Conducting ﬁxed points for inhomogeneous quantum wires:
a conformally invariant boundary theory
N. Sedlmayr,1, ∗ D. Morath,1 J. Sirker,1 S. Eggert,1 and I. Aﬄeck2
1
arXiv:1311.0007v2 [cond-mat.str-el] 25 Jan 2014
Department of Physics and Research Center OPTIMAS,
University of Kaiserslautern, D-67663 Kaiserslautern, Germany
2
Department of Physics and Astronomy, The University of British Columbia, Vancouver, BC V6T 1Z1, Canada
Inhomogeneities and junctions in wires are natural sources of scattering, and hence resistance.
A conducting ﬁxed point usually requires an adiabatically smooth system. One notable exception
is “healing”, which has been predicted in systems with special symmetries, where the system is
driven to the homogeneous ﬁxed point. Here we present theoretical results for a diﬀerent type
of conducting ﬁxed point which occurs in inhomogeneous wires with an abrupt jump in hopping
and interaction strength. We show that it is always possible to tune the system to an unstable
conducting ﬁxed point which does not correspond to translational invariance. We analyze the
temperature scaling of correlation functions at and near this ﬁxed point and show that two distinct
boundary exponents appear, which correspond to diﬀerent eﬀective Luttinger liquid parameters.
Even though the system consists of two separate interacting parts, the ﬁxed point is described by
a single conformally invariant boundary theory. We present details of the general eﬀective bosonic
ﬁeld theory including the mode expansion and the ﬁnite size spectrum. The results are conﬁrmed
by numerical quantum Monte Carlo simulations on spinless fermions. We predict characteristic
experimental signatures of the local density of states near junctions.
PACS numbers: 73.63.Nm, 71.10.Pm, 73.40.-c
I.
INTRODUCTION
Transport in quantum wires is a rich ﬁeld bringing together conductivity experiments1–5 and Luttinger liquid
theory which describes the crucial electron-electron interaction eﬀects in one dimension.6–8 Scattering from a
single impurity or other inhomogeneities, for example,
becomes renormalized by the interaction and can lead to
insulating behavior at low temperatures even for weak
impurities.9–14
In order to determine the conductivity of a onedimensional wire it is necessary to couple it to some leads
or reservoirs, normally a two dimensional electron gas
(2DEG). Such a set up can be most readily described as
an inhomogeneous wire, in which the 2DEGs are modeled as non-interacting wires. In this case the conductance is usually controlled by the parameters of the lead
rather than of the wire,15–28 in contrast to what a naive
calculation on an inﬁnite interacting wire would suggest.
The conductance for perfect adiabatic contacts and wires
can be understood by the decomposition of an electron
into fractional charges.16,29 Additional relaxation processes which take place within the interacting region of
the wire do, however, lead to a resistance which is affected by the wire parameters. The resistance due to
impurity scattering30 or phonon scattering28 within the
interacting wire, for example, will in general depend both
on the Luttinger liquid parameter of the leads and the
wire.
In this paper we consider the intrinsic scattering from
the junctions between the wire and leads, which is generically present due to the abrupt change of parameters
even for otherwise perfect ballistic connections. This
scattering is renormalized by the interaction,30 leading
to a vanishing dc conductance in the low temperature
limit for repulsive interactions within the wire. However,
perfect conductance is still possible by tuning the parameters on the two sides of the junctions as has been
analyzed in detail for a particle-hole symmetric model.27
In this case a line of conducting ﬁxed points in parameter
space exists as only one relevant backscattering operator
is permitted by symmetry which can always be tuned
to zero. Here we generalize to the more experimentally
relevant case where particle-hole symmetry is no longer
present. Even in this more general case we still ﬁnd a line
of conducting ﬁxed points provided the underlying microscopic theory has certain local symmetry properties.
Even though the systems under consideration are inhomogeneous, it is possible to characterize the ﬁxed points
by a single conformally invariant boundary theory with a
characteristic mode expansion and ﬁnite size spectrum.
The results are conﬁrmed by numerical quantum Monte
Carlo (QMC) simulations on spinless fermions. Characteristic experimental signatures for the local density of
states near junctions can be predicted.
For conductivity experiments we must typically consider a system with two junctions, one at each end of
an interacting wire where it is connected to the leads
(e.g. 2DEGs). These junctions are intrinsic sources of
inhomogeneity, but in most cases the junctions do not
inﬂuence each other since the length of the wire is much
larger than the coherence length uβ, where β is the inverse temperature and u the velocity of the collective
excitations. For our purpose to make predictions for the
backscattering and the local behavior near the leads, it
is therefore suﬃcient to analyze one junction between a
lead and a wire.
As an introduction in Sec. II we consider an idealized
2
junction in a non-interacting lattice model and discuss
the applicability of a narrow band approximation. In
Sec. III we start from a microscopic interacting model
and demonstrate how the backscattering terms arise, and
then introduce the general eﬀective bosonic ﬁeld theory.
Focusing on abrupt junctions connecting otherwise homogeneous wires, we examine the renormalization group
ﬂow of perturbing operators in the model. We discuss
the locations of the unstable conducting ﬁxed points in
relation to the symmetry properties of the underlying
microscopic model. Finally in Sec. IV we describe the
conformally invariant boundary theory for the conducting ﬁxed point and the scaling of the local correlation
functions at the boundary. In Sec. V we conclude.
II.
NON–INTERACTING MODELS
Before considering the interacting model it is instructive to analyze the backscattering seen in inhomogeneous
systems of free particles, where exact results are obtainable and can be compared directly with low energy approximations. We start with a lattice model of noninteracting spinless fermions described by the Hamiltonian
†
†
[tj (ψj ψj+1 + H.c.) − Vj ψj ψj ] . (II.1)
ˆ
H0 − µN = −
j
†
ψj creates a particle at site j, tj and Vj are the position
dependent hopping elements and local potential energy
respectively, and N the total particle number. We set =
1 and include in the following the chemical potential µ
in the local potential energy Vj . Generically we consider
situations in which we have two homogeneous regions on
the left (j ≤ j ) and right (j ≥ jr ) side of the wire. In
these asymptotic regions the plane-wave solutions have
the same energy so the parameters are related by
− 2t cos[k a] + V = −2tr cos[kr a] + Vr ,
(II.2)
with k ,r the momenta, V ,r the potential, and t ,r the
hopping on the left ( ) and right (r) side. We have also
introduced the lattice spacing a. We consider a wavefunction incident from the left
ψj =
eik j + Re−ik j , j ≤ j
.
T eikr j ,
j ≥ jr
It is natural to refer to R = 0 as “perfect transmission”,
although this does not necessarily maximize |T |2 . A reasonable deﬁnition of perfect transmission would be maximizing the outgoing current on the right for a given value
of the incoming current from the left, u ; that is, maximizing |T |2 ur /u . Noting that |T |2 ur /u = 1 − |R|2 we
see that the condition for perfect transmission equivalently corresponds to minimizing |R|2 . This can also be
seen by considering the Landauer transmission, see Appendix A.
