In conventional and high transition temperature copper oxide and iron pnictide superconductors, the Cooper pairs all have even parity. As a rare exception, Sr2RuO4 is the first prime candidate for topological chiral p-wave superconductivity, which has time-reversal breaking odd-parity Cooper pairs known to exist before only in the neutral superfluid 3 He. However, there are several key unresolved issues hampering the microscopic description of the unconventional superconductivity. Spin fluctuations at both large and small wavevectors are present in experiments, but how they arise and drive superconductivity is not yet clear. Spontaneous edge current is expected but not observed conclusively. Specific experiments point to highly band-and/or momentum-dependent energy gaps for quasiparticle excitations in the superconducting state. Here, by comprehensive functional renormalization group calculations with all relevant bands, we disentangle the various competing possibilities. In particular we show the small wavevector spin fluctuations, driven by a single two-dimensional band, trigger p-wave superconductivity with quasi-nodal energy gaps.PACS numbers: 74.20.Rp, 71.27.+a Very soon after the discovery of superconductivity in Sr 2 RuO 4 [1], it was proposed that the superconducting (SC) pairing is of unconventional nature [2,3]. Later experiments have provided evidence that the Cooper pair in the SC state is of odd parity [4] with total spin equal to one [5]. Further evidence indicates the superconductivity to be chiral, breaking time reversal symmetry [6,7]. Sr 2 RuO 4 is thus the first prime candidate for a chiral p-wave superconductor [8][9][10][11], an interesting analogue of the neutral superfluid 3 He. It has recently received great interest as by suitable manipulations it may support zero energy Majorana bound states in vortices [12], the building block for topological quantum computing [13]. However, there are a number of outstanding issues associated with the chiral p-wave superconductivity in Sr 2 RuO 4 . First, p-wave spin triplet pairing is expected to be associated with spin fluctuations at small wavevector. However, the spin density wave (SDW) fluctuation observed in Sr 2 RuO 4 is dominated by a large wavevector at higher temperatures and coexist with a feature at small wavevector at lower temperatures. [14] A resolution of this puzzle is vital to understand the superconductivity. Second, one would expect a spontaneous electric current at the edge of the RuO 2 layers as a result of the chiral SC state. The edge current, however, has not been observed conclusively in experiments.[15] One possible reason is the edge current is very fragile and difficult to establish against disorders. Another possibility is a topological cancellation from hole-like and electron-like bands, [16] posing a question as whether the SC state is topologically nontrivial at all. Third, the specific measurement reveals abundance of low energy quasiparticle excitations below the transition temperature. [17] This would point to mu...

We present a comprehensive study of the magnetic properties of the long-range ordered quasi-one dimensional J1-J2 systems with a newly developed torque equilibrium spin-wave expansion approach, which can describe the spin Casimir and magnon decay effects in a unified framework. While the framework does not lose the generality, our discussion will be restricted to two representative systems, each of which has only one type of inter-chain coupling (J3 or J4) and is referred to the J3-or J4-system respectively. In spite of the long-range spiral order, the dynamical properties of these systems turn out to be highly nontrivial due to the incommensurate noncollinear spin configuration and the strong quantum fluctuation effects enhanced by the frustration and lowdimensionality. Both the systems show prominent spin Casimir effects induced by the vacuum fluctuation of the spin waves and related modification of the ordering vector, Lifshitz point's position and sublattice magnetization. In addition to these static properties, the dynamical behaviors of these systems are also remarkable. Significant and spontaneous magnon decay effects are manifested in the quantum corrections to the excitation spectrum, including the broadening of the spectrum linewidth and downward renormalization of the excitation energy. Furthermore, the excitation spectrum appears to be very sensitive to the types of the inter-chain coupling and manifests three distinct features: (i) the magnon decay patterns between J3-and J4-system are very different, (ii) the renormalized spectrum and the overall decay rate of the J3-and J4-systems show very different sensitivity to the magnetic anisotropy, and (iii) there is a nearly flat mode in the renormalized magnon spectrum of the J4-system along the X-M direction. By adjusting the strength of magnetic anisotropy and varying the approximation scheme, it is revealed that these striking distinct features are quite robust and have deep connection with both the spin Casimir and the magnon decay effects. Thus these special consequences of the inter-chain coupling on the spin wave dynamics may be served as a set of probes for different types of inter-chain couplings in experiments. At last, to guide experimental measurements such as inelastic neutron scattering in realistic materials and complement our theoretical framework, we develop the analytical theory of the dynamical structure factor within the torque equilibrium formulism and provide the explicit results of the quasi-one dimensional J1-J2 systems. arXiv:1610.06365v2 [cond-mat.str-el]a 4 c FIG. 1: Crystallographic structure of coupled edge-shared chain magnetic oxides with main intra-and inter-chain couplings J1, J2, J3 and J4 marked by red solid line, green solid arc, orange dotted line, and purple dash dotted line, respectively.

The Casimir effect is a general phenomenon in physics, which arises when the vacuum fluctuation of an arbitrary field is modified by static or slowly varying boundary. However, its spin version is rarely addressed, mainly due to the fact that a macroscopic boundary in quantum spin systems is hard to define. In this article, we explore the spin Casimir effect induced by the zero-point fluctuation of spin waves in a general non-collinear ordered quantum antiferromagnet. This spin Casimir effect results in a spin torque between local spins and further causes various singular and divergent results in the framework of spin-wave theory, which invalidate the standard 1/S expansion procedure. Based on the spin Casimir torque interpretation, we develop a spin-wave expansion approach named as torque equilibrium spin wave theory (TESWT). In this approach, the spin Casimir effect is treated in a self-consistent way, and the spin-wave expansion results are free from singularities and divergences. A detailed spin-wave analysis of the antiferromagnetic spin-1/2 Heisenberg model on a spatially anisotropic triangular lattice is undertaken within our approach. Our results indicate that the spiral order is only stable in the region 0.5 < α < 1.2, where α is the ratio of the coupling constants. In addition, the instability in the region 1.2 < α < 2 is owing to the spin Casimir effect instead of the vanishing sublattice magnetization. Furthermore, our method provides an efficient and convenient tool that can estimate the correct exchange parameters and outline the quantum phase diagrams, which can be useful for experimental fitting processes in frustrated quantum magnets.