Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry

We present a general methodology towards the systematic characterization of crystalline topological insulating phases with time reversal symmetry (TRS).~In particular, taking the two-dimensional spinful hexagonal lattice as a proof of principle we study windings of Wilson loop spectra over cuts in the Brillouin zone that are dictated by the underlying lattice symmetries.~Our approach finds a prominent use in elucidating and quantifying the recently proposed ``topological quantum chemistry" (TQC) concept.~Namely, we prove that the split of an elementary band representation (EBR) by a band gap must lead to a topological phase.~For this we first show that in addition to the Fu-Kane-Mele $\mathbb{Z}_2$ classification, there is $C_2\mathcal{T}$-symmetry protected $\mathbb{Z}$ classification of two-band subspaces that is obstructed by the other crystalline symmetries, i.e.~forbidding the trivial phase. This accounts for all nontrivial Wilson loop windings of split EBRs \textit{that are independent of the parameterization of the flow of Wilson loops}.~Then, we show that while Wilson loop winding of split EBRs can unwind when embedded in higher-dimensional band space, two-band subspaces that remain separated by a band gap from the other bands conserve their Wilson loop winding, hence revealing that split EBRs are at least "stably trivial", i.e. necessarily non-trivial in the non-stable (few-band) limit but possibly trivial in the stable (many-band) limit.~This clarifies the nature of \textit{fragile} topology that has appeared very recently.~We then argue that in the many-band limit the stable Wilson loop winding is only determined by the Fu-Kane-Mele $\mathbb{Z}_2$ invariant implying that further stable topological phases must belong to the class of higher-order topological insulators.Comment: 27 pages, 13 figures, v2: minor corrections, new references included, v3: metastable topology of split EBRs emphasized, v4: prepared for publicatio

Examples of topological spaces

M.S.Robert H. Kasrie

Implementing an Online, Integrative, Multi-Component, Group-Based Cognitive Behavior Therapy (CBT) for the Reduction of Caregiver Burden in Primary Familial Caregivers of Persons with Dementia: A Preliminary Test of Feasibility

The purpose of this study was to examine the preliminary feasibility of an online, manualized, group-based, multi-component, cognitive behavioral therapy (CBT) treatment approach for the reduction of caregiver burden among family caregivers of persons with dementia. This study had five primary hypotheses: (1) the recruitment plan, as outlined within this study, would yield the target number of participants within a 12-month period; (2) the proposed group therapy protocol would produce a positive therapeutic climate, as operationalized by increased levels of perceived social support, perceived group cohesion, positive therapeutic alliance, and positive engagement; (3) this study would maintain adequate participant retention, as operationalized by at least 80% of participants enrolled completing the entire course of treatment; (4) the online CBT group therapy would produce positive levels of satisfaction towards the therapy among group members; and (5) the proposed protocol would not create significant levels of perceived burdensomeness among participants. An additional exploratory hypothesis in this study was that the manualized protocol would produce significant reductions in overall levels of caregiver burden, anxious symptoms, depressive symptoms, and role captivity among participants. Dementia is a growing phenomenon around the world, and more families are choosing to provide informal at-home care for their loved-ones with dementia. While this removes financial burden and strain from institutions, it places a significant amount of stress, burden, and strain (physical, psychological, and financial) on the family caregivers. Caregiver burden, also referred to as caregiver burnout, is defined as the overall impact of the physical, psychological, social, and financial demands of caregiving. Caregiver burden is often associated with increased rates of depression, anxiety, psychotropic drug use, somatic disorders, and physical health concerns. With family members becoming primary caregivers to persons with dementia at increasing rates, so too are the experiences of caregiver burden. While interventions do exist for caregivers experiencing high levels of burden, many of these interventions are either (a) not efficacious, (b) not cost-effective, or (c) not flexible enough to work within the various constraints of caregiving. It is clear that this population is in need of an efficacious and cost-effective treatment approach for the reduction of caregiver burden. Results from this study were ultimately inconclusive for supporting feasibility of the research and treatment protocol as a whole; however, certain aspects of the data did suggest some potential areas for preliminary feasibility such as clinically significant improvement in group cohesion and caregiver burden scores across treatment time. It is strongly recommended that further studies continue to examine the preliminary feasibility of this treatment protocol and to explore areas in which accessibility to this intervention plan may be improved in order to better serve caregivers of persons with dementia

Testing the Efficacy of Transcranial Magnetic Stimulation (TMS) in Treating Depression in Patients with Cognitive Impairment

The purpose of this study is to (1) examine the efficacy of Transcranial Magnetic Stimulation (TMS) in treating depression among individuals with cognitive impairment and (2) to examine if TMS is capable of facilitating cognitive improvements independent of mood improvements. Mild cognitive impairment (MCI) is often seen as a pre-clinical stage to dementia, and depressive disorders are highly prevalent among both MCI and dementia. There is a large body of research that has linked depressive disorders as a prodromal symptom of MCI and the later development of dementia. While some researchers debate whether or not this link between depression and MCI/dementia is a true prodromal relationship, or if depression is independently comorbid with MCI/dementia, it remains clear that these disorders occur together in high prevalence rates. The goal of this study was to determine whether or not Transcranial Magnetic Stimulation (TMS) might demonstrate treatment efficacy in treating depressive symptoms among individuals who meet MCI criteria. TMS has been previously approved by the FDA to treat major depressive disorder (MDD); however, very few research studies have been performed to analyze TMS’ ability in treating MDD among individuals with MCI. By analyzing treatment data from individuals who do and do not meet MCI criteria, TMS does appear to demonstrate positive treatment efficacy for treating depressive symptoms among individuals who meet MCI criteria. TMS also appears to be equally efficacious in treating depressive symptoms among this group in comparison to individuals without MCI. TMS also produces positive changes in neurocognitive functioning, both in the MCI and non-MCI groups; however, the results show that these changes in neurocognitive functioning likely occur as a function of depressive symptom reduction

Transient Moisture Gradient in Fire-Exposed Wood Slab

Improved fire endurance design procedures for heavy timber members may require the input of moisture content-dependent allowable stresses. Therefore, experiments were conducted to characterize the transient moisture gradient in a wood slab subjected to fire on one face. Electrodes and thermocouples embedded within the wood sections monitored moisture and temperature levels at selected locations. Time-moisture content curves indicated an increase in moisture content that appeared to be associated with the vaporization of the bound water in the wood. For the eight slabs tested, the average peak moisture content was 1.26 to 2.00 times greater than the initial moisture content

Quantum Phase Transition in Heisenberg-Kitaev Model

We explore the nature of the quantum phase transition between a magnetically ordered state with collinear spin pattern and a gapless $Z_2$ spin liquid in the Heisenberg-Kitaev model. We construct a slave particle mean field theory for the Heisenberg-Kitaev model in terms of complex fermionic spinons. It is shown that this theory, formulated in the appropriate basis, is capable of describing the Kitaev spin liquid as well as the transition between the gapless $Z_2$ spin liquid and the so-called stripy antiferromagnet. In particular, within a mean field theory, we have a discontinuous transition from the $Z_2$ spin liquid to the stripy antiferromagnet. We argue, however, that subtle spinon confinement effects, associated with the instability of gapped U(1) spin liquid in two spatial dimensions, are playing an important role at the transition. The possibility of an exotic continuous transition is briefly addressed.Comment: 12 pages, 6 figure