Abstract We describe the algorithmic details and a performance evaluation of a Langevin approach to a strongly interacting electron-phonon system, and show it has a near linear scaling with lattice size N s . Many of the limitations of previous attempts to employ such methods to condensed matter lattice Hamiltonians are absent. In particular, the iterative linear algebra solution remains well behaved at strong coupling and low temperatures. The use of Fourier Acceleration is crucial for efficiency, and its use makes the method competitive with the widely-used local update methods, which scale as N s 3 for on-site interactions and N s 4 for long range electron-phonon coupling, even on rather small lattice sizes.
Abstract We describe the algorithmic details and a performance evaluation of a Langevin approach to a strongly interacting electron-phonon system, and show it has a near linear scaling with lattice size N s . Many of the limitations of previous attempts to employ such methods to condensed matter lattice Hamiltonians are absent. In particular, the iterative linear algebra solution remains well behaved at strong coupling and low temperatures. The use of Fourier Acceleration is crucial for efficiency, and its use makes the method competitive with the widely-used local update methods, which scale as N s 3 for on-site interactions and N s 4 for long range electron-phonon coupling, even on rather small lattice sizes.
We present a quantum Monte Carlo (QMC) study, based on the Langevin equation, of a Hamiltonian describing electrons coupled to phonon degrees of freedom. The bosonic part of the action helps control the variation of the field in imaginary time. As a consequence, the iterative conjugate gradient solution of the fermionic action, which depends on the boson coordinates, converges more rapidly than in the case of electron-electron interactions, such as the Hubbard Hamiltonian. Fourier acceleration is shown to be a crucial ingredient in reducing the equilibration and autocorrelation times. After describing and benchmarking the method, we present results for the phase diagram focusing on the range of the electron-phonon interaction. We delineate the regions of charge density wave formation from those in which the fermion density is inhomogeneous, caused by phase separation. We show that the Langevin approach is more efficient than the determinant QMC method for lattice sizes $N\ensuremath{\gtrsim}8\ifmmode\times\else\texttimes\fi{}8$ and that it therefore opens a potential path to problems including, for example, charge order in the three-dimensional Holstein model.
We introduce methodologies for highly scalable quantum Monte Carlo simulations of electron-phonon models, and we report benchmark results for the Holstein model on the square lattice. The determinant quantum Monte Carlo (DQMC) method is a widely used tool for simulating simple electron-phonon models at finite temperatures, but it incurs a computational cost that scales cubically with system size. Alternatively, near-linear scaling with system size can be achieved with the hybrid Monte Carlo (HMC) method and an integral representation of the Fermion determinant. Here, we introduce a collection of methodologies that make such simulations even faster. To combat "stiffness" arising from the bosonic action, we review how Fourier acceleration can be combined with time-step splitting. To overcome phonon sampling barriers associated with strongly bound bipolaron formation, we design global Monte Carlo updates that approximately respect particle-hole symmetry. To accelerate the iterative linear solver, we introduce a preconditioner that becomes exact in the adiabatic limit of infinite atomic mass. Finally, we demonstrate how stochastic measurements can be accelerated using fast Fourier transforms. These methods are all complementary and, combined, may produce multiple orders of magnitude speedup, depending on model details.
We develop a Fermi-Bose bootstrap embedding framework for the ground state of interacting electrons coupled to a phonon mean field. The method combines bootstrap embedding for correlated electrons with a self-consistent coherent-state mean-field treatment for phonons. This method models the interacting electron-phonon problem as a system of correlated electrons traveling in a self-consistently specified potential landscape, allowing for efficient treatment of large lattice systems. Convergence of the methods for fragment size and total system size is demonstrated for the one-dimensional Hubbard-Holstein model for up to 350 sites. Finite-size scaling is performed to extrapolate to the infinite system size. Benchmarking against the density matrix renormalization group for a small 8-site system at half- and quarter-filling shows an orders-of-magnitude runtime advantage. The comparison further reveals that the method performs best in regimes dominated by localization, such as the Mott insulating phase and the strong-coupling tiny polaron regime, where the local embedding ansatz is still valid. However, due to the mean-field treatment for phonons, we find limitations of our methods in the weakly coupled delocalized region and at the Peierls transition, where quantum phonon fluctuations and long-range kinetic correlations become substantial.
We present, in this dissertation, a numerical simulation method to study interacting fermion systems. The general simulation procedures are discussed in connection with a description of a Quantum Monte Carlo simulation algorithm for interacting electrons on lattices. The algorithm presented here has been used to simulate interacting electrons on lattices, and it makes possible the study of substantially larger systems than can be studied by other numerical methods. As long as certain limits of applicability are respected, model Hamiltonians of interacting electrons can be studied without resort to uncontrolled approximations. The method then provides nearly exact solutions to model Hamiltonians of many-body systems, in the sense that the degree of error can be controlled. We also discuss some results obtained from simulations of the extended Hubbard model and and the periodic Anderson model. For the extended Hubbard model in two dimensions, different regions are identified where electron correlation produces antiferromagnetism, charge-density-waves, and singlet pairing superconducting behaviors. We also find regions where transitions from one type of correlation to another occur. For the symmetric periodic Anderson model, we observe the formation of local spin moments at high temperatures and their quenching at low temperatures, as in the single impurity Anderson model. In addition, we find antiferromagnetic interaction between f local moments at low temperatures which are not present in a single impurity problem.
When electrons interact not only with themselves, but also with bosons (phonons or other bosonic quasiparticles), what was previously a ``curse of dimensionality'' for its computational solution now becomes one of infinite size. To deal with this, the authors adapt here a recent method for its stochastic solution, without requiring truncation of the bosonic occupations, and which allows systematic improvability to exact solutions. They apply this to the paradigmatic Hubbard-Holstein model of local phonons, finding a state-of-the-art solution in the thermodynamic limit, as well as to a novel approach in which long-range collective electronic fluctuations are mapped to bosonic quasiparticles to tackle even long-range interacting problems of purely electronic form.
We present the numerically exact ground-state energy, effective mass, and isotope exponents of a one-dimensional lattice polaron, valid for any range of electron-phonon interaction, applying a continuous-time quantum Monte Carlo (QMC) technique in a wide range of coupling strength and adiabatic ratio. The QMC method is free from any systematic finite-size and finite-time-step errors. We compare our numerically exact results with analytical weak-coupling theory and with the strong-coupling $1∕\ensuremath{\lambda}$ expansion. We show that the exact results agree well with the canonical Fr\"ohlich and Holstein-Lang-Firsov theories in the weak and strong coupling limits, respectively, for any range of interaction. We find a strong dependence of the polaron dynamics on the range of interaction. An increased range of interaction has a similar effect to an increased (less adiabatic) phonon frequency: specifically, a reduction in the effective mass.