Topology of superconductors beyond mean-field theory

The study of topological superconductivity is largely based on the analysis of mean-field Hamiltonians that violate particle number conservation and have only short-range interactions. Although this approach has been very successful, it is not clear that it captures the topological properties of real superconductors, which are described by number-conserving Hamiltonians with long-range interactions. To address this issue, we study topological superconductivity directly in the number-conserving setting. We focus on a diagnostic for topological superconductivity that compares the fermion parity $\mathcal{P}$ of the ground state of a system in a ring geometry and in the presence of zero vs. $\Phi_{\text{sc}}=\frac{h}{2e} \equiv \pi$ flux of an external magnetic field. A version of this diagnostic exists in any dimension and provides a $\mathbb{Z}_2$ invariant $\nu=\mathcal{P}_0\mathcal{P}_{\pi}$ for topological superconductivity. In this paper we prove that the mean-field approximation correctly predicts the value of $\nu$ for a large family of number-conserving models of spinless superconductors. Our result applies directly to the cases of greatest physical interest, including $p$-wave and $p_x+ip_y$ superconductors in one and two dimensions, and gives strong evidence for the validity of the mean-field approximation in the study of (at least some aspects of) topological superconductivity.

Introduction: Topological superconductors (TSCs) [1][2][3][4] are of great interest both from a theoretical point of view and for their possible applications to quantum computation.However, most theoretical studies of TSCs rely on simple meanfield Hamiltonians that violate particle number conservation, and over the past few years several authors have expressed concerns about this approach [5][6][7][8][9].In addition, on the experimental side, the interpretation of transport measurements designed to search for Majorana fermions is not yet clear [10][11][12][13][14].To gain a better understanding of these issues, in Ref. 15 we initiated a rigorous study of TSCs in the more realistic number-conserving setting.
A key theoretical concern highlighted in Ref. 15 is the following.In the number-conserving setting one must include long-range interactions to accurately describe real charged superconductors (e.g., to avoid the Goldstone theorem of Ref. 16).However, it is not clear that the topological properties of number-conserving superconductor models with longrange interactions can be correctly captured by mean-field models that violate particle number conservation and have only short-range interactions.
In this paper we address this concern by studying the topological phases of superconductors beyond mean-field theory.Specifically, we study a diagnostic for topological superconductivity that compares the fermion parity P of the ground state [17] of a system in a ring geometry and in the presence of zero vs. π flux of an external magnetic field [1,7,18].Here, π flux is equivalent to one superconducting flux quantum Φ sc = h 2e in units with = e = 1.A version of this diagnostic exists in any spatial dimension and provides a Z 2 invariant ν = P 0 P π for topological superconductivity.The nontrivial phase corresponds to ν = −1, where a "fermion parity switch" occurs between zero and π flux.In this paper we restrict our attention to translation invariant models, and in this case we can apply the flux via a change in the boundary conditions.Finally, we note that ν is only well-defined if the Hamiltonian at zero and π flux possesses a finite parity gap, i.e., a finite energy gap between the ground state and the first excited state with opposite parity.
We study the invariant ν in a general family of numberconserving pairing models of spinless fermions in D spatial dimensions.These models are similar to the reduced BCS model [19] and to Richardson-type models [20,21], and they form a convenient starting point for the study of superconductivity in the number-conserving setting.In special cases exactly solvable versions of these models have already been used to obtain detailed results on p-wave superconductors in D = 1 and p x + ip y superconductors (and more exotic cases) in D = 2 [7,18,[22][23][24][25].In addition, Ref. 18 proved that ν = −1 in an exactly solvable model in D = 1 [26].
