Field-Induced Antiferromagnetism in the Kondo Insulator

The Kondo lattice model, augmented by a Zeeman term, serves as a useful model of a Kondo insulator in an applied magnetic field. A variational mean field analysis of this system on a square lattice, backed up by quantum Monte Carlo calculations, reveals an interesting separation of magnetic field scales. For Zeeman energy comparable to the Kondo energy, the spin gap closes and the system develops transverse staggered magnetic order. The charge gap, however, remains robust up to a higher hybridization energy scale, at which point the canted antiferromagnetism is exponentially suppressed and the system crosses over to a nearly metallic regime. Quantum Monte Carlo simulations support this mean field scenario. An interesting rearrangement of spectral weight with magnetic field is found.

Figure 1

$T=0$ mean field, square lattice. (Left) $B_{\mathrm{c}}$ is a second-order transition; $B^{\prime }$, $B^{\prime \prime }$, and $B^{\prime \prime \prime }$ are crossovers. The singlet phase has hybridization order only. Antiferromagnetic and hybridization order coexist in the striped region. The stippling indicates that the electron moments are directed to opposite the $B$ field. For $|B|\gtrsim B^{\prime \prime \prime }$, the charge gap is exponentially small. Along the line $J<J_{\mathrm{c}}$, $B=0$, the system is Néel ordered (uncanted). (Right) $⟨f^{†}c⟩$, ${\Delta }_{\mathrm{c}}$, $⟨\frac{1}{2}{c}_i^{†}{\sigma }^3{c}_i⟩$, and $⟨(-1{)}^i\frac{1}{2}{c}_i^{†}{\sigma }^1{c}_i⟩$ in the upper half of the phase diagram.

Figure 2

$J/t=3$, $B_{\mathrm{c}}/t\doteq 0.323$. (Top row) Mean field band structure, folded into the reduced Brillouin zone, as a function of applied magnetic field. (Middle) Same, magnified to show the evolution of the band gap. (Bottom) The greyscale (white–black $\leftrightarrow 0–0.5t$) indicates the wave-vector-dependent gap magnitude. For $|B|<B_{\mathrm{c}}$, ${\Delta }_{\mathrm{c}}(\mathit{k})$ is a minimum at $\mathit{k}=0$, and, for $|B|>B_{\mathrm{c}}$, at an expanding ring of points.

Figure 3

Analytically continued data from the one-particle electron Green’s function for $J/t=1.7$, $\beta t=14$ on an $8\times 8$ lattice. Spectral functions $A(\mathit{k},\omega )$ are plotted for wave vectors along high-symmetry lines in the Brillouin zone. (Inset) the peak locations (open circles) are superimposed on the mean field band structure [Eq. (3)] with $V$ chosen to fit the band splitting at $\mathit{k}=(\pi ,0)$. The grey scale indicates the spectral weight (white–black $\leftrightarrow 0–1$).

Figure 4

$J/t=1.7$, $\beta t=14$, $8\times 8$ lattice. Spectral functions $A_{\uparrow }(\mathit{k},\omega )$ are plotted over a range of field values ($0\le B/t\le 1.8$ offset) for a series of wave vectors (identified in the top right inset). Smaller peaks have been scaled by the indicated magnification factor. Note that, instead of shifting smoothly through zero in a magnetic field, the spectral weight jumps across the gap, except for $\mathit{k}$ on the reduced zone boundary. (Top left inset) A partial phase diagram. Electron moments are directed against the field between $B_{\mathrm{c}}$ and $B^{\prime \prime }$.