Boundaries of boundaries: a systematic approach to lattice models with solvable boundary states of arbitrary codimension

We present a generic and systematic approach for constructing D-dimensional lattice models with exactly solvable d-dimensional boundary states localized to corners, edges, hinges and surfaces. These solvable models represent a class of "sweet spots" in the space of possible tight-binding models---the exact solutions remain valid for any tight-binding parameters as long as they obey simple locality conditions that are manifest in the underlying lattice structure. Consequently, our models capture the physics of both (higher-order) topological and non-topological phases as well as the transitions between them in a particularly illuminating and transparent manner.Comment: 19 pages, 12 figure

Extended Bloch theorem for topological lattice models with open boundaries

While the Bloch spectrum of translationally invariant noninteracting lattice models is trivially obtained by a Fourier transformation, diagonalizing the same problem in the presence of open boundary conditions is typically only possible numerically or in idealized limits. Here we present exact analytic solutions for the boundary states in a number of lattice models of current interest, including nodal-line semimetals on a hyperhoneycomb lattice, spin-orbit coupled graphene, and three-dimensional topological insulators on a diamond lattice, for which no previous exact finite-size solutions are available in the literature. Furthermore, we identify spectral mirror symmetry as the key criterium for analytically obtaining the entire (bulk and boundary) spectrum as well as the concomitant eigenstates, and exemplify this for Chern and $\mathcal Z_2$ insulators with open boundaries of co-dimension one. In the case of the two-dimensional Lieb lattice, we extend this further and show how to analytically obtain the entire spectrum in the presence of open boundaries in both directions, where it has a clear interpretation in terms of bulk, edge, and corner states

Topological origin of edge states in two-dimensional inversion-symmetric insulators and semimetals

Symmetries play an essential role in identifying and characterizing topological states of matter. Here, we classify topologically two-dimensional (2D) insulators and semimetals with vanishing spin-orbit coupling using time-reversal ($\mathcal{T}$) and inversion ($\mathcal{I}$) symmetry. This allows us to link the presence of edge states in $\mathcal{I}$ and $\mathcal{T}$ symmetric 2D insulators, which are topologically trivial according to the Altland-Zirnbauer table, to a $\mathbb{Z}_2$ topological invariant. This invariant is directly related to the quantization of the Zak phase. It also predicts the generic presence of edge states in Dirac semimetals, in the absence of chiral symmetry. We then apply our findings to bilayer black phosphorus and show the occurrence of a gate-induced topological phase transition, where the $\mathbb{Z}_2$ invariant changes

On the topological immunity of corner states in two-dimensional crystalline insulators

A higher-order topological insulator (HOTI) in two dimensions is an insulator without metallic edge states but with robust zero-dimensional topological boundary modes localized at its corners. Yet, these corner modes do not carry a clear signature of their topology as they lack the anomalous nature of helical or chiral boundary states. Here, we demonstrate using immunity tests that the corner modes found in the breathing kagome lattice represent a prime example of a mistaken identity. Contrary to previous theoretical and experimental claims, we show that these corner modes are inherently fragile: the kagome lattice does not realize a higher-order topological insulator. We support this finding by introducing a criterion based on a corner charge-mode correspondence for the presence of topological midgap corner modes in n-fold rotational symmetric chiral insulators that explicitly precludes the existence of a HOTI protected by a threefold rotational symmetry.Comment: 10 pages, 5 figures. Accepted for publication in NPJ Quantum Material

Post-glucose-load urinary C-peptide and glucose concentration obtained during OGTT do not affect oral minimal model-based plasma indices

