Transport through constricted quantum Hall edge systems: beyond the quantum point contact
Motivated by surprises in recent experimental findings, we study transport in a model of a quantum Hall edge system with a gate-voltage controlled constriction. A finite backscattered current at finite edge-bias is explained from a Landauer-Buttiker analysis as arising from the splitting of edge current caused by the difference in the filling fractions of the bulk () and constriction () quantum Hall fluid regions. We develop a hydrodynamic theory for bosonic edge modes inspired by this model. The constriction region splits the incident long-wavelength chiral edge density-wave excitations among the transmitting and reflecting edge states encircling it. The competition between two interedge tunneling processes taking place inside the constriction, related by a quasiparticle-quasihole (qp-qh) symmetry, is accounted for by computing the boundary theories of the system. This competition is found to determine the strong coupling configuration of the system. A separatrix of qp-qh symmetric gapless critical states is found to lie between the relevant RG flows to a metallic and an insulating configuration of the constriction system. This constitutes an interesting generalisation of the Kane-Fisher quantum impurity model. The features of the RG phase diagram are also confirmed by computing various correlators and chiral linear conductances of the system. In this way, our results find excellent agreement with many recent puzzling experimental results for the cases of . We also discuss and make predictions for the case of a constriction system with .
pacs:73.23.-b, 71.10.Pm, 73.43.Jn
Despite being a subject of intense experimental and theoretical interest, much is yet to be learnt of the combined effects of electron correlations and impurities on the transport properties of low-dimensional strongly correlated systems. The availability of several non-perturbative theoretical methods for studying the physics of systems in one spatial dimension has, however, allowed for considerable progress to be made for such systems giamarchi (). A physical one-dimensional system ideal for studying these issues are fractional quantum Hall edges (FQHE) wen (); wenint (). Considerable experimental advances have been made in exploring the physics of the edge states changrev () and in confirming many of the theoretical predictions made of the remarkable properties of these systems wenrev (). Several recent experiments have, however, pointed out the need to develop a deeper theoretical understanding of inter-edge quasi-particle tunneling phenomena in FQHE systems with gate-voltage controlled constrictions roddaro1 (); roddaro2 (); chung (); comforti (). These experiments serve as the primary motivation for the models proposed in this work. However, before discussing these experiments, we first present a discussion of the existing theoretical paradigm for the understanding of inter-edge tunneling physics in FQHE systems.
Kane and Fisher kane (); fisher () observed in their classic work that (a) the tunneling between two FQH edges separated by the FQH fluid was akin to the backscattering of electrons by an impurity in a Tomonaga-Luttinger liquid (TLL) and (b) the tunneling between two FQH bubbles separated by vacuum was akin to the tunneling of electrons across a weak-link (infinitely high barrier) between two TLLs. Their perturbative analysis revealed that process (a) was relevant under RG transformations while process (b) was irrelevant, thus suggesting that the low-energy physics of the FQHE tunneling problem was likely to be that of two FQH bubbles separated by vacuum. Both these scenarios are described by the boundary sine-Gordon model gogolin (). In the following years, a quantum Monte-Carlo simulation by Moon etal. moon (), an instanton calculation by Furusaki and Nagaosa furusaki (), a conformal field-theory analysis by Wong and Affleck wong () as well as the exact solution of Fendley etal. fendley () using the thermodynamic Bethe Ansatz method demonstrated that these scenarios were, in fact, correct in their description of the system. Further, they showed that, within the confines of the boundary sine-Gordon model, there was no intermediate fixed point in the RG flow of the backscattering/tunneling couplings in this model. Several works have also analysed the effects of inter-edge interactions oreg (); pryadko (); apalkov (), disorder kane2 () on quasiparticle transport in FQH edge systems. Attention has also been given to tunneling at point-contacts between FQH fluids with different filling-fraction chklovskii () as well as at contacts with Fermi liquid reservoirs chamonfrad (). More recently, attempts have been made at developing a more general theory for the study of critical points in edge tunneling between generic FQH states moore1 (); moore2 ().
