Superconducting state with a finite-momentum pairing mechanism in zero external magnetic field

In the BCS theory of superconductivity, one assumes that all Cooper pairs have the same center-of-mass momentum. This is indeed enforced by self-consistency if the pairing interaction is momentum independent. Here, we show that for an attractive nearest-neighbor interaction, this is different. In this case, stable solutions with pairs with momenta $\mathbf{q}$ and $-\mathbf{q}$ coexist and, for a sufficiently strong interaction, one of these states becomes the ground state of the superconductor. The possibility for a finite-momentum pairing state emerges only for nodal superconductors and is accompanied by a charge order with wave vector $2\mathbf{q}$. For a weak pairing interaction, the ground state is a $d$-wave superconductor.

Figure 1

(Color online) Energy $E=⟨\mathcal{H}⟩$ as a function of pair momentum $\mathbf{q}=\left(q,0\right)$ for different pairing interaction strengths $V$. Calculations were performed for a $384\times 384$ lattice with fixed electron density $\rho =0.8$ and $t^{\prime }=0.3t$. For these parameters, the finite-momentum pairing state becomes the ground state for $V>V_c\approx 2.2t$ with $q=\pi /3$.

Figure 2

(Color online) Density of states $D\left(E\right)$ of the ground-state solutions for interaction strengths $V=2t$ and $V=2.2t$, corresponding to $q=0$ and $q=\pi /3$, respectively. The other parameters are the same as in Fig. 1. The coherence peaks of the $\mathbf{q}=\mathbf{0}$ state are split due to the Van Hove singularity of the two-dimensional tight-binding dispersion.

Figure 3

(Color online) Momentum space properties of the finite-momentum pairing state with $q=\pi /3$ and the same parameters as in Fig. 2 (right panels) and for comparison the $d$-wave superconductor for $q=0$ (left panels). (a) and (b) Occupation probability function $n\left(\mathbf{k}\right)$. (c) and (d) Density of states with zero-energy $\text{Im}G\left(\mathbf{k},\mathbf{k},0-i\delta \right)$ (here, $\delta =0.04t$). (e) and (f) Pair density $P\left(\mathbf{k}\right)$.