A cluster Monte Carlo algorithm for 2-dimensional spin glasses

A new Monte Carlo algorithm for 2-dimensional spin glasses is presented. The use of clusters makes possible global updates and leads to a gain in speed of several orders of magnitude. As an example, we study the 2-dimensional ±J Edwards-Anderson model. The new algorithm allows us to equilibrate systems of size 100² down to temperature T = 0.1. Our main result is that the correlation length diverges as an exponential ( ξ∼e ^2βJ) and not as a power law as T↦T _c = 0. Received 10 January 2001 and Received in final form 29 May 2001     

A new Monte Carlo power method for the eigenvalue problem of transfer matrices

We propose a new Monte Carlo method for calculating eigenvalues of transfer matrices leading to free energies and to correlation lengths of classical and quantum many-body systems. Generally, this method can be applied to the calculation of the maximum eigenvalue of a nonnegative matrix  such that all the matrix elements of Â^k are strictly positive for an integerk. This method is based on a new representation of the maximum eigenvalue of the matrix  as the thermal average of a certain observable of a many-body system. Therefore one can easily calculate the maximum eigenvalue of a transfer matrix leading to the free energy in the standard Monte Carlo simulations, such as the Metropolis algorithm. As test cases, we calculate the free energies of the square-lattice Ising model and of the spin-1/2XY Heisenberg chain. We also prove two useful theorems on the ergodicity in quantum Monte Carlo algorithms, or more generally, on the ergodicity of Monte Carlo algorithms using our new representation of the maximum eigenvalue of the matrixÂ.     

Chaos and Monte Carlo Approximations of the Flip-Annihilation process

The flip-annihilation process is a random particle process with one-dimensional local interaction in discrete time, initially presented by one of us, namely Toom in 2004. Its components are enumerated by integer numbers and every component has two states, “minus” and “plus”. At every time step two transformations occur. The first one, called “flip”, independently turns every minus into plus with probability β. The second one, called “annihilation”, acts thus: whenever a plus is a left neighbor of a minus, both disappear with probability α independently from other components. What is interesting about this process is that it is ergodic for β>α/2 and non-ergodic for β<α ²/250. It is natural to conjecture that there is some transition curve, which we call the true curve and denote by , which separates the areas of ergodicity and non-ergodicity of this process from each other. The estimates, mentioned above, albeit rigorous, leave a large gap between them and the present article’s purpose is to obtain some closer, albeit non-rigorous, approximations of the true curve. We do it in two ways, one of which is a chaos approximation and the other is a Monte Carlo simulation. Thus we obtain two curves, which are much closer to each other than the rigorous estimations. Also we fill in, albeit only numerically, another shortcoming of the rigorous estimation β<α ²/250, namely that it leaves us uncertain whether the true curve has a zero or positive slope at the point α=β=0. Both approximate curves have a positive slope at α=0, as we hoped.     

Entropic Repulsion of the Massless Field with a Class of Self-potentials

We consider the d+1-dimensional effective interface model of gradient type with a quadratic interaction potential and a self-potential. Without the self-potential, the model coincides with the d-dimensional massless Gaussian field. We show that for an arbitrary repulsive self-potential which can be thought as interaction of the interface with a “soft wall”, the field is pushed up at least to the same level when the original Gaussian field is conditioned to be positive everywhere, namely the “hard wall” condition is imposed.     

(3 1)-Dimensional Quantum Mechanics from Monte Carlo Hamiltonian: Harmonic Oscillator

In Lagrangian formulation, it is extremely difficult to compute the excited spectrum and wavefunctions ora quantum theory via Monte Carlo methods. Recently, we developed a Monte Carlo Hamiltonian method for investigating this hard problem and tested the algorithm in quantum-mechanical systems in 1 1 and 2t1 dimensions. In this paper we apply it to the study of thelow-energy quantum physics of the (3 1)-dimensional harmonic oscillator.``     

Quantum monte carlo studies of electron-electron interaction effects in conducting polymers

We discuss recent studies, using the quantum ensemble projector Monte Carlo (EPMC) method, of theoretical models of conducting polymers. Our focus is on the consequences of incorporating direct electron-electron interactions into the “standard” electron-phonon interaction models. Among the observables we examine one energetics of purely dimerized ground states, single solitons, soliton pairs, averaged spin and charge distributions, and local correlation functions.     

