Dual fermionic variables and renormalization group approach to junctions of strongly
interacting quantum wires
Domenico Giuliano1,2 and Andrea Nava1
1

arXiv:1504.00569v2 [cond-mat.str-el] 17 Sep 2015

Dipartimento di Fisica, Universit` della Calabria Arcavacata di Rende I-87036, Cosenza,
a
Italy and I.N.F.N., Gruppo collegato di Cosenza, Arcavacata di Rende I-87036, Cosenza, Italy
2
CNR-SPIN, Monte S.Angelo via Cinthia, I-80126 Napoli, Italy
(Dated: April 15, 2018)
Making a combined use of bosonization and fermionization techniques, we build nonlocal transformations between dual fermion operators, describing junctions of strongly interacting spinful onedimensional 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.
PACS numbers: 71.10.Pm, 72.10.-d, 73.63.Nm

I.

INTRODUCTION

Many-electron systems, such as conduction electrons in a metal, are typically well-described by Landau’s Fermi
liquid theory (LFLT), which is capable of successfully accounting for the main effects of electronic interaction1–4 .
LFLT’s basic assumption is that, close to the Fermi surface, low-energy elementary ”quasiparticle” excitations of the
interacting electron liquid are in one-to-one correspondence with particle- and hole-excitations in the noninteracting
Fermi gas. This corresponds to a nonzero overlap between the quasiparticle wave function in the Fermi liquid and the
electron/hole wave function in the noninteracting Fermi gas, which is reflected by a coefficient ZF of the quasiparticle
peak at the Fermi surface in the spectral density of states that is finite though, in general, < 1 (ZF = 1 corresponds
to the noninteracting limit). The ”adiabatic deformability” of quasiparticles to electrons and/or holes by smoothly
switching off the interaction allows, for instance, to address transport in a Fermi liquid in a similar way to what is
done using scattering approach in a noninteracting Fermi gas, etc.5 .
LFLT is grounded on the possibility of obtaining reliable results by perturbatively treating the electronic
interaction4,6 . This is strictly related to the small rate of multi-particle inelastic processes, in which, due to the
interaction, an electron/hole emits electron-hole pairs. While this is typically the case in systems with spatial dimension d higher than one, in one-dimensional systems, the proliferation of particle-hole pair emission at low energies
leads to a diverging corresponding rate, which makes the quasiparticle peak disappear (ZF = 0). As a result, the
interaction cannot be dealt with perturbatively and one has rather to resort to nonperturbative techniques, allowing
for summing over infinite sets of diagrams7–9 . The breakdown of LFLT in one-dimensional interacting electronic
systems (interacting ”quantum wires” - QW’s) reveals itself in a series of physical contexts, the more remarkable of
which is the transport across an interacting QW in the presence of a constriction, or a weak link, or the conductance
of a junction of QWs, with or without spin10–23 . Moreover, due to the remarkable correspondence between fermionicand spin-bosonic one-dimensional lattice models encoded in the Jordan-Wigner fermionic representation of quantum
spin-1/2 spin operators24, a number of bosonic realizations of one-dimensional models of interacting fermions have
been proposed in bosonic systems, such as quantum spin chains25–30 , quantum Josephson junction networks31–33 , or
topological Kondo-type systems34–38 .
Formally, interacting electrons in one dimension are commonly treated within Tomonaga-Luttinger liquid (TLL)approach39,40 . TLL formalism provides a general description of low-energy physics of a one-dimensional interacting
electronic system in terms of collective bosonic excitations (charge- and/or spin-plasmons). Electronic operators are
realized as nonlinear vertex operators of the bosonic fields41,42 . As for what concerns transport properties, the most
striking prediction of the TLL-approach is possibly the power-law dependence of the conductance on the low-energy
reference scale (”infrared cutoff”), which is typically identified with the (Boltzmann constant times the) temperature,
or with the (Fermi velocity times the) inverse system length (”finite-size gap”) in a dc transport measurement, or
with eV , with V being an applied voltage, in a nonequilibrium experiment. While TLL-formalism poses no particular
constraints on the strength of the ”bulk” interaction within the quantum wires, it suffers of limitations, when used
to describe transport across impurities in an interacting quantum wire, or conduction properties at a junction of
quantum wires (generically referred to, in the following of this section, as quantum impurities in the quantum wire).
Parametrizing the spatial coordinate in each wire with x(≥ 0), within the TLL-framework, a junction of quantum wires

2
is mapped onto a model of K-one dimensional TLLs (one per each wire), interacting with each other by means of a
”boundary interaction” localized at x = 0. Dealing with such a class of boundary problems requires pertinently setting
the boundary conditions on the plasmon fields at x = 0. While in some very special cases the boundary conditions
can be written as simple linear relations between the plasmon fields, in general they cannot. This is a consequence
of the nonlinearity of the relations between bosonic and fermionic fields: even linear conditions among the fermionic
fields are traded for highly nonlinear conditions in the bosonic fields. As a consequence, except at the fixed points of
the boundary phase diagram, where the boundary conditions are ”conformal”, that is, linear in the bosonic fields, the
boundary interaction can only be dealt with perturbatively, with respect to the closest conformal fixed point43,44 . With
very few remarkable exceptions, where the junction model Hamiltonian maps onto some exactly solvable model45–47 ,
the only way for recovering ”global” (i.e., not necessarily in the vicinity of a conformal fixed point) informations about
the phase diagram and the corresponding scaling of the conductance, is by just making educated guesses from the
global topology of the fixed-point manifold of the phase diagram. Recently, the bosonization approach in combination
with zero-temperature numerics has successfully been employed to compute the junction conductance by relating it
to the asymptotic behavior of certain static correlation functions. Correspondingly, the length scale over which the
asymptotic behavior emerges has been worked out, as well48,49 . Also, numerical calculations of the finite-temperature
junction conductance can be performed by using quantum Monte Carlo approach50 or, likely, by implementing some
pertinently adapted version of the finite-temperature density matrix renormalization group approach to quantum spin
chains51,52 .
Alternatively, one may not be required to give up using fermionic coordinates, by employing a systematic renormalization group (RG) procedure to treat the effects of the bulk interaction on the scattering amplitudes at the junction.
Among the various possible ways of implementing RG for junctions of interacting QWs, the two most effective (and
widely used) ones are certainly the poor man’s fermionic renormalization group (FRG)-approach, based upon a systematic summation of the leading-log divergences of the S-matrix elements at the Fermi momentum and typically
yielding equations that can be analytically treated13–15,18 , and the functional renormalization group (fRG)-approach,
based on the functional renormalization group method, leading to a set of coupled differential equations for the vertex
part that, typically, can only be numerically treated53–62 . Both approaches are expected to apply only for a sufficiently
weak electronic interaction in the quantum wires and, in this sense, they are less general than the TLL-approach,
which applies even for a strong bulk interaction. Nevertheless, at variance with the TLL-approach, a RG-approach
based on the use of fermionic coordinates leads to equations for the S-matrix elements valid at any scale and, thus,
it allows for recovering the full scaling of the conductance, all the way down to the infrared cutoff.
Aside from the remarkable merit of providing analytically tractable RG-flow equations, the FRG-approach, when applied to interacting spinful electrons, also accounts for the backscattering bulk interaction, which is usually neglected
in the TLL-framework13,14 . Moreover, it can be readily generalized to describe junctions involving superconducting contacts, at the price of doubling the set of degrees of freedom, to treat particle- and hole-excitations on the
same footing63 . Nevertheless, FRG is known to suffer of limitations, when used to describe the crossover towards
”healed” fixed points, where quantum impurities are renormalized towards boundary conditions corresponding to perfect conduction properties. In particular, this leads to a scaling equation for the conductance which, in the pertinent
asymptotic regime, is not consistent with the one obtained from the bosonization approach. Using the fRG-approach
allows for taking care of this flaw, as the fRG-technique typically takes into account the mutual feedback from all the
single-fermion scattering channels, and not only those from scattering processes between different Fermi points (see
Ref. [59] for a detailed discussion of this point and for a careful comparison between FRG- and fRG-approaches). In
fact, fRG appears to be generically more accurate than FRG (which can be in fact recovered from fRG under suitable
approximations59). Both approaches suffer, however, of the limitation on the bulk interaction, which must be weak,
in order for the technique to be reliably applicable.
To overcome such a limitation, in this paper, we study a junction of spinful interacting QWs by making a combined
use of bosonization and fermionization, that is, we go back and forth from fermionic to bosonic coordinates, and vice
versa, to build ”dual-fermion” representations of the junction in strongly interacting regimes. In resorting from a
bosonic to a fermionic problem, our approach is reminiscent of the refermionization scheme used in Ref. [64] to discuss
the large-distance behavior of the classical sine-Gordon model at the commensurate-incommensurate phase transition.
Specifically, in64 the refermionization allows for singling out at criticality the low-energy two-fermion excitations from
the one-fermion ones and to prove that the latter ones keep gapped along the phase transitions and do not contribute
to the large-distance scaling of the correlations. At variance, in our case it is the second of a two-step process, that
ends up again into a fermionic ”dual fermionic” model for the strongly-interacting system. The guideline to construct
the appropriate novel fermionic degrees of freedom is to eventually rewrite the relevant boundary interactions as
bilinear functionals of the fermionic fields. Specifically, moving from the original fermionic coordinates to the TLLbosonic description of the junction, we are able to warp from the weakly interacting regime to different strongly
interacting regimes. Therefore, at appropriate values of the interaction-dependent Luttinger parameters, we move
back from the bosonic- to pertinent dual-fermionic coordinates, chosen so that the relevant boundary interactions

3
are bilinear functionals of the fermionic fields. Our mapping between dual coordinates is actually preliminary to the
implementation of the RG-approach. Therefore, in principle, it could be equally well applied to extend both FRG
and fRG to the strongly-interacting regime. Since, however, fRG usually requires resorting back to lattice models to
implement a systematic numerical treatment of the flow equations for the interaction vertices, which goes beyond the
scope of this work53–59 , we rather prefer to complement our dual mappings with a pertinently adapted version of the
FRG-approach.
The RG-approach formulated in fermionic coordinates, such as FRG, suffers of the limitation that it requires that
relevant scattering processes at the junction are fully encoded in terms of a single-particle S-matrix. While this is
certainly the case at weak bulk interaction, a strong attractive interaction in either charge-, or spin-channel (or in
both) is known to stabilize phases (RG attractive fixed points) at which two-particle scattering is the most relevant
process at the junction10–12,20,21 . Just because of the way it is formulated, the FRG-approach fails to describe manyparticle scattering processes, even after improvements of the technique that allow to circumvent the constraint of
having a small bulk interaction65–67 . Resorting to the appropriate dual-fermion basis allows us to describe within
the FRG-approach also fixed points stabilized by many-particle scattering processes, as well as fixed points whose
properties have not been mapped out within the TLL-framework in terms of a rotation matrix such as, for instance,
the mysterious-fixed point in the three-wire junction of spinless quantum wires studied in Ref. [21] and its counterpart
in the junction of spinful quantum wires. Moreover, in computing the conductance tensor along the RG-trajectories
connecting fixed points of the phase diagram, we show how our approach, while being consistent with the TLLapproach in the range of parameters where both of them apply, on the other hand allows for complementing the
results of Refs. [10–12,23] about the two-wire and the three-wire junction, with a number of additional results about
the topology of their phase diagram and their conductance properties.
The paper is organized as follows:
• In Sec. II we apply our duality-complemented FRG-approach to a junction of two interacting spinful quantum
wire, discussing the results and comparing them to those obtained within the bosonization approach10–12 ;
• In Sec. III we apply our approach to a junction of three interacting spinful quantum wire. We first compare
our results with those obtained within the bosonization approach23 and, therefore, we show how our technique
allows for mapping out the full crossover of the conductance at the junction even in strongly-interacting regions,
typically not accessible with a fermionic approach;
• We summarize our results and discuss possible further developments of our research in Sec. IV, dedicated to
the concluding remarks of our paper;
• In the various appendixes we review mathematical techniques that are crucial for our derivation. Specifically,
in Appendix A, we review the derivation of the FRG-equations for the S-matrix, in Appendix B, we review the
basic bosonization rules for interacting one-dimensional quantum wires; in Appendix C, we provide some basic
elements of linear transport theory for junctions of one-dimensional quantum wires.
II. DUAL FERMIONIC VARIABLES AND RENORMALIZATION GROUP APPROACH TO THE
CALCULATION OF THE CONDUCTANCE AT A JUNCTION OF TWO SPINFUL INTERACTING
QUANTUM WIRES

To introduce and check the validity of our approach, in this section we discuss a junction of two interacting spinful
quantum wires. This appears to be quite an appropriate place to test our technique: indeed, the two-wire junction has
widely been studied in the past, both within the bosonization approach10–12 , and by means of standard RG techniques
for a weak bulk interaction, either using the FRG-approach13–15,18 , or the fRG-method53–59 . The two-wire junction
of spinful quantum wires is described by the (K = 2) bulk Hamiltonian HBulk = H0 + Hint , with
K

H0 = −iv

L

dx
j=1

σ

0

†
†
ψR,j,σ (x)∂x ψR,j,σ (x) − ψL,j,σ (x)∂x ψL,j,σ (x)

,

(1)

with L being the wire length (eventually sent to infinity at the end of the calculations) and the interaction Hamiltonian
given by

4
K

Hint ≈

j=1

L
0

†
†
†
†
dx gj,1, ψR,j,↑ (x)ψL,j,↑ (x)ψR,j,↑ (x)ψL,j,↑ (x) + gj,1, ψR,j,↓ (x)ψL,j,↓ (x)ψR,j,↓ (x)ψL,j,↓ (x)

†
†
†
†
+ gj,1,⊥ ψR,j,↑ (x)ψL,j,↓ (x)ψR,j,↓ (x)ψL,j,↑ (x) + gj,1,⊥ ψR,j,↓ (x)ψL,j,↑ (x)ψR,j,↑ (x)ψL,j,↓ (x)
K

L

+
j=1

0

†
†
†
†
dx gj,2, ψR,j,↑ (x)ψL,j,↑ (x)ψL,j,↑ (x)ψR,j,↑ (x) + gj,2, ψR,j,↓ (x)ψL,j,↓ (x)ψL,j,↓ (x)ψR,j,↓ (x)

†
†
†
†
+ gj,2,⊥ ψR,j,↑ (x)ψL,j,↓ (x)ψL,j,↓ (x)ψR,j,↑ (x) + gj,2,⊥ ψR,j,↓ (x)ψL,j,↑ (x)ψL,j,↑ (x)ψR,j,↓ (x)

.

