We consider scaling of the entanglement entropy across a topological quantum phase transition for the Kitaev chain model. The change of the topology manifests itself in a subleading term, which scales as ${L}^{\ensuremath{-}1/\ensuremath{\alpha}}$ with the size of the subsystem $L$, here $\ensuremath{\alpha}$ is the R\'enyi index. This term reveals the scaling function ${h}_{\ensuremath{\alpha}}(L/\ensuremath{\xi})$, where $\ensuremath{\xi}$ is the correlation length, which is sensitive to the topological index. The scaling function ${h}_{\ensuremath{\alpha}}(L/\ensuremath{\xi})$ is independent of model parameters, suggesting some degree of its universality.
We consider scaling of the entanglement entropy across a topological quantum phase transition for the Kitaev chain model. The change of the topology manifests itself in a subleading term, which scales as L\ensuremath−1/\ensuremathα with the size of the subsystem L, here \ensuremathα is the R'enyi index. This term reveals the scaling function h\ensuremathα(L/\ensuremathξ), where \ensuremathξ is the correlation length, which is sensitive to the topological index. The scaling function h\ensuremathα(L/\ensuremathξ) is independent of model parameters, suggesting some degree of its universality.
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.
It is generally believed that in spatial dimension d>1, the leading contribution to the entanglement entropy $S=\ensuremath{-}\text{tr}\text{ }{\ensuremath{\rho}}_{A}\text{ }\text{log}\text{ }{\ensuremath{\rho}}_{A}$ scales as the area of the boundary of subsystem A. The coefficient of this ``area law'' is nonuniversal. However, in the neighborhood of a quantum critical point S is believed to possess subleading universal corrections. In the present work, we study the entanglement entropy in the quantum O(N) model in 1<d<3. We use an expansion in $ϵ=3\ensuremath{-}d$ to evaluate (i) the universal geometric correction to S for an infinite cylinder divided along a circular boundary; (ii) the universal correction to S due to a finite correlation length. Both corrections are different at the Wilson-Fisher and Gaussian fixed points, and the $ϵ\ensuremath{\rightarrow}0$ limit of the Wilson-Fisher fixed point is distinct from the Gaussian fixed point. In addition, we compute the correlation length correction to the Renyi entropy ${S}_{n}=\frac{1}{1\ensuremath{-}n}\text{log}\text{ }\text{tr}\text{ }{\ensuremath{\rho}}_{A}^{n}$ in ϵ and large-N expansions. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$, this correction generally scales as N2 rather than the naively expected N. Moreover, the Renyi entropy has a phase transition as a function of n for d close to 3.
We investigate the scaling of the bipartite entanglement entropy across Lifshitz quantum phase transitions, where the topology of the Fermi surface changes without any changes in symmetry. We present both numerical and analytical results which show that Lifshitz transitions are characterized by a well-defined set of critical exponents for the entanglement entropy near the phase transition. In one dimension, we show that the entanglement entropy exhibits a length scale that diverges as the system approaches a Lifshitz critical point. In two dimensions, the leading and subleading coefficients of the scaling of entanglement entropy show distinct power-law singularities at critical points. The effect of weak interactions is investigated using the density matrix renormalization group algorithm.
We introduce a systematic framework to calculate the bipartite entanglement entropy of a compact spatial subsystem in a one-dimensional quantum gas which can be mapped into a noninteracting fermion system. We show that when working with a finite number of particles N, the Rényi entanglement entropies grow as lnN, with a prefactor that is given by the central charge. We apply this novel technique to the ground state and to excited states of periodic systems. We also consider systems with boundaries. We derive universal formulas for the leading behavior and for subleading corrections to the scaling. The universality of the results allows us to make predictions for the finite-size scaling forms of the corrections to the scaling. ar
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 n=2 R\'enyi entanglement entropy of the triangular quantum dimer model via Monte Carlo sampling of Rokhsar-Kivelson- (RK-) like ground-state wave functions. Using the construction proposed by Kitaev and Preskill [Phys. Rev. Lett. 96, 110404 (2006)] and an adaptation of the Monte Carlo algorithm described by Hastings et al. [Phys. Rev. Lett. 104, 157201 (2010)], we compute the topological entanglement entropy (TEE) at the RK point $\ensuremath{\gamma}=(1.001\ifmmode\pm\else\textpm\fi{}0.003)\mathrm{ln}2$, confirming earlier results. Additionally, we compute the TEE of the ground state of a generalized RK-like Hamiltonian and demonstrate the universality of TEE over a wide range of parameter values within a topologically ordered phase approaching a quantum phase transition. For system sizes that are accessible numerically, we find that the quantization of TEE depends sensitively on correlations. We characterize corner contributions to the entanglement entropy and show that these are well described by shifts proportional to the number and types of corners in the bipartition.