Iterative Perturbation Theory Research Articles

Iterated Perturbation Theory for Mott Insulators in a Static Electric Field with Optical Phonons

This article aims to compare the so‐called iterated perturbation theory (IPT) and auxiliary master equation approach (AMEA) impurity solvers for a Mott insulating system driven out of equilibrium by a static electric field. Electronic heat bath and optical phonons are employed as dissipation mechanism of the current‐induced Joule heat that the excited electrons of the lattice experience as the result of the field's driving. Despite its simplicity, the IPT approach yields results that qualitatively are in good agreement with those obtained within the AMEA impurity solver, although fails to reproduce some correlation effects.

Two-particle correlations and the metal-insulator transition: Iterated perturbation theory revisited

Recent advances in many-body physics have made it possible to study correlated electron systems at the two-particle level. In Dynamical Mean-Field theory, it has been shown that the metal-insulator phase diagram is closely related to the eigenstructure of the susceptibility. So far, this situation has been studied using accurate but numerically expensive solvers. Here, the Iterated Perturbation Theory (IPT) approximation is used instead. Its simplicity makes it possible to obtain analytical results for the two-particle vertex and the DMFT Jacobian. The limited computational cost also enables a detailed comparison of analytical expressions for the response functions to results obtained using finite differences. At the same time, the approximate nature of IPT precludes an interpretation of the metal-insulator transition in terms of a Landau free energy functional.

Evolution of the Andreev bands in the half-filled superconducting periodic Anderson model

We employ the periodic Anderson model with superconducting correlations in the conduction band at half filling to study the behavior of the in-gap bands in a heterostructure consisting of a molecular layer deposited on the surface of a conventional superconductor. We use the dynamical mean-field theory to map the lattice model on the superconducting single impurity model with self-consistently determined bath and use the continuous-time hybridization expansion (CT-HYB) quantum Monte Carlo and the iterative perturbation theory (IPT) as solvers for the impurity problem. We present phase diagrams for square and triangular lattice that both show two superconducting phases that differ by the sign of the induced pairing, in analogy to the 0 and $\ensuremath{\pi}$ phases of the superconducting single impurity Anderson model and discuss the evolution of the spectral function in the vicinity of the transition. We also discuss the failure of the IPT for superconducting models with spinful ground state and the behavior of the average expansion order of the CT-HYB simulation.

Scalable and Predictive Spectra of Correlated Molecules with Moment Truncated Iterated Perturbation Theory.

A reliable and efficient computation of the entire single-particle spectrum of correlated molecules is an outstanding challenge in the field of quantum chemistry, with standard density functional theory approaches often giving an inadequate description of excitation energies and gaps. In this work, we expand upon a recently introduced approach that relies on a fully self-consistent many-body perturbation theory coupled to a nonperturbative truncation of the effective dynamics at each step. We show that this yields a low-scaling and accurate method across a diverse benchmark test set that is capable of treating moderate levels of strong correlation effects, and we detail an efficient implementation for applications involving up to ∼1000 orbitals on parallel resources. We then use this method to characterize the spectral properties of the antimalarial drug molecule artemisinin, resolving discrepancies in previous works concerning the active sites of the lowest-energy fundamental excitations of the system.

Development of an efficient impurity solver in dynamical mean field theory for multiband systems: Iterative perturbation theory combined with parquet equations

Although several impurity solvers in the dynamical mean field theory (DMFT) have been proposed, especially in multi-band systems, there are practical difficulties arising from a trade-off between numerical costs and reliability. In this study, we re-interpret the iterative perturbation theory (IPT) as an approximation which captures the strong correlation effects by mimicking the particular frequency structures of the exact full vertex, and extend it such that it can have efficiency and reliability simultaneously by modifying IPT vertex using the parquet equations. We apply this method to several models to evaluate their validity. We confirm that our method shows good agreements with the numerically exact continuous-time quantum Monte Carlo method in the single-site DMFT calculation.

