Two-channel pseudogap Kondo and Anderson models: Quantum phase transitions and non-Fermi liquids
We discuss the two-channel Kondo problem with a pseudogap density of states, , of the bath fermions. Combining both analytical and numerical renormalization group techniques, we characterize the impurity phases and quantum phase transitions of the relevant Kondo and Anderson models. The line of stable points, corresponding to the overscreened non-Fermi liquid behavior of the metallic case, is replaced by a stable particle-hole symmetric intermediate-coupling fixed point for . For , this non-Fermi liquid phase disappears, and instead a critical fixed point with an emergent spin–channel symmetry appears, controlling the quantum phase transition between two phases with stable spin and channel moments, respectively. We propose low-energy field theories to describe the quantum phase transitions, all being formulated in fermionic variables. We employ epsilon expansion techniques to calculate critical properties near the critical dimensions and , the latter being potentially relevant for two-channel Kondo impurities in neutral graphene. We find the analytical results to be in excellent agreement with those obtained from applying Wilson’s numerical renormalization group technique.
The two-channel Kondo effect represents a prime example of non-Fermi liquid behavior arising from a stable intermediate-coupling fixed point.nozieres1980 () Theoretically, its physics is essentially understood, thanks to an exact solution by Bethe ansatz.andrei1984 (); wiegmann1985 () In addition, boundary conformal field theory (CFT) has proved to be a powerful technique to study the low-energy properties of the multi-channel Kondo modelaffleck1991 () allowing, in particular, the calculation of exact asymptotic Green’s functions.ludwig1991 () CFT techniques have also been used to calculate exact crossover Green’s functions.sela () Further, by means of Abelian bosonization and subsequent re-fermionization it has been possible to map the two-channel Kondo problem onto a resonant-level model which reduces to a free fermion form for a particular value of exchange anisotropy.emery1992 ()
On the experimental side, a number of heavy-fermion materials, displaying deviations from Fermi-liquid behavior, have been speculated to realize two-channel Kondo physics arising from non-Kramers doublet ground states of U or Pr ions.coxzawa (); cox1987 (); schiller1998 () However, to our knowledge, there is no unambiguous verification of these proposals. Consequently, various attempts were made to realize the two-channel Kondo effect in nanostructures, and indeed success was reportedpotok () for a setup of a semiconductor quantum dot coupled to two reservoirs.oreg () Very recently, signatures of two-channel Kondo behavior of magnetic adatoms on graphene have been reported,manoharan () and this motivates to discuss two-channel Kondo impurities in non-metallic hosts. In particular, neutral graphene realizes a pseudogap density of states (DOS), with , at low energies.
The single-channel pseudogap Kondo problem has been studied extensively in the context of Kondo impurities in unconventional superconductors. The main difference to the familiar metallic Kondo problemhewson () is the absence of screening at small Kondo coupling , leading to a quantum phase transition upon increasing .withoff (); cassa (); bulla (); tolya2 (); GBI () The universality class of this phase transition changes as function of ,GBI () and the relevant low-energy field theories have been worked out in detail in Refs. VF04, ; FV04, .
Although two-channel Kondo physics has been speculated about in the context of graphene,baskaran () the two-channel pseudogap Kondo model has received little attention. A central question is about the fate and character of the non-Fermi-liquid phase at finite . To our knowledge, the only study of the model has been reported in a brief section of Ref. GBI, , but there only numerical results were given for small bath exponents .GBI_foot ()
The purpose of this paper is to close this gap: We shall investigate the two-channel Kondo and Anderson models with a pseudogap DOS in some detail, using both analytical and numerical renormalization group (RG) techniques. Our main findings for the two-channel Kondo model are:
(A) The overscreened non-Fermi liquid (NFL) phase of the metallic two-channel Kondo model hewson () survives for , albeit with an important modification: It is no longer represented by a line of NFL fixed points (where particle–hole (p-h) asymmetry is marginal), but instead there is only an isolated stable p-h symmetric NFL fixed point, i.e., p-h asymmetry is irrelevant for .GBI_foot () Furthermore, this stable NFL phase is only reached for Kondo couplings larger than a critical coupling, i.e., a boundary quantum phase transition emerges between a local-moment phase with an unscreened spin moment and the NFL phase. In contrast, for the NFL fixed point disappears, leaving only two stable phases with unscreened spin or channel (i.e. flavor) moment, respectively, which are separated by a quantum phase transition.
