The past years have seen a revived interest in the diagrammatic Monte Carlo (DiagMC) methods for interacting fermions on a lattice. A promising recent development allows one to now circumvent the analytical continuation of dynamic observables in DiagMC calculations within the Matsubara formalism. This is made possible by symbolic algebra algorithms, which can be used to analytically solve the internal Matsubara frequency summations of Feynman diagrams. In this paper, we take a different approach and show that it yields improved results. We present a closed-form analytical solution of imaginary-time integrals that appear in the time-domain formulation of Feynman diagrams. We implement and test a DiagMC algorithm based on this analytical solution and show that it has numerous significant advantages. Most importantly, the algorithm is general enough for any kind of single-time correlation function series, involving any single-particle vertex insertions. Therefore, it readily allows for the use of action-shifted schemes, aimed at improving the convergence properties of the series. By performing a frequency-resolved action-shift tuning, we are able to further improve the method and converge the self-energy in a non-trivial regime, with only 3-4 perturbation orders. Finally, we identify time integrals of the same general form in many commonly used Monte Carlo algorithms and therefore expect a broader usage of our analytical solution.
Two weak coupling Continuous Time Quantum Monte Carlo (CTQMC) methods are shown to be equivalent for Hubbard-like interactions. A relation between these CTQMC methods and the Hirsch-Fye Quantum Monte Carlo (HFQMC) method is established, identifying the latter as an approximation within CTQMC and providing a diagrammatic interpretation. Both HFQMC and CTQMC are shown to be equivalent when the number of time slices in HFQMC becomes infinite, implying the same degree of fermion sign problem in this limit.
We describe a controllable and unbiased strong-coupling diagrammatic Monte Carlo technique that is applicable to a wide range of fermionic systems and spin models. Unlike previous strong coupling methods that generally rely on the Grassmannian Hubbard-Stratonovich transformation, our construction is based on Wick's theorem and a recursive procedure to group contractions into effective connected vertices that are non-perturbative in all local physics and can be calculated exactly. The resulting expansion is described by simple diagrammatic rules that make it suitable for systematic treatment vi…
We show that Monte Carlo sampling of the Feynman diagrammatic series (DiagMC) can be used for tackling hard fermionic quantum many-body problems in the thermodynamic limit by presenting accurate results for the repulsive Hubbard model in the correlated Fermi liquid regime. Sampling Feynman's diagrammatic series for the single-particle self-energy we can study moderate values of the on-site repulsion ($U/t \sim 4$) and temperatures down to $T/t=1/40$. We compare our results with high temperature series expansion and with single-site and cluster dynamical mean-field theory.
Using a dual representation of lattice fermion models that is based on spin-charge transformation and fermionisation of the original description, I derive an algorithm for diagrammatic Monte Carlo simulation of strongly correlated systems. This scheme allows eliminating large expansion parameters, as well as large corrections to the density matrix that generally prevent diagrammatic methods from being efficient in this regime. As an example, I compute the filling factor for the Hubbard model at infinite onsite repulsion and compare the results to controllable data obtained from Numerical Linke…
We present a general framework for performing "bold-line" diagrammatic Monte Carlo calculations based on using an analytical partial resummation as a starting point for stochastically summing all diagrams. As an example we solve the one orbital Anderson model using the non-crossing approximation as a starting point. We assess the accuracy of the method by using it to solve the equations of single-site dynamical mean field theory. We establish the validity of the starting approximations and show that the bold method provides a very accurate treatment of the Mott-insulating phase.
