PHYSICAL REVIEW B 68, 035342 ~2003!

Formulas for zero-temperature conductance through a region with interaction

T. Rejec1 and A. Ramsˇak1,21Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia

2Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia~Received 30 January 2003; published 31 July 2003!

The zero-temperature linear response conductance through an interacting mesoscopic region attached tononinteracting leads is investigated. We present a set of formulas expressing the conductance in terms ofpersistent currents in an auxiliary system, namely a ring threaded by a magnetic flux and containing thecorrelated electron region. We first derive the formulas for the noninteracting case and then give argumentswhy the formalism is also correct in the interacting case if the ground state of a system exhibits Fermi liquidproperties. We prove that in such systems, the ground-state energy is a universal function of the magnetic flux,where the conductance is the only parameter. The method is tested by comparing its predictions with exactresults and results of other methods for problems such as the transport through single and double quantum dotscontaining interacting electrons. The comparisons show an excellent quantitative agreement.

DOI: 10.1103/PhysRevB.68.035342 PACS number~s!: 73.23.2b, 73.63.2b

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I. INTRODUCTION

The measurements of the conductivity and the electtransport in general are one of the most direct and sensprobes in solid state physics. In such measurements minteresting new phenomena were signaled, in particularperconductivity, transport in metals with embedded magnimpurities and the related Kondo physics, heavy fermphenomena and the physics of the Mott–Hubbard transiregime. In the last decade technological advances enacontrolled fabrication of small regions connected to leaand theconductance, relating the current through such a rgion to the voltage applied between the leads, also provebe a relevant property of such systems. There is a numbesuch examples, e.g., metallic islands prepared by e-beathography or small metallic grains,1 semiconductor quantumdots,2 or a single large molecule such as a carbon nanotor DNA. It is possible to break a metallic contact and mesure the transport properties of an atomic-size bridgeforms in the break,3 or even measure the conductance osingle hydrogen molecule, as reported recently in Ref. 4all such systems, strong electron correlations are expecteplay an important role.

The transport in noninteracting mesoscopic systemstheoretically well described in the framework of thLandauer–Bu¨ttiker formalism. The conductance is detemined with the Landauer–Bu¨ttiker formula,5–7 where the keyquantity is the single particle transmission amplitudet(«) forelectrons in the vicinity of the Fermi energy. The formuproved to be very useful and reliable, as long as electrelectron interaction in a sample is negligible.

Although the Landauer–Bu¨ttiker formalism provides ageneral description of the electron transport in noninteracsystems, it normally cannot be used if the interactiontween electrons plays an important role. Several approahave been developed to allow one to treat also such systFirst of all, the Kubo formalism provides us with a condutance formula which is applicable in the linear responsegime and has, for example, been used to calculate theductance in Refs. 8 and 9. A much more general appro

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was developed by Meir and Wingreen in Ref. 10. Within tKeldysh formalism they manage to express the conductain terms of nonequilibrium Green’s functions for the samppart of the system. The formalism can be used to treat stems at a finite source-drain voltage and can also be extento describe time-dependent transport phenomena.11 The maintheoretical challenge in these approaches is to calculateGreen’s function of a system. Except in some rare cawhere exact results are available, perturbative approachenumerical renormalization group studies are employed.

In this paper we propose an alternative method for callating the conductance through such correlated systems.method is applicable only to a certain class of systemnamely to those exhibiting Fermi liquid properties, at zetemperature and in the linear response regime. Howevethis quite restrictive domain of validity, the method promisto be easier to use than the methods mentioned aboveshow that the ground-state energy of an auxiliary systeformed by connecting the leads of the original system intring and threaded by a magnetic flux, provides us wenough information to determine the conductance. The madvantage of this method is the fact that it is often mueasier to calculate the ground-state energy~for example, us-ing variational methods! than the Green’s function, which ineeded in the Kubo and Keldysh approaches. The condtance of a Hubbard chain connected to leads was sturecently using a special case of our method and DMRG12,13

and a special case of our approach was applied inHartree–Fock analysis of anomalies in the conductancequantum point contacts.14 The method is related to the studof the charge stiffness and persistent currents in odimensional systems.15–18

The paper is organized as follows. In Sec. II we presthe model Hamiltonian for which the method is applicabIn Sec. III we derive general formulas for the zertemperature conductance through a mesoscopic regionnoninteracting electrons connected to leads. In Sec. IVextend the formalism to the case of interacting electrons.give arguments why the formalism is correct as long asground state of the system exhibits Fermi liquid propertiIn Sec. V convergence tests for a typical noninteracting stem are first presented. Then we support our formalism a

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T. REJEC AND A. RAMŠAK PHYSICAL REVIEW B 68, 035342 ~2003!

with numerical results for the conductance of some ntrivial problems, such as the transport through single adouble quantum dots containing interacting electronsconnected to noninteracting leads. These comparisonscluding the comparison with the exact results for the Andson model, demonstrate a good quantitative agreement. Athe conclusions in Sec. VI we present some more techndetails in Appendix A. In Appendix B we describe the nmerical method used in Sec. V.

II. MODEL HAMILTONIAN

In this section we introduce a general Hamiltonian dscribing a mesoscopic sample coupled to leads as showFig. 1. The Hamiltonian is a generalization of the weknown Anderson impurity model.19 We split the Hamiltonianinto five pieces

H5HL1VL1HC1VR1HR , ~1!

whereHC models the central region,HL andHR describe theleft and the right lead, andVL and VR are the tunnelingcouplings between the leads and the central region. Wealso split the Hamiltonian into a termH (0) describing inde-pendent electrons and a termU describing the Coulomb interaction between them

H5H (0)1U. ~2!

One can often neglect the interaction in the leads andtween the sample and the leads. We assume this is theThen the central region is the only part of the system whone must take the interaction into account

HC5HC(0)1U. ~3!

HereHC(0) describes a set of noninteracting levels

HC(0)5 (

i , j PCs

HC ji(0)dj s

† dis , ~4!

wheredis† (dis) creates~destroys! an electron with spins in

the statei. The states introduced here can have various phcal meanings. They could represent the true single-elecstates of the sample, for example different energy levelsmulti-level quantum dot or a molecule. In this case, the mtrix HC ji

(0) is diagonal and its elements are the single-electenergies of the system. Another possible interpretationHamiltonian~4! is that the statesi are local orbitals at differ-ent sites of the system. In this case, the diagonal matrix

FIG. 1. Schematic picture of the system described by Hamtonian ~1!.

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the off-diagonal matrix elements describe the couplingtween different sites of the system. The sites could havdirect physical interpretation, such as dots in a double qutum dot system or atoms in a molecule, or they could repsent fictitious sites obtained by discretization of a continuosystem. There are other possible choices of basis statethe central region. For example, in a system consistingtwo multilevel quantum dots one could use single-electbasis states for each of the dots and describe the coupbetween the dots with tunneling matrix elements.

The Coulomb interaction between electrons in the samis given by an extended Hubbard-type coupling

U51

2 (i , j PCs,s8

U jiss8nj snis8 , ~5!

where nis5dis† dis is the operator counting the number

electrons with spins at site i. For convenience, we wrotedown only the expression for the Coulomb interaction in tcase, where basis states represent different sites in real sThe expression becomes more complicated if a more genbasis set is used.

We describe the leads or contacts as two semi-infintight-binding chains

HL(R)52t0 (i ,i 11PL(R)

s

cis† ci 11s1h.c., ~6!

wherecis† (cis) creates~destroys! an electron with spins on

site i and t0 is the hopping matrix element between neigboring sites. Such a model adequately, at least for enerlow or comparable tot0, describes a noninteracting, singlmode and homogeneous lead. It would be easy to generthe lead Hamiltonian to describe a more realistic system,example by modeling the true geometry or allowing forself-consistent potential due to interaction between electroHowever, the physics we are interested in, is usuallychanged dramatically by not including these details intomodel Hamiltonian and therefore, we will not discuss thissue into detail.

Finally, there is a term describing the coupling betwethe sample and the leads,

VL(R)5 (j PL(R)

i PCs

VL(R) j i cj s† dis1h.c., ~7!

whereVL(R) j i is the hopping matrix element between statiin the sample and sitej in a lead.

