
v ferent spins are not spatially separated, there is a large
desired spin-polarized current.
1
Coulombforce betweenthem whichcanlead to allkinds
0 The Hubbardmodelcanalsobe mappedto aspinless-
ofcorrelationeffects,e.g.,adrageffect. Inthelastyears,
7 fermionladder. Hence,ourresultsmayalsodescribethat
the interest in spin-dependent transport increased. One
7 situation, but there one might argue that the very spe-
0 keyproblemisthegenerationofaspin-polarizedcurrent,
cific modeling of the Coulomb (on-site) interaction in
4 i.e., a current where only fermions with one of the two
the Hubbard model is unrealistic (and may differ from
0 spin orientations flow. In this context it is important to
other approaches, e.g., Refs. 1,2,3,4). We therefore dis-
/ keep in mind that the spin-polarized current may affect
t cuss two variants (with additional interaction terms) of
a the fermions with the opposite spin orientation. Hence,
m ourmodel. First,wewilldiscussinSec.IVCasituation,
thedrageffectmayplayhereacrucialrˆoleeventhoughit
wherefermionswithdifferentspinsliveondifferentsites.
- isexperimentallynotdirectlyaccessible. (Thisisbecause
d Thefullsystemhasthegeometryofazig-zagchain. Sec-
thedrivingpotentialsareingeneralnotspindependent.)
n ond,weshowintheappendixthataspin-polarizedinter-
o For this “spin-drag” problem the trans-resistivity of actionleadstoequalCis-andTransconductance. Thisis
c
higherdimensionalsystemshas beeninvestigatedinpre- similar to the “absolute” drag result found, e.g., in Ref.
:
v vious publications, e.g., using the Boltzmann equation5 3.
i or the random-phase approximation.6 In this paper we
X
focus on the Transconductance for the Hubbard model.
r
a While most authors used a bosonization approach to II. DEFINITION OF THE SPIN DRAG IN THE
compute the conductance7,8 we will use here, for the HUBBARD MODEL
study of the Transconductance, a Monte Carlo method
whichweintroducedinarecentpaper.9 Thestrategywe
Our model Hamiltonian is the standard Hubbard
followed there was to map our fermion system via the
model (with N sites or atoms)
Jordan-Wigner transformation to a spin system which
canbeanalyzedbyefficientthoughstandardMonteCarlo
H = t c† c +c† c
techniques. We will now extend this method to the one- Hubb n,σ n+1,σ n+1,σ n,σ
X X (cid:16) (cid:17)
dimensional Hubbard model concentrating on the ques- n σ=↑,↓
tion how a spin-polarized current (driven by a voltage µ n +U n n , (1)
n,σ n,↑ n,↓
drop which is assumed to be spin dependent) affects the − X X
n,σ n
2
where n is the occupation number of fermions with III. DESCRIPTION OF THE MONTE CARLO
n,σ
spin σ at site n, and c(†) is the corresponding annihila- METHOD
n,σ
tion (creation) operator. To perform our transport cal-
culations, we will use the approach from Ref. 9. Since Before starting with Monte Carlo we have to cast our
the hopping term does not connect fermions with differ- Hamiltonian in a convenient form.
ent spins, it is natural to consider current and potential Using the Jordan-Wigner transform the Hubbard
operators for each spin orientation separately. Following model can be mapped to a spin ladder. To each oc-
Ref. 9 the potential operators read then: (e being the cupation operator we introduce a spin operator, i.e., we
chargeunitandxbeing thepositionofthevoltagedrop) replace nn,↑ →S2zn+1/2 and nn,↓ →S2zn+1+1/2. If we
expressthe HubbardHamiltonianwiththosenew opera-
P↓ =e n , P↑ =e n . tors we obtain the following spin ladder (with 2N sites):
x n,↓ x n,↑
X X
n>x n>x
The conductance (of a spinless-fermion chain) is the lin- H = J (S+S− +S−S+ )/2+J SzSz +BSz
ear response of one potential operator to another; there- Xn (cid:2) x n n+2 n n+2 z n n+1 n(cid:3)
fore the explicit form of the current operators is not
needed here.9 As we have two potentials, we can define + US2znS2zn+1+U′S2zn+1S2zn+2 , (2)
four transport quantities (conductances) g which de- Xn (cid:2) (cid:3)
ij
scribe the (linear)responseofPi to Pj where i,j ,
where the sites with even number represent spin-up
∈{↑ ↓
. Further details on how to evaluate the g ’s are to be
ij fermions and the sites with odd numbers, spin-down
}
found in section III.
fermions. (For the Hubbard model one has to put
Forthemomentwewilldiscussonlysymmetricmodels, J = 0 = U′. These parameters are used to model the
z
i.e., we have spin-rotational invariance (the only asym-
spin-polarizedinteractionfromtheappendixandthezig-
metric model that we will discuss appears in Sec. IVB);
zag chain from Sec. IVC; see bottom half of Fig. 1. The
thus,weendupwithonlytwodistinctquantities. Wecall
sites 2n and 2n+1 in the ladder represent therefore one
g := g = g the Cisconductance and g := g = g
c ↓↓ ↑↑ t ↓↑ ↑↓ atom of the Hubbard model and interact via an Ising
the Transconductance.
interaction representing the Coulomb force (see Fig. 1).
