We show that the fundamental time-reversal invariant (TRI) insulator exists in $4+1$ dimensions, where the effective-field theory is described by the $(4+1)$-dimensional Chern-Simons theory and the topological properties of the electronic structure are classified by the second Chern number. These topological properties are the natural generalizations of the time reversal-breaking quantum Hall insulator in $2+1$ dimensions. The TRI quantum spin Hall insulator in $2+1$ dimensions and the topological insulator in $3+1$ dimensions can be obtained as descendants from the fundamental TRI insulator in $4+1$ dimensions through a dimensional reduction procedure. The effective topological field theory and the ${Z}_{2}$ topological classification for the TRI insulators in $2+1$ and $3+1$ dimensions are naturally obtained from this procedure. All physically measurable topological response functions of the TRI insulators are completely described by the effective topological field theory. Our effective topological field theory predicts a number of measurable phenomena, the most striking of which is the topological magnetoelectric effect, where an electric field generates a topological contribution to the magnetization in the same direction, with a universal constant of proportionality quantized in odd multiples of the fine-structure constant $\ensuremath{\alpha}={e}^{2}∕\ensuremath{\hbar}c$. Finally, we present a general classification of all topological insulators in various dimensions and describe them in terms of a unified topological Chern-Simons field theory in phase space.

http://absimage.aps.org/image/MAR08/MWS_MAR08-2007-001699.pdf

Topological Field Theory of Time-Reversal Invariant Insulators

It is widely believed that consistent theories of quantum gravity satisfy two basic kinematic constraints: they are free from any global symmetry, and they contain a complete spectrum of gauge charges. For compact, abelian gauge groups, completeness follows from the absence of a 1-form global symmetry. However, this correspondence breaks down for more general gauge groups, where the breaking of the 1-form symmetry is insufficient to guarantee a complete spectrum. We show that the correspondence may be restored by broadening our notion of symmetry to include non-invertible topological operators, and prove that their absence is sufficient to guarantee a complete spectrum for any compact, possibly disconnected gauge group. In addition, we prove an analogous statement regarding the completeness of twist vortices: codimension-2 objects defined by a discrete holonomy around their worldvolume, such as cosmic strings in four dimensions. We discuss how this correspondence is modified in various, more general contexts, including non-compact gauge groups, Higgsing of gauge theories, and the addition of Chern-Simons terms. Finally, we discuss the implications of our results for the Swampland program, as well as the phenomenological implications of the existence of twist strings.

/pdf/non-invertible-global-symmetries-and-completeness-of-the-3mod4d67b4.pdf

Non-Invertible Global Symmetries and Completeness of the Spectrum

The orbital motion of electrons in a three-dimensional solid can generate a pseudoscalar magnetoelectric coupling $\ensuremath{\theta}$, a fact we derive for the single-particle case using a recent theory of polarization in weakly inhomogeneous materials. This polarizability $\ensuremath{\theta}$ is the same parameter that appears in the ``axion electrodynamics'' Lagrangian $\ensuremath{\Delta}{\mathcal{L}}_{EM}=(\ensuremath{\theta}{e}^{2}/2\ensuremath{\pi}h)\mathbf{E}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{B}$, which is known to describe the unusual magnetoelectric properties of the three-dimensional topological insulator ($\ensuremath{\theta}=\ensuremath{\pi}$). We compute $\ensuremath{\theta}$ for a simple model that accesses the topological insulator and discuss its connection to the surface Hall conductivity. The orbital magnetoelectric polarizability can be generalized to the many-particle wave function and defines the 3D topological insulator, like the integer quantum Hall effect, in terms of a topological ground-state response function.

http://absimage.aps.org/image/MAR09/MWS_MAR09-2008-000540.pdf

Magnetoelectric polarizability and axion electrodynamics in crystalline insulators

Recent developments in chiral and spin polarization effects in heavy-ion collisions

We identify infinitely many noninvertible generalized global symmetries in QED and QCD for the real world in the massless limit. In QED, while there is no conserved Noether current for the U(1)_{A} axial symmetry because of the Adler-Bell-Jackiw anomaly, for every rational angle 2πp/N, we construct a conserved and gauge-invariant topological symmetry operator. Intuitively, it is a composition of the axial rotation and a fractional quantum Hall state coupled to the electromagnetic U(1) gauge field. These conserved symmetry operators do not obey a group multiplication law, but a noninvertible fusion algebra. They act invertibly on all local operators as axial rotations, but noninvertibly on the 't Hooft lines. We further generalize our construction to QCD, and show that the coupling π^{0}F∧F in the effective pion Lagrangian is necessary to match these noninvertible symmetries in the UV. Therefore, the conventional argument for the neutral pion decay using the ABJ anomaly is now rephrased as a matching condition of a generalized global symmetry.

/pdf/noninvertible-global-symmetries-in-the-standard-model-1khhwmc4.pdf

Noninvertible Global Symmetries in the Standard Model.

We investigate a higher-group structure of massless axion electrodynamics in (3 + 1) dimensions. By using the background gauging method, we show that the higher-form symmetries necessarily have a global semistrict 3-group (2-crossed module) structure, and exhibit ’t Hooft anomalies of the 3-group. In particular, we find a cubic mixed ’t Hooft anomaly between 0-form and 1-form symmetries, which is specific to the higher-group structure.

/pdf/global-3-group-symmetry-and-t-hooft-anomalies-in-axion-3pspth27e6.pdf

Global 3-group symmetry and ’t Hooft anomalies in axion electrodynamics

Higher-form symmetries and 3-group in axion electrodynamics

Deconfining phase boundary of rapidly rotating hot and dense matter and analysis of moment of inertia

We discuss the counting of Nambu-Goldstone (NG) modes associated with the spontaneous breaking of higher-form global symmetries. Effective field theories of NG modes are developed based on symmetry-breaking patterns, using a generalized coset construction for higher-form symmetries. We derive a formula of the number of gapless NG modes, which involves expectation values of the commutators of conserved charges, possibly of different degrees.

Counting Nambu-Goldstone Modes of Higher-Form Global Symmetries.

We derive the chiral kinetic theory under the presence of a gravitational Riemann curvature. It is well-known that in the chiral kinetic theory there inevitably appears a redundant ambiguous vector corresponding to the choice of the Lorentz frame. We reveal that on top of this conventional frame choosing vector, higher-order quantum correction to the chiral kinetic theory brings an additional degrees of freedom to specify the distribution function. Based on this framework, we derive new types of fermionic transport, that is, the charge current and energy-momentum tensor induced by the gravitational Riemann curvature. Such novel phenomena arise not only under genuine gravity but also in a (pseudo-)relativistic fluid, for which inhomogeneous vorticity or temperature are effectively represented by spacetime metric tensor. It is especially found that the charge and energy currents are antiparallelly induced by an inhomogeneous fluid vorticity (more generally, by the Ricci tensor R0i), as a consequence of the spin-curvature coupling. We also briefly discuss possible applications to Weyl/Dirac semimetals and heavy-ion collision experiments.

/pdf/second-order-chiral-kinetic-theory-under-gravity-and-274d7b69bz.pdf

Yoshimasa Hidaka

Papers

Global 3-group symmetry and ’t Hooft anomalies in axion electrodynamics

Higher-form symmetries and 3-group in axion electrodynamics

Deconfining phase boundary of rapidly rotating hot and dense matter and analysis of moment of inertia

Counting Nambu-Goldstone Modes of Higher-Form Global Symmetries.

Second order chiral kinetic theory under gravity and antiparallel charge-energy flow