We calculate the conductance of a system of two spinless Luttinger liquid wires with different interaction strengths g1 , g2 , 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 dependence on g1 , g2 . The regions of stability of the two fixed points corresponding to conductance G = 0 and G = 1,respectively, are determined.We calculate the conductance of a system of two spinless Luttinger liquid wires with different interaction strengths g1 , g2 , 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 dependence on g1 , g2 . The regions of stability of the two fixed points corresponding to conductance G = 0 and G = 1,respectively, are determined.

Lithuanian Journal of PhysicsJan 1, 20127 citations

Transport properties of a Y junction connecting Luttinger liquid wires

D. N. Aristov, P. Wölfle

We study the transport properties of three Luttinger liquid wires (with possibly different interaction strength), connected through a Y junction, within the scattering state formalism. We first formulate the problem in current algebra language and focus on the case of a symmetric setup, for which the scattering matrix and the matrix of conductances is parametrized by two variables. For these, we derive coupled renormalization group equations, first in a ladder summation up to infinite order in the interaction. The fixed-point structure and the implicit solution of these equations is presented. It is shown that higher-order terms beyond the ladder approximation do not change the scaling behavior near the fixed points. For sufficiently strong attractive interaction, a new fixed point with unusual properties is found.We study the transport properties of three Luttinger liquid wires (with possibly different interaction strength), connected through a Y junction, within the scattering state formalism. We first formulate the problem in current algebra language and focus on the case of a symmetric setup, for which the scattering matrix and the matrix of conductances is parametrized by two variables. For these, we derive coupled renormalization group equations, first in a ladder summation up to infinite order in the interaction. The fixed-point structure and the implicit solution of these equations is presented. It is shown that higher-order terms beyond the ladder approximation do not change the scaling behavior near the fixed points. For sufficiently strong attractive interaction, a new fixed point with unusual properties is found.

arXiv:1105.5883Physical Review BOct 17, 201131 citations

Chiral Y junction of Luttinger liquid wires at strong coupling: Fermionic representation

D. N. Aristov, P. Wölfle

We calculate the conductances of a three-terminal junction setup of spinless Luttinger liquid wires threaded by a magnetic flux, allowing for different interaction strength ${g}_{3}\ensuremath{\ne}g$ 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 classes of contributions in perturbation theory. The resulting general structure of the RG equations may be employed to describe junctions with an arbitrary number of wires and arbitrary interaction strength in each wire. The fixed point structure of these equations (for the chiral Y junction) is analyzed in detail. For repulsive interaction ( $g,{g}_{3}>0$ ) there is only one stable fixed point, corresponding to the complete separation of the wires. For attractive interaction ( $g<0$ and/or ${g}_{3}<0$ ) four fixed points are found, the stability of which depends on the interaction strength. We confirm our previous weak-coupling result of lines of fixed points for special values of the interaction parameters reaching into the strong-coupling domain. We find new fixed points not discussed before, even at the symmetric line $g={g}_{3}$ , at variance with the results of M. Oshikawa et al. [J. Stat. Mech. (2006) P02008]. The pair-tunneling phenomenon conjectured by the latter authors is not found by us.We calculate the conductances of a three-terminal junction setup of spinless Luttinger liquid wires threaded by a magnetic flux, allowing for different interaction strength ${g}_{3}\ensuremath{\ne}g$ 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 classes of contributions in perturbation theory. The resulting general structure of the RG equations may be employed to describe junctions with an arbitrary number of wires and arbitrary interaction strength in each wire. The fixed point structure of these equations (for the chiral Y junction) is analyzed in detail. For repulsive interaction ($g,{g}_{3}>0$) there is only one stable fixed point, corresponding to the complete separation of the wires. For attractive interaction ($g<0$ and/or ${g}_{3}<0$) four fixed points are found, the stability of which depends on the interaction strength. We confirm our previous weak-coupling result of lines of fixed points for special values of the interaction parameters reaching into the strong-coupling domain. We find new fixed points not discussed before, even at the symmetric line $g={g}_{3}$, at variance with the results of M. Oshikawa et al. [J. Stat. Mech. (2006) P02008]. The pair-tunneling phenomenon conjectured by the latter authors is not found by us.

arXiv:1304.5167Physical Review BAug 19, 201325 citations

Transport properties of a two-lead Luttinger-liquid junction out of equilibrium: Fermionic representation

D. N. Aristov, P. Wölfle

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 follows a power law in terms of a scaling variable $\ensuremath{\Lambda}$, which equals the bias voltage $V$ , if $V$ is the largest energy scale compared to temperature $T$ and inverse wire length ${L}^{\ensuremath{-}1}$, and interpolates between these quantities in the crossover regimes.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 follows a power law in terms of a scaling variable $\ensuremath{\Lambda}$, which equals the bias voltage $V$, if $V$ is the largest energy scale compared to temperature $T$ and inverse wire length ${L}^{\ensuremath{-}1}$, and interpolates between these quantities in the crossover regimes.