The purpose of this study was to investigate how renal loss of both C-peptide and glucose during oral glucose tolerance test (OGTT) relate to and affect plasma-derived oral minimal model (OMM) indices. All individuals were recruited during family screening between August 2007 and January 2011 and underwent a 3.5-h OGTT, collecting nine plasma samples and urine during OGTT. We obtained the following three subgroups: normoglycemic, at risk, and T2D. We recruited South Asian and Caucasian families, and we report separate analyses if differences occurred. Plasma glucose, insulin, and C-peptide concentrations were analyzed as AUCs during OGTT, OMM estimate of renal C-peptide secretion, and OMM beta-cell and insulin sensitivity indices were calculated to obtain disposition indices. Post-glucose load glucose and C-peptide in urine were measured and related to plasma-based indices. Urinary glucose corresponded well with plasma glucose AUC (Cau r = 0.64, P < 0.01; SA r = 0.69, P < 0.01), SI (Cau r = −0.51, P < 0.01; SA r = −0.41, P < 0.01), Φdynamic (Cau r = −0.41, P < 0.01; SA r = −0.57, P < 0.01), and Φoral (Cau r = −0.61, P < 0.01; SA r = −0.73, P < 0.01). Urinary C-peptide corresponded well to plasma C-peptide AUC (Cau r = 0.45, P < 0.01; SA r = 0.33, P < 0.05) and OMM estimate of renal C-peptide secretion (r = 0.42, P < 0.01). In general, glucose excretion plasma threshold for the presence of glucose in urine was ~10–10.5 mmol L−1 in non-T2D individuals, but not measurable in T2D individuals. Renal glucose secretion during OGTT did not influence OMM indices in general nor in T2D patients (renal clearance range 0–2.1 %, with median 0.2 % of plasma glucose AUC). C-indices of urinary glucose to detect various stages of glucose intolerance were excellent (Cau 0.83–0.98; SA 0.75–0.89). The limited role of renal glucose secretion validates the neglecting of urinary glucose secretion in kinetic models of glucose homeostasis using plasma glucose concentrations. Both C-peptide and glucose in urine collected during OGTT might be used as non-invasive measures for endogenous insulin secretion and glucose tolerance state

Failing beta-cell adaptation in South Asian families with a high risk of type 2 diabetes

We performed an extended oral glucose tolerance test (OGTT) to investigate the relationship between early and late beta-cell response and type 2 diabetes (T2D) in families of South Asian origin and indigenous Dutch, burdened by T2D. Based on the OGTT, 22 individuals were normoglycemic, 12 glucose intolerant and 23 had T2D in the South Asian families; these numbers were 34, 12 and 18 in the Caucasian families, respectively. The OGTT had 11 blood samplings in 3.5 h for glucose, insulin and C-peptide measurements. Through early and late insulin secretion rate (ISR), the above basal glucose area-under-the-curve after glucose load (glucose disposal) and insulin sensitivity index (ISI), we obtained early and late disposition indices (DI). South Asians on average had lower ISI than Caucasians (3.8 ± 2.9 vs. 6.5 ± 4.7, respectively, P < 0.001), with rapid decline of their early and late DI between normal glucose tolerance versus impaired fasting glucose/impaired glucose tolerance (late DI; P < 0.0001). Adjusted for ISI, age, gender and waist-to-hip ratio, early ISR was significantly associated with glucose disposal in South Asians (β = 0.55[0.186; 0.920]), but not in Caucasians (β = 0.09[-0.257; 0.441]). Similarly, early ISR was strongly associated with late ISR (β = 0.71[0.291; 1.123]; R2 = 45.5 %) in South Asians, but not in Caucasians (β = 0.27[-0.035; 0.576]; R2 = 17.4 %), with significant interaction between ethnicity and early ISR (β = 0.341[0.018; 0.664]). Ordinal regression analyses confirmed that all South Asian OGTT subgroups were homogenously resistant to insulin and solely predicted by early ISR (β = -0.782[-1.922; 0.359], β = -0.020[-0.037; -0.002], respectively), while in Caucasian families both ISI and early ISR were related to glucose tolerance state (β = -0.603[-1.105; -0.101], β = -0.066[-0.105; -0.027], respectively). In South Asian individuals, rapid beta-cell deterioration might occur under insulin resistant conditions. As their early insulin response correlates strongly with both glucose disposal and late insulin response, alterations in beta-cell dynamics may give an explanation to their extreme early onset of T2D, although larger prospective studies are required