Making a combined use of bosonization and fermionization techniques, we build nonlocal transformations between dual fermion operators, describing junctions of strongly interacting spinful one-dimensional quantum wires. Our approach allows for trading strongly interacting (in the original coordinates) fermionic Hamiltonians for weakly interacting (in the dual coordinates) ones. It enables us to generalize to the strongly interacting regime the fermionic renormalization group approach to weakly interacting junctions. As a result, on one hand, we are able to pertinently complement the information about the phase diagram of the junction obtained within bosonization approach; on the other hand, we map out the full crossover of the conductance tensors between any two fixed points in the phase diagram connected by a renormalization group trajectory.
The electrical current through an arbitrary junction connecting quantum wires of spinless interacting fermions is calculated in fermionic representation. The wires are adiabatically attached to two reservoirs at chemical potentials differing by the applied voltage bias. The relevant scale-dependent contributions in perturbation theory in the interaction up to infinite order are evaluated and summed up. The result allows one to construct renormalization group equations for the conductance as a function of voltage (or temperature, wire length). There are two fixed points at which the conductance…
Junctions of multiple one-dimensional quantum wires of interacting electrons have received considerable theoretical attention as a basic constituent of quantum circuits. While results have been obtained on these models using bosonization and Density Matrix Renormalization Group (DMRG) methods, another powerful technique is based on direct perturbation theory in the bulk interactions, combined with the Renormalization Group (RG) and summed in the Random Phase Approximation (RPA). This technique has so far only been applied to the case where finite length interacting wires are attached to non-in…
We develop the renormalization group theory of the conductances of N-lead junctions of spinless Luttinger-liquid wires as functions of bias voltages applied to N independent Fermi-liquid reservoirs. Based on the perturbative results up to second order in the interaction we demonstrate that the conductances obey scaling. The corresponding renormalization group $β$ functions are derived up to second order.
We calculate the zero-temperature impurity entropy of a junction of multiple quantum wires of interacting spinless fermions. Starting from a given single-particle S-matrix representing a fixed point of the renormalization group (RG) flows, we carry out fermionic perturbation theory in the bulk interactions, with the perturbation series summed in the random phase approximation (RPA). The results agree completely with boundary conformal field theory (BCFT) predictions of the ground state degeneracy, and also with known RG flows through the g-theorem.
We propose a Landauer-like theory for nonlinear transport in networks of one-dimensional interacting quantum wires (Luttinger liquids). A concrete example of current experimental focus is given by carbon nanotube Y junctions. Our theory has three basic ingredients that allow to explicitly solve this transport problem: (i) radiative boundary conditions to describe the coupling to external leads, (ii) the Kirchhoff node rule describing charge conservation, and (iii) density matching conditions at every node.
We consider the Y-junction, connecting three quantum wires, in the scattering states formalism. In the absence of fermionic interaction we analyze the restrictions on the values of conductance matrix, imposed by the unitarity of scattering S-matrix. Using the combination of numerical and analytical results, we describe the four-dimensional body of values of reduced conductance matrix. We show that this body touches the unit sphere at six points only, in accordance with Birkhoff-von Neumann theorem. It implies that the Abelian bosonization analysis for the vanishing interaction strength can be …
We calculate the conductances of a three-terminal junction set-up of spinless Luttinger liquid wires threaded by a magnetic flux, allowing for different interaction strength g_3 != g in the third wire. We employ the fermionic representation in the scattering state picture, allowing for a direct calculation of the linear response conductances, without the need of introducing contact resistances at the connection points to the outer ideal leads. The matrix of conductances is parametrized by three variables. For these we derive coupled renormalization group (RG) equations, by summing up infinite …
We calculate the conductance of a system of two spinless Luttinger liquid wires with different interaction strengths g_1, g_2, connected through a short junction, within the scattering state formalism. Following earlier work we formulate the problem in current algebra language, and calculate the scale dependent contribution to the conductance in perturbation theory keeping the leading universal contributions to all orders in the interaction strength. From that we derive a renormalization group (RG) equation for the conductance. The analytical solution of the RG-equation is discussed in depende…
We develop two novel numerical schemes to study the conductance of the two-wire junction of inequivalent Tomonaga-Luttinger Liquids. In the first scheme we use the static current-current correlation function across the junction to extract the linear conductance through a relation that is derived via the bosonization method. In the second scheme we apply a bias and evaluate the time-dependent current across the junction to obtain the current-voltage characteristic. The conductance is then extracted from the small bias result within the linear response regime. Both schemes are based on the infin…
