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arXiv:1808.08973 · Physical review. B./Physical review. B

Langevin simulations of a long-range electron-phonon model

G. G. Batrouni, Richard T. Scalettar

Published Jan 8, 2019• 51 citations
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Abstract

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.

DOI
10.1103/physrevb.99.035114

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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≳8\ifmmode×\else\texttimes\fi8N\ensuremath{\gtrsim}8\ifmmode\times\else\texttimes\fi{}8N\ensuremath≳8\ifmmode×\else\texttimes\fi8 and that it therefore opens a potential path to problems including, for example, charge order in the three-dimensional Holstein model.
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.
The Hubbard-Holstein model describes fermions on a discrete lattice, with on-site repulsion between fermions and a coupling to phonons that are localized on sites. Generally, at half-filling, increasing the coupling ggg to the phonons drives the system towards a Peierls charge density wave state, whereas increasing the electron-electron interaction UUU drives the fermions into a Mott antiferromagnet. At low ggg and UUU, or when doped, the system is metallic. In one dimension, using quantum Monte Carlo simulations, we study the case where fermions have a long-range coupling to phonons, with characteristic range $\ensuremath{\xi}$, interpolating between the Holstein and Fr\"ohlich limits. Without electron-electron interaction, the fermions adopt a Peierls state when the coupling to the phonons is strong enough. This state is destabilized by a small coupling range $\ensuremath{\xi}$ and leads to a collapse of the fermions, and, consequently, phase separation. Increasing interaction UUU will drive any of these three phases (metallic, Peierls, phase separation) into a Mott insulator phase. The phase separation region is once again present in the $U\ensuremath{\ne}0$ case, even for small values of the coupling range.
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.
Recent studies of pairing and charge order in materials such as FeSe, SrTiO3_33​, and 2H-NbSe2_22​ have suggested that momentum dependence of the electron-phonon coupling plays an important role in their properties. Initial attempts to study Hamiltonians which either do not include or else truncate the range of Coulomb repulsion have noted that the resulting spatial non-locality of the electron-phonon interaction leads to a dominant tendency to phase separation. Here we present Quantum Monte Carlo results for such models in which we incorporate both on-site and intersite electron-electron interactions. We show that these can stabilize phases in which the density is homogeneous and determine the associated phase boundaries. As a consequence, the physics of momentum dependent electron-phonon coupling can be determined outside of the trivial phase separated regime.
Based on the canonical Lang-Firsov transformation of the Hamiltonian we develop a very efficient Quantum Monte Carlo algorithm for the Holstein model with one electron. Separation of the fermionic degrees of freedom by a reweighting of the probability distribution leads to a dramatic reduction in computational effort. A principle component representation of the phonon degrees of freedom allows to sample completely uncorrelated phonon configurations. The combination of these elements enables us to perform efficient simulations for a wide range of temperature, phonon frequency and electron-phonon coupling on clusters large enough to avoid finite-size effects. The algorithm is tested in one dimension and the data are compared with exact-diagonalization results and with existing work. Moreover, the new ideas presented here can also be applied to the many-electron case. In the one-electron limit the physics of the Holstein model can be described by a simple variational approach.
The Holstein model describes the interaction between fermions and a collection of local (dispersionless) phonon modes. In the dilute limit, the phonon degrees of freedom dress the fermions, giving rise to polaron and bipolaron formation. At higher densities, the phonons mediate collective superconducting (SC) and charge-density wave (CDW) phases. Quantum Monte Carlo (QMC) simulations have considered both these limits but have not yet focused on the physics of more general phonon spectra. Here we report QMC studies of the role of phonon dispersion on SC and CDW order in such models. We quantify the effect of finite phonon bandwidth and curvature on the critical temperature T_{cdw} for CDW order and also uncover several novel features of diagonal long-range order in the phase diagram, including a competition between charge patterns at momenta q=(π,π) and q=(0,π) which lends insight into the relationship between Fermi surface nesting and the wave vector at which charge order occurs. We also demonstrate SC order at half filling in situations where a nonzero bandwidth sufficiently suppresses T_{cdw}.