The namingconventionscomefromthe physicalinter-
ThehoppingamplitudessatisfyJ =2t,andthestrength
x
pretationofthesecoefficientswhichisthefollowing: Ifwe
of the magnetic field B in Eq. (2) is obtained from the
switchonatacertaintimea(supposedly)spin-polarized chemical potential via the relation B =U/2+U′/2 µ.
potential which acts only on spin-up fermions (i.e., we −
Half filling corresponds therefore to B =0. The two po-
add a time-dependent perturbation of the form tential operatorsP↑,↓ introduced in the previous section
x
VP↑θ(t) can be related to potential operators for the two chains
(e being the charge unit)
withV beingthevoltageamplitudeandθ theHeavyside-
step function) we will find a current of spin-up fermions P↓ =e Sz , P↑ =e Sz .
x 2n+1 x 2n
X X
I↑ =g V. n>x n>x
c
Thenwemayobtainthefourconductancesg introduced
ij
ThisisthedrivecurrentgovernedbytheCisconductance,
in the previous section by computing (i,j , )
but there will also be a current of spin-down fermions ∈{↓ ↑}
~β
I↓ =gtV, gij(ωM)=−ωM/~Z cos(ωMτ)hPxiPyj(iτ)idτ (3)
0
the drag current (governed by the Transconductance).
The latter may be nonzero, even though the spin-down at the Matsubara frequencies ω = 2πM(β~)−1, M
M
fermions do not feel the applied potential. N, and then extrapolating to ω = 0. (The extrapo∈-
The situation of a nonvanishing Transconductance (or lated value should not depend on x or y.9 We chose
drag current) is called Coulomb drag. This problem has x = N/2 = y 1.) For the extrapolation from g(ω )
M
−
been studied, e.g., for coupled spinless-fermion systems to g(ω = 0) we will use a quadratic fit from the first
by bosonization in Refs. 1,2,3 and to second-order per- three Matsubarafrequencies. [We willuse open(OBC’s)
turbation theory in Ref. 4. instead of periodic boundary conditions (PBC’s).] Since
Normally, (since spin-polarized potentials are not the Hamiltonian contains Heisenberg-like interactions as
available)oneisonlyinterestedinthe (full)conductance well as Ising-like interactions, it is advantageous to use
of the Hubbard model, where both fermion species feel the Stochastic Cluster Series Expansion (SCSE) intro-
the same potential, and the full current is the sum of duced in Ref. 10. For the Hubbard model the SCSE
the currents of the spin-up and spin-down fermions. As givesessentiallythesameupdateschemeasthe oneused
is straightforwardto see,the Cis- andTransconductance in Ref. 16. We will explain it now shortly.
give us directly the conductance of the Hubbard model FollowingRef.10wesplittheHamiltonianaccordingto
via the relation g =2(g +g ). H = h,butthistimeintofour-sitesclusters,called
Hubbard c t h∈h
P
3
1 3 5 7 9 considerably.
U U
0 2 4 6 8 IV. NUMERICAL RESULTS
A. Transconductance in the Hubbard model
1 3 5 7
U U
1. Comparison with the Hubbard model
0 2 4 6
InbosonizationtheorytheHubbardmodelisdescribed
FIG.1: ThemodelHamiltonianthatwillbediscussedinthis
by two bosonfields Φ representingthe degreesof free-
paper. Solid lines indicate a full Heisenberg-like interaction ↑,↓
domofdifferentspinorientations. Thecurrentoperators
between the sites; dashed lines stand for sites coupled only
for the spin sectors are then given by J ∂ Φ .7
by a z-z term (Ising-like interaction). The upper part is for ↑,↓ x ↑,↓
∝
U′ = 0. The lower for U′ = U. (Other values of U′ are Theconductancecanbewrittenintheformofacurrent-
not considered.) A plaquette as used in the MC scheme is currentcorrelator7 andmaybe evaluatedintermsofthe
indicated byboldface lines. Luttinger-liquid parameters K of the charge and spin
ρ,σ
field8 Φ = (Φ Φ )/√2. The result is (using the
ρ/σ ↑ ↓
±
linearity of the correlator and the results from Ref. 17)
plaquettes (see Fig. 1). The following operators belong
to the plaquette (containing interactions between the 1 1
sites 2n, 2n+1, 2Pn+2, and 2n+3): gc = 2(Kρ+Kσ) gt = 2(Kρ−Kσ). (4)
h(1) = J S+S− /2 h(2) =J S−S+ /2 When µ = U/2 we are at half filling, where Umklapp
P x 2n 2n+2 P x 2n 2n+2 processes are responsible for a gap in the system.18 The
h(3) = J S+ S− /2 h(4) =J S− S+ /2 (charge)gap∆(U) depends onthe Hubbard repulsionU
P x 2n+1 2n+3 P x 2n+1 2n+3
and is finite for all U.