he BCS-like pairing in ultracold fermionic atomic ($UCFAG$) gases is studied in the model of "isotopic-spin" pairing proposed in 1991 \cite% {Ku-Hof-SSC}. This model assumes a mismatch ($δ$) in chemical potentials of pairing fermionic atoms. It is shown that a $π$-Josephson junction can be realized in $UCFAG$ systems, where the left and right banks $% S$ are the $UCFAG$ superfluids. The weak link $M$ consists from the normal $% UCFAG$ with the finite mismatch $δ$. If the $π$-junction is a part of a closed ring the superfluid mass-current flows spontaneously in the ring, i.e., the time-reversal…
We study four-terminal junction of spinless Luttinger liquid wires, which describes either a corner junction of two helical edges 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 non-equivalent helical edge states and find an agreement with the earlier analysis of this situation. Tunneling from the tip …
We study four-lead junction of semi-infinite wires by using fermionic representation in the scattering state formalism. The model of spinless fermions with short-range Luttinger liquid type interaction is used. We find the renormalization group (RG) equations for the conductances of the system in the first order of fermionic interaction. In contrast to the well-known cases of two-lead and three-lead junctions, we show that the RG equations cannot be expressed solely in terms of conductances. The appearing ambiguity is resolved by choosing a certain sign defined at the level of S-matrix, but hi…
We calculate the conductances of a three-way junction of spinless Luttinger-liquid wires as functions of bias voltages applied to three independent Fermi-liquid reservoirs. In particular, we consider the setup that is characteristic of a tunneling experiment, in which the strength of electron-electron interactions in one of the arms of the junction ("tunneling tip") is different from that in the other two arms (which together form a "main wire"). The scaling dependence of the two independent conductances on bias voltages is determined within a fermionic renormalization-group approach in the li…
It is well known that the pristine bulk of an interacting one-dimensional system in Tomonaga-Luttinger liquid (TLL) phase shows power law suppression of quasi-particle tunneling amplitude for all values of TLL parameter $g$, in the zero energy limit. We perform a density matrix renormalization group (DMRG) study of a fully symmetric Y junction of TLL wires and observe an anomalous enhancement of the tunneling density of states (TDOS) in the vicinity of the junction for both (a) interacting bosons case and (b) interacting fermions case, when $g>1$. We also observe suppression of TDOS for $g<1$ …
We study the conductance of three or more semi-infinite wires which meet at a junction. The electrons in the wires are taken to interact weakly with each other through a short-range density-density interaction, and they encounter a general scattering matrix at the junction. We derive the renormalization group equations satisfied by the S-matrix, and we identify its fixed points and their stabilities. The conductance between any pair of wires is then studied as a function of physical parameters such as temperature. We discuss the possibility of observing the effects of junctions in present day …
Conductance is related to dynamical correlation functions which can be calculated with \textit{time-dependent} methods. Using boundary conformal field theory, we relate the conductance tensors of quantum junctions of multiple wires to static correlation functions in a finite system. We then propose a general method for determining the conductance through \textit{time-independent} calculations alone. Applying the method to a Y-junction of interacting quantum wires, we numerically verify the theoretical prediction for the conductance of the chiral fixed-point of the Y-junction and then calculate…
We analyze the charge and thermal transport at a junction of interacting quantum wires close to equilibrium. Within the framework of Tomonaga-Luttinger liquids, we compute the thermal conductance for a wide class of boundary conditions and detail the physical processes leading to the breakdown of the Wiedemann-Franz law at the junction. We show how connecting external reservoirs to the quantum wires affects the conductance tensors close to the various fixed points of the phase diagram of the junction. We therefore distinguish two types of violation of the Wiedemann-Franz law: a "trivial" one, …
We study the transport properties of a system of three Luttinger liquid wires (with different interaction strengths g, g_3), connected through a Y-junction (main wire and tunneling tip) threaded by magnetic flux, within the scattering state formalism. The scattering matrix and the matrix of conductances are parametrized by three variables. For these we derive coupled RG equations, up to second order in the interaction. The fixed point structure of these equations is analyzed in detail. For repulsive interaction (g, g_3>0) there is only one stable fixed point, corresponding to the complete sepa…
Experimental evidence was recently obtained for topological superconductivity in spin-orbit coupled nano wires in a magnetic field, proximate to an s-wave superconductor. When only part of the wire contacts the superconductor, a localized Majorana mode exists at the junction between superconducting and normal parts of the nanowire. We consider here the case of a T-junction between the superconductor and two normal nanowires and also the case of a single wire with two (or more) partially filled bands in the normal part. We find that coupling this 2-channel Luttinger liquid to the single Majoran…
We present an extended study of anyonic Luttinger liquids wires jointing at a single point. The model on the full line is solved with bosonization and the junction of an arbitrary number of wires is treated imposing boundary conditions that preserve exact solvability in the bosonic language. This allows to reach, in the low momentum regime, some of the critical fixed points found with the electronic boundary conditions. The stability of all the fixed points is discussed.
