Spin-charge separation in an Aharonov-Bohm interferometer
We study manifestations of spin-charge separation (SCS) in transport through a tunnel-coupled interacting single-channel quantum ring. We focus on the high-temperature case (temperature larger than the level spacing ) and discuss both the classical (flux-independent) and interference contributions to the tunneling conductance of the ring in the presence of magnetic flux. We demonstrate that the SCS effects, which arise solely from the electron-electron interaction, lead to the appearance of a peculiar fine structure of the electron spectrum in the ring. Specifically, each level splits into a series of sublevels, with their spacing governed by the interaction strength. In the high- limit, the envelope of the series contains of the order of sublevels. At the same time, SCS suppresses the tunneling width of the sublevels by a factor of . As a consequence, the classical transmission through the ring remains unchanged compared to the noninteracting case: the suppression of tunneling is compensated by the increase of the number of tunneling channels. On the other hand, the flux-dependent contribution to the conductance depends on the interaction-induced dephasing rate which is known to be parametrically increased by SCS in an infinite system. We show, however, that SCS is not effective for dephasing in the limit of weak tunneling. Moreover, generically, in the almost closed ring, the dephasing rate does not depend on the interaction strength and is determined by the tunneling coupling to the leads. In certain special symmetric cases, dephasing is further suppressed. Similar to the spinless case, the high- conductance shows, as a function of magnetic flux, a sequence of interaction-induced sharp negative peaks on top of the classical contribution.
Spin-charge separation (SCS) is a hallmark of non-Fermi-liquid behavior solyom79 (); voit94 (); gogolin98 (); giamarchi04 (). The essence of SCS is that single-electron excitations factorize in space-time into two parts which independently exhibit dynamics of, respectively, the spin and charge degrees of freedom. In one-dimensional (1D) systems, SCS is inherently linked to the decoupling of two types of elementary bosonic spin and charge density excitations which separately carry either spin or charge and propagate with different velocities, and , respectively. Experimental evidence of SCS in nanowires was demonstrated in electron tunneling auslaender05 (); jompol09 (), thermal transport Lorenz02 (), and, more recently, in spin-filtering hashisaka17 () experiments. The effect of SCS is most pronounced in the “spin-incoherent regime” fiete07 (); matveev04 (); fiete04 (); cheianov04 (); hew08 () which is realized in 1D systems with strongly different spin and charge velocities.
One of the key problems that are related to SCS is about the manifestation of SCS in the quantum interference of electron waves. This is the subject of the present paper. We focus on the, perhaps, conceptually simplest device for specifically probing the interference—a single-channel quantum ring tunnel-coupled to the leads (see Fig. 1) and threaded with the magnetic flux . The conductance of the ring exhibits the Aharonov-Bohm (AB) effect bohm59 (); aronov87 (), i.e., changes periodically with the dimensionless magnetic flux , where is the flux quantum,—with a period —because of the interference of electron trajectories winding around the hole. The sensitivity of the phase of an electron wavefunction to the flux enables the design of AB interferometers aronov87 (); AB1 (); yacoby96 (); AB2 (); bykov00 (); bykov00a (); AB3 (); AB4 (); AB5 (); AB6 (); AB7 (); AB8 (); AB9 (); roulleau07 (); roulleau08 (); zhang09 (); weisz12 () that can be tuned by the external magnetic field. The peculiar predictions that we make in this paper appear to be amenable to experimental verification on many-electron nanorings, with a few or single conducting channels, which have already been produced shea00 (); piazza00 (); fuhrer01 (); keyser03 (); zou07 ().
We consider a clean ring without disorder, so that electrons only experience scattering on the contacts—and because of interactions with each other. Consider first the noninteracting limit. Transmission of an electron through the ring can occur along paths with different numbers and different sequences of clockwise and anticlockwise windings between the left and right contacts, characterized by different transmission amplitudes , where is the index for a particular path. The classical and interference contributions to the transmission coefficient are then proportional to and , respectively. The classical contribution does not depend on , so that an efficient manipulation of the AB interferometer by the external magnetic field relies on the existence of the interference contribution.