In this work we introduce the Dual Boson Diagrammatic Monte Carlo technique for strongly interacting electronic systems. This method combines the strength of dynamical mean-filed theory for non-perturbative description of local correlations with the systematic account of non-local corrections in the Dual Boson theory by the diagrammatic Monte Carlo approach. It allows us to get a numerically exact solution of the dual boson theory at the two-particle local vertex level for the extended Hubbard model. We show that it can be efficiently applied to description of single particle observables in a …
We study the interplay of electron-electron and electron-phonon interactions in the course of electron-hole bound state formation for gapped solid state systems. Adapting the essentially approximation-free diagrammatic Monte Carlo method for the calculation of the optical response, we discuss the absorption of light in correlated electron-phonon systems for the whole interaction and phonon frequency regimes. The spectral function obtained by analytical continuation from the imaginary-time current-current correlation function demonstrates the dressing of excitons by a phonon cloud when the coup…
We express the recently introduced real-time diagrammatic Quantum Monte Carlo, Phys. Rev. B 91, 245154 (2015), in the Larkin-Ovchinnikov basis in Keldysh space. Based on a perturbation expansion in the local interaction $U$, the special form of the interaction vertex allows to write diagrammatic rules in which vacuum Feynman diagrams directly vanish. This reproduces the main property of the previous algorithm, without the cost of the exponential sum over Keldysh indices. In an importance sampling procedure, this implies that only interaction times in the vicinity of the measurement time contri…
We develop an approximation-free Diagrammatic Monte Carlo technique to study fermionic particles interacting with each other simultaneously through both an attractive Coulomb potential and bosonic excitations of the underlying medium. Exemplarily we apply the method to the long-standing exciton-polaron problem and present numerically exact results for the wave function, ground-state energy, binding energy and effective mass of this quasiparticle. Focusing on the electron-hole pair bound-state formation, we discuss various limiting cases of a generic exciton-polaron model. The frequently used i…
Diagrammatic Monte Carlo -- the technique for numerically exact summation of all Feynman diagrams to high orders -- offers a unique unbiased probe of continuous phase transitions. Being formulated directly in the thermodynamic limit, the diagrammatic series is bound to diverge and is not resummable at the transition due to the non-analyticity of physical observables. This enables the detection of the transition with controlled error bars from an analysis of the series coefficients alone, avoiding the challenge of evaluating physical observables near the transition. We demonstrate this techniqu…
Using fermionic representation of spin degrees of freedom within the Popov-Fedotov approach we develop an algorithm for Monte Carlo sampling of skeleton Feynman diagrams for Heisenberg type models. Our scheme works without modifications for any dimension of space, lattice geometry, and interaction range, i.e. it is suitable for dealing with frustrated magnetic systems at finite temperature. As a practical application we compute uniform magnetic susceptibility of the antiferromagnetic Heisenberg model on the triangular lattice and compare our results with the best available high-temperature exp…
The diagrammatic Monte Carlo (Diag-MC) method is a numerical technique which samples the entire diagrammatic series of the Green's function in quantum many-body systems. In this work, we incorporate the flat histogram principle in the diagrammatic Monte method and we term the improved version "Flat Histogram Diagrammatic Monte Carlo" method. We demonstrate the superiority of the method over the standard Diag-MC in extracting the long-imaginary-time behavior of the Green's function, without incorporating any a priori knowledge about this function, by applying the technique to the polaron proble…
We introduce a Monte Carlo scheme for sampling bold-line diagrammatic series specifying an unknown function in terms of itself. The range of convergence of this bold(-line) diagrammatic Monte Carlo (BMC) is significantly broader than that of a simple iterative scheme for solving integral equations. With BMC technique, a moderate "sign problem" turns out to be an advantage in terms of the convergence of the process. For an illustrative example, we solve one-particle s-scattering problem. As an important application, we obtain T-matrix for a fermipolaron (one spin-down particle interacting with …
We demonstrate, by considering the triangular lattice spin-1/2 Heisenberg model, that Monte Carlo sampling of skeleton Feynman diagrams within the fermionization framework offers a universal first-principles tool for strongly correlated lattice quantum systems. We observe the fermionic sign blessing---cancellation of higher order diagrams leading to a finite convergence radius of the series. We calculate the magnetic susceptibility of the triangular-lattice quantum antiferromagnet in the correlated paramagnet regime and reveal a surprisingly accurate microscopic correspondence with its classic…
