Loop-cluster Algorithm Research Articles

Finite-size scaling of the high-dimensional Ising model in the loop representation.

Besides its original spin representation, the Ising model is known to have the Fortuin-Kasteleyn (FK) bond and loop representations, of which the former was recently shown to exhibit two upper critical dimensions (d_{c}=4,d_{p}=6). Using a lifted worm algorithm, we determine the critical coupling as K_{c}=0.07770891(4) for d=7, which significantly improves over the previous results, and then study critical geometric properties of the loop Ising clusters on tori for spatial dimensions d=5 to 7. We show that as the spin representation, the loop Ising model has only one upper critical dimension at d_{c}=4. However, sophisticated finite-size scaling (FSS) behaviors, such as two length scales, two configuration sectors, and two scaling windows, still exist as the interplay effect of the Gaussian fixed point and complete-graph asymptotics. Moreover, using the loop-cluster algorithm, we provide an intuitive understanding of the emergence of the percolation-like upper critical dimension d_{p}=6 in the FK-Ising model. As a consequence, a unified physical picture is established for the FSS behaviors in all three representations of the Ising model above d_{c}=4.

Geometric properties of the complete-graph Ising model in the loop representation.

The exact solution of the Ising model on the complete graph (CG) provides an important, though mean-field, insight for the theory of continuous phase transitions. Besides the original spin, the Ising model can be formulated in the Fortuin-Kasteleyn random cluster and the loop representation, in which many geometric quantities have no correspondence in the spin representations. Using a lifted-worm irreversible algorithm, we study the CG-Ising model in the loop representation and, based on theoretical and numerical analyses, obtain a number of exact results including volume fractal dimensions and scaling forms. Moreover, by combining with the loop-cluster algorithm, we demonstrate how the loop representation can provide an intuitive understanding to the recently observed rich geometric phenomena in the random-cluster representation, including the emergence of two configuration sectors, two length scales, and two scaling windows.

Path-integral Monte Carlo method for the localZ2Berry phase

We present a loop cluster algorithm Monte Carlo method for calculating the local Z(2) Berry phase of the quantum spin models. The Berry connection, which is given as the inner product of two ground states with different local twist angles, is expressed as a Monte Carlo average on the worldlines with fixed spin configurations at the imaginary-time boundaries. The "complex weight problem" caused by the local twist is solved by adopting the meron cluster algorithm. We present the results of simulation on the antiferromagnetic Heisenberg model on an out-of-phase bond-alternating ladder to demonstrate that our method successfully detects the change in the valence bond pattern at the quantum phase transition point. We also propose that the gauge-fixed local Berry connection can be an effective tool to estimate precisely the quantum critical point.

Constraint effective potential of the magnetization in the quantum XY model

Using an improved estimator in the loop-cluster algorithm, we investigate the constrainteffective potential of the magnetization in the spin quantum XY model. The numerical results are in excellent agreement withthe predictions of the corresponding low-energy effective field theory. After itslow-energy parameters have been determined with better than per mille precision, theeffective theory makes accurate predictions for the constraint effective potentialwhich are in excellent agreement with the Monte Carlo data. This shows that theeffective theory indeed describes the physics in the low-energy regime quantitativelycorrectly.

High-precision determination of low-energy effective parameters for a two-dimensional Heisenberg quantum antiferromagnet

The 2-d spin 1/2 Heisenberg antiferromagnet with exchange coupling $J$ is investigated on a periodic square lattice of spacing $a$ at very small temperatures using the loop-cluster algorithm. Monte Carlo data for the staggered and uniform susceptibilities are compared with analytic results obtained in the systematic low-energy effective field theory for the staggered magnetization order parameter. The low-energy parameters of the effective theory, i.e.\ the staggered magnetization density ${\cal M}_s = 0.30743(1)/a^2$, the spin stiffness $\rho_s = 0.18081(11) J$, and the spin wave velocity $c = 1.6586(3) J a$ are determined with very high precision. Our study may serve as a test case for the comparison of lattice QCD Monte Carlo data with analytic predictions of the chiral effective theory for pions and nucleons, which is vital for the quantitative understanding of the strong interaction at low energies.

The constraint effective potential of the staggered magnetization in an antiferromagnet

We employ an improved estimator to calculate the constraint effective potential of thestaggered magnetization in the spin quantum Heisenberg model using a loop-cluster algorithm. The first and second momentof the probability distribution of the staggered magnetization are in excellent agreementwith the predictions from the systematic low-energy magnon effective field theory. We alsocompare the Monte Carlo data with the universal shape of the constraint effective potentialof the staggered magnetization and study its approach to the convex effective potentialin the infinite volume limit. In this way the higher-order low-energy parameterk0 is determined from a fit to the numerical data.

