Projector Quantum Monte Carlo Method Research Articles

Gaussian variational method to Fermi-Hubbard model in one and two dimensions

The study of ground-state properties of the Fermi-Hubbard model is a long-lasting task in the research of strongly correlated systems, and is crucial for the understanding of notable quantum phenomena including superconductivity and magnetism. Owing to the exponentially growing complexity of the system, a quantitative analysis usually demands high computational cost and is restricted to small samples, especially in two or higher dimensions. Here, we introduce a variational method in the frame of fermionic Gaussian states, and obtain the ground states of one- and two-dimensional attractive Hubbard models via imaginary-time evolution. We calculate the total energy and benchmark the results in a wide range of interaction strength and filling factor with those obtained via exact two-body results, the density matrix renormalization group based on matrix product states (MPS), and projector Quantum Monte Carlo (QMC) method. For both 1D and 2D cases, the Gaussian variational method presents accurate results for total energy with a maximum systematic error in the intermediate interaction region. The accuracy of these results has negligible dependence on the system size. We further calculate the double occupancy and find excellent agreement with MPS and QMC, as well as the experimental results of cold quantum gases in optical lattices. The results suggest that the Gaussian pairing state is a good approximation to the ground states of attractive Hubbard model, in particular in the strong and weak coupling limits. Moreover, we generalize the method to the attractive Hubbard model with a finite spin-polarization, which can be mapped to the repulsive interaction case via particle-hole transformation, and obtain accurate results of ground state energy and double occupancy. Our work demonstrates the ability of the Gaussian variational method to extract ground state properties of strongly correlated many-body systems with negligible computational cost, especially of large size and in higher dimensions.

The Influence of Long-Range Hopping on Ferromagnetism in the Hubbard Model on the Generalized Shastry-Sutherland Lattice

The projector quantum Monte-Carlo method is used to examine effects of long-range electron hopping on ferromagnetism in the Hubbard model on the generalized Shastry-Sutherland lattice. It is shown that the long-range electron hopping with the exponentially decaying hopping amplitudes $t_{ij}(\alpha ) \sim e^{-\alpha |\mathbf {R_{i}}-\mathbf {R_{j}}|}$ stabilizes the ferromagnetic state for a wide range of electron hopping α, electron interactions U, and electron concentrations n above the half-filled band case n = 1. The critical value of the hopping parameter αc, below which the ferromagnetic state becomes stable, is calculated numerically and the ground-state phase diagram of the model in the n − α plane is presented for physically the most interesting case of the intermediate Coulomb interactions.

Effects of geometrical frustration on ferromagnetism in the Hubbard model on the generalised Shastry-Sutherland lattice

The small-cluster exact-diagonalization calculations and the projector quantum Monte Carlo method are used to examine the competing effects of geometrical frustration and interaction on ferromagnetism in the Hubbard model on the Shastry-Sutherland lattice. It is shown that the geometrical frustration stabilizes the ferromagnetic state at high electron concentrations ($n \gtrsim 7/4$), where strong correlations between ferromagnetism and the shape of the noninteracting density of states are observed. In particular, it is found that ferromagnetism is stabilized only for these values of frustration parameters, which lead to the single peaked noninterating density of states at the band edge. Once, two or more peaks appear in the noninteracting density of states at the band egde the ferromagnetic state is suppressed. This opens a new route towards the understanding of ferromagnetism in strongly correlated systems.

Projector quantum Monte Carlo with averaged vs explicit spin-orbit effects: Applications to tungsten molecular systems

We present a recently developed projector quantum Monte Carlo method for calculations of electronic structure in systems with spin-orbit interactions. The method solves for many-body eigenstates in the presence of spin-orbit using the fixed-phase approximation. The trial wave function is built from two-component spinors and explicit Jastrow correlation factors while the core electrons are eliminated by relativistic effective core potentials with explicit spin-orbit terms. We apply this method to WO and W2 molecules that enables us to build multi-reference wave functions and analyze the impact of both electron correlations and the spin-orbit terms.

