Quantum criticality in the pseudogap Bose-Fermi Anderson and Kondo models: Interplay between fermion- and boson-induced Kondo destruction
We address the phenomenon of critical Kondo destruction in pseudogap Bose-Fermi Anderson and Kondo quantum impurity models. These models describe a localized level coupled both to a fermionic bath having a density of states that vanishes like at the Fermi energy () and, via one component of the impurity spin, to a bosonic bath having a sub-Ohmic spectral density proportional to . Each bath is capable by itself of suppressing the Kondo effect at a continuous quantum phase transition. We study the interplay between these two mechanisms for Kondo destruction using continuous-time quantum Monte Carlo for the pseudogap Bose-Fermi Anderson model with and , and applying the numerical renormalization-group to the corresponding Kondo model. At particle-hole symmetry, the models exhibit a quantum critical point between a Kondo (fermionic strong-coupling) phase and a localized (Kondo-destroyed) phase. The two solution methods, which are in good agreement in their domain of overlap, provide access to the many-body spectrum, as well as to correlation functions including, in particular, the single-particle Green’s function and the static and dynamical local spin susceptibilities. The quantum-critical regime exhibits the hyperscaling of critical exponents and scaling in the dynamics that characterize an interacting critical point. The plane can be divided into three regions: one each in which the calculated critical properties are dominated by the bosonic bath alone or by the fermionic bath alone, and between these two regions, a third in which the bosonic bath governs the critical spin response but both baths influence the renormalization-group flow near the quantum critical point.
pacs:71.10.Hf, 71.27.+a, 75.20.Hr
Quantum criticality is currently being pursued in a number of strongly correlated materials ranging from intermetallic rare earth compounds to insulating magnets and even engineered systems. This interest is spurred in part by the observation that unconventional superconductivity commonly occurs near quantum critical points, as well as by mounting experimental evidence of the inadequacy of the traditional theoretical approach to continuous transitions at absolute temperature . This approach, commonly referred to as the spin-density-wave or Hertz-Millis-Moryia picture, is based on an extension to zero temperature of Landau’s theory of order-parameter fluctuations.Sachdev (1999) The evidence for the breakdown of the spin-density-wave picture is particularly compelling in the context of intermetallic rare-earth compounds near an antiferromagnetic instability. As a result, a major thrust in the quest for a theoretical framework of critical phenomena beyond the Landau-Ginzburg-Wilson paradigm has centered around the question of how Kondo screening can be destroyed at a magnetic-ordering transition.
An extensively discussed alternative to the spin-density-wave picture is local quantum criticality,Coleman et al. (2001); Q. Si, S. Rabello, K. Ingersent and J. L. Smith (2001) in which Kondo screening itself becomes critical precisely at the magnetic-ordering transition. In the vicinity of such a Kondo-destruction quantum critical point (QCP), scaling properties of the order parameter are very different from those predicted by spin-density-wave theory. For example, inelastic neutron-scattering experiments have shown that the dynamical spin (order-parameter) susceptibility exhibits frequency-over-temperature (or ) scaling, and displays a fractional exponent in the frequency and temperature dependences over a wide-range of momentum space.Schröder et al. (2000) In addition, Hall measurements have provided evidence that the single-particle relaxation rate is linear in temperature,Friedemann et al. () a behavior that arises naturally from scaling of the single-particle Green’s function.
A microscopic model of local quantum criticality in heavy-fermion metals has been providedQ. Si, S. Rabello, K. Ingersent and J. L. Smith (2001); Si et al. (2003) through extended dynamical mean-field theory, which maps the Kondo lattice model to a single magnetic impurity coupled to both a fermionic conduction band and to one or more bosonic baths representing collective spin fluctuations. The critical point in the lattice model is then captured in terms of a critical destruction of the Kondo effect in the single-impurity problem. Such a construction gives rise to a QCP that is spatially local but has interacting critical modes in imaginary time, and that correctly captures scaling in the order-parameter susceptibility. While the full extended dynamical mean-field treatment simulates the lattice through self-consistent determination of the band and bath densities of states, valuable insight can be gained by studying Kondo-destruction QCPs in pure impurity problems.
This work investigates the quantum-critical destruction of the Kondo effect in pseudogap variants of the Ising-symmetric (or easy-axis) Bose-Fermi Anderson (BFA) and Bose-Fermi Kondo (BFK) models. In each model, a local degree of freedom couples to a band of conduction electrons having a density of states that vanishes as on approach to the Fermi energy (). The local degree of freedom is also coupled via the component of its spin to a bosonic bath having a density of states proportional to for frequencies up to some cutoff (i.e., for ).
