Disordered Fulde-Ferrel-Larkin-Ovchinnikov state in d-wave superconductors 9 0 0 Youichi Yanase 2 n Department of Physics, University of Tokyo, Tokyo 113-0033, Japan a J E-mail: [email protected] 9 2 Abstract. We study the Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) superconducting state in the disordered systems. We analyze the microscopic model, in which the d- ] n wavesuperconductivityisstabilizedneartheantiferromagneticquantumcriticalpoint, o and investigate two kinds of disorder, namely, box disorder and point disorder, on the c - basisoftheBogoliubov-deGennes(BdG)equation. Thespatialstructureofmodulated r p superconductingorderparameterandthemagneticpropertiesinthedisorderedFFLO u state are investigated. We point out the possibility of “FFLO glass” state in the s . presence of strong point disorders, which arises from the configurational degree of t a freedom of FFLO nodal plane. The distribution function of local spin susceptibility is m calculated and its relation to the FFLO nodal plane is clarified. We discuss the NMR - measurements for CeCoIn . d 5 n o c [ 1. Introduction 1 v FFLO superconductivity was predicted in 1960’s by Fulde and Ferrel [1] and also 7 6 by Larkin and Ovchinnikov [2]. In addition to the U(1)-gauge symmetry, a spatial 6 4 symmetry is spontaneously broken in the FFLO state owing to the modulation of . 1 superconducting (SC) order parameter. After nearly 40 years of fruitless experimental 0 search for FFLO states, recent experiments appeared to give first evidences for such a 9 0 phase [3]. Moreover, FFLO phase is attracting growing interests in other related fields : v such as the cold fermion gases [4] and the high-density quark matter [5]. i X Extensive studies of FFLO state had been triggered by the discovery of r a a novel SC phase at high fields and low temperatures in the heavy fermion superconductor CeCoIn [6, 7]. Possible FFLO states have been discovered also 5 in some organic materials [8, 9, 10, 11, 12]. All of these candidate materials are close to the antiferromagnetic quantum critical point (AFQCP), and then the d-wave superconductivity is expected. Although it has been expected that the AFQCP significantly influences the superconducting state, almost all of the theoretical works on the FFLO state are based on the weak coupling theory and neglect the antiferromagnetism. We have examined the FFLO state near AFQCP by analyzing the two dimensional Hubbard model using the FLEX approximation, and found that Disordered Fulde-Ferrel-Larkin-Ovchinnikov state in d-wave superconductors 2 thed-waveFFLOstateisstableinthevicinityofAFQCPowingtosomestrongcoupling effects [13]. Another intriguing relationship between FFLO superconductivity and antiferro- magnetism has been indicated in CeCoIn . Several experimental results suggest the 5 emergence of a FFLO state in CeCoIn [3, 14, 15, 16, 17]. On the other hand, nuclear 5 magnetic resonance (NMR) and neutron scattering data rather indicate the presence of antiferromagnetic (AFM) order in the high field phase of CeCoIn [18, 19]. The pressure 5 dependence of phase diagram [17] seems to be incompatible with the AFM order in the uniform SC state, because the AFM order is suppressed by the pressure in the other Ce-based heavy fermions [20] while the high field phase of CeCoIn is stabilized by the 5 pressure [17]. Therefore, it is expected that the coexistent state of FFLO supercon- ductivity and AFM order is realized in CeCoIn at ambient pressure, where the AFM 5 moment is induced by the Andreev bound states around the FFLO nodal plane [21]. Another important issue of FFLO superconductivity is the role of disorders. In this paper, we investigate the d-wave FFLO state near the AFQCP in the presence of randomness on the the basis of the mean field BdG equations. The