We present a new open-source program, DCore, that implements dynamical mean-field theory (DMFT). DCore features a user-friendly interface based on text and HDF5 files. It allows DMFT calculations of tight-binding models to be performed on predefined lattices as well as \textit{ab initio} models constructed by external density functional theory codes through the Wannier90 package. Furthermore, DCore provides interfaces to many advanced quantum impurity solvers such as quantum Monte Carlo and exact diagonalization solvers. This paper details the structure and usage of DCore and shows some applications.
We discuss the recently proposed LDA'+DMFT approach providing consistent parameter free treatment of the so called double counting problem arising within the LDA+DMFT hybrid computational method for realistic strongly correlated materials. In this approach the local exchange-correlation portion of electron-electron interaction is excluded from self consistent LDA calculations for strongly correlated electronic shells, e.g. d-states of transition metal compounds. Then the corresponding double counting term in LDA+DMFT Hamiltonian is consistently set in the local Hartree (fully localized limit -…
We present a joint implementation of dynamical-mean-field theory (DMFT) with the pseudopotential plane-wave approach, via Wannier functions, for the determination of the electronic properties of strongly correlated materials. The scheme uses, as input for the DMFT calculations, a tight-binding Hamiltonian obtained from the plane-wave calculations by projecting onto atomic-centered symmetry-constrained Wannier functions for the correlated orbitals. We apply this scheme to two prototype systems: a paramagnetic correlated metal, SrVO3, and a paramagnetic correlated system, V2O3, which exhibits a …
We propose a self-consistent method for electronic structure calculations of correlated systems, which combines the local spin-density approximation (LSDA) and the dynamical mean field theory (DMFT). The LSDA part is based on the exact muffin-tin orbital approach, meanwhile the DMFT uses a perturbation scheme that includes the T matrix with fluctuation exchange approximation. The current LSDA+DMFT implementation fulfills both self-energy and charge self-consistency requirements. We present results on the electronic structure calculations for bulk 3d transition metals (Cr, Fe, and Ni) and for F…
We propose a dynamical mean field approach for calculating the electronic structure of strongly correlated materials from first principles. The scheme combines the GW method with dynamical mean field theory, which enables one to treat strong interaction effects. It avoids the conceptual problems inherent to conventional ``LDA+DMFT'', such as Hubbard interaction parameters and double counting terms. We apply a simplified version of the approach to the electronic structure of nickel and find encouraging results.
The LDA+DMFT approach merges conventional band structure theory in the local density approximation (LDA) with a state-of-the-art many-body technique, the dynamical mean-field theory (DMFT). This new computational scheme has recently become a powerful tool for ab initio investigations of real materials with strong electronic correlations. In this paper an introduction to the basic ideas and the set-up of the LDA+DMFT approach is given. Results for the photoemission spectra of the transition metal oxide La_{1-x}Sr_xTiO_3, obtained by solving the DMFT-equations by quantum Monte-Carlo (QMC) simula…
Dynamical Mean-Field Theory (DMFT) has opened new perspectives for the investigation of strongly correlated electron systems and greatly improved our understanding of correlation effects in models and materials. In contrast to Hartree-Fock-type approximations the mean field of DMFT is dynamical, whereby local quantum fluctuations are fully taken into account. DMFT becomes exact in the limit of high spatial dimensions or coordination number. Using DMFT the dynamics of correlated electron systems can be investigated non-perturbatively at all interaction strengths, electron densities and temperat…
Dynamical Mean Field Theory (DMFT) is a successful method to compute the electronic structure of strongly correlated materials, especially when it is combined with density functional theory (DFT). Here, we present an open-source computational package (and a library) combining DMFT with various DFT codes interfaced through the Wannier90 package. The correlated subspace is expanded as a linear combination of Wannier functions introduced in the DMFT approach as local orbitals. In particular, we provide a library mode for computing the DMFT density matrix. This library can be linked and then inter…
We present a consistent way of treating a double counting problem unavoidably arising within the LDA+DMFT combined approach to realistic calculations of electronic structure of strongly correlated systems. The main obstacle here is the absence of systematic (e.g. diagrammatic) way to express LDA (local density approximation) contribution to exchange correlation energy appearing in the density functional theory. It is not clear then, which part of interaction entering DMFT (dynamical mean-field theory) is already taken into account through LDA calculations. Because of that, up to now there is n…
The spinless Falicov-Kimball model on the simple cubic lattice is analyzed in the Hubbard-I and dynamical mean field (DMFT) approximations. The Matsubara and real frequency itinerant electron Green's functions, the evolution of the system with doping, and the range of phase separation are found in two approximations. At large values of the on-site Coulomb repulsion both approximations give similar results. The phase separation can be also favorable for a more general model, where heavy electrons have a finite bandwidth. This indicates that the phase separation phenomenon is an inherent feature…
