Tunneling conductance of a mesoscopic ring with spin-orbit coupling and Tomonaga-Luttinger interaction

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Tunneling conductance of a mesoscopic ring with spin-orbit coupling and Tomonaga-Luttinger interaction

M. Pletyukhov,¹ V. Gritsev,^2,3and N. Pauget¹

1Institut für Theoretische Festkörperphysik, Universität Karlsruhe, D-76128 Karlsruhe, Germany

2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

3Département de Physique, Université de Fribourg, CH-1700 Fribourg, Switzerland

We study the tunneling current through a mesoscopic two-terminal ring with spin-orbit coupling, which is threaded by a magnetic flux. The electron-electron interaction in the ring is described in terms of a Tomonaga- Luttinger model which also allows us to account for a capacitive coupling between the ring and the gate electrode. In the regime of weak tunneling, we describe how, at temperatures lower than the mean level spacing, the peak positions of the conductance depend on magnetic flux, spin-orbit coupling strength, gate voltage, charging energy, and interaction parameters共charge and spin velocity and stiffness兲.

I. INTRODUCTION

Mesoscopic rings represent an important tool for experi- mental and theoretical studies of various phenomena which take place on a submicrometer scale. The ring geometry al- lows one to probe many interesting theoretical predictions.

One of the most exciting phenomena is the generation of geometric phases which are manifested in the interference patterns of wave packets propagating in the ring. Along with the well-known Aharonov-Bohm 共AB兲 effect¹ which takes place for both spinless and spinful particles, the generation of a spin-dependent phase is also possible. This effect, some- times called the Aharonov-Casher共AC兲effect,²may occur in the transport of electrons when they are subject to suffi- ciently strong spin-orbit共SO兲 coupling. The recent fabrica- tion of HgTe rings³made it possible to directly observe the AC phase. In earlier experiments with other compounds^4–6 the signatures of this effect have been also detected.

In order to probe the AC phase it is necessary to have a tool for manipulating the strength of the spin-orbit coupling.

This is provided by the gate-voltage dependence⁷ of the Rashba SO coupling,⁸which serves as a basis for a construc- tion of a spin field-effect transistor.⁹ Changing the magne- totransport properties of the ring in this way, the experimen- talists are now able to study the AC effect.^3,4

Usually the current through a mesoscopic noninteracting ballistic ring is described theoretically by means of the Landauer-Büttiker scattering matrix theory.¹⁰ Geometric phases arising due to both magnetic flux and SO coupling can be naturally incorporated in this formalism.^11–15 Effects of electron-electron interaction and charging energy are not taken into account in such a consideration. However, they might be important, for example, in small quasi-one- dimensional共quasi-1D兲rings or in arrays of such rings fab- ricated in very recent experiments.^4–6

In the present paper we calculate the linear tunneling con- ductance of the quasi-1D two-terminal ballistic ring with Rashba SO coupling threaded by a magnetic flux. The setup is schematically shown in Fig.1. The spectrum of electrons in the ring is SO-split into two subbands. We will assume electron densities at which only the lowest radial band is

occupied. The electron-electron interaction inside the ring is modeled by the parameters of the Tomonaga-Luttinger liquid 共TLL兲, the leads being noninteracting. Assuming a weak tun- neling between the leads and the ring, we compute the lead- ing term of the Kubo conductance perturbatively expanded in a series of tunneling elements. We mostly follow the ap- proach of Ref.16 where a similar problem for spinless fer- mions was considered. We also make use of the bosonization in order to calculate the required TLL correlation functions.

However, instead of the Matsubara formalism, we apply the Keldysh real-time approach to this quasiequilibrium problem 共cf. Ref.17兲. Such a combination of the Keldysh technique and bosonization appears more efficient for a derivation of asymptotic results at temperatures lower than the mean level spacing of the ring’s spectrum.

After Ref. 18it is known that an electron-electron inter- action strongly renormalizes the height of tunneling barriers between the leads and TLL, and therefore at T= 0 electron transport is suppressed. At finite temperaturesT⫽0 the lin- ear conductance vanishes as a power law of T, while the effective width of a conductance peak grows withT→0. In order to ensure the validity of the weak-tunneling approxi- mation, in our studies we assume a temperature range where the renormalized tunneling rates are smaller than the tem- perature, ⌫˜

l,rⰆT. On the other hand, finite-size effects re- main important at TⰆ␻0, the single-particle level spacing near the Fermi level.