For some models of interacting fermions the known solution to the notorious sign-problem in Monte Carlo (MC) simulations is to work with macroscopic fermionic determinants; the price, however, is a macroscopic scaling of the numerical effort spent on elementary local updates. We find that the {\it ratio} of two macroscopic determinants can be found with any desired accuracy by considering truncated (local in space and time) matices. In this respect, MC for interacting fermionic systems becomes similar to that for the sign-problem-free bosonic systems with system-size independent update cost. W…
Diagrammatic expansions are a central tool for treating correlated electron systems. At thermal equilibrium, they are most naturally defined within the Matsubara formalism. However, extracting any dynamic response function from a Matsubara calculation ultimately requires the ill-defined analytical continuation from the imaginary- to the real-frequency domain. It was recently proposed [Phys. Rev. B 99, 035120 (2019)] that the internal Matsubara summations of any interaction-expansion diagram can be performed analytically by using symbolic algebra algorithms. The result of the summations is then…
We propose a novel approach to nonequilibrium real-time dynamics of quantum impurities models coupled to biased non-interacting leads, such as those relevant to quantum transport in nanoscale molecular devices. The method is based on a Diagrammatic Monte Carlo sampling of the real-time perturbation theory along the Keldysh contour. We benchmark the method on a non-interacting resonant level model and, as a first non-trivial application, we study zero temperature non-equilibrium transport through a vibrating molecule.
The computation of determinants plays a central role in diagrammatic Monte Carlo algorithms for strongly correlated systems. The evaluation of large numbers of determinants can often be the limiting computational factor determining the number of attainable diagrammatic expansion orders. In this work we build upon the algorithm presented in [\emph{Linear Algebra and its applications} 419.1 (2006), 107-124] which computes all principal minors of a matrix in $O(2^n)$ operations. We present multiple generalizations of the algorithm to the efficient evaluation of certain subsets of all principal mi…
Diagrammatic Monte Carlo (DiagMC) is a numeric technique that allows one to calculate quantities specified in terms of diagrammatic expansions, the latter being a standard tool of many-body quantum statistics. The sign problem that is typically fatal to Monte Carlo approaches, appears to be manageable with DiagMC. Starting with a general introduction to the principles of DiagMC, we present a detailed description of the DiagMC scheme for interacting fermions (Hubbard model), as well as the first illustrative results for the equations of state.
We introduce a numerical algorithm to stochastically sample the dual fermion perturbation series around the dynamical mean field theory, generating all topologies of two-particle interaction vertices. We show results in the weak and strong coupling regime of the half-filled Hubbard model in two dimensions, illustrating that the method converges quickly where dynamical mean field theory is a good approximation, and show that corrections are large in the strong correlation regime at intermediate interaction. The fast convergence of dual corrections to dynamical mean field results illustrates the…
We present a simple trick that allows to consider the sum of all connected Feynman diagrams at fixed position of interaction vertices for general fermionic models. With our approach one achieves superior performance compared to Diagrammatic Monte Carlo, while rendering the algorithmic part dramatically simpler. As we consider the sum of all connected diagrams at once, we allow for cancellations between diagrams with different signs, alleviating the sign problem. Moreover, the complexity of the calculation grows exponentially with the order of the expansion, which should be constrasted with the…
We study a model of quasiparticles on a two-dimensional square lattice coupled to Gaussian distributed dynamical fields. The model describes quasiparticles coupled to spin or charge fluctuations and is solved by a Monte Carlo sampling of the molecular field distributions. The non-perturbative solution is compared to various approximations based on diagrammatic perturbation theory. When the molecular field correlations are sufficiently weak, the diagrammatic calculations capture the qualitative aspects of the quasiparticle spectrum. For a range of model parameters near the magnetic boundary, we…
We develop a Monte Carlo scheme for sampling series of Feynman diagrams for the proper self-energy which are self-consistently expressed in terms of renormalized particle propagators. This approach is used to solve the problem of a single spin-down fermion resonantly interacting with the Fermi gas of spin-up particles. Though the original series based on bare propagators are sign-alternating and divergent one can still determine the answer behind them by using two strategies (separately or together): (i) using proper series re-summation techniques, and (ii) introducing renormalized propagators…
We study thermodynamics of the 3D Hubbard model at half filling on approach to the Néel transition by means of large-scale unbiased Diagrammatic Determinant Monte Carlo simulations. We obtain the transition temperature in the strongly correlated regime, as well as temperature dependence of energy, entropy, double occupancy, and the nearest-neighbor spin correlation function. Our results improve the accuracy of previous unbiased studies and present accurate benchmarks in the ongoing effort to realize the antiferromagnetic state of matter with ultracold atoms in optical lattices.