Loop-cluster simulation of the zero- and one-hole sectors of thet−Jmodel on the honeycomb lattice

Inspired by the lattice structure of the unhydrated variant of the superconducting material Na$_x$CoO$_2 \cdot$yH$_2$O at $ x = {1/3}$, we study the $t$-$J$ model on a honeycomb lattice by using an efficient loop-cluster algorithm. The low-energy physics of the undoped system and of the single hole sector is described by a systematic low-energy effective field theory. The staggered magnetization per spin $\widetilde{{\cal M}}_s = 0.2688(3)$, the spin stiffness $\rho_s = 0.102(2) J$, the spin wave velocity $c= 1.297(16) J a$, and the kinetic mass $M'$ of a hole are obtained by fitting the numerical Monte Carlo data to the effective theory predictions.

From an antiferromagnet to a valence bond solid: evidence for a first-order phasetransition

Using a loop–cluster algorithm we investigate the spin- Heisenberg antiferromagnet on a square lattice with exchange couplingJ and an additional four-spin interaction of strengthQ. We confirm the existence of a phase transition separating antiferromagnetism atJ/Q>Jc/Q from a valence bondsolid (VBS) state at J/Q<Jc/Q. Although our Monte Carlo data are consistent with those of previousstudies, we do not confirm the existence of a deconfined quantumcritical point. Instead, using a flowgram method on lattices as large as802, we find evidence for a weak first-order phase transition. Wealso present a detailed study of the antiferromagnetic phase. ForJ/Q>Jc/Q the staggered magnetization, the spin stiffness and the spinwave velocity of the antiferromagnetare determined by fitting Monte Carlo data to analytic results from the systematic low-energyeffective field theory for magnons. Finally, we also investigate the physics of the VBS state atJ/Q<Jc/Q and we show that long but finite antiferromagnetic correlations are still present.

Simulations of [formula omitted] Models for Large N

We discuss simulations of CP ( N − 1 ) models via dimensional reduction of discrete SU(N) quantum spin models. These models can be simulated with an efficient loop cluster algorithm in continuous euclidean time, effective even for very large N. In particular we discuss efforts to compute the mass gap of the CP ( N − 1 ) model.

STUDY ON REGULAR DILUTIONS OF $S=\frac{1}{2}$ IMPURITY IN S=1 CHAINS BY QUANTUM MONTE CARLO SIMULATIONS

We investigate the variant regularly diluted situations of [Formula: see text] in S=1 isotropic antiferromagnetic chains by the quantum Monte Carlo loop cluster algorithm. Our results manifest significant different magnetic properties in the ground state with respect to the odd (even) host S=1 spins in one unit cell with an impurity S=1/2, and the doped system gradually transits to the pure Haldane chain in two different tendencies with the decreasing of the impurity concentrations.

A Quantum Monte Carlo Study on Mixed-Spin Chains of 1/2-1/2-1-1 and 3/2-3/2-1-1**The project supported in part by National Natural Science Foundation of China and the Natural Science Foundation of Zhejiang Province of China

The ground-state and thermodynamic properties of quantum mixed-spin chains of 1/2-1/2-1-1 and 3/2-3/2-1-1 are investigated by a quantum Monte Carlo simulation with the loop-cluster algorithm. For 1/2-1/2-1-1 chain, we find it has two phases separated by an energy-gap vanishing point in the ground-state. For 3/2-3/2-1-1 chain, the numerical results show two energy-gap vanishing points isolated by different phases in its ground-state. Our calculations indicate that all these ground state phases can be understood by means of valence-bond-solid picture, and the thermodynamic behavior at finite temperatures is continuous as a function of parameter .

Thermodynamic and diamagnetic properties of weakly doped antiferromagnets

Finite-temperature properties of weakly doped antiferromagnets as modeled by the two-dimensional t-J model and relevant to underdoped cuprates are investigated by numerical studies of small model systems at low doping. Two numerical methods are used: the worldline quantum Monte Carlo method with a loop cluster algorithm and the finite-temperature Lanczos method, yielding consistent results. Thermodynamic quantities: specific heat, entropy and spin susceptibility reveal a sizeable perturbation induced by holes introduced into a magnetic insulator, as well as a pronounced temperature dependence. The diamagnetic susceptibility introduced by coupling of the magnetic field to the orbital current reveals an anomalous temperature dependence, changing character from diamagnetic to paramagnetic at intermediate temperatures.

