Table Of ContentA Hartree-Fock Study of Persistent Currents in Disordered Rings
A. Kambili∗, C. J. Lambert,
School of Physics and Chemistry, Lancaster University Lancaster LA1 4YB, UK
J. H. Jefferson
Defence Evaluation and Research Agency, Electronic and Optical Material Centre, Malvern, Worcs. WR14 3PS UK
9 (February 1, 2008)
9
9
1
of interest to determine whether or not the results of [4]
n For a system of spinless fermions in a disordered meso-
are contained in a mean-field description using a single
a scopic ring, interactions can give rise to an enhancement of
J Slater determinant. In this paper we address this ques-
the persistent current by orders of magnitude. The increase
tion, by presenting results obtained using the Hartree-
9 in the current is associated with a charge reorganization of
2 Fockmethodforspinlessfermionsinteractingviaashort-
the ground state. The interaction strength for which this
rangeCoulombrepulsion. Inonedimensionweshowthat
reorganization takes place is sample-dependent and the log-
l] averagesovertheensemblearenotrepresentative. Inthispa- theHartree-Fockmethodagreeswiththeexactresultsof
e perwedemonstratethattheHartree-Fockmethodclosely re- [4],reproducingthepredictedbehaviourofthepersistent
-
r producesresultsobtainedbyexactdiagonalization. Forspin- currentaswellasthesample-dependentchargereorgani-
t
s less fermions subject to a short-range Coulomb repulsion U zation. After establishing the validity of the method, we
t. we show that due to charge reorganization the derivative of extendthecalculationtoaquasi-onedimensionalsystem
a the persistent current is a discontinuous function of U. Hav- comprising two parallel chains.
m
ing established that the Hartree-Fock method works well in
The total Hamiltonian for a system of N spinless
- onedimension,wepresentcorrespondingresultsforpersistent
d fermions on a disordered chain (1D) of M sites is
currentsin two coupled chains.
n
Pacs numbers: 71.30.+h, 05.45.+b, 71.10.Pm, 72.15.Rn M M M
o 1
[c Inanormal-metalmesoscopicringthreadedbyamag- H =Xi=1εic†ici−iX,j=1tijc†icj + 2iX,j=1Uijc†icic†jcj (1)
netic flux [1], measured values of the persistent current
1 [2] are two orders of magnitude larger than predicted. The operators c† and c are creation and annihilation
v i i
Thisresultsuggeststhataquantitativetheorymusttreat operators for an electron on site i, the on-site energies
3
4 electron-electron interactions and disorder on an equal εi are random variables uniformly distributed over the
3 footing. Previous studies of spinless fermions concen- interval −W to +W and U is a nearest-neighbour in-
2 2 ij
1 trated on the behaviour of ensemble averagesof the per- teraction of the form
0
sistentcurrentandledtotheconclusionthatrepulsivein-
9 U if j =i±1
teractionscannotsignificallyenhancethecurrent. Itwas U = (2)
9 ij (cid:26) 0 otherwise
/ therefore argued that only systems which include spin
t
a could show such an increase of the current [3]. However, The hopping elements t are restricted to nearest
ij
m more recently it was shown [4] that for one-dimensional neighbours with ti,i±1 = t = 1 except at the ends of
d- systems of spinless fermions interacting through a short- the chain, for which t1N = tN1 = 1 (periodic boundary
n range Coulomb repulsion U, the persistent current does conditions) or t1N = tN1 = −1 (anti-periodic boundary
o increase. This enhancement of the current is accompa- conditions).
c niedbyachargereorganizationofthegroundstatewhich The single-particle Hartree-Fock equation correspond-
v: happensatdifferentvaluesoftheinteractionstrengthU, ing to equation (1) is of the form
i depending on the disorder realization, and therefore the
X ensemble averaged persistent current may not be rele- εiΨn(i)−tii−1Ψn(i−1)−tii+1Ψn(i+1)
r vant. N M
a The results of [4] were obtained using the density ma- + |Ψm(j)|2UijΨn(i)−
trix renormalizationgroup (DMRG) technique [5] which mX=1Xj=1
isessentiallyexactandcontainscorrelationeffects. How- N M
ever, this technique is computationally demanding and Ψ∗m(j)Ψm(i)U Ψn(j)=E Ψn(i) (3)
ij n
not easilyextended to higherdimensions. Therefore it is mX=1Xj=1
whereΨn(i)isthe amplitudeofthenthsingle-particle
wavefunction on site i. The third and fourth terms are
the direct (Hartree) and exhange (Fock) potentials, re-
∗e-mail: [email protected]
spectively.
1
The parametersN,M,t, W,andU werechosento be againobtainsthedelocalizationeffectassociatedwiththe
exactly the same as in [4] so as to directly test the accu- crossoverfrom an Anderson to a Mott insulator.
racy of the method. Here we present results for M =20 The phase sensitivity and the probability distribution
sites and N =10 particles (half-filling) and the strength of the relative increase η are shown in figures 5 and 6,
of the disorder was taken to be large, W =9. At zero U whichagaindemonstratethatinteractionsproduceanin-
the chargedensity ofa givensample is spatially inhomo- creaseinthefluctuationsofthecurrent. However,incon-
geneous due to the presence of such disorder. As U in- trast with a single chain, the average of logD no longer
creasesweobserveareorganizationofthecharge,andfor possesses a local maximum and instead decreases mono-
largeU,obtainahomogeneousconfigurationinwhichthe tonically with increasing U.