(B) The two-channel pseudogap Kondo physics for both and can be fully understood in the language of the two-channel Anderson model, by virtue of a generalization of the approach presented in Ref. FV04, . The low-energy field theory describing the quantum phase transition between the phases with free spin and flavor moments is given by a level crossing of a spin doublet and a flavor doublet minimally coupled to conduction electrons. Similar to the single-channel pseudogap Kondo problem, is found to play the role of an upper-critical dimension, where the hybridization is marginal. For , the transition is a level crossing with perturbative corrections, whereas a non-trivial critical fixed point emerges for . This fixed point is shown to display an emergent spin–channel Z symmetry. As in the single-channel case, none of the quantum phase transitions is described by a Landau-Ginzburg-Wilson-type theory of a bosonic order parameter, instead all are “fermionic” in nature.
The following subsection gives a more detailed summary of our results.
i.1 Summary of results
The two-channel Kondo model with a pseudogap host density of states can be written as , with
Here, we have represented the bath, , by linearly dispersing chiral fermions , where is the channel index. is a spin-1/2 SU(2) spin, is the vector of Pauli matrices, summation over repeated spin indices is implied, and is the conduction electron operator at the impurity site. The spectral density of the fermions follows the power law below the ultra-violet (UV) cutoff ; details of the density of states at high energies are irrelevant for the discussion in this paper. In addition to the Kondo coupling , we have also included a potential scatterer of strength at the impurity site which will be used to tune p-h asymmetry. (An asymmetry of the high-energy part of the DOS would have a net effect similar to non-zero ; for simplicity we will assume in the following that the DOS is p-h symmetric.)
As we shall show below, a comprehensive analysis requires to consider – in addition to the two-channel Kondo model – the two-channel Anderson model, commonly written as withcoxzawa ()
Here, the isolated impurity has four states, i.e., a spin doublet and a channel doublet . Their mass difference, , will play a role as a tuning parameter of the quantum phase transition.
For a given value of the bath exponent , the two-channel pseudogap Kondo and Anderson models display common RG fixed points. The phase diagram and critical behavior depend on , with , , and marking qualitative changes and playing the role of critical “dimensions”. In the following, we describe our central results for the phase diagrams and RG flows, which are partially consistentGBI_foot () with the ones reported in Ref. GBI, . The qualitative behavior is visualized in the RG flow diagrams in Fig. 1 for the two-channel Kondo model and Fig. 2 for the two-channel Anderson model, respectively. In the latter case, a cut through the RG flow at is shown.
The metallic case , has been studied extensively, and a line of infrared stable NFL fixed points governs the behavior at any finite coupling – this is the well-known two-channel (or overscreened) Kondo effect. In the two-channel Anderson model, this line of fixed points can be accessed by varying , i.e., initial parameters with different flow to different fixed points along this lineandrei02 () – note that this flow leaves the plane for (dashed in Fig. 2, all symbols denote the renormalized coupling parameter).
For positive with , the line of stable NFL fixed points collapses to an isolated p-h symmetric NFL fixed point. In addition, the local-moment fixed point (LM) of an unscreened spin moment now becomes stable. In the language of the Kondo model, LM corresponds to , while in the Anderson model it corresponds to , . The phase transition between LM and NFL is controlled by a critical p-h symmetric fixed point (SCR); note that this p-h symmetric fixed point is located outside the plane for the Anderson model shown in Fig. 2. As SCR approaches LM and the critical behavior of SCR is perturbatively accessible for small Kondo coupling . The phase diagram of the Anderson model is mirror symmetric, i.e., there exists also an unscreened channel (or flavor) local-moment fixed point LM at , asc_note () and a corresponding critical intermediate-coupling fixed point SCR at positive .
