Nonanalytic paramagnetic response of itinerant fermions away and near a ferromagnetic quantum phase transition

We study nonanalytic paramagnetic response of an interacting Fermi system both away and in the vicinity of a ferromagnetic quantum phase transition (QCP). Previous studies found that (i) the spin susceptibility $\chi$ scales linearly with either the temperature $T$ or magnetic field $H$ in the weak-coupling regime; (ii) the interaction in the Cooper channel affects this scaling via logarithmic renormalization of prefactors of the $T$, $\mid H\mid$ terms, and may even reverse the signs of these terms at low enough energies. We show that Cooper renormalization becomes effective only at very low energies, which get even smaller near a QCP. However, even in the absence of such renormalization, generic (non-Cooper) higher-order processes may also inverse the sign of $T$, $\mid H\mid$ scaling. We derive the thermodynamic potential as a function of magnetization and show that it contains, in addition to regular terms, a nonanalytic ${\mid M\mid }^3$ term, which becomes $M^4∕T$ at finite $T$. We show that regular $(M^2,M^4,\dots )$ terms originate from fermions with energies of order of the bandwidth, while the nonanalytic term comes from low-energy fermions. We consider the vicinity of a ferromagnetic QCP by generalizing the Eliashberg treatment of the spin-fermion model to finite magnetic field, and show that the ${\mid M\mid }^3$ term crosses over to a non-Fermi-liquid form ${\mid M\mid }^{7∕2}$ near a QCP. The prefactor of the ${\mid M\mid }^{7∕2}$ term is negative, which indicates that the system undergoes a first-order rather than a continuous transition to ferromagnetism. We compare two scenarios of the breakdown of a continuous QCP: a first-order instability and a spiral phase; the latter may arise from the nonanalytic dependence of $\chi$ on the momentum. In a model with a long-range interaction in the spin channel, we show that the first-order transition occurs before the spiral instability.

Figure 1

The field-dependent part of the thermodynamic potential at second order. $k,p,q$ are the four momenta: $k\equiv (\mathbf{k},{\omega }_m)$, etc. Here and in Figs. 2, 6, 11, thin solid lines denote bare Green’s functions.

Figure 2

Diagrams for the thermodynamic potential beyond second order.

Figure 3

Skeleton backscattering diagram. Hatched boxes represent the spin components of the renormalized static vertex, ${\Gamma }_s(\mathbf{k},\mathbf{p},q)$. Thick solid lines depict fully renormalized Green’s functions, given by Eq. (2.13).

Figure 4

(Color online) Function $S_H(f_{s,0},f_{s,1})$, which determines the sign of the magnetic-field dependence of the spin susceptibility calculated to fourth order in the skeleton interaction [cf. Eq. (2.52)], plotted as a function of $f_{s,1}∕f_{s,0}$ for $f_{s,0}=-0.3$ (dashed), $-0.5$ (dotted), $-0.7$ (solid). The curve for $f_{s,0}=-0.3$ was multiplied by a factor of 10 for clarity.

Figure 5

(Color online) Function $S_T(f_{s,0},f_{s,1})$, which determines the sign of the magnetic-field dependence of the spin susceptibility calculated to fourth order in the skeleton interaction [cf. Eq. (2.54)], plotted as a function of $f_{s,1}∕f_{s,0}$ for $f_{s,0}=-0.3$ (dashed), $-0.5$ (dotted), $-0.7$ (solid). The curve for $f_{s,0}=-0.3$ was multiplied by a factor of 10 for clarity.

Figure 6

Renormalization of vertex $\Gamma (k,p;p,k)$ (hatched) in the Cooper channel. Dashed lines are irreducible Cooper amplitudes.

Figure 7

(Color online) Schematic behavior of the thermodynamic potential $\Xi$ as a function of magnetization $M$. Upper curve: the first-order phase transition has not been reached yet but the system exhibits a metamagnetic transition in finite field. Lower curve: the first-order phase transition occurs.

Figure 8

(Color online) Function $V\left(\zeta \right)$ defined by Eq. (4.3). The first-order phase transition occurs when the maximum of $V$ reaches the value of $\delta \times \left(a{k}_F\right)$.

Figure 9

(Color online) Phase diagram of a ferromagnetic phase transition. The lines of second- and first-order transitions (solid and dashed lines, respectively) are separated by a tricritical point.

Figure 10

(Color online) Re-entrant phase diagram of a second-order ferromagnetic phase transition. The two curves correspond to two solutions of Eq. (4.14).

Figure 11

Diagrams for the thermodynamic potential that give rise to the temperature dependence of the spin susceptibility in 3D. Top row: ladder diagrams; bottom row: ring diagrams.

Figure 12

Diagrams for the spin susceptibility generated by differentiating the thermodynamic potential with respect to the magnetic field.