h(5) = C+U/2Sz Sz +U/2Sz Sz +
P 2n 2n+1 2n+2 2n+3
+ U′Sz Sz +J Sz Sz +J Sz Sz
2n+1 2n+2 z 2n+1 2n+3 z 2n 2n+2
+ B/2(Sz +Sz +Sz +Sz ). 2. Numerical simulations
2n 2n+1 2n+2 2n+3
We presentnowMonte Carloresults forthe Transcon-
The set h consists of all h(t) for all plaquettes and all ductance in the Hubbard model Eq. (1) [or Eq. (2) for
t=1,...,5. P P U′ =0].
The heart of the SCSE program is the so called loop We performed simulations for two different chemical
update, where a spin flip of a subset (loop) of all spin potentials: First, µ = U/2 corresponding to half filling
variables is proposed. Since the sites with even and odd and second µ = 0. In the latter case the system is no
numbers form two chains, which are only coupled by a longerathalffilling,buthasaU-dependentfilling,which
z-zinteractionterm,wefindthatthesetofspinvariables is shown in Fig. 11 of the appendix (in the large-U limit
thatwillbeflippedintheloopupdatebelongsentirelyto the system reaches quarter filling).
one of the chains. Therefore, we can view the new algo- We show g and g as a function of U for the two dif-
c t
rithm as making loop updates for each chain separately. ferent µ’s in Fig. 2. The figure shows that the Coulomb
During a loop update for one chain the spin variables of drag is very sensitive to a change in chemical potential.
the other chain remain fixed. The consequence of this is Let us first look at the half-filled case (Fig. 2). If U
that, if we update, e.g., the even chain, then operators is very large the Coulomb repulsion acts as an effective
with superscript i=3,4 can be neglected (are irrelevant projection to those configurations satisfying P↓ = P↑.
−
for the loop construction), and the coupling terms (be- This implies that g + g 0 as U . This con-
c t
→ → ∞
tweenthe chains)reduceto magneticfieldterms (forthe templation is in accordance with Fig. 2. We should
even chain). actually expect from Eq. (4) that for T = 0 we have
It is however advantageous to consider another vari- g +g =K =0becauseofthechargegap∆(U)>0for
c t ρ
ant of the loop update. The construction is similar to all U >0 (cf. Ref. 18). This should lead to a discontinu-
the first variant, but now we propose spin flips for both ous jump at U = 0, because without the Coulomb force
chains, i.e., the spin variables belonging to sites 2n and evidently g = 0 and g is the conductance of uncou-
t c
2n+1areflippedsimultaneously. Thismaybeviewedas pled spinless-fermion chains from Ref. 17. Here we em-
a construction of two parallel loops—one for each chain. phasize that ourmethod is a finite-temperature method,
Since the two loops must be parallel,the number of pos- which means that the conductances calculated by us in-
sible transitions between different plaquette states is re- terpolate smoothly between the values for U = 0 and
duced. This may lead to a less efficient algorithm,13 but U = . The crossover is expected to take place at that
∞
one should note that this parallel-loop update becomes interaction value U which satisfies ∆(U ) = k T. It
T T B
deterministic for the case of B = 0 and hence enhances is therefore interesting to see how g scale with tem-
c,t
theefficiencyofthealgorithm(atleastforthissituation) perature. However, our method gives only access to the
4
∆(U)=kBT 1 3 5 7 9 11
1
U U
B
0.5 0 2 4 Imp 8 10
h] 6
2e/ 0
g[c,t FIG.3: ThemodelHamiltonianwithanimpurityatsiteN/2
g µ=0 for N = 6 in the spin ladder representation from the upper
−0.5 gc µ=0 half of Fig. 1 (for U′ = 0). The site on which the impurity
t
g µ=U/2 potential acts is encircled.
c
g µ=U/2
−1 t
0 1 2 3
U[t] Inthecaseofzeroµ(againFig.2)wherethesystemis
away from half filling there is no charge gap. Hence, K
ρ
FIG. 2: Cis- and Transconductance (filled/empty symbols) isfinite,andsoEq.(4)tellsusthatgc+gt =Kρ doesnot
of the Hubbard model as a function of U for two different decay with U. [Here we note that g +g agrees (within
c t
µ’s (200 sites, T = 0.02t/kB, OBC’s, 2·105 MC sweeps.) error bars) with the values for Kρ available in Ref. 18.]
