We study the low-energy states of the 1D random-hopping model in a strongly disordered regime. The entanglement structure is shown to depend solely on the probability distribution for the length of the effective bonds P ( l b ), whose scaling and finite-size behavior are established using renormalization-group arguments and a simple model based on random permutations. Parity oscillations are absent in von Neumann entropy with periodic boundary conditions, but appear in the higher moments of the distribution, such as the variance. The particle-hole excited states leave the bond structure and the entanglement untouched. Nonetheless, particle addition or removal deletes bonds and leads to an effective saturation of entanglement at an effective block size given by the expected value for the longest bond.
We study the low-energy states of the 1D random-hopping model in a strongly disordered regime. The entanglement structure is shown to depend solely on the probability distribution for the length of the effective bonds P ( l b ), whose scaling and finite-size behavior are established using renormalization-group arguments and a simple model based on random permutations. Parity oscillations are absent in von Neumann entropy with periodic boundary conditions, but appear in the higher moments of the distribution, such as the variance. The particle-hole excited states leave the bond structure and the entanglement untouched. Nonetheless, particle addition or removal deletes bonds and leads to an effective saturation of entanglement at an effective block size given by the expected value for the longest bond.
Abstract The entanglement entropy (EE) distribution of strongly disordered one dimensional spin chains, which are equivalent to spinless fermions at half-filling on a bond (hopping) disordered one-dimensional Anderson model, has been shown to exhibit very distinct features such as peaks at integer multiplications of , essentially counting the number of singlets traversing the boundary. Here we show that for a canonical Anderson model with box distribution on-site disorder and repulsive nearest-neighbor interactions the EE distribution also exhibits interesting features, albeit different than the distribution seen for the bond disordered Anderson model. The canonical Anderson model shows a broad peak at low entanglement values and one narrower peak at . Density matrix renormalization group calculations reveal this structure and the influence of the disorder strength and the interaction strength on its shape. A modified real space renormalization group method was used to get a better understanding of this behavior. As might be expected the peak centered at low values of EE has a tendency to shift to lower values as disorder is enhanced. A second peak appears around the EE value of , this peak is broadened and no additional peaks at higher integer multiplications of are seen. We attribute the differences in the distribution between the canonical model and the broad hopping disorder to the influence of the on-site disorder which breaks the symmetry across the boundary.
The ground state of the uniform antiferromagnetic spin-1/2 Heisenberg chain can be viewed as a strongly fluctuating liquid of valence bonds, while in disordered chains these bonds lock into random singlet states on long-length scales. We show that this phenomenon can be studied numerically, even in the case of weak disorder, by calculating the mean value of the number of valence bonds leaving a block of L contiguous spins (the valence-bond entanglement entropy) as well as the fluctuations in this number. These fluctuations show a clear crossover from a small L regime, in which they behave similar to those of the uniform model, to a large L regime, in which they saturate in a way consistent with the formation of a random singlet state on long-length scales. A scaling analysis of these fluctuations is used to study the dependence on disorder strength of the length scale characterizing the crossover between these two regimes. Results are obtained for a class of models that include, in addition to the spin-1/2 Heisenberg chain, the uniform and disordered critical 1D transverse-field Ising model and chains of interacting non-Abelian anyons.
The entanglement entropy distribution of strongly disordered one dimensional spin chains, which are equivalent to spinless fermions at half-filling on a bond (hopping) disordered one-dimensional Anderson model, has been shown to exhibit very distinct features such as peaks at integer multiplications of ln(2), essentially counting the number of singlets traversing the boundary. Here we show that for a canonical Anderson model with box distribution on-site disorder and repulsive nearest-neighbor interactions the entanglement entropy distribution also exhibits interesting features, albeit different than the distribution seen for the bond disordered Anderson model. The canonical Anderson model shows a broad peak at low entanglement values and one narrower peak at ln(2). Density matrix renormalization group (DMRG) calculations reveal this structure and the influence of the disorder strength and the interaction strength on its shape. A modified real space renormalization group (RSRG) method was used to get a better understanding of this behavior. As might be expected the peak centered at low values of entanglement entropy has a tendency to shift to lower values as disorder is enhanced. A second peak appears around the entanglement entropy value of ln(2),this peak is broadened and no additional peaks at higher integer multiplications of ln(2) are seen. We attribute the differences in the distribution between the canonical model and the broad hopping disorder to the influence of the on-site disorder which breaks the symmetry across the boundary.
