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Rectification in Y-junctions of Luttinger liquid wires Chenjie Wang and D. E. Feldman Department of Physics, Brown University, Providence, Rhode Island 02912, USA (Dated: January 24, 2011) We investigate rectification of a low-frequency ac bias in Y-junctions of one-channel Luttinger liquid wires with repulsive electron interaction. Rectification emerges due to three scatterers in the wires. We find that it is possible to achieve a higher rectification current in a Y-junction than in a single wire with an asymmetric scatterer at the same interaction strength and voltage bias. 1 The rectification effect is the strongest in the absence of the time-reversal symmetry. In that case, 1 the maximal rectification current can be comparable with the total current ∼ e2V/h even for low 0 voltages, weak scatterers and modest interaction strength. In a certain range of low voltages, the 2 rectification current can grow as thevoltage decreases. This leads to a bump in the I-V curve. n a PACSnumbers: 73.63.Nm,73.40.Ei,71.10.Pm J 1 2 I. INTRODUCTION tries. Specifically, we consider a setup of the type shown in Fig. 1. We assume that one of the three terminals is ] kept at zero voltage. ac voltages with amplitudes V and l Recently there was much interest in rectification in al nanoscale systems.1–22 Nonlinear mesoscopic transport γV, γ ∼ 1, are applied to the remaining two terminals. h exhibitsmuchinterestingphysicssuchasapeculiarmag- Inageneralcase,adc-currentisgeneratedineachofthe - netic field dependence of the current16,17 and negative three wires in the junction, Fig. 1. The three currents s are different but can be computed in a similar way. We e differential resistance for the rectification current at low m voltages.18,19 Anothermotivationfortheinvestigationof calculate such rectification dc currents for various types . mesoscopic diode or ratchet23 effect comes from possi- of Y-junctions. We focus on the limit of a low-frequency t ac voltage bias. In order to determine the amplitude of a ble practical applications in nanoelectronics and energy m conversion. Following the pioneering paper by Christen the rectificationeffectinthatlimit, itis sufficientto find - and Bu¨ttiker4 most attention has focused on a simpler the difference of the dc currents at the opposite dc bias d voltages,i.e.,comparethecurrentwhenthepotentialsat case of a Fermi-liquid conductor. At the same time, the n the terminals are time-independent and equal 0, V and diodeeffectrequiresacombinationofspacialasymmetry o γV with the current when the potentials are 0, V and andstrongelectroninteractionsintheconductor. Hence, c − γV (cf. Refs. 18,19). This corresponds to a dc cur- [ onemayexpectastrongerratchetcurrentinstronglyin- − rent generated by low-frequency square voltage waves. teracting Luttinger liquid systems. This expectation has 3 We find that in some classes of Y-junctions the rectifica- been confirmed by a recent study of transport asymme- v triesinone-channelquantumwires.18–20Refs.18–20have tioncurrentishigherthaninalinearwirewiththe same 2 strengthofthe repulsiveelectroninteractionatthe same 5 focusedonaone-channelLuttingerliquidinalinearcon- voltagebias. Inacertainintervaloflowvoltagestherec- 0 ductor in the presence of a single asymmetric scatterer. 5 This is the conceptually simplest situation giving rise to tificationcurrentexhibitsapower-lawdependenceonthe 9. rectification. At the same time, changing geometry may bias: I ∼Vz. The exponent z can be negative. Clearly, suchdependencewithanegativeexponentcannotextend 0 increase asymmetry and hence the rectification current. all the way to zero voltage as I = 0 at V = 0. Hence, 0 Inthispaperweconsideranasymmetricsetupbasedona 1 Y-junctionofthreequantumwireswiththreeimpurities. the rectification currentreachesa maximum at a certain v: We show that a stronger diode effect can be achieved in voltage. Wedemonstratethatinjunctionswithouttime- reversalsymmetry,the maximalrectificationcurrentcan i such system than in a linear Luttinger liquid and recti- X be comparable with the total current e2V/h even for fication is possible even in the case of symmetric point ∼ r scatterers. lowvoltagesandmodestinteractionstrengthinthewires. a In particular, such rectification current can be achieved We focus on the simplest one-channel Y-junctions. in the “island setup”, illustrated in Fig. 1, which can be More complicated Luttinger liquid junctions, such as Y- experimentally realized in quantum Hall systems35. For junctions of single-wallcarbon nanotubes, are also of in- comparison,inalinearwire,therectificationcurrent18,19 terest. In particular, it might be easier to make such is alwaysmuchlowerthane2V/h. The diode effect inY- junctions in a reproducible way. junctionsisnotasstronginthepresenceoftime-reversal Y-junctions are among basic elements of electric cir- symmetry as in its absence. Still, the maximal rectifica- cuits, however, a theoretical investigation of Luttinger tion current is higher than in a linear conductor. liquid Y-junctions24–43 has begun only during the last decade. By now, there is a good understanding of linear The paper is organized as follows: first, after a brief conductance near various fixed points as well as tunnel- qualitative discussion, we describe our setup and for- ing density of states.33,38 In this paper we extend the mulate a model. We then review the properties of Y- previous research to the problem of transport asymme- junctions and derive a general expression for the recti- 2 inthenoninteracting-electronsapproximationinRef.12. At a high voltage bias a strong rectification effect was B found. In this paper, on the other hand, we focus on 2 C the low-voltage regime and, in particular, the universal behavior at low bias. In that limit the density-driven A rectificationeffectisoflittle importanceasitresultsina ±V 1 small rectification current V2 in a noninteracting sys- F ∼ tem. Besides, the density-driven rectification requires a specificwaytoapplybias. Ifweareinterested,forexam- 3D ple, in the transformation of incoming electromagnetic E radiation into a dc current