The (solid) arrow indicates the UT for which the charge gap We still have Kσ = 1, which leads to gc gt = 1 for all
(present at half filling) satisfies ∆(UT) = kBT. (The dotted U (again very well satisfied by the figure)−.
arrow shows U where ∆(U )=k T/2 for comparison.)
T/2 T/2 B Finally, we consider the large-U limit. Inserting
K (U = ) = 0.5 and K 1 from Ref. 18 in Eq.
ρ σ
(4) yields∞g = 0.75 and g = ≡0.25 (units e2/h). These
c t
low-T regime,9 such that we will compare here results results are in accordance with−the figure. (Note that the
for only two different temperatures, kBT = 0.01t and statistical error increases with U such that we cannot
kBT = 0.02t. In the two subsequent simulations for compute g(U) for sufficiently high U in order to extract
the Cis-/Transconductance we did not find any differ- the large-U limit accurately.)
enceatall. Thisimpliesonlyaweaktemperaturedepen-
dence(atlowT)fortheinteractionU whichgovernsthe
T
crossover. Since∆(U)isknownfromanalyticalresults,19
B. Magnetic Impurity
we can calculate the two crossover interactions—defined
by ∆(U ) = 0.02t and ∆(U ) = 0.01t—finding that
0.02 0.01
U 1.25t and U 1.12t do not differ much, as In this subsection we will study the influence of an
0.02 0.01
≈ ≈
expected (they are also both indicated by arrows in Fig. impurity. The modeling of the impurity follows Ref. 7,
2). but for the spin-drag problem it is natural to consider a
Another important consequence of g +g 0 is that spin-dependent impurity, as we will do here. We extend
c t
the signs of the Cis- and Transconductance a→re opposite therefore our Hamiltonian in the following way
or—intermsofthespin-upandspin-downcurrents—that
the induced drag current flows in the opposite direction H =HHubb+BImpnN/2,↑ ,
of the drive current.20
Now we will turn to the spin sector. We have K = i.e., we introduce an (impurity) potential at exactly one
σ
1 by spin-rotational invariance of the Hubbard model18 central site (which acts only on one spin orientation, see
implying [see Eq. (4)] g g =1 for all U which is very Fig. 3).
c t
−
well satisfied by Fig. 2. Fig. 4 shows Cisconductances and Transconductances
Putting the two results for gc gt together, we obtain as a function of the impurity potential BImp at half fill-
g = 0.5e2/h = g valid at hig±h U. This large-U limit ing. (The example is chosen such that the Transconduc-
c t
ofg mayalsobe−computedinsecond-orderperturbation tanceintheunperturbedsystemisrelativelylarge.) The
c
theory. In this approximation the Hubbard model can conductance of the Heisenberg chain with one impurity,
be mapped to a Heisenbergmodel. The operator[onthe which is the large-U limit of gc, is given for comparison.
Hilbert space of the original Hamiltonian Eq. (1)] Px↑, Although the two Cisconductances, g↑↑ and g↓↓, could
which is effectively equal to P↓, is identified with the in principle differ (the model is now asymmetric) they
operator(ontheHilbertspace−ofxtheeffectiveHeisenberg do not in the case of half filling—at least within error
model) P = Tz. [ Tz = (n n )/2 is the bars. Both Cisconductance and Transconductance go—
spin operxator Pforn>fexrmnions; hnere denno,t↑e−d byn,↓T to avoid more or less linearly—to zero as the impurity strength
confusion with the spin operators appearing in Sec. III.] increases.
Applying the results from Ref. 9 the computation of g We note that within errorbars g = g suchthat the
c c − t
reducesthentothecomputationofthespinconductance full conductance of the system
of the Heisenberg model, which equals one half in units
e2/h. g =2(g +g )
c t
5
(V being the voltage amplitude), then the current is
0.5
I =I↑+I↓, I↑ =(g +g )V I↓ =(g +g )V.
0.25 ↑↑ ↑↓ ↓↑ ↓↓
The average spin of a fermion in the current is therefore
h] 0
2e/ (using g↓↑ =g↑↓ =gt)
g [c,t −0.25 g↑↑ I↑ I↓ g↑↑ g↓↓
g = − = −
↓↓
−0.5 g↑↓ S 2I 2(g↑↑+g↓↓+2gt)
g
↓↑
g(xxx,with Impurity) different from zero (see Fig. 6).
