Inchworm Monte Carlo for exact non-adiabatic dynamics. II. Benchmarks and comparison with established methods

Steady-state properties of multi-orbital systems using quantum Monte Carlo

A precise dynamical characterization of quantum impurity models with multiple interacting orbitals is challenging. In quantum Monte Carlo methods, this is embodied by sign problems. A dynamical sign problem makes it exponentially difficult to simulate long times. A multi-orbital sign problem generally results in a prohibitive computational cost for systems with multiple impurity degrees of freedom even in static equilibrium calculations. Here, we present a numerically exact inchworm method that simultaneously alleviates both sign problems, enabling simulation of multi-orbital systems directly in the equilibrium or nonequilibrium steady-state. The method combines ideas from the recently developed steady-state inchworm Monte Carlo framework [Erpenbeck et al., Phys. Rev. Lett. 130, 186301 (2023)] with other ideas from the equilibrium multi-orbital inchworm algorithm [Eidelstein et al., Phys. Rev. Lett. 124, 206405 (2020)]. We verify our method by comparison with analytical limits and numerical results from previous methods.

Stark Many-Body Localization in Interacting Infinite Dimensional Systems

We study bulk particle transport in a Fermi-Hubbard model on an infinite-dimensional Bethe lattice, driven by a constant electric field. Previous numerical studies showed that one dimensional analogs of this system exhibit a breakdown of diffusion due to Stark many-body localization at least up to time that scales exponentially with the system size. Here, we consider systems initially in a spin density wave state using a combination of numerically exact and approximate techniques. We show that for sufficiently weak electric fields, the wave's momentum component decays exponentially with time in a way consistent with normal diffusion. By studying different wavelengths, we extract the dynamical exponent and the generalized diffusion coefficient at each field strength. Interestingly, we find a nonmonotonic dependence of the dynamical exponent on the electric field. As the field increases toward a critical value proportional to the Hubbard interaction strength, transport slows down, becoming subdiffusive. At large interaction strengths, however, transport speeds up again with increasing field, exhibiting superdiffusive characteristics when the electric field is comparable to the interaction strength. Eventually, at the large field limit, localization occurs and the current through the system is suppressed.

Real-Time Simulation of Open Quantum Spin Chains with the Inchworm Method

We study the real-time simulation of open quantum systems, where the system is modeled by a spin chain with each spin associated with its own harmonic bath. Our method couples the inchworm method for the spin-boson model and the modular path integral methodology for spin systems. In particular, the introduction of the inchworm method can significantly suppress the numerical sign problem. Both methods are tweaked to make them work seamlessly with each other. We represent our approach in the language of diagrammatic methods and analyze the asymptotic behavior of the computational cost. Extensive numerical experiments are performed to validate our method.

Shaping Electronic Flows with Strongly Correlated Physics

Nonequilibrium quantum transport is of central importance in nanotechnology. Its description requires the understanding of strong electronic correlations that couple atomic-scale phenomena to the nanoscale. So far, research in correlated transport has focused predominantly on few-channel transport, precluding the investigation of cross-scale effects. Recent theoretical advances enable the solution of models that capture the interplay between quantum correlations and confinement beyond a few channels. This problem is the focus of this study. We consider an atomic impurity embedded in a metallic nanosheet spanning two leads, showing that transport is significantly altered by tuning only the phase of a single local hopping parameter. Furthermore─depending on this phase─correlations reshape the electronic flow throughout the sheet, either funneling it through the impurity or scattering it away from a much larger region. This demonstrates the potential for quantum correlations to bridge length scales in the design of nanoelectronic devices and sensors.

Quantum Monte Carlo Method in the Steady State

We present a numerically exact steady-state inchworm Monte Carlo method for nonequilibrium quantum impurity models. Rather than propagating an initial state to long times, the method is directly formulated in the steady state. This eliminates any need to traverse the transient dynamics and grants access to a much larger range of parameter regimes at vastly reduced computational costs. We benchmark the method on equilibrium Green's functions of quantum dots in the noninteracting limit and in the unitary limit of the Kondo regime. We then consider correlated materials described with dynamical mean field theory and driven away from equilibrium by a bias voltage. We show that the response of a correlated material to a bias voltage differs qualitatively from the splitting of the Kondo resonance observed in bias-driven quantum dots.