Enhancement of the d-wave pairing correlations by charge and spin ordering in the spin-one-half Falicov-Kimball model with Hund and Hubbard coupling

The projector quantum Monte Carlo method is used to examine the effects of the spin-independent Ufd as well as spin-dependent Jz Coulomb interaction between the localized f and itinerant d electrons on the stability of various types of charge/spin ordering and superconducting correlations in the spin-one-half Falicov-Kimball model with Hund and Hubbard coupling. The model is studied for a wide range of f- and d-electron concentrations and it is found that the interband interactions Ufd and Jz stabilize three basic types of charge/spin ordering, namely, i) the axial striped phases, ii) the regular n-molecular phases and iii) the phase-separated states. It is shown that the d-wave pairing correlations are enhanced within the axial striped and phase-separated states, but not in the regular phases. Moreover, it was found that the antiferromagnetic spin arrangement within the chains further enhances the d-wave paring correlations, while the ferromagnetic one has a fully opposite effect.

Topological phase transition in a generalized Kane-Mele-Hubbard model: A combined quantum Monte Carlo and Green's function study

We study a generalized Kane-Mele-Hubbard model with third-neighbor hopping, an interacting two-dimensional model with a topological phase transition as a function of third-neighbor hopping, by means of the determinant projector quantum Monte Carlo method. This technique is essentially numerically exact on models without a fermion sign problem, such as the one we consider. We determine the interaction dependence of the ${Z}_{2}$ topological insulator/trivial insulator phase boundary by calculating the ${Z}_{2}$ invariants directly from the single-particle Green's function. The interactions push the phase boundary to larger values of third-neighbor hopping, thus, stabilizing the topological phase. The observation of boundary shifting entirely stems from quantum fluctuations. We also identify qualitative features of the single-particle Green's function which are computationally useful in numerical searches for topological phase transitions without the need to compute the full topological invariant.

The Fermionic Minus-Sign Problem for the Ground State, Revisited with Higher Order Suzuki–Trotter Decompositions

A theorem by Yoshida which states that the discretization error for the Suzuki–Trotter (ST) decomposition is one order smaller than for the corresponding approximated Hamiltonian has long been overlooked with grave consequences for the accuracy of algorithms which use decomposition schemes. In our error analysis for the ground state energy of the Hubbard Hamiltonian by the projector quantum Monte Carlo method, we have used various orders of ST-decompositions, including for the first time in this field, pseudo-symplectic methods. We show that higher order does not necessarily imply better convergence, and identify the condition to obtain good convergence with higher order ST-decompositions. For the first order ST-decomposition, the ground state energy does not converge to the values for numerical diagonalization, in agreement with Yoshida's theorem, which may have caused confusion in connection with the “fermionic sign problem”. We show how the use of the sign of the fermion determinant is in fact a reweig...

Finite-size scaling and boundary effects in two-dimensional valence-bond solids

Various lattice geometries and boundary conditions are used to investigate valence-bond-solid (VBS) ordering in the ground state of an $S=1/2$ square-lattice quantum spin model---the $J$-$Q$ model, in which four- or six-spin interactions $Q$ are added to the standard Heisenberg exchange $J$. Ground-state results for finite systems (with up to thousands of spins) are obtained using an unbiased projector quantum Monte Carlo method. It is found that great care has to be taken when extrapolating the order parameter to infinite lattice size, in particular, in cylinder geometry. Even though strong VBS order exists in two dimensions, and is established clearly with increasing system size on $L\ifmmode\times\else\texttimes\fi{}L$ lattices (or ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ lattices with a fixed aspect ratio ${L}_{x}/{L}_{y}$ of order 1), only short-range VBS correlations are observed on long cylinders (when ${L}_{x}\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed ${L}_{y}$). The correlation length increases with the cylinder width, until long-range order sets in at a ``critical'' width. This width is very large even when the 2D order is relatively strong. For example, for a system in which the order parameter is $70%$ of the largest possible value, ${L}_{y}=8$ is required for ordering. Extrapolations of the VBS order parameter based on correlation functions (the square of the order parameter) for small $L\ifmmode\times\else\texttimes\fi{}L$ lattices can also be misleading. For a $20%$-ordered system, results for $L$ up to $\ensuremath{\approx}$20 appear to extrapolate clearly to a vanishing order parameter, while for larger lattices the scaling behavior crosses over and extrapolates to a nonzero value (with exponentially small finite-size corrections). The VBS order parameter also exhibits interesting edge effects related the known emergent U(1) symmetry close to a ``deconfined'' critical point, which, if not considered properly, can lead to wrong conclusions for the thermodynamic limit. The observed finite-size behavior for small $L\ifmmode\times\else\texttimes\fi{}L$ lattices and long cylinders is very similar to that predicted for a ${Z}_{2}$ spin liquid. The results therefore raise concerns about recent numerical work claiming ${Z}_{2}$ spin-liquid ground states in 2D frustrated quantum spin systems, in particular, the Heisenberg model with nearest and next-nearest-neighbor couplings. Based on the results presented so far, a VBS state in this system cannot be ruled out.

