Numerical Renormalization Group for Impurity Quantum Phase Transitions: Structure of Critical Fixed Points
ABSTRACT The numerical renormalization group method is used to investigate zero temperature phase transitions in quantum impurity systems, in particular in the particlehole symmetric softgap Anderson model. The model displays two stable phases whose fixed points can be built up of noninteracting singleparticle states. In contrast, the quantum phase transitions turn out to be described by interacting fixed points, and their excitations cannot be described in terms of free particles. We show that the structure of the manybody spectrum of these critical fixed points can be understood using renormalized perturbation theory close to certain values of the bath exponents which play the role of critical dimensions. Contact is made with perturbative renormalization group calculations for the softgap Anderson and Kondo models. A complete description of the quantum critical manyparticle spectra is achieved using suitable marginal operators; technically this can be understood as epsilonexpansion for full manybody spectra.

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ABSTRACT: We consider a quantum impurity model in which a bosonic impurity level is coupled to a noninteracting bosonic bath, with the bosons at the impurity site subject to a local Coulomb repulsion U. Numerical renormalization group calculations for this bosonic singleimpurity Anderson model reveal a zerotemperature phase diagram where Mott phases with reduced charge fluctuations are separated from a BoseEinstein condensed phase by lines of quantum critical points. We discuss possible realizations of this model, such as atomic quantum dots in optical lattices. Furthermore, the bosonic singleimpurity Anderson model appears as an effective impurity model in a dynamical meanfield theory of the BoseHubbard model. Copyright EDP Sciences/Società Italiana di Fisica/SpringerVerlag 2007Physics of Condensed Matter 07/2006; 56(3):199203. · 1.28 Impact Factor
Page 1
arXiv:condmat/0506694v1 [condmat.strel] 27 Jun 2005
Numerical Renormalization Group for Impurity Quantum Phase Transitions:
Structure of Critical Fixed Points
HyunJung Lee∗, Ralf Bulla∗, and Matthias Vojta†
∗Theoretische Physik III, Elektronische Korrelationen und Magnetismus,
Institut f¨ ur Physik, Universit¨ at Augsburg, D86135 Augsburg, Germany and
†Institut f¨ ur Theorie der Kondensierten Materie,
Universit¨ at Karlsruhe, D76128 Karlsruhe, Germany
(Dated: February 2, 2008)
The numerical renormalization group method is used to investigate zero temperature phase tran
sitions in quantum impurity systems, in particular in the particlehole symmetric softgap Anderson
model. The model displays two stable phases whose fixed points can be built up of noninteracting
singleparticle states. In contrast, the quantum phase transitions turn out to be described by inter
acting fixed points, and their excitations cannot be described in terms of free particles. We show
that the structure of the manybody spectrum of these critical fixed points can be understood using
renormalized perturbation theory close to certain values of the bath exponents which play the role
of critical dimensions. Contact is made with perturbative renormalization group calculations for
the softgap Anderson and Kondo models. A complete description of the quantum critical many
particle spectra is achieved using suitable marginal operators; technically this can be understood as
epsilonexpansion for full manybody spectra.
I.INTRODUCTION
Zerotemperature phase transitions in quantum impu
rity models have recently attracted considerable interest
(for reviews see Refs. 1,2,3). These transitions can be
observed in systems where a zerodimensional boundary
with internal degrees of freedom (the impurity) interacts
with an extended bath of fermions or bosons. Examples
of impurity models with nontrivial phase transitions in
clude extensions of the Kondo model where one or two
magnetic impurities couple to fermionic baths1, the spin
boson model describing a twolevel system coupling to
a dissipative environment4,5, as well as socalled Bose
Fermi Kondo models for localized spins interacting with
both fermionic and bosonic baths. Impurity phase tran
sitions are of relevance for impurities in correlated bulk
systems (e.g. superconductors6), for multilevel impuri
ties like Fullerene molecules7, as well as for nanodevices
like coupled quantum dots8or point contacts under the
influence of dissipative noise9,10. In addition, impurity
phase transitions have been argued to describe aspects
of socalled local quantum criticality in correlated lat
tice systems. Here, the framework of dynamical mean
field theory is employed to map, e.g., the Kondo lattice
model onto a singleimpurity BoseFermi Kondo model
supplemented by selfconsistency conditions, for details
see Ref. 11.
Diverse techniques have been used to investigate impu
rity phase transitions, ranging from static and dynamic
largeN calculations12, conformal field theory3, pertur
bative renormalization group (RG)1,6,13and the local
moment approach14,15to various numerical methods. In
particular, significant progress has been made using the
numerical renormalization group (NRG) technique, origi
nally developed by Wilson for the Kondo problem16. The
NRG combines numerically exact diagonalization with
the idea of the renormalization group, where progres
sively smaller energy scales are treated in the course of
the calculation. NRG calculations are nonperturbative
and are able to access arbitrarily small energies and tem
peratures. Apart from static and dynamic observables,
the NRG provides information about the manybody ex
citation spectrum of the system at every stage of the RG
flow. Thus, it allows to identify fixed points through their
fingerprints in the level structure. A detailed understand
ing of the NRG levels is usually possible if the fixed point
can be described by noninteracting bosons or fermions
– this is the case for most stable fixed points of impurity
models, e.g., the strongcoupling (screened) fixed point
of a standard Kondo model. Intermediatecoupling fixed
points, usually being interacting, have a completely dif
ferent NRG level structure, i.e., smaller degeneracies and
nonequidistant levels. They cannot be cast into a free
particle description, with the remarkable exception of the
twochannel Kondo fixed point which is known to have
a representation in terms of free Majorana fermions17.
In general, the NRG fixedpoint spectrum at impurity
transitions is fully universal, apart from a nonuniversal
overall prefactor and discretization effects.
The purpose of this paper is to demonstrate that a
complete understanding of the NRG manybody spec
trum of critical fixed points is actually possible, by uti
lizing renormalized perturbation theory around a non
interacting fixed point. In the softgap Anderson model,
this approach can be employed near certain values of
the bath exponent which can be identified as critical di
mensions. Using the knowledge from perturbative RG
calculations, which yield the relevant coupling constants
Page 2
2
being parametrically small near the critical dimension,
we can construct the entire quantum critical manybody
spectrum from a freefermion model supplemented by a
small perturbation. In other words, we shall perform
epsilonexpansions to determine a complete manybody
spectrum (instead of certain renormalized couplings or
observables). Vice versa, our method allows to identify
relevant degrees of freedom and their marginal couplings
by carefully analyzing the NRG spectra near critical di
mensions of impurity quantum phase transitions.
The paper is organized as follows. In Sec. II we give a
brief introduction to the physics of the softgap Ander
son model and its quantum phase transitions. Sec. III
summarizes the recent results from perturbative RG for
both the softgap Anderson and Kondo models. Section
IV describes the numerical renormalization group (NRG)
approach which is used here to obtain information about
the structure of the quantum critical points. The main
part of the paper is Sec. V in which we discuss (i) the
numerical data for the structure of the quantum critical
points and (ii) the analytical description of these interact
ing fixed points close to the upper (lower) critical dimen
sion r = 0 (r = 1/2). The main conclusions of the paper
are summarized in Sec. VI where we also mention other
problems for which an analysis of the type presented here
might be useful.
II.SOFTGAP ANDERSON MODEL
The Hamiltonian of the softgap Anderson model12is
given by:
?
+
εkc†
H = εf
σ
f†
σfσ+ Uf†
↑f↑f†
?