then clearly the potential os- cillates in all terminals. Thus, in this paper we focus on therectificationmechanismduetogeometricasymmetry. Y-junctions have a “built-in” geometric asymmetry. FIG. 1: Schematic picture of a Y-junction of quantum wires Indeed, let us focus on the current I in one of the three 1 with threeimpurities. Voltagebias±V isappliedtothefirst wiresconnectedbythejunction. ThecurrentI =I +I 1 2 3 wire. Wires2and3areconnectedtotheground. Wecalculate equals the sum of the currents in the other two wires. the dc current in wire 1. One can view AB, CD and EF as Thus, the same current enters the junction through two edgesofan integerquantumHallliquid. Thiscorrespondsto an “island junction”.35 wires and leaves through one only. This means that the two sides of the junction are not equivalent since they correspond to 1 and 2 wires. Thus, we may expect that geometric mechanism of rectification applies to any Y- fication current. Next, we apply the general formalism junction. However, as we demonstrate below, the geo- to the model of three weakly connected wires. Then we metric mechanism only works, if electron interaction is determine the leading contributions to the rectification present or time-reversal symmetry is broken. current at different interaction strengths and matching conditions at the junction. We discover several regimes Indeed, let us consider a time-reversal invariant sys- with different voltage dependences of the current. Fi- temofnon-interactingspinlessfermionsinanN-terminal nally, we discuss how to build a junction with the maxi- mesoscopic junction. We calculate the current between mal rectification current. terminals 1,...,K and K +1,...,N. We compare the currentinthesituationwhenterminals1,...,K arekept at the voltage V and terminals K+1,...,N are kept at II. RECTIFICATION IN MESOSCOPIC zero bias (case 1) with the current in the situation when CONDUCTORS terminals 1,...,K are kept at zero bias and terminals K+1,...,N arekeptatthevoltageV (case2). Fornon- interactingfermionsthecurrentreducestothesumofthe We considera junction of N one-channelwires. In the singleparticlecontributionscorrespondingtoeachenergy subsequentsectionswewillspecializetothecaseofN =3 inthewindow0<E <V. Wethuscomparesuchcontri- and assume that the system is spin-polarized or, equiv- butions for two opposite voltage biases. The wave func- alently, that charge carriers are spinless fermions. In all tionofanelectronincomingfromterminallwiththemo- cases we assume that long range Coulomb interaction in the conductoris screenedby the gates. Insuch situation mentum k is ψk,l = exp(−ikxl)+ Nm=1Slmexp(ikxm), the applied voltage bias affects not only the current but where xm > 0 is the coordinatPe in wire number m also the charge density in the system. and Slm is a unitary scattering matrix. The above no- Rectificationispossible,ifachangeofthe voltagebias tations for the wave function imply that the probabil- sign results in a change of not only the sign but also ity to find an electron with the wave function ψk,l in absolute value of the current. The latter obviously re- wire number m = l is determined by the outgoing wave quires left-right asymmetry in the system. Asymmetry and is proportio6nal to Slm 2. For l = m, the proba- | | mayhavetwoorigins: asymmetrycausedbythewayhow bility is determined by both the incoming and outgo- thebiasisappliedandgeometricasymmetrydueto,e.g., ing waves. The time-reversal symmetry implies that an asymmetric scatterer.18,19 In particular, one can con- ψk∗,l = exp(ikxl)+ Nm=1Sl∗mexp(−ikxm) is also a so- sider a situation in which several terminals are kept at lutionofthe SchrödiPngerequation. Thesolutionismade zero electric potential and the potential of the other ter- of N incoming waves and one outgoing wave. It can be minals changes between +V and V. The charge den- represented as a linear combination of waves ψk,m with − sity is different for the opposite voltage signs. This in differentm. Hence,the outgoingwavesatisfiesthe equa- turn affects the current and leads to rectification. Such tion exp(ikx ) = exp(ikx ) N S∗ S . The absence l l m=1 lm ml density-drivenrectificationispossibleinabroadrangeof of outgoing waves in the chPannels with numbers n = l situationsincludingsystemswithoutelectroninteraction implies that S∗ S = 0. Hence, S∗ = S−1 =6S†, m lm mn in the presence of time-reversal symmetry. It has been i.e., S is a syPmmetric matrix. Now we can compare the investigatedinthecontextofacarbonnanotubejunction contributions to the current from particles with the en- 3 (a) (b) (a) (b) 0 V ρ→(x) ρ←(x) V V V V 0 0 Contact U1 U2 Contact Contact U1 U2 Contact FIG.2: Aschematicpictureofalinearwirewithtwoscatter- 0 V ersofunequalstrength. Chargedensity,averagedoverFiedel (c) (d) 0 V oscillations, follows a“staircase” profile. Thedirectionofthe staircase depends on the voltage sign as seen from figures 2a B B and 2b. V V 0 0 B B ergy E = ~2k2/2m for two opposite signs of the bias. Electrons,incoming from different terminals, are not co- 0 V herent. Hence, in case 1, I K N S 2. 1 ∼ l=1 m=K+1| lm| In case 2, I K N SP2. TPhe currents are 2 ∼ l=1 m=K+1| ml| FIG.3: Charge transport through theboxexhibitsnoasym- equal from the PsymmPetry of the scattering matrix and metries at zero magnetic field, Figs. 3a,b, and is asymmetric hence the diode effect is absent. at a finitefield in shaded areas, Figs. 3c,d. In the presence of electron interaction, the rectification effect becomes possible. This can be understood already from a simple model with two point scatterers of unequal strength in a linear wire (Fig. 2). Similar We assume no electron interactions in the model. rectification mechanisms operate in more complex junc- We consider a box with four holes in its sides. Each tions of Luttinger liquids. We will assume that long- hole corresponds to a terminal in a more realistic de- rangeCoulombinteractionsarescreenedbythegatesand scription of a junction. We assume that all charge carri- thus the charge density depends on the voltage. In Fig. ers,entering the box through the holes, haveexactly the 2a we consider the situation with the incoming