I present a critical overview of so-called "{\it ab initio}" DFT (density fuctional theory) based calculation schemes for the description of the electronic structure, energy spectrum, and optical response for strongly correlated 3$d$ oxides, in particular, crystal-field and charge transfer transitions as compared with an "old"\, cluster model that does generalize crystal-field and ligand-field theory. As a most instructive illustration of validity of numerous calculation techniques I address the prototypical 3$d$ insulator NiO predicted to be a metal in frames of a standard LDA (local density …
ComDMFT is a massively parallel computational package to study the electronic structure of correlated-electron systems (CES). Our approach is a parameter-free method based on ab initio linearized quasiparticle self-consistent GW (LQSGW) and dynamical mean field theory (DMFT). The non-local part of the electronic self-energy is treated within ab initio LQSGW and the local strong correlation is treated within DMFT. In addition to ab initio LQSGW+DMFT, charge self-consistent LDA+DMFT methodology is also implemented, enabling multiple methods in one open-source platform for the electronic structur…
We investigate a photoinduced insulator-metal transition in the Falicov-Kimball model with the dynamical mean-field theory (DMFT) extended to nonequilibrium systems in periodic modulations in time. When the photon energy of the pump light is $Ω\approx U$ ($U$: the interaction strength), a Drude-like peak is found to grow in the optical conductivity spectrum, which is an evidence that the system is driven into a metallic state. During the transition, the band gap does not collapse, whereas the distribution function exhibits a non-monotonic behaviour away from the Fermi distribution. This indica…
The sensitivity of Density Functional Theory plus Dynamical Mean Field Theory calculations to different constructions of the correlated orbitals is investigated via a detailed comparison of results obtained for the quantum material NdNiO$_2$ using different Wannier and projector methods to define the correlation problem. Using the same interaction parameters we find that the different methods produce different results for the orbital and band basis mass enhancements and for the orbital occupancies, with differing implications regarding the importance of multiorbital effects and charge transfer…
Predicting the compositional phase stability of strongly correlated electron materials is an outstanding challenge in condensed matter physics, requiring precise computations of total energies. In this work, we employ the density functional theory plus dynamical mean-field theory (DFT+DMFT) formalism to address local correlations due to transition metal d electrons on compositional phase stability in the prototype rechargeable battery cathode material LixCoO2, and detailed comparisons are made with the simpler DFT+U approach (i.e. the Hartree-Fock solution of the DMFT impurity problem). Local …
This review is devoted to generalization of dynamical mean-field theory (DMFT) for strongly correlated electronic systems towards the account of different types of additional interactions, necessary for correct physical description of many experimentally observed phenomena in such systems. As additional interactions we consider: (1) interaction of electrons with antiferromagnetic (or charge) fluctuations of order parameter in high-Tc superconductors leading to the formation of pseudogap state, (2) scattering of electrons on static disorder and its role in general picture of Anderson-Hubbard me…
Materials with correlated electrons often respond very strongly to external or internal influences, leading to instabilities and states of matter with broken symmetry. This behavior can be studied theoretically either by evaluating the linear response characteristics, or by simulating the ordered phases of the materials under investigation. We developed the necessary tools within the dynamical mean-field theory (DMFT) to search for electronic instabilities in materials close to spin-state crossovers and to analyze the properties of the corresponding ordered states. This investigation, motivate…
We present a computational study of $L$-edge resonant inelastic x-ray scattering (RIXS) in correlated 3$d$ transition-metal oxides using an $ab$ $initio$ method based on local density approximation + dynamical mean-field theory (DMFT). The present method, building on Anderson impurity model with an optimized continuum bath within DMFT, is an extension of the cluster model to include unbound electron-hole pair excitations as well as material-specific charge-transfer excitations with less empirical parameters. We find a good agreement with available experimental data. The relationship between co…
We introduce natural atomic orbitals as the local projector to define the correlated subspace for DFT + DMFT (density functional theory plus dynamical mean-field theory) calculation. The natural atomic orbitals are found to be stably constructed against the number and the radius of basis orbitals. It can also be self-consistently updated inside the DFT+DMFT loop. The spatial localization, electron occupation and the degree of correlation are investigated and compared with other conventional techniques. As a `natural' choice to describe the electron numbers, adopting natural atomic orbitals has…
