Disordered Fulde-Ferrel-Larkin-Ovchinnikov State in d-wave Superconductors

We study the Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) superconducting state in the disordered systems. We analyze the microscopic model, in which the d-wave superconductivity is stabilized near the antiferromagnetic quantum critical point, and investigate two kinds of disorder, namely, box disorder and point disorder, on the basis of the Bogoliubov-deGennes (BdG) equation. The spatial structure of modulated superconducting order parameter and the magnetic properties in the disordered FFLO state are investigated. We point out the possibility of"FFLO glass"state in the presence of strong point disorders, which arises from the configurational degree of freedom of FFLO nodal plane. The distribution function of local spin susceptibility is calculated and its relation to the FFLO nodal plane is clarified. We discuss the NMR measurements for CeCoIn_5.


Introduction
FFLO superconductivity was predicted in 1960's by Fulde and Ferrel [1] and also by Larkin and Ovchinnikov [2]. In addition to the U(1)-gauge symmetry, a spatial symmetry is spontaneously broken in the FFLO state owing to the modulation of superconducting (SC) order parameter. After nearly 40 years of fruitless experimental search for FFLO states, recent experiments appeared to give first evidences for such a phase [3]. Moreover, FFLO phase is attracting growing interests in other related fields such as the cold fermion gases [4] and the high-density quark matter [5].
Extensive studies of FFLO state had been triggered by the discovery of a novel SC phase at high fields and low temperatures in the heavy fermion superconductor CeCoIn 5 [6,7]. Possible FFLO states have been discovered also 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 the d-wave FFLO state is stable in the vicinity of AFQCP owing to some strong coupling effects [13].
Another intriguing relationship between FFLO superconductivity and antiferromagnetism has been indicated in CeCoIn 5 . Several experimental results suggest the emergence of a FFLO state in CeCoIn 5 [3,14,15,16,17]. On the other hand, nuclear magnetic resonance (NMR) and neutron scattering data rather indicate the presence of antiferromagnetic (AFM) order in the high field phase of CeCoIn 5 [18,19]. The pressure 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 5 is stabilized by the pressure [17]. Therefore, it is expected that the coexistent state of FFLO superconductivity and AFM order is realized in CeCoIn 5 at ambient pressure, where the AFM 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]. The d-wave FFLO state in the presence of moderately weak point disorders 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 order parameter has distorted nodes, while more complicated spatial structure indicating 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-inducedantiferromagnetism is a characteristic property of systems near AFQCP. Therefore, the 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.

Formulation
Our theoretical analysis is based on the following model where S i is the spin operator at the site i, n i,σ is the number operator at site i with spin σ, and n i = σ n iσ . The bracket < i, j > and << i, j >> denote the summation over 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 5 and organic superconductors, have quasi-two-dimensional Fermi surfaces. We adopt the unit of energy t = 1, and we fix t /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 i is randomly distributed within [− √ 3W : √ 3W ]. We multiply √ 3 so that the root-mean-square is W = < |W i | 2 > = W . The other is the point disorder where W i = 0 or W i = W . We assume W ε F in the former while W ε F in the latter. Then, the box 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 i = W , determines the randomness in the latter. The chemical potential enters in eq. (2) as µ = µ 0 + 1 2 U n 0 , where n 0 is the number density at U = V = J = H = W = 0. We fix µ 0 = −0.8 for which the electron 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 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 Hartreeterm 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 where The summation of δ is taken over δ = (±1, 0), (0, ±1). The pair potential is obtained as ∆ i, j = (V − J/4) < c i,↑ c j,↓ > −J/2 < c j,↑ c i,↓ > for i = j + δ, and otherwise 0. The thermodynamic average <> is calculated on the basis of the mean field Hamiltonian, eq. (4). The free energy is obtained as where E α is the energy of Bogoliubov quasiparticles. We numerically solve the mean field equations and determine the stable phase by comparing the free energy of self-consistent solutions.
The electron concentration and the magnetization at the site r is obtained as n( r) =< n r,↑ + n r,↓ > and M ( r) =< n r,↑ − n r,↓ >, respectively. The order parameter of superconductivity is described by the pair potential ∆ i, j . The main component of the 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 where 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.

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 5 [3].  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 5 [3,6,7,30,31].
Note that the shape of BCS-FFLO transition line seems to be incompatible with the experimental results for CeCoIn 5 . A large positive slope ∂H BF (T )/∂T > 0, where H BF (T ) is the magnetic field at the BCS-FFLO transition, has been reported in the 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.
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 0a , while the first order transition to the SC state is suppressed by the positive F 0a [32]. Within the mean field theory, the on-site repulsion U and AFM interaction J give rise to the negative and positive F 0a , respectively. The consistency between Fig. 1(b) and experimental 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.