current same speed, perpendicular to the wall in which the hole from the left. Backscattering off two impurities results ismade. Thespeedv =√ 2meV,whereV playsarole, in a “staircase”charge density profile (we have averaged − similar to the electrostatic potential of a terminal in a overFriedeloscillations). The “staircase”goesup asone realistic junction. At V = 0 the speed v = 0 and hence movesfromtherighttotheleft. Fortheoppositevoltage no current is injected. In a Y-junction, charge density, sign,theincomingcurrentarrivesfromtheright,Fig. 2b. injected from each terminal, depends on its electrostatic We again have a “staircase” charge density profile but potential. A comparison between opposite voltage signs now the “staircase” goes down as one moves from the in a realistic junction corresponds in our model to the right to the left. In the absence of electron interactions, comparison between the situations with the charge in- the transmission coefficients must be the same in both jected through the right and left holes (Figs. 3a,c) and cases and rectification is absent. Let us now consider through the upper and lower holes (Figs. 3b,d). In the the effect of electron interactions in a simple mean-field absence of a magnetic field, the current through each Hartree picture. Since long-range Coulomb interactions hole is zero in both situations: currents of the particles, arescreened,therelationbetweenthechargedensityρ(x) injected through the opposite holes, cancel as shown in and the electric potential W(x) is local. Assuming small Figs. 3a,b. Letusnowturnonamagneticfieldinshaded chargedensity variations,we thus find W(x) ρ(x). In- ∼ areas(Figs. 3c,d). Ifthechargecarriersareinjectedfrom comingelectronsarescatteredbyacombinedpotentialof the upper and lower holes then the current remains zero the two impurities and the Hartree potential W(x). The eveninthepresenceofamagneticfield(Fig. 3d). Indeed, latterdepends onthe incomingchargedensityandhence the carriers never enter the region with the field. How- the applied voltage. Hence, for opposite voltage signs, ever, if the carriers arrive from the left and right holes electrons feel different backscattering potentials. This thentheyaredeflectedbythefield(Fig. 3c)andtheelec- resultsindifferenttransmissioncoefficientsandthusrec- triccurrentsarenonzeroinallfourholes. Thisillustrates tification. how magnetic field can result in transport asymmetry. Another mechanism of transport asymmetries comes fromtime-reversalsymmetrybreaking. Thelatterispos- The above arguments make it plausible that the rec- sible in the presence of a magnetic field. We illustrate tification effect is the strongest if both strong electron asymmetrictransportinthe absenceofthe time-reversal interactionispresentandtime-reversalsymmetryisbro- symmetrywithamodelsystemdepictedinFig. 3. While ken. This is confirmed by our calculations below for itisnotarealisticmodelofanymesoscopicconductor,it Y-junctions of quantum wires with and without time- isverysimpleandexhibitsstrongtransportasymmetries. reversalsymmetry. 4 III. Y-JUNCTIONS T = U˜ (Fk†Fk)nexp(in[φk(x=0) φk(x=0)])+h.c., k n O I I − O X n (6) Inthissectionweformulateourmodelandreviewbasic whereU˜ =U exp(iα ),withrealU andα ,areofthe properties of Y-junctions. n n n n n order of the Fourier components of the asymmetric po- We consider a Y-junction with the action tential, k exp(i2nk x)U(x)dx. Note that α can be F F n nonzero eveRn for a symmetric potential U(x) in contrast to the situation considered in Refs. 18,19. For example, L= dt[ L T ], (1) Z X k− X k forU δ(x x0),α1 =2kFx0. Theaboveexpressionfor k=1,2,3 k=1,2,3 ∼ − the backscattering operators T includes multi-particle k backscattering processes.18,19 Such multi-particle contri- where L are the actions of three uniform wires and T k k butions to the action are inevitably generated under the describe three impurities in the wires close to the junc- actionofthe renormalizationgroupby the interplayof a tion point x=0. The above action must be supplied by short-range Coulomb interaction and impurity potential matching conditions for three wires. They will be dis- (see Ref. 19 for a discussion). All prefactorsU have the cussed below. The action of a uniform wire is given by n same order of magnitude and are proportional to U(x). the equation We assume that backscattering amplitudes U have the n same order of magnitude in all three wires. The use of ∞ the fields φ at x = 0 in the backscattering operators is L = dx[iψk†(∂ v ∂ )ψk+iψk†(∂ +v ∂ )ψk k Z I t− F x I O t F x O justified, if the distance from the impurities to the junc- 0 v λ(ψk†ψk+ψk†ψk)2], (2) tion is lower than the scale ~vF/(eV) set by the voltage − F I I O O bias. We assume that Un are sufficiently small so that a perturbativeexpansioninpowersofU canbedeveloped where ψk are the operators of incoming and outgoing n I,O for the calculation of the current. The conditions on U chiral electron fields, v is the effective Fermi velocity, n F will be formulated below. v λ defines the interaction strength. We set ~ = 1 and F Following the standard notation conventions, the the electron charge e = 1 in most of the following text. Hamiltonian, corresponding to Eq. (5), can be written WeassumethatthelongrangepartoftheCoulombforce as is screened by the gates. Hence, the electron density depends onthevoltagebias. The impurityHamiltonians v 1 T = ∞dxU (x)[ψk†ψk +h.c.], (3) Hk = 8π Z dx[g(∂xφkI+∂xφkO)2+g(∂xφkI−∂xφkO)2], (7) k Z k I O 0 where the dimensionless interaction strength g = 1/ 1+2λ/π <1 and v =v /g. whereU (x)isthepotentialoftheimpurityinwirenum- F k Tphe above action alone is not enough to describe the berk,theimpuritybeinglocatedclosetothejunctionat system and matching conditions at the junction are nec- x=0. Weonlytakethebackscatteringpartoftheimpu- essary. As we will see, matching conditions are sub- rity Hamiltonian into account as the forward scattering ject to several restrictions. The most general match- terms do not affect our results. We assume the same ing condition has the form F(φk (x = 0),∂ φk (x = interaction strength and Fermi velocity in each wire. I,O x I,O In order