Two-site Approximation Research Articles

Slave rotor approach to exciton condensation in a two-band system

We have studied exciton formation and condensation in an extended Falicov–Kimball model, going beyond the weak coupling approach, employing a semi-analytical technique: the slave-rotor mean-field theory (SRMF). In this essentially strong coupling theory, charge and spin (or orbital/pseudospin) degrees are treated as independent degrees of freedom, coupled by a local constraint. Using a two-site-extension of SRMF, we capture the effective many body scale beyond conventional mean-field theory. While the formation of excitons is favoured by the interband hybridization , it is strongly influenced by the on-site Coulomb interaction . Beyond a critical hybridization, there is condensation of excitons, leading to a transition from a metal to an excitonic insulator phase. Moreover, the behaviour of excitonic averages differs from the usual Hartree–Fock mean-field theory. Low- results show that excitonic order parameter (Δ) is continuous across the transition both for single as well as two-site approximation, changing to weakly first order one at intermediate for the later. The large- limit shows a continuous transition for two-site analysis but remains first order in the single-site approximation. The slave rotor theory gives a mixed state of excitons and metal in both the analyses. We have also checked the effect of intersite correlation and localized band hopping on the exciton condensation.

Time-dependent Mott transition in the periodic Anderson model with nonlocal hybridization

The time-dependent Mott transition in a periodic Anderson model with off-site, nearest-neighbor hybridization is studied within the framework of nonequilibrium self-energy functional theory. Using the two-site dynamical-impurity approximation, we compute the real-time dynamics of the optimal variational parameter and of different observables initiated by sudden quenches of the Hubbard-U and identify the critical interaction. The time-dependent transition is orbital selective, i.e., in the final state, reached in the long-time limit after the quench to the critical interaction, the Mott gap opens in the spectral function of the localized orbitals only. We discuss the dependence of the critical interaction and of the final-state effective temperature on the hybridization strength and point out the various similarities between the nonequilibrium and the equilibrium Mott transition. It is shown that these can also be smoothly connected to each other by increasing the duration of a U-ramp from a sudden quench to a quasi-static process. The physics found for the model with off-site hybridization is compared with the dynamical Mott transition in the single-orbital Hubbard model and with the dynamical crossover found for the real-time dynamics of the conventional Anderson lattice with on-site hybridization.

Nonequilibrium self-energy functional approach to the dynamical Mott transition

The real-time dynamics of the Fermi-Hubbard model, driven out of equilibrium by quenching or ramping the interaction parameter, is studied within the framework of the nonequilibrium self-energy functional theory. A dynamical impurity approximation with a single auxiliary bath site is considered as a reference system and the time-dependent hybridization is optimized as prescribed by the variational principle. The dynamical two-site approximation turns out to be useful to study the real-time dynamics on short and intermediate time scales. Depending on the strength of the interaction in the final state, two qualitatively different response regimes are observed. For both weak and strong couplings, qualitative agreement with previous results of nonequilibrium dynamical mean-field theory is found. The two regimes are sharply separated by a critical point at which the low-energy bath degree of freedom decouples in the course of time. We trace the dependence of the critical interaction of the dynamical Mott transition on the duration of the interaction ramp from sudden quenches to adiabatic dynamics, and therewith link the dynamical to the equilibrium Mott transition.

Description of spin-crossover in a heterogeneous chain of exchange clusters

Spin transition theory is developed for periodic chains with two and more exchange clusters in a unit cell. A general expression is obtained for the effective magnetic moment of a unit cell μeff with several exchange clusters. For heterospin complexes Cu(hfac)2LR characterized by chains with the head-to-head motif a twosite approximation for the statistical sum of the Cu2+ Jahn-Teller paramagnetic center Cu2+ with spin 1/2 is proposed, within which a comparison with the available experimental data is performed. It is shown that the model developed describes both smooth and abrupt (cooperative) spin-crossover cases for the chains of three-spin exchange clusters.

Anderson localization at large disorder

The localization of one-electron states in the large (but finite) disorder limit is investigated. The inverse participation number shows a nonmonotonic behavior as a function of energy owing to the anomalous behavior of few-site localization. The two-site approximation is solved analytically and is shown to capture the essential features found in numerical simulations on one-, two-, and three-dimensional systems. Further improvement has been obtained by solving a three-site model.

Severe hindrance of viral infection propagation in spatially extended hosts.

