Effects of an Additional Conduction Band on Singlet-Antiferromagnet Competition in the Periodic Anderson Model

TL;DR

This study investigates how adding an extra conduction band influences antiferromagnetic order and quantum criticality in the Periodic Anderson Model, revealing stabilization of AF order and changes in spectral functions through mean field theory and quantum Monte Carlo simulations.

Contribution

It introduces the effects of an additional conduction band on magnetic phases and quantum critical points in the Periodic Anderson Model using combined mean field and quantum Monte Carlo methods.

Findings

Additional band stabilizes AF order at half-filling.

Quantum Monte Carlo confirms AF stabilization via RKKY interaction.

Spectral functions provide insight into AF-singlet competition.

Abstract

The competition between antiferromagnetic (AF) order and singlet formation is a central phenomenon of the Kondo and Periodic Anderson Hamiltonians, and of the heavy fermion materials they describe. In this paper, we explore the effects of an additional conduction band on magnetism in these models, and, specifically, on changes in the AF-singlet quantum critical point (QCP) and the one particle and spin spectral functions. To understand the magnetic phase transition qualitatively, we first carry out a self-consistent mean field theory (MFT). The basic conclusion is that, at half-filling, the coupling to the additional band stabilizes the AF phase to larger $f$ $d$ hybridization $V$ in the PAM. We also explore the possibility of competing ferromagnetic phases when this conduction band is doped away from half-filling. We next employ Quantum Monte Carlo (QMC) which, in combination with finite size scaling, allows us to evaluate the position of the QCP using an exact treatment of the interactions. This approach confirms the stabilization of AF order, which occurs through an enhancement of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. QMC results for the spectral function $A(\textbf{q},\omega)$ and dynamic spin structure factor $\chi(\textbf{q},\omega)$ yield additional insight into the AF-singlet competition and the low temperature phases.

Equations

$$H$$

$$-t\sum _{<\text{i},\text{j}>\sigma }(d_{\text{i}\sigma }^{†}{d}_{\text{j}\sigma }^{}+d_{\text{j}\sigma }^{†}{d}_{\text{i}\sigma }^{})+{ϵ}^d\sum _{\text{i}\sigma }{n}_{\text{i}\sigma }^d$$

$$+U_f\sum _{\text{i}}(n_{\text{i}\uparrow }^f-\frac{1}{2})(n_{\text{i}\downarrow }^f-\frac{1}{2})+{ϵ}^f\sum _{\text{i}\sigma }{n}_{\text{i}\sigma }^f$$

$$-t^{\prime }\sum _{\text{i}\sigma }(c_{\text{i}\sigma }^{†}{d}_{\text{i}\sigma }^{}+d_{\text{i}\sigma }^{†}{c}_{\text{i}\sigma }^{})-V\sum _{\text{i}\sigma }(d_{\text{i}\sigma }^{†}{f}_{\text{i}\sigma }^{}+f_{\text{i}\sigma }^{†}{d}_{\text{i}\sigma }^{})$$

$$\displaystyle\small H=\sum_{\textbf{k}\sigma}\left[{\begin{array}[]{ccc}c^{\dagger}_{\textbf{k}\sigma}&d^{\dagger}_{\textbf{k}\sigma}&f^{\dagger}_{\textbf{k}\sigma}\\ \end{array}}\right]\left[{\begin{array}[]{ccc}\epsilon_{\textbf{k}}&-t^{\prime}&0\\ -t^{\prime}&\epsilon_{\textbf{k}}&-V\\ 0&-V&0\\ \end{array}}\right]\left[{\begin{array}[]{c}c_{\textbf{k}\sigma}\\ d_{\textbf{k}\sigma}\\ f_{\textbf{k}\sigma}\\ \end{array}}\right]$$

$$⟨n_{\text{l}\sigma }^f⟩=\frac{n_f}{2}+\frac{\sigma m_f(-1{)}^{\text{l}}}{2}$$

$$⟨n_{\text{l}\sigma }^f⟩=\frac{n_f}{2}+\frac{\sigma m_f}{2}$$

$$H_{AF}$$

$$-N{n}_c{ϵ}^c-N{n}_d{ϵ}^d-N{n}_f{ϵ}^f$$

$$\displaystyle M^{AF}_{\textbf{k}\sigma}=\left[{\begin{array}[]{cccccc}\eta^{c}_{\textbf{k}}&-t^{\prime}&0&0&0&0\\ -t^{\prime}&\eta^{d}_{\textbf{k}}&-V&0&0&0\\ 0&-V&\eta^{f}_{\textbf{k}}&0&0&-\frac{\sigma m_{f}U_{f}}{2}\\ 0&0&0&\eta^{c}_{\textbf{k}-\textbf{Q}}&-t^{\prime}&0\\ 0&0&0&-t^{\prime}&\eta^{d}_{\textbf{k}-\textbf{Q}}&-V\\ 0&0&-\frac{\sigma m_{f}U_{f}}{2}&0&-V&\eta^{f}_{\textbf{k}-\textbf{Q}}\\ \end{array}}\right].$$

