Towards a Microscopic Theory of the Knight Shift in an Anisotropic, Multiband Type-II Superconductor

A method is proposed to extend the zero-temperature Hall-Klemm microscopic theory of the Knight shift $K$ in an anisotropic and correlated, multi-band metal to calculate $K(T)$ at finite temperatures $T$ both above and into its superconducting state. The transverse part of the magnetic induction ${\bf B}(t)={\bf B}_0+{\bf B}_1(t)$ causes adiabatic changes suitable for treatment with the Keldysh contour formalism and analytic continuation onto the real axis. We propose that the Keldysh-modified version of the Gor'kov method can be used to evaluate $K(T)$ at high ${\bf B}_0$ both in the normal state, and by quantizing the conduction electrons or holes with Landau orbits arising from ${\bf B}_0$, also in the entire superconducting regime for an anisotropic, multiband Type-II BCS superconductor. Although the details have not yet been calculated in detail, it appears that this approach could lead to the simple result $K_S(T)\approx a({\bf B}_0)-b({\bf B}_0)|\Delta({\bf B}_0,T)|^2$, where $2|\Delta({\bf B}_0,T)|$ is the effective superconducting gap. More generally, this approach can lead to analytic expressions for $K_S(T)$ for anisotropic, multiband Type-II superconductors of various orbital symmetries that could aid in the interpretation of experimental data on unconventional superconductors.


I. INTRODUCTION
In nuclear magnetic resonance (NMR) measurements of a nucleus, there is a difference between the resonance frequency of the nucleus when it is in a metal from when it is vacuum or in an insulator. This is known as the Knight shift 1 . Although the temperature T dependence of the Knight shift in a superconductor has long been considered to be a probe of the spin state of the paired electrons [2][3][4] , the only theoretical basis for the experiments was the 1958 assumption of Yosida that the probed nuclear spins could be entirely neglected 5 , and the only quantity of interest was the temperature T dependence of the zero-field limit of the electron spin susceptibility of an isotropic and uncorrelated Type-I superconductor 5 . This led for a BCS singlet-pair-spin superconductor to a T dependence of the Knight shift K S (T ) proportional to x/(1 + x), where x = (β/∆) d∆ dβ , β = 1/T , where we set k B = 1, and ∆ is one-half the BCS energy gap, for which K S (T ) → 0 as T → 0, unlike most experimental results 5 .
For isotropic Type-I superconductors in the Meissner state, crushing the sample to a powder of crystallites the cross-sections of which were less than the magnetic penetration depth was usually found to provide a reasonable method for that conventional theory to be applicable [2][3][4] . In the first years following the BCS theory, the transition metal superconductors were found to behave somewhat differently, as the Knight shift did not vanish as T → 0 6 , and it was thought that surface spin-orbit scattering could explain the near-cancelation of the Knight shift in transition metals 6,7 . But surface impurity spin-orbit scattering could not explain the observed non-vanishing K(0) results observed in clean materials. It is now understood that there is also a component to the Knight [8][9][10][11][12][13][14] . But one of those materials was a quasi-one-dimensional organic superconductor 10,11 , some examples of which often exhibit spin-density waves 15 , and another was the very dirty sodium cobaltate hydrate material [12][13][14] . In the latter example, the upper critical field parallel to that layered compound is Pauli-limited, which normally only occurs when the magnetic field breaks the oppositely-oriented pair spins 16,17 . Since dirt drastically suppresses p-wave superconductivity 18 , the sodium cobaltate hydrate Knight shift results, if correct, are likely to arise from some other mechanism.
Moreover, in highly anisotropic Type-II superconductors, such as the cuprates and heavy fermion materials, other significant breakdowns in the Yosida theory have been found to exist.