In the temperature regime⌫˜

l,rⰆTⰆ␻0the linear conduc- tance is represented by a sequence of resonance peaks when

FIG. 1. The ring threaded by a magnetic flux ⌽ is weakly coupled to the leads through the tunneling barriers t_l and t_r and capacitively coupled to the gate electrode共V_g兲.

Published in "Physical Review B 74(4): 045301, 2006"

which should be cited to refer to this work.

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plotted as a function of gate voltage and/or magnetic flux. In our paper we focus on the problem of how the distribution of the conductance peaks depends on the external parameters 共magnetic flux, SO coupling, gate voltage, charging energy兲 as well as on the parameters of the Tomonaga-Luttinger in- teraction. The perturbative expansion of the linear conduc- tance in tunneling elements is known to break down in the resonance positions. Finding the poles of the leading term we can establish where the conductance peaks are centered.

Thus, the study of electron transport in the TLL ring provides an effective tool of spectroscopy of its many-body states.

Conceptually this is analogous to the study of the tunneling conductance between the two parallel quantum wires¹⁹ which has been realized experimentally.²⁰ We note that a description of a shape of a particular peak is, however, a different problem which is usually tackled in a somewhat different manner共cf., e.g., Refs. 21–23兲, and it will not be addressed in the present context.

In our paper we extensively discuss the importance of the so-called Klein factors and zero modes 共topological excita- tions兲of the bosonized Hamiltonian²⁴ for the description of distribution of conductance peaks. An accurate account of the Klein factors is necessary due to the presence of spin- orbit coupling in the system. The zero-mode sector of TLL decouples from its “continuous” 共bosonic兲 sector and con- tains the whole dependence on external parameters.^16,25 The latter appear in the topological sector after imposing bound- ary conditions. We elaborate on the procedure of averaging the conductance over zero modes in the presence of spin- orbit coupling and obtain analytically asymptotic results for the peak positions at temperatures lower than the mean level spacing. We also reexamine the case of spinless fermions reproducing the result of Ref. 16 and discuss it in further detail.

It is worthwhile to note that the relevance of the topologi- cal modes for a description of mesoscopic phenomena in the TLL rings has been already appreciated in various contexts, including studies of persistent²⁶and Josephson currents²⁵and the study of the AB phase in chiral Luttinger liquids.²⁷ The structure of the topological sector in the presence of SO cou- pling has been recently discussed as well in applications to persistent^28,29 and Josephson³⁰currents.

The paper is organized as follows. In Sec. II we briefly outline the construction of the spectrum of the ring with SO coupling. In Sec. III we summarize the results emerging from an application of the Landauer-Büttiker formalism to this system. They will be further used as a reference in the noninteracting limit. In Sec. IV we present a derivation of the Kubo formula in the real-time approach. Briefly reviewing the bosonization formalism in Sec. V, we derive an expres- sion for the dc conductance to be averaged over zero modes.

The procedure of averaging is performed in Sec. VI. We discuss the interplay of the externally tuned and interaction parameters in the distribution of the conductance peaks, es- pecially focusing on the modification of the Coulomb block- ade due to SO coupling.

II. MESOSCOPIC RINGS WITH RASHBA COUPLING:

DISPERSION RELATIONS

The two-dimensional electron gas with Rashba spin-orbit coupling is described by the Hamiltonian

H= 1

2m^*共p_x²+p_y²兲+␣R共␴xp_y−␴yp_x兲+V共r兲, 共1兲 wherer=

冑

x²+y². The magnetic fieldBis introduced in the kinetic momentum p→p+^e_cA via the gauge potential A

=^B₂共−y,x, 0兲. The radial potentialV共r兲 confining an electron to the ring geometry can be modeled, for example, either by singular isotropic harmonic oscillator or by concentric hard walls.²⁹ For these or any other types of the radial confine- ment the resulting quasi-one-dimensional spectrum␧n␴共k兲is labeled by the radial band indexn= 0 , 1 , . . ., by the angular momentum បk= . . . , −ប, 0 ,ប, . . ., and by the subband index 共chirality兲␴= ±. From now on we will putប= 1.

