Magnetic field induced quantum criticality and the Luttinger sum rule

We show that when there is a sudden transition from a small to a large Fermi surface at a field-induced quantum critical point—similar to what may have been observed in some heavy-fermion compounds—an additional term has to be taken into account in the Luttinger-Friedel sum rule. We calculate this additional term for a local model which has a field-induced quantum critical point (QCP) and show that it changes abruptly at the transition, such that it satisfies a generalized Luttinger-Friedel sum rule on each side of the transition, and characterizes the two Fermi-liquid phases separated by the QCP as a discrete (topological) index.

Figure 1

A plot of $h_c/J$, where $h_c$ is critical field to induce a transition, as a function of the interaction parameter $J$ for a particle-hole symmetric model with $U/\pi \mathrm{\Delta }=5,J_c=2.8272\times {10}^{-5}$, and for an asymmetric model with ${\epsilon }_f/\pi \mathrm{\Delta }=-0.659,U/\pi \mathrm{\Delta }=0.5,J_c=5.4220\times {10}^{-3}$(dashed line).

Figure 2

A plot of the quasiparticle weight factors, $z_{\uparrow }={\tilde{\mathrm{\Delta }}}_{\uparrow }/\mathrm{\Delta }$ and $z_{\downarrow }={\tilde{\mathrm{\Delta }}}_{\downarrow }/\mathrm{\Delta }$, as a function of the magnetic field $h/h_c$ in the region of the transition for the parameter set ${\epsilon }_f/\pi \mathrm{\Delta }=-0.659,U/\pi \mathrm{\Delta }=0.5,J_c=5.4420\times {10}^{-3},J=5.8482\times {10}^{-3}$, where $h_c/\pi \mathrm{\Delta }=0.20795159$.

Figure 3

A plot of the ratios ${\tilde{\epsilon }}_{f,+}/{\tilde{\mathrm{\Delta }}}_{+}({\tilde{\epsilon }}_{f,\uparrow }/{\tilde{\mathrm{\Delta }}}_{\uparrow })$ and ${\tilde{\epsilon }}_{f,-}/{\tilde{\mathrm{\Delta }}}_{-}({\tilde{\epsilon }}_{f,\downarrow }/{\tilde{\mathrm{\Delta }}}_{\downarrow })$ as a function of the magnetic field $h/h_c$ for the parameter set given in Fig. 2.

Figure 4

A plot of the spin-up and spin-down local occupation values $n_{f,+}$ and $n_{f,-}$ from a direct calculation as a function of the magnetic field $h$ for the parameter set of Fig. 2. Also shown are the values of ${\tilde{n}}_{f,+}$ and ${\tilde{n}}_{f,-}$ and the corresponding values shifted by $+\frac{1}{2}$ for $h<h_c$.

Figure 5

A plot of the spin-up and spin-down local spectral densities, ${\rho }_{f,+}$ and ${\rho }_{f,-}$, as a function of the magnetic field $h/h_c$ for the asymmetric parameter set of Fig. 2, showing the sudden increase at the quantum critical point as $h$ increases through $h_c$.

Figure 6

A plot of the spin-up and spin-down local spectral densities, ${\rho }_{f,+}$ and ${\rho }_{f,-}$(${\rho }_{f,+}$ = ${\rho }_{f,-}$), as a function of the magnetic field $h/h_c$ for the particle-hole symmetric parameter set, $U/\pi \mathrm{\Delta }=5,J=3.0354436\times {10}^{-5},J_c=2.8272\times {10}^{-5}$ showing the sudden increase at the quantum critical point as $h$ increases through $h_c$.