Quantum criticality in the two-dimensional periodic Anderson model

TL;DR

This paper investigates the quantum critical behavior of the two-dimensional periodic Anderson model, revealing a phase transition and critical exponents using advanced computational methods.

Contribution

It introduces the application of the dynamical vertex approximation to analyze quantum criticality in the periodic Anderson model, identifying a phase transition and critical exponents.

Findings

Phase transition between antiferromagnetic insulator and Kondo insulator.

Critical exponent γ=2 at quantum critical point.

Different susceptibility behaviors at various temperature regimes.

Abstract

We study the phase diagram and quantum critical region of one of the fundamental models for electronic correlations: the periodic Anderson model. Employing the recently developed dynamical vertex approximation, we find a phase transition between a zero-temperature antiferromagnetic insulator and a Kondo insulator. In the quantum critical region, we determine a critical exponent $\gamma\!=\!2$ for the antiferromagnetic susceptibility. At higher temperatures, we have free spins with $\gamma\!=\!1$ instead, whereas at lower temperatures, there is an even stronger increase and suppression of the susceptibility below and above the quantum critical point, respectively.

Equations

$$\mathcal{H}$$

$$T_K\sim e^{-\frac{1}{{\rho }_{0}J}},$$

$$T_{\mathrm{RKKY}}=\frac{1}{4}{J}^2{\chi }_{0,\mathbf{Q}}^{\omega =0},$$

Full text

Introduction. Quantum phase transitions are exceedingly exciting since, besides the spatial correlations of a classical phase transition, also (quantum) correlations in time become relevant at zero temperature $T$. This changes the universality class, i.e., the critical exponents, and can be best understood when considering imaginary time $\tau$ which is restricted to $\tau\in[0,1/T]$. Hence at any finite $T$, temporal (quantum) correlations are cut off at $1/T$ so that only the spatial correlations remain relevant Sachdev (1999).

Most well studied are, on the experimental side, quantum critical points (QCP’s) in heavy fermion systems v. Löhneysen et al. (2007); Brando et al. (2016) such as CeCu${}_{\text{6-x}}$Au${}_{\text{x}}$ Schröder et al. (2000) and YbRh${}_{\text{2}}$Si${}_{\text{2}}$ Custers et al. (2003); Paschen et al. (2004). Experimentally accessible is the unusual behavior within the quantum critical region at a finite $T$ above the QCP; for a schematics see Fig. 1. The theoretical description of such heavy fermion QCP’s is, however, still in its infancy.

The conventional HertzHertz (1976)-MoriyaMoriya and Kawabata (1973)-MillisMillis (1993) (HMM) theory relies on the consideration of the effective $\phi^{4}$ model for magnetic degrees of freedom and may hence not be applicable for heavy fermion systems with their strong electronic correlations. HMM theory is by construction a (renormalized) weak-coupling approach which is also valid above the upper critical dimension, i.e., for $d_{\rm eff}=d+z>4$. Here, the spatial dimensions $d$ need to be supplemented by a dynamical exponent $z$, which relates the critical behavior of the correlation length in space ($\xi\sim T^{-\nu}$; $\nu$: critical exponent) and time ($\xi_{\tau}\sim T^{-z\nu}$) at the QCP. Other proposals for a solution of the antiferromagnetic (metallic) criticality problem include the fractionalized electron picture Sachdev (2008), the critical quasiparticle theory Senthil (2006), and the strong coupling theory Abrahams et al. (2014), see also Vekić et al. (1995); Hu et al. (2017); Terletska et al. (2011); Haldar et al. (2016); Lenz et al. (2016) for quantum criticality studies employing other methods.