We propose to combine the Floquet formalism for systems in ac fields with the dynamical mean-field theory to study correlated electron systems periodically driven out of equilibrium by external fields such as intense laser light. This approach has a virtue that we can nonperturbatively include both the correlation effects and nonlinear effects due to the driving field, which is imperative in analyzing recent experiments for photoinduced phase transitions. In solving the problem, we exploit a general theorem that the Hamiltonian in a Floquet matrix form can be exactly diagonalized for single-ba…
We generalize the dynamical-mean field (DMFT) approximation by including into the DMFT equations some length scale via a (momentum dependent) ``external'' self-energy Σ_k. This external self-energy describes non-local dynamical correlations induced by short-ranged collective SDW-like antiferromagnetic spin (or CDW-like charge) fluctuations. At high enough temperatures these fluctuations can be viewed as a quenched Gaussian random field with finite correlation length. This generalized DMFT+Σ_k approach is used for the numerical solution of the weakly doped one--band Hubbard model with repulsive…
We give a detailed description of a recently proposed first principles approach to the electronic structure of strongly correlated materials. The method combines the GW approximation with dynamical mean field theory. It is designed to describe Coulomb interactions and screening effects without adjustable parameters, thus avoiding the conceptual problems inherent to LDA+DMFT techniques.
Many-body theories such as dynamical mean field theory (DMFT) have enabled the description of the electron exchange-correlation interactions that are missing in current density functional theory (DFT) calculations. However, there has been relatively little focus on the wavefunctions from these theories. We present the methodology of the newly developed Elk-TRIQS interface and how to calculate the DFT with DMFT (DFT+DMFT) wavefunctions, which can be used to calculate DFT+DMFT wavefunction dependent quantities. We illustrate this by calculating the electron localized function (ELF) in monolayer …
We give a summary of recent progress in the field of electronic structure calculations for materials with strong electronic Coulomb correlations. The discussion focuses on developments beyond the by now well established combination of density functional and dynamical mean field theory dubbed "LDA+DMFT". It is organized around the description of dynamical screening effects in the solid. Indeed, screening in the solid gives rise to dynamical local Coulomb interactions U(w) (Aryasetiawan et al 2004 Phys. Rev. B 70 195104), and this frequency-dependence leads to effects that cannot be neglected in…
We generalize the dynamical-mean field (DMFT) approximation by including into the DMFT equations some length scale via a momentum dependent ``external'' self-energy S(k). This external self-energy describes non-local dynamical correlations induced by short-ranged collective SDW-like antiferromagnetic spin (or CDW-like charge) fluctuations. At high enough temperatures these fluctuations can be viewed as a quenched Gaussian random field with finite correlation length. This generalized DMFT+Sk approach is used for the numerical solution of the weakly doped one-band Hubbard model with repulsive Co…
Phonons are fundamentally important for many materials properties, including thermal and electronic transport, superconductivity, and structural stability. Here, we describe a method to compute phonons in correlated materials using state-of-the-art DFT+DMFT calculations. Our approach combines a robust DFT+DMFT implementation to calculate forces with the direct method for lattice dynamics using nondiagonal supercells. The use of nondiagonal instead of diagonal supercells drastically reduces the computational expense associated with the DFT+DMFT calculations. We benchmark the method for typical …
Dynamical mean field theory (DMFT) combined with the local density approximation (LDA) is widely used in solids to predict properties of correlated systems. In this paper, its application to one of the simplest strongly correlated systems, the hydrogen molecule H$_2$, is demonstrated to develop a parameter-free LDA+DMFT framework. We propose a method to calculate the exact intersection of LDA and DMFT that leads to highly accurate subtraction of the doubly counted correlation in both methods. When the exact double-counting treatment and a good projector to the correlated subspace are used, LDA…
We present a fully self-consistent combined GW and dynamical mean field (GW+DMFT) study of the spectral properties of the extended two-dimensional Hubbard model. The inclusion of the local dynamical vertex stemming from the DMFT self-energy and polarization is shown to cure the problems of self-consistent GW in the description of spectral properties. We calculate the momentum-resolved spectral functions, the two-particle polarization and electron loss spectra, and show that the inclusion of GW in extended DMFT leads to a narrowing of the quasi-particle width and more pronounced Hubbard bands i…
We develop, clarify and test various aspects of cluster methods dynamical mean field methods using a soluble toy model as a benchmark. We find that the Cellular Dynamical Mean Field Theory (C-DMFT) converges very rapidly and compare its convergence properties with those of the Dynamical Cluster Approximation (DCA). We propose and test improved estimators for the lattice self energy within C-DMFT.