↓f↓
?
kσ
kσckσ+ V
kσ
?f†
σckσ+ c†
kσfσ
?. (1)
This model describes the coupling of electronic degrees
of freedom at an impurity site (operators f(†)
fermionic bath (operators c(†)
The felectrons are subject to a local Coulomb repul
sion U, while the fermionic bath consists of a non
interacting conduction band with dispersion εk.
model eq. (1) has the same form as the singleimpurity
Anderson model18but for the softgap model we require
that the hybridization function?∆(ω) = πV2?
an exponent r > 0. This translates into a local con
duction band density of states ρ(ω) = ρ0ωrat low en
ergies.In the numerical calculations we used a band
where this power law extends over the whole band
width D, i.e., from ω = −D/2 to +D/2, and we have
ρ0 = (2/D)r+1(r + 1)/2. However, the universal low
temperature physics to be discussed in the following does
not depend on the details of the density of states at high
energies, and consequently we will use the lowenergy
prefactor of the density of states, ρ0, to represent the
σ ) to a
kσ) via a hybridization V .
The
kδ(ω−εk)
has a softgap at the Fermi level,?∆(ω) = ∆ωr, with
0.000.250.500.75
r
0.00
0.02
0.04
0.06
∆
symmetric case
asymmetric case
strong−coupling
local−moment
FIG. 1:
model in the ph symmetric case (solid line, U = 10−3, εf =
−0.5 · 10−3, conduction band cutoff at 1 and 1) and the ph
asymmetric case (dashed line, εf = −0.4 · 10−3); ∆ measures
the hybridization strength,? ∆(ω) = ∆ωr.
dimensionful energy scale of the problem. Assuming a
particlehole symmetric band, the model (1) is particle
hole symmetric for ǫf= −U/2.
The softgap Anderson model (1) with 0 < r < ∞
displays a very rich behaviour, in particular a continuous
transition between a localmoment (LM) and a strong
coupling (SC) phase.Figure 1 shows a typical phase
diagram for the softgap Anderson model. In the particle
hole (ph) symmetric case (solid line) the critical coupling
∆cdiverges at r =1
(Refs. 19,20). No divergence occurs for ph asymmetry
(dashed line)19.
We now briefly describe the properties of the fixed
points in the softgap Anderson and Kondo models19.
Due to the powerlaw conduction band density of states,
already the stable LM and SC fixed points show non
trivial behaviour19,20. The LM phase has the properties
of a free spin
2with residual entropy Simp = kBln2
and lowtemperature impurity susceptibility χimp =
1/(4kBT), but the leading corrections show rdependent
power laws. The ph symmetric SC fixed point has very
unusual properties, namely Simp = 2rkBln2, χimp =
r/(8kBT) for 0 < r <1
SC fixed point simply displays a completely screened mo
ment, Simp= Tχimp= 0. The impurity spectral function
follows a ωrpower law at both the LM and the asym
metric SC fixed point, whereas it diverges as ω−rat the
symmetric SC fixed point – this “peak” can be viewed
as a generalization of the Kondo resonance in the stan
dard case (r = 0), and scaling of this peak is observed
upon approaching the SCLM phase boundary14,20. At
the critical point, nontrivial behaviour corresponding to
a fractional moment can be observed: Simp= kBCS(r),
χimp = Cχ(r)/(kBT) with CS, Cχ being universal func
tions of r (see Refs. 19,21). The spectral functions at
the quantum critical points display an ω−rpower law
(for r < 1) with a remarkable “pinning” of the critical
exponent.
T = 0 phase diagram for the softgap Anderson
2, and no screening occurs for r >1
2
1
2. In contrast, the ph asymmetric
Page 3
3
III.RESULTS FROM PERTURBATIVE RG
The Anderson model (1) is equivalent to a Kondo
model when charge fluctuations on the impurity site
are negligible. The Hamiltonian for the softgap Kondo
model can be written as
H = J?S · ? s0+
?
kσ
εkc†
kσckσ
(2)
where ? s(0) =?
duction electron density of states follows a power law
ρ(ω) = ρ0ωras above.
kk′σσ′c†
kσ? σσσ′ck′σ′/2 is the conduction
electron spin at the impurity site r = 0, and the con
A. RG near r = 0
For small values of the density of states exponent r,
the phase transition in the pseudogap Kondo model can
be accessed from the weakcoupling limit, using a gener
alization of Anderson’s poor man’s scaling. Power count
ing about the local moment fixed point (LM) shows that
dim[J] = −r, i.e., the Kondo coupling is marginal for
r = 0. We introduce a renormalized dimensionless Kondo
coupling j according to
ρ0J = µ−rj (3)
where µ plays the role of a UV cutoff. The flow of the
renormalized Kondo coupling j is given by the beta func
tion
β(j) = rj − j2+ O(j3).(4)
For r > 0 there is a stable fixed point at j∗= 0 cor
responding to the localmoment phase (LM). An unsta
ble fixed point, controlling the transition to the strong
coupling phase, exists at
j∗= r,(5)
and the critical properties can be determined in a double
expansion in r and j6. Ph asymmetry is irrelevant, i.e.,
a potential scattering term E scales to zero according to
β(e) = re (where ρ0E = µ−re), thus the above expansion
captures the ph symmetric critical fixed point (SCR). As
the dynamical exponent ν, 1/ν = r + O(r2), diverges as
r → 0+, r = 0 plays the role of a lowercritical dimension
of the transition under consideration.
B. RG near r = 1/2
For r near 1/2 the ph symmetric critical fixed point
moves to strong Kondo coupling, and the language
of the ph symmetric Anderson model becomes more
appropriate13. First, the conduction electrons are inte
grated out exactly, yielding a selfenergy Σf= V2Gc0for
the f electrons, where Gc0 is the bare conduction elec
tron Green’s function at the impurity location. In the
lowenergy limit the f electron propagator is then given
by
Gf(iωn)−1= iωn− iA0sgn(ωn)ωnr
where the ωnrselfenergy term dominates for r < 1,
and the prefactor A0is
(6)
A0=πρ0V2
cosπr
2
. (7)
Equation (6) describes the physics of a noninteracting
resonant level model with a softgap density of states.
Interestingly, the impurity spin is not fully screened for
r > 0, and the residual entropy is 2rln2. This precisely
corresponds to the symmetric strongcoupling (SC) phase
of the softgap Anderson and Kondo model19.
Dimensional analysis, using dim[f] = (1 − r)/2 [where
f represents the dressed fermion according to eq. (6)],
now shows that the interaction term U of the Anderson
model scales as dim[U] = 2r − 1, i.e., it is marginal at
r = 1/2. This suggests a perturbative expansion in U
around the SC fixed point. We introduce a dimensionless
renormalized onsite interaction u via
U = µ2r−1A2
0u.(8)
The beta funcion receives the lowest nontrivial contri
bution at twoloop order and reads13
β(u) = (1 − 2r)u −3(π − 2ln4)
For r < 1/2 a noninteracting stable fixed point is at u∗=
0 – this is the symmetric strongcoupling fixed point, it
becomes unstable for r > 1/2. Additionally, for r < 1/2
there is a pair of critical fixed points (SCR, SCR’) located
at u∗2= π2(1 − 2r)/[3(π − 2ln4)], i.e.,
u∗= ±4.22
These fixed points describe the transition between an un
screened (spin or charge) moment phase and the symmet
ric strongcoupling phase13.