The production of large progeny numbers affected by high mutation rates is a ubiquitous strategy of viruses, as it promotes quick adaptation and survival to changing environments. However, this situation often ushers in an arms race between the virus and the host cells. In this paper we investigate in depth a model for the dynamics of a phenotypically heterogeneous population of viruses whose propagation is limited to two-dimensional geometries, and where host cells are able to develop defenses against infection. Our analytical and numerical analyses are developed in close connection to directed percolation models. In fact, we show that making the space explicit in the model, which in turn amounts to reducing viral mobility and hindering the infective ability of the virus, connects our work with similar dynamical models that lie in the universality class of directed percolation. In addition, we use the fact that our model is a multicomponent generalization of the Domany-Kinzel probabilistic cellular automaton to employ several techniques developed in the past in that context, such as the two-site approximation to the extinction transition line. Our aim is to better understand propagation of viral infections with mobility restrictions, e.g., in crops or in plant leaves, in order to inspire new strategies for effective viral control.

Quasicrystals as alloys with short-range order

The electronic structure of quasiperiodic lattices is studied. An alloy theory, including short-range order effects, is used to approximate Fibonacci and Thue–Morse lattices. Short-range order is treated by embedding small clusters in an alloy that itself incorporates a two-site approximation, and the probabilities of these clusters are used to construct an efficient procedure for the calculation of electronic properties. This approach allows easy identification of the contributions of particular clusters to the electronic density of states. As the short-range order is increased via the number of clusters, the density of states can be clearly seen to transition from that of an alloy to that of a quasicrystal. It is shown that the techniques may be applied to other lattices defined by substitution rules.

Solvation potentials for flexible chain molecules in solution: On the validity of a pairwise decomposition

The effects of a solvent on the conformation of a flexible n-site solute molecule can be described formally in terms of an n-body solvation potential. Given the practical difficulty in computing such multibody potentials, it is common to carry out a pairwise decomposition in which the n-body potential is approximated by a sum of two-body potentials. Here we investigate the validity of this two-site approximation for short interaction-site chain-in-solvent systems. Using exact expressions for the conformation of an isolated chain, we construct a mapping between the full chain-in-solvent system and its solvation potential representation. We present results for both hard-sphere and square-well systems with n=5 that show that the two-site approximation is sufficient to completely capture the effects of an explicit solvent on chain conformation for a wide range of conditions (which include varying the solvent diameter in the hard-sphere system and varying the chain-solvent coupling in the square-well system). In all cases, a set of two-site potentials (one for each distinct site-site pair) is required. We also show that these two-site solvation potentials can be used to accurately compute a multisite intramolecular correlation function.

Non-perturbative conserving approximations and Luttinger's sum rule

Weak-coupling conserving approximations can be constructed by truncations of the Luttinger-Ward functional and are well known as thermodynamically consistent approaches which respect macroscopic conservation laws as well as certain sum rules at zero temperature. These properties can also be shown for variational approximations that are generated within the framework of the self-energy-functional theory without a truncation of the diagram series. Luttinger's sum rule represents an exception. We analyze the conditions under which the sum rule holds within a non-perturbative conserving approximation. Numerical examples are given for a simple but non-trivial dynamical two-site approximation. The validity of the sum rule for finite Hubbard clusters and the consequences for cluster extensions of the dynamical mean-field theory are discussed.

Self-energy-functional approach: Analytical results and the Mott-Hubbard transition

The self-energy-functional approach proposed recently is applied to the single-band Hubbard model at half-filling to study the Mott-Hubbard metal-insulator transition within the most simple but non-trivial approximation. This leads to a mean-field approach which is interesting conceptually: Trial self-energies from a two-site single-impurity Anderson model are used to evaluate an exact and general variational principle. While this restriction of the domain of the functional represents a strong approximation, the approach is still thermodynamically consistent by construction and represents a conceptual improvement of the ``linearized DMFT'' which has been suggested previously as a handy approach to study the critical regime close to the transition. It turns out that the two-site approximation is able to reproduce the complete (zero and finite-temperature) phase diagram for the Mott transition. For the critical point at T=0, the entire calculation can be done analytically. This calculation elucidates different general aspects of the self-energy-functional theory. Furthermore, it is shown how to deal with a number of technical difficulties which appear when the self-energy functional is evaluated in practice.

Effects of two-site correlations in the Hubbard model

We propose a theoretical approach, within the framework of the composite operator method, to include the effects of finite cluster correlations into the self-energy of strongly correlated systems. The Hubbard model is analyzed as significant example. The self-energy is rewritten in terms of two-site composite-operator propagators, which are computed by means of a two-site approximation preserving relevant symmetries (e.g., particle–hole symmetry). The involved composite operators describe charge, spin and pair nearest-neighbor correlations and the excitations related to the induced exchange coupling (J=4t2/U). The procedure results in a very rich band structure going well beyond the results of the two-pole approximation.