In general, accurate results cannot be obtained by ignoring states far from the Fermi energy. This can be seen
from the fact that the oﬀ-diagonal components of the T matrix, Tk,k are non-negligible when |k | is not close to
|k|. This implies a non-negligible mixing of low energy
states with high energy ones due to scattering near the
interface. However, in certain limits, a narrow band theory can be used, in which we keep only a narrow band
of states, of width Λ
kF , where kF is the Fermi momentum, and linearize the dispersion relation. This can
be justiﬁed in one of two cases. a) If all potential energy
terms Vi and all hopping terms ti are nearly equal, including the asymptotic ones t ≈ tr . This corresponds to
the adiabatic limit where a local density approximation
suﬃces. b) If there are one or more very weak hopping
terms separating otherwise uniform chains. In this latter
case the ratio t /tr can be arbitrary. These are the limits of weak backscattering or weak tunneling. Starting
with the unperturbed basis of translationally invariant
wave-functions, or wave-functions vanishing at the interface respectively, a small perturbation only mixes states
with energy diﬀerences of order of magnitude of the perturbation.
In these cases we may keep only a narrow band of
states near zero energy and introduce left and right moving ﬁelds in the usual way,
ψ
√j ≈ eikF,x x ψ+ (x) + e−ikF,x x ψ− (x) ,
a
with x = aj a continuous variable and kF x being the
Fermi momentum in the left, kF,x≤aj = kF , or right,
kF,x≥ajr = kF r , of the wire.
Here we want to consider only the simplest model for
a junction while various other types of junctions are discussed in Appendix B. In the simplest model two homogeneous regions are connected at one site such that
(II.3)
tj =
(II.4)
i = { , r}, and current conservation implies
(1 − |R|2 )u = |T |2 ur .
(II.5)
t , j<0
tr , j ≥ 0
Vi =
The region from j to jr is the region of inhomogeneity
describing the junction.
There are two velocities
ui ≡ 2ati sin[ki a] ,
(II.6)
V , j<0
Vr , j > 0
(II.7)
and V0 is kept as a free parameter. The reﬂection amplitude is determined by the Schr¨dinger equation for the
o
central site and results in
R=−
a(V + Vr − 2V0 ) − i(u − ur )
a(V + Vr − 2V0 ) + i(u + ur )
(II.8)
3
The conditions for perfect transmission are therefore
u = ur , and
V0 = (V + Vr )/2 .
(II.9)
When these conditions are satisﬁed, R = 0 and |T |2 = 1.
Curiously, the maximum possible value of |T |2 actually
occurs when V0 = (V + Vr )/2 and ur = 0, in which case
|R| = 1 and |T | = 2. But in this case the current is
actually zero on both sides, so calling this perfect transmission would seem inappropriate. The existence of the
two conditions (II.9) for perfect conductance is related to
the breaking of particle–hole symmetry, see Sec. III A.
Next, we consider the abrupt junction of Eq. (II.7) in
a narrow band approximation setting t = t − δt and
tr = t + δt, with |δt|
t. When δt = 0 we obtain the
usual free, translationally invariant Dirac fermion model,
with uniform velocity u0 = 2at sin kF . Here we treat the
δt term as a perturbation. Using the separation into right
and left moving ﬁelds, Eq. (II.6), the backscattering at
the junction is given by
−1
ˆ
δ H ≈ −a
(2t eikF
a
− V )e2ikF
ja
+ 2tr eikF r a
j=−∞
∞
†
(2tr eikF r a − Vr )e2ikF r ja ψ− ψ+ + H.c. (II.10)
−V0 +
j=1
Since ψ− (x) and ψ+ (x) are assumed to vary slowly on
−1
the scale of kF /r the oscillating terms in the bulk cancel,
leaving only the contributions at x = 0. We may then
write the local backscattering at x = 0 as
†
ˆ
δ H ≈ 2πiλψ− ψ+ (x = 0) + H.c.
(II.11)
with (see also Appendix E)
t
tr
a
−
(II.12)
2π sin[kF a] sin[kF r a]
a
−
(V cot[kF a] − Vr cot[kF r a])
4π
a
=
(t sin[kF a] − tr sin[kF r a]) ,
2π
a
Im λ =
(V + Vr − 2V0 ) ,
4π
Re λ =
where we have used Eq. (II.2) to simplify the real part.
We see that the scattering amplitude λ is real if the local potential energies are equal, V0 = V = Vr . This is
surprising because for any non-zero local potential the
problem is no longer particle-hole symmetric. In App. B
we show that this is a special property of the junction
(II.7) and does not hold in general. Finally, we can use
the fact that we are treating the diﬀerence in hopping
δt perturbatively and approximate kF ≈ kF r ≈ kF in
which case the real part of the scattering amplitude further simpliﬁes,
Re λ = −
u − ur
a δt
sin[kF a] =
,
π
4π
(II.13)
where the diﬀerence in velocities on the two sides of the
junction is given by ur − u = 4aδt sin[kF a]. We see
that for V0 ≈ V ≈ Vr and u ≈ ur , required for the
narrow band approximation to be valid, the result for
the scattering amplitude λ is fully consistent with the
exact result for the reﬂection amplitude (II.8) by using
the general relation R = 4πλ/(ur +u ) between these two
quantities in this limit. In Sec. III A we will discuss how
the narrow band calculation for this type of junction can
be extended to the interacting case using bosonization.
III.
INTERACTING MODEL
As a microscopic interacting model we use the Hamilˆ
ˆ
ˆ
ˆ
tonian H = H0 + HI , where H0 is given by Eq. (II.1)
and
†
†
Uj : ψj ψj :: ψj+1 ψj+1 :
ˆ
HI =
(III.1)
j
for interactions with a position dependent nearest neighbor interaction strength Uj . Normal ordered operators
†
†
†
are given by : ψj ψj := ψj ψj − 0|ψj ψj |0 , with |0 the
ground state. It is assumed that the spatial variation of
ˆ
Uj , tj , and Vj in H0 , is consistent with the narrow band
approximation explained in the preceding section. Later
we will focus on the limiting case of an abrupt jump in
the interaction and hopping parameters at the junction,
as used elsewhere.16,17,27,29,30
In order to ﬁnd the underlying low energy bosonic theory, we ﬁrst need to linearize the spectrum. Analogously
to the normal Luttinger liquid theory,6–8 one can linearize
around the bulk band structure in the left and right regions of the wire.27 Linearization is performed around
the Fermi momenta kF,x for left and right movers:
ψ
√j = ψ(x) =
eiαkF,x x ψα (x) ,
a
α=±
(III.2)
with
the
appropriate
commutation
relations
†
[ψα (x), ψβ (x )]+
= 0 and
ψα (x), ψβ (x )
=
+
δαβ δ(x−x ). Here kF,x is deﬁned by −2t cos kF,x +Vx = 0.
Note that it is not necessary to assume that kF ≈ kF r .