In the first Knight shift measurements on the cuprate YBa 2 Cu 3 O 7−δ , Bennett et al. found that although the Yosida theory appeared to work for the 63 Cu spins in the CuO chains for all field directions, although the orbital contributions are different for each of the three orthogonal applied field directions, and it also appeared to work well for the 63 Cu spins in the CuO 2 planes when the strong constant magnetic field H was applied parallel to the CuO 2 layers. But, when H was applied normal to the CuO 2 layers, no T dependence to the Knight shift was observed in that cuprate 19 . This result was later described by Slichter as possibly being due to a "fortuitous" cancelation of the effect from an isolated planar 63 Cu spin by its interaction with its near-neighbor planar 63 Cu spins 20 . Subsequently, in a number of layered correlated superconductors, the T dependence of the Knight shift probes of the nuclear spins in the layers with the field applied normal to the layers has been observed to vary strongly with field strength, approaching a constant K S (T ) in the large normal field strength limit, as first observed by Bennett et al. [19][20][21][22][23][24] .
Especially in the case of Sr 2 RuO 4 , numerous Knight shift measurements of the 17 O, 99 Ru, 101 Ru, and 87 Sr have all led to temperature-independent Knight shift measurements [25][26][27] , as did polarized neutron scattering experiments 28 . This experiments were all interpreted as evidence for a parallel-spin pair state in that material. However, several upper critical field measurements with the field parallel to the layers showed strong Pauli limiting effects 29,30 , which is inconsistent with a parallel-spin pair state 31,32 . In addition, the fact that T -independent Knight shift measurements were obtained for the field both parallel and perpendicular to the RuO 2 layers is incompatible with any of the crystal point-group-compatible p-wave states. Thus, the only way for a T -independent Knight shift to legitimately arise from a parallel-pair-spin state in both field directions is for the d-vector (the vector describing the components of the three triplet spin states) to rotate with the magnetic field 33,34 . This argument was used to show that while the upper critical field of Sr 2 RuO 4 is strongly Pauli limited for the field applied parallel to the layers, it could possibly be consistent with one or more p-wave helical states, provided that the d-vector is allowed to rotate freely with the magnetic field direction 32 . This means that spin-orbit coupling with the lattice would have to be negligible. However, there is strong evidence that spin-orbit coupling in Sr 2 RuO 4 is very strong at some points on the Fermi surface, ruling out such d-vector rotation possibilities 35  have not yet been made on URhGe and UGe 2 , the latter of which is only superconducting under pressure. In these three ferromagnetic superconductors, there is at least a plausible mechanism for a parallel-spin pair superconducting state, and in URhGe the upper critical field fits the predictions for all three crystal axis directions of a parallel-spin p-wave polar state fixed to the crystal a-axis direction normal to the c-axis Ising ferromagnetic order 40,42 , and there is a reentrant, high field phase that violates the Pauli limit by a factor of 20 40 . In order to obtain further evidence that the classic Yosida interpretation of a T -independent K S (T ) can correctly imply a parallel-spin superconducting state, we urge that 73 Ge, with a strong nuclear moment, (or possibly 103 Rh, with a much weaker nuclear moment) K S (T ) measurements on URhGe be carefully performed in the low-field superconducting phase.

II. THE MODEL
The first microscopic model of the Knight shift at T = 0 in anisotropic and correlated metals was recently presented by Hall and Klemm 43 . This model assumed that the applied magnetic fields probe the nuclear spins, and the spins of the electrons orbiting the nucleus interact with the nucleus via the hyperfine interaction in the form of a diagonal g tensor with two distinct components The assumption D x = D y was made to simplify the calculations, as discussed in more detail in the following. After interacting with the nuclear spins, the orbital electrons can be excited into one of multiple bands, each of which was assumed to have an ellipsoidal Fermi surface of arbitrary anisotropy and shape. The orbital motion of the electrons in each of these bands was constrained by the strong, time-independent part B 0 of the magnetic induction B(t) to be in Landau levels, and the electron spins also could interact weakly with B 0 . It was found that the self-energy due to D z led to the Knight shift, and that due to D x = D y led to the first formulas for the linewidth changes associated with the Knight shift at T = 0. However, since those calculations were made at T = 0, they could not be used to probe the superconducting state. In the following, a method is proposed to do so. In this model, we do not account for competing ferromagnetism or charge-density wave (CDW) or spin-density wave (SDW) formation, at least one of which is normally present in the transition metal dichalcogenides, the organic layered superconductors, the cuprates, the iron pnictides, and the ferromagnetic superconductors. Such competing effects will be the subjects of future studies.
