# Symmetry breaking and physical properties of the bosonic single-impurity Anderson model

###### Abstract

We show how exact diagonalization of small clusters can be used as a fast and reliable impurity solver by determining the phase diagram and physical properties of the bosonic single-impurity Anderson model. This is specially important for applications which require the solution of a large number of different single-impurity problems, such as the bosonic dynamical mean field theory of disordered systems. In particular, we investigate the connection between spontaneous global gauge symmetry breaking and the occurrence of Bose-Einstein condensation (BEC). We show how BEC is accurately signaled by the appearance of broken symmetry, even when a fairly modest number of states is retained. The occurrence of symmetry breaking can be detected both by adding a small conjugate field or, as in generic quantum critical points, by the divergence of the associated phase susceptibility. Our results show excellent agreement with the considerably more demanding numerical renormalization group (NRG) method. We also investigate the mean impurity occupancy and its fluctuations, identifying an asymmetry in their critical behavior across the quantum phase transitions between BEC and ‘Mott’ phases.

###### pacs:

67.85.Bc, 67.85.Jk, 37.10.Gh## I Introduction

Strongly correlated impurity models have played an important role
in the field of condensed matter physics. The Anderson Anderson1961
and the Kondo Kondo1964 single-impurity models were at the
heart of the early investigations into the formation of localized
magnetic moments in metals and their effects on thermodynamic and
transport properties of these materials. An extensive arsenal of theoretical
techniques have been developed in an effort to better elucidate these
and related questions Hewson1993 and a great deal of understanding
has been thereby achieved. More recently, important methods for the
study of *periodic* strongly correlated systems have been developed,
such as the dynamical mean field theory (DMFT) Georges1996
and its extensions, which rely heavily on the knowledge base accumulated
in the analysis of the aforementioned impurity models. Indeed, solving
a single-impurity problem is the most challenging part of the DMFT
algorithm.

Since the advent of the possibility of loading extremely cold atoms onto the effective periodic potential formed by optical lattices Jaksch1998 ; JakschZoller2005 there has been a growing interest in the cross fertilization between these atomic systems and their conventional condensed matter counterparts. For one thing, cold atoms in optical lattices are expected to be very well described by the simplified models used in the condensed matter context, such as the Hubbard model JakschZoller2005 . For solids, by contrast, these models are believed to be at best a bare bones description which hopefully retains the most important physical features. Moreover, the parameters in cold-atom optical lattice systems, such as hopping amplitudes and scattering lengths can often be very flexibly tuned externally, allowing for the thorough investigation of large portions of the phase diagrams. Finally, the quantum statistics can also be switched as bosonic atoms are readily available.

This fruitful interplay between these condensed matter and cold-atom systems has prompted
researchers to attempt to use impurity-model-based approaches as an
analytical tool for cold-atom systems. In particular, the bosonic
version of the dynamical mean field theory (BDMFT) has been developed
byczukvoll08 ; SnoekHofstetterChapter ; Andersetal2011 . In close
analogy with its fermionic counterpart, this method requires the solution
of a bosonic single-impurity Anderson model (B-SIAM). This model has
been directly studied by the powerfull Wilson numerical renormalization group
(NRG) technique and its phase diagram has been determined Lee2007 ; Lee2010 .
Furthermore, applications of BDMFT have also been carried out, using
as impurity solvers exact diagonalization hubeneretal09 ; hutong09
and quantum Monte Carlo andersetal10 ; Andersetal2011 . It should
also be mentioned that impurity-like cold-atom set-ups have been proposed
Recatietal2005 and may also become available (for a realization
in which *different* species occupy the impurity and the bath
regions, see Zipkes2010 ), in which case some version of the
single-impurity model may be directly applicable.