Neural network perturbation theory and its application to the Born series

Deep Learning using the eponymous deep neural networks (DNNs) has become an attractive approach towards various data-based problems of theoretical physics in the past decade. There has been a clear trend to deeper architectures containing increasingly more powerful and involved layers. Contrarily, Taylor coefficients of DNNs still appear mainly in the light of interpretability studies, where they are computed at most to first order. However, especially in theoretical physics numerous problems benefit from accessing higher orders, as well. This gap motivates a general formulation of neural network (NN) Taylor expansions. Restricting our analysis to multilayer perceptrons (MLPs) and introducing quantities we refer to as propagators and vertices, both depending on the MLP's weights and biases, we establish a graph-theoretical approach. Similarly to Feynman rules in quantum field theories, we can systematically assign diagrams containing propagators and vertices to the corresponding partial derivative. Examining this approach for S-wave scattering lengths of shallow potentials, we observe NNs to adapt their derivatives mainly to the leading order of the target function's Taylor expansion. To circumvent this problem, we propose an iterative NN perturbation theory. During each iteration we eliminate the leading order, such that the next-to-leading order can be faithfully learned during the subsequent iteration. After performing two iterations, we find that the first- and second-order Born terms are correctly adapted during the respective iterations. Finally, we combine both results to find a proxy that acts as a machine-learned second-order Born approximation.

Quantum Many-Body Theory for Exciton-Polaritons in Semiconductor Mie Resonators in the Non-Equilibrium

We implement externally excited ZnO Mie resonators in a framework of a generalized Hubbard Hamiltonian to investigate the lifetimes of excitons and exciton-polaritons out of thermodynamical equilibrium. Our results are derived by a Floquet-Keldysh-Green’s formalism with Dynamical Mean Field Theory (DMFT) and a second order iterative perturbation theory solver (IPT). We find that the Fano resonance which originates from coupling of the continuum of electronic density of states to the semiconductor Mie resonator yields polaritons with lifetimes between 0.6 ps and 1.45 ps. These results are compared to ZnO polariton lasers and to ZnO random lasers. We interpret the peaks of the exciton-polariton lifetimes in our results as a sign of gain narrowing which may lead to stable polariton lasing modes in the single excited ZnO Mie resonator. This form of gain may lead to polariton random lasing in an ensemble of ZnO Mie resonators in the non-equilibrium.

Coulomb scattering in the massless Nelson model II. Regularity of ground states

For the massless Nelson model, we provide detailed information about the dependence of the normalized ground states [Formula: see text] of the fiber single-electron Hamiltonians [Formula: see text] on the total momentum [Formula: see text] and the infrared cut-off [Formula: see text]. This information is obtained with the help of the iterative analytic perturbation theory. In particular, we derive bounds of the form [Formula: see text] for some constant [Formula: see text] and a function of the maximal admissible coupling constant [Formula: see text] such that [Formula: see text]. These results hold both in the infrared-regular and infrared-singular cases. They are exploited in part I of this series to construct the two-electron scattering states in the infrared-regular massless Nelson model (in the absence of an infrared cut-off) along the lines of the Haag–Ruelle scattering theory. They should also be relevant to the problem of scattering of two infraparticles in the infrared-singular Nelson model, whose solution is the goal of this series of papers. Although a part of a larger investigation, the present work is written in a self-contained fashion.

Analytical investigation of singularities in two-particle irreducible vertex functions of the Hubbard atom

Two-particle generalized susceptibilities and their irreducible vertex functions play a prominent role in the quantum many-body theory for correlated electron systems. They act as basic building blocks in the parquet formalism which provides a flexible scheme for the calculation of spectral and response functions. The irreducible vertices themselves have recently attracted increased attention as unexpected divergences in these functions have been identified. Remarkably, such singularities appear already for one of the simplest strongly interacting systems: the atomic limit of the half-filled Hubbard model (Hubbard atom). In this paper, we calculate the analytical expressions for all two-particle irreducible vertex functions of the Hubbard atom in all scattering channels as well as the fully irreducible two-particle vertices. We discuss their divergences and classify them by the eigenvalues and eigenvectors of the corresponding generalized susceptibilities. In order to establish a connection to the recently found multivaluedness of the exact self-energy functional $\Sigma[G]$, we show that already an approximation akin to iterated perturbation theory is sufficient to capture, qualitatively, the divergent structure of the vertex functions. Finally, we show that the localized divergences in the disordered binary mixture model are directly linked to a minimum in the single-particle Matsubara Green's function.