In the following sections we discuss the conductanthrough the system introduced above. To derive the condtance formulas, we will need a slightly modified system. Tauxiliary system is a ring formed by connecting the endsthe left and right leads of the original system as shownFig. 2. The ring is threaded by a magnetic fluxF in such away that there is no magnetic field in the region where eltrons move. We can then perform the standard Peierls s

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FORMULAS FOR ZERO-TEMPERATURE CONDUCTANCE . . . PHYSICAL REVIEW B 68, 035342 ~2003!

stitution20 and transform the hopping matrix elements of tHamiltonian~1! according to

t j i →t j i ei (e/\)*xixjA•dx, ~8!

wherexi is the position of sitei andA is the vector potentiadue to the flux, obeying

F5\

ef5 R A•dx. ~9!

Here we defined a dimensionless magnetic fluxf. The en-ergy of the system is periodic inf with a period of 2p anddepends only on value off and not on any details of how thflux is produced. If the original Hamiltonian~1! obeys thetime-reversal symmetry, the energy does not change ifmagnetic field is reversed,

E~2f!5E~f!. ~10!

III. CONDUCTANCE OF A NONINTERACTING SYSTEM

In this section we limit the discussion to noninteractisystems, i.e., we setU50 in Eq. ~2!. In such systems, theLandauer–Bu¨ttiker formula5–7

G5G0ut~«F!u2, ~11!

which relates the zero-temperature conductanceG to thetransmission probabilityut(«F)u2 for electrons at the Fermenergy«F , can be applied. The proportionality coefficienG052e

2/h, is the quantum of conductance. Below we fiderive a set of formulas, which relate the transmission prability, and consequently the conductance, to single-elecenergy levels of the auxiliary ring system introduced in tpreceding section. Then we derive another set of formurelating the conductance to the ground-state energy ofauxiliary system. One of these formulas was derived befin Ref. 14, and a limiting case of another one was discusin Refs. 12 and 13. Here we present a unified approach toproblem, from which these results emerge as special ca

FIG. 2. The sample embedded in a ring formed by joiningleft and right leads of the system in Fig. 1. Magnetic fluxF pen-etrates the ring.

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A. Formulas relating conductance to single-electron energylevels

Let us consider eigenstates of an electron moving oring system introduced in the previous section. We willinterested only in energies of these states and not in thecise form of wavefunctions. The energy of an electron oring penetrated by a magnetic fluxf depends only on themagnitude of the flux and therefore, any vector potential ffilling condition ~9! is good for our purpose. We choosevector potential constant everywhere except between sitei 0and i 011, both in the lead part of the ring as shown in F2. The hopping matrix element between the two sites is tmodified tot0e

if. With no flux penetrating the ring, the electron’s wave function in the lead part of the system isaeiki

1be2 iki , wherek is the electron’s wave vector anda andbare amplitudes determined by properties of the centralgion. If there is a flux through the ring, the wave functionmodified. The Schro¨dinger equations for sitesi 0 and i 011show us that the appropriate form isaeiki1be2 iki for i< i 0and ae2 ifeiki1be2 ife2 iki for i . i 0. The scattering matrixof the central region provides a relation between coefficiea andb,

S be2 ifeikNa D 5S r k tk8tk r k8D S ae2 ife2 ikN

b D . ~12!The elements of the scattering matrix,tk andr k (tk8 andr k8),are the transmission and reflection amplitudes for electrcoming from the left~right! lead, andN is the number ofsites in the lead part of the ring. We added phase face6 ikN to the ‘‘left lead’’ amplitudes to compensate for thphase difference an electron accumulates as it travels throthe lead part of the ring. The scattering matrix defined tway does not depend onN andf, and equals the scatterinmatrix of the original, two-lead system. Equation~12! is ahomogeneous system of linear equations, solvable only ifdeterminant is zero. Using the unitarity property of the sctering matrix, the eigenenergy condition becomes

tk8eif1tke

2 if5eikN1tk

tk8*e2 ikN. ~13!

We assume that the Hamiltonian of the original, two lesystem obeys the time-reversal symmetry and therefore,scattering matrix is symmetric,21 tk5tk8 . Expressing thetransmission amplitude in terms of its absolute value andscattering phase shifttk5utkueiwk, we arrive at the final formof the eigenenergy equation

utkucosf5cos~kN2wk!. ~14!

In Fig. 3 a graphical representation of this equationpresented.

To extract the transmission probabilityutku2, we proceedby differentiating the eigenvalue equation with respectcosf

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T. REJEC AND A. RAMŠAK PHYSICAL REVIEW B 68, 035342 ~2003!

]utku] cosf

cosf1utku52sin~kN1wk!S N ]k] cosf 1 ]wk] cosf D56A12utku2 cos2f

3S N ]k] cosf 1 ]wk] cosf D . ~15!The sign of the last expression depends on weatherk belongsto a decreasing~1! or an increasing~2! branch of the cosinefunction in Eq.~14!, or equivalently, if we are interested ian eigenstate with odd~1! or even~2! n, wheren indexesthe eigenstates from the one with the lowest energy upwLet us choose an eigenstate and consider how the cosponding wave vectork changes as the magnetic fluxf isvaried from 0 top. It is evident that the variation ink is ofthe order of 1/N as the cosine function in the right-hand siof Eq. ~14! oscillates with such a period. Let as assume tthe number of sites in the ring is large enough that transmsion amplitude does not change appreciably in this interv

U]tk]kUpN !1. ~16!Then the derivatives ]k/] cosf, ]utku/] cosf and]wk /] cosf are of the order of 1/N and Eq.~15! simplifiesto

utku56A12utku2 cos2fN]k

] cosf1OS 1ND . ~17!

Introducing the density of states in the leadsr(«)5(1/p)3(]k/]«), which, for example, for a tight-binding leawith only nearest-neighbor hoppingt0 and dispersion«k522t0cosk equals 1/(pA4t022«k2), we finally obtain

] arccos~7ut~«k!ucosf!] cosf

5pNr~«k!]«k

] cosf, ~18!

wheret(«k)5tk . The condition Eq.~16! of validity can alsobe expressed in a form involving energy as a variable

FIG. 3. A graphical representation of the eigenvalue equa~14!. The shaded region represents the allowed values of thehand side of the equation for different values of magnetic flux~forexample, the dashed line shows the values forf5p/4!. The full linerepresents the right-hand side of the eigenvalue equation. Thetem is presented in Fig. 7,N5100.

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r~«!U]t~«!]« U. ~19!

Equation~18! is the central result of this work. It expressethe transmission probabilityut(«)u2 of a sample connected ttwo leads in terms of the variation of single-electron enelevels with magnetic flux penetrating the auxiliary ring sytem. Employing the Landauer–Bu¨ttiker formula Eq. ~11!,this result also provides the zero-temperature conductancthe system. From the derivation it is evident that the methbecomes exact as we approach the thermodynamic lN→`.

In general Eq.~18! has to be solved numerically to obtathe transmission probability on a discrete set of enepoints, one for each energy level of a system. By increasthe system sizeN, the density of these points increases athe errors decrease, as the condition~19! is fulfilled better.We will return to this point in Sec. V where we consider thconvergence issues in detail. Here we present some spcases of Eq.~18! where analytic expressions can be obtainBy averaging the equation over values of fluxf betweenf50 andf5p @note that we may treatut(«k)u andr(«k) asconstant while averaging as the resulting error is of the orof 1/N], we can relate the transmission probability to taverage magnitude of the derivative of a single-electronergy with respect to the flux:

ut~«k!u25sin2S p22 Nr~«k!U]«k]f U D . ~20!Note that it is enough to calculate the energy levels atf50and f5p to calculate the transmission probability au]«k /]fu5(1/p)u«k(p)2«k(0)u. In Fig. 4~a! it is illustratedhow a large variation of single-electron energy as the fluxchanged fromf50 to f5p corresponds to a large condutance and vice versa. The transmission probability can abe calculated from the derivative atf5p/2 resulting in thesecond formula

ut~«k!u25S pNr~«k!]«k]f Uf5p/2

D 2. ~21!Again, Fig. 4~a! shows that there is a correspondencetween a large sensitivity of a single-electron energy toflux at f5p/2, and a large conductance. Finally, we obsethat the curvature of energy levels atf50 and f5p alsogives information of conductance. The third formula read

ut~«k!u25121

11S pNr~«k!]2«k]f2 Uf50,p

D 2 . ~22!B. Formulas relating conductance to the ground-state energy

Above we showed how the flux variation of the energythe last occupied single-electron state allows one to calcuthe zero-temperature conductance through a noninteracsample. The goal of this section is to derive an alternativeof formulas, expressing the zero-temperature conductanc

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FORMULAS FOR ZERO-TEMPERATURE CONDUCTANCE . . . PHYSICAL REVIEW B 68, 035342 ~2003!

terms of the flux variation of the ground-state energyE,which for an even number of electrons in a noninteractsystem is simply a sum of single-electron energies up toFermi energy«F , multiplied by 2 because of the electronspin

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ut~«F!u25sin2S p2\2e Nr~«F!u j ~f!u D . ~31!Only two ground-state energy calculations need to be pformed to obtain the conductance as

\

eu j ~f!u5

1

puE~p!2E~0!u.