As SCR approaches NFL, and the two fixed points disappear for . In the p-h symmetric Kondo model, this implies that the flow is towards LM for any value of , but for large asymmetries, LM may be reached.asc_note () In the Anderson model, the hybridization remains relevant at for , but the flow is towards a single unstable intermediate coupling fixed point (ACR) in the plane, i.e., ACR is p-h asymmetric, but is invariant under the transformation (4). At finite coupling, the transition between the two stable fixed points LM and LM is controlled by ACR.
Finally, as ACR moves toward and for the phase transition becomes a level crossing with perturbative corrections, controlled by the free impurity fixed point (FImp) at .
The bulk of this paper is organized as follows: We start in Sec. II by discussing the relevant impurity models, suitably generalized to higher degeneracies, along with their underlying symmetries. In Sec. III we present selected results from Wilson’s numerical renormalization group (NRG) for the two-channel pseudogap Anderson and Kondo models which illustrate the content of the flow diagrams in Figs. 1 and 2. In particular, we show properties of the non-trivial intermediate-coupling fixed points as function of the bath exponent . Secs. IV and V are devoted to the epsilon expansion studies of the critical fixed points, using the variables of the Kondo model (Sec. IV) and that of the Anderson model (Sec. V). The latter provide access to the physics near the upper-critical dimension . Concluding remarks will close the paper. A discussion of the Majorana representation of the two-channel Kondo problem and its fate in the presence of a pseudogap DOS is relegated to the appendix.
Ii Models, symmetries, and mappings
ii.1 Anderson model
The two-channel Anderson model can be understood as describing the level crossing between two impurity doublets – one spin doublet and one channel (i.e. flavor) doublet – coupled to conduction electrons via a hybridization term. The model features an SU(2) SU(2) symmetry. This can be straightforwardly generalized to SU() SU() symmetry, where is the number of spin degrees of freedom and the number of flavors. The Hamiltonian can be written as:
Here, the conduction electrons transform under a fundamental representation of SU and SU and carry the corresponding spin and flavor indices. indicates a transformation behavior according to the conjugate representation. For , , the Hamiltonian in Eq. (II.1) describes the single-channel Anderson model in the limit of infinite Coulomb repulsion (), studied using RG in Ref. FV04, . In case of a metallic host, , the multi-channel Anderson model is integrable and has been solved using the Bethe Ansatz bolech2005 (); andrei02 (); andrei03 () and the numerical RG method.anders2005 ()
The metallic two-channel Anderson model, i.e. , , has been proposed as a model for the observed non-Fermi liquid behavior of the heavy-fermion superconductor UBe.stewart1983 () In this scenario, the 5f ground state of the U ion is identified as the non-magnetic quadrupolar doublet, while the first excited state is the 5f magnetic doublet.coxzawa () This then can promote a quadrupolar Kondo effect where the quadrupolar doublet is quenched by the hybridization with conduction electrons which carry both magnetic and quadrupolar degrees of freedom.cox1987 () In particular, since the energy difference between the two doublets appears to be small, a mixed valence state is likely requiring the study of the full Anderson model.aliev1995 () Consequently, the model (II.1) with a pseudogap DOS is of potential relevance not only to two-channel impurities in graphene, but also to quadrupolar Kondo impurities in unconventional superconductors.
The Anderson model (II.1) is not particle–hole symmetric for any value of , due to the asymmetric structure of the impurity. However, p-h symmetry is dynamically restored for both inside the NFL/NFL phases and at the critical SCR/SCR fixed points, see Fig. 1.
