Spin waves in a band ferromagnet: spin-rotationally symmetric study with self-energy and vertex corrections

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Correlation effects in a band ferromagnet:
spin-rotationally-symmetric study with
self-energy and vertex corrections
Avinash Singh∗
Max-Planck-Institut fu¨r Physik Komplexer Systeme,
No¨thnitzer str. 38, D-01187 Dresden and
Department of Physics, Indian Institute of Technology Kanpur - 208016
Quantum corrections to the transverse spin-fluctuation propagator are obtained by
including self-energy and vertex corrections to first order within a spin-rotationally-
symmetric inverse-degeneracy (1/N ) expansion scheme which preserves the Gold-
stone mode order by order. A correlation-induced exchange-energy correction is
shown to yield the dominant quantum reduction in the spin stiffness, providing a
quantitative understanding of the ferromagnetic-state stability in terms of simple
lattice-dependent features of energy-band dispersion. The quantum reduction factor
U/W highlights the subtlety in the characteristic competition in a band ferromagnet
between interaction U and bandwidth W .
PACS numbers: 71.10.Fd,75.10.Lp,75.30.Ds,75.40.Gb

2I. INTRODUCTION
The continuous spin-rotation symmetry of a magnetic system is manifested, in the
spontaneously broken-symmetry state, in gapless spin-wave excitations in accordance with
the Goldstone theorem,1 the zero-energy infinite-wavelength mode simply corresponding
to a uniform rotation of all spins. The gapless spin-wave spectrum has particularly im-
portant consequences for low-dimensional (D = 1, 2) magnetic systems, where transverse
spin fluctuations diverge at any finite temperature, resulting in absence of long-range
magnetic order in accordance with the Mermin-Wagner theorem,2 and exponentially large
spin-correlation length in two dimensions.
A quantitative determination of the spin-wave spectrum also allows for various order-
ing, dimension, and lattice-specific investigations — finite-temperature spin dynamics and
reduction of magnetic order with temperature due to thermal excitation of spin waves,3,4,5
estimation of the magnetic transition temperature Tc from the broken-symmetry side,6,7
competing interactions and magnetic stability,8,9 and dispersion of magnetic and elec-
tronic excitations in solids as inferred from inelastic neutron-scattering and angle-resolved
photoemission studies,10,11,12,13,14 References given above illustrate recent spin-wave ap-
plications to the nearly square- and triangular-lattice antiferromagnets such as cuprates
(La2CuO4), multiferroics (HoMnO3), and organic systems κ − (BEDT − TTF)2X, as
well as ferromagnets (Fe,Ni) and magnetic multilayers (Fe/Cr) exhibiting giant magneto-
resistance.
For band ferromagnets such as Fe and Ni, there have been extensive inelastic neutron-
scattering studies in relation with calculations for transverse spin fluctuations in the ran-
dom phase approximation (RPA), which is the lowest-order treatment in which spin ro-
tation symmetry and Goldstone mode are preserved. Various spin-wave features such as
isotropy, stiffness constant, damping and disappearance at higher energy due to interaction
with the continuum of Stoner excitations, persistence for T > Tc, temperature dependence
of dispersion etc. have been discussed,15,16,17,18,19 and also quantitatively compared with
RPA calculations using realistic band structure.20,21 However, despite the extensive study
of magnetic excitations in metals and alloys over the years,22 a spin-rotationally-symmetric
extension of RPA including self-energy and vertex corrections has not been carried out
quantitatively for fcc-type lattices with respect to ferromagnetic-state stability, spin-wave

3and Stoner excitations, damping, and transition temperature.
Some of the related developments beyond RPA are summarized below. The effect of
large spin fluctuations in a nearly ferromagnetic Fermi liquid has been studied in the
context of spin waves in He3 within an extension of the paramagnon model beyond RPA,
where ambiguities of the paramagnon model were shown to be resolved.23 In the context
of self-energy corrections in a band ferromagnet, the importance of vertex corrections in
restoring the spin-rotation symmetry and Goldstone mode has been recognized at a formal
level, and a Ward identity connecting vertex corrections to self-energy corrections has been
derived.24 While spin-wave excitations for arbitrary wave vector were not quantitatively
discussed, the spin-wave stiffness constant was shown to be reduced from its RPA value,
and also compared with earlier studies25,26 in the context of stability of the ferromagnetic
state.27 A variational approach has been used to improve the RPA result for the energy of
long wave length spin-wave modes.28 A spin-wave damping term proportional to q6 due to
scattering off particle-hole excitations has been obtained for a parabolic band.29 Recently
self-energy corrections have been incorporated in a modified RPA approach, although the
q, ω-dependence of vertex corrections was not included.7
In this paper we provide a concrete extension beyond RPA for the transverse spin-
fluctuation propagator. We make use of the inverse-degeneracy (1/N ) expansion within
the generalized N -orbital Hubbard model,30 which provides a systematic diagrammatic
scheme for incorporating quantum corrections while preserving spin-rotation symmetry
and hence the Goldstone mode order-by-order. This spin-rotationally-symmetric scheme
has been applied earlier to examine quantum corrections in the antiferromagnetic state
of the Hubbard model.30 The diagrams include self-energy and vertex corrections, and
physically incorporate effects such as quasiparticle damping, spectral-weight transfer and
coupling of spin and charge fluctuations. We consider the special case of a saturated band
ferromagnet, in which the absence of minority-spin particle-hole fluctuations results in
relative simplification.
Owing to its intrinsically strong-coupling nature, band ferromagnetism has been recog-
nized as a fairly challenging problem, particularly with respect to the estimation of Curie
temperature for the Hubbard model, although considerable progress has been achieved
in the recent past.31 Competition between band and interaction energies, separation of
moment-melting and moment-disordering temperature scales due to strong correlation,

