Magnetic quantum phase transition in a metallic Kondo heterostructure

We consider a two-dimensional quantum spin system described by a Heisenberg model that is embedded in a three-dimensional metal. The two systems couple via an antiferromagnetic Kondo interaction. In such a setup, the ground state generically remains metallic down to the lowest temperatures and allows us to study magnetic quantum phase transitions in metallic environments. From the symmetry point of view, translation symmetry is present in two out of three lattice directions such that crystal momentum is only partially conserved. Importantly, the construction provides a route to study, with negative-sign-free auxiliary-field quantum Monte Carlo methods, the physics of local moments in metallic environments. Our large-scale numerical simulations show that as a function of the Kondo coupling, the system has two metallic phases. In the limit of strong Kondo coupling, a paramagnetic heavy-fermion phase emerges. Here, the spin degree of freedom is screened by means of the formation of a composite quasiparticle that participates in the Luttinger count. At weak Kondo coupling, magnetic order is present. This phase is characterized by Landau-damped Goldstone modes. Furthermore, the aforementioned composite quasiparticle remains intact across the quantum phase transition.

Figure 1

(a) Sketch of Kondo heterostructure, consisting of a two-dimensional array of magnetic impurities (blue dots) and three-dimensional itinerant conduction electrons, modeled by a tight-binding Hamiltonian on a cubic lattice (yellow dots). (b) Three-dimensional Fermi surface of conduction electrons. (c) Projected Fermi surface. (d) Ground-state phase diagram of model in Eq. (1), as extracted from QMC results.

Figure 2

(a) Mean-field order parameters at zero temperature. Here, $t=1, J_{\mathrm{H}}=0.5$, and we set $L=100$. The inset shows the order parameters over a larger range, confirming the analytical result $V=1$ in the strong-coupling limit [Eq. (E14)]. (b) Free energy as function of $J_{\mathrm{K}}$. Both (a) and (b) support a first-order transition.

Figure 3

Local density of states $A_c\left(\omega \right)$ (top) and $A_f\left(\omega \right)$ (bottom) in antiferromagnetic (first column), heavy-fermion (second column) and coexistence (third column) states in mean-field approximation. The corresponding mean-field parameters are given in the title of each column.

Figure 4

Fermion spectral functions $A_c({\mathit{k}}_2,\omega )$ (top row), $A_f({\mathit{k}}_2,\omega )$ (center row), and $A_{\psi }({\mathit{k}}_2,\omega )$ (bottom row) in antiferromagnetic (left column), heavy-fermion (center column) and coexistence (right column) phases in mean-field approximation. Here, we consider the path $\mathit{\Gamma }(0,0)\to \mathit{M}(\pi ,0)\to \mathit{K}(\pi ,\pi )\to \mathit{\Gamma }(0,0)$.

Figure 5

(a) Correlation ratio $R_{\mathrm{c}}(L,J_{\mathrm{K}})$ as function of Kondo coupling $J_{\mathrm{K}}$ from projective QMC for different lattice sizes $L$. (b) Finite-size critical point $J_{\mathrm{K}}^{\mathrm{c}}\left(L\right)$ as function $1/L$. The extrapolation towards the thermodynamic limit indicates a single quantum critical point at $J_{\mathrm{K}}^{\mathrm{c}}=3.019\left(4\right)$.

Figure 6

Spin spectral function $S({\mathit{k}}_2,\omega )$ along a high-symmetry path in the Brillouin zone from finite-temperature QMC for different values of $J_{\mathrm{K}}$, using $\beta =40$ and $L=12$.

Figure 7

Spin-spin correlations $C_f(\mathit{i}-\mathit{j},\tau )$ in impurity layer as function of distance $r$ at equal time $\tau =0$ (left column) and imaginary time $\tau$ at equal position $r=0$ (right column) for different values of $J_{\mathrm{K}}$ in the antiferromagnetic phase (top row), quantum critical regime (center row), and paramagnetic heavy-fermion phase (bottom row), from finite-temperature QMC, using $\beta =L^2/2$. Dashed blue, black, and red lines represent power-law decay functions $f\left(x\right)\sim 1/x, 1/x^2$, and $1/x^4$ for reference. The red solid line at (b), (c), (e), and (f) represent the numerical fitting of the QMC data.

Figure 8

Fermion spectral functions $A_c({\mathit{k}}_2,\omega )$ (top row) and $A_{\psi }({\mathit{k}}_2,\omega )$ (bottom row) from finite-temperature QMC, using $\beta =40$ and $L=12$. The Kondo coupling $J_{\mathrm{K}}$ increases from left to right.

Figure 9

Local density of states $A_c(\omega =0)$ (left) and $A_{\psi }(\omega =0)$ (right) near the Fermi level as function of temperature $T=1/\beta$ for different values of $J_{\mathrm{K}}$ from finite-temperature QMC. Here, we consider $L=10$ and restrict the data to the temperature range with smallest finite-size effects.

Figure 10

Imaginary part of $c$-fermion Green's function $\text{Im}{G}_c^{R_z=0}({\mathit{k}}_2,{\omega }_n)$ as function of Matsubara frequency ${\omega }_n=(2n+1)\pi /\beta$ for (a) ${\mathit{k}}_2=(\pi /2,\pi /4)$ and (b) ${\mathit{k}}_2=(\pi /2,\pi /2)$. Green dots follow Eq. (D5). Purple (orange) dots are obtained from finite-size lattice calculations with linear size $L=8$ and inverse temperature $\beta =40$ with periodic boundary conditions (twisted boundary conditions, averaged over ten different twists). Lines represent guides to the eye.

Figure 11

Conduction-electron local density of states $A_c(\omega =0,T)$ near the Fermi level as function of temperature $T$ for different lattice sizes $L$. In each panel, different colors indicate different system sizes. The Kondo coupling $J_{\mathrm{K}}$ increases from left to right and top to bottom.

Figure 12

Same as Fig. 11, but for the composite-fermion local density of states $A_{\psi }(\omega =0,T)$.

Figure 13

(a) Crystallographic Brillouin zone (blue) and backfolded magnetic Brillouin zone (yellow). (b) Magnon self-energies $\pi$ in one-loop perturbation theory from Eqs. (D9) (top) and (D10) (bottom).

Figure 14

Spin spectral function ${\chi }^{\prime \prime }(\delta {\mathit{p}}_2,\omega )$ in Eq. (D27) along the high-symmetry path $\mathit{M}(\pi ,0)\to \mathit{K}(\pi ,\pi )\to \mathit{\Gamma }(0,0)$. Here, we have used the parameters $t=1, J_{\mathrm{K}}=2.5, J_{\mathrm{H}}=0.5, S=1/2, {\tilde{c}}_B=0.447$, and $\alpha =0.3$.

Figure 15

Examples of the different wave functions on the $c$ sites for system size $L=40$: The dots represent the eigenstates obtained by diagonalizing ${\widehat{H}}_{\text{MF}}$ numerically at $J_{\mathrm{K}}/t=5.1, V/t=0.2$, and ${\epsilon }_{{\mathit{k}}_2}/t=-0.2$. The solid lines are comparisons with the analytic solutions given in the text for this set of parameters. In particular, the extended states in the first line are (a) odd superpositions of Bloch waves and (b) even scattering states with phase shift. The second line shows the two-dimensional states with (c) minimal and (d) maximal energy.