Session M49: Precision Many-Body Physics V: New Algorithms

High-order diagrammatic expansion around BCS: Polarized superfluid phase of the attractive Hubbard model

In contrast to conventional QMC methods, expansions of intensive quantities in series of connected Feynman diagrams can be formulated directly in the thermodynamic limit. Over the last decade, diagrammatic Monte Carlo algorithms made it possible to reach large expansion orders and to obtain state-of-the-art results for various models of interacting fermions in 2 and 3 dimensions, mostly in the normal phase.

We obtained first results inside a superconducting phase, for the 3D attractive Hubbard model [1]. Spontaneous symmetry breaking is implemented by expanding around a BCS Hamiltonian. All diagrams up to 12 loops are summed thanks to the connected determinant algorithm [2] with anomalous propagators. Working on the BCS side of the strongly correlated regime, we observe convergence of the expansion, and benchmark the results against determinant diagrammatic Monte Carlo [3]. In presence of a polarizing Zeeman field (where unbiased benchmarks are unavailable due to the fermion sign problem) we observe a first-order superconducting-to-normal phase transition, and a thermally activated polarization of the superconducting phase. We also discuss the large-order behavior of the expansion and its relation to Goldstone and instanton singularities.

[1] G. Spada, R. Rossi, F. Simkovic, R. Garioud, M. Ferrero, K. Van Houcke, F. Werner, arXiv:2103.12038

[2] R. Rossi, PRL 119, 045701 (2017)

[3] E. Burovski, N. Prokof'ev, B. Svistunov, M. Troyer, PRL 96, 160402 (2006)   

Isometric Tensor Network States on an Infinite Stripe

Contraction of standard 2D tensor network ansatzes relies on approximation schemes, as tensor contractions costs scale exponentially with system size. Recently, the Isometric Tensor Network (isoTNS) ansatz was introduced for 2D finite quantum systems on a square lattice, allowing for exact $\mathcal{O}(1)$ evaluation of expectation values provided we can efficiently move the orthogonality center throughout the network. Here we generalize this isoTNS ansatz to strip geometries, in which the networks are infinite by finite and consist of translationally invariant rows of tensors. We demonstrate several algorithms for the infinite Moses Move (iMM), which moves the orthogonality hypersurface between columns of infinite length in the network. Using these iMM algorithms, we perform imaginary time evolution to identify the ground state of our 2D system, where the cost of optimization scales linearly with strip width rather than exponentially.   

Massively parallel exact diagonalization with symmetries and minimal communication

Scalable, distributed-memory approaches to exact diagonalization require that functions of the Hamiltonian can be applied efficiently to the wavefunction. We present a method that requires a single collective (all-to-all) communication per Hamiltonian application while making use of conserved quantities. The approach is applicable to a wide class of quantum lattice systems with U(1) symmetries and finite-ranged interactions as well as multi-orbital impurity problems. It is less communication intensive than existing approaches, and its parallel scaling is entirely determined by that of an all-to-all communication on the computing platform.   

New formalism for the exact calculation of total energies and associated electronic states of many-body interactions with complexity n4

In this talk, I intend to show a new formalism that should solve the famous many-body problem of quantum mechanics with a complexity varying as a function of n⁴ where n is the finite number of states. There was already an object, the reduced two-body density matrix (2-RDM) that had this complexity but for it to be a quantum state with N electrons, a gigantic number of conditions must be taken into account.

This formalism should considerably reduce the computation time avoiding the exponential complexity of the current exact methods. One can also ask the question of the interest of quantum computers if n⁴ parameters suffice to describe an intricate quantum experimental state.

Based on anti-commutativity relations, we show that we can construct a class of mathematical objects that are isomorphic to many-body wavefunctions but have the advantage of being compactable. Thanks to this new formalism, we show that the 2-RDM of the wavefunctions solutions of a two-body interaction Hamiltonian can be represented exactly thanks to this new set of mathematical objects. We will give also a new geometric interpretation of the 2-RDM in the space of these new mathematical objects.

We will develop the Lagrangian, the corresponding Euler-Lagrange equations and illustrate with numerical examples.   

Floating point precision imaginary time response functions in quantum many-body physics

The dynamics of quantum many-body systems in thermal equilibrium is amenable to the imaginary time formalism. The sequent numerical treatment of imaginary time propagators is central to many forms of perturbative and non-perturbative algorithmic approaches, e.g. perturbation theory, diagrammatic Monte Carlo, embedding methods like dynamical mean-field theory, etc. However, the prevailing techniques for numerical representation and computations in imaginary time, based on equidistant grids, are suffering from poor accuracy and poor scaling at low temperature. This talk will give an overview of the recent progress in numerical representation and computation using highly accurate and compact imaginary time representations, as well as an outlook for applications to real-time Green's functions.   

