Wave function for odd frequency superconductors

We revisit the question of nature of odd-frequency superconductors, first proposed by Berezinskii in 1974. \cite{berezinskii1974} We start with the notion that order parameter of odd-frequency superconductors can be thought of as a time derivative of the odd-time pairing operator. It leads to the notion of the composite boson condensate.\cite{abrahams1995} To elucidate the nature of broken symmetry state in odd-frequency superconductors, we consider a wave function that properly captures the coherent condensate of composite charge $2e$ bosons in an odd-frequency superconductor. We consider the Hamiltonian which describes the equal-time composite boson condensation as proposed earlier in Phys. Rev. B $\textbf{52}$, 1271 (1995). We propose a BCS-like wave function that describes a composite condensate comprised of a spin-0 Cooper pair and a spin-1 magnon excitation. We derive the quasiparticle dispersion, the self-consistent equation for the order parameter and the density of states. We show that the coherent wave function approach recovers all the known proposerties of odd-frequency superconductors: the quasi-particle excitations are gapless and the superconducting transition requires a critical coupling.

Although mainstream discussions of superconudctivity are for even-frequency pairing, there is a growing interest in understanding odd-frequency pairing. The discussion of unconventional pairing (P = 1, T = −1) was initiated by Berezenskii 1 to explain the superfluid phase of 3 He. Although his proposal of triplet odd-frequency pairing could not explain the superfluid phase of 3 He, it certainly motivated a search of other possibilities of the pairing symmetries. Balatsky and Abrahams 3 later extended the concept of odd-frequency pairing to the singlet superconductor (P = T = −1).
Although the realization of the odd-frequency pairing in current systems is still under debate, several reports consider this possibility in a a number of systems. Odd-frequency pairing in the Kondo lattice has been investigated to study superconductivity in heavy-fermion compounds, 5 The proximity effects in a superconductorferromagnet structure 6 , a normal-metal/superconductor junction 7 and diffusive normal metal/unconventional superconductor interface 8 have been attributed to oddfrequency pairing. The p-wave singlet odd-frequency pairing is argued to be a viable pairing in the coexistence region of antiferromagnetism and superconductiv-ity and/or near the quantum critical point in CeCu 2 Si 2 and CeRhIn 5 . 9 In addition, hydrated Na x CoO 2 is suggested to support an s-wave triplet odd-frequency gap 10 . Very recently, Kalas et al. 11 have argued that the bosonfermion cold atom mixture exhibits s-wave triplet oddfrequency pairing above some critical coupling at which the mixture phase separates.
Motivated by the growing interest and possibilities of odd-frequency pairing, here we address the missing part of the odd-frequency superconductivity discussion: what is the wave function of the odd-frequency superconductors? One might wonder how one can even ask this question given that superconducting correlations of two fermion operators in odd-frequency superconductor do not have an equal time expectation value? We assume (pretty safe assumption in fact) that any state, including odd-frequency superconductor, does has a many body wave function that captures superconducting correlations. Any state of matter has an associated wave function |ψ that captures the amplitude distribution of the particles forming this state. Hence we are asking exactly this question about the many body wave function of the odd frequency superconductors. Our wave function builds upon a long discussion ? on the possible order parameter and equal time composite operators that capture superconducting correlations of odd-frequency superconductors in equal time domain.
We propose a BCS-like pairing wave function for an odd-frequency superconductor, and study its consequences for the energy dispersion, superconducting order parameter, and density of states. The wave function, which describes a condensate of a spin-0 Cooper pair and a spin-1 magnon excitation, is consistent with the Hamiltonian suggested earlier in 2 to study odd-frequency superconductivity. We minimize this Hamiltonian with respect to the proposed wave function and derive an expression for the quasiparticle dispersion, a self consistent gap equation and the density of states. We find that a) the quasi-particle dispersion is gapless, b) the gap equation has non zero solution only for a critical value of the coupling, c) the density of states is finite even for an energy less than the gap energy, and d) the density of states is reduced at the gap edge compared to that of the BCS case.
Before introducing the wave function and getting into the details of the minimization of the Hamiltonian, we would like to show that P T = 1 can be obtained by taking P = T = −1 in S = 0 singlet case. Any superconducting order with translational invariance, equilibrium and broken U (1) symmetry would result in an anomalous (Gor'kov) Green's function where α, β are spin indices. We assume that the transition occurs only in a well defined representation. Thus, for S = 0 singlet pairing, we may define and for S = 1 triplet pairing, We now show the properties of F (τ, k) under P and T transformations. For S = 0 from Eq. (2), where θ τ is the Heaviside theta function.
We apply P T to this F : Going back to Eq. 5, we permute µ ↔ ν, All these properties of the Gor'kov function will be reflected in the behavior of the gap function as well. Therefore, the gap function in general is even only under simultaneous transformation: k → −k (P ) and τ → −τ (T ). We recall that P T = 1 is not only satisfied by P = +1, T = +1 but also by P = −1 and T = −1. The former describes the BCS s-wave (even-frequency) pairing whereas the latter describes odd-frequency pairing.

