Quantum phase transitions in the bosonic single-impurity Anderson model

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 2007

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arXiv:cond-mat/0606325v1 [cond-mat.stat-mech] 13 Jun 2006
Quantum Phase Transitions in the Bosonic Single-Impurity Anderson Model
Hyun-Jung Lee1and Ralf Bulla1
1Theoretische Physik III, Elektronische Korrelationen und Magnetismus, Universit¨ at Augsburg, 86135 Augsburg, Germany
(Dated: February 6, 2008)
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.
PACS numbers:
Bose-Einstein condensation phenomena), 03.75.Hh (Static properties of condensates).
05.10.Cc (Renormalization Group methods), 05.30.Jp (Boson systems), 03.75.Nt (Other
The focus of this work is the physics of a bosonic im-
purity state coupled to a non-interacting bosonic envi-
ronment modeled by the Hamiltonian
H = ε0b†b +1
2Ub†b?b†b − 1?
εkb†
kbk+
+
?
k
?
k
Vk
?
b†
kb + b†bk
?
.(1)
The energy of the impurity level (with operators b(†)) is
given by ε0; the parameter U is the local Coulomb re-
pulsion acting on the bosons at the impurity site. The
impurity couples to a bosonic bath via the hybridization
Vk, with the bath degrees of freedom given by the oper-
ators b(†)
k
with energy εk.
We term the system defined by eq. (1) the ‘bosonic
single-impurity Anderson model’ (bosonic siAm), in anal-
ogy to the standard (fermionic) siAm [1] which has a very
similar structure except that all fermionic operators are
replaced by bosonic ones. Furthermore, we do not con-
sider internal degrees of freedom of the bosons, such as
the spin (an essential ingredient in the fermionic siAm).
There are various ways to motivate the study of the
model eq. (1). From a purely theoretical point of view
it is interesting to compare the physics of the fermionic
and bosonic versions of the siAm. Of course, there are
striking differences between these two models: there is
no direct bosonic analog of local moment formation and
screening of these local moments at low temperatures,
characteristics of the fermionic Kondo effect [2].
the other hand, the bosonic model allows for a Bose-
Einstein condensation (BEC), at least in a certain pa-
rameter space (see Fig. 1 below), a phenomenon which is
clearly absent in the fermionic model.
There are certain similarities between the fermionic
and the bosonic model concerning the role of the local
Coulomb repulsion U: increasing the value of U can
induce a quantum phase transition from a phase with
screened spin to a local moment phase in the fermionic
case [3], while it can induce a quantum phase transition
On
from a BEC phase to a ‘Mott phase’ in the bosonic case,
as shown below.
Another motivation for studying the bosonic siAm
comes from a treatment of the Bose-Hubbard model
within dynamical mean-field theory (DMFT) [4].
though such an investigation has not been pursued so
far, it is clear that the effective impurity model onto
which the Bose-Hubbard model is mapped will have a
similar form as eq. (1) (see also Ref. 5). An obvious ap-
plication of such a DMFT treatment would then address
the Mott transition in the Bose-Hubbard model [6] which
must have its counterpart on the level of the effective im-
purity model – and the ‘Mott transition’ on the impurity
level is precisely what we are looking at here.
Finally, a physical system described by the bosonic
siAm could be directly realized in optical lattices, where
the laser fields are tuned in such a way that a single
‘impurity’ site is formed within an approximately un-
perturbed lattice system (‘atomic quantum dots’, see
Refs. 7, 8). Provided the Coulomb repulsion at all sites
except the impurity can be neglected, the corresponding
model can be directly mapped onto the model eq. (1).
Present theoretical studies of atomic quantum dots focus,
however, on a coupling between impurity and excitations
of the superfluid environment [7], for which a description
in terms of the spin-boson model is more appropriate.
For the calculations presented in this paper we use the
numerical renormalization group (NRG) originally devel-
oped by Wilson for the Kondo problem [9]. This method
has been shown to give very accurate results for a broad
range of impurity models, including the fermionic siAm
[1, 10] but also quantum impurities with coupling to a
bosonic bath [11, 12]. Here we employ the bosonic ex-
tension of the NRG (bosonic NRG) to study the model
eq. (1). We shall present ground state phase diagrams
of the bosonic siAm, identify fixed points and discuss
quantum phase transitions between the fixed points. Al-
though far from comprehensive, our results indicate that
the bosonic siAm shows a variety of interesting proper-
Al-

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ties which deserve to be studied in greater detail in the
future.
