Critical Kondo destruction in a pseudogap Anderson model:
scaling and relaxational dynamics
We study the pseudogap Anderson model as a prototype system for critical Kondo destruction. We obtain finite-temperature () scaling functions near its quantum-critical point, by using a continuous-time quantum Monte Carlo method and also considering a dynamical large-N limit. We are able to determine the behavior of the scaling functions in the typically difficult to access quantum-relaxational regime (), and conclude that the relaxation rates for both the spin and single-particle excitations are linear in temperature. We discuss the implications of these results for the quantum-critical phenomena in heavy fermion metals.
pacs:71.10.Hf, 71.27.+a, 75.20.Hr
Continuous zero temperature phase transitions in strongly correlated electronic and atomic models have attracted considerable attention as a new paradigm for addressing the universal features of correlated quantum systems Natphys.08 (). Quantum criticality links two nearby phases and determines the physical properties in a large range of temperature and control parameters, the quantum-critical region, that fans out from the quantum-critical point (QCP). This paradigm is especially pertinent to the understanding of intermetallic rare earth compounds. The phase diagram of these heavy fermion metals close to the border of antiferromagnetism features a QCP, but the associated quantum-critical properties are highly unusual when viewed from the standard description based on Landau’s notion of order-parameter fluctuations SiSteglich.10 (). Especially, inelastic neutron-scattering measurements have shown that the dynamical spin susceptibility in the quantum-critical regime features a linear-in- spin relaxation rate and satisfies a frequency over temperature () scaling Schröder et al. (2000). Very recently, Hall-effect measurements have indicated that the single-particle relaxation rate in the quantum-critical regime is also linear in Friedemann et al. ().
These dynamical scaling and relaxational properties provide important clues to the nature of the heavy-fermion QCP. Yet, theoretically, such real-frequency behavior is difficult to study. Two regimes need to be distinguished: the quantum coherent () and quantum-relaxational () regimes Damle and Sachdev (1997) ( and are set to 1). Calculation methods (such as Monte Carlo simulations) typically work in the imaginary-time domain, and the nonzero Matsubara frequencies () are necessarily in the regime. Extracting the behavior at real frequencies requires an analytical continuation, which is in general a numerically ill-conditioned procedure. The numerical renormalization group operates on the real-frequency axis, but it is not reliable for the quantum-relaxational regime at nonzero temperatures.
In this Letter, we address the dynamical and relaxational properties of the particle-hole symmetric pseudogap Anderson model in both frequency regimes. Our motivations to study this model are multifold. In local quantum criticality for heavy-fermion metals, the critical destruction of the Kondo effect Si et al. (2001); Coleman et al. (2001); v. Löhneysen et al. (2007); Gegenwart et al. (2008) is local in space, and the resulting interacting critical modes are manifested in local correlators which can be studied in quantum-impurity problems. The pseudogap Anderson model is the simplest impurity problem that contains the physics of critical Kondo destruction; it is well known that varying the Kondo coupling yields a QCP Withoff and Fradkin (1990); Gonzalez-Buxton and Ingersent (1998); Ingersent and Si (2002); Vojta (2001); Glossop and Logan (2003a); Glossop et al. (2005b); Fritz et al. (2006), which separates a Kondo-screened Fermi-liquid phase from a Kondo-destroyed local-moment phase. However, a proper understanding of the dynamical scaling at finite temperatures and the associated relaxational behavior is not yet available even in this simplest model. Furthermore, the pseudogap Anderson/Kondo model is relevant in a number of realistic physical settings. It has been invoked in the context of non magnetic impurities in cuprate superconductors Vojta and Bulla (2001). It has also been shown that a judicious tuning of a double quantum-dot system can produce a pseudogap in the effective density of states da Silva et al. (2006). In disordered metals, a novel phase has been attributed to the occurrence of local pseudogaps near the Fermi energy at local-moment sites Zhuravlev et al. (2007). Finally, the pseudogap Kondo model is the appropriate model to describe point defects in graphene Chen.11 ().
