Numerical Renormalization Group at marginal spectral density:
application to quantum tunneling in Luttinger liquids
Axel Freyn1 and Serge Florens1
1

arXiv:1101.1055v3 [cond-mat.str-el] 25 May 2011

Institut N´el, CNRS and Universit´ Joseph Fourier,
e
e
25 avenue des Martyrs, BP 166, 38042 Grenoble, France
Many quantum mechanical problems (such as dissipative phase fluctuations in metallic and
superconducting nanocircuits, or impurity scattering in Luttinger liquids) involve a continuum of
bosonic modes with a marginal spectral density diverging as the inverse of energy. We construct
a Numerical Renormalization Group in this singular case, with a manageable violation of scale
separation at high energy, capturing reliably the low energy physics. The method is demonstrated
by a non-perturbative solution over several energy decades for the dynamical conductance of a
Luttinger liquid with a single static defect.

Bosonic description of fermionic systems, possibly subject to strong interactions, has a long history, ranging
from phase fluctuations in superconducting circuits1,2 following Josephson’s initial ideas, to quantum transport
in metallic grains3 and in strongly correlated materials
near the Mott transition,4 where the phase conjugate
to the electron charge is the relevant physical variable
to understand the interplay of tunneling and Coulomb
blockade. Another interesting example concerns onedimensional electronic wires, the so-called Luttinger liquids (LL), where non-interacting plasmon modes provide a faithful representation of electronic density fluctuations.5–7 Quite remarkably, all these different physical problems share very common features, because one
can describe the electrons by a bosonic variable Φ conjugate to the electronic charge transfered in a nanoscale
junction (or also to the charge density in a wire), via a
phase factor eiΦ . This mapping can be used for instance
to represent the Josephson current in superconducting
junctions (or also the fermionic fields in the bosonization language). This implies in turn that the nature of
the phase dynamics determines the underlying physics:
wild fluctuations of Φ occur for instance in the presence
of strong Coulomb blockade, leading to a rapid decay
of the phase, and implying electronic localization.3,4 In
contrast, phase localization is associated to small fluctuations of the electron charge, characterizing dissipationless
supercurrent1 or Fermi liquid states.4 The intermediate
situation of soft (algebraic) phase decay leads to the wellknown non-Fermi liquid features of a LL.5–7
In all these situations, great complexity arises due to
the coupling of the bosonic mode to static disorder or
dynamical defects, such as discrete Andreev levels8,9 in
superconducting weak links10 ) or magnetic Kondo impurities in metallic junctions11,12 and interacting unidimensional wires.13,14 Focusing the discussion on the case
of impurity effects in LL, but keeping this more general
framework in mind, many technical and physical questions are still open to date, both in the original fermionic
formulation and in the bosonic version of the problem.
On the fermionic side, one needs to handle strong interactions within unidimensional wires together with the pres-

ence of exponentially small energy scales arising from the
impurity,11,15 for which powerful numerical methods have
been developed in the past. The Density Matrix Renormalization Group16 can tackle correlated wires, but on a
linear energy scale only, which does not allow to really
extract critical exponents;17 the Numerical Renormalization Group (NRG)18,19 can however deal with impurity
physics on an exponential energy range, but only for uncorrelated Fermi liquids. A method that could incorporate both virtues would therefore be quite useful, which
is the goal of this Letter.
Using the bosonic language, the description of interacting electrons by non-interacting bosons helps tremendously, but difficulties still arise. Apart from perturbative analysis (or fine tuning of the model parameters to
allow an exact solution),5–7 the analytical bosonization
technique offers limited information on quantum impurity problems, because the physics crosses over from weak
to strong coupling, for instance due to Kondo screening. Actually, all this complexity is already encoded by
a static defect in LL, which drives the conductance from
e2 /h to zero on an energy scale that can be exponentially
small in the backscattering amplitude, a problem that
has triggered substantial work, based on approximate analytical methods15,17,20–23 or numerical techniques on a
linear energy scale, such as quantum Monte Carlo.2,24
The idea we henceforth present here is to use recent
developments of the bosonic NRG25–27 in order to tackle
numerically the phase fluctuation problem in a broad
range of parameters, with possible extensions to dynamical defects. This however faces an immediate and
seemingly intractable difficulty. Quite common to the
Josephson effect in a dissipative environment,9 to quantum tunneling in resistive circuits,12,28–30 or to tunneling into LL,12,15,17,20–23 is the marginal form of the bare
local bosonic spectrum, given by the correlation func2π
0
tion GΦ (iω) = |ω| at imaginary frequency. The key step
in the NRG procedure is the scale separation that results from a logarithmic discretization of the energy band
ωn = ωc Λ−n , with 1 < Λ. A generalized power-law density of states (with exponent s ≥ −1 and high energy cut−1−s s
off ωc ) of the form J(ω) = 2πωc
ω Θ(ωc − ω) can be

