Renormalization group study of a three-dimensional lattice model with directional bonding

We consider a bcc lattice model in which each site is either vacant or occupied by a molecule. The molecules have four symmetrically arranged arms directed towards four of the eight nearest-neighbor sites. Two molecules form a bond if they have bonding arms pointing towards each other and along their line of centers. We introduce bonding energies as well as two-, three-, and four-molecule interactions. The model is studied using a real-space renormalization group method. The form of the pressure-temperature phase diagram is found to be very sensitive to small changes in the relative sizes of the energy parameters. Adjustment of these parameters allows us to obtain a phase diagram which resembles that of the ice-water-steam system. The nature of the transitions between the various ordered phases is examined and the critical exponents are obtained.     

Renormalization by continuous unitary transformations: one-dimensional spinless fermions

A renormalization scheme for interacting fermionic systems is presented where the renormalization is carried out in terms of the fermionic degrees of freedom. The scheme is based on continuous unitary transformations of the Hamiltonian which stays hermitian throughout the renormalization flow, whereby any frequency dependence is avoided. The approach is illustrated in detail for a model of spinless fermions with nearest neighbour repulsion in one dimension. Even though the fermionic degrees of freedom do not provide an easy starting point in one dimension favorable results are obtained which agree well with the exact findings based on Bethe ansatz. Received 21 August 2002 / Received in final form 29 October 2002 Published online 31 December 2002     

U(1)规范理论的定量分析

在Migdal及Migdal-Kadanoff交换方案基础上导出了重整化群方程、并将其应用到Wilson形式的U(1)格点规范理论,给出了强耦合区,弱耦合区及临界耦合点,中间耦合区的数值结果.这些结果与严格的强耦合展开,弱耦合展开及Monte Carlo模拟结果一致.从而说明:本文得到的重整化群方程在讨论规范理论的非微扰特性方面是可行而有效的.     

High density quark matter and the renormalization group in QCD with two and three flavors

We consider the most general four fermion operators in QCD for two and three massless flavors and study their renormalization in the vicinity of the Fermi surface. We show that, asymptotically, the largest coupling corresponds to scalar diquark condensation. Asymptotically the direct and iterated (molecular) instanton interactions become equal. We provide simple arguments for the form of the operators that diagonalize the evolution equations. Some solutions of the flow equations exhibit instabilities arising out of purely repulsive interactions.     

Electron-doping versus hole-doping in the 2D t-t' Hubbard model

We compare the one-loop renormalization group flow to strong coupling of the electronic interactions in the two-dimensional t-t'-Hubbard model with t' = - 0.3t for band fillings smaller and larger than half-filling. Using a numerical N-patch scheme ( N = 32, ..., 96) we show that in the electron-doped case with decreasing electron density there is a rapid transition from a d _{x2 - y2}-wave superconducting regime with small characteristic energy scale to an approximate nesting regime with strong antiferromagnetic tendencies and higher energy scales. This contrasts with the hole-doped side discussed recently which exhibits a broad parameter region where the renormalization group flow suggests a truncation of the Fermi surface at the saddle points. We compare the quasiparticle scattering rates obtained from the renormalization group calculation which further emphasize the differences between the two cases. Received 19 December 2000 and Received in final form 28 February 2001     

A renormalized equation for the three-body system with short-range interactions

We study the three-body system with short-range interactions characterized by an unnaturally large two-body scattering length. We show that the off-shell scattering amplitude is cutoff independent up to power corrections. This allows us to derive an exact renormalization group equation for the three-body force. We also obtain a renormalized equation for the off-shell scattering amplitude. This equation is invariant under discrete scale transformations. The periodicity of the spectrum of bound states originally observed by Efimov is a consequence of this symmetry. The functional dependence of the three-body scattering length on the two-body scattering length can be obtained analytically using the asymptotic solution to the integral equation. An analogous formula for the three-body recombination coefficient is also obtained.     

Improved block-spin transformations,redundant operators and all that

Improved block-spin transformations are studied both numerically and analytically. These are real-space renormalization group transformations that depend on arbitrary parameters which can be adjusted in such a way that a given action reaches the renormalized trajectory after only a few block steps. We demonstrate the role that is played by redundant operators in the optimization. Explicit calculations are done in the gaussian model and the large-N Heisenberg model. In the latter the resulting renormalization group equations are solved analytically close to the fixed point and numerically further away to obtain the renormalized trajectory for various choices of the optimization parameter.     

Quantum spin model of a cubic ferromagnet in a magnetic field

The zero temperature phase diagram of a one-dimensional ferromagnet with cubic single ion anisotropy in an external magnetic field is studied. The mean-field approximation and the density-matrix renormalization group method are applied. Two phases at finite magnetic fields are identified: a canted phase with spontaneously broken symmetry and a phase with magnetization along the magnetic field. Both methods predict that the canted phase exists even for the single-ion anisotropy strong enough to destroy the magnetic order at zero magnetic field. In contrast to the mean-field theory, the density-matrix renormalization group predicts a reentrant behavior for the model. The character of the phase transition at finite magnetic field has also been considered and the critical index has been found. Received 9 May 2000 and Received in final form 5 July 2000     

Renormalization group functions for the radiative symmetry breaking scheme

We obtain the renormalization group (RG) functions for the massless scalar field theory where symmetry breaking occurs radiatively. After obtaining the effective potential for the radiative symmetry breaking scheme by finite transformations for the classical field and coupling constant, we obtain the corresponding RG functions from that of the minimal subtraction (MS) scheme.     

