Duality between different geometries of a resonant level in a Luttinger liquid
We prove an exact duality between the side-coupled and embedded geometries of a single level quantum dot attached to a quantum wire in a Luttinger liquid phase by a tunneling term and interactions. This is valid even in the presence of a finite bias voltage. Under this relation the Luttinger liquid parameter goes into its inverse, and transmittance maps onto reflectance. We then demonstrate how this duality is revealed by the transport properties of the side-coupled case. Conductance is found to exhibit an antiresonance as a function of the level energy, whose width vanishes (enhancing transport) as a power law for low temperature and bias voltage whenever , and diverges (suppressing transport) for . On resonance transmission is always destroyed, unless is large enough.
pacs:71.10.Pm, 73.63.-b, 73.23.Hk
Introduction.— Understanding the behavior of strongly correlated systems has been one of the central themes of condensed matter physics in recent years. Of these, one-dimensional systems stand out as a clear example of non Fermi liquid behavior. When no symmetry is spontaneously broken, the low energy physics of those systems is governed by the bosonic Luttinger liquid (LL) theory (1). This description applies to various experimental realizations, including semiconducting quantum wires, metallic nanowires, and carbon nanotubes; it is also related to the physics of the edges of quantum Hall systems (2) and spin quantum Hall systems (3). An important question, from both fundamental and applicative perspective, is the effect of (randomly or intentionally introduced) impurities on such systems. Particularly interesting are dynamic impurities, e.g., resonant levels which can fluctuate between the occupied and unoccupied states. They can be realized, among other possibilities, as semiconducting quantum dots, metallic grains, or carbon nanotubes or buckyballs. Indeed, much effort has been attracted to the understanding of their effects on transport (1); (4); (5); (6); (7); (8); (9); (10); (13); (11); (12); (14) as well as thermodynamic (15); (17); (16); (18); (19); (20); (21) properties.
An important insight into strongly interacting theories is provided by dualities, i.e., mappings between the properties of a system and those of a different system, usually with reversed coupling strengths. In condensed matter physics, this goes back to the famous Kramers-Wannier duality of the Ising model. Another example, in the context of this work, is the duality between the strong- and weak-coupling limits of a static (potential) impurity in a LL (4). In this work we find a different kind of duality for a level coupled to a LL: an equivalence between the side-coupled and embedded geometries depicted in Fig. 1, valid even in the presence of a finite bias voltage. In this mapping, the LL parameter goes onto and transmittance goes into reflectance, but the strength of the level-lead coupling is unchanged: a strongly (tunnel-) coupled level is mapped onto a strongly coupled level, and vice-versa. In the following, after proving this result, we demonstrate its power by characterizing the transport properties of the side-coupled system for arbitrary strength of the electron-electron interaction, which, to the best of our knowledge, have been previously discussed only for weak interaction (14).
Model.— The system depicted in Fig. 1(a) is described by the Hamiltonian , where, for spinless particles (spin effects will be discussed later):
is the lead bosonized Hamiltonian, expressed in terms of the bosonic fields and , obeying the commutation relation , and where is the interaction parameter ( for repulsion, for attraction) and is the velocity of excitations (1). The level Hamiltonian is , with the level Fermi operator, and its energy. The level and the lead are connected by a tunneling term (effects of level-lead interaction will be considered momentarily):
Here is the tunneling matrix element, and the lead right (left) moving Fermi operators can be expressed in terms of the bosonic fields through , where are chiral right (left) moving fields obeying and are Majorana Fermions ( is the bandwidth).
Duality.— We now turn to the derivation of our central result: the duality between the side-coupled and embedded geometries. Let us define two new bosonic fields, and , in terms of which the interaction parameter is eliminated from the but introduced into . We can also use these fields to write down decoupled (Bogolubov-transformed) right and left moving fields . We then apply a unitary transformation, , where . The transformed Hamiltonian is similar to the original one, except for the addition of a term of the form , as well as a modification of the Fermi operators at the origin to . The Hamiltonian now takes the form , with , and:
This is the Hamiltonian of two chiral LLs (corresponding to the decoupled left and right movers in the original model) with LL parameter , which are symmetrically coupled to a level by both a tunneling term of the same amplitude , and a local charging interaction of strength . But coupling a level to a chiral LL is known to be equivalent to coupling it to the edge of a non-chiral LL (22). This is achieved here by defining  for  as the decoupled right and left moving fields in the new right [left] non-chiral lead. Thus, this result proves the celebrated duality symmetry: a level side-coupled to a LL [Fig. 1(a)] is equivalent a level embedded (in a left-right symmetric manner) between the edges to two LLs [Fig. 1(b)]. The level energy as well as the tunneling matrix element remain unchanged, but is transformed to . In addition, a local level-wire interaction must be included (23); (24).
