Parquet Approximation Research Articles

Parquet approximation for large-Nmatrix Higgs model

The parquet approximation in the matrix Higgs model is considered. We demonstrate analytically that in the large $N$ limit the parquet approximation gives an satisfying agreement with the exact results. It is shown that the parquet planar series can be derived by means of the generating functional.

First-order transition and critical end point in vortex liquids in layered superconductors

We calculate various thermodynamic quantities of vortex liquids in a layered superconductor by using the nonperturbative parquet approximation method, which was previously used to study the effect of thermal fluctuations in two-dimensional vortex systems. We find there is a first-order transition between two vortex liquid phases which differ in the magnitude of their correlation lengths. As the coupling between the layers increases,the first-order transition line ends at a critical point. We discuss the possible relation between this critical end-point and the disappearance of the first-order transition which is observed in experiments on high temperature superconductors at low magnetic fields.

Stability of self-consistent solutions for the Hubbard model at intermediate and strong coupling

We present a general framework how to investigate stability of solutions within a single self-consistent renormalization scheme being a parquet-type extension of the Baym-Kadanoff construction of conserving approximations. To obtain a consistent description of one- and two-particle quantities, needed for the stability analysis, we impose equations of motion on the one- as well on the two-particle Green functions simultaneously and introduce approximations in their input, the completely irreducible two-particle vertex. Thereby we do not loose singularities caused by multiple two-particle scatterings. We find a complete set of stability criteria and show that each instability, singularity in a two-particle function, is connected with a symmetry-breaking order parameter, either of density type or anomalous. We explicitly study the Hubbard model at intermediate coupling and demonstrate that approximations with static vertices get unstable before a long-range order or a metal-insulator transition can be reached. We use the parquet approximation and turn it to a workable scheme with dynamical vertex corrections. We derive a qualitatively new theory with two-particle self-consistence, the complexity of which is comparable with FLEX-type approximations. We show that it is the simplest consistent and stable theory being able to describe qualitatively correctly quantum critical points and the transition from weak to strong coupling in correlated electron systems.

Parquet solution for a flat Fermi surface

We study instabilities occurring in the electron system whose Fermi surface has flat regions on its opposite sides. Such a Fermi surface resembles Fermi surfaces of some high-$T_c$ superconductors. In the framework of the parquet approximation, we classify possible instabilities and derive renormalization-group equations that determine the evolution of corresponding susceptibilities with decreasing temperature. Numerical solutions of the parquet equations are found to be in qualitative agreement with a ladder approximation. For the repulsive Hubbard interaction, the antiferromagnetic (spin-density-wave) instability dominates, but when the Fermi surface is not perfectly flat, the $d$-wave superconducting instability takes over.

Density of states near an Anderson transition in a four-dimensional space. Renormalizable models

Asymptotically accurate results are obtained for the average Green function and density of states of a disordered system for a renormalizable class of models (as opposed to the lattice model examined previously [I. M. Suslov, Zh. Éksp. Teor. Fiz. 106, 560 (1994)]. For N∼1 (where N is the order of perturbation theory), only the parquet terms corresponding to the higher powers of large logarithms are taken into account. For large N, this approximation is inadequate because of the higher rate of increase with respect to N of the coefficients for the lower powers of the logarithms. The latter coefficients are determined from the renormalization condition for the theory expressed in the form of a Callan-Symanzik equation using the Lipatov asymptote as boundary conditions. For calculating the self-energy at finite momentum, a modification of the parquet approximation, is used that allows the calculations to be done in an arbitrary finite logarithmic approximation, including the principal asymptote in N of the expansion coefficients. It is shown that the phase transition point moves in the complex plane, thereby ensuring regularity of the density of states for all energies and avoiding the “false” pole in such a way that the effective interaction remains logarithmically weak.

Low-energy excitations in an incipient antiferromagnet

Abstract We present fully self-consistent calculations in the fluctuation exchange approximation for the half-filled Hubbard model in two dimensions. A non-Fermi-liquid state evolves with decreasing temperature in this self-consistent model of coupled spin fluctuations and quasiparticles. The mean-field phase transition to long-range antiferromagnetic order is suppressed and we find no evidence of a phase transition to long-range magnetic order. We show that the real part of the self-energy at zero energy shows a positive slope and the imaginary part of the self-energy shows a local minimum. The scale of this structure is set by the zero-temperature gap in mean-field theory. The growth of spin fluctuations is reflected in the evolution of sharply peaked structure in the spin fluctuation propagator around zero energy and Q = (π, π). We present calculations for the Hubbard model in one dimension in a two-particle self-consistent parquet approximation. A second-moment sum rule suggests that vertex corrections to the self-energy are responsible for the greater accuracy of the parquet approximation compared with other self-consistent perturbation theories.