A. The Simple Hamiltonian H 0 Since in an NMR experiment, the applied magnetic field can be applied in any direction, we assume the resulting constant magnetic induction B 0 = B 0 (sin θ cos φ, sin θ sin φ, cos θ) = B 0r with respect to the crystalline Cartesian x, y, z axes. We then quantize the spins along B 0 . We thus where where e is the electronic charge, a † i,m I creates a nucleus at the atomic position i of spin I in the subspin state m I = −I, −I + 1, . . . , I − 1, I, b † i,q,σ creates an electron orbiting that nucleus at site i with energy ǫ q and spin-1/2 eigenstates indexed by σ = ±1, where q ∈ (n, ℓ, m) is nominally its weak-spin-orbit local electron orbital quantum number set [or its fully relativistic set (n, j)], ψ † j,σ (r j ) creates an electron or hole with spin eigenstate σ = ±1 at position r j in the j th conduction band ω n = µ n · B 0 , ω e = µ e · B 0 , ω ′ j,e = µ e · g j · B 0 are respectively the Zeeman energies for the probed nucleus, local orbital electrons, and conduction electrons, respectively, where µ n is the nuclear magneton for the probed nucleus (the magnitude of which can be positive or negative), |µ e | = µ B is the Bohr magneton, g j · B 0 defines the quantization axis direction for the anisotropic but assumed diagonal g j tensor in the j th of the N b conduction bands with effective mass m j,ν in the ν th spatial direction, A j,ν (r j ) is the magnetic vector potential at the position r j of the conduction electron in the ν th band, the time independent magnetic induction B 0 = ∇ j,ν × A j,ν is the same in each band, i = √ −1, and we seth = 1. Here we use the previous notation 43 , but rearrange the terms in the overall Hamiltonian in order to properly take account of both the time t and temperature T dependencies essential for probing the superconducting state. We note that for integer or half-integer I, the nuclei would normally be expected to obey Bose-Einstein or Fermi-Dirac statistics, but since different nuclei correspond to different atoms and do not come in contact with one another, that statistics is not expected to be an important feature of the Knight shift.
Equation 1 is the extension to arbitrary nuclear spin I of the bare Hamiltonian studied previously, except that H cond,0 was the time independent part of H A,2 43 . We note that for a diagonal g j tensor, As a starting point, we assume B 0 is uniform in the probed material, but when the material goes into the superconducting state, and B 0 is in an arbitrary direction with respect to the crystal axes, this is only true at the upper critical field H c2 above which the superconductor becomes a normal metal 17,45 . However, in the mixed state for which the time-independent part of the applied

B. The Time-Independent Interaction Hamiltonian H int
We write the time-independent interaction part H i of the Hamiltonian as where where A σ I,m I = I(I + 1) − m I (m I + σ), and depending upon what is calculated, the superconducting pairing interaction may be written either in real space as or in momentum space as Although it appears at first sight to be easier to extend the calculation of the Knight shift into the BCS superconducting state by using H pos sc in order to include the Zeeman terms, we have included the momentum-space pairing interaction H mom sc for p-wave superconductors in magnetic fields 18,42 , for which the simplest single-band parallel-spin pairing interaction We note that σ = ±1 present in A σ I,m I corresponds to the correct matrix elements for raising and lowering the m I value and also corresponds to our description of the spin-1/2 electron spins 43 .
Of course, m I and σ are restricted by −I ≤ m I , m I + σ ≤ I. The first three of these terms were presented previously 43 , except for a slightly different normalization factor proportional to N b , and respectively represent the hyperfine interaction between the nuclear and surrounding orbital electrons, the effective local electron correlation interaction, and the effective Anderson interaction that allows an orbital electron to leave a local atomic site and jump into a conduction band 47 .