After linearization of the free Hamiltonian we ﬁnd
ˆ
H0 = −
−
†
atx eiακx ψα (x)∂x ψα (x) + H.c.
dx
α=±
−
+
2tx e−2iακx − Vx e−2iαkF,x
dx
α=±
†
×ψα (x)ψ−α (x) ,
(III.3)
where the Fermi momenta are determined by
Vx = 2tx cos κ− ,
x
(III.4)
and we have deﬁned κ− = kF,x+a (x + a) − kF,x x and
x
2κ+ = kF,x+a (x + a) + kF,x x. Similarly, one can write
x
4
the linearized interaction as
ˆ
HI =
dxaUx
†
†
: ψα ψα (x) :: ψβ ψβ (x + a) :
α,β=±
†
†
+e−β2ikF,x+a (x+a) : ψα ψα (x) :: ψβ ψ−β (x + a) :
†
†
+e−α2ikF,x x : ψα ψ−α (x) :: ψβ ψβ (x + a) :
+e−α2ikF,x x−β2ikF,x+a (x+a)
(III.5)
†
†
× : ψα ψ−α (x) :: ψβ ψ−β (x + a) :
,
keeping for the moment all of the terms. If the interaction acts homogeneously then many of the terms can be
neglected as they are suppressed by the rapidly oscillating phases. Due to the inhomogeneity in Ux this is no
longer true and all processes could in principle be important. In fact we ﬁnd that umklapp scattering is generically irrelevant under renormalization group (RG) ﬂow,
see Appendix D, and to lowest order the backscattering
only renormalizes the single particle backscattering already present in the non-interacting Hamiltonian.
We bosonize using the local vertex operator31,32
ψα (x) = √
√
1
eiα 4π[φα (x)] .
2πa
(III.6)
We use the following convention: φ(x) = φ+ (x) + φ− (x)
˜
and its adjoint φ(x) = φ+ (x) − φ− (x) with the conjugate
˜
momentum, Π(x) = ∂x φ(x). These ﬁelds obey
i
[φ+ (x), φ− (y)] = − ,
4
iα
[φα (x), φα (y)] =
sgn(y − x), and
4
[φ(x), Π(y)] = iδ(x − y) .
(III.7)
dx
ux
2
1
˜
(∂x φ)2 + gx (∂x φ)2
gx
.
(III.8)
To lowest order we can determine the renormalized velocity
ux ≈ 2atx sin[κ− ] 1 +
x
Ux
sin[κ− ]
x
πtx
,
(III.9)
and the Luttinger parameter
gx ≈ 1 −
Ux
sin[κ− ] .
x
πtx
−
ˆ
H =
1 −i√4πφ(x)−2ikF,x x e−iκx ux
e
− Vx + H.c..
2πi
a sin[κ− ]
x
x=ja
j∈Z
(III.11)
We keep the sum over x = ja here discrete in order to
avoid ambiguity as to what the alternating terms are in
the continuum limit. This also helps the precise calculation of these sums.
A.
(III.10)
An abrupt junction
Let us now focus on the simple junction considered already in the previous section for the non-interacting case
where two semi-inﬁnite wires are joined at x = 0 with
tx<0 = t , tx≥0 = tr , and Ux deﬁned equivalently. The
local potential energy is taken to be uniform, Vj = V ,
except where explicitly said to the contrary. The Fermi
momenta, kF,x , can also be written with a similar structure as kF,x<0 = kF and kF,x≥0 = kF r . In this system
backscattering can be rewritten as
√
ˆ
H ≈ λe−i
4πφ(x=0)
+ H.c.,
−
λ = −i
x
1 −2ikF,x x e−iκx ux
e
− V a . (III.12)
2πa
sin[κ− ]
x
With the help of appendix E, and noting that for an
abrupt jump κ− = kF,x a, we have to lowest order in the
x
interaction
λ ≈
Some further useful formulas for bosonization are given
in App. C.
ˆ
ˆ
ˆ
The full Hamiltonian H = H0 + HI can be rewritten in the bosonic representation as a quadratic Hamiltonian, a local backscatterer, and umklapp scattering:
ˆ
ˆ
ˆ
ˆ
H = Hb + H + HU , see App. C for details. As already
mentioned, away from half-ﬁlling the umklapp scattering
ˆ
term HU becomes a local perturbation conﬁned to the
regions where Uj is varying, and is then irrelevant under
RG ﬂow. It is neglected in the following. We ﬁnd the
quadratic term to be
ˆ
Hb =
The local backscattering from all processes in
Eqs. (III.3) and (III.5) can be summarized in one term
1
t
U
tr
Ur
+
−
−
2π sin[kF a]
π
sin[kF r a]
π
V
−
[cot[kF a] − cot[kF r a]] ,
(III.13)
4π
which generalizes the non-interacting √
result, Eq. (II.12).
As λ is real we ﬁnd that there is no sin[ 4πφ(0)] operator
present at the boundary and the total backscattering is
√
ˆ
(III.14)
H = 2λ cos[ 4πφ(0)] .
√
The perhaps surprising absence of the sin[ 4πφ(0)] operator is connected to the local properties of the Hamiltonian in the vicinity of the boundary, see App. B. As
such there remains only one condition to fulﬁll for the
conducting ﬁxed point: λ = 0 with λ real.
For V = 0 when there is particle-hole symmetry
present, corresponding to the mapping φ → −φ and
√
˜
˜
φ → −φ, it is transparent that sin[ 4πφ(0)] is forbidden. For V = 0 we ﬁnd that λ remains real for the
speciﬁc junction considered—analytically to ﬁrst order in
the interaction U , see Eq. (III.13), and numerically for
all interactions strengths, see below. We do not have a
simple argument why this is the case and App. B shows
that this is in fact not a generic feature of an abrupt
junction.
5
0.54
For this we require the following integral
ρ
β
0.52
τ (x) ≡ 2
√
√
dτ cos[ 4πφ(x, 0)] cos[ 4πφ(0, τ )]
0
β
0.5
dτ e2π[G(x,0;τ )−G(0,0;0)]
=
(III.17)
0
0.48
=
1
T
4πT a
ux
g
¯
ux
2πT x
sinh
2πaT
ux
−gx
P−¯ (z)
g
which has been calculated using the Green’s function in
Appendix D. We introduced
0.46
-40
0
Uℓ = 0 non-interacting
x
40
z ≡ coth
Ur = 1.8tr interacting
2πT x
,
ux
(III.18)
and Pl (z) is the Legendre function. This gives
FIG. 1: (Color online) The full density including Friedel oscillations near the boundary, numerical results (ﬁlled circles)
are ﬁtted to the analytical result of Eq. (III.19) (lines) with
tl = 1.308tr , U = 0, and Ur = 1.8tr . The local potential
energy is V = 0.25tr and tr β = 10. Underneath a schematic
of the system under consideration is shown.
B.
Local density and compressibility
For the system with an abrupt jump in hopping and
interaction strength it is possible to calculate a variety
of properties perturbatively in the boundary operators
using the exact Green’s function for the Hamiltonian
(III.8), see Eq. (D.2) in the Appendix. In addition to
the dc conductance one can also consider local properties such as the local density and compressibility of the
wire. For abrupt changes in parameters the local density
is known to show characteristic oscillations, the Friedel
oscillations33 , which give information about the interacting correlation functions34–36 and the strength of the
backscattering.27,37
The bosonized density operator for the fermions becomes
1
n(x) = n0 (x) − √ ∂x φ(x)
π
√
const.
