For every Matrix Product State (MPS) one can always construct a so-called parent Hamiltonian. This is a local, frustration free, Hamiltonian which has the MPS as ground state and is gapped. Whenever that parent Hamiltonian has a degenerate ground state space (the so-called non-injective case), we construct another 'uncle' Hamiltonian which is also local and frustration free, has the same ground state space, but is gapless, and its spectrum is R + . The construction is obtained by linearly perturbing the matrices building up the state in a random direction, and then taking the limit where the perturbation goes to zero. For MPS where the parent Hamiltonian has a unique ground state (the so-called injective case) we also build such uncle Hamiltonian with the same properties in the thermodynamic limit. arXiv:1210.6613v2 [quant-ph] 24 Nov 20141 The very same year, White introduced the Density Matrix Renormalization Group (DMRG) algorithm [30], which turned out to be extremely successful as a way to derive ground states from 1D gapped local Hamiltonians. It was only realized later that DMRG was a way to find the closest MPS to the target state, indicating that the family of MPS was indeed large enough to describe the low temperature physics of all gapped 1D systems. This was finally proven by Hastings in [13] (see also [28]) and later, with exponentially better parameters, by Arad, Kitaev, Landau and Vazirani in [3].
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 (PEPS), and can be applied to a large class of two-dimensional systems to obtain gapless "uncle Hamiltonians".Introduction.-Since its introduction by Wen in the 80's, topological order has become a central subject of research both in the condensed matter and quantum information communities. The toric code, a many-body spin state originally introduced by Kitaev in the context of topological quantum computing [1], represents a paradigmatic example of a state with topological order. It is the ground state of a local, frustration free Hamiltonian H TC defined on a two-dimensional lattice, whose degeneracy depends on the topology of the space on which it is defined. This Hamiltonian is gapped, and it exhibits (abelian) anyonic excitations. The toric code also possesses long-range entanglement (i.e., it cannot be created by local unitary operations out of a product state), and its entanglement entropy contains a universal part which can serve as a signature of its topological properties. All these properties are robust against local perturbations [2,3]. Apart from that, it can be considered as an error correcting code with non-local encoding but local syndroms, and might therefore be useful as a quantum memory or for fault tolerant quantum computing.The toric code can also be efficiently described in the language of tensor networks. As other states with topological order, it is a Projected Entangled Pair State (PEPS) of very low bond dimension, D = 2 [4,5]. PEPS generalize Matrix Product States (MPS) [6,7] to spatial dimensions higher than one, obey the area law for the entanglement entropy, and are believed to efficiently represent the ground states of local spin and fermionic Hamiltonians in lattices [8,9]. Conversely, for any PEPS one can construct a frustration free parent Hamiltonian for which it is the ground state [5], which allows us to relate a given exotic quantum many-body state to physical Hamiltonians. In fact, H TC is exactly such a parent Hamiltonian for the toric code, and using this construction in the PEPS formalism, one can readily uncover some of its most distinct properties [10]. In the same way, one can build parent Hamiltonians for many other strongly correlated states, such as string-net models [11], the AKLT state [12], resonating valence bond states, and others. In most of these cases, the resulting Hamiltonians are gapped above the ground state space, which makes them robust against local perturbations [13].In this paper, we introduce an alternative way to construct Hamiltonians corresponding to MPS and PEPS, which we term uncle Hamiltonians. The uncle Hamiltonian differs sig-
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