In this review we present results of our theoretical study of charge and spin ordering in strongly correlated electron systems obtained within various generalizations of the Falicov-Kimball model. The primary goal of this study was to identify crucial interactions that lead to the stabilization of various types of charge ordering in these systems, like the axial striped ordering, diagonal striped ordering, phase-separated ordering, phase-segregated ordering, etc. Among the major interactions that come into account, we have examined the effect of local Coulomb interaction between localized and itinerant electrons, long-range and correlated hopping of itinerant electrons, long-range Coulomb interaction between localized and itinerant electrons, local Coulomb interaction between itinerant electrons, local Coulomb interaction between localized electrons, spin-dependent interaction between localized and itinerant electrons, both for zero and nonzero temperatures, as well as for doped and undoped systems. Finally, the relevance of resultant solutions for a description of rare-earth and transition-metal compounds is discussed.
We report results of the study of the recently discovered magnetic resonance in the orbitally ordered phase of CeB6 (the orbital ordering resonance) in a wide frequency range 44-360 GHz. It is found that the g-factor for this resonance increases with frequency from g(44 GHz)~1.55 to g(>250 GHz)~1.7. In addition to the orbital ordering resonance for the frequencies exceeding 200 GHz a new magnetic resonance with the g-factor 1.2-1.3 is detected.
The effects associated with interatomic hoppings of excitons and the excitonic Bose condensate formation in the strongly correlated spin crossover systems are considered in the framework of the effective Hamiltonian for the two-band Kanamori model. The appearance of antiferromagnetic ordering due to the exciton order is found even in the absence of interatomic exchange interaction. The spectrum of excitonic excitations is calculated at various points of the "temperature vs. crystal field" phase diagram. Outside the region of exciton ordering, the spectrum has a gap, which vanishes at the bound…
A phenomenological model accounts for the calculation of incommensurate ordering wave vectors (IC) in several Ce- and U-based f-electron itinerant compounds (CePtSn, CePdSn, CeNiSn, CeRhIn$_5$, URu$_2$Si$_2$, CeNiAsO, CeCu$_{(6-x)}$Au$_x$, UNi$_2$Al$_3$, CeCuSn) as well as in BSSCO, Rb$_2$ZnBr$_4$, MnSi and prototypical IC system Chromium metal. This model is justified by a many-body topological one-dimensional (1-D) Dirac Hamiltonian after center of mass transformation on a chiral set of momenta defined by a local expansion of crystallographic Bragg planes. Implications for the emergence of l…
We investigate the emergence of anti-ferromagnetic ordering and its effect on the helical edge states in a quantum spin Hall insulator, in the presence of strong Coulomb interaction. Using dynamical mean-field theory, we show that the breakdown of lattice translational symmetry favours the formation of magnetic ordering with non-trivial spatial modulation. The onset of a non-uniform magnetization enables the coexistence of spin-ordered and topologically non-trivial states. An unambiguous signature of the persistence of the topological bulk property is the survival of bona fide edge states. We …
The spinless Falicov-Kimball model exhibits outside the particle-hole symmetric point different stable nonhomogeneous charge orderings. These include the well known charge stripes and a variety of orderings with phase separated domains, which can significantly influence the charge transport through the correlated electron system. We show this by investigating a heterostructure, in which the Falicov-Kimball model on a finite two-dimensional lattice is located between two noninteracting semi-infinite leads. We use a combination of nonequilibrium Green's functions techniques with a sign-problem-f…
Collective electronic fluctuations in correlated materials give rise to various important phenomena, such as existence of the charge ordering, superconductivity, Mott insulating and magnetic phases, plasmon and magnon modes, and other interesting features of such systems. Unfortunately, description of these correlation effects requires significant efforts, since they almost entirely rely on strong local and nonlocal electron-electron interactions. Some collective phenomena, such as magnetism, can be sufficiently described by a simple Heisenberg-like models that are formulated in terms of boson…
Predicting the compositional phase stability of strongly correlated electron materials is an outstanding challenge in condensed matter physics. In this work, we employ the DFT+U formalism to address the effects of local correlations due to transition metal d electrons on compositional phase stability in the prototype phase stable and separating materials LixCoO2 and olivine LixFePO4, respectively. We exploit a new spectral decomposition of the DFT+U total energy, revealing the distinct roles of the filling and ordering of the d orbital correlated subspace. The on-site interaction U drives both…
We show how strongly correlated materials could be described within the framework of an excitonic insulator formalism, and delineate the relationship between inter- and intra-band ordering phenomena. Our microscopic model of excitons clarifies the fundamental role of Van-Hove-singularity-nesting in driving both inter- and intra-band ordering transitions, and uncovers an interesting connection with resonating-valence-bond physics.