Parquet approximation in large N matrix theories

A parquet approximation (generalized ladder diagrams) in matrix models is considered. By means of numerical calculations we demonstrate that in the large N limit the parquet approximation gives an excellent agreement with exact results.

Staggered magnetic susceptibility in high-Tc systems: A scaling approach.

A scaling theory is applied in order to calculate the temperature dependence of the staggered magnetic susceptibility associated with the antiferromagnetism observed in high-temperature superconductors. In a logarithmic approximation, only a summation of the leading square logarithmic terms are taken into account, assuming the parquet approximation. We show that scaling theory can be applied to the square logarithmic two-dimensional problem. The scaling method is an improvement compared to the alternative random-phase approximation method. Our scaling result explains the temperature and doping dependence of the nuclear-spin-lattice relaxation time observed in high-${\mathit{T}}_{\mathit{c}}$ systems to satisfaction. However, the results of this paper indicate that the logarithmic parquet approximation may not be sufficient to calculate quantitatively the magnetic susceptibility. Ways to improve the theory are discussed.

Parquet approximation for a disordered two-dimensional electron gas in a strong transverse magnetic field

The density of states and the diagonal magnetoconductivity of a disordered two-dimensional electron gas under quantizing magnetic fields is treated in the parquet approximation. Results for the lowest Landau subband are obtained by considering the lowest-order irreducible vertex diagrams. An analytical solution is derived for a self-consistent approximation, which includes both particle-particle and particle-hole ladder diagrams. The parquet density of states agrees quite well with the exact result obtained by F. Wegner. The dependence on a short-range parameter is discussed. The diagonal magnetoconductivity is calculated in various approximations and the results are compared with each other. Possibilities to improve the calculation of the magnetoconductivity are discussed.

Theory of the quantum hall effect in quasi-one-dimensional conductors

The integer topological invariant, called Chern number, is calculated for a quasi-one-dimensional conductor in the magnetic-field-induced spin-density-wave state. Due to non-zero value of the Chern number the Hall conductivity per layer has the quantized value σ xy = 2 Le 2/ h and the effective action of the system contains so-called Hopf term, which describes topologically nontrivial configurations of the spin-density-wave polarization vector. The dependence of the integer number L on magnetic field H is calculated in the parquet approximation. The theory is applied to the organic conductors (TMTSF) 2X.

Heavy Fermion Superconductivity via Kondo‐Type Pairing

AbstractA consistent quantum field theoretical approach to Kondo‐type pairing in heavy fermion superconducting systems is presented. This pairing mechanism allows both “singlet” and “triplet”, as well as l > 1 pairing, depending on the system parameters. Associated with the superconducting transition is the appearance of hybridization (dielectric gap) between localized f‐states and the conduction band. The parquet approximation is used to calculate the superconducting and excitonic (dielectric) gaps.

New exact diagram approach to the Kondo problem

Described is the physics behind an new exact Feynman diagram approach to the Kondo problem. The method corresponds to a certain self-consistent parquet approximation. Here relatively simple arguments are used to justify the diagram results. The calculation of the static susceptibility and specific heat is described. The cross-over ratio W′=( π e ) 1 2 and Wilson ratio R=2 are given exactly.

Logarithmic Corrections to a Simple Power Law at d = 4 in 1/n Expansion

The 1/n expansion is applied to the study of the logarithmic deviation from the meanfield power law in four dimensions. The results are shown to be consistent with those obtained by parquet approximation and renormalization-group techniques up to order 1/n.

On the static and dynamic behaviour in nematic liquid crystals

Effective static exponents are calculated in the de Gennes model for isotropic-to-nematic phase transition. Applying parquet approximation we show that there are no stable fixed points for s⩾4. The dynamic exponent is evaluated in d⩽6 dimensions with the linear response theory.

Resistance maximum in spin glasses

The temperature ${T}_{m}$ of the resistance maximum in spin-glass magnetic alloys is calculated as a function of the Kondo temperature ${T}_{K}$ for the single impurities and the average Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction strength ${\ensuremath{\Delta}}_{c}$ between different impurities. In addition ${T}_{m}$ is shown to be independent of several other less relevant system parameters. The temperature dependence of the resistivity is obtained in the parquet approximation in the "noise model," where, besides the usual exchange scattering between conduction electrons and impurity spins, there is a transition rate ${\ensuremath{\Delta}}_{c}$ for spin-flip processes due to the RKKY interactions. The result is shown to be valid when ${\ensuremath{\Delta}}_{c}\ensuremath{\gg}{T}_{K}$, and applies to the low-${T}_{K}$ spin-glass systems like $\mathrm{Au}\mathrm{Mn}$, $\mathrm{Ag}\mathrm{Mn}$, $\mathrm{Cu}\mathrm{Mn}$, $\mathrm{Au}\mathrm{Cr}$, and $\mathrm{Au}\mathrm{Fe}$. For example, it permits the recently observed pressure variations of ${T}_{m}$ to be understood in terms of simpler behaviors of the more fundamental system parameters ${\ensuremath{\Delta}}_{c}$ and ${T}_{K}$. Asymptotically for ${\ensuremath{\Delta}}_{c}\ensuremath{\gg}{T}_{K}$ one has ${T}_{m}\ensuremath{\sim}{\ensuremath{\Delta}}_{c}\mathrm{ln}(\frac{{\ensuremath{\Delta}}_{c}}{{T}_{K}})$, which is bigger than the spin-glass freezing temperature ${T}_{0}\ensuremath{\sim}{\ensuremath{\Delta}}_{c}$ identified by the cusp in the magnetic susceptibility.