Quantum criticality below the upper critical dimension for $d_{\rm eff}=3$ ($d=2$, $z=1$) was considered by Chubukov et al. Chubukov et al. (1994) for the Heisenberg model within a $1/N$ expansion and by renormalization-group approaches for Ising symmetry Sachdev (1997); Strack and Jakubczyk (2009). But again, these approaches cannot be straightforwardly extended to include fermionic excitations, which are actually essential regarding the experimental realization of QCP’s in heavy-fermion systems. Despite many promising approaches Sachdev (1999); Si et al. (2001); Coleman (2005); Kopp and Chakravarty (2005); v. Löhneysen et al. (2007); Strack (1859), we hitherto still lack a reliable solution even for the simplest model for heavy fermion QCP’s, the periodic Anderson model (PAM) beyond a mere (conjectured) mapping onto bosonic models.

In this paper, we hence analyze the QCP of the PAM by means of a recently developed method, the dynamical vertex approximation (D$\Gamma$A) Toschi et al. (2007); Katanin et al. (2009). The D$\Gamma$A is, similar as related approaches Rubtsov et al. (2008); Rohringer et al. (2013); Taranto et al. (2014); Ayral and Parcollet (2015); Li (2015), a diagrammatic extension of the dynamical mean field theory (DMFT) Metzner and Vollhardt (1989); Georges and Krauth (1992); Georges et al. (1996); for a recent review see Rohringer et al. (2018). From the DMFT it inherits a reliable and non-perturbative description of (local) temporal correlations. But on top of these, also non-local spatial correlations are taken into account by means of ladder or parquet diagrams, which do not take the bare interaction but the local irreducible or fully irreducible vertex as a building block. These diagrammatic extensions have been successfully employed for studying critical exponents and phenomena in the Hubbard and Falicov-Kimball model Rohringer et al. (2011); Schäfer et al. (2017); Antipov et al. (2014); Hirschmeier et al. (2015); Del Re et al. (2018). We are hence in the fortunate situation that we can revisit quantum criticality in fermionic models thanks to recent methodological progress.

Model and analytical considerations. To arrive at a non-mean-field, non-Gaussian critical behavior we study the PAM in $d=2$ which can be expected to have the same quantum critical exponents as the Heisenberg model, which in turn has a conjectured $z=1$ Chubukov et al. (1994); Troyer et al. (1997). This suggests an effective dimension $d_{\rm eff}=2+1=3$ 111 Note that in Ref. Schäfer et al., 2017 the Hubbard model was studied for $d=3$ which, together with the expected $z=2$ for a metallic antiferromagnetic phase transition, would yield $d_{\rm eff}=5>4$. However, $\nu=1$ instead of the HMM value $\nu=(d+z-2)/(2z)=3/4$ v. Löhneysen et al. (2007) was obtained, because of peculiarities of the Fermi surface, the so-called Kohn lines. This physics, however, can still be understood in terms of Gaussian fluctuations, e.g. in the random phase approximation (RPA) Schäfer et al. (2017).. The Hamiltonian of the PAM reads

[TABLE]

It consists of localized $f$-electrons with creation (annihilation) operators $f_{i\sigma}^{\dagger}$ ($f_{i\sigma}^{\phantom{\dagger}}$), $n_{f,i\sigma}=f_{i\sigma}^{\dagger}f_{i\sigma}^{\phantom{\dagger}}$, interacting through a local Coulomb repulsion $U$ and with a local one-particle potential $\varepsilon_{f}$. Further, there are itinerant $d_{i\sigma}^{\dagger}$ ($d_{i\sigma}^{\phantom{\dagger}}$) electrons with a nearest neighbor hopping $t$, or a corresponding energy-momentum dispersion relation $\varepsilon_{\mathbf{k}}=-2t\left[\cos(k_{x})+\cos(k_{y})\right]$. Finally, there is a hybridization $V$ between both kinds of electrons. In the presented calculations, we fix $U\!=\!4t$ (intermediate-to-strong coupling). We consider the half-filled case $\varepsilon_{f}=-U/2$, for which the PAM maps onto the Kondo lattice model with a coupling $J=8V^{2}/U$ in the limit $U\gg V$. That is, for large $U$, the $f$-electrons form localized spins. This Kondo lattice model shows the famous Doniach Doniach (1977) $T$-$V$ phase diagram, with two competing phases.