The phenomenological description of tunneling between chiral edges outlined above relies on the following scenario. For no backscattering coupling between the two edges of opposite chirality at, say, , we have a system of two chiral 1D systems which are continuous at . This can be seen by consulting Figure (1) given below for the case of the fields and being continuous. Upon introducing a small RG-relevant inter-edge tunnel coupling, we are left at strong backscattering coupling with a system in which the earlier edges are now discontinuous across ; they have, in fact, now become reconnected in a different configuration, with the fields and now being continuous (as can be seen in Figure (1) below). This means that, in order to describe ballistic transport intermediate between these two which is characterised by a finite backscattering of current, one must consider the possibility of the fields describing the chiral edge excitations as being discontinuous across . In doing so, it appears necessary to rely on ideas non-perturbative in nature. Accounting for additional quasiparticle tunneling among the various incoming and outgoing chiral edges is then likely to lead to a non-trivial variation of the boundary sine-Gordon model. Insights on these issues were gained recently in Ref.(epl ()), in the form of a new model for the constriction geometry in quantum Hall system which, while being simple in essence, is clearly beyond the paradigm of the quantum point contact. We aim here to develop the ideas presented in that work, exploring more fully the consequences of such a constriction system.
As will be discussed in the next section, several recent experiments on inter-edge tunneling in FQHE systems show that it is possible to use the voltage of a split-gate constriction to tune the inter-edge transmission to values intermediate to those in the two scenarios described above. Further, they reveal a very interesting evolution of the transmission through the constriction with decreasing inter-edge bias. This will lead us to formulate a simple phenomenological model for the split-gate constriction region. We will then perform a Landauer-Buttiker analysis and compute the conductances of the model. The results of this analysis will be seen to point to some interesting conclusions for transport in the presence of a constriction. It is now well established that the low-energy theory for the dynamics of the gapless long-wavelength excitations on the edges of a FQH system are described by a hydrodynamic continuum chiral TLL theory wen () of propagating density disturbances which are bosonic in nature. Adhering to the spirit of such a hydrodynamic description, we formulate a continuum model for the constricted quantum Hall edge system in section III. In section IV, we introduce local quasiparticle tunneling processes inside the constriction and construct a boundary theory for the problem. In this way, we investigate the RG phase diagram of the system for the various tunnel couplings. We complete the study in section V by computing several chiral correlators and conductances at weak- and strong-quasiparticle tunnel coupling values. We then present a comparison of the results of our model with those obtained from recent experiments in section VI. Here, we will also reflect on the relevance of our model to the case of a constriction with a filling factor of . We end by discussing some finer aspects of the model and outlining some open directions in section VII.
Ii Model for a split-gate constriction
We now propose a simple, phenomenological model for a split-gate constriction created in a quantum Hall system. A schematic diagram of an experimental setup of a FQH bar with a gate-voltage controlled constriction is shown below in Figure (2).
As is indicated in Figure (2), the constriction is created electrostatically in a two-dimensional electron gas (2DEG) quantum Hall system at filling-fraction by the electronegative gating of metallic split-gates. An important effect of the split-gate constriction is to bring the two counter-propagating edges of the Hall fluid in close proximity, allowing for the possibility for quasiparticles to tunnel between them. As discussed earlier, this has been a major focus in the study of the physics of FQHE systems. However, an often neglected effect of the split-gates is that the electric field induced by them reduces the 2DEG density (and hence the filling-fraction of the Hall fluid) in the narrow constriction region; the inter-particle correlations in the constriction are thus likely to increase in strength. We can, therefore, expect the filling-fraction of the FQH fluid in the constriction, , to be a function of as well as the gate-voltage , i.e., , in such a way that (i) for (i.e., no constriction) and (ii) for (i.e., with a constriction). While the filling-factor (for being an odd integer, such that we have only single edge states) can be related to the strength of the inter-edge density-density interactions, , in the bulk of the FQH system wenint (); fisher (); chklovskii ()
no such simple relation exists, at present, for the filling-factor in the constriction, . Clearly, this will need a greater understanding of the role of the gate-voltage in creating the constriction.
ii.1 Surprises from the experiments
We now turn to a discussion of the several puzzling results observed in experiments on transport through split-gate constrictions in integer roddaro1 () and fractional roddaro2 (); chung () quantum Hall systems and outline the several intriguing results observed therein. Working with an experimental setup as shown in Fig.(2), a finite dc bias between the two edges coming towards the constriction is imposed through the source (S) terminal while the drain (D) terminal as well as terminals and are kept grounded.