$$H_F$$

$$-N{n}_c{ϵ}^c-N{n}_d{ϵ}^d-N{n}_f{ϵ}^f$$

$$\displaystyle M^{F}_{\textbf{k}\sigma}=\left[{\begin{array}[]{ccc}\epsilon_{\textbf{k}}+\epsilon^{c}&-t^{\prime}&0\\ -t^{\prime}&\epsilon_{\textbf{k}}+\epsilon^{d}&-V\\ 0&-V&(\frac{n_{f}}{2}-\frac{1}{2}-\frac{\sigma m_{f}}{2})U_{f}+\epsilon^{f}\\ \end{array}}\right].$$

$$⟨\frac{\partial E}{\partial m_f}⟩=⟨\frac{\partial E}{\partial {ϵ}^c}⟩=⟨\frac{\partial E}{\partial {ϵ}^d}⟩=⟨\frac{\partial E}{\partial {ϵ}^f}⟩=0.$$

$$C^f(\text{r})=⟨S_{\text{i}+\text{r},z}^f{S}_{\text{i},z}^f⟩=⟨(n_{\text{i}+\text{r}\uparrow }^f-n_{\text{i}+\text{r}\downarrow }^f)(n_{\text{i}\uparrow }^f-n_{\text{i}\downarrow }^f)⟩$$

$$C^{fd}=⟨{\vec{S}}_{\text{i}}^f\cdot {\vec{S}}_{\text{i}}^d⟩$$

$$⟨m^2{⟩}_{\mathrm{dyn}}$$

$$=\frac{1}{\beta }{\int }_0^{\beta }d\tau ⟨S_{\text{i}z}^f(\tau )S_{\text{i}z}^f(0)⟩$$

$$S^f(\text{k})=\sum _{\text{r}}{e}^{i\text{k}\text{r}}{C}^f(\text{r})$$

$$\frac{3S^f(\text{Q})}{N}=\frac{a}{L}+m_{AF,f}^2$$

$$S_n^f(\text{Q})=\frac{N}{N-n}\sum _{\text{r},|r|>l_c}{e}^{i\text{Q}\text{r}}{C}^f(\text{r})$$

$$G_{\alpha }(\text{q},\tau )={\int }_{-\infty }^{+\infty }d\omega \frac{e^{-\omega \tau }}{1+e^{-\beta \omega }}{A}_{\alpha }(\text{q},\omega )$$

$${\chi }_{\alpha }(\text{q},\tau )$$

Full text

I Introduction

The periodic Anderson Model (PAM) describes the hybridization between mobile ($d$ band) free electrons in a metal with strongly correlated $f$ electrons. The PAM has been extensively studied since its first introductionAnderson61 , and can successfully account for a variety of remarkable $f$-electron (rare-earth and actinide) phenomena including heavy-fermion physicsstewart84 ; lee86 ; Georges96 ; Vidhyadhiraja04 ; Sen16 , valence fluctuationshewson93 ; Shinzaki16 , volume collapse transitionsallen82 ; allen92 ; lavagna83 ; gunnarsson83 ; mcmahan98 ; lipp08 ; bradley12 ; lanata15 and unconventional superconductivitypixley15 .

At low temperatures, as the hybridization strength is varied in the PAM, there is a competition between the RKKY interactionRuderman54 ; Kasuya56 ; Yosida57 , which favors magnetically ordered $f$ band local moments, and the Kondo effectKondo69 ; Kouwenhoven01 , which screens the local moments and induces singlet states. Kondo screening can also occur at the interface between metallic and strongly correlated materialsmannhart05 ; mannhart10 , a situation which has given rise to additional theoretical and numerical investigation of the PAM and its geometrical variantsmillis05 ; freericks06 ; Euverte12 . Here the metallic band is viewed as arising from material on one side of an interface, and the correlated band describes the other side of the interface, as opposed to originating from strongly and weakly correlated orbitals of atoms in a single, homogeneous material.

A natural generalization of models which couple a single conduction band to localized, magnetic, orbitals is to consider similar physics when several conduction bands are present. The new qualitative physics to be explored is how the third band, and the resulting imbalance between the numbers of conduction and loacalized electrons, alters the strong correlation phenomena of the two band PAM: RKKY-induced AF order at weak $V$, the nature of the Kondo gap at strong $V$, and, finally, the position of the AF-singlet transition between these limits.

