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arXiv:1210.1001 · Physical Review B

Networks of quantum wire junctions: A system with quantized integer Hall resistance without vanishing longitudinal resistivity

Jaime Medina, Dmitry Green, Claudio Chamon

Published Jan 30, 2013• 10 citations
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Abstract

We consider a honeycomb network built of quantum wires, with each node of the network having a Y junction of three wires with a ring through which flux can be inserted. The junctions are the basic circuit elements for the network, and they are characterized by $3\ifmmode\times\else\texttimes\fi{}3$ conductance tensors. The low energy stable fixed point tensor conductances result from quantum effects, and are determined by the strength of the interactions in each wire and the magnetic flux through the ring. We consider the limit where there is decoherence in the wires between any two nodes, and study the array as a network of classical three-lead circuit elements whose characteristic conductance tensors are determined by the quantum fixed point. We show that this network has some remarkable transport properties in a range of interaction parameters: It has a Hall resistance quantized at ${R}_{xy}=h/{e}^{2}$, although the longitudinal resistivity is nonvanishing. We show that these results are robust against disorder, in this case nonhomogeneous interaction parameters $g$ for the different wires in the network.

DOI
10.1103/physrevb.87.045128

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We consider a honeycomb network built of quantum wires, with each node of the network having a Y junction of three wires with a ring through which flux can be inserted. The junctions are the basic circuit elements for the network, and they are characterized by 3\ifmmode×\else\texttimes\fi33\ifmmode\times\else\texttimes\fi{}33\ifmmode×\else\texttimes\fi3 conductance tensors. The low energy stable fixed point tensor conductances result from quantum effects, and are determined by the strength of the interactions in each wire and the magnetic flux through the ring. We consider the limit where there is decoherence in the wires between any two nodes, and study the array as a network of classical three-lead circuit elements whose characteristic conductance tensors are determined by the quantum fixed point. We show that this network has some remarkable transport properties in a range of interaction parameters: It has a Hall resistance quantized at Rxy=h/e2{R}_{xy}=h/{e}^{2}Rxy​=h/e2, although the longitudinal resistivity is nonvanishing. We show that these results are robust against disorder, in this case nonhomogeneous interaction parameters ggg for the different wires in the network.
We investigate the transport of correlated fermions through a junction of three one-dimensional quantum wires pierced by a magnetic flux. We determine the flow of the conductance as a function of a low-energy cutoff in the entire parameter space. For attractive interactions and generic flux the fixed point with maximal asymmetry of the conductance is the stable one, as conjectured recently. For repulsive interactions and arbitrary flux we find a line of stable fixed points with vanishing conductance as well as stable fixed points with symmetric conductance (4/9)(e(2)/h).
Within the framework of boundary conformal field theory, we evaluate the conductance of stable fixed points of junctions of two and three quantum wires with different Luttinger parameters. For two wires, the physical properties are governed by a single effective Luttinger parameter for each of the charge and spin sectors. We present numerical density-matrix-renormalization-group calculations of the conductance of a junction of two chains of interacting spinless fermions with different interaction strengths, obtained using a recently developed method [Phys. Rev. Lett. 105, 226803 (2010)]. The numerical results show very good agreement with the analytical predictions. For three spinless wires (i.e., a Y junction) we analytically determine the full phase diagram and compute all fixed-point conductances as a function of the three Luttinger parameters.
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We study the effects of electron-electron interactions on the transport properties of a junction of three quantum wires enclosing a magnetic flux. The wires are modeled as single channel spin-12\frac{1}{2}21​ Tomonaga--Luttinger liquids. The system exhibits a rich phase diagram as a function of the electronic interaction strength, which includes a chiral fixed point with an asymmetric current flow highly sensitive to the sign of the flux and another fixed point where pair tunneling dominates, which is similar to the case of spinless electrons. While in the case of spinless electrons the perturbations that correspond to unequal couplings between the three wires are always irrelevant, we find that, when the electron spin is included, there are small regions in the phase diagram where a current flows only between two of the wires and the third wire is decoupled.
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Inhomogeneities and junctions in wires are natural sources of scattering, and hence resistance. A conducting fixed point usually requires an adiabatically smooth system. One notable exception is ``healing,'' which has been predicted in systems with special symmetries, where the system is driven to the homogeneous fixed point. Here we present theoretical results for a different type of conducting fixed point which occurs in inhomogeneous wires with an abrupt jump in hopping and interaction strength. We show that it is always possible to tune the system to an unstable conducting fixed point which does not correspond to translational invariance. We analyze the temperature scaling of correlation functions at and near this fixed point and show that two distinct boundary exponents appear, which correspond to different effective Luttinger liquid parameters. Even though the system consists of two separate interacting parts, the fixed point is described by a single conformally invariant boundary theory. We present details of the general effective bosonic field theory including the mode expansion and the finite size spectrum. The results are confirmed by numerical quantum Monte Carlo simulations on spinless fermions. We predict characteristic experimental signatures of the local density of states near junctions.