(2)

The various interaction strengths appearing in Eq. (2) are defined as
gj,1, = Vj,↑↑ (2kF ) = Vj,↓↓ (2kF )
gj,2, = Vj,↑↑ (0) = Vj,↓↓ (0)
gj,1,⊥ = Vj,↑↓ (2kF ) = Vj,↓↑ (2kF )
gj,2,⊥ = Vj,↑↓ (0) = Vj,↓↑ (0) ,

(3)

with Vσ,σ′ (k) being the Fourier modes of the two-body interaction ”bulk” interaction potential in the quantum wires
(see Appendix A for the derivation and discussion of Eqs. (2,3).) For a weak bulk interaction, the most relevant
contribution to HB is given by a linear combination of the operators B(j,j ′ ),σ,(X,Y ) (0), defined as
†
B(j,j ′ ),σ,(X,X ′ ) (0) = ψX,j,σ (0)ψX ′ ,j ′ ,σ (0) ,

(4)

with X, X ′ = L, R. Assuming equivalence between the two wires and a spin-symmetric and spin-conserving boundary
interaction, HB can be generically written as
HB =
X,X ′ =L,R

σ

{[τX,X ′ B(1,2),σ,(X,X ′ ) (0) + h.c.] +

µX,X ′ B(j,j),σ,(X,X ′ ) (0)} .

(5)

j=1,2

In addition to the contributions reported in Eq. (5), terms that are quadratic (or of higher order) in the B’s can in
principle arise along RG-procedure, even if they are not present in the ”bare” Hamiltonian. For instance, the simplest
higher-order boundary interaction terms consistent with spin conservation at the junction, H2,0 and H0,2 , are given
by10–12
H2,0 = V2,0
X,X ′ =R,L

H0,2 = V0,2
X,X ′ =R,L

{B(X,X ′ ),↑,(1,2) (0)B(X ′ ,X),↓,(1,2) (0) + h.c.}
{B(X,X ′ ),↑,(1,2) (0)B(X ′ ,X),↓,(2,1) (0) + h.c.} .

(6)

As it can be shown using the bosonization approach, for a weak bulk interaction, higher-order operators such as
those in Eqs. (6) are highly irrelevant operators and, accordingly they are typically ignored and one uses for HB the
formula in Eq. (5). Physically, this means that the relevant scattering processes at the junction consist only of one
single particle/hole scattered into one single particle/hole, such as those drawn in Fig. 1 (a). These processes are
fully described by the single-particle S-matrix, for which the renormalization group equations can be fully recovered
using the technique we review in Appendix A. The symmetry requirements listed above imply that the single-particle
S-matrix takes the block-diagonal form
S(j,σ);(j ′ ,σ′ ) (k) = δσ,σ′ Sj,j ′ (k) ,

(7)

with the S(k)-matrix being given by
S(k) =

rk tk
tk rk

,

(8)

and rk and tk respectively corresponding to the amplitude for the particle to be backscattered in the same wire,
or transmitted into the other wire. In the following we will pose no particular constraints on the rk ’s and the tk ’s,
except that, near the Fermi points, they are quite flat functions of k, without displaying particular features, such as

5
a resonant behavior: accordingly, we assume that the amplitudes are all computed at the Fermi level and drop the k
label (this is a specific case of the general assumptions on the behavior of the S-matrix elements near by the Fermi
surface that we make in Appendix A). To write the RG-equations for the S-matrix elements, one needs the Friedel
matrix F 18 which, in this specific case, is given by
βr
1 0
F = 
0
2
0


0
βr
0
0

0
0
βr
0


0
0 
0 
βr

,

(9)

1
with β = 2πv −g1 − g1⊥ + g2 . Taking into account the symmetries of the S- and of the F -matrix, the RG-equations
for the amplitudes r, t are obtained in the form

β
dr
2
2
r − r |r| − r∗ t2 = βr |t|
=
dℓ
2
dt
2
2
= −βt |r| = −β(t − t |t| ) .
dℓ

(10)

Equations (10) must be supplemented with the RG-equation for the running strength β, which is given by
dβ
1
2
{(g1,⊥ ) + 2g1,⊥ g2,⊥ − g2, + g1, } .
=
dℓ
(2πv)2

(11)

(See Eqs. (2,3) for the definition of the bulk interaction strengths gj,1(2),⊥ , gj,1(2), : here we drop the wire index j as
the interaction strengths are assumed to be the same in each wire.) Equations (10), together with Eq. (11) and Eqs.
(A10) for the running interaction strengths, constitute a closed set of equations, whose solution yields the scaling
functions r(D), t(D). From the explicit formulas for the running scattering amplitudes, one may readily compute
the charge- and the spin-conductance tensors, using the formulas derived in Appendix C 1. As a result, due to the
symmetries of the S-matrix, the charge- and the spin-conductance tensor are equal to each other and both given by
e2
π

Gc (D) = Gs (D) =

− G(D)
G(D)

e2
π

G(D)
− G(D)

,

(12)

2

with G(D) = e |t(D)|2 . An explicit analytical formula can be provided for G(D) in some simple cases such as, for
π
instance, if g1⊥ is fine-tuned to 0. In this case, as it arises from Eq. (11), β keeps constant along the RG-trajectories
and, therefore, one may exactly integrate Eqs. (10) for r(D) and t(D), obtaining
r(D) =

t0 |D/D0 |

−β

t0 |D/D0 |

β

|r0 |2 + |t0 |2 |D/D0 |−2β

t(D) =

2

2

|r0 | + |t0 | |D/D0 |

2β

,

(13)

with r0 , t0 corresponding to the ”bare” scattering amplitudes in Eq. (8). Another case in which an explicit analytical
solution can be provided corresponds to having g1⊥ = g1 = g1 and g2⊥ = g2 = g2 . In this case, the set of Eqs. (A10)
collapse onto a set of two equations for g1 (D), g2 (D) which can be readily integrated, yielding the running interaction
strengths
g1 (D) =

1+

g2 (D) = g2 −

g1
g1
D0
πv ln D
g1
1
g1
+
g1
2
2 1 + πv ln D0
D

,

(14)

and β(D) = [g2 (D) − 2g1 (D)]/(2πv). Once β(D) is known, Eqs. (10) can be integrated, yielding
G(D) =

e2
π

g1
πv

ln

R0 + T0 1 +

g1
πv

T0 1 +

D0
D

ln

3/2
D0
D

|D/D0 |

3/2

2γ

|D/D0 |

2γ

,

(15)

6

a)

b)

c)

1
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1
0
1
0

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FIG. 1: Sketch of possible scattering processes at a two-wire junction (note that incoming particles from wire j can either be
scattered into the same wire, or into a different wire):
(a) Single-particle/single-hole scattering processes. These are determined by Htun in Eq. (5) and are fully described in terms
of a single-particle S-matrix;
(b) Many-body scattering processes in which one particle/one hole is scattered into two particles and one hole/two holes
and one particle. These processes can be induced by boundary interaction Hamiltonians such as those in Eq. (6) and their
proliferation requires resorting to a bosonic Luttinger-liquid description of the junction;
(c) Scattering processes for a particle-particle and for a particle-hole pair. These are again determined by the Hamiltonians
in Eq. (6) and are the only allowed processes in the presence of a strong repulsive (attractive) interaction in the spin (charge)
channel, and vice versa. On pertinently defining new fermionic coordinates, they can still be described in terms of a ”single-pair”
S-matrix.
1
with γ = − g2 + g2 /(2πv), T0 = |t0 |2 , R0 = |r0 |2 . As an example of typical scaling plots for G(D) in the simple
cases discussed before, in Fig. 2 we plot G(D)π/e2 vs. ln(D0 /D), as from Eq. (13) (panel (a)) and from Eq. (15)
(panel (b)), with the values of the parameters reported in the caption. Consistently with the results obtained within
Luttinger liquid framework10–12 , G(D)π/e2 either flows to 0 for an effectively repulsive interaction (β, γ > 0), or to 2
(the maximum value allowed by unitarity), for an effectively attractive interaction (β, γ < 0). For general values of the
interaction strengths, the equations have to be numerically integrated. In Fig. 3, we provide some examples of scaling
of G(D) vs. ln(D0 /D) in the general case. It is important to stress18 that, due to the nontrivial renormalization group
flow of the interaction strengths, the flow of G(D) can be a nonmonotonic function of D for some specific values of the
interaction strengths. It would be interesting to check such a feature in a real life two-wire junction: remarkably, this
prediction is only obtained within the FRG-approach, in which it is possible to account for the flow of the running
interaction strengths, as well.
The possibility of mapping out the full crossover of the conductance as a function of the scale D is possibly the most
important feature of the FRG approach. Yet, since, as we discuss to some extent in Appendix A, the validity of the
FRG technique is grounded on the assumption that all the relevant scattering processes at the junction are described
by the single-particle S-matrix13–15,18 , it breaks down when attempting to recover the full crossover of the conductance
towards fixed points where multi-particle scattering is the most relevant process at the junction, such as the strongly
coupled fixed point stabilized by either H2,0 or H0,2 in Eq. (6)10–12 . Technically, what happens is that, as soon
as boundary operators such as H2,0 or H0,2 become relevant, the proliferation of low-energy many-body scattering
processes such as the one we sketch in Fig. 1 (b) invalidates the single-particle S-matrix description of the junction
dynamics. Nevertheless, some many-body scattering processes can be strongly limited by having, for instance, a
strong repulsive interaction among particles with the same spin and a strong attractive interaction among particles
with the same charge. In the bosonic framework, this corresponds to having values of the Luttinger parameters in Eqs.
(B11) such that gc ≥ 2, gs ≪ 1. Indeed, in this limit on one hand, the strong spin repulsive interaction forbids the
single-particle processes described by HB in Eq. (5) (at small values of the boundary coupling strengths τX,X ′ , µX,X ′
this can be readily seen from the explicit result for the scaling dimension of HB computed within the bosonization
1
1
approach, which is xB,weak = 1 − 2gc − 2gs , which becomes ≫ 1, corresponding to a largely irrelevant operator). On
the other hand, one expects that the strong charge attraction stabilizes tunneling of composite objects carrying zero
spin, such as two-particle pairs, as the one depicted at the left-hand panel of Fig. 1 (c). As a consequence, due to
the fact that these are again one-into-one scattering processes, one expects that it is possible to choose the effective

7

a)

b)

G(D) π/e 2

ln(D 0 /D)

ln(D0 /D)

FIG. 2: (a) Plot of G(D) vs. ln(D0 /D) as from Eq. (13) for T0 = 1 − R0 = 0.4, β = 0.35 (purple curve - corresponding to
an effectively repulsive interaction), and β = −0.35 (blue curve - corresponding to an effectively attractive interaction), with
T0 = |t0 |2 , R0 = |r0 |2 ;
(b) Plot of G(D) vs. ln(D0 /D) as given in Eq. (15) for T0 = 1 − R0 = 0.4, g1 /(2πv) = 0.2, γ = 0.36 (purple curve corresponding to an effectively repulsive interaction), and β = −0.36 (blue curve - corresponding to an effectively attractive
interaction).

b)

a)
G(D) π/e 2

ln(D 0/D)
FIG. 3: (a) Plot of G(D) vs. ln(D0 /D), obtained by numerically integrating Eqs.
g1,⊥ (D0 )/(2πv) = g2,⊥ (D0 )/(2πv) = 0.3, g2, (D0 )/(2πv) = 0, R0 = T0 = 0.5;
(b) Same as in panel (a), but with g2, (0)/(2πv) = 0.3, as well.

ln(D 0/D)
(10,11,A10) for g1, (D0 )/(2πv) =

low-energy degrees of freedom of the system to resort to a single-particle S-matrix in the new coordinates. In fact,
this is the idea behind the dual fermion approach we are going to discuss next. When resorting to dual fermion
coordinates, an important issue is related to whether the vacuum states at a fixed particle number68 for the original
and the dual-fermions are the same. In fact, while dual fermion formalism only captures composite excitations e.g.
two-particle states in the gc ∼ 2, gs ≪ 1-regime, at such values of the parameters, these states are the only ones that
at low energy are effectively able to tunnel across the junction (that is, the only ones whose tunneling is described
by a non-irrelevant operator). So, as long as one is only concerned about states relevant for low-energy tunneling
across the junction (that is, states relevant for the calculation of the dc-conductance tensor of the junction), one can

8
effectively assume that the fixed-particle number vacuum states are the same in terms of the new (dual) and of the
old fermions. The definition of the dual fermion operators strongly depends on the boundary conditions of the various
fields at the junction. Accordingly, in the following we define different dual fermion operators in different regimes of
values of the boundary interaction, and eventually show that, whenever two different sets of dual coordinates apply
to the same region, they yield the same results, as they are expected to.
Let us begin with the weak boundary interaction regime. Referring to the bosonization formulas of Appendix B,
this corresponds to assuming Neumann (Dirichlet) boundary conditions for all the Φ (Θ)-fields in Appendix B and,
in addition, to equating the Klein factors so that ηR,σ,j = ηR,σ,j , ∀σ, j. In this limit, one may respectively rewrite
H0,2 and H2,0 in bosonic coordinates as
H2,0 = v2,0 cos[Φ1,c (0) − Φ2,c (0)]
H0,2 = v0,2 cos[Φ1,s (0) − Φ2,s (0)] ,

(16)

with v2,0 ∝ V2,0 , v0,2 ∝ V0,2 . The scaling dimensions of the operators in Eqs. (16) are respectively given by x2,0 =
2/gc, x0,2 = 2/gs . Thus, in the regime of a strongly attractive interaction in the charge (spin)-channel and strongly
repulsive interaction in the spin (charge)-channel, H2,0 (H0,2 ) may become the most relevant boundary operator at
weak boundary coupling. The strategy of our dual fermion approach consists in defining a novel set of fermionic fields,
in terms of which the operators in Eqs. (16) are realized as bilinears, similar to the B-operators in Eq. (4). To be
specific, let us introduce the center-of-mass and the relative fields in the charge- and in the spin-sector, respectively
given by
1
Φc(s) (x) = √ [Φ1,c(s) (x) + Φ2,c(s) (x)]
2
1
Θc(s) (x) = √ [Θ1,c(s) (x) + Θ2,c(s) (x)] ,
2

(17)

and
1
ϕc(s) (x) = √ [Φ1,c(s) (x) − Φ2,c(s) (x)]
2
1
ϑc(s) (x) = √ [Θ1,c(s) (x) − Θ2,c(s) (x)] .
2
Next, let us perform the canonical transformation to a new set

 √
¯
2 0 0
Φc(s) (x)
1
¯
 Θc(s) (x)   0 √
0
2 √

=
 ϕc(s) (x)   0 0
¯
2
¯
ϑc(s) (x)
0 0 0

(18)

of bosonic fields, defined as


0
Φc(s) (x)
0   Θc(s) (x) 


0  ϕc(s) (x)
1
ϑc(s) (x)
√

.