If the effective ring’s width is much smaller than the ring’s radius, we can neglect the hybridization between the radial bands. We also assume electron densities at which only the lowest radial band 共n= 0兲 is occupied. Thus, we effectively consider the strictly one-dimensional spectrum 共see Fig.2兲which has a parabolic shape and is SO-split into two subbands:²⁹

␧_␴共k兲 ⬅␧_0␴共k兲= 2␲²

m^*L²共k−k_⌽−␴^kR兲². 共2兲 HereLis the ring’s perimeter,k_⌽=⌽/⌽0is a number of flux quanta⌽0threading the ring, and the parameter

kR=

冑

¹4+

冉

^␣R2^m␲^*L

冊

²⁻¹2 共3兲 depends on the Rashba coupling␣R.

Linearizing the spectrum 共2兲 near the Fermi energy, we obtain the four branches

␧_␩␴共k兲=␻0共k−k_␩⁰_␴兲 ⬅␻0共k−␩^kF−k_⌽−␴^kR兲, 共4兲 specified by␩^{= ±}共or␩^=R,L兲and␴= ±. The Fermi angular velocity ␻0=

共

²_L^␲

兲

²_m^kF* defines the level spacing of the spec- trum共4兲, andkFis the Fermi angular momentum in absence of a magnetic field and SO coupling.

III. CONDUCTANCE OF THE MESOSCOPIC RING:

NONINTERACTING ELECTRONS

Let us consider the conductance of the ring attached to the semi-infinite leads 共Fig. 1兲. For noninteracting electrons it can be easily found in the framework of the scattering matrix theory.¹⁰

It is instructive to consider first the case of spinless fer- mions with the two linearization pointsk_R/L⁰ . One finds that in FIG. 2. The lowest radial band of the quasi-1D mesoscopic ring SO-split into two subbands.

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the zero-temperature limit and for the angle␲between the junctions to the leads the dc conductance reads¹⁰ G共k_F,k_⌽兲= e²

2␲

16⑀l⑀rsin²k_F␲^cos2k_⌽␲

关− 2␣l␣r+共1 +␥l␥r兲cos 2␲^kF− 2␤l␤rcos 2␲^k_⌽兴²+共1 −␥l␥r兲²sin²2␲^kF

, 共5兲

where ⑀l/r, ␥l/r= −

冑

1 − 2⑀l/r, ␣l/r= −¹₂共1 +␥l/r兲, and ␤l/r=¹₂共1

−␥l/r兲are the phenomenological parameters describing scat- tering in a T-shaped共leftlor rightr兲junction. The number of flux quanta is given by k_⌽=¹₂共kR0−k_L⁰兲, while the quantity kF=¹₂共kR

0+k_L⁰兲 corresponds to the Fermi momentum at zero flux. It can be replaced bykF→N₀+^⌬␮_␻

0, where⌬␮^{is a dif-} ference between the chemical potential of the leads and the Fermi energy of the ring, and the integerN₀is related to the number 2N₀+ 1 of electrons in the ring at⌬␮= 0. Since the expression共5兲is periodic inkF, the integer part ofkFcan be discarded. Thus, the conductance共5兲actually depends on the fractional part of ^⌬␮_␻

0. For future references we introduce the parameterk_␮=^⌬␮_␻

0−¹₂.

In the weak-tunneling limit ⑀l/rⰆ1 the conductance 共5兲 approximately equals

G⬇ e² 2␲

4⑀l⑀rsin²k_F␲^cos2k_⌽␲ 共cos 2␲^kF− cos 2␲^k_⌽兲²+ 1

4共⑀l+⑀r兲²sin²2␲^kF

.

共6兲 As a functionk_␮ andk_⌽, it represents a sequence of Breit- Wigner resonances. The conductance peaks occur when the resonance condition cos 2␲^kF= cos 2␲^k_⌽is fulfilled—i.e., at the values of the parameters

k_F+k_⌽=n_R, k_F−k_⌽=n_L, 共7兲 wherenR andnL are arbitrary integers. We note that in the weak-tunneling limit the resonance condition 共7兲 remains valid for arbitrary anglexl−xr between the junctions, while the shape of Breit-Wigner resonances is quite sensitive to the value ofxl−xr.