roles of disorder on the FFLO state has been investigated by many authors [22, 23, 24, 25, 26], and it has been shown that the FFLO state is suppressed by the disorders. However, the disorder average is approximately taken in these studies, and therefore, the regular spatial structure is artificially restored. The spatial inhomogeneity is accurately taken into account using the BdG equations adopted in this paper. We focus on the spatial structure of the disordered FFLO state and clarify the relationship with the magnetic properties. The spatial structure of s-wave FFLO state in the presence of weak box disorder has been investigated in ref. [27]. It is expected that the response to the disorder is quite different between the s-wave superconductor and d-wave one, because the s-wave superconductivity is robust against the disorder in accordance with the Anderson’s theorem[28]. Thed-waveFFLOstateinthepresenceofmoderatelyweakpointdisorders has been investigated, and the configuration transition from two-dimensional structure to one-dimensional one has been pointed out [29]. In this paper, we show that the spatial structure of disordered FFLO states significantly depend on the feature of disorders. In case of weak box disorders, the SC orderparameterhasdistortednodes,whilemorecomplicatedspatialstructureindicating the FFLO glass state is induced by the strong point disorders. In the former, the magnetic properties are governed by the spatial nodes of SC order parameters, on which the local spin susceptibility is larger than that in the normal state. On the other hand, the magnetic properties are dominated by the disorder-induced-antiferromagnetism in the latter. It is expected that most of our results are generally applicable to the FFLO state with non-s-wave paring. For example, the spatial structure of SC order parameter is independent of the details of Hamiltonian. On the other hand, the disorder-induced- antiferromagnetism is a characteristic property of systems near AFQCP. Therefore, the Disordered Fulde-Ferrel-Larkin-Ovchinnikov state in d-wave superconductors 3 magnetic properties in the presence of point disorders are significantly affected by the AFQCP. The paper is organized as follows. In §2, we formulate the BdG theory for the microscopic model which describes the d-wave superconductivity near AFQCP. The phase diagram for the magnetic field and temperature in the clean limit is shown in §3. Roles of weak box disorders and strong point disorders are investigated in §4 and §5, respectively. The results are summarized and some discussions are given in §6. 2. Formulation Our theoretical analysis is based on the following model H = H +H (1) 0 I H = t (cid:88) c† c +t(cid:48) (cid:88) c† c +(cid:88)(W −µ)n −gH(cid:88)Sz (2) 0 (cid:126)i,σ (cid:126)j,σ (cid:126)i,σ (cid:126)j,σ (cid:126)i (cid:126)i (cid:126)i <(cid:126)i,(cid:126)j>,σ <<(cid:126)i,(cid:126)j>>,σ (cid:126)i (cid:126)i (cid:88) (cid:88) (cid:88) (cid:126) (cid:126) H = U n n +V n n +J S S , (3) I (cid:126)i,↑ (cid:126)i,↓ (cid:126)i (cid:126)j (cid:126)i (cid:126)j (cid:126)i <(cid:126)i,(cid:126)j> <(cid:126)i,(cid:126)j> where S(cid:126) is the spin operator at the site(cid:126)i, n is the number operator at site(cid:126)i with spin (cid:126)i (cid:126)i,σ σ, and n = (cid:80) n . The bracket <(cid:126)i,(cid:126)j > and <<(cid:126)i,(cid:126)j >> denote the summation over (cid:126)i σ (cid:126)iσ the nearest neighbour sites and next nearest neighbour sites, respectively. We assume a two-dimensional square lattice. The candidate materials for the FFLO state, namely, CeCoIn and organic superconductors, have quasi-two-dimensional Fermi surfaces. We 5 adopt the unit of energy t = 1, and we fix t(cid:48)/t = 0.25. We study two kinds of disorders, which is taken into account in