Risk analysis in investment appraisal based on the Monte Carlo simulation technique by A. Hacura, M. Jadamus-Hacura and A. Kocot

Comment on Eur. Phys. J. B 20, 551 (2001) Since Hertz major work on investment appraisal using the Monte Carlo Simulation technique, the so called “Risk Analysis” has become a standard tool for supporting investment decisions [1,2]. A main problem in investment appraisal is to consider and specify the risk of investment projects in an appropriate way, for enabling consistent project evaluation. In calculating a risky project's net present value (NPV) the major difficulty is to quantify the project's risk for quantifying an appropriate risk adjusted discount rate (RADR). Theoretically not founded risk adjusted discount rates face a lot of critique. Furthermore it is discussed that the incorporation of a constant risk factor into the discount rate makes a certain assumption about the resolution of uncertainty over time [3] and finally that a single net present value could not in general reflect risk properly. Especially in consequence of the last point the proponents of simulation argue that a whole distribution of net present values shows a project's risk better than a single number. In the special issue “Econophysics” of this journal Hacura et al. tried to describe the methodology and use of Monte Carlo Simulation in investment appraisal [4]. The purpose of this comment is to point out three fundamental flaws in that article. Received 29 April 2002 Published online 19 December 2002 RID="a" ID="a"e-mail: robert.obermaier@wiwi.uni-regensburg.de     

The deconfining phase transition in lattice quantum chromodynamics

We present a large-scale Monte Carlo calculation of the deconfining phase transition temperature in lattice quantum chromodynamics without fermions. Using the Wilson action, the transition temperature as a function of the lattice couplingg is consistent with scaling behavior dictated by the perturbativeα function for 6/g²>6.15. Speaker at the conference; on leave from CRIP, Budapest.     

Stochastic approach to nuclear shell model

We propose a Quantum Monte Carlo diagonalization method for solving the quantum many-body interacting systems. Not only the ground state but also low-lying excited states are obtained with their wave functions. Consequently the level structure of low-lying states can be studied with realistic interactions. After testing this method with⁴⁸Cr, we report that the doubly closed shell probability of⁵⁶Ni is shown to be only 53% in a full pf shell calculation, in contrast to the corresponding probability of⁴⁸Ca which reaches 86%. The most recent results on³²Mg are presented. Presented by T. Otsuka at the International Conference on “Atomic Nuclei and Metallic Clusters”, Prague, September 1–5, 1997. This work was supported in part by Grant-in-Aid for Scientific Research (B) (No. 08454058) from the Ministry of Education, Science and Culture.     

Monte Carlo calculations of angular momentum projected many-body matrix elements in the Pairing Plus Quadrupole Model

We extend the recently presented formalism for Monte Carlo calculations of the partition function, for both even and odd particle number systems (Phys. Rev. C 59, 2500 (1999)), to the calculation of many-body matrix elements of the type <ψ| e ^{- βℋ}|ψ> where |ψ> is a many-body state with a definite angular momentum, parity, neutron and proton numbers. For large β such matrix elements are dominated by the lowest eigenstate of the many-body Hamiltonian ℋ, corresponding with a given angular momentum parity and particle number. Emphasis is placed on odd-mass nuclei. Negligible sign fluctuations in the Monte Carlo calculation are found provided the neutron and proton chemical potentials are properly adjusted. The formalism is applied to the J ^π = 0⁺ state in ¹⁶⁶ Er and to the J ^π = 9/2^-, 13/2⁺, 5/2^- states in ¹⁶⁵ Er using the pairing-plus-quadrupole model. Received: 28 April 2000 / Accepted: 20 September 2000     

Diffusion of a Massive Quantum Particle Coupled to a Quasi-Free Thermal Medium

We consider a heavy quantum particle with an internal degree of freedom moving on the d-dimensional lattice _boxclose^d{{\mathbb Z}^d} (e.g., a heavy atom with finitely many internal states). The particle is coupled to a thermal medium (bath) consisting of free relativistic bosons (photons or Goldstone modes) through an interaction of strength λ linear in creation and annihilation operators. The mass of the quantum particle is assumed to be of order λ⁻², and we assume that the internal degree of freedom is coupled “effectively” to the thermal medium. We prove that the motion of the quantum particle is diffusive in d ≥ 4 and for λ small enough.     

First-order transition in a 2D classical XY-model using microcanonical Monte Carlo simulations

We have simulated two-dimensional classical XY-model in a microcanonical ensemble using the Monte Carlo technique. Simulations were carried out on a square lattice having 25, 100 or 900-spins with periodic boundary conditions. The nearest neighbour interaction potential was taken to beV(θ)=2J[1−cos¹⁰⁰(θ/2)]. The system energy, mean square magnetization and vortex-density were calculated as functions of temperature. A sudden change in the system energy, vortex density and mean square magnetization was observed at the first-order transition which is associated with this choice of the nearest neighbour interaction potential. The transition temperature increases with decrease in the system size. It is found that the creation of a vortex-antivortex pair costs more energy during the first-order transition than the energy associated with a tightly bound vortex-antivortex pair.     

Protecting the Conformal Symmetry via Bulk Renormalization on Anti deSitter Space

The problem of perturbative breakdown of conformal symmetry can be avoided, if a conformally covariant quantum field j{\varphi} on d-dimensional Minkowski spacetime is viewed as the boundary limit of a quantum field f{\phi} on d + 1-dimensional Anti-deSitter spacetime (AdS). We study the boundary limit in renormalized perturbation theory with polynomial interactions in AdS, and point out the differences as compared to renormalization directly on the boundary. In particular, provided the limit exists, there is no conformal anomaly. We compute explicitly the one-loop “fish diagram” on AdS₄ by differential renormalization, and calculate the anomalous dimension of the composite boundary field j²{\varphi^2} with bulk interaction kf⁴{\kappa \phi^4}.     