More recently, it has become possible to introduce quenched disorder into optical lattice systems horak98 ; boiron99 ; diener01 ; roth03 ; damski03 ; gimperlein05 ; gavish05 ; massignan06 ; Fallani2007 ; Lucioni11 , which may prove useful for the important problem of the interplay between disorder and interactions Lee1985a . The extension of the BDMFT to the disordered case other , however, requires the solution of a large number of different single-impurity problems for the description of a single disordered sample, as many as the number of lattice sites (for a description of the fermionic version, see Dobrosavljevi'c1998a ). This makes it all but impossible to use such numerically demanding single-impurity solvers as the NRG. Therefore, the feasibility of using the BDMFT to study disordered systems requires the development and characterization of fast yet reliable single-impurity solvers.

One of the goals of this article is to show that the exact diagonalization of small clusters is a good solution to this problem, a procedure which has been successfully applied to the fermionic case CaffarelKrauth1994 ; Kajueter1996 . We will show that it can be efficiently and reliably implemented to solve the B-SIAM by studying its ground state properties. Moreover, with an eye towards its application in a BDMFT calculation, we have thoroughly investigated the appearance of Bose-Einstein condensation (BEC) in this model as a manifestation of the phenomenon of global gauge symmetry breaking. Indeed, the B-SIAM is perhaps the simplest model in which this phenomenon occurs and its reduced Hilbert space provides a unique setting in which to numerically study it. Although this is the standard criterion for the detection of BEC, it was not used in the NRG analysis of the model Lee2007 ; Lee2010 . The same BEC identification which we will show can be done in the B-SIAM will be useful in the analogous task in a BDMFT calculation. Besides, the B-SIAM has extended regions in which BEC is present or absent (the so-called ‘Mott’ phase) even at zero temperature, with quantum phase transitions between them. We will show how global gauge symmetry breaking can be used to analyze this quantum phase transition in a manner analogous to more conventional systems such as magnetic ones, even though the system sizes used are fairly modest.

We should stress that the question of the best or even the correct
criterion for the occurrence of BEC is not devoid of a certain controversy.
Indeed, while the existence of a macroscopic eigenvalue of the one-particle
density matrix seems to be uncontested as a necessary and sufficient
condition for a BEC Yang62 , some objections have been raised,
particularly by Leggett LeggettBook , as to whether the frequently
used (particularly in the condensed matter literature) criterion of
“spontaneously broken gauge symmetry” is an adequate alternative.
Much of Leggett’s objection seems to be aimed at the *physical
basis* of the infinitesimal symmetry-breaking field usually employed
in this criterion, rather than at the validity of the *mathematical
procedure* it is based upon. Since our main interest here is to show
that this criterion is perfectly adequate for a *numerical* investigation
of the less studied case of an impurity model, we should be quite
safe. Furthermore, it seems clear that, at least in the case of unfragmented
condensates, the criterion of a spontaneously broken gauge symmetry
is both a sufficient and a necessary condition for Bose-Einstein condensation
in bosonic systems (for a specific discussion and review, see Yukalov2007 ).

We will show in this article that, even in fairly small clusters with a restricted number of bosonic states, a detailed characterization of the spontaneously broken gauge symmetry of the BEC phase and an accurate determination of the full phase diagram is possible, which is in excellent agreement with the much more demanding numerical renormalization group method. Furthermore, we will analyze how the impurity occupancy and its fluctuations behave within the phases and through the quantum phase transitions between them. In particular, we pinpoint a qualitative difference in the critical behavior of both of these quantities as one crosses the boundary from BEC to ‘Mott’ as compared to going from ‘Mott’ to BEC.

The paper is divided as follows. Section II is devoted to the definition of the model and a summary of known results for the non-interacting as well as the interacting cases. Section III expounds on the criterion of global gauge symmetry breaking as a hallmark of BEC. Details of the numerical procedure are explained in Section IV. Results on the symmetry breaking occurring in the model are presented in Section V.1, whereas other local impurity properties are shown in Section V.2. We draw some final conclusions in Section VI. Some results for the non-interacting limit are relegated to an Appendix.

## Ii The bosonic single-impurity Anderson model

We will focus our attention on the bosonic version of the single-impurity Anderson model Hamiltonian

(1) | |||||

Here, are bosonic annihilation operators for the impurity () and “bath” orbitals (), is the number operator for the impurity, is impurity single-particle energy, and measures the interaction strength between bosons inside the impurity orbital. The next two terms of the Hamiltonian describe, respectively, the bath single-particle orbital energies and the hybridization between impurity and bath states, which occurs with amplitude . We will work in the grand-canonical ensemble at zero temperature and fixed chemical potential . All the single-particle energies in Equation (1) are assumed to be measured with respect to .