Before we come to the results from the bosonic NRG,
let us discuss some general properties and trivial limits
of the model eq. (1).
Similar to other quantum impurity models, the influ-
ence of the bath on the impurity is completely specified
by the bath spectral function
∆(ω) = π
?
k
V2
kδ(ω − εk) .(2)
Here we assume that ∆(ω) can be parametrized by a
powerlaw for frequencies up to a cutoff ωc(we set ωc= 1
in the calculations)
∆(ω) = 2παω1−s
c
ωs, 0 < ω < ωc .(3)
The parameter α is the dimensionless coupling constant
for the impurity-bath interaction. This form of ∆(ω) is
certainly not the most general one and specific applica-
tions (within a bosonic DMFT or for an impurity level
in an optical lattice) will lead to additional structures in
∆(ω).
The NRG calculations can be both performed for
a canonical and a grandcanonical ensemble.
present only results for a grandcanonical ensemble for
which we set the chemical potential of the bath to µ = 0.
This means that a Bose-Einstein condensation of the
whole system (impurity plus bath) can only be induced
by the coupling to the impurity. This can already be
understood from the non-interacting case, U = 0: here
a direct diagonalization of the bosonic siAm shows that
with increasing α, a localized state with negative energy
separates out of the continuum at a critical value α = αc.
The BEC then occurs via populating this localized state,
a feature which can also be observed in the numerical
calculations.
The other trivial limit of the bosonic siAm is the decou-
pled impurity, α = 0. The succession of quantum phase
transitions as shown in Fig. 1 for α = 0 can be easily un-
derstood from the dependence of the many-particle levels
on the parameters ε0 and U. The transition occurs for
ε0/U = −nimpwhen the energies of the states with nimp
and nimp+ 1 electrons are degenerate.
The full phase diagram Fig. 1 is calculated with the
bosonic NRG [12]. In this approach, the frequency range
of the bath spectral function [0,ωc] is divided into in-
tervals [ωcΛ−(n+1),ωcΛ−n], n = 0,1,2,..., with Λ the
NRG discretization parameter (we use Λ = 2.0 for all
the results shown in this paper). The continuous spec-
tral function within these intervals is approximated by a
single bosonic state and the resulting discretized model is
then mapped onto a semi-infinite chain with the Hamil-
tonian
Here we
H = ε0b†b +1
2Ub†b?b†b − 1?+ V
?
b†¯b0+¯b†
0b
?
0 0.10.20.30.4
αωc/U
-2
-1
0
1
2
3
4
-ε0/U
nimp=0
nimp=1
nimp=2
nimp=3
nimp=4
BEC
0
1
2
3
4
FIG. 1: Zero-temperature phase diagram of the bosonic siAm
for bath exponent s = 0.6 and fixed impurity Coulomb in-
teraction U = 0.5. The different symbols denote the phase
boundaries between the Mott phases and the BEC phase. The
Mott phases are labeled by their occupation nimp for α = 0.
Only the Mott phases with nimp ≤ 4 are shown. The NRG
parameters are Λ = 2.0, Nb= 10, and Ns = 100.
+
∞
?
n=0
εn¯b†
n¯bn+
∞
?
n=0
tn
?¯b†
n¯bn+1+¯b†
n+1¯bn
?
. (4)
Here the impurity couples to the first site of the chain via
the hybridization V =
?2α/(1 + s). The bath degrees
of freedom are in the form of a tight-binding chain with
operators¯b(†)
n , on-site energies εn, and hopping matrix
elements tn which both fall off exponentially: tn,εn ∝
Λ−n.
The chain model eq. (4) is diagonalized iteratively
starting with the impurity site and adding one site of
the chain in each iteration. As for the application to the
spin-boson model, only a finite number Nbof basis states
for the added site can be taken into account, and, after
diagonalizing the enlarged cluster, only the lowest-lying
Nsmany-particle states are kept for the subsequent iter-
ations (here we use Nb= 10 − 20 and Ns= 100 − 200).
The main technical difference to the spin-boson model
is that we can use the total particle-number as a con-
served quantity in the Hamiltonian eq. (1). Furthermore,
the renormalization group flow turns out to be consider-
ably more stable as in the calculations for the spin-boson
model so that we can easily perform up to N = 100 it-
erations (a detailed account of the technical details will
appear elsewhere).