to treat the case of strong interaction it is 0),∂x2φkI,O,...) = 0, where F is an arbitrary function. convenienttobosonize44 theactionintermsofthechiral Belowwe willfocus onthe situationin whichthe system fields φk such that ψk = Fk exp( ik x+iφk ), is close to a fixed point. Then only most relevant opera- O,I O/I O/I ± F O/I torsshouldbekeptinthematchingconditionsandhence wheretheoppositesignsshouldbechosenforthein-and all derivatives of the Bose-fields φk can be neglected. out-fields, the commutator I,O Next, we note that the shift of any field φk by a con- I,O [φk (y),∂ φl (x)]= 2πiδ(y x)δ (4) stant φ(x) φ(x)+C is a gauge transformation that O/I x O/I ∓ − kl → does not change the physics of the system. Hence, the and kF plays the role of the effective Fermi momentum matching conditions must be invariant with respect to and determines the average charge density in the wires. suchgauge transformations. This singles out the bound- FI,O are Klein factors, necessary to ensure the proper ary conditions of the form38 Fermi commutation relations. The local densities of in- and out-moving particles in point x are ρO/I = (kF φkO = MkjφjI, (8) ± X ∂xφO/I)/2π. The action now assumes the form j where M is a matrix with real matrix elements. With 1 L = dx [∂ φk(∂ v ∂ )φk this matching condition, it is easy to solve the equations k Z 4π x I t− F x I ofmotionforthechiralfieldsφk intheabsenceofthein- O +∂ φk( ∂ v ∂ )φk vFλ(∂ φk ∂ φk)2]; (5) teraction(λ=0)andimpurities,forarbitraryinitialcon- x O − t− F x O− π x I − x O ditions for fields φk . Substituting the solution into the I,O 5 commutationrelations(4)forin-andoutgoingfields,one finds that they are compatible, providedthat the matrix 1 2 2 M is orthogonal. The same condition for the matching 1 − M =  2 1 2 ; (11) matrixmustholdforageneralcasewithanarbitraryin- 3 − teractionstrength38 sincethe problemcanbe reducedto  2 2 −1 a model of non-interacting bosons by diagonalizing Eq. (7) in each wire (see the next section). b a c Thus, we established that M is a real orthogonal ma- M = a c b , (12) trix. We next demonstrate that the sum of the ele- c b a ments of each of its rows and columns equals one.38   It is again convenient to consider the situation with- where b = α(α+1)/(1+α+α2), a = (α+1)/(1+α+ out electron interaction and tunneling. In that case the α2) and c = α/(1 + α + α2). Eq. (10) corresponds currents of in- and outgoing electrons in each wire are − to three disconnected wires. The cases of α = 0, 1, Ik = v ρk = ∂ φk /2π. Charge conservation − ∞ I/O ∓ F I/O ± t I/O correspond to a junction of two wires and a detached implies that the sum of the currents in all wires is zero thirdwire. Thephysicsis thesameasforonelinearwire at x = 0. Taking into account Eq. (8) one finds that and will not be discussed below. M = 1. Multiplying Eq. (8) by M−1 = MT k kj In the absence of time-reversal symmetry we use the Pand repeating the same argument one also finds that parametrizationfrom Ref. 38: M = 1. Again, electron interactions do not affect j kj tPhe above result38. a b c The origin of the matching conditions can be under- M = c a b , (13) stood if one notes that they can be imposed by adding b c a   to the Hamiltonian a term of the form where a=(1+2cosθ)/3, b(c)=(1 cosθ √3sinθ)/3 − ± and θ = 0,π. The case of b = 1, a = c = 0 corresponds 6 A cos(n[φk(x=0) M φj(x=0)]), (9) to the island setup, depicted in Fig. 1. Note that the − O − kj I matchingmatrix(11)hastheform(13)withθ =π sowe X X k j will consider it with the case of no time-reversalsymmetry. where A is a large constant. Each cosine tends to keep In general, matching matrices contain negative ma- its argument at zero. This can be achieved simultane- trix elements. This means that the incoming current in ously for each cosine only if their arguments commute. one wire may suppress outgoing current in that or other ThishappensforaunitaryM. Thecosinesdescribetun- wires. Such situation is possible in the presence of An- neling between different wires. Thus, charge conserva- dreev scattering, if, for example, a part of the junction tion implies M = 1. This provides an alternative j kj is superconducting. Negative matrix elements were also derivation ofPthe matching conditions. predicted in Y-shaped beam splitters for cold bosonic Since we plan to separately investigate time-reversal atoms45. Note that for noninteracting electrons no neg- invariant and non-invariant systems, we next need to ative matrix elements are allowed46 since in the absence determine what matching conditions satisfy the time- of interaction the elements of the matrix M reduce to reversal symmetry. This is easy as the time-reversal the squares of the absolute values of the elements of the transformation corresponds to the change of variables scattering matrix (cf. Sec. II). φk φk . From Eq. (8) we then see that in time- I/O → O/I reversal-invariant systems M = M−1. Taking into account that M is orthogonal we conclude that in time- IV. ELECTRIC CURRENT reversalinvariantsystemsitis alsosymmetric.42 Theac- tion (1) is always time-reversal invariant. Thus, the be- In order to calculate the current we need to include haviorofthewholejunctionwithrespecttotimereversal a voltage bias. We model Fermi-liquid leads by assum- is determined solely by the symmetry of the matching ing that electron interaction in the wires is zero at large matrix M. distances x from the junction.47–49 We will assume that At this point we are in the position to give a full clas- leads 2 and 3 are kept at a zero voltage. The bias volt- sification of fixed-point matching matrices.42 For time- age V is applied to lead 1. The results do not change reversal invariant systems, it is convenient to use a in a±more general model with the potential V applied parametrization from Ref. 41. We discover three pos- to lead 1, the potential γV, γ 1, applied±to lead 2, sibilities with time reversalsymmetry: and a zero potential app±lied to le∼ad 3. We will use the language of zero γ since it is simpler. Our approach can 1 0 0 be easily extended to a general γ. M = 0 1 0 ; (10) The calculations will be based on a renormalization 0 0 1 group