Evidently, the physics of the model is strongly dependent on the spectral properties of the bosonic bath. It is common practice, particularly in the quantum dissipation literature, to assume a “soft-gap” spectral function for the bath A.J.Leggett1987 . A power-law dependence is usually considered A.J.Leggett1987 ; Lee2007 ; Lee2010 , such that if

(2) |

then

(3) | |||||

(4) |

where is a frequency cutoff, plays the role of a dimensionless coupling constant and is the Heaviside step function. The so-called ohmic case corresponds to , whereas corresponds to the sub-ohmic (super-ohmic) regime.

It is useful to consider first the non-interacting limit, . In this case, the problem can be immediately diagonalized (see the appendix A). The spectrum can be obtained, e.g., from the impurity site Green’s function, and it consists of the roots of the equation

(5) |

In the case of a power-law bath spectral function there is a critical coupling constant (for ), such that there appears a vanishing root for but not for , as shown in the Appendix. At this point, the lowest state of the system is macroscopically occupied, signaling the phenomenon of Bose-Einstein condensation. Further increase of renders the system ill-defined (in the grand-canonical ensemble assumed here), since the lowest state falls below the chemical potential ().

At finite , the BEC only occurs for Lee2007 ; Lee2010 . In this case, for coupling constant values , there is a phase in which the BEC is absent. This phase is adiabatically connected to its counterpart, in which the impurity is decoupled from the bath and which is characterized by an integer occupation of the impurity site. This is very reminiscent of the Mott insulating phases of the Bose-Hubbard model, hence the name “Mott phase”. It should be emphasized, however, that if the impurity occupation deviates from integer values and, in contrast to the Mott insulating case, it does not exhibit plateaus of constant as a function of (see Section V.2 below). Therefore, this terminology is used in a loose sense. The BEC phase is absent for Lee2007 ; Lee2010 .

In the NRG study of references Lee2007 ; Lee2010 , the transition from the BEC to the Mott phase was identified from the vanishing of a gap in the spectrum of low-lying excitations. Indeed, much like in the limit, the splitting-off of an isolated pole from the continuum signals the BEC. We will here, however, explore the criterion of the spontaneous breaking of the (global) gauge symmetry as an alternative signature of this quantum phase transition, in perfect analogy with the case of extended bosonic systems.

## Iii Spontaneous symmetry breaking

We now discuss how to look for the BEC phase transition using the criterion of spontaneously broken gauge symmetry. The original Hamiltonian (1) is invariant under the following global gauge transformation

(6) |

which simply reflects the conservation of total particle number . In order to investigate the spontaneous breaking of this symmetry, one usually introduces a small symmetry breaking field conjugate to the order parameter. In the BEC case, the latter can be taken to be . We thus modify the Hamiltonian of Equation (1) as follows ()

(7) |

The spontaneous symmetry breaking is signaled by a non-zero value of the following limit

(8) |

(and it is a necessary and sufficient condition for the existence of BEC Yukalov2007 ). In Equation (8), .

This can be illustrated in the somewhat artificial non-interacting limit. In this case,

(9) |

Using the results of the Appendix, it can be shown that the ground state expectation values are

(10) | |||||

(11) |

where

(12) |

We note that the BEC is signaled by the vanishing of the lowest single-particle energy (when measured with respect to the chemical potential). From Equation (5), it is clear that when ,

(13) |

and the order parameter as a function of has a diverging slope as a , such that it tends to a constant in the BEC. Besides, the total number of bosons is given by (Appendix)

(14) |

The limit of Equation (8) in this case is given by

(15) |

## Iv Numerical method

We now describe the numerical procedure used in the calculations that follow. The Hamiltonian of Equation (1) will now be defined with a finite number of bath states

(16) | |||||

Note that we have already included the symmetry breaking field and it now acts on both impurity and bath states. Since the condensation occurs in a single-particle state which is a quantum mixture of all -orbitals [see Eqs. (A23) and (A24) for the non-interacting case], it is immaterial whether we couple to all of them or only to . Indeed, we have checked that applying only to the impurity site does not change the results that follow in any significant way.

