Entanglement Hamiltonians in quantum many-body physics

Entanglement is one of the most fascinating features of quantum mechanics. The phenomenon of quantum entanglement refers to a particular type of correlation that is not present in the classical world. It is the most important ingredient behind the concept of quantum computation that may lead to an information technological revolution in the future. However, entanglement also plays a pivotal role in understanding the fundamental principles governing the physics of quantum many-body systems at extremely low temperatures. This realization has led to a vast effort in the research of entanglement properties in the last decades, with a particular emphasis on entanglement measures in low-dimensional systems. In simple terms, entanglement refers to the mechanism, how different parts of a many-body system are coupled in the quantum state under consideration. In a classical equilibrium setting, statistical mechanics dictates us that a small but macroscopic part of a larger total system is described by a thermal ensemble with respect to the subsystem Hamiltonian. In many-body systems at low temperatures, such as in the ground state, quantum correlations dominate over thermal ones and the description changes entirely. One could, however, still consider the state of a subsystem as a kind of thermal state with respect to the so-called entanglement Hamiltonian which is the very focus of this project. In recent years, there has been an increasing interest in the study of entanglement Hamiltonians, both on theoretical and experimental side, and a number of important features has been uncovered. In particular, in a broad class of one-dimensional ground states, the entanglement Hamiltonian of a subsystem resembles the physical Hamiltonian, albeit with a spatially varying inverse temperature. The remarkable simplicity of the entanglement Hamiltonian in the above setting raises numerous questions and opens new lines of research to explore. How does the entanglement Hamiltonian change if an inhomogeneity is present already in the physical Hamiltonian? How can one describe the crossover from the entanglement to the physical Hamiltonian when moving towards higher energies? What happens for 2D fermion systems with a non-trivial Fermi surface? What is the situation if the system is driven out of equilibrium? These are some of the main goals to be addressed in this research project.

Thermodynamics describes the equilibrium properties of macroscopic many-body systems immersed in a heat bath. In particular, the energy fluctuations of the system are characterized by the temperature of the reservoir. If the total system is cooled down to zero temperature, thermal fluctuations vanish completely. However, in ground states of quantum many-body systems, some quantum fluctuations still survive due to entanglement, since two parts of a bipartite system are coupled in the wavefunction. The entanglement Hamiltonian (EH) is an object introduced by thermodynamic analogy, and describes the structure of these quantum fluctuations. It provides the reduced state of a subsystem in an exponential form, however, it is far from obvious whether the EH has any relation to the physical Hamiltonian, as in thermodynamics. Quite remarkably, for a broad range of one-dimensional ground states, the EH has a very simple local structure. In fact, it can be interpreted as if the subsystem was subject to an inhomogeneous temperature, which is very high close to the boundaries but goes to zero towards the bulk, thus shedding light to the origin of the entanglement area law. The goal of this project was to study the EH in free-fermion systems in hitherto unexplored situations, and establish a connection between lattice results and predictions from quantum field theory. We extended the existing results in various directions, such as more complicated subsystem geometries, higher dimensions, inhomogeneous systems, as well as the out-of-equilibrium scenario. Our first main result was to identify a peculiar non-local structure in the EH for two disjoint blocks in a 1D chain, confirming field-theoretical predictions. We also studied half-infinite chains, observing that the EH becomes strictly local even on the lattice, for both gapped and gapless ground states. Considering the EH of a spherical domain in a non-relativistic Fermi gas, we found that locality is asymptotically recovered only in 1D, while in higher dimensions discrepancies at large angular momenta are observed. We also studied models where inhomogeneities are present in the physical Hamiltonian itself. For a hopping chain in a linear potential or a Fermi gas in a harmonic trap, we showed that the subsystem is effectively subject to a linear inverse temperature. Moreover, we also verified that various families of inhomogeneous chains, related to discrete orthogonal polynomials, are described by either a linear or parabolic inverse temperature. Finally, we also considered a nonequilibrium hopping chain after a local quench. In this setup, the EH encodes two different inverse temperatures, associated to the left- and right-moving components of the energy density. The results of the project contribute significantly to a better understanding of the entanglement structure in quantum many-body systems.