arXiv subjects

Olivier Parcollet

Publications and source records attributed to Olivier Parcollet.

At least 37 records · Page 2Linked to original sources

arXiv2023

Automatic, high-order, and adaptive algorithms for Brillouin zone integration

We present efficient methods for Brillouin zone integration with a non-zero but possibly very small broadening factor $η$, focusing on cases in which downfolded Hamiltonians can be evaluated efficiently using Wannier interpolation. We describe robust, high-order accurate algorithms automating convergence to a user-specified error tolerance $\varepsilon$, emphasizing an efficient computational scaling with respect to $η$. After analyzing the standard equispaced integration method, applicable in the case of large broadening, we describe a simple iterated adaptive integration algorithm effective in the small $η$ regime. Its computational cost scales as $\mathcal{O}(\log^3(η^{-1}))$ as $η\to 0^+$ in three dimensions, as opposed to $\mathcal{O}(η^{-3})$ for equispaced integration. We argue that, by contrast, tree-based adaptive integration methods scale only as $\mathcal{O}(\log(η^{-1})/η^{2})$ for typical Brillouin zone integrals. In addition to its favorable scaling, the iterated adaptive algorithm is straightforward to implement, particularly for integration on the irreducible Brillouin zone, for which it avoids the tetrahedral meshes required for tree-based schemes. We illustrate the algorithms by calculating the spectral function of SrVO$_3$ with broadening on the meV scale.

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arXiv2023

Equilibrium Quantum Impurity Problems via Matrix Product State Encoding of the Retarded Action

In the $0+1$ dimensional imaginary-time path integral formulation of quantum impurity problems, the retarded action encodes the hybridization of the impurity with the bath. In this Article, we explore the computational power of representing the retarded action as matrix product state (RAMPS). We focus on the challenging Kondo regime of the single-impurity Anderson model, where non-perturbative strong-correlation effects arise at very low energy scales. We demonstrate that the RAMPS approach reliably reaches the Kondo regime for a range of interaction strengths $U$, with a numerical error scaling as a weak power law with inverse temperature. We investigate the convergence behavior of the method with respect to bond dimension and time discretization by analyzing the error of local observables in the full interacting problem and find polynomial scaling in both parameters. Our results show that the RAMPS approach offers promise as an alternative tool for studying quantum impurity problems in regimes that challenge established methods, such as multi-orbital systems. Overall, our study contributes to the development of efficient and accurate non-wavefunction-based tensor-network methods for quantum impurity problems.

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arXiv2023

Low rank Green's function representations applied to dynamical mean-field theory

Several recent works have introduced highly compact representations of single-particle Green's functions in the imaginary time and Matsubara frequency domains, as well as efficient interpolation grids used to recover the representations. In particular, the intermediate representation with sparse sampling and the discrete Lehmann representation (DLR) make use of low-rank compression techniques to obtain optimal approximations with controllable accuracy. We consider the use of the DLR in dynamical mean-field theory (DMFT) calculations, and in particular, show that the standard full Matsubara frequency grid can be replaced by the compact grid of DLR Matsubara frequency nodes. We test the performance of the method for a DMFT calculation of Sr$_2$RuO$_4$ at temperature $50$K using a continuous-time quantum Monte Carlo impurity solver, and demonstrate that Matsubara frequency quantities can be represented on a grid of only $36$ nodes with no reduction in accuracy, or increase in the number of self-consistent iterations, despite the presence of significant Monte Carlo noise.

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arXiv2022

Sachdev-Ye-Kitaev Models and Beyond: A Window into Non-Fermi Liquids

We present a review of the Sachdev-Ye-Kitaev (SYK) model of compressible quantum many-body systems without quasiparticle excitations, and its connections to various theoretical studies of non-Fermi liquids in condensed matter physics. The review is placed in the context of numerous experimental observations on correlated electron materials. Strong correlations in metals are often associated with their proximity to a Mott transition to an insulator created by the local Coulomb repulsion between the electrons. We explore the phase diagrams of a number of models of such local electronic correlation, employing a dynamical mean field theory in the presence of random spin exchange interactions. Numerical analyses and analytical solutions, using renormalization group methods and expansions in large spin degeneracy, lead to critical regions which display SYK physics. The models studied include the single-band Hubbard model, the $t$-$J$ model and the two-band Kondo-Heisenberg model in the presence of random spin exchange interactions. We also examine non-Fermi liquids obtained by considering each SYK model with random four-fermion interactions to be a multi-orbital atom, with the SYK-atoms arranged in an infinite lattice. We connect to theories of sharp Fermi surfaces without any low-energy quasiparticles in the absence of spatial disorder, obtained by coupling a Fermi liquid to a gapless boson; a systematic large $N$ theory of such a critical Fermi surface, with SYK characteristics, is obtained by averaging over an ensemble of theories with random boson-fermion couplings. Finally, we present an overview of the links between the SYK model and quantum gravity and end with an outlook on open questions.