Tagged-weak <Emphasis Type="Italic">π</Emphasis> method

A new “tagged-weak π method” is proposed for determination of electromagnetic transition probabilities B(E2) and B(M1) of the hypernuclear states with lifetimes of ∼10⁻¹⁰ s. With this method, we are planning to measure B(E2) and B(M1) for light hypernuclei at JLab. The results of Monte Carlo simulations for the case of E2(5/2⁺, 3/2⁺ → 1/2⁺) transitions in _Λ⁷He hypernuclei are presented.     

Statistical and systematic errors in Monte Carlo sampling

We have studied the statistical and systematic errors which arise in Monte Carlo simulations and how the magnitude of these errors depends on the size of the system being examined when a fixed amount of computer time is used. We find that, depending on the degree of self-averaging exhibited by the quantities measured, the statistical errors can increase, decrease, or stay the same as the system size is increased. The systematic underestimation of response functions due to the finite number of measurements made is also studied. We develop a scaling formalism to describe the size dependence of these errors, as well as their dependence on the bin length (size of the statistical sample), both at and away from a phase transition. The formalism is tested using simulations of thed=3 Ising model at the infinite-lattice transition temperature. We show that for a 96×96×96 system noticeable systematic errors (systematic underestimation of response functions) are still present for total run lengths of 10⁶ Monte Carlo steps/site (MCS) with measurements taken at regular intervals of 10 MCS.This paper is dedicated to Jerry Percus on the occasion of his 65th birthday.     

Quantum orientational melting and the phase diagram of a mesoscopic system

The “phase diagram” of a two-dimensional mesoscopic system of bosons is investigated. An example of such a system is a system of indirect magnetoexcitons in semiconductor double quantum dots. Quantum Monte Carlo calculations show the existence of quantum orientational melting. At zero (quite low) temperature, as quantum fluctuations of the particles intensify, two quantum disordering phenomena occur with increasing de Boer parameter q. First, at q≈10⁻³ the system passes to a radially ordered but orientationally disordered state, where different shells of a cluster rotate relative to one another. Then at q≈0.16 a transition to a superfluid state occurs. Pis’ma Zh. éksp. Teor. Fiz. 68, No. 11, 817–822 (10 December 1998)     

Superconductivity as a Kosterlitz–Thouless transition in the two-dimensional Hubbard model

We investigate a superconducting Kosterlitz–Thouless transition in the two-dimensional (2D) Hubbard model using auxiliary quantum Monte Carlo method for the ground state. The pair susceptibility is computed for both the attractive and repulsive Hubbard model. The numerical results show that the s-wave pair susceptibility scales as χ  L² for the attractive case, in agreement with previous quantum Monte Carlo studies. The scaling χ  L² also holds for the d-wave pair susceptibility for the repulsive Hubbard model if we adjust the band parameter t′.     

Quantum melting of mesoscopic clusters

The phase diagram of a two-dimensional mesoscopic system of charges or dipoles, whose realizations could be electrons in a semiconductor quantum dot or indirect excitons in a system of two vertically coupled quantum dots, is investigated. Quantum calculations using ab initio Monte Carlo integration along trajectories determine the properties of such objects in the temperature-quantum de-Boer-parameter plane. At zero (sufficiently low) temperature, as the quantum fluctuations of the particles increase, two types of quantum disordering phenomena occur with increasing quantum de Boer parameter q: first, for q∼10⁻⁵ the systems transform into a radially ordered but orientationally disordered state wherein various shells of the “atom” rotate relative to one another. For much larger q∼0.1, a transition occurs to a disordered state (a superfluid in the case of a system of bosons). Fiz. Tverd. Tela (St. Petersburg) 41, 1856–1862 (October 1999)     

Monte Carlo simulations for two-dimensional quantum systems

The Trotter-Suzuki transformation has been used to obtain the classical representation ford-dimensional lattice systems with boson and fermion degrees of freedom. A Monte Carlo algorithm for the equivalent (d+1)-dimensional classical system is presented. Numerical results are shown for the Heisenberg-spin-glass, the XY model and the spinless fermion lattice gas in two dimensions.     

Cut-and-Permute Algorithm for Self-Avoiding Walks in the Presence of Surfaces

We present a dynamic nonlocal hybrid Monte Carlo algorithm consisting of pivot and cut-and-permute moves. The algorithm is suitable for the study of polymers in semiconfined geometries at the ordinary transition, where the pivot algorithm exhibits quasi-ergodic problems. The dynamic properties of the proposed algorithm are studied in d=3. The hybrid dynamics is ergodic and exhibits the same optimal critical behavior as the pivot algorithm in the bulk.