Monte Carlo Renormalization Group Method Research Articles

Corrections to scaling in the 3D Ising model: A comparison between MC and MCRG results

Corrections to scaling in the 3D Ising model are studied based on Monte Carlo (MC) simulation results for very large lattices with linear lattice sizes up to [Formula: see text]. Our estimated values of the correction-to-scaling exponent [Formula: see text] tend to decrease below the usually accepted value about 0.83 when the smallest lattice sizes, i.e. [Formula: see text] with [Formula: see text], are discarded from the fits. This behavior apparently confirms some of the known estimates of the Monte Carlo renormalization group (MCRG) method, i.e. [Formula: see text] and [Formula: see text]. We discuss the possibilities that [Formula: see text] is either really smaller than usually expected or these values of [Formula: see text] describe some transient behavior which, eventually, turns into the correct asymptotic behavior at [Formula: see text]. We propose refining MCRG simulations and analysis to resolve this issue. Our actual MC estimations of the critical exponents [Formula: see text] and [Formula: see text] provide stable values [Formula: see text] and [Formula: see text], which well agree with those of the conformal bootstrap method, i.e. [Formula: see text] and [Formula: see text].

Clock model interpolation and symmetry breaking in O(2) models

Motivated by recent attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we define an extended-O(2) model by adding a $\gamma \cos(q\varphi)$ term to the ordinary O(2) model with angular values restricted to a $2\pi$ interval. In the $\gamma \rightarrow \infty$ limit, the model becomes an extended $q$-state clock model that reduces to the ordinary $q$-state clock model when $q$ is an integer and otherwise is a continuation of the clock model for noninteger $q$. By shifting the $2\pi$ integration interval, the number of angles selected can change discontinuously and two cases need to be considered. What we call case $1$ has one more angle than what we call case $2$. We investigate this class of clock models in two space-time dimensions using Monte Carlo and tensor renormalization group methods. Both the specific heat and the magnetic susceptibility show a double-peak structure for fractional $q$. In case $1$, the small-$\beta$ peak is associated with a crossover, and the large-$\beta$ peak is associated with an Ising critical point, while both peaks are crossovers in case $2$. When $q$ is close to an integer by an amount $\Delta q$ and the system is close to the small-$\beta$ Berezinskii-Kosterlitz-Thouless transition, the system has a magnetic susceptibility that scales as $\sim 1 / (\Delta q)^{1 - 1/\delta'}$ with $\delta'$ estimates consistent with the magnetic critical exponent $\delta = 15$. The crossover peak and the Ising critical point move to Berezinskii-Kosterlitz-Thouless transition points with the same power-law scaling. A phase diagram for this model in the $(\beta, q)$ plane is sketched. These results are possibly relevant for configurable Rydberg-atom arrays where the interpolations among phases with discrete symmetries can be achieved by varying continuously the distances among atoms and the detuning frequency.

Accurate simulation for finite projected entangled pair states in two dimensions

Based on the scheme of variational Monte Carlo sampling, we develop an accurate and efficient two-dimensional tensor-network algorithm to simulate quantum lattice models. We find that Monte Carlo sampling shows huge advantages in dealing with finite projected entangled pair states, which allows significantly enlarged system size and improves the accuracy of tensor network simulation. We demonstrate our method on the square-lattice antiferromagnetic Heisenberg model up to $32 \times 32$ sites, as well as a highly frustrated $J_1-J_2$ model up to $24\times 24$ sites. The results, including ground state energy and spin correlations, are in excellent agreement with those of the available quantum Monte Carlo or density matrix renormalization group methods. Therefore, our method substantially advances the calculation of 2D tensor networks for finite systems, and potentially opens a new door towards resolving many challenging strongly correlated quantum many-body problems.

Absence of Superconductivity in the Pure Two-Dimensional Hubbard Model

We study the superconducting pairing correlations in the ground state of the doped Hubbard model -- in its original form without hopping beyond nearest neighbor or other perturbing parameters -- in two dimensions at intermediate to strong coupling and near optimal doping. The nature of such correlations has been a central question ever since the discovery of cuprate high-temperature superconductors. Despite unprecedented effort and tremendous progress in understanding the properties of this fundamental model, a definitive answer to whether the ground state is superconducting in the parameter regime most relevant to cuprates has proved exceedingly difficult to establish. In this work, we employ two complementary, state-of-the-art many-body computational methods, constrained path (CP) auxiliary-field quantum Monte Carlo (AFQMC) and density matrix renormalization group (DMRG) methods, deploying the most recent algorithmic advances in each. Systematic and detailed comparisons between the two methods are performed. The DMRG is extremely reliable on small width cylinders, where we use it to validate the AFQMC. The AFQMC is then used to study wide systems as well as fully periodic systems, to establish that we have reached the thermodynamic limit. The ground state is found to be non-superconducting in the moderate to strong coupling regime in the vicinity of optimal hole doping.