In the presence of a metallic conduction band (corresponding to an exponent ), the BFA and BFK models with a sub-Ohmic bosonic bath characterized by a bath exponent feature a second-order quantum phase transition (QPT)Si and Smith (1996); Smith.99 (); Sengupta.00 (); SiSmithIngersent.99 (); Zhu and Si (2002); Zaránd and Demler (2002); Glossop and Ingersent (2005, 2007) between a Kondo-screened (fermionic strong-coupling) phase and a localized (Kondo-destroyed) phase in which the bosons asymptotically suppress spin-flip scattering and a residual impurity moment survives to . For , the Ising-symmetry BFA and BFK models are thoughtZhu and Si (2002); Zaránd and Demler (2002); Glossop and Ingersent (2005, 2007) to share the same critical properties as the corresponding sub-Ohmic spin-boson model, which features an interacting QCP characterized by critical exponents that vary continuously with the bath exponent .
In the absence of the bosonic bath, by contrast, the pseudogap BFA and pseudogap BFK models reduce, respectively, to the pseudogap Anderson and pseudogap Kondo models, in which the depression of the low-energy density of states impedes the formation of a many-body Kondo screening cloud and gives rise to QPTsWithoff.90 (); Bulla.97 (); Gonzalez-Buxton and Ingersent (1998); Ingersent and Si (2002); Fritz.04 (); Glossop et al. (2011) between a Kondo phase and a localized phase in which the impurity exhibits a free spin- at . The QPTs in the pseudogap Anderson and Kondo models belong to the same universality classGonzalez-Buxton and Ingersent (1998); Ingersent and Si (2002); Fritz.04 (); Glossop et al. (2011) and are described by critical exponents that vary with the band exponent . The pseudogap versions of the BFA and BFK models offer fascinating possibilities for nontrivial interplay between two different mechanisms for destruction of the Kondo effect; an interplay that we investigate in this work.
An SU(2)-symmetric version of the pseudogap BFK model (in which the Cartesian components of the impurity spin couple to different bosonic baths sharing the same exponent ) has been studied via perturbative renormalization-group (RG) methods,Vojta and Kirćan (2003); Kirćan and Vojta (2004) while the Ising-symmetric case of bosonic coupling to the component of the impurity spin has been the subject of preliminary investigation using the numerical renormalization group (NRG).Glossop et al. (2008) A spinless variant of the model (coupling the impurity charge to a bosonic bath) has been addressed by perturbative and numerical RG techniques.Chung et al. (2008) In all these previous studies, the order-parameter susceptibility has been found to exhibit an anomalous variation in the quantum-critical regime with an exponent independent of the band exponent . One of the objectives of the present work is to investigate more carefully the universality of this observation.
Another motivation for the present study is to expand our understanding of the conditions that lead to - or dynamical scaling of critical correlation functions near unconventional quantum criticality. It is well known that a quantum impurity model with a bulk component that is conformally invariant can be described by a boundary conformal field theory.Affleck.91 (); Cardy.84 () In any such theory, a conformal mapping can be used to obtain correlators at temperatures from their counterparts. In particular, the zero-temperature two-point correlator of a primary conformal field with scaling dimension exhibits a power-law decay . This gives rise, via a conformal mapping, to the scaling formGinsparg:89 (); Tsvelik:96 (); units ()
The Fourier transform of Eq. (1) can be performed analytically and yields a function of that can be analytically continued to real frequencies providednote1 () that . Therefore, demonstration that a system obeys Eq. (1) with is sufficient to show the presence of scaling, a characteristic feature of interacting QCPs.
In the pseudogap Anderson model, the conduction-band density of states breaks conformal invariance, while in the metallic () BFK model, the conduction band is conformally invariant but the bosonic bath is not. Nonetheless, local correlators of each model have been foundKirchner and Si (2008); Glossop et al. (2011); Pixley et al. (2012) to exhibit a boundary conformal scaling form in imaginary time, consistent with Eq. (1); such properties have been attributed to an enhanced symmetry at the QCPs. It is an interesting question whether the scaling form Eq. (1) also applies to the critical correlators of the pseudogap BFA and BFK models, where the densities of states of the fermionic and bosonic baths both break conformal invariance in the bulk.