arXiv:1408.4914Physical Review BDec 8, 201411 citations

Conductance scaling of junctions of Luttinger-liquid wires out of equilibrium

D. N. Aristov, P. Wölfle

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 $\ensuremath{\beta}$ functions are derived up to second order.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 $\ensuremath{\beta}$ functions are derived up to second order.

arXiv:1802.03927Physical review. B./Physical review. BMay 1, 20182 citations

Chiral Y junction of Luttinger liquid wires at weak coupling: Lines of stable fixed points

D. N. Aristov, P. Wölfle

We calculate the conductances of a Y-junction setup of Luttinger liquid wires threaded by a magnetic flux, allowing for a different interaction strength ${g}_{3}\ensuremath{\ne}g$ in the third wire. The scattering matrix and the matrix of conductances are parametrized by three variables. For these we derive coupled renormalization group (RG) equations, up to second order in the interaction, within the scattering state formalism. The fixed point structure of these equations is analyzed in detail. For a repulsive interaction ( $g,{g}_{3}>0$ ) there is only one stable fixed point, corresponding to the complete separation of the wires. For an attractive interaction ( $g<0$ and/or ${g}_{3}<0$ ) four fixed points are found, whose stability depends on the interaction strength. For special values of the interaction parameters (a) $g=0$ , ${g}_{3}<0$ or (b) ${g}_{3}+{g}^{2}/2=0$ , $g<0$ we find whole lines of stable fixed points. We conjecture that lines of fixed points appear generically at interfaces in the interaction coupling constant space separating two regions with different stable fixed points. We also find that in certain regions of the $g$ - ${g}_{3}$ plane the RG flow is towards a fixed point without chirality, implying that the effect of the magnetic flux is completely screened.We calculate the conductances of a Y-junction setup of Luttinger liquid wires threaded by a magnetic flux, allowing for a different interaction strength ${g}_{3}\ensuremath{\ne}g$ in the third wire. The scattering matrix and the matrix of conductances are parametrized by three variables. For these we derive coupled renormalization group (RG) equations, up to second order in the interaction, within the scattering state formalism. The fixed point structure of these equations is analyzed in detail. For a repulsive interaction ($g,{g}_{3}>0$) there is only one stable fixed point, corresponding to the complete separation of the wires. For an attractive interaction ($g<0$ and/or ${g}_{3}<0$) four fixed points are found, whose stability depends on the interaction strength. For special values of the interaction parameters (a) $g=0$, ${g}_{3}<0$ or (b) ${g}_{3}+{g}^{2}/2=0$, $g<0$ we find whole lines of stable fixed points. We conjecture that lines of fixed points appear generically at interfaces in the interaction coupling constant space separating two regions with different stable fixed points. We also find that in certain regions of the $g$-${g}_{3}$ plane the RG flow is towards a fixed point without chirality, implying that the effect of the magnetic flux is completely screened.

arXiv:1110.1159Physical Review BJul 23, 201215 citations

g,{g}_{3}>0

) there is only one stable fixed point, corresponding to the complete separation of the wires. For attractive interaction (

g<0

and/or

{g}_{3}<0

) four fixed points are found, the stability of which depends on the interaction strength. We confirm our previous weak-coupling result of lines of fixed points for special values of the interaction parameters reaching into the strong-coupling domain. We find new fixed points not discussed before, even at the symmetric line

g={g}_{3}

, at variance with the results of M. Oshikawa et al. [J. Stat. Mech. (2006) P02008]. The pair-tunneling phenomenon conjectured by the latter authors is not found by us.

V

, if

V

is the largest energy scale compared to temperature

T

and inverse wire length ${L}^{\ensuremath{-}1}$, and interpolates between these quantities in the crossover regimes.

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 $\ensuremath{\beta}$ functions are derived up to second order.

g,{g}_{3}>0

) there is only one stable fixed point, corresponding to the complete separation of the wires. For an attractive interaction (

g<0

and/or

{g}_{3}<0

) four fixed points are found, whose stability depends on the interaction strength. For special values of the interaction parameters (a)

g=0

{g}_{3}<0

or (b)

{g}_{3}+{g}^{2}/2=0

g<0

we find whole lines of stable fixed points. We conjecture that lines of fixed points appear generically at interfaces in the interaction coupling constant space separating two regions with different stable fixed points. We also find that in certain regions of the

g

{g}_{3}

plane the RG flow is towards a fixed point without chirality, implying that the effect of the magnetic flux is completely screened.