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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.

[1] A. C. Hewson, The Kondo Problem to Heavy Fermions

(Cambridge Univ. Press, Cambridge 1993).

[2] A bosonic analogue of the Kondo effect can be observed

in the models proposed in S. Florens, L. Fritz, and M.

Vojta, Phys. Rev. Lett. 96, 036601 (2006); these models

are not directly related to the bosonic siAm studied here.

[3] Such a quantum phase transition is seen, for example, in

the soft-gap Anderson and Kondo models, see M. Vojta,

Phil. Mag. 86, 1807 (2006), and references therein.

[4] For the DMFT for fermionic models, see W. Metzner

and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989); A.

Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg,

Rev. Mod. Phys. 68, 13 (1996).

[5] R. Bulla, Phil. Mag. 86, 1877 (2006).

[6] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D.

S. Fisher, Phys. Rev. B 40, 546 (1989).

[7] A. Recati, P. O. Fedichev, W. Zwerger, J. von Delft, and

P. Zoller, Phys. Rev. Lett. 94, 040404 (2005).

[8] D. Jaksch and P. Zoller, Ann. of Phys. 315, 52 (2005).

[9] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).

[10] H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson,

Phys. Rev. B 21, 1003 (1980); ibid. 21, 1044 (1980).

[11] R. Bulla, N.-H. Tong, and M. Vojta, Phys. Rev. Lett. 91,

170601 (2003).

[12] R. Bulla, H.-J. Lee, N.-H. Tong, and M. Vojta, Phys.

Rev. B 71, 045122 (2005).

[13] H.-J. Lee, R. Bulla, and M. Vojta, J. Phys.: Condens.

Matter 17, 6935 (2005).

[14] D. Vollhardt, private communication.

[15] J. K. Freericks and H. Monien, Phys. Rev. B 53, 2691

(1996).