We address these issues through a combination of techniques. For the particle-hole-symmetric pseudogap BFA model, we use a continuous-time quantum Monte Carlo (CT-QMC) method,Gull et al. (2011); J. H. Pixley et al. (2010) which stochastically samples a perturbation expansion in the Anderson hybridization, to probe static and dynamical quantities. We also use the Bose-Fermi extension of the NRG methodGlossop and Ingersent (2005, 2007) to resolve the critical spectrum and extract static critical exponents of the pseudogap BFK model. Our results show that the two models are in the same universality class.
We find that within different ranges of the exponents and , the measured critical exponents are determined by the fermionic band alone, by the bosonic bath alone, or by both the fermions and the bosons. We show that both the single-particle Green’s function and the local spin susceptibility obey the scaling form of Eq. (1) with exponents , proving that each correlator obeys scaling in both the quantum coherent () and the relaxational () regimes. Agreement with NRG results for static quantities confirms the ability of the CT-QMC approach to study quantum-critical properties of models involving bosons.
The remainder of the paper is organized as follows: Section II introduces the pseudogap Bose-Fermi models and Sec. III briefly describes the numerical methods used to solve these models. An overview of the phase diagram in Sec. IV is followed in Sec. V by a detailed description of the quantum-critical properties. These results are summarized and discussed in Sec. VI.
The Bose-Fermi Anderson impurity model with Ising-symmetric bosonic coupling is described by the Hamiltonian
where annihilates a conduction-band electron with wave vector , energy , and spin component with (or ) or (or ); annihilates an impurity electron with energy and spin component ; ; annihilates a boson of energy ; and is the number of unit cells in the host (i.e., the number of distinct points). The other energy scales entering Eq. (II) are the Coulomb repulsion between two electrons in the impurity level, the local hybridization between the impurity level and the conduction band, and the coupling between the component of the impurity spin and the bosonic bath. We focus in this paper on cases corresponding to particle-hole-symmetric impurities, but briefly discuss the effect of breaking this symmetry in Sec. V.4.
Over a wide region of its parameter space, the low-energy properties of can be mappedJ. H. Pixley et al. (2010) via a Schrieffer-Wolff transformation onto the Ising-symmetry Bose-Fermi Kondo Hamiltonian:
where is the Kondo coupling, parameterizes nonmagnetic potential scattering, and
is the on-site conduction-band spin, with being a vector of Pauli spin matrices. For , the potential scattering vanishes for electrons on the Fermi surface, and we can set . Although the bare Kondo exchange term is SU(2) symmetric, the bosonic coupling breaks spin rotational invariance. As a result, the RG description of the BFK model in terms of renormalized couplings requires consideration of an anisotropic exchange .
For both models, we assume a conduction-band (fermionic-bath) density of states
with a power-law pseudogap described by , and a sub-Ohmic bosonic bath specified by
with . The pseudogap density of states leads to a BFA model with the hybridization function , where .
Various limiting cases of Eqs. (II) and (3) have been studied previously. For , the pseudogap Bose-Fermi models simplify to their pure-fermionic counterparts, in which a pseudogap critical point separates Kondo and free-moment phases; at particle-hole symmetry, this critical point exists only for (Ref. Gonzalez-Buxton and Ingersent, 1998). In the absence of the conduction band, and both reduce to the sub-Ohmic spin-boson model in zero transverse field, which has two degenerate ground states in which the bosonic coupling localizes the impurity either in its up- or down-spin configuration. For , the pseudogap BFA and BFK models reduce to their metallic counterparts where the critical properties for Ising symmetry are thoughtZhu and Si (2002); Zaránd and Demler (2002); Glossop and Ingersent (2005, 2007) to coincide with those of the spin-boson model for . In the sections that follow we show that the quantum criticality in the full pseudogap Bose-Fermi models described by Eqs. (II) and (3) falls into one of three distinct types, depending on the values of the bath exponents and , with one of these types being governed by a mixed Bose-Fermi QCP unlike those seen in any of the limiting cases.
iii.1 Continuous-time quantum Monte Carlo
This work uses a form of the CT-QMC method described in Ref. J. H. Pixley et al., 2010. To make suitable for application of the approach, we apply a canonical transformation to eliminate the term linear in bosonic operators. This is achieved by the generator , which transforms Eq. (II) to
where , , and . Physically, the canonical transformation can be viewed as dressing the impurity with a bosonic cloud, which in turn renormalizes , , , and without breaking particle-hole symmetry.
