Mean-field Renormalization Group Research Articles

Diluted and random-bond Ising model and the Mean-field renormalization group (MFRG) method for the interpretation of the magnetic phase diagram of the FexMn0.6−xAl0.4 alloy series

Diluted and random-bond Ising model and the Mean-field renormalization group (MFRG) method for the interpretation of the magnetic phase diagram of the FexMn0.6−xAl0.4 alloy series

Designer flat bands: Topology and enhancement of superconductivity

We construct quasi one-dimensional topological and non-topological three-band lattices with tunable band gap and winding number of the flat band. Using mean field (MF) and exact density matrix renormalization group (DMRG) calculations, we show explicitly how the band gap affects pairing and superconductivity (SC) in a Hubbard model with attractive interactions. We show excellent agreement between MF and DMRG. When a phase twist is applied on the system, a phase difference appears between pairing order parameters on different sublattices, and this plays a very important role in the SC density. The SC weight, $D_s$, on the gapped topological, $W\neq0$, flat band increases linearly with interaction strength, $U$, for low values, and with a slope that depends on the details of the compact localized state at $U=0$. As $U\to 0$ for the gapped non-topological flat band ($W=0$), $D_s$ decays with a power law faster than quadratic but slower than exponential. This indicates that isolated non-topological flat bands are less beneficial to SC. In the gapless case (flat band touching the band above it), we find at low $U$ (both for $W=0$ and $W\neq 0$) that $D_s\propto U^\varphi$ with $\varphi<1$ contrary to the $U{\rm ln}\, ({\rm const.}/U)$ behavior reported in the literature. In other words, $D_s$ increases faster than linearly for low $U$ thus favoring SC at weak interaction more than the gapped case. For systems with touching bands, we observe that the one-body correlation length, $\xi$, diverges as a power law as $U\rightarrow0$, while for the isolated flat band $\xi(U\to 0)$ is a constant smaller than one lattice spacing. Both behaviors are distinct from the exponential divergence of $\xi$ in the dispersive case. Our results re-establish that the BCS mean field and quantum metric alone are insufficient to characterize SC at weak coupling.

Pairing and superconductivity in quasi-one-dimensional flat-band systems: Creutz and sawtooth lattices

We study the pairing and superconducting properties of the attractive Hubbard model in two quasi one-dimensional topological lattices: the Creutz and sawtooth lattices. They share two peculiar properties: each of their band structures exhibits a flat band with a non-trivial winding number. The difference, however, is that only the Creutz lattice is genuinely topological, owing to a chiral (sub-lattice) symmetry, resulting in a quantized winding number and zero energy edge modes for open boundary conditions. We use mean field and exact density matrix renormalization group in our work. Our three main results are: (a) For both lattice systems, the superconducting weight, $D_s$, is linear in the coupling strength, $U$, for low values of $U$; (b) for small $U$, $D_s$ is proportional to the quantum metric for the Creutz system but not for the sawtooth system because its sublattices are not equivalent; (c) conventional BCS mean field is not appropriate for such systems with inequivalent sublattices. We show that, for a wide range of densities and coupling strengths, these systems are very well described by a full multi-band mean field method where the pairing parameters and the local particle densities on the inequivalent sublattices are variational mean field parameters.

Renormalization group analysis of phase transitions in the two-dimensional Majorana-Hubbard model

A lattice of interacting Majorana modes can occur in a superconducting film on a topological insulator in a magnetic field. The phase diagram as a function of interaction strength for the square lattice was analyzed recently using a combination of mean field theory and renormalization group methods, and was found to include second order phase transitions. One of these corresponds to spontaneous breaking of an emergent U(1) symmetry, for attractive interactions. Despite the fact that the U(1) symmetry is not exact, this transition was claimed to be in a supersymmetric universality class when time reversal symmetry is present and in the conventional XY universality class otherwise. Another second order transition was predicted for repulsive interactions with time reversal symmetry to be in the same universality class as the transition occurring in the Gross-Neveu model, despite the fact that the U(1) symmetry is not exact in the Majorana model. We analyze these phase transitions using a modified $\epsilon$-expansion, confirming the previous conclusions.

Effective field theory in larger clusters – Ising model

General formulation for the effective field theory with differential operator technique and the decoupling approximation with larger finite clusters (namely EFT-N formulation) has been derived for several S-1/2 bulk systems. The effect of enlarging this finite cluster on the results for the critical temperatures and thermodynamic properties has been investigated in detail. Beside the improvement on the critical temperatures, the necessity of using larger clusters, especially in nanomaterials has been discussed. Using the derived formulation, applications on the effective field and mean field renormalization group techniques have also been performed.