The last term H sc is responsible for superconducting pairing, and in the form presented allows for pairing between electrons or holes in different bands and with either the same (σ ′ = σ) or different (σ ′ = −σ) spins. In most superconductors, interband pairing is generally considered to be less important than is intraband pairing, but such complications might be important in cases such as Sr 2 RuO 4 , for which two of the bands are nearly identical. For standard BCS pairing, we would , at least in the standard approximation. For parallelspin p-wave superconductors, one cannot assume the paired electrons are at the same location, but different approximations have been found to give reliable results for the upper critical induction in ferromagnetic superconductors 18,32,42,48,49 .
The crucial part of a magnetic resonance experiment arises from the time-dependent field transverse to the stronger constant magnetic field. In a conventional NMR experiment, the timedependent induction B 1 (t) oscillates in the plane normal to the strong, constant magnetic induction ]φ}, whereθ = (cos θ cos φ, cos θ sin φ, − sin θ) andφ = (− sin φ, cos φ, 0) in the same Cartesian coordinates, and in order not to get confused with the time contours, we may choose B 1 (t 0 ) to be alongθ. This is the classic way to obtain a resonance in the power spectrum associated with flipping an electron or proton spin from up to down, or in a spin I nucleus, to obtain a regular pattern of resonance frequencies associated with changes in the multiple Zeeman-like nuclear spin levels. Since one generally takes B 1 ≪ B 0 , this classic case is generally adiabatic 37,38 , and is the simplest case to treat analytically.
For the above classic NMR case of a single angular frequency ω 0 in B 1 (t), we then have where Ω n = µ n B 1 , Ω e = µ e B 1 , and Ω ′ j,e = µ e · g j · B 1 , and A σ I,m I is given by Equation 11. 43 .

III. THE KELDYSH CONTOURS
Following Haug and Jauho 44 and with regard to the contours, Rammer and Smith 50 , we may treat the time and temperature dependence of the particles together in the same formulas, as long as we properly order the time integrations around the appropriate contours. When there is only one type of particle, which we take to be a fermion, the fields at the three-dimensional positions r 1 and r 1 ′ evolve in time according to the simple Hamiltonian H 0 , with its density matrix also involving only H 0 (and the number operatorN in the grand canonical and the Green function is given by the two contour integration paths C and C int , where the operators in H ′ H 0 (τ ) and H int,H 0 (τ ) evolve in time via the easily soluble Hamiltonian H 0 . The Greek letter τ implies that one needs to consider it as being just above or just below the real axis until the contours merge into one. Roman lettering (t) indicates the integrals are on the real axis.
For the case of the time-dependent Hamiltonian H ′ (t) making adiabatic changes, as in the case considered here, the two contours C int and C shown in Figure 1 merge into the left contour C. We use the standard short-hand notation where the particle type, its position, and its energy are still undefined. In order to treat the various time orderings on the contour C, we define in the Heisenberg representation for the full Hamiltonian H, where the ordinary time-ordering operator T and inverse-time-ordering operatorT describe opposite directions in time, as sketched by lines C 1 and C 2 in Figure 2. We note that , so only three of these Green functions are linearly independent.
Here we need to describe three particles, all of which are effectively fermions.

A. Bare Nuclear Contour Green Functions
We first consider the nuclei, which are assumed not to interact with one another, as they are fixed in the crystalline locations, which if there is more than one isotope of a particular type with spin I, may be at a random selection of crystalline sites. In the presence of the constant magnetic induction B 0 , it can be in any one of the 2I + 1 manifold of nuclear Zeeman states, but because each of these local states at the probed nuclear site i can be either unoccupied or singly occupied, this manifold of local nuclear spin states is precisely that of a fermion with (2I + 1) states. Its occupancy in the local state m I on site i in the grand canonical ensemble is therefore easily seen to be where ǫ m I = −ω n m I and µ ncp is the nuclear chemical potential. We then have for the bare nuclear Green functions with H = H 0,n , and G There are only three distinct bare neutron Green functions. This is also true when interactions are included 44 . Although it is somewhat surprising that the nuclear occupation density has the Fermi function form even for integral spin I, this is due to the nuclear Zeeman magnetic level occupancy being either 0 or 1 for each level on a given probed nuclear site.