Interestingly, for and a p-h symmetric bath, the Anderson model displays a spin–channel symmetry, i.e., is invariant under the combined transformation
Here, the spin-carrying impurity states are transformed into the flavor-carrying states and vice versa, i.e., the two SU() sectors are interchanged, together with a p-h transformation.
ii.2 Kondo models
The Anderson model (II.1) has two Kondo limits. On the one hand, for it maps to a -channel SU()-Kondo model, where a spinful impurity is coupled to channels of conduction electrons. For the Hamiltonian reads
where is a spin-1/2 SU(2) spin and . For the impurity spin is in a fundamental representation of SU(). The parameters of the Kondo model (II.2) are related to that of the Anderson model (II.1) through:
The Kondo limit is reached by taking , , keeping fixed. Note that a potential scattering term is always generated.
On the other hand, for the Anderson model can be mapped to a -channel SU()-Kondo model, where represents a SU() impurity which is screened by the spin degrees of freedom of the conduction electrons. Such multi-channel flavor Kondo effect is relevant, e.g., to the charging process of a quantum box, where the flavor degree of freedom is taken by the physical charge.matveev (); LSA03 ()
We note that the multi-channel Kondo model cannot be obtained by a Schrieffer-Wolff transformation from any standard Anderson model (i.e., written with free-electron operators and local Coulomb interaction).
ii.3 Large- limit
The SU() multi-channel Kondo model can be solved in a dynamic large- limit for both fully symmetric (bosonic) and fully antisymmetric (fermionic) representations of SU().page () The fermionic version of this solution, with and , has been generalized to the pseudogap case.mv01 () The large- phase diagram, Fig. 1 of Ref. mv01, , is similar to that of the case, i.e., the overscreened non-Fermi liquid phase survives for small , where it is reached for a certain range of couplings only, while this phase disappears for larger . Also, all leading anomalous dimensions vanish for .
Two qualitative differences between the large- scenario and are worth noting: (i) The critical “dimension” of splits into two in the large- limit, with their values and the detailed behavior depending upon the value of . (ii) The quantum phase transitions in the large- limit are governed by lines of fixed points, with continuously varying exponents as function of the particle–hole asymmetry, in contrast to the isolated critical fixed points SCR and ACR in Figs. 1 and 2. This implies that the scaling dimension of vanishes at criticality as , rendering the large- limit partially singular. We therefore refrain from a detailed discussion of the models (II.1) and (II.2) for large .
In the bulk of the paper, we will focus on a few important observables which characterize the phases and phase transitions of the impurity models under consideration. Those include the correlation-length exponent, the impurity entropy, various susceptibilities, and the conduction-electron T matrix (or impurity spectral function). Their definition is standard, and we refer the reader to Refs. GBI, ; vbs, ; mvrev, ; MKMV, ; FV04, for a detailed exposition. Here we only summarize a few key aspects.
Spin susceptibilities, , are obtained by coupling external magnetic fields both to the bulk and impurity degrees of freedom as explained in detail in Ref. MKMV, . For the impurity part, here, this reads
where is the magnetic field coupling to the impurity spin, while the with are generators of SU(). In the following, we exploit the SU() symmetry and only evaluate the corresponding susceptibility tensor in the -direction choosing the representation .georgi_book () We proceed as usual by calculating the magnetic susceptibilities via the corresponding linear response functions. Note that the impurity susceptibility is composed of
where is the response to , measures the bulk response to the field applied to the bulk, are the cross terms, and denotes the bulk response in the absence of the impurity. Flavor susceptibilities, , can be defined in the Anderson model in analogy to the spin susceptibilities (i.e., with ).
Owing to symmetries, the total magnetization in both the spin and flavor sectors is conserved. This implies that the impurity contributions to the spin and flavor susceptibilities, and , do not acquire anomalous exponents at the intermediate-coupling fixed points, but instead obey Curie laws with (in general) fractional prefactors. In contrast, the local spin and flavor susceptibilities follow anomalous power laws, and , with universal -dependent exponents . We note that a direct calculation of both susceptibilities is only possible in the Anderson model, as the Kondo limit suppresses the local piece of one of the susceptibilities. To shorten notation, we employ the convention and in the following.
