Commensurate and incommensurate spin-density waves in heavy electron systems

Landau’s Fermi liquid (FL) theory has been successful in describing the low energy properties of most normal metals. However, numerous heavy fermion systems¹ display deviations from FL behavior, known as non-Fermi liquid (NFL) properties. The breakdown of the FL can be tuned by alloying (chemical pressure), hydrostatic pressure or the magnetic field. In most cases the systems are close to the onset of antiferromagnetism (AF) and the NFL behavior is attributed to a quantum critical point (QCP).^2,3 The quantum critical behavior of these systems has been studied in Refs. 2 and 4–6.

Previously, we have studied the pre-critical region of a heavy electron band with two pockets,^7,8 one electron-like and the other hole-like, separated by a wave vector Q. A strong repulsive interaction between electrons gives rise to heavy fermion bands

where k is measured from the center of each pocket, and assumed to be small compared to the nesting vector Q. Both bands are parabolic, have different Fermi momenta and for simplicity we assume that the Fermi velocity (or density of states) is the same for both pockets.

In the spirit of the FL theory, there are weak remaining interactions between the heavy quasi-particles left after the heavy particles are formed, given by H_W + H_V + H_U + H_P,^7–10 

Here W is the interaction for particles within the same pocket, V represents the interaction strength for small momentum transfer between the pockets (|q| ≪ |Q|), U corresponds to a momentum transfer of the order of Q, and P refers to a process transferring two particles between the pockets. The latter does not conserve the number of particles of each pocket. p^∗ = Q − G/2 is the vector necessary for the momentum conservation in the H_P interaction. The momentum is automatically conserved for the remaining three scattering amplitudes. For the commensurate case p^∗ = 0, i.e. the Umklapp condition is satisfied in H_P.¹¹ The Hubbard model is obtained by choosing W₁ = W₂ = V = U = P.

Using a renormalization group (RG) approach^7,8 that eliminates the fast degrees of freedom close to an ultraviolet cutoff and rewriting the Hamiltonian in terms of renormalized slow variables, we obtain for p^∗ = 0 that the interaction V induces itinerant AF due to the nesting of the Fermi surfaces of the two pockets. For perfect nesting (electron-hole symmetry between the pockets) an arbitrarily small interaction is sufficient for a ground state with long-range order. The degree of nesting is controlled by the mismatch parameter, $δ=12|kF1−kF2|vF$⁠, where k_F1 (k_F2) is the Fermi momentum of the electron (hole) pocket. In this way the ordering temperature can be tuned to zero, leading to a QCP. The Fermi surface mismatch δ parametrizes the pressure or doping, since both are able to modify the Fermi momenta of the pockets. The QCP gives rise to the desired NFL properties.^7,8,12,13

The commensurate case has been studied for a cylindrical Fermi surface by Chubukov, Efremov and Eremin^11,14 in the context of iron-based superconductors. For certain parameters the model is able to generate a superconductivity dome around the QCP without substantially modifying the NFL properties.⁹ In the incommensurate case the phase space in the scattering process for the transfer of a pair of particles between the pockets is restricted because p^∗ ≠ 0. We investigated the antiferromagnetic and superconducting responses in the neighborhood of the QCP for spherical Fermi surfaces.¹⁰ Four density response functions have to be considered, namely spin and charge density waves with wave vectors Q (incommensurate with the lattice) and G/2 (commensurate), as well as four response functions for superconductivity, namely S, S⁺ with homogeneous order parameters and the corresponding phases with order parameters modulated by G/2 − Q. The latter two resemble superconductivity of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type,¹⁵ but with the oscillations determined by the deviation from commensuration of Q with the lattice rather than tuned by the magnetic field.¹⁰ 

The perturbative corrections to the interaction vertices for p^∗ = Q − G/2 ≠ 0 are given by the diagrams shown in Figs. 1 through 3 of Ref. 9. Note that each vertex diagram has four legs, two in-going with momenta k₁ and k₂ and two outgoing with k₃ and k₄. In analogy to parquet equations, the logarithmic dependence of P can occur in three channels, namely, the zero sound channel (k₁,k₄), the so-called third channel (k₁,k₃) and the Cooper channel (k₁,k₂), which we denote with P_a, P_b and P_c, respectively. The renormalization equations for the six vertex functions are¹⁰ 

Here w = Wρ_F, v = Vρ_F, u = Uρ_F, p_a = ρ_FP_a, p_b = ρ_FP_b and p_c = ρ_FP_c are dimensionless vertices and ρ_F is the density of states (assumed to be equal for both pockets). The logarithmic scaling variable is ξ.

