Abstract The entanglement of different parts of a quantum system is expected to be proportional to the common interface area. Therefore alterations across the interface will lead to changes on the behavior of entanglement entropy. In this work, the effects of bond defects at the boundaries of Kitaev ladders are considered. We find a logarithmic scaling for the ground state entanglement entropy between the two pieces. The prefactor of the logarithm (effective central charge (ECC)) varies continuously with the defect strength. The energy dispersion is also obtained and sharp features in the von Neumann entanglement entropy are observed when bands cross. Phase diagrams for homogeneous Kitaev Hamiltonians with nonzero superconducting paring potential are presented. They show that for chains/legs that are connected to one another through inter-leg hopping, when certain parameters are fine-tuned, the phase transition lines correspond to either single or double gapless modes dispersion. Moreover, even when the defect is turned on, the ECC for ladders with two gapless modes is exactly twice the one for ladders with a single gapless mode. On the other hand, in the absence of superconductivity, we can tune the parameters to obtain homogeneous systems whose number of gapless modes is up to the number of legs of the ladder. Additionally, in this situation, the presence of the bond defect makes the ECC becomes smaller than the number of gapless modes times the ECC of Hamiltonians with one gapless mode. Furthermore, the relationship between the cases with and without superconductivity is presented.
Abstract The entanglement of different parts of a quantum system is expected to be proportional to the common interface area. Therefore alterations across the interface will lead to changes on the behavior of entanglement entropy. In this work, the effects of bond defects at the boundaries of Kitaev ladders are considered. We find a logarithmic scaling for the ground state entanglement entropy between the two pieces. The prefactor of the logarithm (effective central charge (ECC)) varies continuously with the defect strength. The energy dispersion is also obtained and sharp features in the von Neumann entanglement entropy are observed when bands cross. Phase diagrams for homogeneous Kitaev Hamiltonians with nonzero superconducting paring potential are presented. They show that for chains/legs that are connected to one another through inter-leg hopping, when certain parameters are fine-tuned, the phase transition lines correspond to either single or double gapless modes dispersion. Moreover, even when the defect is turned on, the ECC for ladders with two gapless modes is exactly twice the one for ladders with a single gapless mode. On the other hand, in the absence of superconductivity, we can tune the parameters to obtain homogeneous systems whose number of gapless modes is up to the number of legs of the ladder. Additionally, in this situation, the presence of the bond defect makes the ECC becomes smaller than the number of gapless modes times the ECC of Hamiltonians with one gapless mode. Furthermore, the relationship between the cases with and without superconductivity is presented.
The quantum Ising chain of length, L, which is separated into two parts by localized or extended defects is considered at the critical point where scaling of the interface magnetization is nonuniversal. We measure the entanglement entropy between the two halves of the system in equilibrium, as well as after a quench, when the interaction at the interface is changed for time t>0. For the localized defect the increase in the entropy with logL or with logt involves the same effective central charge, which is a continuous function of the strength of the defect. On the contrary for the extended defect the equilibrium entropy is saturated but the nonequilibrium entropy has a logarithmic time dependence the prefactor of which depends on the strength of the defect.
The following thesis focuses on the scaling of entanglement entropy in lower dimensions and is divided into three main parts. Chapter 2 studies the thermal reduced density matrices in fermion and spin systems on ladders. Chapter 3 studies the many-body localization phase transition in a Rokhsar-Kivelson type wave function. Chapter 4 studies the subleading correction term of entanglement entropy in 2+1 dimensional scale invariant systems. Chapter 5 studies the bulk-boundary correspondence in 3 + 1 dimensional topological phases and its entanglement entropy. In chapter 2, we investigate the reduced density matrices for a model of free fermions on a two-leg ladder (gapped by the inter-chain tunneling operator) and in 1/2 spin systems on a ladder with a gapped ground state using exact solutions and several controlled approximations. We calculate the reduced density matrix and the entanglement entropy for a leg of the ladder (i.e. cut made between the chains). In the fermionic system we find the exact form of the reduced density matrix for one of the chains and determine the entanglement spectrum explicitly. Here we find that in the weak tunneling limit of the ladder the entanglement entropy of one chain of the gapped ladder has a simple and universal form dictated by conformal invariance. In the case of the spin system, we consider the strong coupling limit by using perturbation theory and get the reduced density matrix by the Schmidt decomposition. The entanglement entropies of a general gapped system of two coupled conformal field theories (in 1+1 dimensions) is discussed using the replica trick and scaling arguments. We show that 1) for a system with a bulk gap, the reduced density matrix has the form of a thermal density matrix, 2) the long-wavelength modes of one subsystem (a chain) of a gapped coupled system are always thermal, 3) the von Neumann entropy equals to the thermodynamic entropy of one chain, and 4) the bulk gap plays the role of effective temperature. In chapter 3, we construct a family of many-body wave functions to study the many-body localization phase transition. The wave functions have a Rokhsar-Kivelson form, in which the weight for the configurations are chosen from the Gibbs weights of a classical spin glass model, known as the Random Energy Model, multiplied by a random sign structure to represent a highly excited state. These wave functions show a phase transition into an MBL phase. In addition, we see three regimes of entanglement scaling with subsystem size: scaling with entanglement corresponding to an infinite temperature thermal phase, constant scaling, and a sub-extensive scaling between these limits. Near the phase transition point, the fluctuations of the Renyi entropies are non-Gaussian. We find that Renyi entropies with different Renyi index transition into the MBL phase at