General Relativity & Quantum Cosmology

Presented is a discussion of quantum field theory on curved spacetime and of microlocal analysis, with an emphasis on the way that these two areas connected for me personally through a specific problem, namely that of resolving Kay's singularity conjecture for two point functions of a linear scalar field on a globally hyperbolic spacetime. A particular case of this conjecture is presented, namely the translation invariant case on flat Minkowski spacetime, which does not require microlocal analysis. Next, the results of Duistermaat and Hoermander concerning distinguished parametrices of the Klein Gordon equation on a curved spacetime are described, since they lead to the notion of a wave front set (or microlocal) spectral condition, which could be viewed as a remnant of the spectral condition on flat spacetime. This condition on the wavefront set of the two point function has been employed by Brunetti, Koehler and Fredenhagen to develop a method of renormalization on a general curved spacetime, which has been developed further by Hollands and Wald. Other QFT-related topics to which microlocal methods may apply are: Lorentz symmetry breaking models and many body QM models (e.g., the free electron gas in a metal). In the case of vector or spinor models, the polarization set may be used to refine information about the singularities. Similarly, the principal symbol of the two point function, viewed as a Fourier integral operator, is a constant times a canonical half density on the natural Lagrangian submanifold associated with the Klein-Gordon operator, suggesting a tangent space Lorentz invariance property for the free model.

Motivated by spin-wave continuum (SWC) observed in recent neutron scattering experiments in Herbertsmithite, we use Gutzwiller-projected wave functions to study dynamic spin structure factor of spin liquid states on the kagome lattice. As their ground state, spin-1 excited states for spin liquids are represented by Gutzwiller-projected two-spinon excited wave functions. We investigate three different spin liquid candidates, spinon Fermi-surface spin liquid (FSL), Dirac spin liquid (DSL) and random-flux spin liquid (RSL). We find that DSL has no explicit contradiction with experiments. Besides a fractionalized spin moment, DSL has a fractionalized crystal momentum which is also detectable directly in the neutron scattering measurements.

We study non-Fermi liquids that arise at the quantum critical points associated with the spin and charge density wave transitions in metals with the C2 symmetry. We use the dimensional regularization scheme, where a one-dimensional Fermi surface is embedded in 3 − epsilon dimensional momentum space. In three dimensions, marginal Fermi liquids arise at the spin and charge density wave critical points. Below three dimensions, a perturbative anisotropic non-Fermi liquid is realized at the spin density wave critical point, where not only time but also different spatial coordinates develop distinct anomalous dimensions. On the other hand, the perturbative expansion breaks down at the charge density wave critical point immediately below three dimensions.

Motivated by results of DMRG and tensor network simulations on doped $t$-$J$ model on honeycomb lattice, we study superconductivity of singlet and triplet pairing in this model. We show that a superconducting state with coexisting spin-singlet and spin-triplet pairings is induced by the antiferromagnetic order near half-filling. The superconducting state we obtain has a topological phase transition that separates a topologically trivial state and a non-trivial state with Chern number two.

We studied t1-t2-J1-J2 model on Honeycomb lattice at finite doping. We find that when t_1 is very small, the t1-t2-J1-J2 model on Honeycomb lattice may be in a supperconducting phase. Such a supperconducting phase is not driven by the pairing, but by entanglement.

We discuss two basic principles to unify the understanding of both cuprates and iron-based superconductors: (1) the correspondence principle— the short range magnetic exchange interactions and the Fermi surfaces act collaboratively to achieve high Tc superconductivity and determine pairing symmetries; (2) the selective magnetic pairing rule: the superconductivity is only induced by the magnetic exchange couplings from the superexchange mechanism through cation-anion-cation chemical bondings but not those from direct exchange couplings resulted from the direct cation's d-d chemical bondings. These two principles provide a unified explanation why the d-wave pairing symmetry and the s-wave pairing symmetry are robust respectively in cuprates and iron-based superconductors. In the meanwhile, the above two principles can serve as direct guiding rules to search new unconventional high Tc superconductors. We propose that the third classes of unconventional high Tc superconducting candidates in compounds formed by cation-anion trigonal bipyramidal complexes with a d7 filling configuration on the cation ions. Their superconducting states are expected to be dominated by the d+id pairing symmetry. Synthesizing these compounds and verifying the prediction can convincingly establish the high Tc superconducting mechanism and pave a way to design new high Tc superconductors

In underdoped cuprate superconductors, a rich competition occurs between superconductivity and charge density wave order (CDW). Under debate, however, is whether rotational symmetry breaking (nematicity) also plays a central role -- whether it occurs intrinsically and generically or merely as a consequence of other orders. Here we employ resonant x-ray scattering in stripe-ordered (La,X)2CuO4 to probe the relationship between electronic nematicity of the Cu 3d orbitals, structural orthorhombicity of the (La,X)2O4 layers and CDW order. We find distinct temperature dependences of the structural orthorhombicity and the electronic nematicity, with the electronic nematicity, but no structural orthorhombicity enhancement below the onset of CDW order. These results indicate electronic nematicity is an order parameter that is distinct from a purely structural order parameter in underdoped cuprates.

The ground state phase diagram of spin-1/2 J1-J2 antiferromagnetic Heisenberg model on square lattice around the maximally frustrated regime (J2~0.5J1) has been debated for decades. I will discuss some progresses on this old problem based on recent numerical results from density matrix renormalization group(DMRG) and tensor product states algorithms.