Pairing correlations near a Kondo-destruction quantum critical point

Pairing correlations near a Kondo-destruction quantum critical point

Abstract

Motivated by the unconventional superconductivity observed in heavy-fermion metals, we investigate pairing susceptibilities near a continuous quantum phase transition of the Kondo-destruction type. We solve two-impurity Bose-Fermi Anderson models with Ising and Heisenberg forms of the interimpurity exchange interaction using continuous-time quantum Monte-Carlo and numerical renormalization-group methods. Each model exhibits a Kondo-destruction quantum critical point separating Kondo-screened and local-moment phases. For antiferromagnetic interimpurity exchange interactions, singlet pairing is found to be enhanced in the vicinity of the transition. Implications of this result for heavy-fermion superconductivity are discussed.

pacs:
71.10.Hf, 71.27.+a, 75.20.Hr

A quantum critical point (QCP) arises when matter continuously transforms from one ground state to another (1). Whether and how a magnetic QCP underlies unconventional superconductivity in correlated electron systems remains one of the central questions in condensed matter physics (2); (3); (4). At a macroscopic level, a QCP is accompanied by an enhanced entropy (5). At sufficiently low temperatures, in the proximity of a QCP, it is natural for the enhanced entropy to promote emergent phases such as superconductivity. At a microscopic level, however, how quantum criticality drives superconductivity remains an open issue. Developing an understanding of unconventional superconductivity is pertinent to a large list of correlated materials such as iron pnictides, copper oxides, organics, and heavy fermions.

An important opportunity for detailed exploration of this general issue is provided by heavy-fermion metals, in which many QCPs have been explicitly identified (3); (6). Theoretical studies have shown that antiferromagnetic QCPs in a Kondo lattice system fall into two classes. Spin-density-wave QCPs are described in the Landau framework of order-parameter fluctuations (7). The other class of QCPs goes beyond the Landau approach by invoking a critical destruction of the Kondo effect (8); Coleman et al. (2001). Distinctive features of this “local quantum criticality” include scaling in the spin susceptibility and the single-particle spectral function, vanishing of an additional energy scale, and a jump in the Fermi-surface volume. There is mounting experimental evidence for these characteristic properties, e.g., from inelastic neutron-scattering measurements on Au-doped CeCu (10), scanning tunneling spectroscopy on CeCoIn (11), Hall-effect and thermodynamic measurements on YbRhSi (12), and magnetic quantum-oscillation measurements on CeRhIn (13).

Given the considerable advances in the understanding of the unconventional quantum critical behavior of heavy fermions in the normal state, it is clearly important to address its implications for superconductivity. Theoretically, it remains an open question whether a Kondo-destruction QCP promotes unconventional superconductivity (14). To make progress, it is essential to identify simplified models in which this issue can be addressed and insights can be gained. Because an on-site Coulomb repulsion does not favor conventional -wave pairing, this issue can only be studied in models that incorporate correlations among different local-moment sites.

In this work, we propose perhaps the simplest models that support Kondo-destruction physics and allow the study of superconducting correlations: two local moments that interact with each other through a direct exchange interaction and are also coupled both to a conduction-electron band and to a bosonic bath. The models we have considered can be obtained from a cluster generalization of the extended dynamical mean field theory (C-EDMFT) (15) applied to the periodic Anderson model with Ising anisotropy. The critical physics arises from the antiferromagnetic channel, which we will be concerned with. We then arrive at the model defined in Eq. (1) below (15). In the past, significant insights have been gained from single-impurity models, where Kondo-destruction QCPs are characterized by a vanishing Kondo energy scale, an scaling in the local spin susceptibility, and a linear-in-temperature single-particle relaxation rate (43); (17); (18); J. H. Pixley et al. (2010); (42); J. H. Pixley et al. (2010). Such properties are reminiscent of the aforementioned experiments near the antiferromagnetic QCPs of heavy-fermion metals.

