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In type-II superconductors (Fetter & Hohenberg, 1969), magnetic fields penetrate into superconductors as quantized flux lines with flux quanta φ0 = hc/2e, where h is Planck constant, c velocity of light, e electron’s charge. Around a flux line, pair potential ∆(r) of Cooper pair has a vortex structure, where ∆(r) has phase winding 2π reflecting screening super-current around a flux line. At the vortex core, amplitude |∆(r)| is suppressed, and low energy excitations appear within the superconducting gap of the electronic states. Vortex physics makes important roles in the study of unconventional superconductors, because unconventional characters hidden in the uniform superconducting state at a zero field appear around vortices. For example, in superconductors with anisotropic superconducting gap on Fermi surface in momentum space, electronic states around a vortex show real-space anisotropy in the local density of states (LDOS) N(E, r) around vortex core. The vortex core image is observed by scanning tunneling microscopy (STM) (Hess et al., 1990; Nishimori et al., 2004). Around vortex cores, local zero-energy electronic states at Fermi level are seen as star shape with tails extending toward node or weak-gap directions (Hayashi et al., 1996; 1997; Ichioka et al., 1996; Schopohl & Maki, 1995). From the spatial average of the zero-energy states, we can estimate the zero-energy density of states (DOS) N(E = 0), which determines low temperature (T) behaviors of physical quantities. Due to the differences of electronic states around the vortex core, N(E = 0) shows different magnetic field (H) dependences. These H-dependences are studied to identify the pairing symmetry in the experiments for vortex states, such as, electronic specific heat (Moler et al., 1994; Nohara et al., 1999), electronic thermal conductivity, and paramagnetic susceptibility (Zheng et al., 2002). For example, the H-dependence of low temperature specific heat C(H) is often used to distinguish the presence of nodes in the pairing potential. As for Sommerfeld coefficient γ(H) ≡ limT→0 C(H)/T, γ(H) ∝ H in s-wave pairing with full gap, and γ(H) ∝ √ H by the Volovik effect in d-wave pairing with line nodes (Ichioka et al., 1999a;b; Miranovic et al., 2003; Nakai et al., 2004; Volovik, 1993). The curves of γ(H) are expected to smoothly recover to the normal state value towards the upper critical field Hc2. However, in some heavy fermion superconductors, C(H) deviates from these curves. In CeCoIn5, C(H) shows convex curves, i.e., C(H) ∝ Hα (α > 1) at higher fields(Ikeda et al., 2001). This behavior is not understood only by effects of the pairing symmetry. A similar C(H) behavior is observed also in UBe13 (Ramirez et al., 1999). The experimental data of magnetization curve Mtotal(H) in CeCoIn5 show a convex curve at higher fields, instead of a conventional concave curve (Tayama et al., 2002). As an 10

FFLO and vortex states in superconductors with strong paramagnetic effect