Interexcited State Photophysics I: Benchmarking Density Functionals for Computing Nonadiabatic Couplings and Internal Conversion Rate Constants

We present the first benchmarking study of nonadiabatic matrix coupling elements (NACMEs) calculated using different density functionals. Using the S₁ → S₀ transition in perylene solvated in toluene as a case study, we calculate the photophysical properties and corresponding rate constants for a variety of density functionals from each rung of Jacob's ladder. The singlet photoluminescence quantum yield (sPLQY) is taken as a measure of accuracy, measured experimentally here as 0.955. Important quantum chemical parameters such as geometries, absorption, emission, and adiabatic energies, NACMEs, Hessians, and transition dipole moments were calculated for each density functional basis set combination (data set) using density functional theory based multireference configuration interaction (DFT/MRCI) and compared to experiment where possible. We were able to derive simple relations between the TDDFT and DFT/MRCI photophysical properties; with semiempirical damping factors of ∼0.843 ± 0.017 and ∼0.954 ± 0.064 for TDDFT transition dipole moments and energies to DFT/MRCI level approximations, respectively. NACMEs were dominated by out-of-plane derivative components belonging to the center-most ring atoms with weaker contributions from perturbations along the transverse and longitudinal axes. Calculated theoretical spectra compared well to both experiment and literature, with fluorescence lifetimes between 7.1 and 12.5 ns, agreeing within a factor of 2 with experiment. Internal conversion (IC) rates were then calculated and were found to vary wildly between 10⁶-10¹⁶ s^-1 compared with an experimental rate of the order 10⁷ s^-1. Following further testing by mixing data sets, we found a strong dependence on the method used to obtain the Hessian. The 5 characterized data sets ranked in order of most promising are PBE0/def2-TZVP, ωB97XD/def2-TZVP, HCTH407/TZVP, PBE/TZVP, and PBE/def2-TZVP.

Hierarchical equations of motion approach for accurate characterization of spin excitations in quantum impurity systems

Recent technological advancement in scanning tunneling microscopes has enabled the measurement of spin-field and spin-spin interactions in single atomic or molecular junctions with an unprecedentedly high resolution. Theoretically, although the fermionic hierarchical equations of motion (HEOM) method has been widely applied to investigate the strongly correlated Kondo states in these junctions, the existence of low-energy spin excitations presents new challenges to numerical simulations. These include the quest for a more accurate and efficient decomposition for the non-Markovian memory of low-temperature environments and a more careful handling of errors caused by the truncation of the hierarchy. In this work, we propose several new algorithms, which significantly enhance the performance of the HEOM method, as exemplified by the calculations on systems involving various types of low-energy spin excitations. Being able to characterize both the Kondo effect and spin excitation accurately, the HEOM method offers a sophisticated and versatile theoretical tool, which is valuable for the understanding and even prediction of the fascinating quantum phenomena explored in cutting-edge experiments.

Nonadiabatic Dynamics at Metal Surfaces: Fewest Switches Surface Hopping with Electronic Relaxation

A new scheme is proposed for modeling molecular nonadiabatic dynamics near metal surfaces. The charge-transfer character of such dynamics is exploited to construct an efficient reduced representation for the electronic structure. In this representation, the fewest switches surface hopping (FSSH) approach can be naturally modified to include electronic relaxation (ER). The resulting FSSH-ER method is valid across a wide range of coupling strengths as supported by tests applied to the Anderson-Holstein model for electron transfer. Future work will combine this scheme with ab initio electronic structure calculations.