(19)

2

It is worth stressing that the transformation in Eqs. (19) relate to each other bosonic operators at a given position in
real space. Since the correspondence rules between the bosonic and the (original or dual) fermionic fields, summarized
in Appendix B, are local in real space, as well, one concludes that, written in terms of dual fermionic coordinates,
the boundary interaction Hamiltonian HB is still local and that the dynamics far from the junction can be fully
encoded within dual fermion scattering states. Now, assuming gs ≪ 1, gc = 2 + δgc , with |δgc | ≪ 1, we see that
gs ≪ 1 makes H0,2 strongly irrelevant. This fully suppresses spin transport across the junction and, therefore, we
may just focus onto charge transport, ruled by H2,0 . In fact, it appears that single-spinful particle-tunneling processes
are already suppressed against two-particle pair tunneling processes as soon as gs < 2/3. As conservation of spin
symmetry implies gs = 1, in order to realize the condition above one may, for instance, think of two coupled spinless
interacting one-dimensional electronic systems (which could possibly realized as semiconducting quantum wires in the
presence of spin-orbit and Zeeman interactions), with a mismatch in the Fermi momenta that prevents the interaction
from opening a gap in the fermion spectrum. The two channels can, therefore, be regarded as the two opposite spin
polarization, although without any symmetry implying gs = 1. To rewrite this latter operator as a bilinear functional
of fermionic operators, we define the spinless chiral fermionic fields χR,j (x), χL,j (x) as
i

¯

j

i

¯

j

χR,j (x) = ηR,j e 2 [Φc (x)−(−1)
χL,j (x) = ηL,j e 2 [Φc (x)+(−1)

¯
¯
ϕc (x)+Θc (x)−(−1)j ϑc (x)]
¯

¯
¯
ϕc (x)+Θc (x)+(−1)j ϑc (x)]
¯

,

(20)

9
with j = 1, 2 and with ηR,j , ηL,j being real fermionic Klein factors. ”Inverting” the bosonization procedure outlined
in Appendix B into a pertinent re-fermionization to spinless fermions, we find that the bulk Hamiltonian for the
χ-fermions is given by
L

Hc;bulk
−

=
uπδgc
2

−iu

j=1,2
L

j=1,2

0

0

dx {χ† (x)∂x χR,j (x) − χ† (x)∂x χL,j (x)}
R,j
L,j

dx : χ† (x)χR,j (x) :: χ† (x)χL,j (x) :
R,j
L,j

,

(21)

with the velocity u ∝ v. The χL/R,j -fields are the appropriate degrees of freedom to describe pair scattering at the
junction in terms of a single-particle S-matrix. In order to prove that it is so, we note that H2,0 can be regarded as
the bosonic expression for the boundary weak coupling limit of a tunnel Hamiltonian for the spinless fermions, Hc,tun ,
given by
Hc,tun = v2,0 {χ† (0)χ2 (0) + χ† (0)χ1 (0)}
1
2

,

(22)

with χj (0) = χR,j (0) + χL,j (0). While the strong spin repulsion sets the spin conductance tensor to 0, the charge
conductance can nevertheless be different from zero, due to zero-spin pair-tunneling across the junction. Once the
RG-flow for the S-matrix elements describing χ-fermion scattering at the junction has been derived as we did before,
using the formulas we report in Appendix C 1 and the expression of the charge current operator in wire j in terms of
the dual fermionic fields:
√
Jc,j (x) = eu 2 {: χ† (x)χR,j (x) : − : χ† (x)χL,j (x) :} ,
(23)
R,j
L,j
we obtain that the charge conductance tensor scales according to
e2
π

Gc (D) =

− G(D)
G(D)

e2
π

G(D)
− G(D)

,

(24)

with
e2
G(D) =
π

T0 |D/D0 |−

δgc
2

R0 + T0 |D/D0 |−

δgc
2

,

(25)

and the bare reflection and transmission coefficients respectively given by
2
u2 − v2,0
R0 = 2
2
u + v2,0

2uv2,0
T0 = 2
2
u + v2,0

2

2

.

(26)

In Fig. 4, we plot G(D) versus ln(D0 /D) in two paradigmatic cases, respectively corresponding to δgc > 0 and
to δgc < 0. To our knowledge, this is the first example of a full scaling plot of the conductance for a junction of
strongly interacting one-dimensional quantum wires. While, on one hand, this shows the effectiveness of our approach
in describing the crossover of the conductance towards the spin-insulating charge-conducting fixed point, on the other
hand, one has also to prove the consistency of an effective theory strongly relying on the weak boundary coupling
assumption with an RG-flow taking the system all the way down to the perfectly charge-conducting fixed point,
corresponding to the strongly interacting limit of the boundary interaction10–12 . When δgc > 0, the relevance of H2,0
drives the system towards the strongly boundary interaction limit in the charge channel, corresponding to pinning
ϕc (0) and, accordingly, to imposing Neumann boundary conditions on ϑc (0). Since Φc (0) does not appear in the
boundary interaction, one assumes that it still obeys Neumann boundary conditions and, accordingly, that Θc (0) is
pinned at a constant value. We now prove that these boundary conditions are recovered by taking the strongly coupled
limit of Hc,tun in Eq. (22) and using the refermionization rules in Eqs. (20). Indeed, on making the strong-coupling
v
assumption, 2,0 ≫ 1, as from Eqs. (26), one obtains R0 → 0, T0 → 1, that is, the boundary conditions correspond
u
to perfect transmission from wire-1 to wire-2, and vice versa. In terms of the dual fermionic fields, this corresponds
to the conditions

10

G(D) π/e 2

ln(D 0 /D)
FIG. 4: Plot of Gc (D) vs. ln(D0 /D) as from Eq. (25) for T0 = R0 = 0.5 and for δgc respectively equal to 0.6 (blue curve) and
to -0.6 (purple curve).

χR,2 (x) = eiλ χL,1 (−x)
χR,1 (x) = e−iλ χL,2 (−x)

,

(27)

with λ being some nonuniversal phase. From Eqs. (20), one sees that Eqs. (27) imply Dirichlet boundary conditions
¯
¯
¯
at x = 0 for both ϕc (x) and Θc (x), with the dual fields ϑc (x), Φc (x) obeying Neumann boundary conditions. This is
¯
exactly the same result one would obtain working in bosonic variables by sending to ∞ the interaction strength v2,0 in
Eq. (16). Due to the strong repulsion in the spin channel, such a fixed point corresponds to perfect transmission in the
charge channel, but perfect reflection in the spin channel, that is, it must be identified with the non-symmetric chargeconducting spin-insulating phase of Refs. [10–12]. To conclude the consistency check, we note that, on alledging for
additional backscattering contributions to Hc,tun of the generic form µ1 χ† (0)χ† (0) + µ2 χ† (0)χ† (0) (which play
R,1
R,1
L,1
L,
no role at weak coupling) and using again Eqs. (27), one obtains the bosonic operators
¯
˜
Hc,tun ∼ µ cos ϑc (0)

,

(28)

with µ being some nonuniversal constant. Equation (28) corresponds to the bosonic version of the leading boundary
perturbation at the non-symmetric charge-conducting spin-insulating fixed point10–12 .
Our approach also allows for analyzing the complementary situation in which gc ≪ 1 and gs ∼ 2. In this case,
one expects that the strong repulsion in the charge channel and the strong attraction in the spin channel stabilize
single-pair tunneling processes at the junction such as those sketched at the right-hand panel of Fig. 1 (c), that
is, tunneling of particle-hole pairs, with total spin 1. Again, for gs = 2, the S-matrix describes single-particle into
single-particle scattering processes, once it is written in the appropriate basis. To select the pertinent degrees of
freedom, we therefore repeat the refermionization procedure in Eq. (20), by just exchanging the charge- and the
spin-sector with each other. Of course, charge- and spin-conductance are exchanged with each other, compared to
the previous situation and, accordingly, the flow will be towards the charge-insulating spin-conducting fixed point of
Refs. [10–12]. An important remark, however, concerns the effects of a possible residual interaction, which, as we did
before, can be in principle introduced for accounting for gs slightly different from 2. Indeed, a term in the ”residual”
bulk interaction Hamiltonian such as the one ∝ gj,1,⊥ in Eq. (2), once expressed in terms of the fermionic fields in
Eqs. (20) would take the form
2

mj

Hδ =
j=1

dx {χ† (x)χL,j (x) + χ† (x)χR,j (x)} ,
L,j
R,j

(29)

with mj ∝ gj,1,⊥ , which would open a bulk gap in the single-χ fermion spectrum, thus making the whole system behave
as a bulk spin insulator. Therefore, in order to recover the correct physics of the charge-insulating spin-conducting
fixed point, we must assume that all the gj,1,⊥ are tuned to zero, which is typically the case when resorting to the
bosonic approach to spinful electrons10–12 .

11

ψL,2, σ
ψR,1, σ

ψR,2, σ

Φ

ψL,3, σ

ψL,1, σ
ψR,3, σ

FIG. 5: Sketch of a three-wire junction of spinful quantum wires pierced by a magnetic flux Φ.

As we have just shown, resorting to pertinent dual-fermion operators allows for mapping out the full crossover with
the appropriate energy scale of the charge and/or spin conductance of a junction in region of values of the interaction
parameters in which one is typically forbidden to use the standard weak-coupling formulation of either FRG, or fRG.
In the following, we apply our technique to the spinful three-wire junction studied using the bosonization approach
in Ref. [23], and will recover the full crossover of the conductance tensor in regions typically not accessible in the
bosonic formalism.
III. DUAL FERMIONIC VARIABLES AND RENORMALIZATION GROUP APPROACH TO THE
CALCULATION OF THE CONDUCTANCE AT A JUNCTION OF THREE SPINFUL INTERACTING
QUANTUM WIRES

We now consider a three-spinful-wire junction, such as the one we sketch in Fig. 5. Resorting to the appropriate
fermionic variables, we generalize to strongly interacting regions the weak-coupling FRG-approach. As a result, we
map out the full dependence of the conductance tensor on the low-energy running cutoff scale even in stronglyinteracting regions of the parameter space. Eventually, we discuss the consistency of our results about the phase
diagram of the junction with those obtained in Ref. [23], particularly showing how our technique can be used to
recover informations that typically cannot be derived within the bosonization approach used there. Consistently
with Ref. [23], in the following, we make the simplifying assumption that, in the weakly interacting regime, the bulk
interaction is purely intra-wire and is the same in all the three wires. In fact, while this assumption is already expected
to yields quite a rich phase diagram23 , in principle our approach can be readily generalized to cases of different bulk
interactions in different wires, such as the one considered in Ref. [69].
A.

