Simple Ways to improve Discrete Time Evolution

Suzuki-Trotter decompositions of exponential operators like $\exp(Ht)$ are required in almost every branch of numerical physics. Often the exponent under consideration has to be split into more than two operators, for instance as local gates on quantum computers. In this work, we demonstrate how highly optimised schemes originally derived for exactly two operators can be applied to such generic Suzuki-Trotter decompositions. After this first trick, we explain what makes an efficient decomposition and how to choose from the large variety available. Furthermore we demonstrate that many problems for which a Suzuki-Trotter decomposition might appear to be the canonical ansatz, are better approached with different methods like Taylor or Chebyshev expansions. In particular, we derive an efficient and numerically stable method to implement truncated polynomial expansions based on a linear factorisation using their complex zeros.Comment: 10 pages, 3 figures; LATTICE2023 proceedings. arXiv admin note: text overlap with arXiv:2211.0269

The Hubbard Model on the Honeycomb Lattice with Hybrid Monte Carlo

We take advantage of recent improvements in the grand canonical Hybrid Monte Carlo (HMC) algorithm, to perform a precision study of the single-particle gap in the hexagonal Hubbard model, with on-site electron-electron interactions. After carefully controlled analyses of the Trotter error, the thermodynamic limit, and finite-size scaling with inverse temperature, we find a critical coupling of $U_c/kappa=num{3.835(14)}$ and the critical exponent $nu=num{1.181(43)}$ for the semimetal-antiferromagnetic Mott insulator quantum phase transition in the hexagonal Hubbard Model. Based on these results, we provide a unified, comprehensive treatment of all operators that contribute to the anti-ferromagnetic, ferromagnetic, and charge-density-wave structure factors and order parameters of the hexagonal Hubbard Model. We expect our findings to improve the consistency of Monte Carlo determinations of critical exponents. We perform a data collapse analysis and determine the critical exponent $upbeta=num{0.898(37)}$. We consider our findings in view of the $SU(2)$ Gross-Neveu, or chiral Heisenberg, universality class. We also discuss the computational scaling of the HMC algorithm. Our methods are applicable to a wide range of lattice theories of strongly correlated electrons. The Ising model, a simple statistical model for ferromagnetism, is one such theory. There are analytic solutions for low dimensions and very efficient Monte Carlo methods, such as cluster algorithms, for simulating this model in special cases. However most approaches do not generalise to arbitrary lattices and couplings. We present a formalism that allows one to apply HMC simulations to the Ising model, demonstrating how a system with discrete degrees of freedom can be simulated with continuous variables. Because of the flexibility of HMC, our formalism is easily generalizable to arbitrary modifications of the model, creating a route to leverage advanced algorithms such as shift preconditioners and multi-level methods, developed in conjunction with HMC. We discuss the relation of a variety of different methods to determine energy levels in lattice field theory simulations: the generalised eigenvalue, the Prony, the generalised pencil of function and the Gardner methods. All three former methods can be understood as special cases of a generalised eigenvalue problem. We show analytically that the leading corrections to an energy $E_l$ in all three methods due to unresolved states decay asymptotically exponentially like $exp(-(E_{n}-E_l)t)$. Using synthetic data we show that these corrections behave as expected also in practice. We propose a novel combination of the generalised eigenvalue and the Prony method, denoted as GEVM/PGEVM, which helps to increase the energy gap $E_{n}-E_l$. We illustrate its usage and performance using lattice QCD examples. The Gardner method on the other hand is found less applicable to realistic noisy data

Real Time Simulations of Quantum Spin Chains: Density-of-States and Reweighting approaches

We put the Density-of-States (DoS) approach to Monte-Carlo (MC) simulations under a stress test by applying it to a physical problem with the worst possible sign problem: the real time evolution of a non-integrable quantum spin chain. Benchmarks against numerical exact diagonalisation and stochastic reweighting are presented. Both MC methods, the DoS approach and reweighting, allow for simulations of spin chains as long as $L=40$, far beyond exact diagonalisability, though only for short evolution times $t\lesssim 1$. We identify discontinuities of the density of states as one of the key problems in the MC simulations and propose to calculate some of the dominant contributions analytically, increasing the precision of our simulations by several orders of magnitude. Even after these improvements the density of states is found highly non-smooth and therefore the DoS approach cannot outperform reweighting. We prove this implication theoretically and provide numerical evidence, concluding that the DoS approach is not well suited for quantum real time simulations with discrete degrees of freedom.Comment: 16 + 4 pages, 7 figures; code and data available (DOI: 10.5281/zenodo.7164902

