In Ref. 1, a method was given for self-consistently generating sum-of-squares decompositions of quartic fermionic Hamiltonians. Perturbation theory was used to generate a useful choice of cubic operators in this sum-of-squares. On a range of model problems, this method, which is only a fragment of degree-six sum-of-squares, was able to outperform the full degree-four sum-of-squares in both speed and accuracy. Unfortunately for applications, many problems in chemistry have strong density-density interaction terms, as well as moderately strong density-dependent hopping and spin-spin interaction terms, limiting the power of the perturbative choice of the cubic operators. Here we propose a method for generating these decompositions in the presence of these strong interaction terms, hopefully extending the range of applicability of this method.
In Ref. 1, a method was given for self-consistently generating sum-of-squares decompositions of quartic fermionic Hamiltonians. Perturbation theory was used to generate a useful choice of cubic operators in this sum-of-squares. On a range of model problems, this method, which is only a fragment of degree-six sum-of-squares, was able to outperform the full degree-four sum-of-squares in both speed and accuracy. Unfortunately for applications, many problems in chemistry have strong density-density interaction terms, as well as moderately strong density-dependent hopping and spin-spin interaction terms, limiting the power of the perturbative choice of the cubic operators. Here we propose a method for generating these decompositions in the presence of these strong interaction terms, hopefully extending the range of applicability of this method.
The sum-of-squares (SoS) hierarchy is a powerful technique based on semi-definite programming that can be used for both classical and quantum optimization problems. This hierarchy goes under several names; in particular, in quantum chemistry it is called the reduced density matrix (RDM) method. We consider the ability of this hierarchy to reproduce weak coupling perturbation theory for three different kinds of systems: spin (or qubit) systems, bosonic systems (the anharmonic oscillator), and fermionic systems with quartic interactions. For such fermionic systems, we show that degree-4 SoS (called 2-RDM in quantum chemsitry) does not reproduce second order perturbation theory but degree-6 SoS (3-RDM) does (and we conjecture that it reproduces third order perturbation theory). Indeed, we identify a fragment of degree-6 SoS which can do this, which may be useful for practical quantum chemical calculations as it may be possible to implement this fragment with less cost than the full degree-6 SoS. Remarkably, this fragment is very similar to one studied by Hastings and O'Donnell for the Sachdev-Ye-Kitaev (SYK) model.
The earlier proposed method for exact determination of the energy spectra of Hamiltonians H and the quantities , where W is a finite-dimensional perturbation, is extended to treat a still larger class of Hamiltonians. In particular, some two- and three-dimensional problems became amenable to this approach. In addition, the expressions for the density and local densities of states are deduced. As an example, an impurity problem is considered in a simple cubic lattice with short-range and in a linear chain with long-range interactions.
This is a collection of various result and notes, addressing the sum-of-squares hierarchy for spin and fermion systems using some ideas from quantum field theory, including higher order perturbation theory, critical phenomena, nonlocal coupling in time, and auxiliary field Monte Carlo. This paper should be seen as a sequel to Refs. 1,2. Additionally in this paper, we consider the difficulty of approximating the ground state energy of the Sachdev-Ye-Kitaev (SYK) model using other methods. We provide limitations on the power of the Lanczos method, starting with a Gausian wavefunction, and on the power of a sum of Gaussian wavefunctions (in this case under an assumption).
The sum-of-squares method can give rigorous lower bounds on the energy of quantum Hamiltonians. Unfortunately, typically using this method requires solving a semidefinite program, which can be computationally expensive. Further, the typically used degree-4 sum-of-squares (also known as the 2RDM method) does not correctly reproduce second order perturbation theory. Here, we give a general method, an analogue of Wigner's 2n+1 rule for perturbation theory, to compute the order of the error in a given sum-of-squares ansatz. We also give a method for finding solutions of the dual semidefinite program, based on a perturbative ansatz combined with a self-consistent method. As an illustration, we show that for a class of model Hamiltonians (with a gap in the quadratic term and quartic terms chosen as i.i.d. Gaussians), this self-consistent sum-of-squares method significantly improves over the 2RDM method in both speed and accuracy, and also improves over low order perturbation theory. We then explain why the particular ansatz we implement is not suitable for use for quantum chemistry Hamiltonians (due to presence of certain large diagonal terms), but we suggest a modified ansatz that may be suitable, which will be the subject of future work.
The many-body problem cannot in general be solved exactly, and one of the most prominent approximations is to build perturbation expansions. A huge variety of expansions are possible, which differ by the quantity to be expanded, the expansion variable, the starting point, and ideas of how to re-sum or terminate the series. Although much has been discussed and much has been done, some choices were made for historical reasons, in particular, limited computation or storage capacities. The present work aims to examine the justification for different choices made in different contexts, by comparing the components of functionals based on Green's functions on the one side, and on the charge density on the other side. Of particular interest will be the question of how to build an optimal starting point for the approximation of non-local quantities, making use of near- or far-sightedness, and daring to consider models beyond the homogeneous electron gas. This will include the use of connector approximations. We will also discuss why it is a good idea to build functionals of the density.
The subject of perturbation theory in non-relativistic quantum mechanics is introduced. First, perturbation theory for stationary states in the absence of degeneracy is studied. The case of nearby levels, perturbations of a one-dimensional harmonic oscillator, the occurrence of degeneracy, Stark and Zeeman effects are then described in detail. After a brief outline of the variational method, the Dyson series for time-dependent perturbation theory is derived, with application to harmonic perturbations. The Fermi golden rule is also presented. The chapter ends with an assessment of four branches of the subject: regular perturbation theory, asymptotic perturbation theory, and summability methods, spectral concentration and singular perturbations. Approximate methods for stationary states It is frequently the case, in physical problems, that the full Hamiltonian operator H consists of an operator H 0 whose eigenvalues and eigenfunctions are known exactly, and a second term V resulting from a variety of sources (e.g. the interaction with an electric field, or a magnetic field, or the relativistic corrections to the kinetic energy, or the effects of spin). The problems and aims of perturbation theory for stationary states can be therefore summarized as follows. (i) Once the domain of (essential) self-adjointness of the ‘unperturbed’ Hamiltonian H 0 is determined, find on which domain the ‘sum’ H = H 0 + V represents an (essentially) self-adjoint operator (the reader should remember from appendix 4. A that in general, once some operators A and B are given, the intersection of their domains, on which their sum is defined, might be the empty set). […]