Continuum percolation of congruent overlapping polyhedral particles: Finite-size-scaling analysis and renormalization-group method.

The continuum percolation of randomly orientated overlapping polyhedral particles, including tetrahedron, cube, octahedron, dodecahedron, and icosahedron, was analyzed by Monte Carlo simulations. Two numerical strategies, (1) a Monte Carlo finite-size-scaling analysis and (2) a real-space Monte Carlo renormalization-group method, were, respectively, presented in order to determine the percolation threshold (e.g., the critical volume fraction ϕ_{c} or the critical reduced number density η_{c}), percolation transition width Δ, and correlation-length exponent ν of the polyhedral particles. The results showed that ϕ_{c} (or η_{c}) and Δ increase in the following order: tetrahedron < cube < octahedron < dodecahedron < icosahedron. In other words, both the percolation threshold and percolation transition width increase with the number of faces of the polyhedral particles as the shape becomes more "spherical." We obtained the statistical values of ν for the five polyhedral shapes and analyzed possible errors resulting in the present numerical values ν deviated from the universal value of ν=0.88 reported in literature. To validate the simulations, the corresponding excluded-volume bounds on the percolation threshold were obtained and compared with the numerical results. This paper has practical applications in predicting effective transport and mechanical properties of porous media and composites.

Study on the two-dimensional kinetic Ising model with the dynamic Monte Carlo renormalization group method

Using a modified dynamic Monte Carlo renormalization group method, the two- dimensional kinetic Ising model is studied, and the dynamic critical exponent is obtained. The critical temperature of phase transition can be obtained by the renormalization method for the correlation function. In the method we used, the correlation function is replaced with the absolute value of the magnetization, and it is found that the evolution of the absolute value of the magnetization over time satisfies the power-law form. It is found that the value of the dynamic critical exponent tends to be a stable value in the form of a power-law function as the scale of the system increases. The dynamic critical exponent obtained is z≃2.15. When the modified dynamic Monte Carlo renormalization group method is applied to the two-dimensional Glauber model, the obtained dynamic critical exponent is z≃2.25.

6a-B-2 MCRG法による格子QCDのスケーリング解析

6a-B-2 MCRG法による格子QCDのスケーリング解析

Nonlocal correlations in the orbital selective Mott phase of a one-dimensional multiorbital Hubbard model

We study non-local correlations in a three-orbital Hubbard model defined on an extended one-dimensional chain using determinant quantum Monte Carlo and density matrix renormalization group methods. We focus on a parameter with robust Hund's coupling, which produces an orbital selective Mott phase (OSMP) at intermediate values of the Hubbard U, as well as an orbitally ordered ferromagnetic insulating state at stronger coupling. An examination of the orbital and spin-correlation functions indicates that the orbital ordering occurs before the onset of magnetic correlations in this parameter regime as a function of temperature. In the OSMP, we find that the self-energy for the itinerant electrons is momentum dependent, indicating a degree of non-local correlations while the localized electrons have largely momentum independent self-energies. These non-local correlations also produce relative shifts of the hole-like and electron-like bands within our model. The overall momentum dependence of these quantities is strongly suppressed in the orbitally-ordered insulating phase.

Monte Carlo Renormalization Group Method to Study the First-Order Phase Transition in the Complex Ferromagnet

An improved Monte Carlo renormalization group method is established. Based on a new model, as well as the generalized Glauber dynamics, the first-order phase transition is studied in the Erdös–Rényi network, which is used to represent the complex ferromagnet. Both static and dynamic critical exponents can be evaluated simultaneously, and the static critical exponents are evaluated to be independent of the dimension of the system. Thus by using the hyperscaling relation νd* = 2 − α, the effective dimension d* of complex networks can be found.

Tensor network implementation of bulk entanglement spectrum

Many topologically nontrivial states of matter possess gapless degrees of freedom on the boundary, and when these boundary states delocalize into the bulk, a phase transition occurs and the system becomes topologically trivial. We show that tensor networks provide a natural framework for analyzing such topological phase transitions in terms of the boundary degrees of freedom which mediate it. To do so, we make use of a correspondence between a topologically nontrivial ground state and its phase transition to a trivial phase established in [1]. This involved computing the bulk entanglement spectrum (BES) of the ground state, upon tracing out an extensive subsystem. This work implements BES via tensor network representations of ground states. In this framework, the universality class of the quantum critical entanglement Hamiltonian in d spatial dimensions is either derived analytically or mapped to a classical statistical model in d + 1 dimensions, which can be studied using Monte Carlo or tensor renormalization group methods. As an example, we analytically derive the universality classes of topological phase transitions from the spin-1 chain Haldane phase and demonstrate that the AKLT wavefunction (and its generalizations) remarkably contains critical six-vertex (and in general eight-vertex) models within it.