Besides these interesting fundamental questions, such a model is also worthy of investigation as a first step towards experimentsshishido10 on $f$-electron superlattices like CeIn3(n)/LaIn3(m). In these systems, by varying the thicknesses $n,m$ of the different materials, the effective dimensionality can be tuned, and hence the $2D$-$3D$ crossover of Kondo physics and AF order.

Recent theoretical investigations of the effect of immersing a Kondo insulator, or a superlattice thereof, in a 3D metal has been undertaken by Peters et al.peters13 . The focus there was on the evolution of the density of states $\rho(\omega)$ and, especially, features like the Fermi level hybridization gap of the Kondo sheet. A key conclusion is that the Kondo gap is modified to a pseudogap, with quadratically vanishing $\rho(\omega)$ from coupling to the metallic layer, and that the 3D $\rho(\omega)$ of the metallic layer adjacent to the Kondo layer develops 2D features. Changes to $\rho(\omega=0)$ in the singlet phase will be a key feature of our results here.

A bilayer heavy fermion system comprising a Kondo insulator (KI), represented by a symmetric PAM, coupled to a simple metal (M) has been proposed and studied employing the framework of DMFTSen16 at half-filling. The main goal of the work was to determine the ground state phase diagram from a Kondo screened Fermi liquid to a Mott insulating phase as a function of interaction strength and interlayer coupling. More generally, the possibility of the coexistence of spectral functions with distinct behaviors near the Fermi surface, despite the presence of interband hybridization, is the topic of studies of orbitally selective transitionsliebsch04 ; arita05 .

While we focus here on the influence of an additional metallic band on the properties of the PAM, similar extensions to include electron-phonon couplingZhang13 , dilutionSen15 , and $f$-electron hybridizationEuverte13 have similarly explored the ways in which AF-singlet competition can be influenced by the inclusion of further energy scales and degrees of freedom in the Hamiltonian.

In this paper, we employ the determinant quantum Monte Carlo (DQMC) methodBlankenbecler81 ; Loh92 , which provides an approximation-free solution to strong correlations, to study the magnetic structure of the bilayer KI-M system. By finite size scaling, we reliably extract the AF order parameter as a function of the KI hybridization strength $V$, and then build up the magnetic phase diagram in the $V-t^{\prime}$ plane for a representative potential $U_{f}=4$. To understand more precisely the role of nonzero $t^{\prime}$, we begin with a redetermination of the quantum critical point (QCP) of the AF-singlet phase transition of the half-filled PAM, the $t^{\prime}=0$ limit, with higher accuracy than in previous literatureVekic95 . The DQMC work is mainly focused on the particle-hole symmetric (half-filled) limit where there is no sign problem in the simulation. We also implemented a mean-field theory (MFT) calculation bothe at and away from half-filling as a supplement to DQMC. Our work is distinguished from previous workpeters13 ; Sen16 by its consideration of a PAM rather than a coupling to local (Kondo) spins, and its treatment of intersite magnetic correlations which are suppressed in the paramagnetic DMFT used in earlier work.

II Model and Methods

We consider the bilayer KI-M Hamiltonian on a square lattice,

[TABLE]

$t$ is the intralayer hopping parameter, which, for simplicity, we chose to be the same in the uncorrelated $c$ and $d$ bands. $t^{\prime}$ is the interlayer hopping parameter between the $c$, $d$ bands. $V$ is the hybridization strength between the $d$, $f$ bands. $U_{f}$ is the Coulomb repulsion in the $f$ band. Finally, $\epsilon^{\alpha}$ are the orbital energies of the $\alpha=c,d,$ and $f$ bands, and $n^{\alpha}_{\textbf{i}\sigma}\equiv\alpha^{\dagger}_{\textbf{i}\sigma}\alpha^{\phantom{\dagger}}_{\textbf{i}\sigma}$ are the density operators. The model is shown pictorially in Fig. 1, where $f$ and $d$ bands belong to the KI layer, and $c$ band belongs to the metal layer. Within the KI formed by the $d$ and $f$ bands, $V$ controls the competition between antiferromagnetic (AF) and singlet phases.

In this work, we set $t=1$ as our energy scale and mainly consider the particle-hole symmetric limit where $\epsilon^{\alpha}=0$, so each of the three bands is individually half-filled. We also implement the MFT calculation away from half-filling.

At half-filling, the Hamiltonian can be solved exactly in the noninteracting limit ($U_{f}=0$). Unlike the PAM in which $V$ opens a gap at half-filling and which hence is a band insulator there, the KI-M system is metallic at half-filling. This follows from the fact that the Hamiltonian has an odd number of bands (three): the Fermi level lies in the middle of the central band.

This metallic character at half-filling can be made more precise by going to momentum space $\alpha^{\dagger}_{\textbf{k}\sigma}=(1\ \sqrt{N})\,\sum_{\textbf{l}}e^{i\textbf{kl}}\alpha^{\dagger}_{\textbf{l}\sigma}$ for each of the three bands $\alpha=c,d,f$.