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arXiv2022

Learning Feynman Diagrams with Tensor Trains

We use tensor network techniques to obtain high order perturbative diagrammatic expansions for the quantum many-body problem at very high precision. The approach is based on a tensor train parsimonious representation of the sum of all Feynman diagrams, obtained in a controlled and accurate way with the tensor cross interpolation algorithm. It yields the full time evolution of physical quantities in the presence of any arbitrary time dependent interaction. Our benchmarks on the Anderson quantum impurity problem, within the real time non-equilibrium Schwinger-Keldysh formalism, demonstrate that this technique supersedes diagrammatic Quantum Monte Carlo by orders of magnitude in precision and speed, with convergence rates $1/N^2$ or faster, where N is the number of function evaluations. The method also works in parameter regimes characterized by strongly oscillatory integrals in high dimension, which suffer from a catastrophic sign problem in Quantum Monte-Carlo. Finally, we also present two exploratory studies showing that the technique generalizes to more complex situations: a double quantum dot and a single impurity embedded in a two dimensional lattice.

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arXiv2022

Discrete Lehmann representation of imaginary time Green's functions

We present an efficient basis for imaginary time Green's functions based on a low rank decomposition of the spectral Lehmann representation. The basis functions are simply a set of well-chosen exponentials, so the corresponding expansion may be thought of as a discrete form of the Lehmann representation using an effective spectral density which is a sum of $δ$ functions. The basis is determined only by an upper bound on the product $βω_{\max}$, with $β$ the inverse temperature and $ω_{\max}$ an energy cutoff, and a user-defined error tolerance $ε$. The number $r$ of basis functions scales as $\mathcal{O}\left(\log(βω_{\max}) \log (1/ε)\right)$. The discrete Lehmann representation of a particular imaginary time Green's function can be recovered by interpolation at a set of $r$ imaginary time nodes. Both the basis functions and the interpolation nodes can be obtained rapidly using standard numerical linear algebra routines. Due to the simple form of the basis, the discrete Lehmann representation of a Green's function can be explicitly transformed to the Matsubara frequency domain, or obtained directly by interpolation on a Matsubara frequency grid. We benchmark the efficiency of the representation on simple cases, and with a high precision solution of the Sachdev-Ye-Kitaev equation at low temperature. We compare our approach with the related intermediate representation method, and introduce an improved algorithm to build the intermediate representation basis and a corresponding sampling grid.

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arXiv2021

Charge self-consistent electronic structure calculations with dynamical mean-field theory using Quantum ESPRESSO, Wannier90 and TRIQS

We present a fully charge self-consistent implementation of dynamical mean field theory (DMFT) combined with density functional theory (DFT) for electronic structure calculations of materials with strong electronic correlations. The implementation uses the Quantum ESPRESSO package for the density functional theory calculations, the Wannier90 code for the up-/downfolding and the TRIQS software package for setting up and solving the DMFT equations. All components are available under open source licenses, are MPI-parallelized, fully integrated in the respective packages, and use an hdf5 archive interface to eliminate file parsing. We show benchmarks for three different systems that demonstrate excellent agreement with existing DFT+DMFT implementations in other ab-initio electronic structure codes.

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arXiv2021

Planckian Metal at a Doping-Induced Quantum Critical Point

We numerically study a model of interacting spin-$1/2$ electrons with random exchange coupling on a fully connected lattice. This model hosts a quantum critical point separating two distinct metallic phases as a function of doping: a Fermi liquid phase with a large Fermi surface volume and a low-doping phase with local moments ordering into a spin-glass. We show that this quantum critical point has non-Fermi liquid properties characterized by $T$-linear Planckian behavior, $ω/T$ scaling and slow spin dynamics of the Sachdev-Ye-Kitaev (SYK) type. The $ω/T$ scaling function associated with the electronic self-energy is found to have an intrinsic particle-hole asymmetry, a hallmark of a `skewed' non-Fermi liquid.

