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Siu A. Chin

Publications and source records attributed to Siu A. Chin.

At least 19 recordsLinked to original sources
arXiv2026

The Path Integral Monte Carlo Sign Problem Is Not Always NP-Hard: Harmonic Fermions Can Be Solved in Quadratic Time

In the Path Integral Monte Carlo (PIMC) simulation of fermions in a harmonic trap, with and without pairwise harmonic interactions, the partition functions for any discrete number of imaginary time slices (or beads) and for any choice of the short-time propagator can be analytically obtained from the contracted determinant form of the propagator. This work shows that the resulting recursion relation can be reformulated in the $λ$-ring language, yielding a closed-form finite-bead partition function in two dimensions in terms of permutation statistics. This closed-form partition function can be evaluated by a special algorithm in $O(n^2)$ time, providing an exact and numerically stable scheme for reproducing the energies of the original (undoable) fermion PIMC simulation for $n=10^4$ or more fermions. This result provides a concrete framework in which the numerical instability of the sign problem is completely bypassed, and serves as a counterexample to the prevailing view that all truly fermionic PIMC sign problems are NP-hard.

physics.comp-ph↗
arXiv2026

Parafermions in plain sight

We show that for any potential with a discrete energy spectrum, the well-known interpolation between ideal boson and fermion partition functions at discrete values of $ξ=-1/m$ yielded zero temperature ground state energies corresponding to $m$ fermions occupying a single quantum state. The grand canonical partition function in this case can be a result from genuine parastatistics.

cond-mat.stat-mech↗
arXiv2026

Understanding the sign problem from an exact Path Integral Monte Carlo model of interacting harmonic fermions

This work shows that the recently discovered operator contraction identity for solving the discreet Path Integral of the harmonic oscillator can be applied equally to fermions in any dimension. This then yields an exactly solvable model for studying the sign problem where the Path Integral Monte Carlo energy at any time step for any number of fermions is known analytically, or can be computed numerically. It is found that the sign problem is primarily a property of the free fermion propagator, but repulsive/attractive pairwise interaction can shift the sign problem to larger/smaller imaginary time but does not make it more severe than the non-interacting case. More surprisingly, one can prove analytically that the first closed-shell state in $D$ dimension, with $n=D+1$ fermion, has no sign problem at large imaginary time. Direct numerical simulations confirm that this is also true for higher closed-shell states in two and three-dimension. Fourth-order and newly found variable-bead algorithms are used to compute ground state energies of quantum dots with up to 110 electrons and compared to results obtained by modern neural networks.

cond-mat.str-el↗
arXiv2024

Analytical evaluations of the Path Integral Monte Carlo thermodynamic and Hamiltonian energies for the harmonic oscillator

By use of the recently derived $universal$ discrete imaginary-time propagator of the harmonic oscillator, both thermodynamic and Hamiltonian energies can be given analytically, and evaluated numerically at each imaginary time step, for $any$ short-time propagator. This work shows that, using only currently known short-time propagators, the Hamiltonian energy can be optimized to the twelfth order, converging to the ground state energy of the harmonic oscillator in as few as three beads. This study makes it absolutely clear that the widely used second-order primitive approximation propagator, when used in computing the thermodynamic energy, converges extremely slowly with increasing number of beads.

quant-ph↗
arXiv2024

Simple proof that there is no sign problem in Path Integral Monte Carlo simulations of fermions in one dimension

It is widely known that there is no sign problem in Path Integral Monte Carlo (PIMC) simulations of fermions in one dimension. Yet, as far as the author is aware, there is no direct proof of this in the literature. This work shows that the $sign$ of the $N$-fermion anti-symmetric free propagator is given by the product of all possible pairs of particle separations, or relative displacements. For a non-vanishing closed-loop product of such propagators, as required by PIMC, all relative displacements from adjacent propagators are paired into perfect squares, and therefore the loop product must be positive, but only in one dimension. By comparison, permutation sampling, which does not evaluate the determinant of the anti-symmetric propagator exactly, remains plagued by a low-level sign problem, even in one dimension.

physics.comp-ph↗
arXiv2023

Anatomy of Path Integral Monte Carlo: algebraic derivation of the harmonic oscillator's universal discrete imaginary-time propagator and its sequential optimization