Box disorder
We here investigate FFLO state in the presence of box disorders, where the site potential Since we assume W ε F , all of 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 dwave superconductivity ∆ d ( r) in the FFLO state for W = 0.1 and W = 0.3, respectively. For W = 0.1, the spatial structure of SC order parameter is almost regular, which is approximated by ∆ d ( r) = ∆ 0 cos(q f r x ) ( Fig. 2(a)). On the other hand, we see a spatially 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( r) = M ( r)/H for W = 0.1 and W = 0.3, respectively. In both cases, the magnetization M ( r) is induced around the spatial line node of SC order parameter, where ∆ d ( r) = 0. In particular, for a moderate disorder W = 0.3, the spatial distribution of the magnetization M ( 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 ( r) and M ( 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 r = (35,28). The local spin susceptibility χ( r) = M ( r)/H is increased around 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 c cuprates [35,36]. This 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 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   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 susceptibility P (M/H), which is expressed as where <> av denotes the random average. This distribution function is measured by the 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 5 [16], which shows a single and broad peak 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.

Point disorder
We here turn to the point disorder, in which N = 40 × 40 sites are divided into the host sites where W i = 0 and the impurity sites where W i = W . We assume W = 40 ε F so as to give rise to the unitarity scattering. The impurity concentration is fixed to be N imp /N = 0.05, where N imp is the number of impurity sites. We investigated 10 samples 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.  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 systems near the AFQCP, such as high-T c cuprates, organic materials, and heavy fermion systems. A clear experimental evidence for the disorder-induced-antiferromagnetism has been obtained in high-T c cuprates [43,44,45].
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 between these states. The condensation energy is maximum in the "FFLO2" state shown 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 c as a initial state of the mean field equation. The "FFLO1" state in Fig. 7(a) is obtained when the uniform BCS state is chosen to be an initial state. This means that the FFLO3 state can be stabilized as a meta-stable state by decreasing the temperature through T c , while the FFLO1 state may be realized by increasing the magnetic field through the BCS-FFLO transition. The condensation energy is increased by aligning the spatial nodes of SC order parameter on the "dirty region" where the local concentration of impurity sites is large. Many spatial configurations of the SC order parameter have a similar condensation energy because there are many configurations of spatial nodes which match the dirty region. The situation is similar to the vortex glass state which is induced by the configurational degree of freedom of the quantum vortices [46,47]. The analogy with the vortex glass state indicates the possibility of "FFLO glass" state, which is realized by the paramagnetic de-pairing effect in random systems.
To gain the magnetic energy, the spatial node is also induced in the "clean region" at high fields. We found that the local spin susceptibility is enhanced around the spatial nodes in the clean region (Figs. 7(f-j)). The quasiparticle interference effect [35,37] is not visible in the presence of point disorders because the disorderinduced-antiferromagnetism obscures a weak quasiparticle interference effect.
To clarify the local spin susceptibility arising from the disorder-inducedantiferromagnetism and that from the spatial nodes of SC order parameter, we show the M ( r)/H along r = (x, 1) in Fig. 8. The normal state, FFLO2 state, and BCS state are shown for a comparison. As shown in Fig. 6(a), the system is clean at x < 20, while it is dirty at x > 20. Figure 8 shows that the disorder-induced-antiferromagnetism occurs and gives rise to the significant oscillation of the magnetization in the dirty region (x > 20). This is a ubiquitous phenomenon near the AFQCP in the absence of translational symmetry. We see that the disorder-induced-antiferromagnetism is enhanced in the FFLO state as well as in the BCS state. The local spin susceptibility is more significantly affected by the superconductivity in the clean region (x < 20). It is shown that the spin susceptibility is larger in the FFLO2 state than in the BCS state because of the presence of spatial node around x = 15 ( Fig. 7(b)).  Figure 9 shows the distribution function of the local spin susceptibility P (M/H), which is defined as The summation host r is taken over the host sites and therefore the contribution form the impurity sites is eliminated in eq. (8). The double peak structure appears in the BCS state as well as in the FFLO2 state because of the the disorder-inducedantiferromagnetism. This structure vanishes in the FFLO3 state in which many spatial nodes exists in the SC order parameter. These results are incompatible with the NMR measurements for CeCoIn 5 [16,18]. The width of the peak hardly changes through the BCS-FFLO transition in contrast to ref. [16]. This means that the model based on the point disorder is not relevant for CeCoIn 5 . However, the point disorders can be systematically induced by substituting Ce ions by La ions, or In ions by Cd ions. Therefore, it is interesting to investigate the superconducting state in Ce 1−x La x CoIn 5 and CeCoIn 5−x Cd x [48] at high fields. Before closing this section, we briefly discuss the case of dilute impurities. Figure 10(b) shows the distribution function of the local spin susceptibility in the presence of 2% point disorders. We show the result for a typical distribution of impurities which leads to the SC order parameter shown in Figure 10