Summarizing, both (4) and (9) capture the same criti
cal SCR fixed point. This fixed point can be accessed ei
ther by an expansion around the weakcoupling LM fixed
point, i.e., around the decoupled impurity limit, valid for
r ≪ 1, or by an expansion around the strongcoupling SC
fixed point, i.e., around a noninteracting resonantlevel
(or Anderson) impurity, and this expansion is valid for
1/2 − r ≪ 1.
π2
u3+ O(u5). (9)
?
1/2 − r.(10)
IV.NUMERICAL RENORMALIZATION
GROUP
Here we describe the numerical renormalization group
method, suitably extended to handle nonconstant cou
plings?∆(ω) (see Refs. 19,20,22). This method allows a
Page 4
4
nonperturbative calculation of the manyparticle spec
trum and physical properties in the whole parameter
regime of the model eq. (1), in particular in the low
temperature limit, so that the structure of the quantum
critical points is accessible, as discussed in Sec. V.
A detailed discussion of how the NRG can be applied
to the softgap Anderson model can be found in Refs. 19,
20,22. Here we focus on those aspects of the approach
necessary to understand how information on the fixed
points can be extracted.
The NRG is based on a logarithmic discretization of
the energy axis, i.e. one introduces a parameter Λ and
divides the energy axis into intervals [−Λ−n,−Λ−(n+1)]
and [Λ−(n+1),Λ−n] for n = 0,1,....,∞ (see Refs. 16,23).
With some further manipulations the original model can
be mapped onto a semiinfinite chain with the Hamilto
nian
?
?
π
σ
∞
?
H = εf
σ
f†
σfσ+ Uf†
↑f↑f†
↓f↓
+
ξ0
?
?
?
f†
σc0σ+ c†
0σfσ
?
+
σn=0
εnc†
nσcnσ+ tn
?
c†
nσcn+1σ+ c†
n+1σcnσ
??
,
(11)
with
ξ0=
?1
−1
dω?∆(ω) .(12)
For a ph symmetric conduction band, all the onsite en
ergies εnvanish. If, in addition, the power law in?∆(ω)
act expression for the hopping matrix elements tncan be
given20:
extends up to a hard cutoff ωc(we set ωc= 1), an ex
tn = Λ−n/2r + 1
?
for even n and
r + 2
1 − Λ−(r+2)
1 − Λ−(r+1)
?
1 − Λ−(n+r+1)?
1 − Λ−(2n+r+3)?−1/2
×1 − Λ−(2n+r+1)?−1/2?
(13)
tn = Λ−(n+r)/2r + 1
?
for odd n. The semiinfinite chain is solved iteratively by
starting from the impurity and successively adding chain
sites. As the coupling tn between two adjacent sites n
and n +1 decreases as Λ−n/2for large n, the lowenergy
states of the chain with n + 1 sites are generally deter
mined by a comparatively small number Nsof states close
to the ground state of the nsite system. In practice one
retains only these Nsstates from the nsite chain to set
r + 2
1 − Λ−(r+2)
1 − Λ−(r+1)
?
1 − Λ−(n+1)?
×1 − Λ−(2n+r+1)?−1/2?
1 − Λ−(2n+r+3)?−1/2
(14)
050 100
N
0.0
1.0
2.0
3.0
Λ
N/2EN
∆<∆ c
local moment
0 50100
N
∆=∆c
quantum critical
0 50 100
N
∆>∆c
strong coupling
ab
c
FIG. 2:
citations obtained from the numerical renormalization group
for the three different fixed points of the ph symmetric soft
gap Anderson model (exponent r = 0.4). N is the number
of iterations of the NRG procedure, Λ the NRG discretiza
tion parameter. Solid lines: (Q,S) = (1,0), dashed lines:
(Q,S) = (0,1/2).
Flow diagrams for the lowenergy manybody ex
up the Hilbert space for the n+1 site chain, thus prevent
ing the usual exponential growth of the Hilbert space as
n increases. Eventually, after nNRG sites have been in
cluded in the calculation, addition of another site will not
change significantly the spectrum of manyparticle exci
tations; the spectrum is very close to that of the fixed
point, and the calculation may be terminated.
In this way, the NRG iteration gives the manyparticle
energies EN for a sequence of Hamiltonians HN which
correspond to the Hamiltonian eq. (11) by the replace
ment
∞
?
σn=0
−→
N−1
?
σn=0
.(15)
An example for the dependence of the lowest lying en
ergy levels on the chain length (the flow diagram) is
given in Fig. 2c for the softgap Anderson model with
r = 0.4, D = 2, U/D = 10−3and ∆ = 0.0075; the
parameters used for the NRG calculations are Λ = 2
and Ns= 300. The states are labelled by the quantum
numbers Q (which characterizes the number of particles
measured relative to halffilling), and the total spin, S
[solid lines in Fig. 2 are for (Q,S) = (1,0), dashed lines
for (Q,S) = (0,1/2)]. As mentioned above, the energy
scale is reduced in each step by a factor Λ1/2. To allow
for a direct comparison of the energies for different chain
lengths, it is thus convenient to plot ΛN/2EN instead of
the eigenvalues EN of the Nsite chain directly. Note
that here and in the following we use the convention that
the energies shown in the flow diagrams are proportional
to the bandwidth D.
As is apparent from Fig. 2c, the properties of the sys
tem in this case do not change further for chain lengths
nNRG > 120. Without going into details now, one can
Page 5
5
0.0 0.10.20.3 0.40.5
r
0.0
1.0
2.0
3.0
4.0
Λ
N/2EN
QCP
LM
SC
FIG. 3: Dependence of the manyparticle spectra for the three
fixed points of the ph symmetric softgap Anderson model on
the exponent r: SC (black dotdashed lines), LM (blue dashed
lines), and the (symmetric) quantum critical point (red solid
lines). The data are shown for the subspace Q = 1 and S = 0
only.
show that the distribution of energy levels for N > 120
in Fig. 2c is characteristic of the SC phase of the model
(see Sec. V).
If by contrast we choose instead a value of ∆ = 0.006,
we obtain the flow diagram shown in Fig. 2a. Here it
is evident that the fixed point level structure is entirely
different from the SC solution, and indeed this particu
lar pattern is now characteristic of the LM phase of the
model. We can thus conclude, simply from inspection
of the two flow diagrams, that the critical ∆cseparating
the SC and LM phases of the softgap Anderson model
for the model parameters specified, lies in the interval
[0.006,0.0075].
Tuning the value of ∆ to the critical value ∆c, results
in the flow diagram of Fig. 2b. Apparently, the structure
of the fixed point at ∆c neither coincides with the SC
nor with the LM fixed point. It is clear that it cannot be
simply constructed from singleparticle states as for the
SC and LM fixed points. An important observation is
that certain degeneracies present in the SC or LM fixed
points are lifted at the QCP. As shown in the following
section, a further hint on the structure of the QCPs is
given by the dependence of their manyparticle spectra
on the bath exponent r.
V. STRUCTURE OF THE QUANTUM
CRITICAL POINTS
In Fig. 3, the manyparticle spectra of the three fixed
points (SC: dotdashed lines, LM: dashed lines, and QCP:
solid lines) of the symmetric softgap model are plotted
as functions of the exponent r (for a similar figure, see
Fig. 13 in Ref. 19). The data are shown for an odd num
ber of sites only and we select the lowest lying energy
levels for the subspace Q = 1 and S = 0.