This formula was also discussed in Refs. 12 and 13 forcase where the transmission probability is small. The secformula relates the conductance to the persistent currenf5p/2,14,24

ut~«F!u25S p\e Nr~«F! j S p2 D D2

. ~32!

The third formula, corresponding to Eq.~22! in the single-electron case, turns out to be more complicated and giveimplicit relation for ut(«F)u

pNr~«F!]2E

]f2U

min,max

562ut~«F!u

pA12ut~«F!u2arccos~6ut~«F!u!.

~33!

Here the upper and the lower signs correspond to the sederivative at a minimum and at a maximum of the energyflux curve, respectively. Minima~maxima! occur atf50~p!if an odd number of single-electron levels is occupied aat f5p~0! if an even number of levels is occupied. Thsecond derivative in a minimum is proportional to the chastiffnessD5(N/2)]2E/]f2umin of the system.

18,25 We canalso define the corresponding quantity for a maximum asD̃52(N/2)]2E/]f2umax. In general, Eq.~33! has to be solvednumerically. However, in the limit of a very small condutance and in the vicinity of the unitary limit, additional anlytic formulas are valid

ut~«F!u5H 2pr~«F!D, ut~«F!u→0,12

13p

42pr~«F!D, ut~«F!u→1.

~34!

Note that there is a quadratic relation between the condtance and the charge stiffness in the low conductance liThe corresponding formulas for the maximum of the enevs flux curve are

ut~«F!u5H 2pr~«F!D̃, ut~«F!u→0,12 2~2pr~«F!D̃ !

2, ut~«F!u→1.

~35!

A detailed analysis of convergence properties of the formuderived in this section is presented Sec. V.

We stress again that the validity of these formulas is baon an assumption that the number of sites in the ringsufficiently large according to the condition Eq.~19!. Thismeans that ift(«) exhibits sharp resonances, the calculathas to be performed on such a large auxiliary ring systhat in the energy interval of interest~the width of the reso-

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nance! there is a large number of eigenenergies«n . Thent(«);t(«n8), where«n8 is the eigenenergy closest to«. Suchsharp resonances int(«) are expected e.g. in chaotisystems.26,27 The present method might be impractical~butstill correct! in this case.

IV. CONDUCTANCE OF AN INTERACTING SYSTEM

The zero-temperature conductance of a noninteracsystem can thus be determined with the transmission pability obtained from one of the formulas we derived in tpreceding section, and the Landauer–Bu¨ttiker formula. Themain challenge, however, remains the question of the vaity of this type of approach for interacting systems. In thsection we give arguments why the approach is correct foclass of interacting systems exhibiting Fermi liquid propties. In order to reach this goal, we present four essensteps as follows.

Step 1. Conductance of a Fermi liquid system atTÄ0

The basic property that characterizes Fermi liqusystems28 is that the states of a noninteracting systemelectrons are continuously transformed into states of theteracting system as the interaction strength increases fzero to its actual value. One can then study the propertiesuch a system by means of the perturbation theory, regarthe interaction strength as the perturbation parameter.concept of the Fermi liquid was first introduced fotranslation-invariant systems by Landau,29,30 and was lateralso extended to systems of the type we study here.31

The linear response conductance of a general interacsystem of the type shown in Fig. 1 can be calculated fromKubo formula9,32

G5 limv→0

ie2

v1 idP II~v1 id!, ~36!

whereP II(v1 id) is the retarded current-current correlatiofunction

P II~ t2t8!52 iu~ t2t8!^@ I ~ t !,I ~ t8!#&. ~37!

For Fermi liquid systems atT50, the current–current correlation function can be expressed in terms of the GreefunctionGn8n(z) of the system and the conductance is givwith9

G52e2

h U 12 ipr~«F! e2 ikF(n82n8)Gn8n~«F1 id!U2

, ~38!

wheren andn8 are sites in the left and the right lead, respetively. One candefinethe transmission amplitude as

t~«![1

2 ipr~«!e2 ik(n82n)Gn8n~«1 id!, ~39!

and the conductance formula Eq.~38! then reads

G52e2

hut~«F!u2. ~40!

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FORMULAS FOR ZERO-TEMPERATURE CONDUCTANCE . . . PHYSICAL REVIEW B 68, 035342 ~2003!

For noninteracting systems,t(«) defined this way reduces tthe standard transmission amplitude~Fisher-Lee relation33!and Eq.~40! represents the Landauer–Bu¨ttiker formula. Inthe next step, we will show that the transmission amplituEq. ~39! has a direct physical interpretation also for interaing systems, being the transmission amplitude of Fermiuid quasiparticles.

Step 2. Quasiparticle Hamiltonian

In this step, we generalize the quasiparticle approximato the Green’s function, presented for the single-impurAnderson model in Ref. 34, to the case where the interacis present in more than a single site.

In Fermi liquid systems obeying the time-reversal symetry, the imaginary part of the retarded self-energy aT50 vanishes at the Fermi energy and is quadratic forquencies close to the Fermi energy.35,36 Using the Fermi en-ergy as the origin of the energy scale, i.e.,v2«F→v, wecan express this as

Im S~v1 id!}v2. ~41!

Close to the Fermi energy, the self-energy can be expanin powers ofv resulting in an approximation to the Greenfunction,

G21~v1 id!5v12H(0)2S~01 id! ~42!

2v]S~v1 id!

]v Uv50

1O~v2!.~43!

HereH(0) contains matrix elements of the noninteracting pof the Hamiltonian~2!. Note that expansion coefficients areal because of Eq.~41!. Let us introduce the renormalizatiofactor matrixZ as

Z21512]S~v1 id!

]v Uv50

. ~44!

The Green’s function forv close to the Fermi energy cathen be expressed as

G21~v1 id!5Z21/2G̃21~v1 id!Z21/21O~v2!, ~45!where we defined the quasiparticle Green’s function

G̃21~v1 id!5v12H̃ ~46!

as the Green’s function of anoninteracting quasiparticleHamiltonian

H̃5Z1/2@H(0)1S~01 id!#Z1/2. ~47!

Note that factoring the renormalization factor matrix asdid above ensures the hermiticity of the resulting quasipacle Hamiltonian.

Matrix elements ofZ differ from those of an identity matrix only if they correspond to sites of the central region.other cases, as the interaction is limited to the central regthe corresponding self-energy matrix element is zero. Th

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fore, comparing the quasiparticle Hamiltonian to the nonteracting part of the real Hamiltonian, we observe thateffect of the interaction is to renormalize the matrix elemeof the central region Hamiltonian~4! and those corresponding to the hopping between the central region and the le~7!. The values of the renormalized matrix elements depon the value of the Fermi energy of the system.

Let us illustrate the ideas introduced above for the casthe standard Anderson impurity model.34 We calculated theself-energy in the second-order perturbation theapproximation37–39 and constructed the quasiparticle Hamtonian according to Eq.~47!. In Fig. 5 the local spectrafunctions corresponding to both the original interactiHamiltonian and the noninteracting quasiparticle Hamtonian are presented. The agreement of both results is pein the vicinity of the Fermi energy where the expansion~43!is valid.

The reason for introducing the quasiparticle Hamiltoniis to obtain an alternative expression for the conductancterms of the quasiparticle Green’s function. Equation~45!relates the values of the true and the quasiparticle Grefunction at the Fermi energy,

G~01 id!5Z1/2G̃~01 id!Z1/2. ~48!