Entanglement can be quantified in bipartite systems using the entanglement entropy which is, in many cases, bounded by an area law. Nonetheless, some critical 1D systems violate this area law and present logarithmic corrections which are parametrized by the central charge of the associated conformal field theory (CFT). Furthermore, other violations to the area law may appear in random systems and inhomogeneous systems.
In the first part of the work we focus on the properties of entanglement in random hopping models, focusing on the similarities between the CFT predictions for the clean case and the strong disorder renormalization group (SDRG) predictions. We use a combination of methods: exact diagonalization, a transformation of the SDRG and a new tool based on the study of random permutations. All techniques coincide in providing a compelling image, based on a bond-picture.
The second part focus on the engineering of an inhomogeneous fermionic system, in order to obtain a maximal growth of the block entanglement entropy in the ground state. We show that the deformation parameter allows us to interpolate between an uniform model and the strong disorder limit, which is a maximally entangled state called the rainbow state. This state can be seen as a thermo field state.
The last part focus on time-evolution of some valence bond states after a global quench to a homogeneous Hamiltonian in 1D. We focused on the evolution of the entanglement in the rainbow and dimer states. We proposed a generalization of the quasiparticle picture, which we have called the ballistic picture.
We consider a sublattice-symmetric free-fermion model on a one-dimensional lattice with random hopping amplitudes decaying with the distance as $|{t}_{l}|\ensuremath{\sim}{l}^{\ensuremath{-}\ensuremath{\alpha}}$, and address the question how far an analog of the random-singlet state (RSS) conceived originally for describing the ground state of certain random spin chains is valid for this model. For this purpose, we study the effective central charge characterizing the logarithmic divergence of the entanglement entropy (EE) and the prefactor of the distribution of distances between localization centers on the two sublattices, which must fulfill a consistency relation for a RSS. For $\ensuremath{\alpha}>1$, we find by exact diagonalization an overall logarithmic divergence of the entanglement entropy with an effective central charge varying with $\ensuremath{\alpha}$. The large-$\ensuremath{\alpha}$ limit of the effective central charge is found to be different from that of the nearest-neighbor hopping model. The consistency relation of RSS is violated for $\ensuremath{\alpha}\ensuremath{\le}2$, while for $\ensuremath{\alpha}>2$ it is possibly valid, but this conclusion is hampered by a crossover induced by the short-range fixed point. The EE is also calculated by the strong-disorder renormalization group (SDRG) method numerically. Aside from the traditional scheme, we construct and apply a more efficient minimal SDRG scheme having a linear (nearest-neighbor) structure, which turns out to be an accurate approximation of the full SDRG scheme for not too small $\ensuremath{\alpha}$. The SDRG method is found to provide systematically lower effective central charges than exact diagonalization does, nevertheless, it becomes more and more accurate for increasing $\ensuremath{\alpha}$. Furthermore, as opposed to nearest-neighbor models, it indicates a weak dependence on the disorder distribution.
The properties of the entanglement entropy (EE) in one-dimensional disordered interacting systems are studied. Anderson localization leaves a clear signature on the average EE, as it saturates on the length scale exceeding the localization length. This is verified by numerically calculating the EE for an ensemble of disordered realizations using the density matrix renormalization group method. A heuristic expression describing the dependence of the EE on the localization length, which takes into account finite-size effects, is proposed. This is used to extract the localization length as a function of the interaction strength. The localization length dependence on the interaction fits nicely with the expectations.