# 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.

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**ABSTRACT:**In the beginning of the 1970's, Wilson developed the concept of a fully non-perturbative renormalization group transformation. Applied to the Kondo problem, this numerical renormalization group method (NRG) gave for the first time the full crossover from the high-temperature phase of a free spin to the low-temperature phase of a completely screened spin. The NRG has been later generalized to a variety of quantum impurity problems. The purpose of this review is to give a brief introduction to the NRG method including some guidelines of how to calculate physical quantities, and to survey the development of the NRG method and its various applications over the last 30 years. These applications include variants of the original Kondo problem such as the non-Fermi liquid behavior in the two-channel Kondo model, dissipative quantum systems such as the spin-boson model, and lattice systems in the framework of the dynamical mean field theory.Review of Modern Physics 02/2007; · 44.98 Impact Factor - SourceAvailable from: Jian-Xin Zhu[Show abstract] [Hide abstract]

**ABSTRACT:**We study an Anderson impurity embedded in a d-wave superconductor carrying a supercurrent. The low-energy impurity behavior is investigated by using the numerical renormalization group method developed for arbitrary electronic bath spectra. The results explicitly show that the local impurity state is completely screened upon the non-zero current intensity. The impurity quantum criticality is in accordance with the well-known Kosterlitz-Thouless transition.Physical review. B, Condensed matter 04/2008; · 3.77 Impact Factor - SourceAvailable from: arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We consider a quantum impurity model in which a bosonic impurity level is coupled to a non-interacting bosonic bath, with the bosons at the impurity site subject to a local Coulomb repulsion U. Numerical renormalization group calculations for this bosonic single-impurity Anderson model reveal a zero-temperature phase diagram where Mott phases with reduced charge fluctuations are separated from a Bose-Einstein 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 single-impurity Anderson model appears as an effective impurity model in a dynamical mean-field theory of the Bose-Hubbard model. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007Physics of Condensed Matter 07/2006; 56(3):199-203. · 1.28 Impact Factor

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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

Page 2

2

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

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 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

Page 4

4

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

Page 5

5

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