It has been demonstrated in Ref.12that for electrons with nonzero SO coupling and negligible Zeeman splitting the conductance of the mesoscopic ring is given by the sum of the two contributions: G共k_F,k_⌽+k_R兲 and G共k_F,k_⌽−k_R兲. In other words, the net effect of the SO coupling for noninter- acting electrons is the generation of the different effective flux values for the different channels. Therefore, the pattern of the conductance maxima atkR⫽0 is determined by the resonance conditions

k_F+共k_⌽±k_R兲=n_R±, k_F−共k_⌽±k_R兲=n_L±, 共8兲 where n_R± and n_L± are arbitrary integers. Recalling that effectively k_␮=k_F−₂¹, we show in Fig. 3 how the arrange- ment of the conductance peaks is modified by SO coupling.

IV. KUBO FORMULA

In order to take into account effects of the electron- electron interaction on the distribution of conductance peaks, we discuss in this section the Kubo formula for the linear conductance. Although this expression is very standard, we rederive it using the Keldysh formalism. In doing this, we pursue two objectives. First, we would like to have better control of the approximations used共similar to those made in Ref.16兲. Second, we would like to deduce an expression for the conductance in a real-time representation. Its advantage for the ring geometry will be discussed in the next section where the calculation of time-dependent finite-size TLL cor- relation functions is concerned.

In the second-quantized formulation the mesoscopic ring attached to the leads is described by the Hamiltonian

H=Hl+Hr+Hc+HT. 共9兲 The left/right lead is described by a Fermi-liquid Hamil- tonianH_l/r=兰dxc_l/r^†共x兲

共

_2m^p2*−␮

兲

c_l/r共x兲, and the tunneling term is HT=兺l,r关tl/rc_l/r^†共xl/r兲␺共xl/r兲+ H.c.兴. Here c_l/r and ␺ ^{are the} field operators in the leads and in the ring, respectively.

The Hamiltonian of the central part共ring兲 Hc关␺^†,␺兴 can have any interaction term in addition to the kinetic term. In our consideration we will model the electron-electron inter- action in the ring by the Tomonaga-Luttinger liquid which includes only forward-scattering processes 共“density- density”-type interaction兲. In the framework of this model it is also possible to take into account the charging effects.

They originate from a capacitive coupling of the ring to the gate electrode and are described by the Hamiltonian E_c

共

Nˆ

ring−¹_eC_gV_g

兲

², with the charging energy E_c=e²/ 2C_g. HereCg is the gate capacitance andNˆ

ringis the number op- erator of electrons in the ring.

The linear response of the system to an applied time- dependent bias voltage is described by the Kubo formula for the ac conductance:³¹

FIG. 3. Splitting of the conductance peaks共solid lines兲 due to SO coupling. The dashed lines correspond tok_R= 0.

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G共⍀兲= − 1

⍀

冕

_−⬁^{t dt⬘e−i⍀共t⬘−t兲具关Iˆl共t兲,Iˆr共t⬘兲兴典, 共10兲}

where Iˆ_l/r共t兲=ie关tl/rcˆ_l/r^† 共xl/r,t兲␺^ˆ共xl/r,t兲− H.c.兴 is a current op- erator in the left/right junction written in the Heisenberg rep- resentation.

In the weak-tunneling limit the expression 共10兲 can be expanded in a series of HT. We make use of the real-time Keldysh diagrammatic technique, and fort⬎t⬘ ^{we replace} 具关Iˆl共t兲,Iˆr共t⬘^兲兴典by

具TtIˆl共t兲Iˆr共t⬘兲典−具T˜

tIˆl共t兲Iˆr共t⬘兲典 ⬅2iIm具TtIˆl共t兲Iˆr共t⬘兲典. 共11兲 When expressed on the Keldysh contour, it reads

具TtIˆl共t兲,Iˆr共t⬘^兲典=

冓

^˜Ttexp

^冉

ⁱ

^冕

^{−⬁t HT共t⬙兲dt⬙}

^冊

^Il共t兲

⫻Ttexp

冉

⁻ⁱ

^冕

⁻⬁ t

HT共t⬙兲dt⬙

冊

⫻T˜

texp

冉

ⁱ

^冕

⁻⬁ t⬙

HT共t⬙兲dt⬙

冊

^Ir共t⬘兲

⫻T_texp

冉

⁻ⁱ

^冕

−⬁

t⬘

H_T共t⬙兲dt⬙

冊 冔

^{, 共12兲}

where the operators without carets refer to the interaction 共H_T兲representation.