the third term of eq. (2). One is the box disorder in which the site diagonal potential W is randomly √ √ √ (cid:126)i distributed within [− 3W : 3W]. We multiply 3 so that the root-mean-square is (cid:113) W¯ = < |W |2 > = W. The other is the point disorder where W = 0 or W = W. (cid:126)i i i We assume W (cid:28) ε in the former while W (cid:29) ε in the latter. Then, the box F F disorder is regarded as a Born scatterer, while the point disorder gives rise to the unitary scattering. The randomness is represented by W in the former, while the concentration of impurity sites, where W = W, determines the randomness in the (cid:126)i latter. The chemical potential enters in eq. (2) as µ = µ + 1Un , where n is the 0 2 0 0 number density at U = V = J = H = W = 0. We fix µ = −0.8 for which the electron 0 concentration is 0.8 < n < 0.9. The on-site repulsive interaction is given by U, while V and J stand for the attractive interaction and AFM exchange interaction between nearest neighbour sites, respectively. We take into account the AFM interaction J to describe the FFLO state near the AFQCP. The interaction V stabilizes the d-wave superconductivity which we focus on. These features, namely the d-wave superconductivity and AFQCP, can be self-consistently described using the FLEX approximation on the basis of the simple Hubbard model [13]. But here, we assume the interactions V and J for simplicity in order to investigate the inhomogeneous system. With the last term in eq. (2), we include Disordered Fulde-Ferrel-Larkin-Ovchinnikov state in d-wave superconductors 4 the Zeeman coupling due to the applied magnetic field. We assume the g-factor, g = 2. We examine the model eq. (1) using the BdG theory by taking into account the Hartree-terms arising from U and J in addition to the mean field of SC order parameter. The Hartree-term due to the attractive interaction V is ignored because this term does not have any spin dependence which is essential for the following results. The Hartree- term arising from V may lead to the charge order if we assume a large attractive V. However, we ignore this possibility since the charge ordered state is not stabilized in the systems near AFQCP, and that is an artificial consequence of the simplified model in eq. (1). The mean field Hamiltonian is obtained as H = t (cid:88) c† c +t(cid:48) (cid:88) c† c +(cid:88)W n ,− (cid:88) [∆ c† c† +c.c.], (cid:126)i,σ (cid:126)j,σ (cid:126)i,σ (cid:126)j,σ (cid:126)i,σ (cid:126)i,σ (cid:126)i,(cid:126)j (cid:126)i,↑ (cid:126)j,↓ <(cid:126)i,(cid:126)j>,σ <<(cid:126)i,(cid:126)j>>,σ (cid:126)i,σ <(cid:126)i,(cid:126)j> (4) where W = W +U < n > +1Jσ(cid:80) < S > −Hσ −µ. The summation of (cid:126)δ is (cid:126)i,σ (cid:126)i (cid:126)i,σ¯ 2 (cid:126)δ (cid:126)i+(cid:126)δ (cid:126) taken over δ = (±1,0),(0,±1). The pair potential is obtained as ∆ = (V − J/4) < (cid:126)i,(cid:126)j c c > −J/2 < c c > for(cid:126)i =(cid:126)j +(cid:126)δ, and otherwise 0. The thermodynamic average (cid:126)i,↑ (cid:126)j,↓ (cid:126)j,↑ (cid:126)i,↓ <> is calculated on the basis of the mean field Hamiltonian, eq. (4). The free energy is obtained as (cid:88) (cid:88) F = − log[1+exp(−E /T)]+ W α (cid:126)i,↓ α (cid:126)i 1 1 − (cid:88)(U < n > + Jσ(cid:88) < S >) < n > +(cid:88)∆† < c c >, (5) 2 (cid:126)i,σ¯ 2 (cid:126)i+(cid:126)δ (cid:126)i,σ (cid:126)j,(cid:126)i (cid:126)i,↑ (cid:126)j,↓ (cid:126)i,σ (cid:126)δ (cid:126)i,(cid:126)j whereE istheenergyofBogoliubovquasiparticles. Wenumericallysolvethemeanfield α equations and determine the stable phase by comparing the free energy of self-consistent solutions. The electron concentration and the magnetization at the site (cid:126)r is obtained as n((cid:126)r) =< n +n > and M((cid:126)r) =< n −n >, respectively. The order parameter of (cid:126)r,↑ (cid:126)r,↓ (cid:126)r,↑ (cid:126)r,↓ superconductivity is described by the pair potential ∆ . The main component of the (cid:126)i,(cid:126)j pair potential has the d-wave symmetry, although a small extended s-wave component is induced in the inhomogeneous system. The d-wave component of SC order parameter is obtained as ∆d((cid:126)r) = ∆ +∆ −∆ −∆ , (6) (cid:126)r,(cid:126)r+(cid:126)a (cid:126)r,(cid:126)r−(cid:126)a (cid:126)r,(cid:126)r+(cid:126)b (cid:126)r,(cid:126)r−(cid:126)b (cid:126) where (cid:126)a = (1,0) and b = (0,1). The numerical calculation is carried out on the N = 100×100 lattice in the clean limit, and on the N = 40×40 lattice for disordered systems. We have confirmed that qualitatively same results are obtained for 100×100 and 40×40 lattices in the clean limit. Disordered Fulde-Ferrel-Larkin-Ovchinnikov state in d-wave superconductors 5 3. Phase diagram in the clean limit We first determine the phase diagram for the normal, uniform BCS, and FFLO states in the clean limit. We determine the stable state by comparing the free energy of these states. The order of phase transition is numerically determined by analyzing both the order parameter and free energy. The free energy of two phases cross at the first order phase transition. A discontinuous jump of SC order parameter also shows the first order transition. We show that both on-site repulsion U and AFM interaction J are necessary to reproduce the phase diagram of CeCoIn [3]. 5 (a) U=2.2, J=0 (b) U=0.9, J=0.54 (c) U=0, J=0.6 0.3 0.4 0.15 Normal Normal Normal FFLO 0.3FFLO 0.2 H0.1 H H0.2 0.05 Uniform 0.1 Uniform Uniform 0.1 00 0.05 0.1 0.15 0.2 00 0.05 0.1 0.15 0.2 00 0.05 0.1 0.15 0.2 T T T Figure 1. Phase diagram in the clean limit (W = 0) for (a) U = 2.2 and J = 0, (b) U = 0.9 and J = 0.54, and (c) U = 0 and J = 0.6, respectively. Blue solid lines show the first order phase transition to the SC state from the normal state, while blackdashedlinesshowthesecondorderphasetransition. Reddash-dottedlinesshow the second order transition between the uniform BCS state and the FFLO state. We choose V so that the transition temperature at H =0 is around T =0.2. c Figure 1 shows the phase diagram for (a) U = 2.2 and J = 0, (b) U = 0.9 and J = 0.54, and (c) U = 0 and J = 0.6. For the parameters in (b), the second order phase transition occurs between the uniform BCS state and the FFLO state (BCS-FFLO transition). The phase transition from the normal state to the uniform BCS state and FFLO state is first order at the temperature below the tricritical point, which is slightly higher than the end point of the BCS-FFLO transition. A conventional second order superconducting transition occurs above the tricritical point. These features of phase diagram in Fig. 1(b) are consistent with the experimental results for CeCoIn [3, 6, 7, 30, 31]. 5 Note that the shape of BCS-FFLO transition line seems to be incompatible with the experimental results for CeCoIn . A large positive slope ∂H (T)/∂T > 0, where 5 BF H (T) is the magnetic field at the BCS-FFLO transition, has been reported in the BF experiments. This feature does not appear in Fig. 1(b), however that is reproduced by taking into account the self-energy correction arising from the spin fluctuation near the AFQCP [13]. This means that the mean field theory underestimates the stability of FFLO state. This is not important for the spatial structure of FFLO state in the presence of randomness, on which we focus in this paper. Disordered Fulde-Ferrel-Larkin-Ovchinnikov state in d-wave superconductors 6 A serious discrepancy between the theory and experiment is shown in the phase diagram for J = 0 (Fig. 1(a)) and U = 0 (Fig. 1(c)). The FFLO state is completely suppressed for J = 0, while the first order transition to the SC state is suppressed