We study the model by using a continuous-time quantum Monte Carlo approach (CT-QMC) Werner et al. (2006). We determine the full scaling functions at real frequencies and finite temperatures for both the dynamical spin susceptibility and single-electron Green’s function. We achieve this by taking advantage of insights gained from exact calculations at real frequencies and finite temperatures in a dynamical large- limit of the model. The results in the large- limit motivate us to analyze the imaginary-time correlators in the physical model in a way that uncovers the form of a boundary conformally-invariant fixed point. The latter, in turn, can readily be analytically-continued to real frequency at finite temperatures. We establish that both the dynamical spin susceptibility and single-electron Green’s function display an -scaling and contain a linear-in- relaxation rate. As a by-product, we show that the CT-QMC approach, which is based on a high-temperature expansion, can reach low-enough temperatures with enough accuracy to resolve quantum-critical features.
Pseudogap Kondo model in a dynamical large-N limit: To set the stage for the CT-QMC study, we start with the SU(N)SU(M) Kondo model Parcollet and Georges (1998) in the presence of a pseudogap in the limit of large N and M. In what follows, we set . The Hamiltonian is
Here, the spin and channel indices are and , respectively. The conduction electron density of states takes the form:
with being the bandwidth. That this limit has a nontrivial QCP can be seen through the particular form of the perturbative (in ) renormalization group equation Withoff and Fradkin (1990). In the limit of large N and M, the renormalization group beta function becomes with and Zhu et al. (2004). This establishes that the QCP survives the large-N limit and can be accessed perturbatively. To order , the large-N beta function is identical to its N counterpart Withoff and Fradkin (1990); Ingersent and Si (2002) suggesting that the universal critical scaling properties of the QCP are preserved by taking the large-N limit. In this limit, the local degrees of freedom are expressed in terms of pseudofermions and a bosonic decoupling field , where , and is related to the chosen irreducible representation of SU(N) Parcollet and Georges (1998); Cox and Ruckenstein (1993). The large- equations are
By solving the large-N equations in real frequencies for arbitrary and Zhu et al. (2004), the full scaling functions in both, the quantum coherent () and relaxational () regimes are obtained. At the critical coupling , we find that all the correlators display an -scaling. This is demonstrated in Fig. 1(a) for the local singleparticle Green’s function [i.e., the T-matrix, , associated with ], and in Fig. 1(b) the local spin susceptibility , which corresponds to .
A key insight from the large- result is that the scaling functions contain more information beyond scaling per se. They have the particular form associated with a boundary conformally-invariant fixed point, depending on as a power law in Ginsparg (1989). To see this, we obtain the imaginary-time dependence from the real-frequency results via
for . Here, for bosonic/fermionic . Figure 1 shows the (c) Green’s function and (d) susceptibility versus the combination . Both collapse on a single scaling curve in terms of for all (low-enough) . A power-law behavior for is seen over about decades, and the exponents are compatible with those for the frequency dependence.
Pseudogap Anderson model at : Guided by the large-N results, we turn to the scaling functions for and of the particle-hole symmetric pseudogap Anderson model at ; the low-energy properties of this model are identical to its pseudogap Kondo counterpart. To this end, we bring to bear the recently developed hybridization-expansion Monte Carlo method Werner et al. (2006); Rubtsov et al. (2005) on a quantum-critical model. This CT-QMC approach involves a stochastic sampling of a perturbation expansion in the host-impurity hybridization or a weak coupling expansion Prokof’ev et al. (1998); Rubtsov et al. (2005); Werner et al. (2006); Haule (2007). The results are free of any finite-size effects Kirchner et al. (2009).
The Anderson impurity model is defined by where
with , , being the host dispersion, the hybridization, and the impurity level energy. We consider the particle-hole symmetric case where , with being the onsite Coulomb repulsion. The host-impurity coupling is specified by the imaginary part of the hybridization function . As in Eq. (2), we choose . The critical point exists only for Gonzalez-Buxton and Ingersent (1998).
Central to the CT-QMC approach adopted here is the expansion of the partition function in the hybridization term Werner et al. (2006).
We measure the single-particle Green’s function , the local spin susceptibility and powers of the local magnetization where . The static susceptibility is obtained from . Thermalization can be traced by which obeys in the particle-hole symmetric model. We also performed a binning analysis and obtained the integrated autocorrelation time which increases with decreasing temperature but turned out to be small (compared to the number of measurements) at all temperatures. For the lowest temperature considered () we performed Monte Carlo steps for thermalization, Monte Carlo steps between each measurement and measurements. A Monte Carlo step consists of an attempt to remove, insert and shift a segment as described in Reference Werner et al. (2006).