Expanding共11兲to the second order inH_T, we obtain

冕

−⬁ t

dt₁

冕

−⬁ t⬘

dt₁⬘具关关Il共t兲,HT共t1兲兴,关HT共t1⬘兲,Ir共t⬘兲兴兴典

−

冕

_−⬁^{t dt1}

冕

_−⬁^{t1 dt2具关关关Il共t兲,HT共t1兲兴,HT共t2兲兴,Ir共t⬘兲兴典}

−

冕

_−⬁^t⬘dt1⬘

冕

_−⬁^{t1⬘dt2⬘具关Il共t兲,关HT共t2⬘兲,关HT共t1⬘兲,Ir共t⬘兲兴兴兴典.}

The next step is to perform averaging over the leads’

states. While doing this, we meet the following combina- tions: 共a兲 共Gl

R−G_l^A兲共Gr

R−G_r^A兲, 共b兲 G_l^K共Gr

R−G_r^A兲, 共c兲 共Gl R

−G_l^A兲Gr

K, and 共d兲 G_l^KG_r^K. Here G_l/r^R,A and G_l/r^K are the momentum-averaged retarded, advanced, and Keldysh func- tions of the leads in the real-time representation

共G^R−G^A兲l/r共t兲= −i具兵c_l/r共t兲,c_l/r^† 共0兲其典= − 2␲ⁱ␦共t兲␯l/r

V_l/r,

G_l/r^K共t兲= −i具关cl/r共t兲,c_l/r^† 共0兲兴典= − 2␲

␤sinh共␲^t/␤兲

␯l/r

V_l/r, and␯l/r is the density of states in the left/right lead at the Fermi level.

One can straightforwardly prove that the combinations共a兲 and 共b兲 vanish identically. The combination 共c兲 gives the following contribution to the conductance:

G^共c兲共⍀兲=e²⌫l⌫rL²

冕

_−⬁⁰

冕

_−⬁^{0 dt1dt2}2i⍀^e−i⍀␤^{t1共1 −}sinh关^e␲^−it⍀2/^t␤^2兲兴

⫻Re具兵关关␺l共0兲,␺l†共0兲兴,␺r共t₁兲兴,␺r†共t₁+t₂兲其典, 共13兲 where⌫l/r= 2␲␯l/r兩tl/r兩²/共Vl/rL兲 andV_l/r is the volume of the left/right lead.

From Eq.共13兲we derive an expression for the dc conduc- tance共⍀= 0兲at zero temperature:

G^共c兲= e²

2␲^⌫l⌫rL2

冕

_−⬁⁰

冕

_−⬁^{0 dt1dt2}

⫻Re具兵关关␺l共0兲,␺l

†共0兲兴,␺r共t1兲兴,␺r

†共t1+t₂兲其典. 共14兲 Using the operator identities

兵关C,A兴,B其+兵关C,B兴,A其=关C,兵A,B其兴, 共15兲 兵C,兵A,B其其−兵A,兵C,B其其=关关C,A兴,B兴, 共16兲 we rewrite Eq.共14兲,

G^共c兲= e²

4␲^⌫l⌫rL2

冕

_−⬁⁰

冕

_−⬁^{0 dt1dt2}

⫻兵Re具兵关关␺l共0兲,␺l

†共0兲兴,␺r共t1兲兴,␺r

†共t2兲其典 + Re具关关␺l共0兲,␺l

†共0兲兴,兵␺r共t1兲,␺r

†共t2兲其兴典其, 共17兲 and further express

具兵关关␺l共0兲,␺l

†共0兲兴,␺r共t1兲兴,␺r

†共t2兲其典

=具兵兵␺l共0兲,␺r

†共t2兲其,兵␺r共t1兲,␺l

†共0兲其其典 +具关关␺r

†共t2兲,兵␺l

†共0兲,␺r共t1兲其兴,␺l共0兲兴典

−具兵兵␺l†共0兲,兵␺l共0兲,␺r共t₁兲其其,␺r†共t₂兲其典. 共18兲 It is obvious that in the noninteracting limit the only term 具兵兵␺l共0兲,␺r