for U = 0. Thus, the phase diagram near the Pauli-Chandrasekhar-Clogston limit is significantly affected by the electron correlation. These results can be understood on the basis of the Fermi liquid theory. It has been have shown that the FFLO state is suppressed by the negative Fermi liquid parameter F , while the first order 0a transition to the SC state is suppressed by the positive F [32]. Within the mean 0a field theory, the on-site repulsion U and AFM interaction J give rise to the negative and positive F , respectively. The consistency between Fig. 1(b) and experimental 0a results [3, 6, 7, 30, 31] indicates that the local spin fluctuation, which is essential for the formation of heavy fermions [33], coexists with the AFM spin fluctuation in CeCoIn . 5 We adopt the parameters in Fig. 1(b) in the following sections. 4. Box disorder WehereinvestigateFFLOstateinthepresenceofboxdisorders, wherethesitepotential √ √ W is randomly distributed within [− 3W : 3W]. Since we assume W (cid:28) ε , all of i F the sites are weakly disordered. It has been shown that a two-dimensional FFLO state can be stable rather than the one-dimensional FFLO state [3, 34]. This is the case in our calculation in the clean limit (W = 0), however a weak disorder (W = 0.1) stabilizes the one-dimensional FFLO state as shown in Fig. 2. This is qualitatively consistent with the results for moderately weak point disorders [29]. Figures 2(a) and (b) show a typical spatial dependence of the order parameter of d- wavesuperconductivity∆d((cid:126)r)intheFFLOstateforW = 0.1andW = 0.3, respectively. For W = 0.1, the spatial structure of SC order parameter is almost regular, which is approximated by ∆d((cid:126)r) = ∆ cos(q r ) (Fig. 2(a)). On the other hand, we see a spatially 0 f x modulated structure of SC order parameter for W = 0.3 (Fig. 2(b)). Figures 2(c) and (d) show the spatial dependence of local spin susceptibility χ((cid:126)r) = M((cid:126)r)/H for W = 0.1 and W = 0.3, respectively. In both cases, the magnetization M((cid:126)r) is induced around the spatial line node of SC order parameter, where ∆d((cid:126)r) = 0. In particular, for a moderate disorder W = 0.3, the spatial distribution of the magnetization M((cid:126)r) follows the spatial nodes of SC order parameter. In order to illuminate the features of FFLO state, we show the spatial dependences of ∆d((cid:126)r) and M((cid:126)r)/H in the BCS state. Fig. 3(a) shows the SC order parameter at H = 0.18, where the uniform BCS state is stable in the clean limit. We see that the SC order parameter is nearly uniform in the presence of moderately strong disorders W = 0.3, except for the suppression around (cid:126)r = (35,28). The local spin susceptibility χ((cid:126)r) = M((cid:126)r)/H is increased around (cid:126)r = (35,28) because the superconductivity is suppressed there (Fig. 3(b)). We see the checkerboard structure of the local spin susceptibility, which is similar to high-T cuprates [35, 36]. This c Disordered Fulde-Ferrel-Larkin-Ovchinnikov state in d-wave superconductors 7 ∆d ∆d (a) (r) (W=0.1) (b) (r) (W=0.3) 40 0.4 40 0.4 0.3 0.3 35 35 0.2 0.2 30 0.1 30 0.1 0 0 25 -0.1 25 -0.1 20 -0.2 20 -0.2 -0.3 -0.3 15 -0.4 15 -0.4 10 10 5 5 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 (c) M(r)/H (W=0.1) (d) M(r)/H (W=0.3) 40 0.65 40 0.8 0.6 0.7 35 0.55 35 0.5 0.6 30 0.45 30 0.5 25 0.4 25 0.4 0.35 0.3 20 0.3 20 0.25 0.2 15 0.2 15 0.1 10 10 5 5 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 Figure 2. (a) and (b) Typical spatial dependence of the d-wave SC order parameter ∆d((cid:126)r) in the presence of the box disorder for W =0.1 and W =0.3, respectively. (c) and (d) Spatial dependence of the local spin susceptibility M((cid:126)r)/H for W = 0.1 and W = 0.3, respectively. We assume T = 0.02 and H = 