Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension

Trial wave functions that can be represented by summing over locally coupled degrees of freedom are called tensor network states (TNSs); they have seemed difficult to construct for two-dimensional topological phases that possess protected gapless edge excitations. We show it can be done for chiral states of free fermions, using a Gaussian Grassmann integral, yielding $p_x\pm i{p}_y$ and Chern insulator states, in the sense that the fermionic excitations live in a topologically nontrivial bundle of the required type. We prove that any strictly short-range quadratic parent Hamiltonian for these states is gapless; the proof holds for a class of systems in any dimension of space. The proof also shows, quite generally, that sets of compactly supported Wannier-type functions do not exist for band structures in this class. We construct further examples of TNSs that are analogs of fractional (including non-Abelian) quantum Hall phases; it is not known whether parent Hamiltonians for these are also gapless.

Figure 1

Single-particle entanglement spectrum of the $p-ip$ state $|{\psi }_D\rangle$ on an infinite cylinder of circumference $L=200$. We take antiperiodic boundary conditions for the fermions around the cylinder, such that $k_y\in \frac{2\pi }{L}(\mathbb{Z}+\frac{1}{2})$. Here, $|{\kappa }_1|=1$, and we vary the parameter $\lambda \ge 0$.

Figure 2

Same as in Fig. 1, with $\lambda$ fixed ($\lambda =0.8$). We vary $|{\kappa }_1|$.

Figure 3

Particles on the square lattice ${\mathbb{Z}}^2$ (in green) are created by operators $c_{\mathbf{x},\alpha }^{†}$ acting on the vacuum $|0\rangle$, generating a Hilbert (Fock) space of physical states for each site ${\mathcal{H}}_{\mathbf{x}}$. One associates one tensor $T_{\mathbf{x}}\in {\mathcal{H}}_{\mathbf{x}}\otimes V^h\otimes {\left(V^h\right)}^{*}\otimes V^v\otimes {\left(V^v\right)}^{*}$ per site $\mathbf{x}$. The spaces $V^{h,v}$ are finite-dimensional vector spaces, there is one copy of the space $V^{h,v}$ and its dual ${\left(V^{h,v}\right)}^{*}$ per (horizontal or vertical) edge of the lattice, represented by solid dots (left figure). Tracing over all the spaces $V^{h,v}$, namely, using the canonical evaluation map $V\otimes V^{*}\mapsto \mathbb{C}$ on each edge, we get a state in the many-body Hilbert space ${⨂}_{\mathbf{x}\in {\mathbb{Z}}^2}{\mathcal{H}}_{\mathbf{x}}$. In the main text, we used a different version of a TNS, with auxiliary variables living on the sites, as in the right figure; it can, however, be reformulated as a usual TNS.