different points and have different scaling behavior, suggesting a multifractal behavior. In chapter 4, we study the universal scaling behavior of the entanglement entropy of critical theories in 2+1 dimensions. We specially consider two fermionic scale-invariant models, free massless Dirac fermions and a model of fermions with quadratic band touching, and numerically study the two-cylinder entanglement entropy of the models on the torus. We find that in both cases the entanglement entropy satisfies the area law and has the subleading term which is a scaling function of the aspect ratios of the cylindrical regions. We test the scaling of entanglement in both the free fermion models using three possible scaling functions for the subleading term derived from a) the quasi-one-dimensional conformal field theory, b) the bosonic quantum Lifshitz model, and c) the holographic AdS/CFT correspondence. For the later case we construct an analytic scaling function using holography, appropriate for critical theories with a gravitational dual description. We find that the subleading term in the fermionic models is well described, for a range of aspect ratios, by the scaling form derived from the quantum Lifshitz model as well as that derived using the AdS/CFT correspondence (in this case only for the Dirac model). For the case where the fermionic models are placed on a square torus we find the fit to the different scaling forms is in agreement to surprisingly high precision. In chapter 5, we discuss (2+1)-dimensional gapless surface theories of bulk (3+1)-dimensional topological phases, such as the BF theory at level K, and its generalization. In particular, we put these theories on a flat (2+1) dimensional torus T3 parameterized by its modular parameters, and compute the partition functions obeying various twisted boundary conditions. We show the partition functions are transformed into each other under SL(3, Z) modular transformations, and furthermore establish the bulk-boundary correspondence in (3+1) dimensions by matching the modular S and T matrices computed from the boundary field theories with those computed in the bulk. We propose the three-loop braiding statistics can be studied by constructing the modular S and T matrices from an appropriate boundary field theory. We also study the EE for 3 + 1- dimensional topological phase with or without three-loop braiding phase.
We study the scaling of the entanglement entropy in different classes of one-dimensional fermionic quasiperiodic systems with and without pairing, focusing on multifractal critical points/phases. We find that the entanglement entropy scales logarithmically with the subsystem size NA with a proportionality coefficient C, as in homogeneous critical points, apart from possible additional small oscillations. In the absence of pairing, we find that the entanglement entropy coefficient C is nonuniversal and depends significantly and nontrivially both on the model parameters and electron filling, in multifractal critical points. In some of these points, C can take values close to the homogeneous (or ballistic) system, although it typically takes smaller values. We find a close relation between the behavior of the entanglement entropy and the small-q (long-wavelength) dependence of the momentum structure factor S(q). S(q) increases linearly with q as in the homogeneous case, with a slope that grows with C. In the presence of pairing, we find that even the addition of small anomalous terms affects very significantly the scaling of the entanglement entropy compared to the unpaired case. In particular, we focused on topological phase transitions for which the gap closes with either extended or critical multifractal states. In the former case, the scaling of the entanglement entropy mirrors the behavior observed at the critical points of the homogeneous Kitaev chain, while in the latter, it shows only slight deviations arising at small length scales. In contrast with the unpaired case, we always observe $\mathcal{C}\ensuremath{\approx}1/6$ for different critical points, the known value for the homogeneous Kitaev chain with periodic boundary conditions.
We study momentum-space entanglement in quantum spin-half ladders consisting of two coupled critical XXZ spin-half chains, using field theoretical methods developed by Lundgren et al. [Phys. Rev. B 88, 245137 (2013)] and exact diagonalization. When the system is gapped, we analytically find that the momentum-space entanglement Hamiltonian is described by a chiral conformal field theory with a central charge of 2. This is in contrast to entanglement Hamiltonians of various real-space partitions of gapped-spin ladders that have a central charge of 1. When the system is gapless, the entanglement Hamiltonian consists of one gapless mode that is linear in subsystem momentum and one mode with a flat dispersion relation. In the gapless region, we extend the work of Lundgren et al. to include the effect of a certain irrelevant term on the entanglement spectrum. In both the gapless and gapped phases, the momentum-space entanglement entropy obeys a volume law. In the gapless region, we find a subleading constant term in the entanglement entropy which contains information about the underlying field theory of the system. The analytical predictions for the entanglement spectrum are consistent with results from exact diagonalization.
We investigate the finite-size corrections of the entanglement entropy of critical ladders and propose a conjecture for its scaling behaviour. The conjecture is verified for free fermions, for Heisenberg ladders, and for quantum Ising ladders. Our results support that the prefactor of the logarithmic correction of the entanglement entropy of critical ladder models is universal and is associated with the central charge of the one-dimensional version of the models and with the number of branches associated with gapless excitations. Our results suggest that it is possible to infer whether there is a violation of the entropic area law in two-dimensional critical systems by analyzing the scaling behaviour of the entanglement entropy of ladder systems, which are easier to deal with.
We consider the entanglement entropy for the 2D Ising model at the conformal fixed point in the presence of interfaces. More precisely, we investigate the situation where the two subsystems are separated by a defect line that preserves conformal invariance. Using the replica trick, we compute the entanglement entropy between the two subsystems. We observe that the entropy, just like in the case without defects, shows a logarithmic scaling behavior with respect to the size of the system. Here, the prefactor of the logarithm depends on the strength of the defect encoded in the transmission coefficient. We also comment on the supersymmetric case.