Bipartite fluctuations and topology of Dirac and Weyl systems

Bipartite fluctuations can provide interesting information about entanglement properties and correlations in many-body quantum systems. We address such fluctuations in relation with the topology of Dirac and Weyl quantum systems, in situations where the relevant particle number is not conserved, leading to additional volume laws scaling with the quantum Fisher information. In the case of the $p+ip$ superconductor, we build a relation between charge fluctuations and the associated winding numbers of Dirac cones in the low-energy sector. Topological aspects of the Hamiltonian in the vicinity of these points induce long-range entanglement in real space. We specifically extract analytical expressions for the corner contributions to bipartite fluctuations. We highlight their differences and similarities with the corner functions of the entanglement entropy which characterize the underlying conformal field theory. We provide a detailed analysis of such fluctuation properties, including the role of gap anisotropy, and discuss higher-dimensional Weyl analogs.

Figure 1

Derivative of the QFID for the $p+ip$ superconductor as a function of the chemical potential for several choices of pairing order parameters ${\mathrm{\Delta }}_x$ and ${\mathrm{\Delta }}_y$, using Eqs. (38) and (39) directly in the thermodynamic limit. Phase transitions occur at $\mu =-4t$ (from trivial to topological, with one Dirac cone at $\vec{k}=0$) and at $\mu =0$ [between topological phases with opposing Chern numbers, with two Dirac cones at $\vec{k}=(0,\pi )$ and $\vec{k}=(\pi ,0)$].

Figure 2

Schematics of the different possibilities for the subsystem $\mathcal{A}$. Top left: The quadrant chosen to compute the contribution of a single corner: the obtained logarithmic coefficient correspond to $a_{\alpha \alpha }\left(\psi \right)+2a_{\alpha \alpha }\left(\frac{\pi }{2}\right)$. Top right: $\mathcal{A}$ is simply a square subregion of the square lattice. It respects the symmetries of the model. Bottom left: We can also choose a parallelogram generated by the vectors $\vec{u}$ and $\vec{v}$. The area of the unit cell they generate is one, and therefore the parallelogram is a complete cover of the lattice. We note $\psi$ the angle between $\vec{u}$ and $\vec{v}$. Bottom right: Another possible choice for $\vec{u}$ and $\vec{v}$. The parallelogram is still well defined, but as the area spanned by $\vec{u}$ and $\vec{v}$ is equal to 2, all sites in $\mathcal{A}$ are not an integer linear combination of $\vec{u}$ and $\vec{v}$ (only those in red). A properly taken continuum limit also deals with such parallelograms.

Figure 3

Comparison between the corner function of the bipartite charge fluctuations and of the entropy for Dirac fermions, as a function of the kink's angle. On the left graph we represent $\psi a\left(\psi \right)$.