Statistical mechanics of polymer chains grafted to adsorbing boundaries of fractal lattices embedded in three-dimensional space

We study the adsorption problem of linear polymers, immersed in a good solvent, when the container of the polymer–solvent system is taken to be a member of the Sierpinski gasket (SG) family of fractals, embedded in the three-dimensional Euclidean space. Members of the SG family are enumerated by an integer b (2≤b<∞), and it is assumed that one side of each SG fractal is impenetrable adsorbing boundary. We calculate the surface critical exponents γ11,γ1, and γs which, within the self-avoiding walk model (SAW) of polymer chain, are associated with the numbers of all possible SAWs with both, one, and no ends grafted to the adsorbing surface (adsorbing boundary), respectively. By applying the exact renormalization group method, for 2≤b≤4, we have obtained specific values for these exponents, for various types of polymer conformations. To extend the obtained sequences of exact values for surface critical exponents, we have applied the Monte Carlo renormalization group method for fractals with 2≤b≤40. The obtained results show that all studied exponents are monotonically increasing functions of the parameter b, for all possible polymer states. We discuss mutual relations between the studied critical exponents, and compare their values with those found for other types of lattices, in order to attain a unified picture of the attacked problem.

Historical Review on Analytic, Monte Carlo, and Renormalization Group Approaches to Critical Phenomena of Some Lattice Models

Analytic calculations, Monte Carlo (MC) simulations [N. Metropolis and S. Ulam, J. Am. Stat. Asso 44, 335 (1949)] and renormalization group (RG) theory [K. G. Wilson, Phys. Rev. B 4, 3174 and 3184 (1971)] are important methods of statistical physics to study phase transitions and critical phenomena. This paper gives a brief historical review on analytic, MC, and RG approaches to some lattice critical systems in memory of late Professors Shang-keng Ma (1940-1983) and Yu-Ming Shih (1942-2005), who played some key roles in early developments of statistical physics of critical phenomena in Taiwan. The paper first introduces some developments in the study of critical phenomena from late 19th century to early 70s in the 20th century when renormalization group theory was formulated. The paper then reviews some topics on analytic, Monte Carlo, and RG approach to some lattice models, including variational RG approach to the Ising model, slow relaxation of a spin glass model at low temperatures, iterative method for quantum spin models, percolation theory of critical phenomena of Ising-like models and hard-core particle model, histogram Monte Carlo method and histogram Monte Carlo RG method, universal finite-size scaling functions of lattice models, critical slowing down of the Ising model, finite-size corrections for critical lattice models, and self-organized critical systems. Such topics are closely related to the author's own research experience. Finally, the paper outlines some very recent developments, summarizes main results of this paper, and points out some interesting problems for further studies, e.g. why a biological system can maintain in a non-equilibrium state for a very long time.

Phase transition in a two-dimensional Ising ferromagnet based on the generalized zero-temperature Glauber dynamics

At zero temperature, based on the Ising model, the phase transition in a two-dimensional square lattice is studied using the generalized zero-temperature Glauber dynamics. Using Monte Carlo (MC) renormalization group methods, the static critical exponents and the dynamic exponent are studied; the type of phase transition is found to be of the first order.

Universality in the critical dynamics of the three-dimensional large-q Potts model with bond randomness

The critical behavior of the three-dimensional random-bond Potts model for large state numbers with q=16 and q=32 is studied by the finite-time dynamics combining with Monte Carlo renormalization group method. The correlation length and dynamic exponents as well as the critical temperatures are calculated through reducing the influence of finite-size effects. According to the variations of critical exponents with disorder amplitude, we identify the asymptotic critical exponents with ν=0.424(14), z=3.03(12) and find that they are q-independent. The results further confirm that the intrinsic ν, instead of finite-size one, can escape the bound ν>2/d suggested previously.