Specifically, if bothn and n8 are sites in the leads,Gn8n(01 id)5G̃n8n(01 id) as a consequence of the propertiesthe renormalization factor matrixZ discussed above. Equation ~39! then tells us that the zero-temperature conductaof a Fermi liquid system is identical to the zero-temperatconductance of a noninteracting system defined with the qsiparticle Hamiltonian for a given value of the Fermi energ

Step 3. Quasiparticles in a finite system

The conclusions reached in the first two steps are baon an assumption of the thermodynamic limit, i.e., theyvalid if the central region is coupled to semiinfinite leadHere we generalize the concept of quasiparticles to a fi

FIG. 5. TheT50 local spectral function and the correspondiquasiparticle approximation for the Anderson impurity modelshown in Fig. 9. The values of parameters aret150.4t0 , U51.92t0 and «d52U/2. The calculations were performed withithe second-order perturbation theory as described in Appendix

2-7

an

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oor

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T. REJEC AND A. RAMŠAK PHYSICAL REVIEW B 68, 035342 ~2003!

ring system withN sites andM electrons, threaded bymagnetic fluxf. Let us define the quasiparticle Hamiltoniafor such a system,

H̃~N,f;M !5Z1/2@H(0)~N,f!1S~01 id!#Z1/2. ~49!

Here the self-energy and the renormalization factor maare determined in the thermodynamic limit where, asprove in Appendix A, they are independent off and corre-spond to those of an infinite two-lead system.

Suppose now that we knew the exact values of the remalized matrix elements in the quasiparticle Hamilton~49!. As this is a noninteracting Hamiltonian, we could thapply the conductance formulas of the preceding sectiocalculate the zero-temperature conductance of an infitwo-lead system with the same central region and cenregion-lead hopping matrix elements, i.e., of a systemscribed with the quasiparticle Hamiltonian~47!. As shown instep 2, this procedure would provide us with the exact cductance of the original interacting system. However, totain the values of the renormalized matrix elements, oneeds to calculate the self-energy of the system, whichdifficult many-body problem. In the next step, we will shothat there is an alternative and easier way to achievesame goal.

In Fig. 6 we compare the spectral density of an Andersimpurity embedded in a finite ring system to that of the cresponding quasiparticle Hamiltonian~49!. Note that the

FIG. 6. ~a! TheT50 local spectral function as in Fig. 5, but foa ring system withN5400 sites and fluxf53p/4. ~b! The spectralfunction in the vicinity of the Fermi energy~dashed lines! comparedto that corresponding to the quasiparticle Hamiltonian~49!. Boththe spectral density of the interacting system and the matrixments of the quasiparticle Hamiltonian were calculated withinsecond order perturbation theory.

03534

xe

r-

toteal-

--ea

e

n-

spectral density of the quasiparticle Hamiltonian correcdescribes the true spectral density in the vicinity of the Feenergy.

Step 4. Validity of the conductance formulas

In this last step we finally show how to calculate the coductance of an interacting system. In Appendix A we stuthe excitation spectrum of a finite ring system threaded wa magnetic flux and containing a region with interaction. Wshow that

E@N,f;M11#2E@N,f;M #5 «̃~N,f;M ;1!1O~N23/2!,~50!

whereE(N,f;M ) and E(N,f;M11) are the ground-stateenergies of the interacting Hamiltonian for a ring systewith N sites and fluxf, containingM and M11 electrons,respectively, and«̃(N,f;M ;1) is the energy of the firssingle-electron level above the Fermi energy of the finite rquasiparticle Hamiltonian~49!. This estimation allows one touse single-electron formulas of Sec. III A to calculate tzero-temperature conductance for a Fermi liquid system.showed in step 3 that inserting«̃(N,f;M ;1) into these for-mulas would give us the correct conductance. Equation~50!proves, that the same result is obtained if the differencethe ground state energies of an interacting systE@N,f;M11#2E@N,f;M # is inserted into a formula in-stead. The estimated error, which is of the order ofN23/2, isfor a largeN negligible, because it is much smaller than tquasiparticle level spacing, which is of the order of 1/N.

As demonstrated in Sec. III B, the conductance of a ninteracting system can also be calculated from the variaof the ground-state energy with flux through the ring. Tproof of the formulas involved only the properties of a setneighboring single-electron energy levels. We assumedvalidity of single-electron conductance formulas for eachthese levels and made use of the fact that the ground-senergy of the system increases by a sum of the relevsingle-electron energies as the levels become occupiedadditional electrons. For Fermi liquid systems, the firstsumption was proved above. The second assumption, wfor noninteracting systems is obvious, is proved in AppenA. There we show that as a finite number of additional eltrons is added to an interacting system, the succesground-state energies are determined by the single-elecenergy levels of the same quasiparticle Hamiltonian withvery good accuracy~A1!. Therefore, the proof of Sec. III Bis also valid for interacting Fermi liquid systems, providedsystem is the Fermi liquid for all values of the Fermi enerbelow its actual value.

V. NUMERICAL TESTS OF THE METHOD

A. Noninteracting system

In this section we discuss the convergence propertiethe conductance formulas derived in Sec. III. As a test stem we use a double-barrier potential scattering problemsented in Fig. 7. Results of various formulas for differenumber of sites in the ring are presented in Fig. 8. The ex

e-e

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FORMULAS FOR ZERO-TEMPERATURE CONDUCTANCE . . . PHYSICAL REVIEW B 68, 035342 ~2003!

zero-temperature conductance for this system exhibitsharp resonance peak superimposed on a smoother bground conductance. We notice immediately that as the nber of sites in the ring increases, the convergence is genefaster in the region where the conductance is smooth thathe resonance region, which is consistent with the condi~19!. Comparing the results obtained employing differeconductance formulas we observe that the convergence ifastest in both the single-electron and the ground-stateergy case if the formulas of Eqs.~22! and~33! are applied tothe maximum of the energy vs flux curve~or to the minimumin the single-electron case!. Formulas of Eqs.~20! and ~31!expressing the conductance in terms of the difference ofenergies atf50 andf5p converge somewhat slower. Nohowever that in the former case the second derivative ofenergy with respect to the flux has to be evaluated whilethe later, the energy difference is large and because ofthe calculation is much more well behaved. From the coputational point of view there is another advantage ofenergy difference formulas. In this case, all the matrix ements can be made real if one chooses such a vector ptial that only one hopping matrix element if modified by thflux as then the additional phase factor ise6 ip521. Finally,the remaining formulas, employing the slope of the energyflux curve atf5p/2 and the curvature in the minimum of th

FIG. 7. A double barrier noninteracting system. The heightthe barriers is 0.5t0, where t0 is the hopping matrix element between neighboring sites.

03534

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llyinntthen-

e

enat,-e-en-

s

ground-state energy vs flux curve, do not show convergeproperties comparable to those of the formulas discusabove.

B. Anderson impurity model

In 1980s several theories40,41 were put forward proposinga realization of the Anderson impurity model19 in systemsconsisting of a quantum dot coupled to two leads~see Fig.9!. These theories show that the topmost occupied enelevel in a quantum dot with an odd number of electrons cbe associated with the Anderson model«d level and such asystem should mimic the old Kondo problem of a magnespin 12 impurity in a metal host. In recent years signaturesthe Kondo physics in electron transport through quantdots have also been found experimentally.42,43The Andersonmodel is well defined and is an attractive testing groundnew numerical and analytical methods that are developetackle other challenging many-body problems. Therefore,will also take it as a nontrivial example to test results of tconductance formulas we derived in this paper.

There are three distinct parameter regimes of the Andson model. If «d,«F,«d1U with u«d1U2«Fu@D andu«d2«Fu@D, whereD is the coupling of the quantum dot tleads, we are in the Kondo regime. In this regime, a narrKondo resonance is formed in the spectral function at

fFIG. 9. The Anderson impurity model realized as a quantum

coupled to two leads. The dot is described with the energy leve«dand the Coulomb energy of a doubly occupied levelU. t1 is thehopping between the dot and leads.

-veswsheon

for-d

ng

FIG. 8. Exact and approximate zerotemperature conductance vs Fermi energy curfor the system in Fig. 7. The shaded area shothe exact result. The left set of figures shows tapproximations obtained using the single-electrformulas of Sec. III A while the right set of fig-ures corresponds to the ground-state energymulas of Sec. III B. Different curves corresponto different number of sitesN in the ring. In (a1)and (a2) the conductance was calculated usiEqs. ~20! and ~31!, in (b1) and (b2) using Eqs.~21! and~32!, while in the other figures Eqs.~22!and ~33! were used, in (c1) and (c2) applied tothe maximum and in (d1) and (d2) to the mini-mum of energy vs flux curves.