We solve the two-impurity Bose-Fermi Anderson models via a continuous-time quantum Monte Carlo (CT-QMC) approach J. H. Pixley et al. (2010); J. H. Pixley et al. (2010); (38) and using the numerical renormalization group (NRG) (44); (43). We determine the magnetic quantum critical properties and compute pairing susceptibilities across the phase diagram. We find that pairing correlations are in general enhanced near the Kondo-destruction QCP. This suggests a new mechanism for superconductivity near antiferromagnetic quantum phase transitions (QPTs).

The two-impurity Bose-Fermi Anderson models, illustrated in Fig. 1, are defined by Hamiltonians of the form

(1)

Here, destroys an electron on impurity site or with spin or , energy , and on-site Coulomb repulsion ; , and where are the Pauli matrices. The operator destroys a conduction electron with wave vector , spin , and energy that has a hybridization with each impurity, while destroys a boson with energy that couples with strength to the difference of impurity spin components. is the number of values.

To control the interimpurity exchange interaction, we take the limit of infinite impurity separation to ensure the vanishing of the indirect Ruderman-Kittel-Kasuya-Yosida exchange interaction between and . Then impurities and hybridize with linearly independent combinations of band states, and interact only through their coupling to the bosonic bath and via a direct exchange term , either of the Ising form or the Heisenberg form . Ising exchange is naturally obtained from the C-EDMFT approach (15), but including the static Heisenberg interaction allows us to study anisotropic couplings since this breaks the purely Ising coupling of the bosonic bath. We note that integrating out the bosonic bath will induce a retarded antiferromagnetic exchange of Ising symmetry.

Figure 1: (Color online) Schematic representation of the two-impurity Bose-Fermi Anderson models considered in this work. The impurity spins interact via a direct exchange coupling (or ), and couples with strength to a dissipative bosonic bath having dispersion . For very large impurity separation, each impurity effectively hybridizes with strength with its own conduction band of dispersion .

We assume a flat electronic density of states and a sub-Ohmic bosonic density of states

(2)

For the CT-QMC calculations we have used a cutoff function and chosen so that the density of states is normalized to unity. Within the NRG, we use with . In this work we restrict ourselves to the range .

In the absence of the bosonic bath, the pure-fermionic two-impurity Anderson model can be mapped via a Schrieffer-Wolff transformation to a two-impurity Kondo model with a direct exchange interaction (24). In the case of Heisenberg exchange, both the Anderson and Kondo formulations are well studied (25); (26), displaying a critical point at an antiferromagnetic exchange in the presence of particle-hole symmetry; at this point, the static singlet-pairing susceptibility diverges (27). For an Ising , the model possesses a Kosterlitz-Thouless (KT) QPT at between a Kondo-screened phase and an interimpurity Ising-ordered phase (28); (29). Without the conduction band, Eq. (1) reduces to a two-spin boson model; studies of this model with coupled to a spin bath found a QCP separating a delocalized phase and a ferromagnetically localized phase (30); (31).

We have solved Eq. (1) with of Ising form by extending the CT-QMC approach (38); J. H. Pixley et al. (2010); J. H. Pixley et al. (2010). At temperature we determine the staggered Binder cumulant Binder (1981); (42); J. H. Pixley et al. (2010) , where the staggered magnetization with , and the staggered static spin susceptibility . To solve the Heisenberg form of we have used the Bose-Fermi extension (43) of the NRG (44); (45). To measure the pairing correlation between the electrons at different impurity sites, we study dynamic singlet (-wave) and triplet (-wave) pairing susceptibilities

(3)

where , , and orders in imaginary time. The static pairing susceptibilities follow via . Each numerical technique as applied to the models studied here is further described in the Supplemental Material (34), and additional details will be given elsewhere (35).

Figure 2: (Color online) (a) Phase diagram for an Ising determined via CT-QMC from the staggered Binder cumulant. A square marks the Kosterlitz-Thouless QPT and circles denote second-order Kondo-destruction QPTs governed by the QCP at . (b) Phase diagram for a Heisenberg found with the NRG. Squares represent QPTs governed by the critical point at , while circles represent Kondo-destruction QPTs induced by the coupling to the bosonic bath. Kondo-screened (Kondo), interimpurity-singlet (IS), and local-moment (LM) phases all meet at a tricritical point.