Low-temperature properties of ferromagnetic spin chains in a magnetic field

The thermodynamic properties of ferromagnetic spin chains have been analyzed with a variety of microscopic methods over the years: Bethe ansatz, spin-wave theory, Schwinger-boson mean-field theory, Green functions, and renormalization group methods. Surprisingly, in all these different studies, the manifestation of the spin-wave interaction in the low-temperature series for the thermodynamic quantities, in the presence of a finite magnetic field, has been largely neglected. In the present work, we address this problem by following a different path, based on the systematic effective Lagrangian method. We evaluate the partition function up to two-loop order and derive the low-temperature expansion of the energy density, entropy density, heat capacity, magnetization, and susceptibility in the presence of a weak external magnetic field. Remarkably, the spin-wave interaction only manifests itself beyond two-loop order. In particular, there is no term of order ${T}^{2}$ in the low-temperature series of the free energy density. This is the analog of Dyson's statement that there is no term of order ${T}^{4}$ in the low-temperature series of the free energy density in the case of three-dimensional ideal ferromagnets. The range of validity of our series is critically examined in view of the nonperturbatively generated energy gap. We also compare our results with the condensed matter literature and point out that there are some misleading statements.

Blume-Capel spin-glass model for Fe-Mn-Al alloys

The spin-glass Blume-Capel model is studied by using an extended mean-field renormalization group approximation. The effects of a probability distribution of the bilinear interaction coupling with competing interactions have been investigated. The phase diagrams show second-order phase boundaries among ferromagnetic, antiferromagnetic, paramagnetic, and spin-glass phases. On the other hand, tricritical phenomena exist only for certain values of the concentration of ferromagnetic bonds as well as ratios of the ferromagnetic to the antiferromagnetic competitive interactions. An application to the magnetic properties of ternary system Fe-Al-Mn alloys is presented. Good fittings to the experimental data are obtained.

Publisher’s Note: “Mean field renormalization group: A theoretical approach to the Fe1−qAlq in the bcc lattice” [J. Appl. Phys. 111, 113921 (2012)

Publisher’s Note: “Mean field renormalization group: A theoretical approach to the Fe<i>1−q</i>Al<i>q</i> in the bcc lattice” [J. Appl. Phys. 111, 113921 (2012)

Mean field renormalization group: A theoretical approach to the Fe1−qAlq in the bcc lattice

In this work, the site diluted Heisenberg model is applied to study the magnetic properties of the bcc disordered phase of FeAl alloys by employing mean field renormalization group theory. We suggest a new approach to exchange interaction between nearest neighbors of Fe that depends on de powers of the Al (q) instead of the linear dependence proposed in other papers. Excellent agreement with the experimental data in the T – q plane have been obtained, in particular, in the region with anomalous behavior of the alloy concerned.

Site-diluted Blume–Capel model for the Fe–Al disordered alloys

The phase-diagram of a site-diluted spin-1 Blume–Capel model on a simple cubic lattice is studied by using the mean-field renormalization group approach in the pair approximation. The critical temperature has been obtained as function of the vacant sites concentration and the crystal-field. Considering exchange and superexchange interactions between Fe atoms we applied this model to the description of the thermodynamic properties of Fe–Al alloys. The experimental phase diagram has been adjusted with the theoretical parameters and compared with previous theoretical results.

Phase diagram of self-assembled rigid rods on two-dimensional lattices: Theory and Monte Carlo simulations

Monte Carlo simulations and finite-size scaling analysis have been carried out to study the critical behavior in a two-dimensional system of particles with two bonding sites that, by decreasing temperature or increasing density, polymerize reversibly into chains with discrete orientational degrees of freedom and, at the same time, undergo a continuous isotropic-nematic (IN) transition. A complete phase diagram was obtained as a function of temperature and density. The numerical results were compared with mean field (MF) and real space renormalization group (RSRG) analytical predictions about the IN transformation. While the RSRG approach supports the continuous nature of the transition, the MF solution predicts a first-order transition line and a tricritical point, at variance with the simulation results.

Blume-Capel model for(Fe0.65Ni0.35)1−xMnxandFepAlqMnxalloys

A random-bond Blume-Capel model is proposed in order to study the ternary FeNiMn and FeAlMn alloys. The extended mean-field renormalization group approach and a variational procedure based on Bogoliubov inequality for the free energy are employed in the pair approximation. The phase diagrams are obtained for a symmetric distribution as well as for asymmetric distributions compatible to these alloys. The obtained phase diagram for the FeNiMn system is in a much better accord with the experimental data than the previous traditional mean-field renormalization group applied to the Ising model, mainly concerning the spin-glass and antiferromagnetic phases. Better results are also achieved for the order parameter of the FeAlMn alloys.

An extended approach for computing the critical properties in the two-and three-dimensional lattices within the effective-field renormalization group method

In this letter we employing the effective-field renormalization group (EFRG) to study the Ising model with nearest neighbors to obtain the reduced critical temperature and exponents ν for bi- and three-dimensional lattices by increasing cluster scheme by extending recent works. The technique follows up the same strategy of the mean field renormalization group (MFRG) by introducing an alternative way for constructing classical effective-field equations of state takes on rigorous Ising spin identities.