In this paper we prove that for gapped pairing models ν satisfies the relation where s k = sgn( k ), k is the single particle energy dispersion in the pairing model, and K 0 is the set of time-reversal invariant wave vectors in the first Brillouin zone (these satisfy k = −k + G for some reciprocal lattice vector G).This is exactly the result one would obtain for ν by studying the pairing model using a BCS-type mean-field approximation, which reduces the pairing model to a quadratic mean-field model.In addition, the product k∈K0 s k is known to be a Z 2 topological invariant for mean-field models of spinless superconductors in one and two dimensions [1,27].Therefore our result proves that the mean-field approximation correctly predicts the value of ν for these gapped pairing models and, by the definition of ν, for any model that is adiabatically connected to a gapped pairing model.For the precise statements of these results, see Theorem 1 and Corollary 1 below.Previous studies of TSCs with number conservation have mostly focused on one-dimensional systems [7,15,18,[28][29][30][31][32][33][34][35][36][37][38][39][40][41].Important exceptions include Refs.22-25, which considered exactly solvable pairing models in D = 2, and Refs.8 and 9, which considered the braiding statistics of vortices in D = 2. Our rigorous results for all D ≥ 1 should nicely complement these previous studies and serve as a useful guide for future work on this topic.

Number-conserving pairing models of spinless fermions:
We now introduce the pairing models that we study in this paper.The degrees of freedom in these models are spinless fermions c x on the sites x of a Bravais lattice Λ in D spatial dimensions (D ≥ 1), and these operators obey standard anti-commutation relations {c x , c y } = 0 and {c x , c † y } = δ xy .To avoid unnecessary complications, we also assume that the number of unit cells in the lattice in each coordinate direction is an even number.
We consider translation invariant models with two different choices of boundary condition.In the first case, we consider periodic boundary conditions in all coordinate directions, corresponding to the absence of any magnetic flux.In the second case, we consider anti-periodic boundary conditions in a single coordinate direction, and periodic boundary conditions in the remaining D − 1 coordinate directions.This corresponds to the presence of Φ sc = h 2e ≡ π flux through a single hole in the D-dimensional torus on which our system lives.In D = 1 the second case just corresponds to standard anti-periodic boundary conditions.In each case the boundary conditions determine a set K of allowed wave vectors in the first Brillouin zone (BZ).For each allowed wave vector k we can define a fermion operator in reciprocal space via the usual Fourier transform, c k = |Λ| −1/2 x c x e −ik•x , where |Λ| is the number of unit cells in the lattice.
We can always decompose K as K = K 0 ∪K + ∪K − , where the three factors appearing here are as follows.The first set K 0 consists of all time-reversal invariant wave vectors in the first BZ.These wave vectors satisfy k = −k + G for some reciprocal lattice vector G.The remaining factor K + ∪K − denotes any decomposition of the remaining wave vectors into two sets in such a way that, if k ∈ K + , then −k ∈ K − .The significance of the set K 0 is that fermions at the wave vectors in this set do not participate in the pairing interaction in the Hamiltonians that we consider.A crucial point for the remainder of the paper is that in the case of anti-periodic boundary conditions we have K 0 = ∅, i.e., there are no time-reversal invariant wave vectors in the first BZ in the anti-periodic case.
It is helpful to illustrate our notation with an example.Consider a one-dimensional system in a ring geometry with L sites and L even (and with a lattice spacing equal to 1).Then in the periodic case we have K 0 = {0, π} and The Hamiltonian for the pairing models that we consider, for either choice of boundary condition, takes the general form where n k = c † k c k , k is a single particle energy dispersion, and g kk parametrizes the interaction between the pairs at (k, −k) and (k , −k ).Note that H commutes with the total particle number operator We also absorb any chemical potential term −µN into the definition of k .
We make the following assumptions about k and g kk .First, we assume that k is an even function of Next, we assume that g kk takes the factorized form where η k is a complex function of k (the overline denotes complex conjugation), and we assume that η k = 0 for all k ∈ K + .For specific examples of models of this form, which include the cases of p-wave superconductors in D = 1 and p x + ip y superconductors in D = 2, we refer the reader to Refs. 7, 18, 22-25.We note here that, unlike those references, we do not assume any fine-tuning of k or η k that might lead to exact solvability.One benefit of the factorization assumption ( 4) is that it implies that these pairing models also take a sensible form in real space.In this case each individual sum k∈K+ η k c −k c k in the pairing term can be Fourier transformed back to real space, and one finds that in real space the pairing term becomes a long-range pair hopping term.
