On the Nambu fermion-boson relations for superfluidHe3

Abstract

Superfluid 3 He is a spin-triplet (S = 1), p-wave (L = 1) BCS condensate of Cooper pairs with total angular momentum J = 0 in the ground state. In addition to the breaking of U(1) gauge symmetry, separate spin or orbital rotation symmetry is broken to the maximal sub-group, SO(3) S × SO(3) L → SO(3) J . The Fermions acquire mass, m F ≡ ∆, where ∆ is the BCS gap. There are also 18 Bosonic excitations -4 Nambu-Goldstone (NG) modes and 14 massive amplitude Higgs (AH) modes. The Bosonic modes are labeled by the total angular momentum, J ∈ {0, 1, 2}, and parity under particle-hole symmetry, c = ±1. For each pair of angular momentum quantum numbers, J, J z , there are two Bosonic partners with c = ±1. Based this spectrum Nambu proposed a sum rule connecting the Fermion and Boson masses for BCS type theories, which for 3 He-B is M 2 J, + + M 2 J, − = 4m 2 F for each family of Bosonic modes labeled by J, where M J, c is the mass of the Bosonic mode with quantum numbers (J, c). Nambu's sum rule (NSR) has recently been discussed in the context of Nambu-Jona-Lasinio models for physics beyond the standard model to speculate on possible partners to the recently discovered Higgs Boson at higher energies. Here we point out that Nambu's Fermion-Boson mass relations are not exact. Corrections to the Bosonic masses from (i) leading order strong-coupling corrections to BCS theory, and (ii) polarization of the parent Fermionic vacuum lead to violations of the sum-rule. Results for these mass corrections are given in both the T → 0 and T → T c limits. We also discuss experimental results, and theoretical analysis, for the masses of the J c = 2 ± Higgs modes and the magnitude of the violation of the NSR. I. INTRODUCTIONOne of the key features of spontaneous symmetry breaking in condensed matter and quantum field theory is the emergence of new elementary quanta -phonons in crystalline solids, magnons in ferromagnets, the Higgs and gauge bosons of the standard model. In the latter example, spontaneous symmetry breaking (SSB) in the BCS theory of superconductivity played an important role in theoretical models for the mass spectrum of elementary particles. [1][2][3] In BCS superfluids the binding of Fermions into Cooper pairs leads to an energy gap, ∆, in the Fermion spectrum, i.e. Fermions in the broken symmetry phase (Bogoliubov quasiparticles) acquire a mass m F = ∆, while condensation of Cooper pairs leads to the breaking of global U(1) gauge symmetry, the generator being particle number. The latter also implies that the Bogoliubov Fermions are no longer particle number (Fermion "charge") eigenstates, but coherent superpositions of normal-state particles and holes. Charge conservation is ensured by an additional contribution to the charge current -a collective mode of the broken symmetry phase. This massless Bosonic excitation of the phase of condensate amplitude 4,5 is the NambuGoldstone (NG) mode associated with broken U(1) symmetry, and is manifest as a phonon in neutral superfluid 3 He. II. NAMBU'S MASS RELATIONSNambu and Jona...

Vortex phase diagram of rotating superfluid He3−B

“…A recent example is the analysis of the experimentally measured Bosonic collective mode frequencies ("Higgs masses") of superfluid 3 He-B using a time-dependent extension of the strong-coupling GL theory in Ref. 44 , which provided consistent experimental results for the strength of the f-wave pairing interaction in superfluid 3 He over the full pressure range 45 , a material parameter that is important for understanding ground-states and excitations of superfluid 3 He at high pressures and high magnetic fields. Secondly, the extended GL theory is supported by the microscopic strong-coupling pairing theory based on leading order corrections to the weak-coupling BCS theory originating from binary collision scattering between fermionic quasiparticles of the normal phase of liquid 3 He 14 .…”

Corrections to Higgs mode masses in superfluid He3 from acoustic spectroscopy

“…The mode masses, M 2 + and M 2 − , including renormalization due to Fermi liquid and f -wave interactions, were first calculated by Sauls and Serene [4,5] in the weak coupling limit. We combine their result with Eqs.…”

“…Koch and Wölfle noted [27] that these non-trivial corrections to a c J can be estimated using the strong-coupling corrections to the β-parameters of the Ginzburg-Landau (GL) functional. The five β-parameters, β i , are the coefficients of the fourth-order invariants of the order parameter [4,28]. The addition of strong-coupling corrections to β i and therefore a c J is made possible by recent advances in determining the strong-coupling interactions and their temperature dependence [29,30].…”

“…Here we apply a simple procedure along these lines that incorporates mass renormalization due to Fermi liquid, fwave, and strong-coupling effects to both ∆ and a c J . In terms of β-parameters, a c J is given by [4] a + 2 = 2…”

Recent Progress and New Challenges in Quantum Fluids and Solids

“…However, many of them have an excitation gap, and the dispersion for these branches is exactly that of a massive Dirac particle, ω 2 (k) = ω 2 o + c 2 k 2 . This aspect has been discussed in terms of the Higgs mechanism in the Standard Model [34,35]. A recent experiment by a group of researchers at Aalto University and the Landau Institute beautifully demonstrated how a gapless mode (a massless particle) in the B-phase becomes gapped (gains mass) on the energy scale much smaller than the condensation energy, an inter-esting analogy to the Higgs boson discovered in 2012 at CERN at a much lower energy (125 GeV) than the expected energy scale of 1 TeV [36].…”