We introduce the notion of infinitesimal variations of mixed Hodge structures and invariants associated to them. We describe these invariants in the case of a pair $(X,Y)$ with $X$ a Fano 3-fold and $Y$ a smooth anticanonical K3 surface and in more detail in the case when $X$ is a cubic threefold. In this last setting, we obtain a generic global Torelli theorem for pairs.

Rodolfo Aguilar Mark Green Phillip Griffiths

Using the general framework due to Donagi-Markman \cite{DM} and Markushevich \cite{M} we shall derive an expression for the differential of Abel-Jacobi mappings on Fano threefolds. This formula involves information normal to the Lagrangian submanifolds constructed in \cite{DM} and \cite{M}. It may be applied to give new proofs of a number of classical results about these varieties.

Rodolfo Aguilar Mark Green Phillip Griffiths

We introduce a class of embedded CR manifolds satisfying a geometric condition that we call weak $Y(q)$. For such manifolds, we show that dbar-b has closed range on $L^2$ and that the complex Green operator is continuous on $L^2$. Our methods involves building a weighted norm from a microlocal decomposition. We also prove that at any Sobolev level there is a weight such that the complex Green operator inverting the weighted Kohn Laplacian is continuous. Thus, we can solve the dbar-b equation in $C^\infty$.

Phillip Harrington Andrew Raich

We show exactly with an SU(N) interacting model that even if the ambiguity associated with the placement of the chemical potential, $\mu$, for a T=0 gapped system is removed by using the unique value $\mu(T\rightarrow 0)$, Luttinger's sum rule is violated even if the ground-state degeneracy is lifted by an infinitesimal hopping. The failure stems from the non-existence of the Luttinger-Ward functional for a system in which the self-energy diverges. Since it is the existence of the Luttinger-Ward functional that is the basis for Luttinger's theorem which relates the charge density to sign changes of the single-particle Green function, no such theorem exists. Experimental data on the cuprates are presented which show a systematic deviation from the Luttinger count, implying a breakdown of the electron quasiparticle picture in strongly correlated electron matter.

Kiaran B. Dave Philip W. Phillips Charles L. Kane

In Paper II [N. G. Phillips and B. L. Hu, previous abstract] we presented the details for the regularization of the noise kernel of a quantum scalar field in optical spacetimes by the modified point separation scheme, and a Gaussian approximation for the Green function. We worked out the regularized noise kernel for two examples: hot flat space and optical Schwarzschild metric. In this paper we consider noise kernels for a scalar field in the Schwarzschild black hole. Much of the work in the point separation approach is to determine how the divergent piece conformally transforms. For the Schwarzschild metric we find that the fluctuations of the stress tensor of the Hawking flux in the far field region checks with the analytic results given by Campos and Hu earlier [A. Campos and B. L. Hu, Phys. Rev. D {\bf 58} (1998) 125021; Int. J. Theor. Phys. {\bf 38} (1999) 1253]. We also verify Page's result [D. N. Page, Phys. Rev. {\bf D25}, 1499 (1982)] for the stress tensor, which, though used often, still lacks a rigorous proof, as in his original work the direct use of the conformal transformation was circumvented. However, as in the optical case, we show that the Gaussian approximation applied to the Green function produces significant error in the noise kernel on the Schwarzschild horizon. As before we identify the failure as occurring at the fourth covariant derivative order.

Nicholas G Phillips B. L. Hu

We report an implementation of self-consistent Green's function many-body theory within a second-order approximation (GF2) for application with molecular systems. This is done by iterative solution of the Dyson equation expressed in matrix form in an atomic orbital basis, where the Green's function and self-energy are built on the imaginary frequency and imaginary time domain respectively, and fast Fourier transform is used to efficiently transform these quantities as needed. We apply this method to several archetypical examples of strong correlation, such as a H$_{32}$ finite lattice that displays a highly multireference electronic ground state even at equilibrium lattice spacing. In all cases GF2 gives a physically meaningful description of the metal to insulator transition in these systems, without resorting to spin-symmetry breaking. Our results show that self-consistent Green's function many-body theory offers a viable route to describing strong correlations while remaining within a computationally tractable single-particle formalism.

