The scaling of entanglement entropy for the nearest neighbor antiferromagnetic Heisenberg spin model is studied computationally for clusters joined by a single bond. Bisecting the balanced three legged Bethe Cluster, gives a second Renyi entropy and
the valence bond entropy which scales as the number of sites in the cluster. For the analogous situation with square clusters, i.e. two $L times L$ clusters joined by a single bond, numerical results suggest that the second Renyi entropy and the valence bond entropy scales as $L$. For both systems, the environment and the system are connected by the single bond and interaction is short range. The entropy is not constant with system size as suggested by the area law.
The scaling of entanglement entropy is computationally studied in several $1le d le 2$ dimensional free fermion systems that are connected by one or more point contacts (PC). For both the $k$-leg Bethe lattice $(d =1)$ and $d=2$ rectangular lattices
with a subsystem of $L^d$ sites, the entanglement entropy associated with a {sl single} PC is found to be generically $S sim L$. We argue that the $O(L)$ entropy is an expression of the subdominant $O(L)$ entropy of the bulk entropy-area law. For $d=2$ (square) lattices connected by $m$ PCs, the area law is found to be $S sim aL^{d-1} + b m log{L}$ and is thus consistent with the anomalous area law for free fermions ($S sim L log{L}$) as $m rightarrow L$. For the Bethe lattice, the relevance of this result to Density Matrix Renormalization Group (DMRG) schemes for interacting fermions is discussed.
We introduce a deterministic discrete-particle simulation approach, the Linearly-Transformed Particle-In-Cell (LTPIC) method, that employs linear deformations of the particles to reduce the noise traditionally associated with particle schemes. Formal
ly, transforming the particles is justified by local first order expansions of the characteristic flow in phase space. In practice the method amounts to using deformation matrices within the particle shape functions; these matrices are updated via local evaluations of the forward numerical flow. Because it is necessary to periodically remap the particles on a regular grid to avoid excessively deforming their shapes, the method can be seen as a development of Denavits Forward Semi-Lagrangian (FSL) scheme [J. Denavit, J. Comp. Physics 9, 75 (1972)]. However, it has recently been established [M. Campos Pinto, Smooth particle methods without smoothing, arXiv:1112.1859 (2012)] that the underlying Linearly-Transformed Particle scheme converges for abstract transport problems, with no need to remap the particles; deforming the particles can thus be seen as a way to significantly lower the remapping frequency needed in the FSL schemes, and hence the associated numerical diffusion. To couple the method with electrostatic field solvers, two specific charge deposition schemes are examined, and their performance compared with that of the standard deposition method. Finally, numerical 1d1v simulations involving benchmark test cases and halo formation in an initially mismatched thermal sheet beam demonstrate some advantages of our LTPIC scheme over the classical PIC and FSL methods. Benchmarked test cases also indicate that, for numerical choices involving similar computational effort, the LTPIC method is capable of accuracy comparable to or exceeding that of state-of-the-art, high-resolution Vlasov schemes.
The entanglement entropy of $ u=1/2$ and $ u=9/2$ quantum Hall states in the presence of short range disorder has been calculated by direct diagonalization. Spin polarized electrons are confined to a single Landau level and interact with long range C
oulomb interaction. For $ u=1/2$ the entanglement entropy is a smooth monotonic function of disorder strength. For $ u=9/2$ the entanglement entropy is non monotonic suggestive of a solid-liquid phase transition. As a model of the transition at $ u=1/2$ free fermions with disorder in 2 dimensions were studied. Numerical evidence suggests the entanglement entropy scales as $L$ rather than the $L ln{L}$ as in the disorder free case.
The entanglement entropy of two gapless non-interacting fermion subsystems is computed approximately in a way that avoids the introduction of replicas and a geometric interpretation of the reduced density matrix. We exploit the similarity between the
Schmidt basis wavefunction and superfluid BCS wavefunction and compute the entropy using the BCS approximation. Within this analogy, the Cooper pairs are particle-hole pairs straddling the boundary and the effective interaction between them is induced by the projection of the Hilbert space onto the incomplete Schmidt basis. The resulting singular interaction may be thought of as lifting the degeneracy of the single particle distribution function. For two coupled fermion systems of linear size $L$, we solve the BCS gap equation approximately to find the entropy $S approx (w^2/t^2)log{L}$ where $w$ is the hopping amplitude at the boundary of the subsystem and $2t$ is the bandwidth. We further interpret this result based upon the relationship between entanglement spectrum, entropy and number fluctuations.
We explore the possibility of detecting many-body entanglement using time-of-flight (TOF) momentum correlations in ultracold atomic fermi gases. In analogy to the vacuum correlations responsible for Bekenstein-Hawking black hole entropy, a partitione
d atomic gas will exhibit particle-hole correlations responsible for entanglement entropy. The signature of these momentum correlations might be detected by a sensitive TOF type experiment.
The entanglement entropy of the $ u = 1/3$ and $ u = 5/2$ quantum Hall states in the presence of short range random disorder has been calculated by direct diagonalization. A microscopic model of electron-electron interaction is used, electrons are co
nfined to a single Landau level and interact with long range Coulomb interaction. For very weak disorder, the values of the topological entanglement entropy are roughly consistent with expected theoretical results. By considering a broader range of disorder strengths, the fluctuation in the entanglement entropy was studied in an effort to detect quantum phase transitions. In particular, there is a clear signature of a transition as a function of the disorder strength for the $ u = 5/2$ state. Prospects for using the density matrix renormalization group to compute the entanglement entropy for larger system sizes are discussed.
The entanglement entropy of the incompressible states of a realistic quantum Hall system in the second Landau level are studied by direct diagonalization. The subdominant term to the area law, the topological entanglement entropy, which is believed t
o carry information about topologic order in the ground state, was extracted for filling factors nu = 12/5 and nu = 7/3. While it is difficult to make strong conclusions about nu = 12/5, the nu = 7/3 state appears to be very consistent with the topological entanglement entropy for the k=4 Read-Rezayi state. The effect of finite thickness corrections to the Coulomb potential used in the direct diagonalization are also systematically studied.
The entanglement entropy of the incompressible states of a realistic quantum Hall system are studied by direct diagonalization. The subdominant term to the area law, the topological entanglement entropy, which is believed to carry information about t
opologic order in the ground state, was extracted for filling factors 1/3, 1/5 and 5/2. The results for 1/3 and 1/5 are consistent with the topological entanglement entropy for the Laughlin wave function. The 5/2 state exhibits a topological entanglement entropy consistent with the Moore-Read wave function.
An exponential dependence of the fragmentation cross-section on the average binding energy is observed and reproduced with a statistical model. The observed functional dependence is robust and allows the extraction of binding energies from measured c
ross-sections. From the systematics of 75,77,78,79Cu isotope cross-sections have been extracted. They are 636.94 +/- 0.40 MeV, 647.1 +/- 0.4 MeV, 651.6 +/- 0.4 MeV and 657.8 +/- 0.5 MeV, respectively. Specifically, the uncertainty of the binding energy of 75Cu is reduced from 980 keV (listed value in the 2003 mass table of Audi and Wapstra) to 400 keV. The predicted cross-sections of two near drip-line nuclei, 39Na and 40Mg, from the fragmentation of 48Ca are discussed.