In the following, we work with fixed and . This choice places the Anderson impurities at mixed valence with a high Kondo temperature (for ), ensuring a correspondingly high temperature of entry into the quantum critical regime J. H. Pixley et al. (2010). We also take and focus on sub-Ohmic bath exponents [see Eq. (2)] for Ising exchange and for the Heisenberg case.

Figure 3: (Color online) Staggered spin susceptibility in the Kondo phase (circles), at the QCP (triangles), and in the LM phase (squares), for (a) an Ising with , and (b) a Heisenberg with . At the QCP, .

Ising . Figure 2(a) shows the phase diagram for the case of Ising exchange, as obtained using CT-QMC. For , each impurity spin is locked into a Kondo singlet with the conduction band and approaches a constant at low temperatures [e.g., Fig. 3(a)]. Upon increasing and/or , the system passes through a QPT into an Ising-antiferromagnetic local-moment (LM) phase in which the impurity spins are anti-aligned and decoupled from the conduction band, as seen through a Curie-Weiss behavior of the staggered spin susceptibility: [Fig. 3(a)]. The Kondo energy scale vanishes continuously on the Kondo side of the QPT, characteristic of a Kondo-destruction QCP. The staggered Binder cumulant varies from 3 deep in the Kondo phase to 1 far into the LM phase. For fixed , the cumulant near the QCP has a scaling form

(4)

identifying as the point of temperature independence of vs [Fig. 4(a)]. Optimizing the scaling collapse according to Eq. (4) gives a correlation-length exponent [Fig. 4(b)], close to the value found using the NRG for the single-impurity Ising-symmetry Bose-Fermi Kondo model (43).

For , the Ising critical point is KT-like, characterized by a divergence . Consequently, the coupling has a scaling dimension and is relevant for . This dictates a flow away from the KT fixed point along the phase boundary in Fig. 2(a) toward the critical point (34). Tuning to the boundary at fixed , we find that the staggered spin susceptibility diverges as

(5)

with for increasing . These values are consistent with , suggesting that the staggered channel exhibits the same critical properties as the single-impurity Ising-symmetric Bose-Fermi Kondo model (43).


Figure 4: (Color online) (a) Binder cumulant vs for an Ising with , , and at the labeled inverse temperatures . The intersection of curves gives the critical bosonic coupling . (b) A scaling collapse of the same data near according to Eq. (4) yields a correlation-length exponent . (c) Flow of a low-energy NRG eigenstate vs iteration number for a Heisenberg with , , and six values of , where . (d) Low-energy crossover scale from the NRG, fitted to yielding .

Heisenberg . For Heisenberg exchange, the NRG gives the phase diagram shown in Fig. 2(b), based on runs performed with a basis of up to bosons per site of the bosonic Wilson chain and retaining up to many-body eigenstates at the end of each iteration. For small and , the model is in the Kondo phase. Tuning for , we pass through a critical point into an interimpurity singlet (IS) phase, in which the impurity spins are locked into a singlet and decoupled from the conduction band. At the particle-hole-symmetric critical point (25); (26), the staggered spin susceptibility diverges as . Using the corresponding scaling dimension of the staggered impurity spin, along with the scaling dimension of , we determine that the bosonic coupling has scaling dimension and is irrelevant for . Indeed, we find that the NRG spectrum along the phase boundary is independent of for small values of (34), indicating that the critical behavior is governed by the QCP.

For small , tuning the bosonic coupling yields a QPT from the Kondo phase to the LM phase [Fig. 4(c)]. The Kondo energy scale vanishes continuously on approach from the small- side of this Kondo-destruction QCP. At the QCP, the staggered spin susceptibility obeys Eq. (5) with replaced by and [Fig. 3(b)], again consistent with . Nearby, a low-energy crossover temperature (equal to the effective Kondo temperature for ) varies as , yielding for the data shown in Fig. 4(d) a correlation-length exponent . However, we find that (unlike the global phase diagram and the value of the exponent ), the value of is sensitive to the NRG truncation of states. Increasing from 4 to 6 and from to leads to a refinement of our estimate to , within numerical error identical to the value found for the single-impurity Ising-symmetry Bose-Fermi Kondo model (43). We therefore conclude that the Kondo-destruction QCPs for Ising and Heisenberg exchange fall within the same universality class. In both Ising and Heisenberg cases, the Kondo-destruction QCPs are insensitive to breaking of particle-hole symmetry via setting , as well as to a finite impurity separation (35).