We are now in a position to calculate the partition function by expanding in the hybridization . This is similar to the approach in Refs. Werner and Millis, 2007 and Werner and Millis, 2010, except that we couple the bosonic bath to the component of the impurity’s spin rather than to its occupancy. [An extension of this method to the case of an SU(2)-symmetric spin-boson coupling has recently been proposed.Otsuki (2010)] The resulting perturbation expansion is then sampled stochastically using a Metropolis algorithm. Tracing out the fermionic band, which is unchanged by the presence of bosons,Gull et al. (2011) allows the partition function to be recast as , where at perturbation order there are segments along the imaginary time axis, each one defined by a pair of renormalized impurity operators, and . The weight of a particular configuration of segments is , where is the weight of the band fermions, is the weight of the local configuration and is the weight of the bosonic bath. can be obtained as a product of determinants and is calculated in terms of the length and overlap of imaginary time segments; for details see Ref. Gull et al., 2011. If the operators in a given segment configuration are time-ordered, with () indicating that the operator acts at time to create (annihilate) an electron of spin component , then the bosonic weight is , where . Tracing out the bosons and performing the momentum summation yields
To calculate the function we replace the hard cut-off of the bosonic spectral function in Eq. (6) by an exponential cut-off, , which yields
where is the gamma function and is the Hurwitz zeta function.Weiss (2008) Now the local update procedure is identical to that in Ref. Werner and Millis, 2007. In all the CT-QMC calculations reported below, we have considered the low-temperature scaling limit () and consequently dropped the terms from Eq. (III.1). We have checked that retaining or discarding these terms does not change any of the universal features at the QCP such as critical exponents.
We perform an unbiased CT-QMC study of the quantum-critical properties of the pseudogap BFA model by applying finite-temperature scaling to determine the location of the QCP. We measure powers of the local magnetization
where , from which the Binder cumulant
is obtained. Similar to the case without bosonsGlossop et al. (2011); Pixley et al. (2012) we find that swap movesGull et al. (2011) are essential to accurately calculate the Binder cumulant. Near classical phase transitions, the Binder cumulant is a function of system size and temperature, and finite-size scaling allows one to obtain the critical temperature in the thermodynamic limit from classical Monte Carlo simulations of finite systems.Binder (1981) For the current quantum-mechanical problem, the inverse temperature and coupling play the roles of system size and temperature, respectively, so taking the zero-temperature limit allows one to determine the critical coupling . In the vicinity of the critical point, the Binder cumulant obeys the scaling formBinder (1981)
where is the correlation-length exponent defined in Eq. (17) below. Therefore, finite-temperature scaling applied to the Binder cumulant can be used to extract the correlation-length critical exponent.
We also use CT-QMC to measure the single-particle Green’s function [ in zero magnetic field] and the local spin susceptibility , as described in Ref. Gull et al., 2011. For the susceptibility calculations presented in this paper, we have used the segment representation, and have checked that the results are consistent with those calculated using a “matrix” formalism.Gull et al. (2011) The static local spin susceptibility can then be determined as , where we have set the Landé g factor and the Bohr magneton to unity. For the noninteracting problem (), with , we find agreement within numerical accuracy between the exact and CT-QMC results for both the Green’s function and the local spin susceptibility.J. H. Pixley et al. (2010)
The CT-QMC results reported below were obtained for , , , , and for two different bath exponents: and . It is known from previous workGlossop et al. (2011) that in the absence of bosons, the QCP point for and occurs at . Therefore, by fixing and adjusting the bosonic coupling , we are able to disentangle the pseudogap Bose-Fermi QCP from the pure-fermionic pseudogap QCP.
iii.2 Numerical renormalization group
The Bose-Fermi NRGGlossop and Ingersent (2005, 2007) treatment of the Hamiltonian (3) entails three key steps: (i) Partition of the fermionic and bosonic baths described by and into logarithmic bins spanning the energy ranges , where , 2, 3, and is the Wilson discretization parameter. Within each logarithmic bin, the continuum is replaced by a single state, namely, the linear combination of states that couples to the impurity. (ii) Tridiagonalization of the baths, yielding the Hamiltonian
All information about is encoded in and the dimensionless coefficients and , while is parametrized by and the dimensionless quantities and . For a particle-hole-symmetric such as that in Eq. (5), for all . For large values of of , the remaining tight-binding coefficients satisfy and . (iii) Iterative solution of Eq. (III.2) on fermionic chains restricted to sites with , 1, 2, . Due to the faster decay of the bosonic hopping coefficients with increasing index , the bosonic chain is restricted to , meaning that a site is added to this chain only at even values of . As in the conventional (pure-fermionic) NRG, the many-body states of lowest energy are retained to form the basis for iteration . In problems involving bosonic baths, it is also necessary to truncate the Fock space on each site of the bosonic chain. In this work we employed a basis of boson number eigenstates .