A key obstacle hindering the AB interference is interaction-induced dephasing of the electron waves. In the presence of interactions, electrons on the ring form a Luttinger liquid (LL) in the ground state, with SCS being one of the inherent properties of the spinful LL. A “natural expectation” would be that SCS enhances, possibly strongly, dephasing of the AB oscillations. Indeed, the single-electron excitation that is created in the ring after tunneling from the lead splits into the charge and spin components which start propagating with different velocities, as illustrated in Fig. 1. This decomposition of an electron into spatially separated charge and spin pieces is known to increase the decay rate of single-electron excitations in an infinite system to a value of the order of (see, e.g., Refs. gornyi05 (); lehur05 (); yashenkin08 ()), compared to the decay rate of the order of for the spinless case. Here, is the dimensionless constant characterizing the strength of repulsive interaction, which we assume to be small, (experimentally, the value of is controlled by the electrostatic environment; in particular, by the distance to the metallic gate). For spinful electrons, the same decay rate governs dephasing of the quantum interference conductivity correction in an infinite disordered Luttinger liquid yashenkin08 (). One might thus expect a similar enhancement of dephasing by SCS in the spinful ring.
In the present paper, we show that, in fact, in the weakly tunnel-coupled interferometers, the “AB dephasing rate” is insensitive to SCS. Moreover, generically, does not depend on the interaction strength and is determined by the tunneling coupling to the leads. In certain special symmetric cases, dephasing is further suppressed. In particular, this happens for the case of fully isotropic (in spin and chirality spaces) interaction.
To an extent, the insensitivity of to the strength of generic interactions in the weak-tunneling limit is similar to the spinless case. Indeed, apart from Ref. yashenkin08 (), our approach to the SCS effects in the ring geometry builds upon the earlier work on transport of spinless electrons through a quantum ring dmitriev10 (); dmitriev14 (). As was shown there, the dephasing rate in an almost closed spinless ring is given by the total tunneling rate, namely the rate at which the ring exchanges electrons with the leads. In the present work, we find that, generically, the dominant mechanism of dephasing in the spinful ring is the same—the so-called zero-mode (ZM) dephasing dmitriev10 (). This, in turn, means that is determined by the total tunneling rate as in the spinless case, thus vanishing in the limit of weak tunneling. We also find that this mechanism is not effective in the symmetric cases mentioned above.
There is one more point of similarity—now only at the qualitative level—between the spinless and spinful systems: we show that the destructive interference between right- and left-moving electrons leads, in the presence of SCS, to a series of sharp interaction-induced negative peaks in the high- conductance as a function of magnetic flux, bearing resemblance to the interference pattern in the spinless case dmitriev10 (). Importantly, however, the width and depth of the envelope of the AB conductance peaks are strongly modified by SCS.
To be more specific, it is useful to recall the origin of the interference pattern in the high- limit in the case of spinless electrons. To begin with, for noninteracting electrons (on a disorder-free ring weakly tunnel-coupled to the contacts), the sharp antiresonances in the function in the high- limit occur at , where is integer imry (). The antiresonances originate from the destructive interference in tunneling via pairs of quantum levels inside the ring for electrons of opposite chirality. At , the levels of electrons rotating clockwise and anticlockwise are pairwise exactly degenerate and the tunneling amplitudes for two levels in each pair are of opposite sign. The total transmission coefficient at is thus exactly zero for an arbitrary energy of the tunneling electron, which explains the survival of the interference pattern as increases in the noninteracting case. Electron-electron interactions change the picture dramatically. As shown in Ref. dmitriev10 (), interactions between spinless electrons of opposite chirality can be incorporated into an effective magnetic flux dependent on the circular current inside the ring. Tunneling-induced fluctuations of the circular current and, in turn, of the effective flux split the antiresonance at into a series of peaks (“persistent-current blockade”). This is the “interference pattern” for spinless electrons that we referred to in the above.
The picture based on the introduction of the effective flux controlled by the circular current dmitriev10 () is no longer valid in the presence of SCS. It is thus the fate of the persistent-current blockade in the presence of the spin degree of freedom that is one of the subjects of this paper. As already mentioned above, SCS does not wipe out the splitting of the “noninteracting” antiresonance in into a series of sharp peaks. Rather, SCS brings about new physics behind the emergence of the resonant structure and, consequently, modifies its parameters. Moreover, SCS determines the characteristic transparency of the tunnel contacts at which the fine structure in blurs out as the tunneling rate is increased: this occurs when the dephasing rate becomes of the order of the “spin-charge collision rate” , at which the paths of the spin and charge components cross each other. For larger , all resonances overlap and form a single dip in with a width given by the single-particle decay rate in units of the level spacing in the absence of interaction .