Finally, let ∆ 0 and ∆ π be the parity gaps of the Hamiltonian with the two choices of boundary condition/magnetic flux.The invariant ν is only well-defined if both of these gaps are non-zero.In the periodic case we trivially find that ∆ 0 = 0 if k = 0 for any k ∈ K 0 , and so in what follows we always assume that k = 0 for all k ∈ K 0 .
Main results: We now state our main results.Our first result is a formula for ν in gapped pairing models.
Theorem 1.Let H be a pairing Hamiltonian of the form (2) with non-zero parity gaps ∆ 0 and ∆ π .Then for this Hamiltonian ν satisfies the relation where s k = sgn( k ) and K 0 is the set of time-reversal invariant wave vectors in the first Brillouin zone for the case of periodic boundary conditions.
By combining Theorem 1 with the definition of P 0 and P π , we immediately obtain the following corollary.
Corollary 1.Let H 0 be a pairing Hamiltonian for which Theorem 1 applies, and let H 1 be any other translation invariant Hamiltonian such that H s = (1 − s)H 0 + sH 1 has non-zero parity gaps ∆ 0 (s) and ∆ π (s) for all s ∈ [0, 1].Then where ν 0 and ν 1 are the invariants for H 0 and H 1 , respectively.
Theorem 1 shows that for gapped pairing models the invariant ν is equal to the value that one would predict using a BCStype mean-field approximation, namely the value k∈K0 s k .The product k∈K0 s k is known to be a Z 2 topological invariant for quadratic mean-field models of spinless superconductors, and in the mean-field context it was originally derived in Ref. 1 for the case of D = 1 and Ref. 27 for D = 2 (see also Refs.42 and 43 for D > 1).Therefore, Theorem 1 and Corollary 1 prove that the mean-field approximation correctly predicts the value of ν for any translation invariant model of spinless fermions that is adiabatically connected to a gapped pairing model.Note that for Corollary 1 we do not need to assume that H 1 is number-conserving, but here we do assume that H 1 is translation invariant so that we can apply the π flux via a change in the boundary conditions (i.e., by the choice of the set K of allowed wave vectors).
One possible application of Corollary 1 is to predict a topological superconducting phase in realistic Hamiltonians.For example, H 1 might be a Hamiltonian of the form where v xy is a translation invariant density-density interaction in real space (n x = c † x c x ).If v xy favors a superconducting ground state with a finite parity gap then, following the logic of the original BCS paper [19], it is possible that H 1 is adiabatically connected to a gapped pairing Hamiltonian of the form (2). In that case we could then use Corollary 1 to predict whether H 1 supports a topological superconducting phase.
The proof of Theorem 1 relies on a lemma (Lemma 1) that we prove below.To state the lemma, let us first rewrite H in the form H = k∈K0 k n k + H, where The Hamiltonian H only contains the fermions that participate in the pairing interaction, which are exactly the fermions at wave vectors contained in K + ∪ K − .This means that, if we study H on its own, we must restrict our attention to states with a particle number less than or equal to |Λ| − |K 0 |, where |K 0 | is the number of wave vectors in the set K 0 .Note also that, in the anti-periodic case, we have be the ground state of H in the M -particle sector (or a particular ground state if H has a ground state degeneracy in that sector).Finally, let M * , with 0 ≤ M * ≤ |Λ| − |K 0 |, be the (not necessarily unique) integer satisfying The integer M * plays an important role in our proof below, and for this integer we have the following result.
Lemma 1.The integer M * can always be chosen to be even.

Proof of Lemma 1:
We start by proving Lemma 1.The proof is based on the fact that H possesses the property of reflection positivity.In this case the "reflection" one needs to consider is actually inversion in momentum space and takes k to −k.However, we show below that H can be mapped exactly to a Hamiltonian possessing Lieb's reflection positivity in spin space [44].We can then immediately apply the results of Tian and Tang [45] on spinful pairing models to prove Lemma 1.
