Lefschetz thimble structure in one-dimensional lattice Thirring model at finite density

Abstract

We investigate Lefschetz thimble structure of the complexified path-integration in the one-dimensional lattice massive Thirring model with finite chemical potential. The lattice model is formulated with staggered fermions and a compact auxiliary vector boson (a link field), and the whole set of the critical points (the complex saddle points) are sorted out, where each critical point turns out to be in a one-to-one correspondence with a singular point of the effective action (or a zero point of the fermion determinant). For a subset of critical point solutions in the uniform-field subspace, we examine the upward and downward cycles and the Stokes phenomenon with varying the chemical potential, and we identify the intersection numbers to determine the thimbles contributing to the path-integration of the partition function. We show that the original integration path becomes equivalent to a single Lefschetz thimble at small and large chemical potentials, while in the crossover region multi thimbles must contribute to the path integration. Finally, reducing the model to a uniform field space, we study the relative importance of multiple thimble contributions and their behavior toward continuum and low-temperature limits quantitatively, and see how the rapid crossover behavior is recovered by adding the multi thimble contributions at low temperatures. Those findings will be useful for performing Monte-Carlo simulations on the Lefschetz thimbles.

32 pages, 14 figures (typo etc. corrected)

References in corpus (54)

• High density QCD on a Lefschetz thimble?
• Analytic Continuation Of Chern-Simons Theory
• The Complex Langevin method: When can it be trusted?
• Functional integrals for QCD at nonzero chemical potential and zero density
• Simulating full QCD at nonzero density using the complex Langevin equation
• Complex Langevin: Etiology and Diagnostics of its Main Problem
• Can stochastic quantization evade the sign problem? -- the relativistic Bose gas at finite chemical potential
• Monte Carlo simulations on the Lefschetz thimble: taming the sign problem
• Gauge cooling in complex Langevin for QCD with heavy quarks
• Hybrid Monte Carlo on Lefschetz Thimbles -- A study of the residual sign problem
• Controlling complex Langevin dynamics at finite density
• Adaptive stepsize and instabilities in complex Langevin dynamics
• Stochastic quantization at finite chemical potential
• New insights into the problem with a singular drift term in the complex Langevin method
• Complex Langevin Dynamics for chiral Random Matrix Theory
• Determinants of Block Tridiagonal Matrices
• Complex Langevin dynamics in the SU(3) spin model at nonzero chemical potential revisited
• Structure of Lefschetz thimbles in simple fermionic systems
• Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling
• Some remarks on Lefschetz thimbles and complex Langevin dynamics
• Hidden topological angles and Lefschetz thimbles
• An efficient method to compute the residual phase on a Lefschetz thimble
• Stability of complex Langevin dynamics in effective models
• Simulating QCD at nonzero baryon density to all orders in the hopping parameter expansion
• Complex Langevin dynamics for dynamical QCD at nonzero chemical potential: a comparison with multi-parameter reweighting
• Lefschetz thimbles and stochastic quantisation: Complex actions in the complex plane
• Complex Langevin dynamics and other approaches at finite chemical potential
• Complex Langevin dynamics at finite chemical potential: mean field analysis in the relativistic Bose gas
• On the convergence of complex Langevin dynamics: the three-dimensional XY model at finite chemical potential
• Localised distributions and criteria for correctness in complex Langevin dynamics
• Evading the sign problem in the mean-field approximation through Lefschetz-thimble path integral
• Full simulation of chiral Random Matrix Theory at non-zero chemical potential by Complex Langevin
• Thimble regularization at work: from toy models to chiral random matrix theories
• Degenerate distributions in complex Langevin dynamics: one-dimensional QCD at finite chemical potential
• Lefschetz-thimble techniques for path integral of zero-dimensional $O(n)$ sigma models
• A New Look At The Path Integral Of Quantum Mechanics
• Hamilton dynamics for the Lefschetz thimble integration akin to the complex Langevin method
• Thirring model at finite density in 0+1 dimensions with stochastic quantization: Crosscheck with an exact solution
• An improvement in complex Langevin dynamics from a view point of Lefschetz thimbles
• Progress in complex Langevin simulations of full QCD at nonzero density
• Thirring model at finite density in 2+1 dimensions with stochastic quantization
• Complex Langevin method applied to the 2D $SU(2)$ Yang-Mills theory
• Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles
• Lefschetz thimble Monte Carlo for many body theories: application to the repulsive Hubbard model away from half filling
• Simple QED- and QCD-like Models at Finite Density
• Exploring the phase diagram of QCD with complex Langevin simulations
• Adaptive gauge cooling for complex Langevin dynamics
• QCD at nonzero chemical potential: recent progress on the lattice
• Complex Langevin dynamics for SU(3) gauge theory in the presence of a theta term
• Complex Langevin simulation for QCD-like models
• Localised distributions in complex Langevin dynamics
• Extending complex Langevin simulations to full QCD at nonzero density
• Complex Langevin dynamics: criteria for correctness
• Systematic approximation for QCD at non-zero density