Motivated by recent experimental evidence of charge order in the pseudogap phase of cuprates, we perform a variational analysis of charge-neutral, spin-singlet ordering in metals on the square lattice, using a wavefunction with double occupancy projected out. We examine ordering with and without time-reversal symmetry, with arbitrary wavevector and tunable form factor. Depending on parameters, we find d-wave bond density wave ordering with wavevector either parallel to the lattice generators or diagonally oriented, or a ground state which carries a time reversal-breaking pattern of spontaneous…
A combined study of transport, thermodynamic, neutron diffraction, nuclear magnetic resonance measurements and first principles calculation were performed for $β$-V$_2$PO$_5$ single crystal. It was shown to be a semiconductor with a band gap of 0.48 eV, undergoing a charge ordering (unusual V$^{2+}$ and V$^{3+}$) phase transition accompanied by a tetragonal to monoclinic structural distortion at 610 K and a paramagnetic to ferrimagnetic phase transition at 128 K with a propagation vector of $\textbf{k} = 0$. The easy axis is in the monoclinic $ac$ plane pointing 47(9)$^\circ$ away from the mon…
In this letter, we reported the results of NMR study on LiFeAs single crystals. We find a strong evidence of the low temperature spin fluctuations; by changing sample preparation conditions, the system can be tuned toward an spin-density-wave (SDW) quantum-critical point. The detection of an interstitial Li(2) ion, possibly locating in the tetrahedral hole, suggests that the off-stoichiometry and/or lattice defect can probably account for the absence of the SDW ordering in LiFeAs. These facts show that LiFeAs is a strongly correlated system and the superconductivity is likely originated from t…
Enhancement of the electron spin polarization in a correlated two-layer two-dimensional electron system at a total Landau level filling factor of one is reported. Using resistively detected nuclear magnetic resonance, we demonstrate that the electron spin polarization of two closely-spaced two-dimensional electron systems becomes maximized when inter-layer Coulomb correlations establish spontaneous isospin ferromagnetic order. This correlation-driven polarization dominates over the spin polarizations of competing single-layer fractional Quantum Hall states under electron density imbalances.
We show how the onset of a non-Slater antiferromagnetic ordering in a correlated material can be detected by optical spectroscopy. Using dynamical mean-field theory we identify two distinctive features: The antiferromagnetic ordering is associated with an enhanced spectral weight above the optical gap, and well separated spin-polaron peaks emerge in the optical spectrum. Both features are indeed observed in LaSrMnO_4 [Gössling et al., Phys. Rev. B 77, 035109 (2008)]
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…
Motivated by the co-existing charge and spin order found in strongly correlated ladder systems, we study an effective pseudospin model on a coupled two-leg ladder. A bosonisation analysis yields a rich phase diagram showing Wigner/Peierls charge order and Neel/dimer spin order. In a broad parameter regime, the orbital antiferromagnetic phase is found to be stable. An intermediate gapless phase of finite width is found to lie in between two charge-ordered gapped phases. Our work is potentially relevant for a unified description of a class of strongly correlated, quarter-filled ladder systems.
Through resistivity measurements of an organic crystal hosting massless Dirac fermions with a charge-ordering instability, we reveal the effect of interactions among Dirac fermions on the charge transport. A low-temperature resistivity upturn appears robustly irrespectively of pressure and is enhanced while approaching the critical pressure of charge ordering, indicating that the insulating behavior originates from short-range Coulomb interactions. Observation of apparently vanishing gap in the charge-ordered phase accords with the theoretical prediction of the non-topological edge states.