∗
+
sin[2kF,x x + 4πφx ] .
π
√
const.
∗
sin[2kF,x x + 4πφ(x)]
π
,
In order to test the calculations we have developed a
quantum Monte Carlo (QMC) code using a stochastic series expansion (SSE) with directed loops.38,39 In Figs. 1
and 2 we show a comparison of this analytical result with
the outcome of QMC simulations on spinless Fermions.
Even for a very large jump in parameters the ﬁt remains
very good. Note that what is seen in the local density
and compressibility proﬁles, see below, is an interplay between the shape of τ (x) and the incommensurate oscilla∗
tions from sin[2kF,x x]. For the ﬁtting procedure between
the analytical and numerical results there are two parameters. The ﬁrst is the amplitude of the eﬀect due to
the unknown constant in Eq. (III.16) and the cutoﬀs in
the ﬁeld theory. The second is a small oﬀset in position,
ρalt (x − a), due to an eﬀective width of the scattering
¯
center, with a being of the order of the lattice spacing a.
¯
The Luttinger parameters g ,r can be found from Bethe
ansatz.40–43
The local compressibility is deﬁned as
χx = −
(III.15)
As before we keep the local potential energy constant,
Vx = V . The oscillating contribution to the density,
i.e. the Friedel oscillations, which are given by
ρalt (x) ≡
√
const. β
∗
ˆ
dτ sin[2kF,x x + 4πφ(x)]H
π
0
const.
∗
= −λ 2 τ (x) sin[2kF,x x]
(III.19)
π a
ρalt (x) = −
∂ nx
ˆ
∂δV
,
(III.20)
δV =0
analogous to the local susceptibility in a spin chain.35 For
the alternating contribution this yields
∗
χalt ∝ λxτ (x) cos[2kF,x x] .
(III.21)
Unlike the Friedel oscillations in the density this observable remains non-zero even for half-ﬁlling and is therefore
in that particular case a more useful quantity to study.
(III.16)
C.
∗
will be calculated to ﬁrst order in λ. kF,x is the renormalized Fermi momentum at ﬁnite temperatures which can
∗
be found from the bulk density: ρx ≡ n0,x = kF,x /π.
Conducting ﬁxed points
In Sec. III A we have predicted that for the abrupt
junction considered only one parameter needs to be tuned
6
0.5
0.01
V = - 0.4 tr
0.01
x
0.48
0.02
V = - 0.2 tr
(-1) ρalt(x)
ρ
0.46
0
0
(-1) ρalt(x)
0.44
0.4
0
x
10
20
30
40 0
10
20
V = - 0.6 tr
0.02
x
0.42
-40
0
30
V = - 0.8 tr 0.03
0.02
0.01
0.01
0
0
40
FIG. 2: (Color online) The full density including Friedel oscillations near the boundary, numerical results (ﬁlled circles)
are ﬁtted to the analytical result of Eq. (III.19) (lines) with
tl = 1.31tr , U = 0, and Ur = 1.8tr . The local potential
energy is V = 0.75tr and tr β = 10.
in order to ﬁnd a conducting ﬁxed point. The low-order
expansion for λ given by Eq. (III.13) is not suﬃcient
however to ﬁnd the location of the ﬁxed points for the
large interaction strengths we want to consider in general. Only in the limit Ux → 0, where we know the exact
result, can we be conﬁdent of its predictions. An exception is the half-ﬁlled case where we have previously
argued27 that the scattering amplitude λ vanishes for all
interaction strengths if u = ur , with the velocities at
half-ﬁlling known in closed form as a function of the interaction strength from Bethe ansatz.40,41
Instead, at generic ﬁllings, we can ﬁnd the locations
of the solutions t∗ (V ) which solve λ(t = t∗ , V ) = 0,
keeping Ux and tr ﬁxed, by analyzing the local density
or compressibility of the system by QMC simulations described in the preceding subsection. We ﬁnd that, away
from half-ﬁlling, these do not correspond to u = ur . For
λ = 0 the density is determined entirely by the Hamiltonian Eq. (III.8), plus irrelevant perturbations. For λ = 0,
on the other hand, the relevant backscattering term contributes. By plotting the density for diﬀerent t in Fig. 3
we can ﬁnd the places where the leading corrections vanish and λ changes sign,27 which typically can be observed
in the range 5a x 10a. Since we can always identify a
value of hopping where the leading contribution vanishes,
there must be a line of conducting ﬁxed points in parameter space. In turn the existence of a full line of ﬁxed
points demonstrates that there is only one condition for
the conducting ﬁxed point, λ = 0 with real λ. We want
to stress though that even at such a point in parameter
space there are still irrelevant backscattering processes
present which only vanish in the zero temperature limit
β → ∞.
0
10
20
x/a
30 0
10
20
30
x/a
FIG. 3: (Color online) Plotted are the Friedel oscillations for
diﬀerent local potential energies V calculated by QMC simulations, see main text for details, on the right hand side of
the junction (x > 0). We only show the longer wavelength
amplitude of the rapid oscillations. In each panel from top
to bottom: t = 1.3tr for (black) circles, t = 1.4tr for (red)
squares, t = 1.5tr for (green) diamonds, t = 1.6tr for (blue)
up-triangles, and t = 1.7tr for (purple) down-triangles. We
have used everywhere U = 0, Ur = 1.8tr and inverse temperature tr β = 10.
IV.
CONFORMALLY INVARIANT BOUNDARY
THEORY
In the preceding sections it has been demonstrated that
it is possible to ﬁnd an unstable conducting ﬁxed point
in two wires connected at a junction by appropriately
tuning the bulk parameters of the wires. The existence
of this ﬁxed point immediately invites the question of
the nature of the eﬀective low energy theory. Obviously
translational invariance is lost and it is also not possible
to use mirror charges as would be the case for an open
boundary condition. Therefore it is highly non-trivial to
postulate a description in terms of a conformally invariant theory in this case. Nonetheless, as we will show in
this section it is possible to characterize this ﬁxed point
in terms of mode expansions and two eﬀective boundary
Luttinger liquid parameters. Particular attention is paid
to the case of half-ﬁlling where we can pinpoint the ﬁxed
point precisely. This allows convenient numerical checks
of the results.
A.
Mode expansion and ﬁnite size spectrum
In the absence of backscattering at a junction we have
the bosonic Hamiltonian17,27,30
ˆ
H=
dx
1
2
1
˜
(∂x φ)2 + gx (∂x φ)2
gx
(IV.1)
Compared to (III.8) the position, x, was rescaled on
the two sides of the junction such that u , ur → 1.
7
The ﬁelds obey the canonical commutation relation:
˜
[φ(x), ∂y φ(y)] = iδ(x−y). Therefore we have the relation
∂t φ(x) = i[H, φ(x)]
˜
= gx ∂x φ(x) .
