Real-space Renormalization-group Study Research Articles

Ising universality in the two-dimensional Blume-Capel model with quenched random crystal field.

Using high-precision Monte Carlo simulations based on a parallel version of the Wang-Landau algorithm and finite-size scaling techniques, we study the effect of quenched disorder in the crystal-field coupling of the Blume-Capel model on a square lattice. We mainly focus on the part of the phase diagram where the pure model undergoes a continuous transition, known to fall into the universality class of a pure Ising ferromagnet. A dedicated scaling analysis reveals concrete evidence in favor of the strong universality hypothesis with the presence of additional logarithmic corrections in the scaling of the specific heat. Our results are in agreement with an early real-space renormalization-group study of the model as well as a very recent numerical work where quenched randomness was introduced in the energy exchange coupling. Finally, by properly fine tuning the control parameters of the randomness distribution we also qualitatively investigate the part of the phase diagram where the pure model undergoes a first-order phase transition. For this region, preliminary evidence indicate a smoothing of the transition to second-order with the presence of strong scaling corrections.

Many-body localization in a long-range model: Real-space renormalization-group study.

We develop a real-space renormalization-group (RSRG) scheme by appropriately inserting the long-range hopping t∼r^{-α} with nearest-neighbor interaction to study the entanglement entropy and maximum block size for the many-body localization (MBL) transition. We show that for α<2 there exists a localization transition with renormalized disorder that depends logarithmically on the finite size of the system. The transition observed for α>2 does not need a rescaling in disorder strength. Most important, we find that even though the MBL transition for α>2 falls in the same universality class as that of the short-range models, the transition for α<2 belongs to a different universality class. Because of the intrinsic nature of the RSRG flow toward delocalization, MBL phase for α>2 might suffer an instability in the thermodynamic limit while the underlying systems support algebraic localization. Moreover, we verify these findings by inserting microscopic details to the RSRG scheme where we additionally find a more appropriate rescaling function for disorder strength; we indeed uncover a power-law scaling with a logarithmic correction and a distinctly different stretched exponential scaling for α<2 and α>2, respectively, by analyzing system with finite size. This finding further suggests that microscopic RSRG scheme is able to give a hint of instability of the MBL phase for α>2 even considering systems of finite size.

Real space renormalization group study of the two-dimensional super-antiferromagnetic Ising model

The phase diagram of the two-dimensional super-antiferromagnetic (SAF) Ising model in the presence of a magnetic field is investigated within the framework of a real-space renormalization-group (RSRG) approximation. We consider nearest neighbor ferromagnetic interactions along the x direction and antiferromagnetic interactions in the y direction. The main results are depicted in the magnetic field versus temperature phase diagram. A second-order transition line separates a SAF phase from a field induced ferromagnetic (FIF) phase. Our study reveals that the magnetic field induces a phase transition at a single temperature value, thus, we did not find any evidence of reentrant behavior as claimed by some authors.

Facilitated spin models in one dimension: A real-space renormalization group study

We use a real-space renormalization group (RSRG) to study the low-temperature dynamics of kinetically constrained Ising chains (KCICs). We consider the cases of the Fredrickson-Andersen (FA) model, the East model, and the partially asymmetric KCIC. We show that the RSRG allows one to obtain in a unified manner the dynamical properties of these models near their zero-temperature critical points. These properties include the dynamic exponent, the growth of dynamical length scales, and the behavior of the excitation density near criticality. For the partially asymmetric chain, the RG predicts a crossover, on sufficiently large length and time scales, from East-like to FA-like behavior. Our results agree with the known results for KCICs obtained by other methods.

Real Space Renormalization Group Study of theS=1/2 XXZ Chains with Fibonacci Exchange Modulation

Ground state properties of the S=1/2 antiferromagnetic XXZ chain with Fibonacci exchange modulation are studied using the real space renormalization group method for strong modulation. The quantum dynamical critical behavior with a new universality class is predicted in the isotropic case. Combining our results with the weak coupling renormalization group results by Vidal et al., the ground state phase diagram is obtained.

Localization of the Hall resistivity at high magnetic fields: absence of the quantized Hall insulator

We present a real-space renormalization-group study of localization in a two-dimensional hierarchical network model that is designed to represent phase-coherent electron transport in the quantum-Hall regime. Scaling expressions for both the longitudinal and Hall resistivities are derived. In agreement with recent numerical studies, we find that the Hall resistivity is quantized in the metallic phase but diverges in the insulating phase. This suggests that the characteristics of a quantized Hall insulator can occur only in the presence of a strong dephasing mechanism.