[TABLE]

Here $\epsilon_{\textbf{k}}=-2t(\cos(k_{x})+\cos(k_{y}))$. Diagonalizing the Hamiltonian Eq. 9 yields the three energy bands. In general, these bands cross and, at $\epsilon^{\alpha}=0$, are all partially filled. However, in certain limits, e.g. $V=0$ and $t^{\prime}>4t$, band gaps are present. Even so, in these situations the central band is half-filled and the system remains metallic.

The focus of our work will be the implications of the additional metallic ($c$) band on the competition between the RKKY interaction-induced AF and the Kondo regime of screened singlets, both central to the behavior of heavy-fermion materialsstewart84 ; lee86 ; Georges96 and captured in the solution of the PAM ($t^{\prime}=0$). A natural expectation is that, with the increase of $t^{\prime}$, the RKKY interaction between the local $f$ moments is enhanced, due to the additional conduction band channels, while the Kondo energy scale, set by $V$ and $U_{f}$, remains roughly fixed. This should lead to an overall movement of the quantum critical point to larger values of $V$.

To understand the precise effect of $t^{\prime}$ on the AF-singlet transition, we first carried out a self-consistent mean field theory (MFT). We then turned to a more exact, DQMC solution.

III Results: Mean Field Theory

Together with Kondo phases, ferromagnetic, antiferroferromagnetic and mixed order are all possible in the PAM and related Hamiltoniansbernhard15 ; eder16 . In the MFT treatment presented here, we thus consider three possible phases, the AF phase, the ferromagnetic (F) phase and the singlet phase. The AF MFT ansatz is

[TABLE]

where $\sigma$ is spin up ($\uparrow$) or spin down ($\downarrow$) and $m_{f}$ is the $f$ band AF order parameter. While the F MFT ansatz is

[TABLE]

where $\sigma$ follows the same definition and $m_{f}$ is the $f$ band ferromagnetic order parameter. In order to fix the particle densities $n_{c}$, $n_{d}$ and $n_{f}$, the terms $Nn_{c}\epsilon^{c}$, $Nn_{d}\epsilon^{d}$ and $Nn_{f}\epsilon^{f}$ must be subtracted from the original Hamiltonian, Eq. 1 reported in Ref. Costaa .

The AF MF decoupling of the interaction then gives a quadratic Hamiltonian in which momenta ${\bf k}$ and ${\bf k}$ - ${\bf Q}$ (where ${\bf Q}=(\pi,\pi)$) are coupled, resulting in

[TABLE]

where v^{\dagger}_{\textbf{k}\sigma}=\left[{\begin{array}[]{cccccc}c^{\dagger}_{\textbf{k}\sigma}&d^{\dagger}_{\textbf{k}\sigma}&f^{\dagger}_{\textbf{k}\sigma}&c^{\dagger}_{\textbf{k}-\textbf{Q},\sigma}&d^{\dagger}_{\textbf{k}-\textbf{Q},\sigma}&f^{\dagger}_{\textbf{k}-\textbf{Q},\sigma}\\ \end{array}}\right] ,

[TABLE]

In $M^{AF}_{\textbf{k}\sigma}$, $\eta^{c}_{\textbf{k}}$, $\eta^{d}_{\textbf{k}}$ and $\eta^{f}_{\textbf{k}}$ stand for $\epsilon_{\textbf{k}}+\epsilon^{c}$, $\epsilon_{\textbf{k}}+\epsilon^{d}$ and $(\frac{n_{f}}{2}-\frac{1}{2})U_{f}+\epsilon^{f}$ respectively.

On the other hand, the F MF decoupling leads to

[TABLE]

where $u^{\dagger}_{\textbf{k}\sigma}$ stands for \left[{\begin{array}[]{ccc}c^{\dagger}_{\textbf{k}\sigma}&d^{\dagger}_{\textbf{k}\sigma}&f^{\dagger}_{\textbf{k}\sigma}\\ \end{array}}\right] and the matrix is

[TABLE]

Hence, $m_{f}$, $\epsilon^{c}$, $\epsilon^{d}$, $\epsilon^{f}$ and the corresponding MFT phase boundary, are computed self-consistently by minimizing the total ground state energy $E$

[TABLE]

We first explore the AF MFT ansatz at half-filling. Results for $m_{f}$ as a function of $V$ for different couplings $t^{\prime}$ of the KI to the metal are shown in Fig. 2. A sharp QCP is evident at $t^{\prime}=0$ whose location agrees with previous workVekic95 . The evolution of $m_{f}(V)$ is smoother for $t^{\prime}\neq 0$. This difference is associated with the fact that at $t^{\prime}=0$ the KI is a band insulator in the noninteracting limit, whereas, as discussed in the previous section, the KI-M model is metallic. In fact, the density of states $N(E)$ has a van-Hove singularity at the half-filled Fermi surface $E_{\bf k}=0$, for all $V$ with $t^{\prime}\neq 0$, as also occurs in the square lattice half-filled Hubbard model. In an expansion of the free energy, $F(m_{f})$ picks up a $|m_{f}|$ contribution from these $E_{\bf k}=0$ modes, which persists in the thermodynamic limit owing to the divergence of $N(E=0$). This effect pushes $V_{c}$ out to $V=\infty$ in mean field theory: AF order persists for all hybridization strengths.