We study a one-dimensional extended Hubbard model with longer-range Coulomb interactions at quarter-filling in the strong coupling limit. We find two different charge-ordered (CO) ground states as the strength of the longer range interactions is varied. At lower energies, these CO states drive two different spin-ordered ground states. A variety of response functions computed here bear a remarkable resemblance to recent experimental observations for organic TMTSF systems, and so we propose that these systems are proximate to a QCP associated with T=0 charge order. For a ladder system relevant t…
A recent study revealed the dynamics of the charge sector of a one-dimensional quarter-filled electronic system with extended Hubbard interactions to be that of an effective pseudospin transverse-field Ising model (TFIM) in the strong coupling limit. With the twin motivations of studying the co-existing charge and spin order found in strongly correlated chain systems and the effects of inter-chain couplings, we investigate the phase diagram of coupled effective (TFIM) systems. A bosonisation and RG analysis for a two-leg TFIM ladder yields a rich phase diagram showing Wigner/Peierls charge ord…
We study the interplay of spin, orbital and lattice degrees of freedom in a one-dimensional Kugel-Khomskii model coupled to phonons. In the vicinity of the dimer point we analyze the excitation spectrum, mapping the spin and orbital degrees of freedom to bond operators. In particular we study the renormalization of the phonon propagator due to coupling to the orbital and magnetic excitations. Considering both, ferro- and antiferro-orbital ordering we will show that the appearance of orbiton shake-up processes in the phonon spectrum is sensitive to the type of Jahn-Teller distortion. The releva…
It is shown that strongly correlated electrons on frustrated lattices like pyrochlore, checkerboard or kagome lattice can lead to the appearance of closed and open strings. They are resulting from nonlocal subsidiary conditions which propagating strongly correlated electrons require. The dynamics of the strings is discussed and a number of their properties are pointed out. Some of them are reminiscent of particle physics.
Topotactic phase transformation in correlated vanadium dioxide through oxygen vacancy ordering
Key Laboratory of Magnetic Molecules and Magnetic Information Materials of Ministry of Education & School of Chemistry and Materials Science, Shanxi Normal University（山西师范大学化学与材料科学学院、教育部磁性分子与磁性信息材料重点实验室）；Research Institute of Materials Science, Shanxi Key Laboratory of Advanced Magnetic Materials and Devices, Shanxi Normal University（山西师范大学材料科学研究所、山西省先进磁性材料与器件重点实验室）
The properties of a dilute electron gas, coupled to the lattice degrees of freedom, are studied and compared with the properties of an electron gas at half-filling, where spinless fermions with two orbitals per lattice site are considered. The simplest model which includes both the local electron-lattice interaction of the Jahn-Teller type and the electronic correlations is the $E\otimesβ$-Jahn-Teller-Hubbard model. We analyze the formation and stability of Jahn-Teller polarons and bipolarons, respectively. Our approach is based on a hopping expansion in the strong-coupling regime. The results…
Spectra of one-electron and collective excitations in narrow-band ferromagnets with unquenched orbital moments are calculated in various theoretical models. The interaction of spin and orbital excitations with conduction electrons results in the damping of the former which, however, turns out to be rather small; therefore, apart from usual spin waves, well-defined orbitons can exist. Non-quasiparticle states occur in the electron energy spectrum near the Fermi energy due to this interaction. The criteria of stability of the saturated spin and orbital ferromagnetic ordering are considered. Poss…
This is a short review of recent theoretical work on magnetization plateaux in one-dimensional strongly correlated electron systems, i.e. in spin chains, spin ladders or more generally, Hubbard chains and ladders.
Thermodynamic measurements reveal that the Pauli spin susceptibility of strongly correlated two-dimensional electrons in silicon grows critically at low electron densities - behavior that is characteristic of the existence of a phase transition.
We reexamine the Yang-Yang-Takahashi method of deriving the thermodynamic Bethe ansatz equations which describe strongly correlated electron systems of fundamental physical interest, such as the Hubbard, $s-d$ exchange (Kondo) and Anderson models. It is shown that these equations contain some additional terms which may play an important role in the physics of the systems.