The NRG calculation of critical exponents in the vicinity of a QCP is carried out as described previously for the metallic () BFK modelGlossop and Ingersent (2005, 2007) and as illustrated in a preliminary publication on the pseudogap BFK model.Glossop et al. (2008) The local magnetization (equivalent to above) and the static local susceptibility are evaluated as
where is a magnetic field that couples only to the impurity through an additional Hamiltonian term . All results reported below were obtained for , using Wilson discretization parameter and a bosonic truncation parameter , and retaining after each iteration many-body multiplets corresponding to approximately 900 eigenstates.multiplets () Experience from previous studiesGlossop and Ingersent (2005, 2007); Glossop et al. (2008) indicates that critical exponents calculated for these NRG parameter choices are well-converged with respect to discretization errors (induced by the departure of from its continuum value 1) and truncation errors (arising from restricting the values of and ).
Over the range of bosonic exponents , the Bose-Fermi NRG yields critical exponents for the case that reproduce those obtained via NRGVojta.05 () for the spin-boson model having the same exponent . It has been suggested in Refs. Vojta.10, and Vojta.12, that, for , errors associated with the NRG calculation on a finite bosonic Wilson chain lead to unphysical results with nontrivial exponents and hyperscaling. On the other hand, for this same range of , Refs. Kirchner.09, –Kirchner.12, have demonstrated an scaling for both the leading and sub-leading components of the self-energy, which provides evidence for the interacting nature of the fixed point. Since this issue has yet to be fully resolved, we restrict ourselves in this paper to study of the range .
In this section, we describe the two zero-temperature phases of the pseudogap BFA and BFK models that can be accessed by tuning one model parameter while holding all other parameters constant, as illustrated in the case of the BFK model by the horizontal and vertical dashed lines in Fig. 1. We will assume for the moment that the bosonic coupling is the parameter that is varied. It is important to emphasize that, if carried out at any temperature , such a variation invariably produces smooth crossovers in physical properties, represented schematically by the crossing of dashed lines on the vs diagram in Fig. 2. Only at is it possible to drive the pseudogap BFA or BFK model through a QPT at separating a Kondo phase (reached for ) from a localized phase (accessed for ).
The Kondo phase can be characterized by the vanishing (as ) of the long-imaginary-time value of the local spin susceptibility . Correspondingly, the static local susceptibility approaches a constant at low temperatures (see, for example, the squares in Figs. 5 and 9), signaling that the impurity spin is screened. Since these behaviors (and all other universal low-energy properties found for ) prove to be identical to those in the Kondo phase of the pseudogap Anderson and Kondo models, and independent of the bosonic bath exponent , we associate them with an RG fixed point located at renormalized couplings and either or .
In the localized phase, by contrast, approaches a constant in the limit and the static local susceptibility assumes the Curie-law form (triangles in Figs. 5 and 9) characteristic of a free spin whose size is . In this low-temperature regime, the impurity spin is essentially decoupled from the conduction band and follows the fluctuations of the bosonic bath. The asymptotic low-energy properties are governed by an RG fixed point located at renormalized couplings and either or .
The two phases described above are separated by a QPT occurring at bosonic coupling where the Curie constant extrapolates continuously to zero as is decreased. As a result, can serve as an order parameter for the QPT,Gonzalez-Buxton and Ingersent (1998); Ingersent and Si (2002) vanishing throughout the Kondo phase and equaling the Curie constant in the localized phase. However, it is conventional instead to take as the order parameter
which rises continuously from zero on entry into the localized phase.
If one fixes all parameters apart from and in the BFA model [ and in the BFK model], then the function  defines the boundary between the Kondo and localized phases. This boundary, shown for the Kondo case as the solid line in Fig. 1, is anchored at by the QCP of the pure-fermionic pseudogap models. One of the central questions addressed in our work is whether the critical behavior reached by crossing the phase boundary at coincides with or differs from that for .
Our two numerical techniques lend themselves to different approaches for locating the phase boundary. In the CT-QMC treatment of the pseudogap BFA model, the low-temperature limit of the Binder cumulant evolves continuously from in the Kondo phase to in the localized phase. is independent of temperature at the critical bosonic coupling , but not at other nearby values of . Thus, one can find via the intersection of curves vs for different (low) temperatures (see Figs. 3 and 7).
