Strong-coupling Perturbation Theory Research Articles

Holstein polaron

We describe a variational method to solve the Holstein model for an electron coupled to dynamical, quantum phonons on an infinite lattice. The variational space can be systematically expanded to achieve high accuracy with modest computational resources (12-digit accuracy for the one-dimensional polaron energy at intermediate coupling). We compute ground-state and low-lying excited-state properties of the model at continuous values of the wave vector k in essentially all parameter regimes. Our results for the polaron energy band, effective mass, and correlation functions compare favorably with those of other numerical techniques, including the density-matrix renormalization-group technique, the global-local method, and the exact diagonalization technique. We find a phase transition for the first excited state between a bound and unbound system of a polaron and an additional phonon excitation. The phase transition is also treated in strong-coupling perturbation theory.

Strong-coupling perturbation theory for the two-dimensional Bose-Hubbard model in a magnetic field

The Bose-Hubbard model in an external magnetic field is investigated with strong-coupling perturbation theory. The lowest-order secular equation leads to the problem of a charged particle moving on a lattice in the presence of a magnetic field, which was first treated by Hofstadter. We present phase diagrams for the two-dimensional square and triangular lattices, showing a change in shape of the phase lobes away from the well-known power-law behavior in zero magnetic field. Some qualitative agreement with experimental work on Josephson-junction arrays is found for the insulating phase behavior at small fields.

Polaron effective mass, band distortion, and self-trapping in the Holstein molecular-crystal model

We present polaron effective masses and selected polaron band structures of the Holstein molecular crystal model in 1-D as computed by the Global-Local variational method over a wide range of parameters. These results are augmented and supported by leading orders of both weak- and strong-coupling perturbation theory. The description of the polaron effective mass and polaron band distortion that emerges from this work is comprehensive, spanning weak, intermediate, and strong electron-phonon coupling, and non-adiabatic, weakly adiabatic, and strongly adiabatic regimes. Using the effective mass as the primary criterion, the self-trapping transition is precisely defined and located. Using related band-shape criteria at the Brillouin zone edge, the onset of band narrowing is also precisely defined and located. These two lines divide the polaron parameter space into three regimes of distinct polaron structure, essentially constituting a polaron phase diagram. Though the self-trapping transition is thusly shown to be a broad and smooth phenomenon at finite parameter values, consistency with notion of self-trapping as a critical phenomenon in the adiabatic limit is demonstrated. Generalizations to higher dimensions are considered, and resolutions of apparent conflicts with well-known expectations of adiabatic theory are suggested.

Strong-coupling perturbation theory of the Hubbard model

The strong-coupling perturbation theory of the Hubbard model is presented and carried out to order (t/U)^5 for the one-particle Green function in arbitrary dimension. The spectral weight A(k,omega) is expressed as a Jacobi continued fraction and compared with new Monte-Carlo data of the one-dimensional, half-filled Hubbard model. Different regimes (insulator, conductor and short-range antiferromagnet) are identified in the temperature--hopping integral (T,t) plane. This work completes a first paper on the subject (Phys. Rev. Lett. 80, 5389 (1998)) by providing details on diagrammatic rules and higher-order results. In addition, the non half-filled case, infinite resummations of diagrams and the double occupancy are discussed. Various tests of the method are also presented.

Spectral weights in the charge-transfer model

We identify local kinetic energy contributions responsible for the rapid transfer of spectral weights in one-particle spectra. The sum rules derived using the cell method combined with strong-coupling perturbation theory are in excellent agreement with the direct evaluation of spectral weights in the charge-transfer model.

Converging toward a practical solution of the Holstein molecular crystal model

We present selected results for the Holstein molecular crystal model in one space dimension as determined by the Global–Local variational method, including complete polaron energy bands, ground state energies, and effective masses. We juxtapose our results with specific comparable results of numerous other methodologies of current interest, including quantum Monte Carlo, cluster diagonalization, dynamical mean field theory, density matrix renormalization group, semiclassical analysis, weak-coupling perturbation theory, and strong-coupling perturbation theory. Taken as a whole, these methodologies are mutually confirming and provide a comprehensive and quantitatively accurate description of polaron properties in essentially any regime. In particular, this comparison confirms the Global–Local variational method as being highly accurate over a wide range of the polaron parameter space, from the nonadiabatic limit to the extremes of high adiabaticity, from weak coupling through intermediate coupling to strong coupling.

