We study the structure of entanglement in a supersymmetric lattice model of fermions on certain types of decorated graphs with quenched disorder. In particular, we construct models with controllable ground-state degeneracy protected by supersymmetry and the choice of Hilbert space. We show that in certain special limits, these degenerate ground states are associated with local impurities and that there exists a basis of the ground-state manifold in which every basis element satisfies a boundary law for entanglement entropy. On the other hand, by considering incoherent mixtures or coherent superpositions of these localized ground states, we can find regions that violate the boundary law for entanglement entropy over a wide range of length scales. More generally, we discuss various criteria for constructing violations of the boundary law for entanglement entropy and discuss possible relations of our work to recent holographic studies.
We study the structure of entanglement in a supersymmetric lattice model of fermions on certain types of decorated graphs with quenched disorder. In particular, we construct models with controllable ground-state degeneracy protected by supersymmetry and the choice of Hilbert space. We show that in certain special limits, these degenerate ground states are associated with local impurities and that there exists a basis of the ground-state manifold in which every basis element satisfies a boundary law for entanglement entropy. On the other hand, by considering incoherent mixtures or coherent superpositions of these localized ground states, we can find regions that violate the boundary law for entanglement entropy over a wide range of length scales. More generally, we discuss various criteria for constructing violations of the boundary law for entanglement entropy and discuss possible relations of our work to recent holographic studies.
Abstract We investigate the entanglement entropy in the ground state of the supersymmetric fermion lattice model. The Hamiltonian of this model is block diagonal in the basis of a fixed number of particles, so we compute the entanglement entropy for the blocks of the Hamiltonian separately. We explain the way of calculating the correlation matrix and the entanglement entropy for the ground state of supersymmetric Hamiltonian on diverse graphs. Then we show that the entanglement entropy exhibits the volume law scaling in various choices of the number of fermions in regular graphs. Also, we give the analytical formula for the entanglement entropy in one scalable set of strongly regular graphs in terms of their parameters, and we prove that the entanglement entropy satisfies a volume law.
Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.
We derive a general relation between the bosonic and fermionic entanglement in the ground states of supersymmetric quadratic Hamiltonians.For this, we construct canonical identifications between bosonic and fermionic subsystems.Our derivation relies on a unified framework to describe both bosonic and fermionic Gaussian states in terms of so-called linear complex structures J.The resulting dualities apply to the full entanglement spectrum between the bosonic and the fermionic systems, such that the von Neumann entropy and arbitrary Renyi entropies can be related.We illustrate our findings in one-and two-dimensional systems, including the paradigmatic Kitaev honeycomb model.While typically supersymmetry preserves features like area law scaling of the entanglement entropies on either side, we find a peculiar phenomenon, namely, an amplified scaling of the entanglement entropy ("super area law") in bosonic subsystems when the dual fermionic subsystems develop almost maximally entangled modes.
In many-body systems the extent and range of spatial quantum correlations induced by entanglement provide a great deal of information about several qualitative physical properties. One way of studying this information is to examine the scaling behaviour of the ground state entanglement entropy with respect to a scaled version of a distinguished spatial subregion. In various systems the entanglement entropy grows proportionally to the surface area of the subregion which is referred to as an area law. In this thesis we examine the connection between the scaling behaviour of the entanglement entropy and many-body localisation. In recent years it was show that a number of systems, which are known to be in the localised phase, exhibit area laws of the entanglement entropy. It is commonly expected that the entanglement entropies of delocalised ground states do not satisfy area laws, though not many examples of different scaling behaviours have been shown, yet. The aim of this thesis is to provide further examples of violations of area laws in the context of delocalised systems. In three different models we show that the entanglement entropy of the ground states grows at least like a logarithmically enhanced area law. The first part of this thesis, based on joint work with P. Müller and L. Pastur [MPS20], considers the random dimer model. Even though this non-interacting, one-dimensional model is spectrally localised, there exist critical points in its spectrum at which the localisation length diverges. We consider the ground state corresponding to a Fermi energy positioned at one of these critical energies. In the case of small disorder we show a logarithmic lower bound to the expectation of the entanglement entropy. Moreover, we proof a logarithmic lower bound to the finite-volume entanglement en- tropy at these critical points for any disorder strength. In the second part of this thesis, which is based on joint work with P. Müller [MS20], we consider a multi-dimensional continuum Schrödinger operator, which is given by a perturbation of a negative Laplacian by a compactly supported, bounded potential. We establish both an upper and a lower bound to the entanglement entropy corresponding to a positive Fermi energy. These bounds prove that the scaling behaviour of the entanglement entropy is a logarithmically enhanced area law. This is the same scaling behaviour as the one occurring in the case of free fermions, one of the few delocalised systems for which an asymptotic expansion of the entanglement entropy is known. Finally, in the third and last part, based on joint work with C. Fischbacher [FS20], we consider the finite XXZ spin chain with periodic boundary conditions in the Ising phase. We show that for each eigenvalue in the droplet band there exists at least one eigenvector such that the corresponding entanglement entropy grows at least logarithmically, provided the anisotropy parameter ∆ is sufficiently large. In addition, we show a Combes–Thomas estimate for this model, which may be of independent interest.
We calculate numerically the entanglement entropy of free fermion ground states in one-, two-, and three-dimensional Anderson models and find that it obeys the area law as long as the linear size of the subsystem is sufficiently larger than the mean free path. This result holds in the metallic phase of the three-dimensional Anderson model, where the mean free path is finite although the localization length is infinite. Relation between the present results and earlier ones on area law violation in special one-dimensional models that support metallic phases is discussed.
We investigate the scaling of the entanglement entropy in an infinite translational invariant fermionic system of any spatial dimension. The states under consideration are ground states and excitations of tight-binding Hamiltonians with arbitrary interactions. We show that the entropy of a finite region typically scales with the area of the surface times a logarithmic correction. Thus, in contrast with analogous bosonic systems, the entropic area law is violated for fermions. The relation between the entanglement entropy and the structure of the Fermi surface is discussed, and it is proven that the presented scaling law holds whenever the Fermi surface is finite. This is, in particular, true for all ground states of Hamiltonians with finite range interactions.