†共t2兲其,兵␺r共t1兲,␺l

†共0兲其其典 survives, since the other terms vanish due to the fermionic commutation relations. We approximate the dc conductance in the interacting case by this dominant contribution:

G⬇ e²

4␲^⌫l⌫rL2

冕

_−⬁⁰

冕

_−⬁^{0 dt1dt2}

⫻Re具兵兵␺l共0兲,␺r

†共t2兲其,兵␺r共t1兲,␺l

†共0兲其其典. 共19兲 Splitting the four-particle correlator, one can recover the for- mula G⬇共e²/ 2␲兲⌫l⌫rL²兩G^R共␻^{= 0 ,x}l−x_r兲兩² from Ref. 16, whereG^R共␻^{= 0 ,x}l−xr兲 is a zero-frequency retarded Green’s function for interacting electrons in the ring. This approxi- mation physically means that one scattering event is com- pleted before another takes place. In general, the TLL corre- lation functions of any order can be calculated within the bosonization approach, and this approximation can be re- laxed.

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The combination共d兲withG_l^KG_r^Kalso gives a finite contri- bution to the conductance, which, however, vanishes in the noninteracting limit as well. Therefore, we will neglect it on the same ground as we have just neglected the subdominant terms in Eqs.共17兲and共18兲.

V. BOSONIZATION A. Spinless case

In order to compute the four-particle correlator共19兲, we will make use of the bosonization technique.²⁴

Let us consider for simplicity the spinless case. We intro- duce the shorthand notations for the fermionic fields ␺l

⬅␺共xl兲and␺r⬅␺共xr兲, wherexlandxrare the angle coordi- nates of the left and right junctions. In the following we assume thatxl= 0 and xr=␲^.

In the bosonization the fields␺l/rare represented as a sum of the right-共␩^{= +, orR}兲and left-共␩^{= −, orL}兲moving com- ponents,

␺l/r=␺l/r,R+␺l/r,L=F_l/r,R␺_l/r,R^b +F_l/r,L␺_l/r,L^b , 共20兲 and each of ␺l/r,␩ consists of a topological part F_l/r,␩ and a bosonic part␺_l/r,^b _␩commuting with each other:关F,␺^b兴= 0.

The bosonic part is given by

␺_␩^b共x兲= 1

冑

L␣^˜e

−i冑2␲␾_␩共x兲,

␾_␩共x兲=i

兺

k=1

⬁ e^{−␣˜ k/2}

冑

2␲^k共e

i␩^kxb_␩k−e^−i␩kxb_␩^†_k兲, 共21兲 where␣^˜=2␲␣L is a small dimensionless cutoff parameter and the operatorsb_␩k,b_␩^†_ksatisfy the bosonic commutation rela- tion关b_␩k,b_␩^†_⬘k⬘兴=␦_␩␩⬘␦kk⬘.

The topological part is important for the finite-size TLL with periodic boundary conditions. It includes Klein factors F_␩, zero-mode operatorsN_␩, and the linearization pointsk_␩⁰ 共see Fig.2 assumingk_R= 0兲:

F_l/r,␩=F_␩e^{i␩共N␩−k␩0兲xl/r}. 共22兲 The zero-mode operatorsN_␩=N_␩^† take integer values, and the following relations are satisfied:²⁴

关F_␩,N_␩⬘兴=F_␩␦_␩␩⬘^, 共23兲 兵F_␩,F_␩^†_⬘其= 2␦_␩␩⬘^, 共24兲 兵F_␩,F_␩⬘其=兵F_␩^†,F_␩^†_⬘其= 0 for␩⫽␩⬘^. 共25兲 The bosonized TLL HamiltonianHTLL⬅Hc=Hb+H₀con- sists of a “continuous” 共bosonic兲 Hb part and a topological H₀part which are decoupled from each other. Therefore, the factorization of␺l/r,␩ intoF_l/r,␩ and␺_l/r,^b _␩takes place at any time instant:

␺l/r共t兲=F_l/r,R共t兲␺l/r,R

b 共t兲+F_l/r,L共t兲␺l/r,L

b 共t兲, 共26兲 where the time evolutions of ␺_l/r,^b _␩共t兲 and F_l/r,␩共t兲 are gov- erned by Hb and H₀, respectively. By the same reason the

statistical averagings in both bosonic and topological sectors are independent of each other.