The development of polynomial cost solvers for correlated quantum impurity models, with controllable errors, is a central challenge in quantum many-body physics, where these models find applications ranging from nano-science to the dynamical mean-field theory (DMFT). Here we describe how configuration interaction (CI) approximations to exact diagonalization (ED) may be used as solvers in DMFT. CI approximations retain the main advantages of ED, such as the ability to treat general interactions and off-diagonal hybridizations and to obtain real spectral information, but are of polynomial cost. …
We consider a doped Mott insulator in the large dimensionality limit within both the recently developed Extremely Correlated Fermi Liquid (ECFL) theory and the Dynamical Mean-Field Theory (DMFT). We show that the general structure of the ECFL sheds light on the rich frequency-dependence of the DMFT self-energy. Using the leading Fermi-liquid form of the two key auxiliary functions introduced in the ECFL theory, we obtain an analytical ansatz which provides a good quantitative description of the DMFT self-energy down to hole doping level 0.2. In particular, the deviation from Fermi-liquid behav…
While the traditional local-density approximation (LDA) cannot describe Mott insulators, {\it ab-initio} determination of the Hubbard $U$, for example, limits LDA-plus dynamical mean field theory (DMFT) approaches. Here, we attempt to overcome these bottlenecks by achieving fusion of the quantum chemistry (QC) approach with DMFT. QC+DMFT supplants the LDA bandstructure by its QC counterpart as an input to DMFT. Using QC+DMFT, we show that undoped $La_{2}CuO_{4}$ is a $d$-Mott insulator, and qualitatively discuss the circulating current- and incoherent metal phase, at small but finite hole dopi…
The use of effective local Coulomb interactions that are dynamical, that is, frequency-dependent, is an efficient tool to describe the effect of long-range Coulomb interactions and screening thereof in solids. The dynamical character of the interaction introduces the coupling to screening degrees of freedom such as plasmons or particle-hole excitations into the many-body description. We summarize recent progress using these concepts, putting emphasis on dynamical mean field theory (DMFT) calculations with dynamical interactions ("doubly dynamical mean field theory"). We discuss the relation to…
The dynamical mean-field theory (DMFT) is a widely applicable approximation scheme for the investigation of correlated quantum many-particle systems on a lattice, e.g., electrons in solids and cold atoms in optical lattices. In particular, the combination of the DMFT with conventional methods for the calculation of electronic band structures has led to a powerful numerical approach which allows one to explore the properties of correlated materials. In this introductory article we discuss the foundations of the DMFT, derive the underlying self-consistency equations, and present several applicat…
We perform investigation of Hubbard model with interaction between strongly correlated conducting electrons on a lattice with Debye phonons. To solve the problem generalized dynamical mean-field DMFT+Σmethod is employed with "external" self-energy Σ_ph corresponding to electron-phonon interaction. We present DMFT+Σ_ph results for densities of states and kinks in energy dispersions for a variety of model parameters, analyzing the interplay of recently discovered kinks of purely electronic nature and usual phonon kinks in the electronic spectrum.