Chiral symmetry breaking on the lattice: A study of the strongly coupled lattice Schwinger model

We revisit the strong coupling limit of the Schwinger model on the lattice using staggered fermions and the hamiltonian approach to lattice gauge theories. Although staggered fermions have no continuous chiral symmetry, they posses a discrete axial invari ance which forbids fermion mass and which must be broken in order for the lattice Schwinger model to exhibit the features of the spectrum of the continuum theory. We show that this discrete symmetry is indeed broken spontaneously in the strong coupling li mit. Expanding around a gauge invariant ground state and carefully considering the normal ordering of the charge operator, we derive an improved strong coupling expansion and compute the masses of the low lying bosonic excitations as well as the chiral co ndensate of the model. We find very good agreement between our lattice calculations and known continuum values for these quantities already in the fourth order of strong coupling perturbation theory. We also find the exact ground state of the antiferromag netic Ising spin chain with long range Coulomb interaction, which determines the nature of the ground state in the strong coupling limit.

Effect of Dimensionality and Anisotropy on the Holstein Polaron

AbstractWe apply weak-coupling perturbation theory and strong-coupling perturbation theory to the Holstein molecular crystal model in order to elucidate the effects of anisotropy on polaron properties in D dimensions. The ground state energy is considered as a primary criterion through which to study the effects of anisotropy on the self-trapping transition, with particular attention given to shifting of the self-trapping line and the adiabatic critical point. The effects of dimensionality and anisotropy on electron-phonon correlations and polaronic mass enhancement are studied, with particular attention given to the polaron radius and the characteristics of quasi-l-D and quasi-2-D structures. Perturbative results are confirmed by selected comparisons with variational calculations and quantum Monte Carlo data.

Large-order analysis of the convergent renormalized strong-coupling perturbation theory for the quartic anharmonic oscillator

Two hundred coefficients of the renormalized strong-coupling perturbation expansion for the ground and first excited states of the quartic anharmonic oscillator are calculated numerically. The large-order behavior of the perturbation coefficients is analyzed, a general and comparatively simple analytic formula describing their large-order behavior is proposed, and it is shown that this formula is consistent with known results from the divergent weak-coupling expansion. The accuracy of our numerically determined coefficients is checked by summation rules. In particular, if the summation rules are supplemented by the leading terms of our large-order formula, we obtain remarkably accurate results. This independently confirms the correctness of our large-order analysis. It is shown that the renormalized strong-coupling expansion converges---in contrast to other perturbation expansions---for all physically relevant coupling constants.

Polaron band formation in the Holstein model

We present numerical exact results for the polaronic band structure of the Holstein molecular crystal model in one and two dimensions. The use of direct Lanczos diagonalization technique, preserving the full dynamics and quantum nature of phonons, allows us to analyze in detail the renormalization of both quasiparticle bandwidth and dispersion by the electron-phonon interaction. For the two-dimensional case some of our exact data are compared with the results obtained in the framework of a recently developed finite cluster strong-coupling perturbation theory.

Some exact results for the spin-1/2 Falicov-Kimball model in the strong coupling limit

The spin-1/2 Falicov-Kimball model for electronically driven valence and metal-insulator transitions is studied analytically using strong-coupling perturbation theory. It is shown that in the limit of the infinite interaction strength between localized and itinerant electrons the Falicov-Kimball model undergoes two types of discontinuous valence transitions: the insulator-metal transitions from an integer-valence ground state (nf = 1) into an inhomogeneous intermediate-valence ground state (0 < nf < 1) and the insulator-metal transitions from nf = 1 to nf = 0. In addition, we discuss the role of the electron-phonon interaction on the mechanism of valence transitions and we present possible extensions of the exact one dimensional results to higher dimensions.

Single-polaron band structure of the Holstein model.

The single-polaron band structure of the Holstein model in one and two dimensions is studied using resummed strong-coupling perturbation theory. Well-converged results are obtained for phonon frequencies of the order of the hopping integral and strong-to-intermediate electron-phonon coupling. The polaron band structure at intermediate coupling is shown to deviate markedly from that of a nearest-neighbor tight-binding model and is in fact similar in shape to the prediction of weak-coupling self-consistent perturbation theory. \textcopyright{} 1996 The American Physical Society.

Scaling theory of the Kondo screening cloud

A scaling form for the local susceptibility, derived from renormalization group arguments, is proposed. The scale over which the uniform part of this scaling form varies can be viewed as a definition of the Kondo ``screening cloud" $\sim \xi_K$. The proposed scaling form interpolates between Ruderman-Kittel-Kasuya-Yosida (RKKY) results in the high temperature limit, $T\gg T_K$, and Fermi liquid results in the low temperature, long-distance limit, $T\ll T_K$, $r\gg \xi_K$. The predicted form of the Knight shift is longer range at low temperatures where the screening cloud has formed, than at high temperatures where it has not. Using weak and strong coupling perturbation theory combined with large scale density matrix renormalization group (DMRG) results we study the validity of the finite size version of the scaling form at $T=0$. We explicitly extract a length scale proportional to the Kondo length scale, $\xi_K$. The numerical results are in good agreement with the proposed scaling form and confirm the existence of the Kondo screening cloud.