For perfect nesting the renormalization continues until either ξ tends to infinity or a vertex diverges. For mismatched Fermi surfaces the integration of Eqs. (6) and (7) has to be stopped when the Fermi surface mismatch δ is reached. Similarly, the external energy ω, the smearing of the Fermi surface by the temperature and the effects of p∗ have to be taken into account in the renormalization. The effects of δ and p^∗ are different as shown in Fig. 1. One of the pockets has been translated by Q. Hence, the spheres are no longer concentric, but the centers are now separated by the vector p^∗. In the diagram on the left in Fig. 1 p^∗ is smaller than k₁₂ = k_F1 − k_F2 = 2δ/v_F, while for the sketch on the right p^∗ > k₁₂.

The logarithmic scaling variable in Eqs. (6)-(8) then depends on the diagram under consideration. For Eqs. (6) and (7) the logarithmic variables are (a) ξ₁ = ln[D/(|ω| + 2T + δ)], (b) $ξ2=12ln[D/(|ω|+2T+|δ−8α/3|)]+12ln[D/(|ω|+2T+|δ+8α/3|)]$⁠, and (c) $ξ3=12ln[D/(|ω|+2T+|δ−2α/3|)]+12ln[D/(|ω|+2T+|δ+2α/3|)]$⁠, while for the Cooper channel, Eq. (8), ξ contains the following cutoffs: (d) ξ₀ = ln[D/(|ω| + 2T)], (e) ξ₄ = ln[D/(|ω| + 2T + 8α/3)] and (f) ξ₅ = ln[D/(|ω| + 2T + 2α/3)]. Here D is the energy cutoff and α = p^∗2/2m. We have assumed that p^∗ is a sufficiently small quantity to justify a Taylor expansion in powers of p^∗. Odd powers of p^∗ vanish identically and hence the first nonzero contribution is proportional to α and we have neglected fourth order terms. The overall contribution of the integral is logarithmic in the cutoff and the external frequency and the α-term can be re-incorporated into the logarithmic dependence.

In contrast to the commensurate case, where α = 0 and the RG equations could all be integrated analytically, for the incommensurate situation numerical methods have to be employed. The renormalization ends with |ω| = 0 unless a vertex diverges earlier. Note that the Cooper channel involves diagrams with parallel propagators lines referring to the same pocket, so that δ does not play a role. These equations are completely decoupled from Eqs. (6)-(7).

As mentioned above there are four density wave (charge and spin, commensurate and incommensurate) response functions and four superconductivity (S and S⁺, homogeneous and FFLO) correlation functions. These response functions are obtained from the vertex functions via three nested integrations. The definition of the order parameters and the RG equations for the susceptibilities can be found in Ref. 10. Note that the vertex functions for the commensurate case can be obtained from the incommensurate situation in the limit p^∗ → 0. However, the scaling equations for the susceptibilities are different and the commensurate case cannot be recovered as p^∗ → 0.

The inverse of the spin-density and superconductivity correlation functions normalized to their value at ω₀ = 0.1 is shown in Fig. 2 for various values of α. The incommensurate and commensurate spin-density waves (Figs. 2(a) and 2(b)) are driven by the vertices v and p_a, respectively. Due to the repulsive nature of the interactions there are no CDW instabilities.