The weakly interacting regime

For a weak boundary interaction, assuming total spin conservation at the junction, the most relevant boundary
interaction Hamiltonian is a combination of the bilinear operators in Eq. (4). The relevant scattering processes at
the junction are all encoded in the single-particle S-matrix elements, S(j,σ),(j ′ ,σ′ ) (k). Because of spin conservation,
the S-matrix is diagonal in the spin index, that is, S(j,σ),(j ′ ,σ′ ) (k) = δσ,σ′ Sj,j ′ (k). Assuming also that the boundary
Hamiltonian is symmetric under exchanging the wires with each other, the 3 × 3 matrix S(k) takes the form


¯
r t t
¯
S= t r t  .
(30)
¯ t r
t

(Note that, in Eq. (30) we assume that all the amplitudes are computed at the Fermi level and accordingly drop the
index k from the S-matrix elements. This is consistent with the discussion of Appendix A, where we assume that,
close to the Fermi level, the scattering amplitudes are smooth functions of k.) It is worth mentioning that, in writing
Eq. (30), we allowed for time-reversal symmetry breaking as a consequence, for instance, of a magnetic flux φ piercing
the centre of the junction (see Fig. 5). This implies that, in general, the scattering amplitude t from wire j to wire

12
¯
j + 1 is different from the one from wire j to wire j − 1 (t). From Eq. (30) one therefore finds that the F -matrix
1
elements in Eq. (A8) are given by Fj,j ′ = β rδj,j ′ , with again β = 2πv −g1 − g1⊥ + g2 . On applying the FRG
2
formalism of Appendix A, one readily obtains the RG-equations for the independent S-matrix elements, given by
dr
β
2
¯
r − |r| r − 2ttr∗
=
dℓ
2
β
dt
2
¯
= − 2t |r| + t2 r∗
dℓ
2
¯
β ¯ 2
dt
,
= − 2t |r| + t2 r∗
dℓ
2

(31)

which, again, must be supplemented with the RG-equations for the running coupling strengths, Eqs. (A10,11). Since,
as a consequence of spin conservation in scattering processes at the junction, the S-matrix is diagonal in the spin
indices, the charge- and spin-conductance tensors are equal to each other at the fixed points, as well as along the
RG-trajectories obtained integrating Eqs. (31). In particular, using the formalism of Appendix C 1, one obtains


¯
−R(D) + 1 −T (D)
−T (D)
2
e 
¯
−T (D) −R(D) + 1 −T (D)  ,
(32)
Gc,s (D) =
π
¯
−T (D)
−T (D) −R(D) + 1

¯
¯
¯
with R(D) = |r(D)|2 , T (D) = |t(D)|2 , and T (D) = |t(D)|2 . The RG-flow of the scattering coefficients T (D), T (D) is
recovered by solving the set of differential equations
dT
β
=−
dℓ
2
¯
dT
β
=−
dℓ
2

¯
¯
¯
5T − T (1 − T − T ) − T T
¯
¯
¯
5T − T (1 − T − T ) − T T

,

(33)

¯
which are derived from Eqs. (31) by taking into account the unitarity constraint T (D) + T (D) + R(D) = 1. The fixed
points of the boundary phase diagram are, therefore, determined by setting to zero the terms at the right-hand side
of Eqs. (33). From Eq. (32) one may therefore recover the corresponding charge- and the spin-conductance tensors.
Borrowing the labels used in Ref. [23], we obtain the following fixed points:
• The [Nc , Ns ] (”disconnected”) fixed point

¯
This fixed point corresponds to having R = 1 and T = T = 0 which, according to Eq. (32), yields


e2  0 0 0 
0 0 0
Gc = Gs =
π 0 0 0

,

(34)

as it is appropriate for a disconnected junction.
• The χ++ fixed point

¯
This corresponds to R = T = 0, T = 1, which yields


and, accordingly


0 0 1
|Sj,j ′ |2 =  1 0 0 
0 1 0

,



e2  1 0 −1 
−1 1 0
Gc = Gs =
π
0 −1 1
• The χ−− fixed point

¯
This corresponds to R = T = 0, T = 1, which yields

(35)

;

(36)

13

a)

b)

T(D )=T(D )
0

0

T(D )
0

T(D )
0

R(D )

R(D )

0

0

ln(D 0 /D)

ln(D 0 /D)

FIG. 6: Renormalization group flow of the scattering coefficients for a three-wire junction for g1,⊥ = 0, β constant and equal
to -0.3, and different choices of the initial values of the scattering coefficients:
¯
(a) Renormalization group flow corresponding to T (D0 ) = T (D0 ) = 0.1, R(D0 ) = 0.8. The curves corresponding to T (D) and
¯
to T (D) vs ln(D0 /D) collapse onto the single red curve of the graph, while the flow of R(D) vs ln(D0 /D) is described by the
blue curve. As D0 /D grows, the scattering coefficients flow towards the asymptotic values corresponding to the M -fixed point;
¯
(b) Renormalization group flow corresponding to T (D0 ) = 0.1, T (D0 ) = 0.3, R(D0 ) = 0.6. As D0 /D grows, the scattering
coefficients flow towards the asymptotic values corresponding to the χ−− -fixed point.



and, accordingly


0 1 0
|Sj,j ′ |2 =  0 0 1 
1 0 0

,


1 −1 0
e 
0 1 −1 
Gc = Gs =
π −1 0 1
2



(37)

;

(38)

• The M fixed point
1
4
¯
This corresponds to T = T = 9 , R = 9 and has to be identified with the symmetric18 , or with the Griffith70
fixed point of a junction of three interacting wires. One obtains
 1 4 4 
and, accordingly

|Sj,j ′ |2 = 

9
4
9
4
9

9
1
9
4
9

9
4
9
1
9



,

 8

4
−4 −9
e2  94 89
4
−9 9 −9 
Gc = Gs =
π
4
8
4
−9 −9 9

(39)

.

(40)

χ±± must clearly be identified with the ”chiral” fixed points of Ref. [23], where time-reversal symmetry breaking
is maximum, both in the charge and in the spin sector. Along the RG-trajectories connecting two fixed points, the
conductance flow is determined by Eq. (32). The topology and the direction of the RG-trajectories depend on both
β and on the bare values of the S-matrix elements. β scales with ℓ as determined by Eqs. (A10,11), which makes it
necessary to resort to a full numerical integration approach. A set of simplified situations can be realized, however,
where β keeps constant along RG-trajectories. For instance, if g1,⊥ (D0 ) = 0, Eq. (11) implies that β is constant.
In this case, from Eqs. (33) one readily sees that, if β > 0, the boundary flow is towards the N N -fixed point. At
¯
variance, if β < 0 and T (D0 ) > (<)T (D0 ), the boundary flow is towards the χ++ (χ−− )-fixed point. As an example
¯
of possible RG-trajectories that may be realized in this specific case, in Fig. 6 we plot T (D) and T (D) for β constant
and negative, while we draw similar plots in Fig. 7 for constant and positive β and in Fig. 8 for non-constant β (see
the captions for details).
All the analysis we have done so far applies to a junction of three spinful quantum wires for weak bulk interaction.
We now employ the dual-fermion approach to generalize the FRG-technique to regimes corresponding to strong bulk
interactions either in the charge-, or in the spin-channel (or in both of them).

14

a)

b)

T(D )
0

T(D )=T(D )
0

T(D )

0

0

R(D )

R(D )

0

0

ln(D 0 /D)

ln(D 0 /D)

FIG. 7: Renormalization group flow of the scattering coefficients for a three-wire junction for g1,⊥ = 0, β constant and equal
to 0.3, and different choices of the initial values of the scattering coefficients:
¯
(a) Renormalization group flow corresponding to T (D0 ) = 0.35, T (D0 ) = 0.45, R(D0 ) = 0.2;
¯
(b) Renormalization group flow corresponding to T (D0 ) = T (D0 ) = 0.4, R(D0 ) = 0.2. As D0 /D grows, in both cases the
scattering coefficients flow towards the N N -fixed point.

a)

b)

R(D )
0

T(D )
0

T(D )
0

T(D )

R(D )

0

0

T(D )
0

ln(D 0 /D)

ln(D 0 /D)

FIG. 8: Renormalization group flow of the scattering coefficients for a three-wire junction for non-constant β:
¯
(a) Renormalization group flow corresponding to T (D0 ) = 0.2, T (D0 ) = 0.1, R(D0 ) = 0.4 and to g1, (D0 )/(2πv) =
g1,⊥ (D0 )/(2πv) = g2, (D0 )/(2πv) = g1,⊥ (D0 )/(2πv) = 0.4. For these values of the bare parameters the junction is attracted
¯
by the disconnected fixed point (R → 1 while T, T → 0);
¯
(b) Renormalization group flow corresponding to T (D0 ) = 0.4, T (D0 ) = 0.35, R(D0 ) = 0.25 and to g1, (D0 )/(2πv) =
0.3, g1,⊥ (D0 )/(2πv) = −0.2, g2, (D0 )/(2πv) = g1,⊥ (D0 )/(2πv) = 0.2. For these values of the bare parameters the junction
is attracted by the χ++ -fixed point.

B.

Fermionic analysis of the strongly interacting regime at gc ∼ gs ∼ 3

A first regime to which our dual-fermion approach can be successfully applied corresponds to a strong attractive
interaction, both in the charge and in the spin channels. In particular, we assume gc ∼ gs ∼ 3. According to the phase
diagram derived in Ref. [23] within the bosonization approach, in this range of values of the Luttinger parameters
one expects to find a fixed point where paired electron tunneling and Andreev reflection are the dominant scattering
processes at the junction and, in addition, two fixed points with maximally broken time-reversal symmetry, to be
identified with the χ++ and the χ−− -fixed points discussed in the previous subsection. To apply the FRG-approach
to this part of the phase diagram, we have to define the appropriate dual fermion coordinates. To do so, let us set
gc = gs = 3. We therefore note that, though, in general, the charge- and spin-velocities uc and us can be different
from each other, one may easily make them equal by a pertinent rescaling of the real-space coordinate in the chargeand in the spin-sector of the bosonic Hamiltonian in Eq. (B10). As the rescaling does not affect the boundary
interaction (which is localized at x = 0), in the following, without loss of generality, we will assume uc = us ≡ u. In
choosing the appropriate dual fermion coordinates, we use the criterion of mapping the fixed point we recover in the
strongly interacting limit one-to-one onto those of the phase diagram in the weakly interacting regime. Referring to
˜
˜
the bosonization formulas of Appendix B, we define the dual bosonic fields Φc(s),j (x), Θc(s),j (x) in terms of those in

15
Eqs. (B1,B2) as
 ˜
Φc(s),1 (x)
˜
 Φc(s),2 (x)

 Φc(s),3 (x)
˜

Θ
˜
 c(s),1 (x)
Θ
˜ c(s),2 (x)
˜
Θc(s),3 (x)





1
√
3
1
√
3
1
√
3

1
√
3
1
√
3
1
√
3

1
√
3
1
√
3
1
√
3


 
 
 
=
  0 −1 1
 
  1 0 −1

−1 1 0

0
1
−3

1
3
1
√
3 3
1
√
3 3
1
√
3 3

1
3

0
1
−3

1
√
3 3
1
√
3 3
1
√
3 3

1
−3




 Φc(s),1 (x)
  Φc(s),2 (x)

0   Φc(s),3 (x)


1
√   Θc(s),1 (x)
3 3 
1
Θc(s),2 (x)
√ 
3 3 
Θc(s),3 (x)
1
√
1
3

3 3









.

(41)

Consistently with Eqs. (B4), we therefore define the dual fermionic fields as

i ˜
˜
˜
˜
˜
ψR,σ,j (x) = ηR,σ,j e 2 [Φj,c (x)+Θj,c (x)+σ(Φj,s (x)+Θj,s (x))]
i ˜
˜
˜
˜
˜
ψL,σ,j (x) = ηL,σ,j e 2 [Φj,c (x)−Θj,c (x)+σ(Φj,s (x)−Θj,s (x))]

.

(42)

˜
˜
Using Eq. (41) one sees that, when expressed in terms of the Φ and of the Θ-fields, the bulk Hamiltonian in Eq.
(B10) reduces back to the one with gc = gs = 1, which, when expressed in terms of the fermionic fields defined in
Eqs. (42), corresponds to the free Hamiltonian H0,F , given by
3

L

dx

H0,F = −iu

j=1

0

σ

˜
˜
˜†
˜†
ψR,j,σ (x)∂x ψR,j,σ (x) − ψL,j,σ (x)∂x ψL,j,σ (x)

.

(43)

Equation (43) is the striking result of our technique of introducing dual fermion operators: it is a free-fermion
Hamiltonian which describes a system that is strongly interacting in the original coordinates. Based upon the dual
fermion fields in Eqs. (42) one may therefore introduce dual boundary operators analogous to those defined in Eq.
(4), namely, one may set
˜†
˜
˜
B(j,j ′ ),σ,(X,X ′ ) (0) = ψX,j,σ (0)ψX ′ ,j ′ ,σ (0) ,

(44)

and assume that the boundary interaction is realized as a linear combination of the operators in Eq. (44) and/or
of products of two of them. In the absence of additional bulk interaction involving the dual fermion fields, or in
the weakly interacting regime, the most relevant boundary interaction term is realized as a linear combination of the
˜
B-operators only. Therefore, the physically relevant processes at the junction are all encoded within the single-particle
˜
˜
S-matrix elements in the basis of the dual fields, S(j,σ);(j ′ ,σ′ ) . A nontrivial flow for the S-matrix elements is induced
by a nonzero bulk interaction in the dual-fermion theory, that is, by having gc(s) = 3 + δgc(s) , with |δgc(s) |/gc(s) ≪ 1.
˜
The dual interaction Hamiltonian, Hint can be readily recovered using Eqs. (41,42). The result is
3

˜
Hint =
j=1 σ,σ′

3

L

dx ρR,j,σ (x)˜L,j,σ′ (x) +
˜
ρ

gj;(σ,σ′ )
0

L

g(j,j ′ );(σ,σ′ )
j=j ′ =1 σ,σ′

dx ρR,j,σ (x)˜L,j ′ ,σ′ (x) ,
˜
ρ

(45)

0

˜
˜†
with ρR(L),j,σ (x) =: ψR(L),j,σ (x)ψR(L),j,σ (x) :, and
˜
2πu(δgc + δgs )
2πu(δgc − δgs )
δσ,σ′ +
δσ,¯ ′
σ
9
9
8πu(δgc + δgs )
8πu(δgc − δgs )
=−
δσ,σ′ −
δσ,¯ ′ ,
σ
9
9

gj;(σ,σ′ ) =
g(j,j ′ );(σ,σ′ )

(46)

˜
plus terms that do not renormalize the scattering amplitudes. Hint takes the form of the generalized bulk Hamiltonian
in Eqs. (A13,A14). Given the corresponding F -matrix elements reported in Eq. (A16), one may derive the RG˜
equations in the case in which the spin is conserved at a scattering process at the junction, which implies S(j,σ);(j ′ ,σ′ ) =
˜
˜
δσ,σ′ Sj,j ′ , and the boundary interaction is symmetric under a cyclic permutation of the three wires, that is, the Sj,j ′ matrix elements are given by


