We show that projected entangled-pair states (PEPS) in two spatial dimensions can describe chiral topological states by explicitly constructing a family of such states with a nontrivial Chern number. They are ground states of two different kinds of free-fermion Hamiltonians: (i) local and gapless; (ii) gapped, but with hopping amplitudes that decay according to a power law. We derive general conditions on topological free-fermionic projected entangled-pair states that show that they cannot correspond to exact ground states of gapped, local parent Hamiltonians and provide numerical evidence demonstrating that they can nevertheless approximate well the physical properties of topological insulators with local Hamiltonians at arbitrary temperatures.
We show that projected entangled-pair states (PEPS) in two spatial dimensions can describe chiral topological states by explicitly constructing a family of such states with a nontrivial Chern number. They are ground states of two different kinds of free-fermion Hamiltonians: (i) local and gapless; (ii) gapped, but with hopping amplitudes that decay according to a power law. We derive general conditions on topological free-fermionic projected entangled-pair states that show that they cannot correspond to exact ground states of gapped, local parent Hamiltonians and provide numerical evidence demonstrating that they can nevertheless approximate well the physical properties of topological insulators with local Hamiltonians at arbitrary temperatures.
We investigate the topological character of lattice chiral Gaussian fermionic states in two dimensions possessing the simplest descriptions in terms of projected entangled pair states (PEPS). They are ground states of two different kinds of Hamiltonians. The first one, Hff, is local, frustration free, and gapless. It can be interpreted as describing a quantum phase transition between different topological phases. The second one, Hfb, is gapped, and has hopping terms scaling as 1/r3 with the distance r. The gap is robust against local perturbations, which allows us to define a Chern number for the PEPS. As for (nonchiral) topological PEPS, the nontrivial topological properties can be traced down to the existence of a symmetry in the virtual modes that are used to build the state. Based on that symmetry, we construct stringlike operators acting on the virtual modes that can be continuously deformed without changing the state. On the torus, the symmetry implies that the ground state space of the local parent Hamiltonian is twofold degenerate. By adding a string wrapping around the torus, one can change one of the ground states into the other. We use the special properties of PEPS to build the boundary theory and show how the symmetry results in the appearance of chiral modes, and a universal correction to the area law for the zero R\'enyi entropy.
This thesis is divided into two mainly independent parts: In the first part, we derive a criterion to determine when a translationally invariant Matrix Product State (MPS) has long range localizable entanglement, which indicates that the corresponding state has some kind of non-local hidden order. We give examples fulfilling this criterion and eventually use it to obtain all such MPS with bond dimension 2 and 3. In the second part, we show that Projected Entangled Pair States (PEPS) in two spatial dimensions can describe chiral topological states by explicitly constructing a family of such states with a non-trivial Chern number. We demonstrate that such free fermionic PEPS must necessarily be non-injective and have gapless parent Hamiltonians. Moreover, we provide numerical evidence that they can nevertheless approximate well the physical properties of Chern insulators with local Hamiltonians at arbitrary temperatures. We also construct long range, topological Hamiltonians with a flat energy spectrum for which those PEPS are unique ground states. As for non-chiral topological PEPS, the non-trivial, topological properties can be traced down to the existence of a symmetry on the virtual level of the PEPS tensor that is used to build the state. We use the special properties of PEPS to build the boundary theory and show how the symmetry results in the appearance of chiral modes, a ground state degeneracy of the parent Hamiltonian on the torus and a universal correction to the area law for the zero Rényi entropy. Finally, we show that PEPS can also describe chiral topologically ordered phases. For that, we construct a simple PEPS for spin-1/2 particles in a two-dimensional lattice. We reveal a symmetry of the PEPS tensor that gives rise to the global topological character. We also extract characteristic quantities of the edge Conformal Field Theory using the bulk-boundary correspondence.
We report on a class of gapped projected entangled pair states (PEPS) with non-trivial Euler topology motivated by recent progress in band geometry. In the non-interacting limit, these systems have optimal conditions relating to saturation of quantum geometrical bounds, allowing for parent Hamiltonians whose lowest bands are completely flat and which have the PEPS as unique ground states. Protected by crystalline symmetries, these states evade restrictions on capturing tenfold-way topological features with gapped PEPS. These PEPS thus form the first tensor network representative of a non-interacting, gapped two-dimensional topological phase, similar to the Kitaev chain in one dimension. Using unitary circuits, we then formulate interacting variants of these PEPS and corresponding gapped parent Hamiltonians. We reveal characteristic entanglement features shared between the free-fermionic and interacting states with Euler topology. Our results hence provide a rich platform of PEPS models that have, unexpectedly, a finite topological invariant, forming the basis for new spin liquids, quantum Hall physics, and quantum information pursuits.
Tensor Networks are a tool from quantum information theory to describe many-body
quantum systems. It has been shown that states arising from Tensor Networks in 1D (Matrix
Product States - MPS) accurately describe ground states of local, gapped Hamiltonians.
The same project for higher dimensional systems is currently the subject of great efforts.
However, quantum many-body systems in dimension 2 or greater can exhibit much richer
behaviour than their one-dimensional counterparts. One of the hallmarks of this is the
scarcely understood notion of topological order. It has been shown recently, that Projected
Entangled-Pair States (PEPS - Tensor Networks in 2D) can describe chiral topological
states, that lie in the ground state manifold of a certain class of gapless Hamiltonians. In
this work, we try to perturb the Hamiltonian into a gapped chiral phase. We will then
study, whether this gapped, chiral phase can be described in terms of PEPS or how the
PEPS-construction has to be modified to accommodate for this kind of behaviour.
We introduce a family of states, the fermionic projected entangled pair states (fPEPS), which describe fermionic systems on lattices in arbitrary spatial dimensions. It constitutes the natural extension of another family of states, the PEPS, which efficiently approximate ground and thermal states of spin systems with short-range interactions. We give an explicit mapping between those families, which allows us to extend previous simulation methods to fermionic systems. We also show that fPEPS naturally arise as exact ground states of certain fermionic Hamiltonians. We give an example of such a Hamiltonian, exhibiting criticality while obeying an area law.
We study the parton construction of chiral spin liquids (CSLs) using projected Gaussian fermionic entangled pair states (GfPEPSs). First, we show that GfPEPSs can represent generic spinless Chern insulators faithfully with finite bond dimensions. Then by applying the Gutzwiller projection to a bilayer GfPEPSs, spin-21 Abelian and non-Abelian CSLs are obtained for Chern number C=1 and 2, respectively. As a consequence of the topological obstruction for GfPEPSs, very weak Gossamer tails are observed in the correlation functions of the fermionic projected entangled pair state (PEPS) Ans\"atze, suggesting that the no-go theorem for chiral PEPS is universal but does not bring any practical limitation. Remarkably, without finetuning, all topological sectors can be constructed showing the expected number of chiral branches in the respective entanglement spectra, providing a sharp improvement with respect to the known bosonic PEPS approach.