Randomness effects in the quantum phase transition of a model for heavy fermions

We present a real-space renormalization group study of the effects of disorder in an anisotropic Kondo necklace model for heavy fermions. These systems are close to a magnetic quantum critical point and this model contains the relevant physics for this problem. In the pure case, we find a quantum critical point (QCP) at a finite value of the ratio between the Kondo coupling and the hopping term, Kc=(J/W)c, separating an antiferromagnetic phase from a non-magnetic, Kondo-like, phase. The effect of disorder is investigated taking into account the renormalization of the probability distributions P(J/W). It turns out to be an irrelevant perturbation, in the RG sense, at the QCP of the pure system. We are able to disentangle the disorder in J and W and characterize the non-magnetic, disordered, Kondo- like phase as a collection of weakly coupled singlets with random excitation energies.

Real-space renormalization-group study of the two-dimensional Blume-Capel model with a random crystal field

The phase-diagram of the two-dimensional Blume-Capel model with a random crystal field is investigated within the framework of a real-space renormalization group approximation. Our results suggest that, for any amount of randomness, the model exhibits a line of Ising-like continuous transitions, as in the pure model, but no first-order transition. At zero temperature the transition is also continuous, but not in the same universality class as the Ising model. In this limit, the attractor (in the renormalization group sense) is the percolation fixed point of the site diluted spin-1/2 Ising model. The results we found are in qualitative agreement with general predictions made by Berker and Hui on the critical behaviour of random models.

Real-space renormalization-group study of the phase transition in a Gaussian model of fractals

In this paper the phase transition of the Gaussian model on m-sheet fractals (mSG${)}_{\mathrm{l}}$ and (mDH${)}_{\mathrm{l}}$ is investigated by the real-space renormalization-group method, i.e., decimation following a spin rescaling. The latter is introduced to keep the parameter b constant. Fixed points of the renormalization-group transformation are found and discussed. Our results show the existence of different properties of phase transition between the Gaussian model and the Ising model on fractals. In addition, we find that the critical point k=b/4 in a regular Sierpinski gasket is identified, with result of k=b/d (d is the coordination number) in Euclidean space. This indicates that the critical point of the Gaussian model may be uniquely determined by the coordination number whether on homogeneous fractals or translationally invariant lattices.

Different self-avoiding walks on percolation clusters: A small-cell real-space renormalization-group study

We calculate the average number of stepsN for edge-to-edge, “normal,” and indefinitely growing self-avoiding walks (SAWs) on two-dimensional critical percolation clusters, using the real-space renormalization-group approach, with small “H” cells. Our results are of the formN=AL D SAW+B, whereL is the end-to-end distance. Similarly to several deterministic fractals, the fractal dimensionsD SAW for these three different kinds of SAWs are found to be equal, and the differences between them appear in the amplitudesA and in the correction termsB. This behavior is atributed to the hierarchical nature of the critical percolation cluster.

Real-space renormalization-group study of the phase diagram of two coupled quantum spin-1/2 chains.

We investigate the phase diagram of two coupled quantum spin-1/2 chains, a square ladder, allowing for anisotropies both in the interchain and intrachain coupling, using the recently introduced density-matrix renormalization-group ~DMRG! algorithm. We consider open chains up to a length of 23200, while retaining up to 120 states in the truncated basis of the Hilbert space. The phase diagram is discussed in connection with exact results and bosonization analysis. Numerically, we show that the crossover from the critical behavior of the uncoupled disordered spin-1/2 chain to a gapped quantum liquid sets in for infinitesimal interchain coupling, both for ferromagnetic and antiferromagnetic interchain coupling. The Haldane phase of the coupled chains is, even for small interchain couplings, generically described by the Affleck-Kennedy-Lieb-Tasaki model. We also discuss typical shortcomings of the DMRG for some choices of the interaction parameters.

I-T phase diagram of vortices in layered superconductors.