We present a charge and self-energy self-consistent computational scheme for correlated systems based on the Korringa-Kohn-Rostoker (KKR) multiple scattering theory with the many-body effects described by the means of dynamical mean field theory (DMFT). The corresponding local multi-orbital and energy dependent self-energy is included into the set of radial differential equations for the single-site wave functions. The KKR Green's function is written in terms of the multiple scattering path operator, the later one being evaluated using the single-site solution for the $t$-matrix that in turn i…
These are introductory lectures to some aspects of the physics of strongly correlated electron systems. I first explain the main reasons for strong correlations in several classes of materials. The basic principles of dynamical mean-field theory (DMFT) are then briefly reviewed. I emphasize the formal analogies with classical mean-field theory and density functional theory, through the construction of free-energy functionals of a local observable. I review the application of DMFT to the Mott transition, and compare to recent spectroscopy and transport experiments. The key role of the quasipart…
The correlated Kondo insulator state of the plutonium monochalcogenides is investigated using the dynamical mean field theory (DMFT) and the local density approximation +U (LDA+U). The DMFT-dynamical fluctuations lead to a correlated insulator state at elevated temperature, in sharp contrast to the static LDA+U approach that fails to reproduce both the insulating behavior and anomalous lattice constant. The DMFT conversely predicts the experimentally observed anomalous increase of the gap with pressure and explains the lattice constant very well.
The nonequilibrium Green's functions (NEGF) approach is a versatile theoretical tool, which allows to describe the electronic structure, spectroscopy and dynamics of strongly correlated systems. The applicability of this method is, however, limited by its considerable computational cost. Due to the treatment of the full two-time dependence of the NEGF the underlying equations of motion involve a long-lasting non-Markovian memory kernel that results in at least a $N^3_t$ scaling in the number of time points $N_t$. The system's memory time is, however, reduced in the presence of a thermalizing b…
The LDA+DMFT (local density approximation combined with dynamical mean-field theory) computation scheme has been used to study spectral and magnetic properties of FeSi and Fe$_{1-x}$Co$_{x}$Si. Having compared different models we conclude that a correlated band insulator scenario in contrast to Kondo insulator model agrees with FeSi band structure as well as experimental data. Coulomb correlation effects lead to band narrowing of the states near the Fermi level with mass renormalization parameter $m^*\approx 2$ in agreement with the results of angle-resolved photoemission spectroscopy (ARPES).…
We implemented the derivative of the free energy functional with respect to the atom displacements, so called force, within the combination of Density Functional Theory and the Embedded Dynamical Mean Field Theory. We show that in combination with the numerically exact quantum Monte Carlo (MC) impurity solver, the MC noise cancels to a great extend, so that the method can be used very efficiently for structural optimization of correlated electron materials. As an application of the method, we show how strengthening of the fluctuating moment in FeSe superconductor leads to a substantial increas…
Pseudogap physics in strongly correlated systems is essentially scale dependent. We generalize the dynamical mean field theory (DMFT) by including into the DMFT equations dependence on correlation length of pseudogap fluctuations via additional (momentum dependent) self-energy Sigma_k. This self-energy describes non-local dynamical correlations induced by short-ranged collective SDW-like antiferromagnetic spin (or CDW-like charge) fluctuations. At high enough temperatures these fluctuations can be viewed as a quenched Gaussian random field with finite correlation length. This generalized DMFT+…
We discuss interaction of strongly correlated electrons (described within the Hubbard model solved by dynamical mean-field theory (DMFT)) with Debye and Einstein phonons using recently developed DMFT+Sigma computational scheme. Electron-phonon interaction (EPI) is analyzed in adiabatic approximation (assuming the validity of Migdal theorem), allowing the neglect of EPI vertex corrections. This approach is valid for EPI coupling constant L < e_F/w_ph~10, where e_F is Fermi energy and w_ph is Debye or Einstein frequency. For moderate values of L only small changes in the electronic density of st…
We construct a new functional for the single particle Green's function, which is a variant of the standard Baym Kadanoff functional. The stability of the stationary solutions to the new functional is directly related to aspects of the irreducible particle hole interaction through the Bethe Salpeter equation. A startling aspect of this functional is that it allows a simple and rigorous derivation of both the standard and extended dynamical mean field (DMFT) equations as stationary conditions. Though the DMFT equations were formerly obtained only in the limit of infinite lattice coordination, th…