Within the NRG, one can identify the phase boundary of the pseudogap BFK model through examination of the asymptotic low-energy many-body spectrum, the values of thermodynamic properties such as the impurity contribution to the entropy ( in the Kondo phase, and in the localized phase), or the static local spin susceptibility . In the present study of the pseudogap BFK model, rather than calculating as a function of , we have instead determined the critical Kondo coupling for different values of . For given bath exponents and , points on the phase boundary at any are all found to share the same many-body spectrum and the same power laws in the quantum-critical regime (see Fig. 2). This universality strongly suggests that the quantum criticality is governed by a single QCP to which the system flows starting from any point on the curve . However, the upper temperature , which marks the energy scale at which RG flow first brings the system under the influence of the QCP, varies widely along the boundary. In order to provide the most accurate possible account of the quantum-critical properties, we focus below on NRG results obtained for the boundary point , that yields the highest value of for given , and can therefore be assumed to lie closest to the QCP. An advantage of this approach is that for certain pairs we find , making clear that in such cases the QCP is of pure-fermionic character.
In what follows, we write for the BFA model and for the BFK model to denote the system’s distance from the phase boundary, with describing the Kondo phase and describing the localized phase.
V Results Near the Quantum Critical Point
v.1 Static Critical Behavior
At any point lying on the phase boundary between the Kondo and localized phases, the static susceptibility exhibits a temperature dependence
for all temperatures in the quantum-critical window . Here, the exponent is a universal property of the pseudogap Bose-Fermi QCP (i.e., depends only on the bath exponents and ), whereas the crossover temperature and the prefactor can vary from point to point along the phase boundary; as noted above, is highest for the boundary point that lies nearest to the QCP.
Close to but not precisely on the phase boundary, the system enters the quantum-critical regime once the temperature drops below roughly the same scale found for . However, further lowering the temperature produces a second crossover to an asymptotic regime governed either by the Kondo fixed point (for ) or by the localized fixed point (for ); see Fig. 2. Unlike , the lower crossover scale shows significant dependence, and vanishes continuously upon approach to the phase boundary according to
where is the correlation-length exponent.
In the absence of bosons, the exponent necessarily assumes the value found at the QCP of the pure-fermionic pseudogap Anderson and Kondo models. With a constant fermionic density of states (i.e., ) and isotropic, XY, or Ising symmetry of the bosonic couplings, it is knownZhu and Si (2002); Zaránd and Demler (2002) that for reduces to the exponent of the spin-boson model. Based on perturbative RG,Vojta and Kirćan (2003); Kirćan and Vojta (2004) it has been concluded that for a spin-isotropic version of the pseudogap BFK model is independent of , an observation that agrees with asymptotically exact results obtained in the dynamical large- limitParcollet and Georges (1998); Zhu et al. (2004) where the symmetry group of the spin-isotropic pseudogap BFK model is generalized from SU(2) to SU().Zamani et al. (2012) A preliminary NRG studyGlossop et al. (2008) of the easy-axis pseudogap BFK model also found . However, that study considered only exponent pairs for which , a regime in which it is quite plausible that the bosons should dominate the singular part of the spin response. In the present study we have also investigated cases where that offer better prospects for finding fermion-dominated spin dynamics.
In what follows we show using both CT-QMC calculations and the NRG that the static magnetic critical exponent is simply the smaller of the exponents governing the cases of pure-fermionic and pure-bosonic critical Kondo destruction:
We focus primarily on the pseudogap exponent and two values of the bath exponent: and . For the pure-fermionic Kondo model, the critical susceptibility is described by a temperature exponent obtained within the NRG,Ingersent and Si (2002) consistent with the value found using CT-QMC for the corresponding Anderson model.Glossop et al. (2011) (Here and throughout the remainder of the paper, a number in parentheses indicates our estimated uncertainty in the last digit.) For the Bose-Fermi models, therefore, the cases and are representative of the regimes and , respectively. For each of these cases, we present CT-QMC results for the pseudogap BFA model and NRG results for the pseudogap BFK model, finding the critical behavior of the two models to be fully equivalent.
We supplement the detailed results for , and with static critical exponents for a larger set of pairs as obtained for the BFK model using the NRG. In addition to and introduced in Eqs. (16) and (17), we consider exponents and defined via the relationsGlossop and Ingersent (2007)
If the QPT occurs below its upper critical dimension, yielding an interacting QCP, one expects the singular component of the free energy to take the form
where is a scaling function. With this ansatz, the exponents and are related to and by hyperscaling relations
In each of the cases we have studied, the value of is consistent with Eq. (18), and wherever we have tested them, the hyperscaling Eqs. (22) and (23) are well obeyed. Based on the value of [or alternatively, ]—as well as an analysis of the many-body spectrum at the QCP—we are led to subdivide the region of the plane in which into two parts: one in which the fermions appear to play no role in the critical behavior, and another in which bosonic and fermionic fluctuations combine to produce critical behavior unlike that found in either the metallic () BFK model or the pseudogap Kondo model.
v.1.1 Results for ,
Figure 3 shows the variation with bosonic coupling of the Binder cumulant of the pseudogap BFA model for , at different temperatures as calculated using CT-QMC. The intersection of the curves places the QCP at .
