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We propose an index $\mathcal{I}_G$ which characterizes the degree of ingappability, namely the difficulty to induce a unique ground state with a nonvanishing excitation gap, in the presence of a symmetry $G$. $\mathcal{I}_G$ represents the dimension of the subspace of ambient uniquely-gapped in the entire $G$-invariant "theory space". The celebrated Lieb-Schultz-Mattis theorem corresponds, in our formulation, to the case $\mathcal{I}_G=0$ (completely ingappable) for the symmetry $G$ including the lattice translation symmetry. We illustrate the usefulness of the index by discussing the phase diagram of spin-$1/2$ antiferromagnets in various dimensions, which do not necessarily have the translation symmetry., Comment: 3 figures

We propose a geometric {approach to Lieb-Schultz-Mattis theorem for} quantum many-body systems with discrete spin-rotation symmetries and lattice inversion or rotation symmetry, but without translation symmetry assumed. Under symmetry-twisting on a $(d-1)$-dimensional plane, we find that any $d$-dimensional inversion-symmetric spin system possesses a doubly degenerate spectrum when it hosts a half-integer spin at the inversion-symmetric point. We also show that any rotation-symmetric generalized spin model with a projective representation at the rotation center has a similar degeneracy under symmetry-twisting. We argue that these degeneracies imply that {a unique symmetric gapped ground state that is smoothly connected to product states} is forbidden in the original untwisted systems -- generalized inversional/rotational Lieb-Schultz-Mattis theorems without lattice translation symmetry imposed. The traditional Lieb-Schultz-Mattis theorems with translations also fit in the proposed framework., Comment: 6 pages, 7 figures

In rheumatoid arthritis (RA), the radiographic progression of joint space narrowing (JSN) is evaluated using visual assessments. However, those methods are complicated and time-consuming. We developed an automatic system that can detect joint locations and compute the joint space difference index (JSDI), which was defined as the chronological change in JSN between two radiographs. The purpose of this study was to establish the validity of the software that automatically evaluates the temporal change of JSN. This study consisted of 39 patients with RA. All patients were treated with tocilizumab and underwent hand radiography (left and right hand separately) at 0, 6, and 12 months. The JSN was evaluated using mTSS (modified Total Sharp Score) by one musculoskeletal radiologist as well as our automatic system. Software measurement showed that JSDI between 0 and 12 months was significantly higher than that between 0 and 6 months (p< 0.01). While, there was no significant difference in mTSS between 0, 6, and 12 months. The group with higher disease activity at 0 months had significantly higher JSDI between 0 and 6 months than that with lower disease activity (p= 0.02). The automatic software can evaluate JSN progression of RA patients in the finger joint on X-ray.

In rheumatoid arthritis (RA), the radiographic progression of joint space narrowing (JSN) is evaluated using visual assessments. However, those methods are complicated and time-consuming. We developed an automatic system that can detect joint locations and compute the joint space difference index (JSDI), which was defined as the chronological change in JSN between two radiographs. The purpose of this study was to establish the validity of the software that automatically evaluates the temporal change of JSN. This study consisted of 39 patients with RA. All patients were treated with tocilizumab and underwent hand radiography (left and right hand separately) at 0, 6, and 12 months. The JSN was evaluated using mTSS (modified Total Sharp Score) by one musculoskeletal radiologist as well as our automatic system. Software measurement showed that JSDI between 0 and 12 months was significantly higher than that between 0 and 6 months (p< 0.01). While, there was no significant difference in mTSS between 0, 6, and 12 months. The group with higher disease activity at 0 months had significantly higher JSDI between 0 and 6 months than that with lower disease activity (p= 0.02). The automatic software can evaluate JSN progression of RA patients in the finger joint on X-ray.