Dynamical Monte Carlo studies of the three-dimensional bimodal random-field Isingmodel

The nature of the three-dimensional random-field Ising model with a bimodal probabilitydistribution is investigated using finite-time scaling combined with the Monte Carlorenormalization group method, in the presence of a linearly varying temperature. Our resultssupport the existence of a first-order phase transition for this model, obtained throughreducing the influence of finite-size effects. The critical exponents are estimated to beν = 0.74(3), β = 0.27(1), α = − 0.023(5), andγ = 1.48(3) with corrections to the scaling. The Rushbrooke scaling law is satisfied with theseexponents, and in turn identifies the asymptotic critical behavior.

Positive specific-heat critical exponent of a three-dimensional three-state random-bond Potts model

Whereas the stability of a pure critical system is determined by the sign of its specific-heat critical exponent α according to Harrisʼ criterion, whether a d-dimensional dirty system should satisfy ν⩾2/d and α<0 or not has been a controversial issue for several decades, where ν is its correlation-length critical exponent. Here, contrary to recent analytical and numerical results, we find for the three-dimensional three-state random-bond Potts model whose pure version exhibits a first-order phase transition a random fixed point whose ν<2/d and α>0 using a finite-time scaling combining with extended dynamic Monte Carlo renormalization-group method. This suggests further studies are still needed to clarify the issue in three-dimensional systems.

An advanced multi-orbital impurity solver for dynamical mean field theory based on the equation of motion approach

We propose an improved fast multi-orbital impurity solver for the dynamical mean field theory based on equations of motion (EOM) for Green’s functions and a decoupling scheme. In this scheme the inter-orbital Coulomb interactions are treated fully self-consistently, and involve the inter-orbital fluctuations. As an example of the use of the derived multi-orbital impurity solver, the two-orbital Hubbard model is studied for various cases. Comparisons are made between numerical results obtained with our EOM scheme and those obtained with quantum Monte Carlo and numerical renormalization group methods. The comparison shows a good agreement, but also reveals a dissimilarity of the behaviors of the densities of states which is caused by inter-site inter-orbital hopping effects and on-site inter-orbital fluctuation effects, thus corroborating the assertion of the value of the EOM method for the study of multi-orbital strongly correlated systems.

Statistics of semiflexible self-avoiding trails on a family of two-dimensional compactfractals

We have applied the exact and Monte Carlo renormalization group (MCRG) method tostudy the statistics of semiflexible self-avoiding trails (SATs) on the family of plane-filling(PF) fractals. Each fractal of the family is compact, that is, the fractal dimensiondf is equal to 2 for all members of the PF family, which are enumerated by an odd integerb, . Varying values of the stiffness parameters of trails from 1 to0 (so that when s decreases the trail stiffness increases) we calculate exactly (for3 ≤ b ≤ 7) and through the MCRGapproach (for b ≤ 201) the setsof the critical exponents ν (associated with the mean squared end-to-end distances of SATs) andγ (associated with the total number of different SATs). Our resultsshow that critical exponents are stiffness dependent functions, so thatν(s) is a monotonicallydecreasing function of s, for each studied b, whereas γ(s) displays a non-monotonic behavior for some values ofb. On the other hand, by fixing the stiffness parameters, our results show clearly that for highly flexible trails (withs = 1 and 0.9)ν is a non-monotonicfunction of b, while forstiffer SATs (with s ≤ 0.7)ν monotonically decreaseswith b. We also showthat γ(b) increaseswith increasing b,independently of s. Finally, we compare the obtained SAT data with those obtained for the semiflexibleself-avoiding walk (SAW) model on the same fractal family, and for both models we discussbehavior of the studied exponents in the fractal-to-Euclidean crossover region .

Evidences of the Instability Fixed Points of First-Order Phase Transitions

We study the dynamics of the first-order phase transitions in the two-dimensional Potts model driven by a linearly varying temperature using a finite-time scaling with extended dynamic Monte Carlo renormalization group method. It is found that, for sufficiently large lattice sizes, the flows of apparent exponents of different temperature sweep rates upon renormalization show characteristics that are markedly distinct from those of continuous transitions and are argued to result from the instability fixed points involved and provide a method for estimating the associated instability exponents.

Conformal or walking? Monte Carlo renormalization group studies ofSU(3)gauge models with fundamental fermions

Strongly coupled gauge systems with many fermions are important in many phenomenological models. I use the 2-lattice matching Monte Carlo renormalization group method to study the fixed point structure and critical indexes of SU(3) gauge models with 8 and 12 flavors of fundamental fermions. With an improved renormalization group block transformation I am able to connect the perturbative and confining regimes of the N_f=8 flavor system, thus verifying its QCD-like nature. With N_f=12 flavors the data favor the existence of an infrared fixed point and conformal phase, though the results are also consistent with very slow walking. I measure the anomalous mass dimension in both systems at several gauge couplings and find that they are barely different from the free field value.