¯
r t t
˜ ˜ ˜
˜  ˜ ˜ ˜
¯
(47)
S= t r t  .
˜ t r
¯ ˜ ˜
t

16
Summing over both the inter-wire and the intra-wire processes allowed by the bulk interaction, one obtains
γ−α
d˜
r
˜˜˜
¯
{˜ − |˜|2 r − 2ttr∗ }
r
r ˜
=
dℓ
2
˜
γ−α
dt
˜ ˜
˜r
¯
{2t|˜|2 + (t)2 r∗ }
=−
dℓ
2
˜
¯
dt
γ−α
˜r
¯
˜ ˜
{2t|˜|2 + (t)2 r∗ } ,
=−
dℓ
2

(48)

with α = g(j,j ′ );(σ,σ) /(2πu), γ = gj;(σ,σ) /(2πu). Equations (48) are equal with Eqs. (31) in the weakly interacting
case, provided one substitutes the (running) parameter β in Eqs. (31) with the (constant) parameter γ − α. As a
˜
result, the RG-flow of the S-matrix elements is the same as the one obtained for the S-matrix elements. Nevertheless,
due to the nonlinear correspondence between the original and the dual fermionic fields, the result for the conductance
at corresponding points of the phase diagram is completely different. To discuss this point, let us write the current
operators at fixed spin polarization, Jj,σ (x), in terms of the dual fermionic fields as
Jj,σ (x) = eu
X=L,R

{ρX,j−1,σ (x) − ρX,j+1,σ (x)} ,
˜
˜

(49)

with j + 3 ≡ j, and use the formalism of Appendix C 2 to derive the conductance tensor. From Eq. (C19) one
eventually obtains
Gc (D) = Gs (D) = G0 −

3e2
Γ(D)
2π

,

(50)

with



˜
˜
˜
¯
¯
˜
(T (D) + T (D))
−T (D)
−T (D)
4 −2 −2
e 


˜
˜
˜
˜
¯
¯
−2 4 −2  , Γ(D) = 
G0 =

−T (D)
(T (D) + T (D))
−T (D)
π
˜
˜
−2 −2 4
¯
¯
˜
˜
−T (D)
(T (D) + T (D))
−T (D)
2



,

(51)

˜
˜
˜
˜
¯
˜
¯
˜
¯
and T (D) = |t(D)|2 , T (D) = |t(D)|2 . The RG-flow of T (D) and of T (D) is determined by Eqs. (33), with β replaced
by γ−α ∝ δgc +δgs . As a result, for δgc +δgs > 0 the stable fixed point is the ”dual” disconnected fixed point, which we
˜ ˜
˜
dub [Nc , Ns ], as, in bosonic coordinates, it corresponds to imposing Neumann boundary conditions on all the Φj,c(s) (x)fields at x = 0. From Eqs. (50,51) one therefore finds that the corresponding charge- and spin-conductance tensors
are given by Gc = Gs = G0 . This is absolutely consistent with the result provided in Ref. [23] for gc = gs = 3. Indeed,
˜ ˜
from Eqs. (41,42) one sees that, resorting back to the original bosonic fields, the [Nc , Ns ]-fixed points corresponds to
the [Dc , Ds ]-fixed point of Ref. [23], with Dirichlet boundary conditions imposed on the relative fields ϕ1,c(s) (x) =
1
1
√ [Φ1,c(s) (x) − Φ2,c(s) (x)] and ϕ2,c(s) (x) = √ [Φ1,c(s) (x) + Φ2,c(s) (x) − 2Φ3,c(s) (x)], where the charge- and the spin2
6
conductance tensors for gc = gs = 3 are equal to each other and both equal to G0 . To double-check the consistency
˜ ˜
between our dual-fermion FRG-formalism and the bosonization approach, we note that, at the [Nc , Ns ]-fixed point, a
generic linear combination of the boundary operators in Eq. (44) can be expressed, in the original bosonic degrees of
i
iσ
freedom, as a linear combination of the operators Oj,σ (0) = ηL,2,σ ηL,1,σ e− 2 [Θj+1,c (0)+Θj,c (0)]− 2 [Θj+1,s (0)+Θj,s (0)] and
of their Hermitean conjugates, which is the result obtained in Ref. [23] by means of a pertinent application of the
delayed evaluation of boundary conditions (DEBC)-technique20,21 .
˜ ˜
When δgc + δgs < 0, the [Nc , Ns ]-fixed point becomes unstable. As in the weakly interacting case, we see that, if
˜
¯
˜(D0 ) = T (D0 ), the junction flows towards either one of the ”dual-chiral” fixed points, χ++ , χ−− , with the crossover
˜
˜
T
˜
˜
¯
of the conductance tensors with the scale being given by Eq. (50). In particular, if T (D0 ) > T (D0 ), the flow is
˜
˜ = T = 0, T = 1. From Eq. (50), one therefore
¯
˜
towards the infrared stable χ++ -fixed point. This corresponds to R
˜
obtains that the fixed point conductances are given by


e2  1 −2 1  e2 +
1 1 −2 ≡ Qχ .
(52)
Gc = Gs =
π −2 1 1
π
By consistency, one would expect that the χ++ fixed point should be identified with the χ++ fixed point emerging
˜
from the weak interaction calculation of the previous section. However, in order to compare the conductances obtained

17
in Eq. (52) with those of Eq. (36) one has to take into account that formula for the conductance tensor derived
within dual-fermion approach applies to a junction connected to reservoirs with gc = gs = 3. Therefore, to make
the comparison, one has to trade Eq. (52) for a formula for the conductance tensors of a junction connected to
reservoirs with gc = gs = 1, Gc;wl , Gs;wl . As discussed in Appendix C 2, this can be done by using Eq. (C16) with
2gc(s) 2
Gin,c(s) = gc(s) −1 e . The result is
π
Q+
χ
I+
3

2

Gc;wl = Gs;wl =

e
Q+
π χ

−1




1 0 −1
e 
−1 1 0 
=
π
0 −1 1
2

,

(53)

that is, the same result as in Eq. (36). Similarly, one can prove that the chiral χ−− -fixed point, towards which the
˜
˜
˜(D0 ) < T (D0 ), has to be identified with the χ−− -fixed point of Sec. III A.
¯
RG-trajectories flow if T
˜
¯
˜
˜
When T (D0 ) = T (D0 ), the RG-trajectories flow towards a nontrivial fixed point, which we dub M , by analogy to
˜
¯
˜ = 1/9, T = T = 4/9. Therefore, from
˜
the M -fixed point we found in Sec. III A. Such a fixed point corresponds to R
Eq. (50), one finds that the fixed point conductance tensors are given by
 4

2
−2 −3
e2  32 43
e2
2
Gc = Gs =
(54)
− 3 3 − 3  ≡ QM .
π
π χ
2
−2 −3 4
3
3
Performing the same transformation as in Eq. (53), one eventually finds
 4

2
−1
−2 −5
M
2
2
5
5
Qχ
e  2 4
e
2
=
QM I +
Gc;wl = Gs;wl =
−5 5 −5 
π χ
3
π
2
2
−5 −5 4
5

.

(55)

On comparing Eq. (55) with Eq. (40) we now see that, at odds with what happens with the chiral fixed points, the
˜
M and the M -fixed points cannot be identified with each other. While we are still lacking a clear explanation for this
different behavior at different fixed points, we suspect that this shows that, while the conductance at χ++ as well as the
χ−− -fixed points are in a sense universal, that is, independent of the Luttinger parameters (provided one pertinently
takes into account the corrections due to different Luttinger parameters for the reservoirs), the conductance at the
M -fixed point does depend explicitly on the Luttinger parameters. This would definitely not be surprising, as such a
feature would be shared by a similar fixed point such as, for instance, the nontrivial fixed point at a junction between
a topological superconductor and two interacting one-dimensional electronic systems71 . In any case, we believe that
this issue calls for a deeper investigation, which will possibly be the subject of a forthcoming work.
As so far we mainly concentrated around the ”diagonal” in Luttinger parameter plane, that is, at gc ∼ gs , we are
now going to complement our analysis by discussing the regime with gc ∼ 3, gs ∼ 1, together with the complementary
one, gc ∼ 1, gs ∼ 3.
C.

Fermionic analysis of the strongly interacting regime for gc ∼ 3, gs ∼ 1 and gc ∼ 1, gs ∼ 3

We now discuss the ”asymmetric” regime gc ∼ 3, gs ∼ 1. In order to recover the whole procedure, we again note
that it is always possible to separately rescale the real-space coordinate in the bosonic Hamiltonian in Eq. (B10), so
to make the charge- and the spin-plasmon velocities to be both equal to u. Therefore, to actually define the dual
fermion coordinates, let us assume gc = 3, gs = 1. Due to the absence of bulk interaction in the spin sector, we have
˜
˜
no need to transform the Φs,j , Θs,j -fields. At variance, we do trade the fields Φc,j , Θc,j for the fields Φc,j , Θc,j defined
in Eq. (41). Accordingly, we consistently define the dual fermionic fields as
i

˜

˜

i

˜

˜

χR,σ,j (x) = ηR,σ,j e 2 [Φj,c (x)+Θj,c (x)+σ(Φj,s (x)+Θj,s (x))]
χL,σ,j (x) = ηL,σ,j e 2 [Φj,c (x)−Θj,c (x)+σ(Φj,s (x)−Θj,s (x))]

.

(56)

Again, one sees that, when expressed in terms of the dual fermion operators in Eqs. (56), the bulk Hamiltonian in
Eq. (B10) reduces back to the free fermionic Hamiltonian, given by
3

H0,F ;χ = −iu

L

dx
j=1

σ

0

†
χ†
R,j,σ (x)∂x χR,j,σ (x) − χL,j,σ (x)∂x χL,j,σ (x)

.

(57)

18
Having defined the dual fermion operators, we now assume that the leading boundary perturbation is realized as a
linear combination of the dual boundary operators defined as
˜
Bχ;(j,j ′ ),σ,(X,X ′ ) (0) = χ†
X,j,σ (0)χX ′ ,j ′ ,σ (0) ,

(58)

and/or of products of two of them. Just as we have done before, we also assume that the most relevant scattering
processes at the junction are fully described by means of the single-particle S-matrix elements in the basis of the
χ-fields, Sχ;(j,σ);(j ′ ,σ′ ) . Slightly displacing (gc , gs ) from (3,1), that is, setting gc = 3 + δgc , gs = 1 + δgs , with
˜
|δgc |/3, |δgs | ≪ 1, gives rise to an effective interaction Hamiltonian Hχ;int , which takes exactly the same form as Hint
in Eq. (45), and is given by
3

Hχ;int =
j=1 σ,σ′

3

L

L

dxρχ;R,j,σ (x)ρχ;L,j,σ′ (x) +

gχ;j;(σ,σ′ )
0

dxρχ;R,j,σ (x)ρχ;L,j ′ ,σ′ (x) , (59)

gχ;(j,j ′ );(σ,σ′ )
0

j=j ′ =1 σ,σ′

with ρχ;R(L),j,σ (x) =: χ†
R(L),j,σ (x)χR(L),j,σ (x) :, and
8πu(δgc − 3δgs )
8πu(δgc + 3δgs )
δσ,σ′ +
δσ,¯ ′
σ
9
9
4πu(δgc + 3δgs )
4πu(δgc − 3δgs )
δσ,σ′ −
δσ,¯ ′ ,
=−
σ
9
9

gχ;j;(σ,σ′ ) =
gχ;(j,j ′ );(σ,σ′ )

(60)

plus terms that do not renormalize the scattering amplitudes. Making the assumption that the spin is conserved at a
scattering process at the junction, we again obtain that Sχ;(j,σ);(j ′ ,σ′ ) = δσ,σ′ Sχ;(j,j ′ ) , with, for a boundary interaction
symmetric under a cyclic permutation of the three wires, the Sχ -matrix being given by


¯
rχ tχ tχ
¯
Sχ =  tχ rχ tχ  .
(61)
¯
tχ tχ rχ

The RG-equations for the running Sχ -matrix elements are derived in perfect analogy with Eq. (48). The result is
exactly the same, except that now γ − α ∝ δgc − 3δgs . In order to trace out the correspondence between the fixed
point of the boundary phase diagram for the dual-fermion scattering amplitudes and those of the phase diagram for
the original fermion amplitudes, we now discuss the behavior of the charge- and of the spin-conductance tensor along
the RG-trajectories. To do so, we note that, due to the fact that the spin sector of the theory is left unchanged, when
resorting to the dual coordinates, the spin-conductance tensor, when expressed in terms of the scattering coefficients at
the junction, takes the same form as in the noninteracting case, given in Eq. (32). As variance, the charge-conductance
tensor depends on the scattering coefficients as given in Eq. (51). As a result, one obtains


¯
−Rχ (D) + 1
−Tχ (D)
−Tχ (D)
3e2
e2 
¯
−Tχ (D)
−Rχ (D) + 1
−Tχ (D)  ,
Gc (D) = G0 −
(62)
Γχ (D) , Gs (D) =
2π
π
¯ (D)
−T
−T (D)
−R (D) + 1
χ

χ

χ

¯
¯
with Rχ (D) = |rχ (D)|2 , Tχ (D) = |tχ (D)|2 , Tχ (D) = |tχ (D)|2 . We are, now, in the position of mapping out the whole
phase diagram of the spinful junction for gc ∼ 3, gs ∼ 1, including the fixed point manifold, and of tracing out the
correspondence between the fixed points given in terms of the dual fermion amplitudes, and the described in terms of
the original fermionic coordinates23 . First of all, we note that, when δgc − 3δgs > 0, the system is attracted towards
¯
the ”dual disconnected” fixed point [Nχ,c , Nχ,s ], characterized by the scattering coefficients Rχ = 1, Tχ = Tχ = 0.
At such a fixed point, one obtains Gc = G0 , Gs = 0, which enables us to identify [Nχ,c , Nχ,s ] with the [Dc , Ns ]-fixed
point in the phase diagram of Ref. [23], that is, with a spin-insulating fixed point where the most relevant process at
the junction is pair-correlated Andreev reflection in each wire. At variance, when δgc − 3δgs < 0, the junction flows
towards one among the dual χ++ , χ−− , or M -fixed points. In particular, from Eqs. (62), one sees that, at the dual
χ++ -fixed point, Gc is given by Eq. (52), while Gs takes the form provided in Eq. (36). After the correction of Eq.
(53), one eventually finds that Gc and Gs are equal to each other, and both equal to the fixed-point conductance
at the χ++ -fixed point in the original coordinates. Thus, we are eventually led to identify the dual χ++ -fixed point
with the analogous one, realized in the original coordinates. A similar argument leads to the identification of the dual
χ−− -fixed point with the analogous one, realized in the original coordinates. As for what concerns the dual M -fixed
point, after correcting the charge-conductance tensor as in Eq. (55), one finds that, at such a fixed point,

19

Gc;wl

 4
 8


2
4
−2 −5
−4 −9
e2  52 45
e2  94 89
4
− 5 5 − 2  , Gs;wl = Gs =
−9 9 −9 
=
5
π
π
2
4
2
4
4
8
−5 −5 5
−9 −9 9

.