The direct integration of the harmonic oscillator path integral obscures the fundamental structure of its discrete, imaginary time propagator (density matrix). This work, by first proving an operator identity for contracting two free propagators into one in the presence of interaction, derives the discrete propagator by simple algebra without doing any integration. This discrete propagator is $universal$, having the same two hyperbolic coefficient functions for all short-time propagators. Individual short-time propagator only modifies the coefficient function's argument, its $portal$ parameter, whose convergent order is the same as the thermodynamic energy. Moreover, the thermodynamic energy can be given in a closed form for any short-time propagator. Since the portal parameter can be systematically optimized by matching the expansion of the product of the two coefficients, any short-time propagator can be optimized $sequentially$, order by order, by matching the product coefficient's expansion alone, without computing the energy. Previous empirical findings on the convergence of fourth and sixth-order propagators can now be understood analytically. An eight-order convergent short-time propagator is also derived.

physics.comp-ph↗
arXiv2022

The anatomy of Boris type solvers and the Lie operator formalism for deriving large time-step magnetic field integrators

This work gives a Lie operator derivation of various Boris solvers via a detailed study of trajectory errors in a constant magnetic field. These errors in the gyrocenter location and the gyroradius are the foundational basis for why Boris solvers existed, independent of any finite-difference schemes. This work shows that there are two distinct ways of eliminating these errors so that the trajectory of a charged particle in a constant magnetic field is exactly on the cyclotron orbit. One way reproduces the known second-order symmetric Boris solver. The other yields a previously unknown, but also on-orbit solver, not derivable from finite-difference schemes. By revisiting some historical calculations, it is found that many publications do not distinguish the poorly behaved leap-frog Boris solver from the symmetric second-order Boris algorithm. This symmetric second-order Boris solver's trajectory is much more accurate and remains close to the exact orbit in a combined $nonuniform$ electric and magnetic field at time-steps greater than the cyclotron period. Finally, this operator formalism showed that Buneman's cycloid fitting scheme is mathematically identical to Boris' on-orbit solver and that Boris' E-B splitting is unnecessary.

physics.plasm-ph↗
arXiv2021

A fundamental derivation of two Boris solvers and the Ge-Marsden theorem

For a separable Hamiltonian, there are two fundamental, time-symmetric, second-order velocity-Verlet (VV) and position-Verlet (PV) symplectic integrators. Similarly, there are two VV and PV version of exact energy conserving algorithms for solving magnetic field trajectories. For a constant magnetic field, both algorithms can be further modified so that their trajectories are exactly on the gyro-circle. The magnetic PV integrator then becomes the well known Boris solver, while VV yields a second, previously unknown, Boris-type algorithm. Remarkably, the required on-orbit modification is a reparametrization of the time step, reminiscent of the Ge-Marsden theorem.

physics.plasm-ph↗
arXiv2021

Modern light on ancient feud: Robert Hooke and Newton's graphical method

The feud between Robert Hooke and Isaac Newton has remained ongoing even after 300 years, over whether Newton should have acknowledged Hooke's influence on his graphical method of constructing planet orbits, the celebrated Proposition 1, Theorem 1 of the $Principia$. The drama has escalated in recent decades, with a claim that Hooke may have used the same method and obtained an elliptical orbit for a linear force, a feat that some considered Newton never did for the inverse-square force. Modern understanding of Newton's graphical method as a symplectic integrator can now shed light on whether this claim is creditable. This work, based on knowing the Hamiltonian of the symplectic integrator, deduced the analytical orbit corresponding to Newton's graphical construction. A detailed comparison between this analytical orbit and Hooke's drawing shows that it is unlikely that Hooke had used Newton's graphical method and obtained the correct orbit.

physics.hist-ph↗
arXiv2021

Higher-order Breathers as Quasi-rogue Waves on a Periodic Background

We investigate higher-order breathers of the cubic nonlinear Schrödinger equation on an elliptic background. We find that, beyond first-order, any arbitrarily constructed breather is a single-peaked solitary wave on a disordered background. These "quasi-rogue waves" are also common on periodic backgrounds. We assume the higher-order breather is constructed out of constituent first-order breathers with commensurate periods (i.e., higher-order harmonic waves). In that case, one obtains "quasi-periodic" breathers with distorted side-peaks. Fully periodic breathers are obtained when their wavenumbers are harmonic multiples of the background and each other. They are truly rare, requiring finely-tuned parameters. Thus, on a periodic background, we arrive at the paradoxical conclusion that the apparent higher-order rogue waves are rather common, while the truly periodic breathers are exceedingly rare.