On the one hand there is the Kondo effect Hewson (1993): below the Kondo temperature $T_{K}$, the spins, that are free at high $T$ with a Curie susceptibility $\chi\sim T^{-1}$, get screened. In this case a Kondo resonance forms at the Fermi level. In our particle-hole symmetric case of half-filling, this Kondo resonance is however gapped. This can be understood starting from the non-interacting model ($U=0$): the flat $f$-band at the Fermi energy $E_{F}$ hybridizes with the dispersive conduction $d$-band so that a hybridization gap opens at $E_{F}$. That is, we have a band insulator and for a finite $U$ a quasiparticle-(Kondo-)renormalized picture thereof, i.e. a Kondo insulator. For the (single-site) Kondo model

[TABLE]

where $\rho_{0}$ is the non-interacting density of states of the conduction electrons at the Fermi level Sup ; Hewson (1993). For the PAM we get a similar, somewhat enhanced $T_{K}$ Pruschke et al. (2000); Sup .

Competing with the Kondo effect is a magnetic phase, which can be understood as the effective Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling between $f$-electron spins through the conduction electrons. In second order perturbation theory in $J$, the coupling strength and hence the critical temperature is

[TABLE]

where $\chi_{0}$ is the (non-interacting; $V=0$) susceptibility of the conduction electrons and the factor $1/4=S(S+1)/3$ for spin $S=1/2$ corresponds to the mean-field critical temperature. In our case, the maximal coupling appears at the antiferromagnetic (AF) wave vector ${\mathbf{Q}}=(\pi,\pi)$. An AF ordering opens a gap, so that we obtain an AF insulator. Since $T_{K}$ is exponentially small for small $J$ Doniach (1977), $T_{\rm RKKY}$ prevails for small $J$, whereas at large $J$ the Kondo effect wins. Hence, there is a phase transition from an AF to a Kondo insulator at $T_{K}\approx T_{\rm RKKY}$. Hence, the ground state is always insulating. At high temperatures, the $f$-electrons are also gapped and form free spins, but the conducting electrons are itinerant; at $T\gtrsim T_{K}$ the Kondo peak starts to develop but the Kondo insulating gap that is present at lower $T$’s is still smeared out due to strong scattering.

Phase diagram. Fig. 2 presents the actual phase diagram of the PAM as calculated using DMFT and D${\rm\Gamma}$A. Here, we employ the ladder D$\Gamma$A with Moriya-$\lambda$ correction Rohringer and Toschi (2016) which generates spin-fluctuations starting from the local vertex $\Gamma$ calculated for a converged DMFT solution, for further details on the method we refer the reader to Rohringer et al. (2018, 2012); Schäfer et al. (2013); Wentzell et al. (2016); Kaufmann et al. (2017). For the DMFT phase diagram of the Kondo lattice model (and including short-ranged correlations), cf. Otsuki et al. (2009); Martin et al. (2010); Lenz et al. (2017).

Let us start with the DMFT results, which show AF order at small $V$ in the light-green shaded region of Fig. 2. This order breaks down as the Kondo effect sets in and a QCP emerges: there is a $T=0$ phase transition. As we see, the perturbative result, $T_{\rm RKKY}\sim J^{2}\sim V^{4}$ (yellow line), only holds for small $V$; for larger $V$’s DMFT yields a smaller AF transition temperature due to temporal correlations (green line). As we see the AF order breaks down when the DMFT Kondo temperature (blue line, determined from the maximum of the local susceptibility as a function of $T$) becomes of similar amplitude as the DMFT Néel temperature (green line).

The D${\rm\Gamma}$A phase diagram in Fig. 2 is distinctively different. Concomitant with the Mermin-Wagner theorem Mermin and Wagner (1966), AF order is only found at $T=0$ because of strong non-local fluctuations in $d=2$, cf. Katanin et al. (2009) for D${\rm\Gamma}$A fulfilling the Mermin-Wagner theorem for the $2d$ Hubbard model. Nonetheless, we have AF order along the red line in Fig. 2 and Fig. 1, and hence, at $T=0$, a QCP develops at $V_{\rm QCP}\approx 0.91t$.