The discretized Hamiltonian (16) has no conserved quantities since even the total number of bosons is no longer fixed. Since the boson spectrum is unlimited the Hilbert space has infinite dimension. We worked in a cut-off Hilbert space in which there is a maximum number of bosons

(17) |

We therefore numerically diagonalized the Hamiltonian (16) for fixed and .

The parameters uniquely determine the bath spectral function , whose support we will assume is the interval , see Equation (3). Discretizing the latter set into smaller intervals defined by where , and assuming that it follows that

(18) | |||||

(19) |

Although the use of a purely logarithmic mesh would have been enough, we have chosen to use a mixed linear-logarithmic set

(20) |

It is close to the usual logarithmic discretization of the numerical renormalization group at low energies but does a better job at describing the high-energy part of the bath spectrum. The linear discretization is recovered in the limit . In the calculations that follow, we have used .

## V Results

We will now show our results for the bosonic single-impurity Anderson model with a power-law spectral function as in Equation (3), with . This model exhibits two phases: a Bose-Einstein condensed phase (BEC) and a Mott phase, as shown in Lee2007 ; Lee2010 .

### v.1 Symmetry breaking and the phase diagram

We have computed the value of the order parameter as a function of the symmetry breaking field for a coupling to the bosonic bath of and interaction strength . According to the NRG results Lee2007 ; Lee2010 , for these values of and the system may find itself in either the Mott or the BEC phases, depending of the value of . As shown in Figure 1, for , smoothly extrapolates to zero, with a finite slope, as . This is characteristic of a non-condensed phase with no broken global gauge symmetry, the Mott phase of the model. At , however, even though still vanishes in this limit, it does so with a very large slope, effectively infinite within our numerical accuracy. This signals the boundary with the BEC phase and points to a second order phase transition, consistent with the NRG results Lee2007 ; Lee2010 . Inside the BEC phase (), the superfluid parameter has a step discontinuity (again within our numerical accuracy) across the line. The similarity with the behavior of a ferromagnet in a uniform external field is striking, highlighting the common underlying symmetry breaking mechanism in both cases.

By looking for the points of infinite slope of the versus curves one can then map out the phase diagram of the model. We have done so in the versus plane for . In practice, we have set the phase boundary as the point at which the slope reaches . The same phase diagram had been previously obtained with the much more powerful NRG method using the ‘gap closure’ criterion (see Section II) Lee2007 ; Lee2010 . The NRG parameters used were (for a purely logarithmic discretization), 10 to 20 bosonic states for each added site/iteration and 100 to 200 states kept from each iteration to the next. In our exact diagonalization method, we have used and . In Figure 2, a comparison between the results obtained with the two methods is shown. Despite the simplicity of the present procedure and small number of states retained, the agreement is remarkable. We have also verified that the susceptibility criterion we used tracks very closely the opening of the gap discussed in Section II. The assignment of average occupations to the phases, as shown in Figure 2, will be discussed later in Sec. V.2.

Another interesting quantity is the phase susceptibility,

(21) |

As discussed in Section III, this quantity is finite in the absence of BEC and diverges at the critical point at which BEC first appears, see Equations (10), (12) and (13). The analogous quantity in the case of a ferromagnet is the magnetic susceptibility, which is also finite in the paramagnetic phase and diverges at the (second-order) phase transition. Thus, we expect to be finite in the Mott phase and to diverge at the Mott-BEC boundary. That the transition is indeed second-order was confirmed in Refs. Lee2007 ; Lee2010 . This behavior is apparently consistent with the gross features of Figure 1, but it is important to determine how it is affected by our Hilbert space truncation.