approach with the voltage V playing the role of   6 the infrared cutoff. Thus, we will assume that the tem- The left hand side is the outgoing current we want to perature T < V. We would like to emphasize that the find. The first term in the right hand side is linear in role of the voltage does not reduce solely to that of a the voltage bias and cannot contribute to the rectifica- cut-off; otherwise the density-driven rectification would tion current. Thus, we have to calculate only the sec- be lost(see adiscussionafter Eq. (18)). It iswellknown ond contribution to the right hand side. In other words, that in Luttinger liquids a voltage bias can play a more our problem reduces to the calculation of the average prominent role than just a cut-off (see, e.g., Refs. 50– backscattering current whose operator equals I = I , l 53). At the same time, the leading contribution to the where P total current at zero temperature can be estimated by simply setting the renormalization group cutoff to the Il =i[Ul(FOk†FIk)nexp(iαl+in{φkI(0)−φkO(0)})+h.c.,Oˆ] value of the voltage54. Below we find that the rectifi- =i[U (Fk†Fk)nexp(iα +in (M−1) φp(0) φk(0) ) cation current is comparable with the total current in l O I l { kp O − O } X p certain regimes in the absence of the time-reversal symmetry. This certainly means that in those regimes the +h.c.,Oˆ] rectification current can be computed by assuming that =inU (Fk†Fk)nexp(iα +in[ (M−1) φp(0) φk(0)]) thevoltageplaystheroleofacutoffonly(wehoweverdo l O I l kp O − O X p not make such an assumption). [(M−1) δ ]+h.c.. (16) For a general γ, we do not expect the current to de- k1 k1 × − pend significantly on the temperature at T < V. Such Note thatforanyl the currentoperatorexpressesviaall dependence canemergeatparticularvalues ofγ. In par- backscattering operators in all three wires. ticular, in the island setup (Fig. 1), this happens at In order to find the average current I , we will as- γ = 0,1. This can be seen from the Keldysh perturba- h li tion theory54. The infrared cutoff in some integrals in sume that the backscattering operators Tk are absent at t= and are then gradually turned on. Without the the perturbation expansion is set by max(T,γV) and in −∞ operatorsT the systemcanbe viewedasanequilibrium some others by max(T,[1 γ]V). Hence, at γ =0,1 our k − oneinthegroundstateofanappropriateeffectiveHamil- approach applies only for V T in the island setup. ∼ tonian. Inordertofinditweintroduceanotherauxiliary We want to find the current in the first wire. At low field with the structure, similar to φ˜ : frequenciesthecurrentconservesandhenceitissufficient 1 to find the current in lead 1. It is given by the sum φ¯ (x)=φ1( x),x<0;φ¯ (x)= (Mˆ−1) φk(x),x>0. of the chiral incoming and outgoing currents ±vFρI/O. 1 I − 1 X 1k O The incoming current can be found from the Landauer k (17) formulaandislinearinvoltage. Itdoesnotcontributeto IntheabsenceofthebackscatteringoperatorsT ,theop- the rectification current. We thus focus on the outgoing eratorAˆ= ∞ dx∂ φ¯ /2π commuteswiththekHamilto- current in lead 1. It can be found with a generalization −∞ x 1 nian. ItcanRbeunderstoodasthechargeofachiralmode of the approach of Ref. 55. Our approach is also related propagating through the junction. In other words, it is to that of Ref. 56. anadditiveintegralofmotion. Hence,the systemcanbe In what follows we set the temperature T = 0 to sim- described by a Gibbs distribution with an appropriate plify notations. Our method canbe easily generalizedto thermodynamic potentialconjugatedto Aˆ. The physical finite temperatures. Let us introduce an auxiliary field φ˜ (x): meaning of that thermodynamic potential is the applied 1 voltage bias V. At zero temperature, one finds that the φ˜ (x)=φ1(x),x>0;φ˜ (x)= M φk( x),x<0. systemisinthegroundstateoftheeffectiveHamiltonian 1 O 1 1k I − H′ =H VAˆ, where H is the actualHamiltonianofthe X k − Y-junction. (14) The field φ˜ satisfies a simple matching condition The current, i.e., the average of the sum of the oper- φ˜ (+0) = φ˜1( 0). Thus, the auxiliary field is a chi- ators i[Tk,Oˆ], can be now calculated with the Keldysh 1 1 − technique57. It is convenient to apply the interaction ral field propagating through the junction. In the sta- representation H H VAˆ. The interaction represen- tionary regime the average time-derivative of any op- → − erator is zero. Let us now consider the operator Oˆ = tation introduces time-dependence into the operators Tk a dx∂ φ˜ /2π, where the integration extends between and Il according to the rule: −a x 1 Rpoints taken in the noninteracting leads. Its mean- itniognis0t=he cdhOâ/rdgte c=arriie[dHb,Oyˆ]th,ewmheordeethφ˜e1.HaTmhieltoeqnuiaan- exp(in[Xp (M−1)kpφpO(0)−φkO(0)])→ h i h i H = k(Hk+Tk) describes the whole system including exp(iVAˆt)exp(in[φkI(0)− MkpφpI(0)])exp(−iVAˆt) the lePads, wires and scatterers, reduces to the following Xp relation: =exp(in[ (M−1) φp(0) φk(0)+Vt(M δ )]). kp O − O k1− k1 v ρ1(a) = v M ρk(a) +i [ T ,Oˆ] . (15) Xp Fh O i h F X 1k I i hX k i (18) k k 7 After such time-dependence is added into all backscat- V. SCALING DIMENSIONS tering operators, the contribution of the form VAˆ can − be removed from the action by a linear shift of all fields Itisconvenienttousearenormalizationgrouppointof φkO/I. This does not mean, however, that the voltage viewforthecalculationofthecurrent.54Intherenormal- wouldonlyentertheactionthroughthetime-dependence ization group procedure, the coefficient U in the Hamil- l of Tk. Indeed, since the charge density depends on the tonian and the operators Il scale as Ezl−1, where E is voltage, the amplitudes Uk of the operators Tk may get the energy scale and zl the scaling dimension. At the corrections proportional to the small voltage bias. This energy scale E V the renormalization group proce- effect is discussed in Ref. 19. dure stops. Diff∼erent perturbative contributions to the The currentcannow be found with a perturbative ex- currentcanbe estimatedfromthe scatteringtheory.18,19 pansion in powers of Uk from the standard expression The current can be expressed as an infinite sum of the I = 0S( ,0)IˆS(0, )0 , (19) contributions, proportional