Subsystem eigenstate thermalization hypothesis for entanglement entropy in CFT

We investigate a weak version of subsystem eigenstate thermalization hypothesis (ETH) for a two-dimensional large central charge conformal field theory by comparing the local equivalence of high energy state and thermal state of canonical ensemble. We evaluate the single-interval R\'enyi entropy and entanglement entropy for a heavy primary state in short interval expansion. We verify the results of R\'enyi entropy by two different replica methods. We find nontrivial results at the eighth order of short interval expansion, which include an infinite number of higher order terms in the large central charge expansion. We then evaluate the relative entropy of the reduced density matrices to measure the difference between the heavy primary state and thermal state of canonical ensemble, and find that the aforementioned nontrivial eighth order results make the relative entropy unsuppressed in the large central charge limit. By using Pinsker's and Fannes-Audenaert inequalities, we can exploit the results of relative entropy to yield the lower and upper bounds on trace distance of the excited-state and thermal-state reduced density matrices. Our results are consistent with subsystem weak ETH, which requires the above trace distance is of power-law suppression by the large central charge. However, we are unable to pin down the exponent of power-law suppression. As a byproduct we also calculate the relative entropy to measure the difference between the reduced density matrices of two different heavy primary states.Comment: 28 pages, 4 figures;v2 change author list;v3 related subtleties about weak ETH clarified; v4 minor correction to match JHEP versio

Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis

We calculate various quantities that characterize the dissimilarity of reduced density matrices for a short interval of length $\ell$ in a two-dimensional (2D) large central charge conformal field theory (CFT). These quantities include the R\'enyi entropy, entanglement entropy, relative entropy, Jensen-Shannon divergence, as well as the Schatten 2-norm and 4-norm. We adopt the method of operator product expansion of twist operators, and calculate the short interval expansion of these quantities up to order of $\ell^9$ for the contributions from the vacuum conformal family. The formal forms of these dissimilarity measures and the derived Fisher information metric from contributions of general operators are also given. As an application of the results, we use these dissimilarity measures to compare the excited and thermal states, and examine the eigenstate thermalization hypothesis (ETH) by showing how they behave in high temperature limit. This would help to understand how ETH in 2D CFT can be defined more precisely. We discuss the possibility that all the dissimilarity measures considered here vanish when comparing the reduced density matrices of an excited state and a generalized Gibbs ensemble thermal state. We also discuss ETH for a microcanonical ensemble thermal state in a 2D large central charge CFT, and find that it is approximately satisfied for a small subsystem and violated for a large subsystem.Comment: V1, 34 pages, 5 figures, see collection of complete results in the attached Mathematica notebook; V2, 38 pages, 5 figures, published versio

Interplay between Quantum Size Effect and Strain Effect on Growth of Nanoscale Metal Thin Film

We develop a theoretical framework to investigate the interplay between quantum size effect (QSE) and strain effect on the stability of metal nanofilms. The QSE and strain effect are shown to be coupled through the concept of "quantum electronic stress. First-principles calculations reveal large quantum oscillations in the surface stress of metal nanofilms as a function of film thickness. This adds extrinsically additional strain-coupled quantum oscillations to surface energy of strained metal nanofilms. Our theory enables a quantitative estimation of the amount of strain in experimental samples, and suggests strain be an important factor contributing to the discrepancies between the existing theories and experiments

Transport through the intertube link between two parallel carbon nanotubes

Quantum transport through the junction between two metallic carbon nanotubes connected by intertube links has been studied within the TB method and Landauer formula. It is found that the conductance oscillates with both of the coupling strength and length. The corresponding local density of states (LDOS) is clearly shown and can be used to explain the reason why there are such kinds of oscillations of the conductances, which should be noted in the design of nanotube-based devices.Comment: 6 pages, 4 figure

Single-layer behavior and slow carrier density dynamic of twisted graphene bilayer

We report scanning tunneling microscopy (STM) and spectroscopy (STS) of twisted graphene bilayer on SiC substrate. For twist angle ~ 4.5o the Dirac point ED is located about 0.40 eV below the Fermi level EF due to the electron doping at the graphene/SiC interface. We observed an unexpected result that the local Dirac point around a nanoscaled defect shifts towards the Fermi energy during the STS measurements (with a time scale about 100 seconds). This behavior was attributed to the decoupling between the twisted graphene and the substrate during the measurements, which lowers the carrier density of graphene simultaneously