We extend the recently developed real-time Diagrammatic Monte Carlo method, in its hybridization expansion formulation, to the full Kadanoff-Baym-Keldysh contour. This allows us to study real-time dynamics in correlated impurity models starting from an arbitrary, even interacting, initial density matrix. As a proof of concept we apply the algorithm to study the non equilibrium dynamics after a local quantum quench in the Anderson Impurity Model. Being a completely general approach to real-time dynamics in quantum impurity models it can be used as a solver for Non Equilibrium Dynamical Mean Fie…
The dual-fermion approach provides a formally exact prescription for calculating properties of a correlated electron system in terms of a diagrammatic expansion around dynamical mean-field theory (DMFT). Most practical implementations, however, neglect higher-order interaction vertices beyond two-particle scattering in the dual effective action and further truncate the diagrammatic expansion in the two-particle scattering vertex to a leading-order or ladder-type approximation. In this work we compute the dual-fermion expansion for the two-dimensional Hubbard model including all diagram topolog…
We present a technique that enables the evaluation of perturbative expansions based on one-loop-renormalized vertices up to large expansion orders. Specifically, we show how to compute large-order corrections to the random phase approximation in either the particle-hole or particle-particle channels. The algorithm's efficiency is achieved by the summation over contributions of all symmetrized Feynman diagram topologies using determinants, and by integrating out analytically the two-body long-range interactions in order to yield an effective zero-range interaction. Notably, the exponential scal…
We present recent advances in understanding of the ground and excited states of the electron-phonon coupled systems obtained by novel methods of Diagrammatic Monte Carlo and Stochastic Optimization, which enable the approximation-free calculation of Matsubara Green function in imaginary times and perform unbiased analytic continuation to real frequencies. We present exact numeric results on the ground state properties, Lehmann spectral function and optical conductivity of different strongly correlated systems: Frohlich polaron, Rashba-Pekar exciton-polaron, pseudo Jahn-Teller polaron, exciton,…
We present a novel method for precise numerical solution of the irreducible two-body problem and apply it to excitons in solids. The approach is based on the Monte Carlo simulation of the two-body Green function specified by Feynman's diagrammatic expansion. Our method does not rely on the specific form of the electron and hole dispersion laws and is valid for any attractive electron-hole potential. We establish limits of validity of the Wannier (large radius) and Frenkel (small radius) approximations, present accurate data for the intermediate radius excitons, and give evidence for the charge…
We present the first approximation free diagrammatic Monte Carlo study of a lattice polaron interacting with an acoustic phonon branch through the deformation potential. Weak and strong coupling regimes are separated by a self-trapping region where quantum resonance between various possible lattice deformations is seen in the ground state properties, spectral function, and optical conductivity. The unique feature of such polaron is the interplay between long- and short wavelength acoustic vibrations creating a composite phonon cloud and leading to persistent self-trapping due to the existence …
The bold diagrammatic Monte Carlo (BDMC) method performs an unbiased sampling of Feynman's diagrammatic series using skeleton diagrams. For lattice models the efficiency of BDMC can be dramatically improved by incorporating dynamic mean-field theory (DMFT) solutions into renormalized propagators. From the DMFT perspective, combining it with BDCM leads to an unbiased method with well-defined accuracy. We illustrate the power of this approach by computing the single-particle propagator (and thus the density of states) in the non-perturbative regime of the Anderson localization problem, where a g…
Stoner transitions beyond mean-field in two-dimensional electronic systems: a diagrammatic Monte Carlo study
School of Physics and Astronomy and William I. Fine Theoretical Physics Institute, University of Minnesota（明尼苏达大学物理与天文学学院及威廉·I·芬恩理论物理研究所）；Department of Physics, University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校物理系）