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arXiv2021

Minimally Entangled Typical Thermal States Algorithms for Finite Temperature Matsubara Green Functions

We extend finite-temperature tensor network methods to compute Matsubara imaginary-time correlation functions, building on the minimally entangled typical thermal states (METTS) and purification algorithms. While imaginary-time correlation functions are straightforward to formulate with these methods, care is needed to avoid convergence issues that would result from naive estimators. As a benchmark, we study the single-band Anderson impurity model, even though the algorithm is quite general and applies to lattice models. The special structure of the impurity model benchmark system and our choice of basis enable techniques such as reuse of high-probability METTS for increasing algorithm efficiency. The results are competitive with state-of-the-art continuous time Monte Carlo. We discuss the behavior of computation time and error as a function of the number of purified sites in the Hamiltonian.

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arXiv2021

Inter-orbital singlet pairing in Sr$_2$RuO$_4$: a Hund's superconductor

We study the superconducting gap function of Sr$_2$RuO$_4$. By solving the linearized Eliashberg equation with a correlated pairing vertex extracted from a dynamical mean-field calculation we identify the dominant pairing channels. An analysis of the candidate gap functions in orbital and quasiparticle band basis reveals that an inter-orbital singlet pairing of even parity is in agreement with experimental observations. It reconciles in particular the occurrence of a two-component order parameter with the presence of line-nodes of quasiparticles along the c-axis in the superconducting phase. The strong angular dependence of the gap along the Fermi surface is in stark contrast to its quasi-locality when expressed in the orbital basis. We identify local inter-orbital spin correlations as the driving force for the pairing and thus reveal the continuation of Hund's physics into the superconducting phase.

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arXiv2021

Tracking the Footprints of Spin Fluctuations: A MultiMethod, MultiMessenger Study of the Two-Dimensional Hubbard Model

The Hubbard model represents the fundamental model for interacting quantum systems and electronic correlations. Using the two-dimensional half-filled Hubbard model at weak coupling as a testing ground, we perform a comparative study of a comprehensive set of state of the art quantum many-body methods. Upon cooling into its insulating antiferromagnetic ground-state, the model hosts a rich sequence of distinct physical regimes with crossovers between a high-temperature incoherent regime, an intermediate temperature metallic regime and a low-temperature insulating regime with a pseudogap created by antiferromagnetic fluctuations. We assess the ability of each method to properly address these physical regimes and crossovers through the computation of several observables probing both quasiparticle properties and magnetic correlations, with two numerically exact methods (diagrammatic and determinantal quantum Monte Carlo) serving as a benchmark. By combining computational results and analytical insights, we elucidate the nature and role of spin fluctuations in each of these regimes. Based on this analysis, we explain how quasiparticles can coexist with increasingly long-range antiferromagnetic correlations, and why dynamical mean-field theory is found to provide a remarkably accurate approximation of local quantities in the metallic regime. We also critically discuss whether imaginary time methods are able to capture the non-Fermi liquid singularities of this fully nested system.

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arXiv2020

Quantum Quasi-Monte Carlo algorithm for out-of-equilibrium Green functions at long times

We extend the recently developed Quantum Quasi-Monte Carlo (QQMC) approach to obtain the full frequency dependence of Green functions in a single calculation. QQMC is a general approach for calculating high-order perturbative expansions in power of the electron-electron interaction strength. In contrast to conventional Markov chain Monte Carlo sampling, QQMC uses low-discrepancy sequences for a more uniform sampling of the multi-dimensional integrals involved and can potentially outperform Monte Carlo by several orders of magnitudes. A core concept of QQMC is the a priori construction of a "model function" that approximates the integrand and is used to optimize the sampling distribution. In this paper, we show that the model function concept extends to a kernel approach for the computation of Green functions. We illustrate the approach on the Anderson impurity model and show that the scaling of the error with the number of integrand evaluations $N$ is $\sim 1/N^{0.86}$ in the best cases, and comparable to Monte Carlo scaling $\sim 1/N^{0.5}$ in the worst cases. We find a systematic improvement over Monte Carlo sampling by at least two orders of magnitude while using a basic form of model function. Finally, we compare QQMC results with calculations performed with the Fork Tensor Product State (FTPS) method, a recently developed tensor network approach for solving impurity problems. Applying a simple Padé approximant for the series resummation, we find that QQMC matches the FTPS results beyond the perturbative regime.

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arXiv2020

Quantum Quasi-Monte Carlo Technique for Many-Body Perturbative Expansions

High order perturbation theory has seen an unexpected recent revival for controlled calculations of quantum many-body systems, even at strong coupling. We adapt integration methods using low-discrepancy sequences to this problem. They greatly outperform state-of-the-art diagrammatic Monte Carlo. In practical applications, we show speed-ups of several orders of magnitude with scaling as fast as $1/N$ in sample number $N$; parametrically faster than $1/\sqrt{N}$ in Monte Carlo. We illustrate our technique with a solution of the Kondo ridge in quantum dots, where it allows large parameter sweeps.