to the products of different l h | −∞ l −∞ | i combinations of U : l where S(t,t′) is the evolution operator from time t′ to time t and 0 is the ground state of the effective Hamil- I V const (U Vzl−1). (25) tonian H |ViAˆ. To complete the calculation, we need ∼ X ×Yl l − the Green functions determined by the quadratic part Strictly speaking, it is not enough to include only of the action (2) and the matching conditions (8). They backscatteringoperatorsfromthe bare tunneling Hamil- havebeenfoundinRef.38. ABogoliubovtransformation of the form φ = (1/2√g)[(1+g)φ˜ +(1 g)φ˜ ] tonian (6) in the expansion (25). All operators gener- O/I O/I I/O − atedbytherenormalizationgroupmustalsobeincluded. in each wire allows one to obtain free chiral fields φ˜ I/O Their general structure is with correct Luttinger liquid commutation relations.38 The correlation function of the incoming fields at x = 0 Wˆ =W exp(iα +i n [φk(x=0) φk(x=0)]) is given by the free particle relation l l l k O − I X k hφ˜kI(t)φ˜pI(t′)i=−δkpln(i[t−t′]/τc+δ), (20) =Wlexp(iαl+i√g nk[M˜kp−δkp]φ˜pI). (26) XX where τ is the ultraviolet cutoff time of the order of k p c the inverse bandwidth, δ is infinitesimal. The matching Forsimplicityweusethesamenotationα forthephases conditions for the new fields have the form38 l of W and U . Note that it is always sufficient to keep l k φ˜k(x=0)= M˜ φ˜p(x=0); (21) onlytwodifferentvaluesofkinthesuminEq.(26),since O kp I Xp 3k=1[M˜kp −δkp] = 0. This means that each possible Poperator Wˆ can be generated from a product of two p M˜ =[(1+g) (1 g)M]−1[(1+g)M (1 g)]. (22) operatorsT (6)andhenceW U2. SinceKleinfactors − − − − k p ∼ l donotchangethescalingdimensionsofW ,weomitthem In the time-reversal invariant case M˜ = M. In the ab- l in Eq. (26). sence of the time-reversal symmetry, M˜ has the same To complete the calculation of the current we need to generalstructure (13) as M and satisfies the same set of find z . A straightforward calculation based on Eq. (20) constraints but the matrix elements are different: l yields for the operator (26) with an arbitrary choice of 3g2 1+(3g2+1)cosθ n : a˜= − ; (23) k 3(1+g2+(g2 1)cosθ) − z =g (n )2 n n M˜ . (27) l i i j ij 2(1 cosθ √3gsinθ) (cid:16)X −X (cid:17) ˜b(c˜)= − ± . (24) 3(1+g2+(g2 1)cosθ) In the absence of the time-reversal symmetry the above − expression greatly simplifies: At this point one can write an expression for the current as an expansion in powers of backscattering ampli- z =g(1 a˜)[n2+n2+n2 n n n n n n ]. (28) tudes U . Evaluation of the terms of that expansion is l − 1 2 3− 1 2− 2 3− 3 1 l technically difficult and not very informative. Indeed, In the low-voltage regime, the main contribution comes neither the amplitudes nor the ultra-violet cutoff are from the operators with the smallest z . One easily sees l knownexactly. Asaresult,itisonlypossibletoestimate that it equals theorderofmagnitudeofeachcontribution. Atthesame time, such estimation can be performed without explic- z =g(1 a˜) (29) min − itly calculating integrals over the Keldysh contour and will be sufficient to find the leading power dependence and is achieved, if two of the coefficients n =0 with the i of the rectification current on the bias voltage at small remaining one being 1. ± U and low voltages near each of the fixed points. Such In the time-reversal invariant system, the expression l estimation will be the focus of the remaining sections. for the scaling dimensions is more complicated. It can 8 be simplified by setting m = n n and k = n n . current. Thus, we can just set α = 0. Let us now 1 2 2 3 l − − Using the matching matrix (12), one finds: compare the currents at the bias voltages V and V. − In the case with the bias V, we make the change of z =g[k(b+bc+)+c mc]2 = 1+αg+α2(kα−m)2. (30) avaffreicatbltehseφfkIo/rOm→of−tφhekI/Oqu.aTd−rhaistictrapnasrftoromf athtieonacdtoioens n(o5t) We will assume for simplicity that α is irrational. We and the linear matching conditions. Let us also change will thus avoid situations with z = 0 at some k and m. the order of the Klein factors in the tunneling and cur- Certainly, such situations correspond to the operators rent operators using the commutation relation (32) and Wl with no φ-dependence. Such operators cannot affect redefine F →F†,F† →F. Since Ul pairs up with Ul† in transport. thenonzerosecond-ordercontributiontothecurrent,one Therenormalizationgroupprocedureonlyappliesifall can easily see that the phase factors exp( iπ[1 Mkk]) ± − U aresmallattheinitialenergyscale 1/τ andremain in the commutation relation (32) drop out. At the l c ∼ smalluptothescaleV. Inthecasewithouttime-reversal same time, if we omit the phase factors exp( iαl) and ± symmetry,thisconditioncanbeeasilyexpressedinterms exp( iπ[1 Mkk]) from the action and current opera- ± − of z : tors, we discover that the action assumes precisely the min same formas for the voltagebias V. On the other hand, Ulτc <const(Vτc)1−zmin =constV1−g(1−a˜). (31) thecurrentoperatorIl (16)changesitssign. Thismeans that the average of I also changes its sign and hence The above equation assumes that 1 z > 0, i.e., U l − min l the second order term of the order U2V2zl−1 does not is relevant. If 1 z < 0 then the only restriction on l − min contribute to the rectification current. U is U τ < 1. What happens in the presence of the l l c The above argument assumes that the backscattering time-reversal symmetry will be addressed in subsequent amplitudes U areindependent ofthe voltagebias. Since sections. l the injected charge density depends on the voltage in our setup, U can exhibit a weak linear dependence on l VI. RECTIFICATION CURRENT the bias voltage. This means that second order terms can in fact lead to a “density-driven” rectification effect but its amplitude is suppressed by an additional factor The above discussion applies to all contributions to Vτ ,wherethe ultra-violetcutoffτ isoftheorderofthe the current, both even and odd in the voltage bias. We c c inverseband width. Thus, the secondordercontribution areinterested inthe evencontribution,i.e., the rectifica- to the ratchet current scales as tion current I (V) = [I(V)+I( V)]/2. What