Session R21: Precision Many-Body Physics IV: Model Systems and Hamiltonians

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In this talk, I will present recent developments of the diagrammatic Monte Carlo method applied to the Hubbard model. Diagrammatic Monte Carlo methods are based on the construction of a perturbation series for physical observables. They face two main challenges: The first challenge is the accurate stochastic computation of the series coefficients that suffers from the fermionic sign problem. The second difficulty is the resummation of the series whose success depends on the structure of the physical observable expressed in the complex plane of the expansion parameter, e.g. the onsite repulsion U in the Hubbard model. I will discuss how these two issues can be addressed and, in particular, how the freedom to choose the non-interacting starting point of the theory can be used to construct more efficient perturbation series and directly investigate broken-symmetry phases. I will illustrate these developments with calculations performed in different regimes of the Hubbard model.   

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The competition between antiferromagnetism and hole motion in two-dimensional Mott insulators lies at the heart of a doping-dependent transition from an anomalous metal to a conventional Fermi liquid. Condensed matter experiments suggest that charge carriers change their nature within this crossover, but a full understanding remains elusive. We study this regime by preparing a cold fermionic gas in an optical lattice at a temperature around the superexchange energy. It is imaged using a quantum gas microscope with full spin and density resolution allowing the extraction of a wide range of correlators. Crucial to deeper understanding is the capability to calculate higher order correlators as well as common observables from solid states systems such as the spin susceptibility, all of which are studied as a function of doping level. While at low doping the system exhibits magnetic polarons, i.e. holes with a dressed cloud of spins, higher doping leads to the metallic Fermi liquid regime characterised by incommensurate magnetic fluctuations and altered correlations. The crossover is completed for hole dopings around 30%. Several theoretical models are benchmarked and their agreement with the experiment in different doping regimes discussed.
J.Koepsell et al., arXiv:2009.04440 (2020)   

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The Hubbard model is the paradigmatic model of this field, similar to what the drosophila is to genetics. Despite its simplicity, the Hubbard model presents a formidable challenge to computational and theoretical methods alike.

The work I will talk about, borrowing terminology from astrophysics, represents an extensive ’multi-method, multi-messenger’ assessment of the wealth of computational methods that have been developed in recent years to determine the physical properties of the Hubbard model in two spatial dimensions. These methods range from simple mean-field theory to cutting-edge quantum-field theoretical approaches as dynamical mean field theory and its extensions. Each of these methods is compared to two numerically exact benchmarks provided by diagrammatic Monte Carlo and determinantal quantum Monte Carlo. In approaching the weak coupling regime of this model in such a way, I will elucidate the nature and role of magnetic fluctuations present and explain their implications on the theory of metallic materials with strong magnetic correlations. arXiv:2006.10769   

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The high temperature series for the entropy of the 2D t-J model has been calculated to 12th order in β. Extrapolating this series below T≤ J has proven difficult in the past due to the presence of two temperature scales. Using a calculation of the Heisenberg AF entropy¹ a modified series for the entropy per site S of the t-J model can be produced ΔS(n,T) = S_tJ(n,T) - nS_AF(J^*). By adjusting J^* the series for ΔS can be extrapolated to lower temperatures using Pade approximants. This can be done for any density and allows the calculation of ∂S/∂n│_T. Using standard thermodynamics the temperature derivative of the pressure ∂P/∂T│_n = -n∂S/∂n│_T + S can be found, along with the full pressure P by integrating ∂P/∂T│_n and using the known high temperaure values. There is a range of densities and temperatures where ∂P/∂T│_n < 0. This region has a strong overlap with the model parameters where the strongest d-wave pair fluctuations² are found. Since ∂P/∂T│_n = α/κ, where α is the thermal expansion coefficient and κ ≥ 0 is the isothermal compressibility, the strongest d-wave pair fluctuations are found where the t-J model has a negative electronic thermal expansion.

1. W. O. Putikka, J. Phys.: Conf. Series 640, 012046 (2015).
2. W. O. Putikka, M. U. Luchini, PRL 96, 247001 (2006).   

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We have explored previously developed Coupled Cluster Green’s Function (CCGF) (Shee, Zgid JCTC2019) in two different contexts :

a) As a weak correlation method in a QM-QM embedding method, Self Energy embedding Theory (SEET). This has been envisaged to be useful for the complicated open-shell cases. We have shown difficulties in adapting a non GF based method as a weak correlation method.
b) We have tried successfully CCGF as a solver for SEET before. Now it has been further explored to show that if smaller impurities are used the accuracy of it is almost equivalent to Exact Diagonalization (ED) solver, thus smaller impurities will be preferred over the larger ones. Moreover, we can even reach higher accuracy in comparison to the cases where CCSD has been applied for the entire system.   