Effect of pressure on impurity-impurity interactions in dilute Au:Mn, Cu:Mn, and Au:Fe spin-glass alloys

The electrical resistivity of the dilute spin-glass alloys Au-0.10-at.% Mn, 0.15-at.% Mn, 0.10-at.% Fe, 0.13-at.% Fe, and Cu-0.15-at.% Mn has been measured from 1.2 to 40 K at pressures to 100 kbar. In these alloys the cooperative locking-in of the impurity spins at a temperature ${T}_{0}$ leads to a resistivity maximum at ${T}_{max}$. Application of pressure is found to shift ${T}_{max}$ in a manner which is strongly system dependent: $\frac{d{T}_{max}}{dP}g0$ for Au:Mn, $\frac{d{T}_{max}}{dP}\ensuremath{\approx}0$ for Cu:Mn, and $\frac{d{T}_{max}}{dP}l0$ for Au:Fe. These results are shown to be clearly incompatible with the widely held belief that $k{T}_{max}\ensuremath{\approx}{\ensuremath{\Delta}}_{\mathrm{R}\mathrm{K}\mathrm{K}\mathrm{Y}}$, where ${\ensuremath{\Delta}}_{\mathrm{R}\mathrm{K}\mathrm{K}\mathrm{Y}}\ensuremath{\approx}\frac{c{J}^{2}S(S+1)}{{E}_{F}}$ is the average strength of the Ruderman-Kittel-Kasuya-Yosida interaction, and indicate that ${T}_{max}$ is a function of both ${\ensuremath{\Delta}}_{\mathrm{R}\mathrm{K}\mathrm{K}\mathrm{Y}}$ and ${T}_{K}$, the Kondo temperature. This expectation is confirmed in a recent theory of Larsen who introduces an impurity-impurity interaction strength ${\ensuremath{\Delta}}_{c}$ into the parquet approximation of the Kondo resistivity and obtains an explicit expression for ${T}_{max}={T}_{max}({\ensuremath{\Delta}}_{c},{T}_{K})$. It is shown that both the sign and the magnitude of $\frac{d{T}_{max}}{dP}$ for the systems studied here are a natural consequence of both increasing ${T}_{K}$ and ${\ensuremath{\Delta}}_{c}$ in all systems and depend on the relative magnitudes of ${\ensuremath{\Delta}}_{c}$ and ${T}_{K}$. In particular, one would expect to find $\frac{d{T}_{max}}{dP}l0$ in systems such as Au:Fe with relatively high Kondo temperatures. A further result of the analysis is that ${\ensuremath{\Delta}}_{c}(P)\ensuremath{\approx}{\ensuremath{\Delta}}_{\mathrm{R}\mathrm{K}\mathrm{K}\mathrm{Y}}(P)$, lending support to the view that in these systems the long-range RKKY oscillations represent the dominant impurity-impurity interaction mechanism.

The Kondo effect in a magnetic field

AbstractThe integral equation method for the vertex part proposed by Cheung and Mettuck is generalized for the case of a magnetic field. The s‐d part of the diluted magnetic alloy resistance in the linearized parquet approximation is calculated. The agreement with the results obtained previously is discussed.

Vertex function in 4-ϵ dimensional phase transition theory for a λϕ 3 + μϕ 4-model

The vertex functions and self-energy in a 4-ϵ dimensional theory of phase transitions with λϕ 3 and μϕ 4 interactions are determined by summation of diagrams in the parquet approximation.

Parquet approximation for dipolar interactions at critical points

The transition from the mean-field-like to the true critical behavior for a system with dipolar interaction is studied by a parquet graph expansion. It is shown that the dipolar interaction is manifested as an effective increase of the number of the spin-components.

Infrared singularities and small-distance-behaviour analysis

The infrared-singularity structure of the vertex functions of massless-particle φ4 theory is studied. This allows to construct the asymptotic forms of the vertex functions of massive-particle φ4 theory in a simpler and more explicit fashion than in a previous paper. With the help of the parquet approximation introduced by Diatlov, Sudakov, and Martirosian we show that the infrared-singularity structure in a theory with besides the massless particles, massive ones is the same as in the theory with massless particles only. All these results in φ4 theory have analoga in other renormalizable theories.