2
considered both for fermionic31 and bosonic models,25–27
providing the following coupling strength of the states at
energy ωn :
2
γn =

ωn

dω J(ω) = 2π
ωn+1

1 − Λ−(s+1) −n(s+1)
Λ
.
s+1

(1)

2
For all s > −1, the couplings γn decay exponentially
with n, which allows an iterative diagonalization of the
problem: the possibility of building progressively the
Hilbert space from high to low energies is the reason behind the huge success of NRG to solve quantum impurity problems in a linear numerical effort.18,19 However,
the marginal case s = −1 is special in the sense that
2
the couplings γn = 2π log(Λ) do not decay anymore, invalidating clearly the whole scheme. We stress that we
are considering quantum impurity Hamiltonians that depend explicitly on the phase factor eiΦ , and not on the
spatial derivative of the bosonic mode, ∂x Φ. This latter case, which arises for instance in the so-called ohmic
spin-boson model,25 corresponds to the much simpler situation of linear spectrum (s = 1) of the field ∂x Φ, and
can be easily handled by the bosonic NRG.
This complete violation of scale separation for the
marginal case s = −1 seems to disqualify our proposed
extension of the NRG. However, a free electron wire with
constant density of states (described by the standard
fermionic NRG at s = 0) is equivalent to a free bosonic
bath with s = −1 due to the bosonization mapping, so
that one may believe that the marginal situation could be
tackled using some clever variant of the bosonic NRG. In
order to move forward, let us investigate with greater detail the problem of tunneling in LL. The fermionic Hamiltonian reads in terms of second quantized left and right
†
moving electron modes ψL,R (x) at linear position x in
the wire (omitting the electron spin for simplicity):

H =

†
†
†
†
dx[ivF ψL ∂x ψL − ivF ψR ∂x ψR + g2 ψR ψR ψL ψL ]
†
†
−Vbs [ψR ψL + ψL ψR ]|x=0

(2)

where vF is the Fermi velocity, g2 the short-range
Coulomb repulsion between left and right moving electrons, and Vbs the impurity backward scattering amplitude at the x = 0 location of the defect (forward impurity scattering and g4 interaction within a given Fermi
point do not affect the physics, and were discarded.). The
presence of the interaction term g2 clearly prevents a direct fermionic NRG solution of the model, which requires
Fermi liquid leads. Yet, one can use the exact bosonization mapping5–7 to re-express the electronic variables in
terms of non-interacting collective charge density excitations Φ(x) and conjugate field Π(x). After standard
manipulations5–7,15 one obtains
H=

√
dx
[Π(x)]2 + [∂x Φ(x)]2 − v ⋆ cos[ KΦ(x = 0)]⋆
⋆
⋆
8π
(3)

where normal ordering of the cosine operator, which will
be crucial for the rigorous formulation of the NRG algorithm, has been emphasized. We have also introduced
a small backscattering energy scale v ∝ Vbs and the important Luttinger liquid parameter K = [(1 − g2 )/(1 +
g2 )]1/2 ≤ 1, into which all interaction effects have been
encapsulated.
Let us now present how the bosonic NRG25 can be
tailored to address the impurity model (3), which has
the form of a boundary Sine Gordon Hamiltonian. The
derivation of the “star”-NRG follows the usual procedure19,25 by considering the equivalent energy representation in terms of a continuum of canonical bosons a† :
ǫ
ωc
√
dǫ ǫ a† aǫ − v ⋆ cos[ KΦ]⋆ ,
H=
(4)
⋆
⋆
ǫ
0