The scaling form Eq. (12) of the Binder cumulant in the vicinity of the QCP can be used to extract the correlation-length exponent defined in Eq. (17). As illustrated in Fig. 4, we obtain an excellent collapse of data taken at different temperatures with a fitted exponent .
Having found , we are able to establish that the static local susceptibility has the expected temperature dependence in each phase and at the critical coupling, as illustrated in Fig. 5. In particular, for , follows Eq. (16) with over the lowest decade of temperature for which data were obtained: . (For comparison purposes, we note that for , the Kondo temperature is .) To within numerical accuracy, we find that , in agreement with previous perturbative and numerical RG studies.Vojta and Kirćan (2003); Kirćan and Vojta (2004); Glossop et al. (2008)
In the BFK model, the convergence of the NRG many-body spectrum to the critical spectrum is fastest (i.e., the crossover scale is highest) for , with . (The value of was determined to roughly 10 significant figures.)
Figure 6 superimposes the temperature dependence of the critical static local susceptibilities of the Anderson and Kondo models. The NRG results for the Kondo model exhibit small oscillations around the dependence predicted in Eq. (16). Such oscillations, which are periodic in with period , are a known consequence of the NRG band discretizationOliveira.94 () that can be reduced in amplitude by working with smaller values of . Over the two decades of temperature shown in the figure, the NRG data are described by an exponent . However, a fit over the range yields an improved estimate , consistent to within numerical error with the CT-QMC result for the Anderson model. Like the CT-QMC estimate for the Anderson model, this value is consistent with the hypothesis that for spin fluctuations are primarily driven critical by the bosonic bath, and .
The reciprocal of the correlation-length exponent extracted from the crossover in the NRG many-body spectrum is , again consistent with the value obtained for the Anderson model. This value differs from that found in two other cases: in the metallic BFK model,Glossop and Ingersent (2007) , while in the pseudogap Kondo model,Ingersent and Si (2002) . Although the critical spin fluctuations (and hence the exponent ) for seem to be dominated by the bosonic bath, the RG flow away from the critical point described by the exponent is clearly different from that in cases of pure-bosonic or pure-fermionic criticality.
v.1.2 Results for ,
Figure 7 shows the variation of the BFA-model Binder cumulant for , as calculated using CT-QMC. The intersection of curves representing different temperatures places the QCP at . This critical value is larger than that () for because the bosonic interaction between time segments falls off faster with increasing . Figure 4 shows that the Binder cumulant scales according to Eq. (12) with an excellent collapse of data taken at different temperatures with a fitted exponent .
Figure 9 illustrates the temperature dependence of the static local susceptibility in each phase and at the critical coupling. For , follows Eq. (16) with over the lowest decade of temperature for which data were obtained, a clear departure from the behavior seen above for .
In the BFK model, the convergence of the NRG many-body spectrum to the critical spectrum is fastest (i.e., the crossover scale is highest) for , with . The value means that the impurity is entirely decoupled from the bosons, and the QCP must be of pure-fermionic character. Indeed, the asymptotic low-energy many-body spectrum in the quantum-critical regime can be reproduced by taking every possible combination of (i) one state from the spectrum of free bosons, and (ii) one state from the critical spectrum of the pseudogap Kondo model. We summarize this spectral decomposition in the shorthand (BF critical) = (B free) (F critical) and refer to it below simply as F-type criticality.
Figure 10 compares the temperature dependence of the critical static local susceptibilities of the BFA and BFK models. Over the two decades of temperature shown in the figure, the NRG data for the BFK model are described by an exponent , identical to the CT-QMC value for the Anderson model. However, since the QCP corresponds to , we know that the exponent must coincide exactly with that of the pseudogap Kondo model:Ingersent and Si (2002) . This value clearly differs from the one found within the spin-boson-modelVojta.05 () and the metallic BFK model.Glossop and Ingersent (2005)
Similarly, we can be confident that the correlation-length exponent must be identical to that of the pseudogap Kondo model, , a result that is consistent with the CT-QMC value quoted above.