10
−3
10
−2
0.5−r
10
−2
10
−1
10
0
∆E
∆E0:0.501
∆E1:0.501
∆E2:0.636
∆E3:0.489
∆E4:0.543
∆E5:0.276
FIG. 4: Difference ∆E between the energy levels of QCP and
SC fixed points close to r = 1/2 in a doublelogarithmic plot.
The inset shows the values of the exponents obtained from a
fit to the data points.
As usual, the fixed point structure of the strong cou
pling and local moment phases can be easily constructed
from the singleparticle states of a free conduction elec
tron chain. This is discussed in more detail later. Let us
now turn to the line of quantum critical points. What
information can be extracted from Fig. 3 to understand
the structure of these fixed points?
First we observe that the levels of the quantum critical
points, EN,QCP(r), approach the levels of the LM (SC)
fixed points in the limit r → 0 (r → 1/2):
lim
r→0{EN,QCP(r)} = {EN,LM(r = 0)} ,
lim
r→1/2{EN,QCP(r)} = {EN,SC(r = 1/2)} , (16)
where {...} denotes the whole set of manyparticle states.
For r→
EN,QCP(r) deviates linearly from the levels of the LM
fixed point, while the deviation from the SC levels is pro
portional to
in Fig. 4 where we plot a selection of energy differences
∆E between levels of QCP and SC fixed points close to
r = 1/2. The inset shows the values of the exponents
obtained from a fit to the data points. For some levels,
there are significant deviations from the exponent 1/2.
This is because the correct exponent is only obtained in
the limit r → 1/2 (the QCP levels have been obtained
only up to r = 0.4985).
Note that the behaviour of the manyparticle levels
close to r = 1/2 has direct consequences for physical
properties at the QCP; the critical exponent of the local
susceptibility at the QCP, for example, shows a square
root dependence on (1/2−r) close to r = 1/2, see Ref. 19.
In both limits, r → 0 and r → 1/2, we observe that de
generacies due to the combination of singleparticle lev
els, present at the LM and SC fixed points, are lifted at
the quantum critical fixed points as soon as one is mov
ing away from r = 0 and r = 1/2, respectively. This
already suggests that the quantum critical point is inter
0, each individual manyparticle level
?1/2− r for r → 1/2. This is illustrated
Page 6
6
acting and cannot be constructed from noninteracting
singleparticle states.
In the remainder of the paper we want to show how to
connect this information from NRG to the perturbative
RG of Sec. III. We know that the critical fixed point
can be accessed via two different epsilonexpansions6,13
near the two critical dimensions, and, combined with
renormalized perturbation theory, these expansions can
be used to evaluate various observables near criticality.
Here, we will employ this concept to perform renormal
ized perturbation theory for the entire manybody spec
trum at the critical fixed point. To do so, we will start
from the manybody spectrum of one of the trivial fixed
points, i.e., LM near r = 0 and SC near r = 1/2, and eval
uate corrections to it in lowestorder perturbation theory.
This will be done within the NRG concept working di
rectly with the discrete manybody spectra correspond
ing to a finite NRG chain (which is diagonalized numer
ically). As the relevant energy scale of the spectra de
creases as Λ−n/2along the NRG iteration, the strength
of the perturbation has to be scaled as well, as the goal
is to capture a fixed point of the NRG method. This scal
ing of the perturbation follows precisely from its scaling
dimension – the perturbation marginal at the value of r
corresponding to the critical dimension. With the proper
scaling, the operator which we use to capture the differ
ence between the freefermion and critical fixed points
becomes exactly marginal [see eqs. (21) and (37) below].
A.Perturbation theory close to r = 0
Let us now describe in detail the analysis of the devia
tion of the QCP levels from the LM levels close to r = 0
(the case r = 1/2 is discussed in Sec. VB). An effec
tive description of the LM fixed point is given by a finite
chain with the impurity decoupled from the conduction
electron part, see Fig. 5.
??? ???
??? ???
decoupled impurity
...
free conduction electron chain
012N
FIG. 5: The spectrum of the LM fixed point is described
by the impurity decoupled from the free conduction electron
chain.
The conduction electron part of the effective Hamilto
nian is given by
Hc,N=
N−1
?
σn=0
tn
?
c†
nσcn+1σ+ c†
n+1σcnσ
?
. (17)
As usual, the structure of the fixed point spectra depends
on whether the total number of sites is even or odd. To
simplify the discussion in the following, we only consider
a total odd number of sites (the flow diagrams of Fig. 2
are all calculated for this case). For the LM fixed point,
this means that the number of sites, N + 1, of the free
conduction electron chain is even, so N in eq. (17) is odd.
The singleparticle spectrum of the free chain with an
even number of sites, corresponding to the diagonalized
Hamiltonian
?
is sketched in Fig. 6. (Note that the ǫphave to be eval
uated numerically.)
¯Hc,N=
σp
ǫpξ†
pσξpσ, (18)
p=−3
p=−1
p=1
p=3
...
...
ε
ε
ε
ε−3
1
3
0
−1
FIG. 6: Singleparticle spectrum of the free conduction elec
tron chain eq. (18). The ground state is given by all the levels
with p < 0 filled.
As we assume ph symmetry, the positions of the
singleparticle levels are symmetric with respect to 0 with
ǫp= −ǫ−p , p = 1,3,...,N ,(19)
and
?
p
≡
p=N
?
p=−N, p odd
.(20)
Note that an equally spaced spectrum of singleparticle
levels is only recovered in the limit Λ → 1 (see Fig. 6 in
Ref. 17) for the case r = 0.
The RG analysis of Sec. III tells us that the critical
fixed point is perturbative accessible from the LM one
using a Kondotype coupling as perturbation. We thus
focus on the operator
H′
N= α(r)f(N)?Simp·? s0, (21)
with the goal to calculate the manybody spectrum of
the critical fixed point via perturbation theory in H′
small r. The function α(r) contains the fixedpoint value
of the Kondotype coupling, and f(N) will be chosen
such that H′
Nfor
Nis exactly marginal, i.e., the effect of H′
N
Page 7
7
on the manyparticle energies decreases as Λ−N/2which
is the same N dependence which governs the scaling of
the manyparticle spectrum itself. The scaling analysis
of Sec. III, eqs. (3) and (5), suggests a parametrization
of the coupling as
α(r) =µ−r
ρ0
αr,(22)
where ρ0is the prefactor in the density of states, and µ
is a scale of order of the bandwidth – such a factor is re
quired here to make α a dimensionless parameter. Thus,
the strength of the perturbation increases linearly with
r at small r (where µ−r/ρ0= D + O(r) for a featureless
ωrdensity of states).
The qualitative influence of the operator?Simp· ? s0 on
the manyparticle states has been discussed in general in
Ref. 19 for finite r and in Refs. 16,23 for r = 0. Whereas
an antiferromagnetic exchange coupling is marginally rel
evant in the gapless case (r = 0), it turns out to be
irrelevant for finite r, see Ref. 19. This is of course con
sistent with the scaling analysis of Sec. III: the operator
(21) simply represents a Kondo coupling, with a treelevel
scaling dimension of dim[J] = −r. A detailed analysis of
the Ndependence of the operator?Simp·? s0shows that it
decreases as Λ−Nr/2Λ−N/2= Λ−N(r+1)/2with increasing
N. Consequently, we have to choose
f(N) = ΛNr/2. (23)
This result also directly follows from dim[J] = −r: As
the NRG discretization yields a decrease of the running
energy scale of Λ−N/2, the?Simp·? s0term in H′
as Λ−Nr/2. The function f(N) is now simply chosen to
compensate this effect; using eq. (23) the operator H′
becomes exactly marginal.