2-9

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ine

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cal-ia-

T. REJEC AND A. RAMŠAK PHYSICAL REVIEW B 68, 035342 ~2003!

Fermi energy for temperatures below and close to the Kotemperature, which corresponds to the width of the renance. The zero-temperature conductance in the Kondogime reaches the unitary limit of 2e2/h. Letting either«d or«d1U approach the Fermi energy so thatu«d1U2«Fu oru«d2«Fu becomes comparable withD, we enter the mixedvalence regime where the charge fluctuations on the dotcome important. In this regime, the resonance becowider and merges with the resonance corresponding to«d or«d1U levels. More important for our discussion is the fathat the resonance moves away from the Fermi energytherefore, the conductance drops as we enter this regFinally, there are two nonmagnetic regimes, one in which‘‘impurity’’ level is predominately empty, «d2«F@D,known as the empty orbital regime, and the correspondregime where the dot is doubly occupied. In these regimthe conductance drops toward zero.

In Fig. 10 the zero-temperature conductance througquantum dot acting as an Anderson impurity is presentedcompared to exact results of the Bethe ansatz approach44,45

To calculate the conductance, Eq.~31! was used, with theground-state energies atf50 andf5p obtained using thevariational method presented in Appendix B. There are tvariational parameters defining the auxiliary Hamiltoni~B1!, one describing the effective energy level on the dot athe other renormalizing hoppings into the leads. Two diffent variational basis sets were used in calculations. Infirst set, the basis consisted of wave functions~B2!. As aresult of the rotational symmetry in the spin degree of frdom, two of the basis functions may be merged into oTherefore, the basis set consisted of projections of the ailiary Hamiltonian ground stateu0̃& to states with emptyP0u0̃&, singly occupiedP1u0̃&5P↑u0̃&1P↓u0̃& and doublyoccupiedP2u0̃& dot level. In the second basis set, wave funtions P1VP0u0̃&, P0VP1u0̃&, P2VP1u0̃&, and P1VP2u0̃&~B7! were added to those of the first set, withV5VL1VRbeing the operator describing the hopping between the

FIG. 10. The zero-temperature conductance calculated fground-state energy vs magnetic flux in a finite ring system usthe variational method of Appendix B with 3 and 7 basis functioFor comparison, the exact Bethe ansatz result is presented wdashed line. The system shown in Fig. 9 was used, withU50.64t0 and t150.2t0.

03534

o-

re-

e-es

tnde.e

gs,

and

o

d-e

-.x-

-

ot

and the leads~7!. For each position of the«d level relative tothe Fermi energy, we increased the number of sites inring until the conductance converged. The number of sneeded to achieve convergence@see Fig. 11~a!# was the low-est in the empty orbital regime and the highest~about 1000for the system shown in Fig. 10! in the Kondo regime. Thisis a consequence of Eq.~19! as a narrow resonance relatedthe Kondo resonance appears in the transmission probabof the quasiparticle Hamiltonian~47! in the Kondo regime.In the mixed valence regime, the width of the resonanbecomes comparable toD, which is much larger than theKondo temperature and the convergence is thus faster. Inempty orbital regime the resonance moves away fromFermi energy and an even smaller number of sites is neeto achieve convergence.

Let us return to results shown in Fig. 10. Note that etending the variational space from 3 to 7 basis functiosignificantly improves the agreement with the exact resThe remaining discrepancy at the larger basis set can btributed to the approximate nature of the variational methAnother source of error could be the fact that the Bethe

mg.a

FIG. 11. ~a! Results of conductance calculations using Eq.~31!for the system presented in Fig. 10 as the number of sites in theincreases. Note that the convergence is the fastest in the eorbital regime and the slowest in the Kondo regime.~b! Finite-sizescaling analysis of the same results for various values of«d . Withblack dots, the Bethe ansatz values are shown. Energies wereculated using the variational method of Appendix B with 7 vartional basis functions.

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inioouaneonooofi

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d

FORMULAS FOR ZERO-TEMPERATURE CONDUCTANCE . . . PHYSICAL REVIEW B 68, 035342 ~2003!

satz solution assumes there is a constant coupling to annitely wide conduction band. In our case, the conductband is formed by the states in a tight-binding ring, the cpling to which is not constant. However, it is almost constin the energy interval we are interested in, i.e., near the cter of the band. In order to estimate the effect of the noncstant coupling on the conductance, we calculated the dotcupation number within the second order perturbation thefor both the case of a constant coupling and for the casetight-binding ring. We then calculated the conductanceeach case making use of the Friedel sum rule.46 The agree-ment is significantly better than the difference betweenBethe ansatz and variational conductance curves in Fig.Therefore, we believe that the use of Bethe ansatz resuljustified for this particular problem.

In Fig. 11~b! a finite-size scaling analysis of the convegence is presented. Note that for rings with a large numbesitesN, the error scales approximately as 1/N.

C. Double quantum dot

The next logical step after studying individual quantudots is to consider systems of more than one dot. Sinquantum dots are often regarded as artificial atoms becof a similar electronic structure and comparable numbeelectrons in them. By coupling several quantum dots onthus creating artificial molecules. Here we will not go indetail in describing the physics of such systems. Our goato compare results of our conductance formulas to resultother methods for a double quantum dot system presenteFig. 12.

In the calculation we again employed the conductaformula ~31! and calculated the ground-state energies wthe variational approach of Appendix B with the variationbasis set~B2!. In Fig. 13 the zero-temperature conductanfor the case where the inter-dot and the on-site CoulorepulsionsV and U are of the same size, are plotted asfunction of the position of dot energy levels relative to tFermi energy for various values of the inter-dot hopping mtrix elementt2. The same problem in the particle-hole symmetric case«d1(U/2)1V50 was studied recently in Ref. 8The Matsubara Green’s function was calculated withquantum Monte Carlo method and the values on discfrequencies were extrapolated to obtain the retarded Grefunction at the Fermi energy. Then Eqs.~40! and ~39! wereused to calculate the zero-temperature conductance. Thsults are presented in Fig. 14, together with the conductacalculated within the Hartree–Fock approximation andsults of our method. The agreement with the QMC resultexcellent, while the Hartree–Fock approximation givesqualitatively wrong conductance curve, especially at low v

FIG. 12. A double quantum dot system. Each of the dots wenergy level«d and on-site Coulomb repulsionU is coupled to alead with a hopping matrix elementt1. The inter-dot hoppingt2 isalso present as is the inter-dot Coulomb repulsionV.

03534

fi-n-tn--c-rya

n

e0.is

of

lesef

is

isofin

ehleb

-

eten’s

re-ce-isal-

ues of the inter-dot coupling, indicating strong electroelectron correlations in the system. The results of our metfor lower values ofV are also shown in Fig. 14.

VI. SUMMARY AND CONCLUSIONS

We have demonstrated how the zero-temperature contance of a sample with electron-electron correlations aconnected between noninteracting leads can be determThe method is extremely simple and is based on severalmulas connecting the conductance to persistent currents iauxiliary ring system. The conductance is determined ofrom the ground-state energy of an interacting system, win more traditional approaches, one needs to knowGreen’s function of the system. The Green’s function aproaches are often much more general, allowing the trment of transport at finite temperatures and for a fin

h

FIG. 13. The zero-temperature conductance of the systemFig. 12 as a function of the position of the dot energy level«d andinter-dot hopping matrix elementt2. The remaining parameters arU5V5t0 and t150.5t0.

FIG. 14. The zero-temperature conductance of the double qtum dot system of Fig. 12 at«d1(U/2)1V50 as a function of theinter-dot hopping matrix elementt2 for various values of the inter-dot Coulomb interactionV. As a comparison, the Hartree–Fock anquantum Monte Carlo results~Ref. 8! are presented forV/t051.Other parameters are the same as in Fig. 13.

2-11

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et

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a-tntl

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sein

the

of a

ns

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T. REJEC AND A. RAMŠAK PHYSICAL REVIEW B 68, 035342 ~2003!

source-drain voltage applied across the sample, which inmethod is not possible. However, the advantage ofpresent method comes from the fact that the groustate energy is often relatively simple to obtain by varionumerical approaches, including variational methods,could therefore, for zero-temperature problems, be mappropriate.