Similar to , the impurity entropy approaches a universal fractional value as . The conduction-electron T matrix, on the other hand, follows an anomalous power law similar to the local susceptibility, .
At the non-Fermi-liquid fixed point of the familiar metallic two-channel Kondo model (), power laws are replaced by logs, – this also implies that the prefactor of the leading Curie term in vanishes due to an exact compensation.
Iii Selected numerical results
The NRG techniqueNRGrev () is ideally suited to study properties of quantum impurity models, including non-Fermi liquid phases and quantum phase transitions. Initial NRG results for the two-channel pseudogap Kondo model were shown in Ref. GBI, . Here we extend and complement this early analysis by NRG results for the two-channel () Anderson model. We perform explicit calculations for a bath density of states with . Unless otherwise noted, we employ NRG parametersNRGrev () and .
The qualitative behavior of the two-channel Anderson model is summarized in the flow diagram in Fig. 2. In addition to the stable LM/LM phases, these flow diagrams feature three non-trivial fixed points: the stable NFL/NFL fixed points and the critical fixed points SCR/SCR and ACR. Some of their key properties are summarized in Figs. 3 and 4, which show the numerically determined impurity contributions to the spin susceptibility and the entropy, respectively, together with analytical results obtained from the epsilon expansion of Secs. IV and V. These plots nicely show that NFL and SCR approach each other as , while ACR evolves continuously near . The fixed-point properties also show that SCR approaches LM as , with and , and ACR approaches FImp as , with and . Further, the stable NFL fixed point follows and as – the well-known properties of the metallic two-channel Kondo problem.
The disappearance of both NFL and SCR upon increasing beyond implies a discontinuous evolution of the phase diagram as function of . In Fig. 5 we present a cut through the phase diagram of the Anderson model at fixed which illustrates this fact. We note that such a discontinuous evolution occurs if a stable intermediate-coupling fixed point disappears (here NFL); in contrast, if a trivial fixed point changes its nature from stable to unstable, the evolution is continuous, like in the single-channel pseudogap Kondo model.
The correlation-length exponents are displayed in Fig. 6, illustrating that and play the role of lower-critical dimensions for the p-h symmetric transition controlled by SCR, with , whereas is the upper-critical dimension of the ACR transition, with for all (and logarithmic corrections at ).
Fig. 7 shows the temperature evolution of both and for , i.e., slightly below . Here, the flow from ACR to NFL (compare Fig. 1b) is nicely visible at where both and increase along the RG flow. (The large value of at ACR renders the flow very slow.) Remarkably, the fact violates so-called -theorem,gtheorem () which states that the impurity entropy should decrease along the flow. As this theorem applies to conformally invariant systems only, we conclude that the fixed points under consideration are not described by a conformally invariant theory. (The same conclusion can be drawn for the quantum phase transitions of the single-channel pseudogap Kondo problem, see Ref. FV04, . Also, such “uphill flow” may occur in models with long-range interactions, see e.g. Ref. uphill, .)
Iv Weak-coupling expansion for the multi-channel Kondo model
In this section we review the standard weak-coupling expansionpoor () for the multi-channel SU()-Kondo model in Eq. (II.2), extended to a pseudogapped bath density of states.withoff (); GBI () As we show below, this expansion captures the properties of the critical fixed point SCR at small . In principle, it also allows to access the stable NFL fixed point, but this requires a particular limit of large which does not allow to extract quantitative results for the case of interest.
RG equations can be derived, e.g., using the field-theoretic schemebgz (); lars_book () where logarithmic divergencies, occurring for , are replaced by poles in by means of “dimensional” regularization. Doing so, will only enter in the bare scaling dimension of the couplings. To two-loop order, the equations for the renormalized couplings and read
where is the number of equivalent screening channels. Importantly, there is no renormalization of , a result which persists to higher orders.