Now we turn to a discussion of the manybody spec
trum. The relevant ground state of the effective model
for the LM fixed point consists of the filled impurity level
(with one electron with either spin ↑ or ↓) and all the con
duction electron states with p < 0 filled with both ↑ and
↓, as shown in Fig. 6. Let us now focus on excitations
with energy ǫ1+ǫ3measured with respect to the ground
state. Figure 7 shows one such excitation; in this case,
one electron with spin ↓ is removed from the p = −3
level and one electron with spin ↓ is added to the p = 1
level. The impurity level is assumed to be filled with an
electron with spin ↑, so the resulting state has Q = 0 and
Sz= +1/2. In total, there are 32 states with excitation
energy ǫ1+ ǫ3. These states can be classified using the
quantum numbers Q, S, and Sz.
Here we consider only the states with quantum num
bers Q = 0, S = 1/2, and Sz = 1/2 (with excitation
energy ǫ1+ ǫ3) which form a fourdimensional subspace.
As the state shown in Fig. 7 is not an eigenstate of the to
tal spin S, we have to form proper linear combinations to
obtain a basis for this subspace; this basis can be written
in the form:
1
√2f†
N(21) scales
N
ψ1? =
↑
?
ξ†
1↑ξ−3↑+ ξ†
1↓ξ−3↓
?
ψ0?
...
ε
...
−5
−1
−3
1
3
f
FIG. 7: One possible excitation with energy ǫ1+ǫ3 and quan
tum numbers Q = 0 and Sz = +1/2.
ψ2? =
?
√2f†
?
1
√6f†
1
↑
?
?
ξ†
ξ†
1↑ξ−3↑− ξ†
1↓ξ−3↓
?
ψ0?
?
+
2
√6f†
↓ξ†
1↑ξ−3↓
?
ψ0?
ψ3? =
↑
3↑ξ−1↑+ ξ†
?
3↓ξ−1↓
?
ψ4? =
1
√6f†
↑
ξ†
3↑ξ−1↑− ξ†
3↓ξ−1↓
+
2
√6f†
↓ξ†
3↑ξ−1↓
?
ψ0?
(24)
where the state ψ0? is given by the product of the ground
state of the conduction electron chain and the empty im
purity level:
??
The fourfold degeneracy of the subspace (Q = 0,
S = 1/2, Sz= 1/2) of the LM fixed point at energy ǫ1+ǫ3
is partially split for finite r in the spectrum of the quan
tum critical fixed point. Let us now calculate the influ
ence of the perturbation H′
concentrating on the splitting of the energy levels up to
first order. Degenerate perturbation theory requires the
calculation of the matrix
ψ0? =
p<0
ξ†
p↑ξ†
p↓0?cond
?
⊗ 0?imp.(25)
Non the states ψ1?,...ψ4?,
Wij= ?ψiH′
Nψj? , i,j = 1,...4 , (26)
and a subsequent calculation of the eigenvalues of {Wij}
gives the level splitting.
Details of the calculation of the matrix elements Wij
are given in Appendix A. The result is
{Wij} = α(r)f(N)
0
√3
4γ00
√3
4γ −1
0
2β
0
00
0
√3
4γ
00
√3
4γ −1
?α012+ α0−32?.
2β
,(27)
with γ =
The Ndependence of the coefficients α0p (which relate
?α012− α0−32?
and β =
Page 8
8
the operators c0σand ξpσ, see eq. (A8)) is given by
α0p2∝ Λ−Nr/2Λ−N/2, (28)
(see also Sec. III A in Ref. 19). Numerically we find that
γ = −0.1478· Λ−Nr/2Λ−N/2
β = 2.0249· Λ−Nr/2Λ−N/2,
where the prefactors depend on the exponent r and the
discretization parameter Λ (the quoted values are for r =
0.01 and Λ = 2.0). The matrix {Wij}r=0.01then takes
the form
(29)
{Wij}r=0.01= α(r = 0.01)Λ−N/2
−0.064 −1.013
0
×
0−0.06400
00
00−0.064
00−0.064 −1.013
.(30)
Diagonalization of this matrix gives the firstorder cor
rections to the energy levels
∆E1(r = 0.01) = ∆E3(r = 0.01)
= α(r = 0.01)Λ−N/2· (−1.0615)
∆E2(r = 0.01) = ∆E4(r = 0.01)
= α(r = 0.01)Λ−N/2· 0.0004
with
(31)
EN,QCP(r = 0.01,i) = EN,LM(r = 0.01,i)
+ ∆Ei(r = 0.01) ,(32)
(i = 1,...4). Apparently, the fourfold degeneracy of the
subspace (Q = 0, S = 1/2, Sz= 1/2) with energy ǫ1+ǫ3
is split in two levels which are both twofold degenerate.
We repeated this analysis for a couple of other sub
spaces and a list of the resulting matrices {Wij} and the
energy shifts ∆E is given in Appendix A.
Let us now proceed with the comparison of the pertur
bative results with the structure of the quantum critical
fixed point calculated from the NRG. For our specific
choice of the conduction band density of states, the rela
tion (22) yields α(r) = αrD for small r (where µr≈ 1).
Using the corresponding equations for the energy shifts in
Appendix A, we observe that a single parameter α must
be sufficient to describe the level shifts in all subspaces,
provided that the exponent r is small enough so that the
perturbative calculations are still valid. A numerical fit
gives α ≈ 1.03 for Λ = 2.0, (the Λdependence of α is
discussed later, see Fig. 9).
Figure 8 summarizes the NRG results together with
the perturbative analysis for exponents r close to 0. A
flow diagram of the lowest lying energy levels is shown
in Fig. 8a for a small value of the exponent, r = 0.03,
so that the levels of the QCP only slightly deviate from
0 50
N
100
0
2
EN Λ
N/2
r=0.03
QCP
LM
00.030.07
r
0
2
ENΛ
N/2
QCP
LM
pert.
a) b)
FIG. 8: a) Flow diagram of the lowest lying energy levels for
r = 0.03; dashed lines: flow to the LM fixed point; solid lines:
flow to the quantum critical fixed point. b) The deviation of
the QCP levels from the LM levels increases linearly with r.
This deviation together with the splitting of the energy levels
can be explained by the perturbative calculation (crosses) as
described in the text.
those of the LM fixed point. As discussed above, the
deviation of the QCP levels from the LM levels increases
linearly with r, see Fig. 8b. We indeed get a very good
agreement between the perturbative result (crosses) and
the NRGdata (lines) for exponents up to r ≈ 0.07. The
data shown here are for the subspaces (Q = 0, S = 1/2,
Sz = 1/2) and energy 2ǫ1 (the levels at ENΛN/2≈ 1,
see Appendix A1) and (Q = 0, S = 1/2, Sz= 1/2) and
energy ǫ1+ǫ3(the levels at ENΛN/2≈ 2, see the example
discussed in this section).
In the NRG, the continuum limit corresponds to the
limit Λ → 1, but due to the drastically increasing nu
merical effort upon reducing Λ, results for the contin
uum limit have to be obtained via extrapolation of NRG
data for Λ in, for example, the range 1.5 < Λ < 3.0.
Figure 9 shows the numerical results from the NRG cal
culation together with a linear fit to the data: α(Λ) =
0.985+ 0.045(Λ − 1.0). Taking into account the increas
ing error bars for smaller values of Λ, the extrapolated
value α(Λ → 1) ≈ 0.985 is in excellent agreement with
the result from the perturbative RG calculation, which
is directly for the continuum limit and gives α = 1.0.