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 parameters 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 …
We discuss how to extend the impurity entropy to systems with boundary interactions depending on zero-mode real fermion operators (Majorana modes as well as Klein factors). As specific applications of our method, we consider a junction between N interacting quantum wires and a topological superconductor, as well as a Y-junction of three spinless interacting quantum wires. In addition we find a remarkable correspondence between the N=2 topological superconductor junction and the Y-junction. On one hand, this allows us to determine the range of the system parameters in which a stable phase of th…
We study the tunneling density of states (TDOS) for a junction of three Tomonaga-Luttinger liquid wires. We show that there are fixed points which allow for the enhancement of the TDOS, which is unusual for Luttinger liquids. The distance from the junction over which this enhancement occurs is of the order of x = v/(2 ω), where v is the plasmon velocity and ωis the bias frequency. Beyond this distance, the TDOS crosses over to the standard bulk value independent of the fixed point describing the junction. This finite range of distances opens up the possibility of experimentally probing the enh…
We explore the possibility to break time reversal invariance at the junction of quantum wires. The universal features in the bulk of the wires are described by the anyon Luttinger liquid. A simple necessary and sufficient condition for the breaking of time reversal invariance is formulated in terms of the scattering matrix at the junction. The phase diagram of a junction with generic number of wires is investigated in this framework. We give an explicit classification of those critical points which can be reached by bosonization and study the interplay between their stability and symmetry cont…
We investigate rectification of a low-frequency ac bias in Y-junctions of one-channel Luttinger liquid wires with repulsive electron interaction. Rectification emerges due to three scatterers in the wires. We find that it is possible to achieve a higher rectification current in a Y-junction than in a single wire with an asymmetric scatterer at the same interaction strength and voltage bias. The rectification effect is the strongest in the absence of the time-reversal symmetry. In that case, the maximal rectification current can be comparable with the total current $\sim e^2V/h$ even for low vo…
We study junctions of single-channel spinless Luttinger liquids using bosonisation. We generalize earlier studies by allowing the junction to be superconducting and find new charge non-conserving low energy fixed points. We establish the existence of $g \leftrightarrow 1/g$ duality (where $g$ is the Luttinger Liquid parameter) between the charge conserving (normal) junction and the charge non-conserving (superconducting) junction by evaluating and comparing the scaling dimensions of various operators around the fixed points in normal and superconducting sectors of the theory. For the most gene…
The non-linear current-voltage characteristic of a tunnel junction between two Luttinger systems is calculated for an interaction with finite range. Coulomb blockade features are found. The dissipative resistance, the capacitance and the external impedance, which were introduced ad hoc in earlier theories, are obtained in terms of the electron-electron interaction. The frequency dependence of the impedance is given by the excitation spectrum of the electrons.
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 x 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 …
We investigate transport of spinless fermions through a single site dot junction of M one-dimensional quantum wires. The semi-infinite wires are described by a tight-binding model. Each wire consists of two parts: the non-interacting leads and a region of finite extent in which the fermions interact via a nearest-neighbor interaction. The functional renormalization group method is used to determine the flow of the linear conductance as a function of a low-energy cutoff for a wide range of parameters. Several fixed points are identified and their stability is analyzed. We determine the scaling …
We investigate 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).