(i) A current is incident on the constriction from the upper-left edge and is partially transmitted with the transmitted current finally being collected at the terminal 3. The reflected current is collected at terminal 1 and gives rise to a bias-independent longitudinal differential resistance across the constriction at large bias .
(ii) The two-terminal differential conductance is measured at temperatures as low as and gives the transmission coefficient of the constriction (where is the Hall conductance of the bulk; in Ref.roddaro1 () and in Ref.roddaro2 (); chung ()). At sufficiently large values of the gate-voltage and large bias , is observed to saturate with at a value less than unity. Further, is observed to dip sharply and vanish with a power-law dependence on as . A comparison with the theory of inter-edge Laughlin quasiparticle tunneling developed by Fendley etal. fendley () suggests strongly that the constriction transmission is governed by the local filling-factor of the Hall fluid in the constriction, even though this region is likely to be small in extent. This is unexpected for the case of the bulk being in an integer quantum Hall state roddaro1 () where edge transport is understood in terms of noninteracting electron charge carriers.
(iii) A particularly intriguing observation is that of the evolution of the constriction transmission at very low temperatures (e.g., ) as the split-gate voltage is varied in the limit of vanishing inter-edge bias . While shows a zero-bias minimum at sufficiently large , decreasing leads to a bias-independent transmission at a particular value of and then to an enhanced zero-bias transmission for yet lower values of . The same behaviour of the zero-bias transmission is also observed by holding the gate-voltage fixed and lowering the temperature from to . For the case of , the bias-independent transmission is observed at a value of roddaro1 () while for , it is observed at roddaro2 (); chung (). A similar enhancement of the zero-bias transmission at sufficiently weak gate-voltages was also reported for the case of bulk filling-fractions and chung (). The bias-independence as well as the enhancement of the transmission is quite unexpected from the viewpoint of the theoretical framework of edge tunneling described earlier.
(iv) The constriction transmission for a bulk system displayed two dip-to-peak evolutions, with bias independent behaviours observed at and roddaro1 (). This appears to indicate the independent effects of the two edge modes in the system.
(v) Varying considerably the size and shape of the metallic gates (which form the constriction region) did not appear to affect the dip-to-peak nature of the evolution of the constriction transmission with the strength of the gate voltage roddaro3 ().
Let us now consider the probable effects of a split-gate voltage constriction. Clearly, other than promoting the tunneling of quasiparticles between oppositely directed edges (due simply to enhanced wavefunction overlap due to the proximity of the edges), the more noteworthy effect is likely to be the creation of a smooth and long constriction potential, which depletes the local electronic density (and hence lowers the local filling factor) locally from its value in the bulk. Indeed, this led Roddaro and co-workers roddaro1 () to conjecture on the possibility of a small region in the neighbourhood of the constriction with a reduced filling factor () as the cause of their puzzling results (see fig.(2)). This conjecture, however, remained unsubstantiated by the formal analysis of a concrete theoretical model. Thus, their explanations for the system remained suggestive at best and no attempts at unifying the observations at both integer and fractional values of were made. Thus, the pressing questions that remain to be answered are as follows. What drives the gate-voltage tuned insulator-metal transition at vanishing edge-bias in the constriction system (as evidenced by the dip-to-peak evolution with decreasing strength of the gate voltage)? Can purely local interedge quasiparticle tunneling processes, which need an interplay of impurity scattering and electronic correlations kane (), be the sole cause? Is there a symmetry governing the edge-bias independent response of the constriction transmission at a critical value of the constriction filling factor (as seen by tuning the gate voltage)? If the system is indeed critical at this point, what does its gapless theory look like?