The preceding results suggest that in cases where , the physics at and near the QCP is determined primarily by fermionic fluctuations, and that critical properties should coincide with those of the pseudogap Anderson and pseudogap Kondo models.
v.1.3 Results for other
In order to investigate more systematically the different types of quantum criticality exemplified in the cases and , we have studied the particle-hole-symmetric pseudogap BFK model for 23 different pairs spanning the ranges and . Table 1 summarizes the critical properties, one line per pair. On each line, an estimate of —the value of that produces the highest temperature of entry into the quantum-critical regime for , is followed by the assignment of the critical many-body spectrum to one of three categories (F, B, or M) described further below.
The remaining columns of Table 1 list critical exponents: , , and [defined in Eqs. (16), (19), and (17), respectively] are values calculated or inferred for the pair in question. For purposes of comparison, we also list , the reciprocal of the order-parameter exponent at a pure-bosonic QCP with the same , as determined within the metallic BFK model. The exponent has been calculated for all but a small number of pairs, and in all cases its value is consistent with Eq. (18). The order-parameter exponent can generally be evaluated to higher accuracy than (because the latter depends on values obtained by interpolation of data collected at a discrete set of temperature points), and has also been calculated for almost all pairs. By contrast, we have explicitly computed for only about half the pairs. In every case where it can be tested, spanning the full range of bath exponents and , hyperscaling [Eq. (22)] holds to within our estimated numerical uncertainty. For this reason, we can confidently apply Eq. (22) to predict the value of in cases where it has not been computed directly. Finally, we note that wherever it has been calculated, [defined in Eq. (20), but not listed in Table 1] obeys Eq. (23) to high precision.
Based on an examination of the critical spectra and the exponents and (and hence via hyperscaling), we are able to identify three distinct types of quantum criticality:
Fermionic (F)—The asymptotic low-energy critical spectrum exhibits SU(2) spin symmetry and decomposes into a direct product of the spectrum of free bosons with bath exponent and the critical spectrum of the pseudogap Kondo model with band exponent , i.e., (BF critical) = (B free) (F critical). All static critical exponents that have been calculated are identical to those of the pure-fermionic pseudogap Anderson and Kondo models with the same .
Bosonic (B)—The asymptotic low-energy critical spectrum exhibits SU(2) spin symmetry and decomposes into a direct product of the critical spectrum of the spin-boson model with bath exponent and the strong-coupling spectrumphase_shift () of the pseudogap Kondo model with band exponent , i.e., (BF critical) = (B critical) (F strong-coupling). All static static critical exponents that have been calculated in region B are identical to those of the spin-boson model and of the metallic () BFA and BFK models with the same . However, despite the asymptotic decomposition of the low-energy spectrum, the single-particle spectral function has a non-Fermi liquid form (at least for ), as discussed in Sec. V.2. This may be related to the fact that the fermionic strong-coupling spectrum is not only found at the Kondo fixed point where the exchange couplings take renormalized values , but also is approached for , finite in the limit of energy scales much smaller than . We will return to this observation in Sec. V.3.
Mixed (M)—The critical spectrum exhibits broken SU(2) spin symmetry and does not decompose into a direct product of bosonic and fermionic parts. The exponents satisfy but the order-parameter exponent lies between the values for the spin-boson model and the pseudogap Kondo/Anderson models, i.e., .
The three types of quantum criticality are clearly revealed in plots of and vs at fixed . As can be seen in Fig. 11, coincides with until the latter value exceeds its counterpart in the pure-fermionic pseudogap problem. The break in the slope of vs marks the transition from M-type to F-type criticality. Fig. 12 shows that the Bose-Fermi correlation-length exponent equals the bosonic value for sufficiently small (B-type criticality) and equals the fermionic value over precisely the range of where (F-type criticality), but between these regimes, takes values that differ from both and (M-type criticality).
Figure 13 summarizes the type of criticality found at different locations on the – plane, including points studied for the metallic case where the behavior is always of the B type. It is seen that each type of criticality (F, B, or M) occupies a contiguous region. All the results are consistent with there being a boundary between the F and M regions (shown as a solid curve in Fig. 13). As argued at the beginning of this section, such a boundary arises from the assumption that the spin response at the Bose-Fermi critical point is dominated by the bath (bosonic or fermionic) that has the more singular dynamical spin fluctuations, corresponding to the smaller value of . The results in Fig. 13 are also consistent with there being a boundary between the M and F regions (the straight line in the figure). Such a boundary marks the line of equality of the frequency exponents of the bare bosonic propagator and the fermionic particle-hole bubble,Vojta:private () although the significance of this observation in the present context remains to be established.