Another nontrivial result of this work is for the classical part of the tunneling conductance. In a simpleminded approach to the problem, one would think that also the classical transmission through the tunnel-coupled ring is suppressed because of SCS—indeed, for essentially the same reason as in the case of the interference term in the transmission coefficient. The rationale would be that, the tunneling escape from the ring can only happen if the spin and charge components of the single-electron excitation collide in the vicinity of the contacts to the lead, and these collisions are rare (see Fig. 1). We show that this expectation is not true, either—in fact, independently of the strength of tunneling. The subtle point is that the electron-electron interaction between spinful electrons splits each level in the ring into a series of sublevels. This is a direct consequence of SCS. As we demonstrate below, although the tunneling width of the sublevels is indeed suppressed by SCS, this effect is compensated in the conductance by the increase of the number of tunneling channels, as illustrated in Fig. 2.
Let us clarify the last point in more detail. The key ingredient of the underlying physics here is the multiple windings and, consequently, multiple returns to the contacts in the finite-size system. The characteristic time between spin-charge collisions near the contact is given for by (the difference between the charge and spin velocities, and , is linear in for small ). For , the characteristic “dwelling time” during which the spin and charge excitations run together as a whole (and, therefore, can tunnel out of the ring) is much shorter and given by (note a similarity between and the electron lifetime in an infinite system). As a consequence, the tunneling rate is suppressed by the factor , independent of for . On the other hand, as we show below, the spin-charge collisions are, in essence, correlated even if the spin and charge velocities are not commensurate, namely the spin and charge collide periodically, with a period given by . It is this periodicity that leads to the formation of a fine structure in the electron spectrum and to the resulting increase of the number of tunneling channels by a factor of (Fig. 2). As a consequence, the classical transmission coefficient through the ring does not change, and nor does the dephasing rate for the AB oscillations.
Note that the emergence of the SCS-induced fine structure of AB resonances was discussed earlier in Refs. jagla93 (); hallberg04 (); meden08 (); rincon09 () for strongly interacting electrons, with emphasis on the effects of commensurability between and . In particular, Refs. hallberg04 () and rincon09 () considered the “spin-incoherent” limit of the - model (which corresponds to the limit of a strong onsite Hubbard repulsion) for a quantum ring made of a finite number of sites filled with particles, with the ratio being small in the parameter but finite. The suppression of transport through the ring for certain in the limit of strong Hubbard interactions was associated in Ref. rincon09 () with level crossings in the ground state of particles. By contrast, in our continuous Luttinger-liquid model, similar—in this respect—to that of Refs. jagla93 () and meden08 (), the effects are irrelevant for the relation between and —and altogether for the emergence of the SCS-induced structure in the AB resonances.
It is worth noting that Refs. jagla93 (); hallberg04 (); meden08 () presented their results in terms of the zero- conductance averaged over the period of the energy spectrum. Although such a quantity shows interaction-induced splitting of the resonances, this approach does not produce a solution of the finite- problem, not even in the high- limit, where typical excitations have energies much larger than the characteristic level spacing. This is because it does not include important aspects of the finite- dynamics of the system, namely the interaction-induced decay of single-particle excitations at finite and the effect of dephasing. Below, we develop an analytical theory for the high- AB conductance, taking account for both effects.
The paper is organized as follows. Section II covers some of the basic aspects of SCS in an infinite system (Sec. II.1), in the isolated ring (Sec. II.2), and in the ring tunnel-coupled to the leads (Sec. II.3). In Sec. III, we discuss the two-particle dynamical properties of the spin-charge separated ring in the absence (Sec. III.1) and presence (Sec. III.2) of tunneling. In Sec. IV, we calculate the classical (Sec. IV.1) and interference (Sec. IV.2) contributions to the conductance, and the dephasing rate that governs the latter (Sec. IV.4), with the main results for the case of isotropic interactions summarized in Sec. IV.3. Our conclusions are presented in Sec. V. Some of the technical details are placed in the Appendixes.
Below, we study the linear response conductance of the AB interferometer which consists of a spinful LL ring weakly coupled by tunneling contacts to the leads (for details of the tunneling coupling, see Appendix A). We consider a symmetric setup with both point-like contacts having the same tunneling rate and both arms of the interferometer having the same length. We assume that the Coulomb interaction between electrons on the ring is screened by a ground plane and take the interaction to be pointlike. Throughout the paper we focus on the regime of relatively high temperatures
where is the tunneling rate in the absence of electron-electron interactions and is the Fermi energy, in which there emerges the new interesting physics, discussed already at the qualitative level in Sec. I, related to the interplay of SCS and tunneling. Let us estimate for realistic systems. For example, for a single-channel GaAs ring of the radius nm, where is the circumference of the ring, in the range of eV, we find the level spacing at the Fermi level eV. The condition is seen to be easily satisfied for realistic experimental conditions.