II. HAMILTONIAN AND WAVE FUNCTION
When the idea of the odd-frequency pairing was first formulated for the singlet superconductor, an effective spin-independent interaction mediated by phonon was considered. 4 It was realized that this kind of interaction was unphysical for the singlet pairing. 4 The problem was solved by considering spin dependent electron-electron interactions. Odd-frequency pairing posed another problem related to the selection of the order parameter. In the BCS case the order parameter is generated from the expectation value, F (r, t; r ′ , t ′ → t) = ψ(r, t)ψ(r ′ , t) . But for the odd-frequency superconductor the equal-time gap vanishes since the gap is odd in frequency. This problem was solved by taking dF (r, t; r ′ , t ′ )/dt| t→t ′ as the equaltime order parameter. 2 A Hamiltonian having a spin dependent electronelectron interaction was introduced by Abrahams et al. 2 . Using the equation of motion they derived an expression for dF (r, t; r ′ , t ′ )/dt| t→t ′ . It was shown that the equaltime condensate for odd-frequency pairing is the expectation value of the product of a pair operator and a spin excitation operator. In what follows, we adopt this approach, but for an odd frequency s-wave m = 1 triplet phase. We rewrite the Hamiltonian from Ref. 2 in the following form, where ǫ k↑↓ refers to the kinetic energy of the ↑↓ electrons measured from the Fermi energy, ω q is the magnon kinetic energy, and V kl,qp is an attractive interaction which mediates the condensation. c † kσ and c kσ creates and annihilates electrons at the state kσ. S ± describe magnon excitations. Using this Hamiltonian, we propose a BCSlike wave function and study the superconducting state.
The proposed wave function is written as where |0 represents the vacuum for both the electrons and the spin bosons. This wave function describes the superposition of the wave functions having two paired electrons with k+ q 2 and −k+ q 2 momentum and carrying opposite spins and condensed along with spin excitations (S + q ). v kq (u kq ) represent the amplitude of the occupation (or unoccupation) of these electron pairs with the spin excitation.
There are key properties that explain this particular choice of variational function: i) |ψ is a coherent state of composite bosons (c † ) that carry charge 2e; ii) this wave function describes a coherent state that has broken U (1) symmetry associated with superconducting condensate, as can be explicitly verified by using c k → exp(iφ)c k ; iii) Composite boson that condenses is not a simple Cooper pair 2 but contains two fermions and a spin-1 boson; iv) composite boson field has finite expectation value in this state and therefore |ψ is a mean field wave function for the composite condensate. The normalization of the wave function is given by, which implies that To make a next step we need to find the expectation value of the Hamiltonian (Eq. 8) with respect to the wave function (Eq. 9) and minimize it. Then we will proceed to derive the quasi-particle dispersion, density of states, and the self-consistent equation for the order parameter.

III. TOTAL ENERGY AND ITS MINIMIZATION
The calculation of each term in Eq. 8 is shown in Appendix. Using Eqs. A2, B2, C2, D1, the total energy can be written as Following the BCS method, we choose u kq , v kq such that they satisfy the normalization condition so that u kq = sin θ kq and v kq = cos θ kq / S − S + q . Then the expression for the energy reads The minimization of the energy with respect to θ kq gives which can be rewritten as We proceed by defining the two quantities ∆ and E that will turn out to be the gap parameter and the energy of a composite excitation. and where we have introduced the abbreviation Solving the normalization condition and Eq. 17, we can show that, BCS limit can be recovered at any stage of this analysis if we assume that spin correlators are factorized and have a peak at q = 0. This limit corresponds to the condensation of spin field S − S + q = S − q S + q δ q,0 . In this limit additional summation over q drops out and we recover standard BCS logarithm in selfconsistency equation Eq.(16a), along with other features of BCS solution. This limit corresponds to the factorizitation of composite boson into product ψ|c †

IV. ENERGY SPECTRUM
Unlike the BCS case, E kq is not a single-particle excitation energy. Therefore, we shall derive an expression for the energy required to excite an electron from the superconducting ground state. The excited state for an up spin is given by, where we have defined the composite creation operator We calculate the expectation value of the Hamiltonian Eq. (8) with respect to the excited state wave function Eq. (20). The details are in Appendix E. The expectation value can be expressed as Using Eq. (19b), we can rewrite the above equation as, where ∆E ↑ = ψ ↑ |H| ψ ↑ − ψ|H|ψ is the excitation energy of the up spin electrons. ∆E ↑ can also be written Doing the same for the down spin excited state ψ ↓ , we find