−0.75
0 0.4 0.8 1.2 1.6 2
B [t] 0.5
Imp
nt
e
FIG. 4: The Cis- and Transconductance (filled/empty sym- curr 0.4
e
fiblollisn)g.asTahefuinmcptiuornityofacthtseoinmpspuirnit-yupstfreernmgitohnsB.ImNpotaetthhaaltf n in th 0.3
amthniegglhCetidsdcoiowffnendr)u.fcrTtoamhnectethwfeooroTntrehaeonfsspctiohnne-dusuppciftnea-rndmcoeiwosnnasfreger↑mt↑hi(eotnrsisaamngg↓e↓l.e(Tuthrpie)- olarizatio 0.2
conductance of an xxx (Heisenberg) chain with one impu- n p 0.1
rity(which should coincide with thelarge-U limit of theCis- pi
s
conductance) is given for comparison. (µ = U/2, U = 2t, eg. 0
N =192 sites, T =0.02t/k .) n
B
0 2 4 6
B [t]
1 Imp
FIG. 6: The negative average spin polarization (−S) of the
induced current as a function of the impurity strength BImp
0.5 g in the Hubbard model away from half filling. The impurity
↑↑
g acts on spin-up fermions. The dashed curve is obtained by
↓↓
g fitting g↑↑, g↓↓ and gt in Fig. 5 with an exponential ansatz
↑↓ and substituting these fits into the formula for S. (µ = 0,
g
0 ↓↑ U =4t, N =192 sites, T =0.02t/kB.)
An interesting problem is the question whether for
µ = 0 the Cisconductance in the pure (spin-down) sec-
−0.5
0 0.5 1 1.5 2 2.5 torg↓↓ survivesornotwhenweincreaseBImp toinfinity.
B [t] Note that a finite g would imply a total spin polariza-
Imp ↓↓
tion, i.e., = 1/2. The limit B of g can-
Imp ↓↓
S − → ∞
not be taken directly (because of problems with the MC
FIG. 5: The Cis- and Transconductance (filled/empty sym-
simulation), but here we note that there is another way
bols) as a function of the impurity strength BImp in the
to model the impurity. Instead of applying a local mag-
Hubbard model away from half filling. The impurity acts
neticfieldononesite,onecanintroduceaweaklink,i.e.,
on spin-up fermions. Note that the Cisconductance for the
spin-upfermions g↑↑ (triangle up)differs from the oneof the decrease the hopping amplitude for spin-up fermions be-
spin-down fermions g↓↓ (triangle down). The two Transcon- tweenthesitesN/2andN/2+1fromtheinitialvaluetto
ductances are the same. (µ = 0, U = 4t, N = 192 sites, t↑ . These two variants of impurities behave similarly.7
Imp
T =0.02t/kB.) WecomputedtheCisconductancefortheunaffectedspin
orientation for the model with t↑ = 0 (corresponds to
Imp
B = ) at µ = 0, U = 4t, N = 200 sites. We find a
Imp
remains zero after insertion of the impurity. Further- value of∞about [0.79 0.03]e2/h.
more, investigations with our method at different tem- The different beh±avior of the Cisconductance in the
peratures find no sizeable T-dependence. (unaffected) spin-down sector at half filling and away
Inthe caseof zero chemical potential, µ=0,we find a from half filling may be explained as follows: Suppose
splitting ofthe twoCisconductances (see Fig.5). This is B and U are large. The effect of the B term on
Imp Imp
particularly interesting since this finding implies a spin- the fermions is that it forbids occupation of the impu-
polarized current. If we assume a (spin-independent) rity site for one of the two fermion species (in our case
driving potential of the form spin-up fermions). At half filling a spin-up fermion can
hop only from one site to another by exchanging the
V(P↑+P↓) sitewithaspin-downfermion(therearenoemptysites),
6
i.e.,simultaneouslywiththespin-upaspin-downfermion 2
must hop in the opposite direction et vice versa (imply-
ingg = g ). Afermionofacertainspinindexcanthen
c − t 1
only pass the impurity site if accompanied by a fermion
ofopposite spin(which movesinthe opposite direction).
h]
Since the impurity site is forbidden for one of the two 2e/ 0
faenrdmbioonthspCeicsiceosn,dnuoctfaernmceisonmcuasnt gpoastso tzhereoi.mpurity site, g [c,t ggc µµ==UU
Away from half filling the hopping of an spin-up −1 gt µ=0
c
fermion does not necessarily require the hopping of a g µ=0
t
spin-down fermion (the spin-up fermion can hop to an
−2
empty site) and hence the impurity affects only one of 0 1 2 3
the two Cisconductances. U [t]
FIG.7: Cis-andTransconductance(filled/emptysymbols)of
C. Zig-zag chain
thezig-zagchainfortwomagneticfields(120sitesperfermion
species, T =0.02t/k ,U′ =U inEq. (2),OBC’s, 2·105 MC
B
So far we have dealt with a system of two fermion sweeps.)