Nevertheless, a cross-over $V_{c}$ is still evident in Fig. 2, especially for modest $t^{\prime}$. We assign a quantitive value by choosing the point of $m_{f}(V)$ of largest slope, and extrapolating linearly to $m_{f}=0$ as shown. These cross-over values for $V_{c}$ will be compared with the critical hybridization obtained by DQMC in the following section.

To determine the ground state phase away from half-filling, we compare results from both the AF MFT ansatz and the F MFT ansatz, as shown in Fig. 3. We denote the corresponding ground state energies $E_{AF}$ ($E_{F}$) and, for the singlet (paramagnetic) phase, $E_{S}$. In Fig. 3 (a), $E_{AF}-E_{S}$ and $E_{F}-E_{S}$ are shown as a function of the density $n_{c}$ with fixed $n_{d}=1$, $n_{f}=1$, that is, by doping the additional conduction band. We have chosen $t^{\prime}=1$, $V=1$, $U_{f}=4$ and $L=200$. For these parameters, $E_{AF}$ is always lowest for all dencities $n_{c}$. In Fig. 3(b), the optimal $m_{f}$ from the AF MFT ansatz (denoted as $m^{AF}_{f}$) and the optimal $m_{f}$ from the F MFT ansatz (denoted as $m^{F}_{f}$) are shown as a function of density $n_{c}$. Since the ground state phase is AF, red triangular data characterizes the behaviour of the ground state order parameter. While $n_{c}$ is varied greatly from $0.2$ to $1.8$, the magnitude of the order parameter stays in a small range from $0.85$ to $0.89$, showing that the $c$ band has limited effects on the $f$ band magnetic structure.

In Fig. 3(c), $E_{AF}-E_{S}$ and $E_{F}-E_{S}$ are shown as a function of the hybridization strength $V$ with fixed $n_{c}=0.8$, $n_{d}=1$, $n_{f}=1$, $t^{\prime}=1$, $U_{f}=4$ and $L=200$. Below the critical point $V_{c}\approx 1.75$, $E_{AF}$ is lower than $E_{F}$ and $E_{S}$. Above the critical point, the AF order gives way to the singlet phase in a second order phase transition. In Fig. 3(d), the explicit behaviours of $m^{AF}_{f}$ and $m^{F}_{f}$ with respect to $V$ are presented, in agreement with results of Fig. 3(c). Red triangular data points characterize the behaviour of the ground state order parameter. Notably, by moving away from half-filling, the smooth transition observed at the half-filling limit returns to the conventional MFT transition behaviour with order parameter exponent $\beta=1/2$.

IV Results: Determinant Quantum Monte Carlo

In contrast to MFT, DQMC provides an exact treatment of the interactions in the KI-M Hamiltonian. This is accomplished through the construction of a path integral expression for the partition function and the introduction of an auxiliary field to decouple the exponential of the quartic interaction term into a quadratic formBlankenbecler81 . The fermion trace can be done exactly, and the auxiliary field is then sampled to produce measurements of one and two particle correlation functions.

The DQMC method works on lattices of finite spatial extent, necessitating an extrapolation to the thermodynamic limit as described below. A “Trotter error” is also introduced in the separation of the kinetic and potential energy pieces of the Hamiltonian. We work with an imaginary time discretization small enough such that the Trotter error is negligible, i.e. it is less than our statistical sampling errors. All the following DQMC results are presented at the particle-hole symmetric limit.

To explore the magnetic behaviour, we first study the $f$ band real space equal time spin-spin correlation function,

[TABLE]

$C^{f}({\textbf{r}})$ measures the correlation between the $z$ component of a spin on site i with that on a site a distance r away. Although the definition in Eq. 26 only involves the $z$ component, we average all three components (which are equal by rotational symmetry).

