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

Validation of the chinese version of the oral health impact profile for TMDs (OHIP- TMDs-C)

Objectives: The aim of this study was to evaluate the reliability and validity of the the Chinese version of the Oral Health Impact Profile for TMDs (OHIP-TMDs-C). Study Design: The OHIP-TMDs was initially translated and cross-culturally adapted to Chinese following international guidelines; then subsequently validated for the psychometric characteristics of reliability and validity. In total, 156 participants with temporomandibular disorders (TMDs) were recruited to complete the questionnaire. The reliability of the OHIP-TMDs-C was evaluated using internal consistency and test-retest methods. The validity of the OHIP-TMDs-C was analysed by construct validity and convergent validity. Construct validity was determined based on factor analysis, and convergent validity by analyzing the correlation between OHIP-TMDs-C subscale scores and the global rating of oral health question. Results: Cronbach’s alpha value (internal reliability) for the total OHIP-TMDs-C score was 0.917 and the intraclass correlation coefficient (ICC) value (test–retest reliability) was 0.899. Construct validity was determined by factor analysis, extracting five factors, accounting for 78.6% of the variance. All items had factor loadings above 0.40. In terms of convergent validity, the OHIP-TMDs-C subscale was significant correlated to the global oral health rating. Conclusions: The results suggest that the OHIP-TMDs-C has good reliability and validity and thus may be used as a valuable instrument for patients with TMDs in China

The two-loop supersymmetric corrections to lepton anomalous magnetic and electric dipole moments

Using the effective Lagrangian method, we analyze the electroweak corrections to the anomalous dipole moments of lepton from some special two-loop topological diagrams which are composed of neutralino (chargino) - slepton (sneutrino) in the minimal supersymmetric extension of the standard model (MSSM). Considering the translational invariance of the inner loop momenta and the electromagnetic gauge invariance, we get all dimension 6 operators and derive their coefficients. After applying equations of motion to the external leptons, the anomalous dipole moments of lepton are obtained. The numerical results imply that there is a parameter space where the two-loop supersymmetric corrections to the muon anomalous dipole moments may be significant.Comment: Revtex, 45 pages, including 8 fig

Fabrication of three-dimensional microdisk resonators in calcium fluoride by femtosecond laser micromachining

We report on fabrication of on-chip calcium fluoride (CaF2) microdisk resonators using water-assisted femtosecond laser micromachining. Focused ion beam (FIB) milling is used to create ultra-smooth sidewalls. The quality (Q)-factors of the fabricated microresonators are measured to be 4.2x10^4 at wavelengths near 1550 nm. The Q factor is mainly limited by the scattering from the bottom surface of the disk whose roughness remains high due to the femtosecond laser micromachining process. This technique facilitates formation of on-chip microresonators on various kinds of bulk crystalline materials, which can benefit a wide range of applications such as nonlinear optics, quantum optics, and chip-level integration of photonic devices.Comment: 7 pages, 3 figure