B. Bare Orbital Electron Contour Green Functions
For the surrounding orbital electrons, we assume that the magnetic induction B 0 + B 1 (t) is sufficiently weak that it does not change the electronic structure of the orbital electrons or lead to transitions between the orbital electron states and energy levels. Thus, we assume that it only interacts with the orbital electron spins. We note that this is expected to be a good approximation, as the total charge of the nucleus plus its orbital electrons is on the order of one electron charge where µ ecp is the orbital electron chemical potential. It is then easy to show that the bare orbital electron Green functions are In the j th band, the conduction electrons behave as free particles with wave vector k j,|| along the transformedẑ direction, but propagate in Landau orbits indexed by the harmonic oscillator quantum number n j . Thus, we need to requantize the conduction electron fields asψ j,n j ,σ (k j,|| ).
We therefore rewrite the transformedH cond,0 as where n j = 0, 1, 2, . . . are the two-dimensional simple harmonic oscillator quantum numbers of the Landau orbits for band j, k j,|| are the free-particle dispersions along the transformed induction direction, β j (θ, φ) = [g 2 j,xx m j,1 sin 2 θ cos 2 φ + g 2 j,yy m j,2 sin 2 θ sin 2 φ + g 2 j,zz m j,3 cos 2 θ] 1/2 , and is the spatially-transformed Landau degeneracy for a single electron in the j th band. We may then write the conduction electron occupation number as n cond j,k j,|| ,n j ,σ = 1 e β[ε j (n j ,k j,|| )−σω ′ j,e /2−µ cond,cp ] + 1 (41) for a diagonal g j tensor describing the spins of the j th conduction band, where µ cond,cp is the chemical potential of the conduction electrons. We note that all of the bands that cross this conduction electron chemical potential make important contributions to the Knight shift.
The bare conduction electron Green functions can then be found (43) and the contour-ordered and inverse-contour-ordered bare conduction electron Green functions are obtained as in Equations 25 and 26, so that there are only three independent bare conduction electron Green functions.

IV. TRANSFORMATIONS IN TIME OF THE OPERATORS WITH THE BARE HAMILTONIAN
In order to proceed with the perturbation expansions, we first need to transform the nuclear, orbital electronic, and conduction electronic operators in real time, using the bare Hamiltonian in Equation 1. For the nuclear and orbital electronic operators, this is elementary. We have and integrating the elementary differential equation, we immediately find in order to use this in Equation 16. The quantity a † i,m I +σ (t) is instantly obtained from the Hermitian conjugate of Equation 52 and letting m I → m I +σ, and hence a † i,m I +σ (t) = e −iωn(m I +σ)(t−t 0 ) a † i,m I +σ (t 0 ), so that the time-transformed Equation 16 becomes Similarly, for the local orbital electron operators, we have and For the spatially-transformed conduction electron operators, ψ j,n j ,σ (k j,|| , t) = e iH cond,0 (t−t 0 )ψ j,n j ,σ (k j,|| , t 0 )e −iH cond,0 (t−t 0 ) = e −i[ε j (n j ,k j,|| )−σω ′ j,e /2](t−t 0 )ψ j,n j ,σ (k j,|| , t 0 ), (54) so that the time-transformed Equation 44 becomes We note that all three of these transformed Hamiltonians correspond to spin-dependent external field interactions, where the fields are Then, we time transform the difficult (interaction) parts of the full Hamiltonian. The hyperfine and local electron-electron interactions are elementary to transform. We obtain H e,int,H e,0 (t) = 1 2 i,q,σ U qni,q,σ (t 0 )n i,q,−σ (t 0 ).
Of these, only the transverse (D x ) part of the hyperfine interaction picks up a time dependence.