Quasi-stationary distributions for stochastic processes with an absorbing state

We study the long-time behavior of stochastic models with an absorbing state, conditioned on survival. For a large class of processes, in which saturation prevents unlimited growth, statistical properties of the surviving sample attain time-independent limiting values. We may then define a quasi-stationary probability distribution as one in which the ratios p_n(t)/p_m(t) (for any pair of nonabsorbing states n and m), are time-independent. This is not a true stationary distribution, since the overall normalization decays as probability flows irreversibly to the absorbing state. We construct quasi-stationary solutions for the contact process on a complete graph, the Malthus-Verhulst process, Schlogl's second model, and the voter model on a complete graph. We also construct the master equation and quasi-stationary state in a two-site approximation for the contact process, and for a pair of competing Malthus-Verhulst processes.

Dynamical Evolution of Highway Traffic Flow: from Microscopic to Macroscopic

In this paper, a derivation of the macroscopic mean field theory of the cellular automaton (CA) model of highway traffic flow starting from the microscopic dynamical point of view is presented. Starting from an equation describing the time evolution of the Boolean state variable at each site of the basic CA model, and using a two-site approximation for the multi-site correlation functions, a dynamical mapping between the macroscopic average speeds ν(t + 1) and ν(t) at different time can be derived. Mean field results consistent with the simulation data are obtained by considering the attractors of the mapping and their corresponding basins.

Temperature dependence of electronic states in the t-J model.

The temperature dependence of electronic states in the t-J model is studied in the moderately doped region by use of the composite operator method. Self-energy effects are included in two-site approximation. At higher temperature, the density of states shows a two-peak structure, and the coherent peak at the Fermi level increases its intensity with decreasing temperature. The peak at the Fermi level splits into two peaks with further decreasing temperature, corresponding to the electronic excitations from spin-singlet and -triplet states of the nearest-neighbor electrons. Electronic excitations from the singlet states form a narrower band at the Fermi level. This may relate to the spin gap formation observed in high-${\mathit{T}}_{\mathit{c}}$ superconductors. \textcopyright{} 1996 The American Physical Society.

Alternative to Gutzwiller's approach.

An efficient variational procedure to calculate the ground-state properties of the particle-hole symmetric periodic Anderson model is presented. Expectation values are calculated by using an approximate scheme. It is shown that the ground-state energy calculated with the one-site approximation, which essentially is a single-site mean-field approximation, is identical to the corresponding energy obtained by the well-known Gutzwiller wave function using the Gutzwiller approximation. The two-site approximation, in which intersite correlations are explicitly included, is shown to be a better approximation than the Gutzwiller approximation.

Static properties and relaxation dynamics OP the ising mixture

Abstract We use here the technique that allows us to calculate the q-dependent static and dynamic correlation functions of the Ising mixture within cluster two-site approximation. The results qualitatively differ from the ones in the mean field approximation and show some essential properties of the model and approximation itself.

Theory of One-Dimensional Hopping Conductivity

The frequency-dependent conductivity σ(ω) in a one-dimensional hopping-model system, formed of two sublattices with one species of particles, is investigated by means of the three-site approximation of the path probability method (PPM) and the Monte Carlo simulation (MCS). In a strong-coupling case, where the two-site approximation is rather in poor agreement with the MCS, the three-site approximation is satisfactory enough to reproduce the MCS within the present model

Bipolaron ac conductivity in amorphous semiconductors and dielectrics.

We have developed a theory of the alternating-current (ac) relaxation-type conductivity due to small bipolaron (SB) hopping in amorphous semiconductors and insulators that possess deep centers of the dangling-bond type (D centers) with a negative two-electron correlation energy ${\mathit{U}}_{\mathrm{eff}}$. Unlike small po- larons, SB's were treated as essentially three-level systems in the framework of a two-site approximation. To calculate, both numerically and analytically, the real part ${\mathrm{\ensuremath{\sigma}}}_{1}$\ensuremath{\propto}${\mathrm{\ensuremath{\omega}}}^{\mathit{s}}$${\mathit{T}}^{\mathit{n}}$ of the ac hopping conductivity for different temperatures T in a wide range of audio and low radio frequencies \ensuremath{\omega}, the dynamic polariz- ability, and the SB hopping rates for dangling bond pairs have been determined. When the electron tunneling integral corresponding to the smallest intersite separations is greater than the doubled polaron shift, both the polarizability and the hopping rate strongly depend upon the shape and parameters of the ground-state adiabatic potential of a small-size pair of strongly interacting D centers. This intimate pair can be viewed as a stretched or weakened bond. A classification of possible regimes of relaxation-type SB hopping (adiabatic and nonadiabatic, as well as tunnel and activation) has been proposed. Each of these corresponds to a specific temperature dependence of the exponents s and n. A comparison to experimental data on ac losses in chalcogenide glasses and a-${\mathrm{SiO}}_{2}$ has been made.