Summary and discussion
We investigated the disordered FFLO state with focus on the spatial structure of SC order parameter and the magnetic properties. In particular, the d-wave superconductivity near the AFQCP has been investigated in details. It has been shown that the spatial dependence of SC order parameter is relatively simple in case of the box disorder; the FFLO nodal plane is modulated. On the other hand, the spatial structure is complicated in the presence of point disorders. Then, the spatial nodes are strongly pinned to the locally dirty region. There are many configurations in which the nodes are pinned to the point disorders, and therefore a glassy behaviour appears. We called this state "FFLO glass " in analogy with the vortex glass state.
We have shown that the magnetic properties in the SC state significantly depend on the character of disorders. In the FFLO state with box disorders, the magnetic properties are governed by the nodal plane 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 in the BCS state are mainly determined by the quasiparticle interference effect, which gives rise to an oscillation with a small amplitude. A weak disorder-induced-antiferromagnetism is induced by the inhomogeneity of SC order parameters in the BCS state as well as in the FFLO state.
The magnetic properties are dominated by the disorder-induced-antiferromagnetism in the presence of point disorders.
We found that the disorder-inducedantiferromagnetism is enhanced by the superconductivity in both BCS and FFLO states. The local spin susceptibility in the locally clean region is suppressed in the BCS state, while that is increased in the FFLO state owing to the spatial nodes of SC order parameter.
Finally, we examine the distribution function of local spin susceptibility P (M/H) and discuss the NMR measurements for CeCoIn 5 . It has been known that the distribution function shows a double peak structure in the FFLO state in the clean limit [39]. We have shown that the two peaks merge into the single peak in the presence of box disorders and/or point disorders. However, the distribution function is affected in a different way by the box disorder and point disorder. The single peak structure is induced by the box disorders because of the displacement of FFLO nodes, while the disorder-induced-antiferromagnetism is the main cause of the broad single peak in the presence of point disorders. We can distinguish these two cases by analyzing the line width of P (M/H). The line shape of P (M/H) is very broad and its width does not change through the BCS-FFLO transition in case of the point disorder (Fig. 9). This is because the local antiferromagnetism occurs near the point disorders in both BCS and FFLO states (Fig. 8). On the other hand, the line shape is significantly broadened in the FFLO state through the BCS-FFLO transition in case of the box disorders, as shown in Fig. 5. The latter seems to be consistent with the NMR measurement for CeCoIn 5 at the In(1) site [16].
Another NMR spectrum at the In(2) site has shown the double peak structure with large splitting and indicated the AFM order in the high field phase of CeCoIn 5 [18]. A clear experimental evidence for the AFM order has been obtained by the recent neutron scattering measurement [19]. We have shown that the AFM order can occur in the FFLO state near the AFQCP and the phase diagram suggested in refs. [18,19] is consistent with the coexistence of antiferromagnetism and FFLO superconductivity [21]. Although the double peak structure of the NMR line shape is also induced by the disorderinduced-antiferromagnetism (Fig. 9), this is not the case in CeCoIn 5 . The direction of the AFM moment is parallel to the applied magnetic field in case of the disorder-induced-antiferromagnetism. However, the neutron scattering measurement shows the magnetic moment perpendicular to the magnetic field [19], which means the spontaneous symmetry breaking. Therefore, the true long range order of antiferromagnetism seems to occur in the experiments of refs. [18,19].
Our results in this paper suggest that the single peak structure of NMR spectrum at In(1) site, which is less sensitive to the AFM order than In(2) site, can be caused by the disorder. But, the AFM order may be another cause. We are planning to investigate the magnetic properties in the coexistent state of AFM order and FFLO superconductivity. From the experimental point of view, it would be interesting to investigate the pressure effect in CeCoIn 5 . It is expected that the AFM order is suppressed by the pressure while the FFLO superconductivity is enhanced [13]. The latter has been observed in ref. [17]. Therefore, the pure FFLO state may be realized at high pressures.
According to these considerations, the point disorder is not the main source of the randomness in CeCoIn 5 . However, the point disorder can be induced by substituting Ce ions with La ions or In ions with Cd ions. Thus, Ce 1−x La x CoIn 5 and CeCoIn 5−x Cd x [48] will be an intriguing playground of the FFLO superconductivity with point disorders.
In summary, we investigated the FFLO superconducting state in the presence of randomness. The spatial structure of SC order parameter and magnetic properties are clarified in details. It has been proposed that the experimental results on CeCoIn 5 can be understood by assuming the FFLO state realized in the high field phase and taking into account both the AFM spin correlation and a weak disorder.