Jordan J. Phillips Dominika Zgid

The temperature-dependent Matsubara Green's function that is used to describe temperature-dependent behavior is expressed on a numerical grid. While such a grid usually has a couple of hundred points for low-energy model systems, for realistic systems in large basis sets the size of an accurate grid can be tens of thousands of points, constituting a severe computational and memory bottleneck. In this paper, we determine efficient imaginary time grids for the temperature-dependent Matsubara Green's function formalism that can be used for calculations on realistic systems. We show that due to the use of orthogonal polynomial transform, we can restrict the imaginary time grid to few hundred points and reach micro-Hartree accuracy in the electronic energy evaluation. Moreover, we show that only a limited number of orthogonal polynomial expansion coefficients are necessary to preserve accuracy when working with a dual representation of Green's function or self-energy and transforming between the imaginary time and Matsubara frequency domain.

Alexei A. Kananenka Jordan J. Phillips Dominika Zgid

One-body Green's function theories implemented on the real frequency axis offer a natural formalism for the unbiased theoretical determination of quasiparticle spectra in molecules and solids. Self-consistent Green's function methods employing the imaginary axis formalism on the other hand can benefit from the iterative implicit resummation of higher order diagrams that are not included when only the first iteration is performed. Unfortunately, the imaginary axis Green's function does not give direct access to the desired quasiparticle spectra, which undermines its utility. To this end we investigate how reliably one can calculate quasiparticle spectra from the Extended Koopmans' Theorem (EKT) applied to the imaginary time Green's function in a second order approximation (GF2). We find that EKT in conjunction with GF2 yields IPs and EAs that systematically underestimate experimental and accurate coupled-cluster reference values for a variety of molecules and atoms. This establishes that the EKT allows one to utilize the computational advantages of an imaginary axis implementation, while still being able to acquire real axis spectral properties. Because the EKT requires negligible computational effort, and can be used with a Green's function from any level of theory, we conclude that it is a potentially very useful tool for the systematic study of quasiparticle spectra in realistic systems.

Alicia Rae Welden Jordan J. Phillips Dominika Zgid

We report on a strong coupling approach (on-site Coulomb repulsion, U larger than the nearest-neighbour hopping energy |t|) to the Hubbard model. Starting from the Hubbard operators which diagonalize the interaction term, we generate a hierarchy of composite operators from the equations of motion. Using the Hubbard operators as a basis, we are able to compute the associated Green functions including the anomalous Green functions which describe pair formation. We show explicitly that these anomalous Green functions are non-zero in the d_{x^2-y^2} channel; however, the entities that pair up are not single electron-like particles but rather composite excitations (which we call cexons) made out of an electron and a hole on nearest-neighbour sites. Cexons are fermionic in nature as they have spin 1/2 and also have unit charge. Our calculations of the chemical potential reveal that negative compressibility in the 2D Hubbard model and composite excitation pairing are intimately connected, namely, the larger the negative compressibility, the larger the pairing amplitude. Our observation of negative compressibility in the under-doped regime is consistent with phase segregation or stripe formation in the normal state. While pairing ameliorates the negative compresssibility, it does not eliminate it entirely. In addition, we find that the anomalous correlation functions are particle-hole symmetric and exhibit a maximum at a doping level of roughly 10% as measured from half-filling. For U=8|t|, the onset temperature for pair formation is 0.02|t|.

Tudor D. Stanescu Ivar Martin Philip Phillips

We review several of the normal state properties of the cuprates in an attempt to establish an organizing principle from which pseudogap phenomena, broad spectral features, $T-$linear resistivity, and spectral weight transfer emerge. We first show that standard field theories with a single critical length scale cannot capture the $T-$linear resistivity as long as the charge carriers are critical. What seems to be missing is an additional length scale, which may or may not be critical. Second, we prove a generalised version of Luttinger's theorem for a Mott insulator. Namely, regardless of the spatial dimension, the Fermi surface of the non-interacting system is converted into a surface of zeros of the single-particle Green function when the Mott insulator posesses particle-hole symmetry. Only in the presence of particle-hole symmetry does the volume of the surface of zeros equal the particle density. The surface of zeros persists at finite doping and is shown to provide a framework from which pseudogaps, broad spectral features, spectral weight transfer on the Mott gap scale can be understood.

Philip Phillips