Interaction expansion inchworm Monte Carlo

We generalize the inchworm Monte Carlo framework for the interaction expansion. Inchworm Monte Carlo is a method to iteratively group and sample classes of diagrams in diagrammatic perturbation expansions. In the hybridization expansion, the inchworm was shown to drastically delay the onset of the sign problem. Here, we generalize the iterative procedure to the interaction expansion and show application to multi-orbital quantum impurity models with general interaction and hybridization. We clarify convergence properties and highlight similarities and differences to the bare and bold Monte Carlo methods.   

Probingtopologicalspinliquids on a programmable quantum simulator

Quantum phases with topological order, such as quantum spin liquids, have been the focus of explorations for several decades. Such phases feature a number of remarkable properties including long-range quantum entanglement. Moreover, they can be potentially exploited for the realization of robust quantum computation, as exemplified by the paradigmatic toric code model. In this talk, I will show how a programmable quantum simulator based on Rydberg atom arrays can be used to realize and probe quantum spin liquid states. In our approach, atoms are placed on the links of a kagome lattice and coherent evolution under Rydberg blockade enables the transition into frustrated quantum states with no local order.  We detect the onset of a quantum spin liquid phase of the toric code type by measuring topological string operators in two complementary bases. The properties of this state are further revealed using a lattice with non-trivial topology, representing a step towards the realization of a topological qubit.  Our observations open the door to the controlled experimental exploration of topological quantum matter and could enable the investigation of new methods for topologically protected quantum information processing.   

Gaussian Fermionic Matrix Product States for generalized Hartree-Fock in quasi-1D systems

In many approximate approaches to fermionic quantum many-body systems, such as Hartree-Fock and density functional theory, solving a system of non-interacting fermions coupled to some effective potential is the computational bottleneck. In this work, we demonstrate that this crucial computational step can be accelerated using recently developed methods for Gaussian fermionic matrix product states (GFMPS). As an example, we studied the generalized Hartree-Fock method, which unifies Hartree-Fock and self-consistent BCS theory, applied to Hubbard models with an inhomogeneous potential. We demonstrate that for quasi-one-dimensional systems with local interactions, our approach scales approximately linear in the length of the system while yielding a similar accuracy to standard approaches that scale cubically in the system size.   

Sparse-modeling solver for dynamical two-particle response of correlated electrons

Computing dynamical two-particle response functions of correlated materials is a grand challenge in the field of computational material science. The dynamical mean-field theory (DMFT) is one of the most successful theories for computing one-particle response of correlated electrons. Nevertheless, it is notoriously difficult to compute two-particle responses as it involves solving the Bethe-Salpeter equation (BSE), whose computational effort scales unfavourably with inverse temperature and the number of bands.

In this talk, we propose an efficient method for computing the dynamical susceptibility in DMFT. The proposed method extensively uses sparse-modeling techniques, including the intermediate representation [1,2], sparse sampling of vertex functions [3], and sparse convolution techniques for solving BSE [4]. The computational complexity of the proposed method scales only logarithmically with inverse temperature (apart from solving an impurity model).

We numerically demonstrate the efficiency of the method for the Hubbard model near its magnetic transition. First, we show that the local full vertex can be measured on sparse sampling frequencies using the recently proposed improved estimators [5] based on the continuous-time hybridization expansion quantum Monte Carlo method [6]. Then, we solve the BSE using a "sparse-modeling" BSE solver, which computes the dynamical susceptibility at finite bosonic Matsubara frequencies without using a cutoff for fermionic frequencies. We will first present results for the single-orbital Hubbard model on the square lattice. If time allows, we will also show preliminary results for more realistic multi-orbital models. 

[1] H. Shinaoka et al., PRB 96, 035147 (2017).

[2] H. Shinaoka et al., arXiv:2106.12685.

[3] H. Shinaoka et al., SciPost Phys. 8, 012 (2020).

[4] M. Wallerberger*, H. Shinaoka*, A. Kauch, PRR 3, 033168 (2021).

[5] J. Kaufmann et al., PRB 96, 035114 (2017).

[6] P. Werner et al., PRL 97, 076405 (2006).   

Dynamical Variational Monte Carlo as a quantum impurity solver

The theoretical and numerical treatment of strongly-correlated many electron systems remains an outstanding challenge of modern condensed matter physics. The recently developed dynamical Variational Monte Carlo (dVMC) method enables the computation of the Green's function for larger clusters, but has so far been limited to translationally invariant models with periodic boundary conditions. Here we propose a method to relax these constraints. This new approach makes it possible to use dVMC as a quantum impurity solver for embedding techniques such as Cluster dynamical mean field theory (CDMFT).   

Self-Consistent Ab Initio Embedding Results for Real Materials

We present results for self-consistent embedding calculations of strongly correlated real materials. The ab initio simulation of periodic solids with strong correlation is an active area of research, since reliable parameter-free methods exist only for weakly correlated solids. We report results from a fully self-consistent parameter-free ab initio self-energy embedding theory consisting of a weakly correlated environment (treated at the level of GW) and strongly correlated orbitals (treated with Exact Diagonalization). The method is applied to real materials with transition metal d orbitals and rare-earth element f orbitals to obtain information about the spectral function and thermodynamic properties. We also discuss aspects of stability and convergence.