Before we time transform the remaining two interaction Hamiltonians, we first spatially transform the conduction electron operators in the presence of the magnetic field necessary for the NMR experiment. Then we rewrite H e,cond in terms of the spatially-transformed conduction electron fields, which after time-transformation with respect to H e,0 andH cond,0 becomes H e,cond,H e,0 +H cond,0 (t) = i,q,j The most important Hamiltonian for the Knight shift in a superconductor is the pairing interaction H sc , which is position space was written in Equation 10. Since in a Knight shift measurement, the experimenter first measures the Knight shift in the applied field H(t) and hence the induction while the superconductor is in its normal (metallic) state, and then cools the material through its superconducting transition at T c (H), it is clear that the correct formulation for the superconducting pairing interaction must be in momentum space, and more precisely, to account for the pairing of the electrons (or holes) while they are in Landau orbits in the normal state. We therefore first rewrite H sc in a fully spatially-transformed magnetic-induction-quantized form that allows for different pairing interactions, such as those giving rise to various types of spin-singlet and spin-triplet superconductors arising from a multiple-band metal. As a start to understand the orbital motion of the paired superconducting electrons (or holes), we first assume the standard approximation for the evaluation of the upper critical field H c2 that the paired particles of combined charge 2e move together in Landau levels 18,42,52,53 . For a BCS superconductor for there is no need to transform the wave vector dependence of the pairing interaction due to the Landau orbits formed by the strong applied field 18 . Such pairing interactions will be considered elsewhere. Thus, we begin by considering only the simplest case of isotropic intraband pairing of equivalent strength in all of the bands, which after spatial transformation due to the magnetic induction may be written as and transforming this in time using H cond,0 , we havẽ which is independent of t.

V. DYSON'S EQUATIONS FOR THE GREEN FUNCTIONS
For a system with continuous position variables r, the contour C Dyson equation for an adiabatic time-dependent interaction for which C int → C can be written as 44 where Σ is the self-energy and U is an external field. By carefully keeping the order of the times in going about the contour C, one can analytically continue the integrals off the real axis to the real axis. We first need to define the retarded and advanced Green functions, which are Then, letting C dτ 2 G 0 (1, 2)G(2, 1 ′ ) be represented by C = AB, one can analytically continue the appropriate contour-ordered Green function components on the real axis, so that where the integration C dτ 2 → ∞ −∞ dt 2 . Similarly, by representing the double contour integral C dτ 2 C dτ 3 G 0 (1, 2)Σ(2, 3)G(3, 1 ′ ) by D = ABC, one can analytically continue these contour integration paths to the real axis, obtaining 44 We then implement the three Dyson equations for the nuclear, local orbital electron, and conduction electron Green functions. We first consider the Dyson equation for the nuclear Green function. In this case, there are two terms to consider: the external field U n given by Equation 56, and the hyperfine interaction given by Equation 59. However, the hyperfine interaction does not involve two different times, as in the self-energy, which is analogous to the exchange interaction in the electron gas with electron-electron Coulomb interactions. It is instead analogous to the direct interaction with a fermion loop, but in this case, the fermion loop is for the local orbital electrons.
As first shown by Hall and Klemm, the leading self-energy diagrams for the Knight shift and the linewidth changes in a metal at T = 0 are shown in Figure 3. Since the upper critical field has been obtained for anisotropic superconductors with a variety of pairing interactions 18,42,52,53 and also that the most rapid temperature variation, a discontinuity and the gap function, which in real space for intraband pairing only is As discussed in the next subsection, in order to include the temperature dependence of the normal state of the nuclei, the local orbital electrons, and the conduction electrons, we need to quantize the conduction electrons in momentum space and Landau orbits, as was done for the bare conduction electron Green functions in Equations 42 and 43. Although this was never done in upper critical field calculations 17,18,32,42,48,49,52,53 , the reasons given for not doing it were that impurities would broaden the levels, smearing out the Landau level spacings 18,52 . However, with the present quality of some materials, that argument should be reexamined. More important, in order to calculate the upper critical induction B c2 , one requires the paired electrons (or holes) to be in Landau levels 17,18,32,42,48,49,52,53 . However, as discussed in the next section, it is not clear that electrons (or holes) will only pair with other electrons (or holes) in the same Landau orbit. With multiple bands, an electron in one single-particle Landau level corresponding to one conduction band could in principle pair with another electron in a different Landau level corresponding to another band.
So one will have to make some assumptions about the pairing processes to simplify the calculations.