(IV.2)
1
∂x
gx
φ(x, t) = 0
(IV.3)
φ(0− ) = φ(0+ ),
φ(−L) = φ(L)
−
The corresponding Green’s function can be determined
from Eq. (IV.1), see Eq. (D.2). Here we explore other
properties of this boundary condition. We are interested
in the solutions of the classical equation of motion,
2
∂t − gx ∂x
At the boundaries φ(x) and ∂x φ(x)/gx have to be continuous leading to the boundary conditions
∂x φ(−L)
∂x φ(L)
=
g
gr
∂x φ(0 )
∂x φ(0 )
=
,
g
gr
[φ(x), ∂t φ(y)] = igx δ(x − y) .
g if − L < x < 0
gr if 0 < x < L .
φ(x, t) = φ0 +
g Πt Qxgx
¯
+
+
2L
2¯ L
γ
g Πx
¯
1 Qt
˜
˜
+
−
φ(x, t) = φ0 +
γ 2L 2gx L
¯
(IV.4)
∞
l=1
∞
l=1
1 1
1
1
=
+
g
¯
2 g
gr
,
(IV.8)
which describes the conductance.16,27,30 Interestingly, we
ﬁnd in addition a second boundary Luttinger parameter
γ=
¯
1
[g + gr ] ,
2
(IV.9)
which is important for other correlation functions as
we will see below. Π is the ﬁeld conjugate to √ 0 with
φ
[φ0 , Π] = i. As this ﬁeld is periodic, φ0 → φ0 + π, it is
clear that the eigenvalues of the conjugate ﬁeld Π must
√
be 2 πm, where m is √ integer. Q is the ﬁeld conjugate
an
˜
˜
˜
to φ0 and φ0 → φ0 + √4π so that the eigenvalues of the
conjugate ﬁeld Q are πn for integer n.
The classical equation of motion (IV.3) has to follow
from a classical least action principle from which the classical Hamiltonian
2L
H=
0
dx
2
2
(∂t φ) + (∂x φ) .
2gx
(IV.10)
is determined. Substituting the mode expansion into the
Hamiltonian, we may read oﬀ the ﬁnite size spectrum
π
1
n2
E=
− +
+ m2 g +
¯
L
12 4¯
γ
∞
l(me,l + mo,l ) .
l=1
(IV.11)
(IV.6)
This leads to
e−iπlt/L √
gx
√
g cos(πlx/L)ae,l + √ i sin(πlx/L)ao,l + H.c. ,
¯
γ
¯
2πl
√
e−iπlt/L
g
¯
1
√
i sin(πlx/L)ae,l + √ cos(πlx/L)ao,l + H.c. .
gx
γ
¯
2πl
As before we have the boundary Luttinger parameter
(IV.5)
The classical equation of motion (IV.3) has oscillatory solutions as well as solutions linear in x, see Appendix F for
details. We may expand the ﬁeld φ(x) in these solutions,
while respecting the canonical commutation relation
on a ring with circumference 2L where
gx =
+
(IV.7)
Here n and m are arbitrary integers while me/o,l are
non-negative integers corresponding to the eigenvalues of
a† ae/o,l . We have included the universal term in the
e/o,l
ground state energy −cπ/(12L) with c = 1 for a periodic
system of length 2L.
B.
Scaling properties of the conducting ﬁxed point
As usual, since we have imposed the same boundary
condition at both ends, we may read oﬀ the scaling dimensions of all single-valued boundary operators in the
bosonized theory from the ﬁnite size spectrum. The scaling dimensions are
ζm,n =
n2
+ m2 g +
¯
4¯
γ
∞
l(me,l + mo,l ) .
(IV.12)
l=1
Each dimension corresponds to a diﬀerent boundary op√
erator. m2 g corresponds to exp[im 4πφ(0)] with the
¯
m = ±1 operators being the leading relevant operators
at the unstable ﬁxed point. γ /4 is the dimension of the
¯
√ ˜
operators exp[±i π φ(0)], which eﬀectively correspond to
spin operators S ± (x = 0), see below.
To analyze the scaling properties of the system, and
compare the results with numerical calculations, it is convenient to introduce correlation functions for a spin system equivalent to our fermionic system. The mapping
8
g
¯
0.3
gℓ
gr
γ
¯
g
¯
0.1
γ
¯
Cz(τ)
0.1
C+-(τ)
Cz(τ)
gr
gℓ
0.1
0.01
0.01
(a)
β tr=0.5, 1, 2.5, 5, 10, 15, 20, 25
(b)
0.1
1
0.04
0.1
sin[πτ/β]
1
sin[πτ/β]
FIG. 4: (Color online) The scaling of the local spin correlation
functions Cz (τ ) and C± (τ ) at the ﬁxed point: t = 1.518tr ,
U = 0, and Ur = 1.8tr . The magnetic ﬁeld is zero (i.e. V = 0
for the corresponding fermion system) and the temperature
is tr β = 25. (a) Numerical data, black circles, are compared
to the predicted scaling f (τ )ν with f (τ ) ≡ | sin(πτ /β)|−2 and
ν = g . (red curve). As a comparison we also plot, f (τ )ν with
¯
ν = g , gr , γ , see Eq. (IV.20). (b) Numerical data, black
¯
circles, are compared to the predicted scaling f (τ )1/4ν with
ν = γ (red curve). As a comparison we also plot, f (τ )1/4ν
¯
with ν = g , gr , g , see Eq. (IV.19).
¯
between spin operators and fermionic operators is given
by the Jordan-Wigner transformation
†
+
Sj = ψj eiπ
l 0) where the
scaling modulates from the non-interacting result at the
boundary x = 0, to the bulk interacting value far inside
the wire. In contrast to the density or compressibility of
Sec. III B there is no proximity eﬀect near the boundary
in the non-interacting wire.
10
D.
Fixed points and the g-theorem
From the ﬁnite size spectrum, Eq. (IV.11), we may also
read oﬀ the partition function in the scaling limit:
γ
g
Z(β/L) = η −2 e−πβ/L θ3 e−πβ/[2¯ L] θ3 e−πβ2¯/L .
(IV.26)
Here we have introduced the Dedekind eta and Jacobi
theta functions,
∞
η(q) ≡ q 1/24
(1 − q n ) and
n=1
∞
qn
θ3 (q) ≡
2
/2
.
c
i
and hence g > 1. In this case gd < gd so this ﬂow is also
¯
consistent with the g-theorem.
It is also interesting to consider the ﬂow starting from
the insulating ﬁxed point, but with a weakly connected
resonant site in between the two wires: the resonant ﬁxed
point. Then for a range of Luttinger parameters an RG
ﬂow from the resonant to the conducting ﬁxed point is
expected. A necessary condition for the ﬂow from resonant to conducting ﬁxed points is that the tunneling
operators from each chain to the resonant site are relevant, g ,r > 1/2. The ground state degeneracy of the
resonant ﬁxed point is bigger by a factor of 2 than that
of the insulating ﬁxed point due to the 2-fold degeneracy
of the resonant site and
(IV.27)
r
gd = 2(g gr )1/4 .
n=−∞
In the thermodynamic limit, β/L → 0, this becomes
γ /¯eπL/(3β) .