By varying we can tune the model through a QCP. Correspondingly, Figure 2(a) shows that the large- limit of vanishes for small (Kondo-screened phase) and is equal to the Curie constant for large (Kondo-destroyed local-moment phase). To accurately determine we apply finite-temperature scaling to the Binder cumulant, , where plays the role of the system size. We find swap moves between up and down spin segments Werner and Millis (2006) are necessary to accurately measure the Binder cumulant; for the results in Figure 2(b) we performed a swap move every measurements. The nature of the intersection of the data in Figure 2(b) implies that the phase transition is continuous, from the location of the intersection we obtain the critical value of . In the quantum-critical regime the static local susceptibility displays an anomalous -dependent exponent; we find
with in good agreement with numerical renormalization group results Ingersent and Si (2002).
We now discuss the finite-temperature dynamical scaling properties of and . Guided by the large- results, we plot them as functions of in Figure 3. Excellent scaling collapse is observed over about two decades, for all temperatures in the scaling regime. We reach an important conclusion:
for , Figure 3(b). Since , the results for imply that the order-parameter susceptibility shows -scaling. A similar conclusion applies to , as seen in Figure3(a). Our results yield , with the exponent , which is believed to be exact Fritz et al. (2006). The fact that signifies the importance of vertex corrections and in part reflects the interacting nature of the QCP (see below).
The boundary conformally-invariant form of and immediately imply that their dependence on real frequency satisfies scaling and that their relaxation rates, defined in the quantum-relaxational regime, are linear in . Expressed in terms of for a correlator , the relaxation rates and , where and are universal dimensionless constants. Such a linear-in- form is consistent with what has been observed in quantum-critical heavy fermion compounds, for both the single-particle Green function Friedemann et al. () and order-parameter susceptibility Schröder et al. (2000). A linear-in- relaxation rate signifies that the QCP is interacting, i.e., containing a nonzero nonlinear coupling among the critical modes. By contrast, at a Gaussian QCP (whose critical modes do not interact at the fixed point), the relaxation rate will be super-linear-in- because the nonlinear coupling itself vanishes as approaches zero Damle and Sachdev (1997).
It is instructive to compare our study with previous theoretical treatments of the finite-temperature scaling behavior of the pseudogap Anderson/Kondo model. One study Ingersent and Si (2002) is perturbative in , which not only becomes unreliable for finite but also does not allow the study of the single-particle Green’s function. Another study carries out calculations in real frequency at finite temperatures, but relies on the resummation of a perturbation series whose validity for the quantum-critical regime is not clear Glossop et al. (2005b). Yet another study utilizes a Callan-Symanzik approach which requires analytic continuation that is problematic as reflected in the noncommutativity of the resummation and analytic continuation Fritz et al. (2006); it will be important to check whether that procedure yields a that is compatible in analyticity with . As a more specific illustration of our results, we note that is a nonzero constant, which is contrary to both the perturbative results of Reference Glossop et al. (2005b) and the results of the real-frequency Callan-Symanzik resummation for Fritz et al. (2006).
The scaling of the local correlators in terms of suggests that the boundary critical state and the associated boundary operators may be described by their counterparts in an effective model with conformal invariance Kirchner and Si (2008). This is so in spite of the fact that, for our problem, the pseudogap form of the density of states means that the bulk fermionic component of the Hamiltonian lacks conformal invariance. Hence, our results suggest an enhanced conformal symmetry that characterizes the QCP.
Summary. We have obtained the full finite-temperature scaling functions at the local quantum-critical point of the pseudogap Anderson and Kondo models. Using the results directly obtained in real frequency () in the large- limit, and by showing that the imaginary-time local correlators of the physical model have the form of a boundary conformally invariant fixed point, we succeeded in determining the full scaling function in both the quantum coherent and relaxational regimes without using numerically ill-conditioned analytical-continuation schemes. We demonstrated that the Kondo-breakdown QCP features a linear-in- relaxation rate for both spin and single-electron dynamics, which is consistent with the experimental observations in the quantum-critical heavy fermion metals.
We thank L. Fritz, K. Ingersent, M. Vojta and P. Werner for useful discussions. This work has been supported in part by NSF (Grant No. DMR-1006985), the Robert A. Welch Foundation (Grant No. C-1411), the W. M. Keck Foundation, and the Rice Computational Research Cluster funded by NSF.
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