One of the consequences of taking the high- limit (for ) is that the classical effect of Coulomb blockade of charge transport through the ring can be neglected. Moreover, in this limit, the number of tunneling channels in the temperature window around the Fermi level is large, of the order of . Most importantly in the context of SCS, the condition also means that the spin and charge components of the electron propagator inside the ring each have a characteristic spatial extent ( and , respectively) which is much smaller than the distance between the contacts to the leads. That is, the spin and charge excitations propagate ballistically with velocities and and only rarely “collide” with the contacts and with each other (see Fig. 1), in accordance with the picture outlined in Sec. I footnote-park15 (). This essentially simplifies the two-particle dynamic correlation functions and we use this condition extensively from the very beginning.
For a discussion of the general case of an arbitrary relation between and the characteristic level spacing in an isolated finite-length spinful LL, see Ref. mattsson97 () for a piece of the LL between two hard walls (“quantum dot”) and Refs. eggert97 (); pletyukhov06 () for a quantum ring made of it. The role of SCS in transport through a 1D quantum dot in the regime of Coulomb blockade was intensively studied in Refs. kleimann00 (); braggio01 (); cavaliere04 (); cavaliere04a (). The transport properties of a ring weakly coupled to the leads in the presence of SCS were investigated in the limit (and away from the transmission resonances, i.e., in the valleys of Coulomb blockade) in Refs. kinaret98 (); pletyukhov06 (). Focusing on either isolated systems or transport in the low- limit, these studies do not discuss the peculiar interplay between SCS and tunneling that becomes apparent for and is the subject of the present work. As already mentioned in Sec. I, the effect of finite on the AB oscillations in a spinful LL cannot be mimicked by averaging jagla93 (); hallberg04 (); meden08 () the zero- transmission coefficient over energy, which misses the interaction-induced decay of single-electron excitations at finite as well as the ZM dephasing.
We assume that electron-electron backscattering is absent and consider a ring made of a single-channel wire with otherwise generic interactions characterized by four coupling constants: , and giamarchi04 (). For the most part, the paper is focused on the study of a symmetric model, isotropic in chirality and spin spaces, with . Importantly, the isotropic model fully captures the physics of SCS. However, as already mentioned in Sec. I, the ZM dephasing mechanism, which was shown to be dominant in the spinless case dmitriev10 (), is ineffective in the case of full isotropy. For this case, we obtain an analytical result for the conductance in terms of a phenomenologically introduced dephasing rate (which may arise from an external bath). By going beyond this model, we demonstrate that a violation of isotropic symmetry leads to two effects: an additional ZM-induced splitting of the AB resonances and the emergence of ZM dephasing. Remarkably, the dephasing action is then described by an equation analogous to that in the spinless case.
ii.1 SCS: Green functions in an infinite system
We start by considering the fully isotropic (in spin and chirality spaces) model introduced above. We write first the corresponding expression for the coordinate-time Green function per spin in the Matsubara representation in an infinite spinful LL solyom79 (); voit94 (),
where and denote right- and left-moving fermions, respectively (here and below, ):
Here, is the ultraviolet cutoff in energy space,
with being the zero-momentum Fourier component of the interaction potential.
The main (“chiral” or “square-root”) approximation below, which we make following Ref. jagla93 () (but, in contrast to it, not focusing on the case of “commensurate” given by a simple fraction) and Ref. yashenkin08 (), is to keep, in the electron self-energy, only the terms of leading (linear) order in . That is, we retain the difference between and ,
and put to zero in Eq. (1). Note that it is the appearance of the two velocities for each chirality in the single-particle correlator that signifies SCS. Within this approximation, the right and left electrons are chiral, and the charge and spin velocities enter the electron Green function for each chirality in a symmetric way. From the point of view of symmetry in chirality space, the approximation becomes exact when only interactions between electrons of the same chirality are kept (see the discussion of the generic model in Sec. IV.4 below). It is worth noting that the consistency of the square-root approximation within the isotropic LL model requires that the difference of the two velocities is small [Eq. (4)].