How do the measurable properties of the system map under the duality? It is easy to see that the level population and its correlation functions (determining the dynamic capacitance), as well as other thermodynamic properties of the level (e.g., its contribution to the entropy and specific heat) remain invariant under all the transformations performed, and are thus equal for the side-coupled and embedded geometries. Transport properties, however, do change. To see this, let us use a Landauer-type formalism (25); (26); (27). Attaching the side-coupled system at to reservoirs at potentials is equivalent to imposing the boundary conditions on the average decoupled right and left moving densities , with . Summing these two equations, and using current conservation, , we get . Substituting this relation back in the boundary conditions, and rewriting them in terms of the dual variables (in terms of which left movers at become right movers at and vice versa, while ), we find that the same expression holds for the embedded configuration: . In addition, since the average current can be written as , while the average voltage drop at is given by , we see that and are interchanged under the duality transformation ( is the quantum conductance). Now, in a steady state are separately constant for and . Thus, subtracting the boundary condition equations we have , which leads to the relation between the currents in the two geometries. This result is physically clear: when the level-lead coupling is weak, conductance is good for the side-coupled system (no scattering), but is bad for the embedded one (no tunneling) and vice versa.
It should be noted that one could have also included a local interaction between the electrons in the level and those in the lead in our original system. This would amount to adding to the Hamiltonian the term
with the strength of the forward (backward) interaction. Repeating all the transformations, we again obtain the same dual description in terms of the embedded level, with two modifications: (i) the strength of the local level-wire interaction in the embedded geometry is now (23); (ii) there will be an additional term, of the form . Although such a term is not usually included in the bare Hamiltonian of the embedded geometry, it is nevertheless generated by virtual processes in which an electron from one lead hops into the level and then into the other lead and vice-versa. Thus, this term does not add any new physics into the system, and only changes the results presented below quantitatively and not qualitatively (changing, e.g., the exact shapes of phase boundaries but not their weak- and strong-coupling limits, and affecting prefactors but not exponents in power-laws). Finally, it may be added that the duality can be obtained by comparing the Coulomb gas expansions for the two systems. Details will be given elsewhere (28); (27).
Transport properties.— In the rest of this paper we will study the transport properties of the side-coupled system for arbitrary interaction strengths, and show how the duality with the embedded case is revealed. Let us start from a qualitative description. The embedded geometry should behave similarly to the case of a LL with two barriers tuned to resonance (4); (5); (13); (11); (12). Then, for not too strong interactions the conductance is predicted to have a resonance lineshape as a function of . Without interactions () the lineshape is Lorentzian, and its width saturates at low temperature to , being the local density of states. For the width decreases as the temperature is lowered, suppressing conductance for , while for the width increases, so that transport becomes perfect at low enough . By the transmission-reflection duality, we expect to have an anti-resonance for the side-coupled geometry. From the above, for the lineshape is Lorentzian, and its width saturates to at low temperature, as one can immediately verify by a direct calculation. However, by the correspondence, here for the width should increase as is lowered, whereas for it should decrease. These expectations are borne out by Monte-Carlo calculations on a Coulomb-gas representation, to be discussed elsewhere (28). However, as an illustration we plot some of the results in Fig. 2. Thus, in both geometries conductance is suppressed for , and enhanced for , but this is realized by opposite lineshapes in the two configurations. As we now show, these considerations are supported by direct analysis of the side-coupled problem.