Micromechanical Model for Deformation in Solids with Universal Predictions for Stress-Strain Curves and Slip Avalanches

A basic micromechanical model for deformation of solids with only one tuning parameter (weakening epsilon) is introduced. The model can reproduce observed stress-strain curves, acoustic emissions and related power spectra, event statistics, and geometrical properties of slip, with a continuous phase transition from brittle to ductile behavior. Exact universal predictions are extracted using mean field theory and renormalization group tools. The results agree with recent experimental observations and simulations of related models for dislocation dynamics, material damage, and earthquake statistics.

Critical properties of classical [formula omitted] and Heisenberg models: A mean-field renormalization group study

Using mean-field renormalization group (MFRG) and surface-bulk mean-field renormalization group (SBMFRG) methods, we study the critical properties of classical Heisenberg and XY models. We show the exact result that there is no finite temperature phase transition in one dimension and very good values for critical exponents and critical temperatures are obtained for these models on cubic lattice in three dimensions.

Distance traveled by random walkers before absorption in a random medium

We consider the penetration length l of random walkers diffusing in a medium of perfect or imperfect absorbers of number density rho. We solve this problem on a lattice and in the continuum in all dimensions D, by means of a mean-field renormalization group. For a homogeneous system in D > 2, we find that l is similar to max(xi,rho(-1/2)), where is the absorber density correlation length. The cases of D = 1 and D = 2 are also treated. In the presence of long-range correlations, we estimate the temporal decay of the density of random walkers not yet absorbed. These results are illustrated by exactly solvable toy models, and extensive numerical simulations on directed percolation, where the absorbers are the active sites. Finally, we discuss the implications of our results for diffusion limited aggregation (DLA), and we propose a more effective method to measure l in DLA clusters.

Cluster Approximation for the Farey Fraction Spin Chain

We consider the Farey fraction spin chain in an external field h. Utilising ideas from dynamical systems, the free energy of the model is derived by means of an effective cluster energy approximation. This approximation is valid for divergent cluster sizes, and hence appropriate for the discussion of the magnetizing transition. We calculate the phase boundaries and the scaling of the free energy. At h = 0 we reproduce the rigorously known asymptotic temperature dependence of the free energy. For h ≠ 0, our results are largely consistent with those found previously using mean field theory and renormalization group arguments.

Characterization of magnetic phases in the Fe xMn 0.65−xAl 0.35 disordered alloys

Melted alloys of the Fe x Mn 0.65− x Al 0.35 disordered system, 0.25⩽ x⩽0.65, were experimentally studied by Mössbauer spectrometry, vibrating sample magnetometry and AC magnetic susceptibility. All the alloys exhibit the BCC structure with a nearly constant lattice parameter (2.92 Å). Mössbauer studies at room temperature (RT) show that in the 0.25 ⩽ x⩽0.45 range the alloys are paramagnetic (P) while in the 0.50⩽ x⩽0.65 range, they are ferromagnetic. At 77 K, Mössbauer studies show that the alloy with x = 0.25 presents weak magnetic character that is consistent with an antiferromagnetic (AF) behavior due to the high Mn content, while those with 0.30⩽ x⩽0.40 are paramagnetic, and those in the 0.45⩽ x⩽0.65 range are ferromagnetic (F) with a mean field increasing with the Fe content. Hysteresis cycles at RT prove the paramagnetic character of the alloys between x = 0.25 and 0.40 and the ferromagnetic character for x ⩾ 0.45 . Complementary measurements using AC magnetic susceptibility permit a magnetic phase diagram to be proposed, with the P phase for high temperature and all the compositions, the AF phase for low Fe content and at low temperature, the F phase for high Fe content above RT and the spin glass phase for all the compositions and at temperatures lower than 46 K. In addition, the mean field renormalization group (MFRG) method, applied to a random competitive and site dilute Ising model with nearest-neighbor, gives rise to magnetic phase diagram, which fairly agrees with previous experimental one.

Two‐step mean‐field renormalization group results for the large square Ising clusters

AbstractA transfer matrix approach has been worked out to test the predictions of the improved three‐step mean‐field renormalization group approach to square Ising clusters with linear size up to L = 11. Performing the asymptotic analysis, the convergence of the finite‐size critical couplings and the critical exponents towards the exact values is shown.

Mean-field renormalization-group approach to the boson Hubbard model

We study the phase diagram of the boson Hubbard model at T=0 without disorder in a one-site and a two-site mean-field approximations. The two-site mean-field significantly improves the results compared to ordinary mean-field solutions and allows to use a simple scheme of mean-field renormalization group (MFRG). Some results obtained with MFRG are even better including the critical exponents for the density-driven transition between the Mott insulator and superfluid phases. However, when we evaluate critical exponents for transitions through the tip of the insulating lobes (special transitions without density change), we get similar exponents indicating that the MFRG apparently does not distinguish between both type of transitions and places them in the same universality class. The results for the first insulating lobe in d=1 and d=2 are in good agreement with a perturbative method and with the more reliable quantum Monte Carlo results. Our MFRG method provides analytical results for any lobe in any dimension and can be systematically improved if we consider larger clusters.