The entanglement entropy measures quantum correlations and it can be seen as the uncertainty on a quantum state. In one spatial dimension, the entanglement entropy scales as the boundary that divides two subsystems, so an area law has been proposed. However, the entanglement entropy diverges logarithmically at conformally invariant critical points, so the area law does not hold. The purpose of the work is to find a way to get more information about a critical state. The ground state of the Heisenberg XXZ model at criticality is analyzed by means of critical Ising eigenstates. Two ways of analysis are followed: a basis made of Ising eigenstates is built up and used to represent the XXZ ground state, then the Shannon entropy in the new basis is computed; the adiabatic evolution from the Ising ground state to the XXZ ground state. The result is that the Shannon entropy in the Ising basis scales linearly with the length of the system, while a phase transition is encountered during the adiabatic evolution. The conclusion is that there is no net gain in information after the procedure and possibly it is related to the fact the two systems stand in different phases.
The entanglement entropy measures quantum correlations and it can be seen as the uncertainty on a quantum state. In one spatial dimension, the entanglement entropy scales as the boundary that divides two subsystems, so an area law has been proposed. However, the entanglement entropy diverges logarithmically at conformally invariant critical points, so the area law does not hold. The purpose of the work is to find a way to get more information about a critical state. The ground state of the Heisenberg XXZ model at criticality is analyzed by means of critical Ising eigenstates. Two ways of analysis are followed: a basis made of Ising eigenstates is built up and used to represent the XXZ ground state, then the Shannon entropy in the new basis is computed; the adiabatic evolution from the Ising ground state to the XXZ ground state. The result is that the Shannon entropy in the Ising basis scales linearly with the length of the system, while a phase transition is encountered during the adiabatic evolution. The conclusion is that there is no net gain in information after the procedure and possibly it is related to the fact the two systems stand in different phases.
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.
Entanglement is one of the most intriguing features of quantum theory and a main resource in quantum information science. Ground states of quantum many-body systems with local interactions typically obey an “area law” which means that the entanglement entropy is proportional to the boundary length. It is exceptional when the system is gapless, and the area law had been believed to be violated by at most a logarithm over two decades. Recent discovery of Motzkin and Fredkin spin chain models is striking, since these models provide significant violation of the entanglement beyond the belief, growing as a square root of the volume in spite of local interactions. In this paper, we first analytically compute the Rényi entropy of the Motzkin and Fredkin models by careful treatment of asymptotic analysis. The Rényi entropy is an important quantity, since the whole spectrum of an entangled subsystem is reconstructed once the Rényi entropy is known as a function of its parameter. We find nonanalytic behavior of the Rényi entropy with respect to the parameter, which is a novel phase transition never seen in any other spin chain studied so far. Interestingly, similar behavior is seen in the Rényi entropy of Rokhsar–Kivelson states in two dimensions.
The interplay of the constituents of interacting many-body systems may reveal emergent properties on the macroscopic scale which are not inherent to the individual constituents. These properties are expressed in macroscopic observables \ndescribing the state — denoted as the phase of the system. Continuous phase transition between phases are generically manifested in critical behavior, for example, a divergence of a macroscopic observable. The identification of the present phase of a system and the classification of critical phenomena into universality classes are exciting challenges of condensed-matter physics. \nIn this thesis, we use entanglement entropies as macroscopic quantities for the characterization of phases of quantum matter and critical theories. For groundstates of quantum many-body systems the entanglement entropy is a measure of \nthe amount of entanglement between two subsystems. The generic dependence of the entanglement entropy on the size and shape of the subsystems is contained in the well-known boundary law — stating a scaling of the entanglement entropy with the boundary between the two subsystems. We numerically investigate how the coefficient of this dependence reflects quantum phase transitions in simple spin-half bilayer models. \nSubleading terms to the boundary law such as a logarithmic contribution provide universal numbers for the criticality of field theories. We examine free and interacting theories from an entanglement entropy perspective in order to assess the role of the coefficient of the logarithmic correction induced by corners in the subsystems. Beyond its universality, this coefficient also quantifies degrees of freedom of low-lying excitations in the conformal field theory describing a critical point. \nA constant contribution to the boundary law indicates the presence of so-called topological order in the ground state of a many-body system. This extremely useful property can also be identified in classical counterparts of entanglement entropy which we study at the example of various toric code models. To this endeavor, we have designed Monte Carlo techniques which allow for an efficient numerical computation of the constant contribution. In particular, we analyze via entanglement entropies under which conditions remnants of topological order are present in the quantum system at finite temperature and at perturbations from a magnetic field. The major motivation behind this effort is to use topological order for the robust storage of a quantum information — a basic need for the construction of quantum \ncomputers.