Relying on the chiral approximation, which fully captures SCS, allows us to obtain closed analytical expressions for the conductance and the dephasing rate without sacrificing anything of importance as far as the essence of the effect of SCS on the AB oscillations is concerned. Importantly, the neglected higher-order effects, which produce asymmetry between the spin and charge in Eq. (1), correspond to “fractionalization” pham00 (); lehur08 (); steinberg08 (); karzig11 (); calzona15 (); calzona16 (); acciai17 (); brasseur17 () of the charge sector in much the same way as in the spinless case dmitriev10 (), and likewise do not lead to a suppression of the interference. Indeed, the fractionalized charges emitted at one contact, moving in the opposite directions with the same velocity , collide every time they pass by the contacts. They sum up to the “unfractionalized” electron charge at the contact, so that charge fractionalization does not manifests itself in the ring geometry remark1 (). This should be contrasted with spin-charge separated excitations (plasmons and spinons) which move in the same direction with different velocities and, therefore, pass by a contact at the same time only rarely. The dynamical properties of charge fractionalization on the one hand and those of SCS on the other are thus essentially different.
In the real-time representation, the retarded Green functions gornyi07 () corresponding to the square-root approximation read
(in the energy-momentum representation, the expressions for the spin-charge separated Green functions for can be found in Ref. yashenkin08 ()). Note that, within the square-root approximation, the time dependence of the Green functions at , which determines the tunneling density of states, coincides with that for the noninteracting Green functions.
ii.2 SCS: Isolated ring
Let us now discuss the SCS effects in a single-channel quantum ring of length threaded with the magnetic flux . We first consider an isolated (not coupled to the leads) ring. In the finite-size system, in addition to plasmons and spinons, one should take into account homogeneous ZM excitations. Such excitations are characterized by eigenenergies that are determined by the total numbers of right- and left-moving () particles with a given spin projection and the chemical potential (the same for all sorts of particles) fixed by the leads. Under the assumption of full isotropy of interactions in chirality and spin spaces within the ring, the ZM energy is given by pletyukhov06 (); meden08 ():
, and is the Luttinger constant (for the charge sector). The current and spin contributions to are characterized by the “noninteracting” spacing . It is worth noting that the ZM excitations are not factorizable into independent charge and spin parts. The total number of electrons in the ring is controlled by (which, in turn, is determined by ).
The equilibrium value of an observable is averaged over ZM fluctuations (in a grand canonical ensemble of isolated rings) according to
is the sum of the observable for spin-up and spin-down electrons and a given set of . We do not discuss here the spin-orbit and Zeeman couplings, so that spin-rotational symmetry is preserved: . Therefore, below we omit the spin index.
For a given spin projection, the retarded Green function of right movers is given, in the closed ring, by a product
of the ZM factor and the factor which describes excitations with nonzero momenta and, in turn, factorizes into a product of the spin and charge parts (hence “SC”), similarly to Eq. (5). For given , the ZM factor in Eq. (11) is written as
is the variation of the ZM energy with changing by unity. The retarded Green function for left movers is obtainable from by changing and .
Within the chiral approximation (5), the factor reads
is given by the same expression with substituted with . The argument
in corresponds to the paths with multiple revolutions around the ring and the summation is taken over all integer and . Equation (LABEL:g) is obtained from Eq. (5) by “replicating” the spin and charge factors, namely by replacing in the plasmon factor and in the spinon one, and summing over and . Importantly, the spin and charge factors are replicated independently—this follows most directly from the bosonization approach, where each of these factors is determined by an independent bosonic field, spinon or plasmon, which is periodic in real space with the period . In this way, we obtain a double sum over and under the square root sign in Eq. (LABEL:g). Note that Eq. (11) with from Eq. (LABEL:g) reproduces the Green function of a noninteracting closed ring by equating and using the property
In the limit , the calculations can be simplified by noticing that the function [and similarly ] is then sharply peaked in time and space, within the small time and space intervals of width and , respectively. Namely, and . This, in turn, means that the peaks associated with different terms in the sum over in Eq. (LABEL:g) are well separated in space and time, and similarly for the sum over . This allows us to commute the square root of the sum in Eq. (LABEL:g) into a sum of square roots,
and write as fractionalization ()
Equations (11), (12), and (16) define the spin-charge separated electron Green function in a closed ring for and . An important property of the spin-charge factor in the Green functions of right and left movers, which we use below, is
The time Fourier transform of the functions defines the Green functions in the energy-space representation. The explicit form of is derived in Appendix B.
ii.3 SCS: Tunneling conductance of a ring
Now we include the tunnel coupling between the ring and the leads (Fig. 1). Both the ZM and plasmon-spinon SCS factors are modified by the “opening” of the ring. Before we proceed to a discussion of these modifications (this will be done in Sec. III.2), let us see in what combination the ZM and SCS factors enter the conductance through the ring. In what follows, we assume for simplicity that the Fermi velocity in the leads is equal to the velocity of excitations in the noninteracting ring: .
