Numerically exact simulation of photo-doped Mott insulators

The gist: The authors demonstrate how properties of long-lived photo-doped states in Mott insulators can be computed using numerically exact steady state techniques, in particular Quantum Monte Carlo, by using a time-local ansatz for the distribution function with separate Fermi functions for the electron and hole quasiparticles.

Introduction to the Problem

Short light pulses provide intriguing avenues to manipulate material properties on ultrafast timescales Mott insulators are particularly interesting in this regard because a zoo of complex orders can emerge from a perturbed Mott phase A versatile route toward the generation of non-thermal phases involves photo-doping, i.e., the creation of charge carriers such as doublons (doubly occupied sites) and holes in a single-orbital Mott insulator With increasing gap, carrier recombination becomes exponentially slow so that doublon and hole densities are approximately conserved over extended periods Such cold photo-doped phases in correlated electron systems may undergo metal insulator transitions and band reconstruction and potentially even manifest superconducting instabilities. Dynamical Mean Field Theory (DMFT) and its extensions are powerful approaches to study Mott materials The main challenge in extending these methods to the time domain is the solution of a quantum impurity model. Real-time nonequilibrium DMFT simulations based on numerically exact Quantum Monte Carlo (QMC) or matrix-product states have been limited to short times, hindering the study of cold photo-doped states.

Methodology for Steady State Analysis

The authors aim to use potentially numerically exact steady state solvers to investigate slowly evolving (quasi-steady) photo-doped states. They introduce an approach that is not restricted to large U and does not alter the system by additional reservoirs. The distribution function F(ω, t) is taken to be a dynamical variable in quantum kinetic equations. The non-perturbative steady state solvers can be used to solve the many-body problem with any given distribution function Fsteady(ω). An equilibrium state is a special case where fluctuation-dissipation relations guarantee that Fsteady(ω) is equal to the Fermi function f(ω). They introduce a time-localF ansatz (TLFA), taking the steady state solution with F(ω, t) = Fsteady(ω) as an approximate description of the slowly evolving state around time t. This TLFA is validated through real-time simulations and used to simulate complex photodoped phases with non-perturbative techniques.

Model and Simulation Setup

The model considered is the one-band Hubbard model at half filling, with Hamiltonian H = −t˜0 X σ⟨i,j⟩ c† σicσj + U X i ni↑ni↓ − U 2 X σi niσ. The system is solved by means of nonequilibrium DMFT on the Bethe lattice with coordination number z → ∞ and hopping t˜0 = t0/√z. To describe energy dissipation, an external bosonic bath is included via a phonon self-energy Σph(t, t′) = g 2G(t, t′)Dbath(t, t′). The cutoff ωc = 0.2 ≪ U is chosen such that only kinetic energy relaxation of doublons and holes is possible while direct recombination via phonon emission is not.

Validation and Results

The TLFA is validated by solving the model in real-time using DMFT + NCA within the NESSi simulation package. The spectral function A(ω, t) starts from two Hubbard bands with a partially filled gap due to the high initial temperature. As kinetic energy of the doublons relaxes within a few tens of inverse hoppings, the occupied density of states A<(ω, t) ≡ F(ω, t)A(ω, t) concentrates at the lower band edge of the upper Hubbard band. Two peaks emerge in the spectrum at the edges of the Mott gap indicating simultaneous presence of doublon and hole quasiparticles. The distribution function develops two separate quasiparticle chemical potentials for the hole and doublon charge carriers. The TLFA almost perfectly reproduces the real-time spectra, i.e., the distribution function F(ω, t) characterizes the system at time t without further dependence on the history.

Comparison of Numerical Techniques

The spectral function A(ω) obtained for given nex is compared between inchworm QMC and NCA. The main characteristics of the cold photo-doped state, which is the simultaneous hole and electron quasiparticle peak, are thereby validated by the numerically exact data. The Mott gap ∆g for all three approaches is defined by the spectral weight reaching 0.057 which is half of the maximum of the equilibrium spectrum. NCA overestimates the gap ∆g in the photo-doped state, consistent with its behavior in equilibrium. The gap remains robust at large photo-doping nex even in the numerically exact solution which is an important finding supporting the stability of photo-doped orders.

Convergence and Long-Lived States

The convergence of the inchworm QMC data with respect to DMFT iterations is established by showing that chemical potential µ and gap size are stable throughout consecutive DMFT iterations. The inchworm QMC method is based on the hybridization expansion in the coupling between the impurity and its environment, exploiting the causal structure of this expansion. The TLFA can also be evaluated with AMEA, which approximates the impurity problem with a finite open system described by Nb bath orbitals with additional Lindblad dissipators. The difference between F∆ and FG is larger in the inchworm results, which is expected as the smaller gap would imply a faster dynamics so that the state found by the TLFA has a shorter lifetime and eventually thermalizes.

Conclusion

In this work, they have demonstrated how properties of these long-lived photo-doped states can be accessed with numerically exact techniques by using a time-local ansatz for the electronic distribution function. These findings indicate that nonequilibrium steady state formalisms can be used to directly access quasi-stable photo-doped states in Mott insulators. Furthermore, by integrating the steady state ansatz with QBE schemes, they open up new avenues for characterizing the slow dynamics of Mott insulators. This approach has the potential to extend into time scales far beyond the capabilities of existing real-time simulations.

Supplemental Details

The numerical calculations are based on nonequilibrium DMFT, which is a generalization of conventional DMFT for Green’s functions defined on the L-shaped Kadanoff-Baym contour C. The ansatz for the distribution function is given by FT,µex(ω) = Θα(ω)f(ω + µex, T) + (1 − Θα(ω))f(ω − µex, T), where the smooth step-function Θα(ω) = 0.5 1 - tanh(ωα/2). The results for the spectra are insensitive to the precise value of α across a wide range of values. The convergence of the restricted propagators is essentially converged at second order in the hybridization expansion. The GF for second and third order only show minor discrepancies, which are reflected in a small difference in the quasiparticle peak height. The mean of five inchworm DMFT iterations is plotted together with the standard deviation (shaded region).

1. Based on numerically exact steady state techniques, AI systems can compute properties of long-lived photo-doped states in Mott insulators by using a time-local ansatz for the distribution function with separate Fermi functions for the electron and hole quasiparticles.

2. AI can simulate complex photodoped phases using non-perturbative techniques by solving the many-body problem with any given distribution function Fsteady(ω), as demonstrated by adapting FT,µex(ω) = Θα(ω)f(ω + µex, T) + (1 − Θα(ω))f(ω − µex, T).

3. AI can perform gap estimation for photo-doped states using the inchworm QMC method to determine the Mott gap by the spectral weight reaching 0.057, which validates that the Mott gap remains robust at large photo-doping nex even in the numerically exact solution.

4. AI can compare different theoretical approaches by contrasting results, noting that NCA overestimates the gap ∆g in the photo-doped state, while AMEA shows a trend towards the direction of the numerically exact inchworm QMC method.

5. AI can characterize quasi-particle excitations by quantifying their effective temperature, as seen in comparing methods where TG,inch/T∆ is shown to be larger than TG,NCA/T∆, indicating that FG is consistently larger than F∆.