Vertex-corrected tunneling inversion in superconductors

We have generalized the McMillan-Rowell tunneling-inversion program, which extracts the electron-phonon spectral function α2 F(Ω) and the Coulomb pseudopotentialμ * from experimental tunneling data, to include the lowest-order effect of vertex corrections. In so doing, the momentum dependence of the electron-phonon matrix elements are neglected, which is equivalent to using a local approximation. The vertex-corrected strong-coupling perturbation theory is performed on the imaginary axis and then an exact analytic continuation is used to produce the density of states on the real axis. Comparison with the experimental data for Pb indicates that effects of vertex corrections may be observable even in low-temperature superconductors.

The influence of hopping on valence transitions in the Falicov-Kimball model

We use strong-coupling perturbation theory and the extrapolation of small-cluster exact-diagonalization calculations to describe the ground-state properties and possible intermediate-valence transitions of the spinless Falicov-Kimball model with a generalized type of hopping. It is shown that, even for relatively small values of the interaction strength, the Falicov-Kimball model undergoes only a few (discrete) intermediate-valence transitions, and this result is demonstrated to be independent of finite-size effects.

Spectral properties of the Hubbard bands.

Using strong-coupling perturbation theory for the Hubbard model, explicit expressions are obtained for the integrated weights, energy positions, and widths of the upper and lower Hubbard bands separately, both for the optical and photoemission spectrum. In one dimension all expressions can be explicitly evaluated using the large-U Bethe-ansatz wave function. The k\ensuremath{\rightarrow}-dependent moments for the one-particle spectrum are compared with the closely related two-pole ansatz. The strong momentum dependence of the spectra demonstrates the importance of the k\ensuremath{\rightarrow}-dependent correlation functions in the spectral moments, which are often neglected. In order to estimate corrections due to the neglect of higher-order terms (in t/U), all results are compared with numerical-diagonalization data for intermediate-U values. The sum-rule results show a rapid spectral-weight transfer from the high- to the low-energy regime upon doping, similar to what is observed experimentally in one-particle and optical spectra of Cu (high-${\mathit{T}}_{\mathit{c}}$) and Ni compounds. Such fast weight redistributions as a function of the carrier density are a natural consequence of the strong correlations in these materials. The redistribution of intensity in the optical conductivity is connected with the dilution of the spin system by the added holes. For the one-particle spectrum the decrease of weight in the upper Hubbard band away from half filling can be understood by state counting at U=\ensuremath{\infty}, and the effect is strongly enhanced for small doping by both the first- and second-order terms in t/U. Altogether, the one-particle and optical spectra show the importance of the three-site hopping term. Therefore the t-J model does not represent well the spectral properties of the Hubbard model at large U. The local spin order is essential and determines both the k\ensuremath{\rightarrow} dependence of the one-particle spectrum and the intensities in the optical spectrum.

Scattering of glueballs and mesons in compact QED in 2 + 1 dimensions

We study glueball and meson scattering in compact QED 2+1 gauge theory in a Hamiltonian formulation on a momentum lattice. We compute ground state energy and mass, and introduce a compact lattice momentum operator for the computation of dispersion relations. Using a non-perturbative time-dependent method we compute scattering cross sections for glueballs and mesons. We compare our results with strong coupling perturbation theory.

Two Hubbard bands: Weight transfer in optical and one-particle spectra.

We identify local kinetic-energy contributions which are responsible for the rapid weight transfer from the high- to the low-energy scale observed upon doping in strongly correlated materials. Using strong-coupling perturbation theory for the Hubbard model we derive expressions for the weights of the upper and lower Hubbard bands separately and evaluate them explicitly in one dimension. The optical conductivity is found to be a direct probe of the local spin order.

Magnetic properties of the lattice Anderson model

We perform exact diagonalization studies of the one-dimensional lattice Anderson model for various clusters. The ground state energy and spin–spin correlation functions are calculated as functions of Hubbard U, hybridization V, and f level occupancies. For the symmetric case, we compare our results with weak and strong coupling perturbation theory.

Glueball and meson scattering in compact QED2+1

We study glueball and meson scattering in compact QED 2+1 in a Hamiltonian formulation on a momentum lattice. We compute masses, dispersion relations and scattering cross sections and compare our results with strong coupling perturbation theory.