We formalize our approach to transport of interacting electrons through the ring tunnel-coupled to two Fermi reservoirs in terms of the Kubo formula, in which the process of single-electron tunneling across the ring is accompanied by tunneling of other electrons which serve as a dephasing environment for the tunneling electron. In the noninteracting limit, the conductance is written as the energy-averaged single-electron transmission coefficient :
where is the transmission coefficient at energy , the factor of 2 accounts for spin, and
denotes the thermal averaging over with the Fermi distribution function, , characterized by the chemical potential in the leads.
Importantly, as was shown in Ref. dmitriev10 () (see also Appendix A), in the vicinity of , backscattering by the tunnel contacts can be neglected (both in the noninteracting and interacting spinless cases). Within the square-root approximation of Sec. II.1, this is also true for spinful interacting electrons. As a result, the right and left sectors of our model are decoupled (for given ). This allows us to use Eq. (18) (averaged over ZM fluctuations) in which SCS modifies the energy dependence of .
Since we assumed the arms of the interferometer to be of equal length (with the coordinates of the point-like contacts and ), the transmission coefficient is expressible in terms of the retarded Green function in the energy-coordinate representation as
where we used the short-hand notation for the Green function that connects the contacts and takes into account tunneling to the leads. In Eq. (LABEL:15), electron-electron interactions are accounted for within the square-root approximation for the Green functions forming the fermion loop for the density-density response function and through the ZM averaging. The dimensionless bare (noninteracting) amplitudes and in Eq. (LABEL:15) describe tunneling of an electron to and out of the ring, respectively (see Appendix A) footnote-contacts (). If, as assumed, the contacts are point-like and right-left symmetric (which together means time-reversal symmetry for scattering at the contact),
independently of the strength of tunneling. The tunneling rate for the noninteracting ring is given by dmitriev10 ():
To make use of the results obtained in Sec. II.2 for an isolated ring, it is convenient to transform the thermal averaging (19) in Eq. (18) with from Eq. (LABEL:15) into the real-time representation, which leads to
Here, the function describes propagation of an electron from one contact to the other in time , including multiple revolutions around the ring and tunneling at the contacts. The function stems from the averaging of the fermionic loop over the ZM fluctuations. The factorization of the integrand in Eq. (21) relies on the “right-left” symmetry of the model (equal arms and chirality separation). Importantly, in the Green function transformed into the coordinate-time representation, the functions and factorize, similarly to Eq. (11), even in the presence of tunneling, as can be seen from the expressions derived in Appendix B. We stress that, because of the “right-left” symmetry, the factor is the same for both right and left movers [for an isolated ring, this follows explicitly from Eq. (17)].
Because of tunneling, decays with time, as a result of which the integral in Eq. (21) converges at . It is instructive, however, to first examine the behavior of the integrand in a closed ring (for more details, see Sec. III.1). In this case, we have
The next step is to use the parameter to simplify . Because of the factor in Eq. (21), we have . This, for typical deviations of from , allows us to neglect the dependence on in , yielding
Here, we used
which follows directly from the definition (13) of . After setting in , the SCS dynamics in the ring decouples from the ZM dynamics at any point in time and is then encoded in Eq. (21) through the function
In a tunnel-coupled ring, both the functions and are modified. First, is still represented in terms of the single-electron propagator in exactly the same way as in Eq. (27), but is now dressed by tunneling vertices and describes the SCS dynamics with the inclusion of tunneling-induced decay. Second, in the tunnel-coupled ring is given by the average of the same exponential function as in Eq. (25) but the meaning of the averaging is different. Specifically, become functions of , because of the exchange of electrons between the ring and the leads, so that the averaging goes over the equilibrium dynamic fluctuations in the open system rather than over the grand canonical ensemble in which were -independent numbers.
The transmission coefficient (21) at can thus be written as
Further, it can be represented as a sum of the classical (, independent of ) and quantum (, describing the AB interference) contributions:
where and correspond to the first and second terms in the square brackets in Eq. (28), respectively. Note that both contributions are affected by SCS, whose dynamics is encoded in the factor .
