Variational cluster approach to superconductivity and magnetism in the Kondo lattice model

We investigate in detail antiferromagnetic (AF) and superconducting (SC) phases as well as their coexistence in the two-dimensional Kondo lattice model on a square lattice, which is a paradigmatic model for heavy-fermion materials. The results presented are mainly obtained using the variational cluster approximation (VCA) and are complemented by analytical findings for the equations of motion of pairing susceptibilities. A particularly interesting aspect is the possibility to have $s$-wave SC near half-filling as reported by Bodensiek et al. [Phys. Rev. Lett. 110, 146406 (2013)] When doping the system, we identify three regions which correspond to an AF metallic phase with small Fermi surface at weak coupling, an AF metal with a different Fermi surface topology at intermediate coupling, and a paramagnetic metal with a large Fermi surface at strong coupling. The transition between these two AF phases is found to be discontinuous at lower fillings, but turns to a continuous one when approaching half-filling. In the quest for $s$-wave superconductivity, only solutions are found which possess mean-field character. No true superconducting solutions caused by correlation effects are found in the $s$-wave channel. In contrast, we clearly identify robust $d$-wave pairing away from half-filling. However, we show that only by treating antiferromagnetism and superconductivity on equal footing, artificial superconducting solutions at half-filling can be avoided. Our VCA findings support scenarios previously identified by variational Monte Carlo approaches and are a starting point for future investigations with VCA and further approaches such as cluster-embedding methods.

Figure 1

Schematic phase diagram as obtained with VCA using a $3\times 2$ cluster. At half-filling the system is an antiferromagnetic insulator for $J<J_c=1.94t$ and a paramagnetic insulator for $J>J_c$. Away from half-filling there are three phases: a paramagnetic metal close to half-filling at strong coupling, a $d$-wave superconductor, and a coexistence region of antiferromagnet and $d$-wave superconductor. Furthermore, when suppressing SC, within the AF phase at $J/t\sim 1.24$ an additional transition line between AF phases with different Fermi surface topology appears (not shown), as discussed in Sec. 5b.

Figure 2

Quasiparticle gap ${\mathrm{\Delta }}_{\mathrm{qp}}$ at half-filling in the paramagnetic ($J>J_c$) and antiferromagnetic $(J<J_c$) insulator obtained by using a $3\times 2$ cluster. The solutions for the pure paramagnet (PM) without addition of an AF Weiss field as well as the solution when including AF are shown. The quantity plotted amounts to $\mu \left(n=1.001\right)$ (for details see text).

Figure 3

Local density of states [DOS, see Eq. (11)], at large coupling $J/t=8.0$ close to half-filling ($n=0.95$). The two resonances are separated by roughly $3/2J$ and correspond to the breakup of Kondo singlets.

Figure 4

Top: staggered magnetization $m$ for the different cluster sizes and geometries shown. As a reference, values of $J_c/t$ obtained with dual fermions [33], quantum Monte Carlo [38], and DMFT+NRG [62] are shown. Bottom: finite-size extrapolations of the value of the critical coupling $J_c/t$ to the infinite-size cluster limit. The blue (solid) line takes into account only the regularly shaped clusters, as indicated, while the green (dashed) line considers the remaining cluster types. The scaling factor equals the number of intracluster links divided by twice the number of cluster sites.

Figure 5

Staggered magnetization, expectation value of the spin-spin correlator $\langle \vec{S}·\vec{s}\rangle =\partial \mathrm{\Omega }/\partial J$, and local $f$-spin susceptibility ${\chi }_{\text{loc}}^f={\partial }^2\mathrm{\Omega }/\partial h_f^2\left|{}_{h_f\to 0}\right.$ as a function of $J/t$ at half-filling using a $3\times 2$ cluster. Black lines are fits of the spin-spin correlator, blue lines denote their derivatives. They have been fitted with second-order polynomials in the AF and PM regions up to $J_c/t\pm 0.05$. The set of variational parameters includes $\{M_c,t^{\prime }\}$.

Figure 6

Density of states (DOS) at half-filling for several $J/t$ in the antiferromagnetic phase. The rest of the parameters are the same as in Fig. 19 and the scale of $\mathrm{DOS}\left(\omega \right)$ is kept the same for all panels.

Figure 7

Comparison of the PM and the AF solutions at $J=1.6t<J_c^{\mathrm{AF}}$ as a function of electron filling $n$. For $n>n_c\approx 0.91$ (vertical dashed line in the plot), the AF solution is energetically preferred (see upper panel). The lower panel shows the value of the staggered magnetization of the antiferromagnetic solution. For $n<n_c$ the staggered magnetization $m^c$ is zero and the solution is therefore paramagnetic. The dashed lines are fits to the data points and only a guide to the eye.