At the same time, earlier theoretical efforts papastroh (); papamac (); agosta () were unable to provide any simple explanations of these experimental observations. Most notably, the scenario proposed in ref.papastroh () involved the complications of stripe states arising from longer range interactions. However, it failed to present any mechanism in explaining the evolution of with . The same is also true of proposals of line junctions papamac () as well as the effects of inter-edge interactions on quasiparticle tunneling agosta (). Thus, keeping in mind that the theory of refs.kane (); fendley () matches the experiments in only a very restricted parameter regime, the lack of a clear theoretical understanding remained an important problem to be addressed. The creation of a model with an effort towards explaining the puzzles was, therefore, the main motivation of an earlier work epl (). In what follows, this model is first formulated and then analysed in detail.
ii.2 Landauer-Buttiker analysis of transport
Inspired by these experimental findings, we build, in the remainder of this section, a simple phenomenological model of a FQH split-gate constriction with a reduced local filling-fraction. In this way, we aim to provide a qualitative understanding of some of the observations discussed above. Furthermore, certain elements of this simple model will then be employed as input parameters in a more sophisticated theory involving bosonic edge excitations in subsequent sections in providing explanations of some of the other, more puzzling, experimental results. The analysis of this model will be carried out in two ways. The first will involve an explicit calculation of the various Landauer-Buttiker conductances of the measurement geometry. In the second analysis, we will show how the results of the explicit calculation can be derived more simply by making two assumptions of the system at hand.
The central feature of our model is the region of lowered filling factor () assumed to be created by the split gate constriction gates. Let us now estimate the spatial extent, , of the region. This can be done by noting that the transport data taken at a temperature of does not appear to show any interference effects arising from coherence across the entire constriction roddaro1 (). Thus, can be safely assumed to be longer than the thermal length (where is the edge velocity). For a typical komiyama () and , . Clearly, magnetic length , justifying our assumption of the mesoscopic nature of the region.
In a Landauer-Buttiker analysis buttiker1 (), the net currents flowing in the various arms are assumed to satisfy linear relations with the applied voltages (valid for small values of the voltages), with the proportionality factors being the various transmission coefficients for the quantum system. Solving the various linear relations between the currents and voltages gives us the various conductances of the system. It is helpful to use the fact that the net current for voltage probes is zero, and that we have the freedom to set the voltage of one of the terminals to zero (as currents are related to applied voltage differences). Thus, in Fig.(3), we set , and since terminals and are voltage probes, . When put together with the fact that terminals 2 and 3 are grounded, i.e., , this allows us to exclude the current-voltage relation for terminal altogether (i.e., remove one row from the transmission matrix linking the currents and voltages). Thus, we can write the current-voltage relations in matrix form as
where the current and voltage column vectors are and respectively and the transmission matrix is given by
where is the transmission coefficient for the current backscattered from the constriction. We now solve these linear relations. Measuring all voltages with respect to terminal (which we have set to zero), we can see that as , we find . Further, from , we get
The current leaving the circuit at terminal 3 is given by (where is the current transmitted through the constriction region from terminal to terminal ). This gives us
In a similar manner, we can compute the current leaving the circuit at terminal (which, with terminal being grounded, consists entirely of the current backscattered from the constriction) as . Then, from overall current conservation in our circuit, the total injected current is given by , which gives us
This leads us to . This expression for can also be found very simply by noting that the constraint of unitarity for the transmission matrix means that the sum of the elements in every row (or every column) must add up to zero buttiker1 (). We can now also compute the conductance (in units of ) due to the current backscattered from the constriction as
This also gives us the “background” value of the resistance drop across the constriction as
Having carried out the Landauer-Buttiker analysis, we now show how all of
the results obtained therein can be rederived through
a simple analysis of the circuit which relies on essentially two assumptions
on the nature of the system at hand and the conservation of current epl ().
This will allow us to reflect on the simplicity and efficiency of the assumptions.
Thus, let us begin by stating the assumptions made and show
how they lead in a straightforward way to simple relations for several
physical quantities measured in the experiment. These are:
(i) the voltage bias between the two edges of the sample (i.e., the Hall voltage for the system being in a quantum Hall state) is not affected by the local application of a gate-voltage at a constriction as long as the bulk of the system is in an incompressible quantum Hall state with filling-fraction ,
(ii) the two-terminal conductance measured across the constriction is determined by the current transmitted through it, which in turn is governed by the filling-fraction of the Hall fluid in the constriction, . This needs the breakup of the current coming towards the constriction to take place at the boundary and constriction Hall fluid regions (which is sufficiently far away from the center of the constriction region).