species, where the two species reside on the same set
of sites. In contrast to this, the bosonization ap-
proaches considered mostly systems of two coupled the Coulomb interaction mediates an attractive nearest-
spinless-fermionconductors. Wecancompareourresults neighborinteractionforfermionswithequalspinorienta-
with that situation, if we interpret the Hubbard model tion(thisisaconsequenceofthefrustration). Therefore,
asaspinless-fermionladder. Hereoneassumesthateach inasimpleapproximationtheonlyeffectoftheCoulomb
fermionspecieslivesonadifferentconductor[i.e.,thetwo interactionis to renormalizethe Luttinger-liquid param-
indices (n, ) and (n, ) are supposed to label (spatially) eters for the two spin sectors K↑,↓. Since the Luttinger
↑ ↓
different sites; compare Sec. III and upper half of Fig. parameterforaspinless-fermionchainincreaseswiththe
1]. But one should note that for this case the parameter strengthoftheattraction,8weexpectthatK↑,↓ increases
U shouldbe small(since the distance between separated as U increases. Since K↑,↓ gives the conductance of one
conductors is large)and the Coulomb interaction should spin sector17 (which is essentially the Cisconductance)
be long-ranged (not on-site as in the Hubbard model). we have that g increases with U.
c
Hence we are led to the question how a variation in the Onemayalsoinferfromthefigurethatthedependence
interaction term modifies our results. ofg onachemical-potentialshiftisweak. Withinerror
c,t
Toaddressthisquestionweaddanewinteractionterm barsg decreasesonlyslightlyuponshiftingµawayfrom
c
to the Hubbard model half filling.
One should note that in the limit U = the ground
H =HHubb+U nn,↓nn+1,↑ . state is a spin-polarized configuration (see∞Fig. 9). For
X
n µ = U this means that all conductances are zero in the
This Hamiltonian corresponds to Eq. (2) with U′ = U. large-U limit, for onespinsectoris empty andthe other,
completelyfilled. Incontrasttothis,forµ=0oneofthe
One can justify introducing this new interaction term if
two spin sectors may remain conducting. The crossover
thefermionspeciesliveondifferentsiteswhereeachspin-
to the ordered state occurs at values of U larger than
downsite(n, )liesbetweentwospin-upsites,(n, )and
↓ ↑ 3t which may be seen by simulating and comparing the
(n+1, ). This model has therefore the geometry of a
↑ occupation number for different states. For µ = 0 the
frustrated zig-zag chain as depicted in the lower half of
difference in occupation (between the two spin sectors)
Fig. 1.
n = (n n ) /(2N) is shown in Fig. 8; for
One should note that this system has a total of 2N σ h| n n,↑− n,↓ |i
µ=U itPis zero within error bars as long as U 3t. We
sites,N sitesforeachfermionspecies. [Althoughitwould ≤
concludethatforthevaluesofU consideredinFig.7the
be useful to adopt the notion of a system with two cou-
two spin sectors have approximately the same filling.
pled (spinless-fermion) chains, we will keep here the no-
tation of a system of spinfull fermions.]
The results for the spin drag in this model are shown
in Fig. 7. We discuss again two chemical potentials: one V. CONCLUSION
isµ=U implyinghalffilling,theotherisagainµ=0. In
thelattercasethe(mean)occupationnumberpersiten In this paper we discussed the spin drag for the Hub-
ρ
is different from one half (the occupation at half filling) bard model at zero temperature. We found that the
and depends on U. It is shown in Fig. 8. Transconductance is negative—at half filling the Umk-
One sees in Fig. 7 that g growwith the strengthof lappevenenforcesg = g . Inthatrespectoursituation
c,t c t
| | −
the interaction. This may be explained as follows: First, is different from two coupled Tomonaga-Luttinger mod-
7
0.3 interactingsystem(i.e,theLuttinger-liquidparameterK
1/2−n maydifferfromone),intheHubbardmodeleachspinsec-
ρ
n tor alone is representedby a noninteracting fermion sys-
e) σ
sit 0.2 tem. Wewillshowinthisappendixthataspin-polarized
er interactionleadstoapositiveTransconductanceasfound
p
n ( in the bosonization approaches. To this end we will now
o
ccupati 0.1 nnρ== <<||ΣΣn((nnn,↑+−nnn,↓))||>> //((22NN)) discuss the following variant of our Hamiltonian:
o σ n n,↑ n,↓ H =H + J (n 1/2)(n 1/2). (A1)
Hubb z n,σ n+1,σ
− −
X
n,σ
0 Here the new J term breaks the spin-rotational invari-
0 1 2 3 4 z
U[t] ance. Hence Kσ may now be different from one.