But to get a preliminary microscopic idea of how K S (T ) picks up its T dependence below T c , we will first revisit the Gor'kov procedure for deriving the Ginzburg-Landau equations in real space.
In the standard real-space finite temperature formalism, the Gor'kov equations of motion generalized to include multiple ellipsoidally anisotropic bands and their Zeeman energies without interband pairing are 18 where ω n = (2n + 1)π/β is the fermion Matsubara frequency. Letting G (0) j,j ′ ;σ,σ ′ (r j , r ′ j ′ , ω n ) be the solution for the G function in the normal state with ∆ = 0, one can rewrite the above equations for G and F † in the finite temperature formalism as to order ∆ 2 . One can then substitute this in the equation for F † , multiply by the pairing interaction, and obtain a self-consistent equation for ∆ to order ∆ 354, 55 . The coefficients of the two terms proportional to ∆ define the upper critical induction B c2 18,32,42,48,49,52,53 . Functionally integrating the cubic equation with respect to ∆ j,σ,σ ′ (r j , r ′ j ) and by neglecting the field dependence of the resulting term proportional to |∆| 4 , one can obtain the generalized Ginzburg-Landau free energy.
We note that even if one completely neglects the field dependence of the term of order ∆ 3 in the Gor'kov expansion of F † for ∆(r), it is easy to see that this procedure will lead to the following phenomenological general result: where the expectation value is in the grand canonical ensemble, so that the mean-field effective Hamiltonian for the conduction electrons in the superconducting state becomes dk j,|| 2π ψ † j,n j ,σ (k j,|| )[ε j (n j , k j,|| ) − µ cond,cp − σω ′ j,e /2]ψ j,n j ,σ (k j,|| ) where we have included the chemical potential of the conduction electrons. Note that we assume the total momentum of the paired electrons (or holes) is zero, as both are assumed to be on opposite sides of the same Landau orbit, and have opposite momenta in the direction normal to the plane of the Landau orbits. This effective quadratic Hamiltonian can then be diagonalized by a standard Bogoliubov-Valatin transformation 56 , letting ψ j,n j ,↑ (k j,|| ) = u j,n j ,k j,|| γ j,n j ,↑ (k j,|| ) + v j,n j ,k j,|| γ † j,n j ,↓ (k j,|| ), where we require the γ operators to obey independent fermion statistics. Using the standard transformation procedure to eliminate the off-diagonal terms, we then obtain and the diagonalized superconducting Hamiltonian becomes where |∆ j | 2 = ∆ j,−σ,σ ∆ † j,−σ,σ is positive definite for each j value. We note that the quasiparticle dispersions inH sc,cond are nearly identical to the BCS quasiparticle dispersions, as they do indeed have a real energy gap 2|∆|, but there is in addition an effective Zeeman term arising from the difference in the spin up and spin down quasiparticle energies, leading to a magnetic gap function.
Thus, the self-consistent expression from Equation 80 for ∆ j,−σ,σ becomes The road ahead to construct the first microscopic theory of the Knight shift in a superconductor of any type is now clear. The conduction electrons must be quantized in Landau orbits, and this can be done for any number of ellipsoidally anisotropic electron or hole bands, as outlined above.