We compute the spin susceptibility of the two-dimensional Hubbard model away from half-filling, and analyze the impact of frequency dependent vertex corrections as obtained from the dynamical mean field theory (DMFT). We find that the local dynamics captured by the DMFT vertex strongly affects non-local spin correlations, and thus the momentum dependence of the spin susceptibility. While the widely used random phase approximation yields commensurate Néel-type antiferromagnetism as the dominant instability over a wide doping range, the vertex corrections favor incommensurate ordering wave vecto…
This thesis describes the development of the density matrix embedding theory (DMET) and its applications to lattice strongly correlated electron problems, including a review of DMET theory and algorithms (Ch 2), investigation of finite size scaling (Ch 3), Applications to high-temperature superconductivity (Ch 4-6), a framework for finite-temperature DMET (Ch 7).
The dynamical-mean-field method is applied to investigate the transport properties of heterostructures consisting of a strongly-correlated electron system connected to metallic leads. The spectral function inside the correlated region is sensitive to the change of the interaction strength and bias voltage. Because of this sensitivity, current vs voltage characteristics of such heterostructures are rather nonlinear regardless of the detail of the potential profile inside the correlated region. The electronic properties such as the double occupancy are also changed by the bias voltage.
We argue that the Anderson and Kondo models turn out to be irrelevant for the description of some strongly correlated electron systems and we suggest the mechanism for the formation of many-body states (heavy fermions) being an alternative to the Kondo one. This mechanism involves the quantum tunneling of a heavy particle between the states in the double-well potential.
Conventional Paramagnetism - a state with finite magnetic moment per ion sans long range magnetic ordering, but with lowering temperature the moment on each ion picks up a particular direction, breaking rotational symmetry, and results into long range magnetic ordering. However, in systems with competing multiple degrees of freedom this conventional notion may easily breaks and results into short range correlation much above the global magnetic transition temperature. La2CuIrO6 with complex interplay of spins (s =1/2) on Cu site and pseudo-spin (j =1/2) on Ir site owing to strong spin-orbit co…
The effects associated with exciton Bose condensate formation in strongly correlated spin crossover systems are considered within the effective Hamiltonian obtained from the two-orbital Hubbard-Kanamori model. The collective excitations spectrum at various points of the temperature-crystal field phase diagram is calculated. The role of the electron-phonon interaction is discussed. The exciton and magnetic order photoenhancement (induction) in strongly correlated spin crossover systems a new mechanism based on the cooperative effect of electron-phonon and interatomic exchange interactions and t…
This work explores a simple approximation to describe isolated impurity scattering in a strongly correlated metal. The approximation combines conventional one electron scattering theory and the Dynamic Mean Field Theory to describe strong correlations in the host. It becomes exact in several limits, including those of very weak and very strong impurity potentials. Original electronic structure appears at the impurity site when the impurity potential strength is moderate and the host is close to the Mott transition. Our results may provide useful guidance for interpretation of scanning tunnelin…
Strongly correlated Fermi systems are among the most intriguing and fundamental systems in physics. We show that the herbertsmithite ZnCu3(OH)6Cl2 can be viewed as a new type of strongly correlated electrical insulator that possesses properties of heavy-fermion metals with one exception: it resists the flow of electric charge. We demonstrate that herbertsmithite's low temperature properties are defined by a strongly correlated quantum spin liquid made with such hypothetic particles as fermionic spinons which carry spin 1/2 and no charge. Our calculations of its thermodynamic and relaxation pro…
In this work, we present an analytical theory of strongly correlated Wigner crystals (WCs) in the lowest Landau level (LLL) by constructing an approximate, but accurate effective two-body interaction for composite fermions (CFs) participating in the WCs. This requires integrating out the degrees of freedom of all surrounding CFs, which we accomplish analytically by approximating their wave functions by delta functions. This method produces energies of various strongly correlated WCs that are in excellent agreement with those obtained from the Monte Carlo simulation of the full CF crystal wave …
The low-temperature kinetics of the strongly correlated electron liquid inhabiting a solid is analyzed. It is demonstrated that a softly damped branch of transverse zero sound emerges when several bands cross the Fermi surface simultaneously near a quantum critical point at which the density of states diverges. Suppression of the damping of this branch occurs due to a mechanism analogous to that affecting the phonon mode in solids at room temperature, giving rise to a classical regime of transport at extremely low temperatures in the strongly correlated Fermi system.
We study a model of strongly correlated spinless fermions on a kagome lattice at 1/3 filling, with interactions described by an extended Hubbard Hamiltonian. An effective Hamiltonian in the desired strong correlation regime is derived, from which the spectral functions are calculated by means of exact diagonalization techniques. We present our numerical results with a view to discussion of possible signatures of confinement/deconfinement of fractional charges.