We study theoretically the current-voltage characteristics, shot noise, and full counting statistics of a quantum wire double barrier structure. We model each wire segment by a spinless Luttinger liquid. Within the sequential tunneling approach, we describe the system's dynamics using a master equation. We show that at finite bias the non-equilibrium distribution of plasmons in the central wire segment leads to increased average current, enhanced shot noise, and full counting statistics corresponding to a super-Poissonian process. These effects are particularly pronounced in the strong interac…
We study ensembles of fermionic cold-atom quantum wires with tunable transverse mode population and single-wire resolution. From in situ density profiles, we determine the temperature of the atomic wires in the weakly interacting limit and reconstruct the underlying potential landscape. By varying atom number and temperature, we control the occupation of the transverse modes and study the 1D-3D crossover. In the 1D limit, we observe an increase of the reduced temperature $T/T_{F}$ at nearly constant entropy per particle $S/N k_{B}$. The ability to probe individual atomic wires in situ paves th…
We study a junction of three quantum wires enclosing a magnetic flux. This is the simplest problem of a quantum junction between Tomonaga-Luttinger liquids in which Fermi statistics enter in a non-trivial way. We present a direct connection between this problem and the dissipative Hofstadter problem, or quantum Brownian motion in two dimensions in a periodic potential and an external magnetic field, which in turn is connected to open string theory in a background electromagnetic field. We find non-trivial fixed points corresponding to a chiral conductance tensor leading to an asymmetric flow o…
A fermionic disordered one dimensional wire in the presence of attractive interactions is known to have two distinct phases: A localized and a superconducting one depending on the strength of interaction and disorder. The localized region may also exhibit a metallic behaviour if the system size is shorter than the localization length. Here we show that the superconducting phase has a distinct distribution of the entanglement entropy and entanglement spectrum distinct from the metallic regime. The entanglement entropy distribution is strongly asymmetric with Lévy alpha stable distribution (comp…
We study a junction of three quantum wires enclosing a magnetic flux. The wires are modeled 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 fixe…
Applying the Fermi-Bose equivalence and the boundary state formulation, we study the hetero-junction of two quantum wires. Two quantum wires are described by Tomonaga-Luttinger (TL) liquids with different TL parameters and electrons transport between two wires is depicted by a simple hopping interaction. We calculate the radiative corrections to the hopping interaction and obtain the renormalization (RG) exponent, making use of the perturbation theory based on the boundary state formulation. The model exhibits a phase transition at zero temperature. We discuss the critical line which defines t…
We describe a protocol to read out the topological invariant of interacting 1D chiral models, based on measuring the mean chiral displacement of time-evolving bulk excitations. We present analytical calculations and numerical Matrix Product State simulations of interacting Su-Schrieffer-Heeger (SSH) chains, demonstrating how the mean chiral displacement allows to distinguish between topological insulator, trivial insulator and symmetry-broken phases. Finally, we provide an experimental blueprint for realizing a model displaying these three phases and describe how to detect those.
We study a system consisting of a junction of N quantum wires, where the junction is characterized by a scalar S-matrix, and an impurity spin is coupled to the electrons close to the junction. The wires are modeled as weakly interacting Tomonaga-Luttinger liquids. We derive the renormalization group equations for the Kondo couplings of the spin to the electronic modes on different wires, and analyze the renormalization group flows and fixed points for different values of the initial Kondo couplings and of the junction S-matrix (such as the decoupled S-matrix and the Griffiths S-matrix). We gen…
We develop a method to extract the universal conductance of junctions of multiple quantum wires, a property of systems connected to reservoirs, from static ground-state computations in closed finite systems. The method is based on a key relationship, derived within the framework of boundary conformal field theory, between the conductance tensor and certain ground state correlation functions. Our results provide a systematic way of studying quantum transport in the presence of strong electron-electron interactions using efficient numerical techniques such as the standard time-independent densit…
We study charge transport through $N$-lead junctions ($N\geq 3$) of spinless Luttinger liquid wires with bias voltages applied to Fermi-liquid reservoirs. In particular, we consider a Y junction, which is a setup characteristic of the tunneling experiment. In this setup, the strength of electron-electron interactions in one of the arms ("tunneling tip") is different from that in the other two arms (which form together the "main wire"). For a generic single-particle $S$ matrix of the junction, we find that the bias voltage $V$ applied---even symmetrically---to the main wire generates a current …
We study the formation of inhomogeneous magnetization texture in the vicinity of a tunnel junction between two ferromagnetic wires nominally in the antiparallel configuration and its influence on the magnetoresistance of such a device. The texture, dependent on magnetization rigidity and crystalline anisotropy energy in the ferromagnet, appears upon an increase of ferromagnetic inter-wire coupling above a critical value and it varies with an external magnetic field.