The Hamiltonian H is written in terms of spinless fermions labeled by wave vectors k in both sets K + and K − .We now perform a change of variables to "spinful" fermions c k,σ , σ ∈ {↑, ↓}, labeled by wave vectors in K + only.To define these new variables we first decompose η k into magnitude and phase parts as η k = |η k |e iθ k .Then, for any k ∈ K + we define and one can check that these operators obey standard anticommutation relations for spinful fermions.In terms of these new operators H can be written in the form where , and where we used k = −k for k ∈ K + and also rearranged the order of the operators in the pairing term.With this change of variables H now has the form of the pairing Hamiltonians for spinful fermions studied in Ref. 45 (see their Eq.5), and so it possesses Lieb's spin reflection positivity.Therefore we can immediately apply the results of Ref. 45 (specifically,their Eq. 26) to conclude that which proves Lemma 1.
Proof of Theorem 1 (one-dimensional case): We now present the proof of Theorem 1.We first prove the theorem in the case of one dimension, as the proof in this case requires less cumbersome notation.We present the proof of Theorem 1 in higher dimensions in the Supplemental Material.For the proof in the case of D = 1, we take Λ to be a linear chain with L sites and with L even (and we take the lattice spacing equal to 1).With this choice we have K 0 = {0, π} for periodic boundary conditions (zero flux), while K 0 = ∅ for anti-periodic boundary conditions (π flux).
In the anti-periodic case, Lemma 1 and our assumption of a non-zero parity gap ∆ π immediately imply that P π = 1.Therefore it remains to compute P 0 .Our strategy for this is as follows.Consider the system with periodic boundary conditions and let E (N ) 0 be the ground state energy of H in the N -particle sector.We first show that for any N , E (N ) 0 is greater than or equal to the ground state energy E (M * * ) 0 in the particular sector with M * * particles, where the integer M * * is defined in Eq. (18).Using our assumption of a non-zero parity gap ∆ 0 , this then implies that P 0 = (−1) M * * .Finally, we use Lemma 1 to show that (−1) M * * = s 0 s π , and so P 0 = s 0 s π .Combining this relation with the fact that P π = 1 then proves Theorem 1.
To start, let |ψ (N ) 0 be the ground state of H in the Nparticle sector (or one of the ground states if there is a degeneracy in that sector).Using the fact that [H, n 0 ] = [H, n π ] = 0, we can always write |ψ where the four states |ψ , |ψ , and are all annihilated by c 0 and c π , and where the coefficients a b1b2 satisfy b1,b2=0,1 |a b1b2 | 2 = 1.Note, however, that if N = L − 1 then we must have a 00 = 0, and if N = L then we must have a 00 = a 10 = a 01 = 0. On the other hand, if N ≤ L − 2, then all of a b1b2 can be non-zero in general.In what follows we explicitly consider the case where N ≤ L−2 so that a b1b2 = 0 in general, but the inequality ( 16) also holds for N = L−1 and N = L and can be derived in the same way (but setting a 00 = 0 or a 00 = a 10 = a 01 = 0 from the start for the two cases of N = L − 1 and N = L, respectively).