Let us summarize the key points of the method:~1! The ‘‘open’’ problem of the conductance through

sample coupled to semiinfinite leads is mapped on t‘‘closed’’ problem, namely a ring threaded by a magneflux and containing the same correlated electron region.

~2! For a noninteracting sample, it is shown that the zetemperature conductance can be deduced from the variaof the energy of the single-electron level at the Fermi enewith the flux in a large, but finite ring system. The condutance is given with Eq.~18!, or with three simple formulasEq. ~20!, Eq. ~21! and Eq.~22!.

~3! Alternatively, the conductance of a noninteracting stem is expressed in terms of the variation of the ground-senergy with flux, Eq.~24!. Three additional conductance fomulas, Eq.~31!, Eq. ~32! and Eq.~33!, are derived.

~4! The method is primarily applicable to correlated sytems exhibiting Fermi liquid properties at zero temperatuIn order to prove the validity of the method for such systemthe concept of Fermi liquid quasiparticles is extended tonite, but large systems. The conductance formulas giveconductance of a system of noninteracting quasiparticwhich is equal to the conductance of the original interactsystem. The ground-state energy is a universal functionthe magnetic flux and the conductance is the only param@Eqs.~29! and ~30!#.

~5! The results of our method are compared to resultsother approaches for problems such as the transport throsingle and double quantum dots containing interacting etrons. The comparison shows an excellent quantitative agment with exact Bethe ansatz results in the single quandot case. The results for a double quantum dot systemperfectly match QMC results of Ref. 8.

~6! One should additionally point out that in the derivtion presented in this paper we assumed the interaction inleads to be absent. It is clear that this assumption isjustified for all systems. The method cannot be direcapplied to systems where the interaction in the leadsessential, as are, e.g., systems exhibiting Luttinger liqproperties.

~7! The validity of the method is not limited to systemthat do not break the time-reversal symmetry. A generaltion to systems with a broken time-reversal symmetry, sas Aharonov–Bohm rings coupled to leads, is possiblewill be presented elsewhere.47

~8! Another important limitation of the present methodthe single channel approximation for the leads. It mightpossible to extend the applicability of the method to systewith multichannel leads by studying the influence of sevemagnetic fluxes that couple differently to separate channThis way, one might be able to probe individual matrix ements of the scattering matrix and derive conductancemulas relevant for such more complex systems.

03534

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Note added.After the present work was completed thauthors met R. A. Molina and R. A. Jalabert who reportabout their recent unpublished work where an approach slar to our work is presented.13,48

ACKNOWLEDGMENTS

The authors wish to acknowledge P. Prelovs˘ek and X.Zotos for helpful discussions and for drawing our attentionthe problem of persistent currents in correlated systems.acknowledge V. Zlatic´ for discussions related to perturbativtreatment of the Anderson model and J. H. Jefferson for uful remarks and the financial support of QinetiQ.

APPENDIX A: FERMI LIQUID IN A FINITE SYSTEM

In Sec. IV we based the proof of the validity of the coductance formulas for Fermi liquid systems on the assumtion that

E~N,f;M1m!5E~N,f;M !

1(i 51

m

«̃~N,f;M ; i !1O~N23/2!. ~A1!

Here E(N,f;M1m) and E(N,f;M ) are the ground-stateenergies of an interactingN-site ring with fluxf, containingM1m and M electrons, respectively.«̃(N,f;M ; i ) is asingle-electron energy of the ring quasiparticle HamiltonH̃(N,f;M ) as defined in Eq.~49!, with the Fermi energycorresponding toM electrons in the system. The indexi la-bels successive single-electron energy levels above the Fenergy. We assumem to be finite andN approaching thethermodynamic limit. In this Appendix we will give arguments showing that the assumption of Eq.~A1! is indeedvalid. In Appendix A 1 we first express the problem in termof the Green’s function of the system. In Appendix A 2 wstudy the properties of the self-energy due to interactionfinite ring systems and then use this results to completeproof in Appendix A 3.

1. Relation to the Green’s function

Assume we manage to prove Eq.~A1! for m51, i.e.,

E~N,f;M11!5E~N,f;M !1 «̃~N,f;M ;1!1O~N23/2!.~A2!

Then we can use the same result to relate the energysystem withM12 electrons to that withM11 electrons,

E~N,f;M12!5E~N,f;M !1 «̃~N,f;M ;1!

1 «̃~N,f;M11;1!1O~N23/2!. ~A3!Now the matrix elements of quasiparticle HamiltoniaH̃(N,f;M11) andH̃(N,f;M ) differ by an amount of theorder of 1/N. To see this, note that the shift of the Fermenergy as an electron is added to the system is of the ord1/N, producing a shift of the same order in the self-eneand it’s derivative at the Fermi energy, which define the q

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FORMULAS FOR ZERO-TEMPERATURE CONDUCTANCE . . . PHYSICAL REVIEW B 68, 035342 ~2003!

FIG. 15. Second-order self-energy diagram

r

ticinheio

a

ityreen

emnraol

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en’s

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siparticle Hamiltonian through Eq.~49!. As the difference ofthe HamiltoniansDH̃ is small, we can use the first ordeperturbation theory,

«̃~N,f;M11;1!5 «̃~N,f;M ;2!

1^N,f;M ;2uDH̃uN,f;M ;2&

5 «̃~N,f;M ;2!1O~N22!. ~A4!In the last step we made use of the fact that the quasiparHamiltonians differ only in a finite number of sites in andthe vicinity of the central region, and of the fact that tamplitude of the quasiparticle single-electron wave functuN,f;M ;2& is of the order of 1/AN. Thus we have provedEq. ~A1! for m52 and using the same procedure, we cextend the proof to any finitem.

To complete the proof, we still need to show the validof Eq. ~A2!. As a first step, consider the Lehmann repsentation of the zero-temperature central region Grefunction

Gji ~ t,t8!52 iu~ t2t8!^0u@dj~ t !,di†~ t8!#u0& ~A5!

of a ring system characterized withN and f, containingMelectrons,

Gji ~N,f;M ;z!5(n

^0udj un&^nudi†u0&

z2~EnM112E0

M !

1(n

^0udi†un&^nudj u0&

z2~E0M2En

M21!. ~A6!

The first sum runs over all basis states withM11 electrons,while the second sum runs over the states withM21 elec-trons. The difference in the ground-state energies of systwith M11 andM electrons is evidently equal to the positioof the first d-peak above the Fermi energy in the spectdensity corresponding to the Green’s function. In what flows, we will try to determine the energy of thisd-peak.

2. Self-energy due to interaction

To achieve the goal we have set in the preceding sectwe first need to study the structure of the self-energy duinteraction in a finite ring with flux. Let us again consider tLehmann representation~A6! and to be specific, limit our-selves to states above the Fermi energy. Introducingw j

n

5^0udj un& and«n5EnM112En

M , we can express the Greenfunction as

03534

le

n

n

-’s

s

l-

n,to

Gji ~N,f;M ;z!5(n

w jnw i

n*

z2«n. ~A7!

This expression can also be interpreted as a local Grefunction of a largernoninteractingsystem, consisting of thecentral region and a bath of noninteracting energy levels,number of which is equal to the number of multielectrstates withM11 andM21 electrons of the original interacting system. The self-energy due to ‘‘hopping out of tcentral region,’’ which includes both the effects of the inteaction as well as those due to the hopping into the ring,then be expressed as

S j i ~N,f;M ;z!5(n

VjnVniz2«n

, ~A8!

whereVjn are the ’’hopping matrix elements’’ between thcentral region and the ‘‘bath.’’ Thus we have shown that,far as the single-electron Green’s function is concerned,interacting system can be mapped on a larger, but noninacting system.