(63)

Putting together Eqs. (63,55,40) we again see that M -like fixed points are not mapped onto each other, not even
after the correction of Eq. (55). This is again consistent with our previous hypothesis, namely, that the conductance
at the M -fixed point does depend explicitly on the Luttinger parameters and, therefore, it is different in the various
cases we discussed before.
Before concluding this subsection, we point out that the same analysis we just performed in the case gc ∼ 3, gs ∼ 1
does apply equally well to the complementary situation gc ∼ 1, gs ∼ 3, provided one swaps charge- and spin-operators
(and conductances) with each other.
Putting together all the results we obtained using the FRG-approach, one may infer the global topology of the
phase diagram of a spinful three-wire junction and compare the results with those obtained within the bosonization
approach. This will be the subject of the next subsection.
D.

Global topology of the phase diagram from fermionic renormalization group approach

Following Ref. [23], we discuss the main features of the phase diagram of the three-wire junction within various
regions in the gc − gs plane. Let us start from the ”quasisymmetric” region gc ∼ gs . From the results of Sec. III A
1
1
we see that, setting gc = 1 + δgc , gs = 1 + δgs , as long as δgc + δgs < 0 (corresponding to 2gc + 2gs < 1), the system
1
1
flows towards the disconnected [Nc , Ns ]-fixed point. At variance, for δgc + δgs > 0 (that is, for 2gc + 2gs > 1), as
¯
soon as scattering processes from wire j to wires j ± 1 take place at different rates (i.e., T = T ), the RG-trajectories
flow towards either one of the χ++ or χ−− -fixed points. While this is basically consistent with the region of the
¯
phase diagram derived in23 corresponding to gc ∼ gs ∼ 1, in addition, when T = T , we found that the system flows
towards a symmetric fixed point, which we dubbed M , with peculiar, gc , gs -dependent transport properties. Within
the FRG-approach we were able to map out the full crossover of the charge- and spin-conductance tensor between
any two of the fixed points listed above, with some paradigmatic examples shown in the figures of Sec. III A. Keeping
within the quasisymmetric region, in Sec. III B we show that, setting gc = 3 + δgc , gs = 3 + δgs , for δgc + δgs > 0
(that is, for gc + gs > 6), the stable RG-fixed point corresponds to the [Dc , Ds ]-fixed point of Ref. [23] while, as
soon as δgc + δgs < 0 (that is, for gc + gs < 6), the system flows towards either one of the χ++ or χ−− fixed points
in the non-symmetric case, or towards an ”M -like” fixed point in the symmetric case. As discussed above, while,
at both the χ++ and the χ−− -fixed points the conductance tensors for the junction not connected to the leads are
the same, regardless of the value of the Luttinger parameters, at variance, at the M -fixed point they do depend on
gc and gs and, in this sense, they appear to be ”nonuniversal”. Over all, the results we obtained across the region
gc ∼ gs are consistent with a phase diagram where the [Nc , Ns ] and the [Dc , Ds ]-fixed points are respectively stable
1
1
for 2gc + 2gs < 1 and for gc + gs > 6, while, at intermediate values of the Luttinger parameters, depending on the
bare values of the scattering coefficients at the junction, one out of the (universal) chiral χ++ , χ−− -fixed points or
the (nonuniversal) M -fixed point becomes stable. This results already complements the phase diagram of Ref. [23]
by introducing the M -fixed point, which has necessarily to be there, in order to separate the phases corresponding to
χ++ and to χ−− from each other. To push our analysis outside of the gc ∼ gs -region, we discussed the nonsymmetric
case gc ∼ 3, gs ∼ 1. In this case, we found that the manifold of fixed points consists of the [Dc , Ns ] asymmetric
fixed point, at which the junction is characterized by perfect pair-correlated Andreev reflection in each wire, while
it is perfectly insulating in the spin sector, the chiral χ++ , χ−− -fixed points and, again, an M -like fixed point. Also
in this region our results appear on one hand to be consistent with the phase diagram of Ref. [23], on the other
hand to complement it with singling out the M -like fixed point. The complementary regime gc ∼ 1, gs ∼ 3 can be
straightforwardly recovered from the previous discussion by just swapping charge and spin with each other. Aside
from recovering the global phase diagram of the junction, our technique allows for generalizing to strongly-interacting
regimes the main advantage of using fermionic, rather than bosonic coordinates, that is, the possibility mapping out
the crossover of the conductance between fixed points in the phase diagram.
IV.

DISCUSSION AND CONCLUSIONS

In the paper, we generalize the RG approach to junctions of strongly-interacting QWs. In order to do so, we make
a combined use of both the fermionic and the bosonic approaches to interacting electronic systems in one dimension,
which enables us to build pertinent nonlocal transformation between the original fermion fields and dual-fermion
operators, so that, a theory that is strongly interacting in terms of the former ones, maps onto a weakly interacting

20
one, in terms of the latter ones. On combining the dual-fermion approach with the FRG-technique, we are able to
produce new and interesting results, already for the well-known two wire junction. When applied to a Z3 -symmetric
three-wire junction, our technique first of all allows for recovering fundamental informations concerning the topology
of the global phase diagram, as well as all the fixed points accessible to the junction in various regions of the parameter
space. While, in this respect, our approach looks like a useful means to complement the bosonization approach to
conductance properties of junctions of quantum wires, where it appears extremely useful and, in a sense, rather
unique, is in providing the full crossover of the conductance tensors between any two fixed points connected by an
RG-trajectory. The crossover in the conductance properties can be experimentally mapped out by monitoring the
transport properties of the junction as a function of a running reference scale, such as the temperature, or the effective
system size. While the standard FRG approach just yields crossover curves at weak bulk interaction in the quantum
wires13–15,18 , as stated above, our approach extends such a virtue of the fermionic approach to regions at strong values
of the bulk electronic interaction in the wires.
By resorting to the appropriate dual fermionic degrees of freedom, our approach allows for describing in terms of
effectively one-particle S-matrix elements correlated pair scattering and/or Andreev reflection, in regions of values
of the interaction parameters where they correspond to the most relevant scattering processes at the junction. This
allows for envisaging, within our technique, fixed points such as the [Dc , Ds ], or the [Dc , Ns ] one. In fact, due to basic
assumption of the standard FRG-approach that all the relevant processes at the junction are encoded in the singleparticle S-matrix elements, fixed points such as those listed before are typically not expected to be recovered without
resorting to the appropriate dual fermion coordinates, not even after relaxing the weak bulk interaction constraint66 .
While, for simplicity, here we restrict ourselves to the case of a symmetric junction and of a spin-conserving boundary
interaction at the junction, our approach can be readily generalized to a non-symmetric junction characterized, for
instance, by different Luttinger parameters in different wires69 , and/or by a non-spin-conserving boundary interaction.
Also, a generalization of our approach to a junction involving ordinary63, or topological superconductors71,72 is likely to
allow for describing the full crossover of the conductance in a single junction, as well as of the equilibrium (Josephson)
current in a SNS-junction thus generalizing, in this latter case, the results obtained Refs. [73–75] to an SNS-junction
with an interacting central region.
Finally, it is worth stressing that our technique generically complements the RG-approach, so to extend it to
strongly-interacting problems. Very likely, rather than combining it with the FRG-technique, one could work out
the RG-trajectories by using the alternative (and, to some extent, more accurate, though less tractable analytically)
fRG-approach53–59 . Although we believe this is a potentially interesting research topic to pursue, it goes beyond
the scope of this work, where, for the sake of simplicity and analytical tractability, we rather preferred to use the
FRG-technique.
We would like to thank I. Affleck and Z. Shi for enlightening discussions and for sharing with us their unpublished
results about the FRG-approach to a junction of three spinless quantum wires. We acknowledge insightful discussions
with A. Zazunov and E. Eriksson at various stages of completion of this work. A. N. acknowledges financial support
from European Commission, European Social Fund and Regione Calabria.
Appendix A: Derivation of the fermionic renormalization group equations for the S-matrix

In this Appendix we review the derivation of the FRG-equations for the S-matrix elements describing single-particle
scattering at a junction of quantum wires, as discussed in Refs. [13,14,18]. In our paper we compute dc-transport
properties of junctions of quantum wires. In doing so, we describe each wire by only retaining its low-energy, longwavelength excitations about the Fermi points ±kF . Finally, we alledge for the system to present an ”inner” boundary
at x = 0, where the junction is located and the boundary conditions on the fields are determined by the boundary
interaction describing the junction, and an ”outer” boundary at x = L which is just required for introducing a cutoff
length scale L, eventually sent to ∞ at the end of the calculations. As a result, the ”bulk” Hamiltonian for the wires
is realized as HBulk = H0 + HV , with H0 being the free Hamiltonian for noninteracting chiral fermions in Eq. (1),
while the two-body interaction Hamiltonian HV is given by
K

HV =
j=1

1
2

L
σσ′

0

dx dy ρj,σ (x)Vj;σ,σ′ (x − y)ρj,σ′ (y) .

(A1)

Note that, as typically done in this class of problems13–15,18 , in Eq. (A1), we are assuming a purely ”intra-wire”
interaction. In fact, in the analysis of our paper, we had to deal with an emerging bulk interaction Hamiltonian with
nontrivial inter-wire interaction terms. Yet, for the sake of simplicity, we prefer to add inter-wire interactions to the
simplified model Hamiltonian, where a local approximation for the interaction potential is done. Indeed, on assuming

21
a short-range interaction potential effective over a typical length scale λ (as it happens, for instance, in gated wires),
one may approximate Eq. (A1) by setting x ∼ y in the Coulomb potential, so that HV is approximated by Hint in Eq.
(2). In order to derive Eq. (2), we have neglected terms obtained integrating operators proportional to the rapidly
oscillating functions e±2ikF x . Based on an analogous argument, we assume that the umklapp term (∝ e±4ikF x ) does
not appear either (in fact, an additional argument, related to the irrelevance of the corresponding operator, can be
recovered within the bosonization approach to the problem76,77 ). Finally, it is worth mentioning that we also ignore
terms of the form
K

1
2

Hint,g4 =

L

dx [ρL,j,σ (x)ρL,j,σ′ (x) + ρR,j,σ (x)ρR,j,σ′ (x)]

Vj,σσ′ (0)

,

(A2)

0

j=1 σσ′

since they just renormalize the Fermi velocity and the chemical potential76 and, so, do not effectively contribute to
the renormalization of the S-matrix at the junction. Two relevant limiting cases can be recovered from Eq. (2). The
former one corresponds to spinless fermions and is recovered by dropping terms containing fields with a given spin
polarization, say σ =↓. In this case, on setting ψj,R(L),↑ → ψj,R(L) , it easy to verify that Eq. (2) reduces to
K

Hint →

L
0

j=1

†
†
†
†
dx gj,1 ψR,j (x)ψL,j (x)ψR,j (x)ψL,j (x) + gj,2 ψR,j (x)ψL,j (x)ψL,j (x)ψR,j (x)

K

=
j=1

L

(gj,2 − gj,1 )

0

†
†
dx ψR,j (x)ψL,j (x)ψL,j (x)ψR,j (x)

,

(A3)

in agreement with Eq. (6) of Ref. [18]. The latter one corresponds to assuming spinful fermions, but a spin independent
interaction, setting gj,1(2) = gj,1(2),⊥ = gj,1(2) . In this case, Eq. (2) reduces to
L

Hint →

σσ′

0

†
†
†
†
dx gj,1 ψR,j,σ (x)ψL,j,σ′ (x)ψR,j,σ′ (x)ψL,j,σ (x) + gj,2 ψR,j,σ (x)ψL,j,σ′ (x)ψL,j,σ′ (x)ψR,j,σ (x)

, (A4)

in agreement with Eq. (50) of Ref. [18]. As already stated in the main text, FRG approach typically applies to the
case in which the boundary scattering processes are fully described by the single-particle S-matrix elements in Eq.
(7). The first FRG step consists in trading the quartic bulk interaction Hamiltonian for a quadratic one, by means
of a pertinent Hartree-Fock (HF) decomposition of the interaction13,14,18 . After performing the HF decomposition
and making the standard assumption that, as we are dealing with low-energy excitations around the Fermi level, the
S-matrix elements can be taken to be all independent of energy, Hint reduces to
K