Thus, by denoting the current injected into the system from the source terminal as , we know that where is the bulk Hall conductance and is the edge-bias. From assumption (ii), denoting the current transmitted through the constriction as , it is clear that , where is the two-terminal conductance measured across the constriction. Putting these two relations together using assumption (i), we obtain the transmitted current in terms of the incoming current as
Thus, we see that our assumptions give us a very simple relation for the the splitting-ratio for the currents at the constriction (which is simply related to the transmission coefficient of the constriction discussed above for no inter-edge tunneling) as being . Now, from Kirchoff’s law for current conservation, we get the current reflected at the constriction . This then gives the minimum value of the backscattering conductance as
is simply related to the reflection coefficient of the constriction for no inter-edge tunneling, and shows that the effective filling fraction governing is . Now, with the current at terminal , , being the transmitted current , we get (since ), giving . We then find the “background” value of the longitudinal resistance drop across the constriction to be
which arises from the partial reflection and transmission of the incoming edge current due to the mismatch of the filling-fraction in the bulk and constriction regions. The experimentally obtained value for is, in fact, used by the authors of Refs.roddaro1 (); roddaro2 () to determine the value of the constriction filling-factor from eq.(10). Further, we can see that and are simply related by . More generally, the differential longitudinal drop across the constriction is related to (and also experimentally determined in roddaro2 ()) the differential backscattering conductance by the simple relation, as seen earlier
Further, we also check that the Hall conductances measured on the two sides of the constriction are determined by alone
Thus, we see that by allowing for the constriction region to have a reduced filling-fraction () than that of the bulk () and making the two assumptions stated above, we are able (i) to find a simple expression for the splitting-ratio of the current incident on the constriction (or, the zeroth constriction transmission coefficient) as well as (ii) find an expression for the longitudinal resistance drop across the constriction which arises from the breakup of the current.
At the heart of these results lies the fact that a constriction region with a reduced filling-fraction necessitates the transfer of charge from the incoming edge to the opposite outgoing edge via the incompressible bulk. Put another way, it becomes imperative to consider the non-conservation of edge current in studying transport across such a constriction. This is characterised by the presence of a current reflected at the boundary of the bulk and constriction regions in the model setup above. While charge dissipation away from the edge can be modeled in terms of quasiparticle tunneling at multiple point-contact junctions chamonfrad (); ponomarenko (), such a mechanism appears to be incompatible with the experimental finding of an edge bias-independent current reflected from the constriction region. The existence of a narrow gapless region of Hall fluid lying in between the incompressible bulk and constriction Hall fluid regions may well provide an answer: such a gapless region would act as a channel for the current reflected from the constriction region. It is, therefore, tempting to speculate on the possibility of a non-perturbative physical mechanism neto () of a chiral Tomonaga-Luttinger liquid undergoing charge dissipation along a short stretch of its length while in contact with a bath (the gapless region) as being the microscopic origin for the phenomenological model described above.
While there are ways of studying the electrostatic effects of a gate-voltage controlled constriction on the incompressible quantum Hall fluid chklovskii (); papastroh (), we have instead chosen a particularly simple and tractable path for modeling the edge structure which involves very few details pertaining to the bulk. The electrostatic calculations of Ref. papastroh () explore the possibility of edge reconstruction within the constriction region, i.e., long-range interactions between electrons in the quantum Hall ground state giving rise to a set of compressible and incompressible stripes at the edge yoshioka (). In this work, however, we consider only short ranged electron correlations, which cause the formation of the chiral TLL state without any intervening stripe states wen (). Further, we neglect the possibility of the formation of line-junction nonchiral TLLs across the vacuum regions in the shadow areas of the metallic gates papamac (), focusing instead on the transmitted and reflected edge states arising from the nature of the Hall fluid inside the constriction. Thus, we devote our attention to short-ranged electronic correlations which cause the formation of chiral Tomonaga-Luttinger liquid (TLL) edge states (without the intervention of any stripe states papastroh () arising from longer range interactions).