Full-intensity waveform inversion

Many full-waveform inversion schemes are based on the iterative perturbation theory to fit the observed waveforms. When the observed waveforms lack low frequencies, those schemes may encounter convergence problems due to cycle skipping when the initial velocity model is far from the true model. To mitigate this difficulty, we have developed a new objective function that fits the seismic-waveform intensity, so the dependence of the starting model can be reduced. The waveform intensity is proportional to the square of its amplitude. Forming the intensity using the waveform is a nonlinear operation, which separates the original waveform spectrum into an ultra-low-frequency part and a higher frequency part, even for data that originally do not have low-frequency contents. Therefore, conducting multiscale inversions starting from ultra-low-frequency intensity data can largely avoid the cycle-skipping problem. We formulate the intensity objective function, the minimization process, and the gradient. Using numerical examples, we determine that the proposed method was very promising and could invert for the model using data lacking low-frequency information.

Dynamical recovery of SU(2) symmetry in the mass-quenched Hubbard model

We use non-equilibrium dynamical mean-field theory with iterative perturbation theory as an impurity solver to study the recovery of $SU(2)$ symmetry in real-time following a hopping integral parameter quench from a mass-imbalanced to a mass-balanced single-band Hubbard model at half-filling. A dynamical order parameter $\gamma(t)$ is defined to characterize the evolution of the system towards $SU(2)$ symmetry. By comparing the momentum dependent occupation from an equilibrium calculation (with the $SU(2)$ symmetric Hamiltonian after the quench at an effective temperature) with the data from our non-equilibrium calculation, we conclude that the $SU(2)$ symmetry recovered state is a thermalized state. Further evidence from the evolution of the density of states supports this conclusion. At the same time, we find the order parameter in the weak Coulomb interaction regime undergoes an approximate exponential decay. We numerically investigate the interplay of the relevant parameters (initial temperature, Coulomb interaction strength, initial mass-imbalance ratio) and their combined effect on the thermalization behavior. Finally, we study evolution of the order parameter as the hopping parameter is changed with either a linear ramp or a pulse. Our results can be useful in strategies to engineer the relaxation behavior of interacting, quantum many-particle systems.

Coulomb Scattering in the Massless Nelson Model III: Ground State Wave Functions and Non-commutative Recurrence Relations

Let $H_{P,\sigma}$ be the single-electron fiber Hamiltonians of the massless Nelson model at total momentum $P$ and infrared cut-off $\sigma>0$. We establish detailed regularity properties of the corresponding $n$-particle ground state wave functions $f^n_{P,\sigma}$ as functions of $P$ and $\sigma$. In particular, we show that \[ |\partial_{P^j}f^{n}_{P,\sigma}(k_1,\ldots, k_n)|, \ \ |\partial_{P^j} \partial_{P^{j'}} f^{n}_{P,\sigma}(k_1,\ldots, k_n)| \leq \frac{1}{\sqrt{n!}} \frac{(c\lambda_0)^n}{\sigma^{\delta_{\lambda_0}}} \prod_{i=1}^n\frac{ \chi_{[\sigma,\kappa)}(k_i)}{|k_i|^{3/2}}, \] where $c$ is a numerical constant, $\lambda_0\mapsto \delta_{\lambda_0}$ is a positive function of the maximal admissible coupling constant which satisfies $\lim_{\lambda_0\to 0}\delta_{\lambda_0}=0$ and $\chi_{[\sigma,\kappa)}$ is the (approximate) characteristic function of the energy region between the infrared cut-off $\sigma$ and the ultraviolet cut-off $\kappa$. While the analysis of the first derivative is relatively straightforward, the second derivative requires a new strategy. By solving a non-commutative recurrence relation we derive a novel formula for $f^n_{P,\sigma}$ with improved infrared properties. In this representation $\partial_{P^{j'}}\partial_{P^{j}}f^n_{P,\sigma}$ is amenable to sharp estimates obtained by iterative analytic perturbation theory in part II of this series of papers. The bounds stated above are instrumental for scattering theory of two electrons in the Nelson model, as explained in part I of this series.

