Self-consistent optimization of the trial wave-function in constrained path auxiliary field Quantum Monte Carlo using mixed estimators

We propose a new scheme to implement the self-consistent optimization of the trial wave-function in constrained path auxiliary field Quantum Monte Carlo (CP-AFQMC) in the framewok of natural orbitals. In this scheme, a new trial wave-function in the form of Slater determinant is constructed from the CP-AFQMC results by diagonalizing the mixed estimator of the one-body reduced density matrix. We compare two ways (from real and mixed estimators in CP-AFQMC) to calculate the one-body reduced density matrix in the self-consistent process and study the ground state of doped two dimensional Hubbard model to test the accuracy of the two schemes. By comparing the local density, occupancy, and ground state energy we find the scheme in which one-body reduced density matrix is calculated from mixed estimator is computational more efficient and provides more accurate result with less fluctuation. The local densities from mixed estimator scheme agree well with the numerically exact values. This scheme provides a useful tool for the study of strongly correlated electron systems.Comment: close to the published versio

Absence of Spin Liquid Phase in the J1−J2J_1-J_2 Heisenberg model on the Square Lattice

We perform an in-depth investigation of the phase diagram of the $J_1-J_2$ Heisenberg model on the square lattice. We take advantage of Density Matrix Renormalization Group and Fully-Augmented Matrix Product States methods and reach unprecedented accuracy with large bond dimensions. We utilize excited-level crossing analysis to pinpoint the phase transition points. It was believed before that there exists a narrow spin liquid phase sandwiched by the N\'eel antiferromagnetic (AFM) and valence bond solid (VBS) phases. Through careful finite size scaling of the level crossing points, we find a direct phase transition between the N\'eel AFM and VBS phases at $J_2/J_1 = 0.535(3)$, suggesting the absence of an intermediate spin liquid phase. We also provide accurate results for ground state energies for a variety of sizes, from which we find the transition between the N\'eel AFM and VBS phases is continuous. These results indicate the existence of a deconfined quantum critical point at $J_2/J_1 = 0.535(3)$ in the model. From the crossing of the first derivative of the energies with $J_2$ for different sizes, we also determine the precise location of the first order phase transition between the VBS and stripe AFM phases at $J_2/J_1=0.610(5)$.Comment: 4 pages, 4 figures, with supplementary material

On the Magnetization of the 120∘120^\circ order of the Spin-1/2 Triangular Lattice Heisenberg Model: a DMRG revisit

We revisit the issue about the magnetization of the $120^\circ$ order in the spin-1/2 triangular lattice Heisenberg model (TLHM) with Density Matrix Renormalization Group (DMRG). The accurate determination of the magnetization of this model is challenging for numerical methods and its value exhibits substantial disparities across various methods. We perform a large-scale DMRG calculation of this model by employing bond dimension as large as $D = 24000$ and by studying the system with width as large as $L_\mathrm{y} = 12$. With careful extrapolation with truncation error and suitable finite size scaling, we give a conservative estimation of the magnetization as $M_0 = 0.208(8)$. The ground state energy per site we obtain is $E_g = -0.5503(8)$. Our results provide valuable benchmark values for the development of new methods in the future.Comment: 6 pages, 6 figure

Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method

Ground-state properties of the Hubbard model on a two-dimensional square lattice are studied by the auxiliary-field quantum Monte Carlo method. Accurate results for energy, double occupancy, effective hopping, magnetization, and momentum distribution are calculated for interaction strengths of U/t from 2 to 8, for a range of densities including half-filling and n = 0.3,0.5,0.6, 0.75, and 0.875. At half-filling, the results are numerically exact. Away from half-filling, the constrained path Monte Carlo method is employed to control the sign problem. Our results are obtained with several advances in the computational algorithm, which are described in detail. We discuss the advantages of generalized Hartree-Fock trial wave functions and its connection to pairing wave functions, as well as the interplay with different forms of Hubbard-Stratonovich decompositions. We study the use of different twist angle sets when applying the twist averaged boundary conditions. We propose the use of quasirandom sequences, which improves the convergence to the thermodynamic limit over pseudorandom and other sequences. With it and a careful finite size scaling analysis, we are able to obtain accurate values of ground-state properties in the thermodynamic limit. Detailed results for finite-sized systems up to 16 x 16 are also provided for benchmark purposes

Coupling quantum Monte Carlo and independent-particle calculations: Self-consistent constraint for the sign problem based on the density or the density matrix

Quantum Monte Carlo (QMC) methods are one of the most important tools for studying interacting quantum many-body systems. The vast majority of QMC calculations in interacting fermion systems require a constraint to control the sign problem. The constraint involves an input trial wave function which restricts the random walks. We introduce a systematically improvable constraint which relies on the fundamental role of the density or one-body density matrix. An independent-particle calculation is coupled to an auxiliary-field QMC calculation. The independent-particle solution is used as the constraint in QMC, which then produces the input density or density matrix for the next iteration. The constraint is optimized by the self-consistency between the many-body and the independent-particle calculations. The approach is demonstrated in the two-dimensional Hubbard model by accurately determining the ground state when collective modes separated by tiny energy scales are present in the magnetic and charge correlations. Our approach also provides an ab initio way to predict effective interaction parameters for independent-particle calculations

Effective bi-layer model Hamiltonian and density-matrix renormalization group study for the high-Tc superconductivity in La3_{3}Ni2_{2}O7_{7} under high pressure

High-Tc superconductivity has been reported in the single crystal of La$_3$Ni$_2$O$_7$ under high pressure. Based on the electronic structure given from the density functional theory calculations, we propose an effective bi-layer model Hamiltonian including both $3d_{z^2}$ and $3d_{x^2-y^2}$ orbital electrons of the nickel cations. The main feature of the model is that the $3d_{z^2}$ electrons form inter-layer $\sigma$-bonding and anti-bonding bands via the apical oxygen anions between the two layers, while the $3d_{x^2-y^2}$ electrons hybridize with the $3d_{z^2}$ electrons within each NiO$_2$ plane. The chemical potential difference of these two orbital electrons ensures that the $3d_{z^2}$ orbitals are close to half-filling and the $3d_{x^2-y^2}$ orbitals are near quarter-filling. The strong on-site Hubbard repulsion of the $3d_{z^2}$ orbital electrons gives rise to an effective inter-layer antiferromagnetic spin super-exchange $J$. Applying pressure can increase the local coupling strength $J$ and self-dope holes on the $3d_{z^2}$ orbitals with the same amount of electrons doped on the $3d_{x^2-y^2}$ orbitals correspondingly. By performing numerical density-matrix renormalization group calculations on a plaqutte ladder, we find that the charge densities of both orbitals always have uniform distributions. But a spin-density-wave and a spin-orbital density-wave are developed in the small $J$ limit. In the large $J$ limit, both the spin and spin-orbital density waves get suppressed, and the electron pairing instability emerges due to the formation of inter-layer singlets of the $3d_{z^2}$ electrons. The strongest pairing correlation is given by the superconducting pair-density wave on the intra-layer vertical $3d_{x^2-y^2}$ bonds. Our numerical results have provided useful insights in the high-Tc superconductivity in single crystal La$_3$Ni$_2$O$_7$ under high pressure.Comment: 6 pages, 4 figures; some typos are correcte