In addition to the spatial decay of the $f$ band spin correlation function of Eq. 26, we also study the Kondo singlet correlation functionHuscroft99 , defined as:

[TABLE]

where \vec{S}^{f}_{\textbf{i}}=[f^{\dagger}_{\textbf{i}\uparrow}\;f^{\dagger}_{\textbf{i}\downarrow}]\,{\vec{\sigma}}\,\left[\begin{array}[]{c}f^{\phantom{\dagger}}_{\textbf{i}\uparrow}\\ f^{\phantom{\dagger}}_{\textbf{i}\downarrow}\end{array}\right] and \vec{S}^{d}_{\textbf{i}}=[d^{\dagger}_{\textbf{i}\uparrow}\;d^{\dagger}_{\textbf{i}\downarrow}]\,{\vec{\sigma}}\,\left[\begin{array}[]{c}d^{\phantom{\dagger}}_{\textbf{i}\uparrow}\\ d^{\phantom{\dagger}}_{\textbf{i}\downarrow}\end{array}\right] and $\vec{\sigma}$ are the Pauli matrices.

At a KI-M coupling $t^{\prime}=1$, the DQMC result for the $f$ band spin-spin correlation function $C^{f}(\textbf{r})$ shown in Fig. 4 reveals non-zero (long range) AF correlations at hybridization strength $V=1.06$. This value is well above the pure KI ($t^{\prime}=0$) critical point, indicating that AF order is stabilized by $t^{\prime}$. The right panel of Fig. 4 shows the variation of the singlet correlation function $C^{fd}$ with hybridization strength. As $V$ increases, the system switches from a small $C^{fd}$ regime where singlet correlations are absent (the AF phase dominates) to a large $C^{fd}$ regime where Kondo singlets are well formed (and AF correlations are absent). As has been previously notedVekic95 , the position of the most rapid increase in magnitude of $C^{fd}$ gives an approximate location to the AF-singlet QCP.

In addition to the manner in which the spin correlation function decays with spatial separation, the imaginary time evolution also offers a window into the AF-singlet transition. Specifically, the $f$ band dynamic local moment,

[TABLE]

is the integral of the spatially local, unequal time spin correlation function $C^{f}({\bf r}=0,\tau)$. Here $S^{f}_{\textbf{i}z}(\tau)=e^{H\tau}S^{f}_{\textbf{i}z}e^{-H\tau}$. As with our previous equal time $C^{f}({\bf r})$, we average this correlation function over all spin directions to improve statistics. In a situation where the spin operator commutes with the Hamiltonian, e.g. at $V=0$ where one has isolated moments, the instantaneous, $C^{f}({\bf r}=0,\tau=0)$, and dynamic moments $\langle m^{2}\rangle_{\rm dyn}$ are equal. Quantum fluctuations from the hybridization $V$ cause the spin correlation to decay in imaginary time, reducing the dynamic moment.

As seen in Fig. 5 there are two quite different behaviors of $C^{f}({\bf r}=0,\tau)$ when $V$ is non-zero. $C^{f}({\bf r}=0,\tau)$ decays to zero rapidly at $\tau/\beta=0.5$ for $t^{\prime}=1$, while it decays smoothly to a non-zero value at $\tau/\beta=0.5$ for $t^{\prime}=3$. Integrating $C^{f}({\bf r}=0,\tau)$ yields the dynamic local moments shown in the inset to Fig. 5. Increasing $t^{\prime}$ from $1$ to $3$, induces a very large change in $\langle m^{2}\rangle_{\rm dyn}$, which implies the shifting of the system from a Kondo singlet to an AF phase.

The rapid decay of $C^{f}({\bf r}=0,\tau)$ with $\tau$ is associated with the presence of a singlet gap. An alternate way of interpreting the data of Fig. 5 is that by enhancing the AF tendency, increasing $t^{\prime}$ causes the vanishing to the singlet gap and a large increase in $\langle m^{2}\rangle_{\rm dyn}$.

The $f$-band structure factor $S^{f}(\textbf{k})$ is the Fourier transform of the $f$-band equal time spin-spin correlation function $C^{f}(\textbf{r})$, and is defined as:

[TABLE]

We present results for $\textbf{k}=\textbf{Q}$, the AF structure factor, since this is the dominant ordering wave vector at half-filling.

If there is long range AF order in the system, $C^{f}(\textbf{r})$ remains non-zero to large separations ${\bf r}$ and hence the spatial sum to form $S^{f}(\textbf{Q})$ yields a quantity which increases linearly with the system size $N$. Spin wave theoryHuse88 provides the analytic form for the finite size correction

[TABLE]

Here $m_{AF,f}$ is the AF order parameter in the thermodynamic limit and $L=\sqrt{N}$ is the linear lattice size. The correction factor $a$ can be reduced by excluding short range terms C^{f}\big{(}{\bf r}=(0,0)\big{)} from the sum used to build the full structure factor, since spin correlations at short distances are enhanced over the square of the order parameter $m_{AF,f}$. An improved estimator (lower finite size effects) is therefore Varney09 ,

[TABLE]

where $n$ is the number of separations ${\bf r}$ shorter than $l_{c}$ which are excluded from the sum. In our DQMC measurements, we chose $l_{c}=0$ and $n=1$, removing only the fully local spin-spin correlation $C^{f}(0,0)$. (This is the off-vertical scale data point in Fig. 4.)