Quantum critical region. Above this QCP region we expect a quantum critical region as visualized in Fig. 1, with non-Gaussian fluctuations. Hence, we study the AF susceptibility $\chi=\chi^{\omega=0}_{\mathbf{Q}}$ at momentum ${\mathbf{Q}}=(\pi,\pi)$ and its critical behavior around the critical $V_{\rm QCP}$ in Fig. 3. In DMFT, $\chi\sim(T-T_{N})^{-\gamma}\sim(T-T_{N})^{-1}$ see Fig. 3 (upper panels) so that we have a critical exponent $\gamma=1$. This reflects the (bosonic) mean-field critical behavior of DMFT which neglects spatial fluctuations. At high temperatures, it smoothly evolves into the Curie susceptibility $\chi\sim T^{-1}$ of free spins.

In D$\Gamma$A, Fig. 3 (lower panels), we observe a completely different behavior. While at high $T$, we have the same $\gamma=1$ Curie behavior, there is a crossover to $\chi\sim T^{-2}$, i.e., a quantum critical exponent $\gamma=2$ at lower $T$’s. This critical exponent and the related correlation length $\xi\sim T^{-\nu}\sim T^{-1}$ agrees with the conjectured mapping onto a non-linear $\sigma$ model Chubukov et al. (1994); Chakravarty et al. (1988), which also displays antiferromagnetic ordering within an insulating phase (as we have) with a dynamical critical exponent $z=1$ and yields the same $\xi\sim 1/T$ in the quantum critical regime. This yields the critical exponent $\nu=1$ for the correlation length, which happens to be the same critical exponent that one gets if setting the correlation length in time to its cut-off $\xi_{\tau}\sim 1/T$ and accepting that $z=1$. With the Fisher relation $\gamma/\nu=2-\eta$ Fisher (1967), $\gamma\approx 2$ for the susceptibility as observed in Fig. 3 (note that, typically, $\eta$ is vanishingly small even in $d=2$). In the Supplemental Material Sup Section S.III we present an explanation for this critical exponent on the basis of a sum rule.

With increasing dimensionality, we expect the critical exponents at $d\geq 3$ approach their values in HMM theory Sup . Computing quantum critical exponents of strongly correlated electron models such as the PAM was, however, not possible hitherto; quantum Monte Carlo simulations and cluster extensions of DMFT are restricted to too short-ranged correlations.

At the lowest $T$, deviations from this quantum critical behavior are discernible in Fig. 3 (lower panels) and are to be expected as we leave the cone-shaped quantum critical region in Fig. 1. For $V<V_{\rm QCP}$, eventually antiferromagnetic order sets in at $T=0$. Already at finite $T$’s, an exponential increase of the correlation length and the susceptibility with $1/T$ is to be expected Chakravarty et al. (1988). A similar exponential scaling was observed for the Hubbard model Schäfer et al. (2015). Consistently with this description, one observes a deviation to even larger susceptibilities at $V\ll V_{\rm QCP}$ and lowest $T$’s in Fig. 3. For low $T$ and $V>V_{\rm QCP}$, on the other hand, eventually a Kondo insulating phase develops (quantum disordered phase in Fig. 1). For this (renormalized) band insulator, one has $\chi\rightarrow 0$ for $T\rightarrow 0$. In agreement with this, Fig. 3 shows a deviation to smaller susceptibilities at lower $T$’s; a full suppression of the susceptibility because of the Kondo gap will only occur at larger $V$ in the accessible $T$-range.

An intriguing, non-universal aspect is the strong enhancement of the susceptibility in the crossover regime between the $\chi\sim 1/T$ and $\chi\sim 1/T^{2}$ behavior, in particular at $V=0.9$ and $V=0.91$ in Fig. 3. This originates from enhanced antiferromagnetic correlations, which for the periodic Anderson model set in somewhat above $T_{\rm DMFT}$ (see green line in Fig. 3) and then crossover to the quantum critical $\chi\sim 1/T^{2}$ region, however with a much larger quantum critical susceptibility (prefactor thereof) than for a Heisenberg model with the exchange interaction providing the same mean-field transition temperature. For a more detailed discussion see the Supplemental Material Section S.4 Sup .

