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Register a new userFrom real materials to model Hamiltonians with density matrix downfolding
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Due to advances in computer hardware and new algorithms, it is now possible to perform highly accurate many-body simulations of realistic materials with all their intrinsic complications. The success of these simulations leaves us with a conundrum: how do we extract useful physical models and insight from these simulations? In this article, we present a formal theory of downfolding--extracting an effective Hamiltonian from first-principles calculations. The theory maps the downfolding problem into fitting information derived from wave functions sampled from a low-energy subspace of the full Hilbert space. Since this fitting process most commonly uses reduced density matrices, we term it density matrix downfolding (DMD).
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We propose an electron-phonon parameterization which reliably reproduces the geometry and harmonic frequencies of a real system. With respect to standard electron-phonon models, it adds a double-counting correction, which takes into account the lattice deformation as the system is dressed by low-energy electron-phonon processes. We show the importance of this correction by studying potassium-doped picene (K$_3$Picene), recently claimed to be a superconductor with a $T_c$ of up to 18 K. The Hamiltonian parameters are derived from ab-initio density functional theory, and the lattice model is solved by dynamical mean-field theory. Our calculations include the effects of electron-electron interactions and local electron-phonon couplings. Even with the inclusion of a strongly coupled molecular phonon, the Hubbard repulsion prevails and the system is an insulator with a small Mott gap of $approx$ 0.2 eV.
We propose a way of obtaining effective low energy Hubbard-like model Hamiltonians from ab initio Quantum Monte Carlo calculations for molecular and extended systems. The Hamiltonian parameters are fit to best match the ab initio two-body density matrices and energies of the ground and excited states, and thus we refer to the method as ab initio density matrix based downfolding (AIDMD). For benzene (a finite system), we find good agreement with experimentally available energy gaps without using any experimental inputs. For graphene, a two dimensional solid (extended system) with periodic boundary conditions, we find the effective on-site Hubbard $U^{*}/t$ to be $1.3 pm 0.2$, comparable to a recent estimate based on the constrained random phase approximation. For molecules, such parameterizations enable calculation of excited states that are usually not accessible within ground state approaches. For solids, the effective Hamiltonian enables large scale calculations using techniques designed for lattice models.
Charge-density waves are responsible for symmetry-breaking displacements of atoms and concomitant changes in the electronic structure. Linear response theories, in particular density-functional perturbation theory, provide a way to study the effect of displacements on both the total energy and the electronic structure based on a single ab initio calculation. In downfolding approaches, the electronic system is reduced to a smaller number of bands, allowing for the incorporation of additional correlation and environmental effects on these bands. However, the physical contents of this downfolded model and its potential limitations are not always obvious. Here, we study the potential-energy landscape and electronic structure of the Su-Schrieffer-Heeger (SSH) model, where all relevant quantities can be evaluated analytically. We compare the exact results at arbitrary displacement with diagrammatic perturbation theory both in the full model and in a downfolded effective single-band model, which gives an instructive insight into the properties of downfolding. An exact reconstruction of the potential-energy landscape is possible in a downfolded model, which requires a dynamical electron-biphonon interaction. The dispersion of the bands upon atomic displacement is also found correctly, where the downfolded model by construction only captures spectral weight in the target space. In the SSH model, the electron-phonon coupling mechanism involves exclusively hybridization between the low- and high-energy bands and this limits the computational efficiency gain of downfolded models.
Quantum embedding methods have become a powerful tool to overcome deficiencies of traditional quantum modelling in materials science. However while these can be accurate, they generally lack the ability to be rigorously improved and still often rely on empirical parameters. Here, we reformulate quantum embedding to ensure the ability to systematically converge properties of real materials with accurate correlated wave function methods, controlled by a single, rapidly convergent parameter. By expanding supercell size, basis set, and the resolution of the fluctuation space of an embedded fragment, we show that the systematic improvability of the approach yields accurate structural and electronic properties of realistic solids without empirical parameters, even across changes in geometry. Results are presented in insulating, semi-metallic, and more strongly correlated regimes, finding state of the art agreement to experimental data.
Motivated by the existence of exact many-body quantum scars in the AKLT chain, we explore the connection between Matrix Product State (MPS) wavefunctions and many-body quantum scarred Hamiltonians. We provide a method to systematically search for and construct parent Hamiltonians with towers of exact eigenstates composed of quasiparticles on top of an MPS wavefunction. These exact eigenstates have low entanglement in spite of being in the middle of the spectrum, thus violating the strong Eigenstate Thermalization Hypothesis (ETH). Using our approach, we recover the AKLT chain starting from the MPS of its ground state, and we derive the most general nearest-neighbor Hamiltonian that shares the AKLT quasiparticle tower of exact eigenstates. We further apply this formalism to other simple MPS wavefunctions, and derive new families of Hamiltonians that exhibit AKLT-like quantum scars. As a consequence, we also construct a scar-preserving deformation that connects the AKLT chain to the integrable spin-1 pure biquadratic model. Finally, we also derive other families of Hamiltonians that exhibit new types of exact quantum scars, including a $U(1)$-invariant perturbed Potts model.
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