4and presence of charge fluctuations even in the broken-symmetry state due to partially
filled band(s) are some of the non-trivial elements involved. Ferromagnetism in the Hub-
bard model on fcc and bcc lattices has been recently investigated using several different
approaches, such as the dynamical mean field theory (DMFT),32 by incorporating spin and
charge fluctuations in the correlated paramagnet using the fluctuation-exchange (FLEX)
and the two-particle self consistent (TPSC) approximations,33 by systematically improv-
ing the self energy,31 and a modified RPA scheme.7 Ferromagnetism in a diluted Hubbard
model has also been investigated recently,34,35 which is of interest in the context of carrier-
mediated ferromagnetism in diluted magnetic semiconductors such as Ga1−xMnxAs.
Incorporating only the local (Ising) spin excitations, the DMFT approach ignores long-
wavelength spin fluctuations and the k-dependence of self energy. FLEX incorporates
self-energy corrections, but ignores vertex corrections of the same order, thereby breaking
the spin-rotation symmetry. Both DMFT and FLEX are hence not in accordance with
the Mermin-Wagner theorem. While self-energy corrections in the broken-symmetry state
were incorporated in the modified RPA approach,7 the momentum-energy dependence of
vertex corrections was not considered. While FLEX, DMFT, and RPA results for the
behaviour of Curie temperature with band filling are found to be qualitatively similar,
appreciable quantitative differences7,31,33 clearly highlight the need for a spin-rotationally-
symmetric extension.
We consider the generalized N -orbital Hubbard model30
H = −t
∑
〈ij〉,σ,α
(a†iσαajσα + H.c.) +
1
N
∑
i,α,β
(U1a†i↑αai↑αa
†
i↓βai↓β + U2a
†
i↑αai↑βa
†
i↓βai↑α) , (1)
where α, β refer to the degenerate orbital indices and the factor 1/N is included to render
the energy density finite in the N → ∞ limit. In the isotropic limit U1 = U2 = U , the
two interaction terms (density-density and exchange-type with respect to orbital indices)
are together equal to U(−Si.Si + n2i ) in terms of the total spin Si ≡
∑
α ψ
†
iα(σ/2)ψiα
and charge ni ≡
∑
α ψ
†
iα(1/2)ψiα operators, and the Hamiltonian is therefore explicitly
spin-rotationally symmetry.

5II. TRANSVERSE SPIN FLUCTUATIONS
The transverse spin-fluctuation propagator in the broken-symmetry state, which de-
scribes both collective spin-wave and particle-hole Stoner excitations, is given by
χ−+(q, ω) = i
∫
dt eiω(t−t′)
∑
β
∑
j
eiq.(ri−rj)〈ΨG|T[S−iα(t)S+jβ(t′)]ΨG〉 (2)
in terms of the fermion spin-lowering and -raising operators S∓ = Ψ†(σ∓/2)Ψ. The
spin-fluctuation propagator can be expressed as
χ−+(q, ω) = φ(q, ω)
1− Uφ(q, ω) (3)
in terms of the exact irreducible propagator φ(q, ω), which incorporates all self-energy
and vertex corrections. The inverse-degeneracy expansion30
φ = φ(0) +
(
1
N
)
φ(1) +
(
1
N
)2
φ(2) + ... (4)
systematizes the diagrams in powers of the expansion parameter 1/N which, in analogy
with 1/S for quantum spin systems, plays the role of ~. As only the ”classical” term
φ(0) survives in the N → ∞ limit, the RPA ladder series χ0(q, ω)/1 − Uχ0(q, ω) (with
interaction U2) amounts to a classical-level description of non-interacting spin-fluctuation
modes. The bare antiparallel-spin particle-hole propagator
φ(0)(q, ω) ≡ χ0(q, ω) =
∑
k
1
ǫ↓+k−q − ǫ↑−k + ω − iη
, (5)
where ǫσk = ǫk − σ∆ are the Hartree-Fock ferromagnetic band energies, 2∆ = mU is the
exchange band splitting, and the superscript +(−) refer to particle (hole) states above
(below) the Fermi energy ǫF. For the saturated ferromagnet, the magnetization m is equal
to the particle density n.
As collective spin-wave excitations are represented by poles in (3), spin-rotation sym-
metry requires that φ = 1/U for q, ω = 0, corresponding to the Goldstone mode. Since the
zeroth-order term φ(0) already yields exactly 1/U for q, ω = 0, the sum of the remaining
terms must exactly vanish in order to preserve the Goldstone mode. For this cancel-
lation to hold for arbitrary N , each higher-order term φ(n) in the expansion (4) must
individually vanish, implying that spin-rotation symmetry is preserved order-by-order, as
expected from the spin-rotationally-invariant form (U/N )Si.Si of the interaction term