0.24 in (a) and (c), and adopt T =0.02 and H =0.225 in (b) and (d). We fix U =0.9, J =0.54, and V =0.8 in the following results. checkerboard structure is induced by the quasiparticle interference effect [37, 38]. The quasiparticle interference effect occurs in the FFLO state too, however, the spatial dependence due to the quasiparticle interference effect is much smaller than that arising from the inhomogeneous SC order parameter in the FFLO state. To show the spatial dependences more clearly, we show the local spin susceptibility, SC order parameter, and electron concentration along (cid:126)r = (x,1). We see the enhancement of local spin susceptibility around the spatial nodes of FFLO state, in addition to spatial fluctuation in the atomic scale (Figs. 4(b) and (e)). A large spatial dependence in the FFLO state should be contrasted to the small oscillation for x < 23 in the BCS state (Fig. 4(a)). The latter arises from the quasiparticle interference effect. The spatial fluctuation around x = 30 in the BCS state is induced by the inhomogeneity of SC order parameter (Fig. 4(d)). The local spin susceptibility in the normal state is governed by the weak atomic scale oscillation (Fig. 4(c)), which can be regarded as Disordered Fulde-Ferrel-Larkin-Ovchinnikov state in d-wave superconductors 8 ∆d (a) (r) (b) M(r)/H 40 0.45 40 0.6 35 0.4 35 0.5 0.4 0.35 30 30 0.3 0.3 25 25 0.2 0.25 0.1 20 0.2 20 0 15 0.15 15 -0.1 10 10 5 5 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 Figure 3. (a) D-wave SC order parameter ∆d((cid:126)r) and (b) local spin susceptibility at T =0.02 and H =0.18. The disorder potential is the same as in Figs. 2(b) and (d). (a) BCS (H=0.18) (b) FFLO (H=0.225) (c) Normal (H=0.25) 0.7 0.7 0.7 0.6 0.6 0.6 H H H 0.5 0.5 0.5 / / / ) ) ) 10.4 10.4 10.4 , , , x0.3 x0.3 x0.3 ( ( ( M0.2 M0.2 M0.2 0.1 0.1 0.1 0 0 0 0 10 x20 30 40 0 10 x20 30 40 0 10 x20 30 40 (d) BCS (H=0.18) (e) FFLO (H=0.225) (f) FFLO (H=0.225) 0.5 0.4 1.2 0.3 0.4 1 (x,1)0.3 (x,1)00..201 x,1)00..68 d 0.2 d -0.1 ( ∆ ∆ n0.4 -0.2 0.1 -0.3 0.2 00 10 x20 30 40 -0.40 10 x20 30 40 00 10 x20 30 40 Figure 4. Spatial dependences along (cid:126)r = (x,1) for W = 0.3. We assume the same disorder potential as in Figs. 2(b) and (d). Upper panel: Local spin susceptibility in (a) BCS state (T = 0.02 and H = 0.18), (b) FFLO state (T = 0.02 and H = 0.225), and (c) normal state (T =0.2 and H =0.25). Lower panel: D-wave order parameter in(d)BCSstateand(e)FFLOstate. (f)Theelectronconcentrationn((cid:126)r)intheFFLO state. a weak disorder-induced-antiferromagnetism (see §4). We find no clear relationship between the local spin susceptibility and the electron concentration in the FFLO state. The latter is shown in Fig. 4(f). At the last of this section, we show the distribution function of local spin Disordered Fulde-Ferrel-Larkin-Ovchinnikov state in d-wave superconductors 9 (a) W=0.1 (b) W=0.3 8 15 Normal (T=0.2) Normal (T=0.2) Normal (T=0.06) Normal (T=0.06) FFLO ) BCS )6 FBFCLSO H H 10 / M / M4 ( P 5 ( P 2 0 0 0.2 0.4 0.6 0.8 0 0 0.2 0.4 0.6 0.8 M/H M/H Figure 5. Distribution function of local spin susceptibility P(M/H) for (a) W =0.1 and (b) W = 0.3, respectively. We show the results in the BCS state (T = 0.02 and H = 0.18, green dash-dotted line), FFLO state (T = 0.02 and H = 0.225, blue solid line), and normal state at low temperature (T = 0.06 and H = 0.25, red dotted line) and at high temperature (T = 0.2 and H = 0.25, black dashed line). Three and five samples of the disorder potential are taken for the random average in (a) and (b), respectively. susceptibility P(M/H), which is expressed as 1 (cid:88) P(x) =< δ(x−M((cid:126)r)/H) > , (7) av N (cid:126)r where <> denotes the random average. This distribution function is measured by the av spectrum of NMR measurements. Figure 