Correlation functions of dimer operators, the product operators of spins on two adjacent sites, are studied in the spin-$\frac{1}{2}$ XXZ chain in the critical regime. The amplitudes of the leading oscillating terms in the dimer correlation functions are determined with high accuracy as functions of the exchange anisotropy parameter and the external magnetic field, through the combined use of bosonization and density-matrix renormalization group methods. In particular, for the antiferromagnetic Heisenberg model with SU(2) symmetry, logarithmic corrections to the dimer correlations due to the marginally-irrelevant operator are studied, and the asymptotic form of the dimer correlation function is obtained. The asymptotic form of the spin-Peierls excitation gap including logarithmic corrections is also derived., Comment: 11 pages, 8 figures, 1 table + Supplemental Material (7 pages, 6 tables). v2: accepted version, numerical data for finite magnetic field (Figs. 7, 8 and Tables in Supplemental Material) are modified slightly, some references are added

第60回物性若手夏の学校の講義ノート

The conditions for both the stability and the breakdown of the topological classification of gapped ground states of noninteracting fermions, the tenfold way, in the presence of quartic fermion-fermion interactions are given for any dimension of space. This is achieved by encoding the effects of interactions on the boundary gapless modes in terms of boundary dynamical masses. Breakdown of the noninteracting topological classification occurs when the quantum nonlinear sigma models for the boundary dynamical masses favor quantum disordered phases. For the tenfold way, we find that (i) the noninteracting topological classification $\mathbb{Z}^{\,}_{2}$ is always stable, (ii) the noninteracting topological classification $\mathbb{Z}$ in even dimensions is always stable, (iii) the noninteracting topological classification $\mathbb{Z}$ in odd dimensions is unstable and reduces to $\mathbb{Z}^{\,}_{N}$ that can be identified explicitly for any dimension and any defining symmetries. We also apply our method to the three-dimensional topological crystalline insulator SnTe from the symmetry class AII$+R$, for which we establish the reduction $\mathbb{Z}\to\mathbb{Z}^{\,}_{8}$ of the noninteracting topological classification., Comment: 29 pages

A two-dimensional spin-directed Z(2) network model is constructed that describes the combined effects of dimerization and disorder for the surface states of a weak three-dimensional Z(2) topological insulator. The network model consists of helical edge states of two-dimensional layers of Z(2) topological insulators which are coupled by time-reversal-symmetric interlayer tunneling. It is argued that, without dimerization of interlayer couplings, the network model has no insulating phase for any disorder strength. However, a sufficiently strong dimerization induces a transition from a metallic phase to an insulating phase. The critical exponent. for the diverging localization length at metal-insulator transition points is obtained by finite-size scaling analysis of numerical data from simulations of this network model. It is shown that the phase transition belongs to the two-dimensional symplectic universality class of Anderson transition.

A two-dimensional spin-directed Z(2) network model is constructed that describes the combined effects of dimerization and disorder for the surface states of a weak three-dimensional Z(2) topological insulator. The network model consists of helical edge states of two-dimensional layers of Z(2) topological insulators which are coupled by time-reversal-symmetric interlayer tunneling. It is argued that, without dimerization of interlayer couplings, the network model has no insulating phase for any disorder strength. However, a sufficiently strong dimerization induces a transition from a metallic phase to an insulating phase. The critical exponent. for the diverging localization length at metal-insulator transition points is obtained by finite-size scaling analysis of numerical data from simulations of this network model. It is shown that the phase transition belongs to the two-dimensional symplectic universality class of Anderson transition.

We discuss the thermal (or gravitational) responses in topological superconductors and in topological phases in general. Such thermal responses (as well as electromagnetic responses for conserved charge) provide a definition of topological insulators and superconductors beyond the single-particle picture. In two-dimensional topological phases, the Str\v{e}da formula for the electric Hall conductivity is generalized to the thermal Hall conductivity. Applying this formula to the Majorana surface states of three-dimensional topological superconductors predicts cross-correlated responses between the angular momentum and thermal polarization (entropy polarization). We also discuss a use of D-branes in string theory as a systematic tool to derive all such topological terms and topological responses. In particular, we relate the $\mathbb{Z}_2$ index of topological insulators introduced by Kane and Mele (and its generalization to other symmetry classes and to arbitrary dimensions) to the K-theory charge of non-BPS D-branes, and vice versa. We thus establish a link between the stability of non-BPS D-branes and the topological stability of topological insulators., Comment: 16 pages, 2 figures; Submitted to a topical issue of the Comptes Rendus de l Academie des Sciences (CRAS)

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