We will first consider the limit of weak level-lead coupling for arbitrary values of . Then, let us apply the transformation , where , . This eliminates the forward interaction term from the Hamiltonian, at the cost of modifying the tunneling term. In terms of the dimensionless parameters and ( is a short time cutoff), the level-lead coupling terms in now read:
where . The scaling dimensions of and are thus and , respectively. In addition, the vertex operators obey the operator product expansion (1) (), , and similarly for . Substituting this in Cardy’s general formulas gives the RG equations to second order in and (29):
Off-resonance (), the flow of is stopped as soon as . From this point on, the level is locked into one of its two possible states (occupied or empty, depending the sign of ), and the only RG equation left is . This simply means that off-resonance the level acts as a potential scatterer, whose strength for large enough . From these equations we see that (which is generated by terms of second order in even if not present in the original Hamiltonian) is in general relevant for and irrelevant for , as expected for a backscattering term (4). (which directly affects the low energy physics only for =0) is relevant for , where are the solutions of . Whenever any of these two terms is relevant, scattering induced by the level destroys conductance for small and .
For strong level-lead coupling, the forward scattering rapidly converges to its fixed point value [cf. Eq. (9)]. On resonance, the hopping term in the Hamiltonian is more relevant than the backscattering term, so we can concentrate on it in the strong-coupling limit. After the transformation described above it becomes , where we have defined the spin variables , , and (30). Since commutes with , it assumes one of its possible values (). Then take the form of a potential backscattering term, but with replaced by . From the known behavior of the latter problem (4) we can infer that the strong (suppressed transmission) limit is stable for and unstable for . Off resonance only the term is important at low energies (below , where is frozen). In the spin notation it becomes , where for and vice versa. Now behaves like a usual potential backscattering so the strong limit is stable for , unstable for .
Taking all these results together, we can find the phase diagram of the system, plotted in Fig. 3. There are three phases at : (i) for conductance is suppressed both on and off resonance, by a widening anti-resonance. (ii) for but not too large, we obtain a narrowing anti-resonance, so that at low temperatures transport is perfect everywhere except for . (iii) for large enough [ for large , with for small ] perfect conductance is attained even on-resonance. Moreover, concentrating on phases (i) and (ii) (i.e., not too strong attraction), the anti-resonance width scales as , where is the largest infrared cutoff, and where the power is determined by the scaling dimension of the leading correction to the large limit, i.e., tunneling across a barrier at . In the vicinity of the conductance behaves as , while for large it deviates from the perfect value of by a correction proportional to (determined by the scaling dimension of ) (31). As they should, all these results obey the duality relation with those for resonant tunneling (4).
There are two cases in which the side-coupled system can be analyzed by different methods (valid for arbitrary level-lead coupling), and compared with similar calculations for the embedded geometry: (i) The limit of weak electron-electron interactions ( near 1), which has been recently addressed by fermionic perturbative (in the electron-electron interaction) RG methods (14), previously employed to study of the embedded configuration (13); (11); (ii) Exact solution by refermionization at , in analogy with the embedded case at (12). These results can be shown to confirm both the general analysis given above as well as the duality relation (28).
Including spin.— Finally we note that the derivation of the duality symmetry can be easily extended to the spinful case, i.e., the Anderson impurity model coupled to a LL, relevant for the problem of the Kondo effect in a LL (32). Both the charge and spin LL parameters (1) transform as [ () for charge (spin)]. The strength of the charge and spin level-lead interaction in the embedded case is . This means that for , implying spin anisotropy in the wire, we will need to include spin-anisotropic level-lead interaction, marked by nonzero . Similar extension to a many level dot is also possible.
Conclusions.— To conclude, we have shown that for a level coupled to a LL lead there exists a duality symmetry between the side-coupled and embedded geometries, and examined it through a study of transport properties in the two systems. As we have seen, the conductance lineshape behaves in the opposite way in the two configurations, only to lead to the same final result: at low temperature, transport is suppressed for by a narrowing resonance (widening anti-resonance) for the embedded (side-coupled) configuration, and vice-versa for . These findings have an important implication on experiments (9); (10): since in reality electrons repel each other, physical realizations of the systems discussed are limited to . However, the physics of attractive interactions () in each geometry now becomes experimentally accessible through investigation of the behavior for in the dual system.
Acknowledgements.We would like to thank Y. Gefen, I.V. Lerner, A. Schiller, and I.V. Yurkevich for useful discussions. M.G. is supported by the Adams Foundation Program of the Israel Academy of Sciences and Humanities. Financial support from the Israel Science Foundation (Grant 569/07) is gratefully acknowledged.
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