particlesareequallyspaced. Thisreorganizationhappens In summary, the above results demonstrate that the
atdifferentvaluesoftheinteractionstrengthfordifferent effects discussed in [4] are contained in a single Slater-
samples. As an example, figure 1 shows the charge den- determinant ground state and are describable by mean
sity for a single sample in the free electron case (U =0) field Hartree-Fock theory. By extending the analysis to
andforU =20,whichis largeenoughto yielda periodic two chains, we find that the maximum in the average
array of charges. of logD is no longer present, which suggests that this
In addition to the charge density we have studied the feature may be a peculiarity of strictly one-dimensional
phasesensitivityD,whichisameasureofthedelocaliza- systems. ThefactthatHartree-Focktheoryisapplicable
tion effect mentioned above and is defined by in one dimension, where mean field theories are least ac-
curate, indicates that in higher dimensions Hartree-Fock
M
D(U)= ∆E (4) theoryshouldbesufficienttodescribethegroundstateof
2 asystemwithelectron-electroninteractionsanddisorder,
at least in the strong disorder limit.
Here ∆E = (−1)N(E (0) − E (π)) is the difference
g g
in the ground state energy between periodic and anti-
periodic boundary conditions, where the Hartree-Fock
ground state energy is
N M
1
E = [E + Ψ∗n(i)h Ψn(j)] (5)
g 2 n ij [1] M Buttiker, Y Imry, R Landauer, Phys. Lett. 96A, 365
nX=1 iX,j=1
(1983)
with hij =εiδij +tijδij±1. [2] VWCJhGanadllraagsheenrk,hAar,KRleiAnsWasesberb,,PMhyJs.BRraedvy.,LMetBt.K67e,tc3h5e7n8,
Inagreementwith[4]wefindpeaksinlogDatsample-
(1991)
dependent values ofU, associatedwith reorganizationof
[3] T Giamarchi, B Shastry,Phys. Rev. B 51, 10915 (1995)
the ground-state charge density. For positive U, figure 2
[4] PSchmittekert,RAJalabert,DWienmann,JLPichard,
showstheensembleaverageoflogDalongwithresultsfor Phys. Rev.Lett. 81, 2308 (1998)
four individual samples. For negative U the mean field [5] S R White, Phys. Rev. B 48, 10345 (1993)
equationsdonotconvergeandnoresultswereobtainable.
Fig. 2 is in remarkable agreement with the exact results
of [4]. For example the averageof logD, exhibits a local
maximum around U ≈ t. Following [4] we have also
examined the relative increase of the phase sensitivity
with respect to the free fermion case, η = logD(U)−
logD(0), and in agreementwith [4], obtain a log-normal
distribution for η (figure 3).
Having established the validity of Hartree-Fock the-
ory as a method for computing the charge density and
phasesensitivityofone-dimensionalrings,wenowextend
our analysis to two coupled one-dimensional chains. For
a system consisting of two rings with 20 sites in each,
and 20 spinless fermions in total, we again examine the
case of strong disorder, W = 9. In figure 4 we show
the charge density for the two rings. As we can see, the
chargereorganizationthatwaspresentin1Disalsoobvi-
ousfor twochains. For verystrongU the particleslocal-
ize in the odd and the even sites of the first and second
chains respectively, as expected classically. Thus, one
2
U=0
U=20 0.40
1.0
0.8 0.30
ρ 0.6 P
0.20
0.4
0.10
0.2
0.0 0.00
1.0 3.0 5.0 7.0 9.0 11.0 13.0 15.0 17.0 19.0 −4.0 −2.0 0.0 2.0 4.0
i η
FIG.1. ChargedensityforasamplewithN =10particles FIG.3. Probability distributionofη=logD(2)−logD(0)
on a chain (1D) with M = 20 sites for free fermions and for in 1D calculated from 1000 samples. The curveis fitted by a
U =0 and U =20 Gaussian distribution with <η>=−0.53 and σ2=1.37
−3.0 U=0 for chain 1
U=0 for chain 2
1.2
U=20 for chain 1
U=20 for chain 2
−5.0
1.0
−7.0 0.8
gD ρ
o
l
0.6
−9.0 sample 1
sample 2
sample 3 0.4
sample 4
−11.0 average <logD> for 500 samples
0.2
−13.0 0.0
0.0 2.0 4.0 6.0 8.0 10.0 1.0 3.0 5.0 7.0 9.0 11.0 13.0 15.0 17.0 19.0
U i
FIG.2. Phase sensitivity for half-filling (M =20, N =10) FIG.4. ChargedensityforasamplewithN =20particles
in one dimension for four samples. The average has been in two rings (2D) with M = 20 sites in each ring for free
obtained for 500 disorder realizations fermions and for U =0 and U =20
3
−2.0 sample 1
sample 2
sample 3
sample 4
−4.0
average <logD> for 400 samples
−6.0
D
g
lo −8.0
−10.0
−12.0
−14.0
0.0 2.0 4.0 6.0 8.0 10.0
U
FIG.5. Phase sensitivity for half-filling (M =40, N =20)
in two dimensions for four samples. The average has been
obtained from 400 disorder realizations
0.30
0.20
P
0.10
0.00
−7.0 −5.0 −3.0 −1.0 1.0 3.0 5.0
η
FIG.6. Probability distributionofη=logD(2)−logD(0)
in 2D calculated from 1000 samples. < η >= −1.59 and
σ2 =2.2
4