To locate the phase transition point accurately, we measure the structure factor $S^{f}_{n}(\textbf{Q})$ according to Eq. 30, and then extrapolate to get the order parameter $m_{AF,f}$. The results are shown in Fig. 6. Fixing $t=1,U_{f}=4,\beta=25$, then for $t^{\prime}=1$, $m_{AF,f}$ has a negative extrapolation at $V=1.13$, but is positive at $V=1.08$. These bracket the QCP which we estimate to be at $V_{c}=1.10\pm 0.03$. Similarly, for $t^{\prime}=2$, we conclude $V_{c}=1.70\pm 0.04$; for $t^{\prime}=3$, we have $V_{c}=2.58\pm 0.04$; and finally for $t^{\prime}=4$, we find $V_{c}=3.54\pm 0.08$.

Although our focus here is on the KI-M model and ascertaining the effect of additional metallic bands on the AF-singlet transition, we also have determined $V_{c}$ more accurately for the PAM ($t^{\prime}=0$). To our knowledge, the original DQMC resultsVekic95 have not been re-examined. The main panel of Fig. 7 shows the $U_{f}$-$V$ phase diagram, and the AF and singlet regions at $t^{\prime}=0$. The critical points are deduced from the scaling of $S^{f}_{n}(\textbf{Q})$. A representative plot is shown in the inset for $U_{f}=3$. $V=0.90$ (zero intercept) and $V=0.87$ (non-zero intercept), bracket a critical value $V_{c}=0.89\pm 0.02$ at $U_{f}=3$. Similarly, for $U_{f}=4$, we conclude $V_{c}=0.99\pm 0.02$ and for $U_{f}=6$, we conclude $V_{c}=1.18\pm 0.02$. The dotted line showing the MFT results has been discussed in the previous section.

We now consider the phase diagram for the full KI-M Hamiltonian with $t^{\prime}\neq 0$. Here we chose to fix $U_{f}=4$ and focus on the effect of coupling the KI to the metal with $t^{\prime}$. The phase diagram is shown in Fig. 8. With the increase of $t^{\prime}$, the critical value $V_{c}$ increases in both DQMC and MFT calculations, quantifying the degree to which interlayer hopping parameter $t^{\prime}$ enhances the RKKY interaction and stabilizes the AF phase. The increase in $V_{c}$ is quite substantial. In contrast, previous comparisonsheld00 of the PAM with on-site (insulating) and intersite (metallic) $d$ $f$ hybridization did not reveal as great a difference in $V_{c}$. This suggests the increase found here is associated with the van-Hove singularity in the DOS.

V Spectral Function and Dynamic Spin Structure Factor

DQMC is also able to evaluate (real time) dynamic information through analytic continuationjarrell96 of the imaginary time correlation function. Inverting the integral relation,

[TABLE]

yields the spectral function from the one particle Greens functions measured in DQMC. Here $\alpha=c,d,f$ labels the band. The associated densities of states are given by $\rho_{\alpha}(\omega)=\sum_{\textbf{q}}A_{\alpha}(\textbf{q},\omega)$. The low frequency behavior of $\rho_{\alpha}(\omega)$ quantifies the possible existence of Slater, Mott, or hybridization gaps.

The dynamic spin structure factor is similarly related to an imaginary time counterpart which is a generalization of the quantity of Eq. 28 to include intersite correlations,

[TABLE]

In an AF ordered phase, the presence of low energy spin wave excitations leads to a vanishing of the gap in $\textup{Im}\chi_{\alpha}(\textbf{q},\omega)$ at the ordering wave vector (in our case ${\bf Q}=(\pi,\pi)$).

In Fig. 9 and Fig. 10, we show the one particle spectral function $A_{f}(\textbf{q},\omega)$ and the spin spectral function $\textup{Im}\chi_{f}(\textbf{q},\omega)$, which are calculated using the maximum entropy method(Gubernatis, ; Beach, ), for the PAM model and the KI-M model respectively. $\textup{Im}\chi_{f}(\textbf{q},\omega)$ complements the data for the equal time spin and singlet correlators of Fig. 4. In their AF phases (left panels) the PAM and the KI-M models are both characterized by a single particle gap in $A_{f}(\textbf{q},\omega)$. $\textup{Im}\chi_{f}(\textbf{q},\omega)$ has a finite spectral weight (no gap) near $\omega=0$ indicating the presence of low energy spin wave excitations in an AF phase. As has previously been noted in DQMCVekic95 in the singlet phase of the PAM, a spin gap opens in $\textup{Im}\chi_{f}(\textbf{q},\omega)$. Figure 11 gives the momentum integrated $\textup{Im}\chi_{f}(\omega)$. The similarity between the PAM and KI-M is clear, although the singlet phase dynamic spin response of the KI-M is considerably broader, a natural consequence of the larger value of $V$ required to destroy AF order and of the hybridization to an additional band.

