Casimir forces, entanglement and conformal invariance in inhomogeneous quantum chains

Résumé

The main goal of condensed matter physics is to study how the macroscopic behavior of matter arises from a large number of interacting particles. The particles involved in these systems show a quantum nature and, thus, have quantum correlations amongst them: they are entangled. In general, many-body systems are difficult to work with as they can not be described in terms of single particles that behave independently of each other. The description of many-body systems requires a global wavefunction where its coefficients in a given basis belong to a certain Hilbert space whose dimension increases exponentially with the number of constituents of the system. For that reason, this becomes a really difficult task for large sizes. However, some quantum states in the Hilbert space are more relevant than others. Fortunately, there exist certain quantum states that satisfy the so-called area law. Let us, for instance, consider two subregions of the system, A and B. We can measure the quantum correlations between the two regions through the so-called entanglement entropy (EE). One randomly chosen quantum state in the Hilbert space shows a volumetric law for the entanglement entropy, i.e., it scales with the minimum volume between parts A and B. Nevertheless, some quantum states show an entanglement entropy which is proportional to the boundary between the two subsystems. These quantum states are the ground states (GS) of gapped local Hamiltonians which have a finite correlation length. The ground state is nothing else but the lowest possible energy state of the system, which is the perfect scenario for the study of low temperature physics. On the contrary, there are also ground states that violate the area law. The violation of the area law usually takes place when the correlation length is infinite and, thus, long-range quantum correlations become relevant even far away from the border between the two regions we are considering. For these cases, the entanglement entropy may show a logarithmic correction for one-dimensional systems, which is predicted by conformal field theory (CFT). An example of violation of the area law is given by ground states of some fermionic chains with inhomogeneous hopping amplitudes. A fermionic chain is a mathematical one-dimensional object in which each site may be occupied, or not, by a single particle. The sites in the chain are related among them by the so-called hopping amplitudes which can be tuned. If all the hopping amplitudes are equal and homogeneous, the system represents a Dirac fermion in a flat space-time, also called Minkowski space-time, when the number of particles is equal to half of the sites of the chain. This is what we called half-filling. However, these coupling parameters can be position-dependent, i.e., we can have inhomogeneous hopping amplitudes. Inhomogeneous hopping amplitudes possess a geometric interpretation. Two sites that have a strong hopping amplitude can be considered to be within close proximity. On the other hand, two sites that show a weak coupling can be interpreted as sites located spatially far away. For this reason, it is possible to find an appropriate change of coordinates that maps a chain with inhomogeneous hopping amplitudes to a system where certain sites are spatially closer than others but show the same coupling amplitude regardless of their position Figure 1: System (a) represents an inhomogeneous fermionic chain: sites (blue) located towards the right side show stronger hopping amplitudes (red) than those sites placed on the left side. In system (b) all hopping amplitudes represented correspond to a homogeneous dynamic but the geometry of the system has been deformed. along the chain. This is what we call curved space-times. In addition, these chains may be described by a Hamiltonian which is quadratic in the fermionic operators and, thus, can be solved exactly in terms of effective non-interacting fermions, i.e., particles that move independently of each other. In this case, the Hamiltonian can be diagonalized analytically in terms of single-body modes and energies. These systems are called free fermion models and will be used throughout this work. Free fermion models are specially interesting in the analysis of spin systems which are shown to be of great relevance in the study of magnetic properties of materials, phase transitions and critical points. Some spin systems, such as the well-known Ising model, can be mapped to a free fermion model by the so-called Jordan-Wigner transformation. The first part of this thesis provides a review of the knowledge required to follow the work that has been developed, which will be exposed in the second part. Chapter 1 is an introduction to some important concepts such as the Casimir effect and the quantum vacuum, entanglement, the entanglement entropy and the area law. Chapter 2 summarizes the density matrix formalism. The description of both free fermionic models and spin models can be found in Chapter 3. In addition, to conclude this first part, Chapter 4 presents a review of some relevant conformal field theory results that are needed later on. The second part of this document shows the main goals achieved in this thesis which constitute Chapters 5, 6 and 7, respectively: Analysis of Casimir forces in inhomogeneous fermionic chains. The Casimir energy expression for the ground state of a homogeneous fermionic chain shows a non-universal contribution proportional to the system size N, plus finite-size corrections of order O(1/N). These finite-size corrections are fixed by CFT when the system is subject to conformal invariance. In this work, we study how these corrections behave under the deformation of the system, i.e., when inhomogeneous hopping amplitudes are considered. Study of the fermionic density in inhomogeneous chains away from half-filling. The ground state of an inhomogeneous free fermionic chain shows a homogeneous occupation: all sites in the chain have the same probability of being occupied, or empty. However, away from half-filling, a new physical phenomenon appears and it requires a continuum approximation to be explained. Characterization of the ground state entanglement structure of a critical chain in terms of the energetic relations between a subsystem and its environment. In general, a subsystem of a ground state is not in its ground state but presents an excess energy. Part of this excess energy can be extracted via unitary operations, which we call subsystem ergotropy. For one-dimensional systems with conformal invariance, the part of the excess energy that can not be extracted is related to its size and the entanglement entropy of the subsystem. To conclude, Chapter 8 contains the final remarks and the results obtained in this thesis both in English and Spanish.