terms of r − theperturbationexpansioncontributetotherectification current? The answer to this question can be obtained I(2) U2V2zl. (34) ∼ l from symmetry considerations. Inwhatfollowswewillneedthecommutationrelations Beyondthesecondorder,noadditionalgeneralrestric- forthe Kleinfactorsin(6). Using the commutationrela- tionsonpossiblecontributionstotherectificationcurrent tions ψk†(0)ψk(ǫ) = ψk(ǫ)ψk†(0) and [φk(x),φk(y)] = O I − I O I I can be derived. Particular contributions to the ratchet iπsign(x y), an expressionfor ψk in terms of φk and − − O I current can disappear at particular values of the phases the Baker-Hausdorffformula, one finds α and interaction strength g. l Fk†Fk =exp(iπ[1 M ])FkFk†. (32) O I − kk I O Next, some terms of the perturbative expansion (25) are zero identically as only certain combinations of the VII. THREE WEAKLY CONNECTED WIRES vertex operators in T and I produce non-zero results k l after averagingwith respect to the quadratic part of the SofarwediscussedgeneralfeaturesofY-junctions. We action (5). The condition is well known: now determine the rectification current in different se- exp(i c φk(0)) =0 only if for each k c =0. tups. As a warm-up exercise we investigate the simplest hY X lk I i6 X lk situationofthreealmostdisconnectedwires,Fig. 4. This l k l (33) situation corresponds to the matching matrix (10). The Let us now apply the above results to possible con- rectificationcurrentismuchweakerinthislimitthanfor tributions of different orders of the perturbation theory other matching conditions. At the same time, the quali- to the rectification current. Strong limitations emerge tative picture is quite similar. for second order contributions. Indeed, Eq. (33) implies There is no current at all in our original model with that any second order contribution, proportional to an the action (1) and backscattering operators (3) as it de- operator Ul, must also contain its Hermitian conjugated scribes three disconnected conductors. We thus modify operator U†. Hence, the phase factors exp( iα) can- the model in this section. Instead of the backscattering l ± l cel each other and drop out from the expression for the operators (3) we consider weak tunneling between the 9 Interestingly,atlowvoltagesandg >1/2,thethirdorder ±γV contribution exceeds the second order contribution. B The above discussion ignored the issue of the time- 2 C reversal symmetry. Depending on the presence or ab- senceofthemagneticfieldthroughthe junction,thesys- A ±V 1 tem can have or have no time-reversal symmetry.28,29 It F is instructive to investigate the effect of the symmetry breaking. In the presence of the time-reversal symme- 3D try the magnetic flux is zero and all α = 0. Indeed, k E the time-reversal transformation can be represented as φk φk,φk φk,α α . Since φk(0) = φk(0), I → O O → I k → − k I O we conclude that α = α = 0 at zero magnetic field. k k − On the other hand, in the presence of the magnetic field FIG. 4: Schematic picture of a Y-junction of three weakly αk =0. 6 connected wires. Voltage bias ±V is applied to wire 1, and The secondorder contributionto the rectificationcur- bias ±γV to wire 2. Wire 3 is connected to the ground. We rent does not depend on the phases α . Let us com- k calculate thedc current in wire 1. pare the third order contributions for different values of α . Nonzero third order contributions to the cur- k rent originate only from the product of all three op- wires. The tunneling operators have the form erators T or all three operators T† in the perturba- k k 3 tive expansion of the Keldysh expression for the cur- T = Tk; rent (19). The first contribution, I(3), is proportional X 1 k=1 to exp(i α ), I(3)(V) = exp(i α )J , and the Tk =Fk†+1FkUkeiαkexp(iφk(0)−iφk+1(0))+h.c., (35) second coPntkribkution1 is proportionalPtok ekxp(1−i kαk), wweheureseφkth(0e)c=onφvkIe(n0t)io=n φ3kO+(01), Fk1a.reTKhleeincofmacmtourtsatainodn Isi2(g3n)(.VS)im=uletxapn(e−ouisPlyk, wαke)cJh2a.nLgeetφukOs/Ich→an−geφkOth/Ie,Pvcohlatnaggee relations for the Klein factors a≡re F F = F F ,k =q. the order of Klein factors in all operators and redefine k q q k We assume that wires 3 is kept at zero −voltage. 6The Fk →Fk†,Fk† →Fk. This transformationis equivalentto chemical potential of the first wire is V. The second changing the sign of αk in all operators and simultane- ± wireexperiencesthebias γV. Noteadifferentmeaning ously changing the overall sign of the tunneling Hamil- ± oftheamplitudesU andphasesα fromthemodelwith tonian (35) but not the sign of the current operator. k k three weak scatterers discussed above. In Eq. (35), the Hence, the third order current I(3)( V)+I(3)( V) = phasesαk describe the Aharonov-Bohmeffectdue tothe exp(−i kαk)J1 + exp(i kαk)1J2.−We see2 th−at the magnetic field through the junction. On the other hand, currentPis an even functioPn of the voltage in the pres- in the model with weak scatterers,allinformationabout ence of the time-reversal symmetry, i.e., at α = 0. k k themagneticfluxiscontainedinthematchingconditions. The third order current is an odd functionPof the volt- The voltage bias can be included in the action in the age and does not contribute to the rectification effect, if form of the time-dependence of the tunneling operators, α = π/2. Thus, for three weakly connected wires, similartothediscussionabove. Wewillnotneedexplicit mPakgnketic±fieldsuppressesrectification. Wewillseebelow expressionsbelow. Theoperatorofthecurrent,tunneling that typically it has an opposite effect and rectification between wire 1 and wires 2 and 3, is is stronger in the absence of the time-reversal symmetry. Note that in the above example, the ratchetcurrent I =iT iT† iT +iT†. (36) 1− 1 − 3 3 returns to its maximal value at kαk =π. The current can be estimated from a renormalization Let us finally compare our rePsults with the case of a group procedure that stops at the scale E V. At that linear wire with a high asymmetric potential barrier or, scale U U (Vτ )1/g−1, where g < 1∼characterizes equivalently, two weakly connected wires. In that case k k c → the rectification current is given18 by Eq. (37). Thus, at the