B.Perturbation theory close to r = 1/2
To describe the deviation of the QCP levels from the
SC levels close to r = 1/2, we have to start from an effec
tive description of the SC fixed point. This is given by a
finite chain including the impurity site with the Coulomb
repulsion U = 0 at the impurity site and a hybridiza
tion¯V between impurity and the first conduction elec
tron site, see Fig. 10.
Note that the SC fixed point can also be described by
Page 9
9
1.0 1.52.0 2.53.0
Λ
0.0
0.5
1.0
1.5
α
FIG. 9: Dependence of the coupling parameter α on the NRG
discretization parameter Λ.The circles correspond to the
NRG data and the solid line is a linear fit to the data: α(Λ) =
0.985 + 0.045(Λ − 1.0).
the limit¯V → ∞ and finite U which means that impurity
and first conduction electron site are effectively removed
from the chain. This reduces the number of sites of the
chain by two and leads to exactly the same level structure
as including the impurity with U = 0. However, the
description with the impurity included (and U = 0) is
more suitable for the following analysis.
??????
??????
...
012N U=0
V
FIG. 10: The spectrum of the SC fixed point is described by
the noninteracting impurity coupled to the free conduction
electron chain.
The corresponding effective Hamiltonian is that of a
softgap Anderson model on a finite chain with N + 2
sites and εf = U = 0 (i.e., a ph symmetric resonant
level model).
Hsc,N=¯V
?
σ
?
f†
σc0σ+ c†
0σfσ
?
+ Hc,N,(33)
with Hc,N as in eq. (17).
As for the effective description of the LM fixed point,
the effective Hamiltonian is that of a free chain.
cussing, as above, on odd values of N, the total number
of sites of this chain, N + 2, is odd. The singleparticle
spectrum of the free chain with an odd number of sites,
corresponding to the diagonalized Hamiltonian
Fo
¯ Hsc,N=
?
σl
ǫlξ†
lσξlσ, (34)
is sketched in Fig. 11. As we assume ph symmetry, the
positions of the singleparticle levels are symmetric with
respect to 0 with
ǫ0= 0 , ǫl= −ǫ−l , l = 2,4,...,(N + 1) ,
and
(35)
?
l
≡
l=N+1
?
l=−(N+1), l even
. (36)
The ground state of the effective model for the SC fixed
point is fourfold degenerate, with all levels with l < 0
filled and the level l = 0 either empty, singly (↑ or ↓) or
doubly occupied.
...
...
l=−4
l=−2
l=2
l=4
ε
ε
ε
ε−4
−2
2
4
=0 l=0
0
ε
FIG. 11: Singleparticle spectrum of the free conduction elec
tron chain eq. (34). The ground state is fourfold degenerate
with all the levels with l < 0 filled and the level l = 0 either
empty, singly (↑ or ↓) or doubly occupied.
According to Sec. III the proper perturbation to ac
cess the critical fixed point from the SC one is an onsite
repulsion, thus we choose
?
(nfσ = f†
parametrized as
H′
N= β(r)¯f(N)nf↑−1
2
??
nf↓−1
2
?
, (37)
σfσ) with the strength of the perturbation
β(r) = µ2r−1ρ2
0¯V4β
?
1/2 − r .(38)
see Sec. III. Note that ρ2
tureless powerlaw density of states with bandwidth D.
The N dependence of the operator?nf↑−1
choose
0(r = 1/2) = 9/(2D3) for a fea
2
??nf↓−1
2
?
is given by Λ(r−1/2)NΛ−N/2= Λ(r−1)N, so we have to
¯f(N) = Λ(1/2−r)N. (39)
This again follows from the scaling analysis of Sec. III:
the onsite repulsion has scaling dimension dim[U] = 2r−
1. Thus the?nf↑−1
to make H′
2
??nf↓−1
2
?term in H′
N(37) scales
as ΛN(r−1/2), and¯f(N) (39) compensates this behavior
Nexactly marginal.
Page 10
10
We continue with analyzing the lowlying manybody
levels. Similar as above, we focus on one specific example,
these are excitations with energy 2ε2measured with re
spect to the ground state and quantum numbers Q = −1,
S = 0, and Sz= 0. This subspace is twodimensional and
the basis is given by
ψ1? = −ξ†
ψ2? =
0↑ξ†
?
0↓ξ−2↑ξ−2↓ψ0? ,
ξ†
1
√2
2↑ξ−2↑+ ξ†
2↓ξ−2↓
?
ψ0? , (40)
with
ψ0? =
?
l<0
ξ†
l↑ξ†
l↓0? . (41)
(Note that in this definition of ψ0?, the l = 0level is
empty.)
The twofold degeneracy of this subspace is lifted for
r < 1/2 in the spectrum of the quantum critical points.
The matrix Wij= ?ψiH′
Nψj? (i,j = 1,2) is given by
2 − 2κ + κ22√2κ
{Wij} = β(r)¯f(N)αf24
2√2κ 2 + κ2
,
(42)
with κ = αf02/αf22. The Ndependence of the coef
ficients αfl (which relate the operators fµand ξlµ, see
eq. (B4)) is given by
αfl2∝ Λ(r−1)N/2, (43)
Numerically we find that
αf22= 0.0366· (D/¯V )2Λ(r−1)N/2
αf02= 0.0930· (D/¯V )2Λ(r−1)N/2,(44)
where the prefactors depend on the exponent r and the
quoted value is for r = 0.499. The matrix {Wij}r=0.499
then takes the form
{Wij}r=0.499= β(r = 0.499)(D/¯V)4Λ−N/2
0.0094 0.011
×
0.0044 0.0094
, (45)
Diagonalization of this matrix gives the firstorder cor
rections to the energy levels
∆E1(r = 0.499) = β(r = 0.499)(D/¯V)4Λ−N/2· (−0.0023)
∆E2(r = 0.499) = β(r = 0.499)(D/¯V)4Λ−N/2· (0.018)
(46)
with
EN,QCP(r = 0.499,i) =
EN,SC(r = 0.499,i) + ∆Ei(r = 0.499) ,(47)
(i = 1,2). We repeated this analysis for a couple of other
subspaces and a list of the resulting matrices {Wij} and
the energy shifts ∆E is given in Appendix B.
The comparison of the perturbative results with the
numerical results from the NRG calculation is shown in
Fig. 12b. As for the case r ≈ 0 we observe that a single
parameter β is sufficient to describe the level shifts in
all subspaces, provided the exponent r is close enough
to r = 1/2 so that the perturbative calculations are
valid.For Λ = 2.0 we find β ≈ 70 and the Λ → 1
extrapolation results in β(Λ → 1) ≈ 73.0 ± 5.0 (the
error bars here are significantly larger as for the extra
polation of the coupling α). The results from perturba
tive RG, Sec. III, specifically eqs. (8) and (10), yield
β(r) = µ2r−1ρ2
Similar to Fig. 8 above, we show in Fig. 12a a flow
diagram for an exponent very close to 1/2, r = 0.4985,
so that the levels of the QCP only slightly deviate from
those of the SC levels. As discussed above, this devia
tion is proportional to
shown here are all for subspaces with (Q = −1, S = 0,
Sz= 0); the unperturbed energies E of these subspaces
are:
0¯V42π2u∗. This gives β = 83.3.