In Figure 3(a), we show the inverse phase susceptibility as a function of . It can be seen that extrapolates as expected in the limit of : in the Mott phase () it tends to a finite value, whereas it diverges at the Mott-BEC boundary (their position in the phase diagram is shown in Figure 3(b)). Note the scale of the figure and how the susceptibility values are significantly different already for the largest modest used (. Note that the dependence on is extremely weak, since it modifies only slightly the single-particle orbital in which the bosons condense, whereas limits the maximum number of bosons allowed to condense.

The great differences in susceptibility values allow us to determine the phase diagram with fairly good accuracy also by fixing at a small value, say , and to a moderately large, yet numerically feasible value of 5, and scanning and . As an example of this procedure, we show in Figure 4(a) the order parameter as a function of for fixed (this corresponds to the brown vertical line of Figure 3(b)). The BEC phase is clearly demarcated from the Mott phase by a finite value of the order parameter. Note, however, that the actual numerical value of is strongly dependent on the truncation parameter . Indeed, this is clearly demonstrated in Figure 4(b), which shows a similar scan of for different values of . For small values of the order parameter shows pronounced peaks close to the phase boundaries but dips to smaller values deep inside the BEC phase. Only for sufficiently large does one recover the single hump behavior of , with a maximum value deep inside the BEC phase. Note that the criterion of a phase susceptibility of used in Figure 2 is equivalent to a value of the order parameter of in Figure 4(a). This dependence on is not unexpected since, as will be shown later, the fluctuations in the occupancy are larger in the BEC phase. Thus, the truncation at a finite value of the total number of bosons introduces important errors and correspondingly larger values of are required for a good description in this region. This allows us to estimate the error in the determination of the phase boundary as .

Furthermore, for the value of used in Figure 4, the system is always in a BEC phase for . These large values of give rise to large occupancies of the impurity orbital. As a result, the description in this region is very poor for the values of we employed, as can be seen in Figure 4(b).

### v.2 Other observables

As a test of the accuracy of our procedure, we have calculated other local observables of the impurity orbital. Whenever available, we have compared them with the NRG results Lee2007 ; Lee2010 . In Figure 5, the impurity occupancy () is shown for different values of the coupling to the bosonic bath as a function of . Our results are the full lines and the NRG results Lee2007 correspond to the symbols. The regions without any symbol indicate the BEC phases. The agreement is excellent and one can hardly distinguish the two sets of results. As the coupling to the bosonic bath decreases, the occupancy tends smoothly to the step-like behavior of the decoupled impurity, also shown Figure 5 (blue line). It is this ‘adiabatic’ continuity between the Mott phases at and which allows us to ascribe a definite occupancy to the Mott ‘lobes’ of the phase diagram (see Figure 2), even though the occupancy is never exactly an integer for , as can be seen in Figure 5. The excellent agreement shows that our truncated Hilbert space calculation is more than enough for a good description of at least some of the physical properties of the impurity model.

For completeness, in Figure 6 we show our results for both the impurity occupancy (symbols) and the order parameter squared (full line without symbols) as functions of , both in the presence of a small symmetry-breaking field. We have rescaled by a factor of ten for greater clarity. It is clear that the small symmetry-breaking field, although essential to delineate the phases, affects very little the impurity occupancy (compare with Figure 5). In this figure, the absence of symbols in the BEC region is meant to mimic the convention for this phase used in the NRG calculation Lee2007 (see Figure 5 for comparison).

Another striking feature highlighted in Figure 6 is the discrepant behavior of the impurity occupancy at the two borders of the BEC phase. Whereas at the low border exhibits a discontinuity in its first derivative, the behavior at the high border is perfectly smooth. This behavior is generic to all the other BEC phases, as can be seen in Figure 5. We conclude that although shows signs of critical behavior at the low border of the BEC phases, it is not critical at the other one.

Finally, in Figure 7(a), we show the impurity occupancy fluctuation in the () plane with a color scale. The borders between the phases are also shown as the blue lines with symbols. Three vertical cuts across this plot are shown in Figure 7(b), with the BEC region indicated by the closed symbols and the ‘Mott’ phase by the open ones. The occupancy fluctuation is generally larger in the BEC regions as compared to the ‘Mott’ phases, as expected. Indeed, this is compatible with the requirement of larger values of for a better description deep within the BEC phases (see Figure 4b).