Figure 8

Staggered magnetization of the AF solution as a function of electron filling $n$ and $J/t$ using a $3\times 2$ cluster with variational parameters $\mu ,{\mu }^{\prime },t^{\prime }$, and $M_c$. Circles denote data points, their filling is color coded and shows the value of the staggered magnetization $m$. The dashed lines denote transitions. The one on the left label by “1st” indicates a discontinuity of the value of $m$ when keeping $n$ fixed and varying $J/t$. At fillings $n\gtrsim 0.97$, this becomes continuous (not shown). The dashed line to the right labeled by “2nd” indicates the continuous phase transition between AF and PM metal.

Figure 9

Density of states (DOS) for $J/t=1.6$ using the $3\times 2$ cluster with $\{\mu ,{\mu }^{\prime },t^{\prime },M_c\}$ as variational parameters. The electron fillings $0.91<n<1$ correspond to the AF metal phase. The panels show fillings close to the AF insulator ($n=0.99$), close to the PM metal ($n=0.92$), and right in the middle of the phase. The results obtained at $n=0.85$ are deep in the PM metal phase. The DOS has been obtained on a $k$ grid of $100\times 100$ points and an artificial broadening of $\eta =0.05$ has been added.

Figure 10

Characteristic spectral function $A(\mathbf{k},i\eta )$ for the three regimes that can be identified in the AF solution off half-filling, here shown at filling $n=0.90$ with broadening $\eta =0.01$: PM metal with a large Fermi surface (d) and AF metal with small Fermi surface (a). At intermediate coupling, the Fermi surface volume jumps to a much higher value (b), but the system stays AF (c).

Figure 11

(a) Staggered magnetization $m_{\text{AF}}^c$, expectation value of the spin-spin correlator $\langle \vec{S}·\vec{s}\rangle =\partial \mathrm{\Omega }/\partial J$, and local susceptibility ${\chi }_{\text{loc}}^f={\partial }^2\mathrm{\Omega }/\partial h_f^2{|}_{h_f\to 0}$ as a function of $J/t$ at $n=0.95$ using a $3\times 2$ cluster. (b) Staggered magnetization around the $\mathrm{AF}{}_1\text{−}\mathrm{AF}{}_2$ transition for the three indicated clusters.

Figure 12

DOS (top panel) and relative position of the resonances in the DOS (bottom panel) at $n=0.95$ for different coupling strengths in the antiferromagnetic region obtained from the AF solution of the VCA. A change of the peak structure close to the Fermi energy at $\omega =0$ happens at $J/t\gtrsim 1.2$: the main peak splits and an additional side resonance develops. Close to $J_c$ the structure already resembles the three-peak structure known from the Kondo insulating region for $J>J_c$.

Figure 13

SEF and cluster hopping strength $t^{\prime }$ at the stationary point of the SEF with respect to $\mu ,{\mu }_c$, and $t^{\prime }$ as a function of the Weiss field strength $D_s$. Plotted are two solutions for $J/t=2.4$ at $n=0.9$, one corresponding to a maximum with respect to $t^{\prime }$ (filled symbols), the other one corresponding to a minimum (empty symbols). Note the SEF having an extremum in either case at $D_s=0$, indicating no $s$-wave SC order being stabilized.

Figure 14

Anomalous expectation value ${\mathrm{\Phi }}_d$ as a function of $J/t$ at half-filling for the pure SC solution (blue) and when including also AF (green) using a $3\times 2$ cluster. For the latter case, the staggered magnetization is shown in black.

Figure 15

Anomalous expectation value when considering $d$-wave SC. The figure shows ${\mathrm{\Phi }}_d$ obtained using a $3\times 2$ cluster and variational parameters ${\mu }^{\prime },t^{\prime }$, and $D$. The circles are data points, the color code indicates the value of ${\mathrm{\Phi }}_d$ at these points. For $J\le J_{\mathrm{ano}}^{\text{SC}}\approx 0.9$ there is a superconducting solution even at half-filling, but one has to be careful as this region coincides with an anomaly in $t^{\prime }$. The interaction strength $J$ where the anomaly occurs is quite close to the one found in the paramagnetic solution ($J_{\mathrm{ano}}^{\text{PM}}\approx 0.84$).

Figure 16

Results when considering $d$-wave SC and AF away from half-filling obtained using a $3\times 2$ cluster at $J/t=1.8$. The top panel shows the ground-state energy as a function of filling for the various cases indicated. The vertical lines indicate the transition points at which AF order vanishes when considering AF only or AF+SC, respectively. The center and bottom panels show the anomalous expectation value and the staggered magnetization as a function of electron density $n$ for the superconducting, the antiferromagnetic, and the coexisting SC+AF solutions as indicated.