For the coupling parameters in Fig. 2 a QCP is obtained at the critical nesting mismatch δ₀ = 0.155 and α = 0. The curves in Figs. 2(a) and 2(b) are for δ = 0.17, i.e. for a mismatch larger than the critical one. The inverse of the SDW response functions show a re-entrant behavior as a function of α. For α < 0.042 the system is a paramagnetic FL, then with increasing α long-range order is induced and for α > 0.083 the system enters a paramagnetic phase with a strong NFL component. The response functions for commensurate (wave vector G/2) and incommensurate (wave vector Q) are very similar and it is not possible to determine the nature of the long-range order. The re-entrance as a function of α originates from the compensation of δ by α in ξ₂ through the term with |δ − 8α/3| in the vertices v and p_a. Both SDW correlation functions are driven by these vertices. δ represents the Fermi surface mismatch due to concentric (after translating one pocket by Q) spheres. A non-zero p^∗ corresponds to a translation in reciprocal space that leaves the spheres non-concentric. There is then a shortest and largest distance between the spheres, which in terms of energies corresponds to |δ − 8α/3| and δ + 8α/3, respectively.¹⁰ For ξ₃ only one propagator is shifted leading to a smaller energy adjustment. This is schematically shown in Fig. 1.

Fig. 2(c) displays the responses of the system to possible forms of superconducting order. The Cooper channel involves diagrams with parallel propagator lines, so that the correlation functions are independent of the Fermi surface mismatch δ. For α > 0 all superconducting correlation functions remain finite and none of the superconducting phases is realized. For α = 0 the S⁺ response function diverges, giving rise to a superconducting phase at very low T. This superconducting phase is the tail of the superconducting dome around the QCP.⁹ 

As mentioned above, the differential equations satisfied by the susceptibilities are different for α = 0 and α ≠ 0. Hence, the limit α → 0 is different from α = 0; the limit α → 0 is not meaningful since the difference between commensurate and incommensurate disappears and the period of modulation of the FFLO phases becomes infinite (all phases are homogeneous).

In summary, we considered the two-pocket model^7,8 including the transfer of pairs of electrons between the pockets¹¹ and investigated the RG flow in the context of NFL behavior for heavy fermion systems.⁹ The pair transfer usually requires the inclusion of Umklapp processes, i.e. a nesting wave vector commensurate with the lattice. Here we studied the situation where the nesting vector Q is not commensurate with the lattice, the momentum difference being denoted with p^∗. This opens the possibility of more forms of long-range order, i.e. commensurate and incommensurate density waves and superconductivity of the FFLO type, where the modulation of the order parameter is given by p^∗.

The channels for superconductivity (particle-particle or hole-hole loops) do not depend on the Fermi surface mismatch cutoff, and are completely decoupled from the channels for SDW and CDW (particle-hole loops), which are affected by δ. The parameter α = p^∗2/2m rapidly suppresses superconductivity, but may induce spin density waves. CDW are in general not favorable with repulsive interactions.

For fixed α the AF QCP can be tuned by varying δ so that T_N → 0. Here δ parametrizes the applied pressure or doping, while α is determined by the band structure. Pressure affects the band structure of metals and may increase or decrease the energy of the top and bottom of the pockets. Hence, pressure changes k_F1 and k_F2 with changing the number of electrons and consequently the Fermi surface mismatch parameter δ.

Since superconductivity is only viable for very small α, the α = 0 equations should be used. This leads to a superconducting phase of the S⁺ type above the QCP. Superconducting domes of this type have been observed in CeRh₂Si₂,^16,17 CePd₂Si₂ and CeIn₃^18,19 under pressure. Due to the competition between the SDW and the S⁺ order the dome is split into two regions,⁹ one with a pure S⁺ phase and the other with coexisting S⁺ and SDW, in agreement with NMR experiments for CeIn₃,²⁰ for CeCu₂(Si_0.98Ge_0.02)₂,²¹ and CeRhIn₅.²² 

The results are valid in the disordered phase for weak and intermediate coupling. However, since the renormalization group does not allow a return to a weak-coupling fixed point once the system is strongly coupled, the present approach qualitatively describes the entire precritical regime. Some of the properties are quite universal and independent of the type of QCP. The present model is simple enough so that actual calculations could be performed and may provide insights even for more complex physical situations.

The support by the U.S. Department of Energy (Office of Basic Energy Sciences) under grant DE-FG02-98ER45707 is acknowledged.