To further clarify the concepts introduced above, we cculated the self-energy due to interaction within the secoorder perturbation theory. Following the calculations by Hvatić, Šokčević and Zlatić37–39 for the Anderson model, wesum the second order self-energy diagrams shown in Fig.including Hartree and Fock terms into the unperturbHamiltonian. A lengthy but straightforward calculatiowhich we do not repeat here, shows that one can identifystatesn of Eq. ~A8! with three Hartree–Fock single-electrostate indicesq5(q1 ,q2 ,q3) such thatq1 and q2 are abovethe Fermi energy andq3 is below it ~or vice versa!, and aspin indexs. The ‘‘bath’’ energy levels

«qs5«q11«q22«q3 ~A9!

and the ‘‘hopping matrix elements’’ related to the self-enerfor electrons with spins

Vj qs55 (k8U jk8

ss̄w jq1w

k8

q2wk8

q3* , s5s̄,

1

A2 (k8U jk8

ss@w j

q1wk8

q22w jq2w

k8

q1#wk8

q3* , s5s

~A10!

are then expressed in terms of the Coulomb interactiontrix elements~5!, and the Hartree–Fock single-electron engies «q(N,f;M ) and the corresponding wave functionuwq(N,f;M )&. In Fig. 16 the positions ofd-peaks in theimaginary part of the self-energy as a function of magne

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T. REJEC AND A. RAMŠAK PHYSICAL REVIEW B 68, 035342 ~2003!

FIG. 16. Dashed lines show the positionsd-peaks~A9! in the second-order self-energy coresponding to single-electron energy levels ofunperturbed system presented with gray lines.

istththe

ie-antr

inuinrg

ng

d

aonc

th

o

thes

lf-and

r-velstoein-

estto

ies

there

dth

to

ion

d as

de-

flux through the ring are plotted. Note that as the fluxvaried, the positions of the peaks fluctuate by an amounthe order of the single-electron level spacing which is oforder of 1/N. The weights of the peaks also depend onflux. A similar behavior is expected if higher order processare also taken into account.

Finally, let us study the self-energy in the thermodynamlimit. We will show that in this case, the self-energy is indpendent of flux and is equal to the self-energy of the origintwo-lead system, shown in Fig. 1. To prove this statemewe consider a self-energy Feynman diagram for the cenregion decoupled from the ring, which then is obviouslydependent of flux. To calculate the self-energy for the fsystem, one should insert the self-energy due to hoppingthe ring into each propagator of the diagram. The self-enedue to hopping into the ring is

S j i(0)~N,f;z!5(

k

VjkVkiz2«k

, ~A11!

where «k are the single-electron energy levels of the ridecoupled from the central region andVki52cL

ktLi2cRk tRi

is the hopping matrix element between sitei in the centralregion and the single-electron statek in the ring.Vki is ex-pressed in terms of the hopping matrix elementtLi betweenthe sitei and the ring siteL adjacent to the central region anthe single-electron wave functioncL

k5A(2/N11)sink at siteL, whereN is the number of sites in the ring. There is alsosimilar contribution toVki corresponding to the hopping intthe right lead. In the ring system, the right lead wave fution can be expressed in terms of the left lead one ascR

k

5(21)ne2 ifcLk with k5@np/(N11), if one takes into ac-

count the parity of the wave functions and the effect offlux. Thus, Eq.~A11! transforms into

S j i(0)~N,f;z!5S j i

(L)~N;z!1S j i(R)~N;z!

12~ t jL tRie

2 if1t jRtLieif!

N11 (k~21!n sin2 k

z2«k,

~A12!

whereS j i(L)(N;z) andS j i

(R)(N;z) are the self-energies due thopping into the left and the right leads~each withN sites! of

03534

ofees

c

l,t,al-lltoy

-

e

the two-lead system. In the third term, one can performsum over oddn’s and over evenn’s separately. The sumdiffer only in sign in theN→` limit and therefore, this termvanishes. Therefore, in the thermodynamic limit the seenergy due to interaction is the same in both two-leadring systems.

3. Proof of Eq. „A2…

Positions ofd-peaks in the spectral density of the inteacting system correspond to the single-electron energy leof the noninteracting part of the ring Hamiltonian coupledthe ‘‘bath’’ according to Eq.~A8!. These energies can bobtained by solving for zeroes of the determinant of theverse of the ‘‘local’’ Green’s function

det@v12H(0)~N,f!2S~N,f;M ;v1 id!#50.~A13!

What we are going to prove in this section is that the lowpositive solution of this equation corresponds«̃(N,f;M ;1) as required by Eq.~A2!.

We begin by separating the self-energy at frequencclose to the Fermi energy into two contributions, one~S9!due to the ‘‘bath’’ states close to the Fermi energy andother ~S8! of all the other states with energies which aseparated from the expected solution of Eq.~A13! by at leastan amount of the order of the single-electron level spacingD,which is of the order of 1/N. We first estimate the seconterm. Let us divide the frequency axis into intervals of widD, each contributing to the self-energy atuvu,D an amountgiven by

E«

«1D r~«!

v2«d«, ~A14!

where r j i («)5(nVjnVnid(«2«n) if the notation of Eq.~A8! is used. On average, this contribution correspondsthat of a system in the thermodynamic limit wherer~«! is acontinuous function and the magnitude of each contributis at most of the order of 1/N. To see this, let us assumer~«!is proportional to«2 ~41! for all values of« up to a cutoff ofthe order of 1. Such an approximation can be considerethe upper limit of possible values ofr~«! in Fermi liquidsystems, if one does not take into account the rapidly

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e

FORMULAS FOR ZERO-TEMPERATURE CONDUCTANCE . . . PHYSICAL REVIEW B 68, 035342 ~2003!

FIG. 17. The~a! real and~b! imaginary partsof the self-energy of an interacting system in ththermodynamic limit and for N5400 withf53p/4. The system is described in Fig. 5.

gluhe/thlsrm

a

oecofferge-itrg

th

onthth

d-atteth

o-ameofor

en

gy

e in

senen-

gy

useichan

ofemua-tesar-

heults

ys-uld

creasing tails at higher energies, which contribute a negible amount to the self-energy at the Fermi energy. Evaating the above integral, we find that contributions of tintervals close to the Fermi energy are of the order of 1N2

and contributions of the intervals near the cutoff are oforder of 1/N. Using an analogous procedure, we can aevaluate the derivative of the self-energy close to the Feenergy, with contributions

2E«

«1D r~«!

~v2«!2d«. ~A15!

In this case, also contributions corresponding to intervclose to the Fermi energy are of the order of 1/N. If r~«! fora finite N is used instead, there are large fluctuations abthe average value~see the discussion in the preceding stion! with the amplitude of fluctuations of the same ordermagnitude as the average value itself. To estimate the dience between the finite-system’s real part of the self-ene~or its derivative! close to the Fermi energy and the corrsponding quantity for a system in the thermodynamic limwe note that a sum ofN quantities, each of them of the ordeof 1/N with a standard deviation of the same order of manitude, has a standard deviation of the order ofN21/2, andtherefore, we can estimate that foruvu,D

S8~N,f;M ;v1 id!5S~01 id!1O~N21/2!, ~A16!

]S8~N,f;M ;v1 id!

]v Uv

5]S~v1 id!

]v Uv50

1O~N21/2!.~A17!

Note that we do not need to exclude the contribution ofinterval at the Fermi energy~the one corresponding toS9!from self-energies in the right-hand sides of these equatibecause the corresponding contributions are smallerN21/2 as discussed above. Also the errors arising fromfact that the right-hand sides are evaluated atv50 instead ofat v are only of the order of 1/N, as discussed in the preceing section. In Fig. 17 a comparison of the self-energiesfinite N and in the thermodynamic limit is presented. Nothat in the vicinity of the Fermi energy, the real parts of boself-energies coincide.

One can now proceed as in Eqs.~44! and ~47!, definingthe renormalization matrixZ8(N,f;M )5Z1O(N21/2) andthe quasiparticle HamiltonianH̃8(N,f;M )5H̃(N,f;M )

03534

li--

eoi

ls

ut-fr-y

,

-

e

s,ane

a

1O(N21/2) corresponding to the self-energyS8. As shownin the preceding section, the self-energies of an infinite twlead system and the corresponding ring system are the sand therefore, the renormalized matrix elementsH̃(N,f;M ) correspond to those of a two-lead system. Fuvu,D, Eq. ~A13! transforms into

det@v12H̃8~N,f;M !2S̃9~N,f;M ;v1 id!#50,~A18!

where the coupling to the remaining ‘‘bath’’ levels has berenormalized asS̃95Z81/2S9Z81/2. Let us for a moment ne-glect this term in Eq.~A18!. As the differenceDH̃ betweenHamiltoniansH̃(N,f;M ) and H̃8(N,f;M ) is small for alargeN, one is justified to relate their single-electron enerlevels using the first order perturbation formula

«̃8~N,f;M ;1!5 «̃~N,f;M ;1!1^N,f;M ;1uDH̃uN,f;M ;1&

5 «̃~N,f;M ;1!1O~N23/2!. ~A19!In the last step we made use of arguments similar to thosderiving Eq.~A4!.