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We present TeNPy [1], a Python library for the simulation of strongly correlated quantum many body systems with the ansatz of tensor networks, and in particular matrix product states (MPS). The library aims to provide a good balance between code readability, easy implementation of custom models and algorithms, and numerical efficiency for large-scale simulations. After a short overview of the features (and limitations) of the library, we demonstrate how to setup the density matrix renormalization group (DMRG) algorithm for a custom model on a long cylinder geometry as a concrete example. Further, we showcase some applications of TeNPy, present benchmarks, and discuss the roadmap for future developments.
[1] https://github.com/tenpy/tenpy   

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We examine stationary state properties of an impurity particle injected into a one-dimensional quantum gas. For equal masses of the impurity and host particles the problem reduces to a variant of the Gaudin-Young model, which is integrable and admits a formal solution based on the Bethe Ansatz. Obtaining physical observables from the formal solution is still non-trivial: the form-factor summation can be done via a stochastic enumeration based on the Metropolis algorithm.

For unequal masses, the problem is no longer integrable, and Bethe Ansatz approach breaks down. To account for integrability-breaking perturbations, we develop an exact diagonalization procedure in the (truncated) basis of the Bethe Ansatz states for an integrable model. We study steady-state properties for both cases, of a heavy and a light impurity.   

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I will discuss some of the recent developments in many-body computational studies to understand properties of the two-dimensional Hubbard model. As a fundamental model in quantum many-body physics, the Hubbard model has presented a tremendous challenge, with multiple competing tendencies and its properties often the outcome of a delicate balance of their competition and coexistence. This places extermely demanding requirements on numerical approaches, including high precision, excellent predictive power and accuracy, and reliable access to the thermynamic limit. Progress in method development and the combined use of complementary methods have led to significant recent progress. I will introduce some of the methological advances in both ground-state and finite-temperature auxiliary-field quantum Monte Carlo (AFQMC), and discuss joint efforts using AFQMC and other state-of-the-art methods to determine the nature of magnetic and pairing orders in the two-dimensional Hubbard model.   

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Understanding the pairing mechanism responsible for high-temperature superconductivity is one of the primary goals of the condensed matter physics community. The bilayer Hubbard model provides a simple testbed, in which one can study unconventional pairing in a multi-band system. This model can be tuned through a Lifshitz transition, where one of the bands is pushed below the Fermi energy, similar to the situation in monolayer iron-selenide. Here we discuss dynamic cluster approximation (DCA) Quantum Monte Carlo (QMC) calculations of this model for different parameter regimes. In particular, we describe how its pairing correlations evolve as one of the bands becomes incipient, and how this behavior is linked to changes in the dynamical spin and charge fluctuations.   

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Quantum transport simulations are rapidly evolving, including the development of tensor network techniques for many-body calculations. One powerful approach combines matrix product states with extended reservoirs - an open system methodology where relaxation maintains a chemical potential or temperature drop. Due to the presence of external relaxation, this is a simulation analog of Kramers' turnover, with relaxation-controlled currents at both small and large relaxation strength. Only between these regimes does the natural (Landauer or Meir-Wingreen) conductance define the simulation. Here, we demonstrate how anomalous transport can appear when employing this methodology, even at moderate relaxation. While certain features are known to arise from band structure and gap states, we demonstrate that small steady-state currents unveil two anomalies, one due to virtual transitions and the other due to unphysical broadening of the occupied density of states. Moreover, we show how to approximate the optimal relaxation strength by exploiting control over the virtual anomalies, and demonstrate how reservoir discretizations impact simulation. Taken together, our results constrain the computational parameters that are needed to properly represent physical behavior in the continuum limit.   

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Single molecule transistors offer a fascinatingly diverse range of physics beyond the capabilities of Si transistors. Their ultrasmall size, chemical complexity, and electronic interactions constitute a unique playground for exploring the fundamental physics of correlations on the nanoscale, and their transport signatures. Understanding these systems is an essential prerequisite for possible advanced technological applications utilizing their quantum characteristics. In this talk I examine the interplay of symmetry and Kondo effect in a benzene single electron transistor using a combination of numerical renormalization group and generalised Schrieffer Wolff transformation [1]. Depending on the connectivity of the leads and gate voltages, we uncover spin-1/2 and spin-1 Kondo effects, a quantum phase transition to a state with robust molecular magnetism, and destructive quantum interference at an emergent SU(4) symmetry point. The interplay between emergent many-body effects and molecular symmetry is discussed in the context of quantum-boosted device functionality.

1. Kondo blockade due to quantum interference in single-molecule junctions. Nat Commun 8, 15210 (2017)