?1/2 − r, see Fig. 12b. The data
• E = 0: the levels at ENΛN/2≈ 0, see App. B2,
• E = ǫ2: the levels at ENΛN/2≈ 0.8, see App. B3,
• E = 2ǫ2: the levels at ENΛN/2≈ 1.6, see the ex
ample discussed in this section,
• E = ǫ4: the levels at ENΛN/2≈ 1.8, see App. B4,
• E = 3ǫ2: the levels at ENΛN/2≈ 2.4.
We again find a very good agreement between the per
turbative results (crosses) and the NRG data (lines).
Thus we can summarize that our renormalized pertur
bation theory for the NRG manybody spectrum works
well in the vicinity of both r = 0 and r = 1/2. In princi
ple, from the manybody spectrum (and suitable matrix
elements) all other observables like thermodynamic data
and dynamic correlation functions can be calculated. We
note that the convergence radius of the epsilonexpansion
for the levels seems to be smaller than that of the di
rect epsilonexpansion for certain observables like criti
cal exponents and impurity susceptibility and entropy,
see Ref. 13.
VI.CONCLUSIONS
Using the quantum phase transitions in the softgap
Anderson model as an example, we have demonstrated
that epsilonexpansion techniques can be used to deter
mine complete manybody spectra at quantum critical
points. To this end, we have connected information from
standard perturbative RG, which yields information on
critical dimensions and parametrically small couplings,
and from NRG for the manybody spectra of freefermion
Page 11
11
050
N
100
0
2
ENΛ
N/2
r=0.4985
QCP
SC
0.49850.50.495
r
0
2
ENΛ
N/2
SC
QCP
pert.
a) b)
FIG. 12: a) Flow diagram of the lowest lying energy levels
for r = 0.4985; dashed lines: flow to the SC fixed point;
solid lines: flow to the quantum critical fixed point. b) The
deviation of the QCP levels from the SC levels is proportional
to?
(crosses) as described in the text.
1/2 − r. This deviation together with the splitting of the
energy levels can be explained by the perturbative calculation
fixed points.
renormalized perturbation theory for manybody spec
tra of interacting intermediatecoupling fixed points. For
the softgap Anderson model, which features two lower
critical dimensions at r = 0 and r = 1/2, correspondingly
two different approaches can be utilized to capture the
same critical fixed point: Near r = 0 a Kondo term has
to be added to a freefermion chain with a decoupled im
purity, whereas near r = 1/2 an onsite repulsion is used
as a perturbation to the noninteracting Anderson (or
resonantlevel) model. These perturbations lift the large
degeneracies present in the noninteracting spectra, and
accurately reproduce the critical spectra determined in
NRG calculations at criticality.
Together, these can be used to perform
Vice versa, our method will be useful in situations
where the effective lowenergytheory for the critical point
is not known: a careful analysis of the manybody spec
trum near critical dimensions yields information about
the scaling dimension and structure of the relevant oper
ators.
For instance, a plot similar to Fig. 3 can be calculated
for the spinboson model, using the numerical renormal
ization group method as in Ref. 5. Preliminary results
(not shown here) indicate that the manyparticle levels
of the QCP approach the levels of the delocalized (local
ized) fixed point in the limit s → 0 (s → 1), with s the
exponent of the bath spectral function J(ω) ∝ ωs.
We envision applications of our ideas to more complex
impurity models, e.g., with two orbitals or two coupled
spins, as well as to nonequilibrium situations treated
using NRG24.
Acknowledgments
We thank S. Kehrein, Th. Pruschke, and A. Rosch
for discussions and S. Florens, L. Fritz, M. Kir´ can, and
N. Tong for collaborations on related work.
search was supported by the DFG through SFB 484
(HJL, RB) and the Center for Functional Nanostructures
Karlsruhe (MV). MV also acknowledges support from the
Helmholtz Virtual Quantum Phase Transitions Institute
in Karlsruhe.
This re
APPENDIX A: DETAILS OF THE
PERTURBATIVE ANALYSIS AROUND THE
LOCAL MOMENT FIXED POINT
In this Appendix, we want to give more details for the
derivation of the matrix Wij eq. (27) which determines
the splitting of the fourfold degeneracy of the subspace
(Q = 0, S = 1/2, Sz = 1/2) of the LM fixed point at
energy ǫ1+ ǫ3. We focus on the matrix element W12:
W12= ?ψ1H′
The strategy for the calculations can be extended to the
other matrix elements and the other subspaces, for which
we add the perturbative results at the end of this ap
pendix without derivation. The operator?Simp·? s0is de
composed in four parts:
Nψ2? = α(r)f(N)?ψ1?Simp·? s0ψ2? . (A1)
?Simp·? s0 =
1
2S+
+1
2Sz
impc†
0↓c0↑+1
?
2S−
impc†
0↑c0↓
?
imp
c†
0↑c0↑− c†
0↓c0↓
,(A2)
so that W12can be written as
W12= α(r)f(N)1
2[I + II + III − IV] , (A3)
with
I = ?ψ1S+
impc†
0↓c0↑ψ2? ,(A4)
and the other terms accordingly. With the definitions of
ψ1? and ψ2? of eq. (24) we have:
1
√2?ψ0
?
I =
?
ξ†
−3↑ξ1↑+ ξ†
?
−3↓ξ1↓
?
f↑S+
?
impc†
0↓c0↑
×
1
√6f†
↑
ξ†
1↑ξ−3↑− ξ†
1↓ξ−3↓
+
2
√6f†
↓ξ†
1↑ξ−3↓
?
ψ0? .
(A5)
With S+
containing f↑S+
operators, f↑S+
one arrives at
imp= f†
↑f↓ we immediately see that the terms
impf†
impf†
↑drop out. The remaining impurity
↓, give unity when acting on ψ0? so
I =
1
√3[Ia + Ib] , (A6)
Page 12
12
with
Ia = ?ψ0ξ†
Ib = ?ψ0ξ†
−3↑ξ1↑c†
−3↓ξ1↓c†
0↓c0↑ξ†
0↓c0↑ξ†
1↑ξ−3↓ψ0?
1↑ξ−3↓ψ0? .(A7)
To analyze Ia and Ib, the operators c(†)
pressed in terms of the operators ξ(†)
?
with the sums over p and p′defined in eq. (20). This
gives
?
The only nonzero matrix elements of eq. (A9) are for
p = p′= −3:
Ia = α∗
0σhave to be ex
pσ:
?
c0σ=
p′
α0p′ξp′σ , c†
0σ=
p
α∗
0pξ†
pσ, (A8)
Ia =
pp′
α∗
0pα0p′?ψ0ξ†
−3↑ξ1↑ξ†
p↓ξp′↑ξ†
1↑ξ−3↓ψ0? .(A9)
0−3α0−3?ψ0ξ†
= −α0−32.
Similarly, the term Ib gives
?
= α012,
so that in total:
−3↑ξ1↑ξ†
−3↓ξ−3↑ξ†
1↑ξ−3↓ψ0?
(A10)
Ib =
pp′
α∗
0pα0p′?ψ0ξ†
−3↓ξ1↓ξ†
p↓ξp′↑ξ†
1↑ξ−3↓ψ0?
(A11)
I =
1
√3
?−α0−32+ α012?
impc†
.(A12)
The next term II = ?ψ1S−
the combination of impurity operators: f↑f†
f↑from ?ψ1 and f†
The third term III = ?ψ1Sz
1
√12?ψ0
?
where the term with
been dropped. So we are left with four terms
0↑c0↓ψ2? gives zero due to
↓f↑... with
↓f↑= S−
imp.
impc†
0↑c0↑ψ2? gives
?