Figure 17

The same as in Fig. 16 showing the energy (top panel) and the order parameters (center and bottom panels) as a function of filling $n$, this time for $J/t=1.2$.

Figure 18

Values of the order parameters as a function of $J/t$ and $n$ when simultaneously considering $d$-wave SC and AF. The circles are data points. The values for the anomalous expectation value (intensity color coded in red) and for the staggered magnetization (color coded in blue) as obtained using a $3\times 2$ cluster are displayed. The dashed line indicates the transition line at which the AF continuously disappears when increasing $J/t$.

Figure 19

Top: density of states (DOS) at $J/t=1.0$ and half-filling comparing the AF and PM insulator. Bottom: composition of the AF total DOS (dashed line) by staggered up- and down-spin densities ${\tilde{\rho }}_{\uparrow }$ (red) and ${\tilde{\rho }}_{\downarrow }$ (blue), which are explained in the text. The DOS has been obtained on a $k$ grid of $100\times 100$ points by using the $3\times 2$ cluster and an artificial broadening of $\eta =0.05$.

Figure 20

Variation of the cluster hopping parameters inside the $3\times 2$ cluster at half-filling. Using three different cluster hopping parameters $t_1^{\prime },t_2^{\prime },t_3^{\prime }$ as indicated by different colors in the inset instead of only one parameter $t^{\prime }$ leads to a reduction in the free energy, but not to qualitative differences for the paramagnetic (top panel) and antiferromagnetic case (lower panel). For the latter, especially the value $J_c/t$ for the onset of antiferromagnetism and the value of the staggered magnetization do not change much when using anisotropic cluster hopping (black circles) in comparison to the result when varying only $t^{\prime }$ (continuous yellow line).

Figure 21

Variation of the cluster hopping parameters inside the $3\times 2$ cluster at $J/t=1.6$ as a function of electron filling. Variation of three cluster hopping parameters $t_1^{\prime },t_2^{\prime },t_3^{\prime }$ instead of one parameter $t^{\prime }$ leads to a slightly higher critical filling $n_c$ (top panel), where AF sets in, but not to qualitative differences. Lines indicate square-root fits. Especially, the spreading of the different values of the hopping parameters increases when approaching the transition (lower panel), although their mean value is still comparable to $t^{\prime }$.

Figure 22

Electron densities $n_e^{V1/V2}$ calculated according to Eq. (C3) and $n_e^c$ according to Eq. (C2) as a function of coupling strength $J/t$ for the $3\times 2$ cluster. $n_e$ is split into the contribution of the “outer” part of the Brillouin zone ($n_e^{V_1}$) and the “inner” part ($n_e^{V_2}$). The solid horizontal line marks the predefined electron filling $n_e=0.95$, the dashed vertical lines indicate the phase transitions from $\mathrm{AF}{}_1$ to $\mathrm{AF}{}_2$ and from $\mathrm{AF}{}_2$ to PM. In the insets, $\mathrm{\Theta }\left[G\right(\omega =0,\mathbf{k}\left)\right]$ is shown inside the Brillouin zone for small and strong-coupling values. Contributions to $n_e^{V1\left(2\right)}$ are indicated by green (blue) color.

Figure 23

The spectral function at the Fermi energy with an artificial broadening of $\eta =0.01t$, $A(\mathbf{k},\omega =0+i\eta )$ for three coupling strengths in the $\mathrm{AF}{}_1$, $\mathrm{AF},{}_2$ and PM regions (from left to right) at $n=0.90$. We overlay to the data the Fermi surface (top right), the Luttinger surface (bottom left), and the region with ${\zeta }^{*}>0$ (bottom right).

Figure 24

Results for the SC Weiss field $D_c$ and for the anomalous expectation value ${\mathrm{\Phi }}_s$ at half-filling when considering $s$-wave SC. The set of variational parameters includes $\{D,{\mu }^{\prime }\}$ (top panel) and $\{D,{\mu }^{\prime },t^{\prime }\}$ (bottom panel). In the bottom panel, only a $3\times 2$ cluster is considered whereas in the top panel the three indicated cluster sizes are shown. Note the hopping strength within the cluster is found to be zero within error bars and the Weiss field strength corresponds to the one of an isolated Kondo site, $D_c=3/4J$.

Figure 25

Results for the SEF when considering $s$-wave SC and AF order. Comparison of the results as a function of the strength of the superconducting Weiss field with (“SC+AF”) and without (“SC”) including $M_c$ into the set of variational parameters. The staggered magnetization of the SC+AF solution diminishes for intermediate values of $D_s$, hence, the self-energy functional approaches the SC result.