V. DENSITY OF STATES
The density of states (DOS) as a function of energy, N (E), is defined as, where ± corresponds to up and down spins respectively. We numerically calculate the density of states for two cases of the magnon dispersion: 1) ω q = q 2 /2M , and 2) ω q = ω 0 .. We set S − S + q = 1.0, and M = 10m * . The DOS for case 1) is shown in Fig. 1.
In Fig 1a we show the DOS as a function of energy and order parameter. We have set a magnon momentum cutoff, q c = 0.25k F . We see that the DOS can be non-zero for energies less than the superconducting gap parameter; hence the DOS is gapless. The maximum of the density of state is always at the gap edge, but it is highly reduced at the gap edge compared to the BCS case. At E = 0, the DOS can be non-zero for small ∆. The calculation for a smaller q c (not shown in the figure) shows that the gap becomes more prominent in the DOS and spectral weight is transferred to the gap edge, similar to the BCS case. Hence q c → 0 reproduces the BCS results. In Fig.  1b-d we have shown the plane cut of Fig. 1a for different values of ∆. For ∆ = 0.1 (Fig. 1b) we see that the DOS is non-zero for 0.05 < E < ∆. For ∆ = 0.04 (Fig. 1c) we see that the gap is completely closed and the excitations will be gapless. The effect is even bigger for ∆ = 0.02.
We also calculated the DOS using case 2): ω q = q 2 /2M for q c ≥ k F (the Fermi momentum) for a fixed value of ∆ = 0.1. The result is shown in Fig. 2. In this figure we can see that the DOS almost closes the gap when q c = k F . As we increase q c , the gap closes completely. Then the quasiparticle excitations become gapless. A still further increase in q c results in a finite DOS at E = 0. For q c ≥ k F there is no enhancement of the spectral weight at the gap edge.
The calculation of the DOS for ω q = ω 0 also shows the similar density of state as discussed above for both q c = 0.25k F and q c ≥ k F .

VI. SUPERCONDUCTING GAP VS COUPLING CONSTANT
The self-consistent gap equation (Eq. 16a) can be written as, where we have taken Then ∆ kq = ∆ = ∆ lp . The use of a more complicated interaction potential with a momentum dependence. would bring additional calculational complications, which would not change the nature of the results. We denote p 2 /8m * + (ω p /2) S − S + p by f (p). We first perform the energy integral in Eq. 24 as follows, where N (0) is the DOS in the normal state at the Fermi energy, g is the dimensionless coupling N (0)V /2π 2 , ǫ c (p) = ω c + f (p) and ω c = 0.2µ. If we assume spin correlator to have a sharp peak δ q,0 we recover BCS selfconsistency equation from this equation.
In the BCS case the gap equation is 1 = There is a solution for ∆ for an arbitrary small value of N (0)V due to logarithmic divergence of the integral. In our case, in the presence of the magnon, the denominator will have some nonzero value because of the non-zero magnon energy. Then the right hand side can be made equal to 1 only for some critical value of g, as can be seen in the numerical evaluation discussed below. 1.0. The cutoff for the magnon momentum is given by q c = Bk F , where B varies between 0.12 to 0.06 in equal steps of 0.02. The result is shown in Fig. 3. In this figure we can see that a nonzero order parameter requires a critical coupling. The gap equation is again given by Eq. 26 but now We solve Eq. 26 numerically for ∆ as a function of the coupling strength g. We fix the cutoff for the magnon momentum to be 0.1k F . The result for various ω 0 = Cµ where C = 0, 0.02, 0.04, 0.06, 0.08 is shown in Fig. 4. Again, the superconducting transition requires a critical coupling.

VII. MEISSNER EFFECT
The Meissner effect is one of the defining properties of a superconductor. The Meissner effect has been derived for the composite odd-frequency superconductor by Abrahams et al. 2 . Here, we summarize the derivation given in that reference A superconductor shows the Meissner effect when the paramagnetic electrodynamic response is less than the diamagnetic response. The dc response is given by, where A(q) is the Fourier transform of vector potential A(r), N is the electron density, and m is their mass. Q p ij (q) is given by, where k ± = k ± q/2. Q p can be evaluated near the critical temperature T c by perturbation in the order parameter ∆. The relevant Feynman diagrams of the current-current correlation function for the Meissner effect are used. The analytical expression for q → 0 is where, G(k, ω) and D(k, ω) are the electron and magnon propagators. The condition for the Meissner effect is given by Q p − Q n > 0, which signifies the positive superfluid density in the superconductor.
Situations with several models of the magnon propagators are discussed. If the magnon propagator is momentum independent, there is no contribution to Q ij (q) since the momentum summands are odd functions. So a momentum-dependent magnon propagator is used to discuss the Meissner effect. In the case of a static, spatially uniform magnon propagator having factorized form given by, D(q, ν) = −δ q δ ν , the Meissner effect is found (Q p − Q n > 0). For spread-out δ-functions, the sign of Q p − Q n does not change, thus a positive superfluid density with a value between zero and the BCS value. Thus, it is shown that the composite odd-frequency superconductors exhibit the Meissner effect.