ψ0? ,
III =
?
ξ†
−3↑ξ1↑+ ξ†
−3↓ξ1↓
?
1↑ξ−3↓from ψ2? has already
f↑Sz
impc†
0↑c0↑f†
↑
×ξ†
1↑ξ−3↑− ξ†
1↓ξ−3↓
(A13)
2
√6f†
↓ξ†
III =
1
√12[IIIa − IIIb + IIIc − IIId] ,(A14)
with
IIIa = ?ψ0ξ†
IIIb = ?ψ0ξ†
IIIc = ?ψ0ξ†
IIId = ?ψ0ξ†
−3↑ξ1↑f↑Sz
−3↑ξ1↑f↑Sz
−3↓ξ1↓f↑Sz
−3↓ξ1↓f↑Sz
impc†
0↑c0↑f†
0↑c0↑f†
0↑c0↑f†
0↑c0↑f†
↑ξ†
↑ξ†
↑ξ†
↑ξ†
1↑ξ−3↑ψ0? ,
1↓ξ−3↓ψ0? ,
1↑ξ−3↑ψ0? ,
1↓ξ−3↓ψ0? .
impc†
impc†
impc†
(A15)
Following similar arguments as above one obtains
IIIa =1
2
?
p
′α0p2,(A16)
where the p in?
p
′takes the values
p = 1,−1,−5,−7,...− N ,
then
IIIb = IIIc = 0 ,(A17)
and
IIId =1
2
?
p
′′α0p2,(A18)
where the p in?
p
′′takes the values
p = −1,−3,−5,−7,...− N .
This gives for the third term
III =
1
√12[IIIa − IIId]
??
1
4√3
=
1
4√3
p
′α0p2−
?
p
′′α0p2
?
=
?α012− α0−32?
.(A19)
The calculation of IV proceeds very similarly to III and
one obtains
III = −IV , (A20)
so that we finally arrive at
W12 = α(r)f(N)1
2
?α012− α0−32??1
√3?α012− α0−32?
√3+ 0 + 2
1
4√3
?
= α(r)f(N)1
4
.(A21)
We performed a similar analysis for a couple of other
subspaces. Here we list the results from the perturba
tive analysis for three more subspaces together with the
corresponding basis states.
1.Q = 0, S = 1/2, Sz = 1/2, E = 2ǫ1
This subspace has the same quantum numbers Q, S
and Sz as the one discussed above, so that the details
of the calculation are very similar. The differences origi
nate from the position of particles and holes in the single
particle spectrum of Fig. 6. This reduces the dimension
ality of the subspace from four to two.
Page 13
13
The corresponding basis can be written as
ψ1? =
1
√2f†
?
↑(ξ†
1↑ξ−1↑+ ξ†
1↓ξ−1↓)ψ0? ,
ψ2? =
1
√6f†
↑(ξ†
1↑ξ−1↑− ξ†
1↓ξ−1↓) +
2
√6f†
↓ξ†
1↑ξ−1↓
?
ψ0? .
(A22)
The firstorder corrections are given by the 2×2 matrix
{Wij} = α(r)f(N)
0
√3
4γ
√3
4γ −1
2β
, (A23)
with γ = α012− α0−12and β = α012+ α0−12. Due
to the particlehole symmetry of the conduction band we
have α01 = α0−1; therefore, the offdiagonal matrix el
ements vanish and the effect of the perturbation is simply
a negative energyshift only for the state ψ2?:
{Wij} = α(r)f(N)
00
0 −α012
. (A24)
This effect can be seen in the energy splitting of the first
two lowlying excitations in Fig. 8.
2.Q = −1, S = 0, E = −ǫ−1
There is only one configuration for this combination of
quantum numbers and excitation energy:
ψ? =
1
√2(f†
↑ξ−1↑+ f†
↓ξ−1↓)ψ0? . (A25)
The firstorder perturbation keeps the state in this one
dimensional subspace and the energy correction is given
by
∆E = ?ψH′
Nψ? = −3
4α(r)f(N)α0−12.(A26)
3.Q = −1, S = 0, E = −ǫ−3
The difference to the previous case is the position of
the hole in the singleparticle spectrum. The state is now
given by
ψ? =
1
√2(f†
↑ξ−3↑+ f†
↓ξ−3↓)ψ0? ,(A27)
with the energy correction
∆E = ?ψH′
Nψ? = −3
4α(r)f(N)α0−32. (A28)
APPENDIX B: DETAILS OF THE
PERTURBATIVE ANALYSIS AROUND THE
STRONG COUPLING FIXED POINT
The main difference in the calculation of the matrix
elements {Wij} for this case is due to the structure of
the perturbation, see eq. (37). Furthermore the ground
state of the SC fixed point is fourfold degenerate and the
perturbation partially splits this degeneracy, as discussed
in the following.
1.Q = 0, S = 1/2, Sz = 1/2, E = 0
This is one of the four degenerate ground states at the
SC fixed point:
ψ1? = ξ†
0↑ψ0? ,(B1)
with ψ0? defined in eq. (41).
The perturbative correction is given by
?ψ1H′
Nψ1? =1
2β(r)¯f(N)(1 − αf04)(B2)
which corresponds to the energy shift of the ground state:
∆E1=1
2β(r)¯f(N)(1 − αf04) .(B3)
The coefficients αflare defined via the relation between
the operators f(†)
σ
and ξ(†)
lσ:
?
fσ=
l′
αfl′ξl′σ , f†
σ=
?
l
α∗
flξ†
lσ.(B4)
2.Q = −1, S = 0, E = 0
This state is also a ground state in the U = 0 case:
ψ2? = ψ0? .(B5)
The calculation of the firstorder correction for ψ2? gives
Nψ2? =1?ψ2H′
2β(r)¯f(N)(1 + αf04) .(B6)
This means that the ground state including the effect of
the perturbation is given by ψ1? in eq. (B1) and the state
ψ2? appears as an excited state. For a comparison with
the energy levels shown in the NRG flow diagrams, where
the ground state energy is set to zero in each iteration,
we subtract the perturbative correction of the ground
state (∆E1) from the energies of all other excited states.
Subtracting this energy shift from eq. (B6) gives the net
energy correction for the ψ2? state:
∆E2= β(r)¯f(N)αf04.(B7)
Page 14
14
3.Q = −1, S = 0, E = ǫ2
The state corresponding to this subspace is given by:
ψ3? =
1
√2(ξ†
0↑ξ−2↑+ ξ†
0↓ξ−2↓)ψ0? .(B8)
The firstorder correction reads
?ψ3H′
Nψ3? = β(r)¯f(N)
?1
2(1 − αf04) + 3αf02αf−22
?
.
(B9)
Subtracting the energy correction for the ground state
results in
∆E3= 3β(r)¯f(N)αf02αf−22. (B10)
4.Q = −1, S = 0, E = ǫ4
Similarly for the state
ψ4? =
1
√2(ξ†
0↑ξ−4↑+ ξ†
0↓ξ−4↓)ψ0? ,(B11)
the firstorder correction is given by
?ψ4H′
Nψ4? = β(r)¯f(N)
?1
2(1 − αf04) + 3αf02αf−42
?
,
(B12)
and subtracting the energy correction for the ground
state results in:
∆E4= 3β(r)¯f(N)αf02αf−42. (B13)
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