We present a diagrammatic Monte Carlo method for quantum impurity problems with general interactions and general hybridization functions. Our method uses a recursive determinant scheme to sample diagrams for the scattering amplitude. Unlike in other methods for general impurity problems, an approximation of the continuous hybridization function by a finite number of bath states is not needed, and accessing low temperature does not incur an exponential cost. We test the method for the example of molecular systems, where we systematically vary temperature, interatomic distance, and basis set siz…
We calculate the single-particle excitation spectrum and the Landau Fermi liquid parameters for the archetypal model of solids, the three-dimensional uniform electron gas, with the numerically exact variational diagrammatic Monte Carlo method. In the metallic range of density, we establish benchmark values for the wave-function renormalization factor $Z$, the effective mass $m^*/m$, and the Landau parameters $F_0^s$ and $F_0^a$ with unprecedented accuracy, and we resolve the long-standing puzzle of non-monotonic dependence of mass on density. We also exclude the possibility that experimentally…
We introduce the first bold diagrammatic Monte Carlo approach to deal with polaron problems at finite density non-perturbatively, i.e., by including vertex corrections to high orders. Using Holstein model on a square lattice as a prototypical example, we demonstrate that our method is capable of providing accurate results in the thermodynamic limit in all regimes from renormalized Fermi-liquid to single polarons, across the non-adiabatic region where Fermi and Debye energies are of the same order of magnitude. By accounting for vertex corrections the accuracy of theoretical description is incr…
We present a general method to optimize the evaluation of Feynman diagrammatic expansions, which requires the automated symbolic assignment of momentum/energy conserving variables to each diagram. With this symbolic representation, we utilize the pole structure of each diagram to automatically sort the Feynman diagrams into groups that are likely to contain nearly equal or nearly cancelling diagrams, and we show that for some systems this cancellation is exact. This allows for a potentially massive cancellation during the numerical integration of internal momenta variables, leading to an optim…
We describe a general recipe for constructing Metropolis updates for Diagrammatic Monte-Carlo (DiagMC) algorithms, based on the Schwinger-Dyson equations in quantum field theory. This approach bypasses explicit duality transformations, enumeration or classification of diagrams and can be used for lattice quantum field theories with unknown or complicated dual representations (such as non-Abelian lattice gauge theories). DiagMC algorithms constructed in this way can still be plagued by the sign problem, which is, however, completely different from the sign problem in conventional Monte-Carlo si…
We develop numerical tools for Diagrammatic Monte-Carlo simulations of non-Abelian lattice field theories in the t'Hooft large-N limit based on the weak-coupling expansion. First we note that the path integral measure of such theories contributes a bare mass term in the effective action which is proportional to the bare coupling constant. This mass term renders the perturbative expansion infrared-finite and allows to study it directly in the large-N and infinite-volume limits using the Diagrammatic Monte-Carlo approach. On the exactly solvable example of a large-N O(N) sigma model in D=2 dimen…
The major obstacle preventing Feynman diagrammatic expansions from accurately solving many-fermion systems in strongly correlated regimes is the series slow convergence or divergence problem. Several techniques have been proposed to address this issue: series resummation by conformal mapping, changing the nature of the starting point of the expansion by shifted action tools, and applying the homotopy analysis method to the Dyson-Schwinger equation. They emerge as dissimilar mathematical procedures aimed at different aspects of the problem. The proposed homotopic action offers a universal and s…
In this note, we present an efficient algorithm to sample directly the self-energy in the framework of the Connected Determinant technique. The introduction of the formalism of many-variable formal power series is essential to the proof, and more generally it is a natural mathematical tool for diagrammatic expansions.
