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 particle-hole symmetric soft-gap Anderson model. The model displays two stable phases whose fixed points can be built up of non-interacting single-particle 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 many-body 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 soft-gap 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 epsilon-expansion for full many-body spectra.

Full text

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arXiv:cond-mat/0506694v1 [cond-mat.str-el] 27 Jun 2005
Numerical Renormalization Group for Impurity Quantum Phase Transitions:
Structure of Critical Fixed Points
Hyun-Jung Lee∗, Ralf Bulla∗, and Matthias Vojta†
∗Theoretische Physik III, Elektronische Korrelationen und Magnetismus,
Institut f¨ ur Physik, Universit¨ at Augsburg, D-86135 Augsburg, Germany and
†Institut f¨ ur Theorie der Kondensierten Materie,
Universit¨ at Karlsruhe, D-76128 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 particle-hole symmetric soft-gap Anderson
model. The model displays two stable phases whose fixed points can be built up of non-interacting
single-particle 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 many-body 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 soft-gap 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
epsilon-expansion for full many-body spectra.
I. INTRODUCTION
Zero-temperature 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 zero-dimensional boundary
with internal degrees of freedom (the impurity) interacts
with an extended bath of fermions or bosons. Examples
of impurity models with non-trivial 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 two-level system coupling to
a dissipative environment4,5, as well as so-called 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 so-called 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 single-impurity Bose-Fermi Kondo model
supplemented by self-consistency conditions, for details
see Ref. 11.
Diverse techniques have been used to investigate impu-
rity phase transitions, ranging from static and dynamic
large-N 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 non-perturbative
and are able to access arbitrarily small energies and tem-
peratures. Apart from static and dynamic observables,
the NRG provides information about the many-body 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 non-interacting bosons or fermions
– this is the case for most stable fixed points of impurity
models, e.g., the strong-coupling (screened) fixed point
of a standard Kondo model. Intermediate-coupling fixed
points, usually being interacting, have a completely dif-
ferent NRG level structure, i.e., smaller degeneracies and
non-equidistant levels. They cannot be cast into a free-
particle description, with the remarkable exception of the
two-channel Kondo fixed point which is known to have
a representation in terms of free Majorana fermions17.
In general, the NRG fixed-point spectrum at impurity
transitions is fully universal, apart from a non-universal
overall prefactor and discretization effects.
The purpose of this paper is to demonstrate that a
complete understanding of the NRG many-body spec-
trum of critical fixed points is actually possible, by uti-
lizing renormalized perturbation theory around a non-
interacting fixed point. In the soft-gap 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

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being parametrically small near the critical dimension,
we can construct the entire quantum critical many-body
spectrum from a free-fermion model supplemented by a
small perturbation. In other words, we shall perform
epsilon-expansions to determine a complete many-body
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 soft-gap Ander-
son model and its quantum phase transitions. Sec. III
summarizes the recent results from perturbative RG for
both the soft-gap 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.SOFT-GAP ANDERSON MODEL
The Hamiltonian of the soft-gap 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 f-electrons 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 single-impurity
Anderson model18but for the soft-gap 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) = ρ0|ω|rat 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 low-energy
prefactor of the density of states, ρ0, to represent the
σ ) to a
kσ) via a hybridization V .
The
kδ(ω−εk)
has a soft-gap at the Fermi level,?∆(ω) = ∆|ω|r, 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 p-h symmetric case (solid line, U = 10−3, εf =
−0.5 · 10−3, conduction band cutoff at -1 and 1) and the p-h
asymmetric case (dashed line, εf = −0.4 · 10−3); ∆ measures
the hybridization strength,? ∆(ω) = ∆|ω|r.
dimensionful energy scale of the problem. Assuming a
particle-hole symmetric band, the model (1) is particle-
hole symmetric for ǫf= −U/2.
The soft-gap Anderson model (1) with 0 < r < ∞
displays a very rich behaviour, in particular a continuous
transition between a local-moment (LM) and a strong-
coupling (SC) phase.Figure 1 shows a typical phase
diagram for the soft-gap Anderson model. In the particle-
hole (p-h) symmetric case (solid line) the critical coupling
∆cdiverges at r =1
(Refs. 19,20). No divergence occurs for p-h asymmetry
(dashed line)19.
We now briefly describe the properties of the fixed
points in the soft-gap Anderson and Kondo models19.
Due to the power-law 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 low-temperature impurity susceptibility χimp =
1/(4kBT), but the leading corrections show r-dependent
power laws. The p-h 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 SC-LM phase boundary14,20. At
the critical point, non-trivial 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 soft-gap Anderson
2, and no screening occurs for r >1
2
1
2. In contrast, the p-h asymmetric