Iii Dynamics of SCS
The dynamical properties of SCS (see, e.g., Refs. calzona15 (); calzona16 (); acciai17 (); hashisaka17 ()) are described, in the ring geometry, by the function . As mentioned in Sec. II.3, we first examine the behavior of in a closed ring in Sec. III.1. The SCS dynamics in the presence of tunneling will be analyzed in Sec. III.2.
iii.1 SCS dynamics: Isolated ring
defined for , where
with being a step function. Each of the terms in the function is only nonzero within the time interval between and , has square-root singularities at the end points, and is suppressed if the width of the interval
is much larger than as . Note that inside the above interval in the limit except for the narrow regions of width of the order of in the vicinity of the end points.
A useful way of representing in terms of from Eq. (30) is
are the elements of the charge and spin single-particle “fusion” matrices in space at . Note that each of them contains, upon substitution of Eq. (30) in Eq. (27), a double sum over windings; however, in the limit , the contribution of nondiagonal terms in the double sums is small and can be neglected, which is done in Eqs. (35) and (36). This is because of the defining property of and , which is that each of them is characterized by only one velocity: in the former case and in the latter. The SCS dynamics can now be visualized by straightforwardly generalizing the definition of and to arbitrary and thinking of the charge and spin “wave packets” which rotate independently around the ring with velocities and , respectively.
The complex functions and are periodic in and concentrated around and with , respectively. Each of the peaks has a characteristic width of the order of in both the and direction. This is schematically depicted in Fig. 3, where the regions in space in which the charge and spin wave packets are peaked at are marked by black dots. Now, note that there exist such winding numbers for the charge () and spin () that the charge and spin wave packets are peaked at simultaneously. Namely, the spin-charge collision occurs for and obeying
(in Fig. 3, the collision is shown for and ).
While correct in spirit, the above picture of spin-charge collisions is not quite right in one important aspect: it does not yet include the destructive interference between the wave packets, i.e., the cancellation between the different contributions to after the summation over and . For small , the cancellation tends to make the widths of the peaks in shorter compared to . Indeed, in the strictly noninteracting limit, the spin and charge move together, Eq. (37) is always satisfied, and is a strictly periodic series of functions with a period ; specifically,
The factor restricts both and to within the interval between and .
In order to compare Eq. (39) with the noninteracting result, Eq. (38), it is instructive to rewrite the sum over and as a sum over and the difference between the charge and spin winding numbers . Each term in the sum over for given yields a resonant contribution to similar to the delta-function peaks in Eq. (38). One of the modifications brought about by SCS is the broadening of these peaks, as can be seen from the second line of Eq. (39). Namely, the time interval between the exact borders of the “resonance” in that occurs after revolutions of the charge wave packets and revolutions of the spin ones is rewritten as
A more dramatic effect of SCS on the peaks in is a deep periodic modulation of the amplitude of the envelope of the series of peaks, which is a remarkable consequence of the condition (37). These peaks are enumerated by the integer number . For given the equation (“exact spin-charge collision”) has a solution (assuming for a moment that is a continuous variable), where
For integer charge winding numbers close to , the (generically nonzero) interval is small in the sense of the condition (37), provided is sufficiently small. Specifically, for , the characteristic number of peaks around that are not suppressed by the temperature is given by . In the limit of , we have , so that the bunches of peaks centered at are well separated from each other.
With these ingredients, the picture that emerges in the time domain is that of a train of narrow “double-horn” peaks of width
and the nearest-neighbor spacing , which are grouped together in bunches of width of the order of centered at
As we will see below, the subleading term should be kept in Eq. (45) for the calculation of .
The above picture, illustrated in Fig. 4, is in accord with the two time scales and introduced in Sec. I. Specifically, the characteristic bunch width has the meaning of the dwelling time , during which the spin and charge “stick” to each other:
The distance between the bunches has the meaning of the time between consecutive spin-charge collisions
Note that, as is increased, the width of the peaks (44) becomes of the order of at the half-height of the bunches. That is, except for the very center of a bunch, the characteristic width of the peaks in is given by (similar, in this sense, to the peaks in the spin and charge wave packets, illustrated in the cartoon of Fig. 3).
For , it is useful to introduce the (dimensionless) envelope function for a single bunch of peaks by replacing the double-horn peaks with the delta functions and writing in the form
(recall that the double-horn peaks are centered in the middle of the interval between and , see the inset in Fig. 4). The function has the meaning of the integral over the period around the point which belongs to the -th bunch. Using Eq. (31), we obtain in the scaling form
where the shape of the envelope is given by
Doing the integrals (see Appendix C), we have
The envelope of the series of bunches is written as the sum
The picture of the densely packed bunches of peaks describes the case of . In the opposite limit, the width of the bunches becomes smaller than the interpeak distance. Then, a given bunch contains at most one peak or, typically, no peaks at all, depending on the commensurability between and . We leave the discussion of this case and the related commensurability problem out of the scope of the present paper. For the results in the literature on the role of the commensurability in transport through the interacting ring see, e.g., Refs. jagla93 (); hallberg04 (); meden08 (); rincon09 ().