The Shadow Path Integral Ground State Method: Study of Confined Solid4He

We have studied the zero temperature phase diagram of confined 4He by means of a new exact projector Quantum Monte Carlo method: the Shadow Path Integral Ground State (SPIGS). This method is able to compute exact ground state expectation values without extrapolations and with a Shadow Wave Function (SWF) as trial variational state for the importance sampling. Thanks to the ability of SWF in describing the solid phase via spontaneously broken translational symmetry, this is an important extension which opens the possibility to study disorder phenomena in the solid phase by an exact method at zero temperature. We have applied this technique to study 4He confined in a lattice of big static objects with a purely repulsive interaction with 4He atoms. We have studied the equation of state of this system finding that a well defined liquid-solid transition is still present but with an increased freezing pressure. In the case we have studied this pressure goes from 25.2 atm for pure bulk system to about 38 atm. This is similar to what is experimentally found in 4He confined in porous material like vycor. We have found also that the disorder due to the mismatch between the crystal structure and the spherical geometry of the object induces delocalization of 4He atoms. PACS numbers: 05.10.Ln, 67.80.-s.

Coexistence of s-wave superconductivity and antiferromagnetism.

We study the phase diagram of a new model that exhibits a first order transition between s-wave superconducting and antiferromagnetic phases. The model, a generalized Hubbard model augmented with competing spin-spin and pair-pair interactions, was investigated using the projector quantum Monte Carlo method. Upon varying the Hubbard U from attractive to repulsive, we find a first order phase transition between superconducting and antiferromagnetic states.

Delocalizing effect of the Hubbard repulsion for electrons on a two-dimensional disordered lattice

We study numerically the ground-state properties of the repulsive Hubbard model for spin-1/2 electrons on two-dimensional lattices with disordered on-site energies. The projector quantum Monte Carlo method is used to obtain very accurate values of the ground-state charge density distributions with $N_p$ and $N_p+1$ particles. The difference in these charge densities allows us to study the localization properties of an added particle. The results obtained at quarter-filling on finite clusters show that the Hubbard repulsion has a strong delocalizing effect on the electrons in disordered 2D lattices. However, numerical restrictions do not allow us to reach a definite conclusion about the existence of a metal-insulator transition in the thermodynamic limit in two-dimensions.

Disorder and superconductivity: a new phase of bi-particle localized states

We study the two-dimensional, disordered, attractive Hubbard model by the projector quantum Monte Carlo method and Bogoliubov-de Gennes mean-field theory. Our results for the ground state show the appearance of a new phase with charge localization in the metallic regime of the non-interacting model. Contrary to the common lore, we demonstrate that mean-field theory fails to predict this phase and is unable to describe the correct physical picture in this regime.

Ground-state properties of the two-dimensional disordered Hubbard model

We study the ground state of the two-dimensional disordered Hubbard model by means of the projector quantum Monte Carlo method. This approach allows us to investigate the ground-state properties of this model for lattice sizes up to 10\ifmmode\times\else\texttimes\fi{}10, at quarter filling, for a broad range of interaction and disorder strengths. Our results show that the ground state of this system of spin-$\frac{1}{2}$ fermions remains localized in the presence of the short-ranged Hubbard interaction.

Limitations of the stochastic diagonalization and the projector QMC in the 2D Hubbard model

In the many particle physics the exact diagonalization (ED) is limited to small system sizes, like 4×4, on account of the restrictions of the computer memory. Whereas the stochastic diagonalization (SD) and the projector quantum Monte Carlo (PQMC) are not limited to a certain system size the limits in the parameter space of the analyzed physical model are at hoc not clear. For the two-dimensional tt′-Hubbard model we deduce the parameter space accessible to these two algorithms. Therefore we study several properties indicating the quality of the approximation of the ground state received by SD or PQMC algorithm. Furthermore we present strategies to derive a possible failure of one of these two algorithms without the comparison with another algorithm by analyzing the numerical simulations in detail. Therefore we compare the both algorithms with the ED and mainly SD and PQMC among themselves. Additional we present some new results concerning the sign-problem and the appropriate choice of the test wave functions in the PQMC method. This leads to a new numerical algorithm combining SD and PQMC.