interaction strength. The second order contribution to the rectification current can now be easily found and g >1/2therectificationeffectisstrongerinaY-junction than in a linear wire with the same interaction strength. scales as I(2) U2V2/gτ2/g+1, (37) ∼ c VIII. RECTIFICATION EFFECT IN THE where U U 1/τ . ∼ l ≪ c ABSENCE OF THE TIME-REVERSAL The n-th order contribution to the current cannot ex- SYMMETRY ceed (Uτ )n(Vτ )n/g−n+1/τ ,n 3. Thus,theleading c c c ∼ ≥ higher-order contribution corresponds to n=3, Here we address matching matrices of the form (13). I(3) U3V3/g−2τ3/g. (38) In particular, the results of this section are relevant for ∼ c 10 the island setup, Fig. 1. Our analysis also applies to the low voltages and achieves its maximal value of the order time-reversal-invariant situation with the matching ma- of e2V/h at V = V∗ U1/[1−g(1−a˜)]. In the last equa- ∼ trix (11). Below we find that at sufficiently low voltages tion we omitted a power of τ as this does not lead to a c the rectification current scales as Eq. (39) with U being confusion. theimpuritypotentialstrength,V thevoltage,g <1the Finally, let us compare the result with the case of Luttinger liquid parameter, characterizing electrostatic a linear wire with the same interaction strength g in interactions, τc the ultraviolet cutoff of the order of the the presence of an asymmetric impurity. In that case inversebandwidth,anda˜isdeterminedbythematching the maximal rectification current, calculated in the per- conditions according to Eq. (23). turbative regime in Ref. 18, scales as V1+3g < V at Similar to the previous section we need to compare g <1/3(weomitpowersofτ ). Therectificationcurrent c the leading second and third order contributions to the does not exceed V2/τ at g > 1/3. In a Y-junction, at c rectificationcurrent. They are determined by the opera- 1 g(1 a˜)>0therectificationcurrent V atV V∗is tors of the form Tk ∼ exp(i[φkI −φkO]) with the minimal al−ways−greater than in a linear wire. On∼the othe∼r hand, scaling dimension (29). The leading second order con- a negative 1 g(1 a˜) implies g > 3/4. The maximal ta˜riibsugtiivoennsbcyaleEsqa.s(2I3()2)a∼nd(sUaτtics)fi2e(Vs τc1)2/g3(1−aã˜)/τc,1.whTehree srceactleifiscaastionVc−u3gr(r1e−na˜t−)−co2r>resVpo2n.dTshtuhse,nthtoe Uma∼xim1/aτlcraatnido − ≤ ≤ ∼ leadingthirdordercontributioncomesfromthe termsin of the rectification and total currents is always higher in the perturbation expansion proportional to the product aY-junctionthanina linearwireinthe Luttinger liquid of all three T (or all three T†). It scales as regime V 1/τ . k k ≪ c I(3) U3V3g(1−a˜)−2τ3g(1−a˜). (39) ∼ c IX. RECTIFICATION EFFECT IN THE At small U the second order contribution always ex- PRESENCE OF THE TIME-REVERSAL SYMMETRY ceeds higher-order contributions. Interestingly, however, atgreaterU whicharestillwithintheregionofvalidityof the perturbation expansion,the third order contribution Thiscaseismorecomplicatedthanthesituationwith- is leading. out the symmetry. The summary of the results is given Let us show that the third order contribution domi- in section IX.C. We will need to take into account many nates. We will focus on the largest U accessible with different backscattering operators. Thus, it is important l the perturbation expansion: U 1/τ at g(1 a˜) > 1 toclassifythem. Theclassificationreliesonthematching l c when U is irrelevant; and U ∼ (Vτ )1−g(1−−a˜)/τ at conditions at the junction. l l c c ∼ g(1 a˜) < 1 when U is relevant. In all cases the elec- The matching matrix is given by equation (12) where l − tron interaction is repulsive, i.e., g < 1. Let us first the expressions for a = f (α),b = f (α),c = f (α) in 1 2 3 consider the case of g(1 a˜)>1. According to Eq. (23), the parametrization42 are written under the equation. − a˜ 1/3. Hence g(1 a˜) < 4/3. The ratio of the sec- It is always possible to redefine α in such a way that ≥ − − ond order contribution to the third order contribution a = f (α),b = f (α),c = f (α), where (k,l,n) is an k l n scales as (Uτ )−1(Vτ )2−g(1−a˜) (Vτ )2−g(1−a˜) < 1. arbitrarytransposition of (1,2,3). Indeed, the change of c c c ∼ Thus, the third order contribution dominates indeed. the variable α 1/α exchanges f and f ; α (1+ 1 2 → → − The case of g(1 a˜) < 1 is also easy. In the limit α) exchanges f and f ; α α/(1 + α) exchanges 1 3 of U (Vτ )1−g−(1−a˜)/τ we find τ I(2) (Vτ )2 and f and f . Any other transp→osi−tion is a superposition l c c c c 2 3 ∼ ∼ I(3) V I(2). Interestingly, the third order current of the above three. Since M is an orthogonal matrix, may∼becom≫ecomparabletothetotalaccurrent e2V/h at least one of the elements must be negative or zero. ∼ through the junction at g(1 a˜) < 1. Note also that Otherwise,differentrowscannotbeorthogonal. Without − the exponent 3g(1 a˜) 2 in the voltage dependence of loss of generality we may assume that c 0 and hence − − ≤ the dc current (39) is negative at g(1 a˜) < 2/3. This α 0. This can always be achieved by renumbering the − ≥ corresponds to the dc current increase as the ac voltage wires and redefining α. Note that a,b 0. Similarly, ≥ decreases. without loss of generality we may assume that a b. ≥ The above calculation applies for the voltage inter- Hence, α 1. Note that a 2/3. Thus, the ranges ≤ ≥ val 1/τ > V at g(1 a˜) > 1 and 1/τ > V > of the parameters that we consider are: 0 α 1, c c (Uτ )1/[1−g(1−a˜)]/τ at g−(1 a˜) < 1. The left inequal- 1/3 c 0, 0 b 2/3 and 2/3 a 1. ≤ ≤ c c − − ≤ ≤ ≤ ≤ ≤ ≤ ityisdictatedbytheapplicabilityoftheLuttingerliquid As has already been discussed, we need to deal with model. The right inequality is determined by the va- two types of backscattering operators in the action and lidity of the perturbation theory. One cannot calculate the current operator: the rectificationcurrentatlowervoltageswith the above perturbativeapproach. However,itisobviousthatI =0 ( ) U Uexp(in [φk(x=0) φk(x=0)]); ∗ l ∼ k O − I at V = 0. From this we conclude that there is a bump on the voltage dependence of the rectification current at ( ) W U2exp(in [φ1(x = 0) φ1(x = 0)] + g(1 a˜) < 2/3: it grows as a function of the voltage at ∗∗ inl[φ∼3(x=0) φ31(xO=0)]). − I − 3 O − I