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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 soft-gap 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) = ρ0|ω|ras 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 weak-coupling 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 local-moment 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. P-h 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 p-h 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 lower-critical dimension
of the transition under consideration.
B.RG near r = 1/2
For r near 1/2 the p-h symmetric critical fixed point
moves to strong Kondo coupling, and the language
of the p-h symmetric Anderson model becomes more
appropriate13. First, the conduction electrons are inte-
grated out exactly, yielding a self-energy Σf= V2Gc0for
the f electrons, where Gc0 is the bare conduction elec-
tron Green’s function at the impurity location. In the
low-energy limit the f electron propagator is then given
by
Gf(iωn)−1= iωn− iA0sgn(ωn)|ωn|r
where the |ωn|rself-energy term dominates for r < 1,
and the prefactor A0is
(6)
A0=πρ0V2
cosπr
2
. (7)
Equation (6) describes the physics of a non-interacting
resonant level model with a soft-gap 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 strong-coupling (SC) phase
of the soft-gap 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 on-site interaction u via
U = µ2r−1A2
0u.(8)
The beta funcion receives the lowest non-trivial contri-
bution at two-loop order and reads13
β(u) = (1 − 2r)u −3(π − 2ln4)
For r < 1/2 a non-interacting stable fixed point is at u∗=
0 – this is the symmetric strong-coupling 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 strong-coupling 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 weak-coupling LM fixed
point, i.e., around the decoupled impurity limit, valid for
r ≪ 1, or by an expansion around the strong-coupling SC
fixed point, i.e., around a non-interacting resonant-level
(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 non-constant cou-
plings?∆(ω) (see Refs. 19,20,22). This method allows a

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non-perturbative calculation of the many-particle 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 soft-gap 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 semi-infinite 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 p-h symmetric conduction band, all the on-site 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 cut-off 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 semi-infinite 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 low-energy
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 n-site system. In practice one
retains only these Nsstates from the n-site 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)
0 50 100
N
0.0
1.0
2.0
3.0
Λ
N/2EN
∆<∆ c
local moment
0 50100
N
∆=∆c
quantum critical
050100
N
∆>∆c
strong coupling
ab
c
FIG. 2:
citations obtained from the numerical renormalization group
for the three different fixed points of the p-h 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 low-energy many-body 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 many-particle 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 many-particle
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 soft-gap 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 half-filling), 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 N-site 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

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0.00.1 0.20.30.40.5
r
0.0
1.0
2.0
3.0
4.0
Λ
N/2EN
QCP
LM
SC
FIG. 3: Dependence of the many-particle spectra for the three
fixed points of the p-h symmetric soft-gap Anderson model on
the exponent r: SC (black dot-dashed 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 soft-gap 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 single-particle 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 many-particle spectra
on the bath exponent r.
V.STRUCTURE OF THE QUANTUM
CRITICAL POINTS
In Fig. 3, the many-particle spectra of the three fixed
points (SC: dot-dashed lines, LM: dashed lines, and QCP:
solid lines) of the symmetric soft-gap 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 double-logarithmic 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 single-particle 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 many-particle 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 many-particle 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 single-particle 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 many-particle level
?1/2− r for r → 1/2. This is illustrated