Entanglement is a fundamental feature of quantum mechanics and plays an important role in characterizing many-body systems and quantum phase transitions. A quantitative measure of quantum correlations between subsystems is entanglement entropy. In this thesis, we study the entanglement entropy of an interacting quantum field theory, the O(N) model. In local quantum field theories, the leading-order contribution in the scaling of entanglement entropy obeys an area law. This dominant term is governed by short-distance physics near the entangling surface and depends on the ultraviolet regularization, making it non-universal. Beyond this lead- ing scaling behavior, entanglement entropy can contain subleading universal contributions, which encode intrinsic information about the underlying quantum field theory. In this thesis, we calculate one such universal term: the correlation length coefficient r, which appears as a correction to the entanglement entropy evaluated slightly away from the quantum critical point. The analysis is carried out in D = 2 + 1 spacetime dimensions for an infinite cylindrical geometry. We study the model in the vicinity of criticality using the large-N expansion, which enables controlled analytical calculations. Our main result is that the coefficient r scales linearly with the number of field components N . This behavior is consistent with the interpretation of r as a universal quantity associated with the degrees of freedom of the theory. The calculation is based on the replica method and renormalization group arguments, which are presented in detail. Finally, we discuss the physical meaning of the result and possible methods for verifying it, and we review related results for the O(N) model in different geometries.
We study the entanglement entropy scaling of the XXZ chain. While in the critical XY phase of the XXZ chain the entanglement entropy scales logarithmically with a coefficient that is determined by the associated conformal field theory, at the ferromagnetic point, however, the system is not conformally invariant yet the entanglement entropy still scales logarithmically albeit with a different coefficient. We investigate how such an nontrivial scaling at the ferromagnetic point influences the estimation of the central charge c in the critical XY phase. In particular we use the entanglement scaling of the finite or infinite system, as well as the finite-size scaling of the ground state energy to estimate the value of c. In addition, the spin-wave velocity and the scaling dimension are also estimated. We show that in all methods the evaluations are influenced by the nearby ferromagnetic point and result in crossover behavior. Finally we discuss how to determine whether the central charge estimation is strongly influenced by the crossover behavior and how to properly evaluate the central charge.
The valence-bond structure of spin-1/2 Heisenberg antiferromagnets is closely related to quantum entanglement. We investigate measures of entanglement entropy based on transition graphs, which characterize state overlaps in the overcomplete valence-bond basis. The transition graphs can be generated using projector Monte Carlo simulations of ground states of specific Hamiltonians or using importance sampling of valence-bond configurations of amplitude-product states. We consider definitions of entanglement entropy based on the bonds or loops shared by two subsystems (bipartite entanglement). Results for the bond-based definition agree with a previously studied definition using valence-bond wave functions (instead of the transition graphs, which involve two states). For the one-dimensional Heisenberg chain, with uniform or random coupling constants, the prefactor of the logarithmic divergence with the size of the smaller subsystem agrees with exact results. For the ground state of the two-dimensional Heisenberg model (and also N\'eel-ordered amplitude-product states), there is a similar multiplicative violation of the area law. In contrast, the loop-based entropy obeys the area law in two dimensions while still violating it in one dimension---both behaviors in accord with expectations for proper measures of entanglement entropy.