Equality of Bulk Wave Functions and Edge Correlations in Some Topological Superconductors: A Spacetime Derivation

For certain systems, the $N$-particle ground-state wave functions of the bulk happen to be exactly equal to the $N$-point spacetime correlation functions at the edge, in the infrared limit. We show why this had to be so for a class of topological superconductors, beginning with the $p+ip$ state in $D=2+1$. Varying the chemical potential as a function of Euclidean time between weak and strong pairing states is shown to extract the wave function. Then a Euclidean rotation that exchanges time and space and approximate Lorentz invariance lead to the edge connection. This framework readily generalizes to other dimensions. We illustrate it with a $D=3+1$ example, superfluid ${}^3\mathrm{He}$- B, and a $p$-wave superfluid in $D=1+1$. Our method works only when the particle number is not conserved, as in superconductors.

Figure 1

(a) Wave function: The original superconductor with $\mu ={\mu }_{-}>0$ lies in the $x_1-x_2$ plane and evolves in Euclidean time $x_3$ from $-\infty$ to $0^{-}$, projecting out the ground state $|\Phi ⟩$. At $x_3=0^{+}$ the chemical potential drops abruptly to a large negative value ${\mu }^{-}$, leading to the Fock vacuum. (b) Correlation functions: a Lorentz rotation makes $x_1$ the new time and $x_3$ a the spatial coordinate along which the system has an edge at $x_3=0$. The world sheet of the edge lies in the $x_1-x_2$ plane at $x_3=0$.