The bosonic part of the TLL Hamiltonian is given by Hb=2␲^v

L

兺

a=1,2

兺

k=1

⬁

kd_ak^†dak, 共27兲 wherevis the so-called charge velocity共the renormalization of the Fermi velocity v₀⬅^L_2␲^␻0兲. The operators dak, d_ak^† 共a

= 1 , 2兲 are obtained fromb_␩k,b_␩^†_kby the canonical transfor- mation共A1兲. The latter depends on the interaction parameter

␥⁼¹₂共_K¹+K兲, where K is the so-called charge stiffness. For repulsive interactionsK⬍1, while in the noninteracting limit K=␥^{= 1 and v=v}0.

The topological part of the TLL Hamiltonian is

H₀=

兺

_␩ ^{共a0N˜␩2+a1N˜␩N˜−␩兲, 共28兲}

wherea_0,1=^␻₄⁰共^˜␯^±␭兲and

␯

˜=␯^+4Ec

␻0

, ␯^{= v}

Kv₀, ␭=vK

v₀. 共29兲 The topological numbers N˜_␩=N_␩−k_␩ are shifted by k_␩=k_␩⁰ +␦^k␮, where

␦^k␮=

4E_c

冉

^{1eCgVg− 2N0}

冊

^{+ 2⌬␮−␻0}

2^˜␯␻0

共30兲 redefines the linearization pointsk_␩⁰ in order to include the dependence on ⌬␮ and the gate voltage V_g. In the basisN

=NR+NL,J=NR−NL, the HamiltonianH₀acquires the diag- onal form

H₀=␻0

4 关^˜N␯^˜2+␭˜J²兴, 共31兲 whereN˜=N− 2k_␮,˜J=J− 2k_⌽, andk_␮=N₀+␦^k␮. One can ob- serve that the whole dependence on ⌬␮, Vg, and ⌽ is in- cluded in the topological sector.

Using the commutation relations 共23兲 we find the time evolution of the Klein factors:

F_␩共t兲=e^iH0tF_␩e^−iH0t=F_␩e^{−itP␩+ita0}, 共32兲 where

P_␩= 2a₀N˜

␩+ 2a₁N˜

−␩=␻0

2 关^˜N␯^{˜ ±}␭J˜兴. 共33兲 The details of the time evolution of the bosonic fields are presented in Appendix A. In fact, they are not very important for our purpose. We will only exploit the fact that the aver- age of the bosonic fields,

g^b共t;␥兲=具␺lR b共t兲␺rR

b†共0兲典 ⬅ 具␺lL b共t兲␺rL

b†共0兲典, 共34兲 is a periodic function of time which can be expanded in a Fourier series,

http://doc.rero.ch

g^b共t;␥兲=

兺

p=0

⬁

gp共␥兲e^−ip␻t, 共35兲 with frequency ␻^=2␲L^v and real-valued coefficients gp共␥兲.

Note that the summation in Eq.共35兲is performed only over non-negative integers.

The real-time periodicity ofg^b共t;␥兲 is inherited from the spatial periodic boundary conditions. The occurrence of the Fourier series 共35兲 allows us to perform all time integrals explicitly. The analysis of the remaining series is a much simpler task.

Let us make yet another approximation in the spirit of Ref.16. In particular, we split the four-particle bosonic cor- relator in 共19兲, neglecting the anomalous averages 共e.g., 具␺^b␺^b典兲, the left-right mixing共e.g.,具␺L

b␺Rb†典兲, and the vertex corrections共averages of operators at the same spatial point, e.g.,具␺r

b␺r

b†典兲in thebosonic共continuous兲sector. At the same time, we do not split the topological part of the four-particle correlator共unlike has been done in Ref. 16兲 and perform a single averaging of the whole over zero modes.