Orbital effect of the magnetic field in dynamical mean-field theory

The availability of large magnetic fields at international facilities and of simulated magnetic fields that can reach the flux-quantum-per-unit-area level in cold atoms, calls for systematic studies of orbital effects of the magnetic field on the self-energy of interacting systems. Here we demonstrate theoretically that orbital effects of magnetic fields can be treated within single-site dynamical mean-field theory with a translationally invariant quantum impurity problem. As an example, we study the one-band Hubbard model on the square lattice using iterated perturbation theory as an impurity solver. We recover the expected quantum oscillations in the scattering rate and we show that the magnetic fields allow the interaction-induced effective mass to be measured through the single-particle density of states accessible in tunneling experiments. The orbital effect of magnetic fields on scattering becomes particularly important in the Hofstadter butterfly regime.

A multi-orbital iterated perturbation theory for model Hamiltonians and real material-specific calculations of correlated systems

Perturbative schemes utilizing a spectral moment expansion are well known and extensively used for investigating the physics of model Hamiltonians and real material systems. The advantages they offer, in terms of being computationally inexpensive, with real frequency output at zero and finite temperatures, compensate for their deficiencies and offer a quick, qualitative analysis of the system behavior. In this work, we have developed a method, that can be classified as a multi-orbital iterative perturbation theory (MO-IPT) to study N-fold degenerate and non degenerate Anderson impurity models. As applications of the solver, we have combined the method with dynamical mean field theory to explore lattice models like the single orbital Hubbard model, covalent band insulator and the multi-orbital Hubbard model for density-density type interactions in different parameter regimes. The Hund's coupling effects in case of multiple orbitals is also studied. The limitations and quality of results are gauged through extensive comparison with data from the numerically exact continuous time quantum Monte Carlo method (hybridization expansion CTQMC). In general we observe that the agreement with CTQMC results gets better as we move away from particle-hole symmetry. We have integrated MO-IPT with density functional theory based electronic structure methods to study real material systems. As a test case, we have studied the classic, strongly correlated electronic material, SrVO$_3$. A comparison of density of states and photo emission spectrum (PES) with results obtained from different impurity solvers and experiments yields good agreement.

Local moment approach as a quantum impurity solver for the Hubbard model

The local moment approach (LMA) has presented itself as a powerful semi-analytical quantum impurity solver (QIS) in the context of the dynamical mean-field theory (DMFT) for the periodic Anderson model and it correctly captures the low energy Kondo scale for the single impurity model, having excellent agreement with the Bethe ansatz and numerical renormalization group results. However, the most common correlated lattice model, the Hubbard model, has not been explored well within the LMA+DMFT framework beyond the insulating phase. Here in our work, within the framework we attempt to complete the phase diagram of the single band Hubbard model at zero temperature. Our formalism is generic to any particle filling and can be extended to finite temperature. We contrast our results with another QIS, namely the iterated perturbation theory (IPT) and show that the second spectral moment sum-rule improves better as the Hubbard interaction strength grows stronger in LMA, whereas it severely breaks down after the Mott transition in IPT. We also show that, in the metallic phase, the low-energy scaling of the spectral density leads to universality which extends to infinite frequency range at infinite correlation strength (strong-coupling). At large interaction strength, the off half-filling spectral density forms a pseudogap near the Fermi level and filling-controlled Mott transition occurs as one approaches the half-filling. Finally we study optical properties and find universal features such as absorption peak position governed by the low-energy scale and a doping independent crossing point, often dubbed as the "isosbestic point" in experiments.

Dynamical mean-field theory description of the voltage-induced transition in a nonequilibrium superconductor

Using dynamical mean-field theory (DMFT) we study a simplified model for heterostructures involving superconductors. The system is driven out-of-equilibrium by a voltage bias, imposed as an imbalance of chemical potential at the interface. We solve the self-consistent DMFT equations using iterative second-order perturbation theory in the Nambu-Keldysh formalism. We show that the superconducting state is destabilized by voltage biases of the order of the energy gap. We demonstrate that the transition to the normal state occurs through an intermediate bad superconducting state, which is characterized by a smaller value of the order parameter and incoherent excitations. We discuss the energetic balance behind the stabilization of such exotic superconducting state.