Exploiting the Causal Tensor Network Structure of Quantum Processes to Efficiently Simulate Non-Markovian Path Integrals

In the path integral formulation of the evolution of an open quantum system coupled to a Gaussian, noninteracting environment, the dynamical contribution of the latter is encoded in an object called the influence functional. Here, we relate the influence functional to the process tensor-a more general representation of a quantum stochastic process-describing the evolution. Then, we use this connection to motivate a tensor network algorithm for the simulation of multitime correlations in open systems, building on recent work where the influence functional is represented in terms of time evolving matrix product operators. By exploiting the symmetries of the influence functional, we are able to use our algorithm to achieve orders-of-magnitude improvement in the efficiency of the resulting numerical simulation. Our improved algorithm is then applied to compute exact phonon emission spectra for the spin-boson model with strong coupling, demonstrating a significant divergence from spectra derived under commonly used assumptions of memorylessness.

Quantum impurity models coupled to Markovian and non-Markovian baths

We develop a method to study quantum impurity models, small interacting quantum systems bilinearly coupled to an environment, in the presence of an additional Markovian quantum bath, with a generic nonlinear coupling to the impurity. We aim at computing the evolution operator of the reduced density matrix of the impurity, obtained after tracing out all the environmental degrees of freedom. First, we derive an exact real-time hybridization expansion for this quantity, which generalizes the result obtained in the absence of the additional Markovian dissipation and which could be amenable to stochastic sampling through diagrammatic Monte Carlo. Then, we obtain a Dyson equation for this quantity and we evaluate its self-energy with a resummation technique known as the noncrossing approximation. We apply this novel approach to a simple fermionic impurity coupled to a zero temperature fermionic bath and in the presence of Markovian pump, losses, and dephasing.

Simulation Complexity of Open Quantum Dynamics: Connection with Tensor Networks

The difficulty to simulate the dynamics of open quantum systems resides in their coupling to many-body reservoirs with exponentially large Hilbert space. Applying a tensor network approach in the time domain, we demonstrate that effective small reservoirs can be defined and used for modeling open quantum dynamics. The key element of our technique is the timeline reservoir network (TRN), which contains all the information on the reservoir's characteristics, in particular, the memory effects timescale. The TRN has a one-dimensional tensor network structure, which can be effectively approximated in full analogy with the matrix product approximation of spin-chain states. We derive the sufficient bond dimension in the approximated TRN with a reduced set of physical parameters: coupling strength, reservoir correlation time, minimal timescale, and the system's number of degrees of freedom interacting with the environment. The bond dimension can be viewed as a measure of the open dynamics complexity. Simulation is based on the semigroup dynamics of the system and effective reservoir of finite dimension. We provide an illustrative example showing the scope for new numerical and machine learning-based methods for open quantum systems.

Ehrenfest+R dynamics. I. A mixed quantum-classical electrodynamics simulation of spontaneous emission

The dynamics of an electronic system interacting with an electromagnetic field is investigated within mixed quantum-classical theory. Beyond the classical path approximation (where we ignore all feedback from the electronic system on the photon field), we consider all electron-photon interactions explicitly according to Ehrenfest (i.e., mean-field) dynamics and a set of coupled Maxwell-Liouville equations. Because Ehrenfest dynamics cannot capture certain quantum features of the photon field correctly, we propose a new Ehrenfest+R method that can recover (by construction) spontaneous emission while also distinguishing between electromagnetic fluctuations and coherent emission.

Efficient non-Markovian quantum dynamics using time-evolving matrix product operators

In order to model realistic quantum devices it is necessary to simulate quantum systems strongly coupled to their environment. To date, most understanding of open quantum systems is restricted either to weak system–bath couplings or to special cases where specific numerical techniques become effective. Here we present a general and yet exact numerical approach that efficiently describes the time evolution of a quantum system coupled to a non-Markovian harmonic environment. Our method relies on expressing the system state and its propagator as a matrix product state and operator, respectively, and using a singular value decomposition to compress the description of the state as time evolves. We demonstrate the power and flexibility of our approach by numerically identifying the localisation transition of the Ohmic spin-boson model, and considering a model with widely separated environmental timescales arising for a pair of spins embedded in a common environment.

In general the computational requirements for simulating quantum systems scale exponentially but in some cases it is possible to formulate more efficient schemes. Here the authors present an efficient approach to calculate non-Markovian quantum dynamics using matrix product operator methods.