nlin.PS↗
arXiv2019

Riemannian Manifold Hamiltonian Monte Carlo

The paper proposes a Riemannian Manifold Hamiltonian Monte Carlo sampler to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. The method provides a fully automated adaptation mechanism that circumvents the costly pilot runs required to tune proposal densities for Metropolis-Hastings or indeed Hybrid Monte Carlo and Metropolis Adjusted Langevin Algorithms. This allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the Markov chain. The proposed method exploits the Riemannian structure of the parameter space of statistical models and thus automatically adapts to the local manifold structure at each step based on the metric tensor. A semi-explicit second order symplectic integrator for non-separable Hamiltonians is derived for simulating paths across this manifold which provides highly efficient convergence and exploration of the target density. The performance of the Riemannian Manifold Hamiltonian Monte Carlo method is assessed by performing posterior inference on logistic regression models, log-Gaussian Cox point processes, stochastic volatility models, and Bayesian estimation of parameter posteriors of dynamical systems described by nonlinear differential equations. Substantial improvements in the time normalised Effective Sample Size are reported when compared to alternative sampling approaches. Matlab code at \url{http://www.dcs.gla.ac.uk/inference/rmhmc} allows replication of all results.

stat.CO↗
arXiv2019

Structure of numerical algorithms and advanced mechanics

Most elementary numerical schemes found useful for solving classical trajectory problems are {\it canonical transformations}. This fact should be make more widely known among teachers of computational physics and Hamiltonian mechanics. From the perspective of advanced mechanics, there are no bewildering number of seemingly arbitrary elementary schemes based on Taylor's expansion. There are only {\it two} canonical first and second order algorithms, on the basis of which one can comprehend the structures of higher order symplectic and non-symplectic schemes. This work shows that, from the most elementary first-order methods to the most advanced fourth-order algorithms, all can be derived from canonical transformations and Poisson brackets of advanced mechanics.

physics.ed-ph↗
arXiv2019

Solving fermion problems without solving the sign problem: symmetry-breaking wave functions from similarity-transformed propagators for solving 2D quantum dots

It is well known that the use of the primitive second-order propagator in Path Integral Monte Carlo calculations of many-fermion systems leads to the sign problem. In this work, we show that by using the similarity-transformed Fokker-Planck propagator, it is possible to solve for the ground state of a large quantum dot, with up to 100 polarized electrons, without solving the sign problem. These similarity-transformed propagators naturally produce rotational symmetry-breaking ground state wave functions previously used in the study of quantum dots and quantum Hall effects. However, instead of localizing the electrons at positions which {\it minimize} the potential energy, this derivation shows that they should be located at positions which {\it maximize} the bosonic ground state wave function. Further improvements in the energy can be obtained by using these as initial wave functions in a Ground State Path-Integral Monte Carlo calculation with second and fourth-order propagators.

physics.comp-ph↗
arXiv2019

The hardwall method of solving the radial Schrödinger equation and unmasking hidden symmetries

Solving for the bound state eigenvalues of the Schrödinger equation is a tedious iterative process when the conventional shooting or matching method is used. In this work, we bypass the eigenvalue's dependence on the eigenfunction by simply trying out all eigenvalues to a desired accuracy. When the eigenvalue is known, the integration for the eigenfunction is then trivial. At a given energy, by outputting the radial distance at which the wave function crosses zero (the hardwall radius), this method automatically determines the entire spectrum of eigenvalues of the radial Schrödinger equation without iterative adjustments. Moreover, such a spherically symmetric "hardwall" can unmask "accidental degeneracy" of eigenvalues due to hidden symmetries. We illustrate the method on the Coulomb, harmonic, Coulomb+harmonic, and the Woods-Saxon potentials.

physics.comp-ph↗
arXiv2019

Newton's graphical method as a canonical transformation

This work shows that, Newton's Proposition 1 in the {\it Principia}, is an {\it exact} graphical representation of a canonical transformation, a first-order symplectic integrator generated at a finite time-step by the Hamiltonian. A fundamental characteristic of this canonical transformation is to update the position and velocity vectors {\it sequentially}, thereby automatically conserving the phase-volume and the areal velocity due to a central force. As a consequence, the continuous force is naturally replaced by a series of impulses. The convergence of Newton's Proposition 1 in the limit of $Δt\rightarrow 0$ can be proved easily and the resulting error term for the linear and the inverse square force can explain why Hooke was able to the obtain an elliptical orbit for the former but not the latter.

physics.hist-ph↗