First we consider the large-U limit at half filling. The
U termactsthenasaneffectiveprojectiontotheconfigu-
FIG. 8: Occupation per site (n ) and difference between the
ρ
occupations of the two fermion species (n ) for the zig-zag rations with exactly (because of half filling) one fermion
σ
chain. (100 sites per fermion species, PBC’s, µ=0, 104 MC per site, i.e., nn,↑ = 1 nn,↓. We now set-up an ef-
−
sweeps, T =0.1t.) fective (second-order perturbation theory) Hamiltonian.
From the kinetic-energy term we get again a Heisenberg
model with exchange parameter 4t2/U. The J term of
z
the Hamiltonian does not change the configuration (in
the occupation-number basis) and gives therefore a di-
rect energy contribution 2J TzTz to the effective
z n n n+1
HamiltonianwhereTnz =(nn,P↑−nn,↓)/2denotesthespin
of the fermion on site n. The full effective Hamiltonian
reads
FIG. 9: One of two possible ground state configurations for
thezig-zagHamiltonian whenU >>t. Thesitesinthelower H = (4t2/U)(T+T− +T+ T−)/2
row are occupied by spin-up fermions, the sites in the upper eff X n n+1 n+1 n
n
row are empty. In the other ground state, the lower row
is empty, and the upper completely filled with (spin-down) + (2Jz +4t2/U)TnzTnz+1,
fermions. Xn
and is an xxz chain. If the anisotropy is larger than the
hopping amplitude, i.e., if J > 0, this model is gapped
els as considered in Refs. 1,2,3 which do not incorporate z
(implying both a charge and a spin gap in the original
Umklapp. The“absolute-drag”resultoftheformg =g
c t
model). WethereforeexpectthatCis-andTransconduc-
(e.g., from Ref. 3) can only be recovered by introducing
tance go to zero, if we increase U and keep a finite J .
a spin-polarized interaction (see appendix). z
Now we consider a zero chemical potential µ = 0. We
If we assume that a given potential is in general not
expect that this chemical potential shift away from half
spindependent,theonlyrelevantquantityisthefullcon-
fillingclosesthechargegap,butleavesthespingapmore
ductance g = 2(g + g ), which is only nonzero away
c t
or less unaffected. We consider again the large-U limit.
from half filling. Here both spin orientations contribute
InanyconfigurationtheJ termoftheHamiltoniangives
equally to the current. However, the situation changes z
the following contribution for two neighboring sites
when we add a magnetic impurity. Even if the driving
potential is still spin independent, the resulting current J /2 ifthetwositesareoccupiedwithanti-parallelspins,
z
will be (partially) spin polarized, if we are away from −
half filling. In the limit of a large impurity potential the Jz/2 if the two sites are occupied with parallel spins or
current will be fully spin polarized. are both empty,
0 if one site is occupied and the other, not.
APPENDIX A: SPIN-DEPENDENT We assume that there is a spin gapand that the (degen-
INTERACTION—BROKEN SPIN-ROTATIONAL erate)groundstateconfigurationsarethoseforwhichthe
INVARIANCE spins of the particles are ordered antiferromagnetically.
If only these configurations are allowed, the J term can
z
IntheHubbardmodelCis-andTransconductancehave berepresentedasaone-sitepotentialwithacontribution
opposite sign, in sharp contrast to the bosonization re- J /2 for empty/occupied sites. (One obtains the same
z
±
sults (for coupled spinless-fermion chains), where Cis- energy contributions as from the J term of the original
z
andTranscoductancearebothpositive. Thediscrepancy Hamiltonian, if one keeps in mind that eachsite appears
maycomefromthe differentmodelingofthe interaction. in precisely two pairs of neighboring sites.) We can set-
Inthe bosonizationapproacheseachchainis givenby an up the following effective Hamiltonian (this is just the
8
Transconductance in Fig. 10. In the large-U limit we
1
find good agreement with our prediction that g = g =
c t
0.25e2/h which gives credit to the simulation data de-
0.5
spite the large autocorrelationtimes.
h] Here we want to stress once again the remarkable fact
2e/ 0 that the sign of the Transconductance (the direction of
g[c,t g µ=U/2 the induced current) changes when we switch the mag-
c
g µ=U/2 netic field and the spin-polarized interaction on. (The
−0.5 t
g µ=0 Transconductance is for all U negative in Fig. 2 whereas
c
g µ=0
t in the present situation we expect gt = gc = Kρ/2 > 0
−1
0 1 2 3 for T =0,U = .)