Apart from the LM fixed point, , the function in Eq. (9) yields two further zeros, given by . The smaller one corresponds to an infrared unstable fixed point at
which can be perturbatively controlled for and any . We label this fixed point by SCR. As SCR approaches LM. The larger zero predicts an infrared stable fixed point at
Strictly speaking, this fixed point is perturbatively accessible only if the limits and are either taken in this order (this corresponds to ) or together such that is kept fixed. For , where is marginal, this zero of is commonly associated with the line of stable non-Fermi liquid fixed points of the multi-channel Kondo model, which exists for all . For now becomes irrelevant, in agreement with our numerical resultsGBI_foot () which show that the NFL line of fixed points shrinks to a single p-h symmetric NFL fixed point, Fig.1.
iv.1 Observables near criticality
The properties of the p-h symmetric critical fixed point SCR, existing for , can be determined in analogy to the single-channel case, with explicit calculations given e.g. in Ref. MKMV, . Expanding the beta function (9) around the fixed point value (10) yields the correlation length exponent :
The leading-order perturbative corrections to the impurity susceptibility and entropy are given by
Inserting the fixed-point value (10) we obtain
The anomalous exponent of the local susceptibility is given by to leading order, which evaluates to
Finally, the T matrix exponent is (with no factor of , as the T matrix describes the scattering of electrons from one specific channel), resulting in
Note that this result is exact.FV04 ()
V Hybridization expansion for the multi-channel Anderson model
We now turn our attention to the multi-channel Anderson model. As we show below, the variables of the Anderson model will allow us to obtain an essentially complete understanding of the multi-channel pseudogap Kondo effect both for and . A similar conclusion was reached for the single-channel case in Refs. VF04, ; FV04, , and – on a technical level – our work represents a generalization of the calculation for the infinite- Anderson model in Ref. FV04, .
v.1 Trivial fixed points
For vanishing hybridization , the multi-channel Anderson model (II.1) features three trivial fixed points: for the ground state is the spinful -fold degenerate local-moment state (LM) and, analogously, for it is the -fold degenerate flavor local-moment state (LM). In these cases the impurity entropy equals and , respectively. For there are degenerate impurity states, we refer to this as the free-impurity fixed point (FImp), with entropy . The impurity spin susceptibilities are
The corresponding values of follow via Eq. (4) from LMLM. The hybridization term, , is irrelevant at LM and LM for .
v.2 Hybridization expansion and upper critical dimension
In the following we perform an expansion around the FImp fixed point, i.e., around , .
The impurity states are represented by bosonic operators for and fermionic operators for . Single occupancy of the localized levels is enforced by the Hilbert space constraint which will be implemented using a chemical potential . Observables are then calculated as lambda (); costi ()
where denotes the thermal expectation value in the presence of the chemical potential .
Furthermore, we need to introduce chemical-potential counter-terms which cancel the shift of the critical point occurring in perturbation theory upon taking the limit of infinite UV cutoff. Technically, this shift arises from the real parts of the self-energies of the and particles. We introduce the counter-terms as additional chemical potential for the auxiliary particles,
The have to be determined order by order in an expansion in . Note that counter-term contributions in observables in general enter both numerator and denominator in Eq. (19).
In the path integral form the model (II.1) is written as
where is the chemical potential enforcing the constraint exactly. Here, we implicitly sum over and .
The model (21) shows a transition driven by variation of . At the fixed point, tree level scaling analysis shows that
where is the renormalization group energy scale. No renormalizations are needed for the bulk fermions as their self interaction is assumed to be irrelevant. The RG is conveniently performed at criticality, i.e. we assume that is tuned to the critical line and set .
To determine the RG beta function we evaluate the fermionic self-energy up to one-loop order
corresponding to the diagram in Fig. 8a. Here, we have introduced the abbreviated notation for the Matsubara frequencies . In the limit and we obtain
An analogous expression is found for the bosonic self energy depicted in Fig. 8b.