Hint → −

j=1

L

i gj,2, − gj,1, − gj,1,⊥
4π

0

dx
†
†
∗
Sj,j ψL,j,↑ (x)ψR,j,↑ (x) + ψL,j,↓ (x)ψR,j,↓ (x)
x

†
†
− Sj,j ψR,j,↑ (x)ψL,j,↑ (x) + ψR,j,↓ (x)ψL,j,↓ (x)

,

(A5)

for spinful electrons, and to
K

Hint → −

j=1

L

i gj,2, − gj,1,
4π

0

dx
†
†
∗
Sj,j ψL,j (x)ψR,j (x) − Sj,j ψR,j (x)ψL,j (x)
x

,

(A6)

for spinless electrons. The corrections to the amplitude for an incoming/outgoing electron in wire j to become an
outgoing/incoming electron in wire j ′ under the effect of Hint are readily computed using the approach of13,14,18 . The
corresponding corrections to the Sj,j ′ matrix elements are given by
K

dSj,j ′ = −

Sj,i F †
i,i′ =1

i,i′

Si′ ,j ′ − Fj,j ′

dℓ ,

(A7)

with dℓ = d ln(L/λ), λ being a length scale corresponding to the (finite) range of the interaction potential Vj,σσ′ (x)
and the Friedel matrix F defined as the block-diagonal matrix

22
Fj,j ′ =

1
gj,2, − gj,1, − gj,1,⊥ Sj,j δj,j ′
4πv

,

(A8)

in the spinful case, while
Fj,j ′ =

1
gj,2, − gj,1,
4πv

Sj,j δj,j ′ ,

(A9)

in the spinless case. The term ln(L/λ) is due the presence of the x−1 in the integral giving the correction to the
amplitudes within the HF-approximation. Throughout the poor man’s scaling method, one can replace dℓ = d ln(L/λ)
with d ln(D0 /D), with D0 being a high-energy cutoff (∼ bandwidth of bulk electrons) and D being the low-energy
running scale. The RG-equations for the S-matrix must be supplemented with the RG-equations for the interaction
constants g1, , g1,⊥ , g2, , g2,⊥ . The procedure is discussed in detail, for instance, in Ref. [78]. Here, we just provide
the final result for the scaling equations of the bulk interaction strengths, which is78
dgj,2,
dℓ
dgj,2,⊥
dℓ
dgj,1,
dℓ
dgj,1,⊥
dℓ

1
(gj,1, )2
2πv
1
=−
(gj,1,⊥ )2
2πv
1
(gj,1, )2 + (gj,1,⊥ )2
=−
2πv
1
=−
2gj,1,⊥ gj,2,⊥ − gj,2, + gj,1,
2πv
=−

,

(A10)

valid in the weak coupling regime. For a spin-independent interaction, Eqs. (A10) reduce to
dgj,2,
1
=−
(gj,1, )2
dℓ
2πv
dgj,1,
1
=−
(2S + 1)(gj,1, )2 ,
dℓ
2πv

(A11)
(A12)

with S = 0 for spinless electrons and S = 1/2 otherwise7,13,14 . Clearly, the renormalization group equations for the
S-matrix must be supplemented with Eqs. (A10) in the spinful case in which, differently from what happens in the
spinless case, there is a nontrivial flow of the bulk interaction parameters with the scale L/λ.
In the analysis of our paper, we also have to consider a generalized bulk interaction which, for gj,1,⊥ = 0, has a
nonzero inter-wire component, that is, the generalized bulk interaction Hamiltonian reads
HInt = HIntra + HInter

,

(A13)

with
K

L

HIntra =

dx ρR,j,σ (x)ρL,j,σ′ (x)

gj;(σ,σ′ )
0

j=1 σ,σ′
K

L

HInter =

dx ρR,j,σ (x)ρL,j ′ ,σ′ (x)

g(j,j ′ );(σ,σ′ )
j=j ′ =1 σ,σ′

,

(A14)

0

which reduces to HInt in Eq. (A5), provided one identifies the intra-wire interaction strengths as
gj;(↑,↑) = gj;(↓,↓) ≡ gj,2, − gj,1,

gj;(↑,↓) = gj;(↓,↑) ≡ gj,2,⊥ ,

(A15)

and sets g(j,j ′ );(σ,σ′ ) = 0 for j = j ′ . On assuming again that the S-matrix takes the spin-diagonal form in Eq. (7), one
finds that the RG-equations for the S-matrix elements are again those provided in Eq. (A7), but with the F -matrix
elements now given by
Fj,j ′ = {δj,j ′

g(j,j ′ );(σ,σ)
gj;(σ,σ)
+ [1 − δj,j ′ ]
}Sj,j ′ .
4πv
4πv

(A16)

As we discuss in the main text, the modification in Eq. (A16) does not substantially affect the solutions of the RG
equations.

23
Appendix B: Review of bosonization rules for interacting one-dimensional quantum wires

In this Appendix, we review the basic bosonization formulas for one-dimensional interacting electronic systems,
which provide us with the fundamental formal ground, on which we relate when defining the effective fermionic
coordinates in the strongly interacting regimes. The first step towards a fully bosonic description of the junction is
to rewrite in bosonic coordinates the free Hamiltonian H0 in Eq. (1). This is done by introducing K bosonic fields
Φc,j , described by the (charge-sector) Hamiltonian
H0;B;c =

1
4π

K

L

dx
0

j=1

1
(∂t Φc,j )2 + v(∂x Φc,j )2
v

,

(B1)

,

(B2)

and K bosonic fields Φs,j , described by the (spin-sector) Hamiltonian
H0;B;s =

1
4π

K

L

dx
0

j=1

1
(∂t Φs,j )2 + v(∂x Φs,j )2
v

together with the corresponding dual fields Θc,j , Θs,j , which are related to the Φ-fields by means of the cross derivative
relations
∂t Φj,c(s) = v∂x Θj,c(s)
∂t Θj,c(s) = v∂x Φj,c(s) .

(B3)

Therefore, one rewrites the chiral fermionic fields in terms of the bosonic coordinates introduced above as
i

ψR,σ,j (x) = ηR,σ,j e 2 [Φj,c (x)+Θj,c (x)+σ(Φj,s (x)+Θj,s (x))]
i

ψL,σ,j (x) = ηL,σ,j e 2 [Φj,c (x)−Θj,c (x)+σ(Φj,s (x)−Θj,s (x))]

,

(B4)

with ηR,σ,j and ηL,σ,j being real-fermion Klein factors. Note that the vertex operators in Eq. (B4) have been set
consistently with the fact that, as a consequence of Eqs. (B3), one finds that the following linear combinations are
chiral fields
1
φR,c(s),j (x, t) = √ [Φj,c(s) (x, t) + Θj,c(s) (x, t)]
2
1
φL,c(s),j (x, t) = √ [−Φj,c(s) (x, t) + Θj,c(s) (x, t)]
2

.

(B5)

In bosonic coordinates, the chiral fermionic density operators at fixed spin polarization are realized as
1
{−∂xΘj,c (x) + ∂x Φj,c (x) + σ[−∂x Θj,s (x) + ∂x Φj,s (x)]}
8π
1
†
: ψL,σ,j (x)ψL,σ,j (x) : = − {∂x Θj,c (x) + ∂x Φj,c (x) + σ[∂x Θj,s (x) + ∂x Φj,s (x)]} ,
8π

†
: ψR,σ,j (x)ψR,σ,j (x) : =

(B6)

with the double columns : : denoting normal-ordering with respect to the fermionic groundstate. From Eqs. (B6) one
therefore recovers the charge- and the spin-density operators given by
†
†
{: ψR,σ,j (x)ψR,σ,j (x) + ψL,σ,j (x)ψL,σ,j (x) :} → −

ρc,j (x) =
σ

ρs,j (x) =
σ

†
†
σ{: ψR,σ,j (x)ψR,σ,j (x) + ψL,σ,j (x)ψL,σ,j (x) :} → −

1
∂x Θj,c (x)
2π
1
∂x Θj,s (x) ,
2π

(B7)

together with the corresponding current operators realized as
†
†
{: ψR,σ,j (x)ψR,σ,j (x) − ψL,σ,j (x)ψL,σ,j (x) :} →

1
∂x Φj,c (x)
2π

†
†
σ{: ψR,σ,j (x)ψR,σ,j (x) − ψL,σ,j (x)ψL,σ,j (x) :} →

1
∂x Φj,s (x)
2π

Jc,j (x) =
σ

Js,j (x) =
σ

.

(B8)

24
All the previous transformations lead to contributions to the system Hamiltonian that are quadratic in the bosonic
fields. The interaction Hamiltonian shares this property, once one has set to zero the all the terms ∝ gj,1,⊥ , in which
case, on rewriting Hint in Eq. (2) in terms of the bosonic fields, one obtains Hint = Hint,1 + Hint,2 , with
K

Hint,1 =
j=1
K

Hint,2 =
j=1

L

gj,1, − gj,2,
16π 2
gj,2,⊥
16π 2

L
0

0

dx {−(∂x Θj,c (x))2 − (∂x Θj,s (x))2 + (∂x Φj,c (x))2 + (∂x Φj,s (x))2 }

dx {(∂x Θj,c (x))2 + (∂x Φj,s (x))2 − (∂x Θj,s (x))2 − (∂x Φj,c (x))2 } .

(B9)

On adding the terms in Eqs. (B9) to the noninteracting Hamiltonian for the charge sector (Eq. (B1)) plus the one
for the spin sector (Eq. (B2)), one eventually obtains a bosonic Hamiltonian H that is still quadratic, although with
pertinently renormalized coefficients, given by
H=

+

1
4π
1
4π

K

L

dx

gj,c (∂x Φj,c (x))2 +

dx

uj,c

gj,s (∂x Φj,s (x))2 +

0

j=1
K

L

uj,s
j=1

1
gj,c

0

1
gj,s

(∂x Θj,c (x))2
(∂x Θj,s (x))2

,

(B10)

with
gj,1, − gj,2,
4πv
gj,1, − gj,2,
=v 1−
4πv
gj,1, − gj,2,
=v 1+
4πv
gj,1, − gj,2,
=v 1−
4πv

uj,c gj,c = v 1 +
uj,c
gj,c
uj,s gj,s
uj,s
gj,s

− gj,2,⊥
− gj,2,⊥
+ gj,2,⊥
+ gj,2,⊥

.

(B11)

(Note the use of Eqs. (B3) to express H in terms of both the Φ- and the Θ-fields.) When gj,1,⊥ = 0, one can employ
the identities
†
ψR,σ,j (x)ψR,¯ ,j (x) → e−iσ[Φj,s (x)+Θj,s (x)]
σ

†
ψL,σ,j (x)ψL,¯ ,j (x) → e−iσ[Φj,s (x)−Θj,s (x)] ,
σ

(B12)

to express the total additional contribution to Hint , Hint,3 , as
K

L

Hint,3 ∼

Wj cos[2Φj,s (x)] ,

dx
0

(B13)

j=1

with Wj ∝ gj,1,⊥ . To lowest order, the renormalization group equation for Wj is
dWj
2
Wj ,
= 2−
dℓ
gj,s

(B14)

which tells us that the nonlinear interaction term ∝ gj,1,⊥ is irrelevant as long as gj,s < 1 and, accordingly, even
if gj,1,⊥ has not been fine-tuned to 0, it can be dropped out of the effective low-energy, long-wavelength theory. In
general, when using the bosonization, we assume that either this is the case, or that all the gj,1,⊥ have been finetuned to 0. As for what concerns the boundary Hamiltonian HB describing the junction between the quantum wires,
throughout all the paper we have assumed that total spin was conserved at each scattering event at the junction. This
basically implies that HB can be fully as in Eq. (5) of the main text. While for a weak bulk interaction within the
wires the most relevant contribution to HB is typically realized as a linear combinations of (some of) the operators
in Eq. (4), at strong enough attraction either in the charge-, or in the spin-channel (or both), as we discuss in the
paper, the most relevant boundary interaction term can be realized as a product of two B-operators. In any case,

25
all the boundary operators can be rewritten in bosonic coordinates by pertinently employing the bosonization rules
in Eqs. (B6). As the junction naturally defines a (common) boundary for all the quantum wires, Eqs. (B6) must
typically be supplemented with the appropriate boundary conditions for the Φj - and for the Θj -fields. This ”delayed
evaluation of the boundary conditions” (DEBC)-procedure can be typically implemented in the correspondence of a
conformal fixed point20,21 . It has the net effect of making different B-operators in Eq. (4) to ”collapse” onto each
other, thus substantially reducing the number of independent boundary operators that can potentially enter HB and,
thus, strongly constraining the number of relevant scattering processes that can take place in the proximity of a given
fixed point. For instance, the fields Φj,c(s) (x) corresponding to a wire that is disconnected from the junction must
obey Neumann boundary conditions such as ∂x Φj,c(s) (x = 0) = 0 and, correspondingly, the fields Θj,c(s) (x) must
be pinned at x = 0 (Dirichlet boundary conditions). Such boundary conditions only allow for nontrivial boundary
i
operators in the form B(j,j ′ ),σ,(R,L) (0) ∼ exp − 2 (Φc,j (0) − Φc,j ′ (0)) − iσ (Φs,j (0) − Φs,j ′ (0)) (plus the corresponding
2
Hermitean conjugates). Various alternative possible situations are discussed in the paper.
Appendix C: Linear response theory approach to the conductance tensor of a junction of quantum wires

In this section, we review the linear response theory approach to the derivation of the conductance tensor for a
junction of quantum wires. After developing the main formula for computing the conductance tensor within linear
response theory, we will apply it to a number specific cases we analyze in the main text of our paper.
Let Wj be the j-th wire connected to the junction (j = 1, . . . , K). In order to implement linear response theory,
we imagine to connect Wj to a reservoir Rj , which can either be characterized by the same parameters as the wire
to which it is connected, or not (a typical situation corresponds to Wj being an interacting one-dimensional quantum
wire, described as a single Luttinger liquid, connected to a Fermi liquid reservoir79). For the sake of simplicity, we
assume that the parameters characterizing both the wires and the reservoirs are all independent of j. In order to
induce a current flow across the junction, we assume that each reservoir is characterized by an equilibrium distribution
for particles with spin σ, with chemical potential µj,σ = eVj,σ . This induces electric fields {Ej,σ (t)} distributed in
the various branches of the junction, which we account for by introducing a set of uniform vector potentials Aj,σ (t),
one for each wire, such that Ej,σ (t) = −∂t Aj,σ (t). Letting Jj,σ (x) be the current operator for particles with spin σ
in wire-j and letting each wire to be of length L, we may define the ”source” Hamiltonian HSource (t), describing the
coupling to the applied electric fields, given by
K

L

HSource (t) =
δ

σ

j=1

dx Aj,σ (t)Jj,σ (x) .