We turn next to the transition between the IS and LM phases. Fixing at a large value and tuning , the bosonic bath decoheres and destroys the interimpurity singlet state at a QCP, where we find similar critical properties to those on the Kondo-LM boundary: diverging according to Eq. (5) with , and for , a correlation length exponent indistinguishable from the corresponding value found on the Kondo-LM boundary.

In the particle-hole symmetric case that is the focus of this Rapid Communication, the Kondo, IS, and LM phases all meet at a tricritical point, as shown in Fig. 2(b). Generic particle-hole asymmetry is known to turn the Kondo-to-IS transition in Fig. 2(b) into a crossover (25); (26), leaving only a single line of Kondo-destruction QPTs.

Pairing susceptibilities. We now consider the singlet and triplet pairing susceptibilities defined in Eq. (3). For both the Ising and Heisenberg forms of the interimpurity exchange, the static triplet pairing susceptibility (not shown) is reduced by any nonzero value of , , or .

More interesting is the singlet susceptibility, which we illustrate along paths on the - and - phase diagrams that start from and cross the Kondo-LM boundary. In C-EDMFT (15), such trajectories are representative of tuning spin-spin interactions within the lattice model. Figure 5(a) plots vs Ising exchange coupling at a sequence of temperatures along the cut . The pairing susceptibility grows as increases from zero, is peaked for slightly below , and then falls off within the LM phase as the electrons localize and decouple from the conduction band. The singlet pairing susceptibility saturates at temperatures .

Figure 5(b) illustrates the Heisenberg form of , plotting the singlet pairing susceptibility vs along a path that crosses the Kondo-LM boundary. Very much as in the Ising case, rises from and peaks just below .

The enhancement of the static singlet pairing susceptibility near a Kondo-destruction QCP is one of the principal results of this work. Although peaks just inside the Kondo phase, the pairing correlation at the QCP is significantly higher than at . We stress that these results are associated with the critical destruction of the Kondo effect. They differ from those for , where for Heisenberg exchange diverges at the Kondo-IS QPT (27). We have found (by following the path , not shown) that the singlet pairing susceptibility also diverges on crossing the Kondo-IS boundary at some , consistent with the picture that this boundary is governed by the critical point.

Figure 5: (Color online) (a) Static singlet pairing susceptibility vs for an Ising with along the line , which crosses the Kondo-LM phase boundary at . (b) Static singlet pairing susceptibility vs for a Heisenberg with along the line , which crosses the Kondo-LM phase boundary at . In both (a) and (b), is peaked just on the Kondo side of the phase boundary and remains elevated at the QCP over its value for .

The models considered here have both a dynamic (induced by ) and a static ( or ) exchange interaction between the impurities. The combination of the two antiferromagnetic interactions is responsible for both, the existence of a Kondo-destruction QCP and the enhancement of in its vicinity. This behavior is likely to have significant effects in lattice systems. Within C-EDMFT (15), the cluster pairing susceptibility determines the lattice pairing susceptibility, in such a way that the enhanced may give rise to a pairing instability near a Fermi-surface-collapsing QCP of a Kondo lattice (8); Coleman et al. (2001). As such, this would represent a new mechanism for superconductivity in the vicinity of antiferromagnetic order, and would be of considerable interest in connection with the superconductivity observed in the Ce-115 materials (36) and related heavy-fermion superconductors (37).

In summary, we have introduced and solved two variants of the two-impurity Bose-Fermi Anderson model using robust numerical methods. We have mapped out the phase diagrams for these models and shown that each possesses a line of Kondo-destruction QCPs that are insensitive to breaking particle-hole symmetry. The QCPs in the two models belong to the same universality class despite the differing symmetries of the interimpurity exchange interaction. Just as importantly, we have shown that the Kondo-destruction quantum criticality in these models enhances singlet pairing correlations. Our results hold promise for elucidating the superconductivity observed in heavy-fermion metals whose normal state shows characteristics of Kondo-destruction quantum criticality.