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• Cheshire Cat resurgence, Self-resurgence and Quasi-Exact Solvable Systems
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• Bion non-perturbative contributions versus infrared renormalons in two-dimensional $\mathbb{C}{P}^{N-1}$ models
• Fermions at Finite Density in (2+1)d with Sign-Optimized Manifolds
• Lefschetz thimbles decomposition for the Hubbard model on the hexagonal lattice
• Complex saddle points and the sign problem in complex Langevin simulation
• Fast Estimator of jacobians in Monte Carlo Integration on Lefschetz Thimbles
• Finite Density $QE{D}_{1+1}$ Near Lefschetz Thimbles
• Evolution to the Quark-Gluon Plasma
• Multi-flavor massless QED${}_2$ at finite densities via Lefschetz thimbles
• An improvement in complex Langevin dynamics from a view point of Lefschetz thimbles
• Leveraging Machine Learning to Alleviate Hubbard Model Sign Problems
• A complex path around the sign problem
• Phase structure of the twisted $SU(3)/U(1{)}^2$ flag sigma model on $\mathbb{R}\times S^1$
• Gradient flows without blow-up for Lefschetz thimbles
• What is QFT? Resurgent trans-series, Lefschetz thimbles, and new exact saddles
• Phase Unwrapping and One-Dimensional Sign Problems
• Real-time lattice gauge theory actions: unitarity, convergence, and path integral contour deformations
• Taylor expansions on Lefschetz thimbles (and not only that)
• A study of symmetry breaking in a relativistic Bose gas using the contraction algorithm
• On the gauge invariant path-integral measure for the overlap Weyl fermions in $\underline{16}$ of SO(10)
• Lefschetz thimbles in fermionic effective models with repulsive vector-field
• Why is the mission impossible? -- Decoupling the mirror Ginsparg-Wilson fermions in the lattice models for two-dimensional abelian chiral gauge theories
• Complex spectrum of finite-density lattice QCD with static quarks at strong coupling
• Simulating Yang-Mills theories with a complex coupling
• Resurgence and Lefschetz thimble in 3d N=2 supersymmetric Chern-Simons matter theories
• Analytical studies of the complex Langevin equation with a Gaussian Ansatz and multiple solutions in the unstable region
• Settling an old story: solution of the Thirring model in thimble regularization
• Lefschetz Thimble Quantum Monte Carlo for Spin Systems
• Statistical analysis method for the worldvolume hybrid Monte Carlo algorithm
• Instanton gas approach to the Hubbard model
• Lefschetz Thimbles and Quantum Phases in Zero-Dimensional Bosonic Models
• The Lefschetz thimble and the sign problem
• Application of the Lefschetz thimble formulation to the (0+1) dim. Thirring model at finite density
• Performance of Complex Langevin Simulation in 0+1 dimensional massive Thirring model at finite density
• A quantum Monte Carlo method on asymptotic Lefschetz thimbles for quantum spin systems: An application to the Kitaev model in a magnetic field
• Holographic de Sitter Spacetime and Quantum Corrections to The Cosmological Constant
• On a modification method of Lefschetz thimbles
• A saddle-point finder and its application to the spin foam model
• Stable Evaluation of Lefschetz Thimble Intersection Numbers: Towards Real-Time Path Integrals
• Accurate simulation of the finite density lattice Thirring model
• Is it worth the effort to find Lefschetz thimbles? Integration contours with numerically optimal signal-to-noise ratios in simple fermionic toy models
• Path optimization method for the sign problem caused by fermion determinant