(C1)

Let us, now, denote with J(j,σ);I (x, t), HSource;I (t) the operators taken in the interaction representation with respect
to the Hamiltonian without the source term. Within linear response theory, the current of particles with spin σ
evaluated at point x of wire-j is therefore given by
K

∞

Ij,σ (x, t) = i
j ′ =1

L

dt′

−∞

σ′

δ

dx′ D(j,σ);(j ′ ,σ′ ) (x, t; x′ , t′ )Aj ′ ,σ′ (t′ ) ,

(C2)

with
D(j,σ);(j ′ ,σ′ ) (x, t; x′ , t′ ) = θ(t − t′ ) [J(j,σ);I (x, t), J(j ′ ,σ′ );I (x′ , t′ )]

.

(C3)

Equation (C2) does generically apply to any situation, whether the wires are interacting, or not, and whether one
uses a fermionic, or a bosonic representation for the Hamiltonian of the junction. An important remark, however, is
that, in any case, the current must be consistently probed outside of the region across which the electric fields are
applied, that is, in Eq. (C2) one has always to assume that x > L. Another important formula is the Fourier-space
counterpart of Eq. (C2), that is
Ij,σ (x, ω) = −
with

1
ω

K
j ′ =1

L
σ′

δ

dx′ D(j,σ);(j ′ ,σ′ ) (x, x′ ; ω)Ej ′ ,σ′ (ω)

,

(C4)

26
Ij,σ (x, ω) =

dt eiωt Ij,σ (x, t)

Ej ′ ,σ′ (ω) =

dt eiωt Ej ′ ,σ′ (t)

D(j,σ);(j ′ ,σ′ ) (x, x′ ; ω) =

dt eiωt D(j,σ);(j ′ ,σ′ ) (x, t; x′ , 0) .

(C5)

In the following part of this Appendix, we consider some specific applications to various cases we analyze in the
main text. An important observation to make at this point is that, on connecting the wires to the reservoirs, we
basically assume a continuity condition for the current operator at the interface. From the microscopical point of
view, this appears to be the ”macroscopic” counterpart of the ”smooth” crossover in the interaction strength in real
space discussed in the microscopic lattice model considered in Ref. [53]. It would be also interesting to work out
a macroscopic field-theoretical Hamiltonian describing a ”sharp” interface in the microscopic model, but this goes
beyond of the scope of this work. As specified above, in the following we assume current continuity at the interface,
that is, a smooth crossover in the bulk interaction strength.
1.

Junction of noninteracting quantum wires

Due to the absence of multi-particle processes, the conductance tensor for a junction of noninteracting quantum wires
can be fully expressed in terms of only single-fermion S-matrix elements. The bulk Hamiltonian for a noninteracting
junction of K quantum wires is given in Eq. (1). Within each wire the chiral fields can be expressed in terms of their
Fourier modes as
1
ψR,j,σ (x) = √
L
1
ψL,j.σ (x) = √
L

eikx aR,j,σ (k)
k

e−ikx aL,j,σ (k)

,

(C6)

k

with the right-handed and the left-handed chiral modes related to each other by the S-matrix elements as
K

aR,j,σ (k) =

S(j,σ);(j ′ ,σ′ ) (k)aL,j ′ ,σ′ (k)
j ′ =1

.

(C7)

σ′

The current operator in wire-j for particles with spin polarization σ is given by
†
†
Jj,σ (x) = ev{: ψR,j,σ (x)ψR,j,σ (x) : − : ψL,j,σ (x)ψL,j,σ (x) :} ,

(C8)

which, taking into account that the S-matrix is diagonal in the spin indices, implies
D(j,σ);(j ′ ,σ′ ) (x, x′ ; ω) = e2 v 2 δσ,σ′ {D(R,R);j,j ′ (x, x′ ; ω) + D(L,L);j,j ′ (x, x′ ; ω) − D(R,L);j,j ′ (x, x′ ; ω) − D(L,R);j,j ′ (x, x′ ; ω)} .
(C9)
Assuming a thermal distribution for fermions in the reservoirs, one therefore obtains


′
′
i E+E (x−x′ )
−i E+E (x−x′ )
v
v
iδj,j ′
e
e

D(R,R);j,j ′ (x, x′ ; ω) =
+
dE dE ′ f (E)f (E ′ )  ′
4π 2 v 2
E + E + ω + iη E ′ + E − ω − iη


′
′
−i E+E (x−x′ )
i E+E (x−x′ )
v
v
iδj,j ′
e
e

D(L,L);j,j ′ (x, x′ ; ω) =
+
dE dE ′ f (E)f (E ′ )  ′
4π 2 v 2
E + E + ω + iη E ′ + E − ω − iη


′
′
i E+E (x+x′ )
−i E+E (x+x′ )
v
v
e
e
i|Sj,j ′ |2

+
dE dE ′ f (E)f (E ′ )  ′
D(R,L);j,j ′ (x, x′ ; ω) =
4π 2 v 2
E + E + ω + iη E ′ + E − ω − iη


′
′
−i E+E (x+x′ )
i E+E (x+x′ )
v
v
i|Sj ′ ,j |2
e
e
 ,
D(L,R);j,j ′ (x, x′ ; ω) =
(C10)
+
dE dE ′ f (E)f (E ′ )  ′
4π 2 v 2
E + E + ω + iη E ′ + E − ω − iη

27
with Sj,j ′ denoting the S-matrix elements computed at the Fermi level, f (E) being the Fermi distribution function,
and η = 0+ . Since, as stated above, we assume that the current is measured outside of the region over which the
electric fields are applied, in the following we assume x, x′ > 0, 0 < x′ < L and x > L. As a result, integrating in
Eqs. (C10) over dE ′ and closing the integration path over the appropriate half-plane according to what is the phase
factor in the argument of the integrals, one obtains
δj,j ′
2πv 2
′
D(L,L);j,j ′ (x, x ; ω) = 0

ω

D(R,R);j,j ′ (x, x′ ; ω) =

′

dE ei v (x−x ) f (E){f (−E − ω) − f (−E + ω)}

|Sj,j ′ |2
2πv 2
′
D(L,R);j,j ′ (x, x ; ω) = 0 .

ω

′

dE ei v (x+x ) f (E){f (−E − ω) − f (−E + ω)}

D(R,L);j,j ′ (x, x′ ; ω) =

(C11)

Going to the zero-T limit and focusing onto the dc current (ω → 0), one recovers that the current in wire-j for
particles with spin polarization σ, Ij,σ , is given by limit
Ij,σ =

e2
2π

K

{−|Sj,j ′ |2 + δj,j ′ }Vj ′ ,σ

.

(C12)

j ′ =1

The dc-conductance matrix elements G(j,σ);(j ′ ,σ′ ) are therefore given by
G(j,σ);(j ′ ,σ′ ) =

e2
∂Ij,σ
= δσ,σ′
{−|Sj,j ′ |2 + δj,j ′ } ,
∂Vj ′ ,σ
2π

(C13)

The charge- and the spin-conductance tensors Gc , Gs , are respectively defined as
Gc;(j,j ′ ) =

σσ ′ G(j,σ);(j ′ ,σ′ )

G(j,σ);(j ′ ,σ′ ) , Gs;(j,j ′ ) =
σ,σ′

,

(C14)

σ,σ′

which, from Eq. (C13), implies
Gc;(j,j ′ ) = Gs;(j,j ′ ) =

e2
{−|Sj,j ′ |2 + δj,j ′ } .
π

(C15)

When a weak bulk interaction is added to the junction Hamiltonian, using the FRG-approach as we do in Sec. III, we
still use Eq. (C15) for the charge- and the spin-conductance tensors by just replacing the ”bare” S-matrix elements
with the running ones, Sj,j ′ (D). For instance, this led us to write Eq. (32) of the main text. The result one obtains
does actually correspond to the conductance tensor of the ”connected” junction, that is, of the junction of weakly
interacting wires connected to bulk Fermi liquid reservoirs. As pointed out in20,21,79 , the tensor Gc(s);(j,j ′ ) defined in
¯
Eq. (C13) are related to their analogs for the junction disconnected from the leads, Gc(s);(j,j ′ ) , by means of pertinent
generalizations of Eq. (2.7) of Refs. [20,21], that is,
¯
[G−1 ]c(s);(j,j ′ ) = [G−1 ]c(s);(j,j ′ ) + G−1 δj,j ′ ,
in,c(s)
2g

(C16)

2

c(s)
with the interface charge (spin) conductance Gin,c(s) = gc(s) −1 e . From Eq. (C16) one sees that the matrix elements
π
Gc(s);(j,j ′ ) explicitly depend on the bulk interaction parameters via the Luttinger parameters gc(s) (see Fig. 9 for a
sketch of the junction connected to reservoirs with generic values of the Luttinger parameter gL ). Equation (C16)
suggests how to recover the conductance tensor for the junction disconnected from the leads within the FRG-approach:
¯
it is enough to just compute the Gc(s);(j,j ′ ) s and, therefore, to invert Eq. (C16) to derive the Gc(s);(j,j ′ ) s.
We now discuss how Eq. (C16) is modified in the dual models we used to apply the FRG-approach to junctions at
a strong bulk interaction in the wire.

2.

Junction of strongly interacting quantum wires at gc = gs = 3

As Eq. (C4) applies independently of the specific form of the current operators, in order to generalize Eq. (C16) to
the strongly interacting case we discuss in Sec. III B, one has just to compute the D(j,σ);(j ′ ,σ′ ) (x, t; x′ , t′ )-terms

28

g
111
000
gL
11
00
111
000 L
11
00
111
000
11
00
111
000
11
00
111
11
00
111
000
1111111111
0000000000
g 000 L
11
111
000
1111111111
0000000000 u
uL 00
1111111111
0000000000
11
00
111
000
1111111111
0000000000
1111111111
0000000000
11
00
111
000
0000000000
1111111111
1111111111
0000000000
g
11
00
111
000
1111111111
0000000000
1111111111
0000000000
1111111111
0000000000
1111111111
0000000000
1111111111
0000000000
u
1111111111
0000000000
1111111111
0000000000
1111111111
0000000000
1111111111
u
1 0000000000
1111111111
0000000000
1111111111
0000000000
1111111111
0000000000
1111111111
0000000000
1111111111
0000000000
11
00
1111111111
0000000000
1111111111
0000000000
11
00
11111111111
00000000000
111111
000000
1111111111
0000000000
1111111111
0000000000
11
00
11111111111
00000000000
111111
000000
11111111111
00000000000
000000
111111
00000000000
11111111111
111111
000000
11111111111
00000000000
111111
000000
11111111111
00000000000
111111
000000
11111111111
00000000000
111111
000000
11111111111
00000000000
000000
111111
00000000000
11111111111
111111
000000
11111111111
00000000000
111111
000000
11
00
11111111111
00000000000
111111
000000
11
00
11111111111
00000000000
111111
000000
11
00
11111111111
00000000000
000000
111111
11
00
111111
000000
11
L 00
111111
000000
11
00
111111
000000
11
00
111111
000000
11
00
11
00
L

K

2

g

g

u

u

FIG. 9: Sketch of a connected junction of K wires with parameters g, u connected to reservoirs with parameters gL , uL .

in Eq. (C3) using the current operators in the dual theory given in Eq. (49). As a result, one now obtains
D(j,σ);(j ′ ,σ′ ) (x, x′ ; ω) = δσ,σ′ Dj,j ′ (x, x′ ; ω), with
Dj,j ′ (x, x′ ; ω) =
×

e2
2π

dE f (E)[f (−E − ω) − f (−E + ω)]

′
′
ω
ω
˜
˜
2ei u (x−x ) δj,j ′ + ei u (x+x ) [|Sj−1,j ′ −1 |2 + |Sj+1,j ′ +1 |2 ]

ω

ω

˜
˜
− ei u (x−x ) [δj−1,j ′ +1 + δj+1,j ′ −1 ] − ei u (x+x ) [|Sj−1,j ′ +1 |2 + |Sj+1,j ′ −1 |2 ]
′

′

.

(C17)

Following the same steps as in Appendix C 1, from Eq. (C17) one therefore readily derives the dc conductance tensor,
which is given by
G(j,σ);(j ′ ,σ′ ) = δσ,σ′ Gj,j ′

,

(C18)

with
Gj,j ′ =

e2
˜
˜
2δj,j ′ + |Sj+1,j ′ +1 |2 + |Sj−1,j ′ −1 |2
2π

˜
˜
− [δj+1,j ′ −1 + δj−1,j ′ +1 + |Sj−1,j ′ +1 |2 + |Sj+1,j ′ −1 |2 ]

.

(C19)

From Eq. (C19) one therefore computes the conductance tensor at gc = gs = 3. When either gc , or gs (or both)
are different from 3, using Eq. (C19) one recovers the conductance tensor in the case in which gc and gs both
asymptotically tend to 3 in the leads. The charge- and the spin-conductance tensors for the disconnected junction
2gc(s) 2
are obtained using Eq. (C16) with Gin,c(s) = gc(s) −1 e .
π

1
2
3
4

5

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