Φ ≡ Φ(x = 0) =

ωc

√
2

dǫ
0

a† + aǫ
ǫ
√ .
ǫ

(5)

The bosonic fields are then decomposed in Fourier modes
(p ∈ Z, n ∈ N) on each interval ωn+1 < ǫ < ωn of width
dn = (1 − Λ−1 )Λ−n :
a† =
ǫ
n,p

ei2πpǫ/dn †
√
an,p .
dn

(6)

The first NRG approximation consists in neglecting all
p = 0 modes, keeping only the operators a† ≡ a† (this
n
n,0
step becomes exact in the Λ → 1 limit19 ). This leads to
the “star”-Hamiltonian:
+∞

ξn a† an
n

HS =
n=0

√ +∞ γn †
√ (an + an )
− v cos
K
π
n=0
⋆
⋆

⋆
⋆

(7)

with the “impurity” coupling strength already given in
2
Eq. (1) by γn = 2π log(Λ) in the marginal case s = −1.
The typical energy ξn in each shell is defined by:
ξn =

1
2
γn

ωn

dω ω J(ω) =
ωn+1

1 − Λ−1
ωc Λ−n .
log(Λ)

(8)

As a benchmark of the discretization for the marginal
case s = −1, one can easily compute from (7) the resulting approximation for the original Green’s function:
0
GΦ,Λ (iω)

4
=
1 − Λ−1

+∞
n=0

ωc Λ−n
ω2 +

1−Λ−1
log(Λ)

2

(9)
2
ωc Λ−2n

which can be checked to converge exponentially fast at
2π
0
ω ≪ ωc to the exact result GΦ (iω) = |ω| even for Λ = 2
(we keep this standard value from now on). However, despite the clear exponential decay of the energies (8), the
non-decreasing value of the couplings γn implies a violation of scale separation on all shells, and prevents the
solution by iterative diagonalization of Hamiltonian (7).
The first key idea in successfully constructing the
marginal bosonic NRG is to assume that the energy spectrum is also bounded from below:
2π
Θ(ωc − ω)Θ(ω − ωmin ).
(10)
J(ω) =
ω

3
Clearly both the energies ǫn and the couplings γn are not
modified by this choice (γn still do not decay), and they
are just cut off for n > nmin , with ωmin = ωc Λ−nmin ,
so that nothing seems gained naively. We can however
try to pursue with the second step of the standard NRG
procedure, which amounts to the exact mapping on the
Wilson chain.19,25 This simple tridiagonalization procedure of Hamiltonian (7) leads to the following “chain”
form in terms of new canonical bosons b† :
n
+∞

ǫn b† bn + tn (b† bn+1 + b† b† )
n
n
n+1 n

HC =
n=0

η0 K †
(b0 + b0 )
π

−v ⋆ cos
⋆

(11)

⋆
⋆

with the parameter η0 = 2π log(ωc /ωmin ). Clearly, the
impurity part of the chain Hamiltonian (11) breaks down
for ωmin → 0, owing to the divergence of η0 , but one can
check numerically that the construction is valid for nonzero ωmin . The on-site energies ǫn and hoppings tn of the
Wilson chain can indeed be obtained by numerical tridiagonalization of Eq. (7). For the value ωmin = 10−5 of the
lower cutoff, these are plotted together with the star parameters on Fig. 1. The exponential decay of both chain

must also be normal ordered. For a generic operator
O = ⋆ cos[α(b† +b0 )] ⋆ this reads O = cos(αb† ) cos(αb0 )−
⋆
⋆
0
0
sin(αb† ) sin(αb0 ). Using the Fock states m of the
0
bosonic creation operator b† on the initial site n = 0
0
of the Wilson chain, one obtains the matrix elements:
Min(m,p)

mOp =

m!p! Re

k=0

(iα)m+p−2k
. (12)
(m − k)!(p − k)!k!