We introduce a method for summing Feynman's perturbation series based on diagrammatic Monte Carlo that significantly improves its convergence properties. This allows us to investigate in a controllable manner the pseudogap regime of the Hubbard model and to study the nodal/antinodal dichotomy at low doping and intermediate coupling. Marked differences from the weak coupling scenario are manifest, such as a higher degree of incoherence at the antinodes than at the `hot spots'. Our results show that the pseudogap and reduction of quasiparticle coherence at the antinode is due to antiferromagneti…
We propose a diagrammatic Monte Carlo approach for general spin-boson models, which can be regarded as a generalization of the strong-coupling expansion for fermionic impurity models. The algorithm is based on a self-consistently computed three-point vertex and a stochastically sampled four-point vertex, and achieves convergence to the numerically exact result in a wide parameter regime. The performance of the algorithm is demonstrated with applications to a spin-boson model representing an emitter in a waveguide. As a function of the coupling strength, the spin exhibits a delocalization-local…
We propose a Monte Carlo method, which is a hybrid method of the quantum Monte Carlo method and variational Monte Carlo theory, to study the Hubbard model. The theory is based on the off-diagonal and the Gutzwiller type correlation factors which are taken into account by a Monte Carlo algorithm. In the 4x4 system our method is able to reproduce the exact results obtained by the diagonalization. An application is given to investigate the half-filled band case of two-dimensional square lattice. The energy is favorably compared with quantum Monte Carlo data.
We introduce and compare three different Monte Carlo determinantal algorithms that allow one to compute dynamical quantities, such as the self-energy, of fermionic systems in their thermodynamic limit. We show that the most efficient approach expresses the sum of a factorial number of one-particle-irreducible diagrams as a recursive sum of determinants with exponential complexity. By comparing results for the two-dimensional Hubbard model with those obtained from state-of-the-art diagrammatic Monte Carlo, we show that we can reach higher perturbation orders and greater accuracy for the same co…
Recent successes in Monte Carlo methods for simulating fermionic quantum impurity models have been based on diagrammatic resummation techniques, but are restricted by the need to sum over factorially large classes of diagrams individually. We present a fast algorithm for summing over the diagrams appearing in Inchworm hybridization expansions. The method relies on the inclusion-exclusion principle to reduce the scaling from factorial to exponential. We analyze the growth rate and compare with related algorithms for expansions in the many-body interaction. An implementation demonstrates that fo…
We discuss an alternative accurate Monte Carlo method to calculate the ground-state energy and related quantities for Laughlin states of the fractional quantum Hall effect in a disk geometry. This alternative approach allows us to obtain accurate bulk regime (thermodynamic limit) values for various quantities from Monte Carlo simulations with a small number of particles (much smaller than that needed with standard Monte Carlo approaches).
Quantum Monte Carlo (QMC) methods are some of the most accurate methods for simulating correlated electronic systems. We investigate the compatibility, strengths and weaknesses of two such methods, namely, diffusion Monte Carlo (DMC) and auxiliary-field quantum Monte Carlo (AFQMC). The multi-determinant trial wave functions employed in both approaches are generated using the configuration interaction using a perturbative selection made iteratively (CIPSI) technique. Complete basis set full configuration interaction (CBS-FCI) energies estimated with CIPSI are used as a reference in this compara…
We propose a minimal generalization of the celebrated Markov-Chain Monte Carlo algorithm which allows for an arbitrary number of configurations to be visited at every Monte Carlo step. This is advantageous when a parallel computing machine is available, or when many biased configurations can be evaluated at little additional computational cost. As an example of the former case, we report a significant reduction of the thermalization time for the paradigmatic Sherrington-Kirkpatrick spin-glass model. For the latter case, we show that, by leveraging on the exponential number of biased configurat…