Beer Mats make bad Frisbees

In this article we show why flying and rotating beer mats, CDs, or other flat disks will eventually flip in the air and end up flying with backspin, thus, making them unusable as frisbees. The crucial effect responsible for the flipping is found to be the lift attacking not in the center of mass but slightly offset to the forward edge. This induces a torque leading to a precession towards backspin orientation. An effective theory is developed providing an approximate solution for the disk's trajectory with a minimal set of parameters. Our theoretical results are confronted with experimental results obtained using a beer mat shooting apparatus and a high speed camera. Very good agreement is found.Comment: 4 videos in ancillary file

Real-time simulations of quantum spin chains: Density of states and reweighting approaches

We put the density of states (DoS) approach to Monte Carlo (MC) simulations under a stress test by applying it to a physical problem with the worst possible sign problem: the real-time evolution of a nonintegrable quantum spin chain. Benchmarks against numerical exact diagonalization and stochastic reweighting are presented. Both MC methods, the DoS approach and reweighting, allow for simulations of spin chains as long as L=40, far beyond exact diagonalizability, though only for short evolution times t≲1. We identify discontinuities of the DoS as one of the key problems in the MC simulations and propose calculating some of the dominant contributions analytically, increasing the precision of our simulations by several orders of magnitude. Even after these improvements, the DoS is found highly nonsmooth, and therefore, the DoS approach cannot outperform reweighting. We prove this implication theoretically and provide numerical evidence, concluding that the DoS approach is not well suited for quantum real-time simulations with discrete degrees of freedom

Accelerating Hybrid Monte Carlo simulations of the Hubbard model on the hexagonal lattice

We present different methods to increase the performance of Hybrid Monte Carlo simulations of the Hubbard model in two-dimensions. Our simulations concentrate on a hexagonal lattice, though can be easily generalized to other lattices. It is found that best results can be achieved using a flexible GMRES solver for matrix inversions and the second order Omelyan integrator with Hasenbusch acceleration on different time scales for molecular dynamics. We demonstrate how an arbitrary number of Hasenbusch mass terms can be included into this geometry and find that the optimal speed depends weakly on the choice of the number of Hasenbusch masses and their values. As such, the tuning of these masses is amenable to automization and we present an algorithm for this tuning that is based on the knowledge of the dependence of solver time and forces on the Hasenbusch masses. We benchmark our algorithms to systems where direct numerical diagonalization is feasible and find excellent agreement. We also simulate systems with hexagonal lattice dimensions up to $102\times 102$ and $N_t=64$. We find that the Hasenbusch algorithm leads to a speed up of more than an order of magnitude.Comment: Corrected Proof in Press in Computer Physics Communication

Simulating both parity sectors of the Hubbard Model with Tensor Networks

Tensor networks are a powerful tool to simulate a variety of different physical models, including those that suffer from the sign problem in Monte Carlo simulations. The Hubbard model on the honeycomb lattice with non-zero chemical potential is one such problem. Our method is based on projected entangled pair states (PEPS) using imaginary time evolution. We demonstrate that it provides accurate estimators for the ground state of the model, including cases where Monte Carlo simulations fail miserably. In particular it shows near to optimal, that is linear, scaling in lattice size. We also present a novel approach to directly simulate the subspace with an odd number of fermions. It allows to independently determine the ground state in both sectors. Without a chemical potential this corresponds to half filling and the lowest energy state with one additional electron or hole. We identify several stability issues, such as degenerate ground states and large single particle gaps, and provide possible fixes.Comment: 20 pages, 20 figure

The Hubbard model with fermionic tensor networks

Many electromagnetic properties of graphene can be described by the Hubbard model on a honeycomb lattice. However, this system suffers strongly from the sign problem if a chemical potential is included. Tensor network methods are not affected by this problem. We use the imaginary time evolution of a fermionic projected entangled pair state, which allows to simulate both parity sectors independently. Incorporating the fermionic nature on the level of the tensor network allows to fix the particle number to be either even or odd. This way we can access the states at half filling and with one additional electron. We calculate the energy and other observables of both states, which was not possible before with Monte Carlo methods.Comment: Proceedings of the 38th International Symposium on Lattice Field Theory (Lattice 2021