¯ g
Z→
c
gd
=
γ
¯
g
¯
1
4
=
(g + gr )1/2
.
(4g gr )1/4
(IV.29)
This may be compared to the ground state degeneracy
for the insulating ﬁxed point where the junction consists
of the perfectly reﬂecting ends of two quantum wires with
Luttinger parameters g and gr . This ﬁxed point has11
i
gd = (g gr )1/4 ,
Thus the ratio of ground state degeneracies of the resonant to conducting ﬁxed points is
(IV.28)
Apart from the usual bulk free energy, F = −πL/(3β 2 ),
there is also a “ground state degeneracy”, gd , associated
with the two interfaces in the system. The factor for each
interface is
(IV.30)
According to the “g-theorem”, boundary RG ﬂows between ﬁxed points can only occur when gd is reduced
during the ﬂow.51 Therefore, it is interesting to consider
the ratio
√
r
c
¯
gd /gd = 2 g .
CONCLUSIONS
(IV.31)
The g-theorem states that ﬂow from the conducting to
insulating ﬁxed point is only possible when g < 1. This is
¯
consistent with the √
analysis here since g is the dimension
¯
of the operator cos[ 4πφ(0)] which drives the ﬂow. The
ﬂow only takes place when the operator is relevant, cori
responding to g < 1. For suﬃciently large gd the renor¯
malization ﬂow can occur from insulating to conducting
ﬁxed points. As shown in Ref. 11, the fermion operator,
or equivalently spin raising operator, at the end of the
open chain, has scaling dimension 1/(2g ) or 1/(2gr ) as
appropriate. We might expect the ﬂow from insulating
to conducting when the tunneling between the two open
chains is relevant which occurs when
1
1
1
+
= <1
2g
2gr
g
¯
(IV.34)
√
r
c
We can see that gd /gd ≥ 2 whenever g , gr > 1/2 so the
g-theorem is also obeyed by this RG ﬂow. Even when λ
is tuned to zero, corresponding to resonance, the next
√
most relevant operators, exp[±2i 4πφ(0)] will still be
present. This can drive the ﬂow from the conducting
to the resonant ﬁxed points when √ becomes relevant,
it
c
r
¯
i.e. for 4¯ < 1. Since gd /gd = 1/(2 g ) we see that this
g
ﬂow is consistent with the g-theorem as it only occurs
when g < 1/4. Therefore all expected RG ﬂows are con¯
sistent with the g-theorem.
As ﬁrst observed by Kane and Fisher9 in the case g =
gr , there is a range of Luttinger parameters where both
conducting and resonant ﬁxed points are stable. In this
case they are separated by an intermediate unstable ﬁxed
point.
V.
1
c
i
gd /gd = √ .
g
¯
(IV.33)
(IV.32)
In conclusion, we have described a novel conducting
ﬁxed point in inhomogeneous quantum wires. This ﬁxed
point is reached by tuning to zero the amplitude of the
leading backscattering operator at the junction between
two homogeneous parts of the wire. We have, in particular, studied a lattice model of spinless fermions with
nearest neighbor hopping and interaction in the critical regime. For the case of an abrupt junction we have
derived the backscattering amplitude for all ﬁllings in
lowest order in the interaction. For the half-ﬁlled case
it is even possible to give a condition for the vanishing
of the backscattering amplitude valid for all interaction
strengths. The prediction of a conducting ﬁxed point
were numerically conﬁrmed by numerical QMC calculations of the Friedel oscillations in the local density and
compressibility close to the boundary which vanish in
leading order at the ﬁxed point.
11
One of our main results is the derivation of the boundary conformal ﬁeld for this novel unstable conducting
ﬁxed point. The conformally invariant theory for this
case is highly unusual because the two parts of the wire
are governed by diﬀerent bulk Luttinger parameters g
and gr . As a consequence, we ﬁnd that the scaling dimensions of boundary operators are also governed by two
diﬀerent Luttinger parameters given by γ = (g + gr )/2
¯
and g = 2g gr /(g + gr ). We showed, both analytically
¯
and numerically, that γ is controlling the transverse spin
¯
autocorrelation function while g controls the longitudi¯
nal one in the corresponding spin model. Experimentally, a test of the boundary exponents could possibly be
obtained by scanning tunneling microscopy which would
allow one to measure the local density of states which
shows energy scaling with an exponent being determined
by γ and g .
¯
¯
It is convenient to change integration variables to
the ﬁrst integral and 2 in the second, giving:
µL
d 1
[1 − |R( 1 )|2 ]
2π
I = −e
1 (0)
µR
1
in
(A.3)
d 2 u ( 2)
|T ( 2 )|2
2π ur ( 2 )
+e
1 (0)
Since |T |2 u /ur = 1 − |R|2 this can be written as
µL
I = −e
µR
d
[1 − |R( )|2 ] .
2π
Now taking the limit µL → µR ≡
I→
e2
[1 − |R(
2π
F,
2
F )| ](VL
(A.4)
we ﬁnd:
− VR ) .
(A.5)
2
(A.6)
Hence the linear conductance is
Acknowledgments
G=
e2
dI
=
[1 − |R(
dV
2π
F )|
].
J.S., S.E., and N.S. acknowledge support by the Collaborative Research Center SFB/TR49 and the graduate
school of excellence MAINZ. The research of I.A. was
supported by NSERC and CIfAR. We are grateful for
computation time at AHRP.
This is another way of seeing that [1−|R|2 ] is the suitable
measure of the transmission of the interface.
Appendix A: Landauer formula for transmission
In the main text, Sec. II we have considered the simplest possible junction, a jump between two homogeneous
regions, in the non-interacting case. Here we want to
present calculations for more general junctions to study
the inﬂuence on the backscattering term.
The signiﬁcance of the measure of transmission R = 0
can be veriﬁed by considering the Landauer formula.52
Thus we imagine attaching the wire to reservoirs on the
left and right side with diﬀerent chemical potentials µL
and µR . We consider particles emitted from the left reservoir with a thermal distribution with chemical potential
µL = −eVL and from the right reservoir with a thermal distribution and chemical potential µR = −eVR . At
zero temperature the Fermi wave-vectors on left and right
sides, kF and kF r , are given by
− 2t cos kF
,r
−V
,r
= µL,R .
(A.1)
Suppose that the bottom of the band on the left has
higher energy than the bottom of the band on the right.
Then the total current, at zero temperature, is
kF
I = −e
0
dk
u (k)[1 − |R(k)|2 ]
2π
−k2max
+e
−kF r
Appendix B: Non-interacting calculations
1.
Abrupt junction with additional local variation
The calculation of Sec. II can be extended straightforwardly to a more general model where the hopping
amplitude varies near the origin. Suppose, for example,
that the hopping amplitude from site -1 to 0 is t−1 = t
and from 0 to 1 is t0 = tr with the rest as given by
Eq. (II.7). For simplicity we concentrate again on halfﬁlling, Vj = 0. Then we may write the wave-function
for an incoming wave from the left as in Eq. (II.3) with
j = −1 and jr = 1. ψ0 is now a free parameter. Solving
for the reﬂection amplitude as previously gives
(A.2)
dk
u (k)|T (k)|2 .