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arXiv2020

Real-space cluster dynamical mean-field theory: Center focused extrapolation on the one- and two particle level

We revisit the cellular dynamical mean-field theory (CDMFT) for the single band Hubbard model on the square lattice at half filling, reaching real-space cluster sizes of up to 9 x 9 sites. Using benchmarks against direct lattice diagrammatic Monte Carlo at high temperature, we show that the self-energy obtained from a cluster center focused extrapolation converges faster with the cluster size than the periodization schemes previously introduced in the literature. The same benchmark also shows that the cluster spin susceptibility can be extrapolated to the exact result at large cluster size, even though its spatial extension is larger than the cluster size.

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arXiv2020

Linear resistivity and Sachdev-Ye-Kitaev (SYK) spin liquid behavior in a quantum critical metal with spin-$1/2$ fermions

`Strange metals' with resistivity depending linearly on temperature $T$ down to low-$T$ have been a long-standing puzzle in condensed matter physics. Here, we consider a model of itinerant spin-$1/2$ fermions interacting via on-site Hubbard interaction and random infinite-ranged spin-spin interaction. We show that the quantum critical point associated with the melting of the spin-glass phase by charge fluctuations displays non-Fermi liquid behaviour, with local spin dynamics identical to that of the Sachdev-Ye-Kitaev family of models. This extends the quantum spin liquid dynamics previously established in the large-$M$ limit of $SU(M)$ symmetric models, to models with physical $SU(2)$ spin-$1/2$ electrons. Remarkably, the quantum critical regime also features a Planckian linear-$T$ resistivity associated with a $T$-linear scattering rate and a frequency dependence of the electronic self-energy consistent with the Marginal Fermi Liquid phenomenology.

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arXiv2020

Enhancement of Local Pairing Correlations in Periodically Driven Mott Insulators

We investigate a model for a Mott insulator in presence of a time-periodic modulated interaction and a coupling to a thermal reservoir. The combination of drive and dissipation leads to non-equilibrium steady states with a large number of doublon excitations, well above the maximum thermal-equilibrium value. We interpret this effect as an enhancement of local pairing correlations, providing analytical arguments based on a Floquet Hamiltonian. Remarkably, this Hamiltonian shows a tendency to develop long-range staggered superconducting correlations. This suggests the possibility of realizing the elusive eta-pairing phase in driven-dissipative Mott Insulators.

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arXiv2020

Imaginary-time matrix product state impurity solver in a real material calculation: Spin-orbit coupling in Sr$_2$RuO$_4$

Using an imaginary-time matrix-product state (MPS) based quantum impurity solver we perform a realistic dynamical mean-field theory (DMFT) calculation combined with density functional theory (DFT) for Sr$_2$RuO$_4$. We take the full Hubbard-Kanamori interactions and spin-orbit coupling (SOC) into account. The MPS impurity solver works at essentially zero temperature in the presence of SOC, a regime of parameters currently inaccessible to continuous-time quantum Monte Carlo (CTQMC) methods, due to a severe sign problem. We show that earlier results obtained at high temperature, namely that the diagonal self-energies are nearly unaffected by SOC and that interactions lead to an effective enhancement of the SOC, hold even at low temperature. We observe that realism makes the numerical solution of the impurity model with MPS much more demanding in comparison to earlier works on Bethe lattice models, requiring several algorithmic improvements.

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arXiv2019

A Quantum Monte Carlo algorithm for out-of-equilibrium Green's functions at long times

We present a quantum Monte-Carlo algorithm for computing the perturbative expansion in power of the coupling constant $U$ of the out-of-equilibrium Green's functions of interacting Hamiltonians of fermions. The algorithm extends the one presented in Phys. Rev. B 91 245154 (2015), and inherits its main property: it can reach the infinite time (steady state) limit since the computational cost to compute order $U^n$ is uniform versus time; the computing time increases as $2^n$. The algorithm is based on the Schwinger-Keldysh formalism and can be used for both equilibrium and out-of-equilibrium calculations. It is stable at both small and long real times including in the stationary regime, because of its automatic cancellation of the disconnected Feynman diagrams. We apply this technique to the Anderson quantum impurity model in the quantum dot geometry to obtain the Green's function and self-energy expansion up to order $U^{10}$ at very low temperature. We benchmark our results at weak and intermediate coupling with high precision Numerical Renormalization Group (NRG) computations as well as analytical results.

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