We study the dynamical properties of the two-dimensional square lattice Holstein Hamiltonian using numerically exact determinant quantum Monte Carlo simulations. In particular, we report systematic calculations of the model's single-particle spectral function and optical conductivity over a range of phonon energies, electron-phonon coupling strengths, carrier concentrations, and temperatures. In doing so, we map the evolution of the system from a dressed metallic phase to a (bi)polaron liquid/insulator to a charge-density-wave insulator as the carrier concentration is tuned from dilute values to half-filling. We corroborate these results using several equal-time correlation measurements and related proxy quantities reconstructed from time-displaced correlation measurements. This paper is also accompanied by an extensive open data set, covering over 16,640 unique parameter values.
We have performed numerical studies of the Hubbard-Holstein model in two dimensions using determinant quantum Monte Carlo (DQMC). Here, we present details of the method, emphasizing the treatment of the lattice degrees of freedom, and then study the filling and behavior of the fermion sign as a function of model parameters. We find a region of parameter space with large Holstein coupling where the fermion sign recovers despite large values of the Hubbard interaction. This indicates that studies of correlated polarons at finite carrier concentrations are likely accessible to DQMC simulations. We then restrict ourselves to the half-filled model and examine the evolution of the antiferromagnetic structure factor, other metrics for antiferromagnetic and charge-density-wave order, and energetics of the electronic and lattice degrees of freedom as a function of electron-phonon coupling. From this we find further evidence for a competition between charge-density-wave and antiferromagnetic orders at half-filling.
Electron-phonon interactions play a key role in many branches of solid-state physics. Here, our focus is on the transport properties of one-dimensional systems, and we apply efficient real-time matrix-product state methods to compute the optical and thermal conductivities of Holstein chains at finite temperatures and filling. We validate our approach by comparison with analytical results applicable to single polarons valid in the small polaron limit. Our work provides a systematic study of contributions to the thermal conductivity at finite frequencies and elucidates differences in the spectrum compared to the optical conductivity, covering both the Luttinger-liquid and charge-density-wave regimes. Finally, we demonstrate that our approach is capable of extracting the DC conductivities as well. Beyond this first application, several future extensions seem feasible, such as the inclusion of dispersive phonons, different types of local electron-phonon coupling, and a systematic study of drag effects in this electron-phonon coupled system.
We use a recently developed extension of the weak-coupling diagrammatic determinantal quantum Monte Carlo method to investigate the spin, charge, and single-particle spectral functions of the one-dimensional quarter filled Holstein model with phonon frequency ${\ensuremath{\omega}}_{0}=0.1t$. As a function of the dimensionless electron-phonon coupling we observe a transition from a Luttinger to a Luther-Emery liquid with dominant
2kf charge fluctuations. Emphasis is placed on the temperature dependence of the single-particle spectral function. At high temperatures and in both phases it is well accounted for within a self-consistent Born approximation. In the low-temperature Luttinger liquid phase we observe features that compare favorably with a bosonization approach retaining only forward scattering. In the Luther-Emery phase, the spectral function at low temperatures shows a quasiparticle gap that matches half the spin gap, whereas at temperatures above which this quasiparticle gap closes characteristic features of the Luttinger liquid model are apparent. Our results are based on lattice simulations on chains up to
L=20 for two-particle properties and on cluster dynamical mean-field theory calculations with clusters up to 12 sites for the single-particle spectral function.
Over the past several years, reliable quantum Monte Carlo results for the charge density wave transition temperature
Tcdw of the half-filled two-dimensional Holstein model in square and honeycomb lattices have become available for the first time. Exploiting the further development of numerical methodology, here we present results in three dimensions, which are made possible through the use of Langevin evolution of the quantum phonon degrees of freedom. In addition to determining
Tcdw from the scaling of the charge correlations, we also examine the nature of charge order at general wave vectors for different temperatures, couplings, and phonon frequencies, and the behavior of the spectral function and specific heat.
In this thesis, I investigate the transport and dynamical properties of electron-phonon coupled systems using numerical methods. I focus on the one-dimensional Holstein model, which incorporates both polaron and charge density wave (CDW) physics, and use matrix-product-state-based algorithms. In the first part, I present work where we combine purification with local basis optimization to study the finite-temperature properties of the model. This gives access to polaron spectral functions, polaron and bipolaron optical conductivity, and the energy-transport coefficient at finite filling. The second part focuses on the Holstein model coupled to two tight-binding leads. There, we investigate how current is transported through the structure and how CDW states break down. In the last part, we analyze the spread of an electron in a lattice and the decay of CDWs. We aim to understand if a full quantum mechanical treatment of the phonons is necessary or if trajectory-based algorithms are applicable.
The Holstein Hamiltonian describes fermions hopping on a lattice and interacting locally with dispersionless phonon degrees of freedom. In the low-density limit, dressed quasiparticles, polarons and bipolarons, propagate with an effective mass. At higher densities, pairs can condense into a low-temperature superconducting phase and, at or near commensurate filling on a bipartite lattice, to charge density wave (CDW) order. CDW formation breaks a discrete symmetry and hence occurs via a second-order (Ising) transition and therefore at a finite
Tcdw in two dimensions. Quantum Monte Carlo calculations have determined
Tcdw for a variety of geometries, including square, honeycomb, and Lieb lattices. The superconducting transition, on the other hand, in
d=2 is in the Kosterlitz-Thouless universality class and is much less well characterized. In this paper we determine
Tsc for the square lattice for several values of the density $\ensuremath{\rho}$ and phonon frequency ${\ensuremath{\omega}}_{0}$. We find that quasilong-range order sets in at ${T}_{\mathrm{sc}}\ensuremath{\lesssim}t/20$, where
t is the near-neighbor hopping amplitude, consistent with previous rough estimates from simulations which extrapolated to only the temperatures we reach from considerably higher
T. We also show evidence of a discontinuous evolution of the density as the CDW transition is approached at half filling.