Correlation functions from quantum cluster algorithms: application to U(1) quantum spin and quantum link models

We demonstrate how correlation functions for non-diagonal operators can be measured with the loop-cluster algorithm for quantum spin models. We introduce the U(1) quantum link model and present the construction of a cluster algorithm for the model. We further show how an improved estimator for Wilson loops can be realised with the flux-cluster algorithm.

D-theory: field theory via dimensional reduction of discrete variables

A new non-perturbative approach to quantum field theory — D-theory — is proposed, in which continuous classical fields are replaced by discrete quantized variables which undergo dimensional reduction. The 2-d classical O((3) model emerges from the (2 + 1)-d quantum Heisenberg model formulated in terms of quantum spins. Dimensional reduction is demonstrated explicity by simulating correlation lengths up to 350,000 lattice spacings using a loop cluster algorithm. In the framework of D-theory, gauge theories are formulated in terms of quantum links — the gauge analogs of quantum spins. Quantum links are parallel transporter matrices whose elements are non-commuting operators. They can be expressed as bilinears of anticommuting fermion constituents. In quantum link models dimensional reduction to four dimensions occurs, due to the presence of a 5-d Coulomb phase, whose existence is confirmed by detailed simulations using standard lattice gauge theory. Using Shamir's variant of Kaplan's fermion proposal, in quantum link QCD quarks appear as edge states of a 5-d slab. This naturally protects their chiral symmetries without fine-tuning. The first efficient cluster algorithm for a gauge theory with a continuous gauge group is formulated for the U(1) quantum link model. Improved estimators for Wilson loops are constructed, and dimensional reduction to ordinary lattice QED is verified numerically.

Square-Lattice Heisenberg Antiferromagnet at Very Large Correlation Lengths

The correlation length of the square-lattice spin-1/2 Heisenberg antiferromagnet is studied in the low-temperature (asymptotic-scaling) regime. Our novel approach combines a very efficient loop cluster algorithm -- operating directly in the Euclidean time continuum -- with finite-size scaling. This enables us to probe correlation lengths up to $\xi \approx 350,000$ lattice spacings -- more than three orders of magnitude larger than any previous study. We resolve a conundrum concerning the applicability of asymptotic-scaling formulae to experimentally- and numerically-determined correlation lengths, and arrive at a very precise determination of the low-energy observables. Our results have direct implications for the zero-temperature behavior of spin-1/2 ladders.

Simulations of discrete quantum systems in continuous euclidean time

We propose a new method to study path integrals for discrete quantum systems, in which we work directly in the Euclidean time continuum. The method is applied here to the Heisenberg quantum antiferromagnet using a continuous-time version of a loop cluster algorithm. This algorithm is exploited to confirm the predictions of chiral perturbation theory in the extreme low temperature regime, down to T = 0.01 J. A fit of the low-energy parameters of chiral perturbation theory gives excellent agreement with previous results and with experiments.

Simulations of Discrete Quantum Systems in Continuous Euclidean Time.

We propose a new method to study path integrals for discrete quantum systems in which we work directly in the Euclidean time continuum. The method is of general interest. Here it is applied to the Heisenberg quantum antiferromagnet using a continuous-time version of a loop cluster algorithm. This algorithm is exploited to confirm the predictions of chiral perturbation theory in the extreme low temperature regime, down to $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.01J$. A fit of the low-energy parameters of chiral perturbation theory gives excellent agreement with previous results and with experiments.

A Coherent Updating Algorithm for the XYZ Model

A scheme of loop-cluster algorithm is developed for simulating an anisotropic Heisenberg antiferromagnet–the XYZ model. Results of numerical simulations are presented, suggesting that this quantum system may also be studied by using the cluster updating techniques.

An updating scheme for the loop-cluster algorithm for the anisotropic Heisenberg antiferromagnet

A loop-cluster algorithm is proposed to overcome the critical slowing down in simulations of the anisotropic Heisenberg antiferromagnet, the XYZ model. The primary features of the algorithm are presented, which suggest that this quantum system may also be studied as a classical spin system using cluster updating techniques.

Monte Carlo Simulations of the Quantum X-Y Model by a Loop-Cluster Algorithm

The quantum X-Y model of interacting spins on square lattices is simulated by a loop-cluster algorithm. It is shown that the method can be used to simulate the systems efficiently at low temperatures for lattices as large as 32×32×200.