5(a) clearly shows the double peak structure of P(M/H) in the FFLO state for a weak disorder (W = 0.1). A peak around M/H = 0.3 arises from the region where the SC order parameter is large, while the other peak around M/H = 0.55 comes from the Andreev bound states localized around the spatial nodes of SC order parameter. It has been shown that this double peak structure also appears in the FFLO state in the presence of vortex lattice when the Maki parameter is large [39]. As increasing the disorder potential W, the double peak structure of P(M/H) in the FFLO state vanishes, as shown in Fig. 5(b). The width of the peak in P(M/H) is broader in the FFLO state than in the BCS state. These results seem to be consistent with the NMR measurement of CeCoIn [16], which shows a single and broad peak 5 whose position moves to the large M/H in the high field superconducting phase. Note that the peak of P(M/H) in the BCS state moves to the large M/H with increasing the disorder potential W, since the residual DOS is induced by disorders in the d-wave superconductors [40]. This is contrasted to the FFLO state, where the average of local spin susceptibility M/H is slightly affected by the randomness. Disordered Fulde-Ferrel-Larkin-Ovchinnikov state in d-wave superconductors 10 5. Point disorder We here turn to the point disorder, in which N = 40×40 sites are divided into the host sites where W = 0 and the impurity sites where W = W. We assume W = 40 (cid:29) ε (cid:126)i (cid:126)i F so as to give rise to the unitarity scattering. The impurity concentration is fixed to be N /N = 0.05, where N is the number of impurity sites. We investigated 10 samples imp imp for the impurity distribution, and found that the distribution in Fig. 6(a) gives a typical result. We adopt this sample in the following results. ∆d (a) Impurity sites (b) (r) (c) M(r)/H 40 40 0.45 40 2 0.4 35 0.35 35 1.5 0.3 30 30 0.25 30 1 0.2 25 0.15 25 0.5 0.1 y20 20 0.05 20 0 0 15 -0.05 15 -0.5 10 10 10 5 5 10 2x0 30 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 Figure 6. (a) Typical distribution of the impurity sites. We adopt this sample in Figs. 6-9. (b) SC order parameter ∆d((cid:126)r) and (c) local spin susceptibility M((cid:126)r)/H at T =0.02 and H =0.18. Figure 6(b) shows the suppression of SC order parameter around the impurity sites in the BCS state. We see that the local spin susceptibility is significantly enhanced around the impurity sites (Fig. 6(c)). The maximum of the local spin susceptibility is much larger than the spin susceptibility in the normal state of clean systems. This is because of the disorder-induced-antiferromagnetism, which has been investigated in the nearly AFM Fermi liquid state [41], and in the pseudogap state [42] of high-T c cuprates. The disorder-induced-antiferromagnetism is a ubiquitous phenomenon in the systemsneartheAFQCP,suchashigh-T cuprates,organicmaterials,andheavyfermion c systems. Aclearexperimentalevidenceforthedisorder-induced-antiferromagnetismhas been obtained in high-T cuprates [43, 44, 45]. c A complicated spatial structure is realized at high fields, where the FFLO state is stable in the clean limit. Then, the free energy shows a multi-valley structure. There are many local minimum of free energy with respect to the spatial structure of SC order parameter. Figures 7(a-e) show five examples of the self-consistent solutions of BdG equation for the impurity distribution shown in Fig. 6(a). The local spin susceptibility in each solution is shown in Figs. 7(f-j). The difference of condensation energy is small betweenthesestates. Thecondensationenergyismaximuminthe“FFLO2”stateshown in Fig. 7(b) among the solutions obtained by us. However, we obtain the solution of “FFLO3” state shown in Fig. 7(c) when we choose the SC order parameter near T as c