∞
U [t] Occupationinthegroundstate—Sinceweidentifiedthe
groundstateofthe HamiltonianEq.(A1)H(µ=0,U
→
) with the ground state of the xx chain in magnetic
FIG. 10: The same as Fig. 2, but for Jz = 0.8t (here N = ∞
field, we can calculate the occupation per state of this
140). Thesimulations away from halffillingsufferfrom large
autocorrelation times. Hamiltonian in the large U-limit, the result being:
(n )/N =1 arccos(J /[2t])/π.
n,σ z
restriction of the original Hamiltonian to the assumed X −
n,σ
ground state configurations, i.e., zeroth order in U)
This prediction may be tested against a Monte Carlo
[(R+R− +R+ R−)/2+J Rz], simulation. We find good agreement (see Fig. 11).
n n+1 n+1 n z n
X
n
0.6
which is an xx chain in magnetic field, where the “spin”
operator Rnz (this time denoted by R to avoid confusion > 0.5 acos(0.0)/π
with previous spin operators) is defined by Rz = 1/2 <n
if there is a particle on site n and Rnz = +1n/2 i−f site e: 1− 0.4 acos(0.4)/π
n is empty. Since the effective Hamiltonian describes at
st
the charge part of the Hamiltonian the full conductance er 0.3
p
2co(gncd+ucgtta)nfcoertohfethoreigninewalmHoamdeilltsohnoiualndwcohiincchidieswei2t/hhthaes pation 0.2 JJzz==00.8t
the system is noninteracting.8,17 Since we have g = g u
c t c
c 0.1
bytheassumptionofaspingapandEq.(4),therelation o
g =2(g +g ) yields g =0.25e2/h=g .
c t c t 0
In principle the model Eq. (A1) can also be analyzed 0 4 8 12 16 20
with the Monte Carlo method developed in this paper, U [t]
butwefoundthatthesimulationforthiscaseisproblem-
atic: We measured large autocorrelation times for finite FIG. 11: Occupation per state (away from half filling) for
Jz and µ (e.g., for the computation of the compressibil- the Hamiltonian Eq. (A1) H(µ = 0) for different Jz. The
ity). We therefore must restrict ourselves to J 0.8t. predicted high U valuesare given as dashed lines. (500 sites,
z ≤ PBC’s, 105 MC sweeps, T =0.1t.)
For J = 0.8t we present results for the Cis- and
z
1 K. Flensberg, Phys.Rev.Lett. 81 184 (1998). and Strongly Correlated Systems, Camebridge University
2 A. Komnik R.Egger, EPJ condmat/0007443 Press, Camebridge 1998.
3 Yu.D. Nazarov and D. Averin, Phys. Rev. Lett. 81, 653 9 K. Louis and C. Gros, Phys.Rev.B 68, 184424 (2003).
(1998). 10 K. Louis and C. Gros, submitted to Phys. Rev. B
4 N.A.Mortensen,K.Flensberg,andA.P.Jauho,Phys.Rev. cond-mat/0310465.
Lett. 86, 1841 (2001); Phys.Rev. B 65 085317 (2002). 11 A. Dorneich and M. Troyer, Phys. Rev. E 64, 066701
5 K. Flensberg, T.S. Jensen, and N.A. Mortensen, Phys. (2001).
Rev.B 64, 245308 (2001). 12 A.W. Sandvik and J. Kurkij¨arvi, Phys. Rev. B 43, 5950
6 I. d’Amico and G. Vignale, Phys. Rev. B 68, 045307 (1991); A.W. Sandvik,J. Phys. A 25, 3667 (1992).
(2003);ibidem62,4853(2000);Y.Takahashi,K.Shizume, 13 O.F. Sylju˚asen and A.W. Sandvik, Phys. Rev. E 66,
and N.Masuhara, Physica E 10, 22 (2001). 046701 (2002).
7 C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 68, 1220 14 A.W. Sandvik, “Stochastic series expansion method with
(1992); Phys. Rev.B 46, 15233 (1992). operator-loop update”, Phys.Rev.B 59, R14157 (1999).
8 A. Gogolin, A. Nersesyan, and A. Tsvelik, Bosonization 15 R.T.Clay,A.W.Sandvik,andD.K.Campbell,Phys.Rev.
9
B 59, 4665 (1999). MeetingonstronglycorrelatedElectronsystems,Dec1993,
16 P. Sengupta, A. W. Sandvik, and D. K. Campbell, Phys. cond-mat/9412036.
Rev.B 65, 155113 (2002). 19 Y.Umeno,M.Shiroishi,andA.Klu¨mper,Europhys.Lett.
17 W. Apel and T.M. Rice, Phys. Rev.B 26, 7063 (1982). 62 284 (2003).
18 H.J.Schulz,Phys.Rev.Lett.642831(1990);“TheMetal- 20 P. Schlottmann,Phys. Rev.B 69, 035110 (2004).
Insulator Transition in One Dimension” at Los Alamos