However, a more thorough inspection shows that whereas
increases rapidly upon entering the BEC phase through its low
border, it goes through a maximum while still *inside* the BEC
phase. Upon further increasing ,
decreases until it finally crosses the high border
of the BEC phase, where it shows no sign of critical behavior, in
close similarity with the behavior of shown in Figure 6.
We stress that, even though the high transition
does not manifest itself in these quantities, does not mean that
the system does not experience a true phase transition in that region.
Furthermore, the occupancy fluctuation does not vanish and is always
a monotonically decreasing function of in the
‘Mott’ phases for .

## Vi Discussion and conclusions

In this article, we have fully characterized the diverse physical properties of the B-SIAM by using the relatively undemanding method of exact diagonalization of small clusters. Besides, we have shown how the physically motivated criterion of a spontaneously broken gauge symmetry can be used to accurately identify the BEC or ‘Mott’ phases of the B-SIAM, even with a fairly small truncated Hilbert space. This serves as a proof of principle of the criterion in this particular case. Clearly, the detailed quantum critical behavior requires the use of many more states, in which case the NRG method is probably indispensable. However, the exact diagonalization method is accurate enough for the determination of phase diagrams and physical properties, as we have shown. This is particularly important for applications such as the disordered version of BDMFT. In these cases, the use of the more accurate NRG method is prohibitive.

We have also uncovered an unnoticed asymmetry in the critical behavior of local quantities as one goes from ‘Mott’ to BEC and BEC to ‘Mott’. In the former case, both the mean occupancy and its fluctuations exhibit a discontinuity in the first derivative with respect to the impurity energy, whereas the latter transition seems to be completely smooth. This is probably a feature of the single-impurity model only, however, since the incompressible nature of the Mott phase in the lattice case requires both borders to show non-analytic behavior. The single-impurity model, on the other hand, is not incompressible in the ‘Mott’ phase.

## Acknowledgments

This work was supported by CNPq through grants 304311/2010-3 (EM) and 140184/2007-4 (JHW) and by FAPESP through grant 07/57630-5 (EM).

## Appendix A General properties of the non-interacting limit

### a.1 Diagonalization of the non-interacting model

We will first briefly describe the diagonalization of the non-interacting Hamiltonian, Equation (1) with set to zero. The absence of interactions makes this process independent of the statistics, since it amounts to finding the basis of single-particle states that makes the Hamiltonian diagonal. A more detailed calculation, in the language of fermions, can be found in reference Mahan4.2 . We define annihilation operators for the diagonal single-particle states through

(A1) | |||||

(A2) |

so that the non-interacting Hamiltonian can be written in this basis as

(A3) |

The coefficients and have yet to be determined. By taking the commutator of and with the Hamiltonian in the forms (1) and (A18), respectively, and using (A1) and (A2), we find the eigenvalue system of equations for the unknown coefficients

### a.2 Critical coupling for a power-law spectral density

The BEC in the non-interacting model occurs when the lowest eigenvalue (measured with respect to the chemical potential) vanishes. From Equation (A6), this happens when

(A9) |

We would like to analyze this equation in the continuum limit. In this case, we can use the function defined in Equation (2), which satisfies the Kramers-Kronig relation

(A10) |

in which the integral is a principal part. Thus, one may replace the right-hand side of Equation (A9) by

For the power-law spectral function of Equation (3), we are left with an expression for the critical coupling

Note that the equation is not well defined for . We finally find that

### a.3 The non-interacting Hamiltonian in the presence of a symmetry-breaking field

We now consider the non-interacting Hamiltonian in the presence of a symmetry-breaking field, Equation (9). We first define ‘displaced’ operators and

(A11) | |||||

(A12) |

where the parameters can be taken to be real without loss of generality. Inserting these into (9) we end up with

The terms linear in the new ‘displaced’ operators can be eliminated if we choose and to satisfy

(A13) | |||||

(A14) |

Taking from Equation (A14) into (A13) we find

(A15) | |||||

(A16) |

For this choice, we are left with

(A17) |

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