The renormalization factors are determined such that they cancel the pole in the self-energies minimally and render the inverse Green’s function finite. We thus find
The mass counter-terms are given by the real parts of the self-energies
To one-loop order, there is no vertex renormalization of , hence we have =1 at this order (note that a diagram does not exist due to the directed nature of the propagators). The beta function
can now be obtained by taking the logarithmic derivative of Eq. (25). Since we can solve for and finally obtain
to one-loop order. One can also consider the flow away from criticality, i.e. the flow of the renormalized tuning parameter using insertions. The resulting correlation lengths exponent is given in Eq. (38) below.
For the trivial fixed point is unstable, and the critical properties are instead controlled by an interacting fixed point (labelled ACR) at
with anomalous field dimensions
The corresponding RG flow diagram is displayed in Fig. 2c.
ACR describes a quantum phase transition below its upper-critical dimension. As a result, low-energy observables calculated at and near ACR will be fully universal, i.e., cutoff-independent, and hyperscaling is fulfilled.
For all , i.e., above the upper-critical dimension, the phase transition between LM and LM is now controlled by the non-interacting FImp fixed point at . Hence, for all the phase transition is a level crossing with perturbative corrections – this results e.g. in a jump of the order parameter (see below), i.e., the transition is formally of first order. Consequently, hyperscaling is violated, and all observables will depend upon the UV cutoff.
For the marginal case, , we expect a logarithmic flow of the marginally irrelevant hybridization , characteristic of the behavior at the upper-critical dimension. The RG beta function
can be integrated (recall where describes the reduction of the UV cutoff ) to give
with . This result can be used to determine logarithmic corrections to observables.
v.5 Observables near criticality
We start with the correlation-length exponent, , of the ACR fixed point. This exponent describes the vanishing of the characteristic crossover temperature in the vicinity of the critical point . The lowest-order result for , which can be obtained either using the field-theoretic RG scheme via composite operator insertions or using the familiar momentan shell scheme, is
For the transition is a level crossing, formally .
The local susceptibility at the critical point follows the scaling behavior with an anomalous exponent . To obtain the corrections to the tree-level result we introduce a renormalization factor from which one obtains the anomalous exponent according to
We determine by calculating directly using perturbative corrections up to quadratic order in . The corresponding diagrams entering are given in Fig. 9.
In terms of the renormalized coupling constant we find at the energy scale
Furthermore, the denominator receives corrections from the diagrams in Fig. 10, resulting in
The local susceptibility can then be directly obtained by Eq. (19). The renormalization factor is then determined, using minimal subtraction of poles, in an expansion in as
and from this we can directly deduce the anomalous exponent of the local spin suceptibility
The expression for follows by the replacement .
Inside the stable phases LM and LM, can be used to define an order parameter for the quantum phase transition: is finite (zero) for (), similarly is finite (zero) for (). Approaching the critical point, both order parameters vanish continuously according to
for , which follows e.g. from hyperscaling. Note that these order parameters display a jump upon crossing the transition for .
The evaluation of the impurity susceptibility to second order in requires the summation of further diagrams as depicted in Fig. 11. In terms of the renormalized coupling we obtain
Collecting all contributions to to second order in the poles present in the diagrams cancel and the remaining momentum integrals are UV convergent for . Performing these integrals for the impurity susceptibility reads
As above, the expression for follows by the replacement . With the value of the coupling at the ACR fixed point (33) we finally find for , to leading order in ,
with due to the emergent symmetry (4). A comparison to NRG data is in Fig. 3. Note that receives only weak additive logarithmic corrections at ; multiplicative logs as in are absent here. The same applies to below.
The impurity contribution to the entropy can be obtained from the thermodynamic potential by . At the FImp fixed point the entropy is , and the lowest-order correction is computed by expanding the thermodynamic potential in the renormalized hybridization . Note that this correction vanishes for , as there. Here, we write the partition function in the physical sector of the Hilbert space () asMKMV ()
where is the expectation value in the presence of without coupling to the bath. The thermodynamic potential is given by
The correction to due to the coupling to the bath up to quadratic order in has already been calculated in Eq. (V.5.2) which enables us now to directly evaluate Eq. (51). Taking the temperature derivative of the resulting expression and evaluating the remaining integral in the limit and