Using again the fact that [H, Next, using the variational theorem for the ground state of H (e.g., ψ for any M ≤ L − 2, and b1,b2=0,1 |a b1b2 | 2 = 1, we find that To proceed further, we define h 0 = (1 − s 0 )/2 and h π = (1 − s π )/2, and note that h 0 ∈ {0, 1} and likewise for h π .The inequality (15), combined with the constraint b1,b2=0,1 |a b1b2 | 2 = 1, then implies that As we mentioned above, this inequality holds for any value of N , although in our derivation here we considered the case of The next step of the proof is to study the trial state where | ψ(M * ) 0 is the ground state of H in the M * -particle sector (or one of the ground states if H has a degeneracy in that sector).This trial state has a particle number equal to M * * , where we define In addition, the energy of this trial state is equal to From this we can see that the inequality ( 16) can be rewritten in the form On the other hand, using the variational theorem for H in the M * * -particle sector, we have the upper bound Combining this upper bound with our previous lower bound (20) yields an equality for the ground state energy in the M * *particle sector, Then our previous inequality ( 20) can be rewritten as This inequality shows that the ground state energy in any sector of fixed particle number is greater than or equal to the ground state energy in the M * * -particle sector.Combining this inequality with our assumption of a non-zero parity gap ∆ 0 , we then conclude that Finally, we come to the crucial point.Using Lemma 1, which implies that we can take M * to be even, we find that where we used s 0 = (−1) h0 and likewise for s π .This completes the proof of Theorem 1 for the case of D = 1.Discussion: We have proven that the mean-field approximation correctly predicts the value of the Z 2 topological invariant ν = P 0 P π for any translation invariant Hamiltonian that is adiabatically connected to a gapped pairing model of the form (2). We emphasize that this is a large family of models that is likely to contain many realistic models with a superconducting ground state.Our rigorous results give strong evidence that the mean-field approach is reliable, at least for the calculation of bulk topological invariants.As a topic for future work, we propose to search for evidence of exotic Majoranalike excitations in pairing models with interfaces or boundaries, as our results strongly suggest that some kind of interesting gapless excitations should appear at the boundary between two pairing models with opposite values of ν.Such a study would also have a direct impact on future experiments on TSCs, as the number-conserving pairing models are (presumably) a better description of the true experimental situation than the mean-field models.
Supplemental Material for "Topology of superconductors beyond mean-field theory" Matthew F. Lapa Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, IL, 60637, USA

I. PROOF OF THEOREM 1 IN HIGHER DIMENSIONS
In this section we prove Theorem 1 in any spatial dimension D ≥ 1.The logic of the proof is exactly the same as in the D = 1 case in the main text.To start, since K 0 = ∅ in the anti-periodic case (π flux), our assumption of a non-zero parity gap ∆ π again implies that P π = 1.So all that remains is to again calculate P 0 .
We now introduce some notation that will streamline the calculation of P 0 in this higher-dimensional case.Let H be the pairing Hamiltonian with periodic boundary conditions, let E (N ) 0 be the ground state energy of H in the N -particle sector, and let |ψ (N ) 0 be the ground state of H in the N -particle sector (or one of the ground states if H has a degeneracy in that sector).In addition, let k j , for j ∈ {1, . . ., |K 0 |}, be the wave vectors in the set K 0 , and let s j = sgn( kj ) and h j = (1 − s j )/2.
Using the fact that [H, n kj ] = 0 for all j, we can again write |ψ where the 2 |K0| states ψ We then define the trial state where | ψ(M * ) 0 is the ground state of H in the M * -particle sector (or one of the ground states if H has a ground state degeneracy in that sector).This trial state now has a particle number equal to and an energy equal to

(N − |K 0 |
j=1 bj ) b1•••b |K 0 |are annihilated by c kj for all j, and where the coefficientsa b1•••b |K 0 | satisfy b1,...,b |K 0 | =0,1 |a b1•••b |K 0 | | 2 = 1 .(1.2)As in the one-dimensional case, some of the coefficients a b1•••b |K 0 | may be zero depending on the specific value of N , and all of these coefficients can be non-zero for N ≤ |Λ| − |K 0 |.Using this expression for |ψ (N ) 0 and the same arguments as in the D = 1 case in the main text, we obtain the lower bound E

6 ) 0 =
Using the same variational arguments as in the main text, we again find that E (M * * ) E t and that E can again apply our assumption of a non-zero parity gap ∆ 0 , and the result of Lemma 1, to find thatP 0 = (−1) M * * = |K0| j=1 s j = k∈K0 s k .(1.7)This completes the proof of Theorem 1 for a general spatial dimension D ≥ 1. arXiv:2003.05948v1 [cond-mat.supr-con]12 Mar 2020