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arXiv:cond-mat/9707131 · Physical review. B, Condensed matter

Tunneling in quantum wires II: A line of IR fixed points

Frédéric Lesage, Hubert Saleur, P. Simonetti

Published Feb 15, 1998• 15 citations
arXivHTMLPDF

Abstract

In a previous paper, we showed that the problem of tunneling in quantum wires was integrable in the isotropic case ${g}_{\ensuremath{\sigma}}=2$. In the present work, we continue the exploration of the general phase diagram by looking for other integrable cases. Specifically, we discuss in detail the manifold ${g}_{\ensuremath{\rho}}{+g}_{\ensuremath{\sigma}}=2$, where the associated ``double sine-Gordon'' model is integrable. Transport properties are exactly computed. Surprisingly, the IR fixed points, while having complete reflection of charge and spin currents, do not correspond to two separate leads. Their main characteristic is that they are approached along irrelevant operators of dimension ${1+(1/g}_{\ensuremath{\rho}})$ and ${1+(1/g}_{\ensuremath{\sigma}})$, corresponding to transfer of one electron charge but no spin, or one spin $1/2$ but no charge.

DOI
10.1103/physrevb.57.4694

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We show that the problem of impurity tunneling in a Luttinger liquid of electrons with spin is solvable in the spin isotropic case ${(g}_{\ensuremath{\sigma}}=2$, ${g}_{\ensuremath{\rho}}$ arbitrary). The resulting integrable model is similar to a two-channel anisotropic Kondo model, but with the impurity spin in a ``cyclic representation'' of the quantum algebra su(2)q\mathrm{su}{(2)}_{q}su(2)q​ associated with the anisotropy. Using exact, nonperturbative techniques we study the renormalization-group flow, and compute the dc conductance. As expected from the analysis of Kane and Fisher we find that the IR fixed point corresponds to two separate leads. We also prove an exact duality between the UV and IR expansions of the current at vanishing temperature.
In this report we compute the boundary states (including the boundary entropy) for the boundary sine-Gordon theory. From the boundary states, we derive both correlation and partition functions. Through the partition function, we show that boundary sine-Gordon maps onto a doubled boundary Ising model. With the current-current correlators, we calculate for finite system size the ac-conductance of tunneling quantum wires with dimensionless free conductance 1/2 (or, alternatively interacting quantum Hall edges at filling fraction 1/2). In the dc limit, the results of C. Kane and M. Fisher, Phys. Rev. B46 (1992) 15233, are reproduced.
We (re)consider in this paper the problem of tunnelling through an impurity in a quantum wire with arbitrary Luttinger interaction parameter. By combining the integrable approach developed in the case of quantum Hall edge states with the introduction of radiative boundary conditions to describe the adiabatic coupling to the reservoirs, we are able to obtain the exact equilibrium and non-equilibrium current. One of the most striking features observed is the appearance of negative differential conductances out of equilibrium in the strongly interacting regime g ⩽0.2. In spite of the various charging effects, a remarkable form of duality is still observed. New results on the computation of transport properties in integrable impurity problems are gathered in the appendices. In particular, we prove that the TBA results satisfy a remarkable relation, originally derived using the Keldysh formalism, between the order T 2 correction to the current out of equilibrium and the second derivative of this current at T = 0 with respect to the voltage: this is the first direct link between the TBA approach out of equilibrium and the traditional Keldysh one
We study a junction of three quantum wires enclosing a magnetic flux. The wires are modelled as single-channel spinless Tomonaga–Luttinger liquids. This is the simplest problem of a quantum junction between Tomonaga–Luttinger liquids in which Fermi statistics enter in a non-trivial way. We study the problem using a mapping onto the dissipative Hofstadter model, describing a single particle moving on a plane in a magnetic field and a periodic potential coupled to a harmonic oscillator bath. Alternatively we study the problem by identifying boundary conditions corresponding to the low energy fixed points. We obtain a rich phase diagram including a chiral fixed point in which the asymmetric current flow is highly sensitive to the sign of the flux and a phase in which electron pair tunnelling dominates. We also study the effects on the conductance tensor of the junction of contacting the three quantum wires to Fermi liquid reservoirs.
We study the four-terminal junction of spinless Luttinger liquid wires, which describes either a corner junction of two helical edge states of topological insulators or the tunneling from the spinful wire into the helical edge state. We use the fermionic representation and the scattering state formalism, in order to compute the renormalization group (RG) equations for the linear response conductances. We establish our approach by considering a junction between two possibly nonequivalent helical edge states and find an agreement with the earlier analysis of this situation. Tunneling from the tip of the spinful wire to the edge state is further analyzed which requires some modification of our formalism. In the latter case we demonstrate (i) the existence of both fixed lines and conventional fixed points of RG equations, and (ii) certain proportionality relations holding for conductances during renormalization. The scaling exponents and phase portraits are obtained in all cases.