The recent experimental evidence in favor of stripe ordering in ferropnictides [Chuang {\it et. al.}, Science {\bf 327},181 (2010)] points to the new field of study for strongly correlated systems. Here we argue that due to the smallness of the chemical potential in ferropnictides the effects of stripe modulation are likely to be much more pronounced than in the cuprates leading to stronger degree of one-dimensionality. This may allow to treat the striped pnictides as an array of coupled one-dimensional spin liquids making them a good testing ground for various ideas concerning strongly correl…
Coupled transitions between distinct ordered phases are important aspects behind the rich phase complexity of correlated oxides that hinders our understanding of the underlying phenomena. For this reason, fundamental control over complex transitions has become a leading motivation of the designer approach to materials. We have devised a series of new superlattices by combining a Mott insulator and a correlated metal to form ultra-short period superlattices, which allow one to disentangle the simultaneous orderings in $RE$NiO$_3$. Tailoring an incommensurate heterostructure period relative to t…
Various charge pairings in strongly correlated electron systems are interpreted as quantum entanglement of a composite system. Particles in the intermediate phase have a tendency to form the coherent superposition state of the localized state and the itinerant state, which induces the entanglement of both particles in the bipartite subsystems for increasing the entropy of the system. The correction to the entropic Coulomb force becomes an immediate cause of charge pairing.
The success of tensor network approaches in simulating strongly correlated quantum systems crucially depends on whether the many body states that are relevant for the problem can be encoded in a local tensor network. Despite numerous efforts, strongly correlated projective states, fractional quantum Hall states in particular, have not yet found a local tensor network representation. Here we show that one can encode the calculation of averages of local operators in a Grassmann tensor network which is local. Our construction is explicit, and allows the use of physically motivated trial wavefunct…
Experimental results on the metal-insulator transition and related phenomena in strongly interacting two-dimensional electron systems are discussed. Special attention is given to recent results for the strongly enhanced spin susceptibility, effective mass, and thermopower in low-disordered silicon MOSFETs.
Strongly correlated electron systems require the development of new theoretical schemes in order to describe their unusual and unexpected properties. The usual perturbation schemes are inadequate and new concepts must be introduced. In our scheme of calculations, the Composite Operator Method, is possible to recover, through a self-consistent calculation, a series of fundamental symmetries by choosing a suitable Hilbert space.
An exact formula for the temperature dependent Hall number of metals is derived. It is valid for non-relativistic fermions or bosons, with arbitrary potential and interaction. This DC transport coefficient is proven to (remarkably) depend solely on equilibrium susceptibilities, which are more amenable to numerical algorithms than the conductivity. An application to strongly correlated phases is demonstrated by calculating the Hall sign in the vicinity of Mott phases of lattice bosons.
The purpose of this paper is (i) to present a generic and fully functional implementation of the density-matrix renormalization group (DMRG) algorithm, and (ii) to describe how to write additional strongly-correlated electron models and geometries by using templated classes. Besides considering general models and geometries, the code implements Hamiltonian symmetries in a generic way and parallelization over symmetry-related matrix blocks.
The formalism for exactly calculating the retarded and advanced Green's functions of strongly correlated lattice models in a uniform electric field is derived within dynamical mean-field theory. To illustrate the method, we solve for the nonequilibrium density of states of the Hubbard model in both the metallic and Mott insulating phases at half-filling (with an arbitrary strength electric field) by employing the numerical renormalization group as the impurity solver. This general approach can be applied to any strongly correlated lattice model in the limit of large dimensions.
We adapt the Coupled Cluster Method to solid state strongly correlated lattice Hamiltonians extending the Coupled Cluster linear response method to the calculation of electronic spectra and obtaining the space-time Fourier transforms of generic Green's functions. We apply our method to the $MnO_2$ plane with orbital and magnetic ordering, to interpret electron energy loss experimental data, and to the Hubbard model, where we get insight into a possible pairing mechanism.
