Compact numerical solutions to the two-dimensional repulsive Hubbard model obtained via nonunitary similarity transformations

Abstract

Similarity transformation of the Hubbard Hamiltonian using a Gutzwiller correlator leads to a non-Hermitian effective Hamiltonian, which can be expressed exactly in momentum-space representation, and contains three-body interactions. We apply this methodology to study the two-dimensional Hubbard model with repulsive interactions near half-filling in the intermediate interaction strength regime (U/t = 4). We show that at optimal or near optimal strength of the Gutzwiller correlator, the similarity transformed Hamiltonian has extremely compact right eigenvectors, which can be sampled to high accuracy using the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method, and its initiator approximation.Near-optimal correlators can be obtained using a simple projective equation, thus obviating the need for a numerical optimisation of the correlator. The FCIQMC method, as a projective technique, is well-suited for such non-Hermitian problems, and its stochastic nature can handle the 3-body interactions exactly without undue increase in computational cost. The highly compact nature of the right eigenvectors means that the initiator approximation in FCIQMC is not severe, and that large lattices can be simulated, well beyond the reach of the method applied to the original Hubbard Hamiltonian. Results are provided in lattice sizes up to 50 sites and compared to auxiliary-field QMC. New benchmark results are provided in the off half-filling regime, with no severe sign-problem being encountered. In addition, we show that methodology can be used to calculate excited states of the Hubbard model and lay the groundwork for the calculation of observables other than the energy. |e J − e ex | J = 0.0 J = -0.2 J = -0.25 σ 0.0 σ −0.2 σ −0.25 10 3 10 4 10 5 10 6 10 7 N w J = 0.0 J = -0.3 J = -0.5

Easing the Monte Carlo sign problem

“…In an attempt to ease the sign problem of a given Hamiltonian, it is therefore natural to try and improve the average sign. For a few specific models, these improvements have been achieved by different means: For example, one can exploit known physics to find bases with improved average sign (14,22) that are often induced by sparse representations (17,23,24). For particular observables, one can also exploit clever decompositions of the Monte Carlo estimator into clusters with nonnegative sign (25)(26)(27)(28)(29)(30)(31).…”

“…The simplest examples of these choices include local Hadamard, Clifford, or unitary transformations. Most generally, one can allow quasi-local circuits that are efficiently computable ( 6 ), including not only short circuits and matrix product unitaries ( 15 , 16 ) but also invertible transformations ( 17 ).…”

Incremental treatments of the full configuration interaction problem

“…In addition, the use of alternative and approximative FCI solvers as a means to allow for larger (and faster) CASCI calculations remains to be explored within MBE‐FCI. Finally, in the spirit of recent work seeking to revitalize the idea of transcorrelation, 49–54 a natural, albeit non‐trivial extension of the current generation of MBE‐FCI will be to allow for expansions to spawn from a similarity‐transformed Hamiltonian, akin to what is found in equation‐of‐motion coupled cluster theory 55 . Its appealing traits as a correlated zeroth‐order formulation of electron correlation aside, the non‐Hermiticity of the theory will pose entirely new conceptual as well as technical challenges.…”

Similarity transformation of the electronic Schrödinger equation via Jastrow factorization

“…Although the form of this similarity transformed Hamiltonian has been known for a long time [21], to our knowledge it has never been treated in its full form until now. We retain all three-body terms, motivated in part by our recent study of the two-dimensional Hub-bard model using Gutzwiller similatrity transformations [16], in which we show that the 3-body terms do not incur a huge cost in the FCIQMC formalism, and their full treatment enables essentially exact results to be obtained. Furthermore, this study shows that the similarity transformations can help enormously in the study of strongly correlated systems, by significantly compactifying the right eigenvector of the ground state (which are generally highly multiconfigurational otherwise).…”

“…The ST-FCIQMC calculations were performed in standard valence correlation quantum chemical basis sets, cc-pVXZ, X=D,T,Q (with 14, 30 and 55 basis functions respectively). The non-Hermitian nature of the Hamiltonian, together with 3-body interactions, have previously been treated in FCIQMC [15,16] and implemented in the NECI code [27], and were further adapted for the molecular Hamiltonian presented here.…”