We acknowledge useful discussions with Stefan Kirchner, Lijun Zhu, Aditya Shashi, and Ang Cai. This work was supported in part by NSF Grants No. DMR-1309531 and No. DMR-1107814, Robert A. Welch Foundation Grant No. C-1411, the East-DeMarco fellowship (JHP), and the Alexander von Humboldt Foundation. Computer time and IT support at Rice University was supported in part by the Data Analysis and Visualization Cyberinfrastructure funded by the NSF under Grant No. OCI-0959097. J.H.P. acknowledges the hospitality of the Max Planck Institute for the Physics of Complex Systems, and Q.S. acknowledges the hospitality of the the Karlsruhe Institute of Technology, the Aspen Center for Physics (NSF Grant No. 1066293), and the Institute of Physics of Chinese Academy of Sciences.

SUPPLEMENTARY MATERIAL – Pairing Correlations Near a Kondo-Destruction Quantum Critical Point

by: J. H. Pixley, Lili Deng, Kevin Ingersent, and Qimiao Si


In this supplementary material, we describe in more detail the numerical methods used and present the renormalization-group (RG) flow of the two impurity Bose-Fermi Anderson models.

.1 Methods

Two approaches have been used for our study. The first is an extension of the CT-QMC approach (38); J. H. Pixley et al. (2010); J. H. Pixley et al. (2010), for the model with an Ising form of the exchange interaction. After a generalized Lang-Firsov transformation J. H. Pixley et al. (2010, 2010), the CT-QMC performs time-dependent perturbation theory in the hybridization and stochastically sums the resulting series via a Monte-Carlo algorithm. In order to locate the transition via calculations performed at , we compute the staggered Binder cumulant Binder (1981); (42); J. H. Pixley et al. (2010) , where the staggered magnetization with . We also calculate the staggered static spin susceptibility . In the presence of a bosonic coupling to , the Heisenberg form of is beyond the reach of the CT-QMC.

The second approach is the Bose-Fermi extension (43) of the NRG (44). The staggered spin susceptibility is calculated as with an additional Hamiltonian term . The static pairing correlations are obtained by Hilbert transformation of the imaginary part of the dynamical susceptibilities, computed on the real frequency axis in the usual manner (45). The NRG results presented here were obtained using discretization parameter , allowing up to 6 bosons per site of the Wilson chain, and keeping up to 1 300 many-body eigenstates after each iteration.Ó

.2 RG Flow

Figure S1 corresponds to the case with Ising interimpurity exchange . There are two stable fixed points, indicated by filled circles, which govern the Kondo-screened (Kondo) phase and the local-moment (LM) phase. The Kondo fixed point is at , while the LM fixed point is at . On the phase boundary, the RG flow is from the Kosterlitz-Thouless (KT) fixed point toward the Kondo-destruction (KD) fixed point. The KT and KD fixed points are both unstable and are shown by open circles. The direction of the RG flow on the phase boundary reflects the discussion in the main text, i.e., all the points for nonzero on the phase boundary have the same critical behavior as the KD QCP on the axis.

Figure S2 shows the RG flow for the case with Heisenberg interimpurity exchange . In this case, there are three stable fixed points corresponding to the three phases. The Kondo-screened (Kondo), local-moment (LM) and interimpurity-singlet (IS) phases are respectively located at , and , and are marked by filled circles. On the Kondo-LM phase boundary, the RG flow is from the triple-point (where the three phases meet) to the Kondo-destruction (KD) fixed point. Likewise, the RG flow on the Kondo-IS phase boundary is from the triple-point toward the fixed point KI, which separates the Kondo and IS phases on the -axis.

Figure S1: (color online). Schematic RG flow on the - plane for the two-impurity Bose-Fermi Anderson model with Ising interimpurity exchange . Trajectories with arrows represent the flows of the couplings ( and ) with the decrease of energy.
Figure S2: (color online). Schematic RG flow on the - plane for the two-impurity Bose-Fermi Anderson model with Heisenberg inter-impurity exchange .

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