Frequency dependence of the conductivity tensor in disordered gyrotropic systems

In the presence of magnetic field the frequency dependences of symmetrical ( ∂ xx) and antisymmetrical ( ∂ xy) parts of the conductivity tensor in disordered systems with localized electronic states were calculated by the Kubo formula. In calculating ∂ xx ( ∂ xy) the two-site (three-site) approximation was used. The obtained results for Re ∂ xx ( ω) pretty well describe experimental data on amorphous semiconductors and doped and compensated semiconductors. The curve Re ∂ xy ( ω) undergoes the twofold sign inversion in the absorption band region.

Cluster-variation method applied in two-site approximation to the Hubbard model at high temperatures

In order to study the origin of metallic magnetism and the metal-insulator transition (Mott transition), the cluster variation method is applied in a two-site approximation to the Hubbard Hamiltonian ${H}_{\mathrm{Hubbard}}=\ensuremath{\Sigma}{i=1}^{\mathfrak{N}}\ensuremath{\Sigma}{j=1}^{\mathfrak{N}}\ensuremath{\Sigma}{\ensuremath{\sigma}}^{}{t}_{\mathrm{ij}}{c}_{i\ensuremath{\sigma}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j\ensuremath{\sigma}}+U\ensuremath{\Sigma}{i=1}^{\mathfrak{N}}{n}_{i\ensuremath{\uparrow}}{n}_{i\ensuremath{\downarrow}}$, where ${c}_{i\ensuremath{\sigma}}^{\ifmmode\dagger\else\textdagger\fi{}}$, ${c}_{i\ensuremath{\sigma}}$ are, respectively, the creation and destruction operators of an electron with spin $\ensuremath{\sigma}$ (\ensuremath{\uparrow} or \ensuremath{\downarrow}) on lattice site $i$, ${n}_{i\ensuremath{\sigma}}={c}_{i\ensuremath{\sigma}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{i\ensuremath{\sigma}}$, $U>0$ is the strength of the intrasite Coulomb repulsion between electrons having antiparallel spins, $\mathfrak{N}$ is the total number of sites, and ${t}_{\mathrm{ij}}$ is the "hopping" strength. With the restriction of only nearest-neighbor electron hoppings of strength $\ensuremath{-}t (t>0)$, and considering the half-filled-band case, both one- and three-dimensional (simple cubic lattice) results are calculated numerically. This is achieved by solving the basic equilibrium equations for the following two-site expectation values: ${〈{n}_{i\ensuremath{\uparrow}}{n}_{i\ensuremath{\downarrow}}〉}_{0}$, ${〈{n}_{i\ensuremath{\uparrow}}{n}_{j\ensuremath{\uparrow}}〉}_{0}$, ${〈{n}_{i\ensuremath{\uparrow}}{n}_{j\ensuremath{\downarrow}}〉}_{0}$, ${〈{n}_{i\ensuremath{\uparrow}}{n}_{i\ensuremath{\downarrow}}{n}_{j\ensuremath{\uparrow}}〉}_{0}$, ${〈{n}_{i\ensuremath{\uparrow}}{n}_{i\ensuremath{\downarrow}}{n}_{j\ensuremath{\uparrow}}{n}_{j\ensuremath{\downarrow}}〉}_{0}$, ${〈{c}_{i\ensuremath{\uparrow}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j\ensuremath{\uparrow}}〉}_{0}$, ${〈{c}_{i\ensuremath{\uparrow}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j\ensuremath{\uparrow}}{n}_{i\ensuremath{\downarrow}}〉}_{0}$, ${〈{c}_{i\ensuremath{\uparrow}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j\ensuremath{\uparrow}}{n}_{i\ensuremath{\downarrow}}{n}_{j\ensuremath{\downarrow}}〉}_{0}$, ${〈{c}_{i\ensuremath{\uparrow}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{i\ensuremath{\downarrow}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j\ensuremath{\downarrow}}{c}_{j\ensuremath{\uparrow}}〉}_{0}$. These expectation values can in turn be used to compute various thermodynamic quantities, e.g., internal energy, entropy, specific heat, etc. For sufficiently large parametric value of $\frac{U}{t}$, a high-temperature maximum in the specific heat is resolved and is identified as an indication of a gradual metal-insulator transition. For the simple cubic structure, this high-temperature peak disappears, however, as one decreases the value of $\frac{U}{t}$ to values around or below 15. Also, correlation results strongly suggest that the three-dimensional half-filled-band Hubbard model admits an antiferromagnetic ground state.