As we will see in the following sections, such a model of a constriction in a quantum Hall sample allows for considerable progress to be made in developing a (quadratic) effective field theory for the ballistic transport of current in terms of propagating chiral edge density-wave excitations. Interesting consequences for quasiparticle tunneling will then be shown to result from the exponentiation of these quadratic fields, in particular, giving rise to the competition between two RG relevant quasiparticle tunneling operators which determine the fate of the low-bias transmission and reflection conductances through the constriction. In this way, we will show how our model is able to provide a qualitative understanding of the puzzling findings of the experiments mentioned above in a unified manner. While it appears difficult at first to formulate a continuum model describing a scenario of intermediate ballistic transmission of current through such a constriction by a quadratic bosonic field theory similar to that of Wen wen (), we find that considerable progress can be made by understanding the role of matching (or boundary) conditions in such a theory. In this way, we are able to set up in the following section a very general Hamiltonian, as well as action, formalism describing transport through such a constriction system.
Iii Continuum theory for the constriction system
In this section, we develop a continuum theory for the model of the constriction system presented above. However, for the sake of clarity and continuity, we begin by presenting the basic ingredients of Wen’s continuum theory for the infinitely long chiral-Tomonaga Luttinger liquid wen ().
iii.1 Continuum theory for infinite chiral TLL
Wen’s hydrodynamic formulation describes the excitations of such a system in terms of chiral bosonic density wave modes. The Hamiltonian (and the action) is quadratic in the bosonic field (where is the Euclidean time) and has two parameters: the edge velocity and the filling fraction . This is shown below in Fig.(4).
The energy cost for density distortions of the edge of the quantum Hall system were shown by Wen to lead to a Hamiltonian (for, say, the right-moving edge of a Hall bar)
The equal-time (Kac-Moody) commutation relation for the bosonic field is given by
which makes the momentum canonically conjugate to . The edge density distortion is given by and the Hamilton equation of motion gives
This gives us that the density . Further, from the equation of continuity
we find the current density as . Fourier transforming the equation of motion gives us the expected linear dispersion relation for the edge density waves as . From the commutation relations, we obtain the Legendre transformation for the Hamiltonian . This leads to the Euclidean action for the chiral (right moving) TLL as
The Hamiltonian for the left-moving edge density wave is the same as that given above by for , but the density . As the equal-time commutation relation , the action for the left moving edge chiral TLL has a Legendre transformation term .
iii.2 Continuum theory for the constriction edge model
We now formulate a continuum theory for the constriction edge model discussed in section II along the lines of the Wen hydrodynamic description described just above. The aim will, therefore, be to develop a quadratic theory in bosonic fields in an edge model consisting of chiral, current carrying gapless edge-density wave excitations describing ballistic transport through the transmitting and reflecting edges states surrounding the constriction region. This is shown in Fig.(5) below. As discussed earlier, such a model is critically needed in order to describe the experimentally observed scenario of intermediate ballistic transmission through the constriction roddaro1 ().
We take the spatial extent of the constriction region to lie in the range , where is the total system size and is the magnetic length; the external arms meet the internal ones at the four corners of the constriction. From our earlier discussions, it is also evident that governs the properties of the four outer arms while that of the upper and lower (transmitted) arms of the circuit at the constriction. The effective filling factor for the right and left (reflected) arms of the circuit is treated as a parameter to be determined. We focus in this work on the effects of a changing filling fraction, keeping the edge velocity the same everywhere.