Long-lived nonequilibrium states in the Hubbard model with an electric field

We study the single-band Hubbard model in the presence of a large spatially uniform electric field out of equilibrium. Using the Keldysh nonequilibrium formalism, we solve the problem using perturbation theory in the Coulomb interaction $U$. We present numerical results for the charge current, the total energy of the system, and the double occupancy on an infinite-dimensional hypercubic lattice with nearest-neighbor hopping. The system is isolated from an external bath and is in the paramagnetic state. We show that an electric field pulse can drive the system to a steady nonequilibrium state, which does not evolve into a thermal state. We compare results obtained within second-order perturbation theory (SOPT), self-consistent SOPT, and iterated perturbation theory (IPT). We also discuss the importance of initial conditions for a system which is not coupled to an external bath.

Phase diagram of the half-filled ionic Hubbard model

We study the phase diagram of the ionic Hubbard model (IHM) at half-filling using dynamical mean field theory (DMFT), with two impurity solvers, namely, iterated perturbation theory (IPT) and continuous time quantum Monte Carlo (CTQMC). The physics of the IHM is governed by the competition between the staggered potential $\Delta$ and the on-site Hubbard U. In both the methods we find that for a finite $\Delta$ and at zero temperature, anti-ferromagnetic (AFM) order sets in beyond a threshold $U=U_{AF}$ via a first order phase transition below which the system is a paramagnetic band insulator. Both the methods show a clear evidence for a transition to a half-metal phase just after the AFM order is turned on, followed by the formation of an AFM insulator on further increasing U. We show that the results obtained within both the methods have good qualitative and quantitative consistency in the intermediate to strong coupling regime. On increasing the temperature, the AFM order is lost via a first order phase transition at a transition temperature $T_{AF}(U, \Delta)$ within both the methods, for weak to intermediate values of U/t. But in the strongly correlated regime, where the effective low energy Hamiltonian is the Heisenberg model, IPT is unable to capture the thermal (Neel) transition from the AFM phase to the paramagnetic phase, but the CTQMC does. As a result, at any finite temperature T, DMFT+CTQMC shows a second phase transition (not seen within DMFT+IPT) on increasing U beyond $U_{AF}$. At $U_N > U_{AF}$, when the Neel temperature $T_N$ for the effective Heisenberg model becomes lower than T, the AFM order is lost via a second order transition. In the 3-dimensonal parameter space of $(U/t,T/t,\Delta/t)$, there is a line of tricritical points that separates the surfaces of first and second order phase transitions.

A DFT+nonhomogeneous DMFT approach for finite systems

For reliable and efficient inclusion of electron–electron correlation effects in nanosystems we formulate a combined density functional theory/nonhomogeneous dynamical mean-field theory (DFT+DMFT) approach which employs an approximate iterated perturbation theory impurity solver. We further apply the method to examine the size-dependent magnetic properties of iron nanoparticles containing 11–100 atoms. We show that for the majority of clusters the DFT+DMFT solution is in very good agreement with experimental data, much better compared to the DFT and DFT+U results. In particular, it reproduces the oscillations in magnetic moment with size as observed experimentally. We thus demonstrate that the DFT+DMFT approach can be used for accurate and realistic description of nanosystems containing about hundred atoms.

Effects of correlations on honeycomb lattice in ionic-Hubbard model

In a honeycomb lattice the symmetry has been broken by adding an ionic potential and a single-particle gap was generated in the spectrum. We have employed the iterative perturbation theory (IPT) in dynamical mean field approximation method to study the effects of competition between $U$ and $\Delta$ on energy gap and renormalized Fermi velocity. We found, the competition between the single-particle gap parameter and the Hubbard potential closed the energy gap and restored the semi-metal phase, then the gap is opened again in Mott insulator phase. For a fixed $\Delta$ by increasing $U$, the renormalized Fermi velocity $\tilde{v_F}$ is decreased, but change in $\Delta$, for a fixed $U$, has no effects on $\tilde{v_F}$. The difference in filling factor is calculated for various number of $U, ~\Delta$. The results of this study can be implicated for gapped graphene e.g. hydrogenated graphene.