2π
The ﬁrst term is the current emitted by the left reservoir
and partially reﬂected at the interface. The second term
is the current emitted by the right reservoir and partially
transmitted. The maximum wave-vector for the second
integral, k2max > 0, is given by 2 (−k2max ) = 1 (0) since
lower energy incoming particles from the right have zero
transmission probability.
|R|2 =
t2
u
t2
tr2
t2 ur
r
2
−
t2
u
t2
tr2
t2 ur
r
2
+
+ a2
2
t2
t2
−
tr2
t2
r
2
2−
+ a2
2
t2
t2
−
tr2
t2
r
2
2−
. (B.1)
Solving for R = 0 one ﬁnds
(t /t , tr /tr ) =
√
2(cos θ, sin θ)
(B.2)
with tan2 θ = u /ur . Thus maximal conductance can be
achieved for any choice of energy , and thus any value
of u /ur that can occur as is varied.
12
We see that the simple condition u = ur for perfect
conductance is a special result, which only holds for the
“abrupt junction” considered in the main text. In general, the condition u = ur can be regarded as removing
the intrinsic scattering from a sharp jump between bulk
values of the hopping. Additional variation on top of this
will naturally result in scattering and an additional ﬁnetuning is required to reach the conducting ﬁxed point.
Note that the fact that two parameters, t and tr , need
to be adjusted to achieve perfect conductance, in general
is in accord with the renormalization group (RG) viewpoint. For non-zero energy , particle-hole symmetry is
broken so the scattering amplitude λ can be complex.
In the special case = 0 where particle-hole symmetry
holds there is only one condition for perfect conductance
t 2 /t2 = tr2 /t2 and only one parameter needs to be adr
justed.
Now let us consider the case with non-trivial ti
within the narrow band approximation. We use the
parametrization
t = t + δt = t − δt + δt
tr = tr + δtr = t + δt + δtr .
To keep things as simple as possible we choose
t2L , (i ≤ −2)
.
t2R , (i ≥ 0)
(B.8)
There is no particularly simple or natural choice for t2,−1
so it is kept as a free parameter. Let us assume that all
the t1,i are close together and all the t2,i are close together
so that the narrow band approach is applicable. Thus we
write
t1,i =
t1L = t1 − δt1 ,
t1R = t1 + δt1 ,
t2,−1 = t2 + δt .
t2,i =
t2L = t2 − δt2 ,
t2R = t2 + δt2 ,
(B.9)
A simple extension of the previous calculation gives
πλ = −δt1 csc[kF a] − δt2 cot[kF a] + iδte2ikF a
(B.10)
for the back-scattering coupling constant.
As a simpler special case, consider kF = π/2. Now
πλ = −δt1 − δt2 − iδt ,
(B.3)
In this case, there is another backscattering perturbation
term
t1L , (i ≤ −1)
,
t1R , (i ≥ 0)
(B.11)
and λ is complex in this case despite being at half-ﬁlling;
this is natural since t2 breaks particle-hole symmetry at
all ﬁllings.
†
†
ˆ
δ H = −2δt ψ− ψ+ e−ikF a − 2δtr ψ− ψ+ eikF a + H.c.
†
†
−(δt + δtr )2 cos[kF a](ψ− ψ− + ψ+ ψ+ ) (B.4)
Appendix C: Bosonization details
where ψ− and ψ+ are evaluated at x = 0 in all terms.
Focusing on the backscattering term the perturbation becomes
First we note the following useful relations:
†
ˆ
ˆ
δ H + δ H = 2πiλψ− ψ+ (x = 0) + H.c.
(B.5)
u − ur
iδt e−ikF a + iδtr eikF a
+
.
4πa
π
(B.6)
1
†
ψα (x)ψα (x) = ρα (x) ≡ − √ ∂x φα (x),
π
†
ψα (x)∂x ψα (x) = αiπρ2 (x), and
α
iα −iα√4πφ(x)
†
ψα (x)ψ−α (x) =
.
e
2πa
with
λ=
Although the variations in ti are small, they occur over
only three sites, so this is not an adiabatic change. Note
that while we were able to determine λ explicitly in this
model, with all ti nearly equal, it may not be feasible to
do so in all cases. In fact, a reduction to a narrow band
model is not accurate in general, as discussed in the main
text.
2.
Next-nearest neighbor hopping
(C.1)
Due to the inhomogeneous nature of both the Fermi
momentum and the interactions, the 2kF,x oscillating
terms in the interaction can no longer be neglected,
see Eq. (III.5). One ﬁnds a nonzero contribution to
the backscattering around any region of inhomogeneity.
These terms must be treated carefully, as an example we
†
†
can take : ψα ψα (x) :: ψα ψ−α (x + a) :. Direct rearrange†
†
ment gives for ψα ψα (x)ψα ψ−α (x + a) either
†
†
: ψα ψα (x) :: ψα ψ−α (x + a) :
†
†
+ 0|ψα ψα (x)|0 : ψα ψ−α (x + a) :
(C.2)
or
Next-nearest neighbor hopping can also be added to
the Hamiltonian, explicitly breaking particle-hole symmetry. We consider the Hamiltonian
†
†
− : ψα (x)ψ−α (x + a) :: ψα (x + a)ψα (x) :
(C.3)
†
†
− : ψα (x)ψ−α (x + a) : 0|ψα (x + a)ψα (x)|0 ,
which are therefore equal. Then, using
†
†
[−t1,j ψi ψi+1 − t2,i ψi ψi+2 + H.c.] .
ˆ
H0 =
i
(B.7)
†
0|ψα (x + a)ψα (x)|0 ≈ iα/2πa ,
(C.4)
13
Solving this diﬀerential equation subject to the appropriate boundary conditions17,27 gives
and expanding in the cut-oﬀ a this allows us to write
†
†
: ψα ψα (x) :: ψα ψ−α (x + a) :≈
(C.5)
†
†
− 0|ψα ψα (x)|0 : ψα ψ−α (x + a) :
†
− : ψα (x)ψ−α (x + a) :
iα
†
+ ρα (x) + aψα ∂x ψα (x) .
×
2πa
G(x, y; τ ) = φ(x, 0)φ(y, τ )
g
¯
= − ln sinh πT
π
+
(D.2)
|x| |y|
+
− iτ
ux
uy
sinh πT
L[x, y]gx
ln
π
sinh πT
Now keeping only the leading order terms we ﬁnd
√
:
†
ψα ψα (x)
::
†
ψα ψ−α (x
e−iα 4πφ(x)
.
+ a) :≈
4π 2 a2
(C.6)
Similar expressions hold for the other terms.
The bosonized free and interacting Hamiltonians become
−
ˆ
H0 = −
αitx a2 eiακx (∂x φα )2 + h.c.