Let us now discuss the T = 0 phase diagram of the
bosonic siAm, Fig. 1, calculated for fixed U = 0.5 with
the parameter space spanned by the dimensionless cou-
pling constant α and the impurity energy ε0. We choose
s = 0.6 as the exponent of the powerlaw in ∆(ω) (the s-
dependence of the phase diagram is discussed in Fig. 4 be-
low). The phase diagram is characterized by a sequence
of lobes which we label by the impurity occupation nimp
at α = 0 where it takes integer values. We use the termi-
nology ‘Mott phases’ for these lobes, due to the appar-

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ent similarity to the phase diagram of the Bose-Hubbard
model. The Mott phases are separated from the BEC
phase by lines of quantum critical points which termi-
nate (for s = 0.6) at a finite value of α, except for the
nimp= 0 phase where the boundary extends up to infi-
nite α. These transition can be viewed as the impurity
analogue of the Mott transition in the lattice model, since
it is the local Coulomb repulsion which prevents the for-
mation of the BEC state.
The phase diagram Fig. 1 is deduced from the flow of
the lowest-lying many-particle levels which allows quite
generally to identify fixed points and transitions between
these fixed points [10]. Figure 2 shows the flow for s =
0.4, fixed α = 0.007 and U = 0.5, and two values of
ε0very close to the quantum phase transition. The solid
lines belong to the Mott phase with nimp= 2 and display
a crossover between two fixed points: from an unstable
quantum critical point for iteration numbers N ∼ 20−40
to a stable fixed point for N > 70. Analysis of the stable
fixed point shows that it can be described by a decoupled
impurity with occupation nimp = 2 and a free bosonic
bath given by the decoupled chain. (The actual impurity
occupation, however, differs from the integer values, as
discussed below.) The structure of the quantum critical
point, on the other hand, is presently not clear but might
be accessible to perturbative methods as discussed in [13].
The dashed lines belong to the BEC phase where, ap-
parently, something dramatic is happening at N ≈ 55.
The divergence of the energies for all excited states seen
in this plot is due to the formation of a localized state,
split off from the continuum by an energy gap ∆g, very
similar to the behavior for U = 0. This energy gap takes
a finite value which is not renormalized to zero as Λ−N
as the levels of the other fixed points, therefore the di-
vergence of ∆g· ΛN. In the inset of Fig. 2 we plot EN
(instead of ENΛN) so that the development of the gap
∆g≈ 4×10−18appears as a plateau of the first excitated
state. The extremely small value of the gap is due to the
tuning of the parameters very close to the transition (∆g
vanishes at the transition).
Further analysis of the data shows that the ground
state in the BEC phase has an occupation number given
by the maximum boson number used in the iterative di-
agonalization.
The fact that the unstable fixed point separating the
Mott phase from the BEC phase is indeed a quantum
critical point can be deduced from its non-trivial level
structure, as mentioned above, and the behavior of the
crossover scale T∗. Numerically we find that upon vari-
ation of ε0 close to its critical value ε0,c, the crossover
scale vanishes with a powerlaw at the transition, T∗∝
|ε0− ε0,c|νon both sides of the transition, with a non-
trivial exponent ν ≈ 2.50 ± 0.06. This exponent is ob-
served for all lines of quantum critical points for fixed
s = 0.4 and preliminary results show that ν = 1/s for
0 < s < 1.
40506070
N
0
10-17EN
02040
60
80
100
N
0
1
2
3
4
ENΛN
FIG. 2: Flow diagram of the lowest lying many-particle levels
EN versus iteration number N for parameters s = 0.4, α =
0.007, U = 0.5, and two values of ε0very close to the quantum
phase transition between the Mott phase with nimp = 2 and
the BEC phase. Both the quantum critical point and the
Mott phase appear as fixed points in this scheme whereas in
the BEC phase, a gap ∆g opens between the ground state
and the first excited state, see the inset where EN (instead of
ENΛN) is plotted versus N.
Let us now focus on the impurity occupation nimpfor
temperature T = 0. Figure 3 shows the dependence of
nimpon ε0for s = 0.4, U = 0.5, and various values of α.
The symbols indicate that the parameters lie within the
Mott phase whereas the crosses are for the BEC phase.
Both sets of lines form a continuous curve but it is unclear
at the moment whether the data in the BEC phases are
reliable (due to the special features in the flow diagram
discussed above).