The critical behavior of vortices in a stack of weakly coupled superconducting layers in the presence of a current $I$ applied parallel to the layers in zero magnetic field is studied using a rigorous, real-space renormalization group study. We identify three characteristic values of the current at which the correlation length changes its behavior. The characteristic currents are found to have a predominantly linear temperature dependence and are identified as phase lines which divide the $I$- $T$ phase diagram of this system into four regions. We find our results provide a sound physical explanation for electronic transport data taken on various high-temperature superconductors.

Critical behavior of vortices in layered superconductors.

The critical behavior of vortices in layered systems is studied. The vortex interactions that we use approximate those of vortices in a layered superconductor. We do a mathematically rigorous, real-space renormalization-group study on our model to derive the recursion relations. Terms not found by other studies on layered systems have been derived. It is found that one of the new terms contributes to layer decoupling just above the transition temperature, behavior which is consistent with Monte Carlo studies of this system. By analylzing the correlation length, we are able to study the dependence of the transition temperature and of the size of the three-dimensional (3D) critical region on the strength of the interlayer coupling. The size of the 3D critical region above the transition is found to depend on the interlayer coupling in a way which is different than that predicted by others. We study our results in the context of the high-temperature superconductors and find that they are in good accord with experiments on these materials.

Real-space renormalization-group study of the two-dimensional Hubbard model.

The one-band two-dimensional Hubbard Hamiltonian is analyzed using a real-space renormalization-group block method. The renormalization-group method, previously applied to the one-dimensional case has been extended to two dimensions. It is also shown how to avoid the proliferation of new terms during the process if the shape of the blocks and the symmetries of the kept states are chosen conveniently. We characterize the ground state by its energy per site, the gap of charge excitations, the double occupancy, and the effective hopping

On the real-space renormalization-group study of some 2D quantum spin systems

Abstract Within a real-space renormalization-group framework, the critical behaviour of the spin- 1 2 anisotropic quantum Heisenberg model on the three planar lattices has been studied. To examine the stability of the results with respect to changes in the approximation scheme, various divisions of the lattices and various truncation procedures have been used. It has been shown that there is a non-trivial fixed point and corresponding critical temperature for the nearest-neighbour XY model for each two-dimensional lattice irrespective of the approximation used. Concerning the Heisenberg model, to first order in the cumulant expansion there is no non-trivial fixed point, however, it appears in the second- and third-order calculations. This suggests that the two-dimensional Heisenberg model also exhibits some kind of a critical behaviour. In addition, the phase transitions to the many-sublattice structure have been discussed.

Real-space renormalization-group study of the exchange-interaction model.

Through use of an approximation introduced by Suzuki and Takano, the Migdal-Kadanoff (MK) renormalization-group transformation with the rescaling length b=2 is derived for the exchange-interaction (EI) model. Both the standard MK method and its modification, which preserves the free energy in the renormalization transformation, are used to determine critical temperatures and thermal exponents of the EI model on cubic lattices for various spins.

Real space renormalization group study of Cu-O planes with Coulomb repulsion

The electronic properties of a 2D lattice of Cu and O atoms with on site Coulomb interactions are analyzed within a real space renormalization group scheme. It is shown that a new interaction, electron assisted hopping, is generated. It leads to superconductivity under very general conditions.

Real-space renormalization-group study of fractal Ising models.

Ising models on Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski carpets are studied within a real-space renormalization technique. A specific method for decimating spin blocks of any size is proposed as an alternative to the bond-moving prescription, and its accuracy is checked on various examples. From an analysis of the critical couplings we estimate some of the finite-size effects of Monte Carlo simulations of fractals. The exponent \ensuremath{\nu}, computed for a large variety of carpets, is found smaller than in the bond-moving approach but with the same behavior under variations of the fractal parameters. No universality criterion emerges except in the limit of vanishing lacunarity.

The adjoint string at finite temperature

Expectations for the behavior of the adjoint string at finite temperature are presented. In the Migdal-Kadanoff approximation a real-space renormalization group study of the effective Polyakov like action predicts a deconfinement-like crossover for adjoint sources at a temperature slightly below the deconfinement temperature of fundamental sources. This prediction is compared with a Monto Carlo simulation of SU (2) lattice gauge theory on an 8 3×2 lattice.

A renormalisation group study of weak disorder in the Hubbard Hamiltonian

The authors have undertaken a real-space renormalisation group study of the weakly disordered Hubbard Hamiltonian. They find that the presence of a small correlation leads to an unexpected result: an enhancement of delocalisation which becomes more pronounced as the dimensionality increases.