We will now set forth the Hamiltonian formulation of the model. This approach will elucidate the importance of matching (or boundary) conditions in providing a correct and consistent description of the dynamics of the system epl (). We will follow this up by providing the more elegant formulation of the problem based on the action, showing how the information content of the boundary terms is already included in this language.
iii.2.1 Hamiltonian formulation and matching conditions
The energy cost for chiral density-wave excitations that describe ballistic transport in the various arms of the circuit shown in fig.(5) is given by a Hamiltonian where
The densities are, as usual, represented in terms of bosonic fields describing the edge displacement wen ()
The commutation relations satisfied by these fields are familiar
Further, the Hamiltonian equations of motion derived from again describe the ballistic transport of chiral edge density waves
The given above, however, needs to be supplemented with matching conditions at the corners of the constriction for a complete description. From the form of , it is clear that we need two matching conditions at each corner; a reasonable choice is one defined on the fields and one on their spatial derivatives. We choose, for instance, at the top-left corner
where and are the spatial coordinates describing the and arms respectively. Similarly, we choose the following matching conditions at the other three corners as
The equation of continuity leads to the familiar form for the current operator , where . Thus, we can easily see that current conservation at every corner arises from the matching conditions on the bosonic fields . While the transmitting chiral edge modes convey a finite current across the constriction, the reflecting chiral edge modes convey a finite “backscattered” current across the sample. In this way, we formally establish the intermediate ballistic transmission scenario as observed in the experiments. Charge density fluctuations at each corner are described by the matching conditions on . This matching condition is a statement of the conservation of net charge density at each corner. In this way, the two sets of matching conditions together establish the continuity of current and charge density at every corner of the junction system.
Using eqs.(22), we compute the commutation relation
giving us . The commutation relation for similarly yields once again. This is in conformity with our result for from the Landauer-Buttiker calculation. We now demonstrate explicitly that the cases of a perfect Hall bar () and two Hall bubbles separated by vacuum () can be modeled as special limiting cases of the matching conditions (eqs.(22)) given earlier. For , the commutation relation of the reflecting edge states vanishes, killing its dynamics. This can also be understood within a hydrodynamic prescription wen (), where a vanishing effective filling factor (the amplitude of the Kac-Moody commutation relation, eq.(20)) leads to a diverging energy cost for edge charge density fluctuations; the dynamics of the bosonic field characterising such fluctuations is thus completely damped. Thus, the reflecting edge states carry no current, while the transmitting edge states perfectly transmit all incoming current into the outgoing arms on the opposite side of the constriction. The matching conditions eqs.(22) at the four corners are then reduced to
These identifications of the fields and their spatial derivatives lead to the continuity conditions which underpin the hydrodynamic theory of Wen wen (); kane () for the case of the two infinite chiral edges (say, upper and lower) of a Hall bar (with filling factor ), and eq.(20) then reproduces the well-known Kac-Moody commutation relation everywhere along the edges. This is shown in Fig.(6) below.
Similarly, for the case of , the commutation relation for the transmitting edge states vanishes, killing its dynamics: they carry no current, while the reflecting edge states perfectly convey all incoming current into the outgoing arms on the same side of the constriction. Thus, the matching conditions eqs.(22) at the four corners are reduced to
Again, these identifications of the fields and their spatial derivatives lead to the continuity conditions which underpin the hydrodynamic theory of Wen wen (); kane () for the case of the infinite chiral edges (say, left and right) of two distinct Hall bubbles (each with filling factor ) separated by vacuum, and again reproduce the familiar Kac-Moody commutation relations everywhere along the edges. This is shown in Fig.(6) below.
We have, in this way, constructed a family of free theories describing ballistic transport through the constriction at intermediate transmission, with those of complete transmission and reflection representing two special cases. This represents an importance advance in generalising the quantum impurity model of refs.kane (); fendley ().