We devise a number-conserving scheme for the realization of Majorana Zero Modes in an interacting fermionic ladder coupled by Aharonov-Bohm cages. The latter provide an efficient mechanism to cancel single-particle hopping by destructive interference. The crucial parity symmetry in each wire is thus encoded in the geometry of the setup, in particular, its translation invariance. A generic nearest-neighbor interaction generates the desired correlated hopping of pairs. We exhibit the presence of an extended topological region in parameter space, first in a simplified effective model via bosoniza…
We study numerically the universal conductance of Luttinger liquids wire with a single impurity via the Muti-scale Entanglement Renormalization Ansatz (MERA). The scale invariant MERA provides an efficient way to extract scaling operators and scaling dimensions for both the bulk and the boundary conformal field theories. By utilizing the key relationship between the conductance tensor and ground-state correlation function, the universal conductance can be evaluated within the framework of the boundary MERA. We construct the boundary MERA to compute the correlation functions and scaling dimensi…
We consider a metallic wire coupled to two metallic electrodes via two junctions placed nearby. A bias voltage applied to one of such junctions alters the electron distribution function in the wire in the vicinity of another junction thus modifying both its noise and the Coulomb blockade correction to its conductance. We evaluate such interaction corrections to both local and non-local conductances demonstrating non-trivial Coulomb anomalies in the system under consideration. Experiments on non-local electron transport with Coulomb effects can be conveniently used to test inelastic electron re…
We have studied the influence of both exciton effects and Coulomb repulsion on current in molecular nanojunctions. We show that dipolar energy-transfer interactions between the sites in the wire can at high voltage compensate Coulomb blocking for particular relationships between their values. Tuning this relationship may be achieved by using the effect of plasmonic nanostructure on dipolar energy transfer interactions.
A spin-orbit coupled quantum wire, with one end proximate to an s-wave superconductor, can become a topological superconductor, with a Majorana mode localized at each end of the superconducting region. It was recently shown that coupling one end of such a topological superconductor to $two$ normal channels of interacting electrons leads to a novel type of frustration and a quantum critical point when both channels couple with equal strength. We propose an experimental method to access this critical point in a $single$ quantum wire and show its resilience to disorder.
The Josephson current through a 1D quantum wire with Rashba spin-orbit and electron-electron interactions is calculated. We show that the interplay of Rashba and Zeeman interactions gives rise to a supercurrent through the 1D conductor that is anomalous in the sense that it persists in the absence of any phase difference between the two superconducting leads to which it is attached. The electron dispersion asymmetry induced by the Rashba interaction in a Luttinger-liquid wire plays a significant role for poorly transmitting junctions. It is shown that for a weak or moderate electron-electron i…
We present a general formulation of spin-dependent transport through a clean one-dimensional interacting quantum wire or carbon nanotube, connected to non-collinear ferromagnets via tunnel junctions. We show that the low energy description of each junction is given by a conformally-invariant boundary condition representing *exchange coupling*, in addition to a pair of electron tunneling operators. The effects of the exchange coupling are strongly enhanced by interactions, leading to a dramatic suppression of spin accumulation. This is a direct signature of spin-charge separation in a Luttinger…
We consider the possibility of creating an adiabatic transition through a narrow neck, or point contact, between two different quantized Hall states that have the same number of edge modes, such as ν=1 and ν=1/3. We apply both the composite fermion and the Luttinger liquid formalism to analyze the transition. We suggest that using such adiabatic junctions one could build a DC step-up transformer, where the output voltage is higher than the input. Difficulties standing in the way of an experimental implementation of the adiabatic junction are addressed.
The renormalization of conductances in a Y junction of spinless Luttinger-liquid wires additionally coupled to acoustic longitudinal phonons is investigated in fermionic representation. This system corresponds to geometry of a tunneling experiment and exhibits the interplay between the Coulomb repulsion and the attractive retarded interaction mediated by phonons. The retardation effects related to the propagation of phonons through the junction with arbitrary transmission and reflection amplitudes are taken into account. The appearing logarithmic corrections to conductances of the junction are…
We propose and analyze inter-edge tunneling in a quantum spin Hall corner junction as a means to probe the helical nature of the edge states. We show that electron-electron interactions in the one-dimensional helical edge states result in Luttinger parameters for spin and charge that are intertwined, and thus rather different than those for a quantum wire with spin rotation invariance. Consequently, we find that the four-terminal conductance in a corner junction has a distinctive form that could be used as evidence for the helical nature of the edge states.