Collective field formulation of the multi-species Calogero model and its duality symmetries

We study the collective field formulation of a restricted form of the multi-species Calogero model, in which the three-body interactions are set to zero. We show that the resulting collective field theory is invariant under certain duality transformations, which interchange, among other things, particles and antiparticles, and thus generalize the well known strong-weak coupling duality symmetry of the ordinary Calogero model. We identify all these dualities, which form an Abelian group, and study their consequences. We also study the ground state and small fluctuations around it in detail, starting with the two-species model, and then generalizing to an arbitrary number of species.     

Averaging operations for lattice gauge theories

Usually renormalization group transformations are defined by some averaging operations. In this paper we study such operations for lattice gauge fields and for gauge transformations. We are interested especially in characterizing some classes of field configurations on which the averaging operations are regular (e.g., analytic). These results will be used in subsequent papers on the renormalization group method in lattice gauge theories.Research supported in part by the National Science Foundation under Grant PHY-82-03669     

Renormalization for breakup of invariant tori

We present renormalization group operators for the breakup of invariant tori with winding numbers that are quadratic irrationals. We find the simple fixed points of these operators and interpret the map pairs with critical invariant tori as critical fixed points. Coordinate transformations on the space of maps relate these fixed points, and also induce conjugacies between the corresponding operators.     

From low-momentum interactions to nuclear structure

We present an overview of low-momentum two-nucleon and many-body interactions and their use in calculations of nuclei and infinite matter. The softening of phenomenological and effective field theory (EFT) potentials by renormalization group (RG) transformations that decouple low and high momenta leads to greatly enhanced convergence in few- and many-body systems, while maintaining a decreasing hierarchy of many-body forces. This review surveys the RG-based technology and results, discusses the connections to chiral EFT, and clarifies various misconceptions.     

Inverse Monte Carlo renormalization group transformations for critical phenomena

We introduce a computationally stable inverse Monte Carlo renormalization group transformation method that provides a number of advantages for the calculation of critical properties. We are able to simulate the fixed point of a renormalization group for arbitrarily large lattices without critical slowing down. The log-log scaling plots obtained with this method show remarkable linearity, leading to accurate estimates for critical exponents. We illustrate this method with calculations in two- and three-dimensional Ising models for a variety of renormalization group transformations.     

Optimal truncation procedures in renormalization-group calculations

Renormalization or rescaling transformations generally produce more complicated interactions than are present in the initial Hamiltonian. After each rescaling it is necessary to truncate the Hamiltonian to make the next rescaling mathematically tractable. One is faced with the problem of choosing the coupling constants of the truncated Hamiltonian to obtain the best approximation. Following ideas of McMillan, we consider truncation procedures which give lower and upper bounds to the free energy. Conditions for optimal lower- and upper-bound truncations are derived. These optimal truncations are seen to yield exact results for the free energy in both the high- and low-temperature limits. Some of the problems inherent in all renormalization transformations that incorporate an optimal lower- or upper-bound truncation are discussed. Calculations for the twodimensional Ising model based on renormalization transformations which combine decimation and an optimal truncation are described. Even in the simplest approximation in which only nearest-neighbor interactions are retained the free energy is obtained to an accuracy of better than 1% for all temperatures if an optimal truncation rather than an ordinary truncation with no readjustment of the coupling constants is made. However, the simplest calculations involving optimal truncations are less successful in predicting derivatives of the free energy and critical exponents than the free energy itself.     

Statistical derivation of basic equations of diffusional kinetics in alloys with application to the description of diffusion of carbon in austenite

Basic equations of diffusional kinetics in alloys are statistically derived using the master equation approach. To describe diffusional transformations in substitution alloys, we derive the ??quasi-equilibrium?? kinetic equation that generalizes its earlier versions by taking possible ??interaction renormalization?? effects into account. For the interstitial alloys Me-X, we derive an explicit expression for the diffusivity D of an interstitial atom X. This expression notably differs from those used in previous phenomenological treatments. This microscopic expression for D is applied to describe the diffusion of carbon in austenite based on some simple models of carbon-carbon interaction. The results obtained enable us to make certain conclusions about the real form of these interactions and about the scale of the ??transition state entropy?? for diffusion of carbon in austenite.     

Instability of renormalization-group pathologies under decimation

We investigate the stability and instability of pathologies of renormalization group transformations for lattice spin systems under decimation. In particular we show that, even if the original renormalization group transformation gives rise to a non-Gibbsian measure, Gibbsianness may be restored by applying an extra decimation transformation. This fact is illustrated in detail for the block spin transformation applied to the Ising model. We also discuss the case of another non-Gibbsian measure with nicely decaying correlations functions which remains non-Gibbsian after arbitrary decimation.     

Renormalization of Wilson operators in gauge theories

The operators in a Wilson expansion are not in general multiplicatively renormalized in non-Abelian gauge theories. This is because of the renormalization of the gauge transformations themselves. Renormalized fields may be defined, which have the old gauge transformations. Alternatively, a special choice of gauge may be made, in which the gauge transformations are unchanged on renormalization. In any case, one gauge invariant factor appears in the renormalization of the Wilson operators.     

Exact linear renormalization of a one-dimensional system with phase transition

We study by real-space renormalization a class of one-dimensional self-avoiding walks (SAWs) exhibiting a nonzero critical temperature. A linear renormalization transformation is carried out in closed form in a three-parameter subspace of SAW Hamiltonians. We find lines of fixed points along which the degree of localization of the fixed-point interactions varies. The role of the spin rescaling factor in the transformation is explicitly demonstrated.