The unique property of Coulomb interaction in strict one-dimensional (1D) system is revealed that the Coulomb repulsion energy of paired electrons is divergent. As consequences, electrons in 1D system can not doubly occupy the same spatial orbital and are completely localized. Numerical simulation by time dependent Hartree-Fock approximation shows this distinct property. The '0.7 anomaly' in 1D electron gas is fully explained by the property. Its possible contribution to strongly correlated systems is discussed.
Temperature dependent optical spectra are reported for beta-Na0.33V2O5. The sodium ordering transition at T_Na = 240 K, and in particular the charge ordering transition at T_MI = 136 K strongly influence the optical spectra. The metal-insulator transition at \tmi leads to the opening of a psuedogap (\hbarω= 1700 cm-1), and to the appearance of a large number of optical phonons. These observations, and the presence of a mid-infrared band (typical for low dimensional metals) strongly suggests that the charge carriers in beta-Na0.33V2O5 are small polarons.
We propose a new approach to the self-consistency equation, which arises in the problem of the motion of a hole in a quantum antiferromagnet, appropriate to the case of small exchange energy $J$. The functional equation for the Green function is transformed into a differential equation; its solutions are analyzed and compared to the existing numerical calculations. This method allows one to study the limit of $J\to 0$. Application to other strongly correlated electron systems is discussed.
We study the effect of spatial inhomogeneity on the physics of a strongly correlated electron system exhibiting a metallic phase and a Mott insulating phase, represented by the simple Hubbard model. In three dimensions, we consider various geometries, including vacuum-metal-vacuum, a junction between a weakly and a strongly correlated metal, and finally the double junctions metal-Mott insulator-metal and metal-strongly correlated metal- metal. We applied to these problems the self-consistent Gutzwiller technique recently developed in our group, whose approximate nature is compensated by an ext…
The reason for the appearance of two distinct bands in the photoemission of the single CuO$_2$ plane Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6+δ}$ near the Fermi level is discussed on the basis of the self-consistent solution of the two-dimensional t-J model.
We show that the Krylov-Bogoliubov-Mitropolsky averaging in the canonical formulation can be used as a method for constructing effective Hamiltonians in the theory of strongly correlated electron systems. As an example, we consider the transition from the Hamiltonians of the Hubbard and Anderson models to the respective Hamiltonians of the t-J and Kondo models. This is a very general method, has several advantages over other methods, and can be used to solve a wide range of problems in the physics of correlated systems.
By introducing a basis for a novel realization of the SU(4) Lie algebra, we exactly solve a spin-orbital chain with one-ion $L$-$S$ coupling (OILSC) via the Bethe ansatz (BA) approach. In the context of different Landé $g$ factors of the spin and orbital sectors, the OILSC results in rich and novel quantum phase transitions. Some accurate analytical expressions for the critical fields are obtained. Both spin ordering and orbital ordering are found in a gapped singlet phase. The system exhibits many interesting phenomena such as nonvanishing magnetization of the singlet phase, multi-entrance of…
We present a new method to calculate optical properties of strongly correlated systems. It is based on dynamical mean field theory and it uses as an input realistic electronic structure obtained by local density functional calculations. Numerically tractable equations for optical conductivity, which show a correct non-interacting limit, are derived. Illustration of the method is given by computing optical properties of the doped Mott insulator La_{1-x}Sr_xTiO_3.
In this paper, the Bosonic projected variational method is proposed to study the strongly correlated $^{87}{\rm Rb}$ atoms in optical lattice. A global phase diagram is obtained by this method. There exist two characteristic lattice depths $V$ for $^{87}{\rm Rb}$ atoms in optical lattice : one is $% V=9.5E_r$ to label the maximum height of the 'zero-momentum' peak of condensation, the other is the quantum critical point for the superfluid-insulator (SI) transition at $V=12.3E_r$. As a result of strongly correlated effect for lattice Bosons, the suppressed superfluid state is predicted near the…
Complexity in strongly correlated electron systems is analyzed by considering decoherence process between the localized state, |L> and the itinerant state, |I>. The coherent superposition state of a|I> + b|L> decoheres to the pointer states in the proximity of both extremes of the correlation where the symmetry-breaking ground states of the charge pairing emerge. For maximizing the entropy of the system, the superconducting pairing and the spin density wave coexist within the uncertainty principle, which invokes the metastable states as like pseudogap phase and electronic inhomogenity.
