Gapless Hamiltonians for the Toric Code Using the Projected Entangled Pair State Formalism

We study Hamiltonians which have Kitaev’s toric code as a ground state, and show how to construct a Hamiltonian which shares the ground space of the toric code, but which has gapless excitations with a continuous spectrum in the thermodynamic limit. Our construction is based on the framework of projected entangled pair states, and can be applied to a large class of two-dimensional systems to obtain gapless “uncle Hamiltonians.”

Figure 1

Construction of the uncle Hamiltonian. Any MPS and PEPS tensor $A$ induces a corresponding parent Hamiltonian $h_{\mathrm{loc}}$. The uncle Hamiltonian is constructed by perturbing $A\to A+\epsilon P$, computing its parent Hamiltonian $h_{\mathrm{loc}}^{ϵ}$, and finally taking $\epsilon$ to zero. As we show, the resulting uncle Hamiltonian $h_{\mathrm{loc}}^{\prime }=\lim h_{\mathrm{loc}}^{\epsilon }$ can be very different from the parent Hamiltonian $h_{\mathrm{loc}}$.

Figure 2

Graphical description of PEPSs.