iii.2.2 Action formulation
In this subsection, we discuss the action (or Lagrangian) formulation of our problem. We will, in this way, demonstrate how the information content of the matching conditions above is already encoded in the action of the system in the forms of terms involving the local fields which are connected to one another by the matching conditions in the Hamiltonian formalism. Thus, we begin by writing down the action for the constriction model
where the action for the outer incoming and outgoing arms is
and we have normalised the entire action with regards to the bulk filling-fraction . Further, the action for the inner edges is
where, by assuming that the properties of the upper and lower edge transmitted edge states of the constriction are determined by the effective filling-fraction inside the constriction , the quantity is simply given by . The quantity (where is the effective filling-fraction for the reflected edge states on the left and right) will be determined from the analysis presently. It is worth noting that the same information can be obtained from the Hamiltonians (equns.(18)) and commutation relations (equns.(20))together. Finally, the action for the corner nodes is given by
We can now see the effects of these local terms in the action by computing the equations of motion for the various fields from the action. For the sake of brevity, we carry out this exercise at only the upper-left corner. The results obtained from the other three corners are precisely the same. Thus, we first compute the equation of motion of the “outer” field by extremising the action with regards to
where we have suppressed the dependences of the fields on the spatial coordinates for the sake of compactness. From this, we can immediately see the matching conditions on and at given earlier. We now compute the other two equations of motion at the top left corner in the same way. We find, thus,
from which we can see that the currents and are given by
In the above relations, the currents and corresponding densities are those propagated from the incoming arm into the u(pper) and l(eft) edge states respectively. Now, by applying Kirchoff’s law for the conservation of current (or, more generally, the equation of continuity) at the upper left corner junction, , we obtain
which for gives . This, then, gives us the effective filling-fraction of the reflected edge states as . In this way, we can see that the action contains all the information content given by the Hamiltonians together with the matching conditions.
Iv Boundary theory for the constriction system
In this section, we evaluate the role played by local inter-edge quasiparticle tunneling processes deep inside the constriction region in determining the fate of transport through the constriction. In order to do so, we proceed by first integrating out all bosonic degrees of freedom except the few involved in the tunneling processes. In this way, we are left with an effective boundary theory kane (). Given that we have a Gaussian action in terms of the bosonic fields, integrating out various bosonic degrees of freedom can be easily accomplished by performing Gaussian integrations gogolin (); giamarchi (). (another analogous method involves using the solutions to the equations of motion kane (); lalcontacts ()). As this is a very standard procedure, we refer the reader to Refs.(gogolin (); giamarchi ()) for details. We pass instead to presenting the various boundary theories obtained in our model, revealing in turn the two interedge quasiparticle tunneling processes which compete in determining the low energy dynamics of the system.
Now, as long as there is no quantum coherence across the constriction region, it is easily seen that the problem of weak, local quasiparticle tunneling between the upper (u) and lower (it d) edges deep inside the constriction region (at, say, ) is exactly the same as that of local quasiparticle tunneling between the oppositely directed edges of a homogeneous quantum Hall bar with filling fraction kane (). Importantly, the charge and statistics of the quasiparticles undergoing such tunneling processes should be governed by the local filling fraction alone. Thus, in the action formalism presented earlier, such a quasiparticle tunneling process can be added to the action by the term , where the tunnel coupling strength is given by . Integrating out all bosonic degrees of freedom but , we obtain the familiar Kane-Fisher type boundary theory kane ()
Applying a standard RG procedure, we find the RG equation for as
As , the coupling is found to be RG relevant and will grow under the flow to low energies/long lengthscales. Further, this quasiparticle tunneling process will clearly lower the transmission conductance across the constriction (for a source-drain bias as shown in Fig.(2)).
We have, however, at least one other local quasiparticle tunneling process to account for: it is that between the left (l) and right (r) edges of the constriction and is revealed by the generalised quasiparticle-quasihole symmetry of the ground-state in the lowest Landau level (LLL) girvin (); epl (). This symmetry dictates that all properties of a quantum Hall system composed of quasiparticles in a partially filled lowest Landau level and with a filling factor can be equivalently described by those of a quantum Hall system composed of quasiholes and with a filling factor . This simple relation between and can be derived easily for the case of the filling factor (and, hence, electronic density) of the quantum Hall system deviating from a filling factor of (where ) yoshioka (). To see this, first note that by increasing the electronic density of the system, we add quasiparticles for each electron added. Then, for being the original electronic density, the new increased electronic density and the density of quasiparticles,
where is the magnetic length and we have used . This gives us the quasiparticle filling factor as
where is the electronic filling factor. A similar calculation for the case of an equally lowered value of the electronic density can also be carried out. We must now remember that we add quasiholes to the system for every electron removed. Then, by following the same line of arguments, we get the quasihole filling factor as
Thus, we can see that and are related by a quasiparticle-quasihole conjugation transformation. Further, for the case of , and