The energy~A19! can acquire an additional shift becauof the couplingS9. To estimate this shift we first note that ithe worst case scenario, i.e., when there is a single bathergy level which coincides with the quasiparticle enerlevel ~A19!, the coupling matrix elementsVjn in Eq. ~A8!must be at most of the order ofN23/2 for Eq. ~41! to besatisfied in the thermodynamic limit. Then one can makeof the degenerate first order perturbation theory, whshows that the quasiparticle energy level is shifted byadditional amount of the order of 1/N2. This completes theproof of Eq.~A1!.

As a conclusion, in Fig. 18 we present a comparisonthe total densities of states for a finite ring interacting systwithin the second order perturbation theory and in the qsiparticle Hamiltonian approximation. Note that the stanear the Fermi energy are well described with the quasipticle approximation, while the states further away from tFermi energy are split in the interacting case. Similar reswere reported in Ref. 49.

APPENDIX B: VARIATIONAL GROUND-STATE ENERGY

In order to calculate the conductance for interacting stems, we first need to devise a robust method that wo

2-15

oprbebeero

sora

n

egit

cle

st

and-

etset.

ofx-h

s

hel-

thattingry

heria-nd-ingtral

rgy.s of

isluehein

tatetem.by

ns,

rttheto

any

theehe

g

Th

T. REJEC AND A. RAMŠAK PHYSICAL REVIEW B 68, 035342 ~2003!

allow us to efficiently calculate the ground-state energysuch systems. Note that we need a method that wouldvide us with the energy of a system with a very large numof sites in the ring. However, the number should stillfinite, i.e., we must not perform the calculations in the thmodynamic limit. We made use of the projection methodGunnarson and Schnhammer,50–52 introduced originallyto calculate the ground-state energy of the Anderimpurity model, and extended it to treat the more geneHamiltonian~1!.

Let us introduce an auxiliary noninteracting Hamiltonia

H̃5HL1ṼL1H̃C(0)1ṼR1HR , ~B1!

with arbitrary matrix elements describing the hopping btween the leads and the central region, and the central reitself. Note that these are the same matrix elements asones being renormalized in the Fermi liquid quasipartiHamiltonian~47!. Let us also define a Hilbert space spannby a set of 4M basis functions

uca&[Pau0̃&[)i PC

Pa ii u0̃&, ~B2!

whereM is the number of sites in the central region,u0̃& isthe ground state of the auxiliary Hamiltonian~B1! containingthe same number of electrons as there are in the groundof the original Hamiltonian, and

P0i 5~12ni↑!~12ni↓!, ~B3!

Psi 5nis~12ni s̄!, ~B4!

FIG. 18. ~a! The total density of states of an interacting rinsystem within the second-order perturbation theory.~b! Total den-sity of states corresponding to the quasiparticle Hamiltonian.system is described in Fig. 5.

03534

fo-r

-f

nl

,

-onheed

ate

P2i 5ni↑ni↓ ~B5!

are projection operators on unoccupied, singly occupieddoubly occupied sitei. The original Hamiltonian is diagonalized in the reduced basis set introduced above,

Hba5ESba , ~B6!

with Hba5^cbuHuca& being the matrix elements of thHamiltonian andSba5^cbuca& take into account the facthat the basis functions do not form an orthonormal basisThe eigenstate with the lowest energyEH̃ of this eigenvalueproblem is an approximation to the ground-state energythe original Hamiltonian. Varying the parameters of the auiliary Hamiltonian, one can find their optimal values whicminimize EH̃ . The solution of this minimization problem ithe final approximation to the ground-state energy.

Let us consider some simple limits of the problem. In tnoninteracting case whereU50, one can choose the auxiiary Hamiltonian to be equal to the true HamiltonianH̃5H. Then the wave functionuc&5(aPau0̃&5u0̃&5u0& isthe exact ground-state wave function of the system. Noteapplying the same wave function ansatz to the interaccase and allowing the matrix elements of the auxiliaHamiltonian to be renormalized, provides us with tHartree–Fock solution of the problem. Therefore, the vational method introduced above always gives the groustate energy which is lower or equal to the correspondHartree–Fock ground-state energy. In the limit of the cenregion being decoupled from the ring, i.e.,VL5VR50, thevariational method also yields the exact ground-state eneTo prove this statement, let us select the matrix elementH̃ in such a way that in its ground state there arem electronsin the central region. Then the basis set~B2! spans the fullHilbert space form electrons in the central region. As thereno coupling to the states in the ring, solving the eigenvaproblem~B6! provides us with the exact ground state of tproblem with a constraint of a fixed number of electronsthe central region. By varyingH̃, all the possible values ofmcan be tested and the one yielding the lowest ground-senergy corresponds to the correct ground state of the sys

The variational wave function ansatz can be improvedextending the Hilbert space with additional basis functiothe most promising candidates being of type52

ucbl j i sa&5PbV̂l j i sPau0̃&, ~B7!

where V̂l j i s5Vl j i cj s† dis1h.c. andl is a lead index, i.e.,

either L or R. On the other hand, as the size of the Hilbespace increases exponentially with the number of sites incentral region, it might be convenient to limit the basis setthe states obtained by projecting to the central region’s mbody states between which fluctuations are possible.

Finally, we state some technical details concerningevaluation ofHba andSba . It is convenient to express thesmatrix elements only in terms of quantities related to tcentral region and the neighboring sites in the leads. As

e

2-16

nel

il-o

ousieul

to

fw

ra-ans of

d in

arthee

thetheing

f thethe

FORMULAS FOR ZERO-TEMPERATURE CONDUCTANCE . . . PHYSICAL REVIEW B 68, 035342 ~2003!

Sba5^0̃uPbPau0̃&5^0̃uPau0̃&dba , ~B8!

the scalar products between the basis functions are evideexpressed with the central region quantities. The matrixments of the Hamiltonian can be expressed as

Hba5^0̃uPbHPau0̃&5^0̃uH̃PbPau0̃&1^0̃uPbHPau0̃&

2^0̃uH̃PbPau0̃&5ẼSba1^0̃uPb~VL1HC1VR!Pau0̃&

2^0̃u~ṼL1H̃C(0)1ṼR!Pau0̃&dba , ~B9!

where Ẽ is the ground-state energy of the auxiliary Hamtonian H̃. In the second and the third term we made usethe fact that lead HamiltoniansHL andHR commute with thecentral region projectors and therefore, they cancelAgain, we succeeded in expressing the matrix elementterms of central region quantities together with quantitrelated to the neighboring sites in the leads. Similar resare obtained if the extended basis set of Eq.~B7! is used. Thematrix elements in Eqs.~B8! and~B9! need to be calculatedin a noninteracting state. Therefore, we can make use ofWick’s theorem to decompose the expressions into twoperator averages of type^0̃udj s

† disu0̃&. As a huge number oterms is generated in this procedure, the decomposition

,

ev

y

03534

tlye-

f

t.insts

he-

as

performed automatically by symbolic manipulation of opetors. The ground-state energy of the auxiliary Hamiltoniand the two-operator averages can be expressed in termthe single-electron energies«̃k and wave functionsuw̃k& of H̃as

Ẽ52 (k occ.

«̃k , ~B10!

^0̃udj s† disu0̃&5 (

k occ.w̃ j

k* w̃ ik . ~B11!

The sums run only over the single-electron states occupiethe ground stateu0̃&. The eigenvalues«̃k were calculated in abasis in which the Hamiltonian matrix is banded, i.e., linecombinations of local basis functions corresponding toleft lead and right lead sites were introduced to ‘‘move’’ thhopping matrix elements in corners of the matrix close todiagonal. For each eigenvalue, only the components ofeigenvector related to the central region and neighborsites were calculated, again taking the special structure omatrix into account. The procedure used scales withnumber of sites in the ring asO(N2), which allows one totreat systems with up to 10000 sites in the ring.

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