Structural instability in polyacene: A projector quantum Monte Carlo study

We have studied polyacene within the Hubbard model to explore the effect of electron correlations on the Peierls' instability in a system marginally away from one-dimension. We employ the projector quantum Monte Carlo method to obtain ground state estimates of the energy and various correlation functions. We find strong similarities between polyacene and polyacetylene which can be rationalized from the real-space valence-bond arguments of Mazumdar and Dixit. Electron correlations tend to enhance the Peierls' instability in polyacene. This enhancement appears to attain a maximum at $U/t \sim 3.0$ and the maximum shifts to larger values when the alternation parameter is increased. The system shows no tendency to destroy the imposed bond-alternation pattern, as evidenced by the bond-bond correlations. The cis- distortion is seen to be favoured over the trans- distortion. The spin-spin correlations show that undistorted polyacene is susceptible to a SDW distortion for large interaction strength. The charge-charge correlations indicate the absence of a CDW distortion for the parameters studied.

Competition of magnetic ordering in the U- t- model

Using the projector quantum Monte Carlo method, we investigate the magnetic properties of the half-filled two-dimensional U-t- model on a square lattice. In particular, we describe the transition between the usual antiferromagnetic order and a new, antiferromagnetically layered type of order as a function of the diagonal hopping amplitude . In the case of strong correlations U, our results compare favourably to analytical and numerical results on the - model. Furthermore, we investigate the magnetic ground-state properties for medium correlation strength, too. In this regime, as well as in the strongly correlated case, the system undergoes a transition from antiferromagnetic order to layered antiferromagnetic order at .

Evidence for d-wave superconductivity in the repulsive Hubbard model

We perform numerical simulations of the Hubbard model using the projector Quantum Monte Carlo method. A novel approach for finite size scaling is discussed. We obtain evidence in favor of d-wave superconductivity in the repulsive Hubbard model. For U=4, $T_c$ is roughly estimated as $T_c\approx 30K$.

Estimation of properties of low-lying excited states of Hubbard models: A multiconfigurational symmetrized projector quantum Monte Carlo approach

We present in detail the recently developed multi-configurational symmetrized projector quantum Monte Carlo (MSPQMC) method for excited states of the Hubbard model. We describe the implementation of the Monte Carlo method for a multi-configurational trial wavefunction. We give a detailed discussion of issues related to the symmetry of the projection procedure which validates our Monte Carlo procedure for excited states and leads naturally to the idea of symmetrized sampling for correlation functions, developed earlier in the context of ground state simulations. It also leads to three possible averaging schemes. We have analyzed the errors incurred in these various averaging procedures and discuss and detail the preferred averaging procedure for correlations that do not have the full symmetry of the Hamiltonian. We study the energies and correlation functions of the low-lying excited states of the half-filled Hubbard model in 1-D. We have used this technique to study the pair-binding energies of two holes in $4n$ and $4n+2$ systems, which compare well the Bethe ansatz data of Fye, Martins and Scalettar. We have also studied small clusters amenable to exact diagonalization studies in 2-D and have reproduced their energies and correlation functions by the MSPQMC method. We identify two ways in which a multiconfigurational trial wavefunction can lead to a negative sign problem. We observe that this effect is not severe in 1-D and tends to vanish with increasing system size. We also note that this does not enhance the severity of the sign problem in two dimensions.

Magnetization in the two-dimensional Hubbard model: a numerical study

The Projector Quantum Monte Carlo method is used to study the two-dimensional Hubbard model with generalized boundary conditions at half-filling. The convergence of the algorithm depends strongly on the initial trial state. Spin-density waves provide an excellent trial state for the case of weak and of strong correlations. This choice of a trial state with broken symmetry allows us to calculate directly the staggered (or sublattice) magnetization m0 as a function of the on-site repulsion U. The use of general boundary conditions strongly reduces finite size effects in m0.

Projector quantum Monte Carlo simulations of the Hubbard model: Staggered magnetization

We have used the projector quantum Monte Carlo method to study the magnetic properties of the two-dimensional Hubbard model at half-filling as a function of the on-site repulsion U. Applying spin-density waves as initial trial state, we obtain excellent convergence of the algorithm for the case of weak and of strong correlations. This choice allows us to calculate directly the staggered (or sublattice) magnetization m 0 as a function of U.