In the above, we have analyzed the behavior of the function [Eq. (27)], which describes the dynamics of SCS, in a closed ring. Now we turn to the function [Eq. (25)] which encodes the ZM dynamics. Consider behavior of in the vicinity of the -th bunch. Taking at the times prescribed by the delta functions in Eq. (50) and neglecting the terms in the exponent, we get
Averaging the exponential factor in Eq. (55) over the grand canonical ensemble in the closed ring, is obtained as a sharply peaked periodic function (Fig. 5). The height of the peaks equals 1 and their characteristic width is given by . Notice that the peaks in appear at precisely the same times as the peaks in the envelope function and are much broader, for , than the latter, namely . As a consequence, the effect of the ZM dynamics in the spinful case is masked by SCS and can be neglected in the closed ring. This is in contrast to the spinless case dmitriev10 (), where the counterpart of is responsible for important changes in the spectrum of the ring, ultimately giving rise to peculiar interaction-controlled AB oscillations dmitriev10 ().
To summarize the results of this section, we have found that the product of the SCS and ZM factors in Eq. (28) is a periodic function with the period (half of that for spinless electrons) imposed by , see Fig. 5. Importantly, this function (which determines the interference part of the conductance) does not decay with time in a closed ring.
iii.2 SCS dynamics: Tunnel-coupled ring
Now we take into account a finite tunneling coupling of the ring to the leads. In the noninteracting case, in the close vicinity of and for one can neglect backscattering at the contacts (see Appendix A) and then simply replace with shmakov13 (), where is the Green function in a closed ring. This replacement introduces a factor in the integrand of Eq. (21). As we demonstrate below, the tunneling coupling in the SCS case is essentially different in that it cannot be characterized by a single tunneling rate, the same for each energy level.
In order to introduce tunneling into the SCS dynamics, it is convenient to return to the representation, as in the noninteracting case. For and , we neglect backscattering at the contacts and represent in the presence of SCS as follows (see Appendices A and B):
is given by Eq. (11), and a similar expression holds for . One way of thinking about the meaning of Eq. (56) is in terms of its expansion in powers of , which is a sum over winding numbers of the paths that connect the opposite leads and are “damped” by the possibility of tunneling out of the ring.
In the limit of weak tunneling, the main contribution to the conductance in Eq. (18) comes from that are close to the energy levels of the isolated ring. Therefore, we can replace with an auxiliary Green function whose dependence coincides with that of in the vicinity of the poles of . It is notable that the dependence of is similar to that of in Sec. III.1, namely both functions are peaked around the times [Eq. (46)]. It follows that, in contrast to the noninteracting ring, where the poles of the Green function at are characterized by a single quantum number , the poles in the interacting ring are enumerated by a set of two indices, and in the notation of Sec. III.1. Namely the poles occur at , where
To derive this equation, we notice that the spin and charge factors in [Eq. (30)] are periodic functions of time with the periods and , respectively. Therefore, the energy levels of the system can be written as with integer and (see Appendix B). Using Eq. (4) for , we obtain Eq. (58) with
That is, the levels acquire a fine structure because of SCS, with the “sublevels” enumerated by .
As shown in Appendix B, the weight of a sublevel in the Green function vanishes with increasing for . Effectively, for the purpose of visualizing resonant transport through the ring for , one can think of each level of the noninteracting system being split into sublevels. More specifically, the calculation presented in Appendix B yields
where , the structural factor
with from Eq. (59) is expressible in terms of the dimensionless function
and the sublevel broadening is related to by
The function (62) obeys
The physics of SCS manifests itself through the splitting of the energy levels, with the sublevel spacing in Eq. (58), and through the -dependence of and . The dependence of these quantities on does not affect the results for the conductance qualitatively (see Sec. IV.3). Moreover, as we will demonstrate below [Eq. (153)], the classical term in the conductance contains the sum and thus—because of the property (64)—is not sensitive to the dependence of on even quantitatively. Therefore, for the sake of simplicity, in the discussion in the rest of this section and in the beginning of Sec.