Implementing this procedure, we obtain 具␺l共0兲␺r†共t2兲␺r共t1兲␺l†共0兲典

⬇g^b*共t₂兲g^b共t₁兲

⫻_␩

兺

1,␩2

具Fl,␩2共0兲F_r,␩

2

† 共t2兲Fr,␩1共t1兲F_l,␩

1

† 共0兲典z.m., 共36兲 where具¯典z.m.implies averaging over zero modes to be dis- cussed later. Collecting all contributions, we find

G⬇ e²

2␲^⌫l⌫rL2_p

兺

1,p₂=0

⬁

gp₁共␥兲gp₂共␥兲

冕

_−⬁⁰

冕

_−⬁^{0 dt1dt2}

⫻_␩

兺

=±

Re具共e^{it1共␻p1+a0+P␩兲}−e^{−it1共␻p1+a0−P␩兲}兲

⫻共e^{−it2共␻p2+a0+P␩兲}−e^{it2共␻p2+a0−P␩兲}兲

+e^{i共N␩+N−␩兲␲}共e^{i␩t1共␻p1+a0+P␩兲}−e^{−i␩t1共␻p1+a0−P␩兲}兲

⫻共e^{−i␩t2共␻p2+a0+P−␩兲}−e^{i␩t2共␻p2+a0−P−␩兲}兲典z.m.. 共37兲 Introducing

A_␩^±=

兺

p=0

⬁ gp共␥兲

␻^p+a0±P_␩ 共38兲 andA_␩=A_␩⁺+A_␩⁻, we can cast Eq.共37兲into the form

G⬇ e²

2␲^{⌫l⌫rL2具AR2+AL2+ 2ARALcos共N˜R+N˜L+ 2}␦^k␮兲␲典z.m.. 共39兲 We remark that the alteration of the anglexl−xrbetween the junctions would only modify the Fourier coefficientsgp共␥兲in Eq.共38兲as well as the relative phase of the interference term

⬃A_RA_Lin Eq.共39兲. Meanwhile, the poles ofA_␩^± in Eq.共38兲 are not sensitive to the value ofxl−xr.

In order to treat further the expression 共39兲 we need to establish an efficient procedure of averaging over zero modes. But first we are going to discuss the modification of the conductance共39兲caused by the presence of spin degrees of freedom and by spin-orbit coupling.

B. Spinful case

Performing a similar bosonization procedure in the spin- ful case, we obtain the following expression for the dc con- ductance:

G⬇ e²

2␲^⌫l⌫rL2_␴=±

兺

^{具AR2␴+AL2␴}

+ 2AR␴AL␴cos共N˜

R␴+N˜

L␴+ 2␦^k␮兲␲典z.m.. 共40兲 The zero-mode operators N_␩␴ with integer eigenvalues are shifted toN˜_␩

␴=N_␩␴−k_␩␴ byk_␩␴=k_␩⁰_␴+␦^k␮, where

␦^k␮=

4E_c

冉

^{1eCgVg− 4N0}

冊

^{+ 2⌬␮−␻0}

2␯^˜c␻0

. 共41兲 The integerN₀=¹₄兺_␩,␴k_␩⁰_␴is related to the number 4N₀+ 2 of electrons in the ring when the parameters

k_⌽=1 4

兺

␴ 共kR␴−kL␴兲, 共42兲

k_B,R=1

4

兺

_␴ ^{␴共kR␴±kL␴兲 共43兲}

equal zero. The parameter kB vanishes in the absence of a Zeeman interaction. The parameter

k_␮=1

4

兺

_␩,␴^{k␩␴=N0+␦k␮ 共44兲}

contains the dependence on⌬␮andVg.

Like in the spinless case, it is convenient to introduce

␯

˜_c=␯c+8Ec

␻0

, ␯c,s= v_c,s

K_c,sv₀, ␭c,s=v_c,sK_c,s

v₀ , 共45兲 and ␻c,s=^2␲v_L^c,s, and ␥c,s=¹₂

共

K¹_c,s+K_c,s

兲

, which are expressed through the charge and spin velocitiesvc⫽vs, the charge and spin stiffnessesKc⫽Ks, and the charging energyEc.

In Eq.共40兲the rates⌫l and⌫rremain the same as in the spinless case, since we assume that the density of states in the leads is spin independent and equals ␯l/r for each spin component. The spin dependence appears in the functions A_␩␴=A_␩⁺_␴+A_␩⁻_␴,

A_␩^±_␴=_p

兺

c,p_s=0

⬁ gp_c

冉

¹2␥c

冊

^gps

冉

¹2␥s

冊

␻cp_c+␻sp_s+¯a₀±P_␩␴, 共46兲