Ab initio calculation of the electronic properties of materials is a major challenge for solid state theory. Whereas the experience of forty years has proven density functional theory (DFT) in a suitable, e.g. local approximation (LDA) to give a satisfactory description in case electronic correlations are weak, materials with strongly correlated, say d- or f-electrons remain a challenge. Such materials often exhibit colossal responses to small changes of external parameters such as pressure, temperature, and magnetic field, and are therefore most interesting for technical applications. Encoura…
We present an embedding approach based on localized basis functions which permits an efficient application of the dynamical mean field theory (DMFT) to inhomogeneous correlated materials, such as semi-infinite surfaces and heterostructures. In this scheme, the semi-infinite substrate leads connected to both sides of the central region of interest are represented via complex, energy-dependent embedding potentials that incorporate one-electron as well as many-body effects within the substrates. As a result, the number of layers which must be treated explicitly in the layer-coupled DMFT equation …
We study the low temperature transport characteristics of a disordered metal in the presence of electron-electron interactions. We compare Hartree-Fock and dynamical mean field theory (DMFT) calculations to investigate the scattering processes of quasiparticles off the screened disorder potential and show that both the local and non-local (coming from long-ranged Friedel oscillations) contributions to the renormalized disorder potential are suppressed in strongly renormalized Fermi liquids. Our results provide one more example of the power of DMFT to include higher order terms left out by weak…
We present a comparative electronic structure study using DFT and various beyond-DFT (DFT+$U$, $G_0W_0$, DFT+DMFT) methods for ferromagnetic Iron (Fe) to find better approach for describing the spectral properties of correlated magnetic system. The computed value of $U$ ($W$) is $\sim$5.4 ($\sim$0.8) eV. The calculated spectra of all methods are providing good agreement with experimental spectra (ES) for peaks' positions. But, the proper line shape is only found from DFT+DMFT with correct estimation of incoherent states, which depends on $J$ and form of local Coulomb interactions. The estimati…
Quantitative prediction of electronic properties in correlated materials requires simulations without empirical truncations and parameters. We present a method to achieve this goal through a new ab initio formulation of dynamical mean-field theory (DMFT). Instead of using small impurities defined in a low-energy subspace, which require complicated downfolded interactions which are often approximated, we describe a full cell $GW$+DMFT approach, where the impurities comprise all atoms in a unit cell or supercell of the crystal. Our formulation results in large impurity problems, which we treat h…
Hubb_DMFT and Wan2mb_DMFT: Continuous-Time Quantum Monte Carlo Solvers for Single- and Multi-Orbital Hubbard Model
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；M. N. Mikheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences（俄罗斯科学院乌拉尔分支M.N.米凯耶夫金属物理研究所）
We derive a formula for the electric polarization of interacting insulators, expressed in terms of the full Green's and vertex functions. We exemplify this method in the half-filled ionic Hubbard model treated within dynamical mean field theory (DMFT). The electric polarization of a correlated band insulator is determined by the interplay of ionicity and covalency, and both quantities are renormalized by the electron-electron interactions. We introduce quasiparticle approximation to the exact equation for the polarization, and compare the results of this approximation with those of the exact D…
The description of realistic strongly correlated systems has recently advanced through the combination of density functional theory in the local density approximation (LDA) and dynamical mean field theory (DMFT). This LDA+DMFT method is able to treat both strongly correlated insulators and metals. Several interfaces between LDA and DMFT have been used, such as (N-th order) Linear Muffin Tin Orbitals or Maximally localized Wannier Functions. Such schemes are however either complex in use or additional simplifications are often performed (i.e., the atomic sphere approximation). We present an alt…
Motivated by a collection of experimental results indicating the strongly correlated nature of the ferromagnetic metallic state of $CrO_{2}$, we present results based on a combination of the actual bandstructure [3] with dynamical mean-field theory (DMFT) for the multi-orbital case. In striking contrast with LSDA(+U) [3] and model many-body approaches [14], much better semiquantitative agreement with (i) recent photoemission results, (ii) domain of applicability of the half-metal concept, and (iii) thermodynamic and dc transport data, is obtained within a single picture. Our approach has broad…
Optical conductivity of the weakly doped two-dimensional repulsive Hubbard model on the square lattice with nearest and next nearest hoppings is calculated within the generalized dynamical-mean field (DMFT+Σ_p) approach which includes correlation length scale ξinto the standard DMFT equations via the momentum dependent self-energy Σ_p, with full account of appropriate vertex corrections. This approach takes into consideration non-local dynamical correlations induced e.g. by short-ranged collective SDW-like antiferromagnetic spin fluctuations, which (at high enough temperatures) can be viewed a…