The terminology we use here suggests an integer oc-
cupation throughout the Mott phase, as for the Bose-
Hubbard model, but for the bosonic siAm nimpdeviates
from the integer values as soon as the coupling to the
bath is finite, see Fig. 3. This is to be expected since for
a gapless bath spectral function ∆(ω), the charge fluc-
tuations on the impurity site cannot be completely sup-
pressed. Indeed we observe that for increasing the value
of the bath exponent s, the nimp(ε0)-curve gets closer to
the step function. At this point one can speculate about
the possible development of ∆(ω) in a DMFT treatment
of the Mott phase. The self-consistency might generate
a bath spectral function with a gap and the impurity oc-
cupation might then turn into the step function expected
for the lattice model.
The precise shape of the boundaries in the phase di-
agram Fig. 1 depends on the form of ∆(ω) for all fre-
quencies. Here we stick to the powerlaw form eq. (3) and
present the dependence of the phase diagram on the bath
exponent s in Fig. 4. We observe that upon increasing the
value of s, the areas occupied by the Mott phases extend
to larger values of α and significantly change their shape.
A qualitative change is observed when the exponent ap-
proaches s = 1. First of all, the Mott phases appear

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-0.5
0
0.51 1.5
2
-ε0/U
0
0.5
1
1.5
2
nimp(T=0)
α=0
α=0.007
α=0.010
α=0.028
FIG. 3: Impurity occupation nimp as a function of ε0 for
temperature T = 0, s = 0.4, U = 0.5, and various values of α.
The sharp steps for the decoupled impurity α = 0 are rounded
for any finite α. Symbols (crosses) correspond to data points
within the Mott (BEC) phases.
0 0.20.4
0.6
0.8
1
αωc/U
-1
0
1
2
3
4
-ε0/U
s=1.0
s=0.8
s=0.6
s=0.4
FIG. 4: Zero-temperature phase diagram of the bosonic siAm
as in Fig. 1, but now for different values of the bath expo-
nent s. For increasing value of s, the areas occupied by the
Mott phases significantly change their shape and for s = 1 it
appears that each Mott phase extends up to arbitrarily large
values of α.
to extend up to arbitrarily large values of α. Further-
more, the BEC phase which separates the Mott phases
for s < 1 and α > 1 is completely absent for s = 1! We
do not yet have an explanation for this observation and
it would be interesting to find out whether the absence of
the BEC phase is due to the special form of ∆(ω), eq. (3),
or whether it is a generic feature even when ∆(ω) ∝ ω is
only valid for ω → 0.
The case s = 0 (constant bath density of states) turns
out to be difficult to access numerically. An extrapola-
tion of the phase boundaries for values of s in the range
0.1...0.4 to s = 0 is inconclusive, but the Mott phase is
at least significantly suppressed in this limit.
To summarize, we have presented NRG calculations
for the phase diagram and the impurity occupation of a
bosonic version of the single-impurity Anderson model.
The phase diagram contains Mott phases, in which the
local Coulomb repulsion prevents Bose-Einstein conden-
sation, separated by the BEC phase by lines of quan-
tum critical points. The studies presented here are only
the starting point for a comprehensive investigation of
the bosonic siAm and we are planning to calculate, for
example, physical properties at finite temperatures and
dynamic quantities (impurity spectral function and self-
energy). The latter will be of importance for a possible
DMFT for the Bose-Hubbard model, an approach which
has not yet been fully developed due to various concep-
tual problems. One issue is the proper scaling of the
Hamiltonian parameters in the limit of infinite spatial
dimensions [14]. For example, the model on a hypercu-
bic lattice as studied in Ref. 15 requires a scaling of the
hopping matrix elements as 1/d which leads to a static
mean-field theory. In addition, the bosonic DMFT in the
superfluid phase of the Bose-Hubbard model might gen-
erate a more complex impurity model, the bosonic siAm
introduced here would then be applicable only within the
Mott phases of the lattice model.
Finally, it would be interesting to identify situations for
atomic quantum dots in optical lattices which can be de-
scribed by the bosonic single-impurity Anderson model.
We would like to thank Krzysztof Byczuk, Jim Fre-
ericks, Matthias Vojta, and Dieter Vollhardt for helpful
discussions. This research was supported by the DFG
through SFB 484.
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[2] A bosonic analogue of the Kondo effect can be observed
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