The singlet phase of the KI-M is distinguished from the PAM by a non-zero $A_{f}(\textbf{q},\omega)$ at $\omega=0$ and also a very broadly distributed spectral weight $\textup{Im}\chi_{f}(\textbf{q},\omega)$ (Fig. 10, bottom right). The distinction between the singlet phases of the KI-M and the PAM is further confirmed by the dynamic spin structure factors, given by $\textup{Im}\chi_{f}(\omega)=\sum_{\textbf{q}}\textup{Im}\chi_{f}(\textbf{q},\omega)$ for both models, as shown in Fig. 11.

VI Conclusions

A considerable body of existing theoretical and numerical work potthoff95a ; potthoff95b ; okamoto04 ; zenia09 ; ishida12 ; helmes08 ; Euverte12 has examined coupling of a single band Hubbard model to additional conduction electrons as a model of metal-insulator interfaces, and the possibility of penetration of AF and Mott insulator features of strong interaction into the metal, and vice-versa. Qualitative similarities exist between phenomena like singlet formation between electrons in distinct bands and between electrons in two conjoined materials. In this paper we have first shown that within a self-consistent MFT there is a tendency towards expansion of the region of AF stability in a three band extension of the PAM.

We next employed the DQMC method to confirm these findings with an exact, beyond MF, treatment of the correlations, and thereby identify quantitatively the critical $f$ $d$ hybridization in the plane of interaction strength $U_{f}$ and hopping $t^{\prime}$ between the PAM and the metal. In the process, we improved on the previously known $V_{c}$ in the PAM ($t^{\prime}=0$) limit. Our primary observables in the characterization of the phases were the AF structure factor, the singlet correlator, and the dynamical moment, which all provide a consistent picture of the location of the phase boundary. Work within DMFTpeters13 , which focuses on the paramagnetic phase, is complementary to what we have done here.

Although the AF phase of the KI-M is stabilized by contact with the metal, the behavior of $\textup{Im}\chi_{f}(\textbf{q},\omega)$ is not dramatically different from the PAM. $\textup{Im}\chi_{f}(\textbf{q},\omega)$ has a gap $\Delta_{s}$ at low freqencies in the singlet phase, but has non-zero low frequency spectral weight in the AF phase associated with spin-wave excitations.

In contrast, the single particle spectral weight $A_{f}(\textbf{q},\omega)$, and the momentum-integrated density of states, behave differently in the KI-M than the PAM. The PAM has a nonzero charge gap $\Delta_{c}$ in both the AF and singlet phasesVekic95 , with $\Delta_{s}/\Delta_{c}\rightarrow 1$ as $V$ increases to deep in the singlet phase. We find here that for the KI-M there are peaks in $A(\textbf{q},\omega)$ near $\textbf{q}=(\pi,0)$ and $\textbf{q}=(\pi/2,\pi/2)$ and hence also in $\rho_{f}(\omega)=\sum_{\textbf{q}}A_{f}(\textbf{q},\omega)$. We believe this distinction to originate in the fact that even though the AF order is lost, the additional $c$-electrons still strongly interact with the $d$ and $f$ bands of the KI, so that there is no longer an insulating Kondo gap.

The Hamiltonian, Eq. 1 includes $cd$ and $df$ hybridizations. We have also done some studies of the effect of on-site $cf$ hopping. In order to keep the lattice bipartite and avoid a sign problem we have altered the $df$ hybridization to a near-neighbor form also used, for example, in held00 . This change does not shift the critical $V$ from the on-site value, to within our error bars. Having verified this, we then added $cf$ hopping and find that it, also, leaves the critical point at the same value. We conclude that more complex (and realistic) forms of the electronic kinetic energy have little effect on the qualitative and quantitative results of our paper: an enhancement of the regime of AF order.

Our work on a two layer metal-PAM represents a first step in the application of DQMC to the more general investigation of $f$-electron-metal superlattices, where, on the experimental side, dimensionality can be controlledshishido10 . Theoretical and numerical studies within dynamical mean field theorypeters13 ; okamoto08 ; tada13 ; peters14 of these structures have already led to great insight. Including the full spatial structure of each layer, as done in DQMC, makes the full superlattice problem challenging.

Acknowledgements.

We acknowledge N.C. Costa for useful discussions. WH and RTS at UC-Davis were supported by the U.S. Department of Energy under grant number DE-SC0014671. EWH and BM at Stanford/SLAC, primarily for the analytic continuation and its interpretation, were supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Contract No. DE-AC02-76SF00515.