The procedure will be extended for our model of multiple ellipsoidal bands with the Zeeman cou- and γ † j,n j ,σ (k j,|| ), and will modify Equation 58. Then, the Keldysh contour method can be used to perform a microscopic theory of K S (T ) in the superconducting state of an anisotropic, multiband BCS superconductor. Since the conduction electrons are transformed into non-interacting quasiparticles in the superconducting state, the self-energy Σ(2, 3) in Dyson's equation will only apply to the orbital electrons via the Hubbard interaction U q . All other interactions reduce to effective external fields. After a detailed microscopic evaluation of K S (T ) using the contour-extended version of the diagram pictured in Figure 3(a), special attention will be directed at the conditions for a near vanishing of b(B 0 ), which could lead to a T -independent K S (T ), even for a "conventional" superconductor. The linewidth changes can be evaluated in the superconducting state from the contour-extended versions of the diagrams pictured in Figures 3(b,c). Eventually, other superconducting pairing symmetries could also be studied with this technique, although the pairing interaction would have to be transformed as above, including the Landau orbits. Eventually, this could be done for charge-density and spin-density wave systems, for which no theory of the Knight shift is presently available. We note that 2H-TaS 2 has a nodal charge-density wave below 75 K, with a presumably s-wave superconducting state entering below 0.6 K 17,57 , which is very similar to the complex situation in the high-temperature superconductor Bi 2 Sr 2 CaCu 2 O 8+δ , in which the nodal pseudogap (probably charge-density wave) regions and isotropic s-wave superconducting regions break up into spatial domains 58 . These results are consistent with previous c-axis twist Josephson junction experiments on that material 59 . Although the NMR linewidths in that material are too broad to perform Knight shift measurements, they could be done on other materials, such as the dichalcogenides, and also in improved YBa 2 Cu 3 O 7−δ samples.

VII. DISCUSSION AND CONCLUSIONS
We have outlined a procedure to obtain a microscopic theory of the Knight shift in an anisotropic Type-II superconductor. This was based upon the Hall-Klemm microscopic model of the effect at T = 0 43 , for which multiple anisotropic conduction bands of ellipsoidal shapes were included. We considered the simplest magnetic resonance case of B(t) = B 0 + B 1 (t) with |B 1 | ≪ |B 0 | and B 1 · B 0 = 0 with B 1 (t) oscillating at a single frequency ω 0 . For this simple case, the time changes to the system are adiabatic, so that the interaction Keldysh contour C int effectively coincides with contour C depicted in Figure 1, and the integrations can be analytically continued onto the real axis. The procedure can effectively treat any nuclear spin value I. The conduction electrons (or holes) were quantized in Landau orbits in the applied field in the normal state, and the Hamiltonian for a generalized anisotropic, multiband BCS type-II superconductor was diagonalized, allowing for a full treatment of the superconducting state.
We emphasize that by quantizing the superconducting order parameter in the presence of a strong time-independent magnetic induction B 0 , the energy spacings of the Landau orbits can depend strongly upon the direction of B 0 . At very weak B 0 values, the Landau levels primarily give rise to overall anisotropic constant backgrounds of K(T ) and K S (0), with K S (T ) being predominantly governed by the anisotropic Zeeman interactions and D z . But for sufficiently strong B 0 values in anisotropic materials with layered or quasi-two-dimensional anisotropy, the spacings between the Landau energy levels depends strongly upon the direction of B 0 , so that K S (T ) could become independent of T for T ≤ T c , as first observed for B 0 ||ĉ in YBa 2 Cu 3 O 7−δ 19,20 . Such behavior could also arise for quasi-one-dimensional materials in all B 0 directions, although to different degrees for B 0 parallel and perpendicular to the most conducting crystal direction.
Since the crucial interaction for the Knight shift is the hyperfine interaction between the probed nuclei and their surrounding orbital electrons, the symmetry of this interaction can be very important. Generally, the hyperfine interaction can arise from the electrons in any of the orbital levels.
where Γ n = [(ω 0 − ω n ) 2 + Ω 2 n ] 1/2 is the nuclear resonance function and the constants C m I m ′ I depend upon the initial conditions 43 . Those authors found this form to hold for I = 1/2, 1, 3/2, and in second quantization, up to I = 2, so it is likely to hold for arbitrary I. In the adiabatic regime, we have ω 0 ≪ ω n 37,38 , so that there will be a manifold of geometrical phases that will arise with higher I values.
We remark that it is possible to generalize this treatment to more complicated B 1 (t) functions, such as a periodic function of square-wave or triangle-wave shape. This can be represented as a Fourier series, but if the primary angular frequency is ω 0 , terms of higher multiples n of ω 0 can be present, some of which would violate the adiabatic requirement that they be much smaller than the Zeeman energy spacings. Hence, this experiment would make some amount of non-adiabatic changes that could drive the system out of thermal equilibrium, and the two contours discussed above would not coincide, greatly complicating the analysis.