Altogether our results yield the quantum critical region schematically presented in Fig. 1, where we have also inserted the actual $V$ values employed in our calculation, along with the observed exponents of the $T$-dependence of the susceptibility.

Uniform susceptibility. Let us now turn to the (uniform) susceptibility, i.e., $\chi^{\omega=0}_{\mathbf{Q}}$ at momentum ${\mathbf{Q}}=(0,0)$, which has the advantage that it can be measured more directly in experiment. Its $T$-dependence around $V_{\rm QCP}$ is displayed in Fig. 4. At large $T$ it shows, similar as the antiferromagnetic $\chi$, the $1/T$ Curie behavior of free spins. However as the spins get screened through the Kondo effect, the ferromagnetic susceptibility shows a maximum around the $T_{K}$ of Fig. 4, whereas the antiferromagnetic susceptibility in Fig. 3 further grows, signaling the instability toward AF. Below this maximum, the ferromagnetic susceptibility $\chi^{\omega=0}_{{\mathbf{Q}}=(0,0)}$ shows essentially in a $T$-linear behavior in the quantum critical region. Such a behavior has also been reported for a non-linear $\sigma$ model and $1/N$ calculations Chubukov et al. (1994).

Conclusion. Thanks to an advanced many-body method, the D$\Gamma$A, we are finally able to study the phase diagram and even the quantum critical behavior of the PAM, the prime model for heavy fermions, in $d=2$. We find antiferromagnetic order for small hybridizations $V<V_{\rm QCP}$ at $T=0$, consistent with the Mermin-Wagner theorem in D$\Gamma$A. In DMFT, antiferromagnetism breaks down when the Kondo temperature $T_{K}$ exceeds the Néel temperature $T_{N}$, as in the Doniach scenario, giving rise to a QCP. While $T_{N}=0$ in D$\Gamma$A, we still get a comparable $V_{\rm QCP}$, which is 25% smaller in D$\Gamma$A than in DMFT as the latter neglects non-local spin fluctuations.

We identify a quantum critical region with critical exponents $\nu=1$ for the correlation length and $\gamma=2$ for the antiferromagnetic susceptibility, as displayed in Fig. 1; whereas the uniform susceptibility shows a non-critical linear-$T$ dependence. Above the quantum critical region we observe free spins with $\gamma=1$ at high $T$; while at small $T$ the AF susceptibility is exponentially enhanced in the thermally disordered region $V<V_{\rm QCP}$ and suppressed in the quantum disordered, Kondo insulating region $V>V_{\rm QCP}$.

Our work opens a route for studying quantum criticality in various models, which was hitherto only possible for spin models but not for correlated electrons. This removes a blank spot on the map of quantum critical theories, which bears many sophisticated quantum field theoretical considerations, analytical arguments and derivations, but few means to test these numerically in a reliable way.

Acknowledgments. We would like to thank Fakher Asaad and Patrick Chalupa for stimulating discussions. The present work was supported by the European Research Council under the European Union’s Seventh Framework Program (FP/2007-2013) through ERC Grant No. 306447, SFB ViCoM (M.K.,T.S.,K.H.), Austrian Science Fund (FWF) through the Doctoral School “Building Solids for Function” (T.S.), the Erwin-Schrödinger Fellowship J 4266 (SuMo, T.S.) and I 2794-N35 (A.T.), as well as the Russian Federation through theme “Quant” AAAA-A18-118020190095-4 of FASO (A.A.K.). T.S. further acknowledges the European Research Council for the European Union Seventh Framework Program (FP7/2007-2013) with ERC Grant No. 319286 (QMAC) and received funding through the ”Exzellenzstipendium Promotio sub auspiciis praesidentis rei publicae” of the Federal Ministry of Education, Science and Research of Austria. Calculations have been done mainly on the Vienna Scientific Cluster (VSC).