The construction of the impurity term in (11) proceeds
by a truncation of the infinite Fock space on the initial
Wilson site limited to states with occupation number less
than a given N0 , and use of the matrix elements (12).
Typically N0 = 150 ensures a good representation of the
Hamiltonian. Each further site n > 0 of the chain is described by a basis of Nb states25 (we take Nb = 12 here).
At increasing n, the growing size of the total Hilbert
space becomes rapidly unmanageable, and a truncation
to Ntrunc states is required (this approximation is common to all NRG schemes19 ). Typically Ntrunc = 800 was
employed in all further computations, and we also set
ωc = 1 as the basic energy unit.
60
Perturbation theory
Exact (K = 1/2)

100

100

γn
ξn

10−2

NRG

Re[GΦ (ω)]

10−2

10−4

40

ǫn
tn

10−4

20
0

5

n

10

15

0

5

n

10

15

FIG. 1: (color online) Left panel: parameters ξn and γn of
the star-NRG as a function of n for 0 ≤ n ≤ nmin = 16; the
coupling γn does not decay and violates scale separation on all
shells. Right panel: parameters ǫn and tn of the chain-NRG;
scale separation is only broken on the first shell, as seen by the
initial increase of both parameters, before further exponential
decay (shown by dotted lines as guides to the eye).

parameters ǫn and tn that we discover here is clearly a remarkable surprise, that enables the extension of the NRG
to the marginal situation s = −1. This crucial feature
comes at a small price, seen by the first increase of the
chain parameters from site n = 0 to site n = 1. Thus the
maximal violation of scale separation in the star NRG
presents a small remanence in the chain NRG, limited
only to the first shell. Interestingly, the initial jump of
the parameters is just proportional to log(ωc /ωmin), so
that the lower cutoff ωmin can be decreased on exponential scales without paying a huge numerical cost.
A last difficulty due to the unusual form of the impurity Hamiltonian (11) must be addressed. In the
standard NRG,19,25 only linear to quadrilinear operators are present in the Hamiltonian. However, the central physical role played by the phase factor eiΦ leads
to a cosine term at the impurity site, hence to an operator of infinite order, which by the bosonization rules7

0 −4
10

10−3

ω

10−1

100

FIG. 2: (color online). Bosonic correlation function Re[GΦ (ω)]
at real frequency ω for the LL parameter K = 1/2 comparing
(bottom to top) the NRG to the exact result (13), and to
the strong and weak interaction perturbation theory given
respectively by (14) and (15) (these two expressions are by
accident equivalent for K = 1/2, but nonetheless not exact).

In contrast to more complex extensions of impurity
models with dynamical degrees of freedom (such as the
Kondo model in a Luttinger liquid13,14 ), the present impurity problem benefits from several known limits, that
allow to benchmark our numerics. For instance there exists an exact solution for the dynamical conductance5–7,15
(in units of e2 /h) at K = 1/2:
Gexact (ω) =

Kω
1
Ω
ω
Re[GΦ (ω)] = −
atan
2π
2 2ω
Ω

(13)

where Ω = eγ v 2 , with Euler’s constant γ, is the crossover
energy at which the impurity cuts the chain (for K =
1/2). Perturbation theory works also at strong interaction K ≪ 1, in which case the self-consistent harmonic
approximation applies:20
Gstrong (ω) =

1
ω2
2 ω 2 + (Ω⋆ )2

(14)

4
100
10−1
G(ω) 10

−2

NRG K = 0.1 → 0.9
Weak K = 0.6 → 0.9
Weak/Strong K = 0.5
Strong K = 0.1 → 0.4

10−3
10−4
10−4

10−3

100

10−1

ω

FIG. 3: (color online).
Dynamical conductance G(ω)
in units of e2 /h for several values of the LL parameter
K = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 obtained by the
NRG (symbols, bottom to top). Comparison is made for
K ≤ 0.5 to the strong interaction and for K ≥ 0.5 to the weak
interaction limits. Perturbative methods cannot describe accurately the NRG solution beyond their applicability range.
1

with Ω⋆ = 2πKv 1−K the crossover scale. Finally, the
limit of weak interaction 1 − K ≪ 1 is also known from
several approaches:5,6,15,17,21–23
2

Gweak (ω) =

Kω K −2
2

2

ω K −2 + (Ω⋆ ) K −2

.

(15)

Fig. 2 compares our NRG data for K = 1/2 with the ex-

1

2

3
4
5

6

7
8

9
10
11

12

13

14
15

16
17
18

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To conclude, we have established an extension of the
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We thank L. Glazman and V. Meden for useful discussions. Financial support from ERC Advanced Grant
MolNanoSpin No. 226558 is also gratefuly acknowledged.

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