VIII. CONCLUSION
In this paper we propose a BCS-like wave function for the s-wave triplet odd frequency superconductor. Our alternative approach to the odd-frequency superconductivity is based on the earlier discussion on composite bosons 2 . We present the wave function for the odd frequency superconductor |ψ = kq (u kq + v kq c † , , Eq. (9), that explicitly contains only the equal time operators and hence does not involve frequency or time domain. The wave function describes a condensate of a Cooper pair of spin S = 0 and a magnon of spin S = 1. |ψ does describe a coherent state that has nonzero expectation value for the composite boson operator, it captures the charge 2e condensate that breaks gauge symmetry and corresponds to the superconducting state. Naturally, since this |ψ describes odd-frequency superconductor, spatial parity P of this condensate is reversed compared to the even frequency pairing operators that corresponds to BCS condensate. Specifically, for the case we considered of spin triple S = 1 odd frequency condensate the spatial parity of the composite boson c † ↑ (r)c † ↓ (r)S + (r) is P = +1 and hence this order parameter does posess all the quantum numbers inherent to the odd frequency S = 1 superconductor.
We present a simplified model that captures the important features of the strong coupling theory developed for the odd-frequency superconductors and our results agree with the predictions of earlier studies: i) we show that the superconductivity requires a critical coupling. It was argued earlier 1,3 that a critical coupling is necessary in order to get the superconducting transition in the odd frequency superconductor, which we have also shown in this work. ii) we also derive the dispersion relation for the quasiparticles. We determine the density of states of the excitations. The density of states is very different from that of the BCS case. The gapless nature of quasiparticle excitations we find is also in agreement with earlier predictions. The calculation of the density of states shows that it is always higher at the gap edge but its magnitude is highly reduced compared to the BCS case. For a range of parameters, unlike the BCS case, the DOS is finite for energies less than the gap energy and at E = 0 it can be non-zero, hence odd-frequency supercoductor is gapless. We also argues how the BCS result is recovered by taking the magnon operator to condense and momentum cutoff q c = 0. Present discussion would be useful for the equal time formulation of the odd-frequency superconducting state and physical observables related to condensate. It also would be useful in elucidating the nature of condensate in odd-frequency supercondutors.
Work at Los Alamos was supported by US DOE through LDRD and BES. We also acknowledge hospitality of KITP at UC Santa Barbara.
(A1) This is a trivial identity since we can shift k → k + q 2 and get the same result.
We denote the (mn) component of the wave function as, |ψ mn = (u mn + v mn c † The expectation value of the kinetic energy of the up spin elec- (A2) Here we use the normalization condition that ψ * m ′ =k ′ ,q ′ |ψ m =k,q = δ mm ′ δ qq ′ . Then the kinetic energy of the up spin electrons is KE ↑ = kq ǫ k+ q 2 |v kq | 2 S − S + q .

APPENDIX B: KINETIC ENERGY OF DOWN SPIN ELECTRONS
Using the same argument as discussed in appendix A, we rewrite, k ǫ k↓ c † k↓ c k↓ as, (B1) Then the expectation value of the kinetic energy of the down spin electrons Then the kinetic energy of the down spin electrons is

APPENDIX C: MAGNON ENERGY
The expectation value of the magnon kinetic energy KE m = q ψ * |ω q S + q S − q |ψ can be rewritten as which gives,

APPENDIX D: INTERACTION ENERGY
In the calculation of the expectation value of the interaction energy, E I , it is easy to see that the product of only two states, kq and lp give non-zero contribution to the interaction term. All the other states are normalized to unity. Then,

APPENDIX E: ENERGY OF EXCITED STATES
The wave function of an excited state is, Using the procedure of the Appendix A, we calculate the kinetic energy of the up spin electrons ( KE ↑ ) with respect to the excited state wave function: where the restriction on k in the summation is inherited from the restriction imposed on the excited state wave function. ǫ k ′ + q 2 is due to the creation operator c † k ′ + q 2 ↑ which creates an up spin electron having unit probability of occupation in the state of momentum k ′ + q 2 . We rewrite the Eq. E2 in the following form, Proceeding similarly, we show that the kinetic energy of the down spin electrons can be written as, The kinetic energy of the magnon takes the following form, The interaction energy can be written as From Eq. 16a, we can show that We use this relation in the right hand side of Eq. E6. The second term now gives +2v * k ′ q u k ′ q S − S + q ∆ k ′ q , which, using Eq. 17a gives ∆ 2 k ′ q /E k ′ q . Then,