(C.7)
xα
−
xα
iαtx −2iαkF,x x−iα√4πφ(x) −iακ−
V
x −
e
e
π
2tx
,
and
ˆ
HI =
a2 Ux
−
1 − e−2iακx
xα
(∂x φ)2
2π
√
+e
2iαkF,x x
−e
4iακ+
x
e
2iακ−
x
√
eiα2 4πφ(x)
,
(2πa)2
(C.8)
+
|y|
uy
|x−y|
ux
− iτ
.
− iτ
We have introduced 2L[x, y] ≡ 1 + sgn[x] sgn[y] .
The renormalization procedure is done in the standard
manner by expanding the perturbation, exp(− dτ H ),
to ﬁrst order and integrating out the ﬁelds with fast
Fourier components near the band-edge Λ < |k| < Λ. In
order to recover the original form after re-exponentiating
the action we rescale Λτnew = Λ τ , and deﬁne the new
coupling constant λ, as
λ(Λ ) =
Λ
˜
λ(Λ)e−πG> (x=y=τ =0) .
Λ
(D.3)
˜
where G> is the Green’s function after integrating out
the fast modes.
Therefore for the RG equation what we need is the
Green’s function summed over the fast modes. First let
us change variables to r = x−y and R = (x+y)/2. Then,
with u(r, R = 0) = 2ux uy /(ux + uy )|x=−y = 2u ur /(u +
ur ) ≡ u, we have
G(r, R = 0; τ ) = −
2eiα 4πφ(x)
−1
(2πa)2
|x|
ux
g
¯
ln sinh πT
π
|r|
− iτ
u
. (D.4)
This is the same as the Green’s function for a homogeneous case, but with a new velocity and a new Luttinger
parameter:
1
1 1
1
=
+
g
¯
2 g
gr
respectively. Included in this is the irrelevant umklapp
scattering
.
(D.5)
Now we require
ˆ
HU = −
x
√
Ux
a 2 cos 4κ+ + 2 4πφ(x) .
x
π a
(C.9)
For the abrupt jump of Sec. III A the Green’s function,
at λ = 0, can be calculated exactly. We have
(D.1)
m
2
ωm
∂
ux ∂
−
2gx ux
∂x 2gx ∂x
(D.6)
dreikr G(r, R = τ = 0) .
=
Λ <|k|<Λ
Thus integrating out the fast Fourier components results
in a change of the Green’s function,
g
¯
(D.7)
G> ≈ d ln Λ ,
π
which governs the renormalization in the usual manner:
Appendix D: The Green’s function and
renormalization group calculations
eiωm τ Gm (x, y) and
G(k, R = τ = 0)
Λ <|k|<Λ
Away from half-ﬁlling this is only a boundary contribution.
G(x, y; τ ) = T
G> (0, 0; 0) =
Gm (x, y) = δ(x − y) .
1 dλ
= 1−g.
¯
(D.8)
λ d ln Λ
We therefore expect that the eﬀective backscattering
renormalizes as a power law in the temperature R ∝
¯
T g−1 , which in turn aﬀects the conductance and other
physical observables accordingly. This has been conﬁrmed numerically.27
14
Appendix E: Useful sums for the boundary terms
To ﬁnd the coeﬃcients of the backscattering terms several sums are needed. We want
e−2ikF,x x F (x)O(x)
I≡
(E.1)
x=ja
j∈Z
Appendix F: The mode expansion and its
correlation functions
Let’s ﬁrst consider the solutions of the classical equation of motion, Eq. (IV.3), subject to the boundary conditions Eq. (IV.5). There are two types of oscillating
solutions
(1)
φk (x, t) ∼ eikt cos(kx)
in the particular case where we can write
F (x) = F Θ(−x − a) + Fr Θ(x)
≡F (x)
(E.2)
≡Fr (x)
e−2ikF i x Fi (x)
I ≈ O(x = 0)
(2)
(E.3)
x=ja i=1,2
j∈Z
e−2ikF i x
≈ O(x = 0)
x=ja i=1,2
j∈Z
Zi
[Fi (x) − Fi (x + a)] ,
2
with Zi = 1 + i cot[kF i a]. Only a single term of each sum
over x is non-zero and we ﬁnd
iF eikF a
iFr eikF r a
−
.
2 sin[kF a] 2 sin[kF r a]
I ≈ O(x = 0)
≡F (x)
(E.5)
≡Fr (x)
and we can independently change F0 on the central site.
Then with I ≈ Ox=0 λ we ﬁnd
λ = F0 +
iFr e−ikF r a
iF eikF a
−
.
2 sin[kF a] 2 sin[kF r a]
(F.2)
with φ(x = 0) = 0. In addition there are solutions linear
in x and t. The solutions linear in x have the form
φx (x, t) ∼ gx x .
(F.3)
The bosonization formula (III.6) implies that the bosonic
√
ﬁeld φ is periodic with period φ + π. Furthermore we
are considering solutions on a ring with circumference
√
2L thus φ(x) = φ(x + 2L) + πn. The oscillatory solutions of both types therefore must have k = πl/L for
l = 1, 2, 3, . . .. For the solutions linear in x the same
periodicity conditions imply
√
(E.4)
We may also be interested in the case where
F (x) = F Θ(−x − a) + Fr Θ(x − a) +F0 δ(x)
with ∂x φ(x = 0) = 0, and
φk ∼ gx eikt sin(kx)
with Θ(0) ≡ 1. Assuming Ox is slowly varying on a
length scale of a, this allows us to write
(F.1)
(E.6)
φx (x, t) =
πn
xgx .
2¯ L
γ
(F.4)
ˆ
Let’s now consider the mode expansion.Π is canonically
conjugate to φ0 and the normalization of each term is
ﬁxed by requiring the canonical commutation relations
to hold.For g = gr , we may expand in solutions of the
classical equations of motion, while respecting the canonical commutation relations. This leads to the mode expansion given in Eq. (IV.7).
Using the mode expansion we can ﬁrst calculate the
bosonic commutators in the ground state. We ﬁnd that
1 1
˜
˜
Re φ(x, t)φ(x, 0) = −
ln 24 sin[π(t − 2x)/2L] sin[π(t + 2x)/2L] sin2 [πt/2L]
γ 8π
¯
g 1
¯
sin[π(t − 2x)/2L] sin[π(t + 2x)/2L]
+ 2
ln
gx 8π
sin2 [πt/2L]
(F.5)
and
g
¯
ln 24 sin[π(t − 2x)/2L] sin[π(t + 2x)/2L] sin2 [πt/2L]
8π
g2 1
sin[π(t − 2x)/2L] sin[π(t + 2x)/2L]
+ x
ln
.
γ 8π
¯
sin2 [πt/2L]
Re φ(x, t)φ(x, 0) = −
(F.6)
Finally we will also need
1 gx
g
¯
sin[π(t − 2x)/2L]
˜
Re φ(x, t)φ(x, 0) =
+
ln
.
8π γ
¯
gx
sin[π(t + 2x)/2L]
(F.7)
15
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Electronic address: nicholas.sedlmayr@cea.fr
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