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We present a theory of optimal topological textures in nonlinear sigma-models with degrees of freedom living in the Grassmannian $mathrm{Gr}(M,N)$ manifold. These textures describe skyrmion lattices of $N$-component fermions in a quantising magnetic field, relevant to the physics of graphene, bilayer and other multicomponent quantum Hall systems near integer filling factors $ u>1$. We derive analytically the optimality condition, minimizing topological charge density fluctuations, for a general Grassmannian sigma model $mathrm{Gr}(M,N)$ on a sphere and a torus, together with counting arguments which show that for any filling factor and number of components there is a critical value of topological charge $d_c$ above which there are no optimal textures. Below $d_c$ a solution of the optimality condition on a torus is unique, while in the case of a sphere one has, in general, a continuum of solutions corresponding to new {it non-Goldstone} zero modes, whose degeneracy is not lifted (via a order from disorder mechanism) by any fermion interactions depending only on the distance on a sphere. We supplement our general theoretical considerations with the exact analytical results for the case of $mathrm{Gr}(2,4)$, appropriate for recent experiments in graphene.
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We show that the topological central charge of a topological phase can be directly accessed from the ground-state wavefunctions for a system on a surface as a Berry curvature produced by adiabatic variation of the metric on the surface, at least up to addition of another topological invariant that arises in some cases. For trial wavefunctions that are given by conformal blocks (chiral correlation functions) in a conformal field theory (CFT), we carry out this calculation analytically, using the hypothesis of generalized screening. The topological central charge is found to be that of the underlying CFT used in the construction, as expected. The calculation makes use of the gravitational anomaly in the chiral CFT. It is also shown that the Hall conductivity can be obtained in an analogous way from the U($1$) gauge anomaly.
We develop a nonperturbative approach to the bulk polarization of crystalline electric insulators in $dgeq1$ dimensions. Formally, we define polarization via the response to background fluxes of both charge and lattice translation symmetries. In this approach, the bulk polarization is related to properties of magnetic monopoles under translation symmetries. Specifically, in $2d$ the monopole is a source of $2pi$-flux, and the polarization is determined by the crystal momentum of the $2pi$-flux. In $3d$ the polarization is determined by the projective representation of translation symmetries on Dirac monopoles. Our approach also leads to a concrete scheme to calculate polarization in $2d$, which in principle can be applied even to strongly interacting systems. For open boundary condition, the bulk polarization leads to an altered `boundary Luttinger theorem (constraining the Fermi surface of surface states) and also to modified Lieb-Schultz-Mattis theorems on the boundary, which we derive.
We consider extended Hubbard models with repulsive interactions on a Honeycomb lattice and the transitions from the semi-metal phase at half-filling to Mott insulating phases. In particular, due to the frustrating nature of the second-neighbor repulsive interactions, topological Mott phases displaying the quantum Hall and the quantum spin Hall effects are found for spinless and spinful fermion models, respectively. We present the mean-field phase diagram and consider the effects of fluctuations within the random phase approximation (RPA). Functional renormalization group analysis also show that these states can be favored over the topologically trivial Mott insulating states.
Over the last 100 years, the group-theoretic characterization of crystalline solids has provided the foundational language for diverse problems in physics and chemistry. There exist two classes of crystalline solids: nonmagnetic crystals left invariant by space groups (SGs), and solids with commensurate magnetic order that respect the symmetries of magnetic space groups (MSGs). Whereas many of the properties of the SGs, such as their momentum-space corepresentations (coreps) and elementary band coreps (EBRs) were tabulated with relative ease, progress on deriving the analogous properties of the MSGs has largely stalled for the past 70 years due to the complicated symmetries of magnetic crystals. In this work, we complete the 100-year-old problem of crystalline group theory by deriving the small coreps, momentum stars, compatibility relations, and magnetic EBRs (MEBRs) of the single (spinless) and double (spinful) MSGs. We have implemented freely-accessible tools on the Bilbao Crystallographic Server for accessing the coreps of the MSGs, whose wide-ranging applications include neutron diffraction investigations of magnetic structure, the interplay of lattice regularization and (symmetry-enhanced) fermion doubling, and magnetic topological phases, such as axion insulators and spin liquids. Using the MEBRs, we extend the earlier theory of Topological Quantum Chemistry to the MSGs to form a complete, real-space theory of band topology in magnetic and nonmagnetic crystalline solids - Magnetic Topological Quantum Chemistry (MTQC). We then use MTQC to derive the complete set of symmetry-based indicators (SIs) of band topology in all spinful (fermionic) crystals, for which we identify symmetry-respecting bulk and anomalous surface and hinge states. Lastly, using the SIs, we discover several novel non-axionic magnetic higher-order topological insulators.
The electrostatics arising in ferroelectric/dielectric two-dimensional heterostructures and superlatitices is revisited here within a simplest Kittel model, in order to define a clear paradigmatic reference for domain formation. The screening of the depolarizing field in isolated ferroelectric or polar thin films via the formation of 180$^{circ}$ domains is well understood, whereby the width of the domains $w$ grows as the square-root of the film thickness $d$, following Kittels law, for thick enough films ($wll d$). This behavior is qualitatively unaltered when the film is deposited on a dielectric substrate, sandwiched between dielectrics, and even in a superlattice setting, with just a suitable renormalisation of Kittels length. As $d$ decreases, $w(d)$ deviates from Kittels law, reaching a minimum and then diverging onto the mono-domain limit for thin enough films, always assuming a given spontaneous polarization $P$ of the ferrolectric, only modified by linear response to the depolarizing field. In most cases of experimental relevance $P$ would vanish before reaching that thin-film regime. This is not the case for superlattices. Unlike single films, for which the increase of the dielectric constant of the surrounding medium pushes the deviation from the Kittels regime to lower values of $d$, there is